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[ { "title": "Analytical transformed harmonic oscillator basis for three-body nuclei of astrophysical interest: Application to 6 He", "content": "J. Casal, ∗ M. Rodr'ıguez-Gallardo, † and J.M. Arias ‡ Departamento de F'ısica At'omica, Molecular y Nuclear, Facultad de F'ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain (Dated: November 7, 2021) Recently, a square-integrable discrete basis, obtained performing a simple analytical local scale transformation to the harmonic oscillator basis, has been proposed and successfully applied to study the properties of two-body systems. Here, the method is generalized to study three-body systems. To test the goodness of the formalism and establish its applicability and limitations, the capture reaction rate for the nucleosynthesis of the Borromean nucleus 6 He ( 4 He + n + n ) is addressed. Results are compared with previous publications and with calculations based on actual three-body continuum wave functions, which can be generated for this simple case. The obtained results encourage the application to other Borromean nuclei of astrophysical interest such as 9 Be and 12 C, for which actual three-body continuum calculations are very involved. PACS numbers: 21.45.-v, 26.20.-f, 26.30.-k,27.20.+n", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The study of three-body Borromean nuclei is known to be important for astrophysical questions such as stellar nucleosynthesis. Borromean nuclei are three-body systems whose binary subsystems are unbound [1]. One of the Borromean nuclei which has attracted more interest is 12 C ( α + α + α ) due to the relevance of the tripleα reaction in the red giant phase of stars [2]. This process allows the formation of heavier elements in stars, where mainly α particles and nucleons are present, overcoming the A = 5 and A = 8 instability gaps [3]. The production rate of such process has not yet been determined accurately for the entire temperature range relevant in astrophysics [4]. This is due to experimental problems to measure these processes as well as to discrepancies in the theoretical predictions about the structure of 12 C. The formation of 12 C has traditionally been studied as a sequential process [5-7]. But, at low temperatures, the three α particles have no access to intermediate resonances and therefore they fuse directly [4]. The description of this process requires an accurate three-body model. Other Borromean nuclei are also important for nucleosynthesis in different astrophysical scenarios. For instance, massive stars usually end up with the explosion of a supernova and the possible formation of a neutron star. These neutron-rich environments, with low density and high temperature ( hot bubbles ), are an ideal medium for nucleosynthesis by rapid neutron capture (or r -process) [8, 9]. Among these processes one finds the formation of 6 He ( α + n + n ) or 9 Be ( α + α + n ) that could also overcome the A = 5 , 8 gaps [7]. Therefore, as in the case of the triplealpha capture, understanding of these processes requires a very accurate description of the states of 6 He and 9 Be in a three-body model as well as the corresponding electromagnetic transition probabilities. In particular, the 6 He nucleus has a halo structure. The halo nuclei are weakly-bound exotic systems in which one or more particles have a large probability of being at distances far away from typical nuclear radii [10]. A common characteristics of these systems is their small separation energy and hence their large breakup probability. This process can be understood as an excitation of the nucleus to unbound or scattering states that form a continuum of energies [11]. For that reason, the study of weakly bound three-body systems, such as 6 He, demands a proper treatment of the three-body problem with a reasonable description of their continuum structure. In this work, a method, which includes these characteristics, is proposed and then applied to 6 He as a benchmark calculation. It is worth noting that more fundamental fewbody methods can be applied to 6 He considered as a sixnucleon system, such as the Resonating Group [12] or the Lorentz Integral Transform [13, 14] methods. From the theoretical point of view, the treatment of unbound states of a quantum-mechanical system deals with the drawback that the corresponding wave functions are not square-normalizable and their energies are not discrete values. Solving this problem is a difficult task, especially as the number of charged particles increases, since one needs to know the asymptotic behavior of the unbound states. Nevertheless, there are various procedures to address this problem such as the R-matrix method [15-17], not without difficulties. Another approach to solve the continuum problem consists in using the so-called discretization methods. These methods replace the true continuum by a finite set of normalizable states, i.e., a discrete basis that can be truncated to a relatively small number of states and nevertheless provide a reasonable description of the system. Several discretization methods have been proposed [1]. For instance, one can solve the Schrodinger equation in a box [18], being the energy level density governed by the size of the box. As this is larger, the energy level density increases but numerical problems begin to appear. Another method is the binning procedure, used traditionally in the ContinuumDiscretized Coupled-Channels (CDCC) formalism [19]. In this method the continuum spectrum is truncated at a maximum energy and divided into a finite number of energy intervals or bins. For each bin, a normalizable state is constructed by superposition of the scattering states within that interval. This approach requires, first the calculation of the unbound states and then the matching with the correct asymptotic behavior. As mentioned above, the calculation of this asymptotic behavior for a three-body system with charged particles is by no means an easy task. An alternative method to obtain a discrete representation of the continuum spectrum is the pseudostate (PS) method, which consists in diagonalizing the three-body Hamiltonian in a complete set of L 2 wave functions (that is, square integrable). The eigenstates of the Hamiltonian are then taken as a discrete representation of the spectrum of the system. The advantage of this procedure is that it does not require going through the continuum wave functions and the knowledge of the asymptotic behavior is not needed. A variety of bases have been proposed for two-body [20-23] and also for three-body calculations [24-26]. In previous works, a PS method based on a local scale transformation (LST) of the harmonic oscillator (HO) basis has been proposed [27]. When the ground state of the system is known, a useful procedure to discretize the continuum consists in performing a numerical LST that transforms the actual ground-state wave function of the system into the HO ground state. Once the LST is obtained, the inverse transformation is applied to the HO basis, giving rise to the transformed harmonic oscillator (THO) basis. This method has been used to describe the two-body continuum in structure [22] and reactions [28, 29] studies, showing that the THO method together with the CDCC technique is useful to describe continuum effects in nuclear collisions. The method was also applied to 6 He [26, 30], showing that the numerical THO method is appropriate to describe three-body weakly bound systems with a relative small THO basis. In most recent works [11, 23] an alternative prescription to define the LST was proposed, introducing an analytical transformation taken from Karataglidis et al. [31]. This analytical transformation keeps the simplicity of the HO functions, but converts their Gaussian asymptotic behavior into an exponential one, more appropriate to describe bound systems. This analytical THO method has been applied to study two-body systems, providing a suitable representation of the bound and unbound spectrum to calculate structure and scattering observables within the CDCC method [23]. The analytical THO presents several advantages over the numerical THO. (1) It is not needed to know previously the ground-state wave function of the system considered. (2) Due to the analytical form of the transformation, it can be easily implemented in a numerical code. (3) The parameters of the transformation govern the radial extension of the THO basis allowing the construction of an optimal basis for each observable of interest. In this work, we extend the analytical THO method to study three-body systems. We start with the construction of the basis, then we diagonalize the three-body Hamiltonian, and we compute the transition probabilities needed for the calculation of the reaction rate. As a simple example of application, we check the formalism for the Borromean nucleus 6 He. For this, a rich variety of data is available [32-38] and can be used to benchmark theoretical models. Finally, the structure calculation allows us to determine the rate of the radiative capture reaction 4 He + n + n → 6 He + γ . It is known this reaction is not of great astrophysical interest but provides a robust test for our three-body model. In this case, with just one charged particle, one can generate easily the continuum wave functions and our model results can be confronted to actual continuum calculations. The study of this reaction will validate our formalism so as to make it reliable when applied to cases in which such comparisons with the true continuum cannot be easily done. This will be the case of 9 Be, 12 C, or 17 Ne, that are subjects for future research. The manuscript is structured as follows. In Sec. II the analytical THO method for three-body systems is completely worked out: basis, matrix elements, and calculation of transition probabilities. In Sec. III the expressions and concepts involved in the calculation of the radiative capture reactions of three particles into a bound nucleus are discussed. In Sec. IV the full formalism is applied to the case of 6 He. Finally, in Sec. V, the main conclusions of this work are summarized.", "pages": [ 1, 2 ] }, { "title": "II. PS METHOD: ANALYTICAL THO FOR THREE-BODY SYSTEMS", "content": "Jacobi coordinates { x , y } , illustrated in Fig. 1, are used to describe three-body systems [six-dimensional problems]. The variable x is proportional to the relative coordinate between two of the particles and y is proportional to the coordinate from the center of mass of these two particles to the third one, both with a scaling factor depending on their masses [26]. Please, note that there are three different Jacobi systems. From the Jacobi coordinates, one can define the hyperspherical coordinates { ρ, α, ̂ x, ̂ y } , where ρ = √ x 2 + y 2 is the hyper-radius and tan α = x/y defines the hyperangle. The PS method consists in diagonalizing the Hamiltonian of the system of interest in a discrete basis of L 2 functions. Using hyperspherical coordinates, and introducing Ω ≡ { α, ̂ x, ̂ y } for the angular dependence, the state wave functions of that basis can be expressed as Here Y βjµ (Ω) are states of good total angular momentum, expanded in hyperspherical harmonics (HH) [1, 39] Υ l x l y Klm l (Ω) as and β ≡ { K,l x , l y , l, S x , j ab } is a set of quantum numbers called channel. In this set, K is the hypermomentum, l x and l y are the orbital angular momenta associated with the Jacobi coordinates x and y , respectively, l is the total orbital angular momentum ( l = l x + l y ), S x is the spin of the particles related by the coordinate x , and j ab results from the coupling j ab = l + S x . If we denote by I the spin of the third particle, which we assume to be fixed, the total angular momentum j is j = j ab + I . With that notation, χ σ S x is the spin wave function of the two particles related by the Jacobi coordinate x , and κ ι I is the spin function of the third particle. The HH are eigenfunctions of the hypermomentum operator ̂ K 2 , and can be expressed in terms of the spherical harmonics as where P a,b n is a Jacobi polynomial with order n = ( K -l x -l y ) / 2 and N l x l y K is the normalization constant. On the other hand, R iβ ( ρ ) are the hyperradial wave functions, where the label i denotes the hyperradial excitation. The form of these functions depends on the PS method used. Then, the states of the system are given by diagonalization of the three-body Hamiltonian in a finite basis up to i max hyperradial excitations in each channel, being C iβj n the diagonalization coefficients and R nj β ( ρ ) the hyperradial wave function corresponding to the channel β . The label n enumerates the eigenstates.", "pages": [ 2, 3 ] }, { "title": "A. Analytical THO method", "content": "As stated in the introduction, several PS bases have been proposed for three-body studies [24-27]. Here, we use the THO method based on a LST of the HO functions, so the hyperradial wave functions are obtained as Note that, meanwhile the THO hyperradial wave functions depend, in general, on all the quantum numbers included in a channel β , the HO hyperradial wave functions only depend on one of them, the hypermomentum K . The transformation s ( ρ ) is not unique, and in this work we adopt the analytical form of Karataglidis et al. [31], depending on the parameters ξ , γ , and the oscillator length b . The HO hyperradial variable s is dimensionless according to the transformation defined above [Eq. (8)]. In this way, we take the oscillator length b as another parameter of the transformation. The function s ( ρ ) behaves asymptotically as γ b √ ρ 2 and hence the THO hyperradial wave functions obtained behave at large distances as exp ( -γ 2 ρ/ 2 b 2 ). Therefore, the ratio γ/b governs the asymptotic behavior of the THO functions: as γ/b increases, the hyperradial extension of the basis decreases and some of the eigenvalues obtained by diagonalizing the Hamiltonian explore higher energies [11]. That is, γ/b determines the density of PSs as a function of the energy. Concerning the parameter ξ , the authors of Ref. [31] found a very weak dependence of the results on this parameter. Because of that, we have fixed for all calculations ξ = 4 as in Refs. [11, 23]. The freedom to control the hyperradial extension of the THO basis is an advantage of the analytical THO method. Depending on the observable of interest, one is able to choose either a basis with a finer description of the low energy region (close to the breakup threshold) or a basis carrying more information on the high energy spectrum. In Fig. 2 the LSTs for a fixed b and different γ values are presented.", "pages": [ 3, 4 ] }, { "title": "B. Hamiltonian matrix elements", "content": "The three-body Hamiltonian in hyperspherical coordinates is written as The kinetic energy operator is [26, 40] where m is a normalization mass that we take as the nucleon mass and ̂ K 2 (Ω) represents the hyperangular momentum or hypermomentum operator. ̂ T (Ω) does not connect different channels β or states with different total angular momentum j . The Hamiltonian matrix elements have to be calculated between states given by Eq. (1), which separates the hyperradial and hyperangular parts. The hyperradial wave functions are constructed with Eq. (7) and satisfy the same normalization condition as the HO functions in six dimensions [39], For convenience, we introduce the hyperradial wave functions U THO iβ ( ρ ) as which satisfy the orthonormality relationship With these functions, the kinetic energy operator can be re-written as and its matrix elements are [41] where the anti-hermiticity of the derivation operator has been taken into account. The potential energy operator does connect, in general, different channels within the same j . The hyperangular integration is performed by using a set of subroutines from the code FaCE [42] that provides the hyperangular matrix elements V j ββ ' ( ρ ), depending on ρ . These functions are then integrated in the hyperradial variable, obtaining the potential energy matrix elements as Once the kinetic energy and potential matrix elements are computed, the Hamiltonian is diagonalized in a truncated THO basis with i max and the eigenstates of the system are obtained.", "pages": [ 4 ] }, { "title": "C. Transition probabilities B ( Eλ )", "content": "As in Ref. [26], we follow the notation of Brink and Satchler [43]. The reduced transition probability between states of a system is defined as where ̂ Q λ is the electric multipole operator of order λ . When a three-body system with only one charged particle, such as 6 He ( 4 He + n + n ), is considered and the Jacobi system T illustrated in Fig. 1 is used, the operator ̂ Q λ reads as In this expression Z is the atomic number of the system, e is the electron charge, m the mass of the nucleon, a y the reduced mass of the subsystem related by the Jacobi coordinate y and m c the mass of the charged particle (the core in the case of 6 He). The reduced matrix elements of this operator can be expanded in terms of the THO basis obtaining the expression [30, 39] Since n and n ' enumerate the different eigenstates, transition probabilities given by Eq. (17) are a set of discrete values. In order to obtain continuous energy distributions from discrete values, the best option is to do the overlap with the continuum wave functions [44], if they are known. In this case the smoothed THO B ( E 1) distribution must coincide perfectly with the actual continuum B ( E 1) distribution. When the continuum states are not available, it is considered that, in general, a PS with energy ε n is the superposition of continuum states in the vicinity. There are several ways to assign an energy distribution to a PS [45, 46]. In this work, for each discrete value of B ( Eλ )( ε n ), a Poisson distribution D ( ε, ε n , w ) with the following form is assigned, which is properly normalized. The parameter w controls the width of the distributions; as w decreases, the width of the distributions increases. Finally, the B ( Eλ ) distribution is given by the expression The B ( E 1) distribution so obtained can be compared easily in the case of 6 He with the continuum distribution in order to check the smoothing procedure. One can also calculate the sum rules for electric transitions from the ground state (g.s.) to the states ( n, j ) in order to test the completeness of the basis used. Using the Eq. (17) a closed expression is obtained", "pages": [ 4, 5 ] }, { "title": "III. RADIATIVE CAPTURE REACTION RATE", "content": "The formalism introduced above allows calculations of astrophysical interest. As stated in the introduction, some Borromean nuclei are important in the nucleosynthesis processes, and an accurate knowledge of their reaction and production rates in different scenarios is essential to understand the origin of the different elements in the Universe. We focus on radiative capture reactions of three particles, ( abc ), into a bound nucleus A of binding energy | ε B | , i.e., a + b + c → A + γ . The energy-averaged reaction rate for such process, 〈 R abc ( ε ) 〉 , is given as a function of the temperature T by the expression [47] The function F B ( ε, T ) is the Maxwell-Boltzmann distribution and R abc ( ε ) is the radiative capture reaction rate at a certain excitation energy ε . It can be obtained from the inverse photodissociation process [18, 47] and is given by the expression where ε = ε γ + ε B is the initial three-body kinetic energy, ε γ is the energy of the photon emitted, ε B is the groundstate energy, g i are the spin degeneracy of the particles, ν is the number of identical particles in the threebody system, and a x and a y are the reduced masses of the subsystems related to the Jacobi coordinates { x , y } . The photodissociation cross section σ γ ( ε γ ) of the nucleus A can be expanded into electric and magnetic multipoles [18, 48] which are related to the transition probability distributions dB ( O λ ) /dε , for O = E,M . From Eqs. (24) and (25), we write the energy-averaged capture reaction rate expression for the contribution of order λ as This integral is very sensitive to the dB ( O λ ) /dε behavior at low energy and, for that reason, a detailed description of the transition probability distribution in that region is needed to avoid numerical errors. Accordingly to the traditional literature [49], in absence of low energy resonances, the first multipole contribution is the dominant one and the electric contribution dominates over the magnetic one at the same order.", "pages": [ 5 ] }, { "title": "IV. APPLICATION TO 6 HE", "content": "The 6 He nucleus can be explained as a three-body system, formed by an inert α core and two valence neutrons. This is the simplest case to test the formalism developed in this work since there is just one charged particle and the three-body continuum wave functions can be generated easily. Comparison with actual continuum wave functions may serve as a reference for any other calculation. In addition, valuable experimental information is available on the ground state: total angular momentum j π = 0 + , experimental binding energy of 0.975 MeV [50], and rms point nucleon matter radius within 2.5 -2.6 fm [51]. It has also a well-known 2 + resonance at 0.824 MeV over the breakup threshold. To describe 6 He, we use a model Hamiltonian that includes the two-body n -n and α -n potentials, and also a simple central hyperradial three-body force. These potentials are those used in Ref. [26]; the n -α potential taken from Refs. [16, 52], with central and spin-orbit components, and the GPT n -n potential [53] with central, spin-orbit, and tensor components. These two-body potentials are kept fixed for any total angular momentum and parity j π . However, this Hamiltonian does not include all possible potential contributions. To include them effectively, a three-body force is usually introduced. In this work we have used the simple power form The parameters v 3 b , r 3 b , and a 3 b have been chosen to adjust the energy of the 0 + ground state and the position of the known 2 + resonance to the experimental values. In three-body models of halo nuclei, such as 6 He, the Pauli principle treatment is important to block occupied core states to the valence neutrons. That is, Pauli blocking is needed to remove forbidden states, which would disappear under antisymmetrization. This can be taken into account by several methods. In this work, a 'repulsive core' in the s -wave component of the α -n subsystem is introduced with the requirement that the experimental phase shifts are correctly calculated. This method is referred in the literature as the PC method [16]. The radiative capture of two neutrons by an alpha particle producing 6 He is dominated by a dipolar process from the 1 -continuum of 6 He to the 0 + ground state [18]. For 6 He, low-energy dipolar resonances have not been observed, then the electric dipole dominates over the magnetic dipole. A low-energy quadrupole resonance does exist (as mentioned above). We have calculated both dipolar and quadrupolar electric contributions, concluding the quadrupole is several orders of magnitude lower than the dipole. This means that the reaction rate for this capture process is mainly governed by the dipolar electric transition distribution dB ( E 1) /dε of 6 He. Then, to compute this distribution we need to generate the THO basis for states 0 + and 1 -. The 0 + THO basis must provide a well-converged ground state. The 1 -THO basis must have enough states close to the break-up threshold to get a smooth and detailed B ( E 1) distribution in that region. Using the parameters b and γ from the analytical LST one can find the most suitable THO basis for each total angular momentum (0 + and 1 -). The THO bases were truncated at maximum hypermomentum K max = 20, as it was sufficient to have a good description of the system and provide converged results.", "pages": [ 6 ] }, { "title": "A. States j π = 0 +", "content": "The 0 + states are described with an analytical THO basis defined by parameters b = 0 . 7 fm and γ = 1 . 4 fm 1 / 2 , trying to minimize the size of the basis needed to reach convergence of the ground state. We found that a basis with larger γ/b has a too large energy distribution to provide a fast convergence for the ground state. On the other hand, a basis with smaller γ/b has a very large hyperradial extension and does not describe properly the interior region of the potential where the ground state probability is larger. The three-body force parameters are taken as v 3 b = -2 . 45 MeV, r 3 b = 5 fm, and a 3 b = 3 in order to adjust both, the ground-state energy and the matter radius of 6 He. In Fig. 3 we show the first THO hyperradial wave functions for the channel β ≡ { 2 , 0 , 0 , 0 , 0 , 0 } , using the given LST and three-body force parameters. This channel is the most important ground-state channel, with a 78.6% contribution to the total norm. We can see in the figure that as i increases, the functions are more oscillatory and explore larger distances. In Fig. 4 the Hamiltonian eigenvalues for j π = 0 + , for an increasing number of hyperradial excitations, i max , are presented up to 10 MeV. The calculated ground-state is stable, has a binding energy of 0.9749 MeV and a rms point nucleon matter radius of 2.554 fm. Calculations assume an α radius of 1.47 fm. In Table I the groundstate energy ε B and matter radius r mat are shown as a function of the maximum number of hyperradial excitations i max . We observe a fast convergence of this two ground-state observables within this THO basis. The first three hyperradial components of the groundstate wave function for i max = 25 are presented in Fig. 5. The curves match a reference calculation of the groundstate wave function corresponding to the same model Hamiltonian. By reference calculation we mean the procedure presented in Ref. [16] and implemented in the codes FaCE [42] and sturmxx [54], using a suitable basis for bound states, the so-called Sturmian basis. Once the 0 + ground state is obtained, the 1 -states in the continuum have to be generated. However, no reference is available to fix the 1 -three-body force. For the 2 + continuum states there is a resonance experimentally observed at 0.824 MeV over the breakup threshold. Thus, usually the three-body force is fixed to set the 2 + resonance at the experimental value and it is accepted the same three-body force for the 1 -states. So we generate first the THO basis for 2 + states and adjust the position of the 2 + resonance by using a particular threebody force. Then we use the same force for the 1 -states.", "pages": [ 6, 7 ] }, { "title": "B. 2 + states", "content": "The 2 + states were described with a basis defined by b = 0 . 7 fm and γ = 2 . 0 fm 1 / 2 . This basis has a small hyperradial extension and spreads the eigenvalues obtained upon diagonalization at higher energies. This choice allows us to have only one pseudo-state presenting the characteristics of the resonance, since the rest of states are sufficiently above the resonance energy position for medium-size bases. In this way we can adjust the resonance energy, setting the energy of this state to the experimental value. Then, the three-body force parameters are taken as v 3 b = -0 . 90 MeV, r 3 b = 5 fm, and a 3 b = 3. In Fig. 6, the eigenvalues of the Hamiltonian for j π = 2 + states, for an increasing number of hyperradial excitations, are shown. The lowest state is rather stable and close to the energy of the known 2 + resonance, 0.824 MeV. In Fig. 7, we present the probability density for this first 2 + state, compared with the 0 + ground state probability. The contributions of the three most important channels for each one are shown. We can see the PS representing the resonance is a state with a large probability in the interior part, similar to a bound state.", "pages": [ 7 ] }, { "title": "C. 1 -states", "content": "The preceding calculation on 2 + states with the lowlying resonance as reference allows us to select the threebody force ( v 3 b = -0 . 90 MeV, r 3 b = 5 fm, a 3 b = 3) to be included in the calculation of the required 1 -states. To get a well-defined B ( E 1) distribution near the origin, we need, for 1 -states, a basis which has a large hyperradial extension to concentrate many eigenvalues close to the breakup threshold. For this purpose we use a THO basis with b = 0 . 7 fm and γ = 1 . 0 fm 1 / 2 . The eigenvalues of the Hamiltonian for j π = 1 -states are presented for different i max values in Fig. 8. If we compare this 1 -spectrum with the 0 + and 2 + spectra for a fixed i max , it is clear the difference in eigenstates density depending on the extension of the basis, that is, depending on the LST parameters b and γ . Next we can calculate the discrete transition probabilities B ( E 1) from the 0 + ground state to the 1 -eigenstates. We have first checked the completeness of the basis for a given i max comparing the sum of the discrete B ( E 1) transition probabilities with the sum rule Eq. (23). This is given in Table II. The summation con- verges to the exact value given by the sum rule, 1.493 e 2 fm 2 . For the evaluation of the transition probabilities, we use a THO basis with i max = 35 in order to obtain a detailed behavior for the low energy part of the B ( E 1) distribution. In Fig. 9 we show, up to 6 MeV, a reference calculation obtained by using the actual three-body continuum wave functions which, in this simple case, can be computed easily [16] (dash red line). To generate the continuum wave functions we have used the codes FaCE [42] and sturmxx [54] with the same model Hamiltonian. If the smoothing of our THO calculation is done using the overlap with the continuum wave functions, the obtained B ( E 1) distribution is indistinguishable from the reference one. This guarantees that the formalism presented here is working correctly. However, since our interest is to extend this formalism to other systems for which the true continuum wave functions are difficult to obtain, we propose an alternative smoothing procedure following Eqs. (20) and (21). In Fig. 9 the THO distribution for B ( E 1), using this alternative smoothing, is shown (full black line). We have used Poisson distribu- tions with parameter w = 30 √ ε n , such that it ensures a smooth B ( E 1) distribution without spreading it unphysically. Due to the large number of basis states we have near the threshold, the energy dependence of w is convenient to produce a smooth distribution in that region. The total B ( E 1) strength is the same for both calculations (solid and dashed lines) and the behavior is similar, although small differences are observed in the medium energy range. It is also included in Fig. 9 a calculation taken from Ref. [55]. In that work, the hyperspherical adiabatic expansion method is used instead of the HH method. Then, the three-body states are calculated by box boundary conditions, obtaining a discrete spectrum. The discrete B ( E 1) values are smoothed using the finite energy interval approximation. This calculation clearly have a different behavior at low energies. The difference comes from the difficulty to have a large energy level density at low energies solving the problem in a box. It is also apparent that the total B ( E 1) from this calculation is considerably lower than ours. In our calculation the smoothed B ( E 1) energy distribution is very well-defined close to the break-up threshold since we have been able, using the analytical THO, to build a basis for 1 -states concentrating many eigenvalues close to the breakup threshold. In the literature, one can find other B ( E 1) distributions for 6 He using different three-body formalisms. We would like to cite [56] and [57], globally both compare reasonably well with our results but have not been included in Fig. 9 since it is not possible to extract from the plots presented in those publications the detailed behavior at low energies. Without this information, one cannot calculate converged reaction rates below 1-2 GK. It is worth mentioning that the available experimental data [35] (not shown in Fig. 9) differ significantly from all published theoretical calculations. In particular the data do not show the enhancement at energies around 1 MeV. Either new experiment or reanalysis of the existing data is clearly needed. In order to show the convergence of calculations with K max and that K max = 20 is sufficient to provide converged results, we present in Fig. 10 the B ( E 1) distribution for different K max values. In these calculations the same two- and three-body forces are kept fixed. It is clear from the figure that the calculations for K max = 20, 22, and 24 are very close together. Once obtained the B ( E 1) energy distribution, we can finally calculate the reaction rate (Eq. (27)) for the radiative capture reaction α + n + n → 6 He + γ . In Fig. 11, we present the result for the low temperature region of astrophysical interest (0-5 GK). Our calculation is the full black line. In the same figure the reaction rate obtained using the actual three-body continuum wave functions and the corresponding B ( E 1) is represented with a dash red line. The dotted blue line is the calculation of Ref. [55]. We can see from the figure that our calculation agrees very well with the reference calculation for low and high temperatures. In the region between 0.1 and 1.5 GK, there are differences at most by a 3 or 4 factor. These differences with respect to the reference (red dashed line) calculation are more than one order of magnitude in the same temperature region for the calculation of Ref. [55]. This is due to the already referred different behavior of the corresponding B ( E 1) distributions at low energies (below 0.5 MeV). We have checked that this region is crucial for the computation of the reaction rates, especially at low temperatures (below 1-1.5 GK). We have also checked that small differences in the B ( E 1) distributions between 0.5 and 3.5 MeV do not affect the calculated reaction rate provided the same total strength. In Fig. 11, we have also included the results from a sequential model for the radiative capture [7] (dot-dashed orange). This calculation presents the same behavior as ours but is a factor of two larger above 0.2 GK. It is worth mentioning that this sequential calculation assumes first the formation of a dineutron, which is controversial, and then the capture of this by an α particle. An alternative sequential process, presented also in Ref. [7], starts from a neutron capture by the α particle to give 5 He followed by the capture of a second neutron. This provides a reaction rate more than two orders of magnitude smaller in all studied ranges of temperatures. We would like to stress our calculation is based on a full three-body model that makes no assumptions about the reaction mechanism. In this sense all the physical sequential processes are implicitly included.", "pages": [ 7, 8, 9, 10 ] }, { "title": "V. SUMMARY AND CONCLUSIONS", "content": "We have extended the analytical THO method for the study of three-body systems. There are several advantages of the analytical over the numerical THO method: (1) The previous knowledge of the ground-state of the system is not needed. (2) The analytical transformation is easy to be implemented in programming languages. (3) The versatility of the LST depending on the parameters b and γ , allows one to design the best basis for the observable under study. We have applied the formalism to the well-known Borromean nucleus 6 He. This nucleus can be described as an α particle and two valence neutrons. We have seen that the use of the analytical THO method allows a specific basis selection depending on the needs for each angular momentum of the system and on the observable under study. We have calculated a well-converged 0 + ground state and a rather stable 2 + resonant state. For 1 -states we have chosen a basis concentrating many energy levels close to the breakup threshold in order to have a fine description for that region. With these ingredients we have computed the B ( E 1) transition probabilities from the 0 + ground state to the 1 -states. We have checked that the smoothing, using the overlap with the actual continuum wave functions, produce the same B ( E 1). The smoothing using Poisson distributions produces a similar result with small differences in the medium-energy region. In this case, the obtained B ( E 1) distribution is well-defined at low energies (below 0.5 MeV), which is crucial to estimate properly observables such as the reaction rate of the radiative capture α + n + n → 6 He + γ . We have calculated the reaction rate of the radiative capture α + n + n → 6 He + γ from the B ( E 1) distribution for temperatures of astrophysical interest. The result with Poisson smoothing for the B ( E 1) provides a reasonable approach to the continuum reaction rate. However it differs by a factor of 2 from the sequential mechanism presented in Ref. [7], which assumes the dineutron preformation, what is controversial. The differences with the reaction rate calculated in Ref. [55], using also a full three-body model, come from the different behaviors at low energies of the B ( E 1) distributions (below 0.5 MeV) and the different total B ( E 1) strengths. The present results encourage the application of this formalism to more interesting astrophysical cases, such as 9 Be, the tripleα process to produce 12 C, or 17 Ne. In the study of these systems, one of the major problems is the proper treatment of the Coulomb interaction at large distances. However, this problem is absent in the PS methods, such as the analytical THO presented here.", "pages": [ 10 ] }, { "title": "ACKNOWLEDGMENTS", "content": "Authors are grateful to P. Descouvemont, R. de Diego, E. Garrido, and I. J. Thompson for useful discussions and suggestions. This work has been partially supported by the Spanish Ministerio de Econom'ıa y Competitividad under Projects FPA2009-07653 and FIS2011-28738-c0201, by Junta de Andaluc'ıa under group number FQM160 and Project P11-FQM-7632, and by the ConsoliderIngenio 2010 Programme CPAN (CSD2007-00042). J. Casal acknowledges a FPU research grant from the Ministerio de Educaci'on, Cultura y Deporte.", "pages": [ 10 ] } ]

[ { "title": "Confronting measured near and sub-barrier fusion cross-sections for 20 O+ 12 C with a microscopic method", "content": "R. T. deSouza, 1 S. Hudan, 1 V.E. Oberacker, 2 and A.S. Umar 2 1 Department of Chemistry and Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47405, USA 2 Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA (Dated: August 24, 2021) Recently measured fusion cross-sections for the neutron-rich system 20 O+ 12 C are compared to dynamic, microscopic calculations using time-dependent density functional theory. The calculations are carried out on a three-dimensional lattice and performed both with and without a constraint on the density. The method has no adjustable parameters, and its only input is the Skyrme effective NN interaction. While the microscopic DC-TDHF calculations lie closer to the experimental data than standard fusion systematics they underpredict the experimental data significantly. PACS numbers: 26.60.Gj, 25.60.Pj, 25.70.Jj,21.60.-n,21.60.Jz The outer crust of an accreting neutron star provides a unique environment in which nuclear reactions can occur. It has been proposed that the fusion of two neutronrich light nuclei in the outer crust could provide a heat source to ignite thermonuclear fusion of 12 C + 12 C and produce a signature X-ray superburst [1]. To date, however, a limited amount is known either experimentally or theoretically about the fusion of neutron-rich nuclei. Pioneering experiments with heavy nuclei indicate that the fusion below the barrier may be enhanced [2, 3]. Such an enhancement has recently been associated with the importance of neutron transfer channels which effectively lowers the fusion barrier [4]. In the case of fusion of two neutron-rich light nuclei (Z < 20), even less is known. In principle, this is the most promising domain as neutronrich nuclei up to the drip line can be experimentally produced. Recent experimental measurement of near-barrier fusion in the system 20 O + 12 C [5] suggests that the fusion cross-section is enhanced relative to the predictions of the Bass model [6]. As the empirical Bass model is based upon the systematics of known fusion cross-sections near β -stability, it does not include the increased importance of neutron transfer channels for neutron-rich nuclei. The aim of this paper is to directly compare the experimental results with a microscopic approach, namely the time dependent Hartree-Fock (TDHF) theory. The experiment was performed at the SPIRAL1 facility at the GANIL accelerator complex in Caen, France. An 20 O beam with an intensity of 1 -2 × 10 4 p/s impinged on a 100 µ g/cm 2 thick 12 C target. The energy of the beam on target was varied between 1 MeV/A and 2 MeV/A in order to measure the fusion excitation function. Experimental details have been previously published [5] and are summarized here only for completeness. Nuclei produced by fusion subsequently de-excite via evaporation of neutrons and light charged particles (Z ≤ 2) forming evaporation residues. These residues were detected in two segmented silicon detectors located downstream of the target and identified by measuring both their energy and time-of-flight. The annular detectors spanned the angular range 3 . 54 · ≤ θ lab ≤ 21 . 8 · . Due to the presence of a large atomic background in the experiment, a coincidence between an emitted charged particle and the evaporation residue was necessary to distinguish fusion reactions. Statistical model calculations with a Hauser-Feshbach model, evapOR, indicate that, depending on the excitation energy of the compound nucleus formed, approximately 15-25 % of fusion reactions deexcite via emission of at least one charged particle. The time-dependent Hartree-Fock (TDHF) theory provides a useful foundation for a fully microscopic manybody theory of large amplitude collective motion. It is therefore well suited to describing deep-inelastic and fusion reactions [7, 8]. Only in recent years has it become feasible to perform TDHF calculations on a 3D Cartesian grid without any symmetry restrictions and with much more accurate numerical methods [8-13]. In addition, the quality of effective interactions has been substantially improved [14-17]. TDHF theory predicts an energy density functional which is determined by the given effective NN interaction. One may therefore view TDHF as a special case of a time-dependent density functional theory (TDDFT), a concept used in many areas of nuclear physics, condensed-matter physics, and chemistry. Over the past several years, the Density Constrained Time-Dependent Hartree-Fock (DC-TDHF) method for calculating heavy-ion potentials [18] was utilized to calculate fusion cross-sections. We have applied this method to calculate fusion and capture cross-sections above and below the barrier to about 20 systems to date, examples of which can be found in Refs. [19-23]. Recently, we have also investigated sub-barrier fusion between nuclei that occur in the neutron star crust [4]. In all cases, we have found good agreement between the measured fusion cross-sections and the DC-TDHF results. This agreement is rather remarkable given the fact that the only input in DC-TDHF is the Skyrme effective N-N interaction, and there are no adjustable parameters. The TDHF equations for the single-particle wave func- tions can be derived from a variational principle [7]. In the present TDHF calculations we use the Skyrme SLy4 interaction [14] for the nucleons including all of the timeodd terms in the mean-field Hamiltonian [10]. The numerical calculations are carried out on a 3D Cartesian lattice. For the calculations shown in this work, the lattice spans 40 fm along the collision axis and 24 -30 fm in the other two directions, depending on the impact parameter. We first generate very accurate static HF wave functions for the two nuclei on the 3D grid. In the second step, we apply a boost operator to the single-particle wave functions. The time-propagation is carried out using a Taylor series expansion (up to orders 10 -12) of the unitary mean-field propagator, with a time step ∆ t = 0 . 4 fm/c. Presented in Fig. 1 is a contour plot of the mass density during a collision which clearly shows the formation of a neck between the two fragments. This density distribution, shown here for 20 O+ 12 C at E c . m . = 9 . 5 MeV, is representative of collisions for similar systems. As the collision proceeds in the TDHF calculation, transport of protons and neutrons between the two nuclei can be followed within the theory. For larger impact parameters the larger angular momentum of the system leads the two nuclei to separate and a deep-inelastic reaction occurs. For smaller impact parameters the disrupting influence of angular momentum and Coulomb repulsion is insufficient to overcome the nuclear attraction and fusion results. By examining the density distribution as the two nuclei fuse into one within the calculation, one clearly observes the occurrence of a damped dipole resonance and surface waves. Deep inelastic and fusion reactions are the dominant reaction channels in this energy domain. Distinguishing between these two types of reactions is realized by examining the density distribution as a function of time and observing whether one or two large fragments result from the collision. In the absence of a true quantum many-body theory of barrier tunneling, all current sub-barrier fusion calculations assume the existence of an ion-ion potential V ( R ) which depends on the internuclear distance R . Most of the theoretical fusion studies are carried out with the coupled-channels (CC) method [24-27] in which one uses empirical ion-ion potentials (typically Woods-Saxon potentials, or double-folding potentials with frozen nuclear densities). In contrast to the use of these empirical potentials we have adopted a microscopic approach to extract heavy-ion interaction potentials V ( R ) from the TDHF time-evolution of the dinuclear system which describes the dynamics of the underlying nuclear shell structure. In the DC-TDHF approach [18], the TDHF timeevolution proceeds uninhibitedly. At certain times t or, equivalently, at certain internuclear distances R ( t ) the instantaneous TDHF density is used to perform a static Hartree-Fock energy minimization while constraining the proton and neutron densities to be equal to the instantaneous TDHF densities. This means that we allow the single-particle wave functions to rearrange themselves in such a way that the total energy is minimized, subject to the TDHF density constraint. In a typical DC-TDHF run, we utilize a few thousand time steps, and the density constraint is applied every 10 -20 time steps. We refer to the minimized energy as the 'density constrained energy' E DC ( R ). The ion-ion interaction potential V ( R ) is obtained by subtracting the constant binding energies E A 1 and E A 2 of the two individual nuclei In direct TDHF calculations the fusion cross-section is calculated by determining the maximum impact parameter for which fusion occurs and applying the sharp cut-off approximation. For example, in the case of the reaction 20 O+ 12 C at E c . m . = 9 . 5 MeV we find that impact parameters b ≤ b max = 4 . 075 fm result in fusion, while impact parameters b > b max lead to deep-inelastic reactions. Using the sharp cut-off model, the fusion crosssection is given by σ fus = πb 2 max = 52 . 2 fm 2 = 522 mb. In contrast, in DC-TDHF method fusion cross-sections are obtained by integrating the Schrodinger equation for the potential V ( R ) with a coordinate-dependent mass [21]. We begin the comparison of the microscopic calculations with experimental data by examining the well studied system 16 O+ 12 C. Shown in Fig. 2 are four sets of experimental data for the total fusion cross-section along with the corresponding microscopic DC-TDHF calculations. The experimental techniques used to determine the fusion cross-section range from gamma spectroscopy of the charged particle channels (Christensen and Cujec) to direct measurement of the evaporation residues (Eyal and this work). It should be stressed that the data represented by the red filled circles (this work) utilized the same experimental technique and setup as for the 20 O+ 12 C results subsequently presented. For energies E c . m . < 9 . 5 MeV all the experimental data are in agreement. At higher energies however, the cross-sections measured by Christensen et al. slightly exceed that of the three other datasets. We have no explanation at present for the larger cross-sections measured by Christensen et al. Since the cross-section measurements of Eyal, Cujec, and the present work are all in good agreement we take these cross-sections to accurately represent the true fusion cross-section. It is interesting to note that the DC-TDHF calculations also slightly exceed the measured cross-sections for E c . m . > 7 MeV. At the largest energies measured this excess is of the order of 20 %. Indeed, the agreement of DC-TDHF results with the data of Christensen et al. at the highest energies was somewhat surprising since TDHF dynamics for light heavyions at these energies do not properly account for various breakup channels present for these systems, and results in a fusion-like composite system with long-time collective oscillations. In coupled-channel calculations this discrepancy is cured by introducing a small imaginary potential in the vicinity of the potential minimum [27]. Having established the degree of confidence through the comparison of the fusion cross-sections in 16 O+ 12 C, we then calculated fusion in 20 O+ 12 C. While for isolated 16 O and 12 C nuclei the Hartree-Fock (HF) ground state is found to be spherical in agreement with both theory and experiment, for 20 O the HF calculations predict a prolate quadrupole deformation of β 2 = 0 . 25. This deformation is in disagreement with selfconsistent mean field calculations with pairing (SkyrmeHFB) [31] which predict a spherical nucleus. Moreover, the measured energy level spectrum [32] also shows this nucleus to be spherical. In addition, a measurement of the magnetic moment of 20 O [33] also indicates its spherical nature. We attribute this prolate deformation predicted by the HF calculations to the lack of pairing in the method. One unfortunate consequence of the DCTDHF approach in calculating the fusion cross-section lies in the treatment of pairing during the collision process. In unrestricted TDHF calculations, the BCS occupation numbers can be kept frozen during the collision to have correct initial states. This approximation cannot be utilized in the DC-TDHF method because the static HF solution coupled with a constraint on the instantaneous TDHF density for the combined system requires the reevaluation of the occupation numbers for the lowest energy solution. Consequently, calculations with the DC-TDHF method do not include pairing. On qualitative grounds, it can be argued that this omission should result in a slightly larger prediction of the fusion crosssection. The reason is that pairing results in a spherical 20 O nucleus, and the fusion barrier for a spherical nucleus is higher than the lowest barrier for a deformed nucleus. In order to calculate the fusion cross-section for this system within DC-TDHF method we therefore take an average of all initial orientation angles β of the deformed 20 O nucleus, where β is defined as the angle between the internuclear distance vector and the symmetry axis of the deformed nucleus. As the collision occurs, using TDHF dynamics, it is possible to compute the corresponding coordinate dependent mass parameter M ( R ) [21]. At large distance R , the mass M ( R ) is equal to the reduced mass µ of the system. At smaller distances, when the nuclei overlap, the mass parameter generally increases. In order to calculate the fusion cross-section more easily, one can replace the coordinate-dependent mass M ( R ) and the original potential V ( R ) with the constant mass µ and the 'transformed potential' U ( ¯ R ), using a scale transformation [21]. In Fig. 3 we display the transformed potentials U ( ¯ R ) for initial orientation angles β = 0 · , 10 · , ..., 90 · of 20 O. For sufficiently large separation between the two nuclei, R ≥ 9, fm all the transformed heavy-ion potentials are the same regardless of the orientation of the two nuclei. With decreasing distance as the two nuclei come into contact, the heavy-ion potentials differ. For a nucleus with prolate deformation, the orientation angle β = 0 · leads to the lowest potential barrier. This reduction in the potential occurs because the distance between the nuclear surfaces is minimized for this orientation which is important because of the short-range nature of the strong interaction. The Coulomb interaction is accurately calculated by solving the 3D Poisson equation numerically during the collision. We observe that with increasing orientation angle, the barrier height increases and the barrier position is shifted to smaller distances. Displayed in the inset of Fig. 3 is the fusion excitation function associated with the orientations of β =0 · and 90 · as well as the angle averaged result. As one might quali- ely expect, based upon the heavy-ion potentials, for a given E c . m . , the orientation β = 0 · with the lowest potential barrier corresponds to the largest cross-section. In Fig. 4 we compare the microscopic calculations using the DC-TDHF method to the experimental data. The experimental data at E c . m . = 7.35, 9.29, and 15.24 MeV are shown as the filled circles. Due to the atomic background previously mentioned only the fraction of the fusion cross-section associated with subsequent charged particle emission was experimentally measured. In order to compare the microscopic calculations with the experimental data, we therefore calculated the fraction of compound nuclei produced at each incident energy that de-excite via emission of at least one charged particle. To calculate the de-excitation of the fused nuclei we utilized a Hauser-Feshbach statistical model, evapOR. The theoretically predicted fusion cross-section associated with subsequent charged particle channels is depicted by the dashed line. This predicted cross-section clearly underpredicts the experimentally measured cross-section. At the highest energy, E c . m . =15.24 MeV, the predicted cross-section is 60 % of the experimentally measured one while at the lowest energy the predicted cross-section is substantially lower, only 30 % of the experimental value. For reference, we also present, as a dot-dash line, the fusion cross-section predicted by the Bass systematics that is associated with charged particle emission. This crosssection is less than that of the DC-TDHF method most likely reflecting the influence of neutron transfer channels in aiding the fusion process. It should be appreci- ated that not only does the DC-TDHF method predict a larger cross-section at all energies as compared to the Bass systematics, but this increase grows with decreasing E c . m . . This result suggests that the neutron transfer becomes more important in the sub-barrier domain. The result that the DC-TDHF method underpredicts the experimental data by a significant amount is noteworthy. Moreover, it should be noted that the lack of pairing in the DC-TDHF method and the resulting deformation of the 20 O, as previously discussed, acts to increase the predicted cross-section implying that the discrepancy between theoretical prediction and experimental data is at least as large as that evident in Fig. 4. In order to assess the impact of pairing on the measured cross-sections more quantitatively we have performed unrestricted TDHF calculations (no barrier tunneling) at energies above the barrier. These calculations were initialized with BCS/Lipkin-Nogami pairing for 20 O which resulted in a spherical nucleus, consistent with the experimental data. During the collision of the 20 O with the 12 C the BCS occupation numbers are kept frozen. The results of these calculations are presented as the open triangles in Fig. 4. A slight reduction in the total cross-section is evident. This reduction is of the order of 5-20 % with the largest reduction for the lowest energy point calculated. From these calculations one can infer that the inclusion of pairing in the DC-TDHF calculations should result in a slight reduction of the predicted cross-section, thus increasing the discrepancy with the experimental data as anticipated. It is tantalizing to speculate about possible reasons for the discrepancy between the theoretical predictions and the experimental data. In general terms, the discrepancy between the experimental data for the charged particle channels in Fig. 4 (solid points) and the corresponding DC-TDHF calculations can be thought of as originating either from an underprediction of the total fusion crosssection or from an underestimation of the relative importance of the charged particle decay in the de-excitation of the compound nucleus. Either or both of these sources could explain the underprediction of the cross-section. It is therefore important to not only measure the total fusion cross-section but also the cross-section for individual decay channels. Furthermore, since the underprediction exists for energies well above the barrier, E ≈ 9 . 5 MeV, it is not simply a question of enhanced tunneling. Moreover, beyond the overall underprediction of the microscopic method, one observes that the experimental data manifests a slower fall-off for the measured cross-section with decreasing incident energy as compared to the DCTDHF calculations. This experimentally determined energy dependence is highly provocative and it remains to be seen whether the total fusion cross-section also exhibits this slower fall-off. As this fall-off is intimately related to the neutron transfer channels, near and subbarrier fusion of neutron-rich nuclei provides direct access to the extent of the neutron density distribution.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work has been supported by the U.S. Department of Energy under Grant No. DE-FG02-96ER40975 (Vanderbilt University) and DE-FG02-88ER-40404 (IU).", "pages": [ 5 ] } ]

[ { "title": "The non-linear Glasma", "content": "Jurgen Berges 1 , 2 , Soren Schlichting 1 , 3 1 Institut fur Theoretische Physik, Universitat Heidelberg Philosophenweg 16, 69120 Heidelberg 2 ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum Planckstr. 1, 64291 Darmstadt 3 Theoriezentrum, Institut fur Kernphysik Technische Universitat Darmstadt Schlossgartenstr. 9, 64289 Darmstadt We study the evolution of quantum fluctuations in the Glasma created immediately after the collision of heavy nuclei. It is shown how the presence of instabilities leads to an enhancement of non-linear interactions among initially small fluctuations. The non-linear dynamics leads to an enhanced growth of fluctuations in a large momentum region exceeding by far the originally unstable band. We investigate the dependence on the coupling constant at weak coupling using classical statistical lattice simulations for SU (2) gauge theory and show how these non-linearities can be analytically understood within the framework of two-particle irreducible (2PI) effective action techniques. The dependence on the coupling constant is only logarithmic in accordance with analytic expectations. Concerning the isotropization of bulk quantities, our results indicate that the system exhibits an order-one anisotropy on parametrically large time scales. Despite this fact, we find that gauge invariant pressure correlation functions seem to exhibit a power law behavior characteristic for wave turbulence.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The great interest in relativistic heavy ion collision experiments is to a large part driven by its possibility to explore the properties of deconfined strongly interacting matter described by quantum chromodynamics (QCD). The past decades have revealed remarkable properties of the quark-gluon plasma, probably most strikingly its behavior similar to an ideal fluid [1, 2]. However these properties are not directly accessible experimentally as they are encoded in the final particle spectra measured by the detectors at RHIC and the LHC [3-7]. Consequently the extraction of medium properties crucially depends on theoretical input, such as the time when the plasma thermalizes locally, which has to be calculated within an ab-initio approach. The non-equilibrium dynamics of high energy nuclear collision poses a challenging problem in the underlying theory of QCD, which in practice can only be addressed with suitable approximations. In this context a field theoretical framework known as the 'color glass condensate' has been developed, which provides a real time ab-initio description of nuclear collisions at high energies [8-10]. While several properties of the initial state right after the collision can be explored within this approach [11-15], present studies have not yet been able to explain the thermalization mechanism [16, 17]. Here quantum fluctuations may play an important role as they break the longitudinal boost invariance of the system and can be strongly amplified in the presence of plasma instabilities [16-33]. In this paper we investigate the impact of quantum fluctuations in the color glass condensate description of high energy heavy ion collisions. We work at weak coupling, where the presence of plasma instabilities has been established in previous works [16-18], and present results from classical-statistical lattice simulations along with analytic estimates. We investigate in detail the different dynamical stages of the system undergoing an instability and find that in addition to the 'primary' Weibel type instability [16-18], 'secondary' instabilities emerge due to non-linear interactions of unstable modes. This mechanism is very similar to previous observation in non-expanding gauge-theories [19, 20] and cosmological models [34] and can be naturally understood in the framework of two-particle irreducible (2PI) effective action techniques [35]. This paper is organized as follows: In Sec. II we present a short review of the dynamics of nuclear collisions in the color glass condensate (CGC) framework [8]. We comment in particular on recent developments to include quantum fluctuations within an ab-initio approach [9, 10] and show how the discussion in the literature is related to a classical-statistical treatment. In Sec. IV we present results from classical-statistical lattice simulations. We focus on non-linear effects and obtain the relevant growth rates and set-in times of primary and secondary instabilities. We find that our results are rather insensitive to the value of the strong coupling constant α s as long as α s glyph[lessmuch] 1. We also investigate the impact of instabilities on bulk properties of the system such as the ratio of longitudinal pressure to energy density. Here our results indicate that the system remains anisotropic on parametrically large time scales. We summarize our results and conclude with Sec. V.", "pages": [ 1, 2 ] }, { "title": "A. High-energy limit and CGC", "content": "From a non-equilibrium point of view an ab-initio approach to heavy-ion collisions requires to determine the initial density matrix consisting of two incoming nuclei in the vacuum and subsequently solving the initial value problem in quantum chromodynamics. Though this is beyond the scope of present theoretical methods, one may apply suitable approximations in the high energy and weak coupling limit, which make the problem computationally feasible. This is usually discussed in terms of the light-cone coordinates x ± = ( t ± z ) / √ 2, where at sufficiently high energies the incoming nuclei travel close to the light-cone, which is given by x ± = 0. The collision takes place around the time when x + = x -= 0, where the center of mass of the nuclei coincides and an approximately boost invariant plasma is formed after the collision. The plasma dynamics in the forward light-cone ( x ± > 0) is usually discussed in terms of the co-moving coordinates where τ is the proper time in the longitudinal direction and η is the longitudinal rapidity. The metric in these coordinates takes the form g µν ( x ) = diag(1 , -1 , -1 , -τ 2 ) and we denote the metric determinant as g ( x ) = det g µν ( x ). The dynamics of the collision and the geometry of the coordinates is illustrated in Fig. 1. The different colors in the forward light-cone illustrate the dynamics of the longitudinally expanding plasma, which we study in this paper. In the color-glass framework one considers the dynamics of the plasma at mid-rapidity ( η glyph[lessmuch] η Beam ) and the nuclear partons at high rapidities separately. In the eikonal approximation the trajectories of the incoming nuclei are unaffected by the collision, while the dynamics of gluons at mid-rapdity is described by the classical action where ∫ x = ∫ d 4 x √ -g ( x ) and F a µν ( x ) denotes the nonabelian field strength tensor with Lorentz indices µ, ν and color indices a = 1 ...N 2 c -1 for SU ( N c ) gauge theories. In addition, the gauge field A a µ ( x ) is coupled to an eikonal current J µ a ( x ), which is determined by the properties of the nuclear wavefunction at high rapidities. In practice, the separation in fast and slow degrees of freedom is performed by a renormalization group procedure prescribed by the JIMWLK equations [8]. In the high energy limit the eikonal current J µ a ( x ) is given by static color sources on the light-cone and takes the form where δ µ ± is the Kronecker delta in light-cone coordinates and we denote transverse coordinates as x ⊥ = ( x 1 , x 2 ). The color charge densities glyph[rho1] (1 / 2) a ( x ⊥ ), where the superscript (1 / 2) labels the different nuclei, contain all further information about the beam energy, nuclear species and impact parameter dependence. At high collider energies, these have been conjectured to exhibit a universal behavior, which is described by the saturation scale Q s and the value of the strong coupling constant α s [36, 37]. In this work we simply adapt the McLerranVenugopalan (MV) saturation model [38], where the color charge densities of the nuclei are given by uncorrelated Gaussian configurations The model parameter g 2 µ is proportional to the physical saturation scale Q s up to logarithmic corrections and reflects the properties of the saturated wavefunctions of large nuclei [38]. Consequently the current in Eq. (4) is parametrically large, i.e. formally O (1 /g ) in powers of the coupling constant. This makes the problem inherently non-perturbative and we will come back to this aspect when we discuss the impact of quantum fluctuations. In addition to Eq. (5), we impose a color neutrality constraint on the color charge densities such that the global color charge vanishes separately for each nucleus, i.e. By specifying the eikonal current according to Eq. (4), the longitudinal geometry of the collision has effectively been reduced to the collision of two-dimensional sheets and there is no longer a longitudinal scale inherent to the problem. However quantum fluctuations explicitly break the longitudinal boost invariance of the system and may therefore play an important role in the non-equilibrium dynamics right after the collision [10]. Before we turn to a detailed discussion of quantum fluctuations, we will briefly review the classical solution to the particle production process. We will show later, in Sec. II C, how this solution emerges also in the weak-coupling limit of the quantum field theory, where quantum fluctuations can be handled properly.", "pages": [ 2, 3 ] }, { "title": "B. Classical solution", "content": "Neglecting quantum fluctuations for the moment, the absence of a longitudinal scale in Eq. (4) leads to boostinvariant solutions of the classical Yang-Mills field equations In the classical color glass picture, the strong color-fields right after the collision are entirely determined by the continuity conditions on the light-cone ( x ± = 0) [39-42]. Adapting the Fock-Schwinger gauge condition ( A τ = 0), where the classical Yang-Mills action takes the form ( i = 1 , 2) the initial state right after the collision can be specified at τ = 0 + , where the chromo magnetic and electric fields are given by [39-42] Here α (1 / 2) i ( x ⊥ ) are pure gauge configurations which describe the Yang-Mills field outside the light-cone. They are related to the nuclear color charge densities by [3942] and depend on transverse coordinates only. The relations (9) and (10) specify the 'Glasma' initial state at τ = 0 + right after the collision. The time evolution in the forward light-cone can be studied numerically by solving the lattice analogue of the classical evolution equations and has been studied extensively [11-14]. However the longitudinal boost invariance of the system is preserved in this classical evolution and leads to an effectively 2+1 dimensional Yang Mills theory coupled to an adjoint scalar field [11-14]. In order to study the full 3+1 dimensional Yang Mills theory it is therefore crucial to include quantum fluctuations, which break the boost-invariance of the system explicitly.", "pages": [ 3 ] }, { "title": "C. Quantum fluctuations", "content": "The inclusion of quantum fluctuations has recently attracted great attention due to their expected importance in understanding the thermalization process [9, 10]. This concerns in particular the inclusion of vacuum fluctuations in the initial state, which are quantum in origin but evolve classically at sufficiently weak coupling and short enough times [43]. In contrast to most discussions in the literature [9, 10], our analysis is based on the two particle irreducible (2PI) effective action framework [35]. This formalism has been successfully applied to a variety of similar problems in scalar theories [34, 44] and gauge theories [19, 20, 45] and recently been developed for the problem under consideration here [46]. We give a short general introduction to the formalism and show first how the realization proposed in Ref. [10] emerges in this framework. Then we go beyond that discussion and identify sub-leading quantum corrections and describe the non-linear dynamics of instabilities analytically. The quantum evolution equations can be formulated in terms of the expectation values of the gauge field operators ˆ A a µ ( x ) denoted by and the time ordered two-point correlation function The two independent parts of the propagator G ab µν ( x, y ) can be expressed in terms of the spectral and statistical two-point correlation functions which are associated to the commutator and anticommutator Here expectation values are given by the trace over the initial vacuum density matrix in the presence of the eikonal currents. The initial density matrix is specified in the remote past ( t 0 →-∞ ), where the background field A a µ vanishes and the statistical fluctuations F ab µν take the vacuum form (see e.g. Ref. [46]). In contrast, the initial values of the spectral function are entirely determined by the equal time commutation relations, which in temporal gauge ( A 0 = 0) read and are valid at all times. 1 The gauge field expectation values in Eq. (11) correspond to the Glasma background fields, while the spectral and statistical two-point functions contain the quantum fluctuations. The evolution equations for connected one and two-point correlation functions follow from the stationarity of the two particle irreducible (2PI) effective action [47, 48] and form a closed set of coupled evolution equations. The set of equations is given by the evolution equation of the macroscopic field and the evolution equations for spectral and statistical two point correlation functions, which can be written as [35] Here we denote ∫ b a dz = ∫ b a dz 0 ∫ d d z √ -g ( z ) and iG -1 ,µν 0 ,ab [ x ; A ] denotes the free inverse propagator with γ ( x ) = √ -g ( x ) and we introduced the (background) covariant derivative The non-zero spectral and statistical parts of the selfenergy Π ( ρ/F ) [ A,G ] on the right hand side and the local part Π (0) [ G ] on the left hand side make the evolution equations non-linear in the fluctuations. In general they contain contributions from the vertices depicted in Fig. 2, where in addition to the classical three gluon vertex there is a three gluon vertex associated with the presence of a non-vanishing background field. The explicit expressions for the derivatives on the right hand side of Eq. (18) and the self-energy contributions entering Eqns. (19) and (20) have been calculated to three loop order ( g 6 ) in Ref. [45] and the corresponding expressions in co-moving ( τ, η ) coordinates can be found in Ref. [46]. Before we turn to a more detailed discussion of the right hand side contributions of Eqns. (18), (19) and (20), it is insightful to consider first the leading part in a weak coupling expansion. We will see shortly how this recovers the classical solution for the background field, as discussed in Sec. II B, while initial state vacuum fluctuations are already included to leading order in terms of the connected two-point correlation functions. In order to isolate the leading contributions one has to take into account the strong external currents J µ a ∼ O (1 /g ), which induce non-perturbatively large background fields A a µ ( x ) ∼ O (1 /g ). In contrast, the statistical fluctuations F ab µν ( x, y ) originate from initial state vacuum and are therefore initally O (1). The spectral function ρ ab µν ( x, y ) has to comply with the equal time commutation relations (16) and is therefore parametrically O (1) at any time. Considering only the leading contributions in a weak coupling expansion, the evolution equation (18) reduces to its classical form (c.f. Eq. (7)) and the evolution equations for the spectral and statistical two-point correlation functions at leading order read where sub-leading contributions are suppressed by at least a factor of g 2 relative to the leading contribution. It is important to realize that at this order the evolution of the Glasma background fields decouples from that of the fluctuations, i.e. there is no back-reaction from the fluctuations on the background fields. Therefore the dynamics of the background fields remains unchanged and one recovers the classical field solutions discussed in Sec. II B. In addition the evolution of vacuum fluctuations of the initial state is taken into account by Eqns. (24) and (25) to linear order in the fluctuations. To this order the quantum field theory is known to agree with the classical statistical theory [43] and Eqns. (24, 25) can equivalently be obtained by considering the linearized classical evolution equations for small fluctuations 2 (see e.g. Ref. [10]). The linear approximation in Eqns. (24) and (25) yields a major simplification to Eqns. (19) and (20), as one can solve Eqns. (24) and (25) independently from the evolution of the background field. This has been exploited in Ref. [10] to obtain the spectrum of initial fluctuations right after the collision. In turn, the range of validity of the approximation is limited to the domain where fluctuations remain parametrically small. This is, however, not the case in the forward light-cone ( τ > 0), where Eqns. (24) and (25) exhibit plasma instabilities associated to exponential growth of statistical fluctuations [16, 17] for characteristic modes, where ν is the Fourier coefficient with respect to relative rapidity according to In this regime fluctuations can become parametrically large and strongly modify the naive power counting. Since the approximation underlying Eqns. (24) and (25) is not energy conserving, one encounters exponential divergences when stressing Eqns. (24, 25) beyond their range of validity [49]. In this regime it is crucial to include higher order self-energy corrections, which naturally cure the divergencies. The associated power counting is discussed in Sec. II D, where we identify the diagrammatic contributions, which contain the first relevant corrections and analyze their impact on the dynamics of the instability. A systematic way to include self-energy corrections to all orders was outlined in Ref. [10] and will be discussed in more detail in Sec. III on the classical-statistical approximation.", "pages": [ 3, 4, 5 ] }, { "title": "D. Dynamical power counting and non-linear dynamics", "content": "In order to go beyond a fixed-order coupling expansion of the 2PI effective action, one has to develop a power counting scheme which takes into account not only the suppression by the small coupling constant but also the enhancement due to parametrically large fluctuations in the presence of non-equilibrium instabilities. This has been worked out in detail for scalar theories [34, 44] and applies in a similar way to gauge theories [19, 20]. In this power counting scheme self-energy corrections are classified according to powers of the coupling constant g as well as powers of the background field A a µ ( x ) and the statistical fluctuations F ab µν ( x, y ). For a generic self-energy contribution containing powers g n F m A l ρ k , the integers n, m and l yield the suppression factor from the coupling constant ( n ) as well as the enhancement due to a parametrically large background field ( l ) and parametrically large fluctuations ( m ). The 'weight' of the spectral function ( k ) remains parametrically of order one at all times as encoded in the equal-time commutation relations. For parametrically large macroscopic fields A a µ ( x ) ∼ 1 /g one expects a sizable self-energy correction once fluctuations have grown as large as F ab µν ( x, y ) ∼ 1 /g ( n -l ) /m for characteristic modes. The hierarchy emerging from a classification of diagrams in terms of β = ( n -l ) /m is shown in Tab. I. The one loop diagram shown in the upper panel contains two three gluon vertices, which give rise to a suppression factor g 2 (( n -l ) = 2). On the other hand the diagram is enhanced by two statistical propagators in the loop ( m = 2) and we can classify the overall contribution as O ( g 2 F 2 ). Similarly one can analyze the two loop diagrams and the tadpole diagram also shown in Tab. I. The tadpole diagram contains a suppression factor g 2 from the four gluon vertex ( n = 2 , l = 0) and one statistical propagator ( m = 1), such that the overall contribution can be classified as O ( g 2 F ). The two loop diagrams are of order g 4 ( n -l = 4) in the coupling constant and contain at most three statistical propagators ( m = 3). The overall contribution is thus O ( g 4 F 3 ) in the dynamical power counting. The classification in terms of β = ( n -l ) /m shows that the only diagram yielding a contribution to β = 1 is the one-loop diagram in the upper panel of Tab. I. The leading contribution of higher order loop diagrams can be classified as β = 2 L/ ( L + 1), where L ≥ 1 is the number of loops. The tadpole diagram yields β = 2. Finally there are self-energy contributions which contain powers of the spectral function instead of statistical propagators. These give rise to even higher values of β ≥ 2 and are therefore suppressed at sufficiently weak coupling. This hierarchy of diagrams has important consequences for the dynamics of systems undergoing an instability, where initially small fluctuations grow exponentially in time. At early times statistical fluctuations are small and the system is accurately described by the set of equations (23, 24, 25), which give rise to the linear instability regime [16, 17]. At later times, when statistical fluctuations have grown larger, self-energy corrections become important and alter the dynamics of the system. In this regime self-energy contributions with smaller values of β become important at an earlier stage as compared to contributions with higher values of β , as each diagram requires F ab µν ( x, y ) ∼ g -β to yield a significant contribution. The one-loop diagram in the top panel of Tab. I is of order O ( g 2 F 2 ) in the dynamical power counting. The contribution of this diagram becomes of O (1) as soon as statistical fluctuations have grown as large as O (1 /g ), while all higher order self-energy corrections are still suppressed by at least a fractional power of the coupling constant. Accordingly there is a regime where the one loop diagram displayed in the top panel of Tab. I yields the only relevant correction. In order to investigate the impact of the one-loop correction in more detail, we will in the following neglect all contributions to the memory integrals in Eqns. (19, 20) which originate from outside the forward light cone. This assumption is justified at weak coupling, where fluctuations are sufficiently small until the time when they exhibit exponential growth due to the presence of the instability. We can then switch to a discussion in ( τ, η ) coordinates and employ the FockSchwinger ( A τ = 0) gauge condition in the following. The self-energy contributions from the one-loop diagram in Tab. I (top panel) then take the form [46] where x = ( τ x , x ⊥ ) collectively labels the proper time and transverse coordinates, ν is the Fourier coefficient with respect to relative rapidity and Lorentz indices take the values µ = 1 , 2 , η . The three gluon vertices in Eqns. (28) and (29) take the form where in addition to the classical three gluon vertex there is a contribution from the background field. The classical three gluon vertex can be written as (no summation over b and c ), where the derivative operator ˜ ∂ µ,x a = ( ∂ τ x , -∂ x ⊥ , -iτ -2 x ν ) only acts on the propagator with color index a , and the contribution from the background field is given by with In order to investigate the impact of the above self-energy corrections, we first note that the ρρ term in Eq. (28) is a genuine quantum correction, which is absent in the classical statistical theory [43]. On the other hand statistical fluctuations grow exponentially according to Eq. (26) for unstable modes, such that the dominant contribution in Eq. (28) arises from the classical ( FF ) contributions and one can safely neglect the sub-leading quantum corrections. The right hand side of the evolution equations (20) then receive exponentially enhanced contributions from the self-energies (28) and (29), which grow exponentially in (proper) time. This behavior can be verified explicitly by performing a 'memory expansion' of Eq. (20), i.e. evaluating the memory integrals on the right hand side of Eq. (20) around the latest (proper) times of interest [34, 44]. We find that the dominant contribution originates from the statistical self-energy in Eq. (28), whereas the contribution from the spectral self-energy in Eq. (29) is effectively β = 2 in the above classification scheme and thus becomes important only at later times. The modified evolution equations in this regime then take the form where δ τ is the extent in time for which the memory integrals are evaluated. To obtain Eq. (34), we performed a leading order Taylor expansion of the integrand in Eq. (20) around τ z = τ y and made use of the equal time commutation relations in Eq. (16) to estimate the spectral function. The one-loop integral Π ( F ) , which appears on the right hand side of Eq. (34), is dominated by the contributions from unstable modes and is proportional to exp[2Γ 0 √ g 2 µτ ] at equal times. The contribution on the right hand side acts as a source term in the evolution equation of statistical fluctuations. As is well known from various examples of self-interacting quantum field theories, this term leads to a non-linear amplification of instabilities, where 'secondary' instabilities with strongly enhanced growth rates emerge over a large range of momenta. This has been observed in scalar field-theories [34, 44] as well as in non-abelian gauge theories [19, 20] and is a rather generic feature of self-interacting theories undergoing an instability. We will show in Sec. IV that the phenomenon of non-linear amplification also emerges in numerical simulations of the unstable Glasma and plays a crucial role in understanding gauge-invariant observables. The characteristic time scale for non-linear amplification to take place can be infered by comparing the magnitude of the non-linear contributions in Eq. (34) to the contributions of the background field. In the weak coupling limit this time scale is parametrically given by where Γ 0 is the characteristic growth rate of primary instabilities and we assumed F ab µν ∼ O (1) at τ = 0 + for characteristic modes. In addition to Eq. (35) there are sub-leading contributions associated to the delayed set in of primary instabilities and the spectral distribution of statistical fluctuations in the initial state. The prior give rise to a constant contribution √ g 2 µτ Primary , while the latter enter only logarithmically in this estimate. The emergence of secondary instabilities again modifies the power-counting, and one has to take into account also the contributions which originate from modes which exhibit secondary instabilities. Also higher order self-energy corrections become increasingly important as time proceeds and non-linear amplification can repeat itself, until at some point the growth of instabilities saturates and occupancies become as large as O (1 /g 2 ). In this regime every truncation at a fixed loop order breaks down and the problem has to be addressed in a fully non-perturbative way. While in scalar quantum field theories there are different ways to address this problem, involving e.g. large N resummation techniques [50], the most frequently employed approach in gauge theories is the classical-statistical approximation.", "pages": [ 5, 6, 7 ] }, { "title": "III. CLASSICAL-STATISTICAL APPROXIMATION", "content": "In the classical statistical approximation all fluctuations evolve classically and one neglects genuine quantum fluctuations, such as the ( ρρ ) quantum term in Eq. (28). When transforming to the language of expectation values, as employed in the previous section, it can be shown that the prescription resums an infinite subset of diagrams [43], such that the dynamics of fluctuations is treated on equal footing with the background fields. The set of diagrams included in the classical-statistical treatment can be identified as the self-energy corrections, which contain the most powers of the statistical propagator as compared to powers of the spectral function for each topology [43]. Accordingly, this corresponds to resumming the leading effects of the instability to all orders in the coupling constant [10]. In contrast to expansions at fixed loop orders, the classical statistical approximation thus provides a robust approximation scheme, which is particularly well suited for problems involving large statistical fluctuations. However there are problems associated with the Rayleigh-Jeans divergence, which concern the handling of ultra-violet divergences and the approach to thermal equilibrium at late times, which are discussed in more detail in the literature [43]. In the classical-statistical theory observables 〈 O ( x ) 〉 cs are calculated as an ensemble average of classical field solutions A cl [ A τ 0 , E τ 0 ], which individually satisfy the Yang-Mills evolution equations. The canonical field variables A τ 0 and E τ 0 at initial time τ 0 are distributed according to a phase space density functional W [ A τ 0 , E τ 0 ], such that [43] Here O cl [ A τ 0 , E τ 0 ] denotes the fact that the observable is evaluated as a functional of the classical field solution where A cl [ A τ 0 , E τ 0 ] is the classical Yang-Mills field solution with initial conditions A cl = A τ 0 and E cl = E τ 0 at initial time τ 0 . In practice, Eqns. (36) and (37) state that vacuum fluctuations of the initial state are added on top of the background field at initial time, while the subsequent classical evolution keeps track of all nonlinearities. The set of equations (36) and (37) is precisely the same as in Ref. [10], where it has been obtained as a partial resummation scheme of the perturbative corrections due to vacuum fluctuations of the initial state. The procedure outlined in Ref. [10] consist of a hybrid approach, which employs the linearized evolution equations (23, 24, 25) outside the forward light-cone, where fluctuations are small, while switching to a classical-statistical description in the forward light-cone. This has the advantage that the evolution of the background field and the spectrum of fluctuations, which enter the phase-space weight W [ A τ 0 , E τ 0 ], can be obtained analytically. The phase-space average in Eq. (36) can then be taken on the Cauchy surface τ 0 = 0 + , such that only the dynamics in the forward light-cone has to be studied within classicalstatistical lattice simulations. We follow this approach but employ a simpler spectrum of initial fluctuations, as specified in Sec. III B, instead. This simplification is justified at sufficiently weak coupling, where the spectrum of fluctuations is quickly dominated by the growth of primary instabilities rather than the initial spectrum.", "pages": [ 7, 8 ] }, { "title": "A. Coupling dependence", "content": "An important property of the classical-statistical description is the independence of the gauge coupling constant g , in the sense that the classical-evolution equations are invariant under a change of variables whereas the spectral function remains unaffected, as encoded in the equal time commutation relations. 3 With the rescaling (38) the entire coupling dependence in the classical-statistical evolution can be absorbed into the initial conditions at τ = 0 + , while the classical evolution equations become independent of the coupling constant. For the Glasma background fields this can be achieved most efficiently by replacing gA a µ ( x ) → ˜ A a µ ( x ) and gglyph[rho1] (1 / 2) ( x ⊥ ) → ˜ glyph[rho1] (1 / 2) ( x ⊥ ) for all expressions in Sec. II B, where in the MV model prescription Here the model parameter g 2 µ is directly related to the saturation scale Q s without further powers of the coupling constant appearing in the expression [38]. Accordingly the defining equation (39) is indeed independent of the value of the coupling constant. In contrast, the coupling constant g appears explicitly in the initial spectrum of fluctuations, given by and similarly for derivatives at initial time. Here it is important to note that the magnitude of the vacuum fluctuations F ab µν ( x, y ) on the right hand side is independent of the value of the coupling constant. Thus the initial suppression of vacuum fluctuations compared to the boost invariant background fields is the only measure of the strong coupling constant, present in the classical-statistical field theory. We will exploit this fact in Sec. IV B, where we vary the amplitude of initial fluctuations in our simulations to study the coupling dependence of our results.", "pages": [ 8 ] }, { "title": "B. Initial conditions at τ = 0 +", "content": "Within the classical-statistical framework, the initial conditions for the time-evolution in the forward lightcone are given at τ = 0 + by the set of equations (9) and (10), complemented by the spectrum of initial fluctuations. While in general it is necessary to implement the spectrum of fluctuations as specified in Ref. [10], it is also clear that at sufficiently weak coupling many details of the spectral shape of the initial fluctuations become irrelevant due to the presence of instabilities, which quickly dominate the spectrum. Since implementing the spectrum of fluctuations in Ref. [10] is numerically challenging, we will therefore stick to a simpler choice, where in accordance with previous works [16, 17] the statistical fluctuations initially take the form with such that δE a µ ( x ) is an additive contribution to the background field E a µ ( x ) at initial time. The advantage of this construction is that the Gauss constraint is satisfied explicitly for arbitrary functions f ( η ) and e a i ( x ⊥ ). For the numerical simulations we choose where p T and ν are the Fourier coefficients with respect to relative transverse coordinates and relative rapidity. Here b is a (small) number, which regulates the ultraviolet divergence and the dimensionless parameter ∆ controls the initial amplitude of small wave-number fluctuations. While this construction does not respect the details of the spectral composition, the parameter ∆ provides a measure of the coupling constant ∆ 2 ∼ g 2 (see Sec. III A) and we will vary its size in Sec. IV B to study the coupling dependence.", "pages": [ 8, 9 ] }, { "title": "IV. LATTICE RESULTS", "content": "In this section we present results from classicalstatistical lattice simulations of the Glasma evolution in the presence of boost non-invariant fluctuations. While the existence of a non-equilibrium instability has been established in previous simulations [16, 17], we focus on the non-linear regime where unstable modes have grown large enough to significantly alter the dynamics. In contrast to the linear regime, where the initial size of boost non-invariant fluctuations is irrelevant for the dynamics of unstable modes, it is clear that for the non-linear regime the size of the initial fluctuations matters. In view of the spectrum of initial fluctuations obtained in Ref. [10], the ratio of the initial amplitude of fluctuations compared to the amplitude of the (squared) background field is parametrically of the order of the strong coupling constant g 2 . We study this dependence in Sec. IV B by considering different amplitudes of the initial fluctuations, as characterized by the dimensionless parameter ∆ 2 ∼ g 2 (see Sec. III B). We restrict our analysis to weak coupling ( g 2 glyph[lessmuch] 1), where classicalstatistical methods are expected to provide an accurate description of the quantum dynamics on large time scales. The discussion of our results is organized as follows: In Sec. IV A we investigate the dynamics of the instability at weak coupling and show how deviations from the linear regime emerge in terms of secondary instabilities. To further analyze this behavior, we obtain the relevant growth rates for primary and secondary instabilities as well as the corresponding set-in times. Subsequently, in Sec. IV B, we investigate the dependence on the coupling constant by varying the size of initial fluctuations. If not stated otherwise we perform simulations on N ⊥ = 16, N η = 1024 and N ⊥ = 32, N η = 128 lattices and we employ the set of parameters g 2 µ N ⊥ a ⊥ = 22 . 6 and N η a η = 1 . 6 in accordance with Ref. [16]. We study the time evolution of the gauge-invariant pressurepressure correlation function where P L ( x ) is the longitudinal pressure as a function of space and time arguments and < ., . > T denotes average over transverse coordinates and classical-statistical ensemble average. We will frequently employ the Fourier transforms of Π 2 L ( τ, η, η ' ) with respect to relative rapidity, i.e. we consider and usually show results for Π L ( τ, ν ), i.e. the square root of the above expression. The details of our lattice setup are described in more detail in the appendix.", "pages": [ 9, 10 ] }, { "title": "A. The unstable Glasma", "content": "In a first step we study the time evolution of the gauge-invariant pressure-pressure correlator Π L ( τ, ν ) for a fixed value of the amplitude of initial fluctuations ∆ = 10 -10 and b = 0 . 01. The results are shown in Fig. 3 for different rapidity wave numbers ν as a function of time. From Fig. 3 one observes a sequence of different dynamical regimes which are characterized as follows: At very early times √ g 2 µτ glyph[lessorsimilar] 2 one observes a period of rapid initial growth, which is presumably caused by the dephasing dynamics of the strong background fields, that takes place roughly on the same time scale [11-14, 17, 51]. However at weak coupling, i.e. for small fluctuations, this constitutes a rather small effect as the unstable modes exhibit their dominant growth at later times. The rapid initial period is followed by a regime where the Glasma instability [16, 17] is operative and modes with non-zero rapidity wave number exhibit exponential amplification. The instability sets in with a delay for higher momentum modes and the functional form is well described by an exponential of the form exp[Γ( ν ) √ g 2 µτ ], with the momentum dependent growth rate Γ( ν ), as seen for ν = 4 , 12 in Fig. 3. To further investigate this behavior we fit a set of continuous piecewise linear functions to the modes displayed in Fig. 3 in order to obtain the relevant growth rates and set-in times. The results of these fits are shown in Fig. 4 as a function of rapidity wave number ν . From the upper panel of Fig. 4 one observes that the primary set-in times follow a linear behavior, as reported in Ref. [16]. The primary growth rates are shown in the lower panel of Fig. 4. One observes that modes with small rapidity ν 200 wave number exhibit smaller growth rates as compared to modes with higher rapidity wave number, while at large ν the primary growth rates become approximately constant. The numerical values are compatible with the results reported in Ref. [16], where characteristic growth rates were obtained from a convolution of the spectrum. While the primary instability continues to set-in for higher momentum modes, one observes from Fig. 3 that at later times modes with intermediate ( ν = 43 , 71) and small ( ν = 4) rapidity wave number suddenly exhibit much higher growth rates than previously observed. This change in the dynamics becomes evident when shortly after modes with even higher rapidity wave numbers ( ν = 94 , 200) exhibit even stronger growth rates, such that the spectrum extends quickly to the ultra-violet and the instability propagates towards higher momenta. This is precisely the signature of secondary instabilities, where non-linear self-interactions among unstable modes give rise to an amplification of the primary instability. The amplification happens initially in a small momentum region and then quickly propagates outwards to higher momenta. This can be seen in the upper panel of Fig. 4, where we show the set-in times of primary and secondary growth. One also observes that for modes with large rapidity wave number ν > ν c secondary instabilities set-in before the primary instability, such that the growth of high ν modes is solely due to non-linear effects. The numerical value of ν c depends, of course, on the size of the initial fluctuations and we will confirm the non-linear origin of this phenomenon in Sec. IV B, where we investigate in more detail the dependence on the initial amplitude of fluctuations. The dynamical power counting scheme developed in Sec. II D suggests that these secondary instabilities are caused by the one-loop diagram shown in the upper panel of Tab. I, with secondary growth rates as large as twice the primary ones. If we compare the rates of primary and secondary growth, as shown in the lower panel of Fig. 4, we find that this is indeed the case for modes with intermediate ν , which exhibit the earliest non-linear amplification. Modes with higher values of ν exhibit even larger growth rates, which can be attributed to multiple amplification processes as well as higher order corrections. The growth of primary and secondary instabilities in Fig. 3 continues until at some point saturation of the instability sets in and the system has reached nonperturbatively large occupation numbers. In this regime we observe that the process of non-linear amplification continues even after the growth of the leading primary modes has saturated. This has a significant impact also on bulk observables such as the ratio of longitudinal pressure to energy density. Before we turn to a more detailed discussion of this highly occupied regime, we will first investigate the coupling dependence of the non-linear amplification process.", "pages": [ 10, 11 ] }, { "title": "B. Coupling dependence", "content": "In Sec. IV A we have discussed the time evolution of non-boost invariant fluctuations in the Glasma. We have shown that, at weak coupling, primary instabilities of small rapidity wave number fluctuations occur. In turn, these cause secondary instabilities of modes with higher rapidity wave numbers until saturation of the instabilities occurs and one enters the highly over-occupied regime. In this section we investigate in more detail the dependence on the choice of parameters, in particular the impact of the initial amplitude of fluctuations. According to Sec. III A, this can be interpreted as varying the value of the coupling constant g 2 in our simulations, without respecting in detail the spectral shape of initial fluctuations [10]. In order to study the dependence on the initial amplitude of fluctuations, we vary the parameter ∆ in the range of 10 -15 to 10 -5 . The qualitative behavior is the same as observed in Sec. IV A for all values of ∆, i.e. we observe primary instabilities followed by non-linear amplification and subsequent saturation of the growth. Due to the non-linear origin, the time scales for the set in of secondary instabilities and the saturation of growth depend, of course, on the initial amplitude of fluctuations. The results of our analysis are shown in Fig. 5, where we show the characteristic set-in times of secondary instabilities as a function of the initial amplitude ∆ of boost non-invariant fluctuations. The rather weak dependence observed in Fig. 5 stems from the fact that, at early times, the magnitude of non-linear contributions depends exponentially as exp[2Γ 0 √ g 2 µτ ], whereas the dependence on the initial amplitude is just a power. Assuming that non-linear amplification is caused by the one-loop diagram depicted in Tab. I we obtain the parametric estimate (see Sec. II D) where τ SetIn characterizes the set-in time of primary instabilities and Γ 0 is the characteristic primary growth rate. This behavior is reproduced by the lattice data on a qualitative level.", "pages": [ 11 ] }, { "title": "C. Saturated regime", "content": "The evolution of the system in the saturated regime is of great interest, when studying the thermalization pro- cess at weak coupling. In this regime the system exhibits a much slower dynamics and one expects the system to become approximately isotropic on sufficiently large time scales [55-57]. In order to analyze the behavior, we first consider the evolution of the ratio of longitudinal pressure to energy density, as a measure of the bulk anisotropy of the system. The challenge in this analysis comes from the fact that the relevant ultra-violet cutoff associated with longitudinal momentum Λ z ∼ π/ ( τa η ) decreases with (proper) time. Furthermore having a large rapdity cutoff Λ η ∼ π/a η can cause severe problems at early times and a proper renomalization scheme might be needed to ensure physical results. We adress this problem by chosing the initial amplitude of fluctuations very small, such that the overall contribution of fluctuations to the energy density is less than a percent even for the largest cutoffs that we consider. We then vary the lattice spacing a η while keeping N η a η fixed to study the sensitivity to the cutoff. The results are presented in Fig. 6, where we show the ratio of longitudinal and transverse pressure as a function of time. While at early times the longitudinal pressure of the system is consistent with zero, we observe a clear rise of the longitudinal pressure towards later times. In the saturated regime the trend towards isotropization slows down dramatically and the system exhibits a remaining order one anisotropy over a large time scale. The results are insensitive of the longitudinal discretization, as long as the lattice spacing a η is sufficiently small. We also study the spectrum of the pressure-pressure correlation function Π( τ, ν ) in this regime. This is shown in Fig. 7 at different times, as a function of longitudinal momentum p z = ν/τ . From the top panel of Fig. 7, one observes how the spectrum rapidly extends towards higher momenta at times g 2 µτ glyph[similarequal] 1000 -2000, while the amplitude at low momenta decreases. This redistribution is accompanied by the increase in the bulk pressure observed in Fig. 6. At later times g 2 µτ glyph[greaterorsimilar] 2000 the evolution of the spectrum features enhanced contributions from soft modes and a strong fall-off at high momenta. The evolution of the spectrum in this regime proceeds in a much slower way. The lower panel of Fig. 7 shows the spectrum on a double logarithmic plot. One observes that the soft tail of the spectrum can be described by a power-law. This is illustrated by the black dashed line in Fig. 7, which corresponds to the functional form i.e. a power-law with normal UV regulator. This behavior is very similar to wave-turbulence observed in nonexpanding systems [58-60]. The power law features an infrared power-law exponent of β glyph[similarequal] 1 / 3, while the regulator controls the fall-off at high momenta. The shape of the spectrum at later times is qualititive similar. However we already find sizable amplitudes for modes with longitudinal momenta on the order of the transverse lattice cutoff. To avoid discretization errors, one therefore has to consider also much larger lattices in the transverse direction and we expect future simulations to extent the studies of this regime.", "pages": [ 11, 12, 13 ] }, { "title": "V. CONCLUSION", "content": "In this paper we discussed the impact of quantum fluctuations on the non-equilibrium dynamics of the Glasma. The picture that emerges at weak coupling is quite universal and depends only weakly on the details of initial fluctuations. In the weak coupling scenario initially boost non-invariant fluctuations exhibit exponential growth until at some point they have grown large enough for non-linear interactions to become important. At this stage secondary instabilities set in for a large momentum region, which extends to much higher rapidity wave numbers as the primary growth. Both primary and secondary growth continues until saturation of the instability occurs and the system exhibits a much slower dynamics. This scenario is similar to various examples of instabilities in self-interacting quantum field theories [19, 20, 34, 44] and we have shown in Sec. II D how the emergence of secondary instabilities can be studied systematically within the framework of two particle irreducible effective action techniques. At the qualitative level this analysis is sufficient to predict the parametric dependence of the set-in time of secondary instabilities and the estimate can be made quantitative when the microscopic dynamics of the primary instability is described analytically. The set in time of secondary instabilities depends on the growth rates of the primary instability and to a much weaker extent on the size of the initial fluctuations, which is related to the value of the strong coupling constant. We confirmed this qualitative behavior by varying the size of the initial fluctuations in the classical-statistical lattice simulations, without respecting in detail the spectral composition. The latter will be taken into account in future studies, though one does not expect qualitative changes at weak coupling, where the longitudinal spectrum is quickly dominated by the exponential growth of primary instabilities. In contrast, changes in the spectrum of the background field may have a much more significant impact on the dynamics, as they readily alter the dynamics of primary instabilities. In particular this may change the character of the instability from the Weibel type, which is present in the MV model, to the Nielsen Olesen type [51, 52] and it will be important to consider more realistic saturation models in the future. This is important also in view of the applicability at RHIC and LHC energies, where in addition one has to consider much larger values of the strong coupling constant. This is not unambiguous since one encounters conceptual problems concerning the renormalization of the theory as well as the impact of sub-leading quantum corrections. While exploratory studies in scalar field theories have recently obtained promising results [53, 54], we expect future studies of non-abelian gauge theories to expand on this issue. In addition to the unstable regime, we also studied the dynamics of the saturated regime, which is of great interest in the recent debate on the thermalization mechanism at weak coupling and high collider energies [55-57]. In this context, different scenarios involving elastic and inelastic scattering proccesses [55], instability induced isotropization [56] as well as the formation of a transient condensate [57] have been proposed and are currently under investigation. While for non-expanding systems, the results from classical-statistical lattice simulations suggest the occurrence of a turbulent cascade [58-60], there are only few results for expanding systems [17, 33] and we expect more studies in the future. While a dedicated lattice study has the potential to clarify these questions, we restricted our analysis to the characteristic properties of the system at later times. We investigated the ratio of longitudinal and transverse pressure as a measure of the bulk anisotropy of the system. This observable can be used to distinguish between different scenarios which predict a characteristic evolution of this quantity. Our results indicate that the system exhibits an order one anisotropy on large time scales, which is common to the early stages of all the scenarios [55-57]. In order to clearly indentify the onset of an attractor solutions one therefore has to investigate even larger time scales and carefully monitor all discretization errors, which will be addressed in future studies. By investigating the spectrum of the pressure-pressure correlation function, we found evidence for a scaling solution and similar results have also been obtained in Ref. [17], where a different correlation function has been considered. The question whether this behavior is indeed related to the emergence of a turbulent cascade as observed in non-expanding systems [58-60], will also be subject to future studies. Acknowledgment: We thank F. Gelis, Y. Hatta, A. Kurkela, G.D. Moore, M. Strickland and R. Venugopalan for discussions. This work was supported in part by the BMBF grant 06DA9018.", "pages": [ 13 ] }, { "title": "APPENDIX: CLASSICAL-STATISTICAL LATTICE SIMULATIONS", "content": "The classical-statistical lattice simulations are performed in the Hamiltonian framework with the lattice link variables U µ ( x ) which are related to the continuum variables by where a ⊥ and a η are the lattice spacings in the transverse and longutidinal directions and t a denote the generators of the SU (2) gauge group. The SU (2) exponential can be computed as In Fock-Schwinger gauge one finds U τ = 1 and the (dimensionless) electric field variables read Without loss of generality we set the coupling constant g = 1 in the following (see Sec. III A). The (dimensionless) lattice Hamiltonian density H ( τ, x ⊥ , η ) of the system is given by where c E = a 2 /τ 2 and c B = a 2 / ( τ 2 a 2 η ), and the squares of the electric and magnetic field strengths are, for the SU (2) gauge-group, given by where U αβ ( x ) are the standard plaquettes given by where α, β = 1 , 2 , η and we also defined the V glyph[square] αβ plaquettes for later use.", "pages": [ 13, 14 ] }, { "title": "A. Evolution equations", "content": "The Hamiltonian evolution equations on the lattice in co-moving ( τ, η ) coordinates are given by the equation of motion for the link variables (no summation over α = 1 , 2 , η ). Here W glyph[square] α denote plaquettes involving a time link, given by (no summation over α = 1 , 2 , η ) and the coefficients read The update rules for the chromo-electric fields are given by with the coefficients The time evolution can then be computed by alternately solving Eqns. (58) and (61). The Gauss law constraint is conserved by this evolution and reads which is satisfied separately for all x and a . Here we denote h i = 1, hd η = a 2 / ( τ 2 a 2 η ) and", "pages": [ 14 ] }, { "title": "B. Initial conditions on the lattice", "content": "We first generate sets of uncorrelated standard Gaussian random numbers for every position in the transverse plane and every color associated to the color-charge densities of the nuclei, i.e. where ξ ( A ) a ( x t ) are Gaussian random number and the subtraction of the overall color charge R ( A ) a = N -2 ⊥ ∑ x ⊥ ξ ( A ) a ( x ⊥ ) ensures the overall color neutrality constraint. The result is then Fourier transformed to momentum space where we solve the Laplace equation The result is Fourier transformed back to obtain the solution of the Laplace equation in coordinate space. We then proceed by calculating the pure gauge solutions U (1 / 2) ( x ⊥ ) according to Finally the link variables U µ ( x ⊥ , η ) at initial time are obtained as [16, 17] and the electric fields E a µ ( x ⊥ , η ) are given by [16] To generate the boost non-invariant fluctuations we first generate the functions f ( ν ) and e a i ( p ⊥ ) in momentum space according to where ξ ( ν ) are uncorrelated Gaussian random numbers. After performing a Fourier transform to coordinate space, the fluctuations δE a µ ( x ⊥ , η ) are calculated as where f ' ( η ) is the lattice derivative of the function f ( η ) according to and the Gauss constraint is implemented explicitly in Eqns. (75) and (76).", "pages": [ 14, 15 ] }, { "title": "C. Lattice observables", "content": "We will denote the diagonal components of the stressenergy tensor by the energy density glyph[epsilon1] ( x ), the transverse pressure P T ( x ) and the longitudinal pressure P L ( x ). In terms of the electric and magnetic field strengths squared, as defined in Eq. (56), they are given by These quantities are gauge invariant and satisfy the relation glyph[epsilon1] = 2 P T + P L at every position in space and time. In addition to the above quantities, we also study equal time correlation functions where 〈 . 〉 T denotes ensemble average and average over transverse coordinates. We focus on the correlator in Fourier space with respect to relative rapidity, which is given by and usually show results for Π L ( ν ), i.e. the square root of the above expression.", "pages": [ 15 ] } ]

[ { "title": "Decoherence Problem in Ekpyrotic Phase", "content": "Chien-Yao Tseng 1 1 California Institute of Technology, Pasadena, CA 91125 Quantum decoherence and the transition to semiclassical behavior during inflation has been extensively considered in the literature. In this paper, we use a simple model to analyze the same process in ekpyrosis. Our result is that the quantum to classical transition would not happen during an ekpyrotic phase even for superhorizon modes, and therefore the fluctuations cannot be interpreted as classical. This implies the prediction of scale-free power spectrum in ekpyrotic/cyclic universe model requires more inspection.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "From cosmological observations we know that the current universe is to a good approximation flat, homogeneous and isotropic on large scales [1, 2]. It is well known that in standard Big Bang cosmology this requires an enormous amount of fine-tuning on the initial conditions. Two mechanisms are provided to be possible explanations. The first is inflation [3, 4], a period of accelerated expansion occuring between the big bang and nucleosynthesis. The second is ekpyrosis [5-9], a period of ultra-slow contraction before Big Bang/Big Crunch to an expanding phase. Both mechanisms not only manage to address the standard cosmological puzzles but also have the ability to imprint scale invariant inhomogeneities on superhorizon scales via a causal mechanism [3, 5, 10-14]. These inhomogeneities are thought to provide the seeds which later become the temperature anisotropies in the Cosmic Microwave Background and the Large Scale Structure in the universe. This framework of the cosmological perturbation theory is based on the quantum mechanics of scalar fields, where the relevant observable is the amplitude of the field's Fourier modes [15]. Although treated as a quantum mechanical variable, this amplitude is expected to be stochastic variables, characterized by averages of their products, i.e., power spectrum. This interpretation proves to be very accurate in the CMB and Large Scale structure analyses. However, in order to make this stochastic interpretation consistent, the density matrix has to be diagonal in the amplitude basis. This criterion implies that interference terms in the density matrix are highly suppressed and can be neglected [17, 18]. Interference is associated with the coherence of the system, i.e., the coherence in the state between different points of configuration space [19, 20]. A measure of this is the coherence length which gives the configuration distance over which off-diagonal terms are correlated [21]. An isolated system described by the Schrodinger equation cannot lose its coherence; a pure state always remains pure. However, if it is coarse grained, it may evolve from a pure to a mixed state. One way to realize coarse graining is to let the system interact with an environment [19]. The environment consists of all fields whose evolution we are not interested in. The state of the system is obtained by tracing over all possible states of the environment. Now, even if the state describing system plus environment is pure, the state of the system alone will in general be mixed. In the literature, there are various arguments and calculations suggesting that a form of such environment decoherence can indeed occur for inflationary perturbations [21-31]. The coherence length decreases exponentially for wavelengths greater than Hubble radius. Thus perturbations become classical once their wavelength exceeds the Hubble radius. All of these results lend support to the usual heuristic derivation of the spectrum of density perturbations in inflationary models. In this paper, we use a simple model to study whether decoherence can also occur in the ekyprotic phase. We find that the coherence lengths continue increasing even for the modes outside the horizon. Therefore, the heuristic argument that the modes become classical when they leave the horizon is invalid in the ekyprotic phase and requires more careful inspection.", "pages": [ 1 ] }, { "title": "II. THE MODEL", "content": "A crucial question is how to model the environment. Any realistic model will be very complicated and hard to analyze. However, the basic physics should emerge from the simplest models. Hence, we choose a model [21] which can be solved exactly: the system is a real massless scalar field φ 1 , and the environment is taken to be a second massless real scalar field φ 2 interacting with φ 1 through their gradients. The action of system and environment is where g is the determinant of the background metric which is given by and c glyph[lessmuch] 1 is the coupling constant describing the interaction between two fields. Note that this Lagrangian is quadratic in the derivative of the fields and can hence be diagonalized for which the interaction term disappears and the whole Lagrangian becomes a free field theory. If there is no other field or interaction in our universe, this argument is true. However, we suppose there is a hidden interaction such that we can only obeserve the first field φ 1 but not the environment φ 2 . In other words, we assume the environment and the observed system do not form the diagonal basis. This assumption is reasonable since any observed scalar fields (whose reduced density matrix we want) will interact with gravitational perturbations (which is a part of the environment). Then, the canonical momenta π i conjugate to the fields φ i , i = 1 , 2 are where ' · ' denotes derivative with respect to η . This allows us to write the Hamiltonian H as To study decoherence, it is more convenient to use the functional Schrodinger picture[32]. The commutation relation [ φ i ( x ) , π j ( y )] = iδ ij δ 3 ( x -y ) is equivalent to making the replacement π i ( x ) → -i δ δφ i ( x ) . The wave functional Ψ[ φ 1 , φ 2 ] obeys the Schrodinger equation We make a Gaussian ansatz for Ψ to be able to find the vacuum or ground state solution: Note that we have already used the φ 1 ↔ φ 2 symmetry of the Lagrangian. Furthermore, because of the x ↔ y symmetry of the above integration, we have to require Plug Eq. (7) into Schrodinger equation (6), it is not difficult to get we get Here we have already used the relation A ( -k , η ) = A ( k , η ) coming from Eq. (8). Note that A ( k , η ) is only a function of | k | , so we will write it as A k ( η ) from now on. This differential equation can be easily solved by assuming Then Eq. (19) becomes The wave functional can also be expressed in momentum space, where and φ i ( -k ) = φ ∗ i ( k ) for the real scalar field. Because there is no coupling between modes with different k , we will only consider a single wavelength and drop the index k for convenience from now on.", "pages": [ 1, 2, 3 ] }, { "title": "III. THE DENSITY MATRIX AND THE COHERENCE LENGTH", "content": "We now have the wave functional for all modes with single wavelength k . The next step is to calculate the reduced density matrix for φ 1 by tracing out φ 2 . All the above equations come from the comparison of the coefficients in front of φ i ( x ) φ j ( y ). It is easy to see that Eq. (11) and Eq. (10) are equivalent, which is just the result of the symmetry of φ 1 and φ 2 . In order to satisfy Eq. (10)-(12), we have to require B ( x , y , η ) = cA ( x , y , η ), which gives It is more convenient to solve Eq. (16) in momentum space. Upon writing This can be computed from the Gaussian integral: where To determine the coherence length of the reduced density matrix, it is convenient to introduce the new variables: In terms of these variables, the reduced density matrix (26) becomes Because β = 1 -c 2 2 ( A ∗ -A ) is purely imaginary, the third term in the exponential just gives a complex phase. The first term gives the dispersion of the system, the dispersion coefficient σ being The second term describes how fast the density matrix decays when considering the off-diagonal terms. Hence, l c is called the coherence length and is given by", "pages": [ 3, 4 ] }, { "title": "IV. DECOHERENCE IN THE USUAL INFLATION MODEL", "content": "For usual inflation, a ( t ) = e Ht which is equivalent to a ( η ) = -1 Hη . Here, H is the Hubble constant. Eq. (21) then tells us Considering the wave functional (23), we have to require a positive real part of A for obvious reasons. Therefore, we choose c 1 = 0 and Then, Eq. (33) gives us the coherence length 1 : We see that if no interaction is present ( c = 0), the coherence length approaches a constant value. Adding even a small interaction will reduce it to zero (See Fig. 1). Besides, the coherence length starts to decrease exponentially when the wavelength crosses the Hubble radius, which justifies our heuristic derivation in cosmological perturbation theory.", "pages": [ 4, 5 ] }, { "title": "V. DECOHERENCE IN POWER LAW INFLATION AND EKPYROTIC PHASE", "content": "The scale factor behaviors of power law inflation and ekpyrosis are very similar so we consider them at the same time. We list some properties of their scale factors in the Table I. Because both of the power law inflation and ekpyrosis have the same a a , they share the same solution of u k . The differential equation of (21) can be solved exactly by where H (1 , 2) α are Hankel functions, and we have defined As before, we want A k ( η ) to have a positive real part, so we take c 1 = 0, and Eq. (20) tells us Notice that they are the same for both power law inflation and ekpyrotic phase except p glyph[greatermuch] 1 for the former and p glyph[lessmuch] 1 for the latter. We can then use Eq. (33) to calculate the coherence length for both cases. The numerical solutions are plotted in the Fig. 2 and Fig. 3. In order to get the behavior of the coherence length l c when the modes are well outside the Hubble radius, we need the asymptotic form of the Hankel function as x → 0: where α > 0 and Γ( α ) is the Euler gamma function. After some manipulation of algebra, we have as -kη glyph[lessmuch] 1. For power law inflation, p glyph[greatermuch] 1, we have α = 3 2 + 1 p -1 = 3 2 + glyph[epsilon1] , 0 < glyph[epsilon1] glyph[lessmuch] 1. Therefore, where From Eq. (42), it is obvious that if no interaction is present, the coherence length approaches a constant value l 0 . However, even a small interaction will reduce the coherence length to zero just like what happened in the usual inflationary case. As for the ekpyrotic phase, p glyph[lessmuch] 1, and α = 1 2 -p 1 -p = 1 2 -glyph[epsilon1], 0 < glyph[epsilon1] glyph[lessmuch] 1. Use Eq. (41), it is not difficult to get This means the coherence length approaches a nonzero constant value no matter whether the interaction is present or not, in agreement with our numerical results in Fig. 3.", "pages": [ 5, 6, 7 ] }, { "title": "VI. CONCLUSION", "content": "We have studied a simple model with two free scalar fields interacting via a gradient coupling term in three different background spacetime: the usual inflation, the power law inflation, and the ekpyrosis. We also calculate the reduced density matrix and the corresponding coherence length by summing over one of the fields in all three cases. Our results are that if no interaction is present, the coherence length approaches a constant value. Adding even a small interation will reduce it to zero in either usual inflation or power law inflation case. Since this decoherence starts at Hubble crossing, the quantum fluctuations evaluated at kη = -1 give the classical initial density perturbations which become the seeds of inhomogenities of our universe later on. However, this argument does not work for ekpyrosis whose coherence length never hits zero. This means the quantum coherence would not disappear even when the modes leave the horizon. Therefore, the heuristic argument that the quantum fluctuation can become classical for superhorizon modes is not valid for ekpyrotic phase. The implication of our result is that the power spectrum of CMB fluctuations is not directly related to the ekpyrotic phase. Even though at the end of ekpyrosis the scalar field has a scale-invariant power spectrum, it is hard to say anything about what we observe right now, since that depends on the 'classical' initial density perturbations. This puts some doubts on the analyses of the cosmological perturbations in the cyclic/ekpyrotic universe. We derived our results using a very simple model. In principle, if we would like to claim the decoherence phenomenon cannot occur in ekpyrosis, we have to consider all kinds of interactions between systems and environment which is almost impossible to do. However, we believe the basic physics should emerge from simple models. We can easily generalize our analyses to a massive scalar field, and the results wouldn't change too much. We could also consider different kinds of interactions, but we will leave it to the future work. Finally, we model the environment with a scalar field, which is convincing but might be an oversimplified assumption. The environment can also be taken to consist of the short wavelength modes which are coupled to the long wavelength modes via non-linear couplings [22-28]. Hence, this might be another possible way to generate decoherence during ekpyrosis.", "pages": [ 7 ] }, { "title": "VII. ACKNOWLEDGEMENTS", "content": "This work is supported by the U.S. Department of Energy. We especially thank S. M. Carroll for very useful comments and suggestions.", "pages": [ 8 ] } ]

[ { "title": "Inflationary Dynamics with a Non-Abelian Gauge Field", "content": "Kei-ichi Maeda 1, 2, 3, ∗ and Kei Yamamoto 1, 4, † 1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2 APC-AstroParticule et Cosmologie (CNRS-Universit' e Paris 7) , 10 rue Alice Domon et L ' e onie Duquet, 75205 Paris Cedex 13, France 3 Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan 4 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, N-0315 Oslo, Norway We study the dynamics of the universe with a scalar field and an SU(2) non-Abelian Gauge (Yang-Mills) field. The scalar field has an exponential potential and the Yang-Mills field is coupled to the scalar field with an exponential function of the scalar field. We find that the magnetic component of the Yang-Mills field assists acceleration of the cosmic expansion and a power-law inflation becomes possible even if the scalar field potential is steep, which may be expected from some compactification of higher-dimensional unified theories of fundamental interactions. This power-law inflationary solution is a stable attractor in a certain range of coupling parameters. Unlike the case with multiple Abelian gauge fields, the power-law inflationary solution with the dominant electric component is unstable because of the existence of non-linear coupling of the Yang-Mills field. We also analyze the dynamics for the non-inflationary regime, and find several attractor solutions.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The idea of inflation now gives a standard scenario of the early evolution of the universe [1-5]. It solves several difficulties such as the horizon and flatness problems in the Big-Bang cosmology, which has been confirmed by the precision cosmological observations. It also provides us with a prediction on the origin of the observed density fluctuations. Many cosmological models with such a phase of accelerated expansion have been proposed by introducing a scalar field with an appropriate potential (or some alternative fields). However, it is desirable to derive a natural model from a fundamental theory of particle physics without introducing any anonymous field by hand. The most promising candidate for such a fundamental theory is the ten-dimensional superstring theory[6] or eleven-dimensional M-theory[7]. They are hoped to give an interesting explanation for the accelerated expansion of the universe upon compactification to four dimensions. In the low-energy effective field theories of superstrings or supergravity theories, however, there is the so-called no-go theorem, which forbids such an inflating solution if the internal space is a time-independent nonsingular compact manifold without boundary [8]. In order to evade this theorem, we have to violate some of those assumptions. We have three possibilities: Although some models could be promising, many models are still suffering from instability of a dilaton field or moduli fields. In fact, we naturally expect exponential couplings of moduli fields. Without fixing those moduli, many inflationary models are spoiled. An exponential coupling is not always harmful for inflation, however. In fact, we can find a powerlaw inflation[21] with an exponential potential[22, 23]. It also provides the cosmic no hair theorem similar to the slow-roll inflation[24]. In supergravity theories and superstring models, an effective exponential potential V 0 exp[ -αφ ] naturally appears[25-27]. However, their potential is usually so steep that the power exponent of the scale factor cannot be much larger than unity, which makes it difficult to construct an acceptable inflationary model of the universe. For example, we find α = √ 2 and √ 6 for two scalar fields in N = 2, six-dimensional supergravity model with S 2 -compactification[26], and the same is true for two scalar fields in N = 1, tendimensional supergravity model with gaugino condensation [27]. Townsend summarized the possible exponential potentials derived by the compactification of ten- or eleven-dimensional supergravity theories[28]. From flux compactifications, one expects α ≥ √ 6, while we may find √ 2 ≤ α ≤ √ 6 by hyperbolic compactifications. Neither of them offers a flat enough potential for inflation. In the unified theories of fundamental interactions, there naturally exist gauge fields, which may be included in the original action such as the heterotic string theory or can be induced by Kaluza-Klein compactification. In effective four-dimensional theories derived from higherdimensional unified theories, we also expect those gauge fields coupled exponentially to moduli fields such as 1 4 exp[ λφ ] F 2 . Hull and Townsend discussed such a coupling for the case of U(1) gauge fields. They found that the possible values of the coupling in the four-dimensional effective action are λ = 0 , √ 2 / 3 , √ 2, or √ 6 in the context of black holes in the type II string theory compactified on a six torus[29]. In M-theory (eleven-dimensional supergravity) with intersecting branes, the fourdimensional effective action also contains the same moduli couplings to U(1) multiplet [30]. If the strengths of the couplings between gauge fields and a scalar field are similar to that of the scalar self-coupling, the gauge fields may affect the dynamics of the scalar field. In fact there are several discussions about the dynamics of inflation, where supportive roles of gauge fields in realizing accelerated expansion have been observed[31-42]. The effect of the gauge-kinetic coupling on the inflationary dynamics was first discussed in the context of anisotropic inflation[31], assuming a U(1) gauge field coupled to an inflaton field. Since a single U(1) field cannot exist in FriedmannLemaˆıtre-Robertson-Walker (FLRW) isotropic and homogeneous spacetimes, they discussed Bianchi spacetimes as the cosmological model. They specified the scalar potential to be quadratic and chose exp[ cφ 2 ] as the gauge kinetic coupling. They showed that an anisotropic inflationary era may arise as a transient attractor state while the scalar inflaton is slowly rolling. The anisotropy eventually disappears as the scalar field oscillates towards the end of inflation. The observational relic of the anisotropic inflationary era was also discussed[31, 32]. While the chaotic inflation driven by the quadratic potential is phenomenologically interesting as it automatically results in reheating, the form of the inflaton-gauge interaction discussed in [31] may not naturally appear in the unified theories. They also studied the case with an exponential potential and a U(1) gauge field coupled exponentially to the scalar field, which suits the framework of the unified theories better. They found an exact anisotropic inflationary solution, which is an attractor independent of the initial conditions[33]. Since our present universe is almost isotropic, this model must be severely constrained. However, if there exist more than two gauge fields, we find an interesting scenario. Although it requires an artificial assumption that all the gauge fields couple to the inflaton through a common gauge-kinetic function, one can obtain a totally homogeneous and isotropic inflationary solution as an attractor[34]. Since the anisotropic inflation can be found as a transient attractor, we might have a chance to find distinct observational signatures. An important result is that an isotropic powerlaw inflationary solution appears as an attractor even for a steep exponential potential for the inflaton, which is expected from the unified theories of fundamental interactions. While there are certain conditions to be satisfied by the gauge-kinetic coupling constant, they are not so strict as the usual slow-roll conditions and could fall within the reach of the supergravity theories. In the case of U(1) multiplet fields, we usually expect the different gauge-kinetic coupling constants for different fields in the context of the unified theories. However, if we consider a nonAbelian gauge field, it consists of 'multiple' vector fields with a single common gauge-kinetic coupling constant. As a result, the discouraging feature of U(1) multiplet will disappear. The conventional chaotic inflationary model with a non-Abelian gauge field has been studied[35]. Motivated by its phenomenological development and the aforementioned features of high-energy physics, in this paper, we study SU(2) non-Abelian gauge field coupled exponentially to a scalar field with an exponential potential, in order to know whether the non-Abelian gauge field has the similar nice properties as the U(1) multiplet case. We should note that there is also an approach different from the present gauge-kinetic coupling model[41, 42]. They consider an axion field coupled to a non-Abelian gauge field, which is named chromo-natural inflation. It may give another interesting inflationary regime with non-Abelian gauge fields. In the following, we present the basic equations of our system, and obtain power-law solutions in § . III. We find that a power-law inflationary solution is found only for the case of the magnetic component dominance in contrast to the U(1) triplet case, in which both inflationary solutions with electric field and magnetic field are possible. In § . IV, we describe the solutions as fixed points of a dynamical system and analyze their stabilities. In § . V, we perform numerical analysis for the range of coupling constants where fixed points do not exist. It also tells us how the attractor state is achieved", "pages": [ 1, 2 ] }, { "title": "II. BASIC EQUATIONS", "content": "We use the unit of κ 2 = 8 πG = 1. The action we discuss is where is an SU(2) non-Abelian gauge field, which we call the Yang-Mills (YM) field, and g YM is its coupling constant. The coupling to the scalar field f ( φ ) and the scalar potential V ( φ ) are given respectively by α can be set non-negative without loss of generality. We also restrict ourselves to V 0 ≥ 0 since our primary interest here is inflation. Throughout the article, we discuss a flat FLRW spacetime[43], whose metric is given by We assume that the vector potential is given by so that the YM field is taken to be isotropic. This configulation results in both homogeneous electric and magnetic components, which are written in the coordinate basis as with being their comoving field strengths. This is an important difference from U(1) gauge fields, for which we find only the electric component in the above vector potential. The homogeneous magnetic component in U(1) gauge fields is obtained only when we introduce an appropriate inhomogeneous vector potential. As a result, the electric component and magnetic one in the U(1) fields are independent. We can discuss each component separately. In contrast, the YM field always consists of two components in the above isotropic configuration (2.1) and the homogeneous field is found only by a homogeneous vector potential. If one introduces any spatial dependence to the vector potential, the field strengths become inhomogeneous. We should also note that we need more than two U(1) fields with a common coupling to the scalar field as discussed in [34] in order to find an isotropic and homogeneous attractor spacetime. Otherwise, we find an anisotropic universe. For an SU(2) gauge field, this uniform coupling is a necessary consequence of the symmetry. The Einstein equations are where the dots denote the time derivative d/dt , and H = ˙ a/a is the Hubble expansion rate. ρ YM is the YM energy density, which consists of the electric and magnetic parts, i.e., They are defined by The equation of motion for the scalar field is and the equation of motion for the YM field is simply Using the Bianchi identity, Eq. (2.3) is obtained from Eqs. (2.2), (2.6) and (2.7). Hence we take (2.2), (2.6) and (2.7) as the basic equations of our system. The YM equation can be reduced to the first order equations for each energy density as The terms with 4 H come from the radiation-like behavior of the YM field ( ρ rad ∝ a -4 ) and the last terms are from the non-linear interaction in the YM field. In fact, for U(1) triplet fields with a uniform exponential coupling to a scalar field, we find the evolution equations for energy densities by dropping the non-linear interaction terms. As we shall see later, the non-linear terms play a significant role in the dynamics of the model.", "pages": [ 3, 4 ] }, { "title": "III. POWER-LAW SOLUTIONS", "content": "Since we have the exponential potential, we expect a power-law expansion and look for the possibility of power-law inflation. Suppose our solution is given by where p is assumed to be a constant, and a 0 and φ 0 are initial values. The coefficient 2 /α in front of ln t is determined by requiring that the t dependence of ˙ φ 2 and V be the same, in order to satisfy the Hamiltonian constraint (2.2).", "pages": [ 4 ] }, { "title": "A. The case with U(1) triplet fields", "content": "First we consider the case with U(1) triplet fields, which was discussed in [34]. The equations to be solved are and Eqs. (2.2) and (2.6). In our setting (2.1), the magnetic field vanishes. However, if we add an appropriate inhomegeneoes vector potential, a homegeneous magnetic field can appear and the energy densities of the electromagnetic fields satisfy Eqs. (3.3).", "pages": [ 4 ] }, { "title": "1. Dynamics of the scalar field", "content": "Eqs. (3.3) are eaily integrated as where ρ E 0 , ρ B 0 , φ 0 and a 0 are integration constants. Then the equation of the scalar field is reduced to where with /negationslash Although the original potential V is monotonically decreasing, the effective potential (3.6) has a minimum point for λ < 0 and C E = 0, or for λ > 0 and C B = 0. As a result, the scalar field will evolve more slowly than the case only with the exponential potential V . Since there exists the pre-factor a -4 , the minimum point will move and the minimum value will decrease as the universe evolves. Hence we do not have an exponential expansion, but have a power-law expansion whose power exponent is larger than the original power-law expansion driven solely by the potential V . This is the mechanism that a gauge field coupled to a scalar field assists slowing down the motion of the scalar field and inflationary expansion becomes possible even for a steep potential. /negationslash Next we present the explicit power-law solutions. Assuming that the expansion of the universe and the evolution of the scalar field are described by Eqs. (3.1) and (3.2), and the energy densities of the electromagnetic fields are proportional to t -2 , i.e., ρ E = ρ E 0 /t 2 and ρ B = ρ B 0 /t 2 , we find that Eqs.(3.3) become the algebraic equations: There are two cases: ρ B 0 = 0 and ρ E 0 = 0, which we shall discuss separately.", "pages": [ 4 ] }, { "title": "2. The case with the electric field ( E U1 )", "content": "For the case with the dominant electric field ( ρ B = 0), which we shall call the regime E U1 , assuming ρ E = ρ E 0 t -2 ( ρ E 0 : constant), we find three algebraic equations: Eq. (3.8) and which are rearranged into Since the left-hand-sides are positive definite, in order for such a solution to exist, we have to impose the following conditions: The power-law inflation is possible for the range of coupling parameters of λ < -α and α ( α -λ ) > 4, as was shown in [34].", "pages": [ 5 ] }, { "title": "3. The case with the magnetic field ( B U1 )", "content": "For the case only with the magnetic field ( ρ E = 0), which we shall call the regime B U1 , we find the same result by changing the sign of λ . The solution is described by and the existence conditions are The power-law inflation is obtained for the parameter range of λ > α and α ( α + λ ) > 4. Defining the density parameters of each component by Ω E = ρ E / 3 H 2 (the electric field), Ω B = ρ B / 3 H 2 (the magnetic field), Ω V = V/ 3 H 2 (the potential), and Ω K = ˙ φ 2 / 6 H 2 (the kinetic term of the scalar field), we find that those depend on the coupling parameters. We show one example for the power-law inflation with magnetic field in Fig. 1. We find that the magnetic field gives a certain contribution to the expansion of the universe. We will show the stability condition later in § . IVB.", "pages": [ 5 ] }, { "title": "B. The case with YM field", "content": "Now we show the dynamics changes when the YM interaction is turned on. Since we have the non-linear coupling in the YM field, a simple power-law ansatz may not work. But we first look for a solution similar to those found in the U(1) triplet case.", "pages": [ 5 ] }, { "title": "1. The case with the dominant electric component ( E YM )", "content": "If we assume ρ E /greatermuch ρ B but ρ B being nonvanishing due to the interaction between electric and magnetic fields, which we shall call the regime E YM , dropping the term with ρ B , we find the same equations as the U(1) triplet case. As a result, we find the same solution [Eqs. (3.12)-(3.14)] as long as the electric component stays dominant. Under the conditions λ < -α and α ( α -λ ) > 4, we obtain the power-law inflationary solution. Note that an accelerated expansion is possible even if α > √ 2 just as was the case for the U(1) triplet electric type inflation[34]. However, the situation in the case of YM field is not exactly the same as that for the U(1) triplet fields. In the above analysis, we have ignored the magnetic component, which is valid in the U(1) triplet case because the electric and magnetic fields are decoupled. However, the electric and magnetic components are always coupled in the YM field. Then we have to check whether the magnetic component is always negligible or not when it is initially small. Since we assume the magnetic component is initially very small, we can solve the YM equation (2.7) dropping the term with g YM (the magnetic contribution) as where A 0 and A 1 are integration constants. Using this solution, we evaluate the ratio of two energy densities as where ρ B /ρ E | 0 is the initial value. We drop A 0 since we are interested in the asymptotic behavior ( t → ∞ ). As a result, if the magnetic component is initially sufficiently small, we have a power-law inflation just as the case with the Abelian multiple fields, but the contribution of the magnetic component gets larger during the evolution of the universe, and then the inflationary phase eventually ends because of the growth of the magnetic component. The e-folding time when the approximation becomes no longer valid is evaluated as For example, if we assume ρ B /ρ E | 0 = 10 -8 , we find N e-folding ∼ 4 . 6. Hence unless the initial value of the magnetic energy density is extremely small, we do not find a sufficient e-folding number for the inflationary universe.", "pages": [ 5, 6 ] }, { "title": "2. The case with the dominant magnetic component ( B YM )", "content": "We shall call it the regime B YM , when the magnetic component is much larger than the electric one. In this case the situation is not so simple as the U(1) case because we cannot ignore the nonlinear term in (2.8) even if we assume ρ E /lessmuch ρ B . Suppose we have the same solution as the U(1) case. We then evaluate ˙ A/A by the YM equation (2.7), which is now We then find the asymptotic solution as which leads to as t → ∞ . This ratio decays faster than t -1 , at which rate 4 H ± λ ˙ φ evolve in Eq. (2.8). So we can ignore the non-linear term with ˙ A/A , which gives exactly the same equations as the U (1) magnetic case. As a result, we have the power-law solution just the same as in B U1 . This solution is obtained asymptotically, and the non-linear term does not destroy it unlike the regime E YM where the electric components dominate.", "pages": [ 6 ] }, { "title": "3. The case with both components", "content": "If both electric component and magnetic one are of equal magnitude, we cannot ignore either of them. Since it is a non-linearly coupled system, it is difficult to figure out what kind of solutions we expect. Although we need numerical studies, which we will give later, here we shall discuss one simple case, in which we assume the power-law behavior. Suppose that the YM potential A is a powerlaw function as A ∝ t s , and the scale factor and the scalar field are given by Eqs. (3.1) and (3.2). If the energy densities of the electric and magnetic fields are similar, i.e., ρ E ∼ ρ B , we find s = p -1, and then Inserting this behavior into the YM equation (2.7), we find which implies For the power-law inflation ( p > 1), we have As a result, ρ E ∼ ρ B drops faster than ∝ t -2 , which is the scaling of energy density of the scalar field. Hence the contribution of YM field becomes less important as t → ∞ . It appears that it is not possible to find a power-law inflation with a significant residual ρ E ∼ ρ B . Here, we find the power-law expansion only by a scalar field. An accelerated expansion is possible if α < √ 2, as the conventional power-law inflation. In the next section, we will give more details of interesting solutions in the present system including the inflationary solutions we have found and analyze their stability as fixed points in a dynamical system.", "pages": [ 6, 7 ] }, { "title": "A. Dynamical System", "content": "In order to analyze the dynamical behavior of our solutions found in § . III B, we rewrite the basic equations in the form of a first order autonomous system. The inflationary solutions discussed in the previous sections, along with other interesting ones, appear as fixed points in the dynamical system. This allows us to study their local stability and reveal a complicated dynamical behavior that goes beyond the simple power-law timedependence. We shall change the time coordinate from t to the e-folding number N = ln( a/a 0 ), and introduce new variables normalized by the Hubble expansion rate H as Primes denote differentiations with respect to the e-folding number N . We then introduce the density parameter of the YM field as and those of the potential and the kinetic energy of the scalar field as where /pi1 := ˙ φ/H = φ ' . We also use which describes the difference of the fractions of the magnetic and electric components. It enables a unified treatment of electric- and magneticdominant regimes and also makes the asymmetry clear when the YM coupling comes into play. ∆ = 1 and -1 correspond to the regime B YM and E YM , respectively. The Friedmann equation (2.2) now reads The equation for the scalar field (2.6) is where we have used the Friedmann equation (4.8) to eliminate Ω V . The equations for the YM field (2.8) are now where whose evolution equation is This auxiliary quantity Γ is the 'normalized' YM coupling in the sense that the subsystem defined by Γ = 0 corresponds to the dynamical system that describes homogeneous and isotropic U (1) triplet fields. The YM equations (4.10) and (4.11) are rewritten in terms of the denisty parameter Ω YM and the ratio ∆ as where /epsilon1 = sign( AE ). We then find the dynamical system in a closed form. Since the physical interpretation of the normalized vector potential A is not clear, we take Eqs. (4.9), (4.14), (4.15) and (4.13) with the Hamiltonian constraint (= the Friedmann equation) (4.8) as the basic equations to analyze the stability around the fixed points. The drawback is the appearance of /epsilon1 which takes into account the ambiguity inherent to taking square roots. This causes a problem in the numerical study in the next section when the system undergoes oscillations. For this reason, Eqs. (4.10) and (4.11) instead of Eqs. (4.14) and (4.15) are used there.", "pages": [ 7, 8 ] }, { "title": "B. The case with U(1) triplet fields", "content": "Before going into analysis of our system, for an introduction and a comparison, we first summarize the case with the U(1) triplet fields, which was discussed in [34], using the present dynamical variables. To make a clear distinction, we replace Ω YM with Ω U1 . Now ∆ = 1 and -1 correspond to the regimes B U1 and E U1 respectively. In the case with the U(1) triplet fields, the dynamical system is obtained by setting Γ = 0 in the above; /negationslash /negationslash If /pi1 = 0 and Ω U1 = 0, the fixed points are classified into two cases; ∆ = -1 (the case with the electric field) and ∆ = 1 (the case with the magnetic field). In each case, we find two fixed points as follows: Since the density parameters are positive definite, λ ≤ √ 2 / 3 for the fixed point ( a ) to exist. In this case, Ω V = 0, which means that either the potential is absent from the beginning or the potential becomes asymptotically negligible compared with the kinetic term /pi1 2 / 2. From a perturbative analysis, we can check the stability of these fixed points. For the fixed points ( a ), we find that at least the eigenvalue for the perturbation of ∆ is always positive ( ω ∆ = 6 λ 2 ). Hence it is unstable. Hereafter, we use ω to denote eigenvalues with subscripts indicating the variable to which the eigenvalue is associated. The fixed points ( b ) represent the power-law solutions ( E U1 and B U1 ) found in § . III A. The perturbative analysis gives the following three eigenvalues: and the two roots of the quadratic equation from the perturbations of /pi1 and Ω YM . From the existence conditions given by Eq. (3.15) or Eq.(3.19), we have α ( α + λ ∆) -4 > 0. Hence, if and only if λ ∆ > 0 is satisfied, all the three eigenvalues are negative (or the real parts are negative if they are complex). As a result, the solution with the conditions is stable against linear perturbations. More concretely, the stability conditions for the cases with the electric field ( E U1 ) and magnetic field ( B U1 ) are given by, respectively. If λ > 0, the magnetic power-law solution (Eqs. (3.1), (3.2) and (3.4) with (3.16) and (3.18) ) is always an attractor in the parameter range of α ( α + λ ) > 4, while if λ < 0 the electric power-law solution (Eqs. (3.1), (3.2) and (3.4) with (3.12) and (3.14) ) is always an attractor in the parameter range of α ( α -λ ) > 4. For the rest of the parameter space ( α -4 /α < λ < -α +4 /α ), the attractor is a fixed point with Ω U1 = 0, where the scalar field dominates the universe. The fixed point, which we denote S U1 , is given by The perturbative analysis gives the three eigenvalues as Hence the power-law solution driven only by the scalar field is stable if λ ∆ > 0 and α ( α + λ ∆) -4 < 0. Between the two solutions with alternative sings, the stable one is for λ > 0, while for λ < 0. We summarize the result for the U(1) triplet case in Fig. 2 and in Table I:", "pages": [ 8, 9 ] }, { "title": "C. Important Fixed Points in the dynamical system with YM field", "content": "/negationslash Now we move on to include the non-linear YM interaction. The non-trivial fixed points are classified into two cases: Γ = 0 and Γ = 0. In the former case, we find the same fixed points as the U(1) triplet case, although their stability is differ- t as we will show later. The latter case gives new fixed points, which do not exist in the U(1) triplet system. Note that a fixed point may not be found by an exact solution, but can be reached as a certain limit. For example, the fixed points with Γ = 0 would imply either g YM = 0 or He λφ/ 2 = ∞ , neither of which is of interest in our analysis here. - /negationslash However, starting from g YM = 0 and finite H and φ , the system may approach Γ → 0 asymptotically as t →∞ . From the mathematical point of view, those fixed points are well-defined and a part of the dynamical system and we include them in the following analysis.", "pages": [ 9, 10 ] }, { "title": "1. Γ = 0", "content": "In this case, which should reproduce the fixed points of the previous subsection, we can classify the solutions into two cases: Ω YM = 0 and Ω YM = 0. /negationslash In the case with Ω YM = 0, the scalar field energy is dominant. From (4.9), we find either /pi1 2 = 6 or /pi1 = α with ∆ 2 = 1. The former fixed point corresponds to the case that the kinetic energy of the scalar field is dominant, which is unstable against perturbations. The latter fixed points denote the power-law expanding universe with an exponential potential (the counterpart of S U1 ) and will be called S YM . The ratio of the potential energy V to the kinetic energy is (6 -α 2 ) /α 2 . As is well known, these fixed points are attractors if α ≤ √ 6 for the case only with a scalar field. In the present case, because of the YM field, the stability condition changes as follows. The linearized equations for these fixed points give four eigenvalues: Three eigenvalues (4.23), (4.24), and (4.25) are all negative if λ > 0 and λ < -α +4 /α for ∆ = 1, or if λ < 0 and λ > α -4 /α for ∆ = -1. The forth eigenvalue (4.26) becomes negative if λ > α . As a result, the fixed point with ∆ = 1 is stable in the parameter range of α < λ < -α +4 /α . On the other hand, taking λ < α gives instability against the perturbations of Γ. However, as long as Ω YM stays small, the growing Γ does not disturb the evolution of the universe as well as the dynamics of the scalar field since Γ does not appear explicitly in Eqs. (4.8) and (4.9). This is indeed the case when α -4 /α < λ < α . As we shall confirm later in the numerical analysis, the dynamics of the universe is dominated by the scalar field and accurately described by the fixed point discussed here, despite the apparent instability in the eigenvalue ω Γ . The exponentially increasing Γ only triggers a rapid oscillation for the perturbed YM field whose amplitude remains small. In summary, we conclude that these fixed points are stable in the range α -4 /α < λ < -α +4 /α as was found for S U1 . Nevertheless, there is a distinction between S U1 and S YM for λ < α with the dynamics of the YM field being different.", "pages": [ 10 ] }, { "title": "(b) Ω YM = 0", "content": "/negationslash For the case with Ω YM = 0, the YM field plays an important role in the dynamics of the universe. We find ∆ = ± 1 unless /pi1 = 0, for which we do not have any interesting dynamics. /negationslash ∆ = -1 and 1 correspond to the case of the electric component dominance ( E YM ) and that of the magnetic component dominance ( B YM ), respectively. As we have already mentioned, the YM field always consists of both components. Hence these fixed points are reached only asymptotically, if they are stable. Just the same as the U(1) triplet fields, we find two fixed points for each case. However, one of them with /pi1 = -3 λ ∆ is unstable. Hence we discuss the other cases: These fixed points correspond to the solutions with the magnetic component and the electric one found in § . III A 3 and III A 2, respectively. Without the non-linear interaction, these points were symmetric: they were related by the electromagnetic duality and had the same stability properties. The YM coupling skews the symmetry. For the case (1), the eigenvalues are given by and the two roots of the algebraic equation We find all eigenvalues are negative if and only if λ > α and α ( α + λ ) -4 > 0 are satisfied. Since this condition corresponds to the power-law inflationary solution with YM field, we can conclude that the power-law inflationary solution with magnetic component dominance ( B YM -I) is an attractor. The difference from the U(1) multiplet case is that the solutions in the parameter range of 0 < λ < α , which are not inflationary, are no longer an attractor. We will discuss later which asymptotic state we find in this region. On the other hand, in the case with ∆ = -1, the eigenvalues are given by and the two roots of the algebraic equation We find that three eigenvalues are negative for the power-law inflationary solution as long as λ < -α /negationslash and α ( α -λ ) -4 > 0, but one eigenvalue ω Γ , which corresponds to the perturbations of Γ (non-linear interaction term of the YM field), is always positive and does not depend on any parameters. This is the same behavior which we have seen in § .III B 1. As a result, this solution is unstable and the typical instability time scale is O(1) e-folding time since the present time coordinate is N = ln( a/a 0 ) and therefore Γ ∝ exp( ω Γ N ). If the magnetic component is initially sufficiently small, we may find this power-law inflation solution by the electric components in the beginning, but the orbit leaves it just after O (1) e-folding time. We conclude that the power-law inflationary solution with the electric component dominance E YM -I is unstable, contrary to the E U1 . /negationslash Here we assume that Ω YM = 0 because Γ becomes important when the YM field gives nontrivial contribution to the cosmic expansion. Note that ∆ 2 = 1 ( E YM or B YM ) is no longer a fixed point. We find the following two non-trivial fixed points, if λ 2 ≥ 6: The values at both fixed points can be described neatly by the deceleration parameter as Note that Ω V = 0 at these fixed points and they are unique to the YM case. From (4.35) and the positivity of the density parameter Ω YM ≥ 0, we find Hence the power exponent of the scale factor p ( ± ) is which is between a radiation dominant state and a stiff-matter dominant one. The perturbative analysis is common to both of the fixed points λ ≥ √ 6 ( NA + ) and λ ≤ -√ 6 ( NA -). We find the following eigenvalues; which are associated with the eigen vectors δ Ω YM + ( /pi1/ 3) δ/pi1 and δ Γ / Γ -[( q -2)( q +1) / 8 q ] δ ∆, respectively, and the two roots of the quadratic equation Since q is in the range of (4.36), if 0 < λ < α , we find all eigenvalues are negative, which means NA + is stable. On the other hand, ω Ω YM -/pi1 turns positive when λ < 0 and we find NA -is unstable. As a result, we find a stable fixed point in the parameter range of √ 6 < λ < α ( NA + ), which partly takes care of the lost stability of B YM in the region λ < α . We summarize our result for the SU(2) YM field in Fig. 3 and in Table II:", "pages": [ 10, 11, 12 ] }, { "title": "V. NUMERICAL STUDY", "content": "From the above stability analysis, we find there are stable attractors if λ ≥ α ( B YM -I) or α ≥ λ ≥ √ 6 ( NA + ). We also find that a scalar field dominated universe ( S YM ), which is the same as the stable attractor in the model with a scalar field with an exponential potential ( V = V 0 exp( -αφ )), is stable in the parameter range of α -4 /α ≤ λ ≤ -α +4 /α , even though the YM field does not necessarily settle down to its attractor state. As we will show here, it will oscillate in this scalar dominated background. We may also wonder what is the future asymptotic behavior for the other range of the coupling parameters α and λ , i.e., λ > α and λ < √ 6. Numerical calculations give us some insight into this question too. Numerical study also tells us strengths of the stable attractors. Since our stability analysis is based on the linear perturbations, we need numerical analysis to know how the attractor state is achieved from generic initial data. does not vanish. We indeed find the asymptotic value of the power exponent p is 3 instead of 2. We", "pages": [ 12, 13 ] }, { "title": "1. Stable attractors", "content": "We begin with a small value of α for which the conventional power-law inflation is known to occur in the absence of gauge-kinetic coupling, namely α < √ 2. We choose the representative value to be α = 1. We first performed the calculation for λ = 2 (see Fig. 4), which shows the conventional powerlaw inflation with an exponential potential ( S YM -I ). The YM field energy drops quickly. We find that the asymptotic power exponent of the scale factor is 2, which is consistent with the value of the conventional power-law inflation ( p = 2 /α 2 ). When λ > 3, our analysis suggests the powerlaw inflation assisted by the magnetic component of the YM field ( B YM -I) is a stable attractor of the system. Fig.5 confirms this fact as the density parameter for magnetic component stays constant (Ω YM =constant and ∆ = 1) while the scalar potential dominates the energy budget, which implies the universe undergoes accelerated expansion. An important difference between Figs.4 and 5 is that the acceleration is actually stronger when Ω YM deliberately chose the initial condition such that the scalar kinetic energy and the electric component are dominant over the others and the effect of YM coupling is significant. As shown in Fig.5, ∆ approaches unity and Γ decays quickly whereby the system essentially reduces to the U (1) triplet model. We find B YM -I asymptotically. Next, we take a negative λ and confirm the electric-magnetic asymmetry for non-Abelian gauge fields. Fig.6 exhibits two different regimes. In the beginning, the electric energy density grows according to the linear instability caused by the strong gauge-kinetic coupling and the system is attracted towards the power-law inflation assisted by electric component of the YM field ( E YM -I). During that period, however, Γ continues to increase and eventually destroys the inflationary regime at N ∼ 10. The transient inflation E YM -I continues for 5 ∼ 6 e-folding number, which is consistent with our evaluation given in § . III B 1. After that, the universe is dominated by the scalar field while YM field is oscillating. In this case, since α is small enough to cause accelerated expansion by itself, this oscillation phase is also inflating. For comparison, we also show the plots with a smaller value of | λ | (Fig.7). The behavior is similar to Fig. 6, but there is no transient regime of E YM -I. When λ is negative, from the instability of E YM -I, there is a peculiar behavior of rapid oscil- ion at late time between electric and magnetic components, which is not seen for positive λ . Let us turn our attention to the supportive role of gauge fields in realizing inflation. We take α = 2 for which inflation is impossible by the scalar field itself. With λ = 5, we obtain Fig. 8 where ∆ = 1 in the future asymptotic state. The value of Ω V close to unity shows the expansion is accelerated, which can also be seen by the power exponent p > 1. As was investigated in the previous sections, this is due to the interaction between the scalar and YM fields that transfers scalar field energy to magnetic component of the YM field and slows down its rolling down the potential. Note that the velocity of the scalar field is given by ˙ φ = 2ln t/α , which is the same as the conventional power-law inflation. The difference is the values of total energy densities. The effective potential in the present model is given by the YM en- /Minus ergy as well as the scalar potential V (Eq. (3.6)), which gives a larger Hubble expansion rate. As a result, the velocity with respect to the e-folding number N becomes slower as φ ' = ˙ φ/H . For the U (1) gauge fields, the same type of inflation with non-flat potential E U1 -I could have been seen for negative λ because of the electro-magnetic duality. In the present non-Abelian case, however, a negative λ drives not only electric component but also the normalized gauge coupling Γ, by which the inflationary regime is made transient and the final state contains mixture of electric, magnetic and scalar fields (Fig.9). During the transient phase of inflation supported by the electric component of the YM field ( E YM -I), one can see the values of Ω V and Ω YM being the same as the corresponding magnetic inflation. Finally, we confirm the stability of the new nonAbelian fixed point NA + for λ > √ 6 (Fig.10). The convergence is relatively slow and all the dynamical components undergo oscillations. In contrast to the other cases where Γ either diverges or dies away, this parameter region sees convergence to an attractor value, which necessarily means negligible Ω V . Although it is not of interest in the context of inflation, it illustrates a distinct effect of the gauge coupling by forcing the potential term to vanish that would never happen in scalar-U(1) systems.", "pages": [ 13, 14, 15 ] }, { "title": "2. Oscillation of the YM field", "content": "The focus of this subsection is to understand the future asymptotic behavior of the system in the parameter region where the elementary fixed point analysis suggests there is no stable attractor solution. It turns out the nature of the dynamics in this regime is oscillation driven by the gauge coupling. Fig.11 shows the occurrence of scalar-YM oscillation as the future asymptotic state of the dynamical system for α = 4 , λ = -3. As is expected, the potential energy does not play a prominent role here. Ω V and Ω YM appear to converge to finite values although the numerical calculation has not been able to confirm it due to the computational difficulty caused by the rapid oscillation of ∆ and the ever-growing Γ. The behavior is mostly the same for negative λ regardless of λ < -√ 6 or not (Fig.12). The power exponent p of the scale factor is always slightly larger than 1 / 2. In contrast, λ = √ 6 is a threshold value for positive λ since the YM fixed point becomes the attractor above it (see Fig.10). Below the critical value, the asymptotic dynamics is rather analogous to the cases with negative λ , but with a significantly smaller contribution of Ω V . Convergence to an asymptotic value for Ω YM can be seen more clearly here (Fig.13). The power exponent p of the scale factor is slightly smaller than 1 / 2 for λ > 0.", "pages": [ 15 ] }, { "title": "B. Asymptotic spacetime with the oscillation of the YM field", "content": "From our numerical study, we find that the universe still approaches some attractor spacetime but the YM field is oscillating for some parameter range, where we do not find stable attractors. In order to identify such an attractor by an analytic approach, we assume that the time average of ∆, denoted by 〈 ∆ 〉 , does not change so quickly. We then discuss only three equations for /pi1 , Ω YM , and Γ, giving 〈 ∆ 〉 = ∆ 0 (constant). From our numerical analysis, we find the following two typical asymptotic behaviors: Γ increases monotonically in our numerical study. We then do not consider the equation for Γ to find an approximate asymptotic solution. For the case (ii), since there is the Hamiltonian constraint (4.8), /pi1 and Ω YM are not independent. We discuss the possible asymptotic solutions separately: For the case (i), the dynamical equations are where the reduced system gives a 'fixed point' Using this 'fixed point', the equation for Γ is written as This shows a monotonic increase of Γ, which is confirmed by our numerical calculation. The power exponent of the scale factor and the density parameter of the potential are given by From the positivity of density parameters, the following conditions must be imposed: Once we know p and /pi1 of the background spacetime, we can solve the YM equation as shown in Appendix A. Using this solution, we can take an average of ∆. However the background spacetime depends on ∆ 0 , which must be the same as the above averaged value 〈 ∆ 〉 . Hence we need an iterative procedure to find the correct averaged value of ∆ 0 . In Fig. 14, we present our result. As for the stability, we perturb the above two equations, whose eigenvalues are given by the two roots of the quadratic equation From the existence condition (5.4), we find the following stability conditions: for α > 4 / √ 3, where while for α < 4 / √ 3, we have only λ ∆ 0 > -α . The existence condition guarantees that two eigenvalues are negative. As a result, this 'fixed point' is always stable, although Γ diverges monotonically. We expect the universe in the parameter range of (5.4) will evolve into this spacetime with the oscillating YM field. Since λ ∆ 0 > 0, we find that p > 1 / 2, which is consistent with our numerical calculations. Note that for λ = 0, by which we have a scalar field and Yang-Mills field without interaction, we find p = 1 / 2 as we expect. These approximate 'fixed point' solutions seem to explain well our numerical results. For the case (ii), using the relation Ω YM = 1 -/pi1 2 / 6, we consider the following equation for /pi1 : The asymptotic solution can be obtained as a 'fixed point' in this sytem, which is In this background spacetime, we can also solve the YM equations as given in Appendix A. Using this oscillating solution, we evaluate the averaged value ∆ 0 . However, since the background spacetime depends on ∆ 0 , we have to find the correct value of ∆ 0 iteratively. In Fig. 15, we show the result. Although the qualitative behavior coincides with our numerical result (for example, p < 1 / 2 and Γ increases exponentially.), it does not reproduce our numerical result quantitatively For instance, the asymptotic value of Ω YM is ∼ 0 . 7 in this approximation, but the numerical value is ∼ 0 . 4. A possible source of discrepancy is that the oscillating time-scales for ∆ and Ω YM are the same so that one cannot replace ∆ by the constant averaged value ∆ 0 in the analysis of the dynamics of Ω YM and /pi1 , even though the amplitudes of oscillations for those variables is dying away.", "pages": [ 15, 16, 17 ] }, { "title": "VI. CONCLUDING REMARKS", "content": "We have studied an SU(2) non-Abelian gauge field coupled exponentially to a scalar field with an exponential potential, while making a comparison with the U(1) multiplet case. It gives We found that the power-law inflation with the magnetic component of the gauge field ( B YM -I) is possible and it is an attractor of the present system, if λ > α and λ > -α + 4 /α . The transfer of scalar kinetic energy to the gauge fields through the gauge-kinetic coupling makes an inflationary solution possible even for a steep potential such as α > √ 2, which is expected in the unified theories of fundamental interactions. On the other hand, the inflationary solution dominated by the electric component ( E YM -I) turned out to be unstable in contrast to the U(1) multiplet case. It can be a transient if the initial conditions are tuned. The attractor of the system is instead the conventional power-law inflation ( S YM -I) if α < √ 2. The YM field with a small amplitude is oscillating in this background universe. We have also found new fixed points ( NA ± ) in the parameter range of √ 6 < λ < α , which do not exist in the U(1) multiplet case. The fixed point NA + is an attractor, while NA -is unstable. We have also analyzed the non-inflationary regime, where the generic feature appears to be the oscillation of the YM fields ( O ± ). We summarize our result in Fig. 16. One may wonder whether those isotropic inflationary solutions are stable against anisotropic perturbations. Since there exist vector fields ( A (a) µ ), we usually find an anisotropic spacetime just as the case with a single U(1) gauge field. In order to prove the predictive power of the scenario, we have to show that the FLRW universe is obtained as an attractor in anisotropic Bianchi cosmologies. It is also interesting to know whether anisotropic inflation appears in a transient phase and its relic is observable or not. The study of Bianchi universe in the present model is in progress. Another important subject in the present model is a graceful exit from a stable inflationary universe, including a reheating mechanism and a calculation of density fluctuations. In order to leave the power-law inflationary attractor that is a selfsimilar scaling solution, within the context of unified theories of fundamental interactions, we may have the following possibilities, which may also work for the U(1) triplet case: the present scenario for the large value of σ , and the end of inflation arrives when σ gets small. As for the reheating of the universe, for the cases (1) and (2), since we have a potential minimum around which a scalar field will oscillate, we then find the reheating of the universe. It is not clear whether we can find the hot Big Bang state via the particle production assumed in the case (3). The density fluctuations have been calculated for the case of U(1) triplet, which shows the leading order effect of the background gauge fields is consistent with the current observational data[45]. While YM field is expected to give qualitatively similar results at the linear order, there is an interesting prospect of generating non-Gaussianity through the famous chaotic behaviors that are peculiar to the non-Abelian gauge fields[35, 46, 47]. These subjects are to be investigated in future works.", "pages": [ 17, 18 ] }, { "title": "Acknowledgments", "content": "We would like to thank John Barrow, Gary Gibbons, Keiju Murata, Nobuyoshi Ohta, and Paul Townsend for valuable comments. This work was partially supported by the Grant-inAid for Scientific Research Fund of the JSPS (C) (No.22540291). KM would like to thank DAMTP and the Centre for Theoretical Cosmology for hospitality during this work and Clare Hall for a Visiting Fellowship. He would also acknowledge a hospitality of APC, where this work was completed. KY would like to thank the Institute of Theoretical Astrophysics in the University of Oslo for the support and hospitality. Lect. Notes Phys. 738 , 1 (2008) [arXiv:0705.0164 [hep-th]] ; Spain, Jun 4-11, 1984 , ed. F. Del Aguila, et al. (World Scientific, 1984) pp. 123-146; If we have a power-law inflation a ∝ t p ( p > 1), the curvature term drops faster than the Hubble expansion term as k/a 2 ∝ t -2 p and H 2 ∝ t -2 . The curvature term can be ignored asymptotically just as the conventional inflationary scenario. However, it will be important in the non-inflationary universe. 414 (1988); T. Kawabe, S. Ohta, Phys. Rev. D41 , 1983 (1990); T. Kawabe, S. Ohta, Phys. Rev. D44 , 1274 (1991). See also the review. T. S. Byro, S. G. Matinyan, B. Muller, Chaos and gauge field theory [47] Y. Jin and K. Maeda,Phys. Rev. D71 , 064007 Rev. D72 , 103512 (2005). , (World Scientific, Singapore, 1994). (2005); J.D. Barrow, Y. Jin and K. Maeda, Phys.", "pages": [ 18, 19, 20 ] }, { "title": "Appendix A: Oscillation of the Yang-Mills field in the expanding universe", "content": "In some numerical calculations, we have seen the YM field oscillates very rapidly while the background spacetime evolves smoothly. If the energy density of the YM field is much smaller than that of the scalar field, the YM field does not contribute to the evolution of the universe. Even for the case that the YM field energy cannot be ignored, the oscillation of YM field may not directly affect the dynamics of the universe, but its mean value may contribute to the evolution of the universe. The different time-scales of the YM field oscillation and the evolution of FLRW universe may allow us to treat these two separately. Here we find such an oscillation of the YM field, assuming a given background spacetime and evolution of the scalar field. where we find the following equation for Z : Let us discuss the case where η increases as t increases, i.e., we assume that s > 0, or equivalently, 3( p -1) + λp/pi1 0 < 0. Hence this term in Eq. (A6) may drop as η → ∞ ( t → ∞ ). Once we ignore the second linear term, we find a simple non-linear differential equation which solves as where cn( x ; k ) is the Jacobi's elliptic function. Then the YM field is described in terms of the cosmic time t as Using this solution, we can evaluate the asymptotic behavior of the density parameter and the difference between magnetic and electric components of the YM field as Since the above approximate solution contains the parameter s (A5), which depends on the background solution, there are the following two cases: (1) The background is controlled only by the scalar field. The YM field is oscillating in the background, but its energy density is too small to affect the evolution of the universe. (2) The other case is that the averaged value of ∆ 0 as well as Ω YM give an important contribution onto the background. In that case, we need an iterative procedure to find the correct averaged value ∆ 0 , as shown in the main body of the article. Here we present the averaged value of ∆ and the properties of the asymptotic spacetime in the case (1). The results for the case (2) are given in § . VB. For inflation driven by a scalar field, we have Suppose the background is described by the following power-law solution: where p and /pi1 0 are constants, and N = ln( a/a 0 ) is the e-folding time. The equation for the isotropic YM field in this background is given by Changing the variables t and A to η and Z , which are defined by p = 2 /α 2 and /pi1 0 = α . Then the condition 3( p -1) + λp/pi1 0 < 0 is This is always satisfied in the range we consider. The average of ∆ must be taken in terms of the cosmic time t . We show our numerical result in Fig. 17.", "pages": [ 20, 21 ] } ]

[ { "title": "Wald-like formula for energy", "content": "Aaron J. Amsel ∗ Department of Physics and Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, AZ 85287 Dan Gorbonos † no affiliation We present a simple 'Wald-like' formula for gravitational energy about a constant-curvature background spacetime. The formula is derived following the Abbott-Deser-Tekin approach for the definition of conserved asymptotic charges in higher-derivative gravity.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In this paper, we present a simple formula for the computation of global charges (in particular energy) in asymptotically constant-curvature spacetimes for general gravitational theories. This is of particular interest for solutions of higher-derivative theories that approach a constant curvature solution in the asymptotic region. Our formula is similar to Wald's formula for entropy [1-3] in the sense that both formulas involve a derivative of the Lagrangian of the theory with respect to the Riemann tensor. The main difference is that the entropy is computed as an integral over the horizon, while the energy is computed in the asymptotic region (where we regard the solution as a perturbation of the background). Another difference is that Wald's formula involves only first derivatives with respect to curvature, whereas the formula for energy involves second derivatives. Nevertheless, the two formulas should be related by the first law of black hole thermodynamics and its integrated forms, i.e., Komar integrals and the Smarr formula [in some extended form to higher-derivative gravity and (A)dS spacetimes; see, e.g., Refs. [4, 5]]. Proposals for similar Wald-like formulas for the shear viscosity were given in Refs. [6, 7]. This approach gives us a new viewpoint on black hole thermodynamics. Wald's entropy formula was reinterpreted as the Bekenstein-Hawking entropy with an effective gravitational coupling [8] (see also Ref. [9]). This effective coupling comes from the coefficient of the kinetic term of a specific type of metric perturbation in the theory. Equivalently, this specific type of perturbation corresponds to the propagator for the exchange of a graviton between two covariantly conserved sources at the horizon, and the effective coupling comes from this propagator. Thus, Wald's formula not only has an advantage in the computational aspect, but also gives us a microscopic interpretation of the black hole entropy. In a similar way, a Wald-like formula for energy can be naturally interpreted as giving the effective gravitational coupling in the asymptotic region, namely, on the background. This avenue was explored in Ref. [10] for cubic corrections and in Ref. [11] for Lanczos-Lovelock gravity, where the effective gravitational coupling was identified as coming from higher-derivative corrections to the tree-level scattering amplitude between two background covariantly conserved sources via the exchange of a graviton. This is the same effective coupling that appears in the Wald-like energy formula derived below. This viewpoint, for both the entropy and energy, gives us two sides of the microscopic interpretation (in terms of corrections to a graviton exchange amplitude) of higher-derivative corrections to black hole thermodynamics. The derivation of the formula is based on the Abbot-Deser-Tekin (ADT) method for computing energy [12, 13]. We basically reduce their general method to a single formula which only requires substitution of the Lagrangian and the background solution. The story of this method starts with the result of Arnowitt, Deser, and Misner (ADM) [14] for energy in Einstein-Hilbert gravity with asymptotically flat boundary conditions. This was later generalized to spacetimes with a cosmological constant in Ref. [15]. These so-called 'Abbott-Deser (AD) charges' were written in a manifestly covariant way and once again could be expressed as pure surface integrals. The method used to construct the AD charges was then further generalized to arbitrary higher-curvature theories in Refs. [12, 13]. As we will see, the ADT method involves relatively little formalism and is computationally straightforward. In addition, this method has the advantage of not involving any explicit regularization or subtraction of infinities, as required in counterterm methods (see, e.g., Refs. [16, 17]). Unlike Euclidean path integral techniques (e.g., [18]), the ADT framework naturally gives the gravitational mass as an integral at asymptotic infinity, without any need to identify a horizon in the interior. For perturbations that vanish sufficiently fast at asymptotic infinity, the ADT charges are exactly the same as the charges derived using the covariant phase space methods of Refs. [19-21], which in turn differ from the charges of Wald et al. [2, 22, 23] by a surface term proportional to the Killing equations. This paper is organized as follows. In Sec. II, we present the Wald-like formula for energy and explain how to use it. We also give the energy of black hole solutions for two examples: Gauss-Bonnet gravity, and a theory with six-derivative corrections that was previously studied in Ref. [24]. We show that the calculation of energy becomes much shorter and simpler when using the formula. The rest of the paper is devoted to deriving this formula from the ADT method. After a short presentation of the ADT method in Sec. III, we discuss the general structure of the 'effective' stress-energy tensor in higher-derivative gravity in Sec. IV. We then obtain an explicit formula for the stress-energy tensor for a flat background in Sec. V. This expression is generalized to a curved background in Sec. VI, using the effective quadratic curvature method [10, 11, 25-27]. Following Ref. [12], in Sec. VII we complete the computation by deriving the energy from the general expression of the stress-energy tensor. We conclude with a brief discussion of our results and future directions in Sec. VIII.", "pages": [ 1, 2 ] }, { "title": "II. THE FORMULA FOR ENERGY", "content": "Here we present the formula for energy and explain how to use it. Let us consider a general d -dimensional theory of gravity whose action depends on the metric g µν and the curvature (through the Riemann tensor) We will construct the energy for such theories following the approach of Refs. [12, 13, 15]. We assume that the action is invariant under diffeomorphisms. In order to define a gauge-invariant conserved charge we need the presence of an asymptotic Killing symmetry. The charge is then defined relative to a background solution, denoted as ¯ g µν , which admits a Killing vector ¯ ξ µ . We assume that the background is a homogeneous solution, namely, described by an 'effective' cosmological constant Λ, which can be negative, positive, or zero (which is the asymptotically flat case). In addition, the solutions are required to fall off sufficiently fast at infinity relative to the background 1 . For a large class of solutions [which includes the asymptotically Schwarzschild-(A)dS (SdS) solutions defined below], we can write the energy of a generic higher-derivative theory in the same form as applies to Einstein-Hilbert gravity (with a cosmological constant), but with an overall multiplicative factor that depends on the higher-curvature terms. This will later serve as a basis for the interpretation of higher-derivative corrections as effective modifications of the gravitational coupling constant (Newton's constant). The Lagrangian of Einstein-Hilbert gravity with a cosmological constant is where κ is the d -dimensional gravitational coupling constant. The ADT energy for solutions of this theory is denoted by E 0 and is given explicitly in Eq. (3.6). Then, the energy for the general Lagrangian (2.1) is where [The complete energy formula for more general boundary conditions is given in Eq. (7.7).] Here Λ is the effective cosmological constant associated with the background solution ¯ g µν , which in general is distinct from the 'bare' cosmological constant Λ 0 that may appear in the action. The derivative of the Lagrangian with respect to R µν ρσ is performed formally, as if R µν ρσ and g µν are independent, and we impose the same tensor symmetries as R µν ρσ . For example, in the case of Einstein gravity (2.2) we get The ( . . . ) ¯ g notation indicates that the expression in parentheses is to be evaluated on the background spacetime ¯ g µν . The expressions in Eqs. (2.4) and (2.5) are 'projection' tensors that pick out certain coefficients to give the correct energy. [The subscript (1) will be explained later.] While the contractions with the projectors might appear complicated, their main use is to formally write the final formula (2.3). When we take a derivative with respect to the Riemann tensor and evaluate on the (homogeneous) background, we always get an expression of the form where N is some constant coefficient. The projector (2.4) is defined to give precisely the coefficient N when acting on Eq. (2.7). Thus, in practice one often simply reads off the coefficient after computing the derivative on the background, rather than actually performing the contraction with P ρσ µν . Similarly, for the second derivative with respect to the Riemann tensor evaluated on the background, there are in general three terms, and for a general theory we get Another important example is the energy density of black branes in AdS. When the asymptotic behavior of the black brane solution is Let us now look at two examples.", "pages": [ 2, 3, 4 ] }, { "title": "A. Example: Energy with a Gauss-Bonnet term", "content": "We start with the famous case of Gauss-Bonnet gravity. This term is topological (a total derivative) in four dimensions and leads to a ghost-free theory in any number of dimensions. The Lagrangain with the Gauss-Bonnet term reads and we have For a homogeneous background, so the energy density of the black brane is where N 1 , N 2 , N 3 are constants. When the projector P γδ,µν (1) αβ ρσ acts on Eq. (2.8), it picks out the coefficient N 1 , but again, we can also simply read off this coefficient by writing the expression for the second derivative in the above form. For example, if then N 1 = 2 C . In other words, the second derivative term in Eq. (2.3) roughly corresponds to coefficients of terms with the same type of contractions as in Eq. (2.9). A typical example of solutions that fall off sufficiently fast at infinity are asymptotically SdS solutions. The asymptotic behavior of such solutions is where r 0 is a constant. For these solutions, the energy in the case of Einstein gravity is given by and the projection P µν ρσ gives us the coefficient of δ ρ [ µ δ σ ν ] : The second derivative with respect to the Riemann tensor is Since we only require the coefficient of the first type of term, namely δ [ α µ δ β ] ν δ ρ [ γ δ σ δ ] , we get Substituting Eqs. (2.19) and (2.21) into the formula for the energy (2.12), we obtain This agrees with the result in Ref. [13], which was obtained by a longer calculation.", "pages": [ 4, 5 ] }, { "title": "B. Example: Energy with a six-derivative term", "content": "Here we will consider an example with six derivatives of the metric (cubic curvature), The ADT energy of this theory was previously given in Ref. [24], but required a much lengthier calculation. In this example, the first derivative with respect to the Riemann tensor gives and substituting the background metric leads to The second derivative with respect to the Riemann tensor (which in this example is not just a constant) is When we substitute the background metric, the second derivative becomes proportional only to the first term in Ref. (2.8), so the coefficient is just N 1 , and N 2 = N 3 = 0. Substituting this and Eq. (2.25) into the energy formula (2.12), we finally obtain which agrees with the result in Ref. [24].", "pages": [ 5 ] }, { "title": "III. THE ADT METHOD", "content": "In this section we give a brief review of the ADT method. The ADT method is similar in spirit to the LandauLifshitz pseudotensor method for calculating energy [29] in asymptotically flat curved spacetime. In particular, one proceeds by linearizing the equations of motion with respect to a background spacetime. This leads to an effective stress-energy tensor that consists of matter sources and terms higher-order in the perturbation. This tensor turns out to be covariantly conserved and can thus be used to construct a conserved charge associated with an isometry of the background. Let us consider some arbitrary gravitational theory with equations of motion of the form where κ is the gravitational coupling and τ µν is the matter stress-energy tensor. The symmetric tensor Φ µν , which is the analogue of the Einstein tensor, may depend on the metric, the curvature, derivatives of the curvature, and various combinations thereof. Assuming that the action is invariant under diffeomorphisms, we obtain the geometric identity ∇ µ Φ µν = 0 (the generalized Bianchi identity) and the covariant conservation of the stress tensor ∇ µ τ µν = 0. Now, we further assume that there exists a background solution ¯ g µν to the equations (3.1) with τ µν = 0. Then we decompose the metric as where we note that the deviation h µν is not necessarily infinitesimal, but it is required to fall off sufficiently fast at infinity. Asymptotically SdS spacetimes are a typical example meeting this requirement. By expanding the left-hand side of Eq. (3.1) in h µν , the equations of motion may be expressed as where φ ( i ) µν denotes all terms in the expansion of Φ µν involving i powers of h µν , and we have defined the effective stress-tensor T µν . It then follows from the Bianchi identity of the full theory that ¯ ∇ µ φ (1) µν = 0 = ¯ ∇ µ T µν . Suppose that the background spacetime admits a timelike Killing vector ¯ ξ µ , and let Σ be a constant-time hypersurface with unit normal n µ . Then we can construct a conserved energy in the standard way: where ¯ g Σ denotes the determinant of the induced metric on Σ. Because ¯ ∇ µ ( T µν ¯ ξ ν ) = 0, it follows that T µν ¯ ξ ν = ¯ ∇ ν F νµ for some antisymmetric tensor F νµ . The bulk integral (3.4) can therefore be rewritten as a surface integral over the boundary ∂ Σ: where r µ is the unit normal to the boundary. For example, for the Einstein-Hilbert theory (2.2), the explicit expression for the energy is In summary, to apply the ADT method, one linearizes the equations of motion to obtain the stress-energy tensor, and then expresses the conserved current T µν ¯ ξ ν as a total derivative to find the 'potential' F νµ . Note that by construction, the background spacetime ¯ g µν has E = 0.", "pages": [ 6 ] }, { "title": "IV. THE GENERAL STRUCTURE OF THE STRESS TENSOR", "content": "In Ref. [12], it was shown that the most general quadratic curvature theory has a stress tensor that is schematically of the form where and the α i are constants. Here R L µν is the linearized Ricci tensor and R L is the linearized Ricci scalar Note that the tensors G L µν , H ( i ) µν are each divergence free: It was later found in Ref. [24] that the stress tensor for a certain cubic curvature theory took exactly the same form (the only modification was to the values of the coefficients α i ) and it was suggested that this observation might hold more generally 2 . In this section, we will argue that this is indeed the case for any theory of the form (2.1). The basic idea is as follows. We saw in the previous section that the ADT stress tensor is given by the linearized (in h ) equations of motion. This means that T µν only depends on the action to O ( h 2 ). Now, expanding the general action (2.1) to O ( h 2 ) involves expanding the Riemann tensor, which contains terms of the form ¯ ∇ ¯ ∇ h . Hence, this can yield terms of at most four derivatives. This suggests quite generally that the basic form of the stress tensor does not change from Eq. (4.1) even if the action contains more than two powers of the Riemann tensor 3 . We will show that there exists a basis of three different components for the stress-energy tensor [to O ( h 2 ) and up to four derivatives], and that this basis can be chosen to correspond to G L µν , H (1) µν , and H (2) µν . To demonstrate this claim in more detail, we first consider the most general O ( h 2 ) action with two derivatives: For simplicity we work in the case of a flat background, ¯ g µν = η µν . The generalization to a curved background will be discussed later. Varying I 2 with respect to h µν yields the stress tensor If we impose conservation of the stress tensor, we obtain so equating the coefficients to zero gives Substituting these relations into Eq. (4.9) gives T µν = -4 β 1 G L µν . It follows that the Lagrangian yields a conserved stress tensor that is precisely G L µν (in a flat background). Note that this is also just the famous Fierz-Pauli Lagrangian [30]. Now let us repeat the same procedure for the most general O ( h 2 ) action with four derivatives: The corresponding stress tensor is and imposing ∂ µ T µν = 0 gives We see that there are two independently conserved tensors, and substituting these relations into Eq. (4.14) gives T µν = -2 β 5 H (1) µν +2 β 6 H (2) µν . It follows that the Lagrangian yields a conserved stress tensor that is precisely H (1) µν and the Lagrangian yields a conserved stress tensor that is precisely H (2) µν . Thus, we see that, in a flat background, there are at most two possible conserved combinations of terms with four derivatives in the Lagrangian. The above calculation can in principle be repeated for the case of a curved background spacetime, but it becomes complicated by the fact that the covariant derivatives no longer commute. The key point, however, is that commuting two derivatives in a given expression only produces extra terms of lower differential order. Thus, the expressions for L H ( i ) now must include two-derivative and zero-derivative terms, but the highest derivative order (four) terms are the same as in a flat background. Once these are fixed, the two-derivative and zero-derivative terms are chosen by requiring that each H ( i ) µν is separately conserved. A similar argument shows that the unique conserved quantity consisting of two-derivative and 0-derivative terms is G L µν . In other words, the terms with the highest derivatives (four ( i ) L In summary, we have just seen that any O ( h 2 ) action with no more than four derivatives produces a conserved stress tensor with the same structure as Eq. (4.1). Combining this with the argument that any theory L = L ( R µν ρσ , g µν ) expanded to O ( h 2 ) cannot have terms with more than four derivatives, we conclude that any such theory also has a stress tensor of the form (4.1). derivatives in H µν and two derivatives in G µν ) are the same for curved and flat backgrounds.", "pages": [ 6, 7, 8 ] }, { "title": "V. THE GENERAL FORMULA FOR THE STRESS TENSOR (FLAT BACKGROUND)", "content": "In this section we derive an efficient method to extract the coefficients α i in the stress tensor (4.1) given a Lagrangian L = L ( R µν ρσ , g µν ) for a flat background. The generalization for a curved background will be done in the next section. We wish to expand the action √ -g L to second order in the perturbation h . The Lagrangian can be expanded as The variation of the Riemann tensor is where and By convention, indices of Υ αβγ are raised/lowered with the background metric or its inverse. Note that each factor of δR µν ρσ contributes at least one h µν and two derivatives. In the remainder of this section we restrict to the case of a flat background, ¯ g µν = η µν . Now, the terms in the action with two derivatives and two h 's can only arise from expanding the term with one δR µν ρσ , that is Since L is a function only of g µν and R µν ρσ , it follows that ∂ L /∂R µν ρσ evaluated on a homogeneous background can only be a function of ¯ g µν . Furthermore, this quantity has the same symmetries as the Riemann tensor, so it must take the general form for some constant N . Formally, this constant can be expressed as a 'projection' for the projection tensor Inserting Eq. (5.7) into Eq. (5.6), we see that we simply need the expansion of N √ -g R . This is of course just the Einstein-Hilbert action (up to the overall factor N ), whose expansion is well-known to give the Fierz-Pauli action (see, e.g.,Ref. [31]), NL G . The O ( h 2 ) terms in the action with four derivatives can only arise from the term and for this we just need the linear term in δR µν ρσ , Now, the second derivative evaluated on a homogeneous background can only be a function of ¯ g µν , and this quantity must have the same index symmetries as the product of two Riemann tensors. Hence, there can be three independent contributions for some constants N i . This is similar to the statement that there are only three independent curvature invariants formed from contracting two Riemann tensors. Formally, these constants can be expressed by acting with projectors where The coefficients are with The next step is to insert Eq. (5.12) into Eq. (5.10) and use Eq. (5.11). We treat the three contractions separately. The first is analogous to R 2 µνρσ and gives The second contraction is analogous to R 2 and gives The last contraction is analogous to R 2 µν and gives Thus the relevant part of the expanded action is and the corresponding stress tensor is", "pages": [ 8, 9, 10 ] }, { "title": "VI. THE GENERAL FORMULA FOR THE STRESS TENSOR (CURVED BACKGROUND)", "content": "The procedure described previously for a flat background should in principle generalize to a curved background. The calculation becomes cumbersome, however, since the covariant derivatives no longer commute. Instead, we shall adopt a different approach that turns out to be much more straightforward. It was argued in Refs. [10, 11, 25-27] that any higher-curvature theory which is polynomial in the Riemann tensor and its contractions can be reduced to an 'effective quadratic curvature' action with the same propagator. Since the propagator also only depends on the action up to order h 2 , we can adapt this procedure to determine the ADT stress-tensor for a general theory. Consider expanding the Lagrangian of a generic higher-curvature theory of the form L = L ( R µν ρσ ), Here the dots represent terms which are necessarily of order h 3 and therefore are not relevant to the ADT energy. Next we substitute the general expressions for the derivatives of the Lagrangian with respect to the Riemann tensor evaluated on the background, which were previously given in Eqs. (5.7) and (5.12). Using Eq. (2.17) and collecting coefficients of the full Riemann tensor terms, we obtain the effective quadratic theory Here we have defined and the 'bare' cosmological constant for the effective theory is Now, the most general quadratic curvature theory has already been treated in Ref. [13]. The result is that the stress tensor is where G L µν , H (1) µν , and H (2) µν were given in Eqs. (4.2)-(4.4). Furthermore, the effective cosmological constant Λ is fixed by evaluating the equation of motion on the background solution: Substituting the above expressions for ˜ κ, α, β, γ into Eq. (6.8) yields Note that for Λ = 0, this agrees with the result of the previous section.", "pages": [ 10, 11 ] }, { "title": "A. Examples", "content": "Let us look at some examples for the formula (6.10) for the stress-energy tensor. Let us start with the simple theory The coefficients N,N i are computed as described previously by taking derivatives with respect to the Riemann tensor and evaluating on the background AdS solution. We find that and so N 1 = 2 , N 2 = N 3 = 0. Using Eq. (6.10), we obtain which matches the result of Ref. [13]. For a more complicated example, consider the six-derivative theory whose stress tensor was computed explicitly in Ref. [24]. The results are summarized in the following table: where Substituting the above results into Eq. (6.10) gives where This reproduces precisely the stress tensor obtained in Ref. [24].", "pages": [ 11, 12 ] }, { "title": "VII. THE DERIVATION OF THE ENERGY FORMULA", "content": "Given the result (6.10), the final step in the derivation of the energy formula is to write ¯ ξ ν T µν as a total derivative. For this purpose, we follow the steps in Ref. [12]. For the first term, ¯ ξ ν G µν L , the result has already been given in Eq. (3.6). It is straightforward to show that the second term can be written as The third term, ¯ ξ ν H (2) µν , is more complicated and turns out to give an additional contribution of the form ¯ ξ ν G µν L . To see this, we can rewrite Since ¯ ξ ν is a Killing vector, it satisfies Then the last terms of Eq. (7.2) simplify to Using these results, we find that the final form of the conserved energy is In asymptotically SdS spacetimes [see Eq. (2.10)], the last two terms in Eq. (7.7) fall off too fast at large r to contribute and the total energy is given by or in the full explicit form as in Eq. (2.3).", "pages": [ 12, 13 ] }, { "title": "VIII. DISCUSSION", "content": "In this paper, we have derived a simple formula (2.3) for the ADT energy of any gravitational theory of the form L = L ( R µν ρσ , g µν ). We gave a detailed argument that the energy of such a theory takes the same basic form as in quadratic curvature gravity, but with coefficients modified by the higher-curvature terms. The coefficients are given by taking derivatives of the Lagrangian with respect to the Riemann tensor, and in this sense our energy formula is reminiscent of Wald's entropy formula. We have demonstrated in a number of examples that our formula correctly reproduces previous results, but with significantly less computations. For more complicated theories in which following the full ADT procedure would be unmanageable in practice, it seems that our formula could still be applied relatively easily. We note from the final formula for energy (2.3) that only N and N 1 appear, and it would be interesting to understand why this is the case. We also see that in d = 3, the contribution of the second derivative of the Lagrangian completely drops out. In the case of three-dimensional topologically massive gravity (TMG)[32-34], the action contains a gravitational Chern-Simons term so it is not of the form L = L ( R µν ρσ , g µν ). Indeed, the ADT energy for TMG has a different structure than (2.3) [35]. It is also known that Wald's entropy formula has to be modified in TMG, since the Chern-Simons term does not satisfy the diffeomorphism-covariance requirement in the original construction (see, e.g., Refs. [36, 37]). Given the final expression for the energy (7.8), it seems natural to define the effective gravitational coupling as Then the energy can be written succinctly in terms of the Einstein gravity result as This is analogous to the way the entropy was written in Ref. [8] as where A is the black hole area. However, the effective coupling constant also has an interpretation in the tree-level scattering amplitude via the exchange of a graviton. When one looks at a similar process on the background [10], the effective coupling turns out to be the coefficient of G L µν in the stress-energy tensor: The two definitions for the effective coupling coincide for Lanczos-Lovelock gravity, since any Lagrangian of the Lanczos-Lovelock type can be reduced to a Gauss-Bonnet quadratic theory [11]. This coincidence might be related to the fact that higher-derivative theories which are not of the Lanczos-Lovelock type exhibit ghosts and other inconsistencies [38, 39]. In future work, it would be interesting to further understand this ambiguity in the definition of the effective gravitational coupling. Note Added: One day after this paper appeared on the arXiv, the paper [40] appeared with some overlapping material.", "pages": [ 13, 14 ] }, { "title": "IX. ACKNOWLEDGMENTS", "content": "We thank Ramy Brustein for a discussion and Stanley Deser for comments. [10] T. C. Sisman, I. Gullu and B. Tekin, 'All unitary cubic curvature gravities in D dimensions,' Class. Quant. Grav. 28 , 195004 (2011) [arXiv:1103.2307 [hep-th]].", "pages": [ 14 ] } ]

[ { "title": "Formation of scalar hair on Gauss-Bonnet solitons and black holes", "content": "Yves Brihaye ∗ (1) , Betti Hartmann † (2) and Sardor Tojiev ‡ (2) (1) Physique-Math'ematique, Universite de Mons-Hainaut, 7000 Mons, Belgium (2) School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany October 14, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We discuss the formation of scalar hair on Gauss-Bonnet solitons and black holes in 5-dimensional Antide Sitter (AdS) space-time. We present new results on the static case and point out further details. We find that the presence of the Gauss-Bonnet term has an influence on the pattern of soliton solutions for small enough values of the electric charge. We also discuss rotating Gauss-Bonnet black holes with and without scalar hair. PACS Numbers: 04.70.-s, 04.50.Gh, 11.25.Tq", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Most theories of quantum gravity need more than the standard four space-time dimensions to be consistent. String theory is an example of this. The low energy effective action of these models in general reproduces Einstein gravity, but also contains terms that are higher order in the curvature invariants [1]. In five space-time dimensions this is the Gauss-Bonnet (GB) term, which has the property that the equations of motion are still second order in derivatives of the metric functions. Since black holes are thought to be the testing grounds for quantum gravity models it is of course of interest to study the generalisations of known black hole solutions to include higher order curvature corrections. As such explicit spherically symmetric and asymptotically flat black hole solutions in GB gravity are known for the uncharged case [2, 3], for the charged case [4] as well as in Anti-de Sitter (AdS) [5, 6, 7, 8] and de Sitter (dS) space-times [9], respectively. In most cases, black holes not only with spherical ( k = 1), but also with flat ( k = 0) and hyperbolic ( k = -1) horizon topology have been considered. Moreover, the thermodynamics of these black holes has been studied in detail [7, 8, 10] and the question of negative entropy for certain GB black holes in dS and AdS has been discussed [6, 11]. One of the most important results of String theory is surely the AdS/CFT correspondence [12, 13] that states that a gravity theory in ( d +1)-dimensional AdS space-time is equivalent to a conformal field theory (CFT) on the d -dimensional boundary of AdS. This correspondence is a weak-strong coupling duality and can be used to describe strongly coupled Quantum Field Theories on the boundary of AdS by weakly coupled gravity theories in the AdS bulk. An application of these ideas is the description of high temperature superconductivity with the help of black holes and solitons in AdS space-time [14, 15, 16, 17]. In most cases (3 + 1)-dimensional solutions with planar horizons ( k = 0) were chosen to account for the fact that high temperature superconductivity is mainly associated to 2-dimensional layers within the material. The basic idea is that at low temperatures a planar black hole in asymptotically AdS becomes unstable to the condensation of a charged scalar field since its effective mass drops below the Breitenlohner-Freedman (BF) bound [18] for sufficiently low temperature of the black hole hence spontaneously breaking the U(1) symmetry. It was shown that this corresponds to a conductor/superconductor phase transition. Alternatively, solitons in AdS become unstable to scalar hair formation if the value of the chemical potential is large enough. Since solitons do not have a temperature associated to them this has been interpreted as a zero temperature phase transition between an insulator and a superconductor. Interestingly, there seems to be a contradiction between the holographic superconductor approach and the Coleman-Mermin-Wagner theorem [19] which forbids spontaneous symmetry breaking in (2 + 1) dimensions at finite temperature. Consequently, it has been suggested that higher curvature corrections and in particular GB terms should be included on the gravity side and holographic GB superconductors in (3+1) dimensions have been studied [20]. However, though the critical temperature gets lowered when including GB terms, condensation cannot be suppressed - not even when backreaction of the space-time is included [21, 22, 23]. Next to the stability of solutions with flat sections and their application in the context of holographic superconductors it is of course also of interest to discuss the stability of black holes with spherical ( k = 1) or hyperbolic ( k = -1) horizon topology in AdS space-time. In [24] the question of the condensation of a uncharged scalar field on uncharged black holes in (4 + 1) dimensions has been addressed. As a toy model for the rotating case, static black holes with hyperbolic horizons ( k = -1) were discussed. In contrast to the uncharged, static black holes with flat ( k = 0) or spherical ( k = 1) horizon topology hyperbolic black holes possess an extremal limit with near-horizon geometry AdS 2 × H 3 [25, 26, 27]. This leads to the instability of these black holes with respect to scalar hair formation in the nearhorizon geometry as soon as the scalar field mass becomes smaller than the 2-dimensional BF bound. Note that these black holes are still asymptotically AdS as long as the (4 + 1)-dimensional BF bound is fulfilled. These studies were extended to include Gauss-Bonnet corrections [28]. In [29] static, spherically symmetric ( k = 1) black hole and soliton solutions to Einstein-Maxwell theory coupled to a charged, massless scalar field in (4 + 1)-dimensional global AdS space-time have been studied. The existence of solitons in global AdS was discovered in [30], where a perturbative approach was taken. In [29] it was shown that solitons can have arbitrarily large charge for large enough gauge coupling, while for small gauge coupling the solutions exhibit a spiraling behaviour towards a critical solution with finite charge and mass. The stability of Reissner-Nordstrom-AdS (RNAdS) solutions was also studied in this paper. It was found that for small gauge coupling RNAdS black holes are never unstable to condensation of a massless, charged scalar field, while for intermediate gauge couplings RNAdS black holes become unstable for sufficiently large charge. For large gauge coupling RNAdS black holes are unstable to formation of massless scalar hair for all values of the charge. Moreover, it was observed that for large gauge coupling and small charges the solutions exist all the way down to vanishing horizon. The limiting solutions are the soliton solutions mentioned above. On the other hand for large charge the limiting solution is a singular solution with vanishing temperature and finite entropy, which is not a regular extremal black hole [31]. These results were extended to a tachyonic scalar field as well as to the rotating case [32]. Recently, solutions in asymptotically global AdS in (3 + 1) dimensions have been studied in [33]. It was pointed out that the solutions tend to their planar counterparts for large charges since in that case the solutions can become comparable in size to the AdS radius. The influence of the Gauss-Bonnet corrections on the instability of these solutions has been discussed in [34]. In this paper, we are interested in Gauss-Bonnet solitons as well as black holes in (4 + 1)-dimensional AdS space-time. We study the case of solitons in more detail and point out further details in comparison to [34]. Moreover, we also discuss static and rotating Gauss-Bonnet black holes with and without scalar hair. The rotating solutions without scalar hair have been previously studied in asymptotically flat space-time [35, 36, 37] as well as in AdS space-time [38, 37]. Our paper is organized as follows: we present the model in Section 2, while we discuss solitons in Section 3. In Section 4, we present our results for the black holes and we conclude in Section 5.", "pages": [ 1, 2 ] }, { "title": "2 The model", "content": "In this paper, we are studying the formation of scalar hair on electrically charged black holes and solitons in (4 + 1)-dimensional Anti-de Sitter space-time. The action reads : where Λ = -6 /L 2 is the cosmological constant and α is the Gauss-Bonnet coupling with 0 ≤ α ≤ L 2 / 4. α = 0 corresponds to Einstein gravity, while α = L 2 / 4 is the so-called Chern-Simons limit. L matter denotes the matter Lagrangian which reads : where F MN = ∂ M A N -∂ N A M is the field strength tensor and D M ψ = ∂ M ψ -ieA M ψ is the covariant derivative. e and m 2 denote the electric charge and mass of the scalar field ψ , respectively. The coupled gravity and matter field equations are obtained from the variation of the action with respect to the matter and metric fields, respectively, and read where H MN is given by and T MN is the energy-momentum tensor In the following, we want to study rotating Gauss-Bonnet solitons and black holes in (4 + 1) dimensions. In general, such a solution would possess two independent angular momenta associated to the two independent planes of rotation. Here, we will restrict ourselves to the case where these two angular momenta are equal to each other. The Ansatz for the metric reads [39] where θ runs from 0 to π/ 2, while ϕ 1 and ϕ 2 are in the range [0 , 2 π ]. The solution possesses two rotation planes at θ = 0 and θ = π/ 2 and the isometry group is R × U (2). In the following, we will additionally fix the residual gauge freedom by choosing g ( r ) = r 2 . For the electromagnetic field and the scalar field we choose : Note that originally the scalar field is complex, but that we can gauge away the non-trivial phase and choose the scalar field to be real. The coupled, non-linear ordinary differential equations depend on four independent constants: Newton's constant G , the cosmological constant Λ (or Anti-de Sitter radius L ), the charge e and mass m of the scalar field. Here, we set m 2 = -3 /L 2 . The system possesses two scaling symmetries: as well as which we can use to set L = 1 and γ to some fixed value without loosing generality. In the following, we will choose γ = 9 / 40 unless otherwise stated. Note that the metric on the AdS boundary in our model is that of a static 4-dimensional Einstein universe with boundary metric γ µν , µ , ν = 0 , 1 , 2 , 3 given by and is hence non-rotating. This is different to the case of rotating and charged black holes in 4-dimensional AdS space-time, the so-called Kerr-Newman-AdS solutions [40], which possess a boundary theory with non-vanishing angular velocity Asymptotically, the matter fields behave as follows with where Q and Q m are related to the electric and magnetic charge of the solution, respectively. µ is a constant that within the context of the gauge/gravity duality can be interpreted as chemical potential of the boundary theory. For the rest of the paper we will choose ψ -= 0. Within the AdS/CFT correspondence ψ + corresponds to the expectation value of the boundary operator in the dual theory. The metric functions have the following asymptotic behaviour The parameters in the asymptotic expansion can be used to determine the mass and angular momentum of the solutions. The energy E and angular momentum ¯ J are In the following, we have constructed soliton as well as black hole solutions to the equations of motions numerically using a Newton-Raphson method with adaptive grid scheme [41].", "pages": [ 3, 4 ] }, { "title": "3 Solitons", "content": "In the following, we would like to study globally regular, i.e. soliton-like solutions to the equations of motion. As was shown previously in [32] properly rotating solitons do not exist in our model. We will hence concentrate on the static case which corresponds to the limit ω ( r ) ≡ 0, A ( r ) ≡ 0 and g ( r ) = h ( r ) = r 2 . In the static case, it is also convenient to work with a ( r ) ≡ √ b ( r ) /f ( r ). For the remaining functions, we have to fix appropriate boundary conditions which read at the origin, where ψ (0) ≡ ψ 0 is a free parameter, and on the conformal boundary. Note that the condition ψ (0) = ψ 0 can be replaced by fixing the electric charge r 3 φ ' ( r ) | r →∞ = 2 Q . However, we found it more convenient to fix ψ 0 and determine the charge Q in dependence on this parameter. Once fixing the parameters e 2 and α we can construct families of regular solutions labelled by the charge Q or by the mass M . The pattern of solutions turns out to be so involved that neither Q nor M can be used to characterize the solution uniquely. This is shown in Fig.1, where we give the charge Q as function of a (0) and the mass M as function of ψ (0), respectively. We observe that for small values of e 2 several disconnected branches exist. These solutions have the same values of M and Q , but differ in the values of ψ (0) and a (0). We can also formulate this statement differently: for a fixed value of ψ (0) and a (0) more than one solution exists. The different solutions are then characterized by different values of the mass M and charge Q . As an example let us consider the case α = 0, e 2 = 1 . 2. Here, we find that three soliton solutions exist with the same value of the scalar field at the origin ψ (0). The profiles of these solutions are given in Fig. 2 (upper figure) for ψ (0) = 3 . 75. They have masses M = 0 . 29, M = 2 . 15 and M = 418 . 00, respectively. The existence of several disconnected branches had been observed before in other models [29, 33]. Here, we find that the presence of the GB term changes the general pattern in that at sufficiently large α > α cr the two branches with the lowest mass join. This is shown in Fig.3 for ψ (0) = 3 . 75 and e 2 = 1 . 2. For this choice of parameters, we find α cr ≈ 0 . 086. For α > α cr these lowest mass solitons cease to exist. Contrary to that the solution with the highest mass can be deformed all the way to the maximal value of α , the Chern-Simons limit with α = L 2 / 4. This is seen in Fig. 3. /negationslash For larger values of e 2 we find that the qualitative pattern does not change when increasing α . This is shown in Fig.4, where we give the charge Q in dependence on a (0) and the mass M in dependence on ψ (0) for e 2 = 2 and different values of α . For α = 0 only one branch of soliton solutions exists and this persists for α = 0. No new branches appear due to the presence of the GB term. Let us mention here that the appearance of several branches is due to the presentation of data. If we would instead consider to plot asymptotically measurable quantities as e.g. the mass M , the charge Q or the free energy F = M -µQ we would find that the solutions are uniquely described by these parameters. This is see in Fig.5, where we give the free energy F as function of the charge Q . The plot shows that for a given value of Q there is at most ONE solution and that the free energy decreases with increasing Q . For fixed Q we find that the solution with the smallest e 2 has the lowest free energy. If we fix ψ (0) as described above up to three solutions exist of which we believe the lowest mass solution to be the stable one. This solution is always on the same branch of solutions as the global AdS space-time with M = Q = 0. Finally, we show the profiles of the metric and matter functions of the soliton solutions corresponding to large values of α in Fig.6. The solution with α = 0 . 25 corresponds to a hairy, charged Chern-Simons soliton.", "pages": [ 4, 5, 6 ] }, { "title": "4 Black holes", "content": "We are interested in solutions possessing a regular horizon at r = r h . Hence we require such that the metric fields have the following behaviour close to the horizon [35, 36, 37, 38] where ω ( r h ) ≡ Ω corresponds to the angular velocity at the horizon and f 1 , b 1 and h 0 are constants that have to be determined numerically. In addition there is a regularity condition for the metric fields on the horizon given by Γ 1 ( f, b, b ' , h, h ' , ω, ω ' ) = 0, where Γ 1 is a lengthy polynomial which we do not give here. For the matter fields we have to require where Γ 2 is a polynomial expression in the fields which we also do not present here. Using the expansions of the metric functions we find that temperature T H and the entropy S are given by [35] where V 3 = 2 π 2 denotes the area of the three-dimensional sphere with unit radius.", "pages": [ 6 ] }, { "title": "4.1 Static black holes", "content": "We first study the static case with Ω = 0. This has already been considered in [34]. Here we point out further important details in the pattern of solutions.", "pages": [ 6 ] }, { "title": "4.1.1 Exact solutions", "content": "In the case ψ ≡ 0, i.e. when the scalar field vanishes there exists an explicit solution to the equations. This reads [2, 5, 6, 7, 8] where M and Q are arbitrary integration constants that can be interpreted as the mass and the charge of the solution, respectively. Note first that for M = Q = 0 there are two global AdS solutions with effective AdS radius L 2 eff = L 2 / 2 ( 1 ± 1 -4 α/L 2 ) . √ Since we are interested in black hole solutions here which fulfill f ( r h ) = 0 the '+' solution is of no interest in the static case. In the limit α → 0, the metric function f ( r ) of the '-' solution becomes f ( r ) = 1+ r 2 L 2 -M r 2 + γQ 2 r 4 and the corresponding solutions are Reissner-Nordstrom-Anti-de Sitter (RNAdS) black holes.", "pages": [ 7 ] }, { "title": "4.1.2 The α = 0 limit", "content": "We first discuss the limit of vanishing Gauss-Bonnet coupling α = 0. We find that black hole solutions exist for generic values of e 2 , r h and ψ ( r h ) but that the domain of existence for these parameters is limited. It depends crucially on the value of e 2 . For large values of e 2 for which only a single branch of corresponding soliton solutions exist black holes exist for all values of r h > 0 and ψ ( r h ) > 0. In the limit ψ ( r h ) → 0 with r h fixed they approach the RNAdS solution, while in the limit r h → 0 with ψ ( r h ) fixed they approach the corresponding soliton. The situation is more subtle for small values of e 2 , i.e. when disconnected branches of solitons exist. This is shown in Fig. 7 for e 2 = 1 . 2, where we present the temperature and the mass of the solutions for several values of ψ ( r h ). We find that for sufficiently high values of ψ ( r h ) the black holes exist all the way down to r h = 0, where they join the branch of soliton solutions. In this limit the temperature goes to infinity and the mass tends to the corresponding mass of the soliton solution. This is clearly visible for ψ ( r h ) = 3 . 25 and ψ ( r h ) = 4, respectively. On the other hand, for small values of ψ ( r h ) we find that the black holes exist only above a critical value of the horizon radius r h = r ( cr ) h . For r h → r ( cr ) h the temperature of the black hole tends to zero. However, this is not an extremal black hole, but a singular black hole solution. This is apparent when observing that a ( r h ) → 0 in this limit. This is clearly seen for ψ ( r h ) = 1 and ψ ( r h ) = 2 in Fig.7. There should hence be a critical value of ψ ( r h ) at which the transition between the two pattern appears. We find that the determination of the exact numerical value of this critical ψ ( r h ) is quite involved, but we conjecture that it corresponds to the minimal value of ψ (0) at which the disconnected branches of solitons terminate. We further observe that if r h is big enough (typically r h /greaterorsimilar 0 . 5) a connected branch of black holes with scalar hair labelled by ψ ( r h ) exists. This is shown in Fig. 8 for e 2 = 1 and e 2 = 2, respectively. In this figure we show the mass and temperature as function of ψ h ≡ ψ ( r h ) as well as the charge as function of a ( r h ) of these black hole solutions. For ψ ( r h ) → 0, a ( r h ) → 1 these solutions approach the RNAdS solutions with finite values of the mass M , charge Q and temperature T H .", "pages": [ 7, 8 ] }, { "title": "4.1.3 GB black holes", "content": "In the following, we restrict our analysis to a finite number of parameters and construct mostly (unless otherwise stated) black holes with r h = 0 . 5 and e 2 = 1 or e 2 = 2. We can then discuss the pattern of solutions for different values of α . /negationslash We find that for α = 0 and for some values of the parameters e 2 and ψ ( r h ), the black holes can be continued up to the Chern-Simons limit α = L 2 / 4. This is the case e.g. for e 2 = 1, ψ ( r h ) = 3 . 75, r h = 0 . 5 as shown in Fig. 9. For larger values of e 2 (with the same values of ψ ( r h ) and r h ) the black hole ceases to exist at some intermediate value of α where the temperature T H tends to zero. This is shown in Fig.9 for e 2 = 2. To state it differently, for sufficiently large values of e 2 GB black hole solutions with scalar hair exist only up to a critical value of the GB coupling that is smaller than the Chern-Simons limit. For our specific choice of parameters here we find that solutions exist only for α /lessorsimilar 0 . 14.", "pages": [ 8 ] }, { "title": "4.2 Rotating black holes", "content": "In principle, we want to discuss rotating black holes with scalar hair in this paper. However, during our numerical analysis it turned out that some as yet unnoticed features appear also for rotating GB black holes without scalar hair. This is why we will discuss this case first before turning to the case with scalar hair.", "pages": [ 8 ] }, { "title": "4.2.1 Black holes without scalar hair", "content": "In order to understand the pattern of solutions for ψ ≡ 0, it is useful to recall some limiting cases. Non-rotating solutions with Ω = 0 can be constructed for α ∈ [0 , L 2 / 4], Q ∈ [0 , Q m ], where Q m is the maximal charge up (b) M as function of r h to where the solutions exist. For α = 0 the solutions with Q fixed exist up to a maximal value of the rotation parameter Ω = Ω m where they become extremal [37]. For Q = 0 the solutions with a fixed α also exist up a maximal value of Ω as discussed for L 2 = ∞ in [36] and for L 2 < ∞ in [37] . Let us first discuss how the solutions evolve when gradually increasing the horizon velocity Ω. Our numerical results indicate that a branch of solutions can be constructed up to a critical value α = α cr which depends on Ω such that these solutions exist for The critical value of α = α cr up to where rotating GB black holes exist depends both on Ω and Q . We find e.g. for small values of Ω, r h = 0 . 8, Q = 1 that α cr ≈ 0 . 23. The critical phenomenon occurring in the limit α → α cr can be understood by examining the value h ' ( r h ). It turns out that in this limit, the value h ' ( r h ) (which is a positive number for α large enough) increases very rapidly with α . Our numerical results further suggest that this branch cannot be continued for α > α cr . This is shown in Fig.10, where the parameter h ' ( r h ) is given as a function of α for different values of Q and Ω. In order to understand the physical meaning of this pattern we present the temperature T H as function of entropy S in Fig.11. This clearly shows that two branches with a transition between the branches occurring at an intermediate value of α exist. Note that small (large) entropy and large (small) temperature corresponds to α small (large). We observe that for sufficiently large α the entropy increases with increasing T H which would signal thermodynamical stability. Increasing the value of h ' ( r h ) further we were able to construct a second branch of solutions for α < α cr . The two branches are such that they merge at α = α cr . This is shown in Fig. 12 where we present some physical quantities like the mass M and the temperature T H in dependence on α for a rotating GB black hole solution with Ω = 0 . 02, Q = 1 and r h = 0 . 8. Although the numerical construction becomes quite involved we strongly suspect that further branches can be constructed in the region around α cr . These branches, however, cannot be extended to small values of α . Surprisingly, we find that for the values of α for which the two solutions coexist the solutions of the second branch have smaller energy than the solutions on the first branch. This is demonstrated in Fig.12. The discussion above suggests that GB black holes exist only up to a maximal value of the GB coupling. However, we know that for Ω = 0 black holes also exist for all values of α up to the Chern-Simons limit α = L 2 / 4. (b) M as function of Q It is therefore a natural question to attempt to construct the rotating generalizations of these solutions. We therefore considered the static solutions close to α ∼ L 2 / 4 and constructed rotating generalizations of these. We managed to construct another branch, i.e. a third branch of solutions in this region of the parameter. In contrast to the solutions on the other two branches, the solutions of this new branch have h ' ( r h ) < 0. In addition, we were able to construct yet another, i.e. a fourth, branch of solutions that coincides with the third one at α = ˜ α . It is worth pointing out that the branches of solutions have h ' ( r h ) > 0 and h ' ( r h ) < 0, respectively. Hence they are completely disjoint. This phenomenon seems to be specific for charged, rotating solutions in asymptotically AdS. Indeed, a similar study in the case of uncharged black hole [36, 37] reveals the occurrence of a unique branch of rotating black holes. It is also interesting to understand how these solutions behave for large rotation parameter Ω. We find that the main branch gets smoothly deformed by the rotation. The third branch becomes smaller in α and it cannot be extended continuously up to α = L 2 / 4. This suggest that α = L 2 / 4 is itself a critical value for rotating black holes and that charged Chern-Simons black holes cannot rotate. This, however, should be confirmed by an independent analysis which we do not aim at in this paper. We have also studied the influence of the GB term on the solutions. We find that for small α the black hole terminates into an extremal solution at a maximal value of Ω. For larger α our numerical results suggest that several branches of solutions exist that at a critical value of Ω terminate also into an extremal solution.", "pages": [ 8, 10, 11 ] }, { "title": "4.2.2 Black holes with scalar hair", "content": "We find that rotating GB black holes with scalar hair exist for smaller values of r h as compared to the solutions without scalar hair. Our numerical analysis of the solutions for several values of Ω and α suggests the occurrence of a phenomenon similar (although slightly more involved) to the one discussed for black holes without scalar hair. Choosing r h = 0 . 5 and Q = 1 we constructed families of rotating solutions with α > 0. Again, the parameter h ' ( r h ) plays a crucial role in the understanding of the pattern of solutions. This seems to diverge for (at least) four values of the parameter α , say for α = ˜ α k , k = 1 , 2 , 3 , ... . By diverging we mean that it tends to ±∞ for α approaching ˜ α k from the left and the right, respectively. This critical phenomenon seems to be present already for slowly rotating black hole and persists for larger angular momentum. We observe that the critical values ˜ α k depend only weakly on the value Ω. However, when plotting physical quantities as given in Fig. 13 we observe no discontinuities. We therefore believe that the critical values of α correspond to configurations where the coordinate gauge fixing g ( r ) = r 2 becomes accidentally not appropriate to describe the solution. Further study of this phenomenon is currently under investigation.", "pages": [ 11, 14 ] }, { "title": "5 Conclusions", "content": "In this paper we have studied 5-dimensional solitons and black holes in GB gravity coupled to electromagnetic and scalar fields. In the limit of vanishing GB coupling this model reduces to Einstein-Maxwell theory coupled to a scalar field. For vanishing scalar field the static black hole solutions are the RNAdS solution and its GB generalization. For a fixed value of the electric charge Q , these exist for horizon radius larger than a minimal value. At this minimal value the black hole becomes extremal. Black holes with scalar hair exist for smaller values of the horizon extending the domain of existence of the static black hole solutions in the r h -Q -plane. When considering the limit r h → 0 these hairy black hole solutions tend to the corresponding soliton solutions. In this paper, we have been mainly interested in the rotating generalizations of both types of solutions. We find that although solitons cannot be made rotating properly with non-vanishing angular momentum, the hairy black holes can be generalized to rotating solutions characterized by the angular velocity on the horizon Ω. We find that the hairy solutions can rotate up to a maximal value of Ω where we believe that they become singular. Another aspect of our study has been to investigate the effect of the GB term. In the presence of a negative cosmological constant, it is known that static asymptotically AdS solutions exist for α < L 2 / 4 where L denotes the AdS radius and α the GB coupling constant. The limit α = L 2 / 4 corresponds to the Chern-Simons limit. Our numerical result show that some branches of soliton solutions disappear when α is large enough. For the GB black hole solutions we find that these exist for 0 ≤ α ≤ α cr < L 2 / 4. Moreover, for fixed non-vanishing α several rotating solutions can exist with different values of Ω but the same value of the charge Q . It would be interesting to find analytic arguments for our numerical results. This is currently under investigation. Acknowledgments B.H. and S.T. gratefully acknowledge support within the framework of the DFG Research Training Group 1620 Models of gravity . Y.B. would like to thank the Belgian F.N.R.S. for financial support.", "pages": [ 15 ] } ]

[ { "title": "ABSTRACT", "content": "We extend previous results on the reflection and transmission of self-gravitating dust shells across the apparent horizon during quantum dust collapse to non-marginally-bound dust collapse in arbitrary dimensions with a negative cosmological constant. We show that the Hawking temperature is independent of the energy function and that the wave functional describing the collapse is well behaved at the Hawking-Page transition point. Thermal radiation from the apparent horizon appears as a generic result of non-marginal collapse in AdS space-time owing to the singular structure of the Hamiltonian constraint at the apparent horizon. PACS: 04.60.-m, 04.70.Dy, 04.40.-b", "pages": [ 1 ] }, { "title": "Tunneling during Quantum Collapse in AdS Spacetime", "content": "Cenalo Vaz a 1 and Kinjalk Lochan b 2 a Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA b Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "A fundamental expectation of a quantum theory of gravity is that it will cure the problems that plague classical general relativity. One hopes, for example, that singularities get resolved in quantum gravity [1-5], that quantum gravity will provide the theoretcial foundation for cosmic censorship [6] and that it will give a better understanding of the relationship, already predicted on the semi-classical level, between gravity and thermodynamics [7-10]. In the absence of a generally agreed upon framework for such a theory, a useful approach is to quantize simplified classical gravitational models using canonical techniques, in the expectation that this will lead to new ways to look at some of the issues raised above while, at the same time, pointing to what one may expect out of the full theory. In this spirit, we recently developed [11] a novel approach to Hawking evaporation taking place during the collapse of a self-gravitating dust ball. This approach, based on an exact canonical quantization of the non-rotating, marginally bound gravity-dust system [12], exploited the matching conditions that must be satisfied at the apparent horizon by the wave functionals describing the collapse and differs from the traditional approach in which a pre-existing black hole is imagined to be surrounded by a tenuous field and the Bogoliubov transformation of the field operators is computed in the black hole background [13-15]. The geometrodynamic constraints of all LeMaˆıtre-Tolman-Bondi (LTB) models in any dimension, with or without a cosmological constant, are expressible in terms of a canonical chart consisting of the area radius, R , the dust proper time, τ , the mass function, F , and their conjugate momenta [16, 17]. After a series of canonical transformations in the spirit of Kuchaˇr [18], the Hamiltonian constraint can be shown to yield a Klein-Gordon-like WheelerDeWitt equation for the wave-functional. This equation can be solved by quadrature and, in the simplest cases, closed form solutions can be obtained after regularization is implemented on a lattice in a self-consistent manner. Self-consistency requires that the lattice decomposition is compatible with the diffeomorphism constraint [19]. In these models, the dust ball may be viewed as being made up of shells and the wave functional is described as the continuum limit of an infinite product over the shell wave functions. For the special case of marginal collapse with a vanishing cosmological constant in 3+1 dimensions, the Wheeler-DeWitt equation can be solved explicity. We showed in [11] that matching the shell wave-functions across the apparent horizon requires ingoing modes in the exterior to be accompanied by outgoing modes in the interior and, vice-versa, ingoing modes in the interior to be accompanied by outgoing modes in the exterior. In each case the relative amplitude of the outgoing wave is suppressed by the square root of the Boltzmann factor at a 'Hawking' temperature given by T H = (4 πF ) -1 , where F represents twice the mass contained within the shell. Thus the temperature varies from shell to shell, decreasing from the interior to the exterior, but it has the Hawking form for any given shell. Two separate solutions are possible: one in which there is a flow of matter toward the apparent horizon both in the exterior and in the interior, and another in which the flow is away from the horizon, again in both regions. Matter undergoing continual collapse across the apparent horizon is described by a linear superposition of these solutions and then, because ingoing waves in the interior are accompanied by outgoing waves in the exterior, the horizon appears, to the external observer with no access to the interior, to possess a reflectivity given by the Boltzmann factor at the above Hawking temperature. A different interpretation is also possible when the entire shell wave functions are taken into account. Ingoing waves in the exterior must be accompanied by outgoing waves in the interior, whose amplitude is also suppressed by the square root of the Boltzmann factor at the Hawking temperature. We showed that the transmittance of the horizon is unity, whether for waves incident from the exterior or the interior. Thus this outgoing wave in the interior passes through the apparent horizon unhindered but, because its amplitude is suppressed by the Boltzmann factor at the Hawking temperature relative to the ingoing modes in the exterior, the emission probability of the horizon is given by the same factor. The net effect is therefore reminiscent of the quasi-classical tunneling of particles through the horizon in the semiclassical theory [20-24]. The solutions just described relied on explicit solutions for the shell wave functions. These are available only in the case of the marginal models with a vanishing cosmological constant. Our aim here is to extend these results to non-marginally-bound LTB models in arbitrary dimension with a negative cosmological constant. No explicit solutions can be given in this case. Nevertheless, we will show that the results mentioned in the previous paragraphs are indeed generic and that they are a consequence only of the essential singularity of the KleinGordon equation for shells at the apparent horizon. We will then discuss how diffeomorphism invariant wave functionals may be reconstructed out of the shell wave functions. Hawking radiation from the apparent horizon then appears as a consequence of the generic form of the Wheeler-DeWitt equation describing dust collapse and not of any particular solution disucssed in the earlier work. This will provide a novel way to compute the entropy of the final state black hole. The plan of this paper is as follows. In section II we recall some key results for classical dust collapse with a negative cosmological constant in arbitrary dimensions. In section III we present the exact wave functional that is factorizable on a lattice and solves the WheelerDeWitt equation for dust collapse. The (collapse) wave functional can be thought of as an infinite product of shell wave functions, each occupying a lattice site. Matching shell wave functions across the horizon by analytic continuation in section IV, we argue that ingoing waves in one region must be accompanied by outgoing waves in the other. We superpose the two solutions to conserve the flux of shells across the horizon and then reconstruct the wave functional from the shell wave functions by going to the continuum limit. A consequence of matching shells across the apparent horizon is that the amplitude for outgoing waves relative to ingoing ones is given by e -S/ 2 , where S is the Bekenstein-Hawking entropy of the final state black hole. We close with a brief discussion of our results in section V.", "pages": [ 2, 3, 4 ] }, { "title": "II. NON-MARGINAL DUST COLLAPSE WITH Λ = 0 IN d DIMENSIONS", "content": "/negationslash", "pages": [ 4 ] }, { "title": "A. The classical models", "content": "Let us begin by briefly recalling some pertinent facts about spherical dust collapse in the presence of a negative cosmological constant (see, for example, [27] for details). The LTB models describe self-gravitating time-like dust whose energy momentum tensor is T µν = ε ( x ) U µ U ν , where U µ ( τ, ρ ) is the four velocity of the dust particles which are labeled by the ρ and with proper time τ . The line element can be taken to be where R ( τ, ρ ) is the area radius, E ( ρ ) is an arbitrary function of the shell label coordinate, called the energy function, and Ω n is the n = d -2 dimensional solid angle. Einstein's equations in the presence of a negative cosmological constant, which we call -Λ, yield one dynamical equation for the area radius, where the star refers to a derivative with respect to τ and F ( ρ ) is a second arbitrary function of the shell label coordinate, called the mass function. Above, κ d is given in terms of the d -dimensional gravitational constant G d as κ d = 8 πG d . One also finds the energy density, ˜ in terms of F , where the tilde refers to a derivative with respect to the label coordinate, ρ . Specific models are obtained by making choices of the mass and energy functions. For the solutions of (3) to describe gravitational collapse (as opposed to an expansion) one must impose the additional condition that R ∗ ( t, r ) < 0. The solutions to (3) have been explicitly given in [27] and analyzed in detail for the marginally bound case, E ( ρ ) = 0. Each shell reaches a zero area radius, R ( τ, ρ ) = 0, in a finite proper time, τ = τ s ( ρ ), which leads to a curvature singularity. Thus the proper time parameter lies in the interval ( -∞ , τ s ]. In general both naked singularities and black hole end states can form. Trapped surfaces occur when which determines the physical radius, R h , of the apparent horizon. F is positive outside, i.e., when R > R h , and negative inside, when R < R h .", "pages": [ 4, 5 ] }, { "title": "B. The Canonical Formulation", "content": "To develop a canonical formulation of the LTB models, one begins with the spherically symmetric Arnowitt-Deser-Misner (ADM) metric where N ( t, r ) and N r ( t, r ) are, respectively, the lapse and shift functions and the EinsteinHilbert action for a self-gravitating dust ball where U α = -τ ,α for non-rotating dust, whete τ is the dust proper time. The phase space consists of the dust proper time, τ ( t, r ), the area radius, R ( t, r ), the radial function, L ( t, r ), and their conjugate momenta, respectively P τ ( t, r ), P R ( t, r ) and P L ( t, r ). When the ADM metric is embedded in the spacetime described by (1) it becomes possible, through a series of canonical transformations described in detail in [17], to re-express the canonical constraints in terms of a new canonical chart consisting of the dust proper time, the area radius and the mass density function, Γ( r ), defined by and new conjugate momenta, P τ ( t, r ), P R ( t, r ) and P Γ ( t, r ). The energy function is expressible in this chart as and the transformations also absorb a boundary term, which is present in the original chart. The constraints for the dust-gravity system in any dimension are where the prime denotes a derivative with respect to the ADM label coordinate, r . The Hamiltonian constraint in (10) will be seen to contain no derivative terms, which makes it easier to quantize. However, the Poisson brackets of the Hamiltonian with itself vanishes, indicating that the Hamiltonian constraint does not generate hypersurface deformations. Rather, the transformations generated by the Hamiltonian constraint act along the dust flow lines. Of importance in what follows will be the following relationship between the dust proper time and the remaining canonical variables (see, for example, [28]) where a = 1 / √ 1 + 2 E . 3 The positive sign describes a collapsing dust cloud in the exterior and an expanding dust cloud in the interior whereas the negative sign describes an expanding cloud in the exterior and a collapsing cloud in the interior. Integrating on a hypersurface of constant t we have the formal solution where ˜ τ ( t ) is undetermined. This integral can be difficult to solve for arbitrary ( r dependent) mass and energy functions, but when they are both constant beyond some boundary, r b , then the solution may be expressed as In this case we are dealing with the static Schwarzschild-AdS geometry for which 2 P Γ may be associated with the Killing time [18], so (13) gives the relationship between SchwarzschildAdS and Painlev'e -Gullstrand time [29]. Solutions in the absence of a cosmological constant have been given in [19, 28].", "pages": [ 5, 6, 7 ] }, { "title": "III. QUANTUM STATES IN A LATTICE DECOMPOSITION", "content": "When Dirac's quantization condition is used to raise the classical constraints to operator constraints, which act on a wave functional, the Hamiltonian constraint turns into the Wheeler-DeWitt equation and the momentum constraint imposes spatial diffeomorphism invariance on the wave functional. One sees that the second is solved automatically by a wave-functional of the form provided that W contains no explicit dependence on r and where U : R → C is an arbitrary differentiable function of its argument. The wave functional is factorizable on a lattice placed on the real line (eg., see [11]) if U is chosen to be the exponential map for then, taking the lattice spacing to be σ , (14) can be written as where and we have used X i = X ( r i ). Thus we can think of Ψ[ τ, R, F ] as an infinite product of shell wave functions, each occupying a lattice site. Each shell wave function satisfies where A i = A ( R i , F i ) and B i = B ( R i , F i ) are functions capturing the factor ordering ambiguities that are always present in the canonical approach. They can be uniquely determined by requiring the above equation to be independent of the lattice spacing; one finds the general positive energy solutions [17] where ω i = σ Γ i / 2 and the factor e ω i b i amounts to a normalization. Note that shell ' i ' crosses the apparent horizon when F i = 0, which is an essential singularity of the wave equation. From the shell wave functions in (18) one reconstructs the wave functionals with the ansatz (14) and U = exp as where we have set a ( r ) = a ( F ( r )) and b ( r ) = b ( F ( r )) as is required for diffeomorphism invariance.", "pages": [ 7, 8 ] }, { "title": "IV. TUNNELING", "content": "The wave-functions of the previous section are defined in the interior as well as the exterior of the apparent horizon, but the Wheeler-DeWitt equation has an essential singularity at the apparent horizon along the path of integration. In order to match interior to exterior solutions, it is necessary to deform the path in the complex R -plane. The direction of the deformation is chosen so that positive energy solutions decay. While this deformed path does not correspond to the trajectory of any classical particle it represents a tunneling of s -waves across the gravitational barrier represented by the apparent horizon. This is analogous to the quasi-classical tunneling approach employed in semi-classical analyses [20-24].", "pages": [ 8 ] }, { "title": "A. Shell Wave Functions", "content": "From the expression for the phase in (18), the phase velocity of the i th shell wave function is given by Thus the positive sign in (20) describes ingoing waves in the exterior ( F i > 0), whereas it describes outgoing waves in the interior ( F i < 0) and, likewise, the negative sign describes outgoing waves in the exterior and ingoing waves in the interior. A closed form solution for the integrals appearing in (19) and (12) cannot be given when the mass function and/or the energy function and/or the cosmological constant are non-vanishing. We may however analyze their properties near the apparent horizon in the following way. Noting that F i = 0, equivalently R i = R i,h , is a singularity of the integral appearing in the phase, W i , we define the integral by analytically continuing to the complex plane and deforming the integration path so as to go around the pole at R i,h in a semi-circle of radius /epsilon1 drawn in the upper half plane. Let L /epsilon1 denote the deformed path and let S /epsilon1 denote the semi-circle of radius /epsilon1 around R i,h , then Performing the integration from left to right, 4 for R i = R i,h + /epsilon1 we have and, for the integral over the semi-circle, a Laurent series expansion about F i = 0 gives to lowest order This integral is half the integral over a complete circle taken in a clockwise manner, where 2 g i,h = F ' i ( R i,h ) is the surface gravity of the horizon. Therefore we find The expression defining the proper time in (12) involves a similar integral, but taken over a spatial slice. Even so, the same argument can be made for this integral (see [25, 26] for an analogous argument in the semi-classical context). If we assume, moreover, that a ( F ( r )) and P Γ ( r ) are both regular across the horizon, the net result is that the phases in the exterior get matched to the phases in the interior by the addition of a constant imaginary term, Thus we can give two independent solutions with support everywhere in the spacetime: an ingoing wave in the exterior that is matched to an outgoing wave in the interior and an outgoing wave in the exterior that is matched to an ingoing wave in the interior according to The first of these solutions represents a flow towards the apparent horizon both in the exterior as well as in the interior whereas the second represents a flow away from the apparent horizon in both regions. While each solution is self-consistent, neither wave function accurately reflects the physical situation one expects from semi-classical collapse, in which an ingoing shell proceeds all the way to the center and the horizon emits thermal radiation into the exterior at the Hawking temperature. To recover this picture we consider a linear superposition of the two wave functions where A i are complex valued constants. Following [11], we fix these constants by requiring that the current density is constant across the horizon. This implies that | A i | 2 = 1 and we take A i = 1 for every shell, which gives an absorption probability of unity for a shell to cross the apparent horizon from the exterior. We therefore have The second term in the expression for ψ i in the exterior ( F i > 0) is an outgoing wave that, to an external observer, would represent a reflection with relative probability This is precisely the Boltzmann factor for the shell at temperature T i,H = /planckover2pi1 g i,h 2 πk B , where g i,h is the surface gravity of the apparent horizon. This reflected piece is a purely quantum effect, necessitated by the existence of an ingoing wave in the interior, i.e., by requiring the continued collapse of the shell beyond its apparent horizon. This continued collapse is represented by the second term in the expression for ψ i in the interior ( F i < 0). On the other hand, the ingoing wave in the exterior, represented by the first term in the expression for ψ i when F i > 0, is necessarily accompanied by an outgoing wave in the interior occurring with a relative amplitude of e -πω i /planckover2pi1 g h , which is equal to the amplitude for 'reflection' at the apparent horizon. This leads to the alternate picture mentioned in the Introduction, in which the Hawking process can be viewed as an effective emission from the apparent horizon.", "pages": [ 9, 10, 11 ] }, { "title": "B. Phase Transition", "content": "We can use our results above to examine what happens near the Hawking-Page Transition point [30]. It is worth noting that the Hawking temperature is independent of the energy function. This was first noted in [28] in connection with non-marginal LTB models without a cosmological constant. Here we have shown that the result is robust, holding even in the presence of a negative cosmological constant. Now the apparent horizon is given for each shell as the solution of the equation where x i = R i √ Λ is dimensionless, and it is straightforward to show that the surface gravity for each shell is For fixed n and Λ, so using the fact that dx i,h /dF i > 0, which follows directly from (33), we find that dg i,h /dF > 0 when 2 x 2 i,h > n ( n -1). This is the condition for positive specific heat. When 2 x 2 i,h < n ( n -1) the specific heat is negative and the two regimes are separated by the Hawking-Page (phase) transition [30], which occurs at 2 x 2 i,h = n ( n +1). The wave functions of collapse in (31) are well behaved at the transition point.", "pages": [ 11, 12 ] }, { "title": "C. Wave Functionals", "content": "We now turn to the question of how the collapsing shell wave functions described in the previous section may be combined to yield wave functionals. Obviously the superposed wave functions of (31) cannot be directly used for this purpose as they are not the simple exponentials required for diffeomorphism invariance by (14). Instead, we take the continuum limit of the product of the shell wave functions ψ (1) i and ψ (2) i separately to form two corresponding diffeomorphsim invariant wave functionals, Ψ 1 and Ψ 2 . Then, we take a linear combination of these wave functionals to form the full wave functional describing the collapse [11] . Accordingly, the functional equivalent of the superposed wave functions in (31) is Ψ = When use is made of (33) and (34), the relative amplitude, e -π 2 /planckover2pi1 ∫ dr Γ /g h , works out to precisely e -S/ 2 , where S = A h / 4 /planckover2pi1 G d is the Bekenstein-Hawking entropy of the black hole. Thus the ratio of the reflection probability to the probability for absorption is determined only by the entropy of the black hole, and we have recovered the results of [11] in the more general setting of d -dimensional, non-marginal collapse and in the presence of a (negative) cosmological constant.", "pages": [ 12, 13 ] }, { "title": "V. CONCLUSIONS", "content": "In this paper we have used the wave-functionals of an exact midi-superspace quantization of the non-marginally-bound LTB models in the presence of a cosmological constant and in an arbitrary number of spatial dimensions to study the Hawking evaporation process. As in previous works, regularization was performed on a lattice and the wave functionals were shown to be constructed out of wave functions describing individual shells of collapsing dust. The apparent horizon is an essential singularity of the Wheeler-DeWitt equation and the solutions of the latter could only be given by quadrature separately in the exterior and in the interior of the apparent horizon. The central issue discussed here was how the interior and exterior solutions can be matched across the horizon. To accomplish the required matching we defined the integrals appearing in the solution of the Wheeler-DeWitt equation by deforming the integration path in the complex plane to go around the pole at the apparent horizon. This implied that crossing the horizon involved a rotation of the dust proper time in the complex plane and had the effect of introducing an imaginary constant into the phase of the outgoing wave functions. We were then able to show that an ingoing shell wave function in one region is required to be accompanied by an outgoing shell wave function in the other region. The relative amplitude of the outgoing wave function in each case was shown to be given by the square root of the Boltzamann factor at the Hawking temperature appropriate to the shell. The approach in this paper enjoys several advantages over the approach via Bogoliubov coefficents, while also producing an alternative and attractive view of the evaporation process. In the first place, no near horizon expansion of the wave-functional is necessary. Secondly, the approach via Bogoliubov coefficients does not get much beyond the semi-classical level because it is necessary to approximate the mass function in such a way that it represents a massive black hole surrounded by tenuous dust. Hence one is effectively looking at the semi-classical radiation from the event horizon of a static black hole. By contrast, no such approximation to the mass function is necessary here, so we are genuinely examining the radiation from the apparent horizon during collapse. Thirdly, the inner product used in the calculation of the Bogoliubov coefficient is not the one that is uniquely determined by the lattice regularization (see Appendix B of [19]) but one that is determined from the DeWitt supermetric. This can be justified only in the approximation described above because, for this case alone, no measure is uniquely determined by the regularization scheme [31]). In all other cases, the measure is uniquely determined and different from that provided by the DeWitt supermetric. The results we report here are independent of the inner product. If the matter is assumed to undergo continued collapse we showed that the relative probability for the shell wave function to cross the apparent horizon is unity, whether it is incident from the interior or the exterior. This was argued to lead to two pictures of the evaporation process. In the first picture, one takes the point of view of an external observer with no access to the interior. To this observer the horizon appears to possess a non-zero reflectivity. On the other hand, the observer who has access to the entire wave function sees the outgoing exterior portion of the shell wave function as a transmission of an outgoing interior wave across the horizon, which exists because of the collapsing exterior. Thus the horizon can also be thought of as an emitter. We showed that the Hawking temperature is independent of the energy function, i.e., of the initial velocity distribution of the shells, and that the shell wave functions are well behaved at the Hawking-Page transition point during the collapse. Moreover, the relative amplitude for outgoing wave functionals is e -S/ 2 , where S is the Bekenstein-Hawking entropy of the final state black hole even in the presence of the cosmological constant. This generalizes [11], for which closed form solutions were available. The results are therefore generic to dust collapse.", "pages": [ 13, 14, 15 ] }, { "title": "Acknowledgements", "content": "C.V. is grateful to T.P. Singh and to the Tata Institute of Fundamental Research for their hospitality during the time this work was completed. K.L. and C.V. acknowledge useful conversations with T.P. Singh and the partial support of the Templeton Foundation under Project ID # 20768.", "pages": [ 15 ] } ]

[ { "title": "Self-force on an arbitrarily coupled static scalar particle in a wormhole space-time", "content": "Peter Taylor ∗ School of Mathematics, Trinity College, Dublin 2, Ireland (Dated: March 22, 2021) In this paper, we consider the problem of computing the self-force and self-energy for a static scalar charge in a wormhole space-time with throat profile r ( ρ ) = √ ρ 2 + a 2 for arbitrary coupling of the field to the curvature. This calculation has previously been considered numerically by Bezerra and Khusnutdinov [1], while analytic results have been obtained in the special cases of minimal ( ξ = 0) coupling [2] and conformal coupling [1] ( ξ = 1 / 8 in three dimensions). We present here a closed form expression for the static Green's function for arbitrary coupling and hence we obtain an analytic expression for the self-force. The self-force depends crucially on the coupling of the field to the curvature of the space-time and hence it is useful to determine the dependence explicitly. The numerical computation can identify some qualitative aspects of this dependence such as the change in the sign of the force as it passes through the conformally coupled value, as well as the fact that the self-force diverges for ξ = 1 / 2. From the closed form expression, it is straight-forward to see that there is an infinite set of values of the coupling constant for which the self-force diverges, but we also see that there is an infinite set of values for which the self-force vanishes.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Wormholes are topological bridges connecting different universes or distant regions of the same universe. Interest in wormhole space-times dates back to 1916 [3], pre-dating interest in black hole space-times. Its modern popularity is owed primarily to the work of Morris and Thorne [4] who investigated the idea of using so-called 'traversable wormholes' as a means for time travel. Morris and Thorne showed that such space-times require a stress-energy tensor that violates the null energy condition, that is they require the existence of exotic matter. Wormholes have subsequently played a central role in the investigation of causality violation and the status of energy conditions as physical laws of nature. Comprehensive reviews of wormhole physics can be found in Refs.[5, 6]. We will consider a particularly simple, ultra-static, spherically symmetric wormhole geometry described by the metric where d Ω 2 = dθ 2 +sin 2 θ dφ 2 is the line-element on the two-sphere S 2 and r ( ρ ) = √ ρ 2 + a 2 is the profile of the wormhole throat with a minimum radius r (0) = a . The range of the radial coordinate is the entire real line, -∞ < ρ < ∞ , and the throat connects two identical asymptotically flat space-times. The manifold is everywhere smooth with scalar curvature given by We consider the problem of computing the self-force on a scalar charge at rest in the metric described by (1.1), allowing for the arbitrary values of the coupling constant in the wave equation. A similar calculation was considered by Khusnutdinov and Bakhmatov [2] who computed the electrostatic self-force on a charged particle at rest this wormhole space-time. The electrostatic wave equation is equivalent to that of a minimally coupled scalar charge at rest and hence the self-force on a static scalar charge is equal to the electrostatic sef-force, up to an overall sign. Bezerra and Khusnutdinov [1] numerically evaluated the self-force on a static scalar for arbitrary coupling, though we disagree with the overall sign of their results. The electrostatic case was reconsidered by Linet [7] who derived the Green's function in closed form by transforming to isotropic coordinates and expanding about the Euclidean distance in these coordinates, a method first adopted by Copson [8] in deriving the electrostatic potential in the Schwarzschild space-time. The self-force is obtained by taking the gradient of the retarded field which is singular at the particle's location and requires regularization. In order to regularize the self-force, we compute the Detweiler-Whiting singular field, which upon subtraction yields a quantity that is regular at the scalar charge's location. We obtain an analytic expression for the self-force for arbitrary values of the coupling constant, which reveals some expected features such as infinite poles which also occurs in the expression for the self-force [1] for the wormhole with throat profile r ( ρ ) = | ρ | + a , but we also find some unexpected features such as an infinite set of values for which the self-force vanishes.", "pages": [ 1 ] }, { "title": "II. GREEN'S FUNCTION", "content": "The Green's function satisfies the following inhomogeneous wave equation, where glyph[square] is the d'Alembertian wave operator on the metric (1.1), and ξ is the coupling to the scalar curvature which is given by Eq.(1.2). For the charge at rest according to observers moving on integral curves of the Killing vector ∂/∂t , the wave equation reduces to the three-dimensional Helmholtz equation, The Green's function may be given as a mode-sum over separable solutions to the homogeneous equation, where P l ( x ) is the Legendre polynomial, cos γ = cos θ cos θ ' + sin θ sin θ ' cos ∆ φ and g l ( ρ, ρ ' ) satisfies the inhomogeneous radial equation, With the transformation the radial equation takes a more simple form, The general solution may be written as a normalized product of two linearly independent solutions of the homogeneous equation where y < = min { y, y ' } , y > = max { y, y ' } and the normalization constant N is determined by the Wronskian of the two solutions. The boundary conditions on the Green's function at ρ →±∞ require Writing z = i y , it is clear that the solutions of the homogeneous equation (2.6) are associated Legendre functions of pure imaginary order, The multi-valuedness of the associated Legendre functions gives rise to a discontinuity at y = 0. For y ≥ 0, we choose Ψ (2) l ( y ) = Q µ l ( iy ) which vanishes as y → ∞ as required. However, for the solution to be continuous across y = 0, we take the branch obtained from the principal branch by encircling the branch point z = 1 (but not the point z = -1) once. If we denote this branch by Q µ l, 1 ( z ), then it can be shown [9] Then, the function is continuous for -∞ < y < ∞ and satisfies the appropriate boundary condition and hence is the correct choice. The symmetry of the space-time implies that we take our inner solution to be Ψ (1) l ( y ) = Ψ (2) l ( -y ), or given explicitly, This solution clearly satisfies the vanishing boundary condition at y →-∞ ( ρ →-∞ ) and is linearly independent to Ψ (1) l ( y ) in the two regions of the space-time. We can re-write the solution in a more symmetric form using standard results for the Legendre functions [10], yielding The normalization constant is given by So the radial solution may be written as Let us assume, without loss of generality, that y, y ' ≥ 0 (the closed-form expression won't depend on this condition), then the Green's function is In a recent paper [11], we have derived the following summation formula for associated Legendre functions of arbitrary complex order, where and where the radial variable here is real and runs over the range η > 1. We can analytically continue this result by making the transformation η = iy and taking the appropriate branch. We obtain             -            µ cosh iR - µ cosh iR µ cosh e iR 1 / 2 1 / 2 e - µπi - 1 1 / 2 e ( χ ) - 1 = e ( χ ) - 1 ( χ ) y, y y, y ' ' y > 0 , y ≥ 0 , < 0 , ' < 0 or y < 0 , y ' > 0 , where These results may be checked numerically by multiplying both sides by P l ' (cos γ ) and integrating with respect to γ . For y, y ' ≥ 0, we can use the well-known relations between Legendre functions to rewrite the Green's function (2.12) as We can employ the summation formula (2.15) to obtain the following closed form representation of the Green's function for a static scalar charge in our wormhole spacetime where we have used the fact that A similar analysis for the case where y, y ' < 0 and where the two points are in different regions of the space-time yield the same closed form expression as Eq.(2.17). Finally, restoring the variable ρ = y/a gives where We note that this Green's function has the correct Hadamard singularity structure [12] in three dimensions. We also note that the Green's function possesses infinite poles for certain values of the coupling constant given by This is analogous to the infinite poles that arise in the Green's function for throat profile r ( ρ ) = | ρ | + a , as shown in Ref. [1]. For minimal coupling, Eq.(2.19) is understood to mean which agrees with the closed form representation given in Ref. [2] if we take γ = 0.", "pages": [ 1, 2, 3 ] }, { "title": "III. SELF FORCE AND SELF ENERGY", "content": "There are a number of approaches one can take to compute the self-force (see [13] for example), but the most direct way for a static charge is by subtracting the Detweiler-Whiting [14] parametrix which yields a finite quantity upon taking coincidence limits. It is constructed in such a way as to leave, upon subtraction from the Green's function, only the radiative part of the field entirely responsible for the self-force. This approach was adopted in Ref. [15] to compute the self-force on a static scalar charge in Kerr space-time. There it was shown that for a static scalar charge with world-line x ' α = z α ( τ ) and four-velocity u α = u t ' δ α t ' in a general stationary spacetime, the self-force is where G (3) ( x , x ' ) is the zero-frequency mode of the fourdimensional retarded Green's function (modulo a factor of 2 π ) and G DW ( x, x ' ) is the Detweiler-Whiting Green's function given by [13] where ∆ is the Van-Vleck Morrette determinant, r ret is the retarded distance between the field point x and the retarded point x ' = x ret on the charge's world-line and r adv is the advanced distance between x and the advanced point x ' = x adv on the world-line. In the case of minimal coupling, it has recently been shown [16] that the Detweiler-Whiting Green's function for a static charge in a static space-time is equivalent to the direct part of the Hadamard Green's function on the spatial part of the metric up to the order required for regularization, and equal up to all orders for ultrastatic space-times such as the wormhole space-time under consideration. In that case, the appropriate parametrix that must be subtracted to give the correct self-force is particularly simple, where ∆ and the world-function σ are calculated on the three-dimensional metric However, it is clear that this statement cannot be true for arbitrary coupling since the biscalar V ( x, x ' ) appearing in Eq. (3.2) depends on the coupling constant ξ while the direct part of the three-dimensional Hadamard form (3.3) does not, indeed it is purely geometrical. In obtaining an expression for G DW ( x, x ' ), we will use the standard expansions We can use the fact that the space-time is spherically symmetric to set γ = 0, in which case the Van-Vleck Morrette expansion reduces to From the expansion for V ( x, x ' ), the integral in Eq.(3.2) is where ∆ τ = τ adv -τ ret . Coordinate expansions for the quantities r ret , r adv and ∆ τ were computed for a static charge in an arbitrary stationary space-time in Ref. [15], which in the wormhole space-time under consideration, simplify greatly to give Hence the Detweiler-Whiting parametrix is where recall that µ = √ 2 ξ and we have used the expression for the scalar curvature given by Eq.(1.2). We also require an expansion for the closed form Green's function of Eq.(2.19). Taking the scalar particle's location to be ρ ' and the field point to be ρ , then for ρ near ρ ' the Green's function may be expanded as where ∆ ρ = ρ ' -ρ which we assume, without loss of generality, to be positive. The self-energy for a charge q is proportional to the coincidence limit of the radiative field, and is given by For minimal coupling, ξ = 0 ( µ = 0), we arrive at the result of Khusnutdinov and Bahkmatov [2] (which was rederived by Linet [7]), while for conformal coupling ξ = 1 / 8 ( µ = 1 / 2), the selfenergy vanishes, which agrees with the results of Bezerra and Khusnutdinov [1]. The self-energy, like the Green's function, is also divergent for the particular values of the coupling constant given by Eq. (2.21). Turning now to the calculation of the self-force, which we have defined for a static scalar charge in a stationary space-time in Eq.(3.1). For the ultra-static wormhole space-time under consideration, this reduces to with all other components vanishing due to the spherical symmetry. Substituting in the expression for the singular field (3.9) and the Green's function expansion (3.10), we obtain for the self-force Again, we note that the self-force has an infinite number of poles whenever the coupling constant is ξ = n 2 / 2, where n ∈ Z / { 0 } . A similar analysis [1] for throat profile r ( ρ ) = | ρ | + a also exhibits this behaviour for the selfforce. For minimal coupling, we obtain which is in agreement with the electrostatic self-force derived in Ref. [2], modulo the sign of the force since the field for a minimally coupled static scalar in an ultra-static space-time is minus the electrostatic field. Hence the self-force on a static scalar in the wormhole space-time is always repulsive with respect to the throat whereas in the electrostatic case, it is always attractive with respect to the throat. For the conformal coupling in three dimensions , ξ = 1 / 8, we obtain zero self-force which is in agreement with the result of Ref. [1]. In fact, just as there are an infinite set of values for which the self-force diverges, there are also an infinite set for which it vanishes, That the self-force vanishes for ξ = 1 / 8 for a massless field is expected since the spatial section of the metric is conformally flat. However, the existence of an infinite set of values of the coupling constant for which the self-force vanishes is a surprising result, at least to this author. In Fig. 1, we plot the self-force for various values of the coupling constant. The first thing to note is that the overall sign of the self-force differs to the numerical computation of Ref. [1]. So for the field with conformal coupling in the range 0 < ξ < 1 / 8, the self-force is everywhere repulsive with respect to the wormhole throat, not attractive as previously claimed. The self-force vanishes for conformal coupling ξ = 1 / 8 and then becomes attractive for values in the range 1 / 8 < ξ < 1 / 2. The force becomes increasingly attractive with respect to the throat as ξ increases towards ξ = 1 / 2 where it becomes divergent. This cycle then continues, the force is repulsive for 1 / 2 < ξ < 9 / 8, vanishing at ξ = 9 / 8 and attractive with respect to the throat for 9 / 8 < ξ < 2 and diverges for ξ = 2 etc. These poles in the expression for the self-force, which also appear in the closed form representation of the Green's function, can be understood in the context of quantum mechanical scattering theory as the energy of bound states. In this classical context however, the existence of such poles suggests that for certain coupling strengths of the scalar field to the curvature, the charge cannot remain static. For a particular value of the coupling constant, the direction of the self-force in either part of the space-time is independent of the radius at which the scalar charge is being held. For example, for 1 / 8 < ξ < 1 / 2 the force is always attractive with respect to the throat regardless of where in the wormhole space-time the charge is placed, and hence for the coupling strength in this range we may hypothesize that the scalar charges will accumulate in the vicinity of the throat. On the other hand, for ξ < 1 / 8, the self-force is everywhere repulsive with respect to the throat. The magnitude of the self-force for a static scalar is maximized when the charge is held at ρ = ±| a | / √ 3, and the force at this radius is", "pages": [ 4, 5, 6 ] }, { "title": "IV. CONCLUSIONS", "content": "We have obtained an analytic expression for the self-force on an arbitrarily coupled static scalar charge in a wormhole space-time with throat profile r ( ρ ) = √ ρ 2 + a 2 . Analytic expressions have previously been obtained only for minimally and conformally coupled scalar fields, while the case of general coupling had been computed numerically. We find that there are infinite poles in the expression for the self-force corresponding to the values of the coupling constant where ξ = n 2 / 2, for n ∈ Z \\ { 0 } . We also find that there are an infinite set of values of the coupling constant for which the self-force vanishes.", "pages": [ 6 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "I am grateful to Adrian Ottewill and Marc Casals for reading the manuscript and for their insightful suggestions. I am also indebted to Bernard Linet for his friendly correspondence and for pointing out an error in an earlier version of this manuscript. Series and Products (Academic Press, 2000).", "pages": [ 6 ] } ]

[ { "title": "Generalized uncertainty principles and quantum field theory", "content": "Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 Viqar Husain, Dawood Kothawala, and Sanjeev S. Seahra (Dated: June 4, 2018) Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator [ˆ x, ˆ p ] = if (ˆ p ). We apply this deformed quantization to free scalar field theory for f ± = 1 ± βp 2 . The resulting quantum field theories have a rich fine scale structure. For small wavelength modes, the Green's function for f + exhibits a remarkable transition from Lorentz to Galilean invariance, whereas for f -such modes effectively do not propagate. For both cases Lorentz invariance is recovered at long wavelengths. Introduction One of the most important problems in fundamental physics is an understanding of the high energy behaviour of quantum fields. This question is intimately connected with the structure of spacetime at short distances, because the background mathematical structure that underlies quantum field theory (QFT), namely a manifold with a metric, may come into question in this regime. A part of the problem is that the spacetime metric forms a reference not only for defining the particle concept, but also for the Hilbert space inner product; if the metric is subject to quantum fluctuations then its use in an inner product becomes an issue. There are many approaches that have been deployed to probe such questions, including string theory, noncommutative geometry, loop quantum gravity and causal sets. Some of these suggest that the fundamental commutator [ˆ x, ˆ p ] = i of quantum mechanics is modified at high energies. For example, the particular modification with (dimensionful) constant β > 0 has been studied for a number of systems, including the simple harmonic oscillator [1]. It has also been used in the cosmological context to compute modifications to the spectrum of fluctuations in cosmology [2]. Recent experiments have attempted to put constraints on β [3]. However no direct application to QFT has so far been studied. In this paper, we apply the commutator algebra (1) to QFT in flat spacetime for both generic and specific choices of the function f . Our approach involves applying a 3-dimensional spatial Fourier transform to the classical phase space variables, and then enforcing the deformed commutator in k -space. This approach was used for polymer quantization of the scalar field in [4] following work on a Fock-like quantization in [5]. Quantized scalar field We start with the Hamiltonian of a free scalar field in Minkowski space time: where ( φ, π ) satisfy { φ ( t, x ) , π ( t, y ) } = δ (3) ( x -y ). The Fourier modes are with a similar expansion for π ( t, x ); V = ∫ d 3 x is a fiducial volume for box normalization. After a suitable redefinition of independent modes to enforce that φ is real, the Hamiltonian becomes where the k -space canonical variables satisfy the Poisson bracket { φ k , π k ' } = δ k , k ' . The structure of the Hamiltonian is that of a collection of decoupled simple harmonic oscillators labelled by k , therefore the obvious Hilbert space for constructing the quantum theory is a tensor product H = ⊗ k H k . We quantize field theory by representing the modified commutator on the k -space canonical variables: where f is a dimensionless function and M glyph[star] is an energy scale. In the momentum space representation with ψ ( π k ) ∈ H k = L 2 ( I, f -1 dπ k ), the modified commutator is realized by the operator definitions The interval I must be selected such that f ≥ 0 for all π k ∈ I . Although the function f may be arbitrary up to the action of operators still giving L 2 functions, we impose the additional condition that f (0) = 1 to recover the standard commutator for small momenta π k glyph[lessmuch] M 1 / 2 glyph[star] . This enforces the requirement that the effects of deformation are confined to short wavelengths, as we shall see in the following. In this representation, the energy eigenvalue equation H k ψ = E k n ψ reads This can be recast as a conventional time-independent Schrodinger equation, via the change of variables and the definitions This shows that each deformation of the commutator maps uniquely to a potential in the Schrodinger equation governing the canonical variables describing each Fourier mode. The parameter g plays a central role in what follows: large wavelength modes with g glyph[lessmuch] 1 ( k glyph[lessmuch] M glyph[star] ) behave as in standard physics, but small wavelength modes with g glyph[greatermuch] 1 ( k glyph[greatermuch] M glyph[star] ) exhibit exotic behaviour. Free field propagator Given solutions to the eigenvalue problem (7), it is possible to calculate the scalar field propagator. This can be accomplished with a purely quantum mechanics calculation. We begin with spatial Fourier transform of the vacuum two point function, which is given by the matrix element Inserting into this the Heisenberg formula for the evolved operator ˆ φ k ( t ) and assuming a complete energy eigenfunction basis for H k yields where ∆ E k n = E k n -E k 0 , c n = c n ( k ) = 〈 0 k | ˆ φ k | n k 〉 , and the contour C encircles all poles on the positive real axis. This is a direct generalization of the conventional prescription for the Wightman function; other Green functions can be obtained in a similar way. The formulae (11) are general and apply to any quantum theory defined by the modified commutator (5). The only inputs required are the energy differences ∆ E k n and the matrix elements 〈 0 k | ˆ φ k | n k 〉 obtained from solutions of (7). It is illustrative to consider three specific choices of the function f appearing in (5); the corresponding potentials appearing in the Schrodinger equation are shown in Fig. 1. (i) f ( x ) = 1 . This case is the conventional quantization with [ ˆ φ k , ˆ π k ] = i ; the potential in the Schrodinger equation (Fig. 1 is that of the simple harmonic oscillator. Hence, This gives the usual answer for the Green's function We emphasize two key features responsible for the emergence of Lorentz invariance in the final step: (i) the exact cancellation of k factors in the numerator of Eq. (11b), and (ii) the fact that in standard quantization, ∆ E k 1 = k , which gives the combination -ω 2 + k 2 in the denominator. This provides a curious connection between the simple harmonic oscillator and Lorentz invariance, which does not exist for the potentials associated with the deformed commutator. (ii) f ( x ) = 1 + x 2 . This form is motivated by including the gravitational interaction in discussions of the Heisenberg microscope gedanken experiments [6]. It has been widely studied at the quantum mechanics level, see eg. [7], but not in QFT. The commutator algebra is of the form (1) with β > 0. The potential in the Schrodinger equation (7b) is V ( z ) = tan 2 ( z ) / 2 g 2 , and is plotted in Fig. 1. The change of variables introduced in obtaining (7b) implies that the Hilbert space for this case is L 2 ([ -π/ 2 , π/ 2] , dz ). The solution of the eigenvalue problem can be written down analytically in terms of hypergeometric functions [1] or Gegenbauer polynomials. The energy eigenvalues yield We have obtained analytic expressions for c n ( k ), which are plotted in Fig. 2. Note that c 2 n = 0 for all g due to parity. The nonzero matrix elements have the following limiting behaviour From this formula or Fig. 2, we see that for small g , the first matrix element coincides with the f ( x ) = 1 result | c 1 | 2 = (2 k ) -1 . The higher n contributions are suppressed by successive powers of g 2 , therefore the sums in (11) are dominated by the first term. Since ∆ E k n ≈ nk in this regime, the propagator (13) is recovered for k glyph[lessmuch] M glyph[star] . For g glyph[greaterorsimilar] 1, the n > 1 terms in (11) cannot be neglected; each of these contributes a pair of poles at ω = ± ∆ E k n to the Green's function G ( ω, k ) as depicted in Fig. 3. These poles may be interpreted as discrete resonant modes with dispersion relation ω 2 = (∆ E k n ) 2 . Since the residues of the n = 1 poles are always greater than those for n > 1, we call it the 'principal resonance'; its dispersion relation is shown in Fig. 4. Finally, we note that for g glyph[greatermuch] 1, the Green's function reduces to with α n = 64 n 2 /π 2 (4 n 2 -1) 2 and m eff n = M glyph[star] / (4 n 2 -1). The terms in brackets are the Fourier-space propagators of the non-relativistic Schrodinger equation for a free particle of mass m eff n . This result is a curious surprise: the small wavelength limit of the propagator exhibits the Galilean symmetry of Newtonian mechanics, and indicates non-locality in space in this regime. The positive and negative energy poles reflect the fact that this comes from a relativistic theory in the long wavelength limit. (iii) f ( x ) = 1 -x 2 . The commutator algebra in this case is of the form (1) with β < 0. The potential appearing in the Schrodinger equation (7b) is V ( z ) = tanh 2 ( z ) / 2 g 2 which is a vertical translation of the well-known Poschl-Teller potential V ( z ) = -1 2 j ( j + 1)sech 2 ( z ), but with arbitrary amplitude (see eg. [8]). The eigenvalue problem is again analytically solvable. However, there is a crucial difference between this case and the previous two: the fact that V ( z ) has a finite height implies that for any given value of g = k/M glyph[star] , there is a finite number of normalizable energy eigenstates in L 2 ( R , dz ). These states are labelled by integers n = 0 . . . n max , where n max = floor[( √ 4 + g 2 -g ) / 2 g ]. The energy differences ∆ E k n for this case are given by (14) if we substitute g ↦→ -g . There are no normalizable eigenstates in L 2 ( R , dz ) with energy greater than E max = M glyph[star] / 2. Because of their finite number, energy eigenfunctions do not form a complete basis of L 2 ( R , dz ) and the formulae (11) are not directly applicable. However, there is a complete energy eigenfunction basis if we instead use the Hilbert space L 2 ([ -glyph[lscript], glyph[lscript] ] , dz ). With this choice, the 'scattering' energy eigenstates with E > M glyph[star] / 2 are normalizable and discrete, and the sums in (11) are welldefined. Taking the glyph[lscript] → ∞ limit, the scattering states approach a continuum and the Green's function is The integration is over the scattering states, which are labelled by the continuous parameter ν . The energy of a given scattering state is Also, c ( ν ) = lim glyph[lscript] →∞ 〈 ν k | ˆ φ k | 0 k 〉 where | ν k 〉 is an oddparity scattering mode of energy E k ( ν ) with normalization 〈 ν k | ν k 〉 = 2 glyph[lscript] . (Even parity modes do not contribute by symmetry.) We have computed closed form expressions for both | c n | 2 (plotted in Fig. 2) and | c ( ν ) | 2 (plotted in Fig. 5). At small g , the bound state residues match those for the f ( x ) = 1 + x 2 case, as do the energy differences ∆ E k n . Furthermore, we find | c ( ν ) | → 0 as g → 0, which guarantees that (13) is recovered for long wavelengths. Unlike the previous case, the bound state matrix elements go to zero for finite g . In fact, for g ≥ 1 / √ n ( n +1) we have c n = 0. The vanishing of c n at particular values of g = k/M glyph[star] indicates the threshold where the n th eigenstate swtiches from a bound to scattering state, or vice versa. It follows that for g ≥ 1 / √ 2, the only non-zero contribution to Green's function (17) is from the continuum integral. For g glyph[greatermuch] 1, it can be shown that where F ( u ) = 1 + O (1 /u ) is a non-oscillatory expression involving error functions. The τ 3 / 2 factor in the denominator implies that short wavelength modes decay to zero with characteristic timescale M -1 glyph[star] , and the argument of the exponential gives an effective dispersion relation ω = M 3 glyph[star] / 2 k 2 . That is, disturbances of physical size glyph[lessmuch] M -1 glyph[star] will not generate any long distance wave propagation in this model. It is interesting that some of the features of the f ( x ) = 1 -x 2 case appear to naturally incorporate recent ideas on the ultraviolet completion of non-renormalizable theories via a so-called 'classicalization' [9], where propagating quantum degrees of freedom do not exist at short distance scales. Discussion We have shown that if modified canonical commutators are directly implemented in k space, the resulting QFTs exhibit novel short wavelength behaviour: propagation amplitudes have multiple poles, Lorentz invariance is broken, and there is spatial non-locality. The expression for the propagator Eq. (11b) resembles the Lehmann-Kallen spectral representation for an interacting Lorentz invariant scalar field theory of mass m , where ρ ( µ 2 ) is the density of particle resonances of mass µ . Although purely mathematical, this analogy suggests a multi-particle nature of deformed quantization, even for a free theory. (Our result on multiple poles support an argument to that effect in [10] for QFTs with a minimal length scale based on higher derivative theories.) Another interesting outcome of this work is an explicit non-locality in space at short distance scales. This may be seen in at least two distinct ways. One is simply due to Lorentz violation; in the large k regime for the case f = 1 + x 2 , the non-localilty is exactly Newtonian action-at a-distance, as evident from the form of the propagator. The other follows directly from inverse Fourier transformation of the deformed k space commutator, which would lead to non-local terms of the form M -1 glyph[star] ∫ d 3 z ˆ π ( t, y -z ) ˆ π ( t, z ) in the equal time commutator [ ˆ φ ( t, x ) , ˆ π ( t, x + y )]. The Heisenberg equations of motion derived from this commutator would be non-local integro-differential equations. The above observation appears to resonate with the recently proposed hypothesis of relative non-locality [11], in which locality in a postulated curved momentum space leads to non-locality in physical space. In our approach, space non-locality is evident in the above commutator, and the 'relative' part may be connected with the fact that our quantization depends on a choice of time coordinate. Since that hypothesis is motivated by earlier works on deformed Lorentz symmetry, perhaps such a connection is not unexpected, with the caveat that in our approach it arises at the level of a dynamical quantum field, rather than at the kinematical level in [11]. There are several directions for further work using the deformed quantization we have discussed. It is apparent that the approach may be followed for other spin fields in flat space time. Of particular interest is the scalar field on a black hole background; for example, modified dispersion relations of the type computed for the f -case, with vanishing group velocity for short wavelength modes, have been used to study Hawking radiation [12]. For interacting theories, perhaps of most interest for observational consequences is quantum electrodynamics. Of related interest is whether Lorentz violation from this type of deformed quantization gives rise to order one effects due to loop corrections, as discussed in the context of effective field theory (EFT) in [13]. However, many components of conventional quantization are prima-facie absent in our deformed quantization, so it is not clear whether the axioms of EFT apply. Indeed, space nonlocality at short wavelengths appears to complicate the Wilsonian program of integrating out high energy degrees of freedom and capturing their effects in local counterterm; related comments on this appear in [9]. Thus, any arguments on Lorentz violation based on EFT would have to be carefully reformulated before making statements concerning renormalization and loop corrections. This work was supported by an Atlantic Association for Research in the Mathematical Sciences (AARMS) post-doctoral fellowship (D.K.) and NSERC of Canada.", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "Nonequilibrium Thermodynamical Inequivalence of Quantum Stress-energy and Spin Tensors", "content": "F. Becattini, L. Tinti Universit'a di Firenze and INFN Sezione di Firenze, Florence, Italy It is shown that different pairs of stress-energy and spin tensors of quantum relativistic fields related by a pseudo-gauge transformation, i.e. differing by a divergence, imply different mean values of physical quantities in thermodynamical nonequilibrium situations. Most notably, transport coefficients and the total entropy production rate are affected by the choice of the spin tensor of the relativistic quantum field theory under consideration. Therefore, at least in principle, it should be possible to disprove a fundamental stress-energy tensor and/or to show that a fundamental spin tensor exists by means of a dissipative thermodynamical experiment.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In recent years, there has been a considerable interest in theoretical relativistic hydrodynamics and its most general form including dissipative terms [1]. This renewed interest has been mainly triggered by its successful application to the description of the Quark Gluon Plasma dynamical evolution in ultreralativistic heavy ion collisions [2]. Relativistic hydrodynamics can be seen as the theory describing the dynamical behaviour of the mean value of the quantum stressenergy tensor ̂ T µν , that is tr( ̂ ρ ̂ T µν ). This tensor is generally assumed to be symmetric, although in special relativity it does not need to be such if it is accompanied by a non-vanishing rank 3 tensor, the so-called spin tensor ̂ S λ,µν . In fact, in special relativistic quantum field theory, starting from particular stress-energy and spin tensors, different pairs can be generated (and are generally related) by means of a pseudo-gauge transformation [4, 5] preserving the total energy, momentum and angular momentum: ̂ In a previous paper [3] we have shown that indeed different pairs ( ̂ T, ̂ S ) and ( ̂ T ' , ̂ S ' ) are in general thermodynamically inequivalent as they imply different mean values of physical quantities for a rotating system at equilibrium. Particularly, for the free Dirac field, we showed that the canonical and Belinfante (obtained from the canonical one by setting ̂ Φ = ̂ S and ̂ Z = 0 in (1), hence with a vanishing new spin tensor ̂ S ' ) quantum stress-energy tensors result in different mean values for the momentum density and the total angular momentum density. ̂ ̂ ̂ ̂ where ̂ Φ is a rank three tensor field antisymmetric in the last two indices (often called and henceforth referred to as superpotential ) and Z a rank four tensor antisymmetric in the pairs αλ and µν . The thermodynamical inequivalence is (at least in our view) surprising because it was commonly believed that the only physical phenomenon which can discriminate between stress-energy tensors of a fundamental quantum field theory related by a transformation like (1) is gravity, or, in other words, the coupling to a metric tensor. In this paper we reinforce our previous finding by showing that the inequivalence extends to nonequilibrium thermodynamical quantities, specifically entropy production and transport coefficients. In summary, we will show that the use of different stress-energy tensors, related by (1), to calculate transport coefficients with the relativistic Kubo formula leads, in general, to different results. Therefore, at least in principle, an extremely accurate measurement of transport coefficients or total entropy in an experiment where dissipation is involved, would allow to disprove a candidate stressenergy or spin tensor, with obvious important consequences in relativistic gravitational theories. This finding means, in other words, that the existence of a fundamental spin tensor affects the microscopic number of degrees of freedom, or at least on how quickly macroscopic information gets converted into microscopic, namely on entropy generation. The paper is organized as follows: in Sect. II we will extend the framework of the nonequilibrium density operator introduced by Zubarev [6] to the case of a non-vanishing spin tensor. In Sect. III, it will be shown that the nonequilibrium density operator is not invariant under a pseudo-gauge transformation (1), that is it does depend on the chosen couple of stress-energy and spin tensor. In Sect. IV we will provide a general formula for the change of mean values of observables and we will determine how entropy is affected by a pseudo-gauge transformation. In Sect. V we will show that transport coefficients are also modified and, particularly, we will focus on the modification of the Kubo formula for shear viscosity. Finally, in Sect. VI, we will discuss the implications of this finding and draw our conclusions. In this paper we adopt the natural units, with /planckover2pi1 = c = K = 1. The Minkowskian metric tensor is diag(1 , -1 , -1 , -1); for the Levi-Civita symbol we use the convention ε 0123 = 1. We will use the relativistic notation with repeated indices assumed to be saturated. Operators in Hilbert space will be denoted by an upper hat, e.g. ̂ R , with the exception of the Dirac field operator which is denoted with a capital Ψ.", "pages": [ 1, 2 ] }, { "title": "II. NONEQUILIBRIUM DENSITY OPERATOR", "content": "A suitable formalism to calculate transport coefficients for relativistic quantum fields without going through kinetic theory was developed by Zubarev [6, 9], extending to the relativistic domain a formalism already introduced by Kubo [10]. In this approach, a non-equilibrium density operator is introduced which reads [11] 1 : In the formula (2) the possible contribution of a spin tensor is simply disregarded; therefore, the formula is correct only if the stress-energy tensor is the symmetrized Belinfante one (or improved ones, see last section), whose associated spin tensor is vanishing. It is the aim of this Section to find the appropriate extension of the formula (2) with a spin tensor. where ̂ j is a conserved current, the four-vector field β is a point-dependent inverse temperature four-vector ( β = u/T 0 , u being a four-velocity field and T 0 the comoving or invariant temperature) and ξ = µ 0 /T 0 a scalar function whose physical meaning is that of a point-dependent ratio between comoving chemical potential µ 0 and comoving temperature T 0 ; the Z factor is analogous to a partition function, i.e. a normalization factor to have tr ̂ ρ = 1. The operators in the exponential of Eq. (2) are in the Heisenberg representation. It should be stressed that in the formula (2) covariance is broken from the very beginning by the choice of a specific inertial frame and its time. However, it can be shown that the operator ̂ ρ is in fact time-independent [11], namely independent of t ' , so that ̂ ρ is a good density operator in the Heisenberg representation. Using the identity: integrating by parts and taking into account the continuity equations ∂ µ ̂ T µν = ∂ µ ̂ j µ = 0, the operator ̂ Υ in Eq. (2) can be rewritten as: The first term the so-called local thermodynamical equilibrium one, which is defined by the same formula of the global equilibrium [7, 8] with x -dependent four-temperature and chemical potentials, whereas the term dependent on their derivatives is interpreted as a perturbation. At equilibrium, the right hand side should reduce to the known form, which, at least for the most familiar form of thermodynamical equilibrium with β eq = (1 /T, 0 ) = const and ξ eq = µ/T = const is readily recognized in the first term setting β = β eq and ξ = ξ eq : Hence, the two rightmost terms of (4) must vanish at equilibrium. Indeed, the surface term is supposed to vanish through a suitable choice of the field boundary conditions while the third term vanishes in view of the constancy of β eq and ξ eq . However, this is not the case for the most general form of equilibrium; in the most general form (see discussion in ref. [8]), whilst the scalar ξ eq stays constant the four-vector β fulfills a Killing equation, whose solution is [12]: with both the four-vector b eq and the antisymmetric tensor ω eq constant. Therefore: ∂ µ β eq ν = - ω eq µν which in general is non-vanishing, so that the third term on the right hand side of Eq. (4) survives. For instance, for the thermodynamical equilibrium with rotation [8], the tensor ω turns out to be: ω being the angular velocity and T the temperature measured by the inertial frame. In order to find the appropriate generalization of the operator ̂ Υ, let us plug the formula (5) of general thermodynamical equilibrium into the (4): where ∂ µ ξ eq = 0 has been taken into account. For a symmetric stress-energy tensor ̂ T , the last term vanishes, but if a spin tensor is present ̂ T may have an antisymmetric part. Particularly, from the angular momentum continuity equation: so that the last term on the right hand side of Eq. (7) can be rewritten as: The first term on the right hand side of (9) can be integrated by parts, yielding: where the surface term involving ̂ T in Eq. (7) has been rearranged taking advantage of the antisymmetry of the ω tensor. The surface terms in the above equations now are manifestly the total momentum flux, the charge flux and the total angular momentum flux through the boundary. All of these terms are supposed to vanish at thermodynamical equilibrium through suitable conditions enforced on the field operators at the boundary, so that the (11) reduces to: The first term on the right hand side just gives rise to the desired form of the equilibrium operator. For instance, for a rotating system with ω as in Eq. (6) one has [8]: ̂ J being the total angular momentum, which is the known form [13]. Nevertheless, the second term in Eq. (12) does not vanish and, thus, must be subtracted away with a suitable modification of the definition of the ̂ Υ operator. The form of the unwanted term demands the following modification of (2): where ω µν ( x ) is an antisymmetric tensor field which must reduce to the constant ω eq µν tensor at equilibrium. It is easy to check, by tracing the previous calculations, that the equilibrium form of Υ reduces to the desired form: as the spin tensor term in Eq. (12) cancels out. Therefore, the operator (13) is the only possible extension of the nonequilibrium density operator with a spin tensor. where: and the continuity equation for angular momentum has been used. The first term on the right hand side is the new local thermodynamical term whilst the third term can be further expanded to derive the relativistic Kubo formula of transport coefficients (see Appendix A).", "pages": [ 2, 3, 4 ] }, { "title": "III. NONEQUILIBRIUM DENSITY OPERATOR AND PSEUDO-GAUGE TRANSFORMATIONS", "content": "A natural requirement for the density operator (13) would be its independence of the particular couple of stressenergy and spin tensor, because one would like the mean value of any observable ̂ O : to be an objective one 2 . In ref. [3] we showed that even at thermodynamical equilibrium with rotation this is not the case for the components of the stress-energy and spin tensor themselves because they change through the pseudo-gauge transformation (1). However, at equilibrium, ̂ ρ itself is a function of just integral quantities (total energy, angular momentum, charge) which are invariant under a transformation (1) provided that boundary fluxes vanish, so a specific operator ̂ O , including the components of a specific stress-energy tensor, does not change under (1). However, it is not obvious that this feature persists in a nonequilibrium case, in fact we are going to show that, in general, this is not the case. Let us consider the operator ̂ Υ in (13) and how it gets changed under a pseudo-gauge transformation (1) with ̂ Z = 0. The new operator Υ ' reads: where: ̂ is antisymmetric in the first two indices. We can rewrite Eq. (15) as: after integration by parts. Let us now write the general fields β and ω as the sum of the equilibrium values and a perturbation, that is: and work out first the equilibrium part of the right hand side of Eq. (17). As ∂ λ β eq ν = -ω eq λν one has: where we have used the Eq. (16) and the antisymmetry of indices of the superpotential ̂ Φ. By using the Eq. (5), the last expression can be rewritten as: The two surface integrals above are the additional four-momentum and the additional total angular momentum, in the operator sense, after having made a pseudo-gauge tranformation (1) of the stress-energy and spin tensor. If the boundary conditions ensure that the momentum and total angular momentum fluxes vanish (in order to have conserved energy and momentum operators) for any couple ( ̂ T, ̂ S ) of tensors, then the two fluxes in the above equations must vanish as well. Therefore, we can conclude that: where the dependence of δβ and δω on x is now understood. It can be seen that it is impossible to make this difference vanishing in general. One can get rid of the surface term by choosing a perturbation which vanishes at the boundary and the last term by locking the perturbation of the tensor ω to that of the inverse temperature four-vector: but it is impossible to cancel out the term: unless in special cases, e.g. when the tensor ̂ Φ is also antisymmetric in the first two indices. We have thus come to the conclusion that the nonequilibrium density operator does depend, in general, on the particular choice of stress-energy and spin tensor of the quantum field theory under consideration. Therefeore, the mean value of any observable in a non-equilibrium situation shall depend on that choice. It is worth stressing that this is a much deeper dependence on the stress-energy and spin tensor than what we showed in ref. [3] for thermodynamical equilibrium with rotation. Therein, mean values of the angular momentum densities and momentum densities were found to be dependent on the pseudo-gauge transformation (1) because the relevant quantum operators could be varied, but not because the density operator ̂ ρ was dependent thereupon. In fact, at non-equilibrium, even ̂ ρ varies under a transformation (1). Note that, in principle, even the mean values of the total energy and momentum could be dependent on the quantum stress-energy tensor choice although boundary conditions ensure, as we have assumed, that the total energy and momentum operators are invariant under a transformation (1). Again, this comes about because the density operator is not invariant under (1), in formula: /negationslash tr( ̂ ρ ' ̂ P ' µ ) = tr( ̂ ρ ' ̂ P µ ) = tr( ̂ ρ ̂ P µ ) It must be pointed out that the variation of the Zubarev non-equilibrium density operator (22) depends on the gradients of the four-temperature field and it is thus a small one close to thermodynamical equilibrium. In the next Section we will show in more details how the mean values of observables change under a small change of the nonequilibrium density operator, or, in other words, when the system is close to thermodynamical equilibrium.", "pages": [ 4, 5, 6 ] }, { "title": "IV. VARIATION OF MEAN VALUES AND LINEAR RESPONSE", "content": "We will first study the general dependence of the mean value of an observable ̂ O on the spin tensor by denoting by δ ̂ Υ the supposedly small variation, under a transformation (1), of the operator ̂ Υ. This can be either the one in Eq. (22) or the more general (only bulk terms) in Eq. (20). We have: being Z ' = tr(exp[ -Υ -δ Υ]). We can expand in δ Υ at the first order (Zassenhaus formula): hence, with 〈 〉 = tr( ρ ), at the first order in δ Υ: which makes manifest the dependence of the mean value on the choice of the superpotential ̂ Φ. As has been mentioned, close to thermodynamical equilibrium, the operator δ ̂ Υ is 'small' and one can write an expansion of the mean value of the observable ̂ O in the gradients of the four-temperature field, according to relativistic linear response theory [11]. This method, just based on Zubarev's nonequilibrium density operator method, allows to calculate the variation between the actual mean value of an operator and its value at local thermodynamical equilibrium for small deviations from it. In fact, it can be seen from Eq. (22) that the operator δ ̂ Υ, from the linear response theory viewpoint, is an additional perturbation in the derivative of the four-temperature field and therefore the difference between actual mean values at first order turns out be (see Appendix A for reference): where 〈 . . . 〉 0 stands for the expectation value calculated with the equilibrium density operator, that is: Since tr( ̂ ρ 0 [ ̂ Φ λ, 0 ν , ̂ O ]) = tr( ̂ Φ λ, 0 ν [ ̂ O, ̂ ρ 0 ]) the right hand side of (25) vanishes for all quantities commutating with the equilibrium density operator, notably total energy, momentum and angular momentum. Nevertheless, in principle, even the mean values of the conserved quantities are affected by the choice of a specific quantum stress-energy tensor, though at the second order in the perturbation δβ . We now set out to study the effect of the transformation (1) on the total entropy. In nonequilibrium situation, entropy is usually defined as [13] the quantity maximizing -tr( ̂ ρ log ̂ ρ ) with the constraints of fixed mean conserved densities. The solution ̂ ρ LE of this problem is the local thermodynamical equilibrium operator, namely: which - as emphasized in the above equation - is explicitely dependent on time, unlike the Zubarev stationary nonequilibrium density operator (13); of course the time dependence is crucial to make entropy S = -tr( ̂ ρ LE log ̂ ρ LE ) (28) increasing in nonequilibrium situation. In order to study the effect of the transformation (1) on the entropy it is convenient to define: for which it can be shown that, with calculations similar to those in the previous section, the variation induced by the transformation (1) is: As has been mentioned, it is possible to get rid of the surface and the last term in the right hand side of above equation through a suitable choice of the perturbations, but not of the second term. Since δ ̂ Υ LE is a small term compared to ̂ Υ LE we can determine the variation of the entropy (28) with an expansion in δ ̂ Υ LE at first order. First, we observe that (see also Eq. (24)): Retaining only the first order terms in δ ̂ Υ LE , expanding the logarithm for 〈 δ ̂ Υ LE 〉 LE /lessmuch 1 and inserting the original expression of entropy: Therefore, the variation of the total entropy is, to the lowest order, proportional to the correlation between ̂ Υ and δ Υ, which is generally non-vanishing. ̂ We can expand the above correlation to gain further insight. For the δ ̂ Υ LE , let us keep only the second term of the right hand side of Eq. (30): By using the (29) and the (36), the Eq. (35) can be rewritten as: where x and x ' have equal times. The above expression could be further simplified by e.g. approximating the local equilibrium mean 〈 〉 ̂ Υ with the global equilibrium one 〈 〉 0 , but this does not lead to further conceptual insight. The physical meaning of Eq. (37) is that the entropy difference depends on the correlation between local operators in two different space points multiplied by a factor which is at most of the second order in the perturbation δβ . This kind of expression resembles the product of transport coefficients expressed by a Kubo formula times the squared gradient of the perturbation field. Therefore, the difference between entropies suggest that the introduction of a superpotential may lead to a modification of the transport coefficients. We will show this in detail in the next Section.", "pages": [ 6, 7, 8 ] }, { "title": "V. TRANSPORT COEFFICIENTS: SHEAR VISCOSITY AS AN EXAMPLE", "content": "As has been mentioned, a remarkable consequence of the transformation (1) is a difference in the predicted values of transport coefficients calculated with the relativistic Kubo formula, which is obtained by working out the mean value of the stress-energy tensor itself with the linear response theory and the nonequilibrium density operator in Eq. (2). For this purpose, the derivation in ref. [11] must be extended to the most general expression of the nonequilibrium density operator including a spin tensor, that is, Eq. (13); it can be found in Appendix A. The equation (25), yielding the difference of mean values of a general observable under a transformation (1), cannot be straightforwardly used to calculate the mean value of the stress-energy tensor setting ̂ O = ̂ T µν ( y ) because ̂ T µν ( y ) gets transformed itself. It is therefore more convenient to work out the general expression of the Kubo formula and study how it is modified by (1) thereafter. We will take shear viscosity as an example, the transformation of other transport coefficients can be obtained with the same reasoning. Shear viscosity, in the Kubo formula, is related to the spatial components of the symmetric part of the stress-energy tensor. It is worth pointing out that, since a non-vanishing spin tensor can make the stress-energy tensor non-symmetric, there might be a new transport coefficient related to the antisymmetric part of the stress-energy tensor. For the symmetric part of the stress-energy tensor T µν S ≡ (1 / 2)( T µν + T νµ ), using the general formula of relativistic linear response theory (Eq. 70) of Appendix A), the difference δT µν S ( y ) between actual mean value and local equilibrium value reads, at the lowest order in gradients: In order to obtain transport coefficients, a suitable perturbation must be chosen which can be eventually taken out from the integral. Physically, this corresponds to enforcing a particular hydrodynamical motion and observing the response of the stress-energy tensor to infer the dissipative coefficient. The perturbation δβ = 1 /Tδu is taken to be a stationary one and non-vanishing only within a finite region V , at whose boundary it goes to zero in a continuous and derivable fashion. The perturbation δω is also taken to be stationary and it can be chosen either to vanish or like in Eq. (21); in both cases, one gets to the same final result. Let us then set δω = 0 and expand the perturbation δβ = (0 , 0 , δβ 2 ( x 1 ) , 0) dependent on x 1 in a Fourier series (it vanishes at some large, yet finite boundary). Since we want the higher order gradients of the perturbation to be negligibly small (the so-called hydrodynamic limit), the Fourier components with short wavelengths must be correspondingly suppressed. The component with the longest wavelength will then be much larger than any other and, therefore, δβ 2 can be approximately written, at least far from the boundary, as A sin( πx 1 /L ) where L is the size of the region V in the x 1 direction and A is a constant. The derivative of this perturbation reads: where k ≡ π/L . Therefore, by defining k = ( k, 0 , 0) and plugging the last equation in Eq. (38): taking into account that the commutator is purely imaginary. To extract shear viscosity we have to evaluate the stress-energy tensor in y = 0 to make it proportional to the derivative of the four-temperature field in the same point and we have to take the limit L →∞ which implies V →∞ and k → 0 at the same time: where it has been assumed that the integration domain goes to its thermodynamic limit independently of the integrand. Because of the time-translation symmetry of the equilibrium density operator ̂ ρ 0 , the mean value in the integral only depends on the time difference t -t y . Thus, choosing the arbitrary time t ' = t y and redefining the integration variables, the Eq. (40) can be rewritten as: which shows that the mean value δT µν S ( t y , 0 ) is indeed independent of t y , which is expected as δβ is stationary. We can now take advantage of the well known Curie symmetry 'principle' which states that tensors belonging to some irreducible representation of the rotation group will only respond to perturbations belonging to the same representation and with the same components 3 . In our case the Curie principle implies that only the same component of the symmetric part of the stress-energy tensor, i.e. T 12 S , will give a non-vanishing value: From the above expression, a Kubo formula for shear viscosity can be extracted setting δβ = (1 /T ) δu : which, after a little algebra, can be shown to be the same expression obtained in ref. [11]. Because of the rotational invariance of the equilibrium density operator, shear viscosity is independent of the particular couple (1 , 2) of chosen indices. It is worth pointing out that, had we started from Eq. (71) instead of Eq. (70), choosing δω = 0 or like in Eq. (21), we would have come to the same formula for shear viscosity; in the latter case, the third contributing term in Eq. (71) would have been of higher order in derivatives of δβ , hence negligible. Now, the question we want to answer is whether equation (43) is invariant by a pseudo-gauge transformation (1), which turns the symmetric part of the stress-energy tensor into: where: ̂ Ξ being symmetric in the last two indicess. We will study the effect of the transformation on the mean value of the stress-energy tensor in the point y = 0 starting from the formula Eq. (71) instead of Eq. (70) with δω = 0 or like in Eq. (21), which allows us to retain only the first contributing term to δT 12 S (0). The perturbation δβ is taken to be stationary and t ' is set to be equal to t y = 0. Eventually, the appropriate limits will be calculated to get the new shear viscosity. Thus: We can simplify the above formula by noting that the mean value of two operators at equilibrium can oly depend on the difference of the coordinates, so: hence, the Eq. (46) can be rewritten as: We are now going to inspect the two terms on the right-hand side of the above equation. If the hamiltonian is time-reversal invariant, it can be shown (see Appendix B): where n 0 is the total number of time indices among those in the above expression. Similarly, if the hamiltonian is parity invariant, then: where n s is the total number of space indices. Using the last two equations to work out the last term of Eq. (47) one gets: Now, the two terms on the right hand side of (48) can be worked out separately. Using invariance by time-reversal and parity, one has: and, similarly, the second term as: All terms in Eqs. (50) and (51) with a space derivative do not yield any contribution to first-order transport coefficients. This can be shown by, firstly, integrating by parts and generating two terms, one of which is a total derivative and the second involves the second derivative of the perturbation δβ . The total derivative term can be transformed into a surface integral on the boundary of V which vanishes because therein the perturbation δβ is supposed to vanish along with its first-order derivatives. The second term, involving higher order derivatives, does not give contribution to transport coefficients at first order in the derivative expansion. Altogether, the Eq. (48) turns into: which can be further integrated by parts in the time t , yielding: provided that, for general space-time dependent operators O 1 and O 2 with n = 0 , 1, which is reasonable because thermodynamical correlations are expected to vanish exponentially as a function of time for fixed points in space 4 . From Eq. (53) the variation of the shear viscosity can be inferred with the very same reasoning that led us to formula (43), that is: If the first integral is regular, then the ε → 0 limit kills one term and the (54) reduces to: In general, this difference is non-vanishing, leading to the conclusion that the specific form of the stress-energy tensor and, possibly, the existence of a spin tensor in the underlying quantum field theory affects the value of transport coefficients. The relative difference of those values depends on the particular transformation (1), hence on the particular stress-energy tensor. In the next Section a specific instance will be presented and discussed. An important point to make is that the found dependence of the transport coefficients on the particular set of stress-energy and spin tensor of the theory is indeed physically meaningful. This means that the variation of some coefficient is not compensated by a corresponding variation of another coefficient so as to eventually leave measurable quantities unchanged. This has been implicitely proved in Sect. IV where it was shown that total entropy itself undergoes a variation under a transformation of the stress-energy and spin tensor (see Eq. (35)).", "pages": [ 9, 10, 11, 12 ] }, { "title": "VI. DISCUSSION AND CONCLUSIONS", "content": "As a first point, we would like to emphasize that in our arguments space-time curvature and gravitational coupling have been disregarded. On one hand, this shows that the nature of stress-energy tensor and, possibly, the existence of a fundamental spin tensor could, at least in principle, be demonstrated independently of gravity. On the other hand, for each stress-energy tensor created with the transformation (1), it should be shown that an extension of general relativity exists having it as a source, which could not be always possible. An important question is whether a concrete physical system indeed exists for which the transformation (1) leads to actually different values for e.g. transport coefficients, entropy production rate or other quantities in nonequilibrium situations. For this purpose, we discuss a specific instance regarding spinor electrodynamics. Starting from the symmetrized gauge-invariant Belinfante tensor of the coupled Dirac and electromagnetic fields, with associated S = 0: where ∇ µ = ∂ µ -ieA µ is the gauge covariant derivative, one can generate other stress-energy tensors with suitable rank three tensors and then setting ̂ Φ = -̂ S ' where ̂ S ' is the new spin tensor, according to (1). One of the best known is the canonical Dirac spin tensor: ( { } stands for anticommutator) which is gauge-invariant and transforms the Belinfante tensor (56) back to the canonical one obtained from the spinor electrodynamics lagrangian (see also [3] for a detailed discussion). However, this is totally antysimmetric in the three indices λ, µ, ν and thus the variation of ̂ Υ operator (see Eq. (22) as well as transport coefficients, which depend on the symmetrized ̂ Ξ tensor (45) vanish. Nevertheless, other gauge-invariant ̂ Φ-like tensors can be found. For instance, one could employ a superpotential: which is the gauge-invariant version of the one used in ref. [12] to obtain a conserved spin current. This superpotential gives rise to a non-vanishing spin tensor as well as a Ξ tensor (see Eq. 45)): hence a variation of thermodynamics. By noting that the structure of the above tensor is very similar to the Belinfante stress-energy tensor (56), it is not difficult to find a rough estimate of the variation of e.g. shear viscosity induced by the transformation. Looking at Eq. (55) we note that ̂ Ξ 012 mainly differs from ̂ T 012 in Eq. (56) by the factor 1 /m . The last term on the right hand side of Eq. (56) tells us that the dimension of ̂ Ξ is that of a stress-energy tensor multiplied by a time, and therefore this term must be of the order of η /planckover2pi1 /mc 2 τ where τ is the microscopic correlation time scale of the original stress-energy tensor or the collisional time scale in the kinetic language and η the shear viscosity obtained from the original stress-energy tensor. Thus, the expected relative variation of shear viscosity from Eq. (55) in this case is of the order: which is (as it could have been expected) a quantum relativistic correction governed by the ratio ( λ c /c ) /τ , λ c being the Compton wavelength. For the electron, the ratio λ c /c ≈ 10 -21 sec, which is a very small time scale compared to the usual kinetic time scales, yet it could be detectable for particular systems with very low shear viscosity. It is also interesting to note that the 'improved' stress-energy tensor by Callan, Coleman and Jackiw [14] with renormalizable matrix elements at all orders of perturbation theory, is obtained from the Belinfante's symmetrized one in Eq. (56) with a transformation of the kind (1) setting (for the Dirac field and vanishing constants [14]): and requiring ̂ S ' = ̂ S = 0 so that Φ λ,µν = ∂ α Z αλ,µν , hence: which is just the improved stress-energy tensor [14]. It is likely (to be verified though) that the aforementioned modified stress-energy tensors imply a different thermodynamics with respect to the original Belinfante symmetrized tensor. This problem has been recently pointed out in ref. [15]. To summarize, we have concluded that different quantum stress-energy tensors imply different values of nonequilibrium thermodynamical quantities like transport coefficients and entropy production rate. This reinforces our previous similar conclusion concerning differences of momentum and angular momentum densities in rotational equilibrium [3]. The existence of a fundamental spin tensor has, thus, an impact on the microscopic number of degrees of freedom and on how quickly macroscopic information is converted into microscopic. The difference of transport coefficients depends on the particular form of the tensors and in the examined case it scales like a quantum relativistic effect with /planckover2pi1 /c . Therefore, at least in principle, it is possible to disprove a supposed stress-energy tensor with a suitably designed thermodynamical experiment.", "pages": [ 12, 13, 14 ] }, { "title": "Acknowledgments", "content": "We are grateful to F. Bigazzi, F. W. Hehl and D. Seminara for useful discussions and suggestions.", "pages": [ 14 ] }, { "title": "APPENDIX A - Relativistic linear response theory with spin tensor", "content": "We extend the relativistic linear response theory in the Zubarev's approach to the case of a non-vanishing spin tensor. The (stationary) nonequilibrium density operator is written in Eq. (13), with ̂ Υ expanded as in Eq. (14). As has been shown in Sect. II, at equilibrium, only the first term of the ̂ Υ operator survives in Eq. (14); therefore, one can rewrite that equation using the perturbations δβ , δξ and δω which are defined as the difference between the actual value and their value at thermodynamical equilibrium: In fact, we will use a rearrangement of the right-hand-side expression which is more convenient if one wants to work with an unspecified, yet small, δω . Therefore, the above equation is rewritten as: what it can be easily obtained from Eq. (13) integrating by parts in time. /negationslash /negationslash For the sake of simplicity we calculate the linear response with ξ eq = δξ = 0, but it can be shown that our final expressions hold for ξ eq = 0 (in other words with a non-vanishing chemical potential µ = 0). Let us now define: and: so that: with Z = tr(exp[ ̂ A + ̂ B ]). The operator ̂ B is the small term in which ̂ ρ is to be expanded, according to the linear response theory. It can can be rewritten in a way which will be useful later on. Since: then: The perturbation δβ must be chosen such that δβ | ∂V = 0 so that only the bulk term survives in the above equation: At the lowest order in B : and, according to Kubo identity: where the subscript LE stands for Local Equilibrium and implies the calculation of mean values with the local equilibrium density operator (see Sect. IV). Thereby, putting together (61) and (62) and retaining only first-order terms in ̂ B : hence the mean value of an operator O ( y ) becomes: Let us focus on the last term, which, by virtue of (60), contains expressions of this sort: where ̂ X stands for components of either T or S or ∂ 0 S . From the identity: and the observation that correlations vanish for very distant times (check footnote 4), one obtains: ̂ A /similarequal -̂ H/T where ̂ H is the hamiltonian operator (which ought to exists given the chosen boundary conditions). The straightforward consequence of this approximation is that the second term on the right hand side in Eq. (64) can be written as: where we have also taken advantage of the commutation between exp[ ̂ A ] and exp[ ± z ̂ A ]. We now approximate [11] the local equilibrium density operator with the nearest equilibrium operator ̂ ρ 0 in Eq. (26), which also implies that: 〈 ̂ X ( -∞ , x ) 〉 LE /similarequal 〈 ̂ X ( -∞ , x ) 〉 0 = 〈 ̂ X ( t, x ) 〉 0 because the mean value is stationary under the equilibrium distribution. Therefore, the Eq. (64) can be approximated as: and the (63) as: Once integrated, the second term in (65) gives rise to a term which cancels out exactly the 〈 ̂ B 〉 0 〈 ̂ O ( y ) 〉 0 in the equation above, which then becomes: Let us now integrate the last term on the right hand side in z : Hence, putting the last three equations together, we have: so that the Eq. (69) can be finally written: Another useful (equivalent) expression of δ 〈 ̂ O ( y ) 〉 can be obtained starting from the expression (14) of ̂ Υ, where the continuity equation for angular momentum is used from the beginning. Repeating the same reasoning as above, it can be shown that one gets:", "pages": [ 14, 15, 16, 17, 18 ] }, { "title": "APPENDIX B - Commutators and discrete symmetries", "content": "As we have pointed out, these expressions hold when ̂ ρ 0 has a non-vanishing chemical potential. We want to study the effect of space inversion and time reversal on the mean value of commutators like: where ̂ O 1 and ̂ O 2 are physical tensor densities of rank m and n , respectively. The equilibrium density operator ̂ ρ = exp[ -̂ H/T ] /Z is symmetric for space-time translations and rotations, as well as time reversal and parity if the hamiltonian is itself parity and time reversal invariant. The symmetry under this class of transformations allows to simplify the above expression. For any linear unitary transformation ̂ U which commutes with ̂ ρ one has: Taking ̂ U = ̂ T ( a ) with T ( a ) a general translation operator: and so, setting ( a 0 , a ) = ( -t, -x ): Similarly, for a space inversion: The time reversal operator ̂ Θ is antiunitary, thus a point-dependent physical scalar operator ̂ A ( t, x ) transforms as follows: whence, for commutators: Then, for Hermitian operators, what gets changed is the order of the operators besides their time argument. For tensor hermitian observables and time-reversal symmetric hamiltonian, one obtains: where m 0 and n 0 are the number of time indices among µ 1 , · · · µ m and ν 1 , · · · ν n respectively.", "pages": [ 18 ] } ]

[ { "title": "Wilson Line Response of Holographic Superconductors in Gauss-Bonnet Gravity", "content": "Rong-Gen Cai ∗ , Li Li † , Li-Fang Li ‡ , Hai-Qing Zhang § , and Yun-Long Zhang ¶ State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the Wilson line response in the holographic superconducting phase transitions in GaussBonnet gravity. In the black brane background case, the Little-Parks periodicity is independent of the Gauss-Bonnet parameter, while in the AdS soliton case, there is no evidence for the LittleParks periodicity. We further study the impact of the Gauss-Bonnet term on the holographic phase transitions quantitatively. The results show that such quantum corrections can effectively affect the occurrence of the phase transitions and the response to the Wilson line. PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The AdS/CFT correspondence [1-3] provides us a novel method to study the strongly coupled field theories through a weakly coupled gravitational system in one higher dimension. In recent years, the AdS/CFT correspondence has been used to mimic many basic phenomena in condensed matter physics, such as Nernst effect [4, 5], superconductivity [68] and non-fermi liquid [9-11]. Among these works, the holographic superconductivity is the most studied subject, and various aspects of the holographic superconductors have been intensively discussed, such as the response to the magnetic field [12-15], the Meissner effect [16, 17], the behavior of the entanglement entropy [18-21] and the Josephson junction [22-25]. For more details, please refer to [26-28] and references therein. The Little-Parks (LP) effect was discovered in experiments with empty and thin-walled superconducting cylinders subjected to a parallel magnetic field [29]. It states that certain thermodynamic quantities such as the critical temperature T c are periodic functions of the enclosed magnetic flux with period hc/ 2 e where e is the fundamental charge of carriers. While recently it was shown in the condensed matter literatures [30-34] that the LP period would be broken for small enough cylinder, the LP degeneracy and its uplifting have been realized holographically in [35], where the Wilson line W along the compact direction is identified as a quantum hair. They found that the boundary field theory in a deconfined state dual to black brane is insensitive to the quantum hair, thus the Aharonov-Bohm (AB) effect is suppressed and the LP period appears. In contrast, the theory in the confining vacuum dual to the soliton is sensitive to the quantum hair, thus the AB effect is unsuppressed and there is no trace of the LP effect. This encouraging discovery motivates us to study the responses to a Wilson line on a circle in the frame of Gauss-Bonnet gravity. In [35, 36], the effect of the Wilson line has been studied in the Einstein gravity background. It would be interesting to see how the modification of the bulk gravity may influence the response to the Wilson line. The higher curvature corrections originated from the string theory can effectively describe quantum corrections in the bulk. According to the AdS/CFT correspondence, such effects on the gravity side map to 1 / N suppressed corrections in the large N expansion of the boundary field theory. The studies of holographic superconductor with higher curvature corrections [37-42] indicate that such terms can quantitatively affect many properties of the boundary sys- tem. In particular, the ratio of the frequency gap over the critical temperature, i.e., ω g /T c which was found to be universal in Einstein gravity, breaks down. Therefore, in this paper, we are interested in how the Gauss-Bonnet term changes the response to the Wilson line. We study the responses in both AdS black brane background which can mimic the holographic conductor/superconductor transition, and AdS soliton background which can model the holographic insulator/superconductor transition. We work in the probe limit. Our results show that the responses to Wilson line are dramatically different between holographic conductor/superconductor phase transition and holographic insulator/superconductor phase transition. Explicitly, for black brane, the LP periodicity still holds in the Gauss-Bonnet gravity and is independent of the Gauss-Bonnet parameter, while there is no evidence for the LP periodicity in the AdS soliton case. These phenomena are similar to those in the Einstein theory. Furthermore, we analyze the impact of the Gauss-Bonnet term to the systems in detail. We find that with the change of the Gauss-Bonnet parameter, the corresponding physical quantities, such as the condensation and the current, display regular behaviors. In particular, at fixed chemical potential, the behaviors of condensation and current with respect to a χ ( or equivalently Wilson line W ) are much more different in the two systems with the change of the Gauss-Bonnet parameter. Our paper is organized as follows. First, we construct the holographic model in Section. II. We study the responses to the Wilson line in the holographic conductor/superconductor model and the insulator/superconductor model in Section. III and Section. IV, respectively. In the last Section. V, we give some conclusions and discussions.", "pages": [ 2, 3 ] }, { "title": "II. THE HOLOGRAPHIC MODEL", "content": "Let us start from the Einstein-Gauss-Bonnet gravity with a negative cosmological constant coupled to a U (1) gauge field A µ and a charged scalar field Ψ in 5-dimensional spacetime. The action reads with κ 5 the gravitational constant, L the radius of the AdS spacetime, F µν = ∂ µ A ν -∂ ν A µ and D µ = ∂ µ -iA µ . The quadratic curvature term is the Gauss-Bonnet term with α the Gauss-Bonnet parameter. In order to compare with the results in [35, 36], we only consider the case with a massless scalar field. To mimic a boundary system compactified on a circle, we are interested in the geometry with one compact spatial direction labeled as χ with 0 ≤ χ < 2 πR . In this paper, we will work at finite temperature T , corresponding to a compact Euclidean time direction with radius β = 1 /T . The control parameter that we will consider in the holographic superconductor is a Wilson line along the compact direction χ , with a constant non-trivial gauge vector potential a χ . where the integral is calculated along the compact direction and e is the fundamental charge. The Wilson line on the material can be thought to be generated by the axial magnetic flux since the circulation of the gauge potential equals the magnetic flux enclosed by the path. One parameter that can characterize the response of W is the fluxoid number where the integral is done along the compact direction and θ is the phase of the order parameter. In general, we should solve the full coupled equations of motion, which are more complicated in the case with the Gauss-Bonnet term. However, we can get some qualitative features in the so-called probe limit. Indeed, we can see from the action that in the limit κ 2 5 / ˆ g 2 /lessmuch 1, the back reaction of the gauge field and the complex scalar field can be neglected safely. For our case, the simplest ansatz are as follows Near the AdS boundary z → 0, the scalar field and the Maxwell field behave as From the AdS/CFT dictionary, the coefficients above can be related to physical quantities in the boundary field theory. 〈 O 〉 is the vacuum expectation value (VEV) of the dual operator with s the source which is set to be zero to accomplish the spontaneous symmetry breaking of the gauge symmetry. µ and ρ are chemical potential and charge density, respectively. J χ is the VEV of the U (1) current and a χ plays the role of a gauge potential along the compact direction of the boundary material. The free energy F = TS E can be obtained from the AdS Euclidean action S E evaluated with all the bulk fields on shell, which is used to determine which configuration is thermodynamically favorable. In the next two sections, we will study the response to the Wilson line in both five dimensional Gauss-Bonnet-AdS black brane and Gauss-Bonnet-AdS soliton backgrounds. Especially, we concentrate on the quantitative changes of boundary systems after turning on such quadratic curvature corrections in the bulk.", "pages": [ 3, 4, 5 ] }, { "title": "III. HOLOGRAPHIC CONDUCTOR/SUPERCONDUCTOR TRANSITION", "content": "The 5-dimensional Gauss-Bonnet-AdS black brane with a Ricci flat horizon is described by [43] where the horizon is located at z h . The temperature of the black brane is T = 1 πz h . In order to have a well-defined vacuum for the gravity theory, one has to have α ≤ L 2 / 4. The upper bound α = L 2 / 4 is called the Chern-Simons limit. If we further consider the causality constraint from the boundary CFT, there is an additional constraint on the Gauss-Bonnet parameter with -7 L 2 / 36 ≤ α ≤ 9 L 2 / 100 [44-47]. In the AdS/CFT correspondence, the temperature of the black hole is just the one of the dual field theory. In the following numerical calculations we will set L = 1. Assuming the matter field in the form of Eq.(4), we can obtain the equations of motion To solve the above equations of motion, we impose the regularity conditions at the horizon z = z h From the bulk equations of motion Eq.(7) and the boundary conditions Eq.(8), we see that A χ equivalently appears in the combination ( m/R -A χ ) which comes from the local covariant quantity D χ ψ = i ( m/R -A χ ) ψe imχ/R . This implies that the effective action of the boundary system will only depend on local gauge invariant quantities, and thus will display LP periodicity. This observation is confirmed by our subsequent numerical calculations. Considering the scaling symmetry we can adjust the solutions to satisfy z h = 1. And the corresponding scaling invariant variables are a χ R , 〈 O 〉 R 4 , J χ R 3 and so on. The free energy of the system is which is used to determine which configuration is thermodynamically favored, where V 3 is the spatial volume of the black hole spanned by x , y and χ . Fig.1 shows the phase diagrams for the occurrence of superconductivity in the GaussBonnet-AdS black brane case. Here we choose the Gauss-Bonnet parameter as α = 0 . 09, α = 0 . 03, α = 0 . 0001, and α = -0 . 1 in turn. Although we introduce some kind of quantum corrections effectively described by Gauss-Bonnet term in the bulk, we can see from Fig.1 that the phase diagram displays a precise periodicity with period ∆ a χ R = 1 no matter the choice of the Gauss-Bonnet parameter α . When the parameter α vanishes, the system returns to the one in Einstein theory. Following the terminology in [35], we conclude that the AB effects are suppressed for black brane background. However, the introduction of Gauss-Bonnet term can quantitatively affect other physical observables. For different α , µ c is different accordingly, where µ c is the critical chemical potential denoting the occurrence of phase transition when a χ = 0 (without Wilson line). For α = 0 . 09, α = 0 . 03, α = 0 . 0001, and α = -0 . 1, the corresponding critical chemical potentials are µ c R ≈ 7 . 13, µ c R ≈ 6 . 69, µ c R ≈ 6 . 50, and µ c R ≈ 6 . 02, respectively. It is clear that as one increases the Gauss-Bonnet parameter α , µ c increases accordingly, which makes the condensation more difficult to form for vanishing magnetic flux. Comparing with the above four phase diagrams, we can see that, for a given a χ R , the smaller the Gauss-Bonnet parameter α is chosen, the bigger µ/µ c is required to trigger the phase transition. The variation of the critical chemical potential to the gauge potential and fluxoid number m is found to perfectly behave as where µ is the critical chemical potential for non-vanishing a χ R . The coefficients k ( α ) corresponding to different α are listed in Table. I. According to [29, 33], the coefficient k ( α ) would reflect the ratio of the coherence length to the cylinder radius, i.e. k ( α ) ∼ ξ 0 /R . 1 Thus, from Table. I, we may conclude that compared to the Einstein case, the positive Gauss-Bonnet corrections would decrease the coherence length ξ 0 of the Cooper-pair, while the negative corrections increase the coherence length ξ 0 . Since ξ 0 plays the role of the inverse of the mass of the pair, we can find that the quantum a R corrections described by Gauss-Bonnet term can change the effective mass of the charge carriers. Therefore, from the above diagrams, we clearly see that the LP period ∆ a χ = 1 /R still holds in Gauss-Bonnet gravity no matter the choice of the Gauss-Bonnet parameter, and also the period is the same as that in Einstein theory [36]. In the same spirit of [36], the existence of LP period would be understood by that the effective action of the boundary system has no direct dependence on non-local gauge invariants such as the Wilson line W and fluxoid number m . As argued in [36], the existence of LP period implies that the AB effects are also somehow suppressed in our case in the limit N → ∞ with N the number of colors. Although including the Gauss-Bonnet term can not break LP period, this kind of quantum correction does impose its effect on many physical quantities of the dual boundary system. We focus our attention on the impact of different Gauss-Bonnet parameters. Our results show that the Gauss-Bonnet parameter can quantitatively affect the occurrence of phase transition and the response to the Wilson line.", "pages": [ 5, 6, 7, 8, 9, 10 ] }, { "title": "IV. HOLOGRAPHIC INSULATOR/SUPERCONDUCTOR TRANSITION", "content": "The AdS soliton metric in the Gauss-Bonnet gravity reads [48] Obviously there does not exist any horizon in this soliton solution, but a conical singularity at the tip z = z 0 , which obeys f ( z 0 ) = 0. To remove this singularity, the coordinate χ must have a period πz 0 . This asymptotical AdS solution is dual to a boundary theory in the confining vacuum, which is reminiscent of the insulator phase in condensed matter physics. The equations of motion for this system are At the tip z = z 0 , the requirement of regularity on the above set of equations implies the following boundary conditions /negationslash Different from the Gauss-Bonnet black brane case, in the soliton case, the boundary conditions now depend directly on A χ at z 0 . Since we demand a Wilson line in the external gauge field a χ ∼ ln( W ), A χ ( z ) acquires a nontrivial profile which generates a magnetic field F zχ in the bulk. Furthermore, the requirement A χ ( z 0 ) = 0 breaks down the gauge equivalence among different fluxoid sectors. In addition, we can see that the boundary conditions are also sensitive to m . These differences between the black brane and soliton geometry may lead to several distortions of the dual system. Similar to black brane case, we can solve the equations of motion numerically and obtain our simulation results. The free energy of such a system is which is used to find the thermodynamically favored fluxoid sector. Fig. 4 shows us the phase diagrams for the soliton case. Each subgraph corresponds to different α . Our results show that the LP period is destroyed in the Gauss-Bonnet-AdS soliton case. In the rest of this section, we study how physical observables are affected as one tunes the strength of the quantum corrections measured by the Gauss-Bonnet parameter α . Here µ c is the critical chemical potential when a χ vanishes. For α = 0 . 09, α = 0 . 0001 and α = -0 . 19, the corresponding critical chemical potential is µ c R ≈ 1 . 77, µ c R ≈ 1 . 70 and µ c R ≈ 1 . 60, respectively. We can see that µ c increases with the increase of the GaussBonnet parameter, which makes the phase transition from insulator to superconductor more difficult to happen for a χ = 0. Compared with each phase diagram for different α , it is easy to see that the boundary of the phase diagram gradually drops down as one decreases the Gauss-Bonnet parameter α . This implies that at given µ/µ c > 1, the superconducting phase can be more easily destroyed by the applied a χ for smaller α . Fig. 5 shows the evolution of the condensation 〈 O 〉 as a function of a χ at µ = 1 . 5 µ c for different α . The first jump appears when the m = 1 sector becomes thermodynamically favorable. The second jump occurs when the m = 2 sector becomes the ground state for every α . The jumps here indicate that the effective Lagrangian of the boundary theory gets the non-trivial dependence on fluxoid number m . The position for the jumps between different fluxoid domains of different α is listed in Table. II. From this table, we can find that with the increase of α , the occurrence of the jumps move to the right. Further, if we fix a χ R in Fig. 5, we see that the condensation 〈 O 〉 is larger for larger α . This property implies that with the increase of α , the condensation gap becomes higher. This behavior is opposite to the black brane case discussed in the last section. Α/Equal 0.0001 m=0 R Χ a 6 5 4 3 2 1 0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Μ /Slash1 Μ c 6 5 4 3 2 1 0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Μ /Slash1 Μ Α/Equal/Minus 0.19 6 5 4 3 2 1 0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Μ /Slash1 Μ c The behavior of the current J 〈 O 〉 χ ≡ J χ -J vac χ as a function of a χ at fixed µ/µ c is shown in Fig. 6. J χ is the total current in superconducting phase. J vac χ is a normal-phase persistentcurrent presenting in the soliton (superconductor or not), which can be read from normal Insulator m=0 m=1 R Χ a R Χ a Insulator Insulator m=0 Α/Equal 0.03 m=0 c m=0 m=1 m=1 m=0 phase for vanishing ψ . Thus, J 〈 O 〉 χ is the contribution due to the U (1)-breaking condensation. The position for every jump is the same as Fig. 5. Apart from a small range near a χ = 0, for a given a χ , the magnitude of the current increases with the increase of α , which means that the boundary system corresponding to large α is more sensitive to the response to Wilson line.", "pages": [ 10, 11, 12, 13 ] }, { "title": "V. CONCLUSIONS AND DISCUSSIONS", "content": "In this paper, we have studied the magnetic response of holographic superconductor in the Gauss-Bonnet gravity. Concretely we have studied the response to a Wilson line along the compact spatial direction of dual systems. As we know, the Gauss-Bonnet term effectively describes some kind of quantum correction in the bulk. According to AdS/CFT correspondence, such quantum correction maps to the 1 / N corrections in the boundary theory. It is interesting to investigate how much such quantum corrections change the whole picture. Our calculation shows that for a particular Gauss-Bonnet parameter, in the black brane background, the phase diagram and physical quantities , such as condensation and current, with different ( W,m ) but equal a χ -m/R , are degenerate, while in the soliton phase such degeneracy is uplifted. Thus, we find that including the Gauss-Bonnet term does not modify the qualitative features observed in Einstein theory. Although the Gauss-Bonnet term can not break the LP periodicity in black brane background, other physical quantities in the two holographic systems are undoubtedly modified or affected with the change of the Gauss-Bonnet parameter α , which is equivalent to tuning the strength of quantum corrections. We have analyzed the impact of the Gauss-Bonnet parameter α on the response of Wilson line in detail. More specifically, different Gauss-Bonnet parameters affect the involved physical quantities with the following fashion. The critical chemical potential µ c defined by transition point for vanishing a χ increases as one increases the Gauss-Bonnet parameter, which makes the condensation more difficult for vanishing magnetic flux. For a given a χ R , the smaller the Gauss-Bonnet parameter α is chosen, the bigger µ/µ c is required to trigger the phase transition. For the black brane case, the coherence length ξ 0 increases as we lower α . At fixed chemical potential compared to µ c corresponding to each α , the behaviors of condensation and current with respect to a χ (or equivalently to Wilson line W ) are much more different in the two systems with the change of the Gauss-Bonnet parameter α . In the black brane case, we have found that the magnitudes of the condensation and current increase as one lowers the parameter α , which means they are more sensitive to response to Wilson line for smaller α . On the other hand, in the soliton background case, the magnitude of such two physical quantities increases with the increase of α , indicating that the response to Wilson line is more insensitive for smaller α . Unlike the black brane case, there are jumps existing in the evolution of 〈 O 〉 and J 〈 O 〉 χ , which are due to the fact that the topological sectors labeled by ( W,m ) enter the effective field theory of the boundary theory. And the position of the jumps moves to larger chemical potential with the increase of the Gauss-Bonnet parameter α . Finally we point out that this work is done in the probe limit, the most direct improvement of the present analysis is to include the back reaction of the matter sector on the background geometry, but we expect the qualitative picture will not be changed.", "pages": [ 13, 14, 15 ] }, { "title": "Acknowledgments", "content": "We thank Song He and Shingo Takeuchi for helpful discussions, LFL would like to thank A. Salvio for quick correspondence. This work was supported in part by the National Natural Science Foundation of China (No.10821504, No.10975168 and No.11035008), and in part by the Ministry of Science and Technology of China under Grant No. 2010CB833004. LFL was supported by the National Natural Science Foundation of China with grant No.11205226 and China Postdoctoral Science Foundation with grant No. 2012M510563. Gauge/Gravity Duality,' JHEP 0910 , 067 (2009) [arXiv:0903.1864 [hep-th]]. Periodicity in Superconducting Loops,' Nat. Phys. 4 , 112 (2008). (2009) [arXiv:0906.2922 [hep-th]].", "pages": [ 15, 17, 18, 19 ] } ]

[ { "title": "Constraints on the Holographic Dark Energy Model from Type Ia Supernovae, WMAP7, Baryon Acoustic Oscillation and Redshift-Space Distortion", "content": "Lixin Xu ∗ Institute of Theoretical Physics, School of Physics & Optoelectronic Technology, Dalian University of Technology, Dalian, 116024, P. R. China and College of Advanced Science & Technology, Dalian University of Technology, Dalian, 116024, P. R. China In this paper, we use the joint measurement of geometry and growth rate from matter density perturbations to constrain the holographic dark energy model. The geometry measurement includes type Ia supernovae (SN Ia) Union2.1, full information of cosmic microwave background (CMB) from WMAP-7yr and baryon acoustic oscillation (BAO). For the growth rate of matter density perturbations, the results f ( z ) σ 8 ( z ) measured from the redshift-space distortion (RSD) in the galaxy power spectrum are employed. Via the Markov Chain Monte Carlo method, we try to constrain the model parameters space. The jointed constraint shows that c = 0 . 750 +0 . 0976+0 . 215+0 . 319 -0 . 0999 -0 . 173 -0 . 226 and σ 8 = 0 . 763 +0 . 0477+0 . 0910+0 . 120 -0 . 0465 -0 . 0826 -0 . 108 with 1 , 2 , 3 σ regions. After marginalizing the other irrelevant model parameters, we show the evolution of the equation of state of HDE with respect to the redshift z . Though the current cosmic data points favor a phantom like HDE Universe for the mean values of the model parameters in the future, it can behave like quintessence in 3 σ regions. PACS numbers: 98.80.-k, 98.80.Es Keywords: Holographic Dark Energy; Constraint", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The holographic principle says that the number of degrees of freedom in a bounded system should be finite and has relations with the area of its boundary [1]. By applying the so-called holographic principle to cosmology, one derives a relation between vacuum density and a cosmological scale ρ Λ = 3 c 2 M 2 pl L -2 [1-3], where c is a numerical constant and M pl is the reduced Planck Mass M -2 pl = 8 πG . The obtained vacuum energy, dubbed as holographic dark energy (HDE), can push our Universe into an accelerated expansion phase at late time [4, 5]. By taking different cosmological scale, for example the Hubble horizon [1, 2, 6], the event horizon or the particle horizon [3] as discussed by [1-3] and the Ricci scalar [7], one has different HDE model. Based on the idea that gravity as an entropic force [8], a similar DE density was given in [9] where a linear combination of H 2 and ˙ H was also presented, see also [10, 11]. Furthermore generalized HDE models ρ R = 3 c 2 M 2 pl Rf ( H 2 /R ) and ρ h = 3 c 2 M 2 pl H 2 g ( R/H 2 ) were also presented in Ref. [12]. In this paper, we consider the typical HDE model where the future event horizon is taken as a large cosmological scale, i.e. the IR cutoff L = R eh ( a ). This horizon is the boundary of the volume a fixed observer may eventually observe. This model has been confronted by cosmic observations extensively [13-15], for recent results, please see [16] and [17]. In the literature, to the best of our knowledge, only the geometry information which includes the luminosity distance d L from SN Ia, the angular diameter distance D A from BAO and the full information of CMB from WMAP-7yr were used to constrain this model, for examples please see [16] and [17]. As is well known, to discriminate the cosmological models the geometry information is not enough due to the degeneracies between model parameters. It means that different cosmological models can have the same background evolution history. However the dynamical evolution would be very different even if they have the same background evolution. Which is to say the dynamical evolution is important to break the possible degeneracy. Thanks to the measurement of the cosmic growth rate via the redshift-space distortion (RSD) which relates to the evolutionary speed of matter density contrast, now one can constrain the evolutions of the density contrast δ through f ( z ) σ 8 ( z ), where f ( z ) = d ln δ/d ln a is the growth rate of matter and σ 8 ( z ) is the rms amplitude of the density contrast at the comoving 8 h -1 Mpc scale. Here h is the normalized Hubble parameter H 0 = 100 h km sec -1 Mpc -1 . Here we should notice that the growth rate of structure f ( z ) has been used to constrain the dark energy model and to investigate the growth index in the literature, see [18] for examples. However, the observed values of the growth rate f obs = βb are derived from the redshift space distortion parameter β ( z ) and the linear bias b ( z ), where a particular fiducial ΛCDM model is used. It means that the current f obs data can only be used to test the consistency of ΛCDM model. This is the weak point of using f obs data points. Moreover, the measurements of the linear growth rate are degenerate with the bias b or clustering amplitude in the power spectra. To remove this weakness, Song & Percival proposed to use fσ 8 ( z ) which is almost model indepen- provides good test to dark energy models even without the knowledge of the bias or σ 8 [19]. Recently, the observed values of f ( z ) σ 8 ( z ) were provided by the 2dFGRS [20], WiggleZ [21], SDSS LRG [22], BOSS [23], and 6dFGRS [24]. The latest RSD data points were also summarized in [25]. For convenience, we show the data points used in this paper in Table I, see also Table 1 of Ref. [25]. So, the main motivation of this paper is to investigate the effect of model parameter c to fσ 8 ( z ) and to update our previous results by including the current observational data of RSD as well as SN Ia Union2.1, CMB and BAO on constraining the HDE model parameter space. This paper is structured as follows. In section II, we give a very brief review of the HDE model where the radiation is included and the future event horizon is adopted as an IR cut-off. The scalar perturbation evolution equations for a spatially flat FRW Universe will also be presented. In section III, the constraint methodology and results will be presented. We give a summary in section IV.", "pages": [ 1, 2 ] }, { "title": "II. BACKGROUND AND PERTURBATION EVOLUTION EQUATIONS", "content": "The energy density of the HDE is written as [3] The Friedmann equation for a spatially flat FRW universe reads where Ω i = ρ i / 3 M 2 pl H 2 are dimensionless energy densities for radiation, baryon, cold dark matter and HDE respectively. Here the scale factor a has been normalized to a 0 = 1 at present. Combining Eq. (1) and Eq. (2), one obtains the differential equation for Ω h [16] where ' denotes the derivative with respect to x = ln a and E ( x ) = √ Ω r 0 e -2 x +Ω b 0 e -x +Ω c 0 e -x . This equation describes the evolution of dimensionless energy density of HDE with the initial condition Ω h 0 = 1 -Ω r 0 -Ω b 0 -Ω c 0 . Via the conservation equation of the HDE ˙ ρ h + 3 H ( ρ h + p h ) = 0, one has the equation of state (EoS) of the HDE where the definition w h = p h /ρ h is used. In this paper, the HDE is taken as a perfect fluid with the EoS (5), then in the synchronous gauge the perturbation equations of density contrast and velocity divergence for the HDE are written as following the notations of Ma and Bertschinger [26], where the definition of the adiabatic sound speed is used. When the EoS of a pure barotropic fluid is negative, the imaginary adiabatic sound speed can cause instability of the perturbations. To overcome this problem, one can introduce an entropy perturbation and assume a positive or null effective speed of sound. Following the work of [27], the non adiabatic stress or entropy perturbation can be separated out which is gauge independent. In the rest frame of HDE, the entropy perturbation is specified as where c 2 s,eff is the effective speed of sound. Transforming into an arbitrary gauge gives a gauge-invariant form for the entropy perturbations. By using the Eqs (9,) (10) and (11), one can recast Eqs. (6), and (7) into For the HDE, we assume the shear perturbation σ h = 0 and the adiabatic initial conditions. Actually, the effective speed of sound c 2 s,eff is another freedom to describe the micro scale property of HDE in addition to the EoS [28]. And, we should take it as another free model parameter. The sound speed determines the sound horizon of the fluid via the equation l s = c s,eff /H . The fluid can be smooth or cluster below or above the sound horizon l s respectively. If the sound speed is smaller, the perturbation of the fluid can be detectable on large scale. And in turn the clustering fluid can influence the growth of density perturbations of matter, large scale structure and evolving gravitational potential which generates the integrated Sachs-Wolfe (ISW) effects. However, the authors of [28] have shown that current data can put no significant constraints on the value of the sound speed when dark energy is purely a recent phenomenon. For the HDE considered in this paper, it is related to the future event horizon and would not cluster. So we assume the effective speed of sound c 2 s,eff = 1 in this work.", "pages": [ 2, 3 ] }, { "title": "III. METHODOLOGY AND CONSTRAINT RESULTS", "content": "In our previous work [16], we have used the SN Ia Union2, BAO and full information of CMB from WMAP7yr to constrain the model parameter space, where the effects of model parameter c to the CMB power spectrum were also discussed. In Refs. [16], we showed that large values of c increase the tails of CMB power spectrum at large scale, i.e. l < 10, through the integrated SachsWolfe (ISW) effect. Here we will focus on its effects to the fσ 8 ( z ) caused by the different values of c . At first, we modify the CAMB package which is the publicly available code 1 for calculating the CMB power spectrum To obtain the model parameter space from currently available cosmic observations, we use the Markov Chain Monte Carlo (MCMC) method which is efficient in the case of more parameters case. We modified the pub- to include the HDE. We calculate the values of σ 8 at different redshift for the HDE model. We also write a subroutine to calculate the growth rate f ( z ) for the HDE model. The growth rate can be obtained by solving the following differential equation [29] where Here D ( a ) is the amplitude of the growing mode which connects to f ( a ) via the relation f ≡ d ln D/d ln a . Finally, we can obtain the values of fσ 8 ( z ) at different redshift z . To investigate the effects of c to fσ 8 ( z ), we borrow and fix the relevant cosmological values from our previous results obtained in [16] but take the model parameter c varying in a range. The evolution of fσ 8 ( z ) with respect to the redshift z for different values of c is shown in Figure 1. One can read off that the large values of c decrease and increase the values of fσ 8 ( z ) at higher and lower redshifts respectively from the Figure 1. It clues that the fσ 8 data points favor the values of model parameter c in a range of [0 . 69 , 0 . 9]. However, due to the sparseness and relative large error bars of the RSD data points, the current data sets of fσ 8 ( z ) may not give a much tight constraint to the model parameter space. licly available cosmoMC package 2 [30] to include the likelihood coming from the fσ 8 ( z ). We adopted the 7dimensional parameter space the priors for the model parameters are summarized in Table II. Furthermore, the hard-coded prior on the comic age 10Gyr < t 0 < 20Gyr is also imposed. Also, the physical baryon density ω b = Ω b h 2 = 0 . 022 ± 0 . 002 [31] from big bang nucleosynthesis and new Hubble constant H 0 = 74 . 2 ± 3 . 6kms -1 Mpc -1 [32] are adopted. The pivot scale of the initial scalar power spectrum k s 0 = 0 . 05Mpc -1 is used in this paper. The luminosity distance d L from SN Ia Uinon2.1 [33], the angular diameter distance D A and CMB power spectra from WMAP-7yr are used to fix the background evo- lutions. For the details, please see Appendix A. We ran eight chains on the Computational Cluster for Cosmos and stopped sampling when the worst e-values [the variance(mean)/mean(variance) of 1/2 chains] R -1 was of the order 0 . 01. The global fitting results are summarized in the Table II and the Figure 2. Comparing to our previous result c = 0 . 696 +0 . 0736+0 . 159+0 . 264 -0 . 0737 -0 . 132 -0 . 190 [16], we find that SN Union2.1 favors large values of model parameter c = 0 . 737 +0 . 0830+0 . 196+0 . 320 -0 . 0826 -0 . 148 -0 . 202 . When the RSD fσ 8 ( z ) is included, the values of model parameter c are increased to c = 0 . 750 +0 . 0976+0 . 215+0 . 319 -0 . 0999 -0 . 173 -0 . 226 which confirms the analysis as shown in Figure 1. To show the effects of RSD data points fσ 8 ( z ) to constrain the model parameters space, the 2D contour for model parameter Ω m -c is also plotted in Figure 3. From this figure, one can read that the region of Ω m is shrunk when fσ 8 ( z ) data points are employed. But in this case, the 1 , 2 , 3 σ regions of c are enlarged. And the 2D contour diagram moves little to the top right corner direction when fσ 8 ( z ) data points are included. With the mean values listed in the Table II for the case of SN+BAO+CMB+RSD, we plotted the evolutions of the EoS of HDE with respect to the redshift z in Figure 4, where the shadows denote the 1 , 2 , 3 σ regions from the dark to the light respectively. For calculating the 1 σ region, we consider the propagation of the errors for w ( z ) and marginalize the other irrelevant model parameters by the Fisher matrix analysis [34, 35]. If the other irrelevant model parameters are not marginalized, the error bars will be underestimated. The errors are calculated by using the covariance matrix C ij of the fitting model parameters which is an output of cosmoMC . The errors for a function f = f ( θ ) in terms of the variables θ are", "pages": [ 3, 4, 5 ] }, { "title": "IV. SUMMARY", "content": "In this paper, we updated our previous results obtained in Ref. [16] with the replacement of SN Union2 by SN given via the formula [35-37] where n is the number of the variables. In our case, f would be the EoS w ( z ; θ i ) for HDE. And the variables θ i are (Ω b h 2 , Ω c h 2 , c ) for the HDE model. The corresponding 1 σ errors for w ( z ) are given by where ¯ θ are the mean values of the constrained model parameters. For a relative large values of c , the HDE behaves like quintessence at present ( w h | z =0 = -0 . 971 ± 0 . 0777 with 1 σ error). In 2 σ regions, it still has broad space to behave like phantom even in the future. But in 3 σ region, it has the possiblity to behave like quintessence. So based on this result, we still do not know our Universe will be terminated by a cosmic doomsday or not in 3 σ region. Union2.1 and with the addition of RSD data points of fσ 8 ( z ). We showed the effects of model parameter c to fσ 8 ( z ) by fixing the other relevant model parameters and found out that RSD fσ 8 ( z ) data points favor larger values of c . But due to the sparseness and relative large error bars of the RSD data points, the current data sets of fσ 8 ( z ) cannot give a much tight constraint to the model parameter c . A global fitting to the HDE model was performed by combining the full information of CMB from WMAP-7yr, BAO, SN Union2.1, with and without RSD fσ 8 ( z ) data sets via the MCMC method. The results show that RSD data points fσ 8 ( z ) can shrink the model parameter space Ω m efficiently as shown in Figure 3 but cannot constrain the model parameter c very well. When the RSD fσ 8 ( z ) data points are added, the 2D contour diagram moves little to the top right corner direction on the 2D Ω m -c plane as shown in Figure 3. It means that the RSD fσ 8 ( z ) data points favor larger values of c and Ω m . It confirms our previous analysis as shown in Figure 1. To show the evolution of the EoS with errors for HDE with respect to the redshift z , we should margnialize the other irrelevant model parameters. If not the error bars will be under estimated. We marginalized the other irrelevant model parameters by the Fisher matrix analysis. And the evolution of the EoS for HDE in 3 σ region was plotted in Figure 4 by adopting the mean values as shown in Table II. In this figure one can see that HDE behaves like quintessence at present ( w h | z =0 = -0 . 971 ± 0 . 0777 with 1 σ error). In 2 σ region, it has a wide region to behave like phantom. But in 3 σ region, it has possiblities to behave like quintessence. Then one still cannot conclude whether the future Universe will terminated by a cosmic doomsday or not in 3 σ region.", "pages": [ 5, 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "The author thanks an anonymous referee for helpful improvement of this paper. L. Xu's work is supported in part by NSFC under the Grants No. 11275035 and 'the Fundamental Research Funds for the Central Universities' under the Grants No. DUT13LK01. m", "pages": [ 6, 7 ] }, { "title": "Appendix A: SN Ia Union2.1, BAO and CMB", "content": "For the SN Ia, the Uinon2.1 [33] data sets will be used in this paper. The distance modulus µ ( z ) is defined as where ¯ d L ( z ) is the Hubble-free luminosity distance H 0 d L ( z ) /c = H 0 d A ( z )(1 + z ) 2 /c , with H 0 the Hubble constant, and µ 0 ≡ 42 . 38 -5 log 10 h through the renormalized quantity h as H 0 = 100 h km s -1 Mpc -1 . Where d L ( z ) is defined as where E 2 ( z ) = H 2 ( z ) /H 2 0 . Additionally, the observed distance moduli µ obs ( z i ) of SN Ia at z i are where M is their absolute magnitudes. For the SN Ia dataset, the best fit values of the parameters p s can be determined by a likelihood analysis, based on the calculation of where M ' ≡ µ 0 + M is a nuisance parameter which includes the absolute magnitude and the parameter h . The nuisance parameter M ' can be marginalized over analytically [38] as resulting to with where Cov -1 ij is the inverse of covariance matrix with or without systematic errors. One can find the details in Ref. [33] and the web site 3 where the covariance matrix with or without systematic errors are included. Relation (A5) has a minimum at the nuisance parameter value M ' = B/C , which contains information of the values of h and M . Therefore, one can extract the values of h and M provided the knowledge of one of them. Finally, the expression which coincides to Eq. (A6) up to a constant, is often used in the likelihood analysis [38, 39]. Thus in this case the results will not be affected by a flat M ' distribution. It worths noting that the results will be different with or without the systematic errors. In this work, all results are obtained with systematic errors. For BAO data sets, we used the observational results d obs z from SDSS DR7 [40] and A ( z ) from WiggleZ [21]. The observed values of d obs z are gathered in Table A. where d z ≡ r s ( z d ) /D V ( z ), r s ( z d ) is the comoving sound horizon at the baryon drag epoch, D V ( z ) ≡ [(1 + z ) 2 D 2 A cz/H ( z )] 1 / 3 [41, 42]. Here D A ( z ) the angular di- er distance which is defined as For the SDSS DR7 data points, the χ 2 SDSS ( P ) is given as where C -1 is the inverse covariance matrix To calculate r s ( z d ), one needs to know the redshift z d at decoupling epoch and its corresponding sound horizon. We obtain the baryon drag epoch redshift z d numerically from the following integration [43] where R = 3 ρ b / 4 ρ γ , σ T is the Thomson cross-section and x e ( z ) is the fraction of free electrons. Then the sound horizon is where c s = 1 / √ 3(1 + R ) is the sound speed. Also, to obtain unbiased parameter and error estimates, we use the substitution [43] where d z = r s (˜ z d ) /D V ( z ), ˆ r s is evaluated for the fiducial cosmology of Ref. [40], and ˜ z d is redshift of drag epoch obtained by using the fitting formula [44] for the fiducial cosmology For WiggleZ data points, one calculates acoustic parameter A ( z ) introduced by Eisenstein et al. [41] The observed values of A ( z ) are gathered in Table A The corresponding χ 2 WiggleZ is given as z A 0.44 0 . 474 0.60 0 . 442 0.73 0 . 424 obs ( z ) survey and reference ± ± ± 0 . 034 WiggleZ [21] 0 . 020 WiggleZ [21] 0 . 021 WiggleZ [21] where C -1 is the inverse covariance matrix Then the total χ 2 BAO from BAO is written as For the fσ 8 ( z ), the χ 2 fσ 8 ( P ) is given For CMB data set, the temperature power spectrum from WMAP 7-year data 4 [45] are employed. Then one has the total likelihood L ∝ e -χ 2 / 2 , where χ 2 is given as which is used to get the distribution of the model parameter space.", "pages": [ 7, 8, 9 ] } ]

[ { "title": "Interpolation in waveform space: enhancing the accuracy of gravitational waveform families using numerical relativity", "content": "Kipp Cannon, 1, ∗ J.D. Emberson, 1, 2, † Chad Hanna, 3, ‡ Drew Keppel, 4, 5, § and Harald P. Pfeiffer 1, ¶ 1 Canadian Institute for Theoretical Astrophysics, 60 St. George Street, University of Toronto, Toronto, ON M5S 3H8, Canada 2 Department of Astronomy and Astrophysics, 50 St. George Street, University of Toronto, Toronto, ON M5S 3H4, Canada 3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 4 Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-30167 Hannover, Germany 5 Leibniz Universitat Hannover, D-30167 Hannover, Germany Matched-filtering for the identification of compact object mergers in gravitational-wave antenna data involves the comparison of the data stream to a bank of template gravitational waveforms. Typically the template bank is constructed from phenomenological waveform models since these can be evaluated for an arbitrary choice of physical parameters. Recently it has been proposed that singular value decomposition (SVD) can be used to reduce the number of templates required for detection. As we show here, another benefit of SVD is its removal of biases from the phenomenological templates along with a corresponding improvement in their ability to represent waveform signals obtained from numerical relativity (NR) simulations. Using these ideas, we present a method that calibrates a reduced SVD basis of phenomenological waveforms against NR waveforms in order to construct a new waveform approximant with improved accuracy and faithfulness compared to the original phenomenological model. The new waveform family is given numerically through the interpolation of the projection coefficients of NR waveforms expanded onto the reduced basis and provides a generalized scheme for enhancing phenomenological models. PACS numbers: 04.30.-w, 04.25.D-, 04.25.dg", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Developments are currently underway to promote the sensitivity of LIGO and to improve its prospect for detecting gravitational waves emitted by compact object binaries [1, 2]. Of particular interest are the detection of gravitational waves released during the inspiral and merger of binary black hole (BBH) systems. Detection rates for BBH events are expected to be within 0.4-1000 per year with Advanced LIGO [3]. It is important that rigorous detection algorithms be in place in order to maximize the number of detections of gravitational wave signals. The detection pipeline currently employed by LIGO involves a matched-filtering process whereby signals are compared to a pre-constructed template bank of gravitational waveforms. The templates are chosen to cover some interesting region of mass-spin parameter space and are placed throughout it in such a way that guarantees some minimal match between any arbitrary point in parameter space and its closest neighbouring template. Unfortunately, the template placement strategy generally requires many thousands of templates (e.g. [4]) evaluated at arbitrary mass and spin; something that cannot be achieved using the current set of numerical relativity (NR) waveforms. To circumvent this issue, LIGO exploits the use of analytical waveform families like phenomenological models [5, 6] or effective-one-body models [7, 8]. We shall focus here on the Phenomenological B (PhenomB) waveforms developed by [6]. This waveform family describes BBH systems with varying masses and aligned-spin magnitudes (i.e. non-precessing binaries). The family was constructed by fitting a parameterized model to existing NR waveforms in order to generate a full inspiral-mergerringdown (IMR) description as a function of mass and spin. The obvious appeal of the PhenomB family is that it allows for the inexpensive construction of gravitational waveforms at arbitrary points in parameter space and can thus be used to create arbitrarily dense template banks. To optimize computational efficiency of the detection process it is desirable to reduce the number of templates under consideration. A variety of reduced bases techniques have been developed, either through singularvalue decomposition (SVD) [9, 10], or via a greedy algorithm [11]. SVD is an algebraic manipulation that transforms template waveforms into an orthonormal basis with a prescription that simultaneously filters out any redundancies existing within the original bank. As a result, the number of templates required for matched-filtering can be significantly reduced. In addition, it has been shown in [12] that, upon projecting template waveforms onto the orthonormal basis produced by the SVD, interpolating the projection coefficients provides accurate approximations of other IMR waveforms not included in the original template bank. In this paper, we continue to explore the use of the interpolation of projection coefficients. We take a novel approach that utilizes both the analytic PhenomB waveform family [6] and NR hybrid waveforms [13-15]. We apply SVD to a template bank constructed from an analytical waveform family to construct an orthonormal basis spanning the waveforms, then project the NR waveforms onto this basis and interpolate the projection coefficients to allow arbitrary waveforms to be constructed, thereby obtaining a new waveform approximant. We show here that this approach improves upon the accuracy of the original analytical waveform family. The original waveform family shows mismatches with the NR waveforms as high as 0 . 1 when no extremization over physical parameters is applied (i.e., a measure of the 'faithfulness' of the waveform approximant), and mismatches of 0 . 02 when maximized over total mass (i.e., a measure of the 'effectualness' of the waveform approximant). With our SVD accuracy booster, we are able to construct a new waveform family (given numerically) with mismatches < 0 . 005 even without extremization over physical parameters. This paper is organized as follows. We begin in Section II where we provide definitions to important terminology used in our paper. We then compare our NR hybrid waveforms to the PhenomB family and show that a mass-bias exists between the two. In Section III we present our SVD accuracy booster applied to the case study of equal-mass, zero-spin binaries. In Section IV we investigate the feasibility of extending this approach to include unequal-mass binaries. We finish with concluding remarks in Section V.", "pages": [ 1, 2 ] }, { "title": "A. Terminology", "content": "A gravitational waveform is described through a complex function, h ( t ), where real and imaginary parts store the sine and cosine components of the wave. The specific form of h ( t ) depends on the parameters of the system, in our case the total mass M = m 1 + m 2 and the massratio q = m 1 /m 2 . While h ( t ) is a continuous function of time, we discretize by sampling h ( t i ), where the sampling times t i have uniform spacing ∆ t = 2 -15 s. We shall also whiten any gravitational waveform h ( t ). This processes is carried out in frequency space via where S n ( f ) is the LIGO noise curve and ˜ h ( f ) is the Fourier transform of h ( t ). The whitened time-domain waveform, h w ( t ), is obtained by taking the inverse Fourier transform of (1). In the remainder of the paper, we shall always refer to whitened waveforms, dropping the subscript 'w'. For our purposes it suffices to take S n ( f ) to be the Initial LIGO noise curve. Using the Advanced LIGO noise curve would only serve to needlessly complicate our approach by making waveforms longer in the low frequency domain. As a measure of the level of agreement between two waveforms, h ( t ) and g ( t ), we will use their match, or overlap, O ( h , g ) [16-18]. We define where 〈 h , g 〉 is the standard complex inner product and the norm || h || ≡ √ 〈 h , h 〉 . We always consider the overlap maximized over time- and phase-shifts between the two waveforms. The time-maximization is indicated in (2), and the phase-maximization is an automatic consequence of the modulus. Note that 0 ≤ O ( h , g ) ≤ 1. For discrete sampling at points t i = t 0 + i ∆ t we have that where g ∗ ( t ) is the complex conjugate of g ( t ). Without whitening, (3) would need to be evaluated in the frequency domain with a weighting factor 1 /S n ( f ). The primary advantage of (3) is its compatibility with formal results for the SVD, which will allow us to make more precise statements below. When maximizing over timeshifts ∆ T , we ordinarily consider discrete time-shifts in integer multiples of ∆ t , as this avoids interpolation. After the overlap has been maximized, it is useful to speak in terms of the mismatch, M ( h , g ), defined simply as We use this quantity throughout the paper to measure the level of disagreement between waveforms.", "pages": [ 2 ] }, { "title": "B. NR Hybrid Waveforms", "content": "We use numerical waveforms computed with the Spectral Einstein Code SpEC [19]. Primarily, we use the 15orbit equal-mass (mass-ratio q = 1), zero-spin (effective spin χ = 0) waveform described in [13, 20]. In Section IV, we also use unequal mass waveforms computed by [14]. The waveforms are hybridized with a TaylorT3 post-Newtonian (PN) waveform as described in [15, 21] at matching frequencies Mω = 0 . 038 , 0 . 038 , 0 . 042 , 0 . 044 , and 0 . 042 for mass-ratios q = 1 , 2 , 3 , 4 , and 6, respectively. TaylorT4 at 3.5PN order is known to match NR simulations exceedingly well for equal-mass, zero-spin BBH systems [20] (see also Fig. 9 of [15]). For q =1, a TaylorT3 hybrid is very similar to a TaylorT4 hybrid, cf. Figure 12 of [15]. The mismatch between TaylorT3 and TaylorT4 hybrids is below 10 -3 at M = 10 M glyph[circledot] , dropping to below 10 -4 for 15 M glyph[circledot] ≤ M ≤ 20 M glyph[circledot] , and 10 -5 for 20 M glyph[circledot] ≤ M ≤ 100 M glyph[circledot] . These mismatches are significantly smaller than mismatches arising in the study presented here, so we conclude that our results are not influenced by the accuracy of the utilized q = 1 PN-NR hybrid waveform. For higher mass-ratios, the PN-NR hybrids have a larger error due to the post-Newtonian waveform [21]. The error-bound on the hybrids increases with mass-ratio, however, is mitigated in our study here, because we use the q ≥ 2 hybrids only for total mass of 50 M glyph[circledot] , where less of the post-Newtonian waveform is in band. Because NR simulations are not available for arbitrary mass ratios, we will primarily concentrate our investigation to the equal-mass and zero-spin NR hybrid waveforms described above. The full IMR waveform can be generated at any point along the q = 1 line through a simple rescaling of amplitude and phase with total mass M of the system. Despite such a simple rescaling, the q = 1 line lies orthogonal to lines of constant chirp mass [22], therefore tracing a steep gradient in terms of waveform overlap, and encompassing a large degree of waveform structure.", "pages": [ 2, 3 ] }, { "title": "C. PhenomB Waveforms", "content": "Since our procedure for constructing an orthonormal basis begins with PhenomB waveforms, let us now investigate how well these waveforms model the NR waveforms to be interpolated. For this purpose, we adopt the notation h NR ( M ) and h PB ( M ) to represent NR and PhenomB waveforms of total mass M , respectively. We quantify the faithfulness of the PhenomB family by computing the mismatch M [ h NR ( M ) , h PB ( M )] as a function of mass. The result of this calculation for 10 M glyph[circledot] ≤ M ≤ 100 M glyph[circledot] is shown as the dashed curve in the top panel of Figure 1. The mismatch starts off rather high with M≈ 0 . 1 at 10 M glyph[circledot] and then slowly decreases as the mass is increased, until eventually flattening to M≈ 0 . 005 at high mass. The mismatch between NR and PhenomB waveforms can be reduced by optimizing over a mass-bias. This is accomplished by searching for the mass M ' for which the mismatch M [ h NR ( M ) , h PB ( M ' )] is a minimum. The result of this process is shown by the solid line in the top panel of Figure 1. Allowing for a mass bias significantly reduces the mismatch for M glyph[lessorsimilar] 50 M glyph[circledot] . The mass M ' that minimizes mismatch is generally smaller than the mass M of our NR 'signal' waveform, M ' < M over almost all of the mass range considered. Apparently, PhenomB waveforms are systematically underestimating the mass of the 'true' NR waveforms, at least along the portion of parameter space considered here. The solid line in the bottom panel of Figure 1 plots the relative mass-bias, ( M -M ' ) /M . At 10 M glyph[circledot] this value is 0 . 3%, and it rises to just above 1% for 30 M glyph[circledot] . 1 It is useful to consider how this mass bias compares to the potential parameter estimation accuracy in an early detection. For a signal with a matched-filter signalto-noise ratio (SNR) of 8 - characteristic of early detection scenarios - template/waveform mismatches will influence parameter estimation when the mismatch is M ≥ 8 -2 / 2 ∼ 0 . 01 [23]. Placing a horizontal cut on the top panel of Figure 1 at M = 8 -2 / 2, we see that for M glyph[greaterorsimilar] 40 M glyph[circledot] PhenomB waveform errors have no observational consequence; for 15 M glyph[circledot] glyph[lessorsimilar] M glyph[lessorsimilar] 40 M glyph[circledot] a PhenomB waveform with the wrong mass will be the best match for the signal. For M glyph[lessorsimilar] 15 M glyph[circledot] the missmatch between equal-mass PhenomB waveforms and NR (when optimizing over mass) grows to ∼ 0 . 02. Optimization over mass-ratio will reduce this mismatch, but we have not investigated to what degree.", "pages": [ 3 ] }, { "title": "A. PhenomB Template Bank", "content": "We aim to construct an orthonormal basis via the SVD of a bank of PhenomB template waveforms, and then interpolate the coefficients of NR waveforms projected onto this basis to generate a waveform family with improved NR faithfulness. The first step is to construct a template bank of PhenomB waveforms, with attention restricted to equal-mass, zero-spin binaries. An advantage of focusing on the q = 1 line is that template bank construction can be simplified by systematically arranging templates in ascending order by total mass. With this arrangement we define a template bank to consist of N PhenomB waveforms, labelled g i ≡ h PB ( M i ) ( i = 1 , 2 , . . . , N ), with M i +1 > M i and with adjacent templates satisfying the relation: where O ' is the desired overlap between templates and ε is some accepted tolerance in this value. The template bank is initiated by choosing a lower mass bound M 1 = M min and assigning g 1 = h PB ( M 1 ). Successive templates are found by sequentially moving toward higher mass in order to find waveforms satisfying (5) until some maximum mass M max is reached. Throughout each trial, overlap between waveforms is maximized continuously over phase shifts and discretely over time shifts. For template bank construction we choose to refine the optimization over time by considering shifts in integer multiples of ∆ t/ 100. We henceforth refer to our fiducial template bank which employs the parameters M min = 15 M glyph[circledot] , M max = 100 M glyph[circledot] , O ' = 0 . 97, and ε = 10 -12 . The lower mass bound was chosen in order to obtain a reasonably sized template bank containing N = 127 waveforms; pushing downward to 10 M glyph[circledot] results in more than doubling the number of templates. Template waveforms each have a duration of 8 s and are uniformly sampled at ∆ t = 2 -15 s (a sample frequency of 32768 Hz). 508 MiB of memory is required to store this template bank using doubleprecision waveforms.", "pages": [ 3, 4 ] }, { "title": "B. Representation of Waveforms in a Reduced SVD Basis", "content": "The next step is to transform the template waveforms into an orthonormal basis. Following the presentation in [9], this is achieved by arranging the templates into the rows of a matrix G and factoring through SVD to obtain where U and V are orthogonal matrices and Σ is a diagonal matrix whose non-zero elements along the main diagonal are referred to as singular values. The SVD for G is uniquely defined as long as the singular values are arranged in descending order along the main diagonal of Σ . The end result of (6) is to convert the N complexvalued templates into 2 N real-valued orthonormal basis waveforms. The k th basis waveform, u k , is stored in the k th row of U , and associated with this mode is the singular value, σ k , taken from the k th element along the main diagonal of Σ . One of the appeals of SVD is that the singular values rank the basis waveforms with respect to their ability to represent the original templates. This can be exploited in order to construct a reduced basis that spans the space of template waveforms to some tolerated mismatch. For instance, suppose we choose to reduce the basis by considering only the first N ' < 2 N basis modes while discarding the rest. Template waveforms can be represented in this reduced basis by expanding them as the sum where µ k are the complex-valued projection coefficients, The prime in (7) is used to stress that the reduced basis is generally unable to fully represent the original template. 2 It was shown in [9] that the mismatch expected from reducing the basis in this way is Given Σ , (9) can be inverted to determine the number of basis waveforms, N ' , required to represent the original templates for some expected mismatch 〈M〉 . (9) provides a useful estimate to the mismatch in represeting templates from a reduced SVD basis. In order to investigate its accuracy, however, we should compute the mismatch explicitly for each template waveform. Using the orthonormality condition 〈 u j , u k 〉 = δ jk , it is easy to show from (7) that the mismatch between the template and its projection can be expressed in terms of the projection coefficients: This quantity is minimized continuously over phase and discretely over time shifts in integer multiples of ∆ t . Choosing 〈M〉 = 10 -6 , (9) predicts that N ' = 123 of the 2 N = 254 basis waveforms from our fiducial template bank are required to represent the templates to the desired accuracy. In Figure 2 we compare the expected mismatch of 10 -6 to the actual mismatches computed from (10) for each PhenomB waveform in the template bank. The open squares in this plot show that the actual template mismatch has a significant amount of scatter about 〈M〉 , but averaged over a whole remains well bounded to the expected result. The PhenomB template waveforms can thus be represented to a high degree from a reasonably reduced SVD basis. We are of course more interested in determining how well NR waveforms can be represented by the same reduced basis of PhenomB waveforms. Since NR and PhenomB waveforms are not equivalent, (9) cannot be used to estimate the mismatch obtained when projecting NR waveforms onto the reduced basis. We must therefore compute their representation mismatch explicitly. A general waveform, h , can be represented by the reduced basis in analogy to (7) by expressing it as the sum: where µ k = 〈 h , u k 〉 . As before, the represented waveform h ' will in general be neither normalized nor equivalent to the original waveform h . The mismatch between them is where we remind the reader that we always minimize over continuous phase shifts and discrete time shifts of the two waveforms. In Figure 2 we use open circles to plot the representation mismatch of NR waveforms evaluated at the same set of masses M i from which the PhenomB template bank was constructed. We see that NR waveforms can be represented in the reduced basis with a mismatch less than 10 -3 over most of the template bank boundary. This is about a factor of five improvement in what can be achieved by using PhenomB waveforms optimized over mass. Since NR waveforms were not originally included in the template bank, and because a mass-bias exists between the PhenomB waveforms which were included, we can expect that the template locations have no special meaning to NR waveforms. This is evident from the thin dashed line which traces the NR representation mismatch for masses evaluated between the discrete templates. This line varies smoothly across the considered mass range and exhibits no special features at the template locations. This is in contrast to the thin solid line which traces PhenomB representation mismatch evaluated between templates. In this case, mismatch rises as we move away from one template and subsequently falls back down as the next template is approached. The representation tolerance 〈M〉 of the SVD is a free parameter, which so far, we have constrained to be 〈M〉 = 10 -6 . When this tolerance is varied, we observe the following trends: (i) PhenomB representation mismatch generally follows 〈M〉 ; (ii) NR representation mismatch follows 〈M〉 at first and then saturates to a minimum as the representation tolerance is continually reduced. These trends are observed in Figure 3 where we plot NR and PhenomB representation mismatch averaged over the mass boundary of the template bank evaluated both at and between templates. The saturation in NR representation mismatch occurs when the reduced basis captures all of the NR waveform structure contained within the PhenomB basis. Reducing the basis further hits a point of diminishing returns as the increased computational cost associated with a larger basis outweighs the benefit of marginally improving NR match.", "pages": [ 4, 5 ] }, { "title": "C. Interpolation of NR Projection Coefficients", "content": "We now wish to examine the possibility of using the reduced SVD basis of PhenomB template waveforms to construct a new waveform family with improved NR representation. The new waveform family would be given by a numerical interpolation of the projection coefficients of NR waveforms expanded onto the reduced basis. Here we test this using the fiducial template bank and reduced basis described above. The approach is to sample NR projection coefficients, µ k ( x i ), at some set of locations, x i , and then perform an interpolation to obtain the continuous function µ ' k ( x ) that can be evaluated for arbitrary x . The accuracy of the interpolation scheme is maximized by finding the space for which µ k ( x i ) are smooth functions of x . It is reasonable to suppose that the projection coefficients will vary on a similar scale over which the waveforms themselves vary. Hence, a suitable space to sample along is the space of constant waveform overlap. We define this to be the space x = [ -1 , 1] for which the physical template masses are mapped according to: Moving a distance ∆ x = 2 / ( N -1) in this space is thus equivalent to moving a distance equal to the overlap between adjacent templates. In this space, we find the real and imaginary components, R µ k ( x ) and I µ k ( x ), of the complex projection coefficients to be oscillatory functions that can roughly be described by a single frequency. This behaviour is plotted for the basis modes k = 1, 50, and 123 in Figure 4. Another trend observed in this plot is that the projection coefficients become increasingly complex (i.e. show less structure) for higher-order modes. This is a direct result of the increasing complexity of higher-order basis waveforms themselves. We find that the low-order waveforms are smoothest while the high-order modes feature many of the irregularities associated with the multiple frequency components and merger features of the templates. Though they are more complex, higher-order modes have smaller singular values and are therefore less important in representing waveforms. This is evident from the steady decline in amplitude of the projection coefficients when moving down the different panels of Figure 4. We shall use Chebyshev polynomials to interpolate the projection coefficients. These are a set of orthogonal functions where the j th Chebyshev polynomial is defined as The orthogonality of Chebyshev polynomials can be exploited to perform an n th order Chebyshev interpolation by sampling µ k ( x ) at the n +1 so-called collocation points given by the Gauss-Lobatto Chebyshev nodes [24] for i = 0 , 1 , . . . , n . In general, the interpolation will not be exact and some residual, r k ( M ), will be introduced: Here µ k ( M ) is the actual coefficient of h NR ( M ) projected onto the basis waveform u k , and µ ' k ( M ) is the coefficient obtained after interpolation. The new waveform family is expressed numerically as a function of mass through the relation where the subscript 'intp' reminds the reader that this is computed from an interpolation over µ k . An interpolated waveform of total mass M can be compared to the original NR waveform (which we consider to be the 'true' signal), where the latter is expressed as with h ⊥ ( M ) denoting the component of h NR ( M ) that is orthogonal to the SVD basis (i.e. orthogonal to all PhenomB waveforms in the template bank). h NR ( M ) differs from h intp ( M ) by an amount To compute the impact of the various approximations influencing (20), we calculate the overlap between the interpolated waveform, and the exact waveform, O [ h intp ( M ) , h NR ( M )]. To begin this calculation, it is useful to consider the square of the overlap, where we have dropped the explicit mass-dependence and subscripts for convenience. Using 〈 h , h 〉 = 1 and Taylorexpanding the right-hand-side of (21) to second order in δ h , we find To second order in δ h , the mismatch is therefore We note that the right-hand-side of (23) can be written as 1 2 〈 δ h ⊥ , δ h ⊥ 〉 , where δ h ⊥ is the part of δ h orthogonal to h , However, for simplicity, we proceed by dropping the last term in (23): Using (20), this gives We thus see three contributions to the total mismatch: (i) the interpolation error, ∑ N ' k =1 | r k ( M ) | 2 ; (ii) the truncation error from the discarded waveforms of the reduced basis, ∑ 2 N k = N ' +1 | µ k ( M ) | 2 ; (iii) the failure of the SVD basis to represent the NR waveform, | h ⊥ | . The sum of the last two terms, which together make up the representation error, is traced by the dashed line in Figure 2. The goal for our new waveform family is to have an interpolation error that is negligible compared to the representation error. To remove the mass-dependence of interpolation error in (26), we introduce the maximum interpolation error of each mode, This allows the bound to place an upper limit on the error introduced by interpolation. Figure 5 plots R 2 k as a function of mode-number k as well as the cumulative sum ∑ N ' k =1 R 2 k / 2. The data pertains to an interpolation performed using n = 175 Chebyshev polynomials on the reduced SVD basis containing the frist N ' = 123 of 2 N = 254 waveforms. In this case, we find the interpolation error to be largely dominated by the lowest-order modes and also partially by the highest-order modes. Interpolated coefficients for various modes are plotted in Figure 4 and help to explain the features seen in Figure 5. In the first place, interpolation becomes increasingly more difficult for higher-order modes due to their increasing complexity. This problem is mitigated by the fact that high-order modes are less important for representing waveforms, as evidenced by the diminishing amplitude of projection coefficients. Although low-order modes are much smoother and thus easier to interpolate, their amplitudes are considerably larger meaning that interpolation errors are amplified with respect to high-order modes. (26) summarizes the three components adding to the final mismatch of our interpolated waveform family. Their total contribution can be computed directly from the interpolated coefficients in a manner similar to (12): In the case of perfect interpolation for which µ ' k ( M ) = µ k ( M ), (18) and (29) reduce to (11) and (12) respectively, and the total mismatch is simply the representation error of the reduced basis. In Figure 6 open circles show the total mismatch (29) between our interpolated waveform family and the true NR waveforms for various masses. Also plotted is the NR representation error without interpolation and the mismatch between NR and PhenomB waveforms minimized over mass. We see that interpolation introduces only small additional mismatch to the interpolated waveform family, and remains well below the optimized NRPhenomB mismatch. This demonstrates the efficacy of using SVD coupled to NR waveforms to generate a faithful waveform family with improved accuracy over the effectual PhenomB family that was originally used to create templates. This represents a general scheme for improving phenomenological models and presents an interesting new opportunity to enhance the matched-filtering process employed by LIGO.", "pages": [ 6, 7, 8 ] }, { "title": "IV. HIGHER DIMENSIONS", "content": "So far, we have focused on the total mass axis of parameter space. As already discussed, this served as a convenient model-problem, because the q = 1 NR waveform can be rescaled to any total mass, so that we are able to compare against the 'correct' answer. The natural extension of this work is to expand into higher dimensions where NR waveforms are available only at certain, discrete mass-ratios q . In this section we consider expanding our approach of interpolating NR projection coefficients from a two-dimensional template bank containing unequal-mass waveforms. We compute a template bank of PhenomB waveforms covering mass-ratios q from 1 to 6 and total masses 50 M glyph[circledot] ≤ M ≤ 70 M glyph[circledot] . This mass range is chosen to facilitate comparison with previous work done by [12]. For the two-dimensional case the construction of a template bank is no longer as straightforward as before due to the additional degree of freedom associated with varying q . One method that has been advanced for this purpose is to place templates hexagonally on the waveform manifold [25]. Using this procedure we find N = 16 templates are required to satisfy a minimal match of 0.97. Following the waveform preparation of [12], templates are placed in the rows of a matrix G with real and imaginary components filled in alternating fashion where the whitened waveforms are arranged in such a way that their peak amplitudes are aligned. The waveforms are sampled for a total duration of 2 s with uniform spacing ∆ t = 2 -15 s so that 16 MiB of memory is required to store the contents of G if double precision is desired. Application of (6) transforms the 16 complex-valued waveforms into 32 real-valued orthonormal basis waveforms. The aim is to sample the coefficients of NR waveforms projected onto the SVD basis of PhenomB waveforms using mass-ratios for which NR data exists, and then interpolate amongst these to construct a numerical waveform family that can be evaluated for arbitrary parameters. This provides a method for evaluating full IMR waveforms for mass-ratios that have presently not been simulated. To summarize, we take some NR waveform, h NR ( M,q ), or total mass M and mass-ratio q , and project it onto the basis waveform u k in order to obtain Next we apply some two-dimensional interpolation scheme on (30) to construct continuous functions µ ' k ( M,q ) that can be evaluated for arbitrary values of M and q bounded by the regions of the template bank. The interpolated waveform family is given numerically by the form: As before, the interpolation process works best if we can develop a scheme for which the projection coefficients /circledot /circledot /circledot are smoothly varying functions of M and q . Following the procedure described in [12], the complex phase of the first mode is subtracted from all modes: To motivate why (32) might be useful, let us consider modifying the PhenomB waveform family with a parameter-dependent complex phase Φ( M,q ): When constructing a template bank, or when using a template bank, such a complex phase Φ( M,q ) is irrelevant, because the waveforms are always optimized over a phase-shift. However, Φ( M,q ) will appear in the projection coefficients, (30), Therefore, if one had chosen a function Φ( M,q ) with fine-scale structure, this structure would be inherited by the projection coefficients µ k ( M,q ). For traditional uses of waveform families the overall complex phase Φ( M,q ) is irrelevant, and therefore, little attention may have been paid to how it varies with parameters ( M,q ). The transformation (32) removes the ambiguity inherent in Φ( M,q ) by choosing it such that arg ˜ µ 1 ( M,q ) = 0. This choice ties the complex phase to the physical variations of the µ 1 coefficient, and does therefore eliminate all unphysical phase-variations on finer scales. In the leftmost panels of Figure 7 we plot the real part of the smoothed coefficients ˜ µ k ( M,q ) for PhenomB waveforms projected onto the basis modes k = 3 and k = 16. The middle panels show the same thing except using the NR waveforms evaluated at the set of mass-ratios q = { 1, 2, 3, 4, 6 } for which we have simulated waveforms. Obviously, the refinement along the q axis is much finer for the PhenomB waveforms since they can be evaluated for arbitrary mass-ratio, whereas we are limited to sampling at only 5 discrete mass-ratios for NR waveforms. For comparison purposes, the rightmost panels of Figure 7 show the PhenomB projection coefficients coarsened to the same set of mass-ratios for which the NR waveforms are restricted to. We find the same general behaviour as before that low-order modes display the smoothest structure, while high-order modes exhibit increasing complexity. A plausible interpolation scheme would be to sample ˜ µ k for NR waveforms of varying mass for constant mass ratio (i.e. as we have done previously) and then stitch these together across the q axis. Since the projection coefficients in Figure 7 show sinusoidal structure they must be sampled with at least the Nyquist frequency along both axes. However, looking at the middle and rightmost panels it appears as though this is not yet possible given the present set of limited NR waveforms. At best the 5 available mass-ratios are just able to sample at the Nyquist frequency along the q axis for high-order modes. In order to achieve a reasonable interpolation from these projection coefficients the current NR data thus needs to be appended with more mass-ratios. Based on the left panels of Figure 7 a suitable choice would be to double the current number of mass-ratios to include q = { 1.5, 2.5, 3.5, 4.5, 5, 5.5 } . Hence, though it is not yet practical to generate an interpolated waveform family using the SVD boosting scheme applied to NR waveforms, the possibility remains open as more NR waveforms are generated.", "pages": [ 8, 9 ] }, { "title": "V. DISCUSSION", "content": "We have shown that SVD can be used to improve the representation of NR waveforms from a PhenomB template bank. A reasonably reduced SVD basis was able to reduce mismatch by a factor of five compared to PhenomB waveforms optimized over mass. There was also no mass-bias associated with the SVD basis and therefore no optimization over physical parameters required. This occurs because SVD unifies a range of waveform structure over an extended region of parameter space so that any biases become blended into its basis. SVD therefore represents a generalized scheme through which phenomenological waveform families can be de-biased and enhanced for use as matched-filter templates. We were able to calibrate an SVD basis of PhenomB templates against NR waveforms in order to construct a new waveform family with improved accuracy. This was accomplished by interpolating the coefficients of NR waveforms projected onto the PhenomB basis. Only marginal error was introduced by the interpolation scheme and the new waveform family provided a more faithful representation of the 'true' NR signal compared to the original PhenomB model. This was shown explicitly for the case of equal-mass, zero-spin binaries. We proceeded to investigate the possibility of extending this approach to PhenomB template banks containing unequal-mass waveforms. At present, however, this method is not yet feasible since the current number of mass-ratios covered by NR simulations are unable to sample the projection coefficients with the Nyquist frequency. This method will improve as more NR waveforms are simulated and should be sufficient if the current sampling rate of mass-ratios were to double.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "We thank Ilana MacDonald for preparing the hybrid waveforms used in this study. KC, JDE and HPP gratefully acknowledge the support of the National Science and Engineering Research Council of Canada, the Canada Research Chairs Program, the Canadian Institute for Advanced Research, and Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation. DK gratefully acknowledges the support of the Max Planck Society.", "pages": [ 10 ] } ]

[ { "title": "Spin effects on gravitational waves from inspiraling compact binaries at second post-Newtonian order", "content": "Alessandra Buonanno, 1 Guillaume Faye, 2 and Tanja Hinderer 1 1 Maryland Center for Fundamental Physics & Joint Space-Science Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA 2 Institut d'Astrophysique de Paris, UMR 7095 CNRS Universit'e Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France (Dated: June 26, 2018) We calculate the gravitational waveform for spinning, precessing compact binary inspirals through second post-Newtonian order in the amplitude. When spins are collinear with the orbital angular momentum and the orbits are quasi-circular, we further provide explicit expressions for the gravitational-wave polarizations and the decomposition into spin-weighted spherical-harmonic modes. Knowledge of the second post-Newtonian spin terms in the waveform could be used to improve the physical content of analytical templates for data analysis of compact binary inspirals and for more accurate comparisons with numerical-relativity simulations. PACS numbers: 04.30.-w, 04.25.-g", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Coalescing compact binary systems are a key source of gravitational radiation for ground-based gravitationalwave detectors such as the advanced Laser Interferometer Gravitational Wave Observatory (LIGO) [1], the advanced Virgo [2], the GEO-HF [3], the Large Cryogenic Gravitational Telescope (LCGT) (or KAGRA) [4], coming into operation within the next few years, and future space-based detectors [5, 6]. For this class of gravitational-wave sources, the signal detection and interpretation will be based on the method of matched filtering [7, 8], where the noisy detector output is cross correlated with a bank of theoretical templates. The accuracy requirement on the templates is that they remain as much as possible phase coherent with the signal over the hundreds to thousands of cycles of inspiral that are within the detector's sensitive bandwidth. Constructing such accurate templates has motivated a significant research effort during the past 30 years. In the regime where the separation between the two bodies is large, gravitational waveforms can be computed using the post-Newtonian (PN) approximation method [9-11]. In the post-Newtonian scheme, the results are written as an asymptotic expansion in powers of v A /c , with v A being the magnitude of the orbital coordinate velocity v A of body A at a given time. This approximation is physically relevant for v A /c /lessmuch 1, i.e. in the so-called inspiraling regime where the radiation reaction forces, of order ∼ ( v A /c ) 5 are negligible over an orbital period and act adiabatically on a quasiconservative system. In the domain of validity of the post-Newtonian scheme, the separation r ∼ ( Gm A /v 2 ) ∼ ( c/v ) 2 , with m = m 1 + m 2 and v = | v | ≡ | v 1 -v 2 | , remains large with respect to the radii of both compact objects ∼ Gm A /c 2 or, in other words, the bodies can be regarded effectively as point particles. Post-Newtonian waveforms cease to be reliable near the end of the inspiral and the coalescence phase, where numerical-relativity simulations should be used to predict the gravitational-wave signal [12-14]. By combining the information from post-Newtonian predictions and the numerical-relativity simulations it is possible to accurately and analytically describe the gravitational-wave signal during the entire inspiral, plunge, merger and ringdown process [15-23]. For nonspinning binaries, the post-Newtonian expansion has been iterated to 3 . 5PN order beyond the leading Newtonian order in the gravitational-wave phasing [2426]. The gravitational-wave amplitude has been computed through 3PN order [27-30] and the quadrupole mode through 3 . 5PN order [31]. However, black hole binaries could potentially have large spins [32] which may be misaligned with the orbital angular momentum, in which case the precession effects add significant complexity to the emitted gravitational waves [33]. Ignoring the effects of black hole spins could lead to a reduction in the signal-to-noise ratio and decrease the detection efficiency [34, 35] although this should be overcome with phenomenological and physical models [21, 36-43]. To maximize the payoffs for astrophysics will require extracting the source parameters from the gravitational-wave signal using template models computed from the most accurate physical prediction available [44-47]. Spin effects in the waveform are currently known through much lower post-Newtonian order than for nonspinning binaries. More specifically, spin effects are known through 2 . 5PN order in the phase [48-50], 1 . 5PN order in the polarizations for spin-orbit effects [51, 52], 2PN order for the spin 1 -spin 2 effects [51, 53] and partially 3PN order in the polarizations for the tail-induced spin-orbit effects [54]. In this paper, we compute all spin effects in the gravitational-wave strain tensor through 2PN order. This requires knowledge of the influence of the spins on the system's orbital dynamics as well as on the radiative multipole moments. At this PN order, nonlinear spin effects attributable to the spin-induced quadrupole mo- ments of the compact objects first appear. Using results from Ref. [55-58], we derive the stress-energy tensor with self-spin terms and compute the self-induced quadrupole terms in the equations of motion and in the source multipole moments at 2PN order. Our results are in agreement with previous calculations [59-62]. The two main inputs entering our calculation of the gravitational-wave strain tensor through 2PN order are (i) the results of Refs. [50, 51, 59] for the influence of the spins on the system's orbital dynamics, which have also been derived by effective field theory and canonical methods [56, 63-68], and (ii) the spin effects in the system's radiative multipole moments [50]. Recently, the necessary knowledge to compute the waveform at 2.5PN order was obtained using the effective field theory approach [62, 64]. Here we use (i) and (ii) in the multipolar wave generation formalism [69-71] to obtain the waveform for spinning, precessing binaries through 2PN order. To compute the gravitational polarizations from this result, one must specify an appropriate source frame and project the strain tensor onto a polarization triad. For precessing systems, there are several frames that could be employed [8, 35, 51, 72-76]. For nonprecessing binaries with the spins collinear to the orbital angular momentum, the most natural frame is the one used for nonspinning binaries. Therefore, instead of choosing one frame, for simplicity, we specialize to the nonprecessing case and quasicircular orbits and provide the explicit expressions for the gravitational polarizations. Lengthy calculations are performed with the help of the scientific software mathematica R © , supplemented by the package xTensor [77] dedicated to tensor calculus. Our generic, precessing result is available in mathematica format upon request and can be used to compute the polarizations for specific choices of frame. We notice that the 2PN terms in the polarizations, for circular orbits, linear in the spins were also computed in Ref. [78]. However, these results contain errors in the multipole moments, which were corrected in Ref. [50]. For future work at the interface of analytical and numerical relativity, we also explicitly compute the decomposition of the strain tensor into spin-weighted sphericalharmonic modes for nonprecessing spinning binaries on circular orbits. The test-particle limit of these results can also be directly compared with the black-hole perturbation calculations of Refs. [79, 80], and we verify that the relevant terms agree. The organization of the paper is as follows. In Sec. II, we review the Lagrangian for compact objects with selfinduced spin effects [55-57, 61], compute the stressenergy tensor and derive the self-induced spin couplings in the two-body acceleration and source multipole moments [59-62]. In Sec. III we summarize the necessary information about spin effects in the equations of motion and the wave generation necessary for our calculation. In Sec. IV B we calculate the spin-orbit effects at 2PN order in the strain tensor for generic precessing binaries. In Sec. IV C we complete the knowledge of 2PN spin- spin terms by including the spin self-induced quadrupole terms in addition to the spin 1 -spin 2 terms obtained in Ref. [51]. In Sec. IV E we specialize to quasicircular orbits and explicitly give the polarization tensors for nonprecessing systems. Then, in Sec. IV F we decompose the polarizations into spin-weighted spherical-harmonic modes. Finally, Sec. V summarizes our main findings. We use lowercase Latin letters a, b, ..., i, j, ... for indices of spatial tensors. Spatial indices are contracted with the Euclidean metric, with up or down placement of the indices having no meaning and repeated indices summed over. We use angular brackets to denote the symmetric, trace-free (STF) projection of tensors, e.g., T 〈 ij 〉 = STF[ T ij ] = T ( ij ) -1 3 δ ij T kk , where the round parentheses indicate the symmetrization operation. Square parentheses indicate antisymmetrized indices, e.g., T [ ij ] = 1 2 ( T ij -T ji ). The letter L = i 1 ...i /lscript signifies a multi-index composed of /lscript STF indices. The transverse-traceless (TT) projection operator is denoted P TT ijab = P a ( i P j ) b -1 2 P ij P ab , where P ij = δ ij -N i N j is the projector orthogonal to the unit direction N = X /R of a radiative coordinate system X µ = ( cT, X ), where the boldface denotes a spatial three-vector. As usual, g µν represents the space-time metric and g its determinant. The quantity ε ijk is the antisymmetric Levi-Civit'a symbol, with ε 123 = 1, and /epsilon1 µνρσ stands for the Levi-Civit'a four-volume form, with /epsilon1 0123 = + √ -g . Henceforth, we shall indicate the spin 1 -spin 2 terms with S 1 S 2 , the spin 2 1 , spin 2 2 terms with S 2 and the total spin-spin terms with SS. Throughout the paper, we retain only the terms relevant to our calculations and omit all other terms, which either are already known or appear at a higher post-Newtonian order than required for our purposes.", "pages": [ 1, 2 ] }, { "title": "II. MODELING SPINNING COMPACT OBJECTS WITH SELF-INDUCED QUADRUPOLES", "content": "In this section we review the construction of a Lagrangian for compact objects with self-induced quadrupole spin effects [55-57, 61, 81], compute the stress-energy tensor and derive the self-induced spin couplings in the two-body acceleration and source multipole moments. Our findings are in agreement with previous results [59-62].", "pages": [ 2 ] }, { "title": "A. Lagrangian for compact objects with self-induced spin effects", "content": "A Lagrangian for a system of spinning compact objects with nondynamical 1 self-induced quadrupole mo- ments can be obtained by augmenting the Lagrangian for point particles with L S 2 A describing the quadrupolecurvature coupling for each body A . Since the action for body A must admit a covariant representation, the corresponding Lagrangian L S 2 A should be a function of the four-velocity u µ A , the metric g µν , the Riemann tensor R λ ρµν and its covariant derivatives, evaluated at the worldline point y µ A , and the spin variables entering via the antisymmetric spin tensor S µν A . The spin tensor S µν A contains six degrees of freedom. It is well known that in order to reduce them to the three physical degrees of freedom a spin supplementary condition (SSC) should be imposed [82]. This is equivalent to performing a shift of the worldline y µ A . In this paper we specialize to the SSC S µν A p A ν = 0 which is equivalent to S µν A u A ν = 0 since p µ A ≈ m A cu µ A through 2.5PN order. To ensure the preservation of the SSC under the evolution, we follow Ref. [57] and introduce the spin tensor S µν A = S µν A +2 u [ µ A S ν ] λ A u A λ . The spin tensor S µν A automatically satisfies the algebraic identity S µν A u A ν = 0, which provides three constraints that can be used to reduce the spin degrees of freedom from six to three. From the above discussion and Refs. [56, 83], we assume that the Lagrangian of particle A is of the form L S 2 A = L Aµνλρ S µν A S λρ A , where L Aµνλρ is a polynomial in the Riemann tensor and its derivatives, as well as the four-velocity u µ A . As noticed in Ref. [84], any term proportional to ∇ ... R αβ evaluated at point y µ A can be recast into a redefinition of the gravitational field. As a result, the Riemann tensor may be replaced in each of its occurrences by the Weyl tensor C λ ρµν , which can be decomposed into a combination of the gravitoelectricand gravitomagnetic-type STF tidal quadrupole moments G A µν ≡ G µν ( y α A ) ≡ -c 2 R µανβ u α A u β A and H A µν ≡ H µν ( y α A ) ≡ 2 c 3 R A ∗ µανβ u α A u β A with R ∗ µναβ ≡ 1 2 /epsilon1 µνρσ R ρσ αβ . More generally, the multiple space derivatives of C λ ρµν at point y µ A may be expressed in terms of some STF tidal multipole moments G A µ 1 ...µ /lscript and H A µ 1 ...µ /lscript of parity 1 and -1 respectively. However, those higher-order moments will play no role in this paper. Taking into account that the contraction of the velocity vector u ν A with both G A µν and S µν A vanishes, that the spin and tidal multipole tensors are traceless, and that the Lagrangian must obey parity and time-reversal symmetries we obtain [55-57, 83] Here, we have also assumed that the rotating body is axially symmetric and we have replaced S µν A with S µν A since the difference between these spin variables contributes to the equations of motion at O ( S 3 A ), where S A = √ | S µ A S A µ | with S A µ = /epsilon1 ρσνµ S ρσ A p ν A / (2 m A c ). For a neutron star the numerical constant κ A in Eq. (2.1) depends on the equation of state of the fluid [85]. For an isolated black hole κ A = 1 [59, 60], but for a black hole in a compact binary κ A can deviate from 1. However, these deviations occur at PN orders that are much higher than the ones considered here. We notice that the leading contribution κ A = 1 can be obtained by computing the acceleration of body A from Eq. (2.1) in a compact binary for m A /lessmuch m and matching it with the acceleration of a test particle in the gravitational field of a Kerr black hole of mass m [83].", "pages": [ 2, 3 ] }, { "title": "B. Effective stress-energy tensor with self-induced quadrupoles", "content": "The piece of the stress-energy tensor encoding the selfinduced quadrupole dynamics of body A reads by definition where L S 2 A is the Lagrangian (2.1). To determine the action of the operator δ/δg µν , which stands for the usual 'functional derivative' with respect to the field g µν , we need to adopt a specific model for the spin. The rotational state of the extended object A is usually represented by a tetrad of orthonormal vectors e µ Aα ( τ A ) with α ∈ { 0 , 1 , 2 , 3 } along the worldline y µ A with affine parameter τ A . The corresponding angular rotation tensor is then defined as Ω µν A = η αβ e µ Aα De ν Aβ /dτ A . We now make the reasonable physical hypothesis that the rotation of the axially symmetric object takes place about the symmetry axis. The moment of inertia I A along that direction is a 2PN-order quantity ∼ G 2 m 3 A /c 4 for compactness parameters of order 1, whereas Ω µν A ∼ V A /R A , R A being the radius of body A and V A its typical internal velocity, is roughly equal to c 3 / ( Gm A ). In the weak field limit where G goes formally to zero, the spin must satisfy the relation S µν A = I A Ω µν A , as in special relativity [86]. In the presence of a nonnegligible gravitational field, this relation is expected to be modified by nonminimal coupled terms proportional to positive powers of R A µναβ times positive powers of I A and S µν A [83]: Using the definition (2.3) for the spin variables, we compute in a covariant manner the variation of the action Here we use a hat to distinguish the generic spin variable from the one related to our specific spin model. The corrections I A ×O ( ˆ S A /c 2 ) are not relevant for the twobody dynamics in this paper because they correspond to the 4.5PN order when taking into account the factor O (1 /c ) contained in the spin variable. when the metric varies by δg µν ( x ), and find the following quadrupolar piece of the stress-energy tensor where we have indicated with n ∗ A the Dirac-type scalar density n ∗ A ( x µ ) = ∫ dλ A δ 4 ( x µ -y µ A ( λ A )) / √ -g ( x ν ) and, in the last term, we have adopted the convention that symmetrization of indices applies after antisymmetrization. As derived in Ref. [81], the most general form of the effective stress-energy tensor is where τ A is the proper time of the A th worldline at event y µ A with y 0 A = x 0 and the coefficients t µν | λ 1 λ 2 ...λ /lscript A ( τ A ) are the 'skeleton' multipole moments. The latter are not arbitrary but satisfy algebraic constraints imposed by the equation of conservation ∇ ν T µν skel = 0. Let us check that we can indeed recast the total stress-energy tensor, including the monopolar, dipolar and quadrupolar pieces, in the form (2.6). If we add T µν quad to the monopolar and dipolar contributions [49, 81, 87-89] and redefine the spin variable entering the quadrupolar piece as we obtain the total stress-energy tensor in the form where the four-rank tensor J λρµν A takes the following expression in our effective description: Consistently with the approximation already made in the spin model (2.3), we have neglected here the difference of order I A ×O ( ˆ S A /c 2 ) between the spins ˆ S µν A and S µν A in the above formula. The net result is that Eq. (2.9a) matches Eq. (2.6) for /lscript = 0 , 1 , 2 as expected. Moreover, Eqs. (2.9) agree with Refs. [58, 61]. Lastly, the conservation of the stress-energy tensor (2.9a) is equivalent to the equation of motion for the particle worldline, supplemented by the spin precession equation [58]. They read Those equations are in full agreement with the equations of evolution derived from the Dixon formalism truncated at the quadrupolar order [90].", "pages": [ 3, 4 ] }, { "title": "C. Self-induced quadrupole terms in the 2PN binary dynamics and source multipole moments", "content": "Once the stress-energy tensor has been derived, the post-Newtonian equations of motion and the source multipole moments parametrizing the linearized gravitational field outside the system can be computed by means of the usual standard techniques [10]. At 2PN order, the accelerations including the self-spin interactions were obtained in Refs. [59, 60], but the self-induced quadrupole effects in the source multipole moments were never explicitly included in the standard version of the post-Newtonian scheme, although recently they were calculated at 3PN order using effective-field-theory techniques [91]. Here we can use the results of the previous section, which constitutes a natural extension of the standard post-Newtonian approximation for spinning compact bodies [49], and explicitly derive the self-induced quadrupole couplings in the 2PN dynamics and source multipole moments. Henceforth, we define the spin vectors S i A by the relation S A i /c = g A ij S j A , where S A i is the three-form induced on the hypersurface t = const by S A µ . Note that it is S i A /c that has the dimension of a spin, while S i A has been rescaled in order to have a nonzero Newtonian limit for compact objects. In the post-Newtonian formalism for point particles in the harmonic gauge, it is convenient to represent effectively the source by the mass density σ = ( T 00 + T ii ) /c 2 , the current density σ i = T 0 i /c , and the stress density σ ij = T ij . They are essentially the components of the stress-energy tensor rescaled so as not to vanish in the formal limit c → 0 for weakly stressed, standard matter. At 2PN order, the second term in the right-hand side of Eq. (2.9a) does not contribute. From the last term, we obtain the following self-spin contributions: where 1 ↔ 2 represents the counterpart of the preceding term with particles 1 and 2 exchanged, and δ 1 ≡ δ 3 ( x -y 1 ). At 2PN order, the spin 2 part of the equations of motion (2.10a) for, say, the first particle, reduce to The only occurrence of self-spin interactions at 2PN order on the left-hand side of the above equation comes from the gradient of the time component of the metric, g 00 = -1+2 V/c 2 + O (1 /c 4 ), where the Newton-like potential V satisfies /square V = -4 πGσ . Although V coincides with the Newtonian potential U in the leading approximation, it contains higher order corrections, including quadratic-inspin terms coming from the mass density (2.11a), which are smaller than U by a factor O (1 /c 4 ). They read with ∂ i = ∂/∂x i and r 1 ≡ | x -y 1 | , the symbol ∆ -1 holding for the retarded integral operator. Other potentials appear at the 1PN approximation or beyond, but their sources cannot contain a self-induced quadrupole below O (1 /c 4 ); thus they are negligible here. The self-induced spin part of the acceleration a 1 of the first particle is therefore given by Replacement of the Christoffel symbols Γ λ µν and the Riemann tensor by the leading order values with U = Gm 1 /r 1 + Gm 2 /r 2 + O (1 /c 2 ) yields the more explicit result (posing ∂ 1 i ≡ ∂/∂y i 1 ): which agrees with Refs. [59, 60] in the center-of-mass frame, for S i A /c = ε ijk S jk + O (1 /c 3 ). Self-induced quadrupolar deformations of the bodies also produce 2PN-order terms in the source multipole moments I L and J L . Those are defined as volume integrals whose integrands are certain polynomials in the densities σ , σ i and σ ij as well as some gravitational potentials, such as V , that parametrize the metric. Now, since those potentials are multiplied by prefactors of order O (1 /c 2 ) and cannot contain themselves spin 2 interactions below the 2PN order, monomials involving one potential or more may be ignored for the calculation. The remaining sources are linear in the σ variables. With the help of the general formula (5.15) of Ref. [92], it is then immediate to get the self-spin contribution to I L : Inserting expression (2.11a) for σ S 2 and performing a straightforward integration, we arrive at We can show similarly that J L is of order O ( S 2 A /c 2 ). As a result, at the accuracy level required for the 2PN waveform, the only terms quadratic in one of the spins that originate from the source moments come from the quadrupole /lscript = 2, for which we have whereas similar terms in ( I L ) /lscript ≥ 3 or ( J L ) /lscript ≥ 2 lie beyond our approximation. The above correction to the mass quadrupole agrees with that of Porto et al . [91] truncated at 2PN order. It is formally of order O (1 /c 4 ) but, because ˙ S A = O (1 /c 2 ), it is cast to the 3PN order in the waveform expansion given below [see Eq. (4.1)] after the second time derivative is applied. This result was already argued in Ref. [93].", "pages": [ 4, 5 ] }, { "title": "III. TWO-BODY DYNAMICS WITH SPIN EFFECTS THROUGH 2PN ORDER", "content": "The equations of motion in harmonic coordinates for the relative orbital separation x = r n in the center of mass frame are [10] where with m ≡ m 1 + m 2 , ν ≡ m 1 m 2 /m 2 , n = x /r and v = d x /dt . The 2PN acceleration given, e.g., in Ref. [51] will not be needed for our calculation. The spin-orbit terms are [51] where we denote with δ = ( m 1 -m 2 ) /m and The spin 1 -spin 2 interaction terms are [51] As originally computed in Ref. [59] [see Eq. (2.16) above], an additional term due to the influence of the spininduced mass quadrupole moment on the motion arises at 2PN order: Here, q = m 1 /m 2 is the mass ratio and we recall that the parameters κ A characterize the mass quadrupole moments of the bodies. We find that the quadratic spin contribution to the acceleration can be rewritten in a simpler way by introducing the spin variables and S -0 , which is obtained by exchanging the labels 1 and 2 in the above equation. 2 Those variables generalize the quantity S 0 of Ref. [60] in the case where the two bodies are not black holes. In terms of these spin variables the spin-spin part of the acceleration reads The spin precession equations through 2PN order are [51, 94] It is often convenient to use a different set of spin variables S c Ai whose magnitude remains constant and that obey precession equations of the form d S c A /dt = Ω A × S c A . The relationship between the spin variables appearing in the equations of motion above and the constant magnitude spin variables is [50]", "pages": [ 5, 6 ] }, { "title": "A. General formalism", "content": "The gravitational radiation from the two-body system is calculated from symmetric trace-free radiative multipole moments I L and J L using the general formula from Ref. [69] truncated at 2PN order where N is the unit vector pointing from the center of mass of the source to the observer's location and R is the distance between the source and the observer. Here, the superscript ( n ) signifies the n th time derivative, and the transverse-traceless projection operator is with P ij = δ ij -N i N j . The gravitational radiation (4.1) can be rewritten in a post-Newtonian expansion as The 1PN and 1.5PN spin terms are given explicitly in Refs. [51, 52]. The terms in the source multipole moments that are a priori needed to compute the spin-orbit waveform exactly at 2PN order are identified by considering their schematic structure, together with the scalings of Eqs. (4.1) and (3.1a). Specifically, the following pieces are required: ( I Newt abc ) (3) using the 1 . 5PN motion and ( I SO abc ) (3) with a Newt , ( J SO ab ) (2) with the 1PN motion and the spin evolution, ( J 1 . 5PNSO ab ) (2) with a Newt , ( J Newt ab ) (2) with the 1 . 5PN accurate motion, and ( J SO abcd ) (4) with a Newt . For the SS part, we need ( I Newt ab ) (2) with a SS , as the time derivative of I SS ab does not contribute at 2PN order. When we write the waveform in terms of the constant magnitude spin variables, there is an additional contribution to the 2PN spin piece of the waveform coming from J SO ab with a Newt and the 1PN conversion factor in Σ c . The relevant spin contributions to the multipole moments are [50] The nonspinning contributions to the multipole moments that we employed in our calculation are", "pages": [ 6, 7 ] }, { "title": "B. Spin-orbit effects", "content": "Using the multipole moments of Eqs. (4.5) and (4.6) in Eq. (4.1) and substituting the equations of motion (3.1) and (3.3b), we find the following 2PN spin-orbit piece: h 2PNSO ij TT These contributions add linearly to the other known terms in the waveform. Note that in Eq. (4.7) we have already anticipated the transverse-traceless projection and simplified the expression using δ ij TT = N i TT = N j TT = 0 and the interchange identity [51] for any vectors A and B .", "pages": [ 7, 8, 9 ] }, { "title": "C. Spin-spin effects", "content": "Spin-spin terms in the waveform at 2PN order are entirely attributable to the equations of motion; they arise when substituting a SS in the time derivatives of I Newt ab . The second time derivative of the contribution I S 2 ab given in Eq. (2.19) is at least of 3PN order (because of the fact that spins are constant at leading approximation) and therefore vanishes for our calculation. We derive We notice that the spin-orbit contributions at 2PN order are zero for an equal-mass, equal-spin black-hole binary. This is a consequence of the multipoles (4.5) being zero for this highly symmetric binary configuration. The general results (4.7) and (4.9) are available as a mathematica notebook upon request to be used to compute the gravitational polarizations and spherical harmonic modes for precessing binaries for any choice of the source frame and the polarization triad [8, 35, 51, 7276]. Below, we shall derive the polarizations and spinweighted spherical-harmonic modes for the case of nonprecessing compact binaries on circular orbits.", "pages": [ 9 ] }, { "title": "D. Reduction to quasicircular orbits", "content": "We now specialize Eqs. (4.7) and (4.9) to the case of orbits that have a constant separation r in the absence of radiation reaction and for which the precession time scale is much longer than an orbital period. The details of the derivation of the modified Kepler law relating the orbit-averaged orbital angular frequency ω and the orbitaveraged orbital separation are discussed in Ref. [93]. The instantaneous accelerations (3.1) and (3.5) are projected onto a triad consisting of the following unit vectors: n = x /r , the vector /lscript = L N / | L N | orthogonal to the instantaneous orbital plane, where L N = mν x × v denotes the Newtonian orbital angular momentum, and λ = /lscript × n . The orbital separation r and angular frequency ω are decomposed into their orbit averaged piece, indicated by an overbar, and remaining fluctuating pieces, r = ¯ r + δr and ω = ¯ ω + δω . Projecting the equations of motion along λ yields the equality 2 ω ˙ r + ˙ ω r or, equivalently [93] At the 2PN order, r and ω can be replaced by the constants r and ω , respectively, on the right-hand side. The expression for ωr 2 follows from (i) dropping the time derivatives in the above equation, and (ii) adding an integration constant determined by averaging ωr 2 over an orbit. Inserting the result in the projection along n of the equations of motion, and linearizing in δr we find an explicit solution to the differential equation given by Inverting Eq. (4.12b) to write r as a function of ω in Eq. (4.7) and inserting there the expression (4.12a) of ˙ r , we obtain the following spin-orbit terms in the waveform: Here, we have used that and similarly for Σ c . Finally, we derive the 2PN spin-spin terms for circular orbits. They read", "pages": [ 9, 10, 11 ] }, { "title": "E. Polarizations for nonprecessing, spinning compact bodies", "content": "The two polarization states h + and h × are obtained by choosing a coordinate system and taking linear combinations of the components of h TT ij . Using an orthonormal triad consisting of N and two polarization vectors P and Q , the polarizations are Although different choices of P and Q give different polarizations, the particular linear combination of h + and h × corresponding to the physical strain measured in a detector is independent of the convention used. For nonspinning binaries, one usually chooses a coordinate system such that the orbital plane lies in the x -y plane, and the direction of gravitational-wave propagation N is in the x -z plane. When the spins of the bodies are aligned or antialigned with the orbital angular momentum, the system's evolution is qualitatively similar to the case of nonspinning bodies. This case is characterized by the absence of precession of the spins and orbital angular momentum and thus the orbital plane remains fixed in space. However, the effect of the spins gives a contribution to the phase and a correction to the amplitude of the waveform, which we explicitly provide in this subsection. We use the conventions that the z axis coincides with /lscript and the vectors /lscript , N , n , and λ have the following ( x, y, z ) components: where Φ is the orbital phase defined such that at the initial time, n points in the x direction. We use the following polarization vectors: The vector P is the ascending node where the orbital separation vector crosses the plane of the sky from below. With these conventions, Eqs. (4.16) with Eqs. (4.13), specialized to the case where the only nonvanishing spin components are ( Σ c · /lscript ) and ( S c · /lscript ), become Here, the convention for the 2PN spin pieces of the polarizations is analogous to that adopted for the PN expansion of the waveform (4.3), with the expansion coefficients related by Eqs. (4.16) at each PN order.", "pages": [ 11, 12 ] }, { "title": "F. Gravitational modes for nonprecessing, spinning compact bodies", "content": "The gravitational wave modes are obtained by expanding the complex polarization into spin-weighted s = -2 spherical harmonics as where with The modes h /lscriptm can be extracted by computing where the integration is over the solid angle ∫ d Ω = ∫ π 0 sin θdθ ∫ 2 π 0 dφ and using the orthogonality property ∫ d Ω -s Y /lscriptm ( θ, φ ) -s Y /lscript ' m ' ∗ ( θ, φ ) = δ /lscript/lscript ' δ mm ' , where δ /lscript/lscript ' is the Kronecker symbol and the star denotes complex conjugation. Using Eqs. (4.19) and (4.20) in Eq. (4.25) we find the following nonvanishing modes: (4.27c) (4.27d) (4.27e) We have explicitly checked that in the test-mass limit ν → 0, Eqs. (4.27) reduce to the 2PN O ( q ) and O ( q 2 ) terms given in Eqs. (22) of Ref. [80] (see also [79]), after accounting for the factor of ( -i ) m attributable to the different conventions for the phase origin, as explained in Ref. [52]. It is interesting to note from Eq. (4.27b) that in the nonprecessing case, the dominant h 22 mode contains only terms that are quadratic in the spin at 2PN order. By contrast, for precessing binaries, the 2PN spin-orbit terms will give a nonvanishing contribution to the 22mode.", "pages": [ 12 ] }, { "title": "V. CONCLUSIONS", "content": "We have extended the knowledge of the spin terms in the gravitational-wave strain tensor to 2PN accuracy for precessing binaries. Our result includes the spin-orbit as well as the spin 1 -spin 2 and spin 2 1 , spin 2 2 effects. The quadratic-in-spin terms are entirely due to the equations of motion, whereas the 2PN spin-orbit terms come from both the corrections to the orbital dynamics and the radiation field. For a given choice of an orthonormal polarization triad and a source frame, the gravitational-wave polarizations can be obtained by projecting our result for the gravitational-wave strain tensor given in Secs. IV B and IVC orthogonal to the propagation direction. For precessing binaries, there is no preferred unique choice of the source frame [8, 35, 51, 72-76], but in the case that the spins are collinear with the orbital angular momentum, the procedure to obtain the polarizations can be carried out in a similar fashion as for nonspinning binaries. For the nonprecessing case and circular orbits, we provided ready-to-use expressions for the gravitational polarizations in Sec. IV E, which could be directly employed in time-domain post-Newtonian, phenomenolog- ical and effective-one-body-based template models [1923, 51, 52]. In view of the current interest in interfacing analytical and numerical relativity, we also provided the decomposition of the waveform into spin-weighted spherical harmonic modes for nonprecessing binaries and quasicircular orbits. We verified that the test-particle limit of our result reduces to the expressions obtained from black-hole perturbation theory [79, 80]. We noted that for spins collinear with the orbital angular momentum, the dominant h 22 mode of the waveform contains only quadraticin-spin effects since the spin-orbit contributions vanish in this case, although they are nonzero for generic, precessing configurations.", "pages": [ 12, 13 ] }, { "title": "Acknowledgments", "content": "A.B. acknowledges partial support from NSF Grants No. PHY-0903631 and No. PHY-1208881, and NASA Grant NNX09AI81G. A.B. also thanks the Kavli Institute for Theoretical Physics (supported by the NSF Grant No. PHY11-25915) for hospitality during the preparation of this manuscript. T.H. acknowledges support from NSF Grants No. PHY-0903631 and No. PHY1208881, and the Maryland Center for Fundamental Physics. We thank Gilles Esposito-Far'ese, Larry Kidder and Etienne Racine for useful interactions, as well as David Delavaquerie for help in finalizing one of our mathematica codes.", "pages": [ 13 ] }, { "title": "Appendix: USEFUL IDENTITIES", "content": "According to the way the waveform is computed, the result may take various forms, which are not immediately seen to be equivalent. Their difference vanishes because of some dimensional identities valid in three dimensions. They all amount to expressing the fact that a tensor with four antisymmetrized indices must vanish. We shall present here two of such identities, which turned out to be particularly useful for our checks, together with Eqs. (5.2) of Ref. [49]. Let U A = U i A , for A ∈ { 1 , 2 , 3 } , be three vectors of R 3 . The first identity tells us that for any vector U , we must have To show this, we compute ε i ab ε mjk ε mpq U a 1 U b 2 U p 3 U q 4 in two different manners: (i) we group the first two epsilons, which are next expanded in terms of the identity tensor δ i j using the standard formula ε iab ε mjk = 3! δ m [ i δ j a δ k b ] ; (ii) we group the last two epsilons and apply the contracted version of the previous equation: ε mjk ε mpq = 2 δ j [ p δ k q ] . One of the remaining free indices, say k , is finally contracted with U k . The second identity reads: It is proved by contracting the equality U [ a 1 U b 2 U c 3 δ i ] j = 0 with U 1 a U 2 b U 3 c and expanding. As the trace of the left- and side of Eq. (A.2) is identically zero, the nontrivial content of this identity consists of its STF part. Technical Report, LISA science case document, 2007, http://list.caltech.edu/mission_documents .", "pages": [ 13 ] } ]

[ { "title": "Irreversible thermodynamic description of interacting dark energy - dark matter cosmological models", "content": "Tiberiu Harko 1 ∗ and Francisco S.N. Lobo 2 † 1 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom and 2 Centro de Astronomia e Astrof'ısica da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal (Dated: October 30, 2018) We investigate the interaction between dark energy and dark matter in the framework of irreversible thermodynamics of open systems with matter creation/annihilation. We consider dark energy and dark matter as an interacting two component (scalar field and 'ordinary' dark matter) cosmological fluid in a homogeneous spatially flat and isotropic Friedmann-Robertson-Walker (FRW) Universe. The thermodynamics of open systems as applied together with the gravitational field equations to the two component cosmological fluid leads to a generalisation of the elementary dark energy-dark mater interaction theory, in which the decay (creation) pressures are explicitly considered as parts of the cosmological fluid stress-energy tensor. Specific models describing coherently oscillating scalar waves, leading to a high particle production at the beginning of the oscillatory period, and models with a constant potential energy scalar field are considered. Furthermore, exact and numerical solutions of the gravitational field equations with dark energy-dark matter interaction are also obtained. PACS numbers: 04.50.Kd, 04.20.Cv, 04.20.Fy", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The observational data from type Ia supernovae, initially reported in [1], have generated a large theoretical and observational effort for the understanding of the observed present accelerated expansion of the Universe. Subsequent work on type Ia supernovae [2], the cosmic microwave background [3], and baryon acoustic oscillations [4] fully support the initial interpretation of the observational data that the expansion of the Universe is accelerating. The late-time cosmic acceleration is usually assumed to be driven by a fluid/field generically denoted dark energy [5]. Presently very little is known about dark energy, namely, its possible composition or its structure. Two main scenarios have been proposed to explain the nature of the dark energy: a cosmological constant Λ [5], or a scalar field, usually called quintessence [6]. The action for gravity and the scalar field is S = ∫ [ R/ 16 πG -(1 / 2) ∇ α φ ∇ α φ -V ( φ )] √ -gd 4 x , where V ( φ ) is the self-interaction potential [7]. Another of the central issues in modern astrophysics is the dark matter problem (see [8] for an extensive review of the recent results of the search for dark matter). The necessity of considering the existence of dark matter at a galactic and extragalactic scale is required by two fundamental observational evidences: the behavior of the galactic rotation curves, and the mass discrepancy in clusters of galaxies, respectively. On the galactic/intergalactic scale the rotation curves of spiral galax- ies [9-11] provide compelling evidences pointing towards the problems Newtonian gravity and/or standard general relativity has to face at these scales. The behavior of the galactic rotation curves and of the virial mass of galaxy clusters is usually explained by postulating the existence of some dark (invisible) matter, distributed in a spherical halo around the galaxies. The dark matter is assumed to be a cold, pressure-less medium. Many possible candidates for dark matter have been proposed, the most popular ones being the WIMPs (Weakly Interacting Massive Particles) (for a review of the particle physics aspects of the dark matter see [12]). While extremely small, their interaction cross sections with normal baryonic matter, are expected to be non-zero, so that their direct experimental detection may be possible. In this context, cosmological evolution and dynamics are largely dominated by dark energy and dark matter. Dark energy has a repulsive effect, driving the Universe to accelerate, while dark matter is gravitationally attractive. In the standard approach to cosmology there is no interaction between these two components. Since the gravitational effects of the dark energy and of the dark matter are opposite (i.e., gravitational repulsion versus gravitational attraction) and since dark energy is very homogeneously distributed, while dark matter clumps around ordinary matter, one expects that any dynamic interaction between these two dark components of the Universe would be extremely weak, or even negligible. However, the possibility of such an interaction cannot be excluded a priori and, following some early proposals [13], presently interacting dark matter-dark energy models were extensively investigated in the literature [14]. In the standard approach, one may model dark energy as a scalar field with energy density ρ φ and pressure p φ , while dark matter is described as a matter fluid with density ρ DM and pressure p DM , satisfying an equation of state w DM = p DM /ρ DM ≡ 0. By assuming a spatially flat Friedman-Robertson-Walker (FRW) background with scale factor, a ( t ), and allowing for creation/annihilation between the dark energy (scalar field) and the dark matter fluid at a rate Q , the equations describing the variations of the dark energy and dark matter densities ρ φ and ρ DM can be written as [13, 14] respectively, where H ≡ ˙ a/a is the Hubble function. The derivatives with respect to the cosmological time, t , will be indicated in the following by an overdot. The dark energy and the dark matter create/decay into one another via the common creation/annihilation rate ± Q . Hence Q describes the interaction between the two dark components of the Universe. When Q > 0, dark energy is converted to dark matter, while if Q < 0, dark matter is converted to dark energy. Since there is no fundamental theoretical approach that may specify the functional form of the coupling between dark energy and dark matter, presently coupling models are necessarily phenomenological, although one might view some couplings more physical or more natural than others. Hence a large number of functional forms for Q have been proposed, and investigated in the literature, such as Q = ρ 0 crit (1 + z ) 3 H ( z ) I Q ( z ), where z is the redshift, and I Q ( z ) is an interaction function that depends on the redshift [15], or Q ∝ ρ DM ˙ φ , and Q ∝ Hρ DM , respectively [16]. In writing down Eqs. (1) and (2), one assumes the idealised picture of a very quick decay in which the final products reach equilibrium states immediately. However, the dark energy/dark matter creation/annihilation period may be characterised by complicated nonequilibrium processes, with a highly nonequilibrium distribution of the produced particles, subsequently relaxing to an equilibrium state. Thermodynamical systems in which matter creation occurs fits in the class of open thermodynamical systems in which the usual adiabatic conservation laws are modified thereby including irreversible matter creation [17]. The thermodynamics of open systems were first applied to cosmology in [17]. Explicit inclusion of the matter creation in the matter stress-energy tensor in the Einstein field equations leads to a three stage cosmological history, starting from an instability of the vacuum. During the first stage the Universe is driven from an initial fluctuation of the vacuum to a de Sitter phase, and particle creation occur. The de Sitter phase does exist during the decay time of its constituents (second stage) and ends, after a phase transition into the usual FRW Universe. The phenomenological approach of [17] was further discussed and generalised in [18] through a covariant formulation allowing specific entropy variation as usually expected for non-equilibrium processes. Cosmological models involving irreversible matter creation have been considered in [19]. It is the purpose of the present paper to apply the thermodynamics of open systems as developed in [17] and [18] to a cosmological fluid mixture consisting of two components: dark energy, described by a scalar field, and dark matter, modeled as an ordinary matter fluid, in which particle decay and production occur. This situation may be specific to both early and late stages of cosmological evolution. The thermodynamics of irreversible processes as applied to cosmological models with interacting dark energy and dark matter leads to a self-consistent description of the dark energy and dark matter particle creation/annihilation processes, which in turn determine the whole dynamics and future evolution of the Universe. The present paper is organised as follows. In Section II we present, in some detail, due to its important role, the thermodynamical theory of irreversible matter creation processes. The theory is applied to a two-component cosmological fluid with interacting dark energy and dark matter, and the resulting gravitational field equations are written down in Section III. Particular models, and exact and numerical solutions to the field equations are considered in Section IV. In Section V we discuss and conclude our results. Throughout the paper we use a system of units so that 8 πG = c = 1.", "pages": [ 1, 2 ] }, { "title": "II. THERMODYNAMICS OF IRREVERSIBLE COSMOLOGICAL MATTER CREATION", "content": "We consider a cosmological volume element V containing N particles. For a closed system, N is constant, and the corresponding thermodynamic conservation of the internal energy E is expressed by the first law of thermodynamics as [17] where dQ is the heat received by the system during time dt , p is the thermodynamic pressure, and V is any comoving volume. By introducing the energy density ρ defined as ρ = E/V , the particle number density n given by n = N/V , and the heat per unit particle dq , with dq = dQ/N , Eq. (3) becomes Equation (4) is also valid for open systems in which N is time dependent, N = N ( t ).", "pages": [ 2 ] }, { "title": "A. General relativistic covariant formulation of matter creation", "content": "In a general-relativistic framework the basic macroscopic variables which describe the thermodynamic states of a relativistic simple fluid are the energy-momentum tensor T µν , the particle flux vector N µ , and the entropy flux vector s µ . The energy-momentum tensor satisfies the conservation law ∇ ν T µν = 0. By taking into account matter creation the energy-momentum tensor can be written as where the creation pressure takes into account dissipative effects. The particle flux vector is given by N µ = nu µ , where n is the particle number density, and u µ is the four-velocity of the fluid. The particle flux vector satisfied the balance equation where the function Ψ is a particle source for Ψ > 0, and a particle sink for Ψ < 0. In standard cosmology Ψ is usually assumed to be zero. We also introduce the entropy flux s µ , defined as s µ = nσu µ [18], where σ is the specific entropy per particle. The second law of thermodynamics requires that ∇ µ s µ ≥ 0. For an open thermodynamic system with temperature T in the presence of matter creation the Gibbs equation is By using the above equations one can immediately obtain the entropy balance equation as [18] where Θ = ∇ µ u µ is the expansion of the fluid, and the chemical potential µ is given by Euler's relation µ = ( ρ + p ) /n -Tσ . In the following we consider that the particles are created in the space-time in such a way that they are in thermal equilibrium with the already existing ones. Then the entropy production is due only to the matter creation. Moreover, we shall assume for the creation pressure p c the following phenomenological ansatz [17, 18] where α > 0. With this choice we obtain for the entropy balance the equations where ˙ σ = u µ ∇ µ σ = dσ/ds , which, together with Eq. (8) gives for the specific entropy production the relation [18] If we constrain our formalism by requiring that the specific entropy per particle is constant, σ = constant, then Eq. (11) fixes the form of α as α = ( ρ + p ) /n , giving for the creation pressure the expression [18] By taking into account the condition of the constancy of the specific entropy, the Gibbs equation becomes", "pages": [ 2, 3 ] }, { "title": "B. Matter creation in homogeneous and isotropic cosmological models", "content": "In the case of a homogeneous and isotropic space-time we adopt a comoving frame so that the components of the four-velocity are given by u µ = (1 , 0 , 0 , 0). Moreover, we assume that the thermodynamic as well as the geometric parameters are a function of the time t only. Then the derivative of any function f ( t ) with respect to the line element s coincides with the ordinary time derivative, ˙ f = u µ ∇ µ f = df/dt . Moreover, the expansion of the fluid is given by ∇ µ u µ = ˙ V /V . Equation (13) can be written in a number of equivalent forms as where h = ρ + p is the enthalpy (per unit volume) of the fluid, or, equivalently, The Einstein field equations involve the macroscopic stress-energy tensor T µν , which, in the cosmological case, corresponds to a perfect fluid. It is characterised by a phenomenological energy density ρ and pressure ¯ p , and its components are given by In addition to the Einstein field equations we have the Bianchi identities, which lead to ∇ ν T ν µ = 0, and to the relation In the presence of adiabatic irreversible matter creation the appropriate analysis must be performed in the context of open systems. This involves the inclusion of a supplementary creation/annihilation pressure p c , as we may write Eq. (13) in a form similar to Eq. (18), namely [17] where from Eq. (20) it follows that the creation pressure is given by Creation of matter corresponds to a (negative) supplementary pressure p c , which must be considered as part of the cosmological pressure ¯ p entering into the Einstein field equations (decaying of matter leads to a positive decay pressure), The entropy change dS in an open thermodynamic system can be decomposed into an entropy flow d 0 S , and the entropy creation d i S , with d i S ≥ 0. To evaluate dS we start from the total differential of the entropy, where s = S/V ≥ 0 and µn = h -Ts , µ ≥ 0 being the chemical potential. In a homogeneous system d 0 S = 0, but matter creation contributes to the entropy production. From Eqs. (22) and (23) we obtain [17] To complete the problem we need one more relation between the particle number n and V , describing the time dynamics of n as a result of matter creation (decay) processes. This relation is given by Eq. (6), which in the case of a homogeneous and isotropic cosmological model takes the form where Ψ( t ) is the matter creation (or decay) rate (Ψ( t ) > 0 corresponds to particle creation, while Ψ( t ) < 0 corresponds to particle decay) [17, 18]. The creation pressure (20) depends on the matter creation (decay) rate, thereby coupling Eqs. (20) and (25) to each other and, although indirectly, both of them with the energy conservation law (19), which is contained in the Einstein field equations themselves. The entropy production can also be expressed as a function of the matter creation rate as", "pages": [ 3, 4 ] }, { "title": "III. COSMOLOGICAL DYNAMICS IN A UNIVERSE WITH IRREVERSIBLE DARK ENERGY-DARK MATTER INTERACTION", "content": "We shall model the Universe as an open thermodynamical system, consisting of a two-component (dark energy and dark matter) perfect fluid, with the particle number densities denoted by n φ , and n DM , respectively. n φ corresponds to the 'particles' of the scalar field, while n DM is the particle number of the dark matter. We denote the corresponding energy densities by ρ φ and ρ DM , respectively. The stress-energy tensor of the two-component cosmological fluid is given by where u µ = dx µ /ds is the four-velocity, and The energy density and pressure of the dark energy are given by ρ φ = ˙ φ 2 / 2 + U ( φ ) and p φ = ˙ φ 2 / 2 -U ( φ ), respectively, where U ( φ ) is the self-interaction potential. We suppose that neither the particle numbers nor the stress-energy of the components are separately conserved, that is, particle inter-conversion and exchange of energy and momentum between the two components are admitted. The cosmological fluid mixture is characterised by a total energy density ρ = ρ φ + ρ DM , total thermodynamic pressure ¯ p = ¯ p φ + ¯ p DM and a total particle number n = n φ + n DM . We consider that the geometry of the spacetime is described by the flat FRW line element, given by We shall assume that the particle number densities n φ and n DM of each component of the fluid obey the following balance laws, /negationslash /negationslash respectively, where Γ 1 = 0 and Γ 2 = 0 are arbitrary functions. Equations (30) and (31) describe the decay of the dark energy φ -particles, and the creation of the dark matter particles, with a scalar field decay rate and a dark matter creation rate Ψ( t ) ∝ ρ φ . Thus, the dynamics of φ -particle decay and the creation of dark matter is governed in the present model by the scalar field via its energy density. From Eqs. (30) and (31) it follows that the total particle number n obeys the balance equation Hence in the case of an interacting dark matter and dark energy the total particle number conservation occurs only in very special cases, and therefore we shall suppose that generally Γ 1 = Γ 2 . In the framework of the thermodynamics of irreversible processes particle creation and decay gives rise to a decay and a creation thermodynamic pressure, given by /negationslash and respectively, while the total creation pressure becomes Using the results obtained above the complete Einstein gravitational field equations describing the dynamics of a flat FRW spacetime filled with a mixture of interacting dark matter (scalar field) and dark matter can be expressed in the form and where ρ DM = ρ DM ( n DM ) and p DM = p DM ( n DM ). The dynamical evolution of the dark energy and dark matter particles n φ and n DM is given by Eqs. (30) and (31), respectively, while the energy density and pressure of the dark energy is given by where U ( φ ) is the scalar field self-interaction potential. As applied to each component of the cosmological fluid, Eq. (14), the second law of thermodynamics for open systems provides the relationships and respectively. Equation (39), which describes the dynamics of the dark energy during its interaction with dark matter, can be written in an equivalent form as where we have denoted Γ ( φ, ˙ φ, U ) = Γ 1 ρ φ /n φ . Therefore in the framework of the thermodynamics of irreversible processes a friction term in the scalar field Eq. (41) arises naturally, and in a general form, as a direct consequence of the second law of thermodynamics as applied to an open system. Adding Eqs. (39) and (40), the evolution of the total energy density ρ = ρ φ + ρ DM of the cosmological fluid is governed by the equation For the entropy of the newly created matter we obtain Irreversible matter particle creation is an adiabatic process, the produced entropy being entirely due to the increase in the number of fluid particles, there being no increase in the entropy per particle due to dissipative processes.", "pages": [ 4, 5 ] }, { "title": "IV. IRREVERSIBLE DARK ENERGY-DARK MATTER INTERACTION MODELS", "content": "In the present Section we consider, within the framework of irreversible thermodynamics with matter creation/annihilation, a number of specific cosmological models with dark energy-dark matter interaction. As a first case we consider the situation in which the density of the dark matter is much smaller than the energy density of the scalar field. This case corresponds to an Universe dominated by the dark energy component, assumed to be represented by a coherent wave of φ -particles. In this case the kinetic term dominates in the total energy of the scalar field. As a second case we consider a potential energy dominated scalar field. The general dynamics of the cosmological model with interacting dark energy and dark matter is also considered, and the cosmological evolution equations are studied numerically.", "pages": [ 5 ] }, { "title": "A. Coherent Scalar Waves-Dark matter interaction", "content": "We shall consider in the following that the energy density and particle number of the newly created dark matter is much smaller than the energy density and particle number of the corresponding scalar field fluid component, that is the relations ρ DM /lessmuch ρ φ , n DM /lessmuch n φ , and p DM /lessmuch p φ hold. In this case the Universe is dominated by the scalar field energy density, and its evolution is not influenced by the matter content. We shall work throughout with finite values of the fluid quantities at t = t 0 . The coupling between scalar field and dark matter is realised only by means of the balance equation of ordinary matter via the scalar field energy density, and the basic equations describing the dynamics of a flat FRW scalar field filled space-time interacting with a dark matter component are given by and respectively. A homogeneous scalar field oscillating with frequency m φ can be considered as a coherent wave of 'particles' with zero momenta, and with a particle number density given by [20] In other words, n φ oscillators of the same frequency m φ oscillating coherently with the same phase can be described as a single homogeneous wave φ ( t ). Insertion of the energy density of the scalar field given by Eq. (49) in Eq. (46) leads to the condition or, equivalently, Therefore, a homogeneous oscillating scalar field is described in the present model by a Barrow-Saich type potential, with the potential energy of the scalar field proportional to the kinetic one [21]. The energy density of the scalar field becomes ρ φ = ˙ φ 2 , and this relation, obtained naturally in the framework of the present formalism is very similar to the equation ρ φ = 〈 ˙ φ 2 〉 obtained by replacing ˙ φ 2 by its average value per cycle [22]. In this case the equations describing the dynamics of the FRW type spacetime filled by the decaying oscillating homogeneous scalar field in the presence of dark matter creation become respectively. In the following we will assume, for simplicity, that Γ 1 and Γ 2 are constants. By introducing a set of dimensionless variables τ , h and θ DM by means of the transformations Equations (53) and (54) take the form and yield the following general solutions and respectively, where we have denoted θ = θ ( τ DM 0 DM 0 ). The evolution of the scale factor is given by The deceleration parameter q = d (1 /H ) /dt -1 is given by For ( t -t 0 ) /lessmuch Γ -1 1 , i.e., at the start of the oscillatory period corresponding to the dark matter production, the approximate solution of the field equations is given by This phase corresponds to an Einstein-de Sitter expansion, with decaying dark energy, and dark matter creation. During the initial oscillating period of the scalar field dominated FRW Universe there is a rapid increase of its dark matter content. The particle number density increases during a time interval and reaches a maximum value given by Assuming that the scalar field oscillations decay into relativistic dark matter, the energy density and temperature of the dark matter component of the cosmological fluid is given by ρ DM ∼ n γ , T DM ∼ n γ -1 , where γ = 2 corresponds to a stiff (Zeldovich) fluid obeying an equation of state of the form ρ DM = p DM , and γ = 4 / 3 corresponds to a relativistic, radiation like fluid. The entropy produced during dark matter particle creation can be easily obtained from Eq. (43) and is given, in first approximation, by Therefore, for small times, there is a linear increase of the entropy of the dark energy dominated flat spacetime.", "pages": [ 5, 6, 7 ] }, { "title": "B. Constant scalar field potential dark energy and dark matter interaction", "content": "As a second example in the study of the irreversible interaction between dark energy (a scalar field) and dark matter we consider the case in which the scalar field potential may be approximated, at least for a certain time interval, as a positive constant, U ( φ ) = Λ = constant > 0. Therefore, the energy density and the pressure of the scalar field can be written as Moreover, we assume that the change in the scalar field energy density due to the cosmological expansion can be neglected in the scalar field evolution equations, that is, in the energy density and particle number equations of the scalar field, given by Eqs. (39) and (40), the term containing 3 H can be neglected. In this approximation the main contribution to the temporal dynamics of the scalar field is due to its decay into dark matter particles, and not to the cosmological expansion. On the other hand, the feedback of the newly created dark matter particles on the cosmological dynamics cannot be neglected. Within this approximation, from Eq. (30), describing the scalar field particles decay, we obtain first which gives the scalar field particle number density as a function of the energy of the scalar field. By substituting this expression of ρ φ into Eq. (39) gives Taking into account that we obtain the equation describing the dynamics of the scalar field particles as By denoting ˙ n φ = u , n φ = udu/dn φ , Eq. (71) becomes providing where N 1 is an arbitrary integration constant, which can be determined from the initial condition ˙ n φ ( t 0 ) = -Γ 1 ρ φ ( t 0 ) = -Γ 1 ρ φ 0 and n φ ( t 0 ) = n φ 0 as Therefore the general solution of Eq. (71) is given by where α = √ Γ 1 Λ N 1 , and By assuming that the newly created dark matter particles are pressureless, p DM = 0, and the energy density of the dark matter component is ρ DM = m DM n DM , where m DM is the mass of the dark matter particle, then the evolution equation for the dark matter density can be written as By neglecting the effects of the expansion of the Universe the dark matter energy increases as", "pages": [ 7 ] }, { "title": "C. Effects of the cosmological expansion on the dark energy-dark matter interaction", "content": "In the general case, in which the expansion of the Universe is also taken into account, the system of equations describing the irreversible dark energy-dark matter interaction are given by where, for simplicity, we have assumed Γ 1 = Γ 2 = constant. By introducing a set of dimensionless variables ( r φ , N φ , r DM , τ ), defined as the field equations Eqs. (79)-(82) can be written in a dimensionless form as respectively, where The deceleration parameter q in the irreversible interacting dark energy-dark matter model is given by The density parameters Ω φ and Ω DM of the dark energy and of the dark matter are given by and turns to be above 100 GeV. Dark matter particles with masses of the order of keV necessarily produce cored density profiles, while WIMPs ( m ∼ 100 GeV, T d ∼ 5 GeV) inevitably produce cusped profiles at scales about 0.003 pc. Therefore, based on this analysis in the following we adopt a value of 1 keV for the mass of the dark matter particle. Some possible dark matter candidates with masses in this range would be the sterile neutrino, the gravitino, the light neutralino, the majoron etc. [23]. By normalizing the value of the scale factor so that at the present time t pres the scale factor is a ( t pres ) = 1, the redshift z is given by z = (1 -a ) /a . We consider the effects of the dark energy-dark matter interaction in the recent Universe, starting from z = 2, corresponding to t = 0, which fixes the initial value of the scale factor as a (0) = 0 . 33. Moreover, we consider that at z = 2 the universe was composed of an equal amount of dark energy and dark matter so that r φ (0) = r DM (0), and for the numerical calculations we choose r φ (0) = r DM (0) = 1 . 5. The time variations of the scale factor a , of the scalar field particle number N φ , of the scalar field energy r φ , of the dark matter energy density r DM , of the deceleration parameter q , of the density parameter Ω φ , and of the density parameter Ω DM of the dark matter are rep- respectively, and they satisfy the relation Ω +Ω = 1. φ DM The dynamics of the interacting dark energy-dark matter system is determined by two control parameters, λ , and the mass m DM of the dark matter particle. The mass of the dark matter particle, as well as the decoupling temperature, was determined recently in [23]. By evaluating analytically the dark matter galaxy properties, as the halo density profile, the halo radius and the surface density, and by matching them to their observed values one can obtain the decreasing of the phase space density since equilibration till today, the mass of the dark matter particle and the decoupling temperature T d , and the kind of the halo density profile (core or cusp), respectively. The dark matter particle mass turns to be between 1 and 2 keV and the decoupling temperature T d Τ resented, for different values of the parameter λ , and for m DM = 1 keV, in Figs. 1-7. As one can see from Fig. 1, the interacting dark energydark matter filled Universe is an expansionary state, with the rate of the expansion, and the scale factor evolution, strongly dependent on the dimensionless parameter λ , which depends on the ratio of the (constant) scalar field potential Λ, and of the scalar field decay rate Γ 1 . Accel- Τ erated expansion can also be obtained in the framework of the present model, the creation pressure, corresponding to the irreversible decay of the scalar field, and the matter creation, can drive the Universe into a de Sitter type phase. The scalar field particle number decays during the cosmological evolution, as shown in Fig. 2. The decay rate strongly depends on the numerical value of λ . The dimensionless energy of the scalar field r φ , shown Τ Τ in Fig. 3, tends in the large time limit to the value 1, corresponding to ρ φ = Λ, and to a de Sitter type expansion. This shows that the decay of the scalar field is determined and controlled by the kinetic energy term of the field ˙ φ 2 / 2, which is the source of the dark matter particles creation. When the energy and the pressure of the scalar field are dominated by the scalar field potential Λ, ρ φ = -p φ = Λ, then ρ φ + p φ = 0, and from Eq. (39) Τ it follows that ρ φ = constant, and the scalar field energy cannot be converted any more into other type of particles. The energy density of the dark matter particles, presented in Fig. 4, increases in time due to the decay of the φ -particles. The time variation of the deceleration parameter q , presented in Fig. 5, shows that the Universe with irreversibly interacting dark energy-dark matter starts its evolution at z = 2 from a decelerating state, with q > 0, and with an initial value of the decelerating parameter of around q (0) ≈ 0 . 2. For the present choice of the parameters the Universe enters into an accelerating phase, with q < 0, at a redshift of around z ≈ 1 . 6, and reaches a de Sitter type expansionary phase at the present time, corresponding to a = 1, and z = 0, respectively. The time variations of the density parameters of the dark energy and of the dark matter are represented, for different values of λ , in Figs. 6 and 7, respectively. At t = 0 ( z = 2) dark energy and dark matter have the same values of the density parameters, Ω φ (0) = Ω DM (0) = 1 / 2. During the cosmological expansion in the redshift range 2 ≤ z ≤ 0, the density parameter of the dark energy increases to a value of Ω φ ( t pres ) ≈ 0 . 8, while at the same time the density parameter of the dark matter decreases to Ω φ ( t pres ) ≈ 0 . 2. For λ = 1 . 4, Ω φ ≈ 0 . 77, and Ω DM ≈ 0 . 23. These results are consistent with the latest observational determinations of the composition of the Universe, which give Ω φ ( t pres ) ≈ 0 . 73, and Ω DM ( t pres ) ≈ 0 . 228 [24].", "pages": [ 7, 8, 9, 10 ] }, { "title": "V. DISCUSSIONS AND FINAL REMARKS", "content": "In the present paper we have shown that the thermodynamics of the open systems is a valuable tool for describing the interacting dark energy - dark matter phases of the general relativistic cosmological models. As applied to a two component (scalar field and dark matter) cosmological model, the thermodynamics of irreversible processes provides a generalization of the elementary theory of dark energy-dark matter interaction, which envisages that during the dark energy dominated phase of the expansion of the Universe, when the expansion slows down, the energy stored in the zero mode oscillations of the scalar field transforms into particles via single particle decay. Thus the model presented in this paper gives a rigorous thermodynamical foundation, and a natural generalisation, to the theory of dark energy-dark matter interaction. Particle decay (creation) gives rise to a supplementary decay (creation) pressure which has to be included as a distinct part in the stress-energy tensor of the cosmological mixture. We have considered only particle creation in a scalar field (dark energy) dominated Universes, leading to a model in which particle production rate is extremely high at the beginning of the oscillatory period, and afterwards it tends to zero, when the scalar field energy and pressure become dominated by the scalar field potential, assumed to be a constant, and when the kinetic energy of the scalar field becomes negligibly small. Such a dark matter particle production has very important implications on the dynamics and evolution of the Universe. The details of the dark energy-dark matter interaction mechanism depend on the parameters of the particle physics models involved to describe the newly created particles. One of the parameters is the dark matter particle mass, which is a key parameter in the description of the scalar field-dark matter process. Unfortunately presently there is no definite answer giving the value of the mass of the dark matter particle. Particle creation can be related to what is called the arrow of time: something that provides a direction to time, and distinguishes the future from the past. There are two different arrows of time: the thermodynamical arrow of time, the direction in which entropy increases, and the cosmological arrow of time, the direction in which the Universe is expanding. Particle creation introduces asymmetry in the evolution of the Universe, and enables us to assign a thermodynamical arrow of time, which agrees, in our model, with the cosmological one. This coincidence is determined in a natural way by the decay of the scalar field, due to the presence of a friction force between field and matter. In our present approach we have neglected the backreaction of the newly created particles on the dynamics of the Universe, and the effect of the form of the scalar field potential U ( φ ) have not been completely envisaged. These aspects of the thermodynamic theory of the dark energy - dark matter interaction will be the subject of a future work.", "pages": [ 11 ] }, { "title": "Acknowledgments", "content": "FSNL acknowledges financial support of the Funda¸c˜ao para a Ciˆencia e Tecnologia through the grants CERN/FP/123615/2011 and CERN/FP/123618/2011. Rev. D 74 , 103007 (2006); N. J. Nunes and D. F. Mota, Mon. Not. R. Astron. Soc. 368 , 751 (2006); Z.-K. Guo, N. Ohta, and S. Tsujikawa, Phys. Rev. D 76 , 023508 (2007) M. Quartin, M. O. Calvao, S. E. Joras, R. R. R. Reis, and I. Waga, JCAP 0805 , 007 (2008); J. Valiviita, E. Majerotto, and R. Maartens, JCAP -bf 0807, 020 (2008); C. G. Boehmer, G. Caldera-Cabral, R. Lazkoz, and R. Maartens, Phys. Rev. D 78 , 023505 (2008); H. Garcia-Compean, G. Garcia-Jimenez, O. Obregon, and C. Ramirez, JCAP 0807 , 016 (2008); E. N. Saridakis, P. F. Gonzalez-Diaz, and C. L. Siguenza, Class. Quant. Grav. -bf 26, 165003 (2009); T. Barreiro, O. Bertolami, and P. Torres, Mon. Not. R. Astron. Soc. 409 750 (2010); M. Jamil, E. N. Saridakis, and M. R. Setare, Phys. Rev. D 81 , 023007 (2010); J.-H. He, B. Wang, and E. Abdalla, Phys. Rev. D 83 , 063515 (2011); L. P. Chimento and M. G. Richarte, Phys. Rev. D 85 127301 (2012); A. Aviles and J. L. Cervantes-Cota, Phys. Rev. D 83 , 023510 (2011); S. M. R. Micheletti, Phys. Rev. D 85 , 123536 (2012).", "pages": [ 11, 12 ] } ]

[ { "title": "The issue of zeroth law for Killing horizons in Lanczos-Lovelock gravity", "content": "Sudipta Sarkar ∗ and Swastik Bhattacharya † The Institute of Mathematical Sciences, Chennai, India (Dated: December 28, 2018) We study the zeroth law for Killing horizons in Lanczos-Lovelock gravity. We show that the surface gravity of a general Killing horizon in Lanczos-Lovelock gravity (except for general relativity) may not be constant even when the matter source satisfies dominant energy condition. General relativity (GR), being quantum mechanically non-renormalizable, may make sense as a Wilsonian effective theory working perturbatively in powers of the dimensionless small parameter G ( Energy ) D -2 , where G is the D -dimensional Newton's constant. Then the Einstein-Hilbert Lagrangian is the lowest order term (other than the cosmological constant) in a derivative expansion of generally covariant actions for a metric theory, and the presence of higher curvature terms is presumably inevitable. In general, the specific form of these terms will depend on the detailed features of the quantum gravity model. Still, from a purely classical point of view, a natural modification of the Einstein-Hilbert action is to include terms preserving the diffeomorphism invariance and still leading to an equation of motion containing no more than second order time derivatives. Interestingly, this generalization is unique [1, 2] and goes by the name of LanczosLovelock gravity. Lanczos-Lovelock gravity is free from perturbative ghosts [3] and leads to a well-defined initial value formalism [4]. The lowest order Lanczos-Lovelock correction term in space time dimensions D > 4, namely the Gauss-Bonnet term, also appears as a low energy α ' correction in case of heterotic string theory [3, 5]. Hence, it is interesting to pursue various classical and semi classical properties of Lanczos-Lovelock gravity. For example, the striking similarity of the laws of black hole mechanics with thermodynamics was first established in the case of general relativity [6] and a natural question is to ask whether this analogy is a peculiar property of GR or a robust feature of any generally covariant theory of gravity. Studying the properties of black holes in a general Lanczos-Lovelock theory may provide a partial answer to this important question. The equilibrium state version of first law for black holes was established by Wald and collaborators [7, 8] for any arbitrary diffeomorphism invariant theory of gravity. The entropy of the black hole can be expressed as a local geometric quantity integrated over a space-like cross section of the horizon and is associated with the Noether charge of Killing isometry that generates the horizon. It is also possible to write down a quasi stationary version of the second law for Lanczos-Lovelock gravity [9, 10] which proves that the entropy of black holes in Lanczos-Lovelock gravity monotonically increases for physical processes in which the horizon is perturbed by the accretion of positive energy matter and the black hole ultimately settles down to a stationary state. On the other hand, the zeroth law of black hole mechanics which asserts the constancy of surface gravity has two independent versions. First, zeroth law can be established for stationary Killing horizons in GR, provided the matter obeys dominant energy condition [6]. The other version states that the surface gravity is constant on the horizon of a static or stationaryaxisymmetric black hole with the t -φ orthogonality property [11, 13]. The second version is entirely geometrical and independent of the field equation, whereas, the first version does not require the assumption of t -φ orthogonality property, but is only valid for GR, i.e. when Einstein's equation with matter obeying dominant energy condition holds. Motivated by the fact that both the first law and a quasi stationary second law hold true for LanczosLovelock gravity, we study the zeroth law for a general Killing horizon in Lanczos-Lovelock gravity. We would like to see that if one uses Lanczos-Lovelock equation of motion and suitable energy condition, then whether it is possible to prove the constancy of surface gravity without any assumption of extra symmetry. In fact, in this paper, we provide a negative answer to this question and show that the constancy of surface gravity does not hold in general even when the matter source obeys dominant energy condition. The paper is organized as follows: we first review the properties of Killing horizons. Next, we discuss the Lanczos-Lovelock theory and present the main result. Finally, we conclude with further discussions. We adopt the metric signature ( -, + , + , + , ... ) and our sign conventions are same as those of [12]. Also, we note that in general, a Killing horizon is not necessarily an event horizon. But there is a version of rigidity theorem [11] which states that for a static black hole, the static Killing field must be normal to the horizon, whereas for a stationary-axisymmetric black hole with the t -φ orthogonality property, there exists a Killing field which is normal to the event horizon. In case of GR, it is possible to show [14, 15] that the event horizon of a stationary black hole is also a Killing horizon with no assumptions of symmetries beyond stationarity. We are not aware of any such proof for Lanczos-Lovelock gravity. In a D -dimensional space time, a Killing horizon (not necessarily an event horizon) is a null hyper-surface H whose null generators are the orbits of a Killing field ξ a = ( ∂/∂v ) a , which is null on the horizon. Then there exists a function, surface gravity, ' κ ' of the Killing horizon which is defined by the equation, For static black holes, it is possible to provide a physical interpretation of the surface gravity. In that case, surface gravity of the black hole horizon measures the force which must be exerted at infinity to hold an unit mass at horizon. In general case, the surface gravity is the measure of the failure of the Killing time to be the affine parameter along the horizon null generators. From the definition in Eq.(1), it is straightforward to show that the surface gravity is constant along a generator [6, 12], i.e. ξ a ∇ a κ = 0. In general, surface gravity may vary from one generator to the other. Note that, the definition of surface gravity requires the notion of stationarity. There is no notion of surface gravity for a general non stationary dynamical horizon. The real significance of the surface gravity is realized when one consider quantum effects in a space time containing a black hole. The semi classical calculations by Hawking [16] showed that the black hole emits thermal radiation with a temperature (in units with G = c = k = 1), Hawking's result immediately shows the importance of the zeroth law of black hole mechanics as the zeroth law of black hole thermodynamics which states that the Hawking temperature is uniform everywhere on a stationary black hole horizon. This is reminiscent of the zeroth law of thermodynamics which states that the temperature is uniform everywhere in a system in thermal equilibrium. Since the constancy of the surface gravity along a generator of the horizon follows directly from the definition in Eq.(1), we will only discuss the change of ' κ ' from generator to generator. In order to proceed, we first construct a basis { ξ a , N a , e a A } on the Killing horizon where ξ a is the Killing field, N a is another null vector satisfying ξ a N a = -1, e a A , { A = 2 , ..., D -1 } are the ( D -2) space like vectors along the transverse directions and ξ b γ a b = N b γ a b = 0. Here γ a b is the induced metric of any space like slice of the horizon. We decompose the space time metric as g ab = g ⊥ ab + γ ab , where g ⊥ ab = -2 ξ ( a N b ) , is the metric of the two dimensional space orthogonal to any horizon cross section. Also, for stationary space times with a Killing horizon, both the expansion and shear vanish and using Raychaudhuri equation and the evolution equation for shear, we obtain [12, 17] that on the horizon, and We would like to emphasize that in order to derive these relationships, we have only used the fact that the horizon is a Killing horizon with zero expansion and shear without assuming any further symmetry. Next, we would like to study how the surface gravity changes from one generator to the other. For that, we note that from Eq.(1), we can write κ = -ξ a N b ∇ a ξ b and then we obtain [6], So far, all the results are entirely geometrical and no use of the equation of motion is made. Now, if we further assume that the horizon is axisymmetric and possesses t -φ orthogonality property, then it is possible to show [11, 13] that the RHS of Eq.(5) vanishes identically and the surface gravity is constant on the horizon independent of the field equation. Note that, if we assume that the Killing horizon possesses a bifurcation surface, i.e. a ( D -1) dimensional cross section on which the Killing field ξ a vanishes, then γ a b ∇ a κ = 0 on the bifurcation surface and since the surface gravity can not change along the generator, this will establish the constancy of surface gravity on the entire Killing horizon. Hence, for Killing horizons with regular bifurcation surface, the surface gravity is constant irrespective of gravitational dynamics [18]. But the assumption of the existence of bifurcation surface is a strong assumption and it is only applicable to a sub-class of Killing horizons. We would like to know if as in the case of general relativity, the constancy of surface gravity of Killing horizons in Lanczos-Lovelock gravity can be established without these assumptions. Let us now turn our attention to the features of Lanczos-Lovelock gravity. As discussed before, a natural generalization of the Einstein-Hilbert Lagrangian is provided by the Lanczos-Lovelock Lagrangian, which is the sum of dimensionally extended Euler densities, where the α m are arbitrary constants and L D m is the m -th order Lanczos-Lovelock term given by, where R cd ab is the D dimensional curvature tensor and the generalized alternating tensor δ ... ... is totally antisymmetric in both set of indices. For D = 2 m , 16 π L D m is the Euler density of 2 m -dimensional manifold. The Einstein-Hilbert Lagrangian is a special case of Eq. (7) when m = 1. The field equation of Lanczos-Lovelock theory is, G ab + α m E ( m ) ab = 8 πT ab where, and m ≥ 2. For convenience, we have written the GR part (i.e. for m = 1) separately so that the GR limit can be easily verified by setting all α m 's to zero. Lanczos-Lovelock gravity can be regarded as a natural and well behaved extension of general relativity in higher dimensions. The spherically symmetric black hole solution in Lanczos-Lovelock gravity is derived in [19, 20]. The first law of black hole mechanics is studied in [21] and various thermodynamic properties of these black hole solutions are discussed in [22]. Now, using the field equation of Lanczos-Lovelock gravity, we rewrite Eq.(5) as, Next, to simplify the above expression, we would first like to show that for a Killing horizon, E a ( m ) b ξ a ξ b = 0. In order to prove that, we first expand the curvature tensor in the basis { ξ a , N a , e a A } on the horizon. Therefore, we write, Now, as mentioned earlier, we express the space time metric as g ab = -2 ξ ( a N b ) + γ ab . Also, stationarity ensures that some of the components are zero due to conditions Eq.(3) and Eq.(4). Then we can express, We again remind the reader that this expression is valid only on the horizon. We also used the antisymmetry of the generalized alternating tensor δ ... ... . Now any component of a curvature tensor along the direction of the Killing vector in the expression of E a ( m ) b will not contribute when contracted by ξ a ξ b . These constraints ensure that the only surviving contribution will be from the transverse components and as a result we get, where, indices ( A 1 , B 1 , C 1 , ... ) only take transverse values. Next, we explicitly break the alternating tensor as the totally antisymmetric product of the Kronecker delta. For example, we write, Note that, when contracted by ξ a ξ c , the first term vanishes and also the rest of the terms do not contribute if all other indices are projected along the transverse directions. Using this rule of expansion, we immediately see that on the horizon, E a ( m ) b ξ a ξ b = 0. Then, Eq.(3) and the field equation give that on the horizon T a b ξ a ξ b = 0. Now, if the energy-momentum tensor obeys the dominant Energy Condition [12], T a b ξ b will be a non-space like vector. But on the horizon, we have seen that the field equation implies T a b ξ a ξ b = 0. Therefore, to obey dominant energy condition T a b ξ b must be in the direction of ξ a only and as a result ξ a T ac γ c b = 0. So, we ultimately arrive at, From the above equation, setting α m = 0, we can obtain the result of [6], which proves the constancy of surface gravity for GR. We will now show that on the Killing horizon, ξ a E ( m ) ac γ c b does not vanish identically unless one imposes additional constraints on the geometry of the horizon. To prove this, we again consider the expansion of the curvature tensor R c 1 d 1 a 1 b 1 in the basis { ξ a , N a , e a A } on the horizon and use Eq.(3) and Eq.(4). Due to the antisymmetry of the generalized alternating tensor δ ... ... , any component along ξ a 1 or ξ b 1 in the expression Eq.(11), will not contribute when contracted by ξ a . Then, the only non zero contributions of the expansion of R c 1 d 1 a 1 b 1 in the basis { ξ a , N a , e a A } are, We now consider products of two curvatures of the form R c m -1 d m -1 a m -1 b m -1 R c m d m a m b m . Due to the antisymmetry of the generalized alternating tensor δ ... ... , the products of the com- onents along the direction of the Killing vector will not contribute and the non vanishing contributions in the product R c m -1 d m -1 a m -1 b m -1 R c m d m a m b m can be expressed as, Continuing in this way, we can express the product of m -curvature tensors appearing in ξ a E ( m ) ac γ c b as, This entire expression is contracted with the alternating tensor ξ a γ c b δ ... ... . Again, the expansion of the alternating tensor in Eq. (13) ensures that the first term in the above expansion is zero when contracted by ξ a γ c b . Also, using the expansion rule in Eq.(13), it is straightforward to see that the only non-vanishing contribution comes from the last term in Eq.(17), given by, Again, using the rules of expansion for the alternating tensor, it is straightforward to show that the only non zero contribution from this term is of the form, Since, for stationary black holes, both the expansion and shear vanish, we can write [12], R CD AB = ( D -2) R CD AB , where, ( D -2) R CD AB is the intrinsic curvature of the cross section of the horizon. Then, we recall the expression for the equation of motion of a ( m -1)-th order Lanczos- Lovelock theory constructed using intrinsic curvatures of the horizon cross section, which is given by, Using this expression, we finally arrive at, Eq. (21) is our final expression and the right hand side of this equation does not vanish in general. So, unlike general relativity, for any higher order Lanczos-Lovelock theory of gravity, the surface gravity may vary from one generator to another on the horizon. As a result, although the surface gravity is constant along a single generator (i.e. surface gravity is independent of the affine parameter λ of the horizon), it may depend on the angular coordinates and vary as one moves across the generators. In case of general relativity, the constancy of surface gravity for Killing horizons can be proved without any other assumption. Then, the rigidity theorems [14, 15] ensure that every stationary event horizon in GR must be a Killing horizon and this in turn, proves the constancy of the surface gravity for stationary black holes. Our work shows that higher order Lanczos-Lovelock terms do not share this property and if there exits a stationary solution of Lanczos-Lovelock gravity with Killing horizon, which is not axisymmetric with t -φ symmetry, the surface gravity in general will be a function of the angular coordinates on a cross section of the horizon. Let us now consider the special cases. First of all, if the cross section of the horizon is flat, i.e when the horizon topology is planer, all the intrinsic curvature tensors of the horizon cross section vanish and as a result, the surface gravity will not vary from one generator to another. Also, if R pq b r N p ξ q ξ r = 0 on the horizon, the surface gravity is again constant. A possible way to achieve this is to consider a stationary axisymmetric horizon with t -φ isometry. For example, using equation 2 . 18 and 2 . 27 in [17], we can show that if the expansion and shear vanish, as in the case of a stationary black hole, then we have, R pq Br N p ξ q ξ r = ξ a R ac γ c b , for a stationary horizon. Now, a sufficient condition which ensures that ξ a R ac γ c b = 0, is the existence of a stationary and axisymmetric horizon with t -φ isometry [12]. This is basically the result obtained in [13]. Also, so far, we do not have any stationary solution of Lanczos-Lovelock gravity except general relativity. But, at least for general relativity with Gauss Bonnet correction terms, we have explicit spherically symmetric solutions [19, 20]. Given the quasi-linearity of the field equations of Lanczos-Lovelock gravity, it is quite possible that in this theory, a stationary solution will be found. If such solutions exist, our work shows that these black hole solutions will have non-constant surface gravity unless they are axisymmetric with t -φ symmetry. Also, once we consider quantum effects, the surface gravity is proportional to the Hawking temperature of the black hole and hence, if the surface gravity is no longer constant on the horizon and varies from one generator to another, then we can not treat such a stationary black hole as a system in thermodynamic equilibrium. But, there is still a possibility that as in the case of general relativity, all stationary horizons in a LanczosLovelock theory are axisymmetric with t -φ isometry. If that happens, then the zeroth law will be valid automatically. In order to investigate this, one needs to try for a generalization of the strong rigidity theorem [14, 15] for Lanczos-Lovelock gravity. The proof of the rigidity theorem depends on the initial value formulation of Einstein's equations. Since, the field equations of Lanczos-Lovelock gravity are also second order in time, and as a result the initial value formalism is well defined, it is reasonable to expect the validity of rigidity theorem for Lanczos-Lovelock gravity. Another obvious generalization of our work will be to study the zeroth law of Killing horizons in a general diffeomorphism invariant theory of gravity. Although, the techniques used in this work are to some extent specific to only Lanczos-Lovelock gravity, still we can provide some requirements which will ensure the validity of the zeroth law for any diffeomorphism invariant theory. To begin with, let us consider a general diffeomorphism invariant theory of gravity described by an Lagrangian L . Suppose, the field equation of the theory is given by E ab = 8 πT ab , where E ab represents a covariantly conserved, symmetric tensor obtained from the variation of the Lagrangian L . Then, from Eq.(5), we obtain, The zeroth law will hold if on the Killing horizon, following constraints are satisfied, In general, it is difficult to check these constraints for a general gravity theory, but if the above two conditions hold and the matter source obeys dominant energy then that will be enough to ensure the constancy of surface gravity on the Killing horizon of a stationary space time. In case of general relativity, both of these constraints are satisfied and the zeroth law holds true even for a general Killing horizon. For Lanczos-Lovelock theory, the first constraint holds, but the second one is not true in general and as a result, the surface gravity is no more constant on the horizon. In fact, it is also quite possible that the zeroth law does not hold for a general stationary black hole in some class of gravity theories. In that case, this may be useful as a criterion to select a sub class of diffeomorphism invariant actions as preferred theories where a consistent formulation of black hole thermodynamics is possible.", "pages": [ 1, 2, 3, 4, 5, 6 ] }, { "title": "Acknowledgments", "content": "We are especially grateful to Ted Jacobson for detailed comments on a previous draft of this article. We would like to thank Ghanashyam Date and T Padmanabhan for comments and discussions. We also thank the anonymous referee for various helpful suggestions to improve the presentation of the results. (1996) [gr-qc/9507055].", "pages": [ 6 ] } ]

[ { "title": "Stability of Einstein-Aether Cosmological Models", "content": "P. Sandin 1 , 2 ∗ , B. Alhulaimi 2 , and A. Coley 2 † 1 Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut) Am Muhlenberg 1, D-14476 Potsdam, Germany 2 Dept. of Mathematics, Dalhousie University, B3H 3J5 Halifax, Canada June 30, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "We use a dynamical systems analysis to investigate the future behaviour of Einstein-Aether cosmological models with a scalar field coupling to the expansion of the aether and a non-interacting perfect fluid. The stability of the equilibrium solutions are analysed and the results are compared with the standard inflationary cosmological solutions and previously studied cosmological Einstein-Aether models.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Einstein-Aether theory [1, 2] consists of general relativity coupled, at second derivative order, to a dynamical timelike unit vector field - the aether. It is one of several proposed models of early universe cosmology which incorporate a violation of Lorentz invariance [3]. In this effective field theory approach, the aether vector field, u a , and the metric tensor g ab together determine the local spacetime structure. We shall discuss the late time dynamics of Einstein-Aether cosmological models; in particular we explore the impact of Lorentz violation on the inflationary scenario [4], which provides one of the simplest ways to describe various aspects of the physics of the early universe in standard cosmology. More precisely, we study the inflationary scenario in the scalar-vector-tensor theory where the vector is constrained to be unit and time like, and investigate whether an example of the large class of the inflationary solutions proposed [2] are stable when spatial curvature perturbations are considered.", "pages": [ 1 ] }, { "title": "1.1 Einstein-Aether Cosmology", "content": "In an isotropic and spatially homogeneous Friedmann universe with expansion scale factor a ( t ) and comoving proper time t , the aether field will be aligned with the cosmic frame and is related to the expansion rate of the universe. The Einstein equations are generalized by the contribution of an additional stress tensor for the aether field. If the universe contains a single self-interacting scalar field φ (e.g., a scalar inflation which would dominate in any inflationary epoch), with a self interaction potential V that can now be a function of φ and the expansion rate θ = 3˙ a/a = 3 H, then the modified stress tensor is [1] This corresponds to an effective fluid with pressure p and density ρ of the form ρ = 1 ˙ φ 2 + V -θV θ and p = 1 ˙ φ 2 -V + θV θ + ˙ V θ , where V ( φ, θ ). 2 2 The energy-momentum conservation law or Klein-Gordon eqn. is The augmented Friedmann equation (with the addition of the aether stress to the energy density, and where 8 πG = 1 = c , and G is the renormalized gravitational constant) is given by: where k is the curvature parameter, and the Friedmann metric is given by [In [2] k was set equal to zero]. The Raychaudhuri eqn. follows from the differentiation of the Friedmann eqn.", "pages": [ 1, 2 ] }, { "title": "1.2 Exponential Potentials", "content": "Exponential potentials of the form V 0 e -λφ arise naturally in various higher dimensional frameworks, such as in Kaluza-Klein theories and supergravity [5], and the dynamical properties of the positive exponential potentials leading to inflation in the Friedmann-Robertson-Walker (FRW) model have been widely studied [6, 7, 8]. In Einstein-Aether theories there might exist a coupling between the scalar field and the aether field through the aether expansion scalar θ , which requires one to consider more general types of potentials than the standard exponential form. In [2] Barrow proposes an ansatz of the form where V 0 , λ and { a r } are constants, and concludes that there exist power law solutions of the Einstein-Aether-generalized combined Friedmann-Klein Gordon system on the form where B is a solution of the polynomial equation He presents two potentials for which explicit solutions to (9) can be found, namely potentials where a i = 0 ∀ i and potentials where only a 2 is different from zero. This corresponds to, respectively, the ordinary exponential potential and an exponential potential plus a term quadratic in the expansion. We shall consider one of Barrow's examples where the exponential part couples to the expansion scalar.", "pages": [ 2, 3 ] }, { "title": "2 The Model", "content": "We shall study the case: where, for convenience, we have renormalized the constant a 1 (the constant a 2 can be absorbed [1] and will not play an essential role in the dynamical analysis). The potential V ( θ, φ ) can be assumed to be positive definite, but this does not imply that the constants a 1 , a 2 necessarily are positive, as can be seen in the positive definite potential ( e -1 2 λφ -1 2 θ ) 2 where a 1 is negative and a 2 = a 1 2 / 4. If a 1 , a 2 are small, the potential can be thought of as a perturbation of the standard exponential potential. For very large constants a 1 , a 2 we can study non-perturbative generalizations. For the potential (10) the augmented Friedmann equation (3) becomes where we have normalized the equation with a factor proportional to the square of the expansion. The normalized Friedmann equation suggest a suitable set of expansion normalized variables: assuming that V 0 is a positive constant and that a 2 is larger than -1 / 3. In terms of these variables the Friedmann equation assumes the simple form and the Raychaudhuri equation (expressed in terms of the deceleration parameter q ) becomes Using the Raychaudhuri equation one can express the Klein-Gordon equation as a first order ordinary differential equation completely in terms of the expansion normalized variables, and an expansion normalized time: dτ dt = 3 θ -1 : where the constants ¯ a and ¯ λ are defined through The evolution equation for Φ is directly given from the definition of Φ and the Klein-Gordon and Raychaudhuri equations: The equations (15),(16) constitute an autonomous system of first order differential equations. The invariant set Φ = 0 corresponds to a model with a free scalar field, and it divides the two-dimensional (2D) state space into a region with an expanding universe (Φ ≥ 0) and one corresponding to a contracting universe (Φ ≤ 0). The curvature K is determined from the Friedmann equation (13), but an auxiliary evolution equation can be derived from (15),(16): dK/dτ = [4Ψ 2 -2Φ 2 -2¯ a ΨΦ] K , showing that K = 0 is an invariant set that also partitions the state space into two disjoint regions: a bounded negative curvature region (Ψ 2 +Φ 2 < 1), and an unbounded positive curvature region (Ψ 2 +Φ 2 > 1). Note that a 2 does not explicitly appear in the equations; it corresponds to a term in the potential proportional to the square of the expansion and can be absorbed by a rescaling of the expansion scalar, but the expansion normalized system is invariant under such rescalings. Defining | ¯ A | = √ (9 -¯ λ 2 +¯ a 2 ), the equilibrium points ( p i ) of the system are given by The stationary solution p 6 has Φ < 0 for all values of ¯ a ∈ R , ¯ λ ∈ R + , and corresponds to a contracting universe. We shall in the following only consider expanding solutions with vanishing or negative curvature and will therefore ignore this point since it lies outside the region of interest K ≤ 0 , Φ ≥ 0. Points p 1 and p 2 always satisfies these conditions. p 1 is always a saddle, p + 2 saddle or source, and p -2 always a source. We are most interested in the equilibra p 3 , p 4 , and p 5 . The points p 3 , p 4 , and p 5 are contained in this region only for a restricted, partially overlapping, range of values in ( ¯ λ, ¯ a )-space.", "pages": [ 3, 4, 5 ] }, { "title": "Equilibrium Point p 3 :", "content": "The point remains a sink if, in addition to the conditions in the previous section, also Solving for ¯ λ 2 we get For γ = 2 3 this condition reduces to which is the same restriction as before. Since the r.h.s. of (25) is increasing with γ we have that p 3 is a sink in the same range as before for all γ ≥ 2 3 .", "pages": [ 8, 9 ] }, { "title": "Equilibrium Point p 4 :", "content": "The third eigenvalue is negative if which implies For γ = 2 3 this inequality also coincides with with one of the previous restrictions on ¯ λ required for p 4 to be a sink. Larger γ gives a weaker restriction and hence is p 4 a sink in the same range as before if we have γ ≥ 2 3 .", "pages": [ 9 ] }, { "title": "Equilibrium Point p 5 :", "content": "Range of validity: Eigenvalues: Discussion: The point p 5 is a sink when ¯ λ 2 > 1 2 (¯ a 2 +6 -¯ a √ ¯ a 2 +8). Figure 1 shows the relevant properties of the future attractor of the system and the stability properties of the equilibrium pints p 3 , p 4 , p 5 .", "pages": [ 6 ] }, { "title": "3 The Model with Matter:", "content": "If, in addition to the scalar field, there also exists a barotropic perfect fluid with linear equation of state, p = ( γ -1) ρ (satisfying 2 3 < γ < 2), we will get extra terms in the equations corresponding to the fluid energy density. Assuming that there is no transfer of energy between the scalar field and the fluid except through gravitation, we get an additional evolution equation for the fluid energy density coming from the matter energy conservation equation. A normalized energy density is defined through The augmented Friedmann and Raychaudhuri equations become and the evolution equations become: Note that Ω = 0 defines an invariant set of the three-dimensional autonomous system of first order differential equations. The equilibrium points p i with Ω = 0 from the previous section are also equilibrium points of the extended system (22)-(24). Linearization about the points will result in the same eigenvalues as before for the eigendirections contained in the Ω = 0 subspace, but will also pick up a new eigenvalue from the influence of the fluid on the dynamics. The additional eigenvalues are: Equilibrium points p 1 , p 2 , and p 5 all gain an eigenvalue that is negative for all γ > 2 3 . The new eigenvalue transforms the source(s) to saddle(s) but retains p 5 as a sink in the same range as before. How it affects the points p 3 and p 4 is not immediately obvious, but as we shall see below it does not affect the properties of the sinks when γ is in the range ( 2 3 , 2).", "pages": [ 7, 8 ] }, { "title": "3.1 Equilibrium Points with Ω = 0", "content": "glyph[negationslash] There are also additional equilibrium points with Ω = 0: Corresponding to the flat FRW solution is glyph[negationslash] with eigenvalues Hence p 7 is a saddle in the entire range considered, 2 3 < γ < 2.", "pages": [ 9 ] }, { "title": "Equilibrium Point p 8 :", "content": "There is an equilibrium point, p 8 , representing the Matter Scaling Solution [8]. The point p 8 has zero curvature and the deceleration parameter is given by which is negative for γ > 2 3 . Linearization of the system (22) - (24) about the equilibrium point p 8 yields the eigenvalues: where D = 81 γ 2 ¯ λ 2 -108 γ ¯ λ 2 +36 ¯ λ 2 -324 γ 3 -72 γ 2 ¯ a 2 +648 γ 2 +180 γ ¯ a 2 -8¯ a 2 ¯ λ 2 +8¯ a 4 E = √ ¯ a 2 +9 -9(1 -γ ) 2 (36 γ 2 -108 γ +8 ¯ λ 2 -8¯ a 2 ) . The last eigenvalue is negative for all the values of γ that we consider in this paper. The value of µ + can be both positive and negative depending on the values of γ , ¯ a , and ¯ λ . But since the terms under the square root in E always are positive and hence E real, we have that the expression √ D +¯ a E is either positive or purely imaginary. The eigenvalue µ -will therefore always have negative real part, and the point p 8 will always be a saddle.", "pages": [ 9, 10 ] }, { "title": "4 Notes on More General Potentials", "content": "We also briefly take a look at more general potentials. In particular, we consider the potential of the form (5) with a single non-zero a r ( r an arbitrary integer): where a r here is normalized by an appropriate power of V 0 . Defining normalized variables ˜ Ψ and ˜ Φ as before, the normalized Friedmann eqn. becomes where ˜ a ≡ 1 2 (3) r 2 a r and we shall define ˜ λ = 1 √ 2 λ . The normalized Raychaudhuri equation then becomes: and the evolution eqns. become For r = 1 these equations reduce to the equations studied earlier. Let us consider equilibrium points with zero curvature. If K = 0, we obtain from (31) glyph[negationslash] Setting the right-hand sides of (33),(32) to zero to obtain the equilibrium values ˜ Φ , ˜ Ψ and using (34), assuming ˜ Φ = 0, we find after some algebra that ˜ Φ satisfies the polynomial eqn.: whence we obtain It can easily be checked that these values for ˜ Φ, ˜ Ψ do indeed satisfy the zerocurvature condition (34) and are equilibrium values of the system (33). For vanishing ˜ a , we obtain the usual zero-curvature inflationary power-law equilibrium solution P in standard exponential potential cosmology ( ˜ Φ = √ 1 -1 3 ˜ λ 2 , ˜ Ψ = ˜ λ √ 3 ), which is asymptotically stable to the future. Linearization of the system (32), (33) is more complicated with a general power r , but an approximate linearization for small ˜ a around the zero-curvature solution is possible 1 .", "pages": [ 10, 11 ] }, { "title": "5 Discussion", "content": "We have analyzed the dynamical evolution and stability of inflationary solutions of homogeneous and isotropic Einstein-Aether cosmologies containing a scalar field, under the assumption that the scalar field interacts with itself and the aether through a potential given by (10). The potential is of the general form (5) proposed in [2] and more general than the examples explicitly studied therein. We find that the scalar field-aether interaction term only slightly affects the stable solution of the system for small values of the normalized scalar field self interaction coupling ¯ λ , even when the normalized scalar field-eather coupling ¯ a is large. When large scalar field self interactions are considered there is a qualitative change in the future stable equilibrium of the system compared to the ordinary exponential potential case. For negative values of the coupling ¯ a there exists a region (region II, Fig. 1) where the stable equilibrium is spatially flat but non-inflationary, a situation that in the normal case requires fine tuning of ¯ λ . Even larger scalar field self interactions destabilizes the flat equilibrium and drives the state towards a spatially curved equilibrium, like in the normal exponential potential case. When we also include a matter source term in the form of a perfect fluid with linear equation of state that does not couple directly to the other fields, we find equilibrium states corresponding to the usual FRW model and matter scaling solution. The precise value of the normalized field variables of the latter equilibrium depend on the coupling ¯ a , but the qualitative properties like its curvature and deceleration parameter do not. The equilibrium point is a saddle when curvature perturbations are considered, like in the normal case, but even within the flat models it can remain a saddle for some values of ¯ λ and ¯ a , unlike the case with an ordinary exponential potential where the matter scaling solution is a late time attractor. The matter source term do not qualitatively alter the stability of the sinks found earlier, they all obtain a stable manifold of one dimension larger than previously and thus remain sinks. Also potentials with a coupling between the exponential part and the expansion of general order are susceptible to a dynamical systems analysis in the scale invariant variables. The problem of finding a scale invariant solution with zero curvature can be reduced to solving a polynomial equation, and the solutions all reduce to the standard inflationary power-law solution when the coupling becomes arbitrarily small.", "pages": [ 11, 12 ] }, { "title": "Acknowledgments:", "content": "We thank M. Stevens for helpful insights on the general r potential. PS also thanks AAC and the department of mathematics at Dalhousie University for their kind hospitality. AAC is supported by grants from the Natural Sciences and Engineering Research Council of Canada. BA would also like to thank the Government of Saudi Arabia for financial support.", "pages": [ 12 ] } ]

[ { "title": "Black hole motion in Euclidean space as a diffusion process II", "content": "K. Ropotenko State Service for Special Communication and Information Protection of Ukraine, 13 Solomianska str., Kyiv, 03680, Ukraine [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "A diffusion equation approach to black hole thermodynamics in Euclidean sector is proposed. A diffusion equation for a generic KerrNewman black hole in Euclidean sector is derived from the Bloch equation. Black hole thermodynamics is also derived and it is found, in particular, that the entropy of a generic Kerr-Newman black hole is the same, apart from the logarithmic corrections, as the BekensteinHawking entropy of the black hole. In [1], I derived a diffusion equation for a Schwarzschild black hole from the Bunster-Carlip equations and showed that the black hole evolution in Euclidean sector exhibits a diffusion process. Namely I showed that the Bunster-Carlip equations where t is the lapse of asymptotic proper time at spatial infinity and Θ is the lapse of the hyperbolic angle at the horizon, transform to the equation for the probability density ρ = | ψ ( t E , Θ E ) | 2 under Euclidean continuation Θ E = i Θ and t E = it . In general, the Euclidean time coordinate t E behaves like a spatial coordinate. But (3) is an one-dimensional diffusion equation in the temporal Θ E and spatial t E coordinates with the diffusion coefficient D = 2 G ¯ h . So I reinterpreted the Euclidean time coordinate t E as a spatial coordinate and Θ E as a temporal coordinate. In place of Θ E and t E I will hereafter write Θ and τ . After analytical continuation to the cyclic imaginary time, we deal with a quantum system at a finite temperature, so the black hole should be described not by the probability density ρ ( x ) but by the canonical density matrix ρ ( x, x ' ; β ). Determining ρ ( x, x ' ; β ) I found that the entropy of a Schwarzschild black hole is the same, apart from the logarithmic corrections, as the Bekenstein-Hawking entropy. In the literature, the path integral approach is the only approach to black hole thermodynamics in Euclidean quantum gravity. In this note I propose an alternative diffusion equation approach. I derive a diffusion equation for the density matrix of a generic Kerr-Newman black hole in Euclidean sector immediately from the Bloch equation. I also derive black hole thermodynamics and find, in particular, that the entropy of a Kerr-Newman black hole is the same, apart from the logarithmic corrections, as the Bekenstein-Hawking entropy of the black hole. I begin with the Bloch equation [2] for a Schwarzschild black hole at a temperature T H = β -1 . The (unnormalized) canonical density matrix has the form ρ ( β ) = e -βH . Note that here, as in ordinary statistics, the form of the density matrix must be regarded only as a postulate, to be justified solely on the basis of agreement of its predictions with the thermodynamical properties of black holes. As before, I adopt the Euclidean approach and consider the black hole motion in Euclidean sector in the temporal Θ and spatial τ coordinates, so that ρ ( β ) = ρ ( τ, τ ' ; β ) in coordinate representation. For physical applications Θ should be set equal to 2 π , so that β = 2 π/k , where k is the surface gravity. But for the purposes of thermodynamic analysis I shall keep Θ arbitrary and put Θ = 2 π only in some final results. Therefore, (4) reads Next, I define the Hamiltonian of a black hole with the mass M as that of a free particle moving along the coordinate τ and obtain where D = ¯ h/ 2 kM . This is an one-dimensional diffusion equation, and we can write down its solution readily: where the proportionality factor is chosen such that The crucial property of the black hole solutions in Euclidean sector is their periodicity in the imaginary time, τ ∼ τ + β , where β = 2 π/k . Therefore For a linear system of length β = 2 π/k , the integration over τ gives This is the partition function for a Schwarzschild black hole. Differentiating with respect to the β and then putting Θ = 2 π , we obtain the internal energy It is the same, apart from the Hawking temperature, as the mass of the black hole. The entropy of the black hole is given by It is the same, apart from the logarithmic corrections, as the BekensteinHawking entropy S BH = A/ 4 l 2 P . Let us now consider a Kerr-Newman black hole at a temperature T H = β -1 . As is well known, in the near-horizon approximation the Euclidean sector of a Kerr-Newman black hole is similar to that of a Schwarzschild black hole. So, as in the case of a Schwarzschild black hole, one can introduce the corresponding temporal Θ and spatial τ coordinates for a Kerr-Newman black hole. I define the Hamiltonian of a Kerr-Newman black hole as that of a particle moving in the potential U = Ω J + 1 2 Φ Q , where all quantities have the standard meaning. Proceeding exactly as in the derivation of (7), we get the equation where D = ¯ h/ 2 kM and This is also a diffusion equation but with the convection term -bρ . The equation describes the probability flow in Euclidean sector of a Kerr-Newman black hole with both diffusion D∂ 2 ρ/∂τ 2 along the axis τ and the outflow -bρ in a direction perpendicular to the axis. How much diffusion and convection takes place depends on the relative size of the two coefficients D and b . The solution to the equation (at the initial condition (9)) is There is also an integral formula for solutions to (15), known as the FeynmanKac formula; it is an integral over path space with respect to Wiener measure [2]. Since, as in the Schwarzschild case, τ ∼ τ + β , where β = 2 π/k , the trace of (17) leads to This this the partition function for a Kerr-Newman black hole. Differentiating with respect to the β and then putting Θ = 2 π , we obtain the internal energy It is the same, apart from the Hawking temperature, as the mass of the black hole. Finally, the entropy of a Kerr-Newman black hole is given by It is the same, apart from the logarithmic corrections, as the BekensteinHawking entropy of the black hole. The statistical interpretation of the Bekenstein-Hawking entropy remains a central problem in black hole physics. The path integral approach in Euclidean quantum gravity cannot answer what the degrees of freedom are responsible for the entropy. In this note, the black hole motion in Euclidean sector is modeled as an one-dimensional diffusion in the coordinates τ and Θ. A random walk process is the basis of the diffusion. Thus, if the model is correct, then there should exist the black hole constituents with the Boolean degrees of freedom ± Θ or ± τ .", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity", "content": "Casey Tomlin a,b and Madhavan Varadarajan b Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Raman Research Institute, Bangalore-560 080, India a PA 16802-6300, U.S.A b September 21, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "The G Newton → 0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1) 3 gauge theory. The Poisson bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. Motivated by recent results in Parameterized Field Theory and by the search for an anomaly-free quantum dynamics for Loop Quantum Gravity (LQG), the quantum Hamiltonian constraint of density weight 4 / 3 for this U(1) 3 theory is constructed so as to produce a non-trivial LQG-type representation of its Poisson brackets through the following steps. First, the constraint at finite triangulation, as well as the commutator between a pair of such constraints, are constructed as operators on the 'charge' network basis. Next, the continuum limit of the commutator is evaluated with respect to an operator topology defined by a certain space of 'vertex smooth' distributions. Finally, the operator corresponding to the Poisson bracket between a pair of Hamiltonian constraints is constructed at finite triangulation in such a way as to generate a 'generalised' diffeomorphism and its continuum limit is shown to agree with that of the commutator between a pair of finite triangulation Hamiltonian constraints. Our results in conjunction with the recent work of Henderson, Laddha and Tomlin in a 2+1-dimensional context, constitute the necessary first steps toward a satisfactory treatment of the quantum dynamics of this model.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "A key open issue in canonical LQG relates to the definition of the Hamiltonian constraint operator. This operator is constructed as the continuum limit of its finite triangulation approximant [1, 2]. The latter is the quantum correspondent of a classical approximant which is uniquely defined only up to terms which vanish in the classical continuum limit wherein the triangulation of the spatial manifold is taken to be infinitely fine. In contrast to the classical continuum limit, the continuum limit of the quantum operator is not independent of the choice of finite triangulation approximant thus resulting in an infinitely manifold choice in the definition of the quantum dynamics of LQG. On the other hand, a necessary condition for the very consistency of the quantum theory is an anomaly free representation of the constraint algebra. Therefore, one possible way to restrict the choice of quantum dynamics is to demand that the ensuing algebra of quantum constraints is free from anomalies. Unfortunately, irrespective of the specific choice of quantum dynamics made in the current state of art in LQG, the quantum constraint algebra trivializes i.e. the commutator of a pair of Hamiltonian constraints as well as the operator corresponding to their classical Poisson bracket vanish in the continuum limit [3, 4, 5]. While it is remarkable that no obvious inconsistency arises, we believe that the situation is unsatisfactory for reasons we now elaborate. We refer to the commutator between two Hamiltonian constraints as the Left Hand Side (LHS) and the operator corresponding to their Poisson bracket as the Right Hand Side (RHS). While the LHS and the RHS both vanish in the continuum limit, they do so for very different reasons. The LHS vanishes because the second Hamiltonian constraint acts trivially on spin network deformations produced by the action of the first Hamiltonian constraint [3, 4]. In contrast, the RHS vanishes because there are too many powers of the parameter δ in its expression at finite triangulation, the continuum limit being defined by δ → 0. More in detail, the finite triangulation approximant to the RHS is built out of the basic operators of LQG as follows. The curvature is approximated by a small loop holonomy (divided by its area ∼ δ 2 ), the densitized triad by the electric flux through a small surface (divided by its coordinate area ∼ δ 2 ), and, powers of √ q by small region volumes (divided by δ 3 since √ qδ 3 ∼ volume operator). The lower the density of the Hamiltonian constraints in the LHS, the lower is the power of √ q in the RHS, and hence, the higher the overall power of δ in the RHS. For Hamiltonian constraints of density weight one, it is straightforward to see that one obtains an overall power of δ in the RHS which then kills the RHS as δ → 0 irrespective of its finer details . Thus one may expect that the consideration of higher density weight Hamiltonian constraints would yield a non-vanishing RHS with an LHS which still vanishes because of the independence of the successive actions of the Hamiltonian constraint alluded to above. Hence, it could well be the case that the current definitions of the Hamiltonian constraint are anomalous, the anomaly being hidden by the low density weight. 1 Our view that the current set of choices for the quantum dynamics of LQG may be physically incorrect, and that the consideration of higher density constraints is vital to obtain a non-trivial constraint algebra, is supported by recent work on Parameterized Field Theory (PFT) [6] and the Husain-Kuchaˇr model [7]. In these works the physically correct finite triangulation approximants to the constraints involve choices which are qualitatively different from those currently used. Indeed the approximants bear a qualitative similarity with the physically appropriate ones used in 'improved' LQC [8]. Moreover, the non-triviality of the quantum constraint algebra in these works is seen to be directly tied to the kinematically singular nature of the constraint operators which in turn are a consequence of the higher density nature of the constraints [6, 7]. Given this situation, our aim is to use the insights gained from the study of PFT and the Husain-Kuchaˇr model to construct higher density weight constraint operators for LQG which yield a non-trivial anomaly-free representation of the classical constraint algebra. While PFT and the Husain-Kuchaˇr model have proven to be immensely useful, they suffer from one structural oversimplification vis a vis gravity: Their constraint algebras are Lie algebras, unlike the gravitational constraint algebra, which has structure functions. Therefore, before attempting LQG with all its complications, it is advisable to tackle a simpler system whose constraint algebra bears more of a structural similarity with gravity. Just such a system has been proposed recently by Laddha and its quantum dynamics studied in a 2+1-dimensional context in [9, 10]. The system is obtained by replacing, in the phase space description of Euclidean gravity in terms of triads and connections, the triad rotation group SU(2) by the group U(1) 3 . The U(1) 3 model (in 3+1 dimensions) has three Gauss Law constraints, three spatial diffeomorphism constraints and a Hamiltonian constraint. The constraint algebra for the Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. In fact, it turns out that this system is exactly the G N → 0 limit of Euclidean gravity studied by Smolin in [11]. 2 In this work we initiate the investigation of the quantum dynamics of this U(1) 3 model in 3+1 dimensions with a view to obtaining a non-trivial representation of the Poisson bracket between a pair of Hamiltonian constraints. The work entails many new techniques and constructions and for simplicity we shall ignore issues of spatial covariance. Modifications to our constructions which incorporate spatial covariance will be discussed in a future publication [12], this work serving as a necessary precursor to that one. The layout of the paper is as follows. Section 2 describes the classical Hamiltonian formulation of the U(1) 3 model and provides a brief review of the U(1) 3 'charge' network representation which comprises its LQG-type quantum kinematics. In Section 3 we describe the main steps in our considerations so as to provide the reader with overall the logical structure of our work. In Section 4 we motivate and define the action of the Hamiltonian constraint at finite triangulation and compute the action of its commutator (at finite triangulation) on the charge network basis. In the last part of Section 4, we compute the continuum limit of this finite-triangulation commutator. The notion of continuum limits in LQG is a delicate one. In the literature two different definitions of the continuum limit exist, one through the specification of Thiemann's Uniform Rovelli-Smolin (URS) topology [3], and one through the specification of the Lewandowski-Marolf habitat [4, 3]. The continuum limit we use is, roughly speaking, an intermediate between the two, and can best be described in analogy to the case of the URS topology. The URS topology is a topology on the space of operators on the kinematic Hilbert space (the finite-triangulation constraint operators belong to this space) which is defined by a family of seminorms which, in turn, are specified by diffeomorphism-invariant distributions. These distributions do not lie in the kinematic Hilbert space but in the algebraic dual space. 3 The continuum limit is then specified in terms of Cauchy sequences of finite-triangulation operators in this topology. In the present work as well the continuum limit is specified in term of Cauchy sequences of finite triangulation operators. However, the operator topology is defined by a different subspace of the algebraic dual. As we shall see, examples of elements of this subspace are provided by rough analogs of the Lewandowski-Marolf habitat states [4, 6, 7] which we call 'vertex smooth algebraic' states (VSA states). 4 In Section 4, we obtain the continuum limit of the finite-triangulation commutator in the 'VSA' topology under certain assumptions about the space of VSA states. In Section 5 we construct the finite-triangulation operator which corresponds to the RHS. The construction is based on a remarkable classical identity which we derive in Section 5.1. As shown in Appendix B, the identity extends to the case of internal group SU(2) i.e. to the case of gravity and, hence, is of interest in its own right. To our knowledge this identity has not been noticed before. As in Section 4, we evaluate the continuum limit of the finite-triangulation operator for the RHS under certain assumptions on the space of VSA states. Section 6 is devoted to a proof that there exists a large space of VSA states subject to the assumptions of Sections 4 and 5. The final conclusion of our work in Sections 4 and 5 is that the continuum limits of the LHS and RHS agree in the VSA topology induced by the space of VSA states constructed in Section 6. This agreement is what we mean by an anomaly-free representation of the Poisson bracket between a pair of Hamiltonian constraints. Section 7 is devoted to a discussion of our results as well as an elaboration of open issues, the two key open issues being: (i) an improvement of our considerations so as to incorporate diffeomorphism covariance; (ii) the promotion of our VSA topology-based calculations to the context of a genuine habitat. We work with the semianalytic category in this paper so that the Cauchy slice Σ, coordinate charts thereon, its diffeomorphisms and the graphs embedded in it are semianalytic and C k , k /greatermuch 1.", "pages": [ 1, 2, 3, 4 ] }, { "title": "2 The U (1) 3 model", "content": "In Section 2.1 we obtain the Hamiltonian formulation of the U(1) 3 model from that of Euclidean gravity through Smolin's G N → 0 limit [11]. In Section 2.2 we briefly review its quantum kinematics in the polymer representation.", "pages": [ 4 ] }, { "title": "2.1 The Hamiltonian Formulation", "content": "Recall that Euclidean gravity is described, in its Hamiltonian formulation, by the action: Here E a i , A i a are the canonically conjugate densitized triad and SU(2) connection. The curvature of the connection is F i ab := ∂ a A i b -∂ b A i a + /epsilon1 i jk A i a A j b and D a is the gauge covariant derivative so that D a E a i = ∂ a E a i + /epsilon1 ijk A j a E a k . N,N a , Λ i are the (appropriately densitized) lapse, shift and internal gauge Lagrange multipliers. We have set the speed of light to be unity so that G N has dimensions [length][mass] -1 , A i a , Λ i have dimensions [length] -1 and the triad, lapse, and shift are dimensionless so that Equation (2.1) acquires the dimensions of action. Following Smolin, we define the rescaled connection A i a := G -1 N A i a so that the curvature takes the form F i ab = G N ( ∂ a A i b -∂ b A i a + G N /epsilon1 i jk A i a A j b ) and D a E a i = ∂ a E a i + G N /epsilon1 ijk A j a E a k . Rewriting the action in terms of the scaled connection and then setting G N = 0, it is easy to obtain: where are the Gauss law, diffeomorphism, and Hamiltonian constraints of the theory, and where F i ab := ∂ a A i b -∂ b A i a . Note that the Gauss law constraints generate three independent U(1) 3 gauge transformations on the connections A i a , i = 1 , 2 , 3 with gauge-invariant curvature F i ab and that the three electric fields E a i , i = 1 , 2 , 3 are gauge-invariant. Thus, the action (2.2) describes a U(1) 3 theory as claimed. The constraints G [Λ] , D [ /vector N ] , H [ N ] are first class. Their Poisson bracket algebra is The last Poisson bracket (between the Hamiltonian constraints) exhibits structure functions just as in gravity. Working towards a representation of this last Poisson bracket in quantum theory will occupy the rest of this work.", "pages": [ 4, 5 ] }, { "title": "2.2.1 The Holonomy-Flux Algebra", "content": "Let e be a C k , k /greatermuch 1 semianalytic, embedded edge e : [0 , 1] → Σ. An edge holonomy in the j th copy of U(1) is denoted by h e,q j with Here q j is an integer, κ is a constant of dimension [length][mass] -1 and γ is a positive real number. For fixed κ, γ , the edge holonomies for all edges and all values of the 'charges' q j form a complete set of functions of the connection A j a ; i.e., the knowledge of all these holonomies allows the reconstruction of A j a . We fix κ once and for all. We shall see below that γ is a Barbero-Immirizi-like parameter of the theory which labels inequivalent quantum representations. 5 The edge holonomy h e,/vectorq valued in U(1) 3 is defined to be the product of edge holonomies over the three copies of U(1): Given a closed, oriented graph α with N edges, the graph holonomy h α, { /vector q } := h α, { /vector q I | I =1 ,...,N } is just the product of the edge holonomies over the edges of the graph, so that It is easily verified that the graph holonomy h α, { /vector q } is invariant under U(1) 3 gauge transformations if and only if, for every vertex v of the graph α and for each i , where e I v ranges over the edges incident at v and τ ( I v ) is +1 if the edge is outgoing at v and -1 if ingoing. The labels α, { /vector q I | I = 1 , . . . , N } define a colored graph which we refer to as a charge network . A charge network c = c ( α, { /vector q I | I = 1 , . . . , N } ) is closed oriented graph whose edges are 'colored' by representation labels of U(1) 3 ; i.e., each edge e I is colored with the triple of charges ( q 1 I , q 2 I , q 3 I ) := /vector q I . If the charges satisfy Equation (2.14), we shall say that the charge network is gauge-invariant. 6 Thus, graph holonomies are labelled by charge networks and we may write h α, { /vector q } := h c . For future purposes it is useful to write the graph holonomy h c in the form where Here t I is a parameter which runs along the edge e I . Adapting the old terminology of Gambini and Pullin [13], shall refer to c a i ( x ) as a charge network coordinate . The gauge-invariant electric flux E i ( S ) through a two-dimensional oriented surface S is given by integrating the 2-form η abc E a i over S so that The only non-trivial Poisson bracket amongst the holonomy-flux variables is { h c , E i ( S ) } , which is readily computed: Here the graph α ( c ) underlying c is chosen to be fine enough that isolated intersection points of the graph with S are at its vertices and the integer /epsilon1 ( e I , S ) vanishes unless e I intersects S transversely in which case /epsilon1 ( e I , S ) = 1 if e I is outgoing from and above S or incoming to and below S and -1 otherwise. Unless indicated explicitly below, we will always assume that charge network edges are outgoing at vertices or relevant interior edge points.", "pages": [ 5, 6 ] }, { "title": "2.2.2 The Polymer Representation", "content": "An orthonormal basis for the kinematic Hilbert space is provided by 'charge network' states. To every distinct charge network label c we assign the unit norm charge network state | c 〉 ≡ | γ, { /vector q I }〉 . Two charge network states are orthogonal if and only if their charge network labels differ; i.e., if the colored graphs which label them are inequivalent. We denote this inner product between charge network states by where the Kronecker delta δ c ' ,c vanishes unless there is a choice of colored graph underlying c which is identical to a choice of colored graph underlying c ' in which case c = c ' and δ c,c ' = 1. Let the finite span of the charge network states be D . The Cauchy completion of D in the inner product (2.19) yields the kinematic Hilbert space H kin . The holonomy operators act as follows: The charge network c + c ' is defined as follows: Let α be a fine enough closed, oriented graph which underlies both c and c ' . Add the charge labels of c, c ' edgewise to obtain to new charge labels for α . This newly colored graph specifies the charge network c + c ' . The flux operators act as follows: It can be verified that the above operator actions provide a representation of the holonomy-flux Poisson bracket algebra on H kin . Finally note that, as in LQG, we may derive these operator actions by thinking, heuristically, of the charge network states as wave functions which depend on smooth connections via | c 〉 ∼ c ( A ) = h c ( A ) and by seeking to represent the holonomy operators by multiplication and the electric field operators by functional differentiation.", "pages": [ 6, 7 ] }, { "title": "3 Sketch of Overall Logical Structure", "content": "Our purpose in this section is to give the reader a rough global view of the logical structure of our considerations. In Section 3.1 we provide a brief sketch of the main steps in our work. Section 3.2 contains a precise definition of the continuum limit in terms of a topology on the space of operators and indicates the sense in which the implementation of the steps of Section 3.1 establishes the existence of a non-trivial anomaly-free representation of the constraint algebra. In Section 3.3 we briefly describe the various choices made in order to implement the steps of Section 3.1. To avoid unnecessary clutter we shall not worry about overall factors, both dimensional and numerical (only in this section!). As in LQG, we are faced with a tension between the local nature of the constraints of the model (most importantly the dependence on F i ab ) and the non-local and discontinuous nature of some of the basic operators of the quantum theory (namely the holonomy operators). Since there is no way to extract a connection (or curvature) operator out of the holonomy operators due to their discontinuous action with respect to any shrinking procedure applied to the loops which label them, one proceeds in close analogy to Thiemann's seminal work [1]. We fix a one-parameter family of triangulations T δ of the spatial manifold Σ where δ labels the fineness of the triangulation, with δ → 0 being the continuum limit of infinite refinement, construct finite triangulation approximants to the classical constraints, construct the corresponding operators and then take an appropriate continuum limit, the hope being that while individual operators may not possess a continuum limit, the conglomeration of operators which combine to form the constraint does possess a continuum limit.", "pages": [ 7 ] }, { "title": "3.1 Steps", "content": "Step 1. The finite-triangulation Hamiltonian constraint and its continuum limit: Let the Hamiltonian constraint at finite triangulation T δ be C δ [ N ]. C δ [ N ] is a discrete approximant to the Hamiltonian constraint C [ N ] (see, however, the remark after Step 4 below) so that lim δ → 0 C δ [ N ] = C [ N ]. Let the corresponding operator ˆ C δ [ N ] be such that ˆ C δ [ N ] : D → D where D is the finite span of charge network states. Let D ∗ be the algebraic dual to D so that every Ψ ∈ D ∗ is a linear map from D to C . Let | c 〉 be a charge network state. Then for every pair (Ψ , | c 〉 ) we compute the one-parameter family of complex numbers Ψ( ˆ C δ [ N ] | c 〉 ). The continuum limit action of ˆ C δ [ N ] is defined to be Step 2. Finite triangulation commutator and its continuum limit: Let T δ ' be a refinement of T δ so that δ ' < δ . Define a discrete approximant to C [ N ] C [ M ] by C [ N ] δ ' C [ M ] δ . The corresponding operator product is ˆ C [ N ] δ ' ˆ C [ M ] δ . The commutator at finite triangulation is then ˆ C [ N ] δ ' ˆ C [ M ] δ -ˆ C [ M ] δ ' ˆ C [ N ] δ and its continuum limit action is Step 3. RHS at finite triangulation and its continuum limit: Recall that the RHS, D [ /vector ω ], is just the diffeomorphism constraint smeared with a metric-dependent shift. One could define it at finite triangulation by some discrete approximant D δ [ /vector ω ]. Note that the LHS at finite triangulation, by virtue of the quadratic dependence of the commutator on the constraint, depends on the pair of parameters δ, δ ' . Clearly, a better comparison of the LHS and RHS would result if the RHS could also naturally accommodate a commutator description. Remarkably, it so happens that the classical expression for the RHS, can be written as the Poisson bracket between a pair of diffeomorphism constraints with triad dependent shifts. Specifically, we have that D [ /vector ω ] = ∑ 3 i =1 { D [ N i ] , D [ M i ] } where D ( N i ) is the diffeomorphism constraint smeared with the shift N a i which is constructed out of the lapse N and the electric field variable (see Section 5.1). Let D δ [ N i ] be a finite triangulation approximant to D [ N i ]. Then the finite-triangulation RHS operator can be written as ∑ i ˆ D [ N i ] δ ' ˆ D [ M i ] δ -ˆ D [ M i ] δ ' ˆ D [ N i ] δ and its continuum limit action is defined to be Step 4. Existence of the continuum limit for suitable algebraic dual states: We look for a large (infinite-dimensional) subspace D ∗ cont ⊂ D ∗ such that for every Ψ ∈ D ∗ cont and every charge network state | c 〉 , the limits (3.1), (3.2), and (3.3) exist with (3.2) = (3.3). Further we require that (3.2), and (3.3) do not vanish identically for every pair (Ψ , | c 〉 ). Remark: In accordance with Step 1 above we should first find a classical approximant to the classical constraints such that the approximant is built out of small edge holonomies and small surface fluxes (where the notion of smallness is defined by the finite triangulation parameter δ ). We should then replace the classical phase space functions by their quantum counterparts to obtain the constraint operator at finite triangulation. Instead, in Section 4 we directly motivate, through heuristic considerations, finite-triangulation quantum constraint operators. It is desirable that it be shown that these operators correspond to the quantization of classical finite triangulation approximants. Based on our experience with PFT and the HK model, we are fairly sure that this should be easy to do. However since this is one of the first attempts at obtaining a nontrivial representation of the constraint algebra we choose to press on and leave loose ends such as this to be tied up by future work.", "pages": [ 7, 8 ] }, { "title": "3.2 A Note on the 'Topology' Interpretation of the Continuum Limit", "content": "Given any operator ˆ O : D → D and a pair (Ψ , | c 〉 ) with Ψ ∈ D ∗ cont and | c 〉 being a charge network state, we may define the seminorm of the operator ˆ O to be || ˆ O || Ψ ,c = | Ψ( ˆ O | c 〉 ) | . The family of seminorms || || Ψ ,c for every pair (Ψ , | c 〉 ) defines a topology on the vector space of operators from D to itself. It is straightforward to check that the sequences of operators (indexed by δ , δ ' ) defined in the previous section can be interpreted as sequences which are Cauchy in this topology. Of course there is no guarantee that the limit of such a Cauchy sequence is also an operator from D to itself. Indeed, we shall see that the limit is interpretable as an operator from D ∗ cont into D ∗ ; this follows straightforwardly from the fact that every operator ˆ O from D to itself defines an operator ( ˆ O † ) ' on D ∗ by dual action. It is straightforward to see that the successful implementation of Step 4 implies that: The statements (i)-(iii) constitute a precise definition of what we mean by a nontrivial anomalyfree representation of the Poisson bracket between a pair of Hamiltonian constraints. These statements hold in PFT and the Husain-Kuchaˇr model. However, there, one has the stronger statement that the finite-triangulation operators as well as their limits are operators from D ∗ cont to itself; the linear vector space D ∗ cont then acts as a linear representation space which supports a representation of the constraint algebra. Following Lewandowski and Marolf [4], such a representation space is called a habitat . We are optimistic that our considerations here admit a generalization to a habitat-based representation. Indeed, as we shall see briefly in Section 3.3 and in detail later, our choice of D ∗ cont closely mimics that of the habitats of PFT [6] and the Husain-Kuchaˇr model [7].", "pages": [ 8, 9 ] }, { "title": "3.3 Choices", "content": "absent, one would get a trivial result by virtue of the smoothness of f . As discussed in Section 1, the overall factor of δ is tied to the choice of density weight of the constraint. As has been known for a long time, density weight one objects constructed solely out of the phase space variables when integrated with scalar smearing functions typically lead to LQG operators with no overall factors of δ . This is what would happen if we used the density weight one constraint. Hence in order to get an overall factor of δ -1 , we need to multiply the density weight one constraint by √ q 1 / 3 (recall that √ ˆ qδ 3 ∼ volume operator) i.e. we need to consider a Hamiltonian constraint of weight 4 / 3. It is then straightforward to check that the RHS also acquires an overall factor of δ -1 which, as we shall see, also goes into producing a derivative of f in the continuum limit. Thus the higher density weight allows on one hand the moving of vertices caused by the Hamiltonian constraint to manifest nontrivially, thereby giving rise to a nontrivial LHS, and on the other, compensates for the (hitherto) 'too many factors of δ ' in the RHS, thereby leaving an overall factor of δ -1 which is responsible for its non-triviality.", "pages": [ 10 ] }, { "title": "4 The Hamiltonian Constraint Operator at Finite Triangulation and the Continuum Limit of its Commutator", "content": "The Hamiltonian constraint of density weight 4 / 3 smeared with a lapse N (of density weight -1 / 3) is: Note that the last piece of the above expression, defines an electric field-dependent vector field for each i . For reasons which will become clear shortly, we shall refer to N a i as the electric shift . We refer to its quantum correspondent as the quantum shift . In Section 4.1 we detail our choice of regulating structures. In Section 4.2 we construct the quantum shift operator. Since its phase space dependence is solely on the electric field, the operator is diagonalized in the charge network basis. Moreover, due to its dependence on the inverse metric, its action is non-trivial only at vertices. In Section 4.3 we provide heuristic motivation for the action of the constraint operator at finite triangulation. Motivated by previous work in PFT, the Husain-Kuchaˇr model, and LQC [6, 7, 8], as well as by the requirement that constraint move the vertex on which it acts, we assign a key role to the quantum shift in this action. Specifically, using the key classical identity, as motivation, the quantum shift is used to deform the graph underlying the charge network. While the classical electric shift is smooth, by virtue of the discrete 'quantum geometry', the quantum shift is not a smooth vector field and the choice of the deformations it defines is made on the basis of intuition gained by the study of PFT and the Husain-Kuchaˇr model. We detail this choice in Section 4.4 and conclude with the evaluation of the action of the Hamiltonian constraint operator at finite triangulation on the charge network basis. Note that since the quantum shift only acts at vertices of the charge network, the Hamiltonian constraint (as in LQG) also acts only on vertices. In Section 4.5, we evaluate the commutator of two Hamiltonian constraints at finite triangulation on the charge network basis, and in Section 4.6 we compute the continuum limit.", "pages": [ 10 ] }, { "title": "4.1 Choice of Triangulation and Regulating Structures", "content": "Scalar densities of non-trivial weight need coordinate systems (more precisely n -forms in n dimensions) for their evaluation. Since the lapse is no longer a scalar, it turns out that we need to fix regulating coordinate systems to define the finite-triangulation Hamiltonian constraint. Accordingly, once and for all, around every p ∈ Σ we fix an open neighborhood U p with coordinate system { x } p such that p is at the origin of { x } p . When there is no confusion we shall drop the label p and refer to the coordinate patch as { x } . We shall use the regulating coordinate patches to specify the fineness of the triangulation below, to define the quantum shift in Section 4.2 and to specify the detailed graph deformations generated by the Hamiltonian constraint in Section 4.4. An immediate concern is the interaction of this choice of coordinate patches with the spatial covariance of the Hamiltonian constraint. While we shall comment on this issue towards the end of this paper, we shall (as mentioned in Section 1) defer a comprehensive treatment of the issue to Reference [12]. The one parameter family of triangulations T δ are adapted to the charge network on which the finite triangulation approximants act. Specifically, we require that T δ (for sufficiently small δ ) be such that every vertex v of the coarsest graph underlying the charge network is contained in the interior of a cell /triangle δ ( v ) ∈ T δ , and every cell of T δ contains at most one such vertex. The size of /triangle δ ( v ) is restricted to be of O ( δ 3 ) as measured in the coordinate system { x } v .", "pages": [ 11 ] }, { "title": "4.2 The Quantum Shift", "content": "Let ˆ q -1 / 3 act non-trivially at a vertex v of the charge network c . We shall refer to such vertices as non-degenerate. Let { x } denote the coordinate patch at v . Fix a coordinate ball B τ ( v ) of radius τ centered at v , and restrict attention to small enough τ in the following manipulations so that all constructions happen within the domain of { x } . Let ˆ q -1 / 3 τ denote the regularization of ˆ q -1 / 3 using this coordinate ball. From Appendix A (and from our general arguments in and prior to Section 3.3), the eigenvalue of ˆ q -1 / 3 τ for the eigenstate | c 〉 takes the form τ 2 ( /planckover2pi1 κγ ) -1 ν -2 / 3 where ν is a number constructed out of the charges which label the edges of c at v . Treating ˆ E a i as a functional derivative and | c 〉 as a function of the connection, the action of ˆ E a i at the point v naturally decomposes into a sum of contributions per edge [14] ˆ E a i = ∑ I ˆ E aI i with Next, we define the regulated quantum shift, ˆ N a i τ , evaluated at the point v by From Equation (4.4) and the form of the eigenvalue of ˆ q -1 / 3 τ , we obtain with where ˆ e a Iτ is a unit vector which pierces B τ ( v ) at the point where e I intersects it . That is, the point ∂B τ ( v ) ∩ e I has coordinates τ ˆ e a Iτ in the coordinate system { x } . The appearance of { x } , τ remind us that these values refer to a particular choice of coordinates { x } and a parameter τ defining the size of B τ ( v ). We may now take the regulating parameter τ → 0 to obtain with where ˆ e a I is the unit tangent vector at v along the edge e I in the coordinate system { x } .", "pages": [ 11, 12 ] }, { "title": "4.3 Heuristic Operator Action", "content": "We motivate a definition for a finite-triangulation Hamiltonian constraint through the following heuristic arguments. Using Equation (4.3) and by parts integration, the Hamiltonian constraint (4.1) can be written, modulo terms proportional to the Gauss constraints (recall that these constraints are G i = ∂ a E a i ), as: where N a i is the electric shift (4.2). Next, we add a classically-vanishing term which leads to the modified expression: While classically trivial, we shall see in Sections 4.4 and 4.5 that this term ensures that in the quantum theory the second Hamiltonian constraint acts on a vertex displaced by the first one; this is why we add it above. We shall think of gauge-invariant charge network states | c 〉 as wave functions c ( A ) of the connection A i a . We write c ( A ) in the form of a gauge-invariant graph holonomy (see Section 2.2.1): where we recall that the charge network coordinate c a i ( x ) is given by We now seek the action of the quantum correspondent of C ' [ N ] on c ( A ). Accordingly, we replace the electric shift in Equation (4.11) by the eigenvalue of the quantum shift operator (4.8). The eigenvalue is no longer a smooth field but, as part of our heuristics, in what follows below, we shall treat it as a smooth field which is supported only in the cells /triangle δ ( v ) which contain the vertices v of c . Next, we shall think of the remaining electric field operator (corresponding to the right most term in Equation (4.11) as /planckover2pi1 i δ δA j b . We are lead to the following a heuristic operator action: Expanding, where we have written ¯ c a i ≡ -c a i . Since the quantum shifts /vector N i have support only within the cells /triangle δ ( v ) which contain vertices of the charge network, the integral in (4.15) gets contributions only from such cells. If we further decompose the quantum shift N a i into its edge contributions N aI v i (see Equation (4.8); I v signifies that the edges emanate from v ) at each vertex v and think of each of these contributions as being of compact support in /triangle δ ( v ) , the expression (4.15) of the Lie derivative with respect to N a i splits into a sum over edge contributions in each cell /triangle δ ( v ) . We obtain where val( v ) is the valence of v, and Since the kinematics of LQG supports the action of finite diffeomorphisms rather than infinitesimal ones, we approximate the Lie derivative with respect to N aI v i by small, finite diffeomorphisms, ϕ ( /vector N I i , δ ), generated by N aI v i : Hence where the integrand [ · · · ] I is given by We have used the shorthand ϕ i c a j ≡ ϕ ( /vector N I i , δ ) ∗ c a j and dropped the common I . In the above expression, each line consists of terms which are deformed along a single shift minus the undeformed quantity; note also that each square-bracketed pair of terms is O ( δ ). Making all sums explicit, we have, in obvious notation, Since the square bracketed terms are O ( δ ), we may write The reason we exponentiate the square bracket is that each summand (to the right of the summation signs) is proportional to a graph holonomy (minus the identity) so that the right hand side of the above equation defines a linear combination of charge network states. For instance (suppressing some of the v dependence), describes a graph holonomy which lives on a graph deformation of the original graph underlying c multiplied by a graph holonomy which lives in the undeformed vicinity of v . The deformation is confined to the vicinity of the vertex v , moves the vertex v along the I th edge direction and 'flips' the charges on all edges in the vicinity of deformation by the replacements q 2 → -q 3 , q 3 → q 2 , q 1 → q 1 , and the undeformed piece has charges with an inverse flip (see Section 4.4 below). So far all of these manipulations have been formal and we only use the result to motivate our definition of the constraint operator. In the next section we shall discuss these graph deformations at length as they lie at the heart of our proposed action of the Hamiltonian constraint.", "pages": [ 12, 13, 14 ] }, { "title": "4.4 Deformations", "content": "In the previous section we persisted in the fiction that the quantum shift eigenvalue was a smooth function on Σ. In actuality, due to the discrete 'quantum geometry' (in this case the discrete electric lines of force along graphs), the quantum shift vanishes almost everywhere. This contrast between discrete quantum structures and their smooth classical correspondents is a characteristic feature of LQG and the appropriate replacement of the latter by the former in the quantum theory is more of an art than a deductive exercise. Accordingly, we view the manipulations of the last section as motivational heuristics; the precise graph deformations generated by the quantum shift are arrived at by the usual 'physicist mixture' of intuition and mathematical precision. While the details of our choices may suffer from non-uniqueness, we believe that there is a certain robustness to their main features. As a final remark, we note that our considerations are guided by the view that there must be imprints of the graph deformations which survive the action of diffeomorphisms and the possibility that the chosen deformations have analogs in the SU(2) case of gravity. Before turning to the precise form of the deformations we are proposing, we modify the heuristic starting point in two important ways: so that (4.19) is modified to where the integrand [ · · · ] I v ,i δ is given by where ϕc a j ≡ ϕ ( /vector ˆ e I v , δ ) ∗ c a j , and where we have replaced the region of integration /triangle δ ( v ) by Σ by virtue of the compact support of /vector ˆ e I v ( x ). Following a similar line of argument as before, we are lead to the expression We use (4.27) as our starting point rather than (4.22) for the following reasons: The quantum shift depends on the charge q i I (see (4.9)). In the SU(2) case this would correspond to an insertion of a Pauli matrix into the graph holonomy. Exponentiating such an operation to obtain a linear combination of charge networks seems difficult to us, so we leave q i I as an overall factor. Considerations of diffeomorphism covariance [12] lead us to leave the lapse (see (4.9)) as an overall factor as well. We now proceed to define the graph deformations suggested by (i) and (ii) above. Let us restrict attention to the vicinity of a vertex v (in what follows we shall on occasion suppress the subscripts indicative of this specific vertex). We interpret ϕ ( /vector ˆ e I , δ ) to be a 'singular diffeomorphism' which drags the vertex v (and the edges at v ) a distance of O ( δ ) 'almost' (see (ii) above) along the edge e I . We would like this deformation to have support only at the vertex v in the continuum limit. The right hand side of Equation (4.27), apart from the ' -1' term, is then essentially a sum over charge networks obtained by multiplying three different graph holonomies. The first is the original graph holonomy c ( A ) ∼ h c ( A ); the second is a graph holonomy which sits on the deformed graph and has charge flips of the type mentioned at the end of Section 4.3 and the third is a graph holonomy which sits on the original, undeformed graph but has (the inverse) charge flips. The multiplication of the second and third graph holonomies result in non-trivial charges only in the vicinity of the vertex v and multiplication with the first (original) graph holonomy results in a charge network state which lives on the union of the undeformed graph and its deformation with appropriate sums and difference of the charges coming from the three types of terms. In Section 4.4.1 we detail the position of the displaced vertex and in Section 4.4.2 we detail the accompanying deformation of the edges in the vicinity of v . In Section 4.4.3 we describe the charge labels of the charge network alluded to above as arising from the product of three graph holonomies and, finally, display the action of the Hamiltonian constraint operator at finite triangulation on the charge network basis.", "pages": [ 14, 15, 16 ] }, { "title": "4.4.1 Placement of the Translated Vertex", "content": "Let ˙ e b I v ≡ ˙ e b I ( v ) ≡ ˙ e b I be the tangent vector of the I th edge at the vertex v . Fix a Euclidean metric adapted to { x } such that d s 2 = δ ab d x a d x b . Choose some unit (normal) vector ˆ n a I such that We have a circle's worth of these. Picking one as detailed in Appendix C, we use it to single out the point which locates the displaced vertex. Here we choose p > 2 and, as discussed in Appendix C, ˆ n a I is chosen so that v ' I does not lie on the undeformed graph γ ( c ). Also note that the straight line from v to δ ˆ e a I can deviate from the edge e I to O ( δ 2 ) so that v ' I certainly lies within a distance of O ( δ 2 ) from e I . It is in this sense that v ' I lies 'almost' on e I . Finally, for technical reasons (see Section C of the appendix) we choose p /lessmuch k (recall that we use semianalytic, C k structures in this work).", "pages": [ 16 ] }, { "title": "4.4.2 New Edges", "content": "We imagine the deformed graph to be obtained by 'pulling' the original graph in the vicinity of the vertex v 'almost' along the direction of the edge e I . Thus new edges { ˜ e K } are obtained as the image of those parts of the old edges { e K } which are in the vicinity of the vertex v . The new edges connect the displaced vertex to the old edges as follows (see Figure (1)) For e I , we introduce a trivial vertex ˜ v I (on e I ), a coordinate distance 2 δ from v, and adjoin the new C k -semianalytic edge ˜ e I which connects ˜ v I and v ' I . Since we want ˜ e I to 'almost' overlap with (part of) the edge e I , we demand that the transition from ˜ e I to the original edge e I at ˜ v I be C k -violating in a strictly C 1 manner and that the tangent ˙ ˜ e a I at v ' I be proportional to ˙ e a I at v (these vectors are comparable in the coordinate system { x } ), i.e.: /negationslash -kinks. We will refer to such vertices ˜ v I as C 1 -kink vertices or simply as C 1 Next, we introduce a coordinate ball B δ q ( v ) of radius δ q about v . Note: We choose q ≥ 2 , q < p . This choice is important for the technicalities of Appendix C. /negationslash For the remaining edges e J = I , we introduce trivial vertices ˜ v J on all edges e J where they intersect ∂B δ q ( v ); that is, ˜ v J = e J ∩ ∂B δ q ( v ) . At ˜ v J introduce new C k -semianalytic edges ˜ e J which split off from each e J and head off to meet v ' I . These edges also 'almost' overlap (part of) the edge e I , reflecting our 'singular pulling' along of the vicinity of v along the direction of the edge e I . As a result we require that the C k violation at ˜ v J be strictly C 0 . Such vertices ˜ v J will be referred to as C 0 -kink vertices or simply as C 0 -kinks. 8 Since we imagine ˜ e J to be almost along e I , we require that the tangents of new edges ˜ e J at v ' I be 'bunched' around the direction -˙ ˜ e I at v ' I within a cone with apex angle of O ( δ q -1 ) ( q ≥ 2) with respect to { x } ; i.e. Further, since we think of the deformation as some sort of 'singular' diffeomorphism, we require that some subset of diffeomorphism-invariant properties of the graph structure at v be preserved by the new graph structure at the displaced vertex v ' I . In particular we require that if the set of edge tangents { ˙ e a K } at v are such that no triple lie in a plane, the same should be true of the set { ˙ ˜ e a K } at v ' I . Vertices which such properties arise in the study of moduli in knot classes by Grot and Rovelli [15]. Grot and Rovelli call this property 'non-degeneracy'. Accordingly we term such vertices as GR vertices. Thus, we require that the graph deformation preserve the GR nature of its vertices. Conversely, we also require that non-GR vertices do not acquire the GR property under graph deformation. We shall see that the GR property plays a key role in our analysis of diffeomorphism covariance [12]. Since we are thinking of the deformation as a (singular) diffeomorphism, we also require that no new non-trivial vertices are formed other than v ' I ; i.e., the new edges do not further intersect each other or the original graph. This may be explicitly achieved as follows. /negationslash Let the valence of v be M . Consider the M new edges in some order ˜ e I , ˜ e J 1 , ˜ e J 2 , .., ˜ e J M -1 , J k = I . Let ˜ e I be a semianalytic curve which connects ˜ v I with v ' I in accordance with the requirements on its tangents at ˜ v I , v ' I . Let the coordinate plane (in the { x } coordinates) which contains v ' I and which is normal to the direction ˆ ˙ ˜ e a I | v ' I be P . We require that the curve ˜ e I intersects P only at v ' I so that ˜ e I is always 'above' P . As we show in Appendix C, for small enough δ we can always find such an (almost straight in { x } ) curve. Let ˜ e J 1 be a straight line in { x } which connects ˜ v J with v ' I . If no unwanted intersections are produced, then we are done, and ˙ ˜ e J 1 at v ' I is approximately on the O ( δ q -1 ) cone. If the so-constructed ˜ e J 1 happens to produce an intersection at some (isolated because of the semianalyticity of the edges near v , see Appendix C) point other than ˜ v J 1 and v ' I , we can modify it with a bump function 9 so that the intersection is avoided. It is always possible to tune the size of the bump so as to not produce any new unwanted intersection, nor destroy its tangent-near-on-cone property. We continue in this manner constructing each ˜ e J K as a straight line, modifying this line with bumps where necessary. Since the 'bumping' is achieved via semianalytic diffeomorphisms, the new edges remain C k -semianalytic. It remains to show that the GR property (or lack thereof) of v is preserved. First consider the case when v is GR. Then if v ' I is GR we are done. If not, then as discussed further in Appendix C, we assume that the above prescription can be modified in a small vicinity of v ' I , without introducing any C 0 or C 1 kinks, in such a way as to render v ' I GR while still retaining the properties described by equations (4.29), (4.30), and (4.31). Indeed just such a prescription is constructed in detail in Reference [12] and we refer the interested reader to section 5.2 of that work. On the other hand, if v is not GR, we show in Appendix C.3 that a minor modification of the prescription of the previous paragraph ensures that v ' I is also not GR. Before we conclude this section, we note that the above prescriptions at triangulation fineness δ = δ 1 and at δ = δ 2 with δ 2 < δ 1 are not necessarily related by a diffeomorphism. It turns out that for future considerations, such as the construction of the space of VSA states, as well as for our study of diffeomorphism covariance in [12], it is useful to construct prescriptions which are related by diffeomorphisms. In the appendix we show how this can be done in such a way that Equations (4.29), (4.30), and (4.31) continue to hold.", "pages": [ 16, 17, 18 ] }, { "title": "4.4.3 Charges", "content": "Since [ · · · ] I v ,i δ contains the difference between a deformed (and charge-flipped) charge network coordinate and its undeformed relative (but still with flipped charges), e ∫ [ ··· ] Iv,i δ contains the product of the deformed graph holonomy and the inverse of the undeformed relative, and so all edges of the graph holonomy e ∫ [ ··· ] Iv,i δ away from the deformation 'erase' each other. That is, the (colored) graph underlying e ∫ [ ··· ] Iv,i δ itself can be described simply by a gauge-invariant 'pyramid skeleton' consisting of the thin 'star' formed by v and (for all original edges except the I th ) coordinate length δ q edge segments from the original graph that connect v and ˜ v J (for e I , the contribution to the star has coordinate length 2 δ ). The charges on the star are minus the charge-flipped configuration charges; e.g., for the i = 1 deformation, the star carries ( -q 1 , q 3 , -q 2 ) (with respect to an original coloring ( q 1 , q 2 , q 3 ) ) on each of its segments. The remaining edges (which meet v ' ) carry the flipped charges ( q 1 , -q 3 , q 2 ) . This pyramid charge network is multiplied by the original charge network c ( A ) and, in our example of i = 1, the star part of the resulting state carries ( 0 , q 2 + q 3 , q 3 -q 2 ) , which means that v is now a zero-volume vertex (see Appendix A). A similar conspiration of the charges results for the other values of i. Our ˆ q -1 / 3 will now annihilate this vertex (so the action of another Hamiltonian vanishes here). We change notation slightly and drop from here on the prime on ˆ C ' . Equation (4.27) then reads where we have made the regulating parameter δ explicit on the left hand side and dropped the O ( δ ) term. [ · · · ] I v ,i δ stands for the type of deformation described above (with charge flips). The charge configurations on the edges that meet at v ' I for the three quantum shifts /vector N i are We can write this compactly as where ( i ) specifies which shift /vector N ( i ) acted. In the next section we evaluate the action of a second Hamiltonian constraint on the right hand side of Equation (4.32). In doing so it is of advantage to further improve our notation as follows. Iv,i Denote the charge network corresponding to e ∫ /triangle δ ( v ) [ ··· ] δ c ( A ) by c ( i, v ' a I v ,δ = δ ˆ e a I v + δ p ˆ n a I v ) so that Equation (4.32) is written as: The various quantifiers { I v , i, δ } in the argument of c specify the particular edge e I v emanating from v along which the deformation (of magnitude ∼ δ ) was performed, and the particular flipping of the charges via i. Finally note that ∑ I v q i I v = 0 by gauge invariance (all edges outgoing at v ) so that:", "pages": [ 18, 19 ] }, { "title": "4.5 Second Hamiltonian", "content": "We evaluate the action of a second regularized Hamiltonian constraint, smeared with a lapse M on the right hand side of (4.36). Since we are interested in the continuum limit of (the action of VSA dual states on) the commutator, we drop those terms in ˆ C δ ' [ M ] ˆ C δ [ N ] c ( A ) which vanish in the continuum limit upon the antisymmetrization of N and M and 'contraction' with a dual state. The dropped terms are those in which ˆ C δ ' [ M ] acts at vertices not moved by ˆ C δ [ N ]; that is, the only contributions to the commutator will be from terms where ˆ C δ ' [ M ] acts at a vertex newly created by ˆ C δ [ N ]. 10 Consider the term ˆ C δ ' [ M ] c ( i, v ' I v ,δ ). Since v now has vanishing inverse volume, the constraint acts at the displaced vertex v ' I v ,δ as well as on all other vertices of c ( i, v ' I v ,δ ) which have non-vanishing inverse volume. But these other vertices are precisely the non-degenerate vertices of c other than v . As mentioned above, the contributions from these non-degenerate vertices vanish in the continuum limit evaluation of the commutator and so we do not display them here. The deformations generated by the action of ˆ C δ ' [ M ] on c ( i, v ' I v ,δ ) at the vertex v ' I v ,δ are defined in terms of the coordinate patch around v ' I v ,δ (see Section 4.4). We denote this coordinate system by { x ' a ' } v ' Iv,δ or simply by { x ' a ' } δ or just { x ' } when the context is clear. Note: In this work we require that in their region of joint validity { x ' δ } and { x } are related in a non-singular fashion as δ → 0 so that lim δ → 0 { x ' a ' } δ =: { x ' a ' } δ =0 is a good coordinate system. Specifically, we require that the Jacobian matrix J µ ν ' ( x, x ' δ ) := ∂x µ /∂x ν ' δ is continuous in δ with non-vanishing and non-singular determinant. It follows from the Note above that One possible way to construct such a set of coordinate patches is as follows: Since Σ is compact, it can be covered by finitely-many coordinate charts. We pick one such set. Clearly (at least) one chart { x 0 } in this set covers a neighborhood of v with /vectorx 0 ( v ) being the coordinates of v . Rigidly translate { x 0 } by /vectorx 0 ( v ) to obtain obtain { x } . For small enough δ , { x 0 } also covers small enough neighborhoods of the new vertices v ' I v ,δ with /vectorx 0 ( v ' I v ,δ ) being the coordinates of v ' I v ,δ . Rigidly translate { x 0 } by /vectorx 0 ( v ' I v ,δ ) to obtain { x ' } δ . Clearly, this ensures that the Jacobian for the { x } and { x ' } δ charts is unity. 11 From Equation (4.8) the quantum shift eigenvalues at v ' I v ,δ are defined through: Recall that the edges { e J v } at v are deformed to the edges { ˜ e J v } at v ' I v ,δ so that the valence of v and v ' I v ,δ are equal and we may use the same index J v to enumerate the edges at v and their counterparts at v ' I v ,δ . In what follows the primed index a ' denotes components in the { x ' } system and the primed 'hat' superscript, ˆ ' , denotes unit norm as measured in the { x ' } coordinate metric. and computed, via Equation (4.9) to be: where ˆ ˜ e ' a ' J v is the unit tangent to the edge ˜ e J v at v ' I v ,δ , and where we have used the fact that the inverse volume eigenvalue is independent of the charge flips inherent in the i -dependence of c ( i, v ' I v ,δ ) (see Appendix A). The term that survives the antisymmetrization and continuum limit is where the arguments of c denote the deformation and charge flips determined by ˆ C δ ' [ M ] . We detail their form below. /negationslash We distinguish two types of charge network that appear in the sum: J v = I v and J v = I v . Let J v = I v (this situation is depicted in Figure (2)) and focus on the resulting charge network c ( i, i ' , v '' ( I v ,δ ) ,I v ,δ ' ) ) . Following the prescription given above, v ' I v ,δ moves to (with respect to { x ' } with origin at v ' I v ,δ ) for some ˆ n ' satisfying the conditions spelt out in Appendix C. For J v = I v , v ' I v ,δ gets displaced along one of the 'cone directions' ˆ ˜ e ' a ' J v = I v : /negationslash /negationslash The structure of the deformations are as described for the first action, but with δ replaced by δ ' . The particular charge configurations at v '' resulting from each possible sequence of charge flips is summarized in the following table: Thus Above, we have used gauge invariance to set ∑ J v ( i ) q i ' J v = 0. We have also set ν -2 / 3 v ' Iv,δ ≡ ν -2 / 3 v ' Iv ; this follows from the diffeomorphism invariance of the inverse metric eigenvalue (see Appendix A) together with the fact that the deformations at different values of δ are related by diffeomorphisms (see Appendix C.4).", "pages": [ 19, 20, 21, 22 ] }, { "title": "4.6 Continuum Limit", "content": "In this section we evaluate the continuum limit of the commutator between a pair of finite triangulation Hamiltonian constraints under certain assumptions with regard to the properties of the VSA states. In Section 6 we shall construct a large class of VSA states which satisfy these assumptions. As mentioned in Section 3.1, the VSA states are weighted sums over certain bra states. As we shall see in Section 6, the weights are obtained by the evaluation of a smooth complex-valued function f on the non-degenerate 12 vertices of the bra it multiplies. More precisely, all bras in the sum have the same number n of non-degenerate vertices and the evaluation of f : Σ n → C on the n points corresponding to the n non-degenerate vertices of the bra, provides the weight of that bra in the sum: For simplicity we restrict attention to those ¯ c such that there is no symmetry of ¯ c which interchanges its nondegenerate vertices. We will sometimes write Ψ( c ) := (Ψ | c 〉 . In (4.45), Ψ f B VSA is a VSA state, B VSA is the set of bras being summed over, V (¯ c ) denotes the set of non-degenerate vertices of ¯ c, and we have introduced the ¯ c -dependent real number κ ¯ c into the expression. To avoid notational clutter we have suppressed the κ ¯ c dependence in (Ψ f B VSA | . The continuum limit of the commutator is Using Equation (4.40), we first evaluate lim δ ' → 0 Ψ f B VSA ( ˆ C δ ' [ M ] c ( i, v ' I v ,δ )). We have that where we have set ν v ' Iv,δ = ν v Iv and used gauge invariance to drop the last term: Next, we make the following assumptions which will be shown to hold in Section 6: or ∀ δ, δ ' for which c ( i, i ' , v '' ( I v ,δ ) , ( J v ,δ ' ) ) is defined we have that: If (4.50) holds, the right hand side of (4.47) vanishes. We shall see in Section 6 that in this case, the corresponding 'matrix element' for the RHS also vanishes. We continue the calculation in the case that (4.49) holds. We have that: where, once again we have used gauge invariance to append the term f ( v ' I v ,δ ). In addition for notational convenience only displayed the dependence of f on the (doubly and singly) deformed images of v and suppressed its dependence on the undeformed vertices. Using (4.41), (4.42) and the smoothness of f , we obtain where ( ˆ ˜ e ' J v ) a is the component of the unit vector /vector ˆ ˜ e ' J v in the { x } coordinate system. Here the vector /vector ˆ ˜ e ' J v is obtained by normalizing the tangent vector to the edge ˜ e J v at v ' I v ,δ in the { x ' } system (recall, from (4.41), (4.42) that the components of this vector in the { x ' } system are given by ( ˆ ˜ e ' J v ) a ' ). It follows from the above equation in conjunction with (4.44) that Since M is of density weight -1 / 3 we have: Using this, we obtain Next, we use (4.29) to Taylor expand M as: Using the above Equation in (4.56) to evaluate the commutator, we obtain in 'bra-ket' notation: We now compute the δ → 0 limit of the above equation so as to obtain the continuum limit of the commutator. By virtue of the smooth dependence of x on x ' δ (see the note in Section 4.5) the determinant is a continuous function of δ . It remains to compute the δ → 0 limit of {· · · } I v ,δ . /negationslash Since the { x } and { x ' } ≡ { x ' } δ systems are not necessarily the same, we have that ( ˆ ˜ e ' J v ) a is proportional to ( ˆ ˜ e J v ) a | v ' Iv,δ where now the same tangent vector has been normalized in the { x } system. From the Note and equation (4.37) in Section 4.5, in conjunction with Equations (4.31) in Section 4.4.2, we have that that at v ' I v ,δ , for J v = I v Using this in (4.57) together with the smoothness of ∂ a f , we obtain /negationslash Gauge invariance (4.48) then implies that: Finally, from (4.34) it follows that where Up to this point we have refrained from assuming any particular relation between { x ' δ =0 } and { x } in order to exhibit the structure of the calculation as δ → 0. Section 4.1 together with equation (4.37) implies that the Jacobian between the two coordinate systems is the identity: Using this together with (4.63) and (4.59) we obtain the continuum limit of the commutator under the assumption (4.49) to be:", "pages": [ 22, 23, 24, 25 ] }, { "title": "5 RHS", "content": "In Section 5.1 we display a remarkable classical identity which expresses the RHS as the Poisson bracket between a pair of diffeomorphism constraints, each smeared with an electric shift. This implies, that in the quantum theory, we may identify the RHS with commutator between two such constraints. Accordingly, in Section 5.2 we construct the finite triangulation operator corresponding to single diffeomorphism constraint smeared with an electric shift using arguments which parallel those of Section 4. We compute the finite-triangulation commutator between two such operators in Section 5.3. We compute the continuum limit of this commutator in Section 5.4 under certain assumptions (whose validity is demonstrated in Section 6) on the VSA states.", "pages": [ 25 ] }, { "title": "5.1 A Remarkable Identity", "content": "It is straightforward to check that for we have where Let the diffeomorphism generator smeared with the 'electric shift' (see Section 4.3), N a i := q -α NE a i , be denoted D [ /vector N i ]: We shall refer to D [ /vector N i ] as an electric diffeomorphism constraint. The Poisson bracket between a pair of electric diffeomorphism constraints is (summing over the internal index i ): in which we have used where ( E -1 ) i a is the 'inverse' of E b j so that E i a E b i = δ b a , E i a E a j = δ i j . Thus we may write the RHS as In this work we are interested in α = 1 3 (see Equation (4.1)). In Section 5.4 we use this identity to express the RHS operator as the commutator between two finite diffeomorphism operators. As mentioned in Section 3.1 (see Step 3 of that section), this facilitates the comparison of the LHS and RHS operators. Note that this identity trivializes precisely for the case α = 1 2 ; this is the case of Hamiltonian constraints of density weight one considered hitherto in the literature . We take this trivialization as further support for the move away from the density one case. We also note that, as shown in Appendix B, this identity holds for the SU(2) case in 2 + 1 and 3 + 1 dimensions and in all cases trivializes for the density weight one choice.", "pages": [ 25, 26 ] }, { "title": "5.2 The Electric Diffeomorphism Constraint Operator at Finite Triangulation", "content": "We set α = 1 3 in (5.3). Modulo Gauss law terms we have that: where /vector N i is the electric shift of Section 4. This motivates, analogous to (4.14), the following heuristic operator action Following an argumentation similar to that between Equations (4.14)-(4.24) leads us to the finitetriangulation electric diffeomorphism constraint operator action: where we have used gauge invariance to drop the ' -c ' term in the second line and where the charge network coordinate underlying the state c ( v ' I v ,δ ) is given by where ϕ ( /vector ˆ e I , δ ) deforms the graph underlying c in the manner discussed in Section 4.4. More in detail, the graph underlying c ( v ' I v ,δ ) is obtained by removing the segments of the graph underlying c which connect v to the points ˜ v J and adjoining new edges, ˜ e J which connect ˜ v J to the displaced vertex v ' I v ,δ as explained in Section 4.4. The deformed graph is identical to the one shown in Figure (1), but with the dashed edges removed. Also note that since D [ /vector N i ] is constructed by smearing the diffeomorphism constraint with an electric shift, the edges ˜ e J carry the same charges as e J i.e. there are no 'charge flips'.", "pages": [ 26, 27 ] }, { "title": "5.3 Second Electric Diffeomorphism", "content": "We evaluate the action of a second electric diffeomorphism constraint, smeared with the electric shift /vector M i on the right hand side of (5.9). Since we are interested in the continuum limit of (the action of VSA dual states on) the commutator between two electric diffeomorphism constraints, we drop those terms in ˆ D δ ' [ /vector M i ] ˆ D δ [ /vector N i ] c ( A ) which vanish in the continuum limit upon the antisymmetrization of N and M . The dropped terms are those in which ˆ D δ ' [ /vector M i ] acts at vertices not moved by ˆ D δ [ /vector N i ]; that is, the only contributions to the commutator will be from terms where ˆ D δ ' [ /vector M i ] acts at a vertex which has been moved by ˆ D δ [ /vector N i ]. Consider the term ˆ D δ ' [ /vector M i ] c ( v ' I v ,δ ). The constraint acts at the displaced vertex v ' I v ,δ as well as on all other vertices of c ( v ' I v ,δ ) which have non-vanishing inverse volume. But these other vertices are precisely the non-degenerate vertices of c other than v . As mentioned above, the contributions from these non-degenerate vertices vanish in the continuum limit evaluation of the commutator and so we do not display them here. The deformations generated by the action of ˆ D δ ' [ /vector M i ] on c ( v ' I v ,δ ) at the vertex v ' I v ,δ are, as in the case of Hamiltonian constraint of Section 4.4, defined in terms of the coordinate patch { x ' } around v ' I v ,δ . From Equation (4.8), we have that with M a ' J v i ( v ' I v ,δ ) given by The term that survives the antisymmetrization and continuum limit is /negationslash Restoring the sum over vertices we have, modulo terms which vanish upon antisymmetrization in the lapses and the taking of the continuum limit: where we have used gauge invariance to drop the last term in the second line. Here c ( v '' ( I v ,δ ) , ( J v ,δ ' ) ) denotes the charge network state obtained by deforming the state c ( v ' I v ,δ ) by the 'singular' diffeomorphism generated by ˆ D δ ' [ /vector M i ]. The deformation moves the vertex v ' I v ,δ of c ( v ' I v ,δ ) to its new position, v '' ( I v ,δ ) , ( J v ,δ ' ) given by Equation (4.41) when J v = I v and by Equation (4.42) when J v = I v . The structure of the deformations are as described for the first action in Section 5.2, but with δ → δ ' (see Figure (3)).", "pages": [ 27, 28 ] }, { "title": "5.4 Continuum Limit", "content": "In this section we evaluate the continuum limit of the commutator between a pair of finitetriangulation electric diffeomorphism constraints under certain assumptions with regard to the bra set B VSA which underlies the VSA states (see Section 4.6). These assumptions are in addition to Equations (4.49),(4.50) of Section 4.6. The assumptions are as follows: or, ∀ δ, δ ' for which c ( v '' ( I v ,δ ) , ( J v ,δ ' ) ) is defined, we have that If (5.16) holds, it is immediate to see that the continuum limit of the commutator vanishes; from the assumption above, it follows that the LHS also vanishes. We continue the calculation in the case that (5.15) holds (which also means that by assumption, (4.49) holds as well). From Equation(5.13), we have that where, once again, we have used gauge invariance to append the last term. It follows that It follows from Equation (5.14) that Using (4.55) in the above equation we have, where Using (4.58) in (5.21) and antisymmetrizing in the lapses, one obtains (in bra-ket notation): As in Section 4.6, the determinant is a continuous function of δ . It remains to evaluate the δ → 0 limit of {· · · } i,I v ,δ . Using Equation (4.60) in (5.22) together with gauge invariance, one obtains: Using this together with Equations (4.64), (5.23) and (5.6), we obtain the continuum limit of the RHS, in the case where (5.15) holds, to be: which agrees with Equation (4.65).", "pages": [ 28, 29, 30 ] }, { "title": "6 Existence of a Large Space of VSA States", "content": "In this section we show the existence of VSA states which satisfy the assumptions (1)-(2) of Section 4 and (1)-(3) of Section 5. As mentioned in Sections 4 and 5, the VSA states are weighted sums over a set of bras, the weights being vertex-smooth functions. In Section 6.1, we provide a qualitative discussion of the issues which arise in the construction of an appropriate set of VSA states. In Section 6.2 we construct sets of bras and vertex-smooth functions which specify the VSA states of interest. In Section 6.3 we show that these states satisfy the assumptions of Sections 5 and 6. While the states we construct span an infinite-dimensional vector space, they are still of a restricted variety. Specifically, all elements of the sets of bras under consideration have only one non-degenerate 13 vertex. While a generalization to the case of multiple non-degenerate vertices should not be too difficult, we shall leave this for the future. In what follows it is pertinent to recall that in this paper we consider diffeomorphisms which are semianalytic and C k , k /greatermuch 1 , k /greatermuch p . Such diffeomorphisms send a semianalytic edge into a semianalytic edge which is C k . This implies that the first k derivatives along the edge are continuous everywhere and at worst, in any semianalytic chart, there are a finite number of points p i at which the k th i derivative along the edge is discontinuous for some k i > k .", "pages": [ 30 ] }, { "title": "6.1 Discussion of Our Strategy", "content": "While we do ignore issues of diffeomorphism covariance in this paper, we would like to set things up in such a way that issues of diffeomorphism covariance can be potentially addressed. As a result, we require that the set of bras, B VSA , be closed under the action of diffeomorphisms. This, together with a careful study of the assumptions of Sections 4 and 5 imply that the set of bras should be such that whenever it contains any doubly-deformed charge network obtained by two successive Hamiltonian constraint-type deformations, on some charge network | c 〉 , it should also contain (a) all other doubly-deformed charge networks obtained by the action of any two successive Hamiltonian constraint-type deformations on | c 〉 , and (b) all doubly-deformed charge networks obtained by the action of any two successive 'singular' diffeomorphism-type deformations which occur on the RHS. Conversely, if the set contains any doubly-deformed charge network obtained by two successive singular diffeomorphism-type deformations on some charge network | c 〉 , it should also contain (a) all other doubly-deformed charge networks obtained by the action of any two successive 'singular' diffeomorphism-type deformations, and (b) all doubly-deformed charge networks obtained by the action of any two successive Hamiltonian constraint-type deformations on | c 〉 . In suggestive language we call | c 〉 the parent, the single deformations of | c 〉 its children, and its double deformations its grandchildren. Our problem then is to ensure that if any grandchild is present in the bra set, all grandchildren should be present. This in turn implies that one should be able to infer all possible parent charge networks which could yield a given grandchild. This sort of backward inference is direct for the case of Hamiltonian constraint grandchildren because the parent charge network graph is embedded in that of any grandchild, and the charge flips (4.34) are invertible. However, this embedding of parent into grandchild is not available for singular diffeomorphism-type grandchildren, and the bra set needs to be generated via double (Hamiltonian and) singular diffeomorphism deformations of all possible parent charge networks which could produce a specific grandchild. This is what we do. In order to do this we start out with a set of parents from which the output of grandchildren is well-controlled. Specifically, our starting point is a parent which is an n th -generation child of a 'primordial' charge network (by 'primordial' we mean the charge network is itself not generated by the action of any Hamiltonian constraint/singular diffeomorphism-type of deformations on some other charge network). This n th -generation parent is chosen (for concreteness and simplicity) to be obtained from the primordial charge network by n Hamiltonian constraint-type deformations. Our discussion indicates that the charge networks under consideration encode a sort of 'chronological heredity'. As a result, we introduce a suggestive 'causal' nomenclature for certain graph structures of interest in Section 6.2 which go into the construction of B VSA . As mentioned above, in this paper, we restrict attention to primordial charge networks with a singular non-degenerate GR vertex. While there seems to be no barrier to the consideration of multi-vertex primordial charge networks, we shall leave a generalization of our constructions to such charge networks for future work.", "pages": [ 31 ] }, { "title": "6.2 Construction of the VSA States", "content": "Let | c 0 〉 be a charge network with a single non-degenerate GR vertex of valence M, and let | n, /vectorα, c 0 〉 be the state obtained by n successive finite-triangulation Hamiltonian constraint-type of deformations applied to | c 0 〉 . Here, /vector α := { α i | i = 1 , . . . , n } , and each α i is a collection of labels corresponding to the internal charge, vertex, edge, and deformation parameter which go into specification of the Hamiltonian constraint-type deformations. For example, for the state c ( i, i ' , v '' ( I v ,δ ) , ( J v ,δ ' ) ) in Equation (4.40), we have that n = 2 , α 1 = ( i, v, I v , δ ) , α 2 = ( i ' , v ' I v ,δ , J v , δ ' ) and c = c 0 . Let the set of all distinct diffeomorphic images of 〈 n, /vectorα, c 0 | be B [ n,/vectorα,c 0 ] . For every element of this set, we generate a new family of charge networks. In order to do so, for every 〈 c | ∈ B [ n,/vectorα,c 0 ] we now define some graph structures of interest. Note that every 〈 c | ∈ B [ n,/vectorα,c 0 ] has a unique 'final' non-degenerate vertex v ( c ) of valence M which is connected to one C 1 -kink vertex and to M -1 C 0 -kink vertices. Let the I th edge from v , e I , connect v to the C 1 -kink vertex. Let e J = I connect v to the C 0 -kinks. /negationslash Definition: The 1-past of γ ( c ) 14 : The 1-past of γ ( c ), denoted by γ 1-p ( c ), is the (closed) graph obtained by removing the edges e K , K = 1 , .., M from γ ( c ); i.e. Let e K intersect γ 1-p ( c ) at ˜ v K, 1-p on the edge e K, 1-p of γ 1-p ( c ) so that ˜ v I, 1-p , ˜ v J = I, 1-p are the C 1 , C 0 kinks mentioned above. Since | c 〉 is diffeomorphic to | n, /vectorα, c 0 〉 , it follows that the edges e K, 1-p intersect at a GR vertex which we denote by v 1-p ( c ). The following definitions are illustrated in Figures (4)-(8). /negationslash Definition: The future graph of γ 1 -p ( c ) in c : The future graph of γ 1-p ( c ) in c , denoted by γ f 0 1-p , is defined by Thus, modulo the action of diffeomorphisms, γ f 0 1-p is just the nested graph structure produced by the action of a particular Hamiltonian constraint-type deformation which acts on the 'parent' vertex v 1-p ( c ) of the 'parent' charge network based on the graph γ 1-p ( c ). Next, we define a graph structure which is similar to γ f 0 1-p in terms of its 'causal' properties. Definition: A future graph of γ 1 -p ( c ) with respect to c : A graph γ f 1-p ,c is a future graph of γ 1-p ( c ) with respect to c if and only if it has the following properties: /negationslash Then with respect to γ f c , the point ˜ v I, 1-p is a trivalent C 1 -kink vertex and the points ˜ v J = I, 1-p are trivalent C 0 -kink vertices. Note that the future graph of γ 1-p ( c ) in c is a future graph of γ 1-p ( c ) with respect to c but the converse is not necessarily true. In particular the set of tangent vectors at the non-degenerate vertex v f (of a future graph of γ 1-p ( c ) with respect to c ) need not be obtained through the action of a diffeomorphism from the set of tangent vectors at the non-degenerate vertex v of c ; i.e., the two sets may have different moduli [15]. Next, we define a charge network which is identical to c in terms of its causal properties and colorings. Definition: A causal completion of the 1-past of c : A causal completion of the 1-past of c , denoted by c f ( c ), is the charge network based on the graph γ f c (see Equation (6.3)) with charges on γ 1-p ( c ) being the same as those coming from c , and on { e f K } being the same as those on { e K } in c . Note that the definition of the 1-past in terms of the removal of immediate edges from a final non-degenerate vertex to trivalent kink vertices extends naturally to such causal completions and we shall assume that the definition has been so extended. We now use the above definitions to construct B VSA as follows. Consider all distinct causal completions, 〈 c f ( c ) | for every c ∈ B [ n,/vectorα,c 0 ] . Let the resulting set of bras be B 〈 n,/vectorα,c 0 〉 . Consider all possible single Hamiltonian constraint-type deformations (i.e. for all values of ' α ') of elements of B 〈 n,/vectorα,c 0 〉 and take all distinct diffeomorphic images of the resulting set of charge networks. Call the resulting set B [ H 〈 n,/vectorα,c 0 〉 ] . Repeat this procedure again. That is, once again consider all Hamiltonian constraint-type deformations of the elements of this set and then take distinct diffeomorphic images of such deformed charge networks. Call this set B [ H [ H 〈 n,/vectorα,c 0 〉 ]] . Next, we consider deformations of the type encountered in the RHS. Accordingly, denote a double 'singular' diffeomorphism-type of deformation of any state | c 〉 by ˆ D 2 ( β ) | c 〉 . Here β is a label which specifies the vertex at which the deformation takes place, the two edge labels along which the deformations take place and the parameters δ, δ ' which quantify the amount of deformation. For example, for the state c ( v '' ( I v ,δ ) , ( J v ,δ ' ) ) in Equation (5.13), we have that Act by ˆ D 2 ( β ) for all β on elements of B 〈 n,/vectorα,c 0 〉 and then take all distinct diffeomorphic images thereof to form the set B [ D 2 〈 n,/vectorα,c 0 〉 ] . Finally define B VSA as: Note that every element of B VSA has a single 'final' non-degenerate GR vertex of valence M . In terms of our discussion in Section 6.1, | c 0 〉 is a primordial charge network, | n, /vectorα, c 0 〉 is the parent in the n th generation, B [ n,/vectorα,c 0 ] is the set of all diffeomorphic images of this parent. The role of B 〈 n,/vectorα,c 0 〉 is as follows. Recall from Section 6.1 that if a grandchild is present in B VSA , we need to ensure that all possible related grandchildren are present as well. This necessitates the identification of a set of (grand)parents which give birth to all these grandchildren. Since the specific (grand) parent which gives rise to a double singular diffeomorphism grandchild is not embedded in the grandchild, it is difficult (and perhaps impossible) to infer the identity of the specific (grand)parent which gave birth to such a grandchild. The solution is then to accommodate all possible (grand)parents which could conceivably have given birth to the grandchild in question. The set of all possible such (grand)parents is B 〈 n,/vectorα,c 0 〉 . Before we proceed to the next section, we prove a Lemma which will be of use below. Lemma: The set B 〈 n,/vectorα,c 0 〉 is closed under the action of diffeomorphisms; i.e., in the notation we have used above, we have that B 〈 n,/vectorα,c 0 〉 = B [ 〈 n,/vectorα,c 0 〉 ] . Proof: Let 〈 ˆ c | ∈ B 〈 n,/vectorα,c 0 〉 . This means that ˆ c is the causal completion of the 1-past of some charge network c such that 〈 c | ∈ B [ n,/vectorα,c 0 ] . Consider the charge network φ · c obtained by the action of the diffeomorphism φ on c . It is then straightforward to check that φ · ˆ c is a causal completion of the 1-past of φ · c . This implies that 〈 φ · ˆ c | ∈ B 〈 n,/vectorα,c 0 〉 which completes the proof.", "pages": [ 31, 32, 34, 35 ] }, { "title": "6.3 Demonstration of Assumed Properties of VSA States", "content": "The VSA states are constructed as in Sections 4 and 5 by summing over all bras in the set B VSA defined by Equation (6.5), with each bra weighted by the evaluation of a vertex smooth function f : Σ → C on the single non-degenerate vertex of the bra it multiplies. Let | ¯ c 〉 be a charge network state. Then the following cases are of interest: We now consider each of them in turn. Case (a) : First note that | ¯ c 〉 can be reconstructed from | ¯ α 1 , ¯ α 2 , ¯ c 〉 as follows. Let γ (¯ α 1 , ¯ α 2 , ¯ c ) be the graph underlying | ¯ α 1 , ¯ α 2 , ¯ c 〉 . Clearly its 1-past is the graph γ (¯ α 1 , ¯ c ) which underlies the state | ¯ α 1 , ¯ c 〉 . The colors of | ¯ α 1 , ¯ c 〉 can be obtained as follows. Retain the colors from | ¯ α 1 , ¯ α 2 , ¯ c 〉 on those edges in its 1-past which do not emanate from the final vertex v 1-p (¯ α 1 , ¯ α 2 , ¯ c ) of this 1-past. Note that the edges e K, 1-p , K = 1 , .., M emanating from the final vertex v 1-p (¯ α 1 , ¯ α 2 , ¯ c ) of this 1-past each acquire kink vertices,˜ v K, 1-p , in | ¯ α 1 , ¯ α 2 , ¯ c 〉 . The part of e K, 1-p which connects ˜ v K, 1-p to v 1-p (¯ α 1 , ¯ α 2 , ¯ c ) suffers changes of its colors relative to its coloring in | ¯ α 1 , ¯ c 〉 , but the remaining part retains its charges from | ¯ α 1 , ¯ c 〉 . Hence we can read off the coloring of each e K, 1-p in | ¯ α 1 , ¯ c 〉 from this remaining part and hence reconstruct | ¯ α 1 , ¯ c 〉 . The same procedure can then be applied to | ¯ α 1 , ¯ c 〉 to obtain | ¯ c 〉 . At this stage it is useful to introduce 'deformation' operators as follows. Let us indicate the action of a Hamiltonian constraint-type deformation labelled by α on a state | c 〉 (with a single non-degenerate vertex) by ˆ C α | c 〉 . So in this notation we have, for example, that Next, note that the final vertex of | ¯ α 1 , ¯ α 2 , ¯ c 〉 is connected to its 1-past by edges which end on trivalent kinks. It is immediate to see that the edges from the final vertex of any state in B [ D 2 〈 n,/vectorα,c 0 〉 ] end in bivalent kinks. Hence, it must be the case that | ¯ α 1 , ¯ α 2 , ¯ c 〉 ∈ B [ H [ H 〈 n,/vectorα,c 0 〉 ]] . In the 'deformation operator' notation we have this may be written as for some 〈 c | ∈ B 〈 n,/vectorα,c 0 〉 , appropriate deformation labels α ' 1 , α ' 2 and diffeomorphisms φ 1 , φ 2 with ˆ U φ i , i = 1 , 2 being the unitary operators which implement these diffeomorphisms. Since the definition of the 1-past as well as the process of 'unflipping charges' are diffeomorphism invariant, it is straightforward to see that follows that the above equation implies that From the Lemma at the end of Section 6.2, it follows that 〈 ¯ c | ∈ B 〈 n,/vectorα,c 0 〉 . Hence all its double Hamiltonian constraint-type deformations and all its double singular diffeomorphism-type deformations are in B VSA . This immediately implies that the assumptions of Section 4, 5 are satisfied in this case. Case (b) : In terms of the double singular diffeomorphism operators of Equation (6.4) we have that The last part of Section 4.4.2 implies that the graph structure of γ ( ¯ β, ¯ c ) in the vicinity of v '' (¯ c ) is as follows. Each of the M semianalytic C k edges emanating from v '' (¯ c ) ends in a bivalent C r -kink vertex where r = 0 or 1. The remaining semianalytic C k edge from each such kink when followed 'into the past' also ends in a bivalent C r -kink vertex with r = 0 or 1. The remaining semianalytic C k edge at this kink is part of the graph γ (¯ c ) and each of these remaining edges when followed to 'the past' connect to the rest of γ (¯ c ). We denote the part of γ (¯ c ) which connects to the past endpoints of these edges by γ rest (¯ c ). Since | ¯ β, ¯ c 〉 is in B VSA , it has only one non-degenerate vertex of valence M which we denote by v '' (¯ c ), and this vertex is GR. Therefore ¯ c also has a single non-degenerate M -valent vertex, which we denote by v (¯ c ) and, from Section 4.4.2, this vertex must also be GR. In what follows we denote the graphs underlying | ¯ β, ¯ c 〉 , | ¯ c 〉 by γ ( ¯ β, ¯ c ) , γ (¯ c ). To summarize: We have that (see Figure (9)) where where e v '' (¯ c ) , kink K connects v '' (¯ c ) to the first C r ( r = 0 or 1) kink to its past, e kink , kink K connects this kink to the second one and e kink , rest K ∈ γ (¯ c ) connects this second kink to γ rest (¯ c ). Next, note that by virtue of the connection of its non-degenerate vertex two successive bivalent kinks, it must be the case that | ¯ β, ¯ c 〉 ∈ B [ D 2 〈 n,/vectorα,c 0 〉 ] so that for some appropriate diffeomorphism φ , deformation label β and state 〈 c | ∈ B 〈 n,/vectorα,c 0 〉 . Next, note that it is possible to reconstruct the 1-past of | c 〉 from | β, c 〉 by following exactly the same procedure which resulted in obtaining γ rest (¯ c ) from γ ( ¯ β, ¯ c ). Thus any edge emanating from the final (non-degenerate, GR, M -valent) vertex of | β, c 〉 followed 'back in time' connects to a bivalent C 1 - or C 0 -kink which, in turn, connects to another bivalent C 1 - or C 0 -kink, which is then connected to γ 1-p ( c ) by an edge which lies in γ ( c ). Removing the M sets of such triplets of successive edges which connect the final vertex of | β, c 〉 to γ 1-p ( c ) yields γ 1-p ( c ). Since this procedure (of removing the triplets of successive C k semianalytic edges which emanate from the final nondegenerate vertex) is diffeomorphism-invariant, the same procedure applied to ˆ U ( φ ) D 2 ( β ) | c 〉 yields the 1-past of ˆ U ( φ ) | c 〉 . But, using Equation (6.12), this very same procedure resulted in the graph γ rest (¯ c ). Hence we have that where | c φ 〉 := ˆ U ( φ ) | c 〉 . Moreover, from equations (6.12) and (6.11) and the nature of double singular diffeomorphisms, it follows that the edges e kink , rest K , K = 1 , .., M of equation (6.11) are a part of γ (¯ c ) as well as γ ( c φ ). This, together with (6.13), (6.12) and the last definition of Section 6.2, implies that ¯ c is the causal completion of the 1- past of c φ . Since 〈 c φ | ∈ B 〈 n,/vectorα,c 0 〉 by virtue of the Lemma of Section 6.2, this means that 〈 ¯ c | is in B 〈 n,/vectorα,c 0 〉 . Hence, once again all double Hamiltonian constraint as well singular diffeomorphism-type deformations of 〈 ¯ c | are in B VSA in accord with the assumptions of Section 4 and 5. Case (c): Since | ¯ α, ¯ c 〉 is obtained by the action of a single Hamiltonian constraint, each of the M ( C k , semianalytic) edges emanating from its final vertex is connected to a trivalent kink. This, together with 〈 ¯ α, ¯ c | ∈ B VSA implies that 〈 ¯ α, ¯ c | ∈ B [ H [ H 〈 n,/vectorα,c 0 〉 ]] which means that for some 〈 c | ∈ B [ H 〈 n,/vectorα,c 0 〉 ] , some Hamiltonian constraint deformation α 1 and some diffeomorphism φ we have that Using argumentation similar to that for Case (a), it follows that γ (¯ c ) = γ 1-p (¯ α, ¯ c ), that ¯ c can be reconstructed by appropriately coloring γ (¯ c ) through the procedure of retaining the colors of | ¯ α, ¯ c 〉 away from the vicinity of its final degenerate vertex and coloring those edges which emanate from this vertex with the colors of their continuations past the immediate kinks they connect to, and that all this, together with the diffeomorphism invariance of the reconstruction procedure and Equation (6.14), implies that Since B [ H 〈 n,/vectorα,c 0 〉 ] is closed under the action of semianalytic C k diffeomorphisms, it follows that 〈 ¯ c | ∈ B [ H 〈 n,/vectorα,c 0 〉 ] and, hence, that B [ H [ H 〈 n,/vectorα,c 0 〉 ]] contains all single Hamiltonian constraint deformations of 〈 ¯ c | . It is then easy to see that the considerations of Sections 4.1 and 4.6 imply that the continuum limit of the 'matrix element' of a single finite triangulation Hamiltonian constraint operator is well defined and non-trivial i.e. lim δ → 0 Ψ f B VSA ( ˆ C δ ( N ) | ¯ c 〉 ) is well defined and non-vanishing for suitable f, N (by suitable we mean that N and the first derivative of f do not vanish at the final nondegenerate GR vertex of ¯ c ). Note that Equation (6.14) implies that ¯ c has n +1 degenerate GR vertices and that if either of Cases (a) or (b) hold, ¯ c must have n degenerate GR vertices which means that the matrix element for the single Hamiltonian constraint action vanishes for Cases (a) and (b). Cases (a)-(c) exhaust all possibilities of interest and imply that for any VSA state and any charge network state: It is straightforward to see that (i)-(iii) above imply that (i)-(iii) of Section 3.2 hold. In particular point (iii) shows that, as stated towards the end of Section 1, our considerations yield a non-trivial anomaly free representation of the Poisson bracket between a pair Hamiltonian constraints.", "pages": [ 35, 36, 37, 38 ] }, { "title": "7 Discussion", "content": "In any gauge theory, anomalies in the algebra of quantum constraints typically point to a reduction of the number of true degrees of freedom in the quantum theory. The quantization is then unphysical and, depending on the severity of the anomalies, inconsistent. Hence, typically, the viability of a quantum gauge theory is dependent on its support of an anomaly-free representation of the classical constraint algebra. If the gauge arises from general covariance, the constraint algebra has an additional role to play [16]: It encodes spacetime covariance in the Hamiltonian formulation. We elaborate on this additional role below. Any Hamiltonian formulation splits spacetime into space and time. As a result, spacetime symmetries which are manifest in the Lagrangian description are not explicit in the Hamiltonian formulation. For theories in flat spacetime, the availability of preferred inertial times allows the straightforward recovery of spacetime fields from spatial ones. However, in theories of spacetime, such as general relativity (or even in generally-covariant reformulations of field theories on a fixed spacetime, such as PFT), the absence of a preferred time, with respect to which the Hamiltonian theory is to be defined, makes this loss of manifest spacetime covariance more acute. One may then ask the following question: Which structure in the Hamiltonian description of a generally covariant theory encodes spacetime covariance? The answer to this question is provided by the seminal work of Hojman, Kuchaˇr, and Teitelboim (HKT) [16]. In the Hamiltonian description of a generally-covariant theory of spacetime, initial data is prescribed on a spatial slice embedded in spacetime, the spacetime itself emerging out of the dynamics of the theory. HKT note that this dynamics pushes the spatial slice 'forward' in spacetime to the next one. In order that the spatial slices so generated, stack up in a suitably consistent manner so as to yield a spacetime, HKT show that the Poisson bracket algebra of the generators of dynamics must be isomorphic to the commutator algebra of deformations of the spatial slice within the (emergent) spacetime. These deformations may be separated into those which are tangential and those which are normal to the slice. Their algebra has the characteristic structure that the commutator between two tangential deformations is a tangential one, that between a tangent and normal deformation is normal and, most non-trivially, the commutator of two normal deformations is a tangential deformation which depends on the spatial metric on the slice. This is, of course, exactly the structure of the constraint algebra generated by the diffeomorphism and Hamiltonian constraints of general relativity. 15 In particular, the Hamiltonian constraint generates normal deformations and the Poisson bracket between a pair of Hamiltonian constraints is proportional to a diffeomorphism constraint, the proportionality involving a spatial metric-dependent structure function. The generality and robustness of the arguments of HKT lead one to believe that in the quantum theory, any notion of spacetime covariance is predicated on the commutator algebra of the quantum constraints exactly mirroring the classical Poisson bracket algebra, thus providing a deep physical reason for the requirement of anomaly freedom. In this work we studied a generally-covariant model with the same constraint algebra as gravity. We concentrated on the most non-trivial aspect of this algebra, namely the Poisson bracket between two Hamiltonian constraints, and attempted to define the Hamiltonian constraint operator in an LQG-like quantization in such a way that this Poisson bracket was represented in an anomalyfree manner. Note that at a mathematical level, it would be enough to provide a quantization of the RHS such that it agrees with the LHS. However, the simple geometrical picture of spacetime deformations provided by HKT, suggests that, in addition, the RHS operator should generate a deformation akin to a spatial diffeomorphism . The presence of 'quantum geometry'-dependent operator correspondents of the structure functions on the RHS, together with the fact that the quantum geometry is excited along sets of zero measure, unlike the classical ones, suggests that the deformation should be some sort of 'singular, quantum' version of a smooth diffeomorphism rather than a smooth diffeomorphism. As seen in Sections 4 and 5, the choices we have made in the construction of the Hamiltonian constraint and the RHS incorporate this suggestion. The physical viability of these choices can only be determined once a complete quantization of the system is available. Specifically the work here needs to be completed so as to provide: First consider issue (i). The VSA states of Section 6 provide off-shell closure of the commutator between a pair of Hamiltonian constraints. Since B VSA contains entire diffeomorphism classes, it is straightforward to check [4, 7] that the commutator between two diffeomorphism constraints closes without anomalies as well. It is also straightforward to check that the continuum limit actions of the Hamiltonian and diffeomorphism constraints on a VSA state yield derivatives of its vertex-smooth function so that off-shell VSA states obtained from a specific choice of B VSA can be 'moved' on shell by setting the vertex smooth functions to be a constant. Since we have infinitelymany inequivalent choices of the parameters c 0 , /vectorα, n which go into the construction of B VSA , this procedure yields a large class of solutions to the constraints. 16 These solutions may, of course, prove to be unphysical once we attempt the incorporation of issue (ii). However, it seems plausible that the chances of their physical relevance would be enhanced if it could be shown that their off-shell deformations support the closure of the commutator between the Hamiltonian and the diffeomorphism constraint, this being the only remaining part of the constraint algebra. Clearly, showing this is equivalent to the condition that the Hamiltonian constraint is diffeomorphism covariant; i.e., that ˆ U ( φ ) ˆ H [ N ] ˆ U † ( φ ) = ˆ H [ φ ∗ N ] for all (semianalytic C k ) diffeomorphisms φ and all (density -1 3 ) lapses N . As mentioned in Section 1, we have ignored precisely this issue of diffeomorphism covariance in our constructions. While the issue will be studied in a future publication [12], we briefly comment on the problems inherent in generalizing our constructions here to incorporate diffeomorphism covariance. The primary non-covariant structure we use is the regulating coordinate patches. These patches are chosen once and for all in some arbitrary manner. It turns out (as is eminently plausible) that diffeomorphism covariance requires that coordinate patches associated with diffeomorphic vertex structures (by which we mean the graph structure of a charge network in the vicinity of its (GR, non-degenerate 17 ) vertex) should be related by diffeomorphisms. The ensuing problems are two fold: of relating the corresponding coordinate patches through diffeomorphisms, implies that in the calculation of commutators the coordinate patch { x ' a ' } δ (see the second paragraph of Section 4.5) goes bad as δ → 0. This in turn implies that the continuum limit of the commutator between two Hamiltonian constraints blows up due to the x ' -dependence of the calculation (for example, the Jacobian in Equation (4.59) blows up). A solution to both these problems can be found [12]. It turns out that progress on problem (a) is related to the GR property of the non-degenerate vertices of the VSA states and that a possible way out of problem (b) is to enlarge the dependence of the vertex smooth functions to certain additional vertices of the graph and require some additional regularity properties of the ensuing functional dependence [12]. This concludes our comments on the problem of diffeomorphism covariance and its relation to issue (i). Another key open problem with regard to issue (i) has to do with the very definition of the continuum limit we use (see Section 3.2). This definition, while in the spirit of Thiemann's considerations involving the URS topology, is far from conventional [17]. Notwithstanding the fact that it is extremely non-trivial to obtain an anomaly-free representation in the context of this definition of the continuum limit, we believe that a proper resolution of the problem of an anomaly-free off-shell closure of the constraint algebra requires a representation of the latter on some suitable vector space, which, as mentioned towards the end of Section 3.2, we call a 'habitat'. In the case of the Husain-Kuchaˇr model [4, 7] as well as PFT [6], the habitat is spanned by vertex-smooth algebraic states of the type considered here. It is our hope that these states can be suitably generalized (say, to accommodate not only a dependence of the vertex smooth functions on vertices but, perhaps, on other properties of the state at the vertex such as its edge tangents and their charges) so that our calculations are supported on a genuine habitat. An important aspect of such a generalization would be to ensure that not only the commutator, but also the product of two Hamiltonian constraints has a well-defined action. 18 Preliminary calculations suggest that ensuring this (not only in the context of a habitat but also in the VSA topology considerations of this work) requires a slight modification in the definition of the Hamiltonian constraint operator at finite triangulation from the ' δ -1 form of Equation (4.22) to a '2 δ -δ ' form. Next, we turn to issue (ii). The first step towards the construction of Dirac observables is a detailed analysis of the equations of motion of the classical theory. 19 Such an analysis has been initiated by Barbero and Villase˜nor [18] and we hope that their work will stimulate further progress on issue (ii). As a side remark, we note that a detailed understanding of the classical dynamics of the model would also stimulate progress on Smolin's original idea [11] of approaching Euclidean gravity via an expansion in powers of Newton's constant. Besides the open issues (i) and (ii), our work can also be improved upon in the following aspects. We have required that the 'singular' diffeomorphism type deformations of Sections 4 and 5 preserve the GR (or non-GR) nature of the non-degenerate vertex. This is a rather coarse requirement and it would be good to further restrict the deformation so that it preserves a larger subset of diffeomorphism-invariant properties. This would also lead to a tighter and better-motivated prescription for connecting the original graph to the displaced vertex. A tighter prescription would presumably lead to a smaller bra set B VSA . One may even envisage that the current B VSA can be split into 'minimal' subsets. We now turn to a discussion of various novel features of our constructions and considerations. Our exposition will consist of a series of scattered remarks. First, independent of any ramifications for quantum theory, it would be good to understand if there is a deeper reason behind the existence of the remarkable classical identity of Section 5 and Appendix B. Next, as discussed in Section 6, we note a beautiful feature of repeated actions of our Hamiltonian constraint on an 'initial state'; namely that the resulting 'final' state encodes its own 'chronological history' dating back to the initial state. Finally, we note that while there does seem to be a significant freedom in the details of the choices we have made, the class of choices suggested by our considerations of Section 4.1 are qualitatively different from those considered in the standard treatments of the Hamiltonian constraint [1, 4, 2]. Our considerations here rest on a number of new ideas suggested by earlier studies of toy models [6, 7]. A few of them are: The consideration of higher density weight constraints, a continuum limit defined by VSA states, deformations of charge networks which depend on their charge labels, and a Hamiltonian constraint action which is such that a second such action acts on deformations produced by the first. In summary, while there are many open problems and obstructions to be overcome, we believe that there is room for cautious optimism that the considerations of this work and of the recent work [9, 10] present the first necessary steps to define the correct quantum dynamics of this model, and, perhaps, offer hope that the lessons learnt from this and subsequent studies of the model will provide inputs for the much harder context of gravity.", "pages": [ 39, 40, 41, 42 ] }, { "title": "Acknowledgements", "content": "CT is deeply indebted to Alok Laddha for bringing this model to his attention, for numerous extremely useful discussions at every stage of this work, and for general moral support and mentorship. CT is grateful to Miguel Campiglia-Curcho, Martin Bojowald, Przemys/suppresslaw Mal/suppresslkiewicz, and Keith Thornton for useful discussions. CT is supported by NSF grant PHY-0748336 and a Mebus Fellowship, and acknowledges the generous hospitality and friendly working environment provided by the Raman Research Institute. MV thanks Alok Laddha for numerous discussions, for going through several of the arguments in a draft version of this work and for his extreme generosity with regard to his time and mentorship. MV thanks Abhay Ashtekar, Fernando Barbero, and Eduardo Villase˜nor for their constant encouragement and FB and EV for going through a preliminary version of this manuscript. MV thanks Jurek Lewandowski, Christian Fleischhack and, especially, Hanno Sahlmann for help with the semianalytic category.", "pages": [ 42 ] }, { "title": "A A q -1 / 3 Operator", "content": "In this Appendix we derive some Thiemann-like classical identities for negative powers of the metric determinant that we then quantize on H kin . These identities involve a volume operator, which we take to be the Ashtekar-Lewandowski volume operator ˆ V , with SU(2) replaced by U(1) 3 . The construction of ˆ V in the case of U(1) 3 proceeds just as for SU(2), so we direct the reader to [19] for details. Here we merely cite the result in the U(1) 3 case. Given a region R ⊂ Σ , the volume operator ˆ V ( R ) associated to that region, acting on the charge network state | c 〉 is given by Here, ε ( µ ) is a constant which depends on the choice of an integration measure µ on a finitedimensional 'background structure-averaging' space (if one subscribes to a consistency check in the sense of [20], then this factor can be fixed to be equal to one); the sum extends over all vertices v of c contained in the region R . ˆ q AL ( v ) is diagonal in the charge network basis and acts at vertices v of | c 〉 by where each of the three sums (over I, J, K ) extends over the valence of v, with I, J, K labeling (outgoing) edges e I , e J , e K emanating from v. /epsilon1 IJK = 0 , +1 , -1 depending on whether the tangents of e I , e J , e K are linearly dependent, define a right-handed frame (with respect to the orientation of the underlying manifold), or define a left-handed frame, respectively. As in the main text, q i I is the U(1) i charge on the edge e I . Before moving to inverse metric operators, we note two properties of ˆ V that are shared with the SU(2) theory: /negationslash We now turn to the construction of negative powers of the spatial metric determinant at any point in Σ. Let U ⊂ Σ be an open set with coordinate system { x } . Let any p ∈ U have coordinates /vectorx ( p ) = { x 1 , x 2 , x 3 } . Since the analysis below is expressed in the { x } coordinates, we use the notation p ≡ /vectorx ( p ) ≡ x . The first step is to express negative powers of the classical volume in terms of Poisson bracket identities involving quantities which have unambiguous quantum analogs. Classically, the volume V ( R ) of a region R ⊂ Σ is given by ∣ ∣ Let B /epsilon1 ( x ) ⊂ Σ be a coordinate ball of radius /epsilon1 , centered at x . Its volume V /epsilon1 ( x ) is then: It follows that for smooth q ( y ) and some α ∈ R , Now it is straightforward to verify that where we have defined σ := sgn(det E ) , and neglected terms such as δσ δE a i . Using (A.5) we may then write where the first term is O (1). With an eye on quantizing this expression as an operator on H kin , we replace A i a ( x ) with holonomy approximants as follows: Let e I , I = 1 , 2 , 3 be a triplet of edges, each of coordinate length B I /epsilon1 , emanating from the point x (here B I are a triple of dimensionless /epsilon1 -independent numbers). Let their unit tangents, normalized with respect to the coordinate metric be ˆ e a I and let e I be such that the triple of their edge tangents at x is linearly independent. It is easy to check the following identity: where λ ( /vectore ) is given by Here /epsilon1 IJK is antisymmetric with respect to interchange of its indices with /epsilon1 123 = 1 and the argument /vectore := { e 1 , e 2 , e 3 } signifies the dependence of λ on the triplet of edges. Using equation (A.8) and approximating A i a ˆ e a I in terms of the edge holonomies h i I along e I , we obtain: Setting p = 1 3 we arrive at Now we define an /epsilon1 -regularized operator on H kin by taking all quantities to their operator correspondents, {· , ·} → (i /planckover2pi1 ) -1 [ · , · ], and dropping the classical O ( /epsilon1 ) contribution: with B IJK := B I B J B K (the prime in ˆ q ' appears because this operator is not the final one we will employ in the main body). As it stands, this operator is tied to the coordinate system { x } , which should come as no surprise, since the classical quantity is a scalar density with density weight not equal to 1. In keeping with the general philosophy of this work, in which operators on H kin are tailored to the underlying charge networks that they act on, we will choose the holonomy segments of ˆ q '-1 / 3 to partially overlap edges of charge networks (when this is possible). Let us first consider charge network vertices v ∈ c whose edge tangents span at most a plane (we deem these planar (or linear) vertices); this includes interior points of edges. Since there are not three linearly-independent directions defined by the edge tangents of c at v, we should have to choose the extra segment(s) needed for ˆ q ' ( x ) -1 / 3 /epsilon1 by hand, but this choice is arbitrary, since for the ordering shown in (A.12), there will be some factor [( h i I ) -1 , ˆ V 4 / 9 ] acting on | c 〉 , where ( h i I ) -1 overlaps an existing edge of c, and since ˆ V 4 / 9 acts trivially at planar (and linear) vertices, [( h i I ) -1 , ˆ V 4 / 9 ] annihilates | c 〉 (perhaps even more simply, since planar vertices have zero volume, ˆ σ is the zero operator). We henceforth restrict the discussion to charge network vertices with at least one linearly-independent triple of edge tangents. We write equation (A.12) as: (A.13) where B 123 λ ( /vectore ) has absorbed some dimensionless constants and become B ' . Note that λ ( /vectore ) depends on the charge network c through its dependence on the edge triplet /vectore . It also depends on the choice of regulating coordinate patch { x } through its dependence on the unit edge tangents which are normalized with respect to the coordinate metric defined by { x } . Next, define Q to be the dimensionless rescaled eigenvalue of ˆ q AL ( v ) so that ∣ ∣ and let Q i I be the rescaled eigenvalue of ˆ q AL ( v ) when the regulating holonomy h i I is first laid on the edge e I (and Q -i I when ( h i I ) -1 is laid). Let σ be the eigenvalue of ˆ σ (which is also the sign operator of det E ). Then (A.13) acts on | c 〉 by where we have absorbed some numerical factors into B ' to obtain B . We could stop here, but it turns out that this particular form of ˆ q '-1 / 3 is not quite what we want, as it is not invariant under the charge flips produced by the Hamiltonian constraint, a property that we require in the main body of the paper. However, we can modify the preceding construction slightly to obtain another q -1 / 3 operator which is insensitive to the charge flips. Consider the classical expression (A.11). Instead of using inverse holonomies inside the Poisson brackets, suppose we average over combinations in other representations; specifically q i I = ± 1 for each i, I . Making this change and following the remaining steps to arrive at the operator action, we find where The overall factor of 1 8 comes from averaging over the eight different combinations of O ± i I , and the relative signs arise from the classical Poisson bracket identity, depending on whether we choose to put a fundamental representation holonomy, or its inverse, inside the bracket (an odd number of minus superscripts yields a minus sign). We will see in the next section that these eigenvalues are invariant under charge flips. If there is a choice of edge triplets of c at v such that ˆ q ( v ) -1 / 3 /epsilon1 | c 〉 /negationslash = 0 , we term the vertex v as non-degenerate . Henceforth, we restrict attention to charge networks with a single non-degenerate vertex. For the purposes of this paper, this restriction suffices because the continuum limit action of the quantum Hamiltonian constraint and the quantum electric diffeomorphism vanish on all other charge networks. which in turn stems from the fact that B VSA has states with (at most) only a single non-degenerate vertex. We leave a generalization of our considerations to the multi-vertex case for future work. Note that the inverse metric eigenvalue ν -2 3 in Section 4.2 is defined through the equation We now show how to choose the triplet of edge holonomies in (A.12) in such way that this inverse metric eigenvalue is (a) diffeomorphism invariant, and (b) the same for the (single non-degenerate vertex) charge networks c ( i, v ' I v ,/epsilon1 ) , c ( v ' I v ,/epsilon1 ) of Sections 4 and 5. /negationslash In each diffeomorphism class of charge networks [¯ c ] we pick a reference charge network c 0 and a set of diffeomorphisms D [¯ c ] such that for any element c = c 0 , c ∈ [¯ c ] there is a unique diffeomorphism in D [¯ c ] which maps c 0 to c . Our choice of reference charge networks is further restricted as follows. Let [¯ c i ] , i = 1 , 2 , 3 , [ ˆ ¯ c ] , be such that there exist c i ∈ [¯ c i ] , ˆ c ∈ [ ˆ ¯ c ] , and a charge network c with non-degenerate vertex v such that for some I v , /epsilon1 we have that where c ( i, v ' I v ,/epsilon1 ) , c ( v ' I v ,/epsilon1 ) are the deformations of c as defined in Sections 4 and 5. If equation (A.20) holds, we require that the reference charge networks c i 0 , ˆ c 0 for [¯ c i ] , [ ˆ ¯ c ] be chosen such that there exists a charge network c with a single non-degenerate vertex v 0 and some parameter value δ for which it holds that: Next, we choose a triplet of edges for each reference charge network and define the triplet of edges for any c ∈ [ c 0 ] as the image of these edges by that diffeomorphism in D [ c 0 ] which maps c 0 to c . We restrict our choice of edge triplets as follows. Consider the diffeomorphism classes [¯ c i ] , i = 1 , 2 , 3, [ ˆ ¯ c ] and the charge networks c i 0 , i = 1 , 2 , 3, ˆ c 0 , c , subject to equations (A.20), (A.21). The structure of the deformations sketched in Sections 4,5 (and further elaborated upon in Appendix C) permits the identification of the J th v 0 edge emanating from v ' I v ,δ in c (1 , v ' I v ,δ ) with the J th v 0 edges emanating from v ' I v 0 ,δ in c (2 , v ' I v 0 ,δ ) , c (3 , v ' I v 0 ,δ ) and c ( v ' I v 0 ,δ ); this edge is uniquely identified, in the notation of Sections 4,5 as the deformed counterpart of the J th v 0 edge emanating from the vertex v 0 of c . We choose a triplet of edge labels J K v 0 , K = 1 , 2 , 3 and choose the triplet of edge holonomies for c 10 to be along the J K v 0 th edges emanating from v ' I v 0 ,δ . Our choice for the triplet of edge holonomies for the reference charge networks c 20 , c 30 , ˆ c 0 is then restricted to also be along the J K v 0 th edges emanating from v ' I v 0 ,δ in c 20 , c 30 , ˆ c 0 . We do not, however, restrict the choice of the sets of the reference diffeomorphisms in any way. Once we have made choices subject to the above restrictions, let us, for convenience, once again number our edges in such a way that the triplet of (positively oriented) edges for any charge network c is { e 1 , e 2 , e 3 } so that the action of the inverse metric operator is as denoted in equation (A.17). Recall that the parameter B in that equation is, apart from an overall numerical factor, equal to B 123 λ ( /vectore ). Recall, from equation (A.9) that λ ( /vectore ) depends on the triplet of unit edge tangents normalized in the coordinate metric associated with the coordinate patch around the vertex v of the charge network c being acted upon. Hence λ ( /vectore ) varies as the charge network varies over its diffeomorphism class. We choose B 123 so that B 123 λ ( /vectore ) is constant over each diffeomorphism class. Thus, depending on the charge network c ∈ [ c ], we obtain some λ ( /vectore ) and 'compensate' for this λ ( /vectore ) by appropriately varying the edge length parameters B 1 , B 2 , B 3 so that B 123 λ ( /vectore ) = B 1 B 2 B 3 λ ( /vectore ) is constant over [ c ]. Hence the parameter B in equation (A.17) also depends only on [ c ], or, equivalently, on the reference charge network c 0 ∈ [ c ]. Finally, require that choice of B 123 be identical for the reference charge networks related by (A.21). As we shall see now, these choices ensure that the inverse volume eigenvalue has the properties referred to above. From equation (A.19) we have that The factor σ is equal to the sign of the eigenvalue Q in Equation (A.14). From Equation (A.14), it is easy to check that Q is diffeomorphism-invariant. Moreover, it is straightforward to check that Q is also invariant under the 'charge flips' of equation (4.34). This shows that σ is invariant under diffeomorphisms and charge flips. As we showed above, the factor B is invariant under diffeomorphisms. The rest of the expression consists of various combinations of charge labels of c , and as a result of our choice of regulating edge holonomies, is equal to its evaluation on the reference charge network c 0 ∈ [ c ] irrespective of the choice of the set of reference diffeomorphisms , D [ c ] . Thus ν -2 3 is diffeomorphism invariant. In addition, by construction, B is the same for the quadruple of charge networks c ( i, v ' I v ,δ ) , c ( v ' I v ,δ ) which arise from the action of the Hamiltonian constraint and the action of the electric diffeomorphisms on any charge network c . It follows that, since the charges in equation (A.22) for the charge networks of equation (A.20), (A.21) are related by 'charge flips', the next section also establishes that, as assumed in Sections 4 and 5, ν -2 3 is also the same for the diffeomorphism classes of the charge networks of equations (A.20), (A.21).", "pages": [ 42, 43, 44, 45, 46, 47 ] }, { "title": "A.1 Symmetries", "content": "We are interested in the eigenvalues of ˆ q -1 / 3 for a vertex deformed by the Hamiltonian. There is one important property we are looking for: For the LHS and RHS to match in the main calculation, a charge-flipped vertex produced by the Hamiltonian must have the same ˆ q -1 / 3 eigenvalue as the unflipped configuration. Recall the structure of the charge flips: Depending on the value of i appearing in the quantum shift, edges charged in ( q 1 , q 2 , q 3 ) go to First note that the sign eigenvalue σ of ˆ σ is unchanged; each flipped configuration differs in sign in one entry, and there is a transposition of two charges. Also note that | Q | itself is unchanged by similar arguments. Let us now consider Q i I for some fixed I = ¯ I and i =¯ ı : Here the unbarred indices are summed only over unbarred values. What happens to this value under charge flips? We have argued that Q is unchanged under flips, so focusing on the remainder under we find hence /negationslash Notice that for ˜ ı = ¯ ı, (˜ ı ) Q ± ¯ ı ¯ I = Q ± ¯ ı ¯ I , so at least one factor in each term in (A.17) is invariant. (˜ ı ) Q ± ¯ ı =˜ ı ¯ I changes, but it transforms into one of the other Q ± ¯ ı ¯ I such that the eigenvalue of ˆ q -1 / 3 is invariant. In particular, it is immediate to check that The O ± i I also obey these flip rules, so armed with these properties, it is straightforward to expand and verify that it is in fact equal to ˆ q -1 / 3 | c 〉 , and we conclude that ˆ q -1 / 3 has the symmetry property we need. We close this subsection by noting that the eigenvalues of (the symmetrized) ˆ q -1 / 3 at zero volume vertices vanish. Indeed, in the zero volume case Q = 0, we have that the Q ± i I and O ± i I eigenvalues defined above evaluate to In particular, O + i I = O -i I , and since q -1 / 3 goes as we see that the insensitivity of O ± i I to the sign of the representation of the regulating holonomy leads to the vanishing of this quantity.", "pages": [ 47, 48 ] }, { "title": "A.1.1 Non-Triviality", "content": "The eigenvalues q -1 / 3 are rather complicated functions of the charges, and it is not clear a priori whether the symmetrization procedure followed above perhaps leads to an operator action which is trivially zero through some cancellations. Here we attempt to quell this apprehension somewhat by exhibiting a class of states 20 with large non-zero volume, and small but non-zero q -1 / 3 . Let v be a vertex of c from which emanate N + 3 edges, three of which, e 1 , e 2 , e 3 , define the (positively-oriented) coordinate axes of the system we evaluate ˆ q -1 / 3 with respect to, and let these edges have charges q 1 1 = q 2 2 = q 3 3 = N /greatermuch 1 . Let the other charges on these edges be zero and let the remaining N edges be charged as /vector q = ( -1 , -1 , -1) (so that the state is gauge-invariant). Then we can compute /negationslash /negationslash where the terms quadratic and cubic in the remaining edge charges have vanished as they all have identical charges. We notice that as long as the sum over orientation factors is not negative and O ( N ) , then indeed Q ∼ N 3 . One way to ensure this is to demand that the remaining edges be distributed roughly evenly throughout the octants defined by the tangents to e 1 , e 2 , e 3 at v . In this case the sum over K ' of each orientation factor is O (1) (or perhaps vanishing). For the sake of calculation, let us suppose that N is in fact divisible by 8, and consider the case in which N/ 8 of the small-charge edges lie in each octant. Then the sum over orientation factors in (A.31) in fact vanishes, and we have Q = 6 N 3 . We now wish to compute q -1 / 3 for this configuration. We have, for example /negationslash /negationslash so that /negationslash with analogous results for I = 2 , 3 . Then for I = i, and zero otherwise. Thus where ( ∓ ) ijk denotes the product of the (negative of the) signs in the O superscripts, whence and we conclude that ˆ q -1 / 3 constructed above is not trivially vanishing. In fact, if one allows (an N -independent) tuning of the parameter B, this class of states may be considered as satisfying a crude notion of semiclassicality (to leading order in N ), in the sense that if one chooses", "pages": [ 48, 49 ] }, { "title": "B RHS Identity: SU(2)", "content": "Consider the diffeomorphism generator (modulo Gauss constraint) of the SU(2) theory smeared with the electric shift N a i := q -α NE a i , where N has density weight (2 α -1): Here F i ab := 2 ∂ [ a A i b ] + G N /epsilon1 ijk A j a A k b and the connection again has units of [length × G N ] -1 . It is straightforward to compute the Poisson bracket of two such objects, summing over the SU(2) index: where we have used δq/δE a i = q ( E -1 ) i a , with ( E -1 ) i a the matrix inverse of E a i . The U(1) 3 case results by taking G N → 0 . In 2+1 dimensions, this identity also holds in SU(2) and U(1) 3 : where we have used q = E i E i , E i := 1 2 η ab /epsilon1 ijk E a j E b k and E i η ab = /epsilon1 ijk E a j E b k (see [21]).", "pages": [ 50 ] }, { "title": "C.1 Preliminary Remarks", "content": "We use the notation of Section 4. Let B 4 δ ( v ) be the ball of coordinate radius 4 δ , with respect to the metric δ ab associated with the coordinates { x } , centered at v . Our considerations are confined to the interior of this ball for sufficiently small δ . We shall choose δ to be small enough that the boundary of B 4 δ ( v ) intersects the interior of every edge emanating from v once and only once. Let the edge e I be parameterized by the parameter t I such that e I ( t I = 0) = v . Let the interior of the edge be e int I . Let the coordinates of the point e I ( t I ) be denoted by x µ ( t I ) in the coordinate system { x } . Then for small enough δ it follows from the semianalyticity of the edges that the parameterization t I can be chosen in such a way that x µ ( t I ) ∀ I are analytic functions on e int I ∩ B 4 δ ( v ). Accordingly we choose δ small enough that the edges within B 4 δ ( v ) are analytic in the coordinate system { x } except perhaps at v . We assume for simplicity that v resides at the origin of the coordinate patch { x } . We shall often denote the coordinates { x } of a point by the vector /vectorx from the origin to that point. Since the coordinates range in some open subset of R 3 , we freely use the ensuing R 3 structures, such as constant vectors, vectors connecting a pair of points, straight lines, planes, etc. Recall that ˙ e a I ( v ) =: /vector ˙ e I ( v ) is the tangent vector of the I th edge at v . If /vectora is a vector we denote its component perpendicular to /vector ˙ e I ( v ) by /vectora ⊥ . The vector connecting a point P 1 to the point P 2 is denoted as /vector l P 1 P 2 .", "pages": [ 50, 51 ] }, { "title": "C.2 GR-Preserving Deformation", "content": "1. The GR condition: The set of tangent vectors /vector ˙ e K at v is GR if and only if no triplet lies in a plane. It is easy to verify that this condition implies the pair of conditions: 1.1 /vector ˙ e J ⊥ = 0 , J = I . /negationslash /negationslash 1.2 No pair ( /vector ˙ e J 1 ⊥ , /vector ˙ e J 2 ⊥ ) , J 1 = J 2 = I exists such that /vector ˙ e J 1 ⊥ , /vector ˙ e J 2 ⊥ are linearly-dependent. /negationslash /negationslash /negationslash /negationslash where the hatˆ, as usual, denotes the unit vector in the direction of /vector ˙ e J . Equation (4.29) implies that for J = I , /negationslash Here ( /vector ˆ ˙ e J ) ⊥ is the perpendicular component of the unit vector /vector ˆ ˙ e J and we have used (C.1) in the second line. Note that p > q in the last line so that the first term is the leading order term. /negationslash As asserted in Section 4.4.2, the lines /vector l v ' I ˜ v J , J = I , intersect the graph γ underlying the (undeformed) charge network c at most only at a finite number of points. This can be seen from the following argument. If this was not the case, the analyticity of the edges { e K } (see C.1) and the analyticity of the lines { /vector l v ' I ˜ v J } in the chart { x } implies that a segment of some line /vector l v ' I ˜ v J must overlap with a segment of some edge e K in B 4 δ ( v ). Equation (C.2) together with the GR property of v implies that if this overlap happens it must be for K = J = I . But, whereas || /vector ˙ e J ⊥ || / || /vector ˙ e J || is of /negationslash O (1), equation (C.2) implies that || /vector l ˜ v J v ' I ⊥ || / || /vector l ˜ v J v ' I || , is of O ( δ q -1 ) (here || /vectora || refers to the norm of the vector /vectora ). /negationslash We also note that the lines /vector l v ' I ˜ v J , J = I cannot intersect each other (except at v ' I ) since equation (C.2) implies that they have different slopes. Moreover, since || /vector l ˜ v J v ' I ⊥ || / || /vector l ˜ v J v ' I || , is of O ( δ q -1 ) it follows that these lines (and any bumps thereof can be chosen so that they) are always below the plane P (see Section 4.4.2). Hence these lines cannot intersect the curve ˜ e I of Section 4.4.2. Finally, it easy to see that ˜ e I can indeed be constructed in accordance with the requirements of Section 4.4.2. To do so, we join ˜ v I to v ' I by a straight line and apply appropriate semianalytic diffeomorphisms of compact support in the vicinity of ˜ v I , v ' I only to this line so as to bring its tangents at these points in line with /vector ˆ e I ( v ) as required by equation (4.30) and the requirement that ˜ v I be a C 1 kink. It is straightforward to see that this can be achieved in such a way that ˜ e I remains above P . 4. GR property of v ' I : It remains to show that v ' I is GR. Since we are unable to ascertain if v ' I is GR when connected to γ ( c ) as in 3 . above, we seek a suitable modification of 3 . which ensures that v ' I is GR while preserving the key equations (4.29), (4.30), and (4.31). Since the GR property is generic (as opposed to its negation which requires the condition of coplanarity of some triplet to be enforced) we expect that there should be several ways to do this. However, we do not analyse the issue here and point the reader to Reference [12] wherein we present a detailed resolution of the issue, the particular choice of which is motivated by our considerations in that work. Here, we only note that Reference [12] applies semianalytic diffeomorphisms supported away from identity in a small vicinity of v ' I (only) to each edge in turn which renders the edge tangent configuration 'conical' and hence GR [12]. Each such diffeomorphism is of the type encountered in section C.3 below.", "pages": [ 51, 52 ] }, { "title": "C.3 Non-GR Case", "content": "/negationslash /negationslash As in the previous section we choose /vector ˆ n I in a direction such that v ' I is not on γ ( c ) and follow the prescription of Section 4.4.2 to join v ' I to ˜ v J , J = I by straight lines. Note that, as asserted in Section 4.4.2, any such line /vector l v ' I ˜ v J , J = I can intersect any edge e K at most in a finite number of points. To see this assume the contrary. Analyticity of the lines and edges (see C.1) in the { x } coordinates implies that the line /vector l v ' I ˜ v J overlaps with the edge e K . If /vector ˙ e K | v is proportional to /vector ˙ e I | v , analyticity of e K , /vector l v ' I ˜ v J implies that /vector l v ' I ˜ v J is contained in the line which joins v to v ' I along the direction /vector ˙ e I ( v ). From (4.29), no such line exists. If /vector ˙ e K ⊥ ( v ) = 0 then || /vector ˙ e K ⊥ || / || /vector ˙ e K || is of O (1), while equation (C.2) implies that || /vector l ˜ v J v ' I ⊥ || / || /vector l ˜ v J v ' I || , is of O ( δ q -1 ), which, once again, rules out overlap. /negationslash /negationslash /negationslash Next, any possible overlap between the lines { /vector l v ' I ˜ v J , J = I } can be removed by slightly altering the positions of their vertices ˜ v J as follows. Suppose that /vector l v ' I ˜ v J 1 , /vector l v ' I ˜ v J 2 overlap. Their analyticity and the existence of a common end point v ' I imply that one must be contained in the other. Accordingly, assume that /vector l v ' I ˜ v J 1 is contained in /vector l v ' I ˜ v J 2 so that /vector l v ' I ˜ v J 2 passes through ˜ v J 1 . Since ˜ v J = I ∈ ∂B δ q ( v ), it follows that this pair of lines cannot overlap with any other line. If we now move ˜ v J 1 slightly along e J 1 , this overlap is necessarily removed. For, if it were not, then /vector l v ' I ˜ v J 2 would overlap with e J 1 which is ruled out by the arguments of the previous paragraph. Thus, with this modification, the lines { /vector l v ' I ˜ v J , J = I } intersect each other as well as γ ( c ) at most at a finite number of points and these intersections can be removed by appropriate 'bumping' such that the bumps are all below the plane P of Section 4.4.2. Next, we show that ˜ e I may be chosen so as to satisfy the requirements of Section 4.4.2 on its tangents at its end points while intersecting γ ( c ) at most at a finite number of points and while /negationslash /negationslash Next, suppose that the above prescription leads to v ' I being a non-GR vertex. Then we are done. If not, then proceed as follows. First note that since the 'bumping' is supported away from v ' I , it follows that in a small enough neighborhood of v ' I , the edges ˜ e J = I which connect v ' I to ˜ v J are straight lines. Next, pick some J = I . Then it follows from the above discussion, in conjunction with the GR property of v ' I , that in a small enough neighborhood of v ' I , the plane which contains ˜ e J and which is tangent to the direction /vector ˙ e I ( v ) does not intersect any other edge ˜ e K = J = I . Now consider the vector field which generates rotations about the axis passing through v ' I in a direction normal to this plane. Multiplying this vector field with a semianalytic function of small enough support about v ' I yields a vector field of compact support which generates a diffeomorphism that rotates the tangent /vector ˙ ˜ e J ( v ' I ) to the edge ˜ e J at v ' I into a direction exactly anti-parallel to that of /vector ˙ e I ( v ). We apply this diffeomorphism only to the edge ˜ e J . As a result the vertex v ' I loses its GR property since, now, any triplet of tangent vectors containing the tangents to the I th and J th edges at v ' I lie in a plane by virtue of the anti-collinearity of the (outward-pointing) tangents to the I th and the J th edges. /negationslash /negationslash", "pages": [ 52, 53 ] }, { "title": "C.4 Relating Deformations by Diffeomorphisms", "content": "/negationslash In order to implement (c) simultaneously with (a) and (b), we need to ensure that the diffeomorphism which implements (c) is identity in the vicinity of γ ( c ). We find it simplest to proceed as follows. First we define the position of the displaced vertex at parameter δ through Equation (4.29). Thus the set of points v ' I ≡ v ' I ( δ ) (for all positive δ less than δ 0 ) are contained in a plane tangent to /vector ˙ e I ( v ) , /vector ˆ n I . Our strategy is to choose /vector ˆ n I such that this plane does not intersect γ ( c ) except being positioned above the plane P of Section 4.4.2. Connect ˜ v I to v ' I by the straight line /vector l v ' I ˜ v I . Analyticity implies either a finite number of intersections with γ ( c ) or overlap. Let /vector l v ' I ˜ v I overlap some edge e K . As above, if /vector ˙ e K | v is proportional to /vector ˙ e I | v , analyticity of e K , /vector l v ' I ˜ v I implies that /vector l v ' I ˜ v I is contained in the line which joins v to v ' I along the direction /vector ˙ e I ( v ). From (4.29), no such line exists. If /vector ˙ e K ⊥ ( v ) = 0 then || /vector ˙ e K ⊥ || / || /vector ˙ e K || is of O (1). On the other hand a Taylor series expansion along the edge e I locates ˜ v I to O ( δ 2 ) from the line passing through v in the direction of /vector ˙ e I | v , which, together with equation (C.2) implies that || /vector l ˜ v I v ' I ⊥ || / || /vector l ˜ v I v ' I || , is of O ( δ ), which, once again, rules out overlap. The finite number of intersections with γ ( c ) can be removed by appropriate bumping which preserves the location of /vector l v ' I ˜ v I above the plane P of Section 4.4.2. Finally, the edge tangents at the end point v ' I can be aligned with /vector ˙ e I | v , and the end point ˜ v I transformed into a C 1 -kink by appropriate semianalytic diffeomorphisms which are compactly supported in the vicinity of these end points and which are applied only to /vector l v ' I ˜ v I . /negationslash /negationslash at v (and, at most, in a small vicinity of the straight line passing through v in the direction of /vector ˙ e I ( v )). More precisely, we show that this plane is contained in a small angle 'wedge' with axis along the straight line passing through v in the direction of /vector ˙ e I ( v ), and, that this wedge intersects γ ( c ) at most along (a very small neighborhood of) its axis. This enables the construction of an appropriate diffeomorphism which is identity outside this wedge and which implements (c). In order to show the existence of /vector ˆ n I which allows the construction of such a wedge, it is necessary to confine the edges which are in the vicinity of the straight line passing through v in the direction of /vector ˙ e I ( v ) to manageable neighborhoods so that /vector ˆ n I can be chosen to point away from them. In the GR case only the I th edge is of this type, whereas in the non-GR case there may be several edges with tangent at v along /vector ˙ e I ( v ). It turns out that in both cases these edges can themselves be confined to appropriately small neighborhoods. Given the importance of the 'wedge neighborhoods', it is useful to develop some nomenclature to refer to their construction. We do so in Part 2 below. In Part 3, we show how to choose /vector ˆ n I when v is GR and in Part 4, when v is not GR. Having chosen /vector ˆ n I appropriately, we construct, in Part 5, a diffeomorphism which implements (c) while respecting (a). In Part 6 we construct diffeomorphisms which implement (b) while respecting (a) in such a way that they are identity in the vicinity of v ' I ( δ ) so as not to affect the (prior) implementation of (c). In Parts 3 and 4 we do not fix δ = δ 0 . Rather the considerations in these parts assures us of the existence of a small enough δ which can be set equal to δ 0 in Parts 5 and 6. Accordingly, from C.1, our considerations in Parts 3,4 are restricted to the ball B 4 δ ( v ) and, in Parts 5,6 to B 4 δ 0 ( v ). 2. Some useful nomenclature: Consider a pair of linearly-independent vectors /vectora, /vector b . Consider the set of points for all α ∈ R and all β ≥ 0 such that /vectorx ∈ B 4 δ ( v ). Clearly, the set of these points comprises a 'half plane' which is bounded by the line passing through v in the direction of /vectora . We refer to this set of points as the half plane tangent to ( /vectora, /vector b ) with boundary through v along /vectora . Let us denote this half plane as P . Rotate P about its boundary through v along /vectora by ± θ to obtain a pair of half planes which bound a wedge of angle 2 θ . We shall refer to this wedge as the wedge of angle 2 θ associated with P . 3. Detailed choice of /vector ˆ n I for the GR case: Let the coordinates of the edge e I at parameter value t be /vectorx I ( t ). Since e I is C k , we may use the Taylor expansion: with /vectorv I 1 = /vector ˙ e I ( v ). For simplicity we rescale the parameter t so that /vectorv I 1 = /vector ˆ e I ( v ), where as in the main text, /vector ˆ e I ( v ) is unit in the { x } coordinate metric. Let m be the smallest integer less than k such that the pair /vectorv I m , /vectorv I 1 are not linearly-dependent. If no such m exists then we set m = k -1 so that /vectorv I m ⊥ = 0. /negationslash If /vectorv I m ⊥ = 0 then we proceed as follows. Let P I m be the half plane tangent to ( /vectorv I 1 , /vectorv I m ) with boundary through v along /vectorv I 1 . Then equation (C.4) implies that for small enough δ , the edge e I is confined to the wedge W I m ( θ ) with θ of O ( δ ). Hence there is a '2 π -2 θ ' worth of possible choices for /vector ˆ n I such that v ' I does not lie on e I . We choose /vector ˆ n I such that it lies an angle of O (1) away from the set of vectors { /vectorv I m ⊥ , /vectore J ⊥ } , J = I . Clearly, for small enough δ , v ' I also does not lie on the undeformed graph γ ( c ). /negationslash If /vectorv I m = k -1 ⊥ = 0 then we have that all /vectorv I m ⊥ = 0 for m such that 1 < m ≤ k -1. It follows that the edge e I is confined to a very small neighborhood S k of the line through v along the direction ˙ e I ( v ). To define S k , it is useful to rotate the coordinates { x } = ( x, y, z ) so that the z -axis points along ˙ e I ( v ), v being at the origin. Then we define S k through: Since p /lessmuch k , it follows from (4.29) that for small enough δ , v ' I lies outside S k for any choice of /vector ˆ n I . We choose /vector ˆ n I so that it it lies at an angle of O (1) away from the set of vectors { /vectore J ⊥ } , J = I . /negationslash If there are s edges e J i = I , i = 1 , . . . , s such that /vector ˙ e J i ( v ) is proportional to /vector ˙ e I ( v ), then using the C k nature of these edges, we expand the coordinates /vectorx J i ( t i ) of e J i as a Taylor series in the parameter t i so that: with /vectorv J i 1 proportional to /vector ˙ e I ( v ). As in step 3, for simplicity we rescale the parameters t i so that /vectorv J i 1 = /vector ˆ e I ( v ) For each i , let m i be the smallest integer less than k such that /vectorv J i m i is not proportional to /vector ˙ e I ( v ). If /vectorv J i m i ⊥ = 0 ∀ m i = 1 , 2 , . . . , k -1 then set m i = k -1 so that /vectorv J i m i ⊥ = 0. If /vectorv J i m i ⊥ = 0, let P J i be the half plane tangent to ( /vector ˙ e I ( v ) , /vectorv J i m i ) with boundary through v along /vector ˙ e I ( v ). Let W J i ( θ i ) be the wedge of angle 2 θ i associated to this half plane. Using Equation (C.6), we choose θ i of O ( δ ) such that the edge e J i is confined to the wedge W J i ( θ i ). We choose /vector ˆ n I to be such that its angular separation is of O (1) from the wedges W J i ( θ i ) , i = 1 , .., k as well as from the directions along the vectors /vector ˙ e J ⊥ ( v ) , J / ∈ { I, J 1 , . . . , J k } (recall that /vector ˙ e J ⊥ ( v ) , J / ∈ { I, J 1 , . . . , J k } are the perpendicular components of the tangents to the remaining edges e J , J / ∈ { I, J 1 , . . . , J k } at v ). Clearly, this, together with p /lessmuch k ensures that for small enough δ , v ' I does not lie on γ ( c ). /negationslash 5. Moving the displaced vertex and its vicinity : Let P I be the half-plane tangent to ( /vector ˙ e I ( v ) , /vector ˆ n I ) with boundary through v along /vector ˙ e I ( v ). For the purposes of this part, we rotate the coordinate system { x } = { x, y, z } so that /vector ˙ e I ( v ) is along the z -direction and /vector ˆ n I is along the y -direction. Thus P I is a part of the y -z plane. The choice of /vector ˆ n I implies that there exists small enough δ = δ 0 and θ = θ 0 such that wedge of angle 2 θ 0 associated with P I does not intersect γ ( c ) except, at most, inside S k . Denote this wedge by W I ( θ 0 ). Clearly, at deformation parameter δ 0 , the point v ' I ≡ v ' I ( δ 0 ) has coordinates ( y, z ) = ( δ p 0 , δ 0 ). Let the displaced vertex at parameter δ < δ 0 be denoted by v ' I ( δ ). We place v ' I ( δ ) on P I with coordinates ( y ( δ ) , z ( δ )) given by: Let the straight line joining v ' I ( δ 0 ) to v ' I ( δ ) be l δ 0 ,δ . By virtue of the existence of W I ( θ 0 ) and the fact that p << k , there exists a neighborhood of this line which lies within W I ( θ 0 ) but outside S k , and hence does not intersect γ ( c ). Hence, by multiplying the translational vector field along the direction /vector l v ' I ( δ 0 ) ,v ' I ( δ ) by a suitable function of compact support, a vector field can be constructed that generates a diffeomorphism which rigidly translates a small enough neighborhood of v ' I ( δ 0 ) /negationslash 4. Detailed choice of /vector ˆ n I for the non-GR case: If there are no edges at v other than e I with tangent proportional to /vector ˙ e I ( v ), we place v ' I as for the GR case by choosing /vector ˆ n I to be at an angular separation of O (1) from the set { /vectorv I m ⊥ , /vector ˙ e J ⊥ } for the case that /vectorv I m ⊥ = 0 and from the set { /vector ˙ e J ⊥ } when m = k -1, /vectorv k -1 ⊥ = 0. /negationslash to a corresponding neighborhood of v ' I ( δ ) while being identity in a small enough neighborhood of γ ( c ). The rigid translation property ensures that the edge tangents at v ' I ( δ 0 ) and v ' I ( δ ) are identical. It remains to 'scrunch' the edge tangents of all edges except the I th together. Let the coordinates of v ' I ( δ ) be ( x ( v ' I ( δ )) , y ( v ' I ( δ )) , z ( v ' I ( δ ))) and consider the following linear 'anisotropic' scaling transformation G near v ' I ( δ ): It can easily be verified that this transformation scrunches together the tangent vectors at v ' I ( δ ) as required. The transformation G is generated by the vector field v a G = x ( ∂ ∂x ) a + y ( ∂ ∂y ) a . Once again, multiplying /vectorv G by an semianalytic function of compact support yields a vector field which generates a diffeomorphism that generates the transformation (C.8) at v ' I ( δ ) and is identity in a small enough neighborhood of γ ( c ). /negationslash Fix some edge e J . Let the part of the edge e J between ˜ v J ( δ 0 ) and ˜ v J ( δ ) be e J ( δ 0 , δ ). Let U e J ( δ 0 ,δ ) be a small enough neighborhood of e J ( δ 0 , δ ) such that U e J ( δ 0 ,δ ) ∩ γ ( c ) = e J ( δ 0 , δ ) and such that there exists a small enough neighborhood of v ' ( δ ) which does not intersect U e J ( δ 0 ,δ ) . Let F J be a semianalytic function which vanishes outside U e J ( δ 0 ,δ ) and which is unity on e J ( δ 0 , δ ). Let /vectorg J be a semianalytic vector field which, when restricted to e J , coincides with the tangent vector to e J . Then, clearly, the semianalytic vector field F J g J generates a diffeomorphism which moves ˜ v J ( δ 0 ) to ˜ v J ( δ ) while preserving γ ( c ) and the vicinity of v ' ( δ ). /negationslash We note that the generation of deformations at δ < δ 0 as described above preserves the following properties and/or equations which are sufficient for the analysis of Sections 4-6:", "pages": [ 53, 54, 55, 56 ] } ]

[ { "title": "Reissner-Nordstrom Black Holes on a Codimension-2 Brane", "content": "Derrick Kiley ∗ Department of Physics and Engineering, Los Angeles City College, Los Angeles, CA 90029 (Dated: July 12, 2018) Here we derive the exact Reissner-Nordstrom black hole solution on a tensional codimension-2 brane, generalizing earlier Schwarzschild and Kerr results. We begin by briefly reviewing various aspects of codimension-2 branes that will be important for our analysis, including the mechanism of 'offloading' of brane tension into the bulk that is unique to these branes, as well as the explicit construction of the codimension-2 Schwarzschild black hole as a warm-up exercise. We then show that the same methods can be used to find the metric describing the spacetime surrounding an electrically-charged point source threaded by a codimension-2 brane. The presence of the brane tension leads to an amplification of the apparent strength of gravity, as is well-known, and we further find exactly the same enhancement for the apparent strength of the electric field.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In the area of gravitational research, few topics have received more attention than black holes, which nevertheless remain mysterious. Ideas from string theory add another layer of mystery, positing that the Universe could have additional hidden dimensions. Because gravitation can be viewed as the curvature of space-time, gravitational sources should make their effects felt throughout all of these dimensions. Since gravity would thus be a fundamentally higher-dimensional phenomenon, only appearing four-dimensional to our coarse senses, the intrinsic strength of gravity can be different than the effective four-dimensional value we observe. This difference between the fundamental and observed strengths can potentially provide a 'solution' to the hierarchy problem [1] - [4] (although the problem is then shifted to explaining why the extra dimensions are compactified and in what form). String theory further predicts higher-dimensional surfaces ( branes ) that can live in the extra dimensional space (the bulk ). A particularly interesting brane configuration is a codimension-2 brane, in which a D -dimensional brane floats in a D +2-dimensional bulk; for example, a three brane (one time and three space dimensions) living in a six-dimensional space-time. These branes have a unique and useful property. Typically the presence of matter on a brane can curve the brane, as well as the surrounding bulk. However, in the case of a codimension-2 brane, vacuum energy (called the brane tension ) can be 'offloaded' into the bulk, leaving the brane flat. The bulk then acquires the topology of a cone with the tip centered on the brane. The bulk remains locally flat, but gains a conical singularity along the brane in a way that is similar to that in the space surrounding a cosmic string, as is well-known [5]. Let us briefly review how this happens [6]. We begin with the action describing six-dimensional bulk gravity and include a three-brane with matter La- grangian L 4 , where M 6 is the fundamental six-dimensional Planck scale (in units c = /planckover2pi1 ≡ 1). Variation of the action in Eq. (1) yields the six-dimensional Einstein equations Here T µ ν is the four-dimensional brane stress energy tensor localized to the brane by the delta function. The indices M,N run from 0 to 5, while µ, ν run from 0 to 3, and /vectory denotes the extra two bulk coordinates. Here h is the determinant of the bulk components of the metric, √ h = √ -g 6 / √ -g 4 . Tracing Eq. (2) gives the 6D Ricci scalar where T ≡ T µ µ is the trace of the brane stress-energy tensor. We can now insert Eq. (3) back into Eq. (2) to find Eq. (4) can be broken up into bulk and brane pieces (here a and b denote the two bulk coordinates), We now split up the brane stress-energy tensor into a tensional contribution with brane tension λ , and a piece denoting any other matter contributions, τ µ ν , Plugging in Eq. (7) to Eq. (5) gives our final result along the brane, where τ ≡ τ µ µ is the trace of the non-tensional stressenergy tensor on the brane. In the bulk we plug in Eq. (7) to Eq. (6) to find In Eqs. (8) and (9) we see a remarkable result: the brane tension has vanished from the brane equations, but does appear in the bulk equations. This is the 'offloading' of the tension into the bulk, as discussed earlier. We now suppose that the brane contains only tension, such that τ µ ν = 0. Then Eq. (8) reduces to R µ ν = 0, while Eq. (9) becomes Thus, we see that the brane Ricci tensor vanishes, while the bulk Ricci terms contain a delta function spike leading to a conical singularity, as we will now demonstrate. Because the brane Ricci tensor vanishes the Ricci scalar is simply the trace of the two-dimensional Ricci tensor in Eq. (10), R 6 = R 2 where We take for the two-dimensional bulk metric the ansatz g ab = e -2 θ δ ab , where θ is a function of /vectory , such that the bulk coordinates are conformally flat. In this case the Ricci tensor is R 2 = 2 e 2 θ /vector ∇ 2 /vector y θ . Plugging in the metric ansatz to Eq. (11) and using √ h = √ det | e -2 θ δ ab | = e -2 θ we find for Eq. (11) R 2 = ( 2 λ/M 4 6 ) e 2 θ δ (2) ( /vector y ). Comparing both expressions for R 2 we see that Thus, θ is solved by the two-dimensional Green's function. Noting that /vector ∇ 2 /vector y ln ( | /vectory | //lscript ) = 2 πδ (2) ( /vectory ), where /lscript is a scaling constant needed to make the units work out, we finally find that θ = 2 b ln ( | /vectory | //lscript ) where Hence g ab dx a dx b = ( | /vectory | //lscript ) -2 b ( dy 2 1 + dy 2 2 ) . By making the coordinate transformations the metric becomes ds 2 2 = dρ 2 + (1 -b ) 2 ρ 2 dφ 2 , which is almost flat polar coordinates, except for the factor of (1 -b ) 2 in front of the dφ 2 term. This shows the conical singularity; the polar angle φ is rescaled, φ → (1 -b ) φ meaning that it does not run around a full 2 π radians, but rather from 0 to (1 -b ) × 2 π (there is a deficit angle). We see that the net effect of the brane tension is to extract a wedge of angle δ ≡ 2 πb = λ/M 4 6 from the flat bulk space. The edges of the wedge cut are then identified, yielding the conical space. One can also check that this metric represents a bulk conical space by embedding a cone into a seven -dimensional Minkowski space and checking that the resulting 6 D metric comes out the same [7]. Now, in the absence of any non-tensional matter whatsoever the full six-dimensional metric may be written ds 2 6 = η µν dx µ dx ν + dρ 2 + (1 -b ) 2 ρ 2 dφ 2 . Let us rewrite ds 2 6 in spherical coordinates. Noting that η MN dx M dx N = -dt 2 + dr 2 + r 2 d Ω 2 4 , we can include the effect of the conical deficit by writing d Ω 2 4 = d Ω 2 3 + (1 -b ) 2 Π 3 k =1 sin 2 ( θ k ) dχ 2 [6]. Thus, the flat metric, including the brane tension is simply which is flat space, but with a conical deficit. The metric in Eq. (15) explicitly contains the conical singularity and solves R MN = R 6 = 0 away from the brane. The metric makes sense for all values of the brane tension, including for supercritical values when b > 1. In this case the bulk spacetime is compactified into a two-dimensional teardrop shape [8]. Because the brane tension is offloaded into the bulk, finding gravitational solutions is considerably simplified. On the other hand, it appears that the presence of nontensional stress energy on the brane would ruin this simple offloading because of Eq. (9), which includes the matter contributions. However, it is obvious that allowing matter with a vanishing trace of its stress-energy tensor (such as relativistic matter) to live on the brane would keep the bulk geometry simple, even if the brane geometry becomes complicated. Furthermore, as we will see below, the bulk can remain conical, even in the presence of some further brane sources. A particularly important system with vanishing stressenergy is a black hole, which satisfies the vacuum Einstein equations away from the singularity. Generalizing the black hole solutions to higher-dimensions is straightforward for an empty bulk [9] - [10], but exact black hole solutions including a brane are typically difficult to find. Fortunately, in the codimension-2 case the solution is very straightforward and the exact Schwarzschild solution, including a tensional brane, was constructed in [6]. While the Schwarzschild black hole solution is completely characterized by its mass, other black hole solutions exist involving spin and electric (and magnetic) charge. The solution describing a rotating black hole on a codimension-2 brane was made in [11], while the so- lution including electric charge, the Reissner-Nordstrom solution, is the subject of the remainder of this paper.", "pages": [ 1, 2, 3 ] }, { "title": "II. REVIEWING THE SCHWARZSCHILD CASE", "content": "Before turning to the Reissner-Nordstrom solution, let us very briefly review the much simpler Schwarzschild case, originally constructed in [6], and explicitly determine the exact six-dimensional solution. The ReissnerNordstrom solution will proceed along the same lines, and so the aside will be well worth it. We have seen that the conical/Minkowski space in Eq. (15) allows for vanishing Ricci terms off the bulk, but the Minkowski metric is not the only solution of R = 0, as is very well-known. Let us instead try a metric of the form where f is a function of only the radial coordinate in order to maintain isotropy of the space-time, and we explicitly include the brane tension. Working out the Ricci tensor components gives where m and n stand for the remaining coordinates, { θ, φ, ψ, χ } . Every term in Eqs. (17) is independent of the brane tension, except for the contribution from g χχ . Solving for the Ricci scalar also gives a result independent of the brane tension, Choosing R mn = 0 sets rf ' +3 f -3 = 0, such that where C is an integration constant (this solution for f also satisfies all of the other equations, R MN = R 6 ≡ 0 as expected). We can determine the constant C as follows: we write g tt = g 0 tt + h tt = -(1 -h tt ) = -( 1 + Cr -3 ) , setting h tt = -Cr -3 . However, the ADM mass is [10] where Ω 4 ( b ) is the four-dimensional surface area including the brane tension , and M is the mass. Explicitly, Ω 4 = 8 π 2 3 (1 -b ), and so comparing the results gives Note the appearance of the brane tension parameter, b in this expression. Thus, the generalization of the sixdimensional Schwarzschild metric that is threaded by a tensional brane is Eq. (16) with This expression differs from the ordinary result in six dimensions [10] by the addition of the (1 -b ) -1 factor. The metric result in Eqs. (16) and (18) could have easily been found by starting with the ordinary six-dimensional Schwarzschild solution, and then extracting a wedge of deficit angle δ = 2 πb from the bulk coordinates, arriving at the solution immediately. The solution was first constructed in [6], in just this way. The presence of the brane tension in Eq. (18) leads to some interesting effects. In particular, the event horizon r H is found from setting f ( r ) = 0, where r 0 is the ordinary result in six dimensions, if the brane was not present. The larger the brane tension becomes, such that b → 1, then the larger the horizon size grows. This effect, dubbed the 'lightning rod effect,' [6] is conceptually easy to understand; the gravitational field lines from the black hole are confined to the surface of a cone in the bulk dimensions. This keeps the field lines together such that they cannot spread out and dilute as quickly as they would in an otherwise flat space, much like the electric field around a needle can grow very large even though there may only be a small amount of charge. Because the gravity does not dilute as quickly, this enhancement of the event horizon can be viewed in another way. Gravity appears stronger than would be expected based on a naive analysis including the fundamental gravitational scale, M 6 . However, we can define an effective six-dimensional gravitational scale which includes the conical enhancement. Thus, we see that the net effect of the brane tension is to rescale the gravitational scale, amplifying it. Upon compactification of the extra dimensions to a scale ∼ L , then the fourdimensional Plank mass M 2 Pl ∼ L 2 M 4 6eff , and hence the 4D Planck mass that we observe on everyday scales would thus already include any effects from brane tension [6]. The amplification due to the brane tension can make its effects felt in additional ways, for example increasing the lifetime of an evaporating black hole and changing the angular momentum of a spinning black hole [11]. We will also see below that the amplification is not limited to gravity, but can also affect the electrical field of a charged mass for the same reasons as discussed for gravity: the electrical field lines also cannot dilute as quickly, and so electricity appears stronger. Now that we have seen the method for the Schwarzschild solution, we generalize to electrically charged black holes.", "pages": [ 3 ] }, { "title": "III. THE REISSNER-NORDSTR OM SOLUTION", "content": "While the analysis describing an electrically-charged black hole is performed in the same way as in the Schwarzschild case, the algebra is a bit harder since the Ricci scalar does not vanish in this case. We begin with the Einstein-Maxwell Lagrangian where the second term is the new addition. Here F MN is the six-dimensional electromagnetic field tensor, and we use slightly uncommon 'SI-like' units, explicitly including the six-dimensional permeability of free space for later convenience. Variation of the action in Eq. (21) with respect to the metric yields the gravitational equations, while variation with respect to the electromagnetic field A M says that the field tensor must be divergenceless, Once again we can split up Eq. (22) into bulk and brane pieces, taking the brane stress-energy tensor T µ ν = -λδ µ ν to include only tension, where the tension again cancels from the brane equations. These are the equations that we need to solve. Notice that the bulk equations in Eq. (24) contain electromagnetic components, in contrast to Eq. (10). At first sight this would seem to complicate matters considerably since the bulk is curved from the electromagnetic field. However, this curvature is just what one would expect from the ordinary 6D Reissner-Nordstrom metric, as we will see. We begin again with a tensionless metric ansatz , In what follows it will be convenient to briefly make a coordinate transformation, defining a new 'radial' distance, where K is a scaling constant. This transforms the metric into uniform coordinates , G -1 ( R ) ( dr d R ) 2 depend directly on the metric function f ( r ), but their exact forms are not important for this analysis. In these coordinates the bulk metric components are once again conformally flat To include the brane tension, we again try ˜ g ab = e -2 θ g ab , then ˜ R ab = R ab + δ ab /vector ∇ 2 θ , where R ab is the Ricci tensor in the absence of the brane tension. Noting that which we can set equal to δ ab /vector ∇ 2 θ , we once again find Eq. (12)! Thus, the brane tension affects the electricallycharged black hole in precisely the same way as it does the Schwarzschild black hole, with b once again given by Eq. (13). Using the same coordinate transformations in Eqs. (14) gives (upon setting R 2 ≡ /vectorx 2 + ρ 2 , where /vectorx 2 is the three-dimensional brane distance) This is the same metric as the flat case, except for the conformal factor of G ( R ). We can now undo the coordinate transformation to go back to the metric form in Eq. (26), in terms of f ( r ), which now includes the brane tension parameter, b . This form simply gives back Eq. (16) again (although f is different for the electromagnetic case, of course), and the Ricci terms are again given by Eqs. (17). We can determine f ( r ), by solving Eq. (25), but we first need the F PQ F PQ , which we can find from Eq. (23). For our problem, we are only interested in a static, electrically-charged black hole with electric field E ( r ). In this case, F MN = E ( r ) ( δ M t δ N r -δ M r δ N t ) . Then, Eq. (23) gives ∂ r ( √ -g 6 E ) = 0. From the metric in Eq. (26) √ -g 6 ∼ r 4 , such that ∂ r ( r 4 E ) = 0. Hence, E = const /r 4 . To determine the constant, we can use the 6D Gauss's law, ∮ /vector E · d /vector A 4 = Q//epsilon1 6 , where Q is the electric charge, and we again use 'SI-like' coordinates with a generalized 6D permittivity of free space. The effective four-dimensional permeability and permittivities are found in terms of the six-dimensional fundamental values via compactification, µ 0 ∼ µ 6 L 2 and /epsilon1 0 ∼ /epsilon1 6 L -2 where L is again the compactification radius. Then the 6D quantities satisfy µ 6 /epsilon1 6 = µ 0 /epsilon1 0 = c -2 ≡ 1, in our units. Now, solving Gauss's law on a spherical Gaussian surface gives ∮ /vector E · d /vector A 4 = E Ω 4 ( b ) r 4 = Q//epsilon1 6 , where Ω 4 ( b ) is once again the four-dimensional surface area including the brane tension. Thus, we finally find which again contains the brane tension. Then, using Eq. (30) we have Once again, the R mn may be found from Eq. (25) and Eqs. (17), which gives after setting µ 6 /epsilon1 6 = 1 in our units. Solving for f ( r ) gives where C is again an integration constant. Comparing with the ADM mass (or, more simply noting that Eq. (31) must reduce to Eq. (18) when the electric charge goes to zero) gives same results, So, the full (and exact) metric describing an electricallycharged six-dimensional black hole threaded by a tensional brane is Eq. (16) with Notice in the term ∼ r -6 two factors of (1 -b ) -1 appear, whereas only one factor appears in the term ∼ r -3 . This is easy to understand; one factor can be absorbed into the Planck scale, M 2 6eff ∼ M 2 6 (1 -b ), while the other factor rescales the electric field strength, /epsilon1 6eff ∼ (1 -b ) /epsilon1 6 , in the same way (this explains why we chose the 'SIlike' units). As discussed above, this is to be expected since the electric field lines will also not spread out as quickly in the conical bulk as they could in a fully sixdimensional spacetime. Thus, both the gravitational and electrical forces experience the lightning rod effect discussed above. Once again, the horizons are affected by the presence of the brane, in precisely the same way as Eq. (19), r H = (1 -b ) -1 / 3 r 0RN , where r 0RN are the ordinary 'braneless' six-dimensional Reinsser-Nordstrom horizons. This tensional enhancement of the electrical field would also likely lead to a more rapid electrical discharge of the black hole.", "pages": [ 4, 5 ] }, { "title": "IV. CONCLUSIONS", "content": "We have explicitly constructed the exact metric describing an electrically-charged point mass threaded by a tensional codimension-2 brane in a six-dimensional space-time. This analysis demonstrates a solution to a codimension-2 brane system where the matter stressenergy tensor does not vanish in the bulk, unlike the previous tensional brane black hole solutions found in [6] and [11]. We have found that the presence of the brane tension enhances the apparent strength of the gravitational and also electrical fields surrounding the black hole. The previous black hole solutions in [6] and [11] could be constructed simply by starting with the empty-bulk six-dimensional solution, extracting a wedge from an axis of symmetry by rescaling χ → (1 -b ) χ and rescaling the fundamental six-dimensional Planck constant. Here we find that the electrically-charged black hole solutions contain an additional affect owing to the identical tensional enhancement of the electric field. So, in this case, the final metric is a bit more subtle than a simple construction would suggest, although there are no real surprises in the final result, Eq. (32).", "pages": [ 5 ] }, { "title": "V. ACKNOWLEDGMENTS", "content": "I would like to thank N. Kaloper, for interesting and useful comments. th/0601110].", "pages": [ 5 ] } ]

[ { "title": "Finite temperature current densities and Bose-Einstein condensation in topologically nontrivial spaces", "content": "E. R. Bezerra de Mello 1 ∗ , A. A. Saharian 1 , 2 † 1 Departamento de F'ısica, Universidade Federal da Para'ıba 58.059-970, Caixa Postal 5.008, Jo˜ao Pessoa, PB, Brazil 2 Department of Physics, Yerevan State University, 1 Alex Manoogian Street, 0025 Yerevan, Armenia", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate the finite temperature expectation values of the charge and current densities for a complex scalar field with nonzero chemical potential in background of a flat spacetime with spatial topology R p × ( S 1 ) q . Along compact dimensions quasiperiodicity conditions with general phases are imposed on the field. In addition, we assume the presence of a constant gauge field which, due to the nontrivial topology of background space, leads to Aharonov-Bohm-like effects on the expectation values. By using the Abel-Plana-type summation formula and zeta function techniques, two different representations are provided for both the current and charge densities. The current density has nonzero components along the compact dimensions only and, in the absence of a gauge field, it vanishes for special cases of twisted and untwisted scalar fields. In the hightemperature limit, the current density and the topological part in the charge density are linear functions of the temperature. The Bose-Einstein condensation for a fixed value of the charge is discussed. The expression for the chemical potential is given in terms of the lengths of compact dimensions, temperature and gauge field. It is shown that the parameters of the phase transition can be controlled by tuning the gauge field. The separate contributions to the charge and current densities coming from the Bose-Einstein condensate and from excited states are also investigated. PACS numbers: 03.70.+k, 11.10.Kk, 03.75.Hh", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In recent years, there has been a large interest to the physical problems with compact spatial dimensions. Several models of this sort appear in high energy physics, in cosmology and in condensed matter physics. In particular, many of high energy theories of fundamental physics, including supergravity and superstring theories, are formulated in spacetimes having extra compact dimensions which are characterized by extremely small length scales. These theories provide an attractive framework for the unification of gravitational and gauge interactions. The models of a compact universe with nontrivial topology may also play an important role by providing proper initial conditions for inflation [1]. In the models with compact dimensions, the nontrivial topology of background space can have important physical implications in classical and quantum field theories, which include instabilities in interacting field theories [2], topological mass generation [3, 4] and symmetry breaking [4, 5]. The periodicity conditions imposed on fields along compact dimensions allow only the normal modes with suitable wavelengths. As a result of this, the expectation values of various physical observables are modified. In particular, many authors have investigated the effects of vacuum or Casimir energies and stresses associated with the presence of compact dimensions (for reviews see Refs. [6], [7]). The topological Casimir effect is a physical example of the connection between quantum phenomena and global properties of spacetime. The Casimir energy of bulk fields induces a non-trivial potential for the compactification radius of higher-dimensional field theories providing a stabilization mechanism for the corresponding moduli fields and thereby fixing the effective gauge couplings. The Casimir effect has also been considered as a possible origin for the dark energy in both Kaluza-Klein type models and in braneworld scenario [8]. The main part of the papers, devoted to the influence of the nontrivial topology on the properties of the quantum vacuum, considers the vacuum energy and stresses. These quantities are chosen because of their close connection with the structure of spacetime through the theory of gravitation. For charged fields another important characteristic, bilinear in the field, is the expectation value of the current density in a given state. In Ref. [9], we have investigated the vacuum expectation value of the current density for a fermionic field in spaces with an arbitrary number of toroidally compactified dimensions. Application of the general results are given to the electrons of a graphene sheet rolled into cylindrical and toroidal shapes. For the description of the relevant low-energy degrees of freedom we have used the effective field theory treatment of graphene in terms of a pair of Dirac fermions. For this model one has the topologies R 1 × S 1 and ( S 1 ) 2 for cylindrical and toroidal nanotubes respectively. Combined effects of compact spatial dimensions and boundaries on the vacuum expectation values of the fermionic current have been discussed recently in Ref. [10]. In the latter, the geometry of boundaries is given by two parallel plates on which the fermion field obeys bag boundary conditions. The effects of nontrivial topology around a conical defect on the current induced by a magnetic flux were investigated in Ref. [11] for scalar and fermion fields. In the present paper we consider the finite temperature charge and current densities for a scalar field in background spacetime with spatial topology R p × ( S 1 ) q . In both types of models with compact dimensions used in the cosmology of the early Universe and in condensed matter physics, the effects induced by the finite temperature play an important role. The thermal corrections arise from thermal excitations of fluctuation spectrum and they depend strongly on the geometry. As a consequence of this, thermal modifications of quantum topological effects can differ qualitatively for different geometries. The thermal Casimir effect in cosmological models with nontrivial topology has been considered in Refs. [12]. A general discussion of the finite temperature effects for a scalar field in higher dimensional product manifolds with compact subspaces is given in Ref. [13]. Specific calculations are presented for the cases when the internal space is a torus or a sphere. In Ref. [14], the corresponding results are extended to the case in which a chemical potential is present. In the previous discussions of the effects from nontrivial topology and finite temperature, the authors mainly consider periodicity and antiperiodicity conditions imposed on the field along compact dimensions. The latter correspond to untwisted and twisted configurations of fields respectively. In this case the current density corresponding to a conserved charge associated with an internal symmetry vanishes. As it will be seen below, the presence of a constant gauge field, interacting with a charged quantum field, will induce a nontrivial phase in the periodicity conditions along compact dimensions. As a consequence of this, nonzero components of the current density appear along compact dimensions. This is a sort of Aharonov-Bohm-like effect related to the nontrivial topology of the background space. The organization of the paper is as follows. In the next section the geometry of the problem is described and the thermal Hadamard function is evaluated for a complex scalar field in thermal equilibrium. In Section 3, by using the expression for the Hadamard function, the expectation values of the charge and current densities are investigated. Various limiting cases are discussed. Alternative expressions for the charge and current densities are provided in Section 5 by making use of the zeta function renormalization approach. The Section 6 is devoted to the investigation of the Bose-Einstein condensation in the background under consideration. The properties of the vacuum expectation value of the charge density are discussed in Appendix A. Throughout the paper we use the units /planckover2pi1 = c = k B = 1, with k B been Boltzmann constant.", "pages": [ 1, 2, 3 ] }, { "title": "2 Geometry of the problem and the Hadamard function", "content": "We consider the quantum scalar field ϕ ( x ) on background of ( D + 1) dimensional flat spacetime with spatial topology R p × ( S 1 ) q , p + q = D . For the Cartesian coordinates along uncompactified and compactified dimensions we use the notations x p = ( x 1 , ..., x p ) and x q = ( x p +1 , ..., x D ), respectively. The length of the l -th compact dimension we denote as L l . Hence, for coordinates one has -∞ < x l < ∞ for l = 1 , .., p , and 0 /lessorequalslant x l /lessorequalslant L l for l = p +1 , ..., D . In the presence of a gauge field A µ the field equation has the form where D µ = ∂ µ + ieA µ and e is the charge associated with the field. One of the characteristic features of field theory on backgrounds with nontrivial topology is the appearance of topologically inequivalent field configurations [15]. The boundary conditions should be specified along the compact dimensions for the theory to be defined. We assume that the field obeys generic quasiperiodic boundary conditions, with constant phases | α l | /lessorequalslant π and with e l being the unit vector along the direction of the coordinate x l , l = p + 1 , ..., D . The condition (2.2) includes the periodicity conditions for both untwisted and twisted scalar fields as special cases with α l = 0 and α l = π , respectively. In the discussion below we will assume a constant gauge field A µ . Though the corresponding field strength vanishes, the nontrivial topology of the background spacetime leads to the AharonovBohm-like effects on physical observables. In the case of constant A µ , by making use of the gauge transformation with χ = A µ x µ we see that in the new gauge one has A ' µ = 0 and the vector potential disappears from the equation for ϕ ' ( x ). For the new field we have the periodicity condition where In what follows we will work with the field ϕ ' ( x ) omitting the prime. Note that for this field D µ = ∂ µ . As it is seen from Eq. (2.5), the presence of a constant gauge field shifts the phases in the periodicity conditions along compact dimensions. In particular, a nontrivial phase is induced for special cases of twisted and untwisted fields. As it will be shown below, this is crucial for the appearance of the nonzero current density along compact dimensions. Note that the term in Eq. (2.5) due to the gauge field may be written as where Φ l is a formal flux enclosed by the circle corresponding to the l -th compact dimension and Φ 0 = 2 π/e is the flux quantum. The complete set of positive- and negative-energy solutions for the problem under consideration can be written in the form of plane waves: where k = ( k p , k q ), k p = ( k 1 , . . . , k p ), k q = ( k p +1 , . . . , k D ), with -∞ < k i < + ∞ for i = 1 , . . . , p . For the momentum components along the compact dimensions the eigenvalues are determined from the conditions (2.4): with l = p +1 , ..., D . From Eq. (2.8) it follows that the physical results will depend on the fractional part of ˜ α l / (2 π ) only. The integer part can be absorbed by the redefinition of n l . Hence, without loss of generality, we can assume that | ˜ α l | /lessorequalslant π . The normalization coefficient in (2.7) is found from the orthonormalization condition where δ kk ' = δ ( k p -k ' p ) δ n p +1 ,n ' p +1 ....δ n D ,n ' D . Substituting the functions (2.7), for the normalization coefficient we find with V q = L p +1 ....L D being the volume of the compact subspace and The smallest value for the energy we will denote by ω 0 . Assuming that | ˜ α l | /lessorequalslant π , we have We are interested in the expectation values of the charge and current densities for the field ϕ ( x ) in thermal equilibrium at finite temperature T . These quantities can be evaluated by using the thermal Hadamard function where 〈· · · 〉 means the ensemble average and ˆ ρ is the density matrix. For the thermodynamical equilibrium distribution at temperature T , the latter is given by where β = 1 /T . In Eq. (2.14), ̂ Q denotes a conserved charge, µ ' is the related chemical potential and Z is the grand-canonical partition function In order to evaluate the expectation value in Eq. (2.13) we expand the field operator over a complete set of solutions: with ∑ k = ∫ d k p ∑ n q and n q = ( n p +1 , . . . , n D ). Here and in what follows we use the notation for n = ( n 1 , . . . , n l ). Substituting the expansion (2.16) into Eq. (2.13), we use the relations where µ = eµ ' . Note that the chemical potentials have opposite signs for particles ( µ ) and antiparticles ( -µ ). The expectation values for ˆ a k ˆ a + k ' and ˆ b k ˆ b + k ' are obtained from (2.18) by using the commutation relations and the expectation values for the other products are zero. For the Hadamard function we get where the first term in the right-hand side corresponds to the zero temperature Hadamard function: with | 0 〉 being the vacuum state. In order to ensure a positive-definite value for the number of particles we assume that | µ | /lessorequalslant ω 0 , where ω 0 is the smallest value of the energy (see Eq. (2.12)). By using the expressions (2.7) for the mode functions and the expansion ( e y -1) -1 = ∑ ∞ n =1 e -ny , the mode sum for the Hadamard function is written in the form where ∆ x p = x p -x ' p , ∆ x q = x q -x ' q , ∆ t = t -t ' . For the evaluation of the Hadamard function we apply to the series over n r the Abel-Plana-type summation formula [16, 17] (for applications of the Abel-Plana formula and its generalizations in quantum field theory see Refs. [6, 18, 19]) where k r is given by Eq. (2.8). For the Hadamard function we find the expression where n r q -1 = ( n p +1 , . . . , n r -1 , n r +1 , . . . , n D ), k q -1 = ( k p +1 , . . . , k r -1 , k r +1 , . . . , k D ), and The first term in the right-hand side of Eq. (2.23), G (1) p +1 ,q -1 ( x, x ' ), comes from the first term on the right of Eq. (2.22) and it is the Hadamard function for the topology R p +1 × ( S 1 ) q -1 with the lengths of the compact dimensions ( L p +1 , . . . , L r -1 , L r +1 , . . . , L D ). For the further transformation of the expression (2.23) we use the expansion With this expansion the z -integral is expressed in terms of the Macdonald function of the zeroth order. Then the integral over k p is evaluated by using the formula from Ref. [20]. For the Hadamard function we arrive to the final expression where Note that the n r = 0 term in Eq. (2.26) corresponds to the function G (1) p +1 ,q -1 ( x, x ' ). Hence, the part of the Hadamard function in Eq. (2.26) with n r = 0 is induced by the compactification of the r -th direction to a circle with the length L r . /negationslash An alternative expression for the Hadamard function is obtained directly from Eq. (2.21). We first integrate over the angular part of k p and then the integral over | k p | is expressed in terms of the Macdonald function. The corresponding expression is written in terms of the function (2.27) as with the notation and k 2 q is given by Eq. (2.11). Note that the explicit information contained in Eq. (2.26) is more detailed. Both representations (2.26) and (2.28) present the thermal Hadamard function as an infinite imaginary-time image sum of the zero temperature Hadamard function. This is the well-known result in finite temperature field theory (see, for instance, Ref. [21]).", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Charge density", "content": "Having the thermal Hadamard function we can evaluate the expectation value for the current density l = 0 , 1 , . . . , D , by using the formula By making use of the relation ∂ z f ν ( z ) = -zf ν +1 ( z ), from Eq. (2.26) for the charge density ( l = 0) one finds where the prime on the sign of sum means that the term n r = 0 should be taken with the coefficient 1/2. /negationslash An alternative expression for the charge density, more symmetric with respect to the compact dimensions, is obtained by applying the formula As it is seen from Eq. (3.3), the charge density is an even function of the phases ˜ α l and, for a fixed value of the chemical potential, it vanishes in the zero temperature limit. It is a periodic function of ˜ α l with the period equal to 2 π . In the case of zero chemical potential the charge density is zero. In Eq. (3.3), the term with n r = 0 corresponds to the charge density for the topology R p +1 × ( S 1 ) q -1 with the lengths of the compact dimensions ( L p +1 , . . . , L r -1 , L r +1 , . . . , L D ) and the contribution of the terms with n r = 0 is the change in the charge density due to the compactification of the r -th dimension to S 1 with the length L r . By taking into account Eq. (2.6), we see that the charge density is a periodic function of fluxes Φ l with the period equal to the flux quantum. Note that the sign of the ratio 〈 j 0 〉 /e coincides with the sign of the chemical potential. with a, b, c > 0, w n = √ (2 πn + α ) 2 /a 2 + c 2 , to the series over n r in Eq. (3.3). This leads to the expression with the notation (2.29). This formula could also be directly obtained from Eq. (3.2) using the expression (2.28) for the Hadamard function. The form (3.5) for the charge density in the case of topology R p +1 × ( S 1 ) q -1 is also obtained from Eq. (3.3) taking the limit L r →∞ . In the case of Minkowski spacetime one has p = D , q = 0, and from Eq. (3.5) we get with | µ | /lessorequalslant m . The thermodynamic properties of the relativistic Bose gas in this case have been considered in Refs. [22, 23]. If all spatial dimensions are compactified, the corresponding formulas are obtained from Eqs. (3.3) and (3.5) taking p = 0. In particular, from Eq. (3.5) one has where we have used f 1 / 2 ( x ) = √ π/ 2 x -1 e -x . Let us consider some limiting cases of Eq. (3.5). If the length of the l -th compact dimensions is large compared to other length scales, in the sum over n l in Eq. (3.5) the contribution from large values of n l dominates and, to the leading order, we replace the summation by the integration. The corresponding integral is evaluated with the help of the formula /negationslash and from Eq. (3.5) we obtain the expression of the charge density for the topology R p +1 × ( S 1 ) q -1 . If the length of the l -th compact dimension is small compared with the other length scales and L l /lessmuch β , under the assumption | ˜ α l | < π , the main contribution to the corresponding series in Eq. (3.5) comes from the term with n l = 0. The behavior of the charge density is essentially different with dependence whether the phase ˜ α l is zero or not. When ˜ α l = 0, we can see that, to the leading order, L l 〈 j 0 〉 coincides with the charge density in ( D -1)-dimensional space of topology R p × ( S 1 ) q -1 and with the lengths of the compact dimensions L p +1 ,. . . , L l -1 , L l +1 ,. . . , L D . In particular, this is the case for an untwisted scalar field in the absence of a gauge field. For ˜ α l = 0 and for small values of L l , the argument of the Macdonald function in Eq. (3.5) is large and the charge density is suppressed by the factor e -| ˜ α l | β/L l . In the low-temperature limit the parameter β is large and the dominant contribution to the charge density comes from the term n = 1 in the series over n and from the term in the series over n q with the smallest value of ω n q which corresponds to n l = 0, l = p +1 , . . . , D . To the leading order we find with ω 0 given by Eq. (2.12). From Eq. (3.5) it follows that the expectation value of the charge density is finite in the limit | µ | → ω 0 for p > 2 and it diverges for p /lessorequalslant 2. In order to find the asymptotic behavior near the point | µ | = ω 0 , we note that for p /lessorequalslant 2, under the condition β ( ω 0 - | µ | ) /lessmuch 1, the main contribution to Eq. (3.5) comes from the term with n l = 0 ( ω n q = ω 0 ) and in the corresponding series over n the contribution of large n dominates. In this case we can use the asymptotic expression for the Macdonald function for large values of the argument and to the leading order this gives: where Li s ( x ) is the polylogarithm function. For the latter one has Li 0 ( x ) = x/ (1 -x ), Li 1 ( x ) = -ln(1 -x ). By taking into account that Li s ( e -y ) ≈ Γ(1 -s ) y s -1 for | y | /lessmuch 1 and s < 1, one finds the following asymptotic expressions: In the left plot of figure 1 we present the charge density as a function of the parameter ˜ α D / (2 π ) in the D = 4 model with a single compact dimension of the length L D . Note that for an untwisted scalar field this parameter is the flux measured in units of the flux quantum. For the chemical potential and for the length of the compact dimensions we have taken the values corresponding to µ = 0 . 5 m and mL D = 0 . 5. The numbers near the curves correspond to the values of T/m .", "pages": [ 6, 7, 8 ] }, { "title": "4 Current density", "content": "Now we turn to the expectation value of the current density. As it can be easily seen, the components of the current density along the uncompactified dimensions vanish: 〈 j r 〉 = 0 for r = 1 , . . . , p . By making use of Eq. (3.2) and the expression (2.26) of the Hadamard function, for the current density along the r -th compact dimension we get: with r = p + 1 , . . . , D and, as before, the prime means that the term with n = 0 should be taken with the weight 1/2. Note that, unlike to the case of the charge density, the current density does not vanish at zero temperature for a fixed value of the chemical potential. The zero temperature current density is given by the n = 0 term in Eq. (4.1): The features of this current are discussed in detail in Appendix A. For the model with a single compact dimension the general formula reduces to: An alternative expression of the current density is obtained by making use of the formula (2.28) for the Hadamard function in Eq. (3.2): /negationslash From Eqs. (4.1) and (4.4) it follows that, the current density along the r -th compact dimension is an odd periodic function of ˜ α r and an even periodic function of ˜ α l , l = r , with the period equal to 2 π . The current density is an even function of the chemical potential and it does not vanish in the limit of zero chemical potential. In the absence of uncompactified dimensions one has p = 0 and from Eq. (4.4) we get where we have used f -1 / 2 ( x ) = √ π/ 2 e -x . Here we assume that ω 0 > 0. In the case ω 0 = 0 there is a zero mode and the contribution of this mode should be considered separately. /negationslash In a way similar to that for the case of the charge density, we can see that in the limit when the length of the l -th compact dimension is large ( l = r ), the leading term obtained from Eq. (4.4) coincides with the current density in the space with topology R p +1 × ( S 1 ) q -1 with the lengths of the compact dimensions L p +1 ,. . . , L l -1 , L l +1 ,. . . , L D . For small values of L l , l = r , the behavior of the current density crucially depends whether ˜ α l is zero or not. For ˜ α l = 0 the dominant contribution comes from the term with n l = 0 and from the expression given above we can see that, to the leading order, L l 〈 j r 〉 coincides with the corresponding quantity in ( D -1)-dimensional space with topology R p × ( S 1 ) q -1 and with the lengths of the compact dimensions L p +1 ,. . . , L l -1 , L l +1 ,. . . , L D . For ˜ α l = 0 and for small values of L l , the current density 〈 j r 〉 is exponentially suppressed. /negationslash If L r /greatermuch β , the dominant contribution to the series over n in Eq. (4.3) comes from large values of n ∼ L r /β . In this case we can replace the summation by the integration and the corresponding integral is evaluated by using the formula from Ref. [20] (assuming that | µ | < ω n r q -1 ). To the leading order we get /negationslash For a fixed value of L r this formula gives the leading term in the high-temperature asymptotic for the current density. If in addition L r /greatermuch L l , l = r , the dominant contribution comes from the term with n r = 1, n l = 0, and the current density 〈 j r 〉 is suppressed by the factor e -L r √ ω 2 0 r -µ 2 , where /negationslash In order to see the asymptotic behavior of the current density at low temperatures it is more convenient to use Eq. (4.4). Assuming that β ( ω 0 -| µ | ) /greatermuch 1, the dominant contribution to the temperature dependent part comes from the mode with the smallest energy corresponding to n l = 0 and one has In this case the temperature corrections are exponentially small. For p /lessorequalslant 2 the current density, defined by Eq. (4.4), is divergent in the limit | µ | → ω 0 . The corresponding asymptotic is found in a way similar to that for the case of the charge density. To the leading order we have where the asymptotic expressions for 〈 j 0 〉 for separate values of p are given in Eq. (3.11). In the right plot of figure 1 we displayed the current density along the compact dimension x D as a function of ˜ α D / (2 π ) for the D = 4 model with a single compact dimension of the length corresponding /negationslash to mL D = 0 . 5. The numbers near the curves are the values of T/m and for the chemical potential we have taken the value µ = 0 . 5 m .", "pages": [ 8, 9, 10, 11 ] }, { "title": "5 Zeta function approach", "content": "The expectation values of the charge and current densities can be evaluated directly from Eq. (3.1) by using zeta function techniques (see, for instance, Ref. [24]). First we consider the current density.", "pages": [ 11 ] }, { "title": "5.1 Current density", "content": "Substituting the expansion (2.16) for the field operator and by making use the expression (2.7) for the mode functions, for the current density along compact dimensions one finds the following expression with k r = (2 πn r + ˜ α r ) /L r and r = p +1 , . . . , D . The first term in the square brackets corresponds to the current density at zero temperature. The s = + / -terms are contribution coming from the particles/antiparticles. For the further transformations it is convenient to write Eq. (5.1) in the form In the special case p = 0 this formula is reduced to Eq. (4.5). In the representation (5.2), the zero temperature part corresponds to the n = 0 term. The divergences are contained in this part only. The components of the current density along uncompact dimensions vanish. As the next step, in Eq. (5.2) we use the integral representation This allows us to write the expectation value of the current density in the form Now we apply to the series over n the Poisson summation formula where ˜ g ( y ) = ∫ + ∞ -∞ dxe -iyx g ( x ). For the function corresponding to the series in Eq. (5.4) one has ˜ g ( y ) = √ πe y 2 / 4 -iysµ . After the integration over s we get the expression The current density defined by Eq. (5.6) can be written as with the partial zeta function where n r q = ( n p +1 , . . . , n r -1 , n r +1 , . . . , n D +1 ) and k 2 q is given by Eq. (2.11). Hence, in order to find the renormalized value for the current density we need to have the analytic continuation of the zeta function (5.8) at the point s = 1. The analytic continuation can be done in a way similar to that we have used in Ref. [9] for the zero temperature fermionic current. We first integrate over the momentum along the uncompactified dimensions: Next, the direct application of the generalized Chowla-Selberg formula [25] to the series in Eq. (5.9) leads to the following expression where L r q = ( L p +1 , . . . , L r -1 , L r +1 , . . . L D +1 ), ˜ α q = (˜ α p +1 , . . . , ˜ α r -1 , ˜ α r +1 , . . . ˜ α D +1 ), with and /negationslash The contribution of the second term on the right-hand side of Eq. (5.10) to the current density is finite at the physical point. The analytic continuation is required for the part with the first term only. That is done, by applying the summation formula (2.22) to the series over n r . The further transformations are similar to that we have used in deriving Eq. (2.26) and we get The prime on the summation sign in Eq. (5.10) means that the term n r q = 0 should be excluded from the sum and we use the notation for the vectors c = ( c 1 , . . . , c l ) and b = ( b 1 , . . . , b l ). Note that in Eq. (5.10), cos( n r q · α q ) can also be written as cosh( n D +1 µβ ) ∏ D l = p +1 , = r cos( n l ˜ α l ). The right-hand side of Eq. (5.14) is finite at the point s = 1. Now, substituting Eq. (5.10) into Eq. (5.7) and using Eq. (5.14), we find the following expression for the current density /negationslash An alternative representation for the expectation value of the current density is obtained if we apply the formula (3.4) to the series over n r in Eq. (5.15). Under the condition | µ | /lessorequalslant m , this leads to the following expression Note that in the limit T → 0 and L l → ∞ , l = r , the second term in the right-hand side of this formula vanishes. The first term presents the current density at zero temperature in the model with a single compact dimension (see Eq. (4.2) for a special case p = D -1). where ˜ α r q -1 = (˜ α p +1 , . . . , ˜ α r -1 , ˜ α r +1 , . . . ˜ α D ), L q = ( L p +1 , . . . , L D ), and g 2 n q ( L q ) is defined by Eq. (5.13). In particular, for a massless field and for zero chemical potential, µ = 0, from (5.16) we get The equivalence of two representations for the current density, Eqs. (4.1) and (5.16), can be seen by using the relation where n = ( n 1 , . . . , n l ), α = ( α 1 , . . . , α l ), and w 2 n = ∑ l i =1 (2 πn i + α i ) 2 /a 2 i + c 2 . This relation has been proved in Ref. [17] by using the Poisson's resummation formula. Note that the formula (3.4) is a special case of Eq. (5.18). An expression for the current density, convenient for the discussion of the high-temperature limit, is obtained from Eq. (5.16), by applying to the series over n D +1 the formula (3.4) under the assumption | µ | /lessorequalslant m . This leads to the following expression At high temperatures the dominant contribution comes from n D +1 = 0 term and to the leading order we have The corrections to this leading term are exponentially small. The equivalence of two representations, Eqs. (4.6) and (5.20), for the leading order term can be seen by using the relation (5.18).", "pages": [ 11, 12, 13, 14 ] }, { "title": "5.2 Charge density", "content": "Now we turn to the evaluation of the charge density by using the zeta function approach. Similar to the case of Eq. (5.1), we have the following mode sum The zero temperature part in the charge density vanishes due to the cancellation between the contributions from the virtual particles and antiparticles. The corresponding contributions to the finite temperature part have opposite signs due to the opposite signs of the charge for particles and antiparticles. Introducing the expectation values for the numbers of the particles and antiparticles (per unit volume of the uncompactified subspace), the charge density is written as 〈 j 0 〉 = e 〈 N + -N -〉 /V q . In Eq. (5.22), the upper/lower sign corresponds to particles/antiparticles. Note that in the current density the contributions from particles and antiparticles have the same sign (see Eq. (5.1)). This is due to the fact that, though the charges have opposite signs, the opposite signs have the velocities as well, v (+) r = k r /ω for particles and v ( -) r = -k r /ω for antiparticles (see the phases in the expression (2.7) for the mode functions). The expression for 〈 N ± 〉 is obtained from Eq. (3.5) by the replacement 2 e sinh( nµβ ) /V q → e ± nµβ . The expression (5.21) for the charge density may be written in the form For the further transformation of this expression we use the relation As a result, the expectation value of the charge density is presented in the form Substituting Eq. (5.3), by the transformations similar to that we have used in the case of the current density, one finds where the corresponding zeta function is defined as with ω k defined by Eq. (2.11). After the integration over over the momentum along uncompact dimensions, the function (5.27) is written in the form where n q +1 = ( n p +1 , . . . , n D +1 ) and L D +1 , ˜ α D +1 are defined by Eq. (5.11). The application of the generalized Chowla-Selberg formula [25] to Eq. (5.28) gives with L q +1 = ( L p +1 , . . . , L D +1 ) and ˜ α q +1 = (˜ α p +1 , . . . , ˜ α D +1 ). The prime on the summation sign in Eq. (5.29) means that the term with n l = 0, l = p +1 , . . . , D +1, should be excluded from the sum. Substituting Eq. (5.29) into Eq. (5.26), for the charge density one finds the expression Note that the first term in the right-hand side of Eq. (5.29) does not depend on temperature and the corresponding contribution in Eq. (5.25) vanishes. This expression for the charge density is valid for the region | µ | /lessorequalslant m . The equivalence of the representations (3.5) and (5.30) in this region is proved by using the formula (5.18). In Eq. (5.30), the term with n l = 0, l = p +1 , . . . , D , coincides with the corresponding charge density in Minkowski spacetime ( p = D , q = 0) given by Eq. (3.6). Note that by the replacement 2 e sinh( nµβ ) /V q → e ± nµβ in Eq. (5.30), we can obtain the corresponding formula for 〈 N ± 〉 . An alternative expression for the charge density, convenient for the investigation of the hightemperature limit, is obtained from Eq. (5.30) if we first separate the part corresponding to 〈 j 0 〉 (M) and then apply to the series over n in the remained part the formula (3.4). This leads to the following expression: /negationslash As before, the prime means that the term with n l = 0, l = p + 1 , . . . , D , should be excluded from the sum. At high temperatures the dominant contribution to the second term in the right-hand side comes from the term with n = 0: The higher order corrections to this asymptotic expression are exponentially small. Hence, similar to the case of the current density, the topological part of the charge density is a linear function of the temperature in the high-temperature limit. In order to find the asymptotic expression for the part 〈 j 0 〉 (M) at high temperatures, we use the integral representation Substituting this into Eq. (3.6) and changing the orders of integration and summation, the summation is done explicitly and to the leading order we get where ζ R ( x ) is the Riemann zeta function. This result has been obtained in Ref. [23]. As it is seen from Eq. (5.32), for D > 2 the Minkowskian part dominates in the high temperature limit and one has 〈 j 0 〉 ≈ 〈 j 0 〉 (M) .", "pages": [ 14, 15, 16 ] }, { "title": "6 Bose-Einstein condensation", "content": "In this section we consider the application of the formulas given before for the investigation of the Bose-Einstein condensation (BEC). This phenomena for a relativistic Bose gas of scalar particles in topologically trivial flat spacetime has been discussed in Refs. [22, 23, 26]. The investigation of the critical behavior of an ideal Bose gas confined to the background geometry of a static Einstein universe is given in Refs. [27] for scalar and vector fields. BEC in higher dimensional spacetime with S N as a compact subspace has been considered in Ref. [28]. The case of an ultrastatic (3+1)-dimensional manifold with a hyperbolic spatial part is analyzed in Ref. [29]. The background geometry of closed Robertson-Walker spacetime is discussed in Ref. [30]. Thermodynamics of ideal boson and fermion gases in anti-de Sitter spacetime and in the static Taub universe have been considered in Refs. [31, 32]. In the high-temperature limit, BEC in a general background has been discussed in Refs. [33, 34, 35]. Recently, BEC on product manifolds, when the gas of bosons is confined by anisotropic harmonic oscillator potential, has been investigated in Ref. [36]. Note that in the literature two criteria have been considered for BEC (see, for instance, the discussion in Refs. [37]). In the first one the existence of critical temperature T c > 0 is assumed for which the chemical potential becomes equal to the single particle ground state energy. The derivative ∂ T µ for a fixed value of a conserved charge is discontinuous at the critical temperature and the condensation corresponds to a phase transition. By the second criterion, one assumes the existence of a finite fraction of particle density in the ground state and in states in its neighborhood at T > 0. In this case the presence of a phase transition is not required and thermodynamical functions can be continuous. In particular, in Ref. [23] it has been shown that for massive particles there is no BEC in dimensions D /lessorequalslant 2 if one follows the first criterion. In the discussion of previous sections we have considered the charge and current densities as functions of the temperature, chemical potential and the lengths of compact dimensions. From the physical point of view it is more important to consider the behavior of the system for a fixed value of the charge. We will denote by Q the charge per unit volume of the uncompactified subspace, Q = V q 〈 j 0 〉 . From Eq. (3.5) for this quantity one has For a fixed value of the charge, this relation implicitly determines the chemical potential as a function of the temperature, lengths of the compact dimensions and the charge. For high temperatures the chemical potential determined from Eq. (6.1) tends to zero. Hence, at high temperatures we always have solution with | µ | < ω 0 . The further behavior of the function µ ( T ) with decreasing temperature is essentially different in the cases p > 2 and p /lessorequalslant 2. For p > 2 the expression (6.1) is finite in the limit | µ | → ω 0 . We denote by T c the temperature at which one has | µ ( T c ) | = ω 0 for a fixed value of the charge. It is the critical temperature for BEC. The formula with β c = 1 /T c , determines the critical temperature as a function of the charge, of the lengths of the compact dimensions, and of the parameters ˜ α l . Simple asymptotic formulas for the critical temperature are obtained for low and high temperatures. At low temperatures, ω 0 β c /greatermuch 1, the dominant contribution in Eq. (6.2) comes from the mode with the smallest energy corresponding to n l = 0. By using the asymptotic expression for the Macdonald function for large values of the argument, from Eq. (6.2) to the leading order one finds This regime is realized for values of the charge corresponding to | Q/e | /lessmuch ω p 0 . At high temperatures, by taking into account that to the leading order 〈 j 0 〉 ≈ 〈 j 0 〉 (M) and by using the asymptotic expression (5.34), one finds This asymptotic corresponds to | Q/e | /greatermuch V q ω D 0 . In figure 2 we display the critical temperature as a function of the charge density in the D = 4 model with a single compact dimension ( p = 3, q = 1). The graphs are plotted for mL D = 0 . 5 and for different values of ˜ α D / (2 π ) (numbers near the curves). As it is seen, for fixed lengths of the compact dimensions, the critical temperature for the phase transition can be controlled by tuning the value for the gauge potential. 0 At temperatures T < T c , the equation (6.1) has no solutions with | µ | < ω 0 . The consideration in this region of temperature is similar to the standard one for the BEC in topologically trivial spaces. We note that the expression (6.1) does not include the charge corresponding to the states with k p = 0. At temperatures T < T c the expression (6.1) with | µ | = ω 0 determines the charge corresponding to the states with k p = 0. We denote this charge by Q 1 : /negationslash For the charge corresponding to the Bose-Einstein condensate at the state k p = 0 one has Q c = Q -Q 1 . This charge vanishes at T = T c . At low temperatures, by making use of Eq. (6.3), for the corresponding charges below the critical temperature one finds For high temperatures we use the asymptotic formula (6.4) with the results: In particular, Eq. (6.7) coincides with the corresponding result in Minkowski spacetime [23]. In figure 3, we plot the chemical potential as a function of the temperature in the D = 4 model with a single compact dimension ( p = 3, q = 1). The left and right plots correspond to ˜ α D = 0 and ˜ α D = π/ 2 respectively. For the length of the compact dimension we have taken the value corresponding to mL D = 0 . 5 and the numbers near the curves correspond to the values of the parameter m 1 -D Q/e . Note that for T < T c the scalar field acquires a nonzero ground-state expectation value ϕ c . The latter can be found in a way similar to that used in Ref. [33] (see also Refs. [22, 23] ): with Q and Q 1 given by Eqs. (6.1) and (6.5). Having the chemical potential from Eq. (6.1) for T > T c and taking | µ | = ω 0 for T < T c , we can evaluate the current density for a fixed value of the charge by using Eq. (4.4). Note that at high temperatures the chemical potential tends to zero and the leading term in the corresponding asymptotic expansion for the current density is obtained from Eq. (5.20) with µ = 0. For T < T c , the formula (4.4) gives a part of the current density due to the excited states only. In addition to this, there is a contribution due to the condensate, given by the expression with r = p + 1 , . . . , D and | ˜ α r | /lessorequalslant π . For the expectation value of the total current density one has 〈 j r 〉 t = j r c + 〈 j r 〉 . /negationslash In figure 4 we plot the expectation values of the total current density (left plot, full curve) and of the particle and antiparticle numbers in the states with k p = 0, 〈 N ± 〉 (right plot), versus the temperature for mL D = 0 . 5, ˜ α D = π/ 2 and Q/e = 0 . 5 m D -1 in the D = 4 model with a single compact dimension. For the corresponding critical temperature from Eq. (6.2) one finds T c ≈ 0 . 63 m . In the left plot we have separately presented the contributions to the current density coming from particles (dot-dashed line) and antiparticles (large dashed line). The linear dependence at high temperatures is clearly seen. On the left panel we have also plotted the separate contributions to the current density from the excited states, 〈 j D 〉 / ( em D ) (dashed line), and from the condensate, j D c / ( em D ) (dotted line). Note that for the current density given by Eq. (4.2) we have 〈 j D 〉 0 = 2 . 32 em D . The total current and its first derivative with respect of the temperature are continuous functions at the point of the phase transition. The latter is not the case for the separate parts coming from the condensate and from the excited states. For p /lessorequalslant 2 the charge defined by Eq. (6.1) diverges in the limit | µ | → ω 0 and for finite value of charge density the point | µ | = m cannot be reached. The corresponding asymptotic expressions for the charge and current densities are given by Eqs. (3.11) and (4.9). Thus, for p /lessorequalslant 2 there is no BEC by the first criterion given above. In particular, this is the case in the model with compact space corresponding to p = 0. This result for spaces of finite volume in general has been obtained in Ref. [35]. As before, for p /lessorequalslant 2 and for a fixed value of the charge Q , the dependence of the chemical potential on the temperature is determined by Eq. (6.1). In the limit β →∞ (low temperatures) and for a fixed value of | µ | /negationslash = ω 0 , the expression on the right-hand side of Eq. (6.1) tends to zero. From here we conclude that for a fixed value of Q we should have | µ | → ω 0 for β →∞ . The corresponding asymptotic behavior is found in a way similar to that we have used for Eq. (3.10) and is given by the same expression. Solving with respect to the chemical potential, we find the following asymptotic expressions in the limit T → 0. In figure 5 we have plotted the chemical potential versus temperature in the D = 3 model with a single compact dimension ( p = 2, q = 1) for ˜ α D = 0 and for a fixed value of the charge density corresponding to m 1 -D 〈 j 0 〉 /e = 0 . 5. The numbers near the curves correspond to the values of the parameter L D m . As it is seen, though for a finite value of L D the derivative ∂ T µ ( T ) is a continuous function, in the limit L D → ∞ it tends to the corresponding function in the D = 3 model with trivial topology ( p = 3, q = 0) for which the function ∂ T µ ( T ) is discontinuous at the point T = T c .", "pages": [ 16, 17, 18, 19, 20 ] }, { "title": "7 Conclusion", "content": "In the present paper we have investigated the finite temperature expectation values of the charge and current densities for a complex scalar field, induced by nontrivial spatial topology. As an example for the latter we have considered a flat spacetime with an arbitrary number of toroidally compactified dimensions. This allowed us to escape the problems related to the curvature and to extract pure topological effects. The periodicity conditions along compact dimensions are taken in the form (2.2) with general constant phases. As special cases the latter includes the periodicity conditions for untwisted and twisted fields. In addition, we have assumed the presence of a constant gauge field. By performing a gauge transformation, the gauge field is excluded from the field equation. However, this leads to the shift in the phases appearing in the periodicity conditions given by Eq. (2.5). In the evaluation of the expectation values for the charge and current densities we have used two different approaches which allowed us to obtain alternative representations for the corresponding expectation values. In the first approach we evaluated the thermal Hadamard function by using the Abel-Plana-type summation formula for the series over the momentum along a compact dimension. The corresponding expression is given by Eq. (2.26). The n r = 0 term in that formula corresponds to /negationslash the Hadamard function for the topology R p +1 × ( S 1 ) q -1 and, hence, the n r = 0 part is the change in the Hadamard function due to the compactification of the r -th direction. An alternative representation for the Hadamard function is given by Eq. (2.28). /negationslash /negationslash Given the Hadamard function, the expectation values of the charge and current densities are evaluated by making use of Eq. (3.2). The charge density is given by two equivalent representations, Eqs. (3.3) and (3.5). The explicit information contained in Eq. (3.3) is more detailed. The term with n r = 0 in this representation corresponds to the charge density for the topology R p +1 × ( S 1 ) q -1 with the lengths of the compact dimensions ( L p +1 , . . . , L r -1 , L r +1 , . . . , L D ) and the contribution of the terms with n r = 0 is the change in the current density induced by the compactification of the r -th dimension. The charge density is an even periodic function of the phases ˜ α l with the period equal to 2 π . The sign of the ratio 〈 j 0 〉 /e coincides with the sign of the chemical potential. If the length of the l -th compact dimension is small compared with the other length scales and L l /lessmuch β , the behavior of the charge density is essentially different with dependence whether the parameter ˜ α l is zero or not. For ˜ α l = 0, to the leading order, L l 〈 j 0 〉 coincides with the charge density in ( D -1)-dimensional space with topology R p × ( S 1 ) q -1 and with the lengths of the compact dimensions L p +1 ,. . . , L l -1 , L l +1 ,. . . , L D . For ˜ α l = 0 the charge density is suppressed by the factor e -| ˜ α l | β/L l . At low temperatures and for a fixed value of | µ | < ω 0 , the charge density is suppressed by the factor e -( ω 0 -| µ | ) /T . For a fixed temperature and in the limit | µ | → ω 0 the charge density is finite for p > 2 and it diverges for p /lessorequalslant 2. The corresponding asymptotic behavior in the latter case is given by Eq. (3.11). In the high-temperature limit and for D > 2 the Minkowskian part dominates in the charge density with the leading term given by Eq. (5.34). In the same limit, the topological part of the charge density is a linear function of the temperature. /negationslash For the expectation value of the current density along the r -th compact dimension we have derived representations given by Eqs. (4.1) and (4.4). The components along uncompactified dimensions vanish. The current density along the r -th compact dimension is an odd periodic function of ˜ α r and an even periodic function of ˜ α l , l = r , with the period equal to 2 π . The current density is an even function of the chemical potential. Unlike to the case of the charge density, the current density does not vanish at zero temperature for a fixed value of the chemical potential. The corresponding expression is given by Eq. (4.2) and the properties are discussed in Appendix A. For small values of L l , l = r , and for ˜ α l = 0, to the leading order, the quantity L l 〈 j r 〉 coincides with the r -th component of the current density in ( D -1)-dimensional space with topology R p × ( S 1 ) q -1 and with the lengths of the compact dimensions L p +1 ,. . . , L l -1 , L l +1 ,. . . , L D . For ˜ α l = 0 and for small values of L l , l = r , the current density 〈 j r 〉 is exponentially suppressed. At a fixed temperature and for p /lessorequalslant 2 the current density is divergent in the limit | µ | → ω 0 . The leading term in the corresponding asymptotic expansion is related to the charge density by Eq. (4.9). For a fixed value of the chemical potential | µ | < ω 0 and at low temperatures the finite temperature corrections are given by Eq. (4.8) and they are exponentially small. In the limit of high temperatures, the current density is a linear function of the temperature. /negationslash /negationslash In Section 5, we have derived alternative representations for the expectation values of the charge and current densities by using the zeta function approach. In both cases, by applying to the corresponding zeta functions the generalized Chowla-Selberg formula, Eqs. (5.16), (5.19) and Eqs. (5.30), (5.31) are obtained for the current and charge densities respectively. At high temperatures, the leading term in the asymptotic expansion of the current density is given by Eq. (5.20) with the linear dependence on the temperature and the next corrections are exponentially small. For the charge density, for D > 2 the leading term in the high-temperature expansion coincides with the corresponding charge density in ( D + 1)-dimensional Minkowskian spacetime. The leading term in the correction induced by nontrivial topology linearly depends on the temperature and the following corrections are exponentially suppressed. The Bose-Einstein condensation is discussed in section 6. For a fixed value of the charge, the relation (6.1) determines the chemical potential as a function of the temperature, of the lengths of compact directions and of the phases in the periodicity conditions. For high temperatures the chemical /negationslash /negationslash potential tends to zero. With decreasing temperature the chemical potential increases and for p > 2 one has | µ ( T ) | = ω 0 at some finite temperature T = T c . The critical temperature for BEC, T c , is determined by Eq. (6.2). Simple expressions are obtained for low and high temperatures, Eqs. (6.3) and (6.4), respectively. At temperatures T < T c one has | µ | = ω 0 and Eq. (6.1) determines the charge corresponding to the states with k p = 0 and the remained charge corresponds to the charge of the condensate. At low and high temperatures the charges are given by simple expressions (6.6) and (6.7). Similar to the charge density, for T < T c the current density is the sum of two parts. The first one is the contribution of excited states and is given by Eq. (4.4) with | µ | = ω 0 . The second part is due to the condensate and it is presented by Eq. (6.9). The total current and its first derivative with respect to the temperature are continuous functions at the critical temperature. For p /lessorequalslant 2, the point | µ | = ω 0 cannot be reached for finite value of charge density. For a fixed value of the charge, we have | µ | → ω 0 in the limit T → 0. The corresponding asymptotic behavior is given by Eq. (6.10). In this case the thermodynamical functions are continuous and there is no phase transition at finite temperature. /negationslash", "pages": [ 20, 21, 22 ] }, { "title": "8 Acknowledgments", "content": "E.R.B.M. thanks Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico (CNPq) for partial financial support. A.A.S. was supported by CNPq.", "pages": [ 22 ] }, { "title": "A Vacuum expectation value of the current density", "content": "In this Appendix we give some properties of the zero temperature current density given by Eq. (4.2). An alternative expression is obtained from Eq. (5.16) taking the limit β →∞ . In this limit the term with n = 0 survives only and we get: /negationslash Let us consider the behavior of the zero temperature current density in some limiting cases. First we consider the limit when the length of the r -th compact dimension, L r , is much larger than the other length scales. The behavior of the current density in this limit crucially depends whether ω 0 r , defined by (4.7), is zero or not. In the first case, which is realized for ˜ α l = 0, l = r , and m = 0, the dominant contribution in Eq. (4.2) for large values of L r comes from the modes with n l = 0, l = r , for which ω n r q -1 = ω 0 r = 0. The corresponding expression is obtained from Eq. (4.2) taking the limit ω n r q -1 → 0 and to the leading order we have /negationslash For ω 0 r = 0 and for large values of L r , the main contribution to the zero temperature current density comes from the mode n r = 1, n l = 0, l = r , and from Eq. (4.2) one finds /negationslash In this case we have an exponential suppression. Now we discuss the asymptotic of the current density for small values of L r . In this limit it is more convenient to use Eq. (A.1). First we separate the term n l = 0, l = r , in Eq. (A.1) and use the /negationslash /negationslash /negationslash asymptotic expression of the Macdonald function for small values of the argument. For the remained part in Eq. (A.1), the dominant contribution comes from large values of n r and, to the leading order, we replace the summation over n r by the integration. The corresponding integral involving the Macdonald function is evaluated by using the formula from Ref. [20]. In this way it can be seen that the contribution of the mode with a given n r q -1 is suppressed by the factor exp( -g r √ ˜ α 2 r /L 2 r + m 2 ), where g r = ∑ D l = p +1 , = r n 2 l L 2 l . As a result, we see that the dominant contribution to the current density is due to the modes with n l = 0, l = r , and to the leading order we get This leading term does not depend on the mass and on the lengths of the other compact dimensions. As it is seen from Eq. (A.1), the expression in the right-hand side of Eq. (A.4) coincides with the current density for a massless scalar field in the space with topology R D -1 × S 1 .", "pages": [ 22, 23 ] } ]

[ { "title": "Gravitational Lensing by Phantom Black holes", "content": "Galin N. Gyulchev 1 ∗ , Ivan Zh. Stefanov 2 †", "pages": [ 1 ] }, { "title": "Abstract", "content": "In some models dark energy is described by phantom scalar fields (scalar fields with 'wrong' sign of the kinetic term in the lagrangian). In the current paper we study the effect of phantom scalar field and/or phantom electromagnetic field on gravitational lensing by black holes in the strong deflection regime. The black-hole solutions that we have studied have been obtained in the frame of the Einstein-(anti-)Maxwell-(anti-)dilaton theory. The numerical analysis shows considerable effect of the phantom scalar and electromagnetic fields on the angular position, brightness and separation of the relativistic images. PACS numbers: 95.30.Sf, 04.20.Dw, 04.70.Bw, 98.62.Sb Keywords: Relativity and gravitation; Gravitational lensing; Classical black holes; Phantom black holes; Einstein-Maxwell-dilaton theory; Dark energy", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Modern observational programs including type Ia SNe, cosmic microwave background anisotropy and mass power spectrum suggest that the universe is dominated by mysterious matter termed dark energy (DE) which has negative pressure and violates the energy conditions [1, 2, 3]. Considerable efforts are made to study the nature of DE. Different effective models of dark energy have been proposed in literature (See [4] and [5] for recent exhaustive reviews). In some of them the possibility of describing DE by phantom fields is considered. The natural questions arises whether local manifestations of DE at astrophysical scale can be observed. Exact solutions describing neutron stars containing DE have been obtained in [6]. There have been also some recent efforts in that direction. In [7] the effect of DE on the structure and on the spectrum of qusinormal frequencies of neutron stars has been studied. Mixed stars containing both dark energy and ordinary matter have been presented in a number of papers (See [6] and references therein). Solutions describing black holes coupled to phantom fields have also been found. To our knowledge the first solutions of phantom black holes have been obtained by Gibbons and Rasheed [8]. These solutions were later elaborated by Cl'ement et al. [9, 10] and Gao Zhang [11] for higher dimensions. Regular black holes coupled to phantom scalar field have been reported by Bronnikov [12]. Recent interest in phantom black holes have been connected with the study of their thermodynamics and the possibility of phase transitions [13]. Similar study has been presented in [14] for black holes with phantom electromagnetic field or the so-called anti-Reissner Nordstrom black hole. In this solution the charged term in the metric has an opposite sign with respect to the corresponding term of the standard Reissner Nordstrom black hole. Other works in the field of theories with phantom dilaton and phantom Maxwell field have considered gravitational collapse of a charged scalar field [15] and also light paths in black-hole spacetims [16]. As we have already mentioned gravitational waves and the frequencies of quasinormal ringing in particular can provide rich information for the structure of compact astrophysical objects and thus can serve as a powerful tool for studying the local manifestation of DE. Another possibility could be provided by gravitational lensing especially in the strong deflection regime. There has been considerable effort for the theoretical study of gravitational lensing in the strong deflection regime (For more details on the matter we refer the reader to [17] and references therein). In his papers [18, 19] Bozza proposed a method for the calculation of the deflection angle in the regime of strong deflection in the particular case when both the observer and the gravitational source lie in the equatorial plane † . His method has gained popularity due to is simplicity and has been applied to study the gravitational lensing caused by different exotic, compact objects. The particular cases in which both the scalar field and the electromagnetic field have cannonical form, i.e. the EMD black hole has been already reported by Bhadra [21]. The lensing by EMD black holes with de-Sitter and anti-de-Sitter asymptotics have been studied by [22] and [23], respectively. In the last two cases the scalar field has a non zero potential. Lensing in the strong field regime by black holes coupled to electromagnetic field has been considered also in [24, 25, 26, 27, 28, 29]. Black holes with opposite sign of the charge term in the metric (as in the case of antiReissner Nordstrom black hole) have been applied to model the object in the center of our galaxy - Sgr A* and their lensing has been studied in [30] and [31]. In these black holes, however, the charge is tidal and does not have electromagnetic origin. Lensing by black holes with tidal charge gas been also considered in [32]. One of the aims of the current paper is to study the effect of phantom scalar field (phantom dilaton) on gravitational lensing. In the presence of exotic matter such as phantom fields wormholes may exist. Lensing by different wormholes, for example the Ellis's and the Janis-Newman-Winicour's (JNW) wormholes, has attracted significent research interest [33][44]. JNW naked singularities (naked singularities coupled to canonical massless scalar field) acting as gravitational lens have been considered by Virbhadra et al. [45, 46, 47]. The lensing of the JNW solution in the context of scalar-tensor theories has been studied by Bhadra [48]. Generalization with inclusion of rotation has been made in [49]. Our goal is apply the apparatus of gravitational lensing by black holes in the strong deflection limit to study the possible local manifestation of dark energy. For this purpose we model DE with phantom dilaton and phantom electromagnetic field. We compare the characteristics of relativistic images of four black holes: the standard Einstein-Maxwell black hole (EMD); the Einstein-anti-Maxwell-dilaton black hole which has a phantom electromagnetic field (EMD) ‡ ; the Einstein-Maxwell-anti-dilaton black hole which has a phantom dilaton (EMD); and the Einstein-anti-Maxwell-anti-dilaton black hole in which both the dilaton and the electromagnetic field are phantom (EMD).", "pages": [ 2, 3 ] }, { "title": "2 Phantom black holes", "content": "When phantom dilaton and/or phantom electromagnetic field is considered the action of Einstein-Maxwell-dilaton theory is generalized to the following form R denotes the Ricci scalar curvature, ϕ is the dilaton, F is the Maxwell tensor and the constant α determines the coupling between the dilaton and the electromagnetic field. For the usual dilaton the dilaton-gravity coupling constant η 1 takes the value η 1 = 1 while for phantom dilaton η 1 = -1. Similarly, the Maxwell-gravity coupling constant η 2 takes the values η 2 = 1 and η 2 = -1 in the Maxwell and anti-Maxwell case, respectively.", "pages": [ 3 ] }, { "title": "2.1 Einstein Maxwell Dilaton black holes", "content": "The line element of the EMD black hole § is where the parameter γ = (1 -α 2 ) / (1 + α 2 ) has been introduced for convenience. It varies in the interval [ -1 , 1] for α ∈ ( -∞ , ∞ ), so stronger coupling corresponds to lower values of γ . The solutions for the dilaton and the Maxwell field are For the magnetically charged solution the metric is the same but the sign of the scalar field ϕ must be reversed and the Maxwell field becomes F = P sin θdθ ∧ dφ . The parameters r + and r -are interpreted as an event horizon and an inner Cauchy horizon, respectively. The ADM mass M and the charge Q can be expressed by r + and r - Relations (4) can be inverted to express the horizons in terms of the ADM mass M and the charge Q The equation for r + (or r -) obtained from (4) is biquadratic. The solutions are grouped in two couples. The couple which contains the largest of all four roots is chosen. The same choice is made in the other three classes of solutions considered in this paper. The two horizons merge at and for lower values of ( Q/M ) 2 the solution describes a naked singularity. In the limit γ → 1 the solution restores the Reissner-Nordstrom black hole. The charge is switched off when one of the two parameters r + and r -is equal to zero. In the latter case, the Schwarzschild black hole is recovered with r + = 2 M corresponding to the event horizon. In the former case, the EMD solution reduces to the Janis-Newman-Winicour solution also known as the Fisher solution - a fact that was noticed for the first time by Virbhadra [52]. In this case, at r -= 2 M/γ a singularity is reached and γ ∈ [0 , 1]. In the current work we will restrict our considerations to gravitational lensing of black holes. That is why we have chosen the Schwarzschild black hole as a reference. The gravitational lensing by the central object of the JNW spacetime has been studied in [45, 46].", "pages": [ 3, 4 ] }, { "title": "2.2 Einstein anti-Maxwell Dilaton black holes", "content": "In the case of EMD black hole the line element is again (2). The solutions for the dilaton and the anti-Maxwell field are The ADM mass M and the anticharge Q are The 'horizons' expressed in terms of the ADM mass M and the anticharge Q are The parameter r + is positive and is interpreted as an event horizon while r -is a negative and can be considered as a singularity which is never reached since the singularity at r = 0 is reached before that. Hence, these black holes have the same causal structure as the Schwarzschild black hole. Again, there is restriction for the parameter ( Q/M ) The limit γ → 1 corresponds to the anti-Reissner-Nordstrom black hole (a Reissner-Nordstrom black hole black hole with imaginary charge). ( Q/M ) is unbound for positive γ . Again, the particular solutions with zero electric charge are the Janis-Newman-Winicour solution and the Schwarzschild solution.", "pages": [ 4, 5 ] }, { "title": "2.3 Einstein Maxwell anti-Dilaton black holes", "content": "The line element of the EMD black hole is The solutions for the dilaton and the Maxwell field are When γ > 0, 0 ≤ r -≤ r + , so the causal structure is the same as for the EMD case. For γ < 0, however, r -≤ 0 ≤ r + and the black hole has the same causal structure as in the EMD case. The ADM mass M and the charge Q are expressed by r + and r -in the following way Relations (13) can be inverted to express the 'horizons' in terms of the ADM mass M and the charge Q For r + and r -to be real the following relation must hold Here in the limit γ → 1 the Reissner-Nordstrom black hole is restored. For r -= 0 the Schwarzschild black hole is restored. If we put r + = 0 and substitute γ = 1 /κ the metric obtains the form This is the anti-Fisher or anti-JNW solution since κ ∈ [ -1 , ∞ ). Lensing in this spacetime has been studied in [40].", "pages": [ 5, 6 ] }, { "title": "2.4 Einstein anti-Maxwell anti-Dilaton black holes", "content": "In the case of EMD black hole the line element is given again by (11). The solutions for the dilaton and the anti-Maxwell field are When γ > 0, r -≤ 0 ≤ r + and the causal structure is Schwarzschild-like. For γ < 0, however, 0 ≤ r -≤ r + and the black hole has two horizons, an event horizon and an inner Cauchy horizon. The ADM mass M and the anticharge Q are Relations (17) can be inverted to express the'horizons' in terms of the ADM mass M and the charge Q Unlike all three cases discussed above in the current case there are no restrictions for ( Q/M ) 2 . The limit γ → 1 corresponds, again, to the anti-Reissner-Nordstrom black hole. The particular solutions with zero electric charge are the anti-JNW solution and the Schwarzschild solution.", "pages": [ 6 ] }, { "title": "3 Gravitational lensing in the strong field limit", "content": "Following Bozza's notation we can express the metric of the general static spherically symmetric spacetime in the form where we have introduced the new variable x = r/M . The deflection angle can be expressed as [45] where and here x 0 represents the minimum distance from the photon trajectory to the gravitational source. The deflection angle diverges when the denominator of the above expression turns to zero i.e. at the points where the following relation C ' ( x ) C ( x ) = A ' ( x ) A ( x ) holds. We use prime ( .. ) ' to denote the derivative with respect to x . The largest root of this equation gives the radius of the photon sphere. For more details on photon surfaces we refer the reader to [53, 54, 46] Here and bellow the following convention has been chosen F m = F | r 0 = r m where F is an arbitrary quantity. Acoording to Bozza's method [18, 19] the integral (21) is split in two parts - regular I R ( x 0 ) and divergent I D ( x 0 ) In explicit form In this formulas the following quantities have been introduced. The new variable facilitates the numerical integration since it maps the open interval [ x ps , ∞ ) to the closed interval [0 , 1]. The function is responsible for the divergence of the integrand. As the photon sphere is approached, i.e. when η → 0 the leading order term of the integrand is ( √ β ps η ) -1 . The coefficient in the expansion is The expansion shows that divergence of the deflection angle is logarithmic [18, 19] where D OL denotes the distance between observer and gravitational lens. The impact parameter is The strong field limit coefficients a and b are expressed as, Since the spacetimes under consideration are asymptotically flat we can take advantage of the strong deflection limit lens equation [56] where D LS is the lens-source distance, D OL is the observer-lens distance and η is the source angular position, as seen from the lens. We will be interested also in the following observables. Under the assumption u ps /lessmuch D OL , one can show that up to terms of second order in u ps /D OL the angular separation between the lens and the n -th relativistic image is where We are considering only prograde photons and this is what pro stands for. It is usually considered that only the first relativistic image can be observed separately and all other relativistic images would be packed together at angular position θ ∞ . The angular separation between the first relativistic image and the rest of the relativistic images is [18] The third observable that is usually considered is the ratio between the magnitude of the first image µ 1 and the total magnitude of all other relativistic images ∑ ∞ n =2 µ n which in terms of stellar magnitudes is All observable quantities mentioned above are plotted in the paper for different values of the charge Q/M and the metric parameter γ and under the following assumptions. We consider the massive dark object Sgr A ∗ in the center of our Galaxy as a lens. The observer is positioned at distance D OL = 8 . 33 kpc from the lens. For the lens-source distance, following [20], we have taken D LS = 0 . 005 D OL , D LS = 0 . 05 D OL and D LS = 0 . 5 D OL . According to [57] the lens mass is M = 4 . 31 × 10 6 M /circledot , so M/D OL ≈ 2 . 47 × 10 -11 . As in the Schwarzschild case ¶ our results are practically insensitive to the angular source position η , and the source distance D LS . For simplicity we will present the results for η = 0. In this specific case the relativistic images are observed as Einstein rings [46]. We should also mention that significant information about the properties of the object acting as a gravitational lens can be obtained from the time delay, however such study is not in the scope of the present work. Expression for the time delay in general static spherically symmetric spacetime can be found in [47].", "pages": [ 6, 7, 8, 9 ] }, { "title": "3.1 Photon sphere", "content": "For both solutions with canonical scalar field, EMD and EMD, the expression for the photon sphere is where x + = r + /M and x -= r -/M , r + and r -are the parameters of the corresponding black-hole solution. The photon sphere x ps , the event horizon x + and the inner horizon x -of the EMD and EMD black holes are displayed on Fig. 1. In the EMD case for γ < 0 the photon sphere and the event horizon merge when ( Q/M ) = ( Q/M ) crit . This situation has been recently discussed in [55]. In the EMD case we can see that ( Q/M ) is restricted from above only when γ < 0. The photon sphere and the event horizon do not merge for any value of ( Q/M ) in this case. The inner 'horizon' x -is behind the central singularity and is not present on the figure. For the solutions with phantom scalar field, EMD and EMD, the photon sphere takes the form The photon sphere x ps , the event horizon x + and the inner horizon x -of the EMD and EMD black holes are displayed on Fig. 2. In the EMD case ( Q/M ) is restricted from above. When γ < 0 the inner 'horizon' x -is behind the central singularity and is not present on the figure. There are no constraints on ( Q/M ) for the EMD black hole. In non of the two cases with phantom scalar field the photon sphere and the event horizon merge.", "pages": [ 9, 10 ] }, { "title": "3.2 Einstein Maxwell Dilaton black holes", "content": "The lens parameters a , b and u ps for EMD case are given on Fig. 3. The observables are given on Fig. 4. The dashed line represents the critical curves of the parameters. For example, the critical curve for a is defined as a crit ( γ ) = a (( Q/M ) crit , γ ), where ( Q/M ) crit is the critical value of ( Q/M ) for the corresponding class of black-hole solutions. The critical curves of all other quantities in the paper are defined analogously and are represented by thin dashed lines. The regions beyond the critical curves on the figures correspond to naked singularities and are outside the scope of the current research. In our discussion we will take the Schwarzschild black hole as a reference. The values of the different quantities corresponding to that case are presented by a straight grey line on the figures which we term 'reference line'. The first observation we can make is that on both Fig. 3 and Fig. 4 all curves converge to the value for the Schwarzschild black hole at γ = -1 for arbitrary value of Q/M - a fact with no trivial explanation. The lens parameter a is monotonous function of γ . The slope is positive and becomes more significant with the increase of the electric charge Q/M . The parameter b has a different behavior. Initially it increases with γ but then it passes through a maximum and then decreases. The branch with negative slope becomes very steep as Q/M is increased. Initially the EMD value of b is higher than the Schwarzschild but for high enough values of γ the situation changes. The lowest value of b is obtained at ( Q/M ) crit and γ = 0 which is the value of the coupling in string theory [51]. For the EMD black holes the critical impact parameter u ps is lower than the Schwarzschild case for all non-zero values of Q/M . As Q/M increase u ps decreases. This effect, however, is compensated when stronger coupling and respectively lower value of γ is considered. As we can see from Fig. 4 with the increase of Q/M the relativistic images are attracted towards the black hole, they become less bright and the separation between them increases. The most demagnified image is obtained for ( Q/M ) crit and γ = 0. The dependence on γ becomes more pronounced for higher values of Q/M . All three observables are monotonous functions of γ . The slope of θ pro 1 and r m as functions of γ are negative, while the slope of s pro 1 is positive. In the case of stronger coupling the effect of the electric charge is suppressed. As a result, when γ → -1 for all values of Q/M the relativistic images of the EMD black hole have the same angular position, brightness and separation as those of the Schwarzschild black hole.", "pages": [ 10, 11 ] }, { "title": "3.3 Einstein anti-Maxwell Dilaton black holes", "content": "The results for the EMD case are presented on Fig. 5 and Fig. 6. On all of the graphics for the EMD black hole the curves end on the critical curves, before the value γ = -1 is reached. Beyond the critical curves the object is not a black hole anymore. Unlike the previously discussed case, the lens parameter a decreases when Q/M is increased. a is monotonous function of γ and as in the EMD case the slope of the curves is positive. Here the stronger coupling enhances the effect of the electric charge. In the EMD case b is monotonous function of γ but its behavior is again more complex than that of a . The slope of the curves is negative. For high enough values of γ with the increase of Q/M , b decreases. With the decrease of γ , however, the curves cross the reference line and the value of b becomes higher than that for the Schwarzschild black hole. The behavior of u ps is converse to that of a - higher charge leads to higher values. The effect of the charge is enhanced when the coupling is stronger. What are the effect of the phantom electromagnetic field and the dilaton on the observables? The effect of the phantom electric charge is to repel the relativistic images from the optical axis. The slope of the curves for θ pro 1 is negative. It is negligible for low values of Q/M . The stronger coupling leads to a more pronounced effect of the phantom electric charge. The qualitative behavior of the curves for r m is identical but the curves are much steeper. The separation between the images s pro 1 has a converse behavior. It is lower for the images that a farther from the optical axis. The slope of s pro 1 is positive.", "pages": [ 11, 12 ] }, { "title": "3.4 Einstein Maxwell anti-Dilaton black holes", "content": "The lens parameters and the observable in the case of EMD are presented on Fig. 7 and Fig. 8, respectively. Here again the point γ = -1 is reached only when Q/M = 0. Otherwise the curves end on the critical lines. The lens parameter a is a monotonous function of γ . For Q/M = 0 it is higher than the reference value. The negative slope here means that the effect /negationslash of the electromagnetic field is enhanced when γ is decreased. The behavior of b and u ps is converse - they decrease as Q/M is increased. For b the effect of a stronger coupling is to invoke a stronger effect of Q/M . The critical impact parameter is almost independent of γ as we can see from the almost flat curves. As a result of that, the value of the angular position of the images θ pro 1 is also slightly dependent on γ . The images are attracted to the optical axis with the increase of Q/M . They become less bright as the electric charge is increased and this effect is more significant for higher coupling. The behavior of s is converse - it is higher for higher Q/M", "pages": [ 12, 13 ] }, { "title": "3.5 Einstein anti-Maxwell anti-Dilaton black holes", "content": "/negationslash Fig. 9 and Fig. 10 represent the results for the last case - the EMD black hole. As we mentioned above, in this case there are no restrictions for the electric charge so no critical curves occur on the graphics. Here, just as in the EMD case, all curve converge to the Schwarzschild line when γ = -1. The lens parameter a is lower when Q/M is increased. The effect of the phantom electric field, however, is suppressed in the strong coupling regime. Again, b is not monotonous. It is lower than the Schwarzschild value for all values of Q/M = 0 and γ = -1. With the decrease of γ , b initially decreases. Then, it passes through a minimum and converges to the reference line. The negative slope becomes more steep with the increase /negationslash of Q/M . The critical impact parameter u ps has behavior opposite to that of a . It is higher for higher values of Q/M . Its dependence on γ is insignificant for high enough values of γ but the curves become very steep as the point γ = -1 is approached. Due to the phantom electromagnetic field the relativistic image are observed at higher angular position θ pro 1 . The dependence of θ pro 1 on γ is almost negligible everywhere but in the vicinity of γ = -1. Again the images that are observed farther from the optical axis are also brighter. The slope of the curve for r m is bigger than that of the previous graphic when equal values of Q/M are considered. The separation between the first and second relativistic images s pro 1 has odd behavior. For low values of Q/M it is a monotonous function of γ . For decreasing γ , s increases. For high enough values of Q/M as γ is decreased the curves for s cross the reference line and becomes higher than the value for Schwarzschild. Then it has a local maximum and finishes on the Schwarzschild line at γ = -1.", "pages": [ 13, 14 ] }, { "title": "4 Comparison between the four cases and summary of the results", "content": "In this section we will compare between the four cases - (EMD), (EMD), (EMD) and (EMD) - for black holes with same mass M and electric charge Q . For all of the discussed cases on the same plot the photon sphere x ps is presented on Fig. 11, the lens parameters a and b - on Fig. 12, the impact parameter u ps and the angular position θ pro 1 are on Fig. 13, and the other two observables, r m and s pro 1 , are given on Fig. 14. On all graphics in the current section M = 1 and Q = 0 . 8. For most of the quantities the curves corresponding to black hole with canonical electromagnetic field lay on one side of the reference line while those corresponding to phantom electromagnetic field - on the other. Exception from this behavior is observed for the photon sphere x ps and for the lens parameter b . For weak coupling ( γ close to 1) both black holes with canonical electromagnetic field EMD and EMD have photon spheres with smaller radii than the Schwarzschild black hole while the black holes with phantom electromagnetic field have bigger radii. The situation changes when γ is decreased. The curve for EMD case does not remain below the reference line but crosses it and diverges as γ = -1 is approached. The curve for EMD case also crosses the reference line but downwards and disappears when the critical value of γ corresponding to Q/M = 0 . 8 is reached. It is important to note that in none of the cases the photon sphere converges to the reference line in the limit γ = -1. As we can see from Fig. 12 for both black holes with canonical electromagnetic field, EMD and EMD, a is higher than the Schwarzschild value in the whole interval of admissible values of γ , while for the solutions with phantom electromagnetic field, EMD and EMD, it is lower. What is the role of the parameter γ responsible for the coupling between the dilaton and the Maxwell field? Let us first consider the couple of black hole solutions with canonical scalar field EMD and EMD. As it can be seen from Fig. 12 for lower values of γ , corresponding to stronger coupling, a has lower values. In the phantom scalar field case (see the curves for the EMD and the EMD solutions) on the contrary - the stronger coupling leads to higher values of a . As a result, for the EMD and EMD black holes the stronger coupling suppresses the effect of the Maxwell field and the curves for a converge to the reference line corresponding to the Schwarzschild black hole, while for the EMD and EMD black holes the effects of the two parameters Q/M and γ enhance each other and the curves diverge from the reference line. /negationslash The curves for the lens parameter b have a more complex behavior. At γ = 1 for Q/M = 0 for all four of the considered black holes b has lower values than for the Schwarzschild black hole. As the coupling is increased (and respectively γ is decreased) for both solutions with canonical scalar field the curves cross the reference line and b takes higher values. For the case of phantom scalar field in the whole interval of admissible values of γ the values of b remain lower than those of the Schwarzschild case. As for the previously discussed parameter the curves for b in the EMD and EMD cases converge to the Schwarzschild line at γ = -1. In these cases b has one extremum - a maximum in the former case and a minimum in the latter case. The qualitative behavior of the curves for the impact parameter u ps and for the observables θ pro 1 , r m and s pro 1 is similar to that of the curves for a in a sense that for the EMD and EMD black holes the curves converge to corresponding Schwarzschild values, while for the other couple of black holes, EMD and EMD, the curves diverge from them. All of these quantities are monotonous functions of γ . For all four solutions the following behavior is observed. Images that are closer to the optical axis are dimmer but better separated while those that are farther - on the contrary. In all cases s pro 1 has a converse behavior to that of θ pro 1 and r m - when the latter increase the former decreases. From the studied cases we can conclude that the canonical electromagnetic field attracts the relativistic images towards the optical axis while the phantom electromagnetic field repels them. In the case of canonical scalar field the higher coupling repels the images form the optical axis while in the phantom scalar field case it attracts them. In the limit of infinitely strong coupling for any value of Q/M the EMD and the EMD black holes become practically indistinguishable from the Schwarzschild black hole on the bases of observations for the angular position, the magnification and the separation of the relativistic images.", "pages": [ 14, 15, 16, 17 ] }, { "title": "Acknowledgments", "content": "Partial financial support from the Bulgarian National Science Fund under Grant DMU 03/6 is gratefully acknowledged. The authors would like to thank prof. S. Yazadjiev for the fruitful discussions and the anonymous referee for the valuable remarks.", "pages": [ 17 ] } ]

[ { "title": "Michael P. Salem", "content": "Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA 94305, USA Abstract: The volume fractions of vacua in an eternally inflating multiverse are described by a coarse-grain rate equation, which accounts for volume expansion and vacuum transitions via bubble formation. We generalize the rate equation to account for bubble collisions, including the possibility of classical transitions. Classical transitions can modify the details of the hierarchical structure among the volume fractions, with potential implications for the staggering and Boltzmann-brain issues. Whether or not our vacuum is likely to have been established by a classical transition depends on the detailed relationships among transition rates in the landscape.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "We might live in an eternally inflating multiverse. Eternal inflation occurs whenever a sufficiently large volume is in a state sufficiently close to a vacuum in which the energy density is positive and the decay rate is smaller than the Hubble rate [1], and/or whenever a sufficiently homogeneous field configuration evolves in a sufficiently flat, positive interaction potential [2, 3]. These statements contain a number of qualifications, so it is worth noting that the first set of conditions is satisfied by the observed state of the local universe, if the so-called dark energy is due to vacuum energy, while the second set is similar to the initial conditions implicit in the simplest models of slow-roll inflation, though with the inflaton further up its potential [4]. When it occurs, eternal inflation generates an endless spacetime in which every phase of vacuum takes place in a fractal mosaic of widely separated domains [5, 6], the various vacua being attained either by bubble formation [7, 8], by stochastic diffusion [9], and/or by other processes [10, 11, 12, 13, 14, 15]. In an eternally inflating multiverse, the fraction of the global spacetime volume occupied by any of the various vacua is not directly observable. Nevertheless, these volume fractions might be relevant to understanding the local conditions in our universe. For instance, the proper theory of initial conditions might be a theory of the multiverse as a whole, with our local 'initial' conditions-i.e. the conditions describing the onset of slow-roll inflation in the particular phase of vacuum that gives rise to our universe-being selected according to their prevalence in the global spacetime. It is also possible that these volume fractions express a holographic dual to bulk spacetime physics [16, 17, 18, 19, 20, 21]. In a multiverse where vacuum transitions occur predominantly via bubble formation, the volume fractions occupied by the various vacua are described by a rate equation [6, 22]. An important caveat is that the predictions of the rate equation depend on the choice of global time foliation. This is because different global time foliations explore different regions in the diverging spacetime at different rates, while the exponential expansion of eternal inflation ensures that most of the total volume is near the boundary at any finite time cutoff. This expresses the measure problem of eternal inflation (for some recent reviews, see for example [23, 24, 25, 26]) Resolving the measure problem is of fundamental importance to (eternal) inflationary cosmology, but it is tangential to the thrust of this paper. This paper concerns another shortcoming of the rate equation, which is that it ignores bubble collisions. Semiclassical vacuum transition rates are exponentially suppressed, and a contribution to the rate equation from a bubble collision should involve a product of two such rates (one for each bubble in the collision), and so one might argue that bubble collisions can be ignored by expanding in powers of transition rates. However, these rates are exponentially staggered, meaning the product of two transition rates could be much larger than another transition rate. Moreover, over the course of its evolution each bubble collides with a diverging number of other bubbles. This divergence is regulated by the aforementioned measure, but it is not a priori clear how this resolution will play out. Although the detailed phenomenology of the rate equation is rather technical, for those very familiar with the literature (including the standard notation and assumptions) our conclusions are simple to state. (Those less familiar with the literature will find the conclusions of this paragraph explained more thoroughly in the main text.) After including the effects of bubble collisions, the rate-equation transition matrix becomes where γ ik/lscriptj is related to the average volume fraction in vacuum i in the causal futures of collisions between bubbles of vacua k and /lscript , when these bubbles nucleate in vacuum j . The effects of bubble collisions are most significant when the first two terms in (1.1) are zero but (because of classical transitions) the third term is not. Meanwhile, the components of the dominant eigenvector s i of M ij are, to leading order, where the sum covers the sequences of transitions that connect the dominant vacuum '1' to the vacuum i using the fewest number of upward transitions (and 'leading order' refers to an expansion in these upward transition rates). Again, the effects of classical transitions can be significant because the first term in each factor can be zero when the second term is not. Note that the dominant vacuum-defined as the positive-energy vacuum with the smallest decay rate-still dominates the volume fraction. In particular, even if the dominant vacuum can set up classical transitions to another positive-energy vacuum, the volume fraction of the latter is still much less than unity. On the other hand, the detailed hierarchical structure among the components of s i can be modified by the existence of classical transitions. The detailed hierarchical structure among the volume fractions of the various vacua in the landscape are relevant to the so-called staggering issue, which concerns the competition between anthropic selection for small (magnitude) vacuum energies and cosmological selection for large volume fractions when attempting to explain the observed size of the cosmological constant; see for example [27, 28, 29, 30]. It is also relevant to the so-called Boltzmann-brain issue, which concerns the likelihood for observers to arise in an extremely low-entropy Hubble volume such as we observe, as opposed to in a relatively high-entropy Hubble volume such as describes the distant future [31, 32, 33, 34, 35]. One might also take interest in the likelihood that our vacuum was created by a classical transition, as opposed to by semiclassical bubble formation. Although we discuss these issues, a conclusive investigation requires a detailed understanding of the landscape, and is beyond the scope of this paper. The remainder of this paper is organized as follows. In Section 2 we review the construction of the rate equation, ignoring bubble collisions, while in Section 3 we include bubble collisions. We predominantly work in terms of a scale-factor-time foliation, though we briefly explain how to translate the results into those of a lightcone-time foliation. The phenomenology of the rate equation is studied in Section 4. We begin with a review of a simple toy landscape, ignoring bubble collisions, and then we explore the toy landscape while including the effects of some representative classical transitions. We extend our results to a more general landscape in Section 4.3, where we briefly discuss the staggering issue. In Section 4.4 we discuss the Boltzmann-brain issue and in Section 4.5 we discuss the likelihood of a classical transition in our past. Finally, we draw our conclusions in Section 5.", "pages": [ 2, 3, 4 ] }, { "title": "2. Rate equation without bubble collisions", "content": "To begin, we reconstruct the standard rate equation, adopting the usual assumptions [6, 22]. In particular, we take the spacetime to be everywhere (3+1)-dimensional (the rate equation in a transdimensional multiverse is studied in [36]), we assume that all vacuum transitions occur via semiclassical bubble formation, we coarse-grain over the time scales of any transient cosmological evolution between epochs of vacuum-energy domination, we assume there are no vacua with precisely zero vacuum energy, 1 and we ignore bubble collisions. In the next section we include bubble collisions, but the other assumptions are held throughout the analysis. The rate equation describes the volume fractions of vacua in terms of some global time variable t . To establish the global time foliation, we start with a large, spacelike hypersurface Σ 0 on which we set t = 0. Note that it is not necessary for Σ 0 to be a Cauchy surface: if Σ 0 intersects an eternal worldline (that is, a worldline that never sees a vacuum energy density less than zero), then the rate equation describing the future evolution of Σ 0 will possess an attractor solution that is independent of the detailed orientation and distribution of vacua on Σ 0 . This implies that the volume fractions are independent of the choice of Σ 0 . Suppose the total physical volume of Σ 0 in vacuum i is V i . To construct the rate equation, we first compute the change in physical volume in vacuum i over a time interval ∆ t , i.e. where Σ ∆ t denotes the hypersurface of constant t = ∆ t . The coarse-grain approach explores the limit of large ∆ t to compute the various contributions to ∆ V i , but then expands in ∆ t /lessmuch 1 to construct a differential equation. In the context of bubbles with positive vacuum energy densities-henceforth referred to as dS bubbles-this means that the rate equation ignores the transient cosmological evolution between epochs of vacuum energy domination. In the context of bubbles with negative vacuum energy densities-henceforth referred to as AdS bubbles-it means the rate equation ignores the cosmological evolution altogether. 2 Although the coarse-grain rate equation does not by itself provide an accurate assessment of the physical volume fractions in AdS vacua, it is convenient to track the volume fractions in these vacua anyway. We do this by simply conserving comoving volume during transitions to AdS vacua, and ignoring the subsequent evolution of the volume.", "pages": [ 4, 5 ] }, { "title": "2.1 Scale-factor time", "content": "We first take t to be the scale-factor time [37, 38, 39, 40, 41], where τ is the proper time evaluated along a geodesic congruence orthogonal to Σ 0 , and H is the local Hubble rate, in particular we can take H ≡ (1 / 3) u µ ; µ in terms of the four-velocity field u µ along the congruence. Although a precise definition of scale-factor time involves some subtleties in the treatment of locally contracting spacetime regions [39, 40, 41], these can be ignored in the coarse-grain analysis, which smears over such regions. Before proceeding, it is helpful to collect some facts about bubble formation. Consider a region in some dS vacuum i with cosmological constant Λ i . On scales that are small compared to the curvature radius, the line element can be written where d Ω 2 is the line element on the unit 2-sphere, and the absolute value is inserted to establish a general definition for when we consider negative values of Λ i . Now suppose an initially pointlike bubble nucleates at time τ = τ nuc . To facilitate future reference we take the hypersurface τ = τ nuc to coincide with the aforementioned Σ 0 , at least in the vicinity of the bubble. (A diagram is provided in Figure 1.) The bubble wall of a point-like bubble expands at the speed of light. Therefore, the comoving radius of the bubble at times τ ≥ τ nuc is The bubble expands so as to subtend a finite comoving volume (32 π/ 3) e -3 H i τ nuc in the limit τ →∞ . Note that this comoving volume corresponds to a physical volume (4 π/ 3) H -3 i on the hypersurface Σ 0 . Therefore, the loss of physical volume in vacuum i in the future evolution of Σ 0 due to the nucleation of this bubble is equivalent (in the limit τ → ∞ ) to the loss of physical volume that results from simply ignoring the would-be future evolution of a physical volume (4 π/ 3) H -3 i on the hypersurface Σ 0 . Now suppose that this bubble is a dS bubble. The coarse-grain analysis ignores dynamics on time scales smaller than the time scale of vacuum domination; therefore the line element in the bubble can generically be written where j labels the vacuum in the bubble, and it is implicit that we focus on bubble FRW times τ /greatermuch H -1 j . Importantly, surfaces of constant FRW time τ in the bubble are not surfaces of constant scale-factor time t in the global foliation. In particular, it can be shown that the change in scale-factor time between the hypersurface Σ 0 (corresponding to τ = τ nuc ) in vacuum i and a hypersurface of constant τ in the bubble is [39] Therefore, the induced metric on Σ ∆ t when it overlaps with the bubble is The coarse-grain approximation explores the limit where the first term in brackets dominates over the second (because the coarse-grain approximation implies for example ∆ t > H i /H j ). Accordingly, the physical three-volume of the intersection of Σ ∆ t and the bubble is Technically, the integrand in (2.9) is invalid at large values of ξ , for which the hypersurface Σ ∆ t explores times τ /lessorsimilar H -1 j and the above results receive corrections. Nevertheless, for sufficiently large values of ∆ t the physical volume on Σ ∆ t is dominated by regions for which τ /greatermuch H -1 j , and the above approximations are accurate. Recall that we modeled the initial bubble as point-like. A realistic bubble has some nonzero initial radius, and the bubble wall has zero initial velocity. However, the bubble wall accelerates, its velocity approaching the speed of light. Therefore, in the limit of large ∆ t , our results for point-like initial bubbles coincide with the results for more realistic bubbles. An important exception to this rule is an 'upward' transition from lower to higher vacuum energy, for which the bubble wall fills the horizon [8]. Lacking any clearer guidance, we simply use the above results for these vacua as well. We now return to the rate equation. Breaking the calculation of ∆ V i up into parts, we first compute the change in physical volume in comoving regions that begin and remain in vacuum i over the interval ∆ t . This is given by the definition of scale-factor time: for any dS vacuum i . As mentioned above, we simply ignore the change in physical volume in comoving regions that begin and remain in the same AdS vacuum, as these vacua collapse into a big-crunch singularity on time scales that are small compared to the coarse-graining. Hence, for these vacua we take ∆ V i = 0. Next, we compute the change in physical volume due to vacuum decay in comoving regions that begin in vacuum i . Referring to the above analysis, we note that when i is a dS vacuum then the effect of such decays is equivalent to removing physical volume (4 π/ 3) H -3 i for each bubble nucleation, at the time of its nucleation. The number of such nucleations in an interval ∆ t is equal to the four-volume in vacuum i in that interval times the decay rate, where Γ ji denotes the transition rate from vacuum i to vacuum j , per unit physical threevolume per unit proper time, and we have used ∆ t = H i ∆ τ in vacuum i . The corresponding change in volume in vacuum i is therefore This result assumes that i is a dS vacuum. Although AdS vacua can also decay [42], when they do so they transition exclusively to other AdS vacua, and therefore in the coarse-grain approach these transitions can be ignored (i.e. we set Γ ij = 0 for AdS j ). Finally, we compute the change in physical volume due to transitions to vacuum i from comoving regions that begin in some other vacuum. The number of such transitions is computed in analogy to (2.11), but now with reference to the transition rate from some (dS) vacuum j to vacuum i , summing over the relevant vacua j . In the limit of large ∆ t , each such transition generates a physical volume (2.9) in vacuum i , when i is a dS vacuum. This can be interpreted as the immediate creation of a physical volume (4 π/ 3) H -3 j , times a growth factor e 3∆ t which is already accounted for in (2.10). Thus we write for a dS vacuum i . For AdS vacua i , the analysis surrounding (2.9) no longer applies. Note however that the physical volume created in i due to transitions from a given dS vacuum j in (2.13) is precisely the same as the physical volume lost in j due to transitions to a given vacuum i in (2.12) (which requires exchanging the indices i ↔ j ). Since both of these expressions have the volume expansion factors stripped away, this equivalence expresses the conservation of comoving volume in vacuum transitions. Extending this principle to AdS vacua indicates that we should use (2.13) when describing the creation of volume in vacua i for dS and AdS vacua i . We make an error for AdS vacua since we ignore transitions from one AdS vacuum to another; however since these transition rates are exponentially suppressed next to the time scales of the AdS big crunches, this error is small. Combining results and taking the infinitesimal limit, we obtain where the first term is understood to apply only when i is a dS vacuum, and we have defined the dimensionless decay rates It is convenient to also define the volume fractions f i , where Since the dS vacua dominate the physical volume in the future evolution of Σ 0 , we can write d dt ∑ j V j = 3 ∑ j V j , which means the rate equation can be written where the transition matrix is M ij ≡ κ ij -δ ij ∑ k κ ki . The solution to (2.17) is [22] where the f (0) i are constants reflecting the initial conditions, q > 0 is (minus) the smallestmagnitude eigenvalue of M ij , s i is the corresponding eigenvector-called the dominant eigenvector-and the ellipses denote terms that fall off faster than e -qt . Although we have presented the solutions for both dS and AdS vacua, as we have remarked the coarse-grain rate equation does not reliably assess the volume fractions in the AdS vacua.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "2.2 Lightcone time", "content": "The above analysis is sensitive to the choice of scale-factor time as the global time parameter t . Another popular choice is lightcone time [18, 19], defined according to where V 0 [ I + ( p )] is the volume on Σ 0 subtended by the subset of an initially uniform geodesic congruence orthogonal to Σ 0 that intersects the causal future I + ( p ) of the point p at which the lightcone time is being evaluated. One can think of the geodesic congruence as a tool to project the asymptotic comoving volume of I + ( p ) back onto Σ 0 . Thus, as one considers points p progressively further to the future of Σ 0 , this projection covers a progressively smaller volume on Σ 0 , and the lightcone time increases according to negative one third of the logarithm of this (another, equivalent definition is given in [19]). Rather than repeat the full analysis of the previous subsection, we simply refer to the results of [21]. The crucial difference in comparison to scale-factor time is in the analogue of (2.7), i.e. the change in lightcone time between Σ 0 and a hypersurface of fixed FRW time τ in a bubble that nucleates on Σ 0 . In the case of lightcone time, this is [21] 3 Proceeding as in the previous section, the full consequence of this difference is to multiply (2.9) by a factor of H 3 i /H 3 j , which corresponds to multiplying the second term in (2.14) by a factor of H 3 j /H 3 i . Thus, we recover a rate equation of the form (2.14) if we work in terms of the rescaled volumes ˜ V i = H 3 i V i . Likewise we define Since in the coarse-grain approach the various H i are simply constants, this gives an equation of the form (2.17), from which the solutions can be read off.", "pages": [ 9, 10 ] }, { "title": "3. Including bubble collisions", "content": "The calculation of Section 2 neglects the effects of bubble collisions, which we now address. /negationslash Consider the collision between one bubble of vacuum j and another bubble of vacuum k , the two bubbles having nucleated in vacuum i (see in Figure 2). We assume that both j and k have smaller vacuum energies than i . When the bubbles collide, either the bubble walls annihilate (if j = k ), or one or two domain walls form after the collision. In the case of one domain wall, the causal future of the collision contains both vacua j and k , their relative volume fraction determined by the trajectory of the domain wall. In the case of two domain walls, the domain walls contain some different vacuum /lscript = j, k . If the vacuum energy of /lscript is larger than the vacuum energies of j and k , the domain walls accelerate toward each other. Nevertheless, if the vacuum energy of /lscript is less than the vacuum energy of i , the domain walls do not necessarily collide before future infinity. This is called a classical transition to vacuum /lscript [11, 12, 44, 45, 46]; in this case the causal future of the collision contains vacua j , k , and /lscript , with the relative volume fractions determined by the trajectories of the domain walls. We use λ /lscriptkj i to denote volume fraction in vacuum /lscript in the causal future of a collision between bubbles of vacua k and j which nucleated in vacuum i . As remarked above, for any given collision this volume fraction depends on the trajectories of domain walls. These depend on the model-dependent interaction potential governing the tunneling fields, as well as on the collision-dependent placement of the colliding bubbles. We are uninterested in these details, and instead simply take the λ /lscriptkj i as given, taking the given quantities to average over the relative placement of colliding bubbles (further details are presented below).", "pages": [ 10, 11 ] }, { "title": "3.1 Volume corrections from bubble collisions in scale-factor time", "content": "Consider a dS bubble of vacuum j , which nucleates in some vacuum i at t = 0. According to (2.9), the physical volume in this bubble on the constant scale-factor hypersurface Σ ∆ t is for large ∆ t . However, some fraction of this volume is not actually in vacuum j , because it resides in the causal future of collisions between the bubble of vacuum j and other bubbles. To account for this, we first compute the total volume in the causal futures of these collisions, so that we can subtract this volume of j from the rate equation. We then discuss the volume of each vacuum that should be put back into the rate equation so as to reflect the vacuum composition of the causal futures of these collisions. Part of the calculation is laid out nicely in [47], and we begin by translating the relevant result into our notation. To do so, consider a 2-sphere of radius ξ on a constant FRW time slice in the bubble geometry (2.6). According to the symmetries of the collision, if the causal future of a collision intersects this 2-sphere, the intersection corresponds to a disk on the 2sphere. This disk subtends a certain solid angle, and we are interested in the total solid angle subtended by summing over the causal futures of all bubble collisions. This is the calculation performed in [47], and the resulting solid angle (in the limit of late FRW time slices) is where where the sum runs over all vacua k into which the vacuum i can decay (with Λ k ≤ Λ i ). We have used tan( T co ) = ( H i /H j ) tanh( H j τ/ 2) → H i /H j in converting from the notation of [47] in the context of the coarse-grain focus on the late-FRW-time limit in the bubble. To obtain the physical volume on Σ ∆ t that intersects these causal futures, we integrate over 2-spheres of radii ξ , but using the solid angle (3.2) instead of the usual 4 π , and using the induced metric (2.8) to switch from a constantτ to a constantt hypersurface. This gives where the second expression defines F , which is a dimensionless function of R ≡ H i /H j and κ i ≡ ∑ k κ ki . The integral in F can be evaluated in terms of hypergeometric functions, but the result is not very illuminating. Instead we focus on the situation where R /greatermuch 1, and we assume κ i /lessmuch 1. Then we can approximate where we have kept only the leading-order terms in H i /H j and κ ij . Combining with (3.4), we find that the physical volume in the causal future of these bubble collisions is Note that the volume subtracted from a bubble of vacuum j due to a collision with a bubble of vacuum k is not symmetric with respect to the indices j and k . In other words, when a bubble of vacuum j collides with a bubble of vacuum k , the causal future of the collision intersects a different 'would-be' volume on Σ ∆ t in the bubble of vacuum j , than the 'would-be' volume on Σ ∆ t that it intersects in the bubble of vacuum k . There is no reason these two volumes should have been the same, because neither of them represents the actual geometry in the causal future of the collision; instead they represent the volume that would have been there had these bubbles not collided, and this counterfactual volume is different between the two bubbles because the two bubbles have different Hubble rates. After we subtract the volume (3.8) from a bubble of vacuum j , we must specify how much volume of each type of vacuum to put back in its place. We write the volume of vacuum /lscript on the hypersurface Σ ∆ t in the causal futures of the collisions between a bubble of vacuum j and other bubbles of vacuum k , for all k , as where i labels the vacuum in which the bubbles of vacua j and k nucleate. Of course, all of the microphysics is contained in the λ /lscriptkj i , which are also understood to reflect the average volume fractions after considering the various possible relative placements of the colliding bubbles that contribute to each term in (3.9). The other factors in (3.9) are chosen so that if collisions with bubbles of a given vacuum k always produce a domain wall that runs along the causal future of the collision into the bubble of vacuum k -that is, from the perspective of the bubble of vacuum j , collisions with bubbles of vacuum k have no effect on the interior of the bubble of vacuum j -then λ /lscriptkj i = δ /lscriptj (this returns all of the volume that had been removed from the bubble of vacuum j , but none of the volume that had been removed from the bubbles of vacuum k ). The result (3.8) assumes that both i and j are dS vacua. Since we ignore transitions out of AdS vacua, this assumption about i is sufficient. On the other hand, it is important to keep track of the case where the vacuum j is AdS. This is because it is possible for the collision between two AdS bubbles to create a classical transition to a dS vacuum. 4 Guided by the results of Section 2 in which conservation of comoving volume implied that after stripping away the volume expansion factors the same expressions could be used for transitions to dS and AdS vacua, we assume that (3.8) can be used to describe the volume in the causal future of bubble collisions for both dS and AdS vacua j , modulo the volume expansion factor e 3∆ t . Likewise, we use (3.9) for both dS and AdS vacua /lscript (and j and k ). Although it is possible that using (3.8) for AdS vacua j introduces an error, technically this error can be removed by an appropriate choice for the factors λ /lscriptkj i in (3.9). Our analysis is unconcerned with such details; in our analysis it is only important whether λ /lscriptkj i is precisely zero or not.", "pages": [ 11, 12, 13 ] }, { "title": "3.2 Modifying the rate equation in scale factor time", "content": "We can now compute the change in volume in vacuum i , ∆ V i , after a scale-factor time interval ∆ t , due to bubble collisions. We first consider the loss of volume in bubbles of vacuum type i due to collisions with other bubbles. This is The first three terms in the first expression give the number of bubbles of vacuum i that nucleate in vacuum j in the time ∆ t , according to (2.11). The other terms in this expression give the volume in each such bubble that is in the causal future of collisions with bubbles of vacuum k , given by (3.8), but where we have removed the expansion factor e 3∆ t , which in the coarse-grain analysis is accounted for by the factor of 3 V i in (2.14). Next we consider the change in volume in vacuum i , due to the physical volume in vacuum i in the causal future of bubble collisions. This can be written The first three terms in the first expression give the number of bubbles of type /lscript that nucleate in vacuum j in the time ∆ t , according to (2.11). The other terms in this expression give the volume in vacuum i in the causal futures of the collisions between each such bubble of vacuum /lscript and bubbles of vacuum k , given by (3.9), but where we have removed the expansion factor e 3∆ t , which in the coarse-grain analysis is accounted for by the factor of 3 V i in (2.14). Putting everything together and taking the infinitesimal limit, we obtain In terms of the volume fractions f i defined in (2.16), we have where now where for later convenience we have defined Note that conservation of comoving volume implies that the sum over rows in any column of M ij must be zero. This implies a useful constraint on the γ ik/lscriptj , namely", "pages": [ 13, 14 ] }, { "title": "3.3 Lightcone time", "content": "The previous two subsections worked in terms of scale-factor time. However, the only place where the actual definition of the global time parameter enters is in the induced metric used in (3.4) (and results, such as (3.1), taken from Section 2). Meanwhile, in Section 2.2 we found that the only effect of using lightcone time instead of scale-factor time corresponded to multiplying the second term in (2.14) by H 3 j /H 3 i . The same applies here with respect to the second term in (3.12), and because of the modification to (3.4) we multiply the fourth and fifth terms in (3.12) by H 3 j /H 3 i as well. It is easily checked that if we define ˜ f i and ˜ V i as in (2.21) and above it, we obtain with M ij given by (3.14). Therefore, given a solution in terms of scale-factor time, we can again read off the solution in terms of lightcone time.", "pages": [ 15 ] }, { "title": "4. Phenomenology", "content": "To develop intuition for the phenomenology of the rate equation, we study a simple toy model of the landscape [22, 48, 40]. The model can be represented by the diagram The numbers label different vacua, while the arrows indicate the direct transitions that are allowed among the vacua. (We use the phrase 'direct transition' to designate semiclassical bubble formation via quantum tunneling through one potential barrier, which we take to be the dominant form of vacuum transition, aside from perhaps classical transitions, which are discussed below.) For concreteness, we assume the vacuum energies of this model obey That is, the labels '1,' '2,' and '3' designate dS vacua, while the other labels designate AdS vacua. We also assume that '1' corresponds to the dS vacuum with the smallest decay rate.", "pages": [ 15 ] }, { "title": "4.1 Toy model without bubble collisions", "content": "We first ignore bubble collisions. Then this model is similar to models studied elsewhere in the literature [22, 48, 40]. Seeking the late-time attractor solution-that is, the dominant eigenvector-we insert the ansatz f i , ˜ f i = s i e -qt into the rate equation, obtaining where we focus on the volume fractions of the dS vacua, and κ i ≡ ∑ j κ ji . This system of equations can be solved algebraically, but the solution is complicated and not very enlightening. Since κ 21 and κ 23 are transition rates from lower to higher vacuum energy, they are expected to be exponentially suppressed relative to the other transition rates. We therefore solve the above system of equations by expanding in κ 21 and κ 23 , treating them as similar in order for the purpose of the expansion. At zeroth order the only nonzero dS component of s i is s 1 , which we can set to unity. The corresponding value of q is where the second equality implicitly evaluates κ 1 at zeroth order, a notational shorthand that we also employ below to help simplify expressions. The other dS components of s i become relevant at first order, and are given by The zeroth-order solutions for s 1 and q receive first-order corrections, but these are subdominant and so we ignore them. We see that the dS components of s i are dominated by '1'-the dS vacuum with the smallest decay rate-which is accordingly called the dominant vacuum (this assumes that the decay rates are not tuned so as to make q very near to κ 2 or κ 3 , in which case there could be a set of degenerate dominant vacua [40]). Intuitively, the dominant vacuum dominates the asymptotic volume of spacetime due to its small decay rate, and the volume fractions of all of the other vacua reflect in part the relative likelihood of transitioning to them from the dominant vacuum. In particular, s 2 carries a factor of κ 21 , reflecting the transition from '1' to '2,' while s 3 carries the factor κ 21 κ 32 , reflecting transitions first from '1' to '2,' then from '2' to '3.' These factors also contain total decay rates in the denominator, so they only correspond to suppression factors when a transition rate in the numerator is relatively suppressed. This applies to κ 21 in the first-order terms in (4.7), and it applies to contributions to the s i that arise due to transitions through '3,' which are suppressed relative to the above contributions.", "pages": [ 15, 16 ] }, { "title": "4.2 Effects of bubble collisions", "content": "The above discussion summarizes previous work. Our goal is to understand the implications of allowing for bubble collisions. Collisions that result in a single domain wall, and therefore merely shift the volume fractions of the vacua involved in the collision by order-unity factors, are not consequential in the context of the qualitative dynamics described above. However, the possibility of classical transitions might change the above conclusions in more dramatic ways. To explore this, we first assume that bubbles of vacua '4' can collide and produce a classical transition to vacuum '3.' We denote this with the diagram Note that this transition requires that Λ 3 < Λ 1 [12, 45]. Although the landscape (4.1) permits other bubble collisions, for the moment we assume that these do not produce classical transitions so that we can ignore them. Then the full set of nonzero γ ijk/lscript can be taken to be The first term accounts for the subtraction of volume in the causal future of the collisions that would be in vacuum '4' but for the collision, while the second term accounts for the restoration of this volume but in vacuum '3,' a consequence of the classical transition. In a realistic model we expect some of the causal future of the collision to be in '4,' which means that γ 4441 should be less negative and γ 3441 should be smaller. However, these effects change our results by factors that are negligible next to the exponential staggering of decay rates, and so for simplicity we ignore them. Including these collisions and their effects, the rate equation gives where again we focus on the dS components of s i . As before, the solution is more transparent if we expand it in terms of κ 21 and κ 23 . At zeroth order, we find where we have normalized the zeroth-order solution so that ∑ i s i = 1, where the sum runs over dS vacua i , and q is unchanged from before: q = κ 1 . The leading-order contribution to s 2 still appears at first order in the expansion, where s 1 and s 3 refer to the zeroth-order quantities. Evidently, the hierarchical structure of the dominant eigenvector is qualitatively changed relative to before. In particular, while the dominant vacuum '1' still dominates the dS components of s i , the component s 3 now contains a product of two downward transition rates from '1,' as opposed to an upward transition rate. Note that since '1' is by definition the dS state with the smallest decay rate, we still expect s 3 to be exponentially suppressed relative to s 1 . Nevertheless, the size of the suppression is dramatically reduced. Meanwhile, the volume fraction of '2' is still suppressed by an upward transition rate, but this can come from '1' or from '3.' Although the results are changed from when we ignored bubble collisions, our intuition for the solution is maintained. In particular, the dS vacuum with the smallest decay rate dominates the asymptotic volume fraction, and the volume fractions of the other dS vacua reflect the relative likelihood of transitioning to them from the dominant vacuum. In the case of '3,' this likelihood is enhanced by the possibility for classical transitions. This in turn enhances the volume fraction of '2,' insofar as it can be reached by transitions from '3.' The classical transition represented by (4.8) is not the only possibility in the landscape (4.1). Another possibility occurs when bubbles collide in vacuum '2.' We now focus on the possibility that collisions between bubbles of vacua '5' can result in a classical transition to '3,' represented by the diagram and ignore all other possible collisions. Then the full set of nonzero γ ijk/lscript can be taken to be where the discussion below (4.9) applies here as well. The resulting rate equation gives We again expand in terms of κ 21 and κ 23 . The zeroth order dS components are the same as without classical transitions; namely the only nonzero component is s 1 , which we can set to unity. The other dS components of s i become relevant at first order, and are given by Note that s 2 and s 3 are still suppressed by an upward transition rate out of '1.' Therefore, the effects of classical transitions do not change the qualitative expectations for the dominant eigenvector described above. On the other hand, it is possible for the component s 3 to be enhanced relative to before, depending on the relative size of γ 3552 κ 2 52 and κ 32 .", "pages": [ 16, 17, 18 ] }, { "title": "4.3 Generalization of the toy model", "content": "The toy landscape model (4.1) is simple, but the intuition developed above extends to more general landscapes. Indeed, it is possible to compute the components of the dominant eigenvector s i for any number of vacua and for any set of transitions among them, by expanding in the off-diagonal elements of the transition matrix M ij [29, 49]. As before, we assume the dS vacuum with the smallest decay rate is unique and denote it as '1.' The (unnormalized) dS components of s i are then, to leading order, where q is the decay rate of '1' and the sum covers the sequences of transitions that connect '1' to i using the fewest number of upward transitions. In the second expression we have exploited the fact that q is the smallest decay rate to write κ i -q ≈ κ i , which allows for the terms to be rearranged so as to express factors as branching ratios. We expect the dS vacuum with the smallest decay rate to also possess a small vacuum energy (for some discussion of this matter see [40]). This ensures that transitions from '1' to other dS states are either transitions to larger vacuum energy or transitions that involve a very small change in vacuum energy compared to the expected size of the barrier. The rates associated with such transitions are exponentially suppressed relative to the rates associated with transitions to AdS vacua with much larger magnitude vacuum energies [7]. Therefore, we expect the rightmost term in (4.21) to be exponentially small, while all of the other factors are less than one by definition. This maintains the notion of '1' as the state that dominates the volume fractions, i.e. the dominant vacuum. It is straightforward to generalize the above result to include the effects of bubble collisions. In particular, the effect of bubble collisions is simply to add the (small) quantity ∑ k,/lscript γ ik/lscriptj κ kj κ /lscriptj to the quantity κ ij corresponding to each off-diagonal perturbative element of the transition matrix M ij . We can then read off the results from the analyses of [29, 49]: Of course, it is not necessary for both terms in a given factor to be nonzero, and in fact the consequences of bubble collisions are most significant when the first term in at least one of these factors is zero, as with the classical transitions described above. Note that the dS vacuum with the smallest decay rate, vacuum '1,' still dominates the volume fractions. To elaborate, we repeat the argument given above for the case where bubble collisions are ignored, which holds unless the first term in the numerator of the rightmost factor of (4.23) is zero. This occurs when a dS vacuum i can be reached via classical transitions caused by bubble collisions in '1.' In this case, each contribution to (4.23) contains a factor of the form κ i 1 /q , which is necessarily less than one, in addition to a factor of the form γ ijk 1 κ k 1 , which we expect to be much less than one. The latter expectation arises because although any γ ijk 1 involves a ratio of Hubble rates (squared), which in principle could be very large, the factor κ k 1 is expected to be doubly exponentially small, as it is less than the decay rate per Hubble volume of the longest-lived dS state. Evidently, classical transitions can modify the detailed hierarchical structure among the components of the dominant eigenvector. For instance, a given ratio s i /s j could be exponentially large or small depending on whether one ignores classical transitions or not. Moreover, the existence of classical transitions can modify the distribution of components of s i as a function of their size. To illustrate these conclusions, we organize the vacua in the landscape according to the number N up of upward transition rates that appear in the numerators of the factors in the corresponding components of s i . For instance, a vacuum i for which the sequences of transitions { p a } in the summation of (4.23) contain three upward transitions has N up = 3. Accounting for classical transitions can allow for sequences of transitions that reduce N up relative to otherwise, for a given vacuum in the landscape. Since upward transition rates are exponentially suppressed relative to downward transition rates, this exponentially increases the volume fractions of certain vacua (relative to not including classical transitions), and it modifies the distribution of vacua as a function of volume fraction. This is relevant to the staggering issue, which concerns the competition between anthropic selection for a small (magnitude) cosmological constant Λ and cosmological selection for a large volume fraction when attempting to explain the observed value of Λ [27, 28, 29, 30]. To elaborate, imagine organizing the various vacua in the landscape into bins according to the size of Λ, and consider the total volume fraction represented by the vacua in each bin. The landscape explanation of the observed value of Λ assumes that for a bin size ∆Λ less than the observed value of Λ, the total volume fraction in each bin is roughly independent of Λ for small | Λ | . The validity of this assumption depends on the number of vacua in the landscape as well as the distribution of these vacua as a function of volume fraction. If the number of vacua is too small, then the number of vacua in each bin will be small and their total volume fraction will vary wildly from bin to bin. If the distribution of vacua as a function of volume fraction falls too steeply, then the total volume fraction in each bin will be dominated by the volume fractions of a few vacua, and again the total volume fraction will vary wildly from bin to bin. In either case, the anthropic suppression associated with a set of vacua with Λ many orders of magnitude larger than the value we observe could be compensated by a much larger volume fraction occupied by these vacua. The quantitative analysis is model dependent and further discussion of this issue is beyond the scope of this paper.", "pages": [ 18, 19, 20 ] }, { "title": "4.4 Boltzmann brains", "content": "The volume fractions represented by s i are also relevant to the Boltzmann-brain issue, which concerns the likelihood for an observer to arise after reheating from a relatively low-entropy inflationary state, as opposed to after a quantum fluctuation from a relatively high-entropy vacuum-energy-dominated state [31, 32, 33, 34, 35]. The former observers are called normal observers and the latter observers are called Boltzmann brains. To discuss this issue, it is helpful to first summarize the case where bubble collisions are ignored. With respect to the scale-factor cutoff measure, the ratio of Boltzmann brains to normal observers can be approximated by [40] where κ BB i is the Boltzmann-brain formation rate per unit Hubble volume in vacuum i , n NO ij is the peak number of normal observers per unit Hubble volume in a bubble of vacuum i that nucleates in vacuum j , and the sums are understood to run over only dS vacua. To analyze (4.24), first note that Boltzmann-brain formation rates are generically doubly exponentially small. For example, simply demanding that a Boltzmann brain possess at least the information content of a human brain gives the upper bound κ BB i /lessorsimilar e -10 16 [40]. Second, note that any transition rate out of a dS vacuum i must exceed the recurrence rate in i , i.e. κ ij > e -3 π/G Λ j . Therefore, assuming that there exists a sequence of transitions that connects the dominant vacuum '1' to a given vacuum i such that no vacuum between '1' and i has G Λ /lessorsimilar -1 / ln( κ BB i ), we can use (4.21) to write [39] where we use the arithmetic of double exponentials (whereby if x is a double exponential and | ln( y ) | < | ln( x ) | , then xy ∼ x/y ∼ x ) to ignore the various factors of κ jk /κ k in s i next to κ BB i , applying the principle of detailed balance to write κ BB i κ j 1 = κ BB i e 3 π/G Λ j -3 π/G Λ 1 κ 1 j ∼ e -3 π/G Λ 1 κ BB i . Finally, assuming that there exists a sequence of transitions that connects '1' to a vacuum i with | ln( n NO ij ) | < | ln(max { κ BB k } ) | , such that no vacuum between '1' and i has G Λ /lessorsimilar -1 / ln( κ BB i ), we can apply the same techniques to the denominator of (4.24) to obtain Note that e 3 π/G Λ 1 sets the scale for the recurrence time in the dominant vacuum. Therefore, if it is possible for Boltzmann brains to form in the dominant vacuum, the first term in brackets is much greater than one and Boltzmann brains dominate over normal observers. Consequently, we assume that the dynamics of the dominant vacuum are insufficient to support Boltzmann brains-a plausible assumption considering the complex dynamics that underlies our existence. Then, normal observers dominate if we furthermore assume that in each vacuum the decay rate is larger than the Boltzmann-brain formation rate. It is straightforward to repeat this analysis including the effects of bubble collisions. The denominator of (4.24) essentially expresses the bubble formation rate, and generalizes to where n NO ik/lscriptj is defined in analogy to n NO ij and the sums over k and /lscript are understood to include AdS vacua. The subsequent analysis is unchanged except for the possibility that Boltzmann brains form in a dS vacuum that can be reached via classical transitions caused by collisions between AdS bubbles in the dominant vacuum. (This affects the analysis because we cannot use detailed balance to rewrite κ j 1 in terms of κ 1 j for an AdS vacuum j .) Accounting for this additional possibility, we find where we have assumed that any vacua reached by classical transitions from bubble collisions in the dominant vacuum do not produce normal observers (as would be the case if for example these transitions do not establish a period of slow-roll inflation followed by reheating). Note that if normal observers are produced in these vacua, then we simply obtain (4.26) again, since factors of the form e 3 π/G Λ 1 γ /lscriptj k 1 κ j 1 κ k 1 cancel between the numerator and denominator of (4.24) (according to the algebra of double exponentials). As with (4.26), to avoid Boltzmann-brain domination we must assume that in each vacuum the decay rate is larger than the Boltzmann-brain formation rate, and that the dominant vacuum does not support Boltzmann brains. Given these assumptions, the last term in the brackets does not exceed e 3 π/G Λ 1 κ BB i for any of the relevant vacua i . Therefore, we still avoid Boltzmann-brain domination unless G Λ 1 < -1 / ln( κ BB i ) < 10 -16 . While such small values of Λ are presumably very uncommon in a Planck-scale landscape, the dominant vacuum '1' is specially selected for the smallness of its decay rate, and it seems plausible that this might also select for an extremely small value of Λ. Accepting this as a possibility, the next question is how plausible is it that '1' can set up a classical transition to a dS vacuum /lscript that supports Boltzmann brains, bearing in mind that vacua with sufficiently complex degrees of freedom and interactions are presumably rare, and the classical transition only occurs if Λ /lscript < Λ 1 . To explore this question, we consider a large landscape of N vacua to arise in a roughly log( N )-dimensional configuration space [50, 51, 52]. Thus, we assume that each vacuum can directly transition to roughly log( N ) nearby states. Classical transitions are permitted based on special relationships among local minima in the vacuum configuration space, such that each unique pair of vacua involved in a collision can classically transition to at most one unique state [12, 44, 46]. Hence, there are at most roughly log 2 ( N ) distinct states that can be reached by collisions consequent the direct transitions mentioned above. In the context of string theory, it is commonly argued that log( N ) ∼ O (1000) or so [50, 51]. Based on these considerations, it seems very unlikely that the dominant vacuum could set up a classical transition to a vacuum i with 0 < G Λ i < -1 / ln( κ BB i ) < 10 -16 , let alone such a vacuum that can also support Boltzmann brains. On the other hand, it is not evident that we should only consider direct transitions out of the dominant vacuum. Naively, if '1' transitions to i at the rate κ i 1 and if i transitions to j at the rate κ ji , then '1' transitions to j at roughly the rate κ ji κ i 1 . Although κ ji κ i 1 is exponentially suppressed relative to κ i 1 , the factor e 3 π/G Λ 1 in the last term in the brackets of (4.28) is doubly exponentially large, and it seems possible that it could take an exponential number of factors of transition rates to cancel it. If so, classical transitions set up by the dominant vacuum could access a significant fraction of the landscape, presumably including many dS vacua capable of supporting Boltzmann brains. On the other hand, if these classical transitions also access a dS vacuum capable of supporting normal observers, then granting the assumption that κ BB i < κ i for all dS vacua i , these normal observers will dominate over all of the Boltzmann brains reached in this way. Recently, it has been argued that the number of vacua capable of supporting Boltzmann brains is 'only' exponentially (as opposed to doubly exponentially) larger than the number of vacua capable of forming normal observers [53]. (Specifically, [53] argues that the number of bubbles that feature a significant amount of slow-roll inflation is 'only' exponentially suppressed relative to the number that do not.) If so, it would seem to take a fine-tuning for the dominant vacuum to set up classical transitions that reach enough vacua to include one supporting Boltzmann brains, but not enough so as to also include one that forms normal observers.", "pages": [ 20, 21, 22, 23 ] }, { "title": "4.5 Probability to observe a classical transition", "content": "So far our discussion has focused on global volume fractions. On the other hand, if the initial conditions for the phase of slow-roll inflation believed to describe our past were established by a classical transition-as opposed to by the formation of a single bubble-then this could have consequences for observational cosmology, at least if slow-roll inflation did not last too long. Indeed, the collision geometry retains an SO(2,1) symmetry [54], and so the phenomenological consequences should resemble those of the 'anisotropic bubble nucleation' scenario studied in [55, 56]. We leave further exploration of these signatures to future work, and here focus on assessing the relative likelihood of this possibility. One way to approach this problem is to compare the rate of classical transitions to our vacuum to the rate of bubble nucleations of our vacuum. The results of Section 3 allow us to go a step further, since there we have computed the physical volume on a fixed scale-factor time hypersurface in the causal futures of these events. In short, these are given respectively by the fourth and second terms on the right-hand side of the physical-volume rate equation, (3.12). Using V i ∝ s i e (3 -q ) t , we write the ratio of the volume in i in the causal future of a classical transition to the volume in i in the causal future of a bubble nucleation Evidently, R depends on the detailed microphysics of the landscape. In particular, while the sums over vacua j in the numerator and denominator both cover all dS vacua j , for almost all of these vacua the transition rates κ ij are zero, while the vacua j for which the rates are nonzero are in general different in the numerator and denominator. Since the components of the dominant eigenvector s i are exponentially staggered, R could be enormous or zero depending on the detailed structure of a large set of transition rates in the landscape (zero corresponds to the case where classical transitions cannot create vacuum i ). It is amusing to consider a highly idealized situation. Suppose that classical transitions to our vacuum can only be caused by bubble collisions in one type of vacuum, and suppose that bubbles of our vacuum can only nucleate in one type of vacuum, and suppose that the Hubble rates and the relevant decay rates and the components of the dominant eigenvector for these vacua are the same: H 0 , Γ, and s 0 respectively. Finally, suppose that the classical transition converts the entire causal future of the collision into vacuum i . In this situation, (3.16) and (3.15) imply that ∑ k,/lscript λ ik/lscriptj = 1. Putting all of this together, we find This expression is familiar from the bubble-collision literature, and gives the typical number of bubble collisions in an observer's past lightcone (see for example [26]). In this context, H i is the Hubble rate during slow-roll inflation in the observer's bubble, while H 0 and Γ are respectively the Hubble rate and vacuum decay rate outside of the bubble. The congruence between these two results is not surprising: the idealized situation that we have established above is equivalent to comparing the volume in the causal future of bubble collisions to the volume not in the causal future of bubble collisions, assuming that the volume in the causal future of two collisions is double-counted, etc. This is equal to the number of bubble collisions in an observer's past lightcone. The appearance of the inflationary Hubble rate H i in the bubble-collision result, compared to what one might suggest should be the Hubble rate associated with late-time cosmological constant domination in the relative probability (4.30), is due to the former focusing on observers like us who arise at finite time while the latter (interpreted as above) implicitly focuses on hypothetical observers at future infinity.", "pages": [ 23, 24 ] }, { "title": "5. Conclusions", "content": "We have updated the rate equation describing the volume fractions of vacua in the multiverse to account for bubble collisions. The rate equation refers to a global time foliation, and we have presented our analysis in terms of a scale-factor-time foliation, also providing a couple of brief remarks on how to translate the results into those of a lightcone-time foliation. As in the case where bubble collisions are ignored, the intuition for the asymptotic attractor solution to the rate equation revolves around the dS vacuum with the smallest decay rate. Owing to its stability, this vacuum dominates the volume fraction of the multiverse-hence it is called the dominant vacuum-and the relative volume fractions of any other vacua are understood in terms of the relative likelihoods of the sequences of transitions that connect the dominant vacuum to them. The most significant effect of bubble collisions stems from the possibility for classical transitions to vacua that are otherwise not involved in the collision. These transitions should be included along with bubble nucleations in the sequences of transitions referred to above. Thus, the existence of classical transitions modifies the detailed hierarchical structure among components of the dominant eigenvector. Although the volume fractions of many vacua might be exponentially larger on account of including classical transitions, the volume fractions of all dS vacua remain exponentially suppressed relative to the volume fraction of the dS vacuum with the smallest decay rate. Thus, the notion of this vacuum as the dominant vacuum is maintained. On the other hand, any modifications to the detailed hierarchical structure among components of the dominant eigenvector have relevance to the staggering and Boltzmann-brain issues. A conclusive investigation of these issues is beyond the scope of this paper, yet we have discussed plausible circumstances under which accounting for classical transitions does not reveal a Boltzmann-brain problem. In particular, Boltzmann-brain domination occurs only if the dominant vacuum has a sufficiently small vacuum energy and can set up classical transitions to a dS vacuum that supports Boltzmann brains but not to any dS vacua that form normal observers. We have also explored the likelihood that our local Hubble volume was established by a classical transition, as opposed to a semiclassical bubble formation. We found that the relative likelihood of these possibilities depends on the detailed relationships among transition rates in the landscape.", "pages": [ 24, 25 ] }, { "title": "Acknowledgments", "content": "The author thanks Adam Brown, I-Sheng Yang, and Claire Zukowski for valuable discussions. The author is also grateful for the support of the Stanford Institute for Theoretical Physics.", "pages": [ 25 ] }, { "title": "References", "content": "[56] P. W. Graham, R. Harnik, and S. Rajendran, Observing the Dimensionality of Our Parent Vacuum , Phys.Rev. D82 (2010) 063524, [ arXiv:1003.0236 ].", "pages": [ 28 ] } ]

[ { "title": "Neutrino physics from future weak lensing surveys", "content": "R. Ali Vanderveld 1 and Wayne Hu 1, 2 1 Kavli Institute for Cosmological Physics, Enrico Fermi Institute, University of Chicago, Chicago, IL 60637 2 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 (Dated: March 22, 2019) Given recent indications of additional neutrino species and cosmologically significant neutrino masses, we analyze their signatures in the weak lensing shear power spectrum. We find that a shear deficit in the 20-40% range or excess in the 20-60% range cannot be explained by variations in parameters of the flat ΛCDM model that are allowed by current observations of the expansion history from Type Ia supernovae, baryon acoustic oscillations, and local measures of the Hubble constant H 0 , coupled with observations of the cosmic microwave background from WMAP9 and the SPT 2500 square degree survey. Hence such a shear deficit or excess would indicate large masses or extra species, respectively, and we find this to be independent of the flatness assumption. We also discuss the robustness of these predictions to cosmic acceleration physics and the means by which shear degeneracies in joint variation of mass and species can be broken.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "As our cosmological observations become ever more precise, our ability to probe smaller scales continues to advance, thereby allowing us to study the physics of structure formation beyond the standard cold dark matter paradigm. In particular, we are now able to use cosmology to learn about neutrino properties, including the sum of their masses M ν and the effective number of species N eff , both of which imprint their signatures on the small-scale matter power spectrum. Massive neutrinos act as hot or warm dark matter, thereby suppressing structure formation below their thermal freestreaming scale, while adding or subtracting relativistic species changes the ratio of the acoustic and damping angular scales of the cosmic microwave background (CMB) [1, 2]. The former is currently constrained to be M ν glyph[greaterorsimilar] 0 . 05 eV by solar, atmospheric, and laboratory experiments [3-5], and (roughly) M ν glyph[lessorsimilar] 0 . 6 eV from cosmology [6-9]. As for N eff , recent oscillation and reactor experiments [10, 11] lend support to the sterile neutrino interpretation of the LSND electron antineutrino appearance result [12] and M ν glyph[greaterorsimilar] 0 . 4 eV while other neutrino results inhibit a simple global explanation (see e.g. Ref. [13] for a recent review). Meanwhile, recent observations of the CMB damping tail [14-17], Sunyaev Zel'dovich-selected clusters [18], and baryon acoustic oscillations (BAO) [19] provide further hints of extra neutrino species. In this paper we explore how weak gravitational lensing fits into this picture, in light of forthcoming lensingoptimized large-area surveys such as with the groundbased Dark Energy Survey (DES) [20] and Large Synoptic Survey Telescope (LSST) [21], from the balloon-borne High Altitude Lensing Observatory [22], or from space with Euclid [23] and the Wide-Field Infrared Survey Telescope ( WFIRST ) [24]. Weak lensing, whereby the images of distant galaxies are distorted by the gravitational field of matter in the foreground, can be a powerful cosmological probe provided that we have sufficient systematics control. By extracting the weak lensing shear and its evolution with redshift we are able to robustly map out the gravitational potential of the Universe and how it changes with time. The power spectrum of this 'cosmic shear' is directly related to the underlying matter power spectrum. Despite promising results, e.g. [25-29] (see Refs. [30-32] for reviews), the current constraining power of cosmic shear is very limited. Since we do not know the intrinsic shapes of individual galaxies, we must average them over finite patches of sky, making weak lensing a necessarily statistical measure whose constraining power is directly related to sky coverage [33, 34]. Future data sets will be optimized in this respect, but in the meantime it is particularly timely to determine our expectations. We can robustly test any given cosmological model class by exploiting consistency relations between observables pertaining to the expansion history and those pertaining to structure growth [35-38]. Given one, coupled with a class of cosmological models with tunable parameters, we can predict the other and then compare our predictions to data. If the data points lie significantly outside of the prediction contours, then the model class in question is falsified. For instance, just one cluster that is massive and at high-enough redshift could falsify all ΛCDM and quintessence models if its mass and redshift fall significantly outside of what we predict based on Type Ia supernovae (SNe), BAO, local measurement of the Hubble constant ( H 0 ), and the CMB [37]. In this way we can take advantage of the wealth of data already in hand from the CMB and distance measures to predict what we expect for these anticipated future weak lensing observations. Our analysis here builds upon [38] to explore the effects of neutrinos. In what follows we will add neutrinos to this prediction framework, to explore how their masses and number of species change weak lensing observables. We find that, for a fixed CMB, these two properties shift the cosmic shear power spectrum in opposite directions; adding relativistic species amplifies the shear power, whereas endowing the neutrinos with nonzero masses reduces power. Given that quintessence can only decrease the amount of power [38], the former provides qualitatively distinct predictions. This paper is organized as follows. We review our methodology in § II, including the data sets we use, our Markov-Chain Monte Carlo (MCMC) analysis, and the calculation of posterior probability distributions for cosmic shear observables. We then discuss our results in § III, including predictions for four cases - three massless neutrinos, three massive neutrinos, a variable number of massless neutrinos, and both variable mass and number. We conclude in § IV.", "pages": [ 1, 2 ] }, { "title": "II. METHODOLOGY", "content": "We describe here the data sets we use and our procedure for predicting the cosmic shear power spectrum under various assumptions for the total neutrino mass M ν and effective number of species N eff . Our methodology is similar to that of Refs. [35-38].", "pages": [ 2 ] }, { "title": "A. Data sets", "content": "We use the redshifts, luminosity distances, and systematic uncertainty estimates of the Union2 Type Ia SN sample [39]. This sample includes 557 SNe out to a redshift z = 1 . 12, where all light curves have been uniformly reanalyzed using the SALT2 fitter [40]. We use measurements of the BAO feature from Ref. [41] (which includes data from SDSS and the 2degree Field Galaxy Redshift Survey), the WiggleZ Dark Energy Survey [42], and the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) [19, 43]. These measurements extend out to z = 0 . 73 and are reported as distances relative to the sound horizon, D V ( z ) /r s , where D V ( z ) ≡ [(1 + z ) 2 D 2 A ( z ) cz/H ( z )] 1 / 3 , D A is the angular diameter distance, H ( z ) is the Hubble expansion rate, and r s is the sound horizon at last scattering. Even though these data sets have some overlap in both area and redshift, we treat all three as independent due to the different bias and type of galaxies that are targeted in each sample. Unlike Refs. [35-38], we use CMB results from the most recent, 9-year release from the WMAP satellite (WMAP9) [44], computing the CMB angular power spectra using the code CAMB [45, 46]. We now further add the publicly available 2500 square degree release of the SPT measurement of the CMB damping tail [16] over the multipole range 650 < l < 3000, as these smaller-scale peaks are sensitive to neutrino physics. Per Ref. [16], we treat the SZ and point-source contributions as (three) additional nuisance parameters, choosing the same Gaussian priors for each. Finally, we use the combined H 0 estimate from Ref. [47], of H 0 = 73 . 8 ± 2 . 4 km / s / Mpc. This measurement strengthens our constraint on N eff .", "pages": [ 2 ] }, { "title": "B. Parameter sets", "content": "We use this data set to probe flat ΛCDM models with varying f ν and N eff . The former is the fraction of the dark matter density in the form of massive neutrinos where and M ν is the sum of the neutrino masses. The parameter N eff is the so-called effective number of neutrino species [48]: where ρ R is the energy density in relativistic species and ρ γ is the energy density of photons. We increase (decrease) N eff by adding (subtracting) massless species. The default is three species with degenerate masses. Unfortunately this means that N eff < 3 . 046 leads to a negative number of massless species, an unphysical situation which is treated by CAMB as a negative energy density; we follow the SPT analyses [15-18] and ignore this since N eff > 3 . 046 is highly favored and indeed these unphysical cases make up no more than a few percent of the samples in our MCMC chains. We then set the primordial helium abundance Y p from the physical baryon density Ω b h 2 by requiring 'big bang nucleosynthesis consistency' [49], such that where S depends on the number of neutrino species The total parameter set we use is where Ω DM h 2 is the present physical dark matter density relative to the critical density, τ is the reionization optical depth, θ A is the angular size of the acoustic scale at last scattering, n s is the spectral index of the power spectrum of initial fluctuations, and A s is the amplitude of the initial curvature power spectrum at k p = 0 . 05 Mpc -1 . All other parameters, including the Hubble constant H 0 = 100 h km / s / Mpc, the present total matter density Ω m , the dark energy density Ω DE , and the amplitude of the matter power spectrum today σ 8 , can be derived from this set. We will study four different cases: (1) f ν = 0 and N eff = 3 . 046, i.e. standard flat ΛCDM, (2) f ν is allowed to vary, N eff = 3 . 046, (3) N eff is allowed to vary, f ν = 0, and (4) both f ν and N eff allowed to vary. For a given set of parameters θ that defines the cosmological model class in question, we use the CosmoMC code [50, 51] to sample from the joint posterior distribution, where L ( x | θ ) is the likelihood of the dataset x given the model parameters θ and P ( θ ) is the prior probability density. For the standard ΛCDM parameters we use the same priors as in [38] (flat priors that are wide enough to not limit our constraints), and we similarly choose wide priors for f ν and N eff that are informed by the current limits from data as summarized in Refs. [18, 52]. In particular we choose the extremely conservative prior 1 . 047 < N eff < 10 . 0, as in the SPT analyses.", "pages": [ 2, 3 ] }, { "title": "C. Weak lensing observables", "content": "We can compute the posterior probability distribution for any derived statistic from the joint posterior distribution of the cosmological parameters. In particular, in order to compute the cosmic shear power spectrum, we must first compute the comoving angular diameter distance D and the nonlinear matter power spectrum ∆ 2 NL . In a flat universe (curvature Ω K = 0), the former is equal to the comoving radial coordinate and is related to the cosmological parameters through Here the Hubble expansion rate is where the total matter density is and the contribution from radiation is assumed to be negligible. We compute the z = 0 linear matter power spectrum ∆ 2 L ( k ; 0) using CAMB. The linear matter power spectrum at earlier redshifts then depends on the growth function of linear density perturbations. Massive neutrinos suppress growth in a scale-dependent manner, and we model this using the Eisenstein and Hu [53] fitting function, where T ( k, z ) and D 1 ( z ) are given by their Eqs. (7) and (8), respectively, which we have modified according to Ref. [54] to improve accuracy in the case of three massive neutrinos. Note that D 1 ( z ) corresponds to the standard scale-independent growth function in the absence of neutrinos, and the scale-dependent effects of their freestreaming are encoded in T ( k, z ). Comparing to results from CAMB for nonzero redshifts, this fitting formula typically reproduces the growth to better than 1% for all k and z we use here to compute the shear. We compute the full nonlinear matter power spectrum at a given redshift using the Halofit fitting function [55] (see Ref. [38] for a summary), modified for the effects of massive neutrinos [52]. The original Halofit fitting functions have been found to only be accurate (even for the flat ΛCDM model) at up to the 5-10% level compared with N -body results, for instance with the Coyote Universe project [56-58]. We find that whether or not we use the massive neutrino modification [52] leads to errors of order a few percent for the small neutrino masses considered here. These systematic errors are smaller than the statistical errors arising from our data sets in the same regime [38], and so we expect our results to be fairly robust to them. Likewise, for a wide range of baryonic effects, systematic shifts are at most comparable to current statistical errors [38]. The shear (or equivalently the convergence) power spectrum is then equal to where k ≈ l/D in units of Mpc -1 in the Limber approximation and we have defined the geometric lensing efficiency factor The efficiency factor weights according to the source distribution in a given survey, n ( z ), normalized such that ∫ ∞ 0 n ( z ) dz = 1. Here we use the model with parameters ( z 0 , α, β ) = (0 . 555 , 1 . 197 , 1 . 193) for a simplified model ground-based survey, such as CFHTLS or DES, with an approximate median redshift of 0 . 8.", "pages": [ 3 ] }, { "title": "III. RESULTS", "content": "The histograms in Fig. 1 illustrate what happens to our predictions for our six flat ΛCDM parameters { Ω b h 2 , Ω DM h 2 , τ, θ A , n s , ln A s , f ν , N eff } when f ν and/or N eff are allowed to vary. All of our results from here on are presented with the following color-coding: We further summarize our constraints on M ν and N eff in Table I for each of these four cases. In the most general case (4), we constrain M ν < 0 . 67 eV at the 95% confidence level and we find N eff = 3 . 71 ± 0 . 35. These constraints are in agreement with the current state-ofthe-art as seen in the literature for cosmological probes, e.g. Ref. [59]. On the other hand, since the M ν constraints are based on the CMB and expansion history measurements rather than growth measurements (see e.g. [60]), they are less robust to generalizations of the flat ΛCDM model, e.g. the addition of spatial curvature (see below). We find that allowing these small neutrino masses does not significantly change any of our parameter constraints, whereas allowing additional sterile neutrino species does. In particular, we see the physical dark matter density Ω DM h 2 is increased while the power spectrum of primordial fluctuations gains an enhancement in both the tilt n s and amplitude A s [61]. In Fig. 2 we show how the degeneracies between N eff and Ω DM h 2 or n s tighten with the addition of the SPT CMB data, which provides several more peaks in the small-scale regime. As has been noted elsewhere, e.g. [59], we find that the inclusion of an H 0 prior strengthens our constraints on N eff . However, we still get meaningful results without it due to our inclusion of SPT data. The damping scale test provides constraints which are independent of low-redshift dynamics, and therefore the specifics of the dark energy model. In Fig. 3 we show how the z = 0 linear matter power spectrum prediction contours shift with the addition of massive neutrinos, additional neutrino species, or both. The top-left panel shows the baseline (i.e. three massless neutrinos) prediction, and the top-right and bottom panels are plotted with respect to the maximum likelihood baseline model prediction, with color-coding as before. For plotting purposes we follow the usual convention of taking P ( k ) = (2 π 2 /k 3 )∆ 2 ( k ), with k in units of h Mpc -1 . We see that endowing neutrinos with mass serves to suppress structure growth, in accordance with conventional wisdom, despite the similar parameter predictions as seen in Fig. 1. We further see that allowing for additional sterile neutrino species serves to enhance structure on small scales. This is because of an N eff -n s degeneracy. A larger N eff suppresses power in the high glyph[lscript] CMBspectrum due to damping which then allows a compensating increase in n s or the highk primordial power spectrum. Given the preference for additional species seen in Table I, we find that this significantly shifts our P L contours up for large k . We show the resulting 2D cosmic shear power spectra in Fig. 4 for our model ground-based weak lensing survey, again with the top-left panel showing the baseline prediction, and the top-right and bottom panels plotted with respect to the maximum likelihood baseline model prediction. In the context of the flat ΛCDM model, an observed deficit of small-scale cosmic shear of between 20 -40% would indicate finite neutrino mass and could not be explained by other currently allowed cosmological parameter variations. An observed excess of 20 -60% would indicate extra neutrino species and comes from the freedom to raise the tilt due to the N eff -n s degeneracy in the CMB. Cosmic shear measurements provide a means of breaking this degeneracy. We illustrate the issue in Fig. 5, where we plot our constraints on the cosmic shear power at l = 1000 vs n s for the case where N eff is varied but f ν = 0. The case where both f ν and N eff are allowed to vary is harder to distinguish in that the two effects can partially compensate for each other. On the other hand, a further breaking of the N eff -n s degeneracy is expected from the Planck survey [62], thereby allowing these mixed cases to be better separated with cosmic shear. It is interesting to note that while the addition of SPT data tightens the predictions for the shear power spectrum it actually weakens and shifts the predictions on the growth function, as we show in Fig. 6 for z = 0. This reflects a mild tension between this data set and the BAOmeasurements caused by its improved measurement of Ω DM h 2 . Unlike the similar tension between BAO and H 0 for the flat ΛCDM model, this tension is not alleviated by allowing N eff (or f ν ) to vary. On the other hand, constraints on the growth function are not the dominant source of error for shear predictions and so this tension is not relevant for our purposes. Beyond the flat ΛCDM model, there are other possibilities that can explain a deficit or excess of small-scale shear. When the dark energy equation of state is generalized to allow quintessence, only a deficit can arise due to the restriction that w ≥ -1 [38]. Hence these cases can masquerade as massive neutrino models unless further information on the shape and redshift dependence of the power spectrum is obtained. Without neutrino number changes, an excess cannot be explained by quintessence and hence would indicate more exotic cosmic acceleration physics with enhanced forces in the dark sector. Again, the expected improvements from the Planck survey will help distinguish between these possibilities. Likewise we have also tested the robustness of these results to dropping the flatness assumption. Allowing for curvature significantly degrades our constraints on M ν , where we find the 95% limits expand to M ν < 1 . 68 eV and M ν < 1 . 88 eV for the cases when f ν only is varied and for when both f ν and N eff are varied, respectively. On the other hand, our constraints on N eff are not significantly different then those of the flat case. We further find that our high-redshift growth function constraints are weakened, but since the confidence contours are still well within the 1% range we find that there is not a significant effect on the shear predictions. Indeed our shear predictions are qualitatively the same.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "IV. DISCUSSION", "content": "Inspired by recent evidence for massive neutrinos and the possibility of additional species, we have provided an analysis of the signatures of such 'nonstandard' neutrino physics on the weak lensing shear power spectrum. By using observations of the expansion history from Type Ia SNe, BAO, and local measures of H 0 , coupled with observations of the CMB, we can predict future structuregrowth observables such as those from weak lensing. From doing so for our four different scenarios - the standard case with three massless neutrinos, three massive neutrinos, any number of massless neutrinos, and three massive neutrinos with any number of massless neutrinos - we can look for signatures that cannot be mimicked by any currently allowed variation in the other parameters of the flat ΛCDM model. We present only results for a representative model ground-based survey here, but the results for any survey configuration will be qualitatively the same. For our most general case, where we vary both neutrino mass and number of species, we find parameter constraints that are consistent with the current literature. Using only distance measures and the CMB, we are able to constrain M ν < 0 . 67 eV at the 95% confidence level and N eff = 3 . 71 ± 0 . 35. Such variations from the standard neutrino parameters allow changes in the predicted shear power spectrum that cannot be mimicked by other flat ΛCDM parameters. For example shear deficit in the 20-40% range would indicate neutrino masses near saturation of current bounds with N eff ∼ 3 whereas shear excess in the 20-60% range would indicate extra neutrino species and lower masses. The latter is because of a partial degeneracy between raising N eff and spectral tilt of the primordial power spectrum n s (see Fig. 5). It has been noted previously that generalizing from the flat ΛCDM model to quintessence can only serve to reduce the matter power spectrum [37, 38], within the context of the standard (three massless) neutrino scenario. Quintessence effects can therefore mimic the deficit predicted by massive neutrinos without further information from the shape and redshift dependence of the power spectrum. A future measurement of excess cosmic shear could provide supporting evidence for extra neutrino species. An excess could also be explained by more exotic acceleration physics that enhances structure growth. Since the neutrino effect comes from the primordial power spectrum, improved measurements from the Planck satellite would help distinguish these options should an excess be found.", "pages": [ 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Bradford Benson, Michael Mortonson, and Kyle Story for useful conversations, and Antonio Cuesta for assistance with BAO data. RAV and WH acknowledge the support of the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and an endowment from the Kavli Foundation and its founder Fred Kavli. WHacknowledges additional support from DOE contract DE-FG02-90ER-40560 and the Packard Foundation. [21] http://www.lsst.org/ . Phys.Rev.Lett. 80 , 5255 (1998), arXiv:astro-ph/9712057 [astro-ph].", "pages": [ 7, 8, 9 ] } ]

[ { "title": "Observational constraints from SNe Ia and Gamma-Ray Bursts on a clumpy universe", "content": "Nora Bret'on and Ariadna Montiel. Departamento de F'ısica, Centro de Investigaci'on y de Estudios Avanzados del I. P. N., Apartado Postal 14-740, 07000 M'exico D.F., Mexico. The luminosity distance describing the effect of local inhomogeneities in the propagation of light proposed by Zeldovich-Kantowski-Dyer-Roeder (ZKDR) is tested with two probes for two distinct ranges of redshifts: supernovae Ia (SNe Ia) in 0 . 015 ≤ z ≤ 1 . 414 and gamma-ray bursts (GRBs) in 1 . 547 ≤ z ≤ 3 . 57. Our analysis is performed by a Markov Chain Monte Carlo (MCMC) code that allows us to constrain the matter density parameter Ω m as well as the smoothness parameter α that measures the inhomogeneous-homogeneous rate of the cosmic fluid in a flat ΛCDM model. The obtained best fits are (Ω m = 0 . 285 +0 . 019 -0 . 018 , α = 0 . 856 +0 . 106 -0 . 176 ) from SNe Ia and (Ω m = 0 . 259 +0 . 028 -0 . 028 , α = 0 . 587 +0 . 201 -0 . 202 ) from GRBs, while from the joint analysis the best fits are (Ω m = 0 . 284 +0 . 021 -0 . 020 , α = 0 . 685 +0 . 164 -0 . 171 ) with a χ 2 red = 0 . 975. The value of the smoothness parameter α indicates a clumped universe however it does not have an impact on the amount of dark energy (cosmological constant) needed to fit observations. This result may be an indication that the Dyer-Roeder approximation does not describe in a precise form the effects of clumpiness in the expansion of the universe. PACS numbers: 98.80.-k, 98.80.Jk, 95.36.+x, 95.35.+d", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "One of the most important problems in modern cosmology is the coincidence in the order of magnitude between the inferred energy density of the source driving the acceleration and the energy density of matter. Since the most evident qualitative change in the late universe is the formation of nonlinear structures, one cannot help noticing that cosmic acceleration and the formation of large structure occur, coincidently, in near epochs. If the recent formation of large structures could be identified as the driver for cosmic acceleration, the coincidence would be explained in a natural way. In fact there are theoretical results that prove that gravitational collapse can lead acceleration, see for example [1]. Moreover, in Refs. [2] and [3], it is argued that the influence of inhomogeneities on the effective evolution history of the universe is encoded in backreaction terms, and that these terms can mimic dark energy; in this context see also Ref. [4] for a review on different approaches to the dynamical aspects of backreaction effects in recent eras. It has also been addressed by several authors (see for instance Refs. [4-6], that the process of averaging and solving the Einstein equations is non-commutative and that maybe if we do it carefully, the accelerated expansion lately observed can be explained without the need of dark energy. On the other hand, most cosmological observations probe quantities related to light propagation such as the redshift, the angular diameter distance, the luminosity distance or image distortion. The prevailing model ΛCDM assumes that this propagation occurs through a homogeneous perfect fluid; but on scales smaller than 100 h -1 Mpc it is not reliable to consider that the universe can be modeled by a FRW cosmology. The incorporation to the FRW model of local inhomogeneities have been addressed by several authors since the sixties, namely, works by Zel'dovich [7], Bertotti [8], Kantowski [9] and, more recently, by Dyer and Roeder [10], again by Kantowski [11] and by Demianski [12]. The general idea is to consider the propagation of light rays in a fluid that may clump. The proportion of clumped matter respect to the homogeneous fluid is measured by the smoothness parameter. Departing from the Sachs expression for a congruence of null geodesics, neglecting twist and shear effects, and including in the matter content the smoothness parameter, they arrive to an equation for the angular diameter distance, that can be connected, via the Etherington relation, to the observable luminosity distance, resulting then the so called Zeldovich-KantowskiDyer-Roeder (ZKDR) luminosity distance. In this paper we consider observations of SNe Ia in the range of 0 . 015 ≤ z ≤ 1 . 414 [13] and GRBs in the range of 1 . 547 ≤ z ≤ 3 . 57 [14] to test the ZKDR distance and constrain the smoothness or clumpiness parameter. For our analysis we employ the Markov Chain Monte Carlo (MCMC) technique, it leads us to obtain better constraints than previous analysis [15]. Furthermore, the GRBs sample is a very improved one with a significative diminished dispersion in the data [14]. From the joint analysis, SNe Ia and GRBs, the best fits are (Ω m = 0 . 284 +0 . 021 -0 . 020 , α = 0 . 685 +0 . 164 -0 . 171 ) with a χ 2 red = 0 . 975, indicating a clumped universe. Despite this fact, the amount of the dark energy (cosmological constant) needed to fit observations is not less than in the homogeneous FRW model. This may be an indication that the Dyer-Roeder approach is missing something in its physical description and should be revised, see Ref. [16] and references therein for a further discussion. In the literature the ZKDR luminosity distance has been probed also with Hubble parameter measurements. Similarly, we include this test using the data reported in Ref. [17], obtaining from the joint analysis (SNe Ia, GRBs and Hubble) the best fits: Ω m = 0 . 275 +0 . 019 -0 . 018 and α = 0 . 821 +0 . 110 -0 . 129 with a χ 2 red = 0 . 974 that improves the previous analysis reported in Ref. [18]. The paper is organized as follows: in Sec. II we briefly sketch how to obtain the ZKDR luminosity distance from the null geodesic congruence in a clumped universe, using an exact solution of a hypergeometric equation; in Sec. III the observational data and statistical method are described. In Sec. IV we discuss the results and finally in the last Section concluding remarks are given.", "pages": [ 1, 2 ] }, { "title": "II. THE ZKDR LUMINOSITY-DISTANCE", "content": "Let us consider the light propagation in the geometric optics approximation. The two optical scalars, expansion and shear, θ and σ , respectively, satisfy the Sachs propagation equations [19], where a dot denotes differentiation with respect to the affine parameter λ and k µ is the null vector field tangent to the light ray. Notice that since we shall address the RW geometry, the Weyl tensor is zero and the shear vanishes. In [20] it is shown that neglecting the Weyl focusing seems to be correct in our universe. The expansion θ is related to the relative change of an infinitesimal surface area A of the beam's cross section by Since the angular diameter distance D A is proportional to √ A , using D A instead of √ A and considering the affine parameter λ as a function of the redshift z , it is obtained (4) where we have used Einstein's equations to replace R αβ in terms of T αβ . A solution of the last expression is related to the angular diameter distance if it satisfies the initial conditions that make the wavefront satisfy Euclidean geometry when leaving the source: where H 0 is the present value of the Hubble parameter and c is the velocity of light. In order to take into account local inhomogeneities in the distribution of matter, Dyer and Roeder [10] considered that the matter density ρ m should be interpreted as the average density of the intergalactic regions where the observed light has traveled. Then to measure this effect they introduced a phenomenological parameter α = (1 -ρ clumps /ρ ) called the clumpiness or smoothness parameter, which quantifies the amount of matter in clumps relative to the amount of matter uniformly distributed. Its maximum value α = 1 corresponds to the homogeneous FRW case, while for a totally clumped universe α = 0; an intermediate value 0 < α < 1 indicating a partially clumped universe. Assuming an energy-momentum tensor of the form T µν = αρ m u µ u ν + ρ Λ g µν , where u µ is the four-velocity of a comoving volume element, ρ m is the matter energy density and ρ Λ = Λ / 8 πG is the vacuum energy density associated to the cosmological constant, Eq. (4) can be rewritten in the form We shall consider a flat (zero curvature), homogeneous and isotropic geometry of Robertson-Walker (RW), including only dark matter (dust) and dark energy components. The corresponding Hubble parameter is given by On the other hand, transforming the affine parameter to the observable redshift by using and an appropriate change of variable, Eq. (6) can be transformed into a hypergeometric equation where the parameter ν is related to the clumpiness parameter, α , by The range for ν is 0 ≤ ν ≤ 2, where ν = 0( α = 1) corresponds to a FRW fluid, while ν = 2( α = 0) to a totally clumped universe. Since (1 -Ω m ) < 1, the variable in Eq. (10) satisfies ζ > 1; thus to make it compatible with the defined range for the argument of the hypergeometric functions, we transform to the inverse argument, ζ -1 < 1, using properties of the hypergeometric functions. To connect the angular diameter distance D A with the observable luminosity distance d L , we use the Etherington relation [21], d L = D A (1 + z ) 2 . Incorporating also the initial conditions, the luminosity distance we used for the observational tests is given by [22], A few comments are in order: In Eq. (1), for the rate of expansion of the null congruence, the Ricci tensor, R αβ , makes null rays converge. Then it is reasonable to expect that diminishing R αβ would result into reducing the convergence of the null bundle, i.e. there would be a dimming of light rays. This effect can be done by diminishing the density of matter ρ m through the introduction of the smoothness parameter, ρ m ↦→ αρ m , 0 ≤ α ≤ 1. The plots in Fig. 1 for the ZKDR luminosity distance versus the redshift for different values of the smoothness parameter, show that this actually happens so: as α is smaller, d L grows faster. Have in mind that it is a naive reasoning, because neither Eq.(1) nor Eq.(3) are linear equations. Notice that in this approach the luminosity distance, Eq. (12), does not totally capture the effect of inhomogeneities in the expansion rate, since in the derivation of d L are assumed the same expressions for H ( z ) and λ ( z ) than in the homogeneous FRW model, Eqs. (7)-(8). However, the effects of local inhomogeneities seem to be more complex to be described only through the smoothness parameter. In this regard, in Ref. [16] it is quantified the density probability distribution as a function of the beamwidth of supernovae and it is discussed a modified version of the DR approximation. It is shown that even for Gpclenght beams of 500 kpc diameter, nonlinear corrections appear to be non-trivial, being not even clear whether underdense regions lead to dimming or brightening of sources (see Ref. [23]). It was already noted by Linder in Ref. [24] that the smoothness parameter is a function of the redshift. This dependence is also considerer in Ref. [25], where a generalized Dyer-Roeder distance is introduced. However as was recognized in Ref. [26], this should be considered carefully because if the smoothness parameter changes with the redshift, the luminosity distance expression will depend on several free parameters with difficult physical interpretation. A more natural dependence of the smoothness parameter with the redshift, coming from weak lensing, is presented in Ref. [26]. In Ref. [27] it was addressed the properties of cosmological distances in inhomogeneous quintessence cosmology. Although the DR approach has been criticized by several authors, the contrast of d L versus observational data so far has not lead to conclusive results, in the sense of totally discarding this model. Therefore it is important to elucidate its validity and scope in the description of the universe. So for the time being, we look for useful information from the simplest approach that considers α being constant.", "pages": [ 2, 3 ] }, { "title": "A. Samples and statistical method", "content": "We have used the updated observational data from Union2.1 supernovae data set reported in Ref. [13] and the distance modulus of long Gamma-ray Bursts reported in Ref. [14] to constrain the two free parameters of the model, the matter density Ω m , and the smoothness parameter ν (related to α through Eq. (11)). In order to do that, we used a Markov Chain Monte Carlo (MCMC) code to maximize the likelihood function L ( θ i ) ∝ exp[ -χ 2 ( θ i ) / 2] where θ i is the set of model parameters and the expression for χ 2 ( θ i ) depends on the dataset used. The MCMC methods (completely described in Refs. [28-30] and references therein) are wellestablished techniques for constraining parameters from observational data. To test their convergence, here we follow the method developed and fully described in Ref. [31]. Additionally, we have tested the ZKDR luminosity distance with the observational Hubble parameter data given in Ref. [17].", "pages": [ 3, 4 ] }, { "title": "B. Supernovae Type Ia (SNe Ia)", "content": "Hitherto we know one of the cosmological observations highly capable to measure directly the expansion rate of the universe are the Supernovae Type Ia. In this work, we use the updated compilation released by the Supernova Cosmology Project (SCP): the Union2.1 compilation which consists of 580 SNe Ia [13]. The Union2.1 compilation is the largest published and spectroscopically confirmed SNe Ia sample to date. Constraints from the SNe Ia data can be obtained by fitting the distance moduli µ ( z ). The distance modulus can be calculated as where µ 0 = 42 . 38 -5 log 10 h , h is the Hubble constant H 0 in units of 100 km s -1 Mpc -1 , and d L ( z j , θ i ) is the ZKDR luminosity distance given by Eq. (12); θ i denotes the vector model parameters (Ω m , ν ). The χ 2 function for the SNe Ia data is given by where the σ 2 µ corresponds to the error on distance modulus for each supernova. The parameter µ 0 in Eq. (13) is a nuisance parameter since it encodes the Hubble parameter and the absolute magnitude M , so it is more convenient to marginalize over it. Here, we maximize the likelihood by minimizing χ 2 with respect to µ 0 , as suggested in [32, 33]. Then one can rewrite the χ 2 function as where The minimization over µ 0 gives µ 0 = c 2 /c 3 . So the χ 2 function takes the form Since ˜ χ 2 SN = χ 2 SN ( µ 0 = 0 , θ i ) (up to a constant), we can instead minimize ˜ χ 2 SN which is independent of µ 0 .", "pages": [ 4 ] }, { "title": "C. Long gamma-ray bursts (LGRBs)", "content": "Previous analysis have shown that the ZKDR distance at short distances (redshifts in the range 0 . 1 ≤ z ≤ 1 . 7) does not make a difference respect to the ΛCDM model; it is then appealing to test the model for larger distances, like the ones of the GRBs, that extend the redshift range as far as 8.1. However, the problem is that GRBs appear not to be standard candles and to extract cosmological information it is necessary calibrate them for each cosmological model tested. There have been several efforts to calibrate the correlations between the luminosity and spectral properties of GRBs in a cosmology-independent way and some proposals to use SNe Ia measurements to calibrate them externally are given in Refs. [34-36]. More recently, in Ref. [14], it has been estimated the distance modulus to long gamma-ray bursts (LGRBs) using the Type I Fundamental Plane, a correlation between the spectral peak energy E p , the peak luminosity L p , and the luminosity time T L ≡ E iso /L p , where E iso is the isotropic energy. Basically the calibration was done in this way: first, the Type I Fundamental Plane of LGRBs was calibrated using 8 LGRBs with redshift z < 1 . 4 and SNe Ia (Union2) in the same redshift range by a local regression method, to avoid any assumption on a cosmological model; then this calibrated Type I Fundamental Plane was used to measure the distance modulus to 9 high-redshift LGRBs (see Ref. [14] for calibration's details). We used the 9 calibrated LGRBs reported in Table 2 of Ref. [14] to derive constraints on Ω m and ν from the ZKDR luminosity distance. The χ 2 function for the GRBs data is defined by where µ ( z j ) = 5log 10 [ d L ( z j , θ i )] + 25. Notice that we have used the standard expression of χ 2 given through the observed distance moduli just to be consistent with the way in which the calibration was done. We have fixed H 0 as 73 . 8 ± 2 . 4 from Ref. [37].", "pages": [ 4 ] }, { "title": "D. Hubble parameter observations", "content": "We use the compilation of Hubble parameter measurements estimated with the differential evolution of passively evolving early-type galaxies as cosmic chronometers, in the redshift range 0 . 09 ≤ z ≤ 1 . 75 recently updated in Ref. [17] but first reported in Ref. [38]. This approach consists in the measurement of the differential age evolution of these chronometers as a function of redshift, in this way a direct estimate of the Hubble parameter H ( z ) = -1 / (1 + z ) dz/dt glyph[similarequal] -1 / (1 + z )∆ z/ ∆ t is obtained, providing a reliable differential quantity ∆ z/ ∆ t , with many advantages in minimizing common issues and systematic effects. Compared with other techniques, it provides a direct measurement of the Hubble parameter, and not of its integral, in contrast to SNe Ia or angular/angle-averaged BAO. Observed values of H ( z ) can be used to estimate the free parameters of the model and also the best-value for H 0 by minimizing the quantity where σ 2 H are the measurement variances. H 0 has been fixed from Ref. [37], H 0 = 73 . 8 ± 2 . 4. The vector of model parameters, θ i , in our case will be θ i = (Ω m , ν ). Noteworthy that constraining the free parameters of this model via H ( z ) is controversial because this is not a direct comparison with the ZKDR luminosity distance, but we have calculated H ( z ) from the expression connecting in this way H ( z ) with the parameter ν (or α ). We note that we are mixing the homogeneous FRW H ( z ) with the inhomogeneous proposal, however a joint analysis with other probes makes sense and has been addressed in previous works [18].", "pages": [ 5 ] }, { "title": "IV. RESULTS", "content": "The results of our analysis are shown in Tables I-VIII. We made the analysis with the parameter ν , but in several tables are included the results for both parameters, ν used by Kantowski, and α related with ν by Eq.(11). In Tables I and II are displayed the best fits for Ω m , ν (or α ) from the SNe Ia and GRBs probes reported in Ref. [13] and Ref. [14], respectively. Table I corresponds to the analysis assuming a Gaussian prior on Ω m = 0 . 266 ± 0 . 029 from WMAP-7 years [39]. Fig. 2 shows the corresponding contour levels in the (Ω m , ν ) plane. Table II is the same as Table I but without assuming any prior. The inclusion of a prior does not have a dramatic effect, but there is an improvement in the GRBs analysis, that goes from a χ 2 red = 0 . 84 without prior to χ 2 red = 0 . 877 when the prior is assumed. The values of the smoothness parameter show a degree of clumpiness in the cosmic fluid, however it does not have an impact on reducing the proportion of dark energy, that remains over the 70% of the total density; more precisely Ω Λ = 1 -Ω m = 0 . 716 +0 . 020 -0 . 021 from the joint analysis of SNe Ia and GRBs. These results make us think that the DR approach does not describe the effects of clumpiness in a reliable way. Tables III and IV correspond to the 2 σ confidence level constraints of the cosmological parameters, Ω m and α with and without the prior on Ω m , respectively. Table IV, showing the 2σ confidence level for Ω m and α without assuming a prior on Ω m , can be directly compared with the results given in Ref. [15] (that we include for completeness in table V). In Ref. [15] it is confronted the ZKDR luminosity distance versus SNe Ia and GRBs probes with the samples data 557 SNe Ia Union2 [40] and the 59 Hymnium GRBs [41]. Using a χ 2 minimization, their results, shown in Table V, did not constrain the smoothness parameter. The best fits obtained by combining the SNe Ia and GRBs probes are Ω m = 0 . 27 and α = 1, with a χ 2 = 568 . 36 or χ 2 red = 0 . 927. Particularly, for GRBs, using a better sample and the MCMC method, we obtained a reliable constraint with a χ 2 red = 0 . 877 from GRBs only, and χ 2 red = 0 . 973 from the joint analysis without assuming the prior on Ω m and χ 2 red = 0 . 975 from the joint analysis assuming the prior on Ω m . In a previous work done by Busti et al. [18], a chi- m square analysis was performed with Hubble parameter measurements reported in Ref. [42] and in Ref. [43]. Our analysis provides better constraints, that are summarized in Table VI; while in Table VII we make the comparison between the results obtained in Ref. [18] and ours. Considering that the samples of SNe Ia and GRBs correspond to disjoint intervals of redshift, the different values of α for each probe can be interpreted as a dependence of the clumpiness parameter α on the redshift (see Table VIII).", "pages": [ 5, 6 ] }, { "title": "V. CONCLUSIONS", "content": "We probed the ZKDR luminosity distance proposed by Dyer and Roeder (DR), that includes the effect of local inhomogeneities through the smoothness parameter α , with data from SNe Ia [13] and GRBs [14]. Additionaly we tested d L with the direct Hubble measurements reported in Ref. [17]. Better data and a refined statistics for GRBs allow us to probe the ZKDR luminosty distance in a range of higher redshifts with confident results. Indeed, our analysis improves, in great deal, previous works that did not succeed in constraining the smoothness parameter. We can summarize the presented results as follows: 1) We have probed in a reliable way two distinct redshift ranges, in fact two disjoint intervals of z : the SNe Ia range, 0 . 015 ≤ z ≤ 1 . 414 and 1 . 547 ≤ z ≤ 3 . 57 for GRBs. The obtained results, two different values of α in each interval, strongly suggest a dependence of the clumpiness parameter on the redshift, see Table VIII for the values of α and the corresponding redshift range. Since the redshift intervals are disjoint, the different values of α show a kind of evolution in the smoothness parameter with respect to the redshift. In principle, assuming that this approach describes the clumpiness effects with accuracy, one would expect that nearby regions should correspond to more clumpiness, then to smaller values of α . In other words, the structure formation process leads to a more locally inhomogeneous universe and thus, the smoothness parameter should evolve from homogeneity ( α = 1) to total clumpiness ( α = 0). However, the test turns out on the contrary: the smoothness parameter evolves toward homogeneity, being α = 0 . 587 +0 . 201 -0 . 202 for the range 1 . 547 ≤ z ≤ 3 . 57 and a greater value, α = 0 . 856 +0 . 106 -0 . 176 for the SNe Ia range, 0 . 015 ≤ z ≤ 1 . 414. These results show that the Dyer-Roeder equation for d L should be revised, as several authors have suggested [16]. 2) Our results indicate the existence of clumpiness, because the values of α differ from the homogeneous FRW model ( α = 0 . 856, α = 0 . 587 and α = 0 . 895 from SNe Ia, GRBs and Hubble, respectively), however this does not have a clear impact in the amount of dark energy needed to fit the observations. According to SNe Ia, Hubble and GRBs probes, the amounts of dark energy that fit to observations are: Ω Λ = 0 . 715 +0 . 018 -0 . 019 , Ω Λ = 0 . 732 ± 0 . 023 and Ω Λ = 0 . 741 ± 0 . 028, respectively (with prior on Ω m ); in other words, the average matter density increases in the range of small z (Ω m = 0 . 285 +0 . 019 -0 . 018 , from SNe Ia compared with the value in the range of GRBs, Ω m = 0 . 259 +0 . 028 -0 . 028 ). In conclusion, our results indicate that DR approach models in an incorrect way the effect of local inhomogeneities in the propagation of light. In Section II we have noticed that the model is formulated with at least two questionable assumptions: that the Hubble parameter and the affine parameter as a function of the redshift have the same expressions than in the homogeneous FRW model. So far, the evidence indicates that a modification of the Dyer-Roeder equation is necessary to describe in a reliable way the backreaction effects due to local inhomogeneities. It is worth then to test with observational data some of the improved models that try to mend the drawbacks of the DR model; for instance in Ref. [26] it is considered a correction in the analogous of the smoothness parameter coming from density fluctuations along the line of sight.", "pages": [ 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "A. M. acknowledges financial support by CONACyT (Mexico) through a PhD scholarship; partial support from CONACyT-Mexico through project 166581 is acknowledged.", "pages": [ 7 ] } ]

[ { "title": "Quasiperiodic oscillations and Tomimatsu-Sato δ = 2 space-time", "content": "Ivan Zh. Stefanov 1 ∗ , Galin G. Gyulchev 2 † , Stoytcho S. Yazadjiev 3 ‡ 1 Department of Applied Physics, Technical University of Sofia 8, St. Kliment Ohridski Blvd., 1000 Sofia, Bulgaria Department of Physics, Biophysics and Roentgenology, Faculty of Medicine, St.Kliment Ohridski University of Sofia 1, Kozyak Str., 1407 Sofia, Bulgaria 3 Department of Theoretical Physics, Faculty of Physics, St.Kliment Ohridski University of Sofia 5, James Bourchier Blvd., 1164 Sofia, Bulgaria June 26, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "We model the spacetime of low-mass X-ray binaries with the Tomimatsu-Sato δ = 2 (TS2) metric and study the properties of the orbital and the epicyclic frequencies. The numerical analysis shows that the properties of the characteristic frequencies of oscillation do not differ qualitatively from those of the Kerr black hole. Estimates for the angular momenta of the three stellar mass black hole candidates GRO 1655-40, XTE 1550-564 and GRS 1915+105 are made with the application of the nonlinear resonance model. We find agreement between the predictions based on the 3 : 2 nonlinear resonance model for a TS2 background and the current estimates found in the literature.", "pages": [ 1 ] }, { "title": "PACS numbers:", "content": "Keywords: Tomimatsu-Sato space-time, quasiperiodic oscillations, epicyclic frequencies, microquasars, black holes, singularities, angular momentum", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Quasiperiodic oscillations (QPO) in the X-ray flux from low-mass (galactic) binaries has attracted considerable research interest recently due to their potential to be applied as a tool for probing the space-time around black holes and massive stars and testing gravity in strong field regime [1, 2, 3, 4, 5]. One of the major possible application of QPOs which makes them so attractive is the measurement of the angular momentum of the central object in X-ray binaries. There are very few methods for this measurement [6]: continuum fitting of thermal spectra [7, 8, 9], determining the shape of the gravitationally redshifted wing of Fe line[10] and QPOs [11]. Three types of QPO are usually observed: low-frequency QPOs (few tens of Hertz), intermediate frequency QPOs and high frequency QPOs (up to kilo Hertz). While the former two types of QPOs are believed to have astrophysical origin, i.e. they are related with the physics of the accretion disc, the high frequency QPOs are believed to depend on/reflect the structure of space-time in the vicinity of the central object - a black holes or a neutron star. Some properties of the QPOs support that idea: high-frequency QPOs depend very weakly on the X-ray flux and in several sources the frequencies of the QPOs occur in pairs whose ratio is 3 : 2. The physical mechanism of the production of these QPOs is not know. One of the major hypothesis is proposed by the nonlinear resonance model (NRM) according to which QPOs are related to the characteristic frequencies of oscillation - epicyclic and orbital - of test particles orbiting around the central object and the appearance of nonlinear resonances between them. This idea has initially been proposed by Aliev and Galtsov [14, 15]. It has reappeared and further elaborated in the works of Abramowicz, Kluzniak and collaborators [16, 17, 18]. Simplicity makes the nonlinear resonance model very attractive, however, it has some significant deficiencies [20]. The model does not propose an excitation mechanism for the QPOs. The origin of the coupling between the different characteristic frequencies is not clear. The effect from astrophysical complication such as turbulence flow and magnetic instability have not been assessed and cannot be simply included in the model. All QPO models are purely dynamical and do not concern the emission mechanism of X-rays. A major difficulty of the NRM is the dissonance between its predictions for the angular momenta of the observed black holes candidates and the measurements based on spectral continuum fitting [19, 20]. Non of the different versions of the NRM can explain the observed angular momenta of the three black-hole candidates GRO 1655-40, XTE 1550-564 and GRS 1915+105. In the current paper we will address only the last one of the problems. According to Bambi [12] the dissonance between the predictions of the NRM and the measurements for the angular momenta mean that one of the following four possibilities must be true: i ) The NRM is wrong ii ) continuum-fitting method does not provide reliable estimates of the angular momenta iii ) both techniques do not work correctly iv ) the central objects of the observed stellar-mass black hole candidates are not Kerr black holes. Here we will consider only the fourth possibility. The effect of deviations from Kerr space-time on the QPOs in the frame of RNM has already been studied in several papers [13], [12], [5], [21]. In all of these cases the deviation from Kerr space-time is described by an additional continuous free parameter ( /epsilon1 , tidal charge, etc.) which leads to degeneracy of the results in the sense that for fixed of the central object mass and value of the deviation parameter there is a whole interval of values of the angular momentum which are admissible by the observations. Due to this degeneracy, in order to be able to determine the value of the deviation parameter we need independent measurements of the mass and the angular momentum. One of the few exact solutions that describe the exterior field of a stationary rotating axially symmetric objects is the Tomimatsu-Sato solution. The deviation parameter δ in this solution takes discrete values ( δ = 1 corresponding to Kerr solution δ = 2 corresponding to an exotic object) which allows the degeneracy to be avoided . The aim of the present paper is to apply the NRM to the space-time of Tomimatsu-Sato with δ = 2 (TS2), to study the properties of the characteristic frequencies and to check if the dissonance problem can be resolved in such background. The paper is organized as follows. The Tomimatsu-Sato spacetime is briefly presented in Section 2. The formulas that we use for the calculation of the epicyclic frequencies are obtained in Appendix A and applied in Section 3. In Section 4 NRM model is sketched. The results for the estimates of the angular momenta of the three black hole candidates are in Section 5. The sum-up of the results is in Section 6", "pages": [ 2, 3 ] }, { "title": "2 Tomimatsu-Sato space-time", "content": "The Kerr-Tomimatsu-Sato family is usually presented in the canonical Weyl-Papapetrou form of the metric for stationary, axisymmetric space-time[22] where [23] The functions A, B and C are polynomials of the prolate spheroidal coordinates x and y defined by In explicit form for the Kerr space-time δ = 1 these polynomials are In the Tomimatsu-Sato case δ = 2 , A, B and C are given by The Kerr-TS family has three free parameters σ, p and q which are related to the mass M and the angular momentum J of the central object as The properties of the TS2 space-time have been studied in several papers [24, 28, 29, 30]. A simple sketch of the structure of TS2 can be found in [31]. We will just briefly mention its main properties. The TS2 space-time has two spherically symmetric Killing horizons symmetrically distributed with respect to the equatorial plane at ρ = 0 and z = σ ( x = 1 and y = 1) connected by a rod-like conical singularity which carries all the gravitational mass. The Killing horizons are everywhere regular except at the points of connection with the rod-like singularity. In the equatorial plane there is a ring singularity at the roots of B ( x, y = 0) = 0 with zero Komar mass and infinite circumference. The metric component g φφ diverge on the ring singularity and is negative between the ring singularity and the central rod-like singularity. The latter means that there is causality violation and closed time-like curves in that region. The formula for the quadrupole momentum of the Kerr-Tomimatsu-Sato family has been correctly derived in [30] Due to its peculiar causal structure the TS2 space-time is discarded as a candidate to describe the final stage of gravitational collapse. The more appealing physical interpretation is the one given by Gibbons and Russell-Clark [24] that it describes the exterior field of a specific star whose prolateness is bigger than that of the Kerr black hole. The stability of TS2 has not been studied up to now.", "pages": [ 3, 4 ] }, { "title": "3 Characteristic epicyclic and orbital frequencies", "content": "The epicyclic frequencies of relativistic compact objects to our knowledge have been derived for the first time in [14]. General formulas for the vertical and radial epicyclic frequencies can be found in [25] and [26]. In appendix A of the current paper we present a slightly different derivation. In this derivation the decoupling of the vertical and the radial modes is not assumed but rather follows from the reflectional symmetry of the background metric (For more details on reflectional symmetry see [27].). The vertical and radial epicyclic frequencies, respectively, are ν 2 ρ = D ρ ρ / (2 π ) and ν 2 z = D z z / (2 π ). The matrix D is defined in formulas (29) and (30) form Appendix A. For the particular background metric, the TS2 spacetime, the off-diagonal components vanish, D z ρ = 0 and D ρ z = 0. For the Keplerian frequency Ω K we apply the formula Plus '+' refers to prograde orbits and minus ' -', to retrograde. For the calculation of the characteristic frequencies of the TS2 space-time we apply a package linear algebra in Maple . The formulas are too cumbersome to be displayed explicitly. In some of the particular cases in which the resonance model was applied that we cited in the Introduction it was found that the characteristic frequencies may have rather odd behavior qualitatively different from the case of Kerr black hole. Vertically unstable modes have been reported in [21, 12], multiple (rather than one) extrema of the radial frequencies - in [32, 21], violation of the ν k > ν θ > ν r inequality, where ν θ and ν r are the frequencies of the vertical and the radial modes respectively, has been shown to exist in [12, 32, 21], absence of radially unstable modes has been found in [21]. Unlike the cases discussed above there are no peculiarities in the properties of the epicyclic frequencies of the TS2 space-time. Qualitatively their behavior is the same as in the case of Kerr black hole. The radial frequency ν ρ has only one extremum (maximum). It increases as the singularity is approached, passes through a maximum and becomes zero at ρ = ρ ISCO 1 before the singularity is reached. The frequency ν ρ is presented on the left panel of figure (1) for q = -0 . 9; 0; 0 . 9. The same behavior is observed for the whole interval of admissible values of q . As we can see from the representative cases shown on the figure for positive angular momenta of the central object the frequencies are higher than for the negative angular momenta. The vertical epicyclic frequency ν z and the orbital frequency ν K increase monotonically as the singularity is approached (See the right panel of figure (1) and figure (2) ). For these two frequencies the effect of q is opposite in comparison to the radial frequency case. Negative angular momenta invoke higher frequencies while positive angular momenta - lower frequencies. All three frequencies ν ρ , ν z and ν K are displayed on figure (3) for q = 0 . 9. For corotating orbits ν ρ < ν z < ν K . As we have already mentioned the TS2 space-time is a significant deformation of Kerr space-time. The properties of these two space-times differ significantly even when q = 0. On figure (4) ν ρ for q = 0 is shown and as it can be seen ν ρ is lower in the TS2 case. The radius of the TS2 ISCO is higher than the one of Kerr. The differences of ν z and ν K in the two cases are indistinguishable on the graphics.", "pages": [ 4, 5 ] }, { "title": "4 Nonlinear resonance model for high frequency QPOs", "content": "According to the NRM resonances occur at the orbits for which the relation is satisfied for integer ' m ' and ' n '. The values of ' m ' and ' n ' vary in the different resonance models. When ' q ' is fixed (9) can be solved for the radii of the resonance orbits r nm . For the observed twin peak high-frequency QPOs the upper ν U and the lower ν L frequency are in 3 : 2 ratio so within the model they are expressed in the following way where the epicyclic frequencies are evaluated at the resonance orbits. The small integers { m 1 , m 2 , n 1 , n 2 } are chosen appropriately so the ν U : ν L = 3 : 2 ratio is satisfied. For the forced oscillation model ( m,n ) = (3 , 2). In the parametric resonance model either ( m,n ) = (3 , 1) or ( m,n ) = (2 , 1). In the keplerian resonance model the resonance condition is, respectively, where ( m,n ) = (3 , 2); (3 , 1); (2 , 1). The upper and the lower twin peak QPO frequencies are Again the radial epicyclic frequency and the orbital frequency are evaluated at the resonance orbits.", "pages": [ 5, 8 ] }, { "title": "5 Estimates for the angular momenta of black-hole candidates in microquasars", "content": "In this section we apply different nonlinear resonance models to estimate the angular momenta of the central object in three microquasars GRO 1655-40, XTE 1550-564 and GRS 1915+105. For these black holes candidates we have also dynamical measurement of the mass. For all three of them two high-frequency QPOs have been observed and the ratio between the upper ν U and the lower frequency ν L is 3 : 2 (See Table. 1 ). Two high-frequency QPOs have been observed also in the X-ray binary H 1743-322, however, there is no measurement for its mass. The results from the numerical calculation for the estimates of the angular momenta of the microquasars based on the models in which nonlinear resonance occurs between the radial and vertical epicyclic frequencies are presented in Table. 2 and for resonance between the radial epicyclic frequency and the orbital frequancy - in Table. 3. In general we can say that when the central object of the microquasar is modeled by the central object of TS2 space-time the estimates for the angular momenta are lower than in the case when the central object is modeled by a Kerr black hole. The estimates in the latter case can be found in [17]. One of the most significant problems of the nonlinear resonance model is the discrepancy between the estimates for the angular momenta based on the observed QPOs and the estimates obtained thought analysis of the observed thermal spectra of the objects' accretion disks. In non of the resonance models there is agreement between QPOs and thermal spectra for all three microquasars. For example, the measured value of angular momentum of GRS 1915+105 coming from the continuum-fitting method is in agreement with the 3 : 2 epicyclic resonance model but the value predicted by the same resonance model for GRO 1655-40 does not match the spectral fitting estimate. It would be interesting to know whether such discrepancy is still present when TS2 solution is used to describe the space-time in the vicinity of the microquasar. In order to be able to legitimately compare between the estimates of the resonance model of QPOs and those of the thermal spectra fitting the experimental data in the later case should be reanalyzed with TS2 background. In the absence of such analysis we could compare the estimates of the resonance model for TS2 with the estimates of the spectral continuum-fitting and other methods for Kerr. The continuumm-fitting method [8] gives the following values for GRO 1655-40, a ∈ (0 . 65 ÷ 0 . 75). The angular momentum of XTE 1550-564 has been estimated in [9] to be a ∈ (0 . 29 ÷ 0 . 52). These values have been obtained through the combination of the results from the continuum-fitting method and the Feline method. In literature two conflicting estimates for GRS 1915+105 can be found (See also [36]). The first one is given in [7] a > 0 . 98. The estimate given in [37] is a ∼ 0 . 7. From tables 2 and 3 we can see that the 3 : 2 Keplerian resonance is consistent with these values for all three microquasars if the alternative estimate for the angular momentum of GRS 1915+105 [37] is chosen.", "pages": [ 8, 9 ] }, { "title": "6 Sum-up", "content": "In the current paper we have studied the properties of the three characteristic frequencies of a particle propagating alone a circular orbit in the in the equatorial plane of Tomimatsu-Sato space-time - the radial ν ρ and the vertical ν z epicyclic frequencies and the orbital (Keplerian) frequency ν K . For the calculation of the frequencies we have followed a variation of the method proposed [14]. The formulas are presented in appendix A of the current paper. The numerical analysis shows that for the whole interval of admissible values of the free parameter q corresponding to angular momentum per unit mass the quantitative behavior of the frequencies are the same as in the case of Kerr black hole. We have also applied the resonance mode to estimate the angular momenta of three microquasars GRO 1655-40, XTE 1550-564 and GRS 1915+105. The central object here is modeled by the central object in the Tomimatsu-Sato δ = 2 space-time. The results are presented in Table. 2 and Table. 3. The estimated angular mementa of the three microquasars are considerably lower than in the case when the central object is modeled by a Kerr black hole. The 3 : 2 keplerian NRM seems to be in agreement with the some of the current estimates for the angular momenta of three black hole candidates. However, future studies should take into account that the continuum-fitting method is model dependent and in the cases cited here it was supposed that the central object is a Kerr black hole. In conclusion, the results presented here do not exclude the Tomimatsu-Sato δ = 2 space-time as a model of the space-time of microquasars.", "pages": [ 10 ] }, { "title": "A Derivation of the formula for the epicyclic frequencies", "content": "We are interested in circular orbits in space-times with two Killing vectors ξ t and ξ φ - stationary (or static in particular) and axially symmetric . Due to the invariance of the space-time with respect to translations in time and rotations with respect to an axis the law of conservation of energy and the law of conservation of angular momentum hold for point particles. We will use adapted coordinates in which the two Killing vectors have the form ξ t = ∂/∂t and ξ φ = ∂/∂φ . The solutions whose QPOs we will study are presented Weyl-Papapetrou ( t, ρ, z, φ ) or Boyer-Lindquist ( t, r, θ, φ ) coordinates. The derivation of the epicyclic frequencies presented here is valid also if prolate spheroidal coordinates are used to present an axially symmetric solution. By flat orbits here we mean orbits that remain in a plane and in particular in the equatorial plane. In the coordinate systems we have chosen these orbits are described as ρ = const and z = const or alternatively r = const and θ = const. For convenience we will adopt the following notation. Small Latin letters a, b, c, d, s, m, n.... will be used to denote the r and θ (respectively the ρ and z ) coordinates. Capital Latin letters A, B, C, D, S, M, N.... represent the other two coordinates, those related with the energy and angular momentum conservation laws, t and φ respectively. We will also term them cyclic coordinates since the lagrangian of point particles does not depend on them. Greek letter will denote all coordinates α, β, γ, δ, σ, µ, ν... = t, r, θ, φ ( t, ρ, z, φ ). In that notation flat, circular orbits are expressed as ˙ x a = 0 where the dot denotes the derivative with respect to the affine parameter of the orbits λ . Due to stationarity and axial symmetry for the metric we have g µν = g µν ( x a ). It can be proved that all quantities that are function of the x a coordinates only remain constant on the chosen type of orbits. Let f = f ( x a ). Then since ˙ x a = 0. In particular For the particular case of flat, circular orbits x a = 0 while in general for the cyclic coordinates x A = 0 holds. So Due to (16) and (17), (14) can be rewritten as Taking into account that we can easily show that some of the Christoffel symbols vanish. /negationslash /negationslash /negationslash /negationslash We have taken into account that g AB = 0 for any A and B , g ab = 0 for a = b , g Ab = 0 = g aB and for the inverse metric g AB = 0 for A any B , g ab = 0 for a = b , g Ab = 0 = g aB . /negationslash /negationslash The M components of (14) simplifies considerably. From (18) and (20) so Setting the constant to zero is equivalent to choosing the difference between initial phases of δ ˙ x M and δx σ to be π/ 2. Since ( d/dt ) = ˙ t -1 ( d/dλ ) from (22) it follows that Ω K is the Keplerian frequency L and E are the angular momentum and the energy of the particle, respectively. Now let us obtain the equation that governs the m component of the perturbations It contains both the δx m and the δx P perturbations. The δx P can be decoupled if (22) is used Finally, where we have introduced the following matrix Dividing the upper equation in ˙ t we obtain or In the general case the horizontal ( r or ρ ) and the vertical ( θ or z ) perturbations are coupled. When the 'elasticity' matrix D is diagonal the epicyclic frequencies are No summation over the repeated index m is implied.", "pages": [ 10, 11, 12 ] }, { "title": "Acknowledgements", "content": "This work was partially supported by the Bulgarian National Science Fund under Grant No DMU 03/6.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "MIFPA-12-39 November 2012", "pages": [ 1 ] }, { "title": "Inflection Points and the Power Spectrum", "content": "Sean Downes and Bhaskar Dutta Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "Inflection point inflation generically includes a deviation from slow-roll when the inflaton approaches the inflection point. Such deviations are shown to be generated by transitions between singular trajectories. The effects on the power spectrum are studied within the context of universality classes for small-field models. These effects are shown to scale with universality parameters, and can explain the anomalously low power at large scales observed in the CMB. The reduction of power is related to the inflection point's basin of attraction. Implications for the likelihood of inflation are discussed.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Inflation is a leading theoretical explanation for the origin of our universe[1-3]. In addition to resolving many longstanding problems with big bang cosmology, it also motivates the observed, nearly scale invariant spectrum of density perturbations. Such primordial perturbations are widely believed to have seeded the formation of structure, which is still being uncovered at the largest scales. As a possible explanation of these perturbations, inflation also offers a quantitative description of the observed temperature anisotropies in the cosmic microwave background (CMB). Inflation utilizes a scalar field to drive exponential expansion of the spacetime metric. Quantum perturbations of an accelerating background are violently stretched to cosmic scales, where they exit the particle horizon, become nondynamical and wait for the horizon to catch up. The once-quantum fluctuations reenter the horizon as a random gravitation potential for the constituents of the cosmic fireball - the aftermath of the big bang. The density perturbations can therefore be seen today as thermal fluctuations in the CMB. It is the detailed study of these ancient photons that has made the case for inflation particularly compelling[4]. Despite being a successful theoretical paradigm, inflation has resisted a concrete embedding into known physical theories. At present, it seems that neither string theory nor particle physics have any distinguished role or a priori need for inflation. Yet there are dozens of incarnations of inflation, each manifested in hundreds of models. Inflation, it would seem, stands embarrassed by its riches. Its agent - the hypothetical scalar field dubbed the 'inflaton' - insists on remaining anonymous. In the past decade there have been attempts to uncover its identity. One major approach involves search for nongaussianities in the CMB [5, 6], formalized through the so-called effective field theory of inflation[7]. The idea being that nonlinear effects in the CMB can be tied to nonlinear terms in the scalar potential, which may shed light on the inflaton's identity. A complimentary approach was recently proposed in [8]. Here, model independent details of how inflation embeds into models derived from string theory and particle physics were codified in a set of universality classes. For the inflection point inflation scenario, this universal behavior was shown to relate phenomena as disparate as density perturbations and supersymmetry breaking. The theoretical technology afforded by V. I. Arnold's Singularity Theory [9] gave rise to scaling relations between physical observables. In this work we investigate the effect of such scaling relations on the linear perturbations. We shall see that universality works in concert with attractor dynamics [10] to leave an observable footprint on the angular power spectrum of the CMB. In particular, we shall see how the inflaton temporarily leaves slow roll, giving rise to an observable reduction in power. Put a different way, nontrivial dynamics of the inflaton can reduce the power at scales where such modes left the horizon. This leads to definite observational consequences. Indeed, at the largest scales accessible there is a sharp, anomalous decrease in power in the CMB. Possible origins for these effects have been studied previously [11-15], but generally in the context of the minimal 'just enough' inflation. This work substantially generalizes this approach. Taken at face value, these considerations are speculative. The azimuthal modes of the power spectrum are taken to be independent measurements. Therefore, large scale harmonic modes means low statistics, and therefore a theoretical limit on the precision of the angular power spectrum. This uncertainty, known as cosmic variance , can accommodate the anomalously low power at large angles. Nevertheless, one may hope that the same physics which yields a thermal power spectrum with low power at large scales may also render a better global fit to the cosmological data. The remainder of this work is organized as follows: in Sec. II we review a few relevant features from slow roll inflation. In Sec. III we describe the systematic analysis of the power spectrum and present the results. In Sec. IV we conclude.", "pages": [ 3, 4 ] }, { "title": "II. SELECTIONS FROM SLOW-ROLL INFLATION", "content": "We begin our discussion by developing a few relevant details from slow-roll inflation. First, we develop the attractor dynamics using the formalism in [10]. Next, we discuss the conditions for the inflaton trajectory to temporarily fall out of slow-roll. In particular, we emphasize the transition from chaotic to inflection point inflation. Finally, we review the 'ensemble of universes' given by the set of all possible trajectories of the gravity-scalar system; it is the framework for performing our systematic analysis of the power spectrum in Section III. This is the basin of attraction in phase space; we demonstrate how it varies with the catastrophe parameters of [8].", "pages": [ 4, 5 ] }, { "title": "A. Slow-roll and other singular trajectories", "content": "We work in natural units, with c = glyph[planckover2pi1] = M P = 1 / √ 8 πG = 1. The field equations which govern the Fridemann-Lemaˆıtre-Robertson-Walker-scalar system are given by, Asubscript φ denotes a partial derivative with respect to φ . Dots corresponds to time derivatives. To separate these coupled equations and explicitly manifest the attractor dynamics, we parametrize time with the (suitably normalized) scalar factor, which leads to relations like where primes denote derivatives with respect to N . The normalization a 0 is the value of the scale factor today. With this parametrization, the field equations can be rewritten, The field equation (4) admits three 'singular' trajectories were the right hand side vanishes. The two solutions, φ ' = ± √ 6, corresponds to kinetic domination of the energy density. These are 'fast-roll' solutions and are typically dynamical repulsors. The other, 'slow-roll' trajectory - call it φ glyph[star] - solves, This is only an approximate solution, since which does not vanish in general. However, the condition that φ glyph[star] approximate a solution to (4) - that φ '' glyph[star] nearly vanishes - agrees with the familiar slow roll conditions, Indeed, the vanishing of (8) is the more precise statement of (9). Despite the failure of φ glyph[star] to solve the field equation, it is still a dynamical attractor, at least so long as φ glyph[star] is less than √ 6. The details of the attractor/repulsor trajectories are understood [16, 17], and were reviewed in this framework in [10]. The small deviations from a full solution to (7) has direct consequences for those φ that do solve (4). The resulting deviation from φ glyph[star] leads to consequences in the power spectrum, and is the principal focus of this work.", "pages": [ 5, 6 ] }, { "title": "B. Deviations from slow-roll", "content": "A convenient parametrization of the deviation from the slow-roll trajectory φ glyph[star] is given by the variable Ξ, Simple algebra reveals that, So Ξ vanishes for all 'singular' solutions of (4), both slow-roll and fast-roll. This is important to bear in mind, as Ξ also enters directly into the mode equations for the linear perturbations, and can therefore impact the power spectrum when it is of order unity. Quantifying this impact with respect to the scaling phenomena observed in [8] is the central focus of this paper. We will discuss this and (11) in more detail in the next section. As φ transitions between different singular trajectories, Ξ can become large. To build an intuition for such effects, we consider three examples of large Ξ: when the φ starts from rest, when φ starts from a fast-roll trajectory, and when it transitions from chaotic to inflection point inflation. All three can may lead to observable effects on the power spectrum. This has been studied, for instance, in [11-13].", "pages": [ 6, 7 ] }, { "title": "Field velocity and Ξ", "content": "We begin with the chaotic inflation scenario. Consider a field slowly rolling in a quadratic potential, V ∝ φ 2 . If φ starts at rest, it must first accelerate towards the slow-roll trajectory, (7). Thus, a nearly vanishing field velocity rarely qualifies as slow-roll. Indeed, if the field starts from rest, Ξ can be quite large as the field rapidly accelerates toward the slow-roll trajectory. When φ ' vanishes, Ξ diverges. This is illustrated numerically in Fig. 1. Alternatively, the inflaton may start near the fast-roll condition, φ ' ≈ -√ 6. In this case, Ξ starts off nearly vanishing, but spikes when the field transitions to slow-roll, as shown in Fig. 2. Near N = 50, slow-roll ends and φ accelerates towards the minimum. These two examples involved chaotic inflation. There are other cases of interest, like a transition between chaotic and inflection point inflation.", "pages": [ 7 ] }, { "title": "Transition from chaotic to inflection point inflation", "content": "Chaotic inflation solves the slow roll conditions (9) with large V , which necessitates large field excursions. Inflection point inflation satisfies (9) by a vanishingly small V φ and V φφ . In this sense, inflection point inflation is a misnomer. V must possess a degenerate critical point; its first derivative must also vanish at an inflection point. Inflation then occurs over a small field excursion in the vicinity of the degenerate critical point. Since the first two derivatives vanish, the Taylor expansion near the 'inflection point' has the form, These two inflationary scenarios are not mutually exclusive. As a result of attractor dynamics, reviewed in the next subsection, the inflaton can transition from a period of chaotic to inflection point inflation. As illustrated in Fig. 3, Ξ spikes as φ approaches the inflection point. This transition from chaotic to inflection point potentials was first studied in [18] and later in [8]. Physically, trajectories close to slow-roll in the chaotic regime must brake abruptly at the inflection point. The 'braking' in φ glyph[star] is too strong to be physical, leading instead to a spike in Ξ. It is an example of the failure of an exact solution φ to match φ glyph[star] mentioned in Sec. II A. The cubic coupling at the inflection point is the leading term near the critical point, as seen in (12). This coupling strongly influences Ξ. As seen in Fig. 4, the amplitude of Ξ decreases inversely to this coupling. In inflection point models, this is the most important contribution to Ξ. Since Ξ can be quite large as the field approaches the inflection point, there is the possibil- ty for an observable effect in the power spectrum. We investigate this in the next section. Before that, we lay the groundwork for a systematic analysis by discussing the inflection point's basin of attraction.", "pages": [ 8, 9, 10 ] }, { "title": "C. The basin of attraction", "content": "The attractor dynamics of slow-roll inflation are well known, particularly in the chaotic inflation scenario [19, 20]. What is less well known is that inflection points are very efficient attractors [10, 21]. We review the basin of attraction for the simplest class of inflection point models; the generalization is straight forward. Models of inflection point inflation fall into universality classes [8], depending on the number of parameters in the potential. The canonical representative of the class of twoparameter models ( A 3 ) is, The condition on the parameters for a degenerate critical point ('inflection point') is This gives an inflection point at φ = α , and a minimum - a vacuum suitable for reheating - at φ = -3 α . After shifting the origin of field space to coincide with the inflection point, what remains is a family of potentials, parametrized by α . Thus, for two-parameter models of inflection point inflation, a solution is specified by a choice of α and an initial point in phase space. For any choice of three such numbers, some solutions will asymptotically approach the inflection point and come to rest. Others will overshoot. Owing to the solitonic nature of these trajectories, the boundary of the basin of attraction in this space is a transcendental function. Despite this, such an analysis can be carried out numerically [10]. For fixed α , the basin of attraction can be seen in Fig 5. GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230> GLYPH<230>GLYPH<230>GLYPH<230>GLYPH<230>", "pages": [ 10, 11 ] }, { "title": "Finite e-foldigs", "content": "So far we have studied idealized potentials with exactly degenerate critical points. The attractor behavior gives rise to infinitely many e-foldings within the basin of attraction described. Physically, inflation must end, so the critical points must not be precisely degenerate. As discussed is [10], this is good, otherwise the subset of viable couplings would be of measure zero. Inflation ends due to the presence of an additional, small term in the potential. In (13), this means a slight shift in b . This can be modelled by a slightly deformed potential, The constant term has been deformed slightly to keep the minima approximately zero. The number of e-foldings associated to the inflection point is approximately [22], Note that arbitrarily many e-foldings of slow-roll inflation may exist prior to the contribution of inflection point. Physically, of course, we are interested in the the point where the observed perturbations in the CMB entered the horizon - last sixty or so e-foldings. So long as λ 1 glyph[lessmuch] α 3 , as is typically required to achieve sufficiently many e-foldings to match observations, the basins of attraction discussed above reasonably models the physically viable solutions. In the following section, we shall focus on solutions within this basin.", "pages": [ 11, 12 ] }, { "title": "Critical Couplings", "content": "We close this section with a technical aside. Once again, we consider the case where λ vanishes. Not all values of α in (14) give rise to a finite basin of attraction. Below a critical value, α C ∼ 0 . 66, the basin is simple a point; overshooting the inflection point is inevitable. This was first observed in the one-parameter ( A 2 ) case [18], and was generalized in [10] which also mapped out the basin of attraction. In the next section we shall exclusive focus on couplings well inside the basin, so that we may sensibly talk about sufficient inflation. Despite that, it is worth pausing to mention the interesting behavior near these critical couplings. We saw in Fig. 4 that α decreases, the amplitude of deviation Ξ grows. This deviation occurs as the field brakes while approaching the inflection point. As α approaches α C from above, the required braking becomes substantial. This is depicted in Fig. 6. With significant breaking comes significant deviation from slow-roll. The deviation finds a lower bound 1 at Ξ = -3, as depicted in Fig. 7. The duration at which Ξ remains near this value increases rapidly near α C . As one might expect, this behavior actually occurs for any point on the boundary of the basin of attraction . From (11), we see that the linear perturbations in this regime obey, Since φ ' ∝ φ ∝ exp( -3 N ) in this regime, this rapidly approaches For perturbations well outside the horizon - small k -the six dominates the last term. This represents an underdamped harmonic oscillator, with frequency √ 6, nothing close to scale invariance. All this unstable behavior near α C , leads us to conjecture that the semiclassical (mean field) theory is ill-defined near this special value of the coupling. We avoid such critical points in our present work, and shall leave its study for future investigation.", "pages": [ 12, 13 ] }, { "title": "Mode Equations", "content": "The Mukhanov-Sasaki variable, u corresponds to fluctuations of the gravitational field. Upon Fourier transforming, the modes u k obey [23] equations of the form, Here the functions f u are given by A solution to (18) becomes simple upon making a simple ansatz, This reduces to an equation for Z M , Upon making the identification, the functions Z M solve Bessel's equation. We now apply these solutions to the study of the power spectrum of density perturbations.", "pages": [ 14 ] }, { "title": "Power Spectrum", "content": "The comoving distance to the last scattering surface, r glyph[star] ∼ 14 Gpc, sets the scale of interest for the power spectrum. η k is the conformal time when the modes u k leave the horizon, η glyph[star] is the time when the modes associated to the scale r glyph[star] left the horizon. Thus, those mode which leaves before η glyph[star] have already frozen out. From that fact we can write the most general solution to (21), Where the H ( i ) M are Hankel functions (not to be confused with the Hubble parameter), and M is defined by (22). The power spectrum is proportional to the squared modulus of u k , evaluated at η glyph[star] , and is traditionally presented in terms harmonic modes observable in the sky, the angular power spectrum,", "pages": [ 14, 15 ] }, { "title": "B. Slow Roll and Its Deviations", "content": "Let us examine the dependence of P ( k ) on Ξ more closely. From (22) and (19), one finds that the indices of the relevant Hankel functions are, Deep in the slow roll regime, the contribution from f u nearly vanishes, leaving M ≈ ± 1 / 2. At these special values, spherical Hankel functions emerge, These are just plane waves, e ± iη . As such, when integrated over to find C glyph[lscript] , one finds a flat spectrum, that is, for all glyph[lscript] , consistent with the Harrison-Zeldovich spectrum. The physical power spectrum depends on the trajectory of the inflaton. To linear order, each mode evolves independently. The power spectrum is a slice through the set of the modes at a fixed conformal time η glyph[star] . Features in the trajectory of φ which occur after η glyph[star] will be imprinted on the power spectrum. To understand this, we deconstruct the quantity x k , which we define as the argument of the linear perturbations (23), That is, the modes are, with A transition from chaotic to inflection point inflation, shown in Fig. 3, can distorts x k , as illustrated in Fig. 8. Features like that that happen at conformal time ˜ η can appear in the power spectrum at ˜ k ∼ aH ∣ ∣ ∣ η =˜ η . To be observed, such features should occur shortly before the the modes relevant for the CMB left the horizon. If the feature occurs too early, it will be pushed too far to the red part of the spectrum to be measured. This may require a fortuitous coincidence, but such signatures - low power at large scales - are observed in the CMB power spectrum. In any case, we now turn to quantifying these possibilities by examining parameters and the initial conditions. We now discuss how these features yield such a reduction of power. The important, x k dependent-quantity for this analysis is M , the index of the Hankel function, (26). As previously mentioned, during slow-roll, M asymptotes towards 1 / 2. A spike in both Ξ and x k , as happens when φ approaches the inflection point, temporarily pushes M through zero and into imaginary values. This dramatically alters the behavior of absolute value of the mode wavefunctions, | u k | 2 . As illustrated in Fig. 9, the absolute values of √ xH (1) M ( x ) and √ xH (2) M ( x ) are identical for real M . When M passes to imaginary values, they ramify, with √ xH (1) M ( x ) exponentially growing and √ xH (2) M ( x ) decaying. This fact motivates the choice A k = 0 in (23), consistent with a Bunch-Davies vacuum [24]. When the transition is over, φ finds itself in slow-roll, M again approaches 1 / 2, and the wavefuctions resume their standard 1 /k 3 behavior. The upshot of all this is dramatic reduction of power, demonstrated by the huge 'bight' in power spectrum during the transition. Such a feature is manifest in the red curve of Fig. 9. As we shall now see, this 'bight' has direct implications for the angular power spectrum.", "pages": [ 15, 16, 17 ] }, { "title": "C. Scanning the basin of attraction", "content": "We now quantify the effect of a transient deviation from slow-roll inflation on the power spectrum of primordial density perturbations. In particular, we focus on the effect of a transition from chaotic to inflection point inflation. We demonstrate this by examining the angular power spectrum, C glyph[lscript] , as a function of the catastrophe parameter α in the A 3 model (see Sec. II C). The set up First we connect the theoretical ideas discussed so far to observational quantities. In particular, we need to establish the various scales in the system to carryout a numerical analysis. To that end, we begin with the normalization of the power spectrum, ∆ 2 , which is related to the wavefunction of the linear modes [23] by The C glyph[lscript] 's from Eqn. (25) can be reparametrized using the fact that k = aH , so that The argument of j glyph[lscript] is dimensionless. For generic choice of units, it is given by In the vicinity of the inflection point, Here V 0 represents the overall scaling of the potential, which though largely irrelevant for the background dynamics (c.f. Eqn. (4)), is important for the perturbations. Simple algebra reveals that where N 0 normalizes the scale factor to unity today. Therefore, if the transition occurred at N k ∼ 15, as in Fig. 3, observability requires Low power at large scales The primordial power spectrum appears to be very close to scale invariant. The sharp peaks and troughs observed in the CMB are well explained by baryon acoustic oscillations [25]. Power at the very largest scales, however, appears to be anomalously low. Taken together, this suggests that the universe inflated fairly regularly from η glyph[star] until the end of inflation. Any nontrivial dynamics must - and could - have occurred just prior to η glyph[star] . For definiteness, we focus on the transition from chaotic to inflection point inflation, although similar arguments apply for similar features. The suppression of power at low scales depends on two quantities. The time between the transition and η glyph[star] and the strength of the cubic coupling during inflation, α . The former quantity is related to V 0 and the total number of e-foldings, as can be inferred from (29). We define an effective parameter β to absorb these dependencies. that is, Increasing β corresponds to either raising the scale of inflation or lowering the initial total number of e-foldings, i.e. N 0 = log a 0 . This can be related to the initial conditions for φ , and therefore the basin of attraction. α and β have qualitatively different effects on the angular power spectrum. In particular, the effects of α are largely independent of N 0 - the total number of e-foldings of inflation. We now study the effects of varying both α and β in detail.", "pages": [ 17, 18, 19 ] }, { "title": "Changing α and β", "content": "We begin by considering changes in α . As expected from (23), the 'bight' in ∆ tracks the deviation of Ξ from zero. Similarly, their magnitudes are correlated, as one can see by comparing Fig. 4 and Fig. 10. The α dependence of a nonzero Ξ is somewhat clearer to analyze, and we do so by defining a deviation parameter Ω, Ω rapidly falls to zero as α deviates from α C ≈ 0 . 659. This behavior is plotted by the solid red curve in Fig. 11. The integral is taken over the feature associated to the transition. For example, if the transition occurs at N = 15, we integrate from around N = 5 to around N = 30, since both endpoints are well within the slow-roll regime. Of course, such a transition can occur at any value of N . A larger value of Ω corresponds to a more violent and sustained deviation from slow-roll and a larger 'bight' in the power spectrum. Ω has an approximate power law dependence on α , which is represented by the dashed curve in Fig. 11. The actual scaling α dependence oscillates slowly between powers of -1 . 8 and -1 . 3, but the qualitative behavior is clear. The temporary reduction in ∆ can also be seen in the angular power spectrum. The relation between them can be seen by comparing Fig. 10 and Fig. 12. When these features in ∆ occur on the largest observable scales, the the first few modes of the angular power spectrum are suppressed. Fig. 12 illustrates this for α = 0 . 7 , 0 . 8 , 1 . 2 and 1 . 6. The largest effect occurs for α = 0 . 7, which is close to α C . The values of α (and colors) are linked across both of these plots. We now turn to β , which determines which range of N dominate the angular power spectrum - the C glyph[lscript] integrals. The integrands, dC glyph[lscript] , are the oscillating curves in Fig. 13. Large values of β shifts the curves to the left - to larger scales. These integrals are enveloped by ∆ 2 , which does not change with β . These are the light, dashed curves seen in Fig. 13. As we just discussed, α changes the shape of the envelope. In particular, close to the critical value, α C , a larger 'bight' is removed from the envelope. * * To explain the anomaly in the CMB angular power spectrum, β must take a value close to that in Fig. 13c. That is, the associated power reduction must be focused on the low multipole moments. Since no such power reduction appears seems to exist at smaller scales, to be physical it may also be smaller. While it is fixed by these considerations, from (30) we see that there is still a degenerancy between V 0 and a 0 . Since a 0 depends on the total expansion of the universe before today, it can be related to the initial conditions. In particular, for fixed V 0 we can see how β varies over the entire basin of attraction. The amount of inflection point inflation depends on λ , and therefore independent of the initial conditions. However, the length and duration of the approach to the inflection point gives rise to a prior history of chaotic inflation. Call this 'extra' inflation N ex . It scales by β by This is the effect, illustrated in Fig. 14, which we now quantify. Any initial field velocity will drastically reduce N ex , but starting from a larger value of φ will increase it. Each curve in both Fig. 14a and Fig. 14b corresponds to a different initial velocity: φ ' = 0 , -1 . 9 , -2 . 4. These are the red, black and orange curves, respectively. Note that the maximum possible φ ' is -√ 6 ∼ -2 . 449, as this corresponds to kinetic energy domination. In Fig. 14b, the", "pages": [ 19, 20, 21 ] }, { "title": "Angular Power Spectrum For Various a", "content": "basin of attraction is sketched for reference. The important effect to observe is that N ex grows rapidly as the initial conditions are chosen deeper inside the basin of attraction. The larger velocities are have little effect on N ex , as can be seen in Fig. 14a. Indeed, N ex is only strongly suppressed exponentially close to the boundary of the basin of attraction, which is shown in Fig. 5.", "pages": [ 22 ] }, { "title": "Implications for the measure problem", "content": "A number of studies have attempted to put a measure on phase space. The standard Liouville measure suggests that early-time kinetic domination is by far the most likely scenario. While including the couplings amongst the random variables enhances the likelihood of inflection point inflation to 1 /N 3 , this measure still favors a substantial initial velocity. This suggests that a larger β is far more likely. In short, N ex ∼ 0. In this case (30) together with (16) gives, For β fixed by the CMB, and fixed α , the scale of inflection point inflation is fixed by its duration, Note that λ must be sufficiently small to generate sufficient expansion for the observable universe. We also stress here that this analysis is extremely sensitive to the details of the chosen measure on phase space, although it is interesting to see further implications of the measure used in [26]. More generally, one must appeal to Fig. 14.", "pages": [ 22, 23 ] }, { "title": "IV. CONCLUSION", "content": "The dynamics of the inflaton have a rich structure despite the generic predictions for the cosmic microwave background. Advances in observational and theoretical technologies have increased our sensitivity to the effects of nontrivial dynamics on the cosmological perturbations. Small-field models of inflation generically involve temporary deviations from the slow-roll, attractor trajectory. In this work we have quantified these deviations and shown how they may affect the primordial density perturbations. Crucially, these effects arise in a model independent fashion. Since these effects only depend on the local structure of the potential, they scale with the same universality parameters discussed in early work [8]. We demonstrated analytically how sufficiently large deviations from slow-roll change the structure of the wavefunction for the perturbations, and explicitly how this reduced the power in their spectrum. We closed by relating both the couplings and the initial conditions to the strength and timing of this power suppression. While this completes the systematics for how low power at large scales may have arisen with a chosen universality class, we also discussed how it informs the likelihood of inflation. This sudden change in the power spectrum may have important implications for nongaussianities, particularly in the context of multifield models where the dynamics have a richer structure. It would be interesting understand the relation, if any, between the strength and shape of nongaussianities and the universality parameters. Finally, the deviation from slow-roll is most dramatic near critical values of the couplings. This fact leads to a curious saturation of the Ξ parameter, and has an extreme effect on the perturbations. Itzhaki and Kovetz [18] showed that the background has the properties of a second order phase transition. More generally, this feature occurs along the entire boundary of the basin of attraction. As Ξ is maximized for an extended period, it stands to give the strongest effect on primordial nongaussianities. From a field theoretic perspective, such a dramatic effect near the vicinity of a nontrivial fixed point alone warrants further investigation.", "pages": [ 24, 25 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "The authors wish to thank Eiichiro Komatsu, Ely Kovetz and Todd Zapata for helpful discussions. S.D. would also like to thank the Physics department at Northeastern University for their hospitality where parts of this work was completed. This work is supported in part by the DOE grant DE-FG02-95ER40917 and the Mitchell Institute for Fundamental Physics and Astronomy.", "pages": [ 25 ] } ]

[ { "title": "Impact of Higher-order Modes on the Detection of Binary Black Hole Coalescences", "content": "Larne Pekowsky, 1 James Healy, 1 Deirdre Shoemaker, 1 and Pablo Laguna 1 1 Center for Relativistic Astrophysics and School of Physics Georgia Institute of Technology, Atlanta, GA 30332 The inspiral and merger of black-hole binary systems are a promising source of gravitational waves for the array of advanced interferometric ground-based gravitational-wave detectors currently being commissioned. The most effective method to look for a signal with a well understood waveform, such as the binary black hole signal, is matched filtering against a library of model waveforms. While current model waveforms are comprised solely of the dominant radiation mode, the quadrupole mode, it is known that there can be significant power in the higher-order modes for a broad range of physically relevant source parameters during the merger of the black holes. The binary black hole waveforms produced by numerical relativity are accurate through late inspiral, merger, and ringdown and include the higher-order modes. The available numerical-relativity waveforms span an increasing portion of the physical parameter space of unequal mass, spin and precession. In this paper, we investigate the degree to which gravitational-wave searches could be improved by the inclusion of higher modes in the model waveforms, for signals with a variety of initial mass ratios and generic spins. Our investigation studies how well the quadrupole-only waveform model matches the signal as a function of the inclination and orientation of the source and how the modes contribute to the distance reach into the Universe of Advanced LIGO for a fixed set of internal source parameters. The mismatch between signals and quadrupole-only waveform can be large, dropping below 0.97 for up to 65% of the source-sky for the non-precessing cases we studied, and over a larger area in one precessing case. There is a corresponding 30% increase in detection volume that could be achieved by adding higher modes to the search; however, this is mitigated by the fact that the mismatch is largest for signals which radiate the least energy and to which the search is therefore least sensitive. Likewise, the mismatch is largest in directions from the source along which the least energy is radiated. PACS numbers: 04.25.D-, 04.25.dg, 04.30.Db, 04.80.Nn", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The merger of a binary black hole (BBH) system has long been considered a strong source of gravitational waves for ground and space based gravitational wave observatories. These mergers are characterized by 15 parameters, 9 intrinsic to the black-hole systems (2 blackhole masses, 2 spin vectors and eccentricity) and 6 extrinsic to the source (binary orientation vector, sky position and distance). The LIGO and Virgo detectors have recently completed a joint run during which inspiral horizon distances exceeded 40 Mpc [1] and new upper limits have been placed on the rates of such events [2]. These observatories are currently being upgraded and when the new design sensitivities are achieved they will have ranges up to ten times greater and hence volumes 1000 times greater. By the end of this decade LIGO and Virgo, along with GEO, will be joined by KAGRA in Japan and possibly the proposed LIGO India, greatly increasing not only the range of the global network but also the ability to recover information about the sources [3]. When the theoretical model of the gravitational waveform is well understood, the most effective method to search and recover a gravitational wave signal is matched filtering against a library of model waveforms called a template bank [4]. The ability of such a templated search to detect signals is dependent on four factors: ctor. Throughout this paper we use the targeted aLIGO zero-detuned, high-power [5] sensitivity curve. In this paper we will be concerned with the last two points. For the BBH systems potentially observable by ground-based detectors, astrophysical processes place few constraints on the intrinsic physical parameters that characterize the emission of radiation from these cataclysmic events, thus placing the burden on source models to cover nearly the full compliment of physical parameters. Rigorous requirements from matched filtering place an additional burden on the source models. In order for the model waveforms to match potential signals to within a given tolerance, we need not only enough waveforms to cover the parameter space but also each waveform must represent nature effectually enough to ensure the signal does not fall through cracks in the template bank and faithfully enough to recover the source parameters. One source of mismatch with nature is the truncation of the spherical harmonic series in which we have decomposed the model waveform. Current template waveforms are only of the dominant, quadrupole mode, although we know that generic signals will have many excited harmonics present when detected. Fig.(1) shows the ratio of several non-dominant modes to the dominant mode for two non-spinning systems, note that for the system where the masses of the component holes are not equal the next-to-leading mode is within an order of magnitude of the quadrupole mode, suggesting that accounting for additional modes may be important for detection, especially as the mass-ratio strongly deviates from one and generic spins are explored. This paper builds on previous work by ourselves and other authors. In [6, 7], we conducted a preliminary study on higher modes for spinning, equal-mass systems comparing numerical relativity waveforms containing the largest five harmonics to an equal-mass non-spinning system of just the dominant mode. We found that for low spins, the non-spinning dominant mode was an effective model waveform. McWilliams et al [8] found that over a range of the source orientations, the equal-mass waveform was effective at detecting moderate mass ratios over source orientations. Brown et al[9] is exploring the value added of higher modes in EOBNR models of unequalmass waveforms. In this paper we investigate the degree to which inclusion of additional terms of the spherical harmonic series to template waveforms could improve matchedfilter based searches. We use numerical relativity (NR) waveforms as both signal and template, and we consider both unequal masses and some generic spins generated by the Maya code. We study how well the quadrupoleonly model waveform matches the signal as a function of the inclination and orientation of the source and determine how the volume reach of advanced LIGO depends on the inclusion/exclusion of non-dominant harmonics in the model waveforms. We concentrate on system masses greater than 100 M glyph[circledot] to give the NR portion of the waveform prominence and negating the need for post-Newtonian information. Our findings show that for non-precessing signals up to 65% of source orientations can be missed when using only the quadrupole mode, implying a 30% gain in detection volume which could be achieved by including higher modes. For our most precessing case when using the quadrupole mode only the loss of source orientations is 83% and the potential gain in volume over which such systems could be detected is again 30%. These potential gains in volume are mitigated by the fact that the mismatch is largest for signals which radiate the least energy and to which, therefore, the search is therefore least sensitive. Likewise, the mismatch is largest in directions from the source along which the least energy is radiated. Finally, we do a preliminary investigation into how the series truncation might impact parameter estimation by exploring a potential degeneracy between mass and inclination for full waveforms in the last section of this paper. We proceed as follows: in § II we introduce our methodology for matched filtering, and in § III the NR waveforms used in all of our studies. In § IV we consider various aspects of the overlaps between the dominant mode and the higher modes. In § V we examine the volume of the universe accessible to advanced detectors using quadrupole-only waveforms and hypothetical ideal waveforms containing most of the modes, for several cases. We conclude in § VI that the smallest overlaps are obtained for systems and source orientations which radiate the least total power, and hence have the smallest accessible volumes even when an ideal waveform is used. In this section we also present a first look at the implications of higher modes for parameter estimation. Conventions: Throughout this paper we adopt the following conventions. We denote the Fourier transform of a function g ( t ) with a tilde, as ˜ g ( f ). We characterize the mass ratio of a BBH system by q = m 1 /m 2 with m 1 ≥ m 2 . The relation of the source to the detector is specified by five angles. Two ( ι, φ ) place the detector in coordinates centered at the source, it is these angles in which the decomposition into spherical harmonics is performed. Two ( θ, ϕ ) place the source in the sky of the detector. The final angle, ψ , determines the relative rotation between these two coordinate systems, we associate ψ with the source because in what follows we will treat it similarly to ι and φ . We define these angles in fig.(2). The final parameter connecting the source and detector is the distance between them, we will be concerned with the maximum distance at which the source can be detected and will determine this value in what follows.", "pages": [ 1, 2 ] }, { "title": "II. MATCHED-FILTER SEARCHES FOR GRAVITATIONAL WAVES", "content": "The response of an interferometric detector is described by an antenna pattern [10], (1) Following [11] we rewrite this in the more convenient form where and For reference we show the antenna pattern in fig.(3). For gravitational waves, the intrinsic characteristics of a source are fully encapsulated in the polarization strains h + and h × . When an incoming gravitational wave is incident on the detector the strains give rise to a signal s given by where we have used eqn.(2) and defined The output of the detector is then s + n , where n is the noise of the detector. Following standard practice we incorporate the noise only as S n ( f ) and do not add it to the signal in what follows. We will take h in eqn.(3) to be the output of a numerical simulation, to be discussed in the following section. We now briefly review some of the data analysis framework employed in current LIGO/Virgo searches, and which will be used throughout this paper. An inner product on the space of real, time-dependent waveforms A ( t ) and B ( t ), with respect to a given noise curve described by a power spectral density S n ( f ), is In stationary, Gaussian noise, the optimal measure of the presence of a gravitational wave signal that matches a model waveform, called a template , is the signal-to-noiseratio (SNR) denoted by ρ , with and where we are studying the response of a single detector to one polarization, typically taken to be h + . We note in passing that in a multi-detector search the data streams from all instruments will be filtered against the same h + , and that the source angles ι, φ will be the same at all detectors. However, the orientations of the different detectors will provide different values of ψ , making the detectors sensitive to different combinations of the polarization. In addition each detector's F 0 will have a different dependence on θ, ϕ providing coverage of regions of the sky to which any one detector might be insensitive. The signal will arrive at an unknown time which we identify as the time of coalescence and denote t 0 . We assume the template waveform h is a good approximation to the signal s , and search for the signal at all times by shifting the template. This has the effect in the Fourier domain of changing ˜ h ( f ) to ˜ h ( f ) exp( -2 πift 0 ). The signal will also have an unknown phase at the time of coalescence, corresponding to the value of φ in fig.(2), which we denote φ 0 . This introduces an additional factor of exp(2 πiφ 0 ). In practice, this leads the SNR to be evaluated as where the absolute value removes the dependence on the unknown phase. Eqn.(6) may be evaluated by a single complex inverse Fourier transform, and the maximization over t 0 is then accomplished by finding the maximum of the resulting time series. Eqn.(6) is only an exact calculation of the SNR if ( h + | h × ) = 0 [12], which is not true in general; however, we expect the errors introduced by this approximation to be small. Note that, by eqn.(3), the dependence on the SNR of the detector angles may be factored out in eqn.(6). Note also that F 0 (0 , 0) = 1. These imply that, given the SNR of a signal at θ = ϕ = 0, we know the SNR of a signal in the same orientation at all other sky positions. Related to the SNR is the match or overlap obtained by normalizing both waveforms The overlap is a measure from 0 to 1 of how well the template matches the signal, an overlap of 1 indicates that the template is an exact match to the signal and anything lower than one is a diminished match. Gravitational-wave strain falls off as the the reciprocal of the distance between source and detector. It follows from eqn.(5) that the SNR falls off in the same way, while the normalization removes the distance dependence of the template. Henceforth we place the signal s in eqn.(5) at 1 Mpc from the detector and denote the resulting SNR as ρ 1Mpc . We also choose a threshold SNR , a value above which indicates the presence of a signal in the data. We will take this to be 5.5, the threshold used in current LIGO/Virgo searches. The choice of this value is motivated by the behavior of the noise in the detector [13]. The distance at which a signal would have an SNR of 5.5 is then We now consider two templates, h ideal which exactly matches the signal and h which in some way approximates the signal. We can determine the fraction of the available distance that is lost by using the approximate template as: The first equality follows from eqn.(8), the second from dividing both numerator and denominator by the common factor ( s | s ) 1 / 2 and the third from the fact that when the template exactly matches the signal the overlap is 1. The overlap therefore measures the fraction of the SNR lost by using an incorrect template, and equivalently the fraction of the distance lost. As the universe is approximately uniform at distances accessible to even initial LIGO [2], the event rate is approximately equal to the cube of the range, although this will also depend on the antenna pattern. However, we note that the overlap does not give the value of r ideal . As an extreme example, if r ideal is sufficiently small that the number of expected events per observation time is close to zero, then the fractional loss of range implied by a low overlap is inconsequential.", "pages": [ 3, 4 ] }, { "title": "III. THE BINARY BLACK HOLE COALESCENCE WAVEFORMS", "content": "This paper uses NR waveforms covering the late inspiral, merger and ringdown for a variety of mass ratios and spins. All of the NR simulations used in this study were produced with GATech's Maya code [14-19]. The Maya code uses the Einstein Toolkit [20] which is based on the CACTUS [21] infrastructure and CARPET [22] mesh refinement. We use sixth-order spatial finite differencing and extract the waveforms at a finite radius of 75 M , where M is a code unit set to unity and can be scaled to any physical mass scale. All grids have 10 levels of refinements unless noted below. We use 28 simulations in this paper and group them according to their initial parameters in Table I. Grid details, including outer boundary and resolution on the finest are also shown. The simulations can be separated into three groups: non-spinning, equal-mass with aligned spin, or unequal-mass with precessing spin. For the simulations with q > 4, we used the coordinate-dependent gauge term as described in Refs. [23] and [24]. For the q = 10 and q = 15 simulations, initial parameters in Ref. [25] were used. These simulations ( q > 4) have an extra level of refinement for 11 levels total, with the exception of q = 6 and q = 15. These have 10 levels and 12 levels, respectfully. The output of all simulations is the Weyl Scalar, Ψ 4 , decomposed into spin-weighted spherical harmonics. Simulations are performed in a coordinate system which we will denote the source-centric frame, to distinguish it from the detector-centric frame we will employ subsequently. See fig.(2) for the definition of the angles used in this frame. In terms of these angles the decomposition /circledot /circledot is: This is related to the strain measured by gravitationalwave observatories as The quadrupole mode is given by ( glyph[lscript], | m | ) = (2 , 2) . Throughout this paper we work in the frequency domain, and therefore avoid the integration to strain since ˜ h = ˜ Ψ 4 / ( -4 π 2 f 2 ).", "pages": [ 4, 5 ] }, { "title": "IV. OVERLAP", "content": "We start by examining the relative importance of the non-dominant modes in a waveform comparison. The full waveform involves factors of the spherical harmonics and the amplitudes of the modes (see eqn.(10)). When the amplitudes of the higher modes are vanishingly small, they can be ignored; however, as we have already noted in fig.(1), the relative amplitudes grow in strength with mass ratio. glyph[negationslash] In fig.(4) we plot the overlap of each mode against (2,2) individually. If all modes matched well against (2,2) it would suggest that a template containing only this mode would be a good match to the full signal, regardless of the source orientation; however, we find that not to be the case. In both the q = 1 and q = 4 case, the overlap between (2,2) and the next most dominant modes is poor, below 0.6. Furthermore, although the inner product, eqn.(4), and the decomposition into modes, eqn.(10), are themselves linear, the maximization over time and phase introduces nonlinearities. In particular, defining h others = ∑ l,m =2 , 2 h lm , the sum is only linear if the inner products maximize at the same time. If not, there will be a 'tension' in the modes and the combined SNR will be less than the sum of the individual SNRs, i.e. glyph[negationslash] To quantify this we plot the time series of both SNRs on the right-hand side of eqn.(12) in fig.(5). The two series peak at notably different times, and at the peak of the h 22 series the h other series has dropped by 38% thus we can conclude that the non-linearities are important, and we cannot use the linear approximation. While fig.(4) shows that the (2,2) mode is not an effective representation of the other modes, how well does the (2,2) mode cover the sky of the source? The overlap between the full-mode waveform and the (2,2) mode is a function of the angles centered at the source, ( ι, φ ). The (2,2)-only template depends on the angles through a single factor, 2 Y 22 ( ι, φ ), which is canceled by the normalization; and, therefore, we simplify the overlap by placing this waveform at ι = φ = 0. We also place both waveforms optimally in the sky of the detector, at θ = ϕ = 0, and choose ψ = 0. We will generalize this momentarily. Fig.(6) shows the resulting overlaps for five cases, the non-spinning q = 1 and q = 7, and the precessing cases from tab.(I). At ι = 0 , π the waveform is dominated by the (2 , 2) modes, the overlap approaches 1.0 at these points. Equation (9) then implies that there is no loss of distance incurred by searching with the (2,2)-only template for systems that are oriented face-on with respect to the detector. We can further quantify this by determining the faction of surface area over which the overlap falls below 0.97%, where this value is motivated by the allowed 3% loss of SNR from using a discrete set of templates [2]. Tab.(II) lists this value for several simulations, along with the the average, median and lowest overlaps as further measures of the impact of the higher modes. a single detector when the intrinsic parameters are kept fixed to the signal: the q=1 case is well served with a (2,2)-only waveform over all source angles. The higher the mass ratio, the worse a (2,2)-only waveform does in matching the signal, and this fraction of angles over which the match does poorly increases. Furthermore, a precessing system is badly served by a (2,2) waveform. We will explore this matter further in future work. We now generalize the previous results to include other values of the detector-centric angles ( θ, ϕ ) and ψ . Consider two templates: h 22 which, as in current searches, contains only the (2 , 2) mode of the NR waveform optimally oriented ( θ = ϕ = ι = φ = ψ = 0), and a perfect template h ideal which exactly matches the signal. In fig.(7) we show the overlap between the signal and h 22 for several non-spinning systems. Each colored line on the graph represents a system mass ratio, moving along the line gives different system masses. As we move from top to bottom, we are moving from q=1 to q=15. The difference in colors along the line give the overlap value. The plot shows that for higher mass ratios the total power is distributed into the higher modes and the match drops accordingly. This is consistent with [8, 9]. Now consider the q = 4 non-spinning system, scaled to to 100 M glyph[circledot] and placed at a distance of 1Gpc from the detector, and examine the overlap between the signal and both templates. We randomly choose values for all angles and plot results with respect to ι , which has the most significant dependence. The results are shown in fig.(8), which illustrates that at ι = 0 , π the variation of the additional angles do not affect the overlap, while the spread in results widens towards ι = π/ 2. This again shows that the (2,2) mode only captures a face-on source orientation and misses the source as its inclination increases toward the edge-on case. This would imply that the higher modes are essential for detecting non-optimally oriented signals, but how far away can a single detector see these cases? We quantify how important the modes will be in terms of SNR and volume reach in the next section.", "pages": [ 5, 6, 7 ] }, { "title": "V. SNR AND VOLUME", "content": "As noted at the end of § II , the overlap is equal to the fractional loss in distance to which a signal can be detected, but this value should be viewed in light of the maximum possible distance. This maximum distance depends on three factors: (1) the total energy radiated by the source, (2) the ability of the template to extract energy of the signal from the background noise and (3) the location of the source in the sky of the detector. For example, in the plane of the detector along the lines 45 degrees to the arms, the response goes to zero. Along these lines the loss in range implied by a low overlap is irrelevant for a single detector. In this section we consider the accessible distances, noting the influence of all three factors. We start with fig.(9), which shows the radiated energy and distances accessible using the h ideal templates, as a function of the source orientation. As expected, the range tends to be lowest where the least power is radiated, although the energy and distance plots are not identical due to weighting by the noise curve. The energy, and hence distance, plots have the same general shape as those corresponding in fig.(6), indicating that the overlaps between the signal and h 22 are lowest at orientations where the energy and distance reach of the ideal template are also lowest. This is due to the fact that ι ι the higher modes not only have poor matches with (2 , 2), as shown in fig.(4), but they also contain less power, as shown in fig.(1). Fig.(9) shows that orientations where the higher modes dominate have both low matches with h 22 and lower ranges. This indicates that the fractional loss in distance incurred by using the incorrect template is greatest where the best possible range is smallest. Finally, in order to characterize the performance of different templates by a single number with physical significance we calculate the spatial volume to which the search is sensitive. The distance to which a signal can be seen depends on all five angles, but from eqn.(3) and the comments at the end of § II the dependence on the detector-centric angles may be factored out Since there is no preferred orientation we define an average visibility range , R , by averaging the distances over the orientation angles ι, ϕ, ψ : We evaluate this average by choosing random values for ι, cos( φ ) , ψ uniform in (0 , 2 π ) , ( -1 , 1) , (0 , 2 π ) respectively. The average visibility distance as a function of the detector-centric angles is therefore and the volume of the Universe to which a given template is sensitive is therefore The remaining integral may be done numerically, yielding a value ≈ 3 . 687. The volumes for different waveforms, using the h 22 and h ideal templates are summarized in table III. The trend is for lower mass ratios and higher aligned spins to correspond to both larger absolute volumes and smaller relative differences by including higher modes in the template. The larger volumes correspond directly to the increased total energy radiated by such systems, which is shown in fig.(10). Finally, as another way of quantifying the difference rad between the templates, in fig.(11) we show histograms of the visibility ranges over the complete set of orientations at θ = ϕ = 0. Using h ideal shifts the ranges from lower to higher values somewhat, but does not increase the maximum distance, which occurs for face-on systems which are dominated by (2,2). These results include three precessing q = 4 , a = 0 . 6 systems. In all cases the accessible volume is less than that for the q = 4 non-spinning system. As might be expected from the non-precessing cases the volume decreases as the spin becomes anti-aligned with the angular momentum and less total energy is radiated. However, at least for the systems considered here, this dependence becomes smaller than our uncertainties when the angle between the orbital angular momentum and the spin of the larger hold exceeds 150 degrees.", "pages": [ 7, 8, 9 ] }, { "title": "A. Error analysis", "content": "Because we choose random values in evaluating the average eqn.(14) we are able to determine the error in the results as the standard deviation between several runs. Due to the computational expense of complete runs we instead estimate this by choosing one sky position. We show the SNR histograms obtained by 900 runs of θ = ϕ = π/ 3 for two waveforms in fig.(12). In both cases the error is on order of 0 . 5%. Since V = r 3 and r has an error δr , then δV = √ (( dV/dr ) δr ) 2 . Here we have δV/V = / ⋃⊗∫ ⋃⊗∫ 3 δr/r . The error for the results in tab.(III) is then on order 1 . 5%. There are also uncertainties associated with the choice of extraction radius and resolution. We show the volumes obtained using the q = 4 systems and h ideal template for several value of both parameters in tab.(IV). The variation is on the order of 1 . 5%, and our two sources of uncertainty are comparable, and small enough that they do not effect our conclusions.", "pages": [ 9 ] }, { "title": "VI. CONCLUSIONS", "content": "As can be seen from table III there are two conflicting trends as the mass ratio increases. As the total radiated energy is reduced, the volume drops. Conversely, as the fraction of this energy is distributed into higher modes the benefit gained by using the ideal template increases. The energy radiated, and hence volume, increase with spin. Together, these results imply a strong bias towards the detection of equal-mass, aligned-spin systems when averaged over the sky. This conclusion is consistent with [26, 27], while adding the fact that the inclusion of higher modes is not important for detecting these systems. We expect that a search using (2,2) IMRPhenB aligned-spin templates will perform well, this will be tested as part of the ongoing NINJA2 project [28]. For non-spinning systems with q glyph[greaterorsimilar] 3 and the mildly precessing systems considered here, the inclusion of higher modes in the template can improve the volume reach of the single detector. Whether or not this translates to an increase in detection rate depends on the unknown underlying rates of such systems. Put another way, the inclusion of higher modes in templates will allow the advanced detector network to better measure or bound these unknown rates. There are, however, some caveats. First, we stress that the template used for the rightmost column of table III exactly matches the signal, that is, it assumes we exactly know the signal for which we are looking in advance. To the extent that matched filtering is the optimal detection statistic any approximate inclusion of higher mode information will of necessity do worse. Furthermore, there are potential downsides to including higher modes in the templates. Such an addition would require increasing the number of templates. This entails a corresponding increase in the computational cost of the search. In addition, these additional templates may respond to glitches in the detector, raising the number of 'background' events and increasing the SNR at which a signal would need to be observed in order to confidently claim a detection. Concerns such as this lead to changing the mass range in the S6 search from 35 M glyph[circledot] to 25 M glyph[circledot] -the templates at the higher mass end produced sufficient numbers of background triggers to impair the ability to detect lower-mass systems [1]. It would be undesirable to allow a search for systems to which the detector network is comparatively insensitive to impact the ability to detect equal-mass and aligned-spin systems. We also note that, at present, it is not known how to construct a template bank of precessing signals. Further studies are needed to determine the right strategy for detecting both mildly and heavily precessing systems. We have not yet considered spinning systems with q > 1. Such simulations are available for spins up to 0.6 and mass ratios up to 7, however, we defer their analysis to future work. For spins aligned with the angular momentum the volumes accessible will certainly be larger than the non-spinning counterparts. It is possible that the dependence on higher modes will be preserved in these cases, leading to a potentially large volume increase by using templates that include higher modes. We have so far considered only a single detector. Additional detectors will provide better sky coverage, effectively increasing the value of the integral in eqn.(16). Furthermore, as noted in the introduction, detectors oriented differently are sensitive to different polarizations, it is therefore conceivable that the inclusion of higher modes in templates would have more impact on the range of the network as a whole than on any one detector. We have also not considered other aspects of the full search, such as signal-based vetoes. The effect of such vetoes is being studied in [29]. One important aspect of gravitational-wave detection we have also not considered is the fact that the data is filtered against a bank of templates with different parameters. For the initial detection it is acceptable for the signal to be picked up by a template with the wrong parameters; once the detection has been confirmed more computationally expensive parameter estimation codes can be run. While this freedom can not raise the volume accessible to h ideal , as it is already a perfect match to the signal, it is quite possible that maximization over a bank of h 22 templates will lead to larger average SNRs and hence volumes. In this case the fractional gain by going to an approximation of h ideal may be even smaller. This last point leads to the question of the importance of higher modes in parameter estimation. We expect higher modes to be important here; as a simple example the difference between a signal at ι = ψ = π/ 4 and one at ι = ψ = 0 is entirely encapsulated in the mode content. We expect that there are degeneracies between the orientation parameters and intrinsic parameters, we intend to investigate this further in subsequent studies. However we present a preliminary result in fig.(13), which shows that 〈 s ( ι, ψ ) | h 22 〉 can be increased by maximizing over the mass M of the template, at the cost of misestimating the mass. The increase in overlap is most pronounced at ι = π/ 2, where the higher modes are most significant. Correspondingly the mass which maximizes the overlap deviates the most from the true value at this point. This suggests a degeneracy between mass and higher mode content. One possible explanation is that the higher modes contain more power at higher frequencies, as do lower-mass systems. We will explore this possibility in our follow-up studies.", "pages": [ 9, 10, 11 ] }, { "title": "VII. ACKNOWLEDGMENTS:", "content": "Work supported by NSF grants 0914553, 0941417, 0903973, 0955825. Computations at Teragrid TG- ι ι PHY120016, CRA Cygnus cluster and the Syracuse University Gravitation and Relativity cluster, which is supported by NSF awards PHY-1040231, PHY-0600953 and PHY-1104371. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. We also would like to thank Stephen Fairhurst and Duncan Brown for their comments.", "pages": [ 11 ] } ]

[ { "title": "Gravitationally Driven Electromagnetic Perturbations of Neutron Stars and Black Holes", "content": "Hajime Sotani 1 , Kostas D. Kokkotas 2 , 3 , Pablo Laguna 4 , Carlos F. Sopuerta 5 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 2 Theoretical Astrophysics, University of Tubingen, IAAT, Auf der Morgenstelle 10, 72076, Tubingen, Germany 3 Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece 4 Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA 5 Institut de Ci'encies de l'Espai (CSIC-IEEC), Facultat de Ci'encies, Campus UAB, Torre C5 parells, Bellaterra, 08193 Barcelona, Spain. (Dated: April 22, 2022) Gravitational perturbations of neutron stars and black holes are well known sources of gravitational radiation. If the compact object is immersed in or endowed with a magnetic field, the gravitational perturbations would couple to electromagnetic perturbations and potentially trigger synergistic electromagnetic signatures. We present a detailed analytic calculation of the dynamics of coupled gravitational and electromagnetic perturbations for both neutron stars and black holes. We discuss the prospects for detecting the electromagnetic waves in these scenarios and the potential that these waves have for providing information about their source. PACS numbers: 04.30.-w, 04.40.Nr, 95.85.Sz", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Multi-messenger astronomy has arrived. Already astro-particle observations (neutrinos and cosmic rays) are complementing traditional electromagnetic observations. The third pillar is almost ready with near future gravitational-wave observations by interferometric detectors like LIGO, Virgo, GEO600 and LCGT [1, 2]. This new astronomy will enable multi-channel observations of astrophysical phenomena such as γ -ray bursts, supernovae, or flaring magnetars, unveiling an unprecedented view of the nature of the source and its environment. An important component in many astrophysical phenomena is strong magnetic fields, as demonstrated by the active role they play in the accretion processes of low-mass X-ray binaries and GRBs [3]. The presence of strong magnetic fields opens up the possibility for interesting effects. Among them, which is the central topic of this work, is the coupling between electromagnetic and gravitational emissions that could yield synergistic multi-messenger observations. In particular, it is important to assess the conditions in which electromagnetic and gravitational emissions influence each other. There are already hints for such scenario. It is believed that the flare activity of magnetars seems to be associated with starquakes [4]. These quakes are responsible not only for dramatic perturbations and rearrangements of the magnetic field, but also for the breaking of the neutron star crust and internal motions, possibly resulting in the emission of gravitational waves. Detailed studies of magnetar flare activity have revealed a number of features in the afterglow, which can be associated with the crust oscillations as well as with Alfv'en waves propagating from the core towards the surface [5-17]. The link or coupling between electromagnetic radiation and gravitational waves have been investigated for some cases. One of them looked at the propagation of gravitational waves linearly coupled to an external magnetic field [18]. It was shown that this configuration triggers magneto-hydrodynamics waves in the plasma [19-23]. Furthermore, the linear nature of the coupling limits the electromagnetic waves to low frequencies, in the best case a few tenths of kHz, which will be easily absorbed by the interstellar medium or plasma. In order to produce high frequency and detectable electromagnetic waves, non-linear couplings are needed, requiring much stronger gravitational waves. In most of these studies, the gravitational waves were assumed to propagate on a flat space-time background. This is a reasonable assumption when the interaction between the gravitational and electromagnetic waves takes place far from the source. There have been only very few attempts to treat the electromagnetic-gravity coupling in the strong field regime [24]. The aim of this work is to study the interaction of electromagnetic and gravitational waves in the vicinity of magnetized neutron stars or black holes immersed in strong magnetic fields using perturbation theory, paying particular attention to how gravitational modes drive the excitation of electromagnetic perturbations. Our work also includes estimates of the energy transferred between the gravitational and electromagnetic sectors. As expected, we find that the excited electromagnetic waves have roughly the same frequency as the driving gravitational waves, i.e., of the order of a few kHz. Electromagnetic waves at these low frequencies can be easily absorbed by the interstellar medium. As a consequence, one needs to associate them with secondary emission mechanisms (e.g., synchrotron radiation) in order to be able to trace the effects of gravitational waves on the strong magnetic fields. The later process can be studied following the mechanisms described in [19-23], and there is work in progress for the special case of strong gravitational fields. This article is organized as follows: Section II gives details of the space-time background configuration. In Sec. III, we review the general form of the perturbation equations, their couplings, and the angular dependences of the various types of electromagnetic and gravitational perturbations. In Sec. IV, we reduce the equations to the particular case of dipole electromagnetic perturbations driven by the quadrupole gravitational mode for the case of a neutron star background. In Sec. V, we do the same as in Sec. IV but for the case of a black hole and consider both the case of axial and polar gravitational perturbations. In Sec. VI we show numerical results or dipole electromagnetic waves driven by quadrupole gravitational waves with axial parity for both neutron stars and black holes. Conclusions are given in Sec. VII. We adopt geometric units, c = G = 1, where c and G denote the speed of light and the gravitational constant, respectively, and the metric signature is ( -, + , + , +).", "pages": [ 1, 2 ] }, { "title": "II. EQUATIONS FOR THE BACKGROUND", "content": "The background space-times we are considering (neutron stars and black holes) are governed by the EinsteinMaxwell equations, which read: The tensors that appear in these equations are: The Einstein tensor G µν , the Faraday antisymmetric tensor F µν , the electromagnetic four-current J µ , the energy-momentum tensor of the matter fluid T µν , and the energy-momentum tensor of the electromagnetic field is E µν . The energy-momentum tensors are explicitly given by where ρ stands for the energy-density, p for the pressure, and u µ for the four-velocity of the matter fluid. The presence of a magnetic field could in principle induced deformations to the neutron star or black hole we are considering. However, even for astrophysically strong magnetic fields, B ∼ 10 16 G , as in the case of magnetars, the energy of the magnetic field E B is much smaller than the gravitational energy E G , by several orders of magnitude. In fact, E B / E G ∼ 10 -4 ( B/ 10 16 [ G ]) 2 . Therefore, in setting up the background space-time metric, one can ignore the magnetic field. That is, the background metric has the form where the functions ν ( r ) and λ ( r ), in the interior of a neutron star, are determined by the well-known TolmanOppenheimer-Volkoff (TOV) equations (see, e.g. [25]) and the matter fluid four-velocity u µ = ( e -ν/ 2 , 0 , 0 , 0). In the exterior of a neutron star, and in the case of a black hole, they are determined by the standard Schwarzschild solution: e -λ = e ν = 1 -2 M/r .", "pages": [ 2 ] }, { "title": "A. A Dipole Background Magnetic Field: Exterior region", "content": "Next, we compute the magnetic field for both the neutron star and the black hole. We consider first the exterior (vacuum) solution. In this case, the component of Maxwell equations given by Eq. (2.4) is automatically satisfied. The magnetic field is then obtained by solving the remaining Maxwell equations, Eqs. (2.3), which in vacuum reads with F µν = A ν,µ -A µ,ν . Since the background space-time is static, it is natural to assume that the magnetic field is also static. In addition, we require the magnetic field to be axisymmetric and poloidal, which has a dependence on the polar coordinate, θ . From the relation between the magnetic field, the matter fluid velocity u µ , and the field strength where /epsilon1 µναβ is the complete antisymmetric tensor, determined by the convention /epsilon1 0123 = √ -g . It is not difficult to show that the only non-vanishing component of the vector potential A µ is the φ -component, which we will denote as A (ex) φ . Therefore, the vacuum Maxwell equation (2.8) in the Schwarzschild background becomes Expanding A (ex) φ in vector spherical harmonics as we rewrite Eq. (2.11) as The solution of this equation for the dipole case ( l M = 1) has the form [26] where µ d is the magnetic dipole moment for an observer at infinity. With the solution of Eq. (2.14) and Eq. (2.10), the coefficients of the magnetic field in Eq. (2.9) are given by: Notice that in the limit r →∞ ,", "pages": [ 2, 3 ] }, { "title": "B. A Dipole Background Magnetic Field: Interior region", "content": "We assume that the magnetic field inside the star is also axisymmetric and poloidal, with current J µ = (0 , 0 , 0 , J φ ) [27, 28]. The ideal MHD approximation is also adopted, i.e. infinite conductivity σ , which leads to E µ = F µν u ν = 0, as follows from the relativistic Ohm's law Therefore, the vector potential A µ is similar to that for the exterior magnetic field, i.e. A µ = (0 , 0 , 0 , A (in) φ ). The counterpart equation to Eq. (2.11) but for the interior is Expanding both, the vector potential A (in) φ and the current J φ , in vector spherical harmonics, one gets which can be use to rewrite Eq. (2.19) as It is only feasible to obtain numerical solutions to Eq. (2.22), even for the dipole case ( l M = 1), since among other things the coefficients are also computed numerically from the TOV equations. In addition, when prescribing j 1 ( r ), it must satisfy an integrability condition (see [29, 30] for details). We adopt a current with a functional form [28]: where f 0 is an arbitrary constant. In addition, we should impose the following regularity condition at center of the neutron star, where α c is also an arbitrary constant. These arbitrary constants, f 0 and α c , are determined by from the matching conditions at the surface of the star, namely that a 1 and a 1 ,r are continuous across the stellar surface. Finally, once we have the numerical solution for a 1 ( r ), the magnetic field is obtained from with With the magnetic field determined both in the interior and exterior regions, the Faraday tensor for the background field becomes", "pages": [ 3, 4 ] }, { "title": "III. PERTURBATION EQUATIONS", "content": "We consider small perturbations of both the gravitational and electromagnetic fields, which can be described as ˜ g µν = g µν + h µν , (3.1) where g µν and F µν are the background quantities derived in the previous section. The tensors h µν and f µν denote small perturbations, i.e. h µν = δg µν and f µν = δF µν . Linearization of the Einstein-Maxwell equations yields From Eq. (3.5), we find that the electromagnetic perturbations are driven by the gravitational perturbations via the term containing δ ( -g ) 1 / 2 in the right hand side. On the other hand, for simplicity, we omit the back reaction of the electromagnetic perturbations on the gravitational perturbations, i.e. we set δE µν = δ ( E µν ; ν ) = 0 in Eqs. (3.3) and (3.4). This simplification is based on the assumption that the energy stored in gravitational perturbations is considerably larger than that in electromagnetic perturbations, which are typically driven by the former. On the other hand, in the giant flares of SGR 1806-20 and SGR 1900+14 [31-33], whose peak luminosities are in the range of 10 44 -10 46 ergs s -1 , the dramatic rearrangement of the magnetic field might lead to emission of gravitational waves. Nevertheless, recent non-linear MHD simulations [34-38] do not support these expectations. The first two perturbative equations, Eq. (3.3) and Eq. (3.4), have been studied extensively in the past, in the absence of magnetic fields, both for stellar and black hole backgrounds (see, e.g. [39-44]). Thus, in this article we use the perturbation equations derived in earlier works, and we derive the analytic form of the perturbation equations for the electromagnetic field together with their coupling to the gravitational perturbations. The metric perturbations h µν , in the Regge-Wheeler gauge [39], can be decomposed into tensor spherical harmonics in the following way where H 0 ,lm , H 1 ,lm , H 2 ,lm and K lm are the functions of ( t, r ) describing the polar perturbations , while h 0 ,lm and h 1 ,lm describe the axial ones. On the other hand, the tensor harmonic expansion of the electromagnetic perturbations, f µν , for the Magnetic multipoles (or axial parity ) are given by while the expansion for the Electric multipoles (or polar parity ) can be written as Hereafter, the quantities describing the magnetic- and electric-type electromagnetic perturbations will be denoted with the indices (M) and (E), respectively. We point out that h µν is a symmetric tensor, while both f (M) µν and f (E) µν are anti-symmetric tensors, i.e. f (M) µν = -f (M) νµ and f (E) µν = -f (E) νµ . From the perturbed Maxwell equations, Eqs. (3.6), we can obtain the following relations connecting the above perturbative functions: where Λ ≡ l ( l +1). Notice that f (M) 23 ,lm and ˜ Ψ, defined as are gauge invariant variables (see Eq. (II-27) in Ref. [45] and Eq. (II-11) in Ref. [46]).", "pages": [ 4, 5 ] }, { "title": "A. Perturbations of a Dipole Magnetic Field: Exterior region", "content": "In the exterior vacuum region, we adopt the condition δJ µ = 0. With this condition, the perturbed electromagnetic fields will be determined via the linearized form of Maxwell's equations, Eqs. (3.5), (assuming that J µ = δJ µ = 0) together with the perturbed Maxwell equation (3.6). Equation (3.13) for µ = t and µ = r can be written down as where the indices ' A ' and ' P ' stand for axial and polar gravitational perturbative quantities, and obviously ' I ' stands for the t and r components of Eq. (3.13). The coefficients of Eq. (3.14) have the following expressions One can decompose the equations above for a specific mode with fixed harmonic numbers ( l, m ), by multiplying with Y ∗ lm and integrating over the two-sphere, i.e. In a similar way, from the two remaining equations, i.e. Eq. (3.13) for µ = θ and µ = φ , one gets the relations where These equations lead to an extra set of evolution equations for a specific mode ( l, m ) by multiplying with Y ∗ lm and integrating over the two-sphere: lm where the coefficients are given by χ = 1 2 B 2 h 1 ,lm , (3.35)", "pages": [ 5, 6, 7 ] }, { "title": "B. Perturbations of a Dipole Magnetic Field: Interior region", "content": "In the stellar interior, because we have adopted the ideal MHD approximation for which F µν u ν = 0, the components of the perturbed electromagnetic field tensor are determined by using the perturbed Maxwell equation (3.6), i.e. where δu µ is the perturbed fluid 4-velocity, defined as From Eq. (3.37) one can get the following equations where the coefficients A lm and B lm are functions of the perturbed electromagnetic fields, while ˜ A lm , ˜ B lm , and ˜ C lm are functions of the perturbed matter fluid 4-velocity. The expressions for these coefficients are By multiplying Eqs. (3.39), (3.40), and (3.41) with Y ∗ lm and integrating over the two-sphere we can obtain the following system of equations that depends only on r where Finally, we should compute Eqs. (3.23), (3.28), and (3.29) for the exterior region, and Eqs. (3.47), (3.48), and (3.49) for the interior region of the star. From this system of equations, we can see the specific couplings between the electromagnetic and gravitational perturbations. For example, an electromagnetic perturbation of specific parity with harmonic indices ( l, m ) depends on the gravitational perturbations of the same parity with ( l, m ) as well as the gravitational perturbations of the opposite parity with ( l ± 1 , m ). In other words, for the special and simpler case of axisymmetric perturbations ( m = 0), we arrive at the following conclusions: 1) Dipole electric (polar) electromagnetic perturbations will be driven by axial quadrupole gravitational perturbations, and 2) Dipole magnetic (axial) electromagnetic perturbations will be driven by polar quadrupole and radial gravitational perturbations. These two types of couplings will be discussed in detail in the next sections.", "pages": [ 7, 8 ] }, { "title": "C. Junction conditions for perturbed electro-magnetic fields", "content": "In order to close the system of equations derived in the previous subsection, we should impose appropriate junction conditions on the stellar surface. Such junction conditions for the perturbed electromagnetic fields can be derived from the conditions where n µ is the unit outward normal vector to the stellar surface, while q ν µ is the corresponding projection tensor associated with n µ . These junction conditions lead to the following set of equations:", "pages": [ 8 ] }, { "title": "IV. DIPOLE PERTURBATIONS OF A MAGNETIC FIELD ON A STELLAR BACKGROUND", "content": "In the previous section, we provided the general form of the perturbative equations. In order to focus on a simple case, we only consider axisymmetric perturbations ( m = 0) in this section. In this way, the various couplings become less complicated. Under these conditions, we study the excitation of dipole electric perturbations driven by axial gravitational ones and dipole magnetic perturbations driven by polar gravitational ones. These perturbative modes are actually the most important ones from the energetic point of view.", "pages": [ 8 ] }, { "title": "A. Dipole Electric Perturbations driven by Axial Gravitational Perturbations", "content": "Here, we consider only dipole 'electric type' perturbations driven by quadrupole axial gravitational perturbations. Since we neglect the back reaction of electromagnetic perturbations on the gravitational ones, the quadrupole axial gravitational perturbations of a spherically symmetric star can be described by a single wave equation [40, 47], which is given by where Note that r ∗ is the tortoise coordinate defined as r ∗ = r +2 M ln( r/ 2 M -1). Since there are no fluid oscillations if the matter is assumed to be described as a perfect fluid (unless we introduce rotation), the spacetime only contains pure spacetime modes, i.e. the so-called w -modes [42, 47, 48]. In this case, the axial component of the fluid perturbation, U lm , has the form U lm = -e -ν/ 2 h 0 ,lm /r 2 , while the component of h 0 ,lm is computed from the equation which is used later to simplify the coupling terms between the two types of perturbations. On the other hand, in the same way as in the case of electromagnetic perturbations in the exterior region, Eqs. (3.11) and (3.23) for I = 1, and Eq. (3.28) lead to three simple evolution equations for the three perturbation functions f (E) 12 , 10 , f (E) 01 , 10 , and f (E) 02 , 10 : where S (1) 20 and S (2) 20 are the source terms describing the coupling of the electromagnetic perturbations with the gravitational ones, and are given by In order to derive second-order wave-type equations for the electromagnetic perturbations, we introduce a new function: Ψ lm = Ψ lm ( t, r ), given by With this variable, the above evolution equations can be written as From this system of evolution equations, one can construct a single wave-type equation for the 'electric' perturbations where the source term S (E) 20 is given by Without the coupling term, this wave equation outside the star is the well-known Regge-Wheeler equation for electromagnetic perturbations. It should be pointed out that Ψ is not a gauge-invariant quantity while the function ˜ Ψ given by Eq. (3.12) is a gauge invariant variable, where both variables Ψ and ˜ Ψ can be related to each other via the evolution equation (4.11), i.e. where Finally, the electromagnetic perturbations in the interior region are determined from Eqs. (3.47), (3.48), and (3.11), i.e.", "pages": [ 8, 9, 10 ] }, { "title": "B. Dipole Magnetic Perturbations driven by Polar Gravitational Perturbations", "content": "As it was mentioned earlier in Sec. III for the case of axisymmetric perturbations, the 'magnetic (axial) type' perturbations of the electromagnetic field with harmonic index l are driven by polar gravitational perturbations with harmonic index l ± 1. Here, we consider the axisymmetric perturbations ( m = 0) for the dipole ( l = 1) electromagnetic fields, which are driven by quadrupole ( l = 2) gravitational perturbations. For the description of the perturbations of the spacetime and the stellar fluid, we adopt the formalism derived by Allen et al . in [44]. In this formalism, the perturbations are described by three coupled wave-type equations, in such a way that two equations describe the perturbations of the spacetime and the other one the fluid perturbations. In addition to these three wave equations, there is also a constraint equation. The two wave-type equations for the spacetime variables are where F lm , S lm , and H lm are given by while δp lm is the perturbation in the pressure, n ≡ ( l -1)( l +2) / 2, and C s is the sound speed. On the other hand, the wave equation for the perturbed relativistic enthalpy H lm , describing the fluid perturbations, is This third wave equation (4.25) is valid only inside the star, while the first two are simplified considerably outside the star, which can be reduced to a single wave-type equation, i.e. the Zerilli equation (see [44] and § VB). Finally, the Hamiltonian constraint, can be used for setting up initial data and monitoring the evolution of the coupled system. Regarding the quadrupole gravitational perturbations, the perturbation equation for the 'magnetic type' dipole in the exterior region is obtained from Eq. (3.29) as where Φ lm ≡ f (M) 23 ,lm and In order to derive the wave equation (4.27), we have used Eq. (3.10) and the ( r, φ )-component of the perturbed Einstein equations, i.e. e -ν ˙ H 1 -H ' 0 + K ' -ν ' H 0 = 0. We remark that the wave equation (4.27) without the source terms is the same as the one derived in [49, 50]. In addition, the other components of the electromagnetic perturbations, f (M) 12 , 10 and f (M) 02 , 10 , can be determined with Φ 10 via the relation (3.10). Finally, from Eq. (3.49) and Eq. (3.10), we can obtain the equation that determines the dipole 'magnetic type' perturbations for the interior region: where the perturbations of the fluid velocity, R 20 and V 20 , in the source term are given by", "pages": [ 10, 11 ] }, { "title": "V. PERTURBATIONS OF DIPOLE MAGNETIC FIELD ON A BH BACKGROUND", "content": "The perturbations of a dipole magnetic field on a Schwarzschild black hole background are described by the same set of perturbation equations as in the exterior region of the star except for the boundary conditions, i.e. the boundary conditions for the neutron star imposed on the stellar surface are Eqs. (3.53) - (3.55), while for the black hole case one should impose the pure ingoing wave conditions at the event horizon. Then, even in the case of the black hole background, we observe the same coupling of the various harmonics of the electromagnetic and gravitational perturbations as for the neutron star background. That is, for the axisymmetric perturbations, the 'electric' dipole ( l = 1) perturbations of the electromagnetic fields will be driven by axial quadrupole ( l = 2) gravitational perturbations, while the 'magnetic' dipole ( l = 1) perturbations of the electromagnetic fields will be driven by polar quadrupole ( l = 2) gravitational ones. In this specific case, our study is similar to the work in [24], although they use a different formalism.", "pages": [ 11 ] }, { "title": "A. Dipole Electric Perturbations driven by Axial Gravitational Perturbations (BH)", "content": "The axial quadrupole ( l = 2) gravitational perturbations are described by the Regge-Wheeler equation where In accordance with the results of Section IV A, the perturbations of the electromagnetic fields will be described by a single wave equation, that is, the Regge-Wheeler equation for electromagnetic perturbations, give by where the source term becomes of the same form as in Section IV A: Ψ lm = e ν f (E) 12 ,lm .", "pages": [ 11, 12 ] }, { "title": "B. Dipole Magnetic Perturbations driven by Polar Gravitational Perturbations (BH)", "content": "The equation describing the 'magnetic' type perturbations driven by the gravitational perturbations is the same equation as the one derived for a neutron star background (see Eq. (4.27)), that is where Φ lm = f ( M ) 23 ,lm , and the source term is also of the same form as in Eq. (4.28). The perturbative equation for the spacetime variables can be written in the form of the Zerilli equation where Λ 1 ≡ ( l +2)( l -1) / 2. Meanwhile, in the same way as for the neutron star background, one can also adopt F lm and S lm as the perturbation variables for the spacetime. In this case, the two wave equations simplify to become which have to be supplemented with the Hamiltonian constraint equation Note that there are useful relations between the perturbation variables ( F lm , S lm ) and the Zerilli function ( Z ), i.e. which can be used in constructing initial data (since the Zerilli function is gauge invariant and unconstrained), or for the extraction of the Zerilli function.", "pages": [ 12 ] }, { "title": "VI. APPLICATIONS", "content": "As an application, we consider the case in which dipole 'electric type' perturbations are driven by axial gravitational ones and present numerical results. First, we study the coupling on a Schwarzschild black hole background and later on the background of spherical neutron stars, as discussed in § VA and in § IVA, respectively. The more complicate cases that involve the driving of 'magnetic type' electromagnetic field perturbations driven by polar gravitational ones will be discussed elsewhere in the future.", "pages": [ 12 ] }, { "title": "A. Perturbations on a Black-Hole Background", "content": "In order to calculate the waveforms in the black hole background, we need to modify the background magnetic field near the event horizon. The reason for this is that the solution for a dipole magnetic field in vacuum diverges at the event horizon (see Eqs. (2.15) and (2.16)). In fact, the isolated black hole cannot have magnetic fields due to the no hair theorem. But, according to the simulations of the accretion onto the black hole, the magnetic field can reach almost to the event horizon, because the accreting matter will fall into the black hole with infinite time [51, 52]. Thus, we adopt a simple modification of the dipole magnetic field near the event horizon, that is, we set B 1 ( r ) = B 1 (6 M ) and B 2 ( r ) = B 2 (6 M ) for r ≤ 6 M , where the position at r = 6 M corresponds to the innermost stable circular orbit for a test particle around the Schwarzschild black hole. The magnetic dipole moment µ d is identified with the normalized magnetic field strength B 15 , defined as B 15 ≡ B p / (10 15 [ G ]), where B p is the field strength at r = 6 M and θ = 0. We assume vanishing electromagnetic perturbations i.e., Ψ 10 = ∂ Ψ 10 /∂t = 0 at the initial time slice t = 0, while the initial gravitational perturbations, X 20 , are prescribed in terms of a Gaussian wave packet. Under these initial conditions, the electromagnetic waves will result from the coupling to the gravitational ones. In the numerical calculations, we adopt the iterated Crank-Nicholson method [53] with a grid choice of ∆ r ∗ = 0 . 1 M and ∆ t = ∆ r ∗ / 2 (see [54] for the dependence of the choice of ∆ r ∗ and ∆ t on the waveforms). The energy emitted in the form of either gravitational ( E GW ) or electromagnetic waves ( E EM ), is estimated by integrating the luminosity ( L (A) GW ,l ) for the axial gravitational waves and for electric type electromagnetic waves ( L (E) EM ,l ), which are described by the following formulae [45, 46] ∣ ∣ In practice, we can find a relation between the energy emitted in gravitational and electromagnetic waves for a given initial spacetime perturbation, which has the form: where α is a 'proportionality constant'. In the simulation that we describe we set the magnetic field strength to the value B p = 10 15 Gauss. Fig. 1 shows the waveform of the gravitational wave observed at r = 2000 M , the amplitude is normalized to correspond to an emitted energy of E GW ≈ 1 . 8 × 10 49 ( M/ 50 M /circledot ) ergs. On the other hand, the waveforms of electromagnetic waves driven by the gravitational waves are shown in Fig. 2. From this figure, we can observe somewhat complicated waveforms of electromagnetic waves due to the coupling with the gravitational waves. From the specific waveforms, one can estimate the value of the proportionality constant in the relation (Eq. (6.3)) to be α = 8 . 02 × 10 -6 . This efficiency might not be very high, but the radiated energy of gravitational waves can reach ∼ 10 51 ergs for a black hole formation due to the merger of a neutron stars binary (see, e.g. [55]). In this case the strength of the magnetic field can be amplified by the Kelvin-Helmholz instability to reach values of the order of 10 15 -17 Gauss [56]. Although this is not an ideal situation for the black hole case we are considering in this paper, if one adopts the above efficiency for the case of a black hole formed after merger, one can expect that energies of the order of ∼ 10 46 -50 ergs can be emitted in the form of electromagnetic waves which can be potentially driven by the gravitational field perturbations. Furthermore, in Fig. 3, we show the Fast Fourier Transform (FFT) of the electromagnetic waveforms shown in Fig. 2, where for comparison we also add the frequencies of the quasinormal modes for l = 1 electromagnetic waves (dashed line) and for l = 2 gravitational waves (dot-dash line) radiated from the Schwarzschild black hole [45]. From this figure, one can obviously see that the driven electromagnetic waves have two specific frequencies corresponding to the l = 1 quasinormal mode of electromagnetic waves and the l = 2 quasinormal mode of gravitational waves. This means that it might be possible to see the effect of gravitational waves via observation of electromagnetic waves. However, electromagnetic waves with such a low frequencies could be coupled/absorbed by the interstellar medium (and/or accretion disk around the central object) during the propagation and it will be almost impossible to directly detect the driven electromagnetic waves. The only possible way to see the driven electromagnetic waves is the observation of indirect effects, such as synchrotron radiation.", "pages": [ 13 ] }, { "title": "B. Perturbations on a Neutron-Star Background", "content": "In order to examine the coupling between the emitted gravitational and electromagnetic waves in a neutron star background, we adopt the same initial conditions as for the black hole case, i.e. the electromagnetic perturbations are set to zero and the initial gravitational perturbations are approximated by an ingoing Gaussian wave packet. In the numerical calculations, we adopt a grid spacing of ∆ r = R/ 200 and a time step ∆ t/ ∆ r = 0 . 05, where R is the stellar radius. For the background stellar models, we adopt the polytropic equation of state (EOS) of the form P = Kρ Γ . Then, one can get the waveforms of the reflected gravitational waves and the induced electromagnetic ones. As an example, we show results for a stellar mode with Γ = 2 and K = 200 km 2 . Fig. 4 shows the waveforms of the gravitational waves (solid line) and the electromagnetic waves (dotted line) observed at r = 300 km, where we adopt two stellar models with different compactness M/R (see Table I for the stellar properties). Compared with the fast damping of gravitational waves, one can see the long-term oscillations in the electromagnetic waves, which can be driven not only by the quasinormal ringing of gravitational waves but also during the tail phase of the gravitational waves. For the waveforms shown in Fig. 4, the FFT is plotted in Fig. 5, where the left and right panels correspond to the FFT of the gravitational and electromagnetic waves, respectively. From this figure, one can see the same features as in the case of a black hole. Namely, the FFT of the electromagnetic waves driven by the gravitational waves has two specific frequencies, i.e. one is the proper electromagnetic oscillation (1st peak in the right panel of Fig. 5) and the other one is the oscillation corresponding to the gravitational waves (2nd peak in the right panel of Fig. 5). We remark that electromagnetic waves with such low frequencies could be absorbed by the interstellar medium and then, their direct detection is almost impossible. Namely, we should consider the secondary emission mechanism such as a synchrotron radiation. Maybe, the plasma around the central object will be excited after receiving the energy from the electromagnetic waves driven by the gravitational waves and move along with the magnetic field lines. Anyway, such a secondary emission mechanism will be discussed somewhere. Furthermore, we find that as in the case for a black hole, the relationship between the emitted energies of gravitational and electromagnetic waves can be described by Eq. (6.3), even for neutron stars, if B p is considered as the magnetic field strength at the stellar pole ( r = R and θ = 0). In practice, for the specific stellar models in Fig. 4, the proportionality constant becomes α = 1 . 61 × 10 -5 and 4 . 37 × 10 -6 for the particular stellar models with M/R = 0 . 162 and 0.237, respectively. In order to see the dependence on the stellar properties, we study a variety of stellar models with different stiffness of the equation of state and with different central densities, radii, and masses, which are given in Table I. As a result, we find that the proportionality constant α can be written as a function of the stellar compactness, which is almost independent of the stellar models and the adopted equation of state. In fact, in Fig. 6 we plot the values of α for various stellar models, where the circles, diamonds, and squares correspond to the results for the stellar models characterized by (Γ , K ) = (2 , 100), (2,200), and (2.25,600). From this figure, one can see that the proportionality constant α depends strongly on the stellar compactness, as expected, with typical values ranging from 10 -6 up to ∼ 10 -4 .", "pages": [ 14, 15, 16 ] }, { "title": "VII. CONCLUSION", "content": "/negationslash We have considered the coupling between gravitational and electromagnetic waves emitted by compact objects, i.e. black holes and neutron stars. We have derived a coupled system of equations describing the propagation of gravitational and electromagnetic waves. In our study we have investigated the driving of electromagnetic perturbations via their coupling to the gravitational ones. However, for simplicity, we have neglected the back reaction from the electromagnetic waves on the gravitational waves, because the magnetic energy of the compact objects, even for magnetars, is quite small as compared with the gravitational energy. We found that the electromagnetic waves of specific parity with harmonic indices ( l, m ) can be coupled to gravitational waves of the same parity and with harmonic indices ( l, m ) (for m = 0) and harmonic indices ( l ± 1 , m ), for every value of m . In particular, our findings lead to the result that, for the axisymmetric perturbations, i.e., m = 0, the dipole electric electromagnetic waves will be driven by axial quadrupole gravitational waves, while the dipole magnetic electromagnetic waves will be driven by polar gravitational waves. As an application of our perturbative framework, we presented numerical calculations for the case in which dipoleelectric electromagnetic waves are driven by the axial gravitational ones, both for the case of a black hole and a neutron star background. We found that the emitted energy in electromagnetic waves driven by the gravitational waves is proportional to not only the emitted energy in gravitational waves but also to the square of the strength of the magnetic field of the central object. For the case of a black hole background, the ratio of the emitted energy of the electromagnetic waves to that of the gravitational waves is around 8 × 10 -6 ( B p / 10 15 G) 2 , where B p is the magnetic field strength at r = 6 M . On the other hand, in the case of a neutron star background, we find that this proportionality constant can be written as a function of the stellar compactness. Although we have considered only the case of axial gravitational waves and the associated induced electromagnetic waves, the polar oscillations also play an important role in extracting the information about the neutron star structure since in the case of non-rotating stars, the matter oscillations are typically coupled to the polar gravitational waves. This is a direction that we are currently investigating.", "pages": [ 16, 17 ] }, { "title": "Acknowledgments", "content": "H.S. is grateful to Ken Ohsuga for valuable comments. This work was supported by the German Science Foundation (DFG) via SFB/TR7, by Grants-in-Aid for Scientific Research on Innovative Areas through No. 23105711, No. 24105001, and No. 24105008 provided by MEXT, by Grant-in-Aid for Young Scientists (B) through No. 24740177 provided by JSPS, by the Yukawa International Program for Quark-hadron Sciences, and by the Grant-in-Aid for the global COE program 'The Next Generation of Physics, Spun from Universality and Emergence' from MEXT. C.F.S. acknowledges support from contracts FIS2008-06078-C03-03, AYA-2010-15709, and FIS2011-30145-C03-03 of the Spanish Ministry of Science and Innovation, and contract 2009-SGR-935 of AGAUR (Generalitat de Catalunya). P.L. acknowledges the support from NSF awards 1205864, 0903973 and 0941417. [47] S. Chandrasekhar and V. Ferrari, Proc. Roy. Soc. London A432 , 247 (1991).", "pages": [ 17, 18 ] } ]

[ { "title": "Hidden Symmetries and Geodesics of Kerr spacetime in Kaluza-Klein Theory", "content": "Alikram N. Aliev Yeni Yuzyıl University, Faculty of Engineering and Architecture, Cevizlibaˇg-Topkapı, 34010 Istanbul, Turkey", "pages": [ 1 ] }, { "title": "Goksel Daylan Esmer", "content": "Istanbul University, Department of Physics, Vezneciler, 34134 Istanbul, Turkey (Dated: October 16, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "The Kerr spacetime in Kaluza-Klein theory describes a rotating black hole in four dimensions from the Kaluza-Klein point of view and involves the signature of an extra dimension that shows up through the appearance of the electric and dilaton charges. In this paper, we study the separability properties of the Hamilton-Jacobi equation for geodesics and the associated hidden symmetries in the spacetime of the Kerr-Kaluza-Klein black hole. We show that the complete separation of variables occurs only for massless geodesics, implying the existence of hidden symmetries generated by a second rank conformal Killing tensor. Employing a simple procedure built up on an 'effective' metric, which is conformally related to the original spacetime metric and admits a complete separability structure, we construct the explicit expression for the conformal Killing tensor. Next, we study the properties of the geodesic motion in the equatorial plane, focusing on the cases of static and rotating Kaluza-Klein black holes separately. In both cases, we obtain the defining equations for the boundaries of the regions of existence, boundedness and stability of the circular orbits as well as the analytical formulas for the orbital frequency, the radial and vertical epicyclic frequencies of the geodesic motion. Performing a detailed numerical analysis of these equations and frequencies, we show that the physical effect of the extra dimension amounts to the significant enlarging of the regions of existence, boundedness and stability towards the event horizon, regardless of the classes of orbits.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "As is known, one of the most attractive features of the ordinary Kerr spacetime, which describes a family of rotating black holes in general relativity (GR), is its separability structure. The Hamilton-Jacobi equation for geodesics admits a complete separation of variables [1], despite the fact that the spacetime possesses only two global isometries generated by two commuting Killing vector fields. Clearly, the separability structure implies an extra integral of motion that in turn signals the existence of hidden symmetries in the spacetime. The authors of [2] showed that this is indeed the case. The Kerr spacetime possesses hidden symmetries generated by a second rank symmetric Killing tensor, rendering the HamiltonJacobi equation for geodesics completely integrable. Moreover, the Kerr spacetime provides full quantum separability in both the Klein-Gordon equation [3] and the Dirac equation [4, 5]. While the existence of the Killing tensor plays a crucial role for the separability of the Klein-Gordon equation, the situation with the Dirac equation is more subtle. In addition to the Killing tensor, the Kerr spacetime also admits a second rank antisymmetric Killing-Yano tensor which can be thought of as a 'square root' of the Killing tensor [6]. It is the Killing-Yano tensor that lies behind the separability of the Dirac equation in the Kerr background [7]. Thus, in a sense, the usual 'square root' relationship between the Dirac and Klein-Gordon equations turns out to be echoed in the structure of hidden symmetries of the Kerr spacetime. In recent years, there has been an active interest in hidden symmetries of higherdimensional black hole spacetimes. In [8, 9], it was shown that the spacetime of higherdimensional rotating black holes given by the Myers-Perry metric [10], which generalizes the Kerr metric to all higher dimensions, admits both the Killing and Killing-Yano tensors. In other words, the hidden symmetries of the Kerr spacetime survive for the Myers-Perry spacetime in higher dimensions as well. To gain some insight into the origin of hidden symmetries generated by the Killing-Yano tensor, the authors of [11] managed to relate them to a new kind of supersymmetry , appearing in the worldline supersymmetric mechanics of spinning point particles in the Kerr background. In a recent work [12], it was shown that similar analysis based on the viewpoint of worldline supersymmetric mechanics also remains true for the higher-dimensional spacetime of the Myers-Perry black holes. The hidden symmetries of general rotating charged black holes in five-dimensional minimal gauged supergravity [13] as well as various black hole solutions of supergravity and string theories have been studied in a number of works (see, for instance, [14-19] and references therein). Intriguing generalizations of the Kerr spacetime have also been studied in Kaluza-Klein theory. In a relatively simple setting, the Kerr solution in Kaluza-Klein theory describes a rotating black hole in four dimensions from the Kaluza-Klein point of view and involves the signature of an extra dimension. This shows up through the appearance of the electric and dilaton charges, though the dilaton charge is not an independent parameter. That is, the solution satisfies the coupled Einstein-Maxwell-dilaton field equations, which are obtained from the Kaluza-Klein reduction of Einstein gravity in five dimensions. The procedure of obtaining such a solution is well known [20-22] and amounts to boosting a four-dimensional 'seed' solution under consideration in the fifth dimension with a subsequent Kaluza-Klein reduction to four dimensions. The most general solution for rotating black holes in KaluzaKlein theory was obtained in [23, 24] by employing a solution generation technique based on the use of hidden symmetries of the Einstein field equations. Recently, intriguing developments have also been towards exploring the physical effects of black holes in four and higher dimensions. Observations of rapidly rotating black holes (with the angular momenta approaching the Kerr bound in GR) in some X-ray binaries [25, 26] have sparked the old theoretical question of bona fide spacetime geometry around the black holes. In light of this, many investigators have studied gravitational effects of black holes both in GR and beyond it, focusing in some cases on the imprints of the extra dimension in our physical world (for instance, see Refs.[27-34] and references therein). The purpose of the present paper is two-fold: Firstly, we examine the separability structure and the hidden symmetries of the rotating black hole in the Kaluza-Klein framework (the Kerr-Kaluza-Klein black hole), where it carries the imprint of the extra fifth dimension through the electric and dilaton charges. We consider the Hamilton-Jacobi equation for a massive (uncharged) particle in the background of this black hole and show that the complete separation of variables occurs only for the vanishing mass of the particle, in contrast to the case of the original Kerr black hole in GR. This implies that the black hole spacetime under consideration possesses hidden symmetries generated by a second rank conformal Killing tensor. Next, we construct the explicit form for the conformal Killing tensor by employing a nice procedure built up on an effective metric, which is conformally related to the original spacetime metric and admits the separability structure due to the Killing tensor. Such a procedure of constructing the conformal Killing tensor was earlier used in [35] as well. Secondly, we explore the geodesic motion of the uncharged massive particle in the equatorial plane of the Kerr-Kaluza-Klein black hole. Using the Hamilton-Jacobi and geodesic equations, we study the effects of the extra fifth dimension on the properties of the circular motion around this black hole. We show that the extra dimension has its greatest effect in enlarging the regions of existence, boundedness and stability of the circular motion towards the event horizon, regardless of the classes of orbits. The outline of the paper is as follows: In Sec.II we describe a theoretical framework for the Kerr-Kaluza-Klein black hole. This includes a brief recalling the construction of the pertaining spacetime metric, the description of its physical properties as well as the properties of the Hamilton-Jacobi equation in this spacetime. In Sec.III we introduce a procedure that builds up on the use of an effective metric, conformally related to the original spacetime metric and admitting the Killing tensor. Here we show that such a procedure enables one to construct the conformal Killing tensor for the Kerr-Kaluza-Klein spacetime under consideration. In Sec.IV we study the properties of the circular and quasicircular (epicyclic) motions in the equatorial plane of both static and rotating Kaluza-Klein black holes. In both cases, we present defining equations for the boundaries of the existence regions as well as for the boundaries of the regions of boundedness and stability of the circular motion. Here we also present the results of a detailed numerical analysis of these equations. Next, using the general theory of the epicyclic motion, earlier developed in [36, 37], we give the analytical expressions for the orbital, radial and vertical epicyclic frequencies and perform the numerical analysis of these expressions. In Sec.V we conclude with the discussion of our results.", "pages": [ 2, 3, 4 ] }, { "title": "II. THE KERR-KALUZA-KLEIN BLACK HOLE", "content": "We begin by recalling briefly the construction of the exact solution that represents a rotating black hole in Kaluza-Klein theory, namely the Kerr-Kaluza-Klein black hole with the Maxwell and dilaton fields. The details of the construction can be found in the original paper [21] as well as in a recent paper [22], including a NUT parameter as well. At the first step, the procedure of obtaining this solution amounts to adding an extra spacelike flat dimension to the usual Kerr solution of four-dimensional GR. Thus, in the Boyer-Lindquist coordinates we have the five-dimensional metric given by where the parameters M and a determine the mass and the angular momentum of the solution. Next, one needs to boost this metric in the fifth dimension by and with the velocity of the boost v = tanh α . Clearly, the boosted metric will satisfy the vacuum equations of five-dimensional GR. Putting this metric into the standard KaluzaKlein form we compactify the extra fifth dimension, identifying the four-dimensional metric and the associated potential one-form A and the dilaton field Φ, which are given by Here we have used the notation We see that for the vanishing boost velocity, α → 0, the Maxwell and dilaton fields vanish and the metric in (5) reduces to the original Kerr solution. It is straightforward to check that solution (5), accompanied with the Maxwell and dilaton fields given in (6), satisfies the equation of motion derived from the four-dimensional action of Kaluza-Klein theory where F = dA . We recall that this action is obtained from the five-dimensional Einstein action for the metric in the form given by (4). (See Ref.[22] for details).", "pages": [ 4, 5 ] }, { "title": "A. Physical Properties", "content": "It is easy to see that the spacetime in (5) admits two commuting Killing vectors ξ ( t ) = ∂/∂t and ξ ( φ ) = ∂/∂φ , which reflect its time-translational and rotational invariance. Calculating the various scalar products of these vectors, we arrive at the metric components in the form On the other hand, as follows from metric (5), the boosting and dimensional reduction procedures do not change the location of the event horizon. It is still determined by the largest root of the equation ∆ = 0, which is given by implying that the horizon exists provided that a ≤ M . As for the physical parameters of the metric, the total mass, angular momentum and the total electric charge, they can be determined by evaluating the corresponding Komar integrals and the flux integral over a 2-sphere at spatial infinity, respectively. This has been done in works [20-22]. Writing these parameters in terms of the boost velocity v , we have It should be noted that the dilaton charge is not independent as it can be expressed in terms of the other parameters [20, 22]. Clearly, the ultrarelativistic limit v → 1 implies the vanishing of the 'seed' (unboosted) mass M as well, thus keeping the physical mass M fixed. Another important feature of spacetime (5) arises from its dragging properties, which can easily be understood by introducing a family of locally nonrotating observers. These observers move on orbits with constant r and θ and with a four-velocity u µ , obeying the condition u · ξ ( φ ) = 0. From this condition, we find that the coordinate angular velocity of these observers is given by At large distances, we have the following expansion for the angular velocity which reveals the dragging property of metric (5) in the φ -direction, vanishing at spatial infinity. This expansion also confirms the physical angular momentum of the metric, given in (11). Meanwhile, as follows from equation (12), towards the event horizon the angular velocity increases, approaching its constant value at r = r + . Thus, we have It is not difficult to show that the corotating Killing vector defined as ξ ( t ) + Ω H ξ ( φ ) is tangent to the null surface of the horizon. That is, the quantity Ω H is nothing but the angular velocity of the horizon. We note that the angular velocity of the extreme horizon, a = M , diverges in the ultrarelativistic limit v → 1 as the horizon radius in this limit shrinks to zero, by equations (10) and (11). Therefore, in the following we will focus only on the physically acceptable values of the boost velocity, i.e. on those obeying the condition v < 1. In summary, the spacetime metric in (5) generalizes the Kerr solution of general relativity to include the signature of the extra fifth dimension that in four dimensions shows up through the appearance of the Maxwell and dilaton fields. In other words, it describes a rotating black hole from the Kaluza-Klein point of view, whose physical properties were briefly described above.", "pages": [ 6, 7 ] }, { "title": "B. The Hamilton-Jacobi Equation", "content": "Let us now consider the geodesic motion of a massive (uncharged) particle in spacetime (5) of the Kerr-Kaluza-Klein black hole. The Hamilton-Jacobi equation governing the geodesic motion is given by where λ is an affine parameter. Since the spacetime under consideration possesses two commuting timelike and spacelike Killing vectors, one can assume that the action S admits the following representation Here F ( r, θ ) is an arbitrary function of two variables, the constants of motion correspond to the mass m , energy E and to the angular momentum L of the particle. If we now substitute this action along with the contravariant metric components into equation (15), it is straightforward to show that the latter can be put in the form We note that separation of r and θ variables in this equation does not occur due to the presence of the factor B on its right-hand side (the explicit form of B is given in Eq. (7)). On the other hand, such a separation occurs for the particle of zero mass ( m = 0), impliying the existence of a new conserved quantity, quadratic in 4-momentum, along the null geodesics. This is due to the fact that the spacetime metric in (5) admits hidden symmetries, generated by a second rank symmetric conformal Killing tensor [2], which give rise to the new nontrivial integral of motion. Below, we explore the hidden symmetries and present the explicit form for the conformal Killing tensor.", "pages": [ 7, 8 ] }, { "title": "III. HIDDEN SYMMETRIES", "content": "We have seen that one of the salient features of the spacetime metric in (5) is that it does not allow for the complete separation of variables in the Hamilton-Jacobi equation for massive particles. Interpreting this fact in terms of pertinent hidden symmetries, one concludes that the spacetime does not admit hidden symmetries, which are generated by a second rank symmetric Killing tensor K µν , in contrast to the original Kerr spacetime. We recall that the Killing tensor obeys the equation where the operator ∇ denotes the covariant differentiation and the round brackets stand for symmetrization over the indices enclosed. Let us now assume that such a Killing tensor exists for an effective metric h µν , which is conformally related to the original metric g µν in (5) as follows where Ω is a smooth scalar function. It is straightforward to show that the associated Christoffel symbols γ λ µν and Γ λ µν for the metrics h µν and g µν , respectively, are related as Meanwhile, for the covariant derivatives of a second rank symmetric tensor P µν we find the relation where and the operator D denotes covariant differentiation with respect to the metric h µν , the comma stands for the partial derivative. It is also straightforward to show that for the tensor P µν defined as where K µν is the Killing tensor in the metric h µν , equation (22) reduces to the form in which we recognize the defining equation for the conformal Killing tensor P µν [2]. It is worth noting that for the one-form I = I µ dx µ being exact, the conformal Killing tensor goes over into the Killing tensor (see equation (19)), implying the representation where f is a scalar function. With all this in mind, we turn now to the effective metric h µν in (20). Choosing the function Ω in the form we see that the Hamilton-Jacobi equation in the effective metric admits a complete separation of variables. In other words, in this case under consideration the factor B on the right-hand side of equation (18) disappears and the resulting equation allows for separation in the r and θ variables. Thus, for the action S in the form we arrive at two independent ordinary differential equations where K is a constant of separation. It is evident that the separability occurs due to the existence of the new quadratic integral of motion K = K µν p µ p ν , which is guaranteed by the existence of the irreducible Killing tensor K µν in the effective metric h µν , thereby confirming our assumption made above. Using this fact in equation (30) and taking into account the relation m 2 = -h µν p µ p ν , we obtain the explicit form of the Killing tensor. It is given by For the vanishing boost parameter, α → 0, this expression agrees with that obtained in [2] for the ordinary Kerr spacetime. Next, using equations (20) and (24) along with equation (27), we calculate the nonvanishing components of the 'current-vector' in (23). As a consequence, we find that the associated current one-form is given by In obtaining this expression we have also used the K r r and K θ θ components of the Killing tensor in (31). We see that this expression vanishes for a = 0 and in this case, as follows from equation (25), the metric in (5) admits the reducible Killing tensor. However, in the general case it admits the conformal Killing tensor defined in (24). Performing some calculations, we find the explicit form for the conformal Killing tensor where for the Killing tensor K µν is given by which is obtained by lowering the contravariant indices of the tensor in (31) with respect to the metric h µν . It is straightforward to verify that the conformal Killing tensor (33) satisfies equation (25) with the current-vector given in equation (32).", "pages": [ 8, 9, 10, 11 ] }, { "title": "IV. GEODESIC MOTION IN THE EQUATORIAL PLANE", "content": "In this section, we restrict ourselves to the description of geodesics in the equatorial plane of the Kerr-Kaluza-Klein black hole. As we have mentioned above, such a black hole transmits the imprint of the extra fifth dimension into four-dimensional spacetime through the appearance of the electric and dilaton charges. As a consequence, its physical parameters become substantially different from those of the original Kerr black hole, as given in (11). Clearly, the effect of the extra dimension would also change the properties of observable orbits near the black hole. To get some insight into this issue, it is useful to explore the equatorial geodesic motion in metric (5). For this motion, θ = π/ 2, from equation (18) it follows that ∂F/∂θ = 0 and where B 0 denotes the value of B in (7) taken on the equatorial plane i.e. With this in mind, using action (16) in the equation we obtain the following equations of motion in the equatorial plane where the effective potential in the radial equation (40) is given by When the right-hand side of equation (40) vanishes, the geodesic motion occurs in circular orbits. The energy and the angular momentum of these orbits are given by the simultaneous solutions of the equations Meanwhile, the region of stability of the circular orbits is governed by the inequality where the case of equality refers to the innermost stable orbits. It is worth to note that one can also provide an intriguing description of the equatorial motion in black hole spacetimes by invoking the geodesic equation where Γ µ αβ are the Christoffel symbols of the spacetime under consideration and the parameter s is supposed to be the proper time along the geodesics. Such a description possesses some simplifying advantages, in particular, when exploring the quasiequatorial motion by using the method of successive approximations. In this approach, the circular motion in the equatorial plane is described at the zeroth-order approximation. The position four-vector of the circular orbits is given by where Ω 0 is the orbital frequency of the motion and it is determined by the µ = 1 component of equation (44) on the equatorial plane. Meanwhile, the quasicircular or epicyclic motion occurs due to small perturbations about the circular orbits and it is the subject to the first-order approximation scheme. Substituting the associated deviation vector for small perturbations into the geodesic equation (44), we perform an appropriate expansion in ξ µ , restricting ourselves only to the first-order terms. As a consequence, we obtain the following equation where the quantities γ µ α and ∂ a U µ are taken on a circular orbit r = r 0 , θ = π/ 2 and we have passed to the coordinate time t , instead of the proper time s . After a simple algebra, we have Next, writing down the components of equation (47), it is not difficult to show that the epicyclic motion consists of two decoupled oscillations in the radial and vertical directions, which are governed by the equations respectively. It also follows that the conditions Ω 2 r ≥ 0 and Ω 2 θ ≥ 0 determine the stability of the circular motion against small oscillations. Thus, in this framework the description of the equatorial motion in black hole spacetimes can be performed in terms of three fundamental frequencies, the orbital frequency Ω 0 , the radial Ω r and the vertical Ω θ epicyclic frequencies. We note that the general description of the epicyclic motion in the spacetime of stationary black holes was first given in works [36, 37]. Some details of this description can also be found in recent works [30, 33]. Below, we calculate the physical parameters of the geodesic motion occurring in both equatorial and off-equatorial planes of spacetime (5), using the frameworks of the Hamilton-Jacobi equation as well as the geodesic equation described above. To make the things more transparent, it is instructive to consider the cases of static and rotating black holes separately. In what follows, to figure out the effects of the extra dimension, we will express all quantities of interest only in terms of the boost velocity v and the physical mass of the black hole. This also makes the things much simpler than expressing them in terms of the electric and dilaton charges.", "pages": [ 11, 12, 13, 14 ] }, { "title": "A. The static case", "content": "Setting in equation (41) the rotation parameter a to zero and solving the simultaneous equations in (42), we find that the energy and the angular momentum of the circular motion around a Schwarzschild-Kaluza-Klein black hole are given by It is not difficult to see that the region of existence of the circular motion extends from infinity up to the limiting photon orbit, whose radius is governed by the vanishing denominator of (51). Thus, in terms of the boost velocity v and the physical mass M , we have the equation the largest root of which is given by It follows that for v = 0, the radius of the limiting photon orbit r ph = 3 M as for the original Schwarzschild black hole, while it moves towards the event horizon with the growth of v and we find that for v /similarequal 0 . 95, r ph /similarequal 0 . 7 M . (We recall that for v = 0, we have M = M , as follows form Eq.(11)). It is clear that not all circular orbits in the region of existence are bound. The radius of bound circular orbits obeys the inequality r > r mb , where the radius of the innermost bound orbit r mb is given by the largest root of the equation In obtaining this equation we have used the condition E 2 = m 2 , writing the result in terms of the physical mass of the black hole. We note that for v = 0, r mb = 4 M , while for v /similarequal 0 . 95, we find that r mb /similarequal 0 . 8 M . As for the region of stability of the circular motion, its boundary is determined by the equation which is obtained by using equations (51) and (52) in (43). It follows that for v = 0, the radius of the innermost stable circular orbit r ms = 6 M , while for v /similarequal 0 . 95, we have r ms /similarequal 1 . 3 M . We note that in the ultrarelativistic limit, v → 1, the radius of the limiting photon orbit as well as the radii of the innermost bound and stable orbits shrink to zero, merging with the singular horizon r + = 0. The results of a detailed numerical analysis of equations (54)- (56) are plotted in Figure 1. We note that with increasing the boost velocity v , the regions of existence, boundedness and stability of the circular motion essentially enlarge towards the event horizon, thereby clearly showing up the physical effects of the extra fifth dimension. (In this figure and in the following ones we take M = 1 that makes all physical quantities of interest dimensionless). Another quantity of physical interest is the binding energy of the innermost stable circular orbit. Using expression (51), it is not difficult to show that for r ms /similarequal 1 . 3 M and v /similarequal 0 . 95, the binding energy per unit mass of a particle is or nearly 16.3% of the particle rest-mass energy. That is, the energy-release process in the vicinity of the Schwarzschild-Kaluza-Klein black hole is potentially much more efficient than for the original Schwarzschild black hole, for which it is about 5.72% of the rest-mass energy. We turn now to the description of the equatorial motion in terms of the orbital and epicyclic frequencies. From the µ = 1 component of equation (44) on the equatorial plane, θ = π/ 2, we find that the orbital frequency is given by where Ω s = M 1 / 2 /r 3 / 2 is the Kepler frequency and It is not difficult to show that using this expression for the orbital frequency in the normalization condition for the four-velocity of the particle g µν u µ u ν = -1, with equation (45) and E = mu 0 in mind, we obtain the same expression for the energy of the circular motion as that given (51). Next, using equations in (49) and (50) it is straightforward to show that the vertical epicyclic frequency Ω θ is precisely the same as the orbital frequency, Ω 2 θ = Ω 2 0 , while for the radial epicyclic frequency we find that where the function h ( r, M , v ) is the same as that given on the left-hand side of equation (56). It follows that the circular motion is always stable against small oscillations in the vertical direction ( Ω 2 θ ≥ 0 ), while the boundary of the stability region in the radial direction is given by the condition Ω 2 r = 0, resulting in the same equation as in (56). In Figure 2 we plot the radial epicyclic frequency as a function of the radii of circular orbits for given values of the boost velocity. We see that with increasing the boost velocity, the location of the maximum moves towards the event horizon of the black hole. Furthermore, for the large enough value of the boost velocity, the pertaining highest value of the radial epicyclic frequency is significantly greater compared to that for the original Schwarzschild black hole, v = 0.", "pages": [ 14, 15, 16, 17 ] }, { "title": "B. The rotating case", "content": "In this case, the simultaneous solution of the equations in (42), determining the energy and the angular momentum of the circular motion turns out to be very formidable. Therefore, we appeal to the geodesic equation (44), in which case one gains some simplifying advantages. Substituting in this equation the Christoffel symbols for metric (5), with equation (45) in mind, we find that its µ = 0 , 2 , 3 components become trivial, while the µ = 1 component yields the defining equation for the orbital frequency Ω 0 of the circular motion. Solving this equation, we obtain that where we have used the notation The upper sign in the numerator of (61) refers to the direct orbits (the motion of the particle is corotating with respect to the rotation of the black hole), whereas the lower sign corresponds to the retrograde, counterrotating motion of the particle. Meanwhile, from the normalization condition g µν u µ u ν = -1 , we find that the energy and the orbital frequency of the circular motion are related by where the components of the metric tensor are given in equation (9) with θ = π/ 2. From this equation it follows that the radius of the limiting photon orbit is governed by the equation Next, we substitute in this equation the expression for the orbital frequency given in (61) and express the result in terms of the physical mass of the black hole. Solving the resulting equation numerically, we find that in the limit of the extremal rotation, a → M , and for v = 0, we have r ph /similarequal 1 . 23 M ( a = 0 . 98 M ) for the direct motion, while r ph /similarequal 4 M ( a = M ) for the retrograde motion just as for an extreme Kerr black hole. On the other hand, for v = 0 . 95 and for the the rotation parameters as given above, we find that r ph /similarequal 0 . 22 M and r ph /similarequal 0 . 92 M for direct and retrograde orbits, respectively. It is also not difficult to show that the radius of the innermost bound orbits is given by the equation which is obtained from equation (64) with E 2 = m 2 . Again, substituting expression (61) into this equation and performing the similar numerical analysis as in the case of (65), we find that for v = 0, r mb /similarequal 1 . 50 M (direct orbits, a = 0 . 95 M ) and r mb /similarequal 5 . 83 M (retrograde orbits, a = M ). Meanwhile, with the rotation parameters as given above and with v = 0 . 95, we have r mb /similarequal 0 . 27 M for direct orbits and r mb /similarequal 0 . 18 M for retrograde orbits. As in the static case, to explore the stability of the circular motion in the radial and vertical directions we need to know the explicit expressions for the pertaining epicyclic frequencies given in equations (49) and (50). Using in equation (49) the components of the Christoffel symbols for metric (5) and performing straightforward calculations, we find that the radial epicyclic frequency is given by where For the vanishing rotation parameter, a = 0, this expression agrees with that given in (60) for the static black hole, whereas for v = 0, it goes over into the expression for the original Kerr black hole [36, 37]. (See also works [30, 33]). The boundary of the stability region in the radial direction is determined by the equation Ω 2 r = 0. Writing this equation in terms of the physical mass of the black hole, we apply a numerical analysis to explore its solutions in the extremal limit of rotation a → M . In particular, we find that for the direct motion and for a = 0 . 95 M , the radii of the innermost stable orbits r ms /similarequal 1 . 93 M ( v = 0) and r ms /similarequal 0 . 4 M ( v = 0 . 95). Meanwhile, for the retrograde motion and for a = M we have r ms = 9 M ( v = 0) and r ms /similarequal 1 . 95 M ( v = 0 . 95). For an extreme Kerr-Kaluza-Klein black hole, the results of the full numerical analysis of the boundaries of the circular motion are plotted in Figure 3. The curves clearly show that as the boost velocity increases the radius of the limiting photon orbit as well as the radii of the innermost bound and the innermost stable orbits essentially enlarge towards the event horizon, both for direct and retrograde motions. It is also of interest to explore the dependence of the radial epicyclic frequency on the radii of circular orbits. In Figure 4 we illustrate this dependence in the limit of the extremal rotation, a → M , and for different values of the boost velocity. We see that with increasing the boost velocity, the locations of the maxima shift towards the event horizon for both direct and retrograde orbits. Accordingly, the pertaining values of the radial epicyclic frequency become significantly higher (especially for the retrograde motion) compared to the case of the original Kerr black hole, v = 0. Similarly, using in equation (50) the associated Christoffel symbols for metric (5) it is not difficult to show that the vertical epicyclic frequency is given by where It is easy to show that for v = 0 this expression coincides with that for the ordinary Kerr spacetime, earlier obtained in [36, 37]. A detailed numerical analysis of expression (69) shows that it is always nonnegative in the regions of existence and radial stability of the circular motion. In other words, the circular motion is stable with respect to small perturbations in the vertical direction. In Figure 5 we plot the vertical epicyclic frequency as a function of the radii of direct orbits in the field of an extreme Kerr-Kaluza-Klein black hole. We note that for the direct motion, the vertical epicyclic frequency attains its highest value in the near-horizon region. Again, the growth of the boost velocity results in moving the location of the maxima towards the event horizon, thereby significantly increasing the maximum value of the vertical epicyclic frequency. It is also interesting that the location of the maxima lies in the region of the radial stability of the motion. For instance, solving the equation ∂ Ω θ /∂r = 0 numerically, we find that for a = 0 . 95 M and v = 0 . 95, the location of the maxima is given by r max /similarequal 0 . 42 M , which is greater than the radius of the pertaining innermost stable circular orbit. To conclude this subsection, we wish to calculate the binding energies of the innermost stable circular orbits in the field of the extreme Kerr-Kaluza-Klein black hole, for both direct and retrograde motions. Using expression (64) and performing some numerical calculations, with equation (61) in mind, we find that for the direct motion in contrast to the binding energy E binding /similarequal 19 % of a particle in the Kerr field with a = 0 . 95 M and r ms /similarequal 1 . 93 M . Similarly, for the retrograde motion we obtain that while in the Kerr field with a = M and r ms /similarequal 9 M , we have E binding /similarequal 3 . 7 %. Thus, our analysis shows that the rotating Kaluza-Klein black holes are more energetic objects, compared to the original Kerr black holes, in the sense of the potential energy-release process in their vicinity.", "pages": [ 17, 18, 19, 20, 21, 22, 23 ] }, { "title": "V. CONCLUSION", "content": "The remarkable property of rotating black holes in Kaluza-Klein theory is that they involve the imprint of the extra dimension through the appearance of additional charges in the spacetime metric. In the most simple setting, it is the Kerr spacetime that from the Kaluza-Klein point of view carries the signature of the extra fifth dimension by acquiring the electric and dilaton charges. In this paper, we have examined the separability structure of the Hamilton-Jacobi equation for geodesics and the pertaining hidden symmetries in the spacetime of the Kerr-Kaluza-Klein black hole. We have shown that in the general case of massive geodesics, the Hamilton-Jacobi equation does not admit the complete separation of variables, whereas such a separability occurs for massless geodesics. This fact implies the existence of hidden symmetries in the spacetime, which are generated by a second rank conformal Killing tensor. Next, we have employed a simple framework based on the effective metric which has the following properties: (i) it is conformally related to the original spacetime metric under consideration, (ii) it admits the Killing tensor, rendering the associated Hamilton-Jacobi equation for massive geodesics completely separable. With this framework, we have constructed the explicit expression for the conformal Killing tensor. We have also examined the properties of the geodesic motion in the equatorial plane of the Kerr-Kaluza-Klein black holes, using the frameworks of both the Hamilton-Jacobi and geodesic equations. In order to make the description more transparent, we have considered the cases of static and rotating black holes separately. For both cases, we have obtained the analytical expressions for the energy and angular momentum/orbital frequency of the circular motion as well as we have derived the defining equations for the boundaries of the regions of existence, boundedness and stability of the motion. In order to gain some simplifying advantages, we have also invoked the description of the geodesic motion in terms of three fundamental frequencies: The orbital frequency, the radial and vertical epicyclic frequencies and we have obtained the associated analytical expressions for these frequencies. Next, applying a numerical analysis, we have found that the greatest effect of the extra fifth dimension amounts to the significant enlarging of the regions of existence, boundedness and stability towards the event horizon, regardless of the classes of orbits. Furthermore, it turns out that for the large enough values of the boost velocity, the locations of the maxima of the epicyclic frequencies essentially shift towards the event horizon, thereby resulting in much greater values of these frequencies, compared to those for the original Schwarzschild/Kerr black holes, respectively. Finally, we have explored the binding energy of the innermost stable circular orbits for both the static and rotating Kaluza-Klein black holes. It is interesting that for these black holes the energy-release process in their vicinity turns out to be potentially much more efficient than for the ordinary Schwarzshild and Kerr black holes of general relativity. It should be emphasized that throughout the paper we have focused on the physical aspects of our description. Of course, it would also be of interest to explore possible astrophysical implications of our results, especially in the context of high frequency quasiperiodic oscillations observed in some black hole binaries. This is an intriguing task for future work.", "pages": [ 23, 24 ] }, { "title": "VI. ACKNOWLEDGMENTS", "content": "One of us (A. N. Aliev) thanks Ekrem C¸alkılı¸c and H. Husnu Gunduz for their invaluable encouragement and support. He also thanks the Scientific and Technological Research Council of Turkey (T UB ˙ ITAK) for partial support under the Research Project No. 110T312. The work of G. D. E. is supported by Istanbul Univesity Scientific Research Project (BAP) No. 9227.", "pages": [ 24 ] } ]

[ { "title": "Dirac prescription from BRST symmetry in FRW space-time", "content": "Francesco Cianfrani ∗ Instytut Fizyki Teoretycznej, Uniwersytet Wroc/suppresslawski, pl. M. Borna 9, 50-204 Wroc/suppresslaw, Poland, EU.", "pages": [ 1 ] }, { "title": "Giovanni Montani †", "content": "ENEA - C.R, UTFUS-MAG, Via Enrico Fermi 45, 00044 Frascati, Roma, Italy, EU and Dipartimento di Fisica, Universit'a di Roma 'Sapienza', Piazzale Aldo Moro 5, 00185 Roma, Italy, EU A procedure to define the BRST charge from the Noether one in extended phase space is given. It is outlined how this prescription can be applied to a Friedmann-Robertson-Walker space-time with a differential gauge condition and it allows us to reproduce the results of [20]. Then we discuss the cohomological classes associated with functions in extended phase space having ghost number one and we recover the frozen formalism for classical observables. Finally, we consider the quantization of BRST-closed states and we define a scalar product which implements the superHamiltonian constraint. PACS numbers: 04.60.-m,04.60.Kz", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The realization of a quantum theory for the gravitational field represents one of the most challenging issue in theoretical physics. In view of the up-to-now lack of experimental data which can guide us in the realization of a final theory, the current attempts (as for instance Loop Quantum Gravity [1] and the Asymptotic Safety Scenario in Quantum Gravity [2]) are based on extending to geometrodynamics some ideas (as quantization of the holonomyflux algebra and Wilsonian renormalization approach, respectively) which have found a fruitfull application in the description of fundamental interactions. In this work, we will take BRST symmetry as our guide to analyze the fate of time parametrization invariance in a quantum theory for the Friedmann-Robertson-Walker model. BRST simmetry [3] plays a prominent role in the standard paradigm of fundamental interactions. In fact, the development of a meaningful path-integral formulation for a Yang-Mills theory requires to fix a gauge condition, which implies that the original gauge symmetry is broken. Nevertheless, the extension of phase-space via the introduction of additional variables allows us to recover a formulation with a residual global invariance, i.e. BRST symmetry, associated with nihilpotent transformations. It is such a symmetry which implies that transition amplitudes do not depend on the adopted gauge-fixing condition, such that all the nice properties ( in primis renormalization) found through a perturbative expansion hold on a non-perturbative level too [4]. Moreover, it is possible to trace back the origin of BRST symmetry to the peculiar features of the phase-space for constrained systems [5]. In fact, in the presence of first-class constraints observables are defined by a two-step procedure: i) they must be restricted to the constraint hypersurfaces, ii) they must be constant along the gauge orbits. These two requirement can be satisfied by enlarging phase-space via the introduction of some Grassmanian variables and by defining observables as functions belonging to the first cohomological class of a differential operators s implementing i) and ii). It is possible to associate a canonical action to s , such that a proper BRST charge Ω can be defined. This is the case on a Hamiltonian level, the Batalin-Fradkin-Vilkowsky (BFV) [6] theory, as well as on a Lagrangian level, the Batalin-Vilkovinsky (BV) [7] framework. This two formulations have been proved to be perturbatively equivalent [8]. However, in general s contains some additional higher order (in ghost number) terms, such that one must consider an iterative expansion in the ghost number and look for a solution order by order (see for instance [9]). Only if the gauge symmetry is simple (as in the case of a Lie algebra) an explicit expression is known for the charge. For gravity in a 3+1 representation, the gauge group is the product of 3-diffeomorphisms and time reparametrizations. The gauge algebra is open and the definition of a proper BRST charge has still technical issues, such that it has been accomplished only on a perturbative level [10] or in 2+1 dimensions [11]. The applications of the BFV formalism in minisuperspace has been realized in the seminal work of Halliwell [12], where the Wheeler-DeWitt equation has been recovered from a path integral formulation. In [13] the BV approach has been adopted to develop a proper BRST invariant Lagrangian for a Bianchi IX model in the presence of a differential gauge condition. Thanks to the presence of time derivatives of the lapse function in the total Lagrangian, the Hamiltonian in extended phase space is free from constraints and it can be defined simply by a Legendre transformation. By the direct inspection of the equations of motion, it was inferred the expression of the BRST charge in a Bianchi IX model. However, the definition of the BRST charge and transformations for more general metrics is hampered by the fact that the equations of motion are too complex for a direct inspection. Then, it has been suggested that BRST symmetry does not hold on a quantum level in absence of asymptotic states (as for a closed FRW model). This lead to the materialization of a reference frame, whose cosmological implications have been discussed in [14]. In this work, starting from the Hamiltonian formulation developed in [13], we outline how to find out the proper BRST charge. In a theory as GR, one deals with Lagrangian containing second order derivatives, which can be avoided by perfoming some partial integrations and discarding boundary contributions. Of course, this procedure does not affect the classical equations of motion, which are evaluated by considering vanishing variations at the boundary, thus also the classical symmetries are untouched. However, the resulting action can be not explicitly invariant under the original symmetries and the variation can provide some boundary contributions. These boundary contributions will enter the definition of the conserved charge. We will show that this is the case for a FRW model, in which the first order action is not BRST-invariant, but a boundary contribution arises. By properly accounting for these additional terms, the conserved charge can defined and its expression coincide with the one given in [13]. Then, we will characterize the BRST cohomological classes for functions having ghost number one. We will show how it is possible to choose proper functions Φ rep along the orbits generated by the BRST charge such that the closure condition fixies the dependence from the lapse function, while exact forms are obtain via the Poisson action of the superHamiltonian. This achievements ensure that a 1-1 correspondence between classical observables and BRST cohomological classes exists and that the Poisson action of the physical Hamiltonian in extended phase space vanishes. Finally, we will quantize the system by considering Φ rep as wave functions. We will find out a proper definition of the scalar product, which implements the superHamiltonian constraint mimicking the procedure defined in [5]. This way, after integrating out the ghosts and gauge variables, we will infer an expression for the scalar product in the kinematical Hilbert space which reproduces the result of refined algebraic quantization [16, 17], thus also the Dirac prescription for the quantization of constrained systems [15]. In particular, the manuscript is organized as follows. In sec.2 we review the prescriptions given by Dirac, by refined algebraic quantization and by the BRST formulation for the canonical quantization of constrained systems. In sec.3 the Hamiltonian formulation of the FRW model is presented in extended phase space. Sec. 4 is devoted to establish the relationship between the Noether charge and the one in the presence of boundary contributions, so giving a new derivation for the BRST charge in the FRW case. The cohomological classes are discussed in sec.5, while in sec.6 the quantization of the associated system is defined and the scalar product implementing the superHamiltonian constraint is defined. Brief concluding remarks follow in sec.7.", "pages": [ 1, 2 ] }, { "title": "2. DIRAC PRESCRIPTION FROM BRST COHOMOLOGICAL CLASS", "content": "The observables in a theory with some first-class constraints G a ( q, p ) = 0 are defined as those phase-space functions O ( q, p ) which are invariant under the Poisson action of the constraints, i.e. the relations { G a , O } = 0 hold. The quantization of such a theory is based on the Dirac prescription [15], i.e. the physical states ψ phys are those ones for which ˆ G a being the operators associated with the phase-space functions G a . An equivalent formulation on a quantum level is obtained in the so-called refined algebraic quantization [16, 17], in which one works with generic states ψ = ψ ( q ) and implements the condition G a = 0 in the scalar product as follows µ being a proper measure. In order to get a finite result for the scalar product (2), some gauge-fixing conditions χ a = 0 have to be implemented via the insertion of δ -functions and of some factors ensuring the invariance under the gauge choice in the measure µ [5, 18]. This can be an intriguing point. In what follows, we will concentrate on how to implement the restriction to the hypersurface where the constraints G a = 0 hold and we assume the δ -functions of the gauge-fixing condition and the proper factors to be contained into the measure µ . a Hennaux and Teitleboim [5] pointed out how the characteration of observales and of physical quantum states can be inferred from the BFV formulation in extended phase space. In particular, observables are elements of a proper BRST cohomological class, while it can be defined a scalar product which reproduces Eq.(2). In extended phase-space one introduces the ghosts θ a and the antighosts ¯ θ a associated with the constraints G a = 0. Let us consider also the so-called nonminimal sector, in which one treats also the Lagrangian multipliers λ a , implementing first-constraints in the Hamiltonian, on equal footing as others variables and their associated conjugate momenta b a are introduced. Hence, the additional conditions b a = 0 are present, together with the associated couple of ghosts-antighosts variables ρ a , ¯ C a . According with the BFV formulation [6], the total BRST charge reads where { G a , G b } = C c ab G c , while . . . denotes some terms of higher order in the anti-ghost number of the minimal sector. Let us now consider the following functions of configuration variables having maximum ghost number in the minimal sector where the product extend over all the ghosts θ a , while φ ( q ) is a function of configuration variables only (it does not depend on λ a ). The crucial property of the functions (4) is the following which is due to the fact that Ψ already contains all the available ghosts and ( θ a ) 2 = 0. The Poisson brackets with the charge gives { Ω , Φ } = {- θ G , Φ } + { ρ b a , Φ } + { ¯ θ a C a bc θ θ + . . . , Φ } = where in the second line we used the relation (5). Henceforth, the functions (4) are BRST closed if φ does not depend on the Lagrangian multipliers λ a , i.e. Two different BRST-closed functions Φ 1 = φ 1 Π a θ a and Φ 2 = φ 2 Π a θ a belong to the same cohomological class (Φ 1 = Φ 2 + { Ω , Ψ } ) if there exists a third function of configuration variables only ψ = ψ ( q ) such that for some a . This means that each cohomological class { φ } is made of the gauge orbits of phase-space functions, i.e. It can be demonstrated that such cohomological classes are isomorphic to the cohomological classes of the states with minimum ghost number in the minimal sector, i.e. The functions (10) are closed if and they coincide with cohomological classes since no exact state of the kind (10) exists. Therefore, the elements of the BRST cohomological class H 1 (Ω) are functions of the gauge orbits only, thus they are in 1-1 correspondence with the observables of the theory. On a quantum level, the identification of proper quantum states can be done according with the procedure implemented in non-Abelian gauge theories [19]. In fact, in such models the imposition of the invariance under the choice of the gauge fixing implies BRST invariance for amplitudes. These amplitudes are evaluated between asymptotic states, which are taken from free field theory. Hence, having proper in- and out- Hilbert spaces, the requirement of BRST invariance for amplitudes becomes a restriction of the space of admissible states, i.e. the space of BRST closed states. a a b c For gravitational systems like the FRW model, we do not have at our disposal a so-clear picture for quantization and in some cases ( k = 1) we cannot define asymptotic states at all. This feature leads to conjecture that BRST symmetry cannot be implemented on a quantum level [13]. Here we take the opposite point of view and starting from BRST closed states (7), we look for a proper definition of the scalar product implementing the restriction to the cohomology classes (9). This can be realized as follows [5] (square brackets denote the commutator) where ˆ K reads[22] ˆ P a being the operators associated with the conjugate momenta to θ a . In fact, one finds such that the scalar product (12) becomes The term -i Π a ˆ P a ρ a transforms the ghosts Π a θ a inside Ψ 2 into Π a ρ a , so the Berezin integration over dθ a dρ a gives non-vanishing finite results, while the integration over λ a provides the restriction to the subspace for which G a = 0, i.e. Therefore, the definition of the scalar product (12) provides the restriction to the constraint hypersurfaces where the constraints G a = 0 holds and it reproduces the results of the refined algebraic quantization (2) and of the Dirac prescription. This procedure is rather formal, because we do not specify the space where φ 's live, the integration measure (which contains the δ -functions over some gauge-fixing conditions) and the complex structure. We are going to apply it to the FRW case.", "pages": [ 2, 3, 4 ] }, { "title": "3. BRST CHARGE IN FRW MODEL", "content": "The metric tensor for FRW models is given in spherical coordinates { t, r, θ, φ } by N and a being the lapse function and the scale factor, respectively, which depend on the time variable t only, while k = 1 , 0 , -1 for a closed, flat and open Universe, respectively. The Lagrangian density takes the following expression where ˙ a denotes the derivative of a with respect to the time coordinate t . One sees that the conjugate momentum to the lapse function N is constrained to vanish By a Legendre transformation one finds the following expression for the Hamiltonian in which the superHamiltonian H is given by π a being the conjugate momenta to a . The conservation of the condition (19) implies the secondary constraint such that the total Hamiltonian is constrained to vanish and no physical evolution occurs (frozen formalism). The observable of the theory are those phase space functions which commute with the constraints (19) and (22), i.e. The presence of the constraints (19) and (22) is due to the fundamental gauge symmetry of the FRW dynamical system, which is the invariance under time parametrizations, i.e. where η = η ( t ) is an infinitesimal parameter. In order to give a well-defined formulation in extended phase space the following gauge condition has been considered in [20] such that, once the ghost θ and the antighost ¯ θ are introduced, the Lagrangian density in extended phase space containing only first-derivatives reads λ being a Lagrangian multiplier which imposes the gauge conditions (25), while π = λ + ˙ ¯ θθ . The analysis of equations of motion performed in [13] gives the following expression for the BRST charge ρ being the conjugate momentum to ¯ θ , while H denotes the total Hamiltonian in extended phase space, which can be written as in which ¯ ρ denotes the conjugate momentum to θ and ˜ H can be obtained from H (21) by replacing π a with π ∂F ∂a + π a , i.e. It is worth noting the difference between the charge (27) and the expression (3) obtained in the BFV model, in particular as soon as the Lagrangian multiplier N and λ 's are concerned. This fact reflects the non-trivial mixing between gauge and physical degrees of freedom which takes place in the case of gravitational systems.", "pages": [ 4, 5 ] }, { "title": "4. BRST CHARGE", "content": "Let us consider a dynamical system, described by some fields φ and a Lagrangian density L ( φ, ∂φ ), and a group of infinitesimal transformations The variation of the action under the transformations above, when evaluated on classical trajectories, reads where in the last line a partial integration occurs and the Euler-Lagrange equations are used. The requirement of gauge invariance is usually implemented by imposing δS = 0, from which one infers the conservation of the Noether charge Q = ∫ [ δL δ∂ 0 φ δφ -θ 0 L ] d 3 x by discarding spatial boundary contributions. However, it is possible to weaken such a condition and to require the variation of the action to be a boundary term, i.e. The transformations for which the condition (32) holds are actual symmetries on a classical level. This is due to the fact that the equations of motion are evaluated by performing variations of the field which vanish at the boundary. Hence, a conserved charge Ω can still be defined and it differs from the expression Q (which is not a Noether charge since the action is not invariant) by a term D 0 , i.e. In fact, it can be verified from the relations (31) and (32) that where we discard the spatial boundary contributions. In the following, we will outline how this is the case for the BRST symmetry in FRW models and we will infer the expressions for the corresponding charge.", "pages": [ 5, 6 ] }, { "title": "4.1. FRW model", "content": "Let us now consider the FRW model. The BRST transformations associated with time translations (24) can be obtained by replacing the infinitesimal parameter η with bθ , b being a constant Grassmanian parameter. The behavior of N and a is the following one The transformation of the ghost can be deduced from the fact that θ behaves as an infinitesimal displacement of the time coordinate, i.e. as the time-like component of a vector field. Hence, the variation of θ gives while for λ and ¯ θ we fix in analogy with the BRST transformations in the Yang-Mills case, the following relations It can be explicitly verified that the transformation defined by Eqs (35), (36) and (37) is nihilpotent. From the evaluation of the on-shell variation of the Lagrangian density (26), one finds that the variation of the gravitational part vanishes when the equations of motion for N and a hold, while the total variation reads The total variation of the action is given by summing to the variation above the term due to the transformation from t ' to t . This gives a contribution ∫ ∂ 0 ( θL ) dt , which evaluated on classical trajectories reduces to Therefore the full variation of the action reads Hence, the transformations (35), (36) and (37) act as a symmetry for classical trajectories in extended phase space, because the variation of the action is a just boundary term which does not contribute to the equations of motions. Moreover, these transformations are nihilpotent and they constitutes the proper BRST transformations of the FRW model. At this point, the conserved charge can be evaluated from the Noether charge Q = -θH -πN ˙ θ -πρ , by summing πρ so finding the following expression, This expression coincides on-shell with the one obtained in [13] and, in fact, it generates the transformations (35), (36) and (37). This result outlines how the direct evaluation of the on-shell variation of the action allows us to infer the right expression for the BRST charge Ω starting from the Noether charge Q .", "pages": [ 6, 7 ] }, { "title": "5. COHOMOLOGICAL CLASSES", "content": "Let us now discuss the cohomological classes of the BRST charge (27) having ghost number one, H 1 (Ω). In the FRW case, these functions in extended phase space can be written as φ θ and φ ρ being arbitrary functions of a and N . The requirement of BRST invariance implies that The exact functions Θ with total ghost number one can be obtained from a generic function ϕ ( a, N ) as follows and two functionals Φ 1 and Φ 2 belong to the same orbit if there exists ϕ such that Φ 1 = Φ 2 + { Ω , ϕ } . Let us partially fix an element Φ rep within each BRST orbit by the condition which can be realized starting from a generic function (42) by summing Θ = { Ω , ϕ } with As soon as Eq.(45) holds, the relation (43) implies that BRST invariant functions read φ ( a ) being an arbitrary function of the scale factor. The exact functions of the kind (47) can be obtained by the Poisson action of the charge Ω on the ghost zero functions This can be seen by the following explicit calculation Henceforth, the requirement (45) does not fix uniquely an element within each cohomological class H 1 (Ω), which are formed by Therefore, the BRST-cohomological classses (50) are determined by the equivalence class of functions φ ( a ) under the Poisson action of H . Hence they are in 1-1 correspondence with the classical observables of the FRW model (23). The Hamiltonian flow of the functions (50) in extended phase space is generated by the extended Hamiltonian (28). The Poisson bracket with Φ cc gives and one can take it back to a element Φ rep by summing Θ = { Ω , ϕ } with This way one obtains Therefore, the BRST-cohomological classses (50) do not evolve under the action of the physical Hamiltonian in extended phase-space.", "pages": [ 7, 8 ] }, { "title": "6. QUANTIZATION IN EXTENDED PHASE-SPACE", "content": "The quantization of the FRW model in extended phase space can be done as in the BFV case by defining proper states and a scalar product. Let us consider as quantum states the BRST-closed functions (47) (which we denote by Φ) and let us fix the operator ordering with momenta to the right. The complex structure is the standard complex conjugation, while ghosts θ and ρ are real. The momenta are defined as -i times the (left) derivatives operators of the corresponding variables. We look for the extension of the scalar product (12), i.e. where µ ( a, N ) is an un-specified measure. We cannot reproduce exactly the procedure described in section (2) because of the nontrivial mixing of gauge and physical degrees of freedom. Let us keep for moment ˆ K as a generic operator having ghost number -1, i.e. ˆ k 1 , ˆ k 2 being two arbitrary operators. The operator [ ˆ K, ˆ Ω] acts on states (47) as follows It is worth noting how the last term in the expression above is able to provide a nonvanishing result for the scalar product (55). By squaring the operator [ ˆ K, ˆ Ω] one gets Let us make now some assumptions about the operators ˆ k 1 and ˆ k 2 : for simplicity we take ˆ k 1 = -Ii , while we fix ˆ k 2 as follows where the first term is obtained by evaluating -N [ ˆ ˜ H,N ] and by avoiding the piece containing the operator ˆ π . This way, the following relations hold and generically one has (see the appendix) Therefore, the exponential within the scalar product (55) reads and after performing the integration over θ and ρ only the second term gives a nonvanishing contribution so getting By defining the measure µ ( a, N ) = -µ ( a ) /N in order to avoid the factor N coming from Φ 1 , the scalar product above can be written as in (16), i.e. Therefore, it is obtained the same Hilbert space structure as in the case the Dirac prescription for the quantization of constrained systems is used. In fact, there is no restriction on the form of the measure term µ ( a ) and one can write it as a δ -function over a gauge-fixing condition times the proper factors which ensure the invariance under the choice of the gauge-fixing function itself. Hence, the expression (65) is the starting point to define a proper scalar product for the FRW model just like the case in which the Dirac prescription for the quantization of constrained systems is used [5].", "pages": [ 8, 9 ] }, { "title": "7. CONCLUSIONS", "content": "In this work we derived the BRST charge associated with FRW space-time by a Noether-like analysis in extended phase space. The total Lagrangian has been defined according to the BV method for differential gauge fixing conditions. These conditions allowed us to reintroduce missing velocities and to have a well-defined Hamiltonian formulation in extended phase space [20]. Nihilpotent BRST transformations were defined thanks to the invariance under time parametrizations of the original formulation. The final expression for the action has been analyzed, finding that its variation under BRST transformations provided some time-boundary contributions. We accounted for these contributions and we could achieve the expression of the BRST charge for the considered systems. Then, we characterized the BRST cohomological classes in the case of functions with ghost number one. We chose proper elements within each BRST orbit, such that the closure condition fixied the dependence from N . This is the counterpart of the imposition of primary constraints in the Dirac approach. Then, the construction of equivalence class of closed forms modulo exact ones ensured the invariance under the action of the secondary constraint. We also investigate the physical evolution of cohomological classes under the action of the total Hamiltonian. We found that such an action vanished. This achievement confirms that the frozen formalisms extends to observables in extended phase space. Hence, the BRST formulation identifies the right degrees of freedom even in the case of a gravitational system, in which there is a nontrivial interplay between gauge (the lapse function) and physical (the scale factor) degrees of freedom (which is reflected into the expression of the BRST charge (27)). Finally, we considered the quantization of the FRW model in extended phase space. We demonstrated how a suitable scalar product could be defined according with the procedure described in [5] (the only difference being the form of the function K ) such that the vanishing of the superHamiltonian operator was implemented. This achievement outlines how the equivalence between a quantum formulation in extended phase space and other approaches to the quantization of constrained systems, as refined algebraic quantization and the Dirac prescription, can be realized. Moreover, our findings can be thought as the canonical counterpart of the outcomes of [12], where a path integral formulation for a FRW model with a differential gauge fixing condition is discussed and the restriction to propagators implementing in a proper way the condition H = 0 is obtained. The extension of this analysis to more complex space-times will be the subject of forthcoming investigations, aimed to test to what extend the BRST framework can be used to implement diffeomorphisms invariance on a quantum level. The extension of the procedure to infer the conserved charge will require more algebraic manipulations with respect to the FRW case, since the Lagrangian in extended phase-space is more complex and the variation of the gravitational Lagrangian (which here vanishes on-shell) can give some additional contributions. However, such a Lagrangian will be obtained by discarding some boundary contributions from the Einstein-Hilbert one, whose associated action is invariant under space-time transformations. Therefore, even though the variation of the gravitational Lagrangian will contribute to the total variation of the action, the additional terms will still take the form of some boundary contributions and the definition of the conserved charge can be given as in Eq.(33). A different apporach is discussed in [21], where it is chosen to work with a Lagrangian containing second-order derivatives such that the whole action is invariant and the conserved charge is simply the Noether one.", "pages": [ 9, 10 ] }, { "title": "Acknowledgment", "content": "The authors wish to thank the anonymous referees, whose remarks allowed them to enhanced the quality of the paper. The work of F.C. was supported by funds provided by 'Angelo Della Riccia' foundation and by the National Science Center under the agreement DEC-2011/02/A/ST2/00294. This work has been realized in the framework of the CGW collaboration (www.cgwcollaboration.it).", "pages": [ 10 ] }, { "title": "Appendix", "content": "Let us demonstrate the relation which we have already verified for n = 1 , 2 (60), (61). Let us assume that Eq.(66) holds and let us evaluate By using the expression (59) for ˆ k 2 one finds where in the last line we used the fact that the expression (29) for ˆ ˜ H contains powers of π up to the second order, thus the operator [[ ˆ ˜ H,N ] , N ] commutes with N . The second term in the last line of Eq.(67) contains the following object where in the third line we still used the fact that [[ ˆ ˜ H,N ] , N ] commutes with N . By collecting togheter the results (68) and (69) one sees that and which probes eq.(66). I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B , 69 , 309(1977).", "pages": [ 10, 11 ] } ]

[ { "title": "I-Sheng Yang ∗", "content": "ISCAP and Physics Department Columbia University, New York, NY, 10027 , U.S.A. AND IOP and GRAPPA, Universiteit van Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands Abstract: The usual (type A) thin-wall Coleman-de Luccia instanton is made by a bigger-than-half sphere of the false vacuum and a smaller-than-half sphere of the true vacuum. It has a the standard O (4) symmetric negative mode associated with changing the size of false vacuum region. On the other hand, the type B instanton, made by two smaller-than-half spheres, was believed to have lost this negative mode. We argue that such belief is misguided due to an over-restriction on Euclidean path integral. We introduce the idea of a 'purely geometric junction' to visualize why such restriction could be removed, and then explicitly construct this negative mode. We also show that type B and type A instantons have the same thermal interpretation for mediating tunnelings.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Coleman and de Luccia wrote down the instanton solution that became the paradigm of first order phase transitions with gravity [1]. When the critical bubble is much smaller than the de Sitter radius of the parent vacuum, the CDL instanton has almost the entire 4-sphere of the false vacuum, and a small bubble of the true vacuum. It is very similar to the Coleman instanton in flat space [2]. In the thin-wall approximation, the instanton geometry contains a kink at the domain wall between the true and false vacua, as shown in Fig.1. In the conventional analysis, there is a negative mode corresponding to moving the domain wall (together with the kink), which is similar to changing the bubble size in the flatspace version. Interestingly, a continuous parameter change from the above 'type A' instanton leads to the 'type B' instanton. As depicted in Fig.1, the type B instanton contains two smaller-than-half spheres. Such solution is troublesome in two aspects. First, it appears to lose the usual negative mode corresponding to the change of bubble size [3]. Second, having less than half of the false vacuum de Sitter 4-sphere makes it difficult to interpret the phase transition as nucleating a bubble. The existence and uniqueness of the negative mode is a criterion for the instantons to mediate vacuum transitions. For the Coleman instanton without gravity, it was proved in [4]. Considering the gauge theory nature of gravity, the CDL instanton needs certain appropriate mode reduction process, otherwise there will be spurious modes [5-7]. A tentative existence plus uniqueness proof was presented in [8] 1 . Using similar techniques, numerical thick-wall examples of both type A and type B instantons are shown to have exactly one negative mode [9-11]. Unfortunately, such framework provides only the existence but cannot clearly demonstrate which physical deformation the negative mode corresponds to. No one has explicitly constructed the physical deformation of the negative mode for type B instantons. Thus the sharp contrast between type A and type B instantons is not fully resolved. In this paper, we will stick to the thin-wall approximation and introduce the idea of a 'purely geometric junction'. We explain why this feature is allowed in off-shell configurations of Euclidean path integral. Employing this feature, we can explicitly construct the physical deformation corresponding to the negative mode for type B instantons. Just like for the type A instantons, the deformation is still the change of bubble size. 2 In light of this, we see no reason to treat them differently. In fact, we will show that in the thermal interpretation [12], it represents the phase transition similarly to how a type A instanton represents the reverse transition [13]. The structure of this paper goes like the following. In Section 2, we review the basic solution of a thin-wall CDL instanton and the missing negative mode for type B. In Section 3 we justify the usage of purely geometric junctions and explicitly construct this negative mode. In Section 4 we discuss how to interpret the type B instanton as mediating phase transitions. Finally we summarize and conclude in Section 5. In Appendix A we provide the simplest thick wall construction to address possible concerns and further justify our usage of purely geometric junctions.", "pages": [ 2, 3, 4 ] }, { "title": "2. Two Types of CDL Instantons", "content": "Consider a scalar field with the following potential, We have a true vacuum and a false vacuum at with energy difference roughly ∆ V , and a domain wall separating them with tension given by Given that ∆ V glyph[lessmuch] λ 2 φ 4 0 , a false vacuum background can nucleate a thin-wall bubble of true vacuum, which will then expand and complete the phase transition. The rate of this nucleation process is given by (keeping only the exponent) S stands for the Euclidean action: S F for the background false vacuum and S I for the instanton solution that contains a bubble of true vacuum. The 4D Euclidean configuration of the instanton solution is a 3-sphere of domain wall, filled with the true vacuum and surrounded by the false vacuum. One can easily write down the action difference, It has a maximum at Varying r is an unique negative mode that signifies this configuration being the leading saddle point contribution that mediates the phase transition. Including gravity, the Euclidean action becomes: where the matter action is For simplicity, in this paper we will focus on the scenario that both vacua has positive energy by adding a constant term to V , such that The Euclidean action of the false vacuum configuration is simply The two terms in the bracket are the field and gravity contributions, and the last factor stands for the 4-volume of a 4-sphere with radius R F . The instanton is the matching of two 4-spheres with radii R T and R F at some junction radius r , where the domain wall resides. Its action is given by The last two terms are the combined contribution of field and gravity from two 'shells' of the true and false vacua. The first term is the field contribution at the junction, namely the domain wall tension times the wall area. The second term is the gravitational contribution from the junction 3 , It takes the plus sign when the corresponding side is a portion of 4-sphere that contains a full equator 3-sphere, namely the bigger portion. Otherwise it takes the minus sign when the corresponding side is a smaller portion. The same ambiguity appears in V ( R F , r ), which means the volume of a partial 4-sphere bounded by a 3-sphere of radius r and can be either the bigger or the smaller side. Fortunately as shown in [1,14] that all these ambiguous terms can be combined to show that without ambiguity, the instanton action is extremized at where r c is the critical bubble size in flat space, given by Eq. (2.7). When r c glyph[lessmuch] R F , the instanton contains a small portion of the true vacuum and a large portion of the false vacuum, r e ∼ r c , and varying r is a negative mode just as in the case without gravity. This is the more standard case and called the type A instanton. An interesting behavior arises when we tune the potential to increase r c , for example by reducing ∆ V . Eq. (2.14) shows that r e will eventually become inversely proportional to r c instead. At the same time the instanton becomes two smaller-than-half portions of spheres. This is what we call the type B instanton. These two cases are drawn in Fig.1. 4 At the level of solving the equation of motion, namely finding the critical point of the Euclidean action, these is no dramatic change between the two cases and Eq. (2.14) is always valid. However when one varies the action around this critical point by changing r , it corresponds to a negative mode for type A but a positive mode for type B [3]. The type B instanton still has an action bigger than S F (and S T ), so people tend to believe that it has at least one negative mode 5 . Since the above analysis is restricted to O (4) symmetry and thin-wall approximation, the common intuition is to go beyond either or both of them. Maybe the disappearance of the radial negative mode signifies the emergence of many more subtle negative modes, and condensing them leads to a thick-wall or less symmetric solution that has only one negative mode. However, these suggestions are all motivated by the apparent 'disappearance' of the O (4) symmetric negative mode. We will explicitly show that such a mode actually still exists.", "pages": [ 4, 5, 6 ] }, { "title": "3. Recovering the Negative Mode", "content": "Let us reconsider what happened in the extremizing process described in the previous section. At the extremum, we have which is the Israel junction condition [15], namely the integrated Einstein equations across a co-dimension one delta function. It tells us how the tension of the domain wall determines the angle of the geometric kink. While looking for the negative mode, one varies the position of the domain wall and the geometric kink together to values other than r e . Those will be off-shell configurations in the path integral, and Eq. (3.1) will not hold. This is totally fine, since the full equations of motions are with the metric Only Eq. (3.3), the constraint equation, should hold for off-shell configurations. Eq. (3.1) comes from the delta-function integral with glyph[epsilon1] → 0 and ρ ( ¯ ξ ) = r . Its value only involves the ρ '' term in R . So not solving the junction condition just means not solving Eq. (3.4) but still obeys the constraint Eq. (3.3). What we will do next is qualitatively the same as the usual CDL variation described above. In the variation of CDL radial mode, one goes through configurations where the gravity contribution to the junction, Eq. (2.13), does not match the matter contribution from the domain wall tension. For the same token, we shall be allowed to do the following. Put a geometric junction where there is no domain wall. It only contributes gravitationally, as if there is a zero tension domain wall. The constraint equation demands only that in the true (false) vacuum region, the geometry has to be a portion of the 4-sphere with R T ( R F ). 6 With that in mind, let us consider the following solution parametrized by two radii, r g and r w . At r g there is a purely geometric junction, and at r w we have the usual domain wall with tension σ separating the true and false vacuum. When the purely geometric junction is in the true vacuum region, we have When the junction is in the false vacuum region, just switch T and F in the above equation. All the 4-volume functions V here are referring to the smaller portion without an equator 3-sphere, which should be obvious from Fig.2. When r g = r m = r e , this is exactly the critical solution of a type B CDL instanton. Now we can fix r g = r e and start to vary r m to either side, as shown in Fig.2. We see that the action always decreases 7 . We can further shrink r m to zero to eliminate the true or false vacuum portion, and smooth out the purely geometric junction. The action is strictly decreasing during the entire process and recovers S T or S F . The analysis of radial mode in [3] was restricted to a single geometric junction that always sticks with the domain wall. This unnecessary restriction led to a bias that for type B instantons, 'changing the bubble size' is actually changing the total size of the entire instanton. As a hindsight, such deformation has no reason to be the relevant negative mode for a tunneling process. The negative mode should represent two directions that the instanton rolls toward either the true or the false vacuum. The physical deformation we show in Fig.2 is exactly doing that. It is 'really' changing the bubble size-shrinking the false/true vacuum region while expanding the other. The fact that this deformation corresponds to a negative mode should not be very surprising.", "pages": [ 7, 8, 9 ] }, { "title": "4. The Thermal Interpretation", "content": "Brown and Weinberg [12] provided a very accurate picture to interpret how the type A CDL instanton mediates the vacuum transition. Instead of taking the Euclidean 4sphere as the global geometry, they described it as a horizon 3-volume times a compact coordinate from the finite temperature. One side of the equator is the horizon volume before tunneling, and the other side is the same horizon volume after tunneling. This interpretation clarified a few confusions. For example, without an exactly thin wall, the 'false vacuum region' of the instanton will not be identical to the same portion of the false vacuum 4-sphere. If one takes the instanton as a global geometry, it is unsatisfying that nucleating a bubble requires changes far away, out of causal contact from the bubble. In the thermal interpretation this has a clear explanation. With nonzero temperature, the transition is not purely quantum, but always thermally assisted, as depicted in Fig.3. The horizon volume of the false vacuum always needs to be thermally excited, even just a little bit, to the configuration that is the left hand side slicing of an instanton, then the quantum tunneling starts. Note that not only the field configuration of the instanton is slightly away from the pure vacuum, so is the geometry. This is also straight forward since the gravitational back reaction from a non-vacuum state leads to a non-vacuum geometry 8 . Realizing this fact also means that we can accept the reverse tunneling being mediated by the same instanton, just in the reverse direction. It is a dramatic fluctuation from the true vacuum to the initial condition of this reverse tunneling, in terms of both the field configuration and the geometry. From the horizon volume of the true vacuum, the fluctuation leads to a bubble of true vacuum surrounded by the false vacuum, and a much reduced horizon size 9 . But that is just what is has to be and most of the suppression in the tunneling rate is indeed a thermal factor. With these in mind, the type B instanton mediates tunneling in the same way, only that both directions require a dramatic thermal fluctuation. From a horizon volume of a vacuum, we need a thermal fluctuation up to a smaller volume surrounded by a domain wall before the quantum tunneling starts. This is shown in Fig.4.", "pages": [ 9, 10 ] }, { "title": "5. Conclusion", "content": "We explicitly constructed the negative mode for type B CDL instantons. It is the same radial negative mode as changing the bubble size in the type A instantons. This natural physical deformation was not considered in earlier literature due to an unnecessary restriction of the geometry. We removed such restriction by introducing a purely geometric junction in the off-shell configurations of the path integral. We argued that they satisfy exactly the same principles for gravitational path integral as the original CDL mode analysis. We also provided simple thick-wall analysis in the Appendix to further justify this novel usage. Our result agrees with the numerical thick wall examples in [9-11]. Although a full analysis including thick-wall effects and less symmetries is still lacking, we believe the conceptual difference between type A and type B instantons is eliminated. The type A instantons, being similar to the Coleman instantons in flat space, has been widely accepted as the correct saddle point for the tunneling. The same should be true for the type B instantons. In the thermal interpretation, we provided the conceptual unification of how both types of instantons mediate upward and downward tunnelings.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "I am especially grateful that Erick Weinberg has planted this problem deeply in my mind, and the discussions we shared to sharpen the argument. I also thank Adam Brown, Bartek Czech, Ben Freivogel, George Lavrelashvili, Jean-Luc Lehners, Neil Turok, and Xiao Xiao for stimulating discussions. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO).", "pages": [ 11 ] }, { "title": "References", "content": "[16] J. Hartle and R. Sorkin, 'BOUNDARY TERMS IN THE ACTION FOR THE REGGE CALCULUS,' Gen.Rel.Grav. 13 (1981) 541-549. [17] G. Hayward, 'Gravitational action for space-times with nonsmooth boundaries,' Phys.Rev. D47 (1993) 3275-3280.", "pages": [ 13 ] }, { "title": "A. Thick Wall Constructions", "content": "The idea of a 'purely geometric junction' appeared much earlier, and the calculation of their contribution to the action is well-known [16,17]. However it might be the first time that they play a crucial role in evaluating the off-shell value of Euclidean action. Certain level of scrutiny is warranted. Two reasonable concerns were brought to our attention independently and separately by Brown, Freivogel, Weinberg and Xiao. Here we provide the thick wall justification of our thin wall calculation to address these concerns.", "pages": [ 13 ] }, { "title": "A.1 Thick-Wall Field Profile", "content": "A great deal of subtleties in gravitational path integral come from the constraint, Eq. (3.3). We imagined a purely geometric junction of zero thickness and avoided any explicit consequence from the constraint. One might worry that we are implicitly violating the constraint thus the configurations studied are not allowed in the path integral. The specific objection goes like the following. The Objection. Consider a purely geometric junction that connects a shell up to some ¯ ξ with its mirror image. The radius of this junction is of course smaller than R . Now imagine a thick wall version of this, there must be a place that ρ ' = 0 because it changes sign, and this must happen at some value close to ¯ ρ < R . Therefore, the constraint equation, Eq. (3.3), demands that at this point, the field could not have stayed in the vacuum. The idea of a 'purely geometric junction' is wrong since a nontrivial field profile is necessary. The Answer. It is certainly true that any thick wall geometric junction cannot be pure-certain field profile must accompany it to obey the constraint. However, all we cared about was to evaluate its contribution to the action. When the accompanying field profile contributes to a small correction to the value of this purely geometric junction, our method is still valid. That is indeed the case when the potential allows thin-wall approximation. First we expand Eq. (2.1), including the uplift to de Sitter, near one vacuum. where m 2 = 8 φ 2 0 λ 2 and the definition of φ is shifted. Instead of directly matching two shells, we will insert a narrow segment in the middle. We will replace the coordinate ξ by x within this segment, where x = 0 sits the middle. Note that we do not need to obey Eq. (3.2), so basically we are just inventing a field configuration that solves Eq. (3.3) for our purpose. The field configuration we want should be continuous in φ ' . The most na¨ıve description is from the acceleration, Namely, φ smoothly increases from 0 to φ t during a short interval 2 m -1 , stays at that value for 2 L , then decreases back to zero, as shown in Fig. 5. The value of φ t determines the geometry during the middle 2 L interval. The purpose of this purely geometric junction is to hold the place of the domain wall in the on-shell configuration. The required ρ 0 is given by Here V top is the top of the potential barrier, and φ ' ins is the field velocity there in the instanton solution. Directly comparing Eq. (A.6) and (A.7), we already see that φ t does not need to reach the top of the potential barrier. Actually, for a potential allowing the thin-wall approximation, φ ' ins is large in the sense that Thus the required φ t in Eq. (A.6) is far away from the top of the potential barrier and remains in the region that the approximation, Eq. (A.3), is valid. The thin-wall requirement also means R glyph[greatermuch] m -1 . Combined with the fact that during the 2 m -1 interval when φ changes, the relevant change in Eq. (3.3) is bounded, we know ρ does not change too much during this interval. This means the geometry of the inserted segment, Eq. (A.5), matches to the two shells, Eq. (A.1), roughly by Now we can calculate the action contribution of this middle segment. We have integrated by part to get the boundary term that exactly equals to the contribution from a purely geometric junction. Now obviously, the middle range L is just a place holder. We can take ρ 0 → ¯ ρ , such that L → 0. This extra term is an integral similar to other terms in the action, but with a small integration range, 2 m -1 glyph[lessmuch] R . Therefore we can see that typically, namely for an order one geometric junction, ρ 0 ∼ R , the extra integral is a small correction to the purely geometric term. This argument will not apply in two extreme cases. First when the bubble (which this purely geometric junction is supposed to hold place for) is originally small, ρ 0 → m -1 , then L g,wall itself becomes small and the integral term is not negligible. However this limit means the domain wall thickness is comparable to ρ 0 , which is exactly when the thin-wall approximation breaks down. For that case a more complete thick-wall analysis is needed and our approach was never meant to be valid anyway. The other limit is when the purely geometric junction happens to be very mild, ρ 0 → R and L g,wall is again close to zero. In that case the field contribution will not be negligible. That is actually crucial in the next section to resolve a paradox. Here we are satisfied that away from these two extremes, an order one purely geometric junction is an appropriate approximation of a thick wall object that satisfies the constraint equation.", "pages": [ 13, 14, 15, 16 ] }, { "title": "A.2 Resolving an Apparent Paradox", "content": "Another way to see potential problems with the purely geometric junction is the following paradox.", "pages": [ 16 ] }, { "title": "The Paradox.", "content": "Consider the vacuum solution of the potential given by Eq. (A.3). It is a 4-sphere with radius R . Now imaging that we develop a purely geometric junction on the equator. For a convex type of junction that both sides are smaller than half of the 4-sphere, the action will be higher; for the concave type that both sides are bigger than half of the 4-sphere, the action will be lower. (which eventually splits into two 4spheres) So it seems like the purely geometric junction introduces a fictitious instability. It is actually a 3rd order marginal instability as shown in Fig.6.", "pages": [ 16 ] }, { "title": "The Answer.", "content": "As hinted in the previous section, the field contribution to a very mild junction cannot be ignored. If we try to fluctuate these junctions from nothing, we have to keep track of the total action in the thick wall analysis. We will then see that the field contribution is lower order and positive in the direction of this marginal instability, therefore cures it. Consider attaching two semi-4-spheres to a middle band, with a waist radius ρ w < R , as the smooth version of a concave geometric junction. As a small fluctuation, this band should last for an interval L glyph[lessmuch] R , dips only a little bit ∆ ρ = ( R -ρ w ) glyph[lessmuch] R , and involves a small field fluctuation m 2 φ 2 w glyph[lessmuch] 3 M 2 p /R 2 . The thick wall action contribution of the waist is We have again integrated by part, but this time the boundary term is zero because the waist connects to the equator of two hemispheres where ρ ' = 0. According to the constraint, Eq. (3.3), we can rewrite the integrand as We can see that the φ ' 2 term is positive definite and may be the cure we want. In order to prove that in general it will, we should make the assumption to minimize it. Namely, we minimize the number of wiggles in φ profile such that it monotonically increases to φ w in the middle, then monotonically decreases back to zero. At the matching points to the two shells, and at the middle of the waist, the above quantity is zero since all derivatives are zero. So we can estimate this integrand by how it 'grows' from the matching point to the waist. Plugging the above estimators into the integrand, we get We can see that in the limit ∆ ρ → 0, this is a positive definite quantity. So the thick wall contribution cures the apparent marginal instability in the thin-wall analysis.", "pages": [ 16, 17, 18 ] } ]

[ { "title": "A proposed proper EPRL vertex amplitude", "content": "Jonathan Engle ∗ Department of Physics, Florida Atlantic University, Boca Raton, Florida, 33431 USA June 18, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "As established in a prior work of the author, the linear simplicity constraints used in the construction of the so-called 'new' spin-foam models mix three of the five sectors of Plebanski theory as well as two dynamical orientations, and this is the reason for multiple terms in the asymptotics of the EPRL vertex amplitude as calculated by Barrett et al. Specifically, the term equal to the usual exponential of i times the Regge action corresponds to configurations either in sector (II+) with positive orientation or sector (II-) with negative orientation. The presence of the other terms beyond this cause problems in the semiclassical limit of the spin-foam model when considering multiple 4-simplices due to the fact that the different terms for different 4-simplices mix in the semiclassical limit, leading in general to a non-Regge action and hence non-Regge and nongravitational configurations persisting in the semiclassical limit. To correct this problem, we propose to modify the vertex so its asymptotics include only the one term of the form e iS Regge . To do this, an explicit classical discrete condition is derived that isolates the desired gravitational sector corresponding to this one term. This condition is quantized and used to modify the vertex amplitude, yielding what we call the 'proper EPRL vertex amplitude.' This vertex still depends only on standard SU (2) spin-network data on the boundary, is SU (2) gauge-invariant, and is linear in the boundary state, as required. In addition, the asymptotics now consist in the single desired term of the form e iS Regge , and all degenerate configurations are exponentially suppressed. A natural generalization to the Lorentzian signature is also presented.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "At the heart of the path integral formulation of quantum mechanics [1, 2] is the prescription that the contribution to the transition amplitude by each classical trajectory should be the exponential of i times the classical action. The use of such an expression has roots tracing back to Paul Dirac's Principles of Quantum Mechanics [3], and is central to the successful derivation of the classical limit of the path integral, using the fact that the classical equations of motion are the stationary points of the classical action. The modern spin-foam program [4-6] aims to provide a definition, via path integral, of the dynamics of loop quantum gravity (LQG) [4,6-8], a background independent canonical quantization of general relativity. The only spin-foam model to so far match the kinematics of loop quantum gravity and therefore achieve this goal is the so-called EPRL model [9-12], which, for Barbero-Immirzi parameter less than 1 is equal to the FK model [13]. In loop quantum gravity, geometric operators have discrete spectra. The basis of states diagonalizing the area and other geometric operators are the spin-network states . The spin-foam path integral consists in a sum over amplitudes associated to histories of such states, called spin-foams . Each spinfoam in turn can be interpreted in terms of a Regge geometry on a simplicial lattice. The simplest amplitude provided by a spin-foam model is the so-called vertex amplitude which gives the probability amplitude for a set of quantum data on the boundary of single 4-simplex. The semiclassical (i.e. large quantum number, equivalent to /planckover2pi1 → 0) limit [14] of the EPRL vertex amplitude, however, is not equal to the exponential of i times the Regge action as one would desire, but includes other terms as well. 1 As a consequence, when considering multiple 4-simplices, the semiclassical limit of the amplitude has cross-terms, each of which consists in the exponential of a sum of terms, one for each 4-simplex, equal to the Regge action for that 4-simplex times differing coefficients, yielding what can be called a 'generalized Regge action' [16, 17]. The stationary point equations of this 'generalized Regge action' are not the Regge equations of motion and hence not those of general relativity, whence general relativity will fail to be recovered in the classical limit. As presented in the recent work [18, 19], the extra terms causing this problem correspond to different sectors of Plebanski theory, as well as different orientations of the space-time. These various sectors and orientations are present in the spin-foam sum because the so-called linear simplicity constraint - the constraint which is also used in the Freidel-Krasnov model [13] - allows them. In this paper, we propose a modification to the EPRL vertex amplitude which solves this problem. We begin by deriving, at the classical discrete level, a condition which isolates the sector corresponding only to the first term in the asymptotics, the exponential of i times the Regge action. We call this sector the 'Einstein-Hilbert' sector, because it is the sector of Plebanski theory in which the BF action reduces to the Einstein-Hilbert action. More specifically, this sector consists in configurations which are either in (what is called) Plebanski sector (II+) with positive space-time orientation, or (what is called) Plebanski sector (II-) with negative orientation. 2 This condition is then appropriately quantized and inserted into the expression for the vertex, leading to a modification of the EPRL vertex amplitude. The resulting vertex continues to be a function of a loop quantum gravity boundary state and hence may still be used to define dynamics for loop quantum gravity . It furthermore remains linear in the boundary state and fully SU (2) invariant - two conditions forming a nontrivial requirement restricting the possible expressions for the vertex. It is also in a precise sense Spin (4) invariant. Lastly, as is shown in the final section of this paper, for a complete set of boundary states, the asymptotics of the vertex include only a single term, equal to the exponential of i times the Regge action, enabling the correct equations of motion to dominate in the classical limit. We call the resulting vertex amplitude the proper EPRL vertex amplitude . A natural generalization to the Lorentzian case is presented in section 4.4. A summary of these results can be found in [20]. We begin the paper with a review of the classical discrete framework underlying the spin-foam model and derive the condition isolating the Einstein-Hilbert sector. Then, after briefly reviewing the existing EPRL vertex amplitude, the definition of the new proper vertex is introduced. The last half of the paper is then spent proving the properties summarized above. We then close with a discussion. 2 In a prior version of this article, the sector corresponding to the first term in the asymptotics was mischaracterized as the (II+) sector, whereas in fact it is the combination of sectors stated here. This mistake was due to an error in the prior work [18] which was corrected in [19]. The correction of this error did not at all change the proper vertex or its motivation rooted in the semiclassical limit, but only changed the interpretation in terms of Plebanski sectors and orientations.", "pages": [ 1, 2 ] }, { "title": "2.1.1 Generalities", "content": "We use the same definitions as in [18]. Let τ i := -i 2 σ i ( i = 1 , 2 , 3), where σ i are the Pauli matrices. For each element λ ∈ su (2), λ i ∈ R 3 shall denote its components with respect to the basis τ i . Let I denote the 2 × 2 identity matrix. We also freely use the isomorphism between spin (4) := su (2) ⊕ su (2) and so (4), ( J -, J + ) ≡ ( J i -τ i , J i + τ i ) ↔ J IJ ( I, J = 0 , 1 , 2 , 3), explicitly given by J i + and J i -are called the self-dual and anti-self-dual parts of J IJ . Furthermore, we remind the reader [14] of the explicit expression for the action of Spin (4) = SU (2) × SU (2) group elements on R 4 . For each V I ∈ R 4 define Then the action of G = ( X -, X + ) is given by", "pages": [ 3 ] }, { "title": "2.1.2 Discrete classical framework", "content": "Spin-foam models of quantum gravity are based on a formulation of gravity as a constrained BF theory , using the ideas of Plebanski [21]. In the continuum, the basic variables are an so (4) connection ω IJ µ and an so (4)-valued two-form B IJ µν , which we call the Plebanski two-form , where lower case greek letters are used for space-time manifold indices. The action is with F := d ω + ω ∧ ω the curvature of ω , /star the Hodge dual on internal indices I, J, K . . . , κ := 8 πG , and γ ∈ R + the Barbero-Immirzi parameter. If B IJ µν satisfies what we call the Plebanski constraint [22,23], it must be one of the five forms which we call Plebanski sectors . Here /epsilon1 IJKL denotes the internal Levi-Civita array, and η µνρσ denotes the Levi-Civita tensor of density weight 1. In sectors (II ± ), the BF action reduces to a sign times the Holst action for gravity [24], the Legendre transform of which forms the starting point for loop quantum gravity [7,24]. In spin-foam quantization, one usually introduces a simplicial discretization of space-time. However, in this paper we concern ourselves with the so-called 'vertex amplitude', which may be thought of as the transition amplitude for a single 4-simplex. For clarity, we thus focus on a single oriented 4simplex S . The EPRL model has also been generalized to general cell-complexes [12]; however because we use the work [14], and because we introduce formulae that, so far, apply only to 4-simplices, we restrict the discussion to the case of a 4-simplex. In S , number the tetrahedra a = 0 , . . . , 4, 3 and let ∆ ab denote the triangle between tetrahedra a and b , oriented as part of the boundary of a . One thinks of each tetrahedron, as well as the 4-simplex itself, as having its own 'frame' [10]. One has a parallel transport map from each tetrahedron to the 4-simplex frame, yielding in our case 5 parallel transport maps G a = ( X -a , X + a ) ∈ Spin (4), a = 0 , . . . , 4. The continuum two-form B is then represented by the algebra elements B ab = ∫ ∆ ab B , where each element is treated as being 'in the frame at a .' For each ab , in terms of self-dual and anti-self-dual parts, these elements are related to the momenta conjugate to the parallel transports (see section 3.2) by [9,18] We call B ab and J ab the canonical bivectors due to their role in the canonical theory in section 3.2. /negationslash From the discrete data { B IJ ab , G a } one can reconstruct the continuum two-form B IJ µν as follows. Fix a flat connection ∂ µ on the 4-simplex S , such that S is the convex hull of its vertices as determined by the affine structure defined by ∂ µ ; we say such a flat connection is adapted to S . The choice of such a connection is unique up to diffeomorphism and hence is a pure gauge choice (see appendix A). If the data { B IJ ab , G a } satisfy (1.) closure, ∑ b = a B IJ ab = 0, and (2.) orientation, G a /triangleright B ab = -G b /triangleright B ba , then it has been proven [18,25] that there exists a unique two-form field B IJ µν on the manifold S , constant with respect to ∂ µ , such that /negationslash for all a = b . Here the left hand side is the parallel transport of the bivectors B IJ ab to the '4simplex frame', henceforth denoted B ab , and /triangleright here and throughout the rest of the paper denotes the adjoint action. Both closure (1.) and orientation (2.) are imposed in the EPRL vertex in the sense that violations are suppressed exponentially [14]. In addition, the EPRL model imposes (3.) linear simplicity , where N I := (1 , 0 , 0 , 0), as a restriction on the allowed boundary states for each 4-simplex, as shall be reviewed in the quantum theory below. From (2.8), it follows that the continuum two-form B IJ µν defined by (2.7) is in Plebanski sector (II+), (II-) or (deg) [18]. We represent this sector by a function ν ( B µν ), defined to be +1 if B µν is in (II+), -1 if B µν is in (II-), and 0 if B µν is degenerate. If ν ( B µν ) = 0, B µν furthermore defines an orientation of S , which can either agree or disagree with the fixed orientation of S used to define form integrals. We represent this dynamically defined orientation by its sign relative to the fixed orientation ˚ /epsilon1 µνρσ of S : /negationslash where, for convenience, sgn( · ) is defined to be zero when its argument is zero. Because the only arbitrary choice in the construction of B IJ µν , that of the flat connection ∂ µ , is unique up to diffeomorphism, a diffeomorphism which, when chosen to preserve each face of S , must be orientation preserving, and because each Plebanski sector as well as the dynamically determined orientation is invariant under such diffeomorphisms, the functions ν ( B µν ( {B ab } , ∂ )) and ω ( B µν ( {B ab } , ∂ )) are independent of the choice of connection ∂ µ adapted to S , so that one can write simply ν ( {B ab } ) and ω ( {B ab } ). (For a more detailed derivation of this fact, see appendix A.) This reviews the sense, established in [18], in which the classical constraints imposed quantum mechanically in the EPRL model admit the three distinct, well-defined Plebanski sectors (II+), (II-), and (deg), as well as two possible dynamical orientations. 4", "pages": [ 3, 4, 5 ] }, { "title": "2.1.3 Reduced boundary data", "content": "The set of canonical bivectors B IJ ab satisfying linear simplicity is parameterized by what we call reduced boundary data -one unit 3-vector n i ab per ordered pair ab , and one area A ab per triangle ( ab ) - via From (2.6) and (2.10), the generators of internal spatial rotations in terms of the reduced boundary data are The corresponding bivectors in the 4-simplex frame then take the form /negationslash We call (2.12) the 'physical' bivectors reconstructed from A ab , n ab , G a . In terms of the reduced boundary data, closure and orientation become the conditions ∑ b = a A ab n ab = 0 and X ± a /triangleright n ab = -X ± b /triangleright n ba .", "pages": [ 5 ] }, { "title": "2.1.4 Reconstruction theorem", "content": "In addition to reconstructing the 2-form field B IJ µν from the bivectors B ab = B phys ab ( A ab , n ab , G a ), one can also reconstruct a geometrical 4-simplex in R 4 . This will be needed in the present paper. Let M denote R 4 as an oriented manifold, equipped with the canonical R 4 metric. A geometrical 4-simplex σ in M is the convex hull of 5 points, called vertices, in M , not all of which lie in the same 3-plane. We define a numbered 4-simplex σ to be a geometrical 4-simplex with tetrahedra numbered 0 , . . . 4. Given a numbered 4-simplex in M , the associated geometrical bivectors ( B geom ab ) IJ are defined as ( B geom ab ) IJ := A (∆ ab ) ( N a ∧ N b ) IJ | N a ∧ N b | , where A (∆ ab ) is the area of the triangle ∆ ab shared by tetrahedra a and b , and N I a is the outward unit normal to tetrahedron a , ( N a ∧ N b ) IJ := 2 N [ I a N J ] b , and | X IJ | 2 := 1 2 X IJ X IJ . /negationslash A set of reduced boundary data { A ab , n ab } is nondegenerate if, for each a , the span of the vectors n ab with b = a is three dimensional. We call two sets of SU (2) group elements { U 1 a } , { U 2 a } equivalent , { U 1 a } ∼ { U 2 a } , if ∃ Y ∈ SU (2) and five signs /epsilon1 a such that For the proof of the following partial version of theorem 3 in [14], see [14,18]. Theorem 1 (Partial version of the reconstruction theorem) . Let a set of nondegenerate reduced boundary data { A ab , n ab } satisfying closure be given, as well as a set { G a } ⊂ Spin (4) , a = 0 , . . . , 4 , solving the orientation constraint, such that { X -a } /negationslash∼ { X + a } . Then there exists a numbered 4-simplex σ in R 4 , unique up to inversion and translation, such that for some µ = ± 1 , with µ independent of the ambiguity in σ . The sign µ in the above theorem is uniquely determined by the data { A ab , n ab , G a } . In fact, as shown in [19], it is equal to the product of the sign corresponding to the Plebanski sector ν ( B phys ab ( A ab , n ab , G a )) and the sign of the orientation ω ( B phys ab ( A ab , n ab , G a )). Recall we have defined the Einstein-Hilbert sector to include two-forms B µν which are either in Plebanski sector (II+) with positive orientation or in Plebanski sector (II-) with negative orientation. The continuum two form B µν reconstructed from the bivectors { B phys ab ( A ab , n ab , G a ) } will thus be in the Einstein-Hilbert sector (in which case we also say the bivectors are in the Einstein-Hilbert sector) if and only if µ = νω = +1.", "pages": [ 5, 6 ] }, { "title": "2.2 Explicit classical expression for the geometrical bivectors, and the restriction to the Einstein-Hilbert sector", "content": "We now come to the new part of the classical analysis. Lemma 1. Let { A ab , n ab , G a } be given satisfying the hypotheses of theorem 1 and let σ be the numbered 4-simplex guaranteed to exist by this theorem. Let { N I a } denote the outward pointing normals to the tetrahedra of σ . Then for some set of signs α a . Proof. We first note that where (2.14) was used in the first line, and (2.1) was used in the last line. Since this holds for all b , it follows that G a · N is proportional to N a ; as both of these vectors are unit, the the coefficient of proportionality must be ± 1 for each a . /squaresolid For the following theorem and throughout the rest of the paper, let ̂ = denote equality modulo multiplication by a positive real number. Theorem 2. Let { A ab , n ab , G a } be given satisfying the hypotheses of theorem 1 and let σ be the numbered 4-simplex guaranteed to exist by this theorem. Then where β ab ( { G a ' b ' } ) ≡ β ba ( { G a ' b ' } ) := -sgn [ /epsilon1 ijk ( G ac · N ) i ( G ad · N ) j ( G ae · N ) k /epsilon1 lmn ( G bc · N ) l ( G bd · N ) m ( G be · N ) n ] (2.17) with { c, d, e } = { 0 , . . . , 4 }\\{ a, b } in any order, and sgn is defined to be zero when its argument is zero. Proof. Let { N I a } be the outward pointing normals to the tetrahedra of σ . Then they satisfy the four-dimensional closure relation (see appendix B) where V a > 0 is the volume of the a th tetrahedron, implying /negationslash Thus where { α a } are the signs in lemma 1. Therefore where β ab ( { G a ' b ' } ) is as in (2.17). We thus have Throughout this paper, let β ab ( { G a ' b ' } ) be defined by (2.17), and for convenience we define ˜ B geom ab ( G a ' ) := β ab ( { G a ' b ' } )( G a · N ) ∧ ( G b · N ), the right hand side of (2.16). Because the expression ( G · N ) i used above will appear often, it is useful to stop for a moment to prove some facts about it. From (2.2) and (2.3), from which one obtains the alternate expression The meaning of this latter expression in turn is made clear in the following definition. Definition 1. Given g ∈ SU (2) not equal to ± I , there exists a unique unit vector n [ g ] i ∈ R 3 and α [ g ] ∈ (0 , 2 π ) satisfying We call n [ g ] i the proper axis of g . In terms of the above definition, one has /squaresolid Lemma 2. Let { A ab , n ab , G a } be given satisfying the hypotheses of theorem 1 and let σ be the numbered 4-simplex thereby guaranteed to exist. Then Proof. Starting from (2.14) and theorem 2, /squaresolid We now come to the classical condition isolating the Einstein-Hilbert sector. Theorem 3. Let a set of nondegenerate reduced boundary data { A ab , n ab } satisfying closure be given, as well as a set { G a } ⊂ Spin (4) , a = 0 , . . . 4 solving the orientation constraint. Then B phys ab ( A ab , n ab , G a ) is in the Einstein-Hilbert sector (that is, µ = ων = +1 ) iff for any one pair a, b .", "pages": [ 6, 7, 8 ] }, { "title": "Proof.", "content": "for all t . Taking d d t of both sides, /squaresolid Applying this to the family of operators ˆ O t = P S ( n t · ˆ L ) on V j -,j + and the states | n t ; j -, j + , k, m 〉 , and to the family of operators ˆ O t = P S ( n t · ˆ L ) on V k and the states | n t ; k, m 〉 , yields the following. Corollary 11. For any variation δ of n , any j -, j + , k , any m ∈ {-k, -k + 1 , . . . , k } , and any set S ⊂ R , one has and /squaresolid", "pages": [ 28 ] }, { "title": "3.1 Notation for SU (2) and Spin (4) structures.", "content": "Let V j denote the carrying space for the spin j representation of SU (2), and ρ j ( g ) , ρ j ( x ) the representation of g ∈ SU (2) and x ∈ su (2) thereon, with the j subscript dropped when it is clear from the context. Let ˆ L i := iρ ( τ i ) denote the generators in each of these representation according to the context. Let /epsilon1 : V j × V j → C denote the invariant bilinear epsilon inner product, and 〈· , ·〉 the Hermitian inner product, on V j [4, 14]. These inner products determine an antilinear structure map J : V j → V j by /epsilon1 ( ψ, φ ) = 〈 Jψ,φ 〉 . J commutes with all group representation matrices, so that it anticommutes with all generators. Let V j -,j + = V j -⊗ V j + denote the carrying space for the spin ( j -, j + ) representation of Spin (4) ≡ SU (2) × SU (2), and ρ j -,j + ( X -, X + ) := ρ j -( X -) ⊗ ρ j + ( X + ) the representation of ( X -, X + ) ∈ Spin (4) thereon, again with the subscript dropped when it is clear from the context. ˆ J i -:= iρ ( τ i ) ⊗ I j + and ˆ J i + := iI j -⊗ ρ ( τ i ) are then the anti-self-dual and self-dual generators respectively, so that ˆ L i := ˆ J i -+ ˆ J i + are the generators of spatial rotations on V j -,j + . Define the bilinear form /epsilon1 : V j + ,j -× V j + ,j -→ C by /epsilon1 ( ψ + ⊗ ψ -, φ + ⊗ φ -) := /epsilon1 ( ψ + , φ + ) /epsilon1 ( ψ -, φ -), and the antilinear map J : V j -,j + → V j -,j + by J : ψ + ⊗ ψ -↦→ ( Jψ + ) ⊗ ( Jψ -), so that As in the case of the SU (2) representations, all Spin (4) representation operators commute with J , and all generators anticommute with J . Lastly, let ι j -,j + k denote the intertwining map from V k to V j -⊗ V j + , scaled such that it is isometric in the Hilbert space inner products.", "pages": [ 8, 9 ] }, { "title": "3.2 Canonical phase space, kinematical quantization, and the EPRL vertex", "content": "In the general boundary formulation of quantum mechanics [4], one associates to the boundary of any 4-dimensional region a phase space , whose quantization yields the boundary Hilbert space of the theory for that region. In the present case, the region is the 4-simplex S . The boundary data consists in the algebra elements B ab and J ab in the frame of each tetrahedron a , and for each pair of tetrahedra a, b one has a parallel transport map G ab from b to a , related to the G a introduced in section 2.1.2 by G ab = ( G a ) -1 G b . These boundary data form a classical phase space isomorphic to the cotangent bundle over any choice of ten independent parallel transport maps G ab = ( X + ab , X -ab ), Γ = T ∗ ( Spin (4) 5 ) = T ∗ (( SU (2) × SU (2)) 5 ), which for simplicity we choose to be the ten with a < b . For a < b , J ab = ( J -ab , J + ab ) and J ba = ( J -ba , J + ba ) respectively generate right and left translations on G ab . The boundary Hilbert space of states H Spin (4) ∂S is the L 2 space over the ten G ab = ( X -ab , X + ab ) ∈ Spin (4) with a < b . The momenta operators ˆ J ± ab and ˆ J ± ba then act by i times right and left invariant vector fields, respectively, on the elements X ± ab , and, in terms of these, ˆ L i ab := ( ˆ J -ab ) i +( ˆ J + ab ) i . One can define an overcomplete basis of H Spin (4) ∂S , the projected spin-network states (see [26,27]), each element of which is labeled by four spins j ± ab , k ab , k ba and two states ψ ab ∈ V k ab , ψ ba ∈ V k ba per triangle: When acting on such a state, the operators ˆ L i ab , ˆ L i ba act specifically on the irreducible representation (irrep) vectors ψ ab , ψ ba : /negationslash /negationslash In terms of the projected spin-network overcomplete basis, the linear simplicity constraint, when quantized as in [9], is equivalent to /negationslash for all a = b . The projected spin networks satisfying linear simplicity are thus parameterized by one spin k ab and two states ψ ab , ψ ba ∈ V k ab per triangle ( ab ), the same parameters specifying a generalized SU (2) spin-network state of LQG: Because j ± ab = 1 2 | 1 ± γ | k ab are always half-integers, one deduces that only certain values of the spins k ab are allowed; let K γ be this set of allowable values, and let H γ ∂S be the span of the SU (2) spin-networks (3.6) with { k ab } ⊂ K γ . One has an embedding where here, and throughout the rest of the paper, we set Due to (3.3) and (3.4) (and because the SU (2) spin-networks satisfy a similar property), this embedding in fact intertwines the spatial rotation generators ˆ L i ab in the Spin (4) and SU (2) theories. Through the embedding ι , the operators ˆ L i ab in the SU (2) theory thus have the same physical meaning as the corresponding operators in the Spin (4) boundary theory. Having reviewed the above, the EPRL vertex for a given LQG boundary state Ψ LQG { k ab ,ψ ab } ∈ H γ ∂S ⊂ H LQG ∂S is then", "pages": [ 9, 10 ] }, { "title": "4.1 Definition", "content": "Let us consider the structure of the original EPRL vertex amplitude (3.9): The integration over the group elements G a can, in a precise sense, be interpreted as a 'sum over histories' of parallel transports from the tetrahedra frames to the 4-simplex frames. This integration over the G a 's inside the vertex amplitude can be thought of as a remnant of the process of integrating out the discrete connection used to obtain the initial BF spin-foam model (see [28]). Furthermore, in the semiclassical analysis [14], one sees that the G a 's over which one integrates in (3.9) play precisely the role of such parallel transports. Given this interpretation of the G a 's, in order to impose the desired restriction to the Einstein-Hilbert sector, one must restrict the discrete history data G a so that they satisfy the inequality (2.25): Normally one would do this by inserting into the path integral where Θ is the Heaviside function, defined to be zero when its argument is zero. However, in the integral (3.9), it is not the classical quantity L i ab that appears, but rather states ψ ab in irreducible representations of the corresponding operators ˆ L i ab . 5 As noted in equations (3.3) and (3.4), ˆ L i ab acts on ψ ab via the SU (2) generators ˆ L i . Therefore, we partially 'quantize' the expression (4.2) by replacing L i ab with the generators ˆ L i , yielding the following G a -dependent operator acting in the spin k ab representation of SU (2): where P S ( ˆ O ) denotes the spectral projector onto the portion S ⊂ R of the spectrum of the operator ˆ O . Inserting (4.3) into the face factors of (3.9) we obtain what we call the proper EPRL vertex amplitude : Let us stop for a moment and remark on the properties of this vertex amplitude. First, as the EPRL vertex, it depends on an SU (2) spin network boundary state and hence may be used to construct a spin-foam model for loop quantum gravity . It is linear in the SU(2) boundary state, as required for the final spin-foam amplitude to be linear in the initial state and antilinear in the final state. Furthermore, as we will show in the next subsection, it is invariant under SU(2) gauge transformations. Finally, and most importantly, as we will show in the next section, its asymptotics only include the single term e iS Regge , as desired. Throughout the rest of this paper, the notation P ba ( { G a ' b ' } ) introduced in (4.3) will also refer to the projector acting in the spin ( s -ab , s + ab ) representation of Spin (4), defined by the same expression (4.3). In each statement using the notation P ba ( { G a ' b ' } ), either the context will determine which projector is intended, or the statement will hold for both projectors. Finally, let us briefly note two ways to rewrite the proper vertex: (1.) It may at first appear arbitrary that the projector was inserted on the right side of each face factor in equation (4.4). However, in fact, one can put the projector (appropriately transformed) anywhere in each face-factor, and the vertex amplitude doesn't change. See appendix D. (2.) We note that, using equation (3.1), one has the following equivalent expression for the proper vertex:", "pages": [ 10, 11 ] }, { "title": "4.2 Proof of invariance under SU (2) gauge transformations", "content": "Theorem 4. The proper EPRL vertex is invariant under arbitrary SU (2) gauge transformations at the tetrahedra. Proof. Let { k ab , ψ ab } be the data for a given spin network on the boundary, and let five SU (2) elements h a , one at each tetrahedron, be given. We wish to show A (+) v (Ψ { k ab ,ρ ( h a ) ψ ab } ) = A (+) v (Ψ { k ab ,ψ ab } ). First, define ˜ G ab := ( h a , h a ) -1 · G ab · ( h b , h b ). Then From this and the SO (3) invariance of /epsilon1 ijk , it follows that We thus have where lemma 10 has been used in the second line, and (4.6) and (4.7) have been used in the third. Using (4.8), we finally have where we have used in the third line the intertwining property of ι s -ab s + ab k ab and in the second to last line the right and left invariance of the Haar measure. /squaresolid", "pages": [ 11, 12 ] }, { "title": "4.3 Spin (4) invariance", "content": "As mentioned in section 2, in defining the classical discrete variables { G a , B ab } , one thinks of each tetrahedron as having its own 'frame'. Concretely, this is manifested in the fact that there exists a local Spin (4) gauge transformation acting at each tetrahedron. Given a choice of Spin (4) group element H a at each tetrahedron a , one has the following gauge transformation: The definition of the proper vertex (4.4) makes key use of a fixed internal direction N I = (1 , 0 , 0 , 0). This vector is used to impose the simplicity constraints (2.8) at each tetrahedron, and superficially breaks the above Spin (4) gauge symmetry. Furthermore, in order to embed LQG states into BF states solving simplicity, the proper vertex uses the map ι s -s + k , which is defined using a specific embedding h : g ↦→ ( g, g ) of SU (2) into Spin (4) via the symmetry condition This use of h also seems to break the above Spin (4) symmetry. The fixed vector N I and embedding h are related by the fact that the SO (4) action of every element in the image of h preserves N I . (The original EPRL vertex amplitude uses these two exact same extra structures [9,11].) Spin (4) acts on the unit vector N I by its SO (4) action, while it acts on the map ι s -s + k via for Λ ∈ Spin (4). The transformed map (Λ · ι ) j -,j + k : V k → V j -,j + still satisfies a symmetry condition similar to (4.10), but with a different embedding (Λ · h ) : SU (2) → Spin (4): where (Λ · h )( g ) := Λ h ( g )Λ -1 . In this section we consider what happens when, in the definition of the proper vertex, the unit vector N I and the map ι s -s + k are replaced, at each tetrahedron a , by their transformation under an arbitrary Spin (4) element Λ a . The resulting, a priori possibly modified proper vertex amplitude we denote by { Λ a } A (+) v . An arbitrary Spin (4) gauge transformation { H a } then acts on { Λ a } A (+) v via We shall prove that the generalized proper vertex { Λ a } A (+) v is in fact independent of { Λ a } , and so is trivially invariant under the above action and in this sense is Spin (4) invariant at each tetrahedron. This result is similar to that in [29]. We begin by noting how to write A (+) v in a way that makes its dependence on N I explicit, which then allows us to write down explicitly the generalized proper vertex { Λ a } A (+) v , after which we prove its independence of { Λ a } . From the first line of equation (D.5), The above projector P ba ( { G a ' b ' } ) on V s -ab ,s + ab can be written with with { c, d, e } = { 0 , . . . , 4 } \\ { a, b } in any order. This immediately yields the following expression for the generalized proper vertex: where with", "pages": [ 12, 13, 14 ] }, { "title": "4.4 Lorentzian generalization", "content": "We close this section by noting that there is an obvious generalization of the expression (4.4) of the proper vertex to the Lorentzian signature. In the Lorentzian EPRL model [9,30], one uses the unitary representations of SL (2 , C ), which are labeled by a real number ρ together with an integer n . Denote the carrying space for such representations by V Lor ρ,n , and let ρ ( G ) denote the representation thereon of G ∈ SL (2 , C ). V Lor ρ,n decomposes into an infinite direct sum of irreducible representations of SU (2): where in the sum k is incremented in steps of 1. The analogue of the embedding ι s -s + k : V k → V s -,s + in the Lorentzian case is the embedding I k : V k → V Lor 2 γk, 2 k mapping V k into the lowest k component of V Lor 2 γk, 2 k in the sum (4.18). The elements in the image of this embedding satisfy a quantization of the simplicity constraints just as those of ι s -s + k do in the Euclidean case [9]. Furthermore, just as one has the invariant bilinear form /epsilon1 on V s -,s + , related to the Hermitian inner product via the antilinear map J , so too one has an invariant bilinear form β on V Lor ρ,n , related to the Hermitian inner product on V Lor ρ,n via an antilinear map J in the same way [31]. For simple representations, ( j + , j -) = ( s -, s + ), ( ρ, n ) = (2 γk, 2 k ), /epsilon1 and β furthermore have the same (anti-)symmetry properties: /epsilon1 ( ψ, φ ) = ( -1) 2 k /epsilon1 ( φ, ψ ), β ( ψ, φ ) = ( -1) 2 k β ( φ, ψ ). In terms of these structures, the expression for the Lorentzian EPRL vertex amplitude is exactly analogous to the Euclidean expression (3.9) [9,31]: /negationslash the only notable difference being that one of the group integrations is dropped in order to ensure finiteness of the amplitude [32, 33]. One can then modify this vertex amplitude in exactly the same way as was done in the Euclidean case, to yield a Lorentzian version of the proper EPRL vertex: /negationslash where with with { c, d, e } = { 0 , . . . , 4 } \\ { a, b } in any order, and where G · N denotes the SO (1 , 3) action of G on N I = (1 , 0 , 0 , 0). Though this generalization of the proper vertex to the Lorentzian signature is natural, and it is difficult to imagine how otherwise to generalize to this case, nevertheless one should justify this generalization more systematically, by quantizing an appropriate classical condition isolating the Lorentzian Einstein-Hilbert sector. One should also check whether the above generalization has the required semiclassical limit, as we will prove below is the case for the Euclidean proper vertex. Henceforth in this paper, unless otherwise indicated, 'proper vertex' shall again always refer to 'Euclidean proper vertex.'", "pages": [ 14, 15 ] }, { "title": "5 Asymptotics", "content": "In the following we state and prove the asymptotics of the proper vertex, using key results from [14].", "pages": [ 15 ] }, { "title": "5.1 Statement of the formula", "content": "It will be useful for later purposes to define the following before defining coherent states. Definition 2. Given any unit n i ∈ R 3 , let | n ; k, m 〉 denote the eigenstate of n · ˆ L in V k with eigenvalue m , and | n ; j -, j + , k, m 〉 the eigenstate of ˆ L 2 and n · ˆ L in V j -.j + with eigenvalues k ( k +1) and m , with phase fixed arbitrarily for each set of labels. Definition 3. Given a unit 3-vector n , a spin j , and a phase θ , we define the corresponding coherent state as The θ argument represents a phase freedom, and will usually be suppressed. Additionally, when the spin is clear from the context, it will be omitted. Such coherent states were first used in quantum gravity by Livine and Speziale [34]. We call an assignment of one spin k ab ∈ K γ and two unit 3-vectors n i ab , n i ba to each triangle ( ab ) in S a set of quantum boundary data . Given such data, the corresponding boundary state in the SU (2) boundary Hilbert space of S is where the θ ab are any phases summing to θ modulo 2 π . The phase θ will usually be suppressed. The state Ψ { k ab ,n ab } so defined is a coherent boundary state corresponding to the classical reduced boundary data A ab = A ( k ab ) := κγk ab and n ab . /negationslash /negationslash When { A ( k ab ) , n ab } is nondegenerate and satisfies closure, we also say that { k ab , n ab } is nondegenerate and satisfies closure. In this case, for each tetrahedron a , there exists a geometrical tetrahedron in R 3 , unique up to translations, such that { A ( k ab ) } b = a and { n i ab } b = a are the areas and outward unit normals, respectively, of the four triangular faces, which we denote by { ∆ t ab } b = a . If these five geometrical tetrahedra can be glued together consistently to form a 4-simplex, we say that the boundary data { k ab , n ab } is Regge-like . For such data, there exists a set of SU (2) elements { g ab = g -1 ba } , unique up to a Z 2 lift ambiguity [14], such that the adjoint action of each g ab on R 3 maps (1.) ∆ t ab into ∆ t ba , and (2.) n ba into -n ab . These group elements can be used to completely remove the phase ambiguity in the boundary state (5.2), by requiring the phase of the coherent states to be chosen such that g ab | n ba 〉 k ab = J | n ab 〉 k ab , where J is as defined in section 3.1. The resulting boundary state Ψ { k ab ,n ab } is called the Regge state determined by { k ab , n ab } , and is denoted by Ψ Regge { k ab ,n ab } . /negationslash The following theorem, as theorem 1 in [14], uses the fact that, because the boundary data { k ab , n ab } determine the geometry of all boundary tetrahedra, it also determines the geometry of the 4-simplex itself [14,35], and hence, in particular, the dihedral angles Θ ab ∈ [0 , π ] via the equation N a · N b = cos Θ ab where N a and N b are the outward pointing normals to the a th and b th tetrahedra, respectively. Theorem 6 (Proper EPRL asymptotics) . If { k ab , n ab } is boundary data representing a nondegenerate Regge geometry, then, in the limit of large λ , where N is independent of λ and the error term is bounded by a constant times λ -13 . If { k ab , n ab } does not represent a nondegenerate Regge geometry, then A (+) v (Ψ λk ab ,n ab ,θ ) decays exponentially with large λ for any choice of phase θ . To prove this theorem, in manner similar to [14], we cast the proper vertex in appropriate integral form A (+) v = ∫ d µ ( x ) e S γ< 1 ( x ) and A (+) v = ∫ d µ ( x ) e S γ> 1 ( x ) , separately for the cases γ < 1 and γ > 1, where S γ< 1 and S γ> 1 are 'actions'. We then determine the critical points for each action. In proving this theorem, we are interested in critical points whose contributions are not exponentially suppressed. For this reason, we define the term 'critical point' to mean points where the action is stationary and its real part is nonnegative . If a point in the domain of integration is such that the real part of the action is an absolute maximum and is nonnegative, we shall say it is a maximal point .", "pages": [ 15, 16, 17 ] }, { "title": "5.2 Integral expressions and critical points", "content": "In the following, whenever we say the words 'critical points' with no other qualification, we refer to critical points of the proper EPRL vertex (4.4).", "pages": [ 17 ] }, { "title": "5.2.1 The case γ < 1", "content": "The relevant integral form of the proper vertex in this case is where The action S γ< 1 is, as in [14], generally complex. The two conditions that determine critical points are maximality and stationarity. In both proving the equations for maximality and checking stationarity, it will be simplest to reuse the results in [14]. This will highlight the simplicity of the additional steps necessary for the present modification. Recall from [14] that the action for γ < 1 for the original EPRL model is For the purpose of the following lemmas and the rest of this section, it is convenient to define a set of group elements together with boundary data { G a , k ab , n ab } to satisfy proper orientation if, for all a < b , β ab ( { G a ' b ' } )tr( τ i X -ba X + ab ) n i ba > 0. Lemma 3. Given boundary data { k ab , n ab } , { G a } is a maximal point of S γ< 1 iff orientation and proper orientation are satisfied. Proof. From (5.5), where the Cauchy-Schwarz inequality has been used in the second line, the fact that J , ι k ab , and ρ ( G ab ) are norm preserving and that || | n ab 〉|| = 1 have been used in the last equality, and || | n ba 〉|| = 1 has been used in the last inequality. /negationslash We now proceed to prove that exp(Re S γ< 1 ) = 1 iff orientation and proper orientation are satisfied. ( ⇐ ) Suppose orientation and proper orientation are satisfied. From equation (52) in [14], it follows that, for each a = b , there exists λ ba such that X -ba X + ab = exp( λ ba n ba · τ ), so that tr( τ i X -ba X + ab ) ̂ = : n [ X -ba X + ab ] i = ± n i ba . Proper orientation then implies that the sign in this equation is β ab ( { G a ' b ' } ) for all a < b . By definition of | n ba 〉 , it follows that | n ba 〉 is an eigenstate of β ab ( { G a ' b ' } ) n i [ X -ba X + ab ] ̂ L i with maximal, and in particular positive, eigenvalue, whence P ba ( { G a ' b ' } ) | n ba 〉 = | n ba 〉 , for all a < b . But this in turn implies that exp(Re S γ< 1 ) = exp(Re S EPRL γ< 1 ) = 1, where the last equality follows from orientation, as proven in section V.A.2 of [14]. ( ⇒ ) Suppose exp(Re S γ< 1 ) = 1. Then both inequalities in (5.7a) are equalities. In particular, this implies /negationslash for all a < b , which in turn implies that exp(Re S EPRL γ< 1 ) = exp(Re S γ< 1 ) = 1, which, from section V.A.2 in [14], implies orientation. As argued above, this implies that, for all a = b , n i ba = ξ ba n [ X -ba X + ab ] i for some ξ ba = ± 1. From the definition of | n ba 〉 , one then has β ab ( { G a ' b ' } ) n [ X -ba X + ab ] i ˆ L i | n ba 〉 = β ab ( { G a ' b ' } ) ξ ba k ab | n ba 〉 . Equation (5.7b) then implies the eigenvalue in the foregoing equation is positive for all a < b , so that ξ ba = β ab ( { G a ' b ' } ), whence β ab ( { G a ' b ' } ) n [ X -ba X + ab ] · n ba = 1 ≥ 0, proving proper orientation. /squaresolid Lemma 4. Let boundary data { k ab , n ab } be given, and suppose { G a } is a maximal point of S γ< 1 . Then it is also a stationary point of S γ< 1 iff closure is additionally satisfied. Proof. If δ is any variation of the group elements G a , from (5.5), (5.6) and the fact that { G a } is maximal, one has From lemma 9.c, Taking the variation of both sides and using the result in (5.8), From lemma 3, as { G a } is a maximal point, orientation and proper orientation are satisfied. From orientation, where lemma 9.b was used in line 1 and k ab = s -ab + s + ab , was used in lines 2 and 4. Furthermore, from orientation and proper orientation, by the same argument used in lemma 3, we have that | n ba 〉 k ab is an eigenstate of P ba ( { G a ' b ' } ) with eigenvalue 1, so that, by equation (5.9), ι s -ab s + ab k ab | n ba 〉 k ab is also an eigenstate of P ba ( { G a ' b ' } ) with eigenvalue 1. This, together with (5.11), via corollary 11 in appendix C, implies that the second term in (5.10) is zero. As proven in [14], using the fact that orientation is satisfied, the remaining term in (5.10) is zero iff closure is satisfied. /squaresolid Theorem 7. Given boundary data { k ab , n ab } , { G a } is a critical point of S γ< 1 iff closure, orientation, and proper orientation are satisfied.", "pages": [ 17, 18 ] }, { "title": "5.2.2 The case γ > 1", "content": "For this case, we derive from scratch an expression for the proper vertex analogous to (18) and (19) in [14]. In doing this, we use the spinorial form of the irreps of SU (2). Let A,B,C, · · · = 0 , 1 denote spinor indices. The carrying space V j can then be realized as the space of symmetric spinors of rank 2 j (see, for example, [4]). Let n A denote the spinor corresponding to the coherent state | n 〉 1 2 . As in [14, 36], the key property of coherent states we use is that, in their spinorial form, the higher spin coherent states are given by From the relation (3.8) between k and s + , s -for a given triangle, one deduces for γ > 1 that s + = s -+ k . For this case, the explicit expression for ι s -s + k in terms of symmetric spinors is given in equations (A.12) and (A.13) of [4] 6 . Let v A 1 ··· A 2 k ∈ V k be given. For γ > 1, one has where the symmetrization is over the A indices only. In order to impose the symmetrization over the A indices, similar to [14], on the left of each ι s -s + k , acting in the self-dual part of the codomain, we insert a resolution of the identity on V s + into coherent states: where d m is the measure on the metric 2-sphere normalized to unit area, and d s := 2 s +1. In spinorial notation where m † A := ( 1 2 〈 m | ) A . Starting from equation (4.4) with ψ ab = | n ab 〉 k ab = n A 1 ab · · · n A 2 k ab ab , writing out all spinor indices explicitly, we insert two resolutions of the identity (5.15) into each face factor in (4.4), one after each ι s -s + k . Denote the integration variables m ab and m ba respectively for the left and right insertions. Writing out the /epsilon1 -inner product in terms of alternating tensors /epsilon1 AB , using m † A = -/epsilon1 AB ( Jm ) B , simplifying, and then writing the final expression again in terms of Hermitian inner products, one obtains where Recall from [14] that the action for γ > 1 for the original EPRL model is 7 /negationslash Lemma 5. Given boundary data { k ab , n ab } , { G a , m ab } is a maximal point of S γ> 1 iff orientation and proper orientation are satisfied and m ab = n ab for all a = b . Proof. From (5.17), where the Cauchy-Schwarz inequality, the fact that J and ρ ( X ± ab ) are norm preserving, and || | n ab 〉|| = || | m ab 〉|| = 1 have been used in the first inequality, and || | n ba 〉|| = 1 has been used in the last inequality. /negationslash We now proceed to prove that exp(Re S γ> 1 ) = 1 iff orientation and proper orientation are satisfied and m ab = n ab for all a = b . /negationslash ( ⇐ ) Suppose orientation and proper orientation are satisfied and m ab = n ab for all a = b . By the same argument used in lemma 3, it follows that P ba ( { G a ' b ' } ) | n ba 〉 = | n ba 〉 for all a < b . But this in turn implies that exp(Re S γ> 1 ) = exp(Re S EPRL γ> 1 ) = 1, where the last equality follows from orientation and m ab = n ab , as proven in section V.A.2 of [14]. ( ⇒ ) Suppose exp(Re S γ> 1 ) = 1. Then both inequalities in (5.19a) are equalities. In particular it follows for all a < b , which in turn implies that exp(Re S EPRL γ> 1 ) = exp(Re S γ> 1 ) = 1, which, from section V.A.2 in [14], implies orientation and m ab = n ab . Furthermore, by the same argument used in lemma 3, (5.19b) also implies proper orientation. /squaresolid Lemma 6. Let boundary data { k ab , n ab } be given, and suppose { G a , m ab } is a maximal point of S γ> 1 . Then it is also a stationary point of S γ> 1 iff closure is additionally satisfied. Proof. If δ is any variation of G a and m ab , from (5.17) and (5.18) one has /negationslash Because { G a , m ab } is a maximal point, from lemma 5, orientation and proper orientation are satisfied, and m ab = n ab for all a = b . It follows that | n ba 〉 k ab = | m ba 〉 k ab and, by the same argument used in lemma 3, both are eigenstates of P ba ( { G a ' b ' } ) with eigenvalue 1, so that by corollary 11 in appendix D, the second term above is zero. As proven in [14], because orientation is satisfied and m ab = n ab for all a = b , it follows that the remaining term in (5.20) is zero iff closure is satisfied. /squaresolid /negationslash /negationslash Theorem 8. Given boundary data { k ab , n ab } , { G a , m ab } is a critical point of S γ> 1 iff closure, orientation, and proper orientation are satisfied, and m ab = n ab for all a = b .", "pages": [ 19, 20, 21 ] }, { "title": "5.3 Proof of the asymptotic formula", "content": "Using the above results, we proceed to prove theorem 6. Before getting into the details of the proof, we summarize its general structure. As already mentioned, the critical point equations for the proper vertex integrals (5.4) and (5.16) have a set of symmetries (5.21), of which the global Spin (4) symmetry is the only continuous one. In order to apply the stationary phase method to calculate the asymptotics, the critical points must be isolated, and hence this continuous symmetry must be removed. As in [14], we do this by performing the change of variables ˜ G a := ( G 0 ) -1 G a for a = 1 , . . . 4. Then G 0 no longer appears in the integrand, so that the G 0 integral drops out. Upon removing the tilde labels, the remaining integrand is the same as the original integrand except with G 0 replaced by the identity. In what follows G a = ( X -a , X + a ) shall denote these 'gauge-fixed' group elements, with G 0 ≡ id, in terms of which the continuous symmetry has been removed. The proof then has two steps, the first of which has already been done in theorems 7 and 8 above: (1.) prove that the critical points of proper EPRL are precisely the subset of critical points of original EPRL at which proper orientation is satisfied. (2.) prove that, given a set of SU (2) boundary data { k ab , n ab } , the critical points of original EPRL at which proper orientation is satisfied are all equivalent and are precisely the critical points which give rise to the asymptotic term (5.3) in the original EPRL asymptotics [14]. Because proper orientation is satisfied, the projector P ba will act as the identity, and the value of the proper EPRL action at these critical points will be the same as the value of the original EPRL action at these points, yielding precisely the asymptotic behavior (5.3) claimed. Let us begin by reviewing the results from theorems 7 and 8. The critical point equations for γ < 1 and γ > 1 are equivalent: the only difference is that for γ > 1 one integrates over extra variables, m ab , which, however, come with the critical point equations m ab = n ab , eliminating them. This allows us to effectively consider both the γ < 1 case and γ > 1 case simultaneously in the following. As given in theorems 7 and 8, the remaining critical point equations are and for all a < b . The first of these, (5.22), is of the same form for both { X + a } and { X -a } : One therefore proceeds by finding the solutions { U a } to (5.24) for a given set of SU (2) boundary data { k ab , n ab } , and then from these one constructs the solutions { G a } to (5.22), and then one checks which among these, if any, solves (5.23) in order to determine the critical points of the vertex integral. /negationslash The solutions to (5.24) have already been analyzed by [14]. To use the results in this reference, one needs the notion of a vector geometry : A set of boundary data { k ab , n ab } is called a vector geometry if it satisfies closure and there exists { h a } ⊂ SO (3) such that ( h a /triangleright n ab ) i = -( h b /triangleright n ba ) i for all a = b . The authors of [14] then proceed by considering separately the three cases in which the boundary data (i.) does not define a vector geometry (ii.) defines a vector geometry which is, however, not a nondegenerate 4-simplex geometry, and (iii.) defines a nondegenerate 4-simplex geometry. We use this same division and consider each of these three cases in turn.", "pages": [ 21, 22 ] }, { "title": "Case (i.): Not a vector geometry.", "content": "In this case, as proven in [14], there are no solutions to (5.24) and hence no solutions to (5.22), and hence no critical points. The vertex integral therefore decays exponentially with λ . Case (ii.): A vector geometry, but no nondegenerate 4-simplex geometry. In this case, as proven in [14], there is exactly one solution to (5.24), up to the equivalence (2.13). The only solution to (5.22) is therefore ( X -a , X + a ) = ( U a , /epsilon1 a Y U a ). But then X -ba X + ab = ± I , so that this solution fails to satisfy condition (5.23), so that there are no critical points. The vertex integral therefore decays exponentially with λ . Case (iii.): A nondegenerate 4-simplex geometry. In this case, as proven in [14], (5.24) has two inequivalent solutions { U 1 a } and { U 2 a } , so that there are four inequivalent solutions to (5.22): ( X -a , X + a ) = ( U 1 a , U 1 a ) , ( U 2 a , U 2 a ) , ( U 1 a , U 2 a ) , ( U 2 a , U 1 a ). Neither ( U 1 a , U 1 a ) nor ( U 2 a , U 2 a ), nor any solution equivalent to these, satisfies (5.23), again because X -ba X + ab = ± I . It remains only to consider the solutions Because 1 X -ab 1 X + ba = ( 2 X -ab 2 X + ba ) -1 , the proper axes n [ 1 X -ab 1 X + ba ] i , n [ 2 X -ab 2 X + ba ] i defined in (2.22) are equal and opposite, so that From this one deduces which gives us /negationslash for all a = b . Because { U 1 a } /negationslash∼ { U 2 a } , we have { 1 X + a } /negationslash∼ { 1 X -a } and { 2 X + a } /negationslash∼ { 2 X -a } , so that both { 1 G a } and { 2 G a } satisfy the hypotheses of lemma 2, implying that neither side of (5.28) is zero. It follows that exactly one of β ab ( { α G a ' b ' } )tr( τ i α X -ab α X + ba ) n i ba , α = 1 , 2, is positive, so that exactly one of { 1 G a } , { 2 G a } satisfies proper orientation and so is a critical point. Furthermore, at this one critical point, by theorem 3, the µ arising in the reconstruction theorem (theorem 1 here and theorem 3 in [14]) is 1. Because, at this point, the value of the action (5.5) (respectively (5.17)) for the proper vertex is equal to the value of the action (5.6)(respectively (5.18)) for the original vertex, from the analysis of [14], this one critical point gives rise to precisely the desired asymptotics stated in theorem 6.", "pages": [ 22, 23 ] }, { "title": "6 Conclusions", "content": "The original EPRL model, as shown and emphasized in [18,19], due to the fact that it is based on the linear simplicity constraints, necessarily mixes three of what we call Plebanski sectors as well as two dynamically determined orientations. This mixing of sectors was identified as the precise reason for the multiplicity of terms in the asymptotics of the EPRL vertex calculated in [14]. Furthermore, when multiple 4-simplices are considered, asymptotic analysis thus far [16,17] indicates that critical configurations contribute in which these sectors can vary locally from 4-simplex to 4-simplex. The asymptotic amplitude for such configurations is the exponential of i times an action which is not Regge, but rather a sort of 'generalized Regge action'. The stationary points of this 'generalized' action are not in general solutions to the Regge equations of motion, and thus one has sectors in the semiclassical limit which do not represent general relativity. In this paper, a solution to this problem is found. We began by deriving a classical discrete condition that isolates the sector corresponding to the first term in the asymptotics - what we have called the Einstein-Hilbert sector, in which the BF action is equal to the Einstein-Hilbert action including sign. Equivalently, this is the sector in which the sign of the Plebanski sector (II ± ) matches the sign of the dynamical orientation. By appropriately quantizing this condition and using it to modify the EPRL vertex amplitude, we have constructed what we call the proper EPRL vertex amplitude. This vertex amplitude continues to be a function of SU (2) spin-network data, so that it may continue to be used to define dynamics for LQG. We have shown that the proper vertex is SU (2) gauge invariant and is linear in the boundary state, as required to ensure that the final transition amplitude is linear in the initial state and antilinear in the final state. It is furthermore Spin (4)invariant in the sense that, similar to the original EPRL model [29], it is independent of the choice of extra structures used in its definition which seem to break Spin (4) symmetry. Finally, it has the correct asymptotics with the single term consisting in the exponential of i times the Regge action. Two interesting further research directions would be (1.) to justify the Lorentzian signature generalization given in equation (4.20), via a quantization of the Lorentzian Einstein-Hilbert sector, and to verify that (4.20) also has the desired single-termed semiclassical limit and (2.) to generalize the present work to the amplitude for an arbitrary 4-cell, which might be used in a spin-foam model involving arbitrary cell-complexes, similar to the generalization [12] of Kami'nski, Kisielowski, and Lewandowski. The first of these tasks should be straightforward. The second, however, seems to require a new way of thinking about the discrete constraint (2.25) used to isolate the Einstein-Hilbert sector. For, the β ab sign factor involved in this condition uses in a central way the fact that there are 5 tetrahedra in each 4-simplex. Lastly, it is important to understand if and how the graviton propagator calculations [37-41], and spin-foam cosmology calculations [42-44] will change if the presently proposed proper vertex is used in place of the original EPRL vertex. In the case of the graviton propagator, only the leading order term in the vertex expansion has thus far been calculated [40]. To this order, only one 4-simplex is involved, and one does not expect the use of the proper vertex to change the results, because the coherent boundary state used in this work already suppresses all but the one desired term (5.3) in the asymptotics. However, higher order terms in the propagator may very well be affected by the use of the proper vertex. We leave this and similar such questions for future investigations.", "pages": [ 23, 24 ] }, { "title": "Acknowledgements", "content": "The author thanks Christopher Beetle for invaluable discussions and for pointing out a simpler proof for theorem 2, Carlo Rovelli and Antonia Zipfel for remarks on a prior draft, Alejandro Perez and Abhay Ashtekar for encouraging the author to finish this work, and the quantum gravity institute at the University of Erlangen-Nuremburg for an invitation to give a seminar on this topic, which led to the correction of an important sign error. This work was supported in part by the NSF through grant PHY-1237510 and by the National Aeronautics and Space Administration through the University of Central Florida's NASA-Florida Space Grant Consortium.", "pages": [ 24 ] }, { "title": "A Well-definedness of orientation and Plebanski sectors", "content": "As in section 2 in the main text, we let B ab denote the bivectors G a /triangleright B ab in the 4-simplex frame. Throughout this appendix we assume that G a and B ab satisfy closure, orientation, and linear simplicity, implying corresponding restrictions on B ab . 8 As mentioned in section 2, for each choice of flat connection ∂ µ adapted to S , the discrete variables {B ab } determine a unique continuum two form B µν ( {B ab } , ∂ ) via the conditions ∂ σ B µν ( {B ab } , ∂ ) = 0 and ∫ ∆ ab ( S ) B ( {B cd } , ∂ ) = B ab . This continuum two form in turn determines a dynamical orientation of S , as well as determining one of three Plebanski sectors, represented respectively by the functions ω ( B µν ) and ν ( B µν ), defined in section 2.1.2, taking values in { 0 , 1 , -1 } . We here prove that the orientation and Plebanski sector of B µν ( {B ab } , ∂ ) are independent of the choice of ∂ µ adapted to S . (In the paper [18,19], a slightly different but equivalent way of reconstructing B µν was used. The well-definedness of orientation and Plebanski sector of B µν as reconstructed there was proven in that paper. We here prove it anew for the new present reconstruction of B µν , for completeness.) In the following, we denote the vertex of S opposite each tetrahedron a ∈ { 0 , . . . , 4 } by p a . We also use the term 'face' in the general sense of any lower dimensional simplex which forms part of the boundary of a higher dimensional simplex. Bold lower case Greek letters, µ , ν = 0 , 1 , 2 , 3, shall be used to label different coordinates of a given coordinate system. Lemma 7. Given a flat connection ∂ µ adapted to S , there exists a unique coordinate system x µ such that (1.) ∂ µ is the associated coordinate derivative operator and (2.) the values of the coordinates at the five vertices of S , p 0 , p 1 , p 2 , p 3 , p 4 , are respectively given by Furthermore, the range of possible values of the 4-tuple of coordinates ( x µ ) coincides precisely with the linear 4-simplex in R 4 with these five vertices, which we refer to as the 'canonical 4-simplex' in R 4 . We call { x µ } 'the coordinate system determined by ∂ µ '. Proof. For each µ = 0 , 1 , 2 , 3, let V µ µ denote a vector in T p 4 S tangent to the edge p 4 p µ , pointing away from p 4 . This vector is unique up to scaling by a positive number. Fix this scaling freedom by first parallel transporting V µ µ along the edge p 4 p µ , and then requiring that the affine length of p 4 p µ with respect to V µ µ be 1. Because the connection ∂ µ is flat, one can use ∂ µ to unambiguously parallel transport V µ µ to all of S , yielding a vector field V µ µ on S for each µ = 0 , 1 , 2 , 3. Because the vectors { V µ µ ( p 4 ) } at p 4 were chosen linearly independent, the vectors { V µ µ ( p ) } at each point p ∈ S form a basis of T p S . Let { λ µ µ ( p ) } denote the basis dual to { V µ µ ( p ) } at each p . For each µ , the resulting one form λ µ µ then satisfies ∂ µ λ µ ν = 0, implying ∂ [ µ λ µ ν ] = 0; because S is simply connected, this implies that, for each µ , there exists a function x µ , unique up to addition of a constant, such that λ µ µ = ∂ µ x µ . Fix this freedom in each x µ by requiring x µ ( p 4 ) = 0. Because λ µ µ = ∂ µ x µ are everywhere linearly independent, { x µ } forms a good coordinate system on S . Furthermore, from V µ µ ∂ µ x ν = V µ µ λ ν µ = δ ν µ , one has V µ µ = ( ∂ ∂x µ ) µ . Because ∂ µ annihilates λ µ µ = ∂ µ x µ (and V µ µ = ( ∂ ∂x µ ) µ ), ∂ µ is the coordinate derivative operator for { x µ } . /negationslash Consider the differential equation V µ µ ∂ µ x ν = δ ν µ for a given fixed µ . Because V µ µ is tangent to the edge p 4 p µ , this equation dictates how to evolve each of the four coordinates x ν along p 4 p µ from its starting value x ν = 0 at p 4 , thereby determining its value at p µ . For ν = µ this implies x ν = 0 at p 4 p µ . For ν = µ , the differential equation simply expresses that x µ is an affine coordinate for V µ µ along p 4 p µ , so that, by construction, x µ = 1 at p µ . Now, the coordinates x µ provide an embedding Φ of S into R 4 , Φ : p ↦→ x µ ( p ). By construction the point p 4 maps to (0 , 0 , 0 , 0), whereas, as just shown, the points p 0 , p 1 , p 2 , p 3 map to the points (1 , 0 , 0 , 0), (0 , 1 , 0 , 0), (0 , 0 , 1 , 0), (0 , 0 , 0 , 1). Because ∂ µ is adapted to S , S is the convex hull of its vertices as determined by the affine structure defined by ∂ µ . But this affine structure is the same as that defined by the coordinates x µ , so that Φ[ S ] is the convex hull, in R 4 , of the points (A.1). That is, Φ[ S ] is the linear 4-simplex in R 4 with vertices (A.1). /squaresolid For the purposes of the following, the action of a diffeomorphism ϕ on a derivative operator ∂ µ is defined by ( ϕ · ∂ ) µ λ ν := ( ϕ -1 ) ∗ ∂ µ ( ϕ ∗ λ ν ). The resulting action of ( ϕ · ∂ ) on a general tensor t α...γ ρ...σ is then given by ( ϕ · ∂ ) µ t α...γ ρ...σ := ϕ · ( ∂ µ ( ϕ -1 · t α...γ ρ...σ )) where ϕ · denotes the left action of ϕ on the tensor in question (thus, push-forward for contravariant indices and pull-back via ϕ -1 for covariant indices [45]). Lemma 8. Given any two flat connections ∂ µ , ˜ ∂ µ adapted to S , there exists an orientation preserving diffeomorphism ϕ : S → S mapping each face of S to itself, and mapping ∂ µ to ˜ ∂ µ .", "pages": [ 24, 25 ] }, { "title": "B Four dimensional closure", "content": "The following property is mentioned, for example, in [15,46,47]. Theorem 10. For any geometrical 4-simplex σ in R 4 , where the sum is over tetrahedra, and V t and N I t are the volume and outward normal to tetrahedron t . /squaresolid Proof. Define 3 /epsilon1 I to be the three-form on R 4 with components ( 3 /epsilon1 I ) JKL = /epsilon1 I JKL . Then Thus, Let t /epsilon1 denote the volume form for t , so that for each t , Pulling back JKL to tetrahedron t , it follows that which combined with B.2 yields the result. /squaresolid", "pages": [ 26, 27 ] }, { "title": "C Properties of embeddings and projectors", "content": "Recall | n ; k, m 〉 denotes the eigenstate of n · ˆ L in V k with eigenvalue m , and | n ; j -, j + , k, m 〉 the eigenstate of ˆ L 2 and n · ˆ L in V j -.j + with eigenvalues k ( k +1) and m ,", "pages": [ 27 ] }, { "title": "D Expression for vertex with projectors on the left", "content": "Lemma 12. For each unit n i ∈ R 3 , g ∈ SU (2) , k , and m , there exists θ ∈ R 2 π Z such that Proof. /squaresolid Lemma 13. For any S ⊆ R , and in any irrep of Spin (4) , and any v i ∈ R 3 , Proof. Let v i =: λn i with λ ≥ 0 and n i unit. Using that J anticommutes with ˆ L i , for any n and k , whence for some { θ m } ⊂ R 2 π Z , so that, for all m , /squaresolid Theorem 12. The vertex amplitude (4.4) can also be written with the projector, appropriately transformed, moved to anywhere in each face factor: Proof. One starts from (4.5), and uses lemma 9.c, lemma 10, the hermicity of orthogonal projectors, and lemma 13 in succession, as well as using the fact that if X i j denotes the adjoint action of X = SU (2) then X i j τ j = X -1 τ i X . /squaresolid", "pages": [ 29 ] } ]

[ { "title": "Ayan Chatterjee", "content": "∗ Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India. Bhramar Chatterjee † and Amit Ghosh ‡ Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, INDIA. (Dated: November 16, 2018) In completely local settings, we establish that a dynamically evolving black hole horizon can be assigned a Hawking temperature. Moreover, we calculate the Hawking flux and show that the radius of the horizon shrinks. PACS numbers: The laws of black hole mechanics in general relativity are remarkably analogous to the laws of thermodynamics [1]. This analogy is exact when quantum effects are taken into account. Indeed, Hawking's semiclassical analysis establishes that quantum mechanically, a stationary black hole with surface gravity κ radiates particles to infinity with a perfect black body spectrum at temperature κ/ 2 π [2]. Consequently, asymptotic observers perceive a thermal state and assign a physical temperature to the black hole. The precise match to thermodynamics is complete when the thermodynamic entropy of the black hole is identified with a quarter of its area [3]. The original calculation of Hawking is independent of the gravitational field equations. It relies only on the behavior of quantum fields in a specific spacetime geometry describing a stationary black hole formed due to a gravitational collapse. Over the years, several other techniques have been developed to study spontaneous particle emission and the Hawking temperature for more general spacetimes. For example, the Hartle-Hawking proposal [5] and the Euclidean approach [6] have been extensively used to associate thermal states to spacetimes with bifurcate Killing horizons. In fact, it has been established that in any globally hyperbolic spacetime with bifurcate Killing horizon, there can exist a vacuum thermal state at temperature κ/ 2 π which remains invariant under the isometries generating the horizon [7]. Although these constructions are elegant, they are quite restrictive, inapplicable even for spacetimes with superradiance [7]. These formulations also do not indicate how such a thermal state may arise as a result of some version of physical process. In addition, their existence requires knowledge of global structure of spacetime. As a result, they do not appear very useful to study thermal properties of local horizons. On the other hand, the laws of black hole mechanics apply equally well to black hole horizons which can been proved using only local geometrical properties of null surfaces, without any assumptions on the global development of the spacetime in which the horizon is embedded [8-11]. It has also been established that such horizons can be assigned an entropy proportional to the area of the local horizon [12, 13]. Thus, it seems to be a reasonable physical expectation that even with a local definition of black hole horizon one should be able to establish the analogy to thermodynamics. More precisely, such horizons should have a temperature of κ/ 2 π . Incidentally, this question has been investigated in a semiclassical approach which treats Hawking radiation as a quantum tunneling phenomenon [14]-[18]. The method involves calculating the imaginary part of the action for the (classically forbidden) process of s-wave emission, from inside and through the horizon (see [19] for more details). Using the WKB-approximation the tunnelling probability for such a classically forbidden trajectory is calculated to be, Γ = e -2Im S where, S is the classical action of the trajectory to leading order in /planckover2pi1 . This is equated to the Boltzmann factor e -βE to extract the inverse Hawking temperature β . The main advantage of this formalism is that the calculations involve only the local geometry and hence can be applied to any local horizon. Indeed, tunnelling method has been applied to local dynamical black hole horizons and the temperature is found to be κ/ 2 π where κ is the dynamical surface gravity [20, 21]. Still there are some problems with the method itself and some issues which have not been addressed in this treatment of dynamical horizons. First, the approach depends heavily on the semiclassical approximation and though it is argued that this remains valid near the horizon, it would be better to devise a more general formalism which does not rely on WKB-like approximations. Secondly, in calculating the imaginary part of the semiclassical action S from the HamiltonJacobi equation, a singular integral appears with a pole at the horizon. While for the static case the result is standard, for the dynamical horizon it is not clear how the integration is to be performed since the position of the horizon changes in a dynamical process. Lastly, in all these treatments of radiation from dynamical horizons the evolution of the horizon itself is never addressed. In other words, it is not clear how the horizon loses area due to emission of a flux of radiation. The local formalisms of black hole horizon should be able to address these issues. In this paper, a formalism is developed to establish two basic issues. First, that one can associate a temperature to local dynamical horizons without the need for any WKB-like approximation schemes. Second, that there exists a precise relation between the radiation emitted by the horizon and area loss, i.e., flux of outgoing radiation through the horizon in between two partial Cauchy slices exactly equals the difference of radii of the sphere that foliates the horizon at those two instances. We elucidate our arguments as follows. To calculate temperature for local dynamical horizons, we begin by considering the Kodama vector field [22]. For dynamical spacetimes, this vector field provides a preferred timelike direction and is parallel to the Killing vector at spatial infinity which we assume to be flat. We can construct well-behaved positive frequency field modes on both sides of the horizon by considering the Kodama vector field but the outgoing modes exhibit logarithmic singularities on the horizon under some approximation. However, if considered as distribution valued, these modes can be interpreted as horizon crossing and the probability current for these modes remain well defined. The Hawking temperature is determined if one equates the conditional probability, that modes incident on one side is emitted to the other side, to the Boltzmann factor [23, 24], Since this method does not depend on the entire evolution of the field modes in the spacetime, it is ideally suited for our purpose. To evaluate the Hawking flux, we recall that there are two well known (and related) definitions of local black hole horizon, the future outward trapping horizon (FOTH) [8, 9] and the Dynamical Horizon [25, 26] (or its equilibrium version called the isolated horizon). In these local settings, black hole horizons are a stack of apparent horizons which, under suitable energy conditions, are either null or spacelike. As such, energy flux can only remain on the surface or flow into the horizon. In order that matter fields flow out off such a surface requires that the surface must be timelike in some affine interval. However, to achieve a timelike evolution of the horizon, some energy conditions need to be violated. This is only natural since Hawking radiation necessarily associates, with the thermal emission of particles, a positive flux of energy flowing to infinity (we shall assume that the spacetime is asymptotically flat) and a corresponding flux of negative energy flowing into the black hole (this negative energy flux can also be motivated by the fact that the expectation values of stress energy tensor of quantum fields generically violate energy conditions). In this process the horizon looses area and energy. The plan of the paper is as follows: First, we will discuss the geometrical setup which is based on future outer trapping horizon (FOTH). Next, we show that how the Hawking temperature is proportional to the dynamical surface gravity associated with the Kodama vector. Finally, we will calculate the flux of energy radiated in a dynamical process. We begin with definitions. We follow the conventions of [9]. Consider a four dimensional spacetime M with signature ( -, + , + , +). A three-dimensional submanifold ∆ in M is said to be a future outer trapping horizon (FOTH) if 1) It is foliated by a preferred family of topological two-spheres such that, on each leaf S , the expansion θ + of a null normal l a + vanishes and the expansion θ -of the other null normal l a -is negative definite, 2) The directional derivative of θ + along the null normal l a -(i.e., L l -θ + ) is negative definite. Thus, ∆ is foliated by marginally trapped two-spheres. According to a theorem due to Hawking, the topology of S is necessarily spherical in order that matter or gravitational flux across ∆ is non-zero. If these fluxes are identically zero then ∆ becomes a Killing or isolated horizon. Even though our arguments will remain local, for definiteness, we choose a spherically symmetric background metric where both f and r are smooth functions of x ± . The expansions of the two null normals are θ ± = (2 /r ) ∂ ± r respectively where ∂ ± = ∂/∂x ± . In this coordinate system, the second requirement for FOTH translates to ∂ -θ + < 0 on ∆. Let the vector field t a = l a + + hl a -be tangential to the FOTH for some smooth function h . Then the Raychaudhuri equation for l a + and the Einstein equation implies where T ++ = T ab l a + l b + and T ab is the energy momentum tensor. Several consequences follow from this equation. First, the FOTH is degenerate (or null) if and only if T ++ = 0 on ∆. In that case, the FOTH is generated by l a + . Degenerate FOTH is not interesting for Hawking radiation because this implies ∂ + r = 0. As a consequence, the area, A = 4 πr 2 of S , and the Misner-Sharp energy for this spacetime, given by E = 1 2 r , also remains unchanged. Secondly, since t 2 = -2 he -f , a FOTH becomes spacelike if and only if T ++ > 0 and is timelike if and only if T ++ < 0. For a timelike FOTH, several consequences follow. Here, L t r < 0, and hence, ∆ is timelike if and only if the area A and the Misner-Sharp energy E decreases along the horizon. This is also expected on general grounds since the horizon receives an incoming flux of negative energy, T ++ < 0. As we have emphasized before, in the dynamical spacetime (2) the Kodama vector field plays the analog role of the Killing vector. For this spacetime, it is given by The surface gravity is defined through K a ∇ [ b K a ] = κK b and is k = -e f ∂ -∂ + r . The FOTH condition ∂ -θ + < 0 implies k > 0. Let us now determine the positive frequency modes of the Kodama vector. It is easy to see that any smooth function of r is a zero-mode of the Kodama vector. Once, a zero-mode is obtained, other positive frequency eigenmodes are evaluated using Here, Z ω are the eigenfunctions corresponding to the positive frequency ω . For simplification, let us introduce new coordinates, y = x -and r and two new functions, ¯ Z ω ( y, r ) = Z ω ( x + , x -) and G ( y, r ) = e f ( ∂ + r ). As a result, the eigenvalue equation (5) reduces to Integrating and transforming back to old coordinates, the above equation gives where F ( r ) is an arbitrary smooth function of r and the subscript r under the integral sign denotes that while doing the integration r is kept fixed. To evaluate the integral in (7), we multiply the numerator and the denominator by ( ∂ -θ + ) and use the fact that for any fixed r surface, e f ( ∂ -θ + ) = -2 k/r , (although the strict interpretation of k as the surface gravity holds only for surfaces with θ + = 0, it exists as a function in any neighbourhood of the horizon). Thus, in some neighbourhood of the horizon we get We now assume (this is the only assumption we make in this calculation) that during the dynamical evolution k is a slowly varying function in some small neighbourhood of the horizon (the zeroth law takes care of it on the horizon, but we also assume it to hold in a small neighbourhood of the horizon). This gives where the spheres are not trapped 'outside the trapping horizon' ( θ + > 0) and fully trapped 'inside' ( θ + < 0). These are precisely the modes which are defined outside and inside the dynamical horizon respectively but not on the horizon. Now we have to keep in mind the modes (9) are not ordinary functions, but are distribution-valued. Comparing with the spherically symmetric static case [24], we find for /epsilon1 → 0 + for the choice λ = -iω/k . The distribution (10) is welldefined for all values of θ + and λ , and it is differentiable to all orders. The modes Z ∗ ω are given by the complex conjugate distribution. We wish to calculate the probability density in a single particle Hilbert space for positive frequency solutions across the dynamical horizon A straightforward calculation gives, apart from a positive function of r , The conditional probability that a particle emits when it is incident on the horizon from inside is, This gives the correct Boltzmann weight with the temperature k/ 2 π , which is the desired value. We now show that as the horizon evolves, the radius of the 2-sphere foliating the horizon shrinks in precise accordance with the amount of flux radiated by the horizon. To study the flux equation, consider new coordinates, ( x + , x -) ↦→ ( θ + , ˜ x -) where ˜ x -= x -. On FOTH, ( ∂ -θ + ) / ( ∂ + θ + ) is equal to -( ∂ -∂ + r ) / (4 πr T ++ ) and negative definite. As a result, the derivatives are related to each other by It is not difficult to show that ˜ ∂ -is proportional to the tangent vector t a to the FOTH. Observe that the normal one-form to ∆ must be proportional to ( dr -2 dE ), which on the horizon is equal to the one-form In arriving at the above identity we have made use of two Einstein's equations [9] and energy equations As a result, the normal vector n a is proportional to so that the tangent vector t a = ∂ a + + h∂ a -, which is clearly proportional to (14). So ˜ x -, θ, φ are natural coordinates on FOTH. The lineelement (2) induces a line-element on ∆ Consequently, the volume element on ∆ is given by dµ = √ 2 e -f h -1 r 2 sin θ d ˜ x -dθdφ . We can now calculate the flux of matter energy that crosses the dynamical horizon-it is an integral on a slice of horizon bounded by two spherical sections S 1 and S 2 where ˆ n a is the unit normal vector and K a is the Kodama vector. Using spherical symmetry, eqn. (14) and eqn. (16), we get Making use of the Einstein equation (16) on the horizon and (14), we get where r 1 , r 2 are respectively the two radii of S 1 , S 2 . Since the area is decreasing along the horizon, r 2 < r 1 where S 2 lies in the future of S 1 . As a result, the outgoing flux of matter energy radiated by the dynamical horizon is positive definite (and the ingoing flux of matter energy is negative definite). The flux formula (23) differs from that given in [25]. Since the Kodama vector field provides a timelike direction and is null on the horizon, it seems more appropriate to use K a for the dynamical horizon. The derivation of Hawking temperature and the flux law depends on two assumptions. First, that the Kodama vector exists in the spacetime. For spherically symmetric spacetimes, the Kodama vector field exists unambiguously and the Misner-Sharp energy is well defined. For more general spacetimes, a Kodama-like vector field is not known, however, one can still define some mass for such cases that reduces to the Misner-Sharp energy in the spherical limit [27]. The second assumption, the existence of a slowly varying k can also be motivated for large black holes. In such cases, the horizon evolves slowly enough so that the surface gravity function should vary slowly in some small neighbourhood of the horizon. Alternatively, we can conclude that the Hawking temperature for a dynamically evolving large black hole is k/ 2 π if the dynamical surface gravity is slowly varying in the vicinity of the horizon. The set-up described in this paper can be further developed to model dynamically evaporating black hole horizons through Hawking radiation, analytically as well as numerically. Over the years, several models have been constructed which study radiating black holes, formed in a gravitational collapse, based on the imploding Vaidya metric with a negative energy-momentum tensor, show that a timelike apparent horizon forms due to violation of energy conditions [28]. However, such models are based on global considerations of event horizons, while local structures like that used in [29] might be useful for a better understanding of Hawking radiation and computations of quantum field theoretic effects (see also [30]). It is also interesting to speculate on the extension of the present method for other diffeomorphism invariant theories of gravity. While the zeroth and the first law hold for any arbitrary such theory, the second law has only been proved for a class of such theories [31]. If the present formalism can be extended to other theories of gravity, it will lend a support to the existence of the area increase theorem for such theories. While more interesting and deeper issues can only be understood in a full quantum theory of gravity, the present framework can elucidate the suggestions of [32] and provide a better understanding of the Hawking radiation process. [arXiv:gr-qc/9710089].", "pages": [ 1, 2, 3, 4 ] } ]

[ { "title": "Electrically charged dilaton black holes in external magnetic field", "content": "Stoytcho S. Yazadjiev ∗ Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgaria and Theoretical Astrophysics, Eberhard-Karls University of Tubingen, Tubingen 72076, Germany", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the present paper we construct a new solution to the Einstein-Maxwelldilaton gravity equations describing electrically charged dilaton black holes immersed in a strong external magnetic field and we study its properties. The black holes described by the solution are rotating but with zero total angular momentum and possess an ergoregion confined in a neighborhood of the horizon. Our results also show that the external magnetic field does not affect the black hole thermodynamics.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Dilaton black holes have been extensively studied in various aspects during the last two decades. Nevertheless, there still remain open problems which are not solved up to now. The present paper is devoted to one such problem, namely the construction of exact solutions describing electrically charged dilaton black holes in a strong external magnetic field and the study of their properties. In contrast to the exact solutions describing uncharged dilaton black holes in external magnetic field, the construction of electrically charged black holes in external magnetic field is much more difficult. The reason hides behind the fact that such solutions have to be in principle stationary due to the contribution of electromagnetic circulating momentum flux E ∧ B in the energymomentum tensor. It is well known, however, that the construction of rotating black holes in Einstein-Maxwell-dilaton (EMd) gravity is extremely difficult for arbitrary dilaton coupling parameter. In fact, the only known rotating black hole solution in EMd gravity is the black hole with Kaluza-Klein coupling α = √ 3 [1],[2]. The finding of rotating EMd black holes with arbitrary dilaton coupling parameter is another yet unsolved problem. In view of the described difficulties we will consider the EMd gravity with α = √ 3. The study of black holes in a magnetic field (the so-called magnetized black holes) has a long history starting with the classical papers by Wald [3] and Ernst [4]. Afterwards the magnetized black holes within general relativity and other alternatives theories were studied by many authors in various aspects ranging from pure theoretical studies to astrophysics [5]-[23]. The thermodynamics of the magnetized black holes is of special interest. The naive expectation was that a thermodynamic description of these configurations would include also the value of the background magnetic field as a further parameter. However, more detailed studies showed that this was not the case for a static electrically uncharged solutions in four dimensions and some rotating black holes in higher dimensions - the external magnetic field only distorts the horizon geometry without affecting the thermodynamics [19], [20], [23]. The thermodynamics of electrically charged or/and rotating magnetized black holes is not fully studied and it is not clear whether the magnetic field affects it or not. The situation with the thermodynamics of electrically charged and/or rotating magnetized black holes seems very complicated in the light of the very recent result of [27]. Unexpectedly, it was discovered in [27] that the magnetized electrically charged Reissner-Nordstrom and Kerr-Newman black holes are not asymptotic to the static Melvin solution [24]. Even more surprising, it was shown in [27] that, in the general case, the ergoregion of the magnetized Reissner-Nordstrom and Kerr-Newman black holes extends all the way from the horizon to infinity. The natural question is whether a similar behavior occurs for other electrically charged black holes. In the present paper we give also a partial answer to this question for some black holes in EMd gravity.", "pages": [ 1, 2 ] }, { "title": "2 Construction of the exact solution", "content": "The field equations of the four-dimensional Einstein-Maxwell-dilaton gravity are given by where ∇ µ and R µν are the Levi-Civita connection and the Ricci tensor with respect to the spacetime metric g µν . F µν is the Maxwell tensor and the dilaton field is denoted by ϕ , with α being the dilaton coupling parameter governing the coupling strength of the dilaton to the electromagnetic field. We shall consider spacetimes admitting a spacelike Killing field η with closed orbits. The dimensional reduction of the field equations along the Killing field was performed in [25]. So we present here only the basic steps skipping the details which can be found in [25]. Since the Maxwell 2-form F is invariant under the flow of the Killing field there exist potentials Φ and Ψ defined by d Φ = i η F and d Ψ = -e -2 αϕ i η /star F such that where The twist ω of the Killing field η , defined by ω = /star ( dη ∧ η ), satisfies the equation Hence we conclude that there exists a twist potential χ such that ω = dχ +2Ψ d Φ -2Φ d Ψ. The projection metric γ orthogonal to the Killing field η , is defined by In local coordinates adapted to the Killing field, i.e. η = ∂/∂φ , we have The 1-form W = W a dx a is closely related to the twist ω which can be expressed in the form The dimensionally reduced EMd equations form an effective 3-dimensional gravity coupled to a nonlinear σ -model with the following action where R ( γ ) is the Ricci scalar curvature with respect to the metric γ ab , X A = ( X,χ, Φ , Ψ , ϕ ) and G AB can be viewed as a metric on an abstract Riemannian manifold N with local coordinates X A and its explicit form is given by What is important for the aim of the present paper is the fact that the Riemannian space ( N , G AB ) is a symmetric space for the critical coupling α = √ 3. In fact N is an SL (3 , R ) /O (3) symmetric space and therefore its metric can be written in the form where S is a symmetric SL (3 , R ) matrix explicitly given by [26] The group of symmetries SL (3 , R ) can be used to generate new solutions from known ones via the scheme where Γ ∈ SL (3 , R ). In the present paper we consider seed solutions corresponding to the matrix and transformation matrices in the form with B being an arbitrary real number. In the particular case when the seed solution is a pure Einstein black hole solution, i.e. Ψ 0 = 0 and ϕ 0 = 0, the above transformation gives the solution describing uncharged black holes in external magnetic field [23]. The physical meaning of the parameter B is the asymptotic strength of the external magnetic field. The new solutions which can be generated from the seed and the transformation matrices under consideration, are encoded in the matrix S = Γ S 0 Γ T and the explicit form of their potentials is as follows In order to find the solution describing an electrically charged dilaton black hole in external magnetic field we choose the seed solution to be the electrically charged, static and spherically symmetric dilaton black hole solution with α = √ 3 [28],[29], namely The relation between the mass and the charge of the seed solution with the parameters r + and r -is given by the formulae Using the definition of the potential Ψ we find that the potential Ψ 0 corresponding to the seed solution is given by The only two quantities that should be found in order to obtain the new solution describing an electrically charged dilaton black hole in an external magnetic field, are the rotational 1-form W and the Maxwell 2-form F . The rotational 1-form W can be found by reversing eq.(9) and taking into account that for the new solution ω = -4Φ d Ψ 0 . After some algebra with differential forms we obtain The Maxwell 2-form can be found in the same way by using eq.(4) and the result is or equivalently the gauge potential is given by Now we can present the solution in a fully explicit form: In the particular case when q = 0 the solution reduces to the solution describing the Schwarzschild-dilaton black hole with α = √ 3 in an external magnetic field. It is interesting to note that our 4-dimensional EMd solution corresponds to a pure vacuum solution to the 5-dimensional Einstein gravity which can be found by performing a Kaluza-Klein uplifting via the equation In completely explicit form we have It is also worth noting that there is one more method for generating our EMd solution. It can be generated via a twisted Kaluza-Klein reduction of the 5-dimensional uplifted seed solution along the Killing field V = B ∂ ∂φ + ∂ ∂x 5 . Indeed, introducing the new coordinate φ ∗ = φ -Bx 5 , which is invariant under V , we find that (36) transforms to (35) and the further Kaluza-Klein reduction obviously gives our EMd solution.", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "3 Properties of the solution", "content": "In the present section we investigate some of the basic properties of the solution constructed in the previous section. First we note that our solution is free from conical singularities and the periodicity of φ is the usual one ∆ φ = 2 π . In order to see that the parameter B is indeed the asymptotic magnetic field let us calculate /vector B 2 on the axis of axial symmetry. We have which shows that B is the asymptotic magnetic field strength in the limit r →∞ . There is a Killing horizon at r = r + where the Killing field K = ∂ ∂t +Ω H ∂ ∂φ becomes null. Here Ω H = 2 qB r + is the angular velocity of the horizon. The metric induced on the horizon cross section is By applying the Gauss-Bonnet theorem one can show that surface of the event horizon is topologically a 2-sphere. The horizon area is and evidently it coincides with the horizon area of the seed solution. As in the uncharged case, the external magnetic field deforms the horizon but preserves the horizon area. In fact the geometry of the horizon cross section deviates from that of the round 2-sphere. A simple inspection of (38) reveals that the polar circumference ( φ = const ) is greater than the circumference about the equator ( θ = π 2 ). Therefore, for the solution under consideration the magnetic field elongates the black hole along the magnetic field and the black hole is prolate in shape. The physical electric charge is given by and coincides with that of the seed solution. The ergoregion for the solution under consideration is determined by the region where g ( ∂ ∂t , ∂ ∂t ) = g tt is positive. The explicit form of g tt is /negationslash It is easy to see that very close to the horizon we have g tt > 0 for sin θ = 0 and g tt = 0 for sin θ = 0 and r = r + . Also, it is not difficult to see that for r ≥ r + + r -it holds that g tt < 0. Therefore, we conclude that there exists an ergoregion confined in a compact neighborhood of the horizon, in contrast with the magnetized ReissnerNordstrom solution for which the ergorgeon extents to infinity [27]. The boundary of the ergoregion, i.e. the ergosurface, is defined by g tt = 0 which reduces to a cubic equation in r , namely r -r + = ( r + + r --r )( r -r -) 2 B 2 sin 2 θ . Solving this cubic equation for r we can find the equation r ( θ ) of the ergosurface. The explicit form of r ( θ ) is too cumbersome to be presented here. What is important is that the ergosurface is qualitatively the same as the ergosurface of the Kerr solution. The surface gravity κ associated with the Killing field K can be calculated via the well-known formula where λ = g ( K,K ). The direct computation gives the following result which is just the surface gravity of the seed solution. Therefore, the surface gravity is not affected by the external magnetic field for the solution under consideration. The inspection of the electric field shows that /vector E 2 → 0 for r →∞ . The same holds for the norm of the twist of the Killing vectors η = ∂ ∂φ and ξ = ∂ ∂t . As a whole, our solution is asymptotic to the dilaton-Melvin solution with α = √ 3. This can be easily seen from its explicit form.", "pages": [ 7, 8 ] }, { "title": "4 Thermodynamics", "content": "The study of the thermodynamics of the black holes in external magnetic fields is difficult because of the asymptotic structure. Since the spacetime is not asymptotically flat a substraction procedure is needed to obtain finite quantities from integrals divergent at infinity. The natural choice for the substraction background in our case is the dilaton-Melvin background. To calculate the mass we use the quasilocal formalism [23]. Here we give for completeness a very brief description of the quasilocal formalism. The spacetime metric is decomposed into the form with N and N i being the lapse function and the shift vector. The decomposition means that the spacetime is foliated by spacelike surfaces Σ t of metric h µν = g µν + u µ u ν , labeled by a time coordinate t with a unit normal vector u µ = -Nδ µ 0 . The spacetime boundary consists of the initial surface Σ i ( t = t i ), the final surface Σ f ( t = t f ) and a timelike surface B to which the vector u µ is tangent. The surface B is foliated by 2-dimensional surfaces S r t , with metric σ µν = h µν -n µ n ν , which are the intersections of Σ t and B . The unit spacelike outward normal to S r t , n µ , is orthogonal to u µ . In order to have well-defined variational principle we must consider the extended EMd action with the corresponding boundary terms added: Here K is the trace of the extrinsic curvature K µν of Σ t i,f and Θ is the trace of the extrinsic curvature Θ µν of B , given by where ∇ µ and D ν are the covariant derivatives with respect to the metric g µν and h µν , respectively. The quasilocal energy M and the angular momentum J i are given by Here k = -σ µν D ν n µ is the trace of the extrinsic curvature of S r t embedded in Σ t . The momentum variable p ij conjugated to h ij is given by The quantity ˆ Π j is defined by The quantities with the subscript '0' are those associated with the background. After long calculations, for the quasilocal energy of the black hole we find which is evidently independent from the external magnetic field and coincides with the mass of the seed solution. In the same way, for the angular momentum we obtain /negationslash Even more, the angular momentum J S r t associated with every surface S r t is zero. So we have a rotating black hole (i.e. Ω H = 0) while the total angular momentum is zero. This, at first sight strange result, means that the gravitational contribution to the angular momentum is exactly compensated by the opposite in sign angular momentum of the electromagnetic field. Furthermore, the following Smarr-like relation is satisfied where the potential Ξ H is in fact the corotating electric potential evaluated on the horizon and given by On the basis of the results obtained so far we can conclude that the external magnetic field does not affect the thermodynamics of the charged dilaton black holes.", "pages": [ 8, 9, 10 ] }, { "title": "5 Discussion", "content": "In the present paper we constructed a new solution to the EMd gravity equations describing charged dilaton black holes in external magnetic field for dilaton coupling parameter α = √ 3. The basis properties of the solution and its thermodynamics were studied. The black holes described by the solution are rotating but with zero total angular momentum and possess an ergoregion confined in a neighborhood of the horizon. Our results also show that the external magnetic field does not affect the black hole thermodynamics. The natural generalization of this work is to consider electrically charged and rotating dilaton black holes immersed in an external magnetic field and to study their thermodynamics. Especially, it is interesting whether the thermodynamics of the rotating solutions depends nontrivially on the external magnetic field. Other interesting questions are the asymptotic structure at infinity and the compactness or non-compactness of the ergoregion. These problems are currently under investigation and the results will be presented elsewhere.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "The author is grateful to the Research Group Linkage Programme of the Alexander von Humboldt Foundation for the support of this research and the Institut fur Theoretische Astrophysik Tubingen for its kind hospitality. He also acknowledges partial support from the Bulgarian National Science Fund under Grant DMU-03/6.", "pages": [ 11 ] } ]

[ { "title": "Negative Energy Seen By Accelerated Observers", "content": "L. H. Ford ∗ Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA † Thomas A. Roman Department of Mathematical Sciences, Central Connecticut State University, New Britain, Connecticut 06050, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "The sampled negative energy density seen by inertial observers, in arbitrary quantum states is limited by quantum inequalities, which take the form of an inverse relation between the magnitude and duration of the negative energy. The quantum inequalities severely limit the utilization of negative energy to produce gross macroscopic effects, such as violations of the second law of thermodynamics. The restrictions on the sampled energy density along the worldlines of accelerated observers are much weaker than for inertial observers. Here we will illustrate this with several explicit examples. We consider the worldline of a particle undergoing sinusoidal motion in space in the presence of a single mode squeezed vacuum state of the electromagnetic field. We show that it is possible for the integrated energy density along such a worldline to become arbitrarily negative at a constant average rate. Thus the averaged weak energy condition is violated in these examples. This can be the case even when the particle moves at non-relativistic speeds. We use the Raychaudhuri equation to show that there can be net defocussing of a congruence of these accelerated worldlines. This defocussing is an operational signature of the negative integrated energy density. These results in no way invalidate nor undermine either the validity or utility of the quantum inequalities for inertial observers. In particular, they do not change previous constraints on the production of macroscopic effects with negative energy, e.g., the maintenance of traversable wormholes. PACS numbers: 03.70.+k,04.62.+v,05.40.-a,11.25.Hf", "pages": [ 2 ] }, { "title": "I. INTRODUCTION", "content": "It is well known that quantum field theory allows for the existence of negative energy density, which constitute local violations of the weak energy condition. For a recent review, see Ref. [1]. Negative energy density can arise either from boundaries, as in the Casimir effect, from background spacetime curvature, or from selected quantum states in Minkowski spacetime. The last possibility will be the focus of the present paper. It is possible to create states, such as a squeezed vacuum state of the quantized electromagnetic field, in which the energy density at a given spacetime point is arbitrarily negative. However, the duration of the negative energy is strongly constrained by quantum inequalities [2-9]. These are restrictions on a time averaged energy density measured by an observer. (Time averaging is essential, as there is no analogous restriction on spatial averages [10].) Let us consider the case of inertial observers in Minkowski spacetime, with four velocity u µ . If 〈 T µν 〉 is the expectation value of the normal ordered stress tensor operator in an arbitrary quantum state, then quantum inequalities take the form Here τ is the observer's proper time, f ( τ ) is a sampling function with characteristic width τ 0 , and d is the number of spacetime dimensions. The dimensionless constant C 0 depends upon the form of the sampling function, and is typically small compared to unity. In the limit τ 0 →∞ , Eq. (1) becomes the averaged weak energy condition which states that the integrated energy density along an inertial worldline is non-negative. The essence of a quantum inequality is that there is an inverse relation between the magnitude and duration of negative energy density. These relations place strong constraints on the effects of negative energy for violating the second law of thermodynamics [2], and for maintaining traversable wormholes [11] or warpdrive spacetimes [12]. A more general quantum inequality for arbitrary worldlines has been proven by Fewster [13]. However, this inequality is often very difficult to evaluate explicitly and can be very weak. There are some known examples where the integrated energy density along a non-inertial world line can be arbitrarily negative. One example comes from the FullingDavies moving mirror model in two spacetime dimensions [14, 15]. A mirror with increasing proper acceleration to the right can emit a steady flux of negative energy to the right. An inertial observer could only see this negative energy for a finite time before being hit by the mirror, and the integrated energy density seen will be consistent with Eq. (1). However, an accelerated observer who stays ahead of the mirror can see an arbitrary amount of negative energy. This example suffers from two unrealistic features: it can only be formulated in two spacetime dimensions, and it requires an observer with ever increasing proper acceleration. A second example was provided by Fewster and Pfenning [16], who analyzed the case of a uniformly accelerating observer in the Rindler vacuum state. This state has negative energy everywhere within the Rindler wedge. An observer with constant acceleration can also see an arbitrary amount of negative energy. However, the constant acceleration requires the observer to move arbitrarily close to the speed of light and hence have an unlimited source of energy. It is also not clear whether the Rindler vacuum is a physically realizable state. The main purpose of this paper is to construct some more realistic examples of accelerated motion in which the observer can have arbitrarily negative integrated energy density. We will consider observers who undergo sinusoidal motion in the presence of a squeezed vacuum state of the quantum electromagnetic field. We find that even in the case of non-relativistic motion, it is possible for the integrated energy density in such an observer's frame to grow negatively at a constant rate in time. In Sect. II, we consider a squeezed vacuum state for a single plane wave mode, and motions both perpendicular and parallel to the direction of propagation of the wave. In Sect. III, we repeat the analysis for the lowest mode in a resonant cavity in a squeezed vacuum state. In Sect. IV, we address a possible physical effect of accumulating negative energy density, in the form of defocussing of a congruence of accelerated worldlines. Our results are summarized and discussed in Sect. V. In particular, we argue that the results in this paper neither contradict, nor diminish the utility of, the usual quantum inequalities proven for inertial observers. Throughout this paper, units in which ¯ h = c = 1 will be used. Electromagnetic quantities are in Lorentz-Heaviside units.", "pages": [ 3, 4 ] }, { "title": "II. OSCILLATIONS THROUGH A PLANE WAVE", "content": "Let us first evaluate the stress tensor components for a single mode plane wave in a squeezed vacuum state of the electromagnetic field. The electromagnetic stress tensor is given in terms of the field strength tensor as Its spatial components are the energy density is and the energy flux in the i -direction is Write the electric and magnetic field operators in terms of photon creation operators ˆ a † k λ and annihilation operators ˆ a k λ as and Assume that the excited mode is a plane wave propagating in the z -direction, with polarization in the x -direction. Then its mode functions take the form E k λ = ˆ x E , and B k λ = ˆ y B , where Here V is the quantization volume and Ω = | k | is the angular frequency of the wave. Quadratic operators are assumed to be normal ordered with respect to the Minkowski vacuum state, so where ˆ a is the annihilation operator for the excited mode. Similarly, The quantum state is taken to be a single mode in which case where r is the 'squeeze parameter' and δ is a phase parameter. The nonzero components of the stress tensor are given by We see from Eqs. (12) and (13) that the energy density can be periodically negative in the lab (i.e., inertial) observer's frame, but the positive energy density always outweighs the negative energy density, in accordance with the quantum inequalities. The energy density in the inertial frame has its minimum (most negative) value when the cosine term in Eq. (12) is one, so Thus the maximally negative energy density is bounded below, and occurs for large r . However, in this limit the maximally positive energy density is unbounded and grows as e 2 r . In the opposite limit, where r glyph[lessmuch] 1, the energy density is approximately oscillatory However, there is also a positive non-oscillatory term of order r 2 .", "pages": [ 4, 5, 6 ] }, { "title": "A. Perpendicular Motion", "content": "Now consider a non-geodesic observer who moves on a path which is perpendicular to the direction of propagation of the wave. Let this path be defined by where | A | < 1, and v y = v z = 0, and ω is the angular oscillation frequency of the observer's motion, and where we have chosen z = 0. Then and the observer's four-velocity (as measured in the lab frame) is where u t = γ = dt/dτ . The integrated energy density along the accelerated observer's worldline is where the integrand is Here we used the facts that 〈 T tx 〉 = 〈 T xx 〉 = 0 and γ 2 dτ = γ dt . If we expand to first order in r , the result is The numerator of this expression describes the fact that, for small squeeze parameter, the inertial frame stress tensor components are nearly sinusoidal. The denominator describes the effect of going to the non-inertial frame. If we can arrange that the γ factor has its maximum value when the numerator is negative, then accelerated observer will see net negative energy. This situation occurs when ω = Ω and when δ = π , as illustrated in Fig. 1. We will make this choice throughout the remainder of this subsection. In this case, the integrated energy density becomes If we perform the integration on t and multiply by the quantization volume, we get where F (Ω t, A 2 ) and E (Ω t, A 2 ) are elliptic integrals of the first and second kind, respectively. As a specific example, let us plot I V for r = 0 . 01, A = 0 . 9, and in units where Ω = 1. Since, strictly speaking, the energy density is inversely proportional to V , we want to make a graph of I V as a function of τ , i.e., a graph of the integrated energy density, multiplied by the quantization volume, seen by the accelerated observer as a function of his proper time. The relation between τ and t is τ = ∫ dt/γ , which is If we plot Eq. (23) against Eq. (24) for our chosen parameters, we get Fig. 2. Now let us examine our expression for I in the A glyph[lessmuch] 1 limit. If we expand the Lorentz factor to second-order in A , we obtain In this limit, the difference between dt and dτ will be O ( A 2 ). If we use Eq. (25) in Eq. (22) to calculate I , we find: The sinusoidal terms will eventually be dominated by the linear term, but this can take many cycles, so we keep the O ( A 0 ) sinusoidal term, but drop the O ( A 2 ) sinusoidal terms. Therefore, our two leading order terms are Here A 2 glyph[lessmuch] 1, so the oscillating term is larger in magnitude until T > 4 / ( A 2 Ω). After this, the linear term dominates. However, we should recall that there is positive r 2 term in Eq. (15). This term will give a contribution to I of Ω r 2 T/V , and is negligible only if we require that glyph[negationslash] Nonetheless, accumulating negative energy density, can occur for arbitrarily small velocities. For any A = 0, we can find a value of r which satisfies Eq. (28). Then eventually the first term in Eq. (27) will dominate.", "pages": [ 6, 7, 8, 9 ] }, { "title": "B. Parallel Motion", "content": "We now consider the case of the accelerating observer moving parallel to the direction of the propagation of the wave. In the lab frame, we have 〈 T tt 〉 = 〈 T zz 〉 = 〈 T tz 〉 . The accelerated observer's three-velocity and position, respectively, are and so where u t = γ = dt/dτ . Therefore, we have that where (1 -v ) / (1 + v ) is a linear Doppler shift factor (as opposed to the transverse Doppler factor in the perpendicular case). As a result, since dτ = γ -1 dt = √ 1 -v 2 dt . Here the observer is moving in the direction of wave propagation, so we can no longer set z = 0. Now the energy density in the inertial frame is given by Eq. (12), with z = -( A/ω ) cos( ω t ), so In this case, we find accumulating negative energy density for ω = 2Ω, and δ = -π/ 2. The integral in Eq. (33) can be done analytically for small A , as will be discussed below, but for more general A , it can only be performed numerically. As an example, let us choose the case where r = 0 . 01, δ = -π/ 2, A = 0 . 9, and ω = 2, in units where Ω = 1. In Fig. 3, we graph IV against the observer's proper time, which will again be given by Eq. (24). Now we wish to consider the non-relativistic limit, and work to first order in v and hence in A . To this order, dτ ≈ dt , so where the energy density in the accelerating frame is If we expand Eq. (34) to first order in A , the result is where we have used the fact that Next we evaluate the energy density in the accelerating frame to first order in A and set ω = 2Ω to find This expression reveals that we can have growing negative energy density if δ = -π/ 2 and r glyph[lessmuch] 1. In this case, we may write where order A oscillatory terms have been dropped. If which is the analog of Eq. (28), the integrated energy density grows negatively as The latter asymptotic form holds for In the parallel motion case, the rate of growth of the negative integrated energy density is first order in A , as compared to second order in the perpendicular motion case treated in the previous subsection. This is due to the fact that in the parallel case, there is a linear Doppler shift, whereas in the perpendicular case the Doppler shift is transverse.", "pages": [ 9, 10, 11, 12 ] }, { "title": "A. The Perpendicular Case", "content": "We now consider the case of a particle oscillating in a closed cavity with dimensions a , b , and d aligned along the x , y , z axes respectively, where b < a < d . The modes in this cavity were discussed in Ref. [17]. With the condition that b < a < d , the lowest frequency mode is the TE mode with p = l = 1 , m = 0, where the frequency of the mode is given by and the non-zero components of the electric and magnetic fields are where the electric field is taken to be polarized in the y -direction. This mode is independent of y . Here C is a real normalization constant, given by For the case where only a single mode j is excited, the normal ordered expectation values of the squared fields are and where In this case, we can have the particle moving in the y -direction, and located in the center of the cavity in the other directions, so that This considerably simplifies the mode functions, leading to The only non-zero components of the stress tensor which we will need are 〈 T tt 〉 and 〈 T yy 〉 , which become where Because the direction of oscillation of the particle is in the y -direction, where v y = A sin ωt . The integrand of I is Let We will assume that A glyph[lessmuch] 1, and expand to second order in A . Therefore, if we use Eq. (52) in Eq. (55), and set ω = Ω, we have that If we now expand the right-hand side of Eq. (57) to second order in r , set δ = 0, integrate from 0 to T , and drop oscillatory terms in A 2 , we obtain where we have also dropped a higher order A 2 r 2 term. The positive first term is negligible compared to the second when The middle negative linear growing term will dominate the sinusoidal term when In this case, the restrictions on r and T are the same as those for perpendicular motion in the plane wave case. In these limits, we therefore have negative energy density which grows linearly as If we use Eqs. (46) and (53), we can write the previous equation as where V = abd is the volume of the cavity. Compare this result with the first term in Eq. (27), the corresponding rate for perpendicular motion in a plane wave mode. If we identify the cavity volume in the former with the quantization volume in the latter, then they differ only by a factor of two.", "pages": [ 12, 13, 14 ] }, { "title": "B. The Parallel Case", "content": "In this subsection, we will consider the case of a particle oscillating in a cavity along the z -axis, in the limit where A glyph[lessmuch] 1, and work to first order in A . We take and where z = z 0 corresponds to the equilibrium position of the particle, and the last term corresponds to relativistic corrections. We will choose the x -position of the particle to be The energy density in the particle's frame is Here We need to calculate 〈 T tt 〉 and 〈 T tz 〉 using the mode functions in Eq. (45), and then expand the result to second order in r . The result for 〈 T t ' t ' 〉 may be written as where and and where C 2 is once again given by Eq. (46). The integrated energy density may be written as where and As in the case of parallel motion in the plane wave case, with the appropriate choices for ω, Ω , r and δ , we expect to get a linearly growing negative term, a term which is first order in r and sinusoidal in time, and a positive r 2 term. The first and second of these terms will arise from F 2 and F 3 , while the third term will arise from F 1 . We also expect that we will find a non-trivial effect in first order in A . Let us first examine the terms involving F 3 in Eq. (68). These terms both involve the product v z F 3 , and are hence already of order A . Thus we may use Eq. (64) to write where we have set z 0 = d/ 4. (As it turns out, the linearly growing term we want will come from the F 3 term in I 2 , so we cannot choose z 0 = d/ 2.) A similar situation applies to F 1 , which contributes only to an order r 2 term. This is a positive, growing term which we need only to zeroth order in A . For this purpose, we may evaluate F 1 at z = z 0 : Thus, for estimating the order r 2 term, we may use where F 1 has the value in Eq. (76). The negatively growing term comes from I 2 , which involves F 2 , so we need to expand the latter to first order in A , using Eq. (64), as The I 2 term will be maximally negative when ω = 2Ω and δ = 0. In this case, a short calculation yields where oscillatory, order A terms have been dropped. Note that 3 -a 2 /d 2 > 0 because a < d . Therefore, the integrated energy density becomes, We see that the negative linearly growing term will dominate the sinusoidal term when and the positive, order r 2 term, when In this case, we find that the integrated energy density in the particle's frame grows negatively as where we have used the definition of C 2 and the fact that V = abd is the volume of the cavity. Compare this result with Eq. (42), the corresponding rate for parallel motion in a plane wave mode. If we identify the cavity volume in the former with the quantization volume in the latter, then they differ only by the factor in the square brackets. If a and d are of the same order of magnitude, then Eq. (44) tells us that Ω ∼ O (1 /a ) ∼ O (1 /d ), and this factor is of order unity.", "pages": [ 15, 16, 17 ] }, { "title": "IV. EFFECTS OF THE NEGATIVE ENERGY ON FOCUSSING", "content": "In this section, we will treat one possible effect of the accumulating negative energy along a particle's worldline. It is well-known that the attractive character of gravity, with ordinary matter as a source, leads to focussing of null and timelike geodesics. One expects that negative energy densities might have the opposite effect, and produce defocussing through repulsive gravitational effects.", "pages": [ 17 ] }, { "title": "A. Raychaudhuri Equation", "content": "The effect of gravity on a congruence of timelike worldlines is described by the Raychaudhuri equation. In our case, we allow the worldlines to be non-geodesics, so the equation takes the form [18] Here u α and a β = u α ∇ α u β are the 4-velocity and 4-acceleration of the congruence, and σ αβ and ω αβ are the shear and vorticity tensors. Also, θ = ∇ α u α is the expansion, and R αβ is the Ricci tensor. The last term in Eq. (84) is the acceleration term, which vanishes for geodesics. We will assume a hypersurface orthogonal congruence, in which case the vorticity tensor vanishes, ω αβ = 0. In addition, we assume that the shear and expansion are sufficiently small, that the terms quadratic in those quantities may be neglected. In this case, the Raychaudhuri equation becomes where ˙ θ ac = ∇ β a β is the acceleration term, and the Ricci tensor term describes the effects of gravity. Next we assume that an electromagnetic field is both the cause of the acceleration and the sole source of the gravitational field. Particles with rest mass m and electric charge q obey the equation of motion where the field strength tensor, F βρ , is assumed to obey the source free equation We can now write the acceleration term as The covariant derivative of the 4-velocity may be expressed as [19] when ω βα = 0. However, all terms on right hand side of this expression, except for the last, are symmetric tensors which vanish when contracted into the antisymmetric field strength tensor. Thus we obtain The electromagnetic stress tensor, given in Eq. (3) is tracefree, so the Einstein equations become where glyph[lscript] p is the Planck length, and Newton's constant is G = glyph[lscript] p 2 , in units where ¯ h = c = 1. We may write where we have used Eq. (86). We may use this expression to evaluate the Ricci tensor term in the Raychaudhuri equation and write", "pages": [ 17, 18, 19 ] }, { "title": "B. Fields Producing Acceleration", "content": "In previous sections, we assumed a prescribed sinusoidal motion, but did not explicitly give the electromagnetic fields which would produce this motion. Here we will concentrate on the case of motion parallel to a plane wave mode, which was treated in Sec. II B. In particular, we consider the case of non-relativistic motion along the z -direction, as described by Eq. (29), with A glyph[lessmuch] 1. This motion can approximately be produced by a plane wave with polarization in the z -direction. Here we will consider a classical electromagnetic wave propagating in the y -direction, with electric field E = E c ˆz , and magnetic field B = E c ˆ x , where To order A , only the electric field determines the motion of the particle, with the magnetic force contributing in order A 2 . Because the motion of the particle is only in the z -direction, we may set y = 0. In this case, the energy density of the classical wave, in the laboratory frame, is In addition to this classical field, the particle is also subjected to the quantum fields associated with the squeezed vacuum state mode. These fields potentially produce a fluctuating force on the particle, which we wish to include. Let E q and B q be the terms in Eqs. (7) and and where E is defined in Eq. (9). We will treat the velocity of the particle due to the quantum electric field as an operator in the photon state space, v q = v q ˆ x , where v q will be evaluated explicitly below. There is a third effect which we will not include explicitly. This the effect of the emitted radiation by the particle. There will be an average radiation reaction force which will slightly change the trajectory of the particle for a given classical field. However, this is normally very small and will be neglected. There will also be a shot noise effect, an uncertainty in the particle's momentum due to the statistical uncertainty in the number of photons emitted. This effect depends primarily on the classical field driving the average motion and not upon the quantum electric field. Hence it, and the radiation reaction force, would cancel in any experiment which compares particle motion with and without the quantum electric field. In addition, this momentum uncertainty grows as the square root of the mean number of photons radiated, and hence as the square root of time. Here we are interested in effects which grow linearly in time. We will now compute the components of the acceleration four-vector in the lab rest frame, taking account of both the classical and quantum parts of the electromagnetic field and of the particle's four-velocity. The acceleration four-vector satisfies In the non-relativisitic limit, the four-velocity is where v c = A sin( ωt ) . The non-zero components of the field strength tensor are and those obtained by antisymmetry of F ρα . The components of a ρ become We can now form the scalar a ρ a ρ , expand it to first order in the velocities, dropping v 2 c , v 2 q and v c v q terms, and take its expectation value in the squeezed vacuum state. The result is Let us examine each term on the right-hand side of this expression. The classical energy density, which is the same to first order in velocity in the lab frame and in the particle rest frame, is just E 2 c . Because the classical wave is propagating in the y -direction, and all the motion is in the x and z directions, this is the perpendicular motion case, with respect to the classical wave. Thus, to order v , T t ' t ' c ≈ T tt c , since T t ' t ' c ≈ T tt c + O ( v 2 ). The expectation value of the quantum energy density in the lab frame is 〈 E 2 q 〉 , and is given explicitly by Eq. (12). This quantity in the particle rest frame is 〈 E 2 q 〉 (1 -2 v c ). The final term is the contribution of the velocity fluctuations to the acceleration. For both the classical and quantum electromagnetic fields, we have assumed plane waves, for which E 2 = B 2 , and hence F µν F µν = 0. Thus we may drop the last term in Eq. (93), and write mean rate of change of the expansion as", "pages": [ 19, 20, 21 ] }, { "title": "C. Velocity Fluctuations and Defocussing", "content": "The fluctuating part of the velocity, v q , is determined by Eq. (101): where the term proportional to v c on the right hand side is due to the magnetic force produced by B y . Note that time derivative here is a total derivative, and we need to account for both the explicit time dependence and the implicit dependence through z ( t ): recalling that v c = dz/dt . The solution to Eq. (104) becomes where we have used Eqs. (9) and (96). Note that the effects of the magnetic force and of the implicit time dependence cancel one another. We may compute 〈 E q v q 〉 = 〈 : E q v q : 〉 in a squeezed vacuum state to find Here we used in the squeezed vacuum state. We may use Eq. (94) with y = 0, and set δ = -π/ 2 to write We will work only to first order in A , which means that we can ignore the z -dependence (see Eqs. (30) and (38)) in the above expression, which will contribute in order A 2 . When we set ω = 2Ω, and drop oscillatory terms, then we have In the small r limit, this becomes which is to be compared with the same limit for the squeezed state energy density in the accelerated frame, The latter quantity is just the order r , non-oscillatory term in Eq. (40). We see that both terms have the same form and same sign, and both contribute to defocussing, although the effect of the quantum velocity fluctuations is four times that of the negative energy density in this limit. If we combine these terms, as well as the time average of the classical energy density, Eq. (95), evaluated at ω = 2Ω, then Eq. (102) for the mean squared acceleration becomes The positive term is the focussing effect of the classical energy density, and the negative term is the combined defocussing effect of the negative energy density and the velocity fluctuations. These two terms depend upon different combinations of parameters, and it seems possible to arrange for the defocussing effect to dominate. Note that the gravitational effect, from the Ricci tensor, is ∝ glyph[lscript] p 2 in Eq. (103). The part without glyph[lscript] p 2 is a pure acceleration effect from the acceleration term. However, both effects have the same functional form here.", "pages": [ 21, 22, 23 ] }, { "title": "V. SUMMARY AND DISCUSSION", "content": "The key result of this paper is that an accelerated observer undergoing sinusoidal motion in space can observe an average constant negative energy density, so the integrated energy density grows negatively in time in this observer's frame. This is contrast to an inertial observer, in whose frame the energy density is more constrained by quantum inequalities. We considered a squeezed vacuum state for both a plane wave and a standing wave in a cavity. The case in which growing integrated negative energy is possible is when the squeeze parameter is small, r glyph[lessmuch] 1. In this case, the energy density in an inertial frame is almost sinusoidal, with the positive energy outweighing the negative energy only in order r 2 . The effect of the periodic motion of the accelerated observer is to introduce Doppler shift factors which enhance the negative energy compared to the positive energy. The accelerated observer then sees the negative energy blueshifted and the positive energy redshifted. In the cases of perpendicular motion treated in Sect. II A and III A, the effect is a transverse Doppler shift, and is hence of order A 2 , where A is the oscillation amplitude. For the parallel motion cases in Sects. II B and III B, the effect is a linear Doppler shift, leading to an effect of order A . It is possible to have growing negative integrated energy density even for arbitrarily slow motion, which means arbitrarily small A . However, for a given A , the squeeze state parameter r is constrained by relations such as Eqs. (28) and (41), which limit the rate of growth. Note that non-relativistic motion is not a requirement for growing negative energy, and the numerically integrated results depicted in Figs. 1 and 3 are for relativistic motion, but small squeeze parameter. We studied a model which gives an operational meaning to integrated negative energy density in the form of defocussing of bundle of worldlines. In Sect. IV, we analyzed the Raychaudhuri equation for the expansion along a bundle of accelerated worldlines. The motivation for this study is that positive energy leads to attractive gravitational effects and hence focussing, so negative energy should do the opposite. This expectation was born out in our results. However, the situation is complicated by the need to include an acceleration term in the Raychaudhuri equation, and the effects of the fluctuating velocity of the accelerating charged particles in a fluctuating electromagnetic field. In the cases which we examined, the gravitational effects and the acceleration effects have the same functional form. The effect treated in this paper bears a superficial resemblance to the effect treated in Ref. [20], which is a linearly growing or decreasing mean squared velocity of a charged particle undergoing sinusoidal motion near a mirror. The latter effect can be interpreted as non-cancellation of anti-correlated quantum electric field fluctuations. A charge at rest in the Casimir vacuum produced by the mirror is subjected to field fluctuations which can give or take energy from the charge for a time consistent with the energy-time uncertainty principle, but this effect will be cancelled by a subsequent anti-correlated fluctuation. The sinusoidal motion upsets this cancellation, and allows the mean squared velocity to grow or decrease, depending upon the phase of the oscillation. The effect discussed in the present paper also involves linear growth, but does not have an obvious interpretation in terms of non-cancelling fluctuations. The natural interpretation seems to be in terms of Doppler shifts which can be arranged to enhance negative energy and suppress positive energy. A topic for future research is to study further the connection between these two effects. Another topic is to understand to relation between the growing integrated negative energy and the general worldline quantum inequality of Fewster [13]. This inequality is difficult to evaluate explicitly for the sinusoidal worldline considered here. In this case, the inequality must be weak enough to allow the linear growth found here, but it might provide insight into the allowed behavior in situations more general than we have treated. A further question of interest is the possible physical consequences of accumulating negative energy beyond those discussed in Sect. IV. A possible detection model for negative energy was proposed in Ref. [17], in which negative energy can suppress the decay rate of atoms in excited states. The atoms in this model are moving along inertial worldlines, but it might be possible to devise a more general model involving non-inertial motion. Let us also stress that our results do not in any way invalidate or diminish the implications of the quantum inequality bounds for inertial observers . The strength of a quantum inequality bound may depend on the particular observer chosen, but the validity of the bound does not. As an example, suppose one is using a quantum inequality, applied to the motion of a particular inertial observer, to determine constraints on the geometry of a traversable wormhole. Let us further assume that in this case, the quantum inequality provides a very strong constraint. Now suppose one looks at the same problem from the point of view of, say, a different inertial or an accelerating observer and finds a much weaker bound. The weakness of the latter bound does not invalidate the strength of the previous bound. The observer whose motion provides the strongest quantum inequality bound implies the strongest constraint on the geometry of the wormhole. The latter cases simply yield true but weaker bounds.", "pages": [ 23, 24, 25 ] }, { "title": "Acknowledgments", "content": "One of us (TR) would like to thank Werner Israel for a discussion, many years ago, of the moving mirror problem. The authors would also like to thank participants in the Beyond conference in January 2013 for stimulating comments. This work was supported in part by the National Science Foundation under Grants PHY-0855360 and PHY-0968805.", "pages": [ 25 ] } ]

[ { "title": "Action principle for the connection dynamics of scalar-tensor theories", "content": "Zhenhua Zhou ∗ , Haibiao Guo, Yu Han and Yongge Ma † Department of Physics, Beijing Normal University, Beijing 100875, China A first-order action for scalar-tensor theories of gravity is proposed. The Hamiltonian analysis of the action gives the desired connection dynamical formalism, which was derived from the geometrical dynamics by canonical transformations. It is shown that this connection formalism in Jordan frame is equivalent to the alternative connection formalism in Einstein frame. Therefore, the action principle underlying loop quantum scalar-tensor theories is recovered. PACS numbers: 04.50.Kd, 04.20.Fy, 04.60.Pp.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Modified gravity theories have recently received increased attention in issues related to the 'dark Universe' and nontrivial tests on gravity beyond general relativity (GR). Since 1998, a series of independent astronomic observations implied that our Universe is currently undergoing a period of accelerated expansion [1]. This causes the 'dark energy' problem in the framework of GR. It is thus reasonable to consider the possibility that GR is not a valid theory of gravity on a galactic or cosmological scale. A simple and typical modification of GR is the so-called f ( R ) theory of gravity [2]. Besides f ( R ) theories, a well-known competing relativistic theory of gravity was proposed by Brans and Dicke in 1961 [3], which is apparently compatible with Mach's principle. To represent a varying 'gravitational constant', a scalar field is nonminimally coupled to the metric in Brans-Dicke theory. To be compared with the observational results within the framework of broad class of theories, the Brans-Dicke theory was generalized by Bergmann [4] and Wagoner [5] to general scalartensor theories (STT). Scalar-tensor modifications of GR are also popular in unification schemes such as string theory (see, e.g., [6] [7] [8]). Note that the metric f ( R ) theories and Palatini f ( R ) theories are equivalent to the special kinds of STT with the coupling parameter ω = 0 and ω = -3 2 respectively [2], while the original Brans-Dicke theory is the particular case of constant ω and vanishing potential of φ . In the past two decades, a nonperturbative quantization of GR, called loop quantum gravity (LQG), has matured [9] [10] [11] [12]. It is remarkable that both f ( R ) theories and STT can be nonperturbatively quantized by extending the LQG techniques [13] [14] [15]. Thus LQG is extended to more general metric theories of gravity [16, 17]. The background independent quantization method relies on the key observations that these theories can be cast into the connection dynamical formulations with the structure group SU (2). The connection dynamical formulation of f ( R ) theories and STT were obtained by canonical transformations from their geometrical dynamics [13] [14] [15]. However, the action principle for above connection dynamics of either f ( R ) theories or STT is still lacking, although the first-order action for the connection dynamics in Einstein frame of STT was proposed in [18]. The purpose of this paper is to fill out this gap. We will propose a first-order action for general STT of gravity, which includes f ( R ) theories as special cases. The connection dynamical formalism will be derived from this action by Hamiltonian analysis. It turns out that this connection dynamics is exactly the same as that derived from the geometrical dynamics by canonical transformations. Moreover, the equivalence between this connection formalism in Jordan frame and the alternative one in Einstein frame will be proved. Hence, loop quantum STT, as well as loop quantum f ( R ) theories, have got their foundation of action principle. Throughout the paper, we use the Latin alphabet a , b , c ,. . . , to represent abstract index notation of spacetime[19], capital Latin alphabet I , J , K ,. . . , for internal Lorentzian indices, and i , j , k ,. . . , for internal SU (2) indics. The other convention are as follows. The internal Minkowski metric is denoted by η IJ = diag ( -1 , 1 , 1 , 1). The Hodge dual of a di ff erential form FIJ is denoted by /star FIJ = 1 2 /epsilon1 IJKLF KL , where /epsilon1 IJKL is the internal LiviCivital symbol. The antisymmetry of a tensor AIJ is defined by A [ IJ ] = AIJ -AJI .", "pages": [ 1 ] }, { "title": "II. EQUATIONS OF MOTION", "content": "In order to get the Lagrangian formalism of connection dynamics of STT proposed in [15], let us first consider the following first-order action on a 4-dimensional spacetime M , where e = det ( e I a ) is the determinant of the right-handed cotetrad e I a , ¯ Ω ab IJ = ¯ ∂ [ a ¯ ω IJ b ] + ¯ ω IK [ a ¯ ω J b ] K is the curvature of the S L (2 , C ) spin connection ¯ ω IJ a , V ( φ ) is the potential of the scalar field φ with φ satisfying φ > 0, K ( φ ) is an arbitrary function of φ , and γ is an arbitrary real number. The variation of action (1) with respect to ¯ ω IJ a gives Here the generalized derivative operator ¯ D a is defined as where ¯ Γ c ab is a torsion-free a ffi ne connection. From Eq.(2) we have (see [21] for details) which tells us that the spin connection ¯ ω IJ a is compatible with tetrad e I a . On the other hand, taking account of Eq.(4), the variation of action (1) with respect to the tetrad e I a gives where Gab is the Einstein tensor of e I a and ¯ ∇ a is the covariant derivative operator compatible with gab . Finally, taking account of Eq.(4), the variation of action (1) with respect to the scalar field φ gives where a prime over a function represents a derivative with respect to the argument φ . We define a new function Then it is straightforward to transform Eqs. (5) and (6) into the form in [15]. Hence the first-order action (1) gives exactly the equations of motion of STT.", "pages": [ 1, 2 ] }, { "title": "III. HAMILTONIAN ANALYSIS", "content": "Let the spacetime M be topologically Σ × R for some 3manifold Σ . One introduces a foliation of M and a timeevolution vector field t a in it. t a can be decomposed with respect to the unit normal vector n a of Σ as where N and N a are lapse function and shift vector respectively. In the (3 + 1)-decomposition of M , it is convenient to make a gauge fixing nI : = n a eaI = (1 , 0 , 0 , 0) in the internal space [20]. In a coordinate system adopted to the (3 + 1)- decomposition, the Lagrangian density in Eq.(1) reads where a dot over a letter represents a derivative with respect to the time coordinate, and we have defined and ¯ K i t : = t a ¯ K i a , ¯ ω i t : = t a ¯ ω i a are the time component of ¯ K i a and ¯ ω i a , E is the square root of the determinant of the spatial metric qab : = gab + nanb , E a I : = q a b e b I , ω IJ a : = q b a ¯ ω IJ b K i a : = q b a ¯ K i b are the spatial component of e a I , ¯ ω IJ a and ¯ K i a respectively, D a is the spatial SO (1 , 3) generalized covariant derivative operator reduced from ¯ D a and corresponds to a SO (1 , 3)-valued spatial connection 1-form ω i j a , ∂ a is the flat derivative operator on Σ reduced from ¯ ∂ a , N : = N / E is the densitized lapse scalar of weight -1, and ˜ E a i : = EE a i is the densitized spatial triad of weight 1. Recall that the unique torsion-free SO (3) generalized covariant derivative operator annihilating E a i is defined as: where Γ b ac and Γ j ai are respectively the Levi-Civita connection and the spin connection on Σ . For convenience we define Let C i a : = ω i a -Γ i a . We further define new variables: Then by using the definitions (10) and (15), the connection components ω io a can be rewritten as: Note that we have the identity where the curvature R j ab is defined as Note also that the two constraint equations with respect to the Lagrangian multipliers ¯ K j t and ¯ ω j t are equivalent to We will denote Ω j , Λ j as the corresponding Lagrangian multipliers. Then the Lagrangian density (9) can be expressed as: where Cij : = Cai ˜ E a j and C : = δ i j Cij . Since the variation of the action with respect to Cij gives the Lagrangian density (22) can be reduced to where By Legendre transformation, the momentum conjugate to the configuration variables A i a and φ are defined respectively as The fundamental Poisson brackets read It should be noted that the second-class constraints appeared in the Hamiltonian analysis have been solved by the partial gauge fixing. In the case when K /nequal 0, the corresponding Hamiltonian reads where the Gaussian, vector and scalar constraints read respectively as: In the special case when K = 0, it is easy to see from Eq.(27) that there is a primary constraint which is called the conformal constraint in [15]. Thus the Hamiltonian becomes where the scalar constraint reads It is obvious that the above Hamiltonian formulations in both cases coincide with those in [15]. On the other hand, as pointed out in [18], the following first-order action can give a connection dynamics of STT in Einstein frame. We now show that the Hamiltonian formalism of action (37) is equivalent to the one which we just derived from action (1), because they are related to each other by a canonical transformation. In the case when K /nequal 0, the Hamiltonian corresponding to action (37) is a linear combination of first-class constraints as where with and ˆ F i ab and ˆ R i ab standing for the curvature of ˆ A i a and ˆ Γ i a respectively, i.e., Here ˆ Γ i a is the SU (2) spin connection satisfying where ˆ Γ c ab is the Christo ff el connection determined by the spatial metric with ˆ E : = 1 / det ( ˆ E a i ). The fundamental Poisson brackets are To do the canonical transformation, we first define Then we further define Using Eqs. (47) and (48), we can get the Poisson brackets between new variables as Taking account of Eq.(7), the constraints (39), (40) and (41) can be written in terms of new variables, up to Gaussian constraint, as where F i ab : = ∂ [ aA i b ] + /epsilon1 i jk A j a A k b . It is obvious that these constraints coincide with our results as well as those in [15]. Similarly, it is easy to get the same conclusion in the special case when K = 0.", "pages": [ 2, 3, 4 ] }, { "title": "IV. CONCLUDING REMARKS", "content": "As candidate modified gravity theories, STT provide the great possibility to account for the dark Universe and some fundamental issues in physics. The nonperturbative loop quantization of STT is based on their connection dynamical formalism obtained in Hamiltonian formulation in [15]. The achievement in this paper is to set up an action principle for the connection dynamics of STT in Jordan frame. Since f ( R ) theories of gravity can be regarded as the special kinds of STT, our action principle is also valid for the connection dynamics of f ( R ) theories. To get the action principle, we first show that the first-order action (1) gives the right equations of motion for general STT. Then a detailed Hamiltonian analysis is done to this action. By a partial gauge fixing, the internal S L (2 , C ) group of the theory is reduced to SU (2), and the second-class constraints are solved. Thus we obtain a first-class Hamiltonian system with a SU (2) connection as a configuration variable. This Hamiltonian formalism is exactly the same as the one in [15] derived from the geometrical dynamics by canonical transformations. On the other hand, the directly corresponding Hamiltonian connection formulation of action (37) is in Einstein frame, while as shown in [15] the natural connection formulation obtained by canonical transformations in Hamiltonian framework is in Jordan frame. However we have shown that they are equivalent to each other at classical level. Nevertheless, the ambiguity, whether one should start with the Jordan frame or Einstein frame to quantize STT, still exits. Besides providing the action principle for connection dynamics of STT, actions (1) and (37) also lay the foundation of spinfoam pathintegral quantization of STT. We leave this issue for future study.", "pages": [ 4, 5 ] }, { "title": "Acknowledgments", "content": "This work is supported by NSFC (No. 10975017 and No. 11235003) and the Fundamental Research Funds for the Central Universities. (2004).", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "1", "pages": [ 1 ] }, { "title": "Slowly rotating black holes in Hoˇrava-Lifshitz gravity", "content": "Enrico Barausse 1, 2 and Thomas P. Sotiriou 3 Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada 2 Institut d'Astrophysique de Paris, UMR 7095 du CNRS, Universit'e Pierre & Marie Curie, 98bis Bvd. Arago, 75014 Paris, France 3 SISSA, Via Bonomea 265, 34136, Trieste, Italy and INFN, Sezione di Trieste (Dated: October 8, 2018) In a recent paper we claimed that there there are no slowly rotating, stationary, axisymmetric black holes in the infrared limit of Hoˇrava-Lifshitz gravity, provided that they are regular everywhere apart from the central singularity. Here we point out a subtlety in the relation between Einsteinæther theory and the infrared limit of Hoˇrava-Lifshitz gravity which was missed in our earlier derivation and drastically modifies our conclusion: our earlier calculations (which are otherwise technically correct) do not really imply that there are no slowly rotating black holes in HoˇravaLifshitz gravity, but that there are no slowly rotating black holes in the latter that are also solutions of Einstein-æther theory and vice versa. That is, even though the two theories share the static, spherically symmetric solutions, there are no slowly rotating black holes that are solutions to both theories. We proceed to generate slowly rotating black hole solutions in the infrared limit of HoˇravaLifshitz gravity, and we show that the configuration of the foliation-defining scalar remains the same as in spherical symmetry, thus these black holes are expected to possess a universal horizon.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In Ref. [1] we considered slowly rotating, stationary, axisymmetric black holes which are regular everywhere apart from the central singularity, and we claimed that such solutions do not exist in the infrared limit of Hoˇrava-Lifshitz (HL) gravity. This claim is actually incorrect. As we explain below, our derivation was implicitly using the relation between the infrared limit of HL gravity and Einstein-aether theory (æ-theory). However, a subtle point in this relation has been missed, which effectively implies that we were implicitly requiring our solutions to be solutions of æ-theory with a hypersurfaceorthogonal aether configuration. The purpose of this note is threefold: a) to clarify how this oversight affected our calculations; b) to reinterpret our existing results, given that the calculations of Ref. [1] are actually technically correct and lead to some meaningful statements about ætheory; c) to correct our claims about HL gravity, point out that there are indeed slowly rotating black holes, and briefly discuss their characteristics. In what follows we use the notation and definitions of Ref. [1] unless specified otherwise.", "pages": [ 1 ] }, { "title": "II. A SUBTLETY IN THE RELATION BETWEEN HL GRAVITY AND Æ -THEORY", "content": "Einstein-aether theory is described by the action where g is the determinant of the metric g µν , ∇ µ is the associated covariant derivative, R is the Ricci scalar of this metric, c 1 to c 4 are dimensionless parameters, and the aether u µ is assumed to satisfy the constraint This constraint can be enforced either by considering restricted variations of the action which respect it, or by adding to the action explicitly the Lagrange multiplier ζ ( u µ u µ -1). It has been shown in Ref. [2] that æ-theory is equivalent to the infrared limit of HL gravity if the aether is assumed to be hypersurface orthogonal before the variation. Locally, hypersurface orthogonality can be imposed through the condition where T is a scalar field that defines a foliation. Choosing T as the time coordinate one selects the preferred foliation of HL gravity, and the action (1) reduces to the action of the infrared limit of HL gravity, whose Lagrangian is denoted as L 2 and given in eq. (2) of Ref. [1], and the correspondence of the parameters of the two theories is given in eq. (6) of the same paper. We consider now the corresponding field equations. Obviously, the equations of HL gravity will not be the equations of æ-theory with the aether expressed as in eq. (4), as the equivalence only holds if the hypersurfaceorthogonality condition is imposed at the level of the action. Let us consider the variation. Adding to action (1) the Lagrange multiplier in order to enforce the unit constraint leads to and then variation yields where Then the field equations are being the æ-theory field equation for the metric and the aether respectively. Contracting eq. (9) with u µ and using the unit constraint, the equation can be re-written as These are the exact same equations one would obtain if, instead of the Lagrange multiplier, one had performed a restricted variation that implies the unit constraint (as in Ref. [2]). If the hypersurface-orthogonality condition is imposed at the level of the action then the variation of u µ has to be expressed in terms of δT and δg µν , i.e. So, for (covariantized) HL gravity or hypersurface orthogonal æ-theory one has δS h.o. = ∫ √ -g ( E µν δg µν +Æ µ δu µ ) = ∫ √ -g [ ( E µν -1 2 Æ λ u λ u µ u ν ) δg µν (15) -∇ ν ( 1 √ g αβ ∂ α T∂ β T ( δ ν µ -u ν u µ )Æ µ ) δT ] , which leads to the equations The unit constraint is automatically satisfied, and this is why it is not present in the action or the field equations. It should be clear then that eq. (16) is identical to eq. (11) and hence, if one just starts with the ae-theory metric field equation and just imposes that the aether is given by eq. (4), then one obtains the HL gravity metric field equation. Obviously, the aether equation (12) is different than the equation of motion for T , eq. (17). 1 Above we performed the variation keeping u µ fixed in æ-theory. Things become much more subtle if one attempts to keep u µ fixed instead. In particular, starting from eq. (6) and taking into account that one has where E µν and Æ µ are the same quantities as above. Then the equations are and eliminating ζ leads to It is straightforward to see that eq. (23) differs from eq. (11). Of course, this does not mean that the dynamics of æ-theory depends on what is kept fixed in the variation. One can use eq. (24) to show that However, this subtlety is crucial for the relation with HL gravity. In particular, one cannot start from eq. (23), assume that the aether is given by eq. (4) and obtain eq. (16), as was possible when the æ-theory equations were obtained with u µ kept fixed. This was exactly what was missed in Ref. [1], where we used in our calculation eq. (23) and (4) instead of eqs. (11) and (4), as an equivalent to eq. (16).", "pages": [ 1, 2 ] }, { "title": "III. RE-INTERPRETATION OF EARLIER RESULTS", "content": "When the aether is hypersurface orthogonal, eq. (16) and eq. (23) differ only by terms that vanish when the aether's equation, eq. (24) is satisfied. This implies that by considering in Ref. [1] the system of eqs. (23), (4) and (17), we effectively considered solutions of æ-theory for which the aether is hypersurface orthogonal (at the level of the solution). Therefore, our earlier results imply that no such solutions exist, and Æ-theory can only admit slowly rotating black hole solutions where the aether is not hypersurface orthogonal. In HL gravity, instead, u µ is by construction a hypersurface orthogonal vector. This implies that, even though the two theories share the static, spherically symmetric black hole solutions [1, 3], they do not share any slowly rotating solution. Another interesting implication of the fact that ætheory does not admit slowly rotating solutions where the aether is hypersurface orthogonal is that slowly rotating black hole spacetimes in æ-theory do not have a preferred foliation (but just a preferred frame locally). The existence of the universal horizons found in Refs. [4, 5] for static, spherically symmetric black holes appears to be related with the existence of a preferred foliation.", "pages": [ 2, 3 ] }, { "title": "IV. SLOWLY ROTATING BLACK HOLES IN HL GRAVITY", "content": "Having re-interpreted the results of Ref. [1], it should now be clear that the claim that there are no slowly rotating black holes in HL gravity is not supported. So, in the rest of the paper we re-address the question of finding slowly rotating solutions in HL gravity. As discussed in Ref. [1] the most general ansatze for the metric and u µ for a slowly rotating, stationary, axisymmetric black hole are without loss of generality 2 where /epsilon1 is the book-keeping parameter of the expansion in the rotation, and A ( r ), B ( r ) and f ( r ) are given by the static, spherically symmetric seed solutions (see. Ref. [4]). Note in particular that hypersurface orthogonality implies u ϕ = 0 (at least to O ( /epsilon1 )) [1]. In Ref. [1] we found it convenient to perform a field redefinition to the spin-0 metric g ' αβ = g αβ +( s 2 0 -1) u α u β ( s 0 being the speed of the spin-0 mode) and to the rescaled aether vector u ' α = s -1 0 u α [6] and perform the analysis in terms of the redefined fields. The analysis for HL gravity actually turns out to be slightly simpler in the original variables g αβ and u µ , so we will avoid the field redefinitions here. The key difference from the analysis of Ref. [1] is the issue discussed above. As a result, the rφ and tφ components of the Einstein equations get modified and become which replace eqs. (12) and (13) of our paper. Equation (11) of Ref. [1], which is the θϕ component of the Einsten equations, instead remains unchanged at linear order in /epsilon1 and is still given by (Also, note that the equation of motion of T , eq. (17), is identically satisfied at linear order in /epsilon1 .) Just as in Ref. [1], here k 0 and q 0 are complicated functions of the couplings c i , as well as of A , f and B and their derivatives, but they must evaluate identically to zero when one uses the spherically symmetric static solution, because Ω =const must be a solution to the field equations [1]. Combining eqs. (29)-(30) we then immediately obtain while from eq. (31), as in Ref. [1], one can conclude that if the black-hole horizon is to be regular and located at r = r H , one must have Ω( r H , θ )= constant. Alternatively, one can observe that Ω( r, θ ) = Ω( r ) is the only solution to eq. (29) that is regular at the poles θ = 0 , π . We can then integrate eq. (32) and obtain where J and Ω 0 are integration constants. In particular, because with asymptotically flat boundary conditions one has B ∼ 1 far from the black hole, a comparison to the slowly rotating Kerr metric highlights that J plays the role of the spin of the black hole, while Ω 0 can be eliminated from the metric with a coordinate change φ ' = φ -Ω 0 t/ 2.", "pages": [ 3 ] }, { "title": "V. CONCLUSIONS", "content": "Our claim, in Ref. [1], that no slowly rotating regular black-hole solutions exist in HL gravity was incorrect. Such solutions actually exist and are given by eq. (33), as our amended analysis indicates. It goes beyond the scope of this manuscript to study the characteristics of these slowly rotating black holes, their deviations from the slowly rotating Kerr spacetime and any related astrophysical implications. We will address this question in a separate publication. It is, however, obvious, that the existence of these solutions implies that there is no a priori tension between the prediction of HL gravity and astrophysical evidence for the existence of spinning black holes, in contrast with our previous claim. It is worth noting that the configuration of the foliations defining scalar T (or u µ ) in these solutions receives no correction at first order in the rotation, and hence it is effectively the same as in their spherically symmetric seed solution. Hence, we expect these solutions to possess a universal horizon. The results found in Ref. [1], when appropriately reinterpreted, imply that the slowly rotating solutions of HL gravity are not solutions of æ-theory and that the latter has no slowly rotating solution with a hypersurfaceorthogonal aether configuration. Hence, æ-theory does not have slowly rotating black holes with a preferred foliation. Even though the two theories share the static, spherically symmetric, asymptotically flat solutions, they do not share slowly rotating solutions.", "pages": [ 3, 4 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are indebted to Ted Jacobson for private communications which contributed critically to understanding the subtlety in the relation between the two theories and its implications, discussed in the first section of the manuscript. EB acknowledges support from a CITA National Fellowship while at the University of Guelph, and from the European Union's Seventh Framework Programme (FP7/PEOPLE-2011-CIG) through the Marie Curie Career Integration Grant GALFORMBHS PCIG11-GA-2012-321608 while at the Institut d'Astrophysique de Paris. T.P.S. acknowledges financial support from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement n 306425 'Challenging General Relativity', from the Marie Curie Career Integration Grant LIMITSOFGR2011-TPS Grant Agreement n 303537, and from the 'Young SISSA Scientist Research project' scheme 20112012.", "pages": [ 4 ] } ]

[ { "title": "Adel Awad 1", "content": "Centre for Theoretical Physics, Zewail City of Science and Technology, Sheikh Zayed, 12588, Giza, Egypt. Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, EGYPT", "pages": [ 1 ] }, { "title": "Abstract", "content": "We use phase space method to study possible consequences of fixed points in flat FLRW models. One of these consequences is that a fluid with a finite sound speed, or a differentiable pressure, reaches a fixed point in an infinite time and has no finite-time singularities of types I, II and III described in hep-th/0501025 . It is impossible for such a fluid to cross the phantom divide in a finite time. We show that a divergent dp/dH , or a speed of sound is necessary but not sufficient condition for phantom crossing. We use pressure properties, such as asymptotic behavior and fixed points, to qualitatively describe the entire behavior of a solution in flat FLRW models. We discuss FLRW models with bulk viscosity η ∼ ρ r , in particular, solutions for r = 1 and r = 1 / 4 cases, which can be expressed in terms of Lambert-W function. The last solution behaves either as a nonsingular phantom fluid or a unified dark fluid. Using causality and stability constraints, we show that the universe must end as a de Sitter space. Relaxing the stability constraint leads to a de Sitter universe, an empty universe, or a turnaround solution that reaches a maximum size, then recollapses.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Various cosmological observations [1, 3, 4, 5] have provided us with a strong evidence for accelerating expansion of the universe. Although, the component that causes this acceleration is not known yet, the best model that fits dark energy and other cosmological data is the ΛCDM model, which is a Friedmann-Lematre-Robertson-Walker (FLRW) universe with a cosmological constant. Unfortunately, this model does not provide a physical picture of dark energy. In order to describe dark energy in the realm of general relativity, we need to consider some exotic fluid with an unusual equation of state that has a negative pressure and violates the strongenergy condition (see for example [9, 10]). The existence of this exotic fluid not only opens the door for reexamining the constituents of our universe but also evades some nonsingularity theorems through relaxing the strong-energy condition. This revives interest in nonsingular cosmologies, especially those describing the universe in early and late times. Having a more general class of equations of state, that violates the strong-energy condition, creates a wider class of singularities beyond that of Big Bang and Big Crunch [6, 7, 8]. In FLRW models these singularities are classified as follows [25]: It is of interest to build cosmological models, which are free from the above singularities. In the last decade, there have been several proposals to describe dark energy/matter or dark energy alone as a single barotropic fluid in FLRW models with various equations of state (EoS) such as; Chaplygin gas, Van der Waal, linear, and quadratic EoS[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Notice that, most of these models have fixed points, or de Sitter solutions. In this article we use a phase space method to study general solutions of single fluid FLRW models with fixed points and a pressure p ( H ), where H is the hubble parameter. We discuss possible consequences of having these fixed points. Some of these consequences are; (i) if we describe our universe as a single component fluid and model the late time acceleration by a future fixed point, then the resulting cosmology does not have future-time singularities of types I, II and III described in [25], (ii) cosmologies with a future and a past fixed points are free of types I, II and III singularities, (iii) one can use a simple argument to show the phantom divide [27, 28], or in a single fluid FLRW models it is impossible for a physical solution to cross the phantom divide line in a finite time, and (iv) in these models, the only way to get bounce solutions is to have a nonvanishing pressure as ρ → 0. The phase space method can be used to construct nonsingular late time model, in particular, unified dark fluid (UDF) and dark energy models. We use this qualitative analysis to describe the entire behavior of a FLRW cosmology with bulk viscousity η ∼ ρ r , presenting two exact solutions, with r = 1 and r = 1 / 4 expressed in terms of Lambert-W function. The last solution describes either a nonsingular phantom dark energy or a unified dark fluid model. At the end of section 4 we list possible scenarios for the future of our universe.", "pages": [ 2, 3 ] }, { "title": "2 Flowing to a Fixed Point", "content": "In this section, we use a phase space method in flat FLRW models with fixed points to argue that a fluid with a continuous and differentiable pressure always reaches a fixed point in an infinite time and has no finite-time singularities of types I, II and III described in [25]. Let us start with a FLRW universe, with an equation of state where the pressure p ( H ) is a continuous function of the hubble parameter H . Using the unit convention, 8 πG = c = 1, Einstein field equations lead to Friedmann equation and Raychaudhri equation For energy-momentum conservation, we obtain which is not independent of Eqn. (3). In this work we are interested in the flat FLRW case, the spatially curved case will be discussed elsewhere [30]. For a flat FLRW universe, Eqn.(3) becomes which can be expressed in terms of a dimensionless hubble parameter and time h = H/H ∗ and τ = c H ∗ t , as follows Where H ∗ is a parameter that depends on the equation of state parameters and c is some number. A solution, h ( τ ) of Eqn. (6) is subject to an initial condition h (0) = h 0 . Motivated by the fact that many of the proposed models for dark energy and unified dark matter/energy models do have fixed points, we assume the existence of fixed points for Eqn.(6), which are the zeros of the function F ( h ) = -1 / 2 ( ρ + p ). Let us call the zeros of F ( h ), h 1 , h 2 , ... , where h 1 < h 2 < h 3 .. . Fixed points are classified according to their stability as follows; stable, unstable, or halfstable, depending on the sign of their tangents as shown in Figure (1) 2 . Unstable fixed points are represented by arrows emanating out of them and stable points are represented by arrows pointing toward them as shown in Figure (1). Half-stable points are stable from one side and unstable from the other side or vice versa. The arrows determine how the solution develops with time. Notice that, a fixed point satisfies equation (6), i.e., h ( τ ) = h 1 , therefore, it is a solution. This constant solution is nothing but a de Sitter space. If the system started from an initial value, h 0 = h 1 , i.e., at a fixed point, then it will remain at this point forever. But when the value of h 0 is close to that of h 1 , the solution h ( τ ) develops towards h 1 , if h 1 is a stable fixed point, or away from it, if the point is unstable. This technique has been used in literature to study particular equations of state and their solutions (e.g., [18]), but here we try to keep our discussion general assuming a pressure p ( H ). Using this tool, one can determine how a solution behaves upon knowing the nature of these fixed points. For example, if we start with a value for h 0 between the stable and the unstable fixed points in Figure (1), then the solution will develop to the left, i.e., will take values with smaller h till it reaches the stable point. Therefore, the solution in this case interpolates between two de Sitter spaces. If h 0 takes values larger than the unstable fixed point in Figure(1), then the solution develops to the right and might reach a singularity if the time to reach h →∞ is finite. Fixed points and the asymptotic behavior of F ( h ) enable us to predict the behavior of the system without knowing the form of the solution. But, in order to have a reliable qualitative description of a solution we have to know how long it takes to reach a fixed point or a singularity, i.e., a point where h → ∞ . Using the argument below we will see that the continuity and differentiability of the pressure, p ( h ) determine if the time to reach a fixed point is finite or infinite. Also, in the coming section we use the asymptotic behavior of F ( h ) to determine the time taken by a solution to reach a point where F ( h ) is diverging. We are going to use pressure properties to describe the behavior of a solution qualitatively without the need for exact or approximate solutions. Here we argue that for a flat FLRW fluid with a pressure p ( h ), which satisfies; i) F ( h ) is continoues and differentiable, and there exists a unique solution h ( τ ) which is defined for times τ > 0, and has no future finitetime singularities of types I, II and III. Notice that, the first assumption is needed to ensure the existence of a unique local solution around some initial value h 0 by the existence-uniqueness theorem. In addition, it leaves the solution free from type-II singularities, which might contradict casuality since the sound speed dp/dρ = c 2 s will diverge as well. The second assumption ensures the extendibility of this local solution to all values τ > 0 while keeping the hubble parameter bounded. Let us start our argument by integrating equation (6), assuming an initial value h (0) = h 0 , then one obtains /negationslash Since F ( h 0 ) = 0, its either F ( h 0 ) > 0 or F ( h 0 ) < 0, let us choose F ( h ) > 0, and h 0 < h 1 from ii). As a result G ( h ) is a monotone near h 0 . For any solution φ ( τ ), we have Since G ( h ) is a monotone near h 0 , the above relation can be inverted where G -1 is the inverse map of G . This is a local solution by construction (unique since F ( h ) is differentiable by the existence-uniqueness theorem) that can be extended by looking for a maximal interval in which G ( h ) is a monotonic function. Since F ( h 0 ) > 0 and remains positive for the values h 0 < h < h 1 , then G ( h ) is a monotone in this interval. This imply that the maximum interval in which we can extend the solution is [ h 0 , h 1 ]. The solution can be defined for all values τ > 0 if Now, let us show that τ + = ∞ , when i) and ii) are given. Since F ( h ) is differentiable, the slope of the tangent at any point is finite. Let us choose a number M < F ' ( h ) , ∀ h ∈ [ h 0 , h 1 ]. The existence of M enables us to define a linear function Y(h) such that where Y ( h ) = M ( h -h 1 ). This leads to therefore, We have shown that for the first-order system of Eqn.(6) there exists a unique solution, h ( τ ) defined for times τ > 0 if the above two assumptions are satisfied. Also, it takes the solution an infinite time to reach the future fixed point h 1 . It is clear that the solution is bounded, i.e., h ( τ ) ∈ [ h 0 , h 1 ] for times τ > 0. Therefore, the density, ρ ( τ ) is bounded for times τ > 0, as a result, there is no future singularities of type I and III. The pressure p = -2 F ( h ) -3 h 2 H ∗ 2 is bounded too in this interval, since F ( h ) and h are bounded 3 , which means, no future singularities of type II in these spacetimes. As a result this class of FLRW solutions are free from future finite-time singularities of type I, II and III . The above argument can be generalized by relaxing the differentiability condition, in this case we have one of the following;", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Consequences of Fixed Points", "content": "Here we discuss consequences of fixed points and how can we use this dynamical method to describe the entire behavior of a single fluid in flat FLRW without knowing the form of the solution.", "pages": [ 7 ] }, { "title": "3.1 Direct Consequences", "content": "Consequences of using phase space method to study fixed points can be listed as follows; i) If the late-time behavior of our universe is described by a single fluid, as in unified dark fluid models, and the late-time acceleration is developing towards a de Sitter universe, then there is no future-time singularity of types I, II and III in this solution. Since the pressure p ( h ), is a differentiable function of h , or at most has finite discontinuities, it takes the universe infinite time to reach the de Sitter space. iii) As one might notice, the solutions we have so far, still, admit a weaker type of singularity, namely, type-IV in the classification given in [25]. But even these weaker singularities can be avoided by requiring p ( h ) to be a smooth function, i.e., a C ∞ function. One can show that as follows; The n-time derivative of h ( τ ) can be written as; which means, unless F ( h ) or one of its derivatives (up to the ( n -1)th derivative) is divergent, h ( n ) is always bounded in [ h 2 , h 1 ]. This leads to the conclusion: In a flat FLRW universe, if a) p ( h ) is a smooth function, i.e., arbitrarily differentiable, and b) F ( h ) has a future and a past fixed point, then the spacetime is free from singularities of types, I, II, III and IV [25] when h 0 ∈ [ h 2 , h 1 ]. /negationslash iv) A single fluid in flat FLRW with a pressure p ( h ) admits bouncing solutions only if there are no fixed points between h = 0 and h = h 1 , i.e., F (0) = 0. These solutions either have a bounce or turnaround at h = 0 depending on sign of F (0). At h = 0, a/a = F (0), therefore, if F (0) > 0 it will be a bounce and for F (0) < 0 it will be a turnaround. a single fluid FLRW cosmology it is impossible for a causal solution to go from a region where ω ( h ) < -1 to another with ω ( h ) > -1. In other words, a solution in one of the mentioned regions has no access to the other region. Let us explain this in more details using the analysis we have in section 2. First, let us assume that the pressure p ( h ) is differentiable. In this case, if a solution approaches a fixed point, where w ( h ) = -1, starting from a region where w ( h ) < -1, then, it will spend an infinite time to reach it, as a result it will never cross it. If p ( h ) is not differentiable, then p ' ( h ) has either a finite or an infinite discontinuity. If the discontinuity is finite, the time to reach the crossing point or the fixed point is infinite, as we showed in the previous section, therefore, the crossing will not occur. If p ' ( h ) has an infinite discontinuity, a solution will reach w ( h ) = -1 in a finite time, but in this case the solution is not causal. Although, the solution is not causal, it is not clear if it is going to cross the phantom divide or not. To further investigate this case, let us list the conditions on F ( h ): iii) τ = ∫ h 1 h 0 1 /F ( h ) < ∞ . A class of functions which satisfy the above conditions is F ( h ) = F 0 ( h 1 -h ) s , where 0 < s < 1 and F 0 > 0. A solution is given by the following expression; where, h (0) = h 0 and τ ∗ = ( h 1 -h 0 ) 1 -s /F 0 (1 -s ). It is clear from Eqn. (15) that the solution stays at the fixed point for τ ≥ τ ∗ . Also, notice that h 1 is a stable fixed point. One can see that by assuming a small perturbation away from h 1 , i.e., h ( τ ) = h 1 + δ ( τ ), it leads to As a result, if we extend the definition of F ( h ) for h > h 1 , e.g., F ( h ) = -F 0 ( h -h 1 ) s as shown in Figure (2)-a), we will not have a phantom crossing. But if we define F ( h ) = ± F 0 ( h 1 -h ) s , i.e., a double valued function as in Figure (2)-b), the solution can cross the phantom divide in a finite time. Therefore, in addition to its infinite discontinuity F ' ( h ), need to be a double valued function to have a phantom crossing solution. Our conclusion is that a general causal solution of any flat FLRW model, with a continuous pressure p ( H ), can not cross the phantom divide line in a finite time. It is clear that infinite discontinuity of F ( h ) is necessary but not sufficient for a phantom crossing.", "pages": [ 7, 8 ] }, { "title": "3.2 Describing a Solution Qualitatively", "content": "As we mentioned earlier, to have a complete qualitative description of a general solution in flat FLRW cosmology we need to know the fixed points as well as the asymptotic behavior of F ( h ). The later property enables us to determine the time to reach a point where F ( h ) →±∞ , starting from some initial value h (0) = h 0 . Let us first show the relation between the asymptotic behaviors of F ( h ) and finite-time singularities. Asymptotic Behavior of F ( h ) and Singularities: The dimensionless Hubble parameter h , in a flat FLRW cosmology is controlled by a one-dimensional phase space evolution function F ( h ). In this dynamical system a solution develops towards either a fixed point, where F ( h ) → 0, or a point where F ( h ) →±∞ . A solution approaching a point where F ( h ) →±∞ does not necessarily mean that the it has a finite-time singularity. It is crucial to know how fast F ( h ) reaches infinity. This enables us to determine if the singularity is reached in a finite time or not [32]. Considering the integral it is easy to see that if lim h →±∞ F ( h ) ∼ h s , where 0 ≤ s ≤ 1, the integral diverges. As a result, the solution takes an infinite time to reach the singular point, therefore, it has no finite-time singularities. One can observe that if F ( h ) grows as a linear function or slower 5 , as h →±∞ , the solution will have no finite-time singularities. One can show this rigourously following the same argument in section 2. An example of a singular asymptotic behavior is a quadratic function F ∼ h 2 , which leads to a Big Bang singularity. It is intriguing to notice that there is a class of F ( h ) that grows faster than a linear function, but still, does not have finite-time singularities. An example of this is F ( h ) ∼ h ln h , or h ln h ln ln h, and so on [31]. These functions grow faster than a linear function, but lead to time τ ( h ), that depends on h logarithmically, therefore, leads to nonsingular solutions. In general, if F ( h ) can be expressed as F ( h ) = g/g ' , where g ( h ) is any function such that g ( h ) →∞ as h →±∞ , then τ ( h ) will diverge logarithmically as h →±∞ , which leads to a nonsingular solution. To conclude, except for F ( h ) with a special form (as in the above mentioned cases), the solution reaches a point where F ( h ) → ±∞ in a finite time if F ( h ) grows faster than a linear function. Qualitative Description: In this section we take the pressure p ( h ) to be a continuous function of h , or at most has finite discontinuities. Therefore, we are not going to allow any infinite discontinuities for p ' ( h ), since this leads to a divergent speed of sound that violates causality. One can qualitatively describe a solution as follows; The same analysis can be followed backward in time but with past fixed points and points where F ( h ) →±∞ . The solution has no finite time singularities if it has no future and no past time singularities.", "pages": [ 8, 9, 10 ] }, { "title": "4 Examples and Application", "content": "In this section we use the qualitative method developed in the previous sections to describe the general behavior of viscous fluids in FLRW models and compare it with exact solutions. The last exact solution with r = 1 / 4 describes either as a nonsingular phantom fluid or a unified dark fluid. At the end of this section we use the phase space method, causality and stability constraints to list possible future scenarios of the universe.", "pages": [ 10 ] }, { "title": "4.1 Examples: Nonsingular viscous fluids", "content": "Consider the following equation of state (EoS) where η ( ρ ) = η 0 ρ r . The above EoS can describe a fluid with bulk viscosity η ( ρ ) (see e.g., [33]), a polytropic fluid [37], or a fluid with adiabatic particle production (see e.g., [36]). Although these different interpretations of the above EoS produce the same dynamics, their thermodynamics could be different. Several solutions for the above EoS with different values of r , including the cases discussed here, are known in the literature, see for example [35]. Here we are going to discuss two viscous solutions, the one with r = 1 which is well known in the literature [34] and another with r = 1 / 4, which is less known and express them in terms of Lambert-W function. This clearly shows how the density and the scale factor behave as functions of time. Now using the above pressure in Eqn. 3 we obtain Taking h = H/H ∗ and τ = c H ∗ t , where H ∗ = ( γ 3 r η 0 ) 1 2 r -1 and c = 3 γ/ 2 we get Notice that we have two fixed points, h 1 , 2 = 0 , 1, for this equation. The nature of these fixed points depends on the value of r . Solutions with r > 1 / 2 behave differently compared to those with r < 1 / 2. In the first case F ( h ) is negative during its interpolation between the two fixed points, while the reverse is true for the second case. It is interesting to notice that the first case describes a nonsingular universe which has an EoS parameter w ( ρ ) ≥ -1 since F ( h ) = -1 / 2 ( ρ + p ) while the second case describes a nonsingular universe dominated by a phantom component with EoS parameter w ( ρ ) ≤ -1. It is interesting to notice that fluids with r < 1 / 2 and h 0 > 1 have a similar behavior to that of a generalized Chaplygin-gas cosmology, i.e., for large scale factor, it behaves as a cosmological constant and for small scale factor it behaves as a fluid with EoS p = ( γ -1) ρ . One can see that by writing Eqn.(4) as where η ' = √ 3 η 0 solving the above equation for a general viscous fluid with r -1 / 2 = -s < 0, we get If ρ 0 > ( η ' /γ ) 1 /s = ρ ∗ , the integration constant C 1 is positive, sice then for small a , i.e., ( a 0 /a ) 3 γs [( ρ 0 /ρ ∗ ) s -1] >> 1, we get which describe a fluid with an EoS p = ( γ -1) ρ , and for large a we get which describes an empty space with a cosmological constant ρ ∗ . Another interesting feature of these models is that they show that a normal matter, i.e., γ > 0, with bulk viscosity, behaves as a phantom matter. Furthermore, in the r < 1 / 2 case these solutions are nonsingular. To show these features, let us start with the pressure which leads to an effective EoS parameter since γ > 0, and H ∗ /H > 1, we always have w eff ≤ -1, which breaks all energy conditions. Since H ( t ), in this case, interpolates between two fixed points ( H ∗ and 0) and the pressure p ( H ) is continuous and differentiable, then according to the argument in section 2, the solution is nonsingular and takes an infinite time to reach a fixed point. We are going to see an explicit example of this behavior for the r = 1 / 4 case. Fluids with r=1 : This case was first discussed in [34] as nonsingular solution for a viscous fluid cosmology. In literature this solution usually expressed in terms of time as a function of the scale factor or the density. Here we express the energy density and the scale factor as functions of time in terms of Lambert W-function. First, let us analyze this case qualitatively taking subsection 3.2 in consideration. The asymptotic behavior has the form F ( h ) ∼ h 3 which leads to singular solutions unless there are fixed points. F ( h ) has two fixed points h 1 = 0 and h 2 = 1, as shown in Figure (4). The first is half-stable point and the second is an unstable point. These points divide possible solutions into three types; i) a solution where h ∈ ( -∞ , 0], if h 0 < h 1 , ii) a solution where h ∈ [0 , 1], if h 2 > h 0 > h 1 , and iii) a solution where h ∈ [1 , ∞ ), if h 2 < h 0 . Notice that, in case b), since p ( h ) is differentiable then by the argument in section 2, it takes the solution an infinite time to reach h 1 starting from some initial value, h 0 , where h 1 < h 0 < h 2 . The same is true if we calculate the time taken by the solution to go from h 2 to h 0 . Therefore, for h 1 < h 0 < h 2 the solution is nonsingular and interpolates smoothly between h 1 , and h 2 . If h 0 < h 1 or h 0 > h 2 the solution has a finite time singularity since the asymptotic behavior of F ( h ) ∼ h 3 is growing faster than a linear behavior. For h 0 > h 2 the solution describes a universe that starts from a de Sitter space and endes with a Big Rip singularity. The above equations can be solved exactly in terms of Lambert W-function, which is the solution of the equation W e W = x . The hubble parameter and the scale factor in terms of the time 't' are given by Having a ( t 0 ) = a 0 , and ρ ( t 0 ) = 3 H 0 2 at t = t 0 , we get where β is given by It is crucial at this point to know the sign of β since it controls the behavior of the W-function. The h 0 = H 0 /H ∗ < 1 initial value corresponds to a positive β , which leads to a smooth behavior in all times for the energy density. For the scale factor, the moments at which it either diverges or goes to zero are when t = + ∞ and t = -∞ respectively. Therefore, we have no finite-time singularities in this case. In early times, this model describes an empty universe with a cosmological constant Λ ∼ H ∗ 2 , which evolves to a universe with an EoS p = ( γ -1) ρ in late times. The behavior of H ( t ) and a ( t ) as a function of time t , is shown in Figure(5), which is clearly monotonic. If the initial value h 0 = H 0 /H ∗ > 1 ( β < 0 in the case), then we have a singularity of type-I, or a Big Rip singularity. In this case the effective EoS parameter w eff = p/ρ < -1, therefore the fluid is phantom. Notice that, if h 0 < h 2 , the r < 1 / 2 cases describe nonsingular phantom solutions. One can modify the above model to accommodate the late-time acceleration by adding a cosmological constant Λ. The exact solution of this modified model is not easy to get, but if Λ /H ∗ 2 = λ << 1, or the cosmological constant of the early times is larger than that of late times, then the fixed points become This model describes a universe that interpolates between two de Sitter spaces one with large cosmological constant in early times and another with small cosmological constant in late times which can model the inflation and late time acceleration periods. Fluid with r=1/4: Here we discuss the solution for the r = 1 / 4 case, which can be expressed in terms of Lambert W-function. As we mentioned above, cases with r < 1 / 2 are interesting since they have a unified dark fluid behavior when h 0 > h 2 . If h 0 < h 2 , the solution describes a nonsingular phantom matter with w eff ≤ -1. To analyze this case qualitatively, let us start with the asymptotic behavior of F ( h ) which has the form F ( h ) ∼ h 2 . It clearly leads to singular solutions unless we have fixed points. F ( h ) has two fixed points h 1 = 0 and h 2 = 1, as shown in Figure (8). The first is an unstable point and the second is a stable point. These points divide possible solutions into two types; i) a solution where h ∈ [0 , 1], if h 2 > h 0 > h 1 , and ii) a solution where h ∈ [1 , ∞ ), if h 2 < h 0 . Notice that, in the first case, i), since p ( h ) is differentiable then by the argument in section 2, it takes the solution an infinite time to reach h 1 starting from some initial value, h 0 , where h 1 < h 0 < h 2 . The same is true if we calculate the time to go from h 2 to h 0 , therefore, the solution is nonsingular and interpolates smoothly between h 1 , and h 2 . The second case ii) is singular, since the asymptotic behavior F ( h ) ∼ h 2 is growing faster than a linear behavior. In fact, this solution describes a universe that starts from a finite-time singularity in the past (Big Bang type) and evolves to a de Sitter space after an infinite time. Eqn.(19) for r = 1 / 4 case has an exact solution which has the following form Using initial conditions, the integration constants are where Notice that β ' is positive for h 0 < 1 and negative for h 0 > 1. For initial value h 0 < 1 one expects a nonsingular solution which interpolates monotonically between Minkowski space and de Sitter space. The Hubble parameter H ( t ) and the scale factor a ( t ), as functions of time 't', are shown in Figure(7). As we mentioned in the beginning of the section, this class of solutions shows how a viscous fluid with a usual EoS (i.e., γ > 0) behaves as phantom component. Similar to the r = 1, one can consider adding a small cosmological constant. A small cosmological constant Λ /H ∗ 2 = λ << 1 leads to the following fixed points The solution with initial value h 1 < h 0 < h 2 described a universe filled with a phantom matter interpolating between a de Sitter space with a small cosmological constant at early times and another with a large cosmological constant in late times.", "pages": [ 11, 12, 13, 14, 15, 16 ] }, { "title": "4.2 Fate of the Universe", "content": "Here we list all possible future scenarios of the universe as a single fluid in FLRW cosmology without assuming any fixed points but imposing the causality and stability constraints. It is known that, if p > -1 / 3 ρ , destiny of the universe is tied to geometry (see for example [38]) and the value of k is important to predict the fate of the universe. On the other hand, if p < -1 / 3 ρ destiny is not tied to geometry but controlled by the behavior of the energy density ρ . This can be shown if we consider the known mechanical model for a , by rewriting Friendmann equation as Taking the EoS p = wρ leads to For large a , if w > -1 / 3, k controls the existence of vanishing velocities, or turning points, but for w > -1 / 3, the potential gets a small contribution from k compared to that coming from a 2 ρ . This breaks the connection between the geometry and destiny. In fact, this simple argument is also suggesting that our universe will keep on expanding because of the domination of the dark energy component. We will see next that this is not generally the case. Here we are going to use the above constraints to list possible scenarios for the future of the universe. Let us model our universe using a general single barotropic fluid, which is a reasonable assumption since dark energy is dominating in late times. First, it is easy to show that we are in a region in the phase space (i.e., ˙ H -H space ) where ˙ H < 0 6 . It is known that the deceleration parameter has changed sign from positive to negative as the universe evolved from a matter dominating era to a dark energy dominating era. To see how this crossing happened consider first the zero acceleration curve which is given by then, as an example of dark energy, consider an accelerating universe with a cosmological constant and matter By plotting both functions -H 2 and -1 / 2 (3 H 2 -ρ Λ ) in Figure(8) one can see how this crossing happened, i.e., going from a < 0 to a > 0. The point at which the acceleration vanishes can be used as a reference point to draw the constraints on possible evolutions of the universe. Using the causality and stability constraints, dp/dρ ≤ 1 and dp/dρ ≥ 0, we get 0 ≥ dp ( H ) /dH ≤ 6 H which in turn leads to -3 H ≥ dF ( H ) /dH ≤ -6 H . Integrating this inequality leads to -3 / 2 H 2 + C 1 ≥ F ( H ) ≤ -3 H 2 + C 2 , where the integration constants C 1 and C 2 are fixed by the initial values of H and ˙ H at the reference point. Notice that, C 1 and C 2 have to be positive numbers, otherwise crossing the zero-acceleration curve will not occur. The last inequality constrains the future behavior of F ( H ) to lie between these two parabolas, as a result, F ( H ) must meet the H -axes in a future time and ends as a de Sitter universe after infinite time. In addition, this evolution starting from the point of zero-acceleration till the end of time is nonsingular since it ends with de Sitter universes. As one might notice, although, the causality constraint is essential for any physical model. It is not clear if we should insist on having the stability constraint, since we do not know the physics of the dark energy component. Now, if we relax the stability constraint, dp/dρ ≥ 0 and allow the pressure derivative to go negative. From the above phase space diagram, it is clear that the universe either ends as an empty universe or hits the negative ˙ H -axes. The last possibility is interesting since according to the discussion in subsection 3.1 it describes a turnaround behavior, therefore, the universe in a future finite time reaches a maximum size, then, recollapses, since the hubble parameter H changes sign.", "pages": [ 16, 17, 18 ] }, { "title": "5 Conclusion", "content": "In this work, we used a phase space method to study possible consequences of having fixed points in a single fluid flat FLRW models. Some of these are; (i) if we describe our universe as a single component fluid with a future fixed point, then the resulting cosmology does not have future-time singularities of types I, II and III in [25], (ii) cosmologies with a future and a past fixed points are free of of types I, II and III singularities, (iii) one can use a simple argument to show the phantom divide [27, 28], or in a single fluid FLRW models it is impossible for a physical solution to cross the phantom divide line in a finite time, and (iv) in these models, the only way to get bounce solutions is to have a nonvanishing pressure as ρ → 0. This method can be used to construct nonsingular late-time models, in particular, unified dark fluid and dark energy models. We use this method to qualitatively describe any flat FLRW model with fixed points. We discussed FLRW cosmology with bulk viscosity η ∼ ρ r , and presented two exact solutions with r = 1 and r = 1 / 4, which are expressed in terms of Lambert-W function. The last solution describes either a nonsingular phantom dark energy or a unified dark fluid model. The phantom solution is interesting since it shows how a viscous normal fluid behaves very similar to a phantom matter without Big Rip singularities. In addition, it interpolates between two de Sitter spaces with small and large cosmological constants. Possible future scenarios of our universe include; a de Sitter space, an empty universe with vanishing cosmological constant, or a turn a round solution that reaches a maximum size, then collapses.", "pages": [ 18, 19 ] }, { "title": "Acknowledgement", "content": "I would like to thank P. Argyres, S. Das, A. Shapere, E. Lashin and A. El-Zant for several discussions and comments.", "pages": [ 19 ] } ]

[ { "title": "Seeking Inflation Fossils in the Cosmic Microwave Background", "content": "Liang Dai, Donghui Jeong, and Marc Kamionkowski Department of Physics & Astronomy, Bloomberg Center, The Johns Hopkins University, Baltimore, MD 21218, USA (Dated: June 17, 2018) If during inflation the inflaton couples to a 'fossil' field, some new scalar, vector, or tensor field, it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background (CMB) temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fossils can be distinguished geometrically from those of scalars through the parity of the BiPoSHs they induce. However, the two-dimensional nature of the CMB sky does not allow vectors to be distinguished geometrically from tensors. We estimate the detectability of a signal in terms of the scalar-scalarfossil coupling for scalar, vector, and tensor fossils, assuming a local-type coupling. We comment on a divergence that arises in the quadrupolar BiPoSH from the scalar-scalar-tensor correlation in single-field slow-roll inflation.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Despite the observational success of the inflation paradigm, the nature of the inflationary epoch is still poorly understood. The single-field slow-roll model, being the simplest inflation model, might not be the whole story of the physics behind inflation. An abundance of inflation models introduce new fields, which are often coupled to the inflaton [1, 2]. Candidates for those new degrees of freedom include extra scalar fields in extensions of the simplest inflation model [3]. They might also be primordial vector fields that drive or impact inflation [4], or even new vectorial degrees of freedom arising from modified theories of gravity [5]. Even in the simplest single-field slow-roll inflation, tensor metric perturbations [6] couple to the inflaton [7-10]. Fortunately, the cosmic microwave background (CMB) provides a special window to probe physics at very early moments, by encoding non-trivial primordial correlations beyond Gaussianity in its anisotropy pattern. By measuring the primordial bispectra, trispectra, and possibly higher-order correlation functions from its temperature and polarization anisotropies [11-15], various inflation models will hopefully be distinguished, and new physics might be revealed. Recently, Ref. [16] proposed a generic parameterization of the primordial bispectra involving the scalar metric perturbation Φ (the gauge-invariant Bardeen potential [17]) and a new field h from inflation. The new field is dubbed an inflation fossil , a hypothesized primordial degree of freedom that no longer interacts or very weakly interacts during late-time cosmic evolution. The only observational effect of an inflation fossil might therefore be its imprint in the primordial curvature perturbation. Inflation fossils can be those extra fields that are introduced in a variety of alternatives to the single-field slowroll model. Different models may be distinguished by the spin of the new field. To treat different possibilities for the spin in a model-independent fashion, in Ref. [16] the fossil field is parameterized by a symmetric traceless tensor field. Since a symmetric traceless tensor contains a longitudinal scalar (L), two divergence-free vectors (V), and two divergence-free traceless tensors (T), the parameterization allows for all three possibilities for the spin. A given realization of the fossil field causes two different scalar perturbation modes to correlate with each other (panel (a) of Fig. 1). Scalar, vector, and tensor fossils give rise to geometrically distinct cross-correlations. For a stochastic fossil background, connected trispectra that correlate four differents modes of Φ are generated (panel (b) of Fig. 1) when an ensemble average over all fossil-field realizations is performed. Such a scenario of scalar four-point correlations but without scalar threepoint correlations can be sought in galaxy surveys where the correlations from scalar, vector and tensor fossils can be disentangled geometrically [16]. In this paper we study the CMB signatures of fossil fields. Correlations between different Fourier modes of the curvature perturbation induce couplings between different CMB-temperature spherical-harmonic coefficients. We parametrize these cross-correlations in terms of bipolar spherical harmonics (BiPoSHs). In the presence of some realization of the fossil field, the CMB appears non- Gaussian with nonvanishing BiPoSHs, or equivalently, a nonzero four-point correlation function, or trispectrum. Here we first calculate the BiPoSH coefficients from a specific realization of the fossil field, and then find the BiPoSH power spectra for a stochastic background for the fossil field. We write down the minimum-variance CMB estimators for the amplitudes of the fossil-field power spectra. As an example, we numerically examine the case of a local-type scalar-scalar-fossil bispectrum with a scale-free fossil spectrum. That case can be described by two parameters, the amplitude of the bispectrum, and the normalization of the fossil power spectrum. We find that the dominant effects from a local-type bispectrum are modulations of small-angular-scale correlations on large angular scales. We evaluate the sensitivities for the reduced amplitude of the fossil background, a combination of the two parameters, for different fossil-field spin. Our results show that the sensitivity achievable by Planck is more than an order of magnitude better than that by current galaxy-clustering surveys and is comparable to that of larger next-generation surveys. This paper is organized as the following. In Sec. II we first review the parametrization of the effect of fossil fields on scalar perturbations. Since this paper is concerned with the calculation of observables on a spherical sky, we recast the parametrization in Ref. [16] in terms of total-angular-momentum (TAM) waves [18]. The TAM formalism makes the rotational symmetry of the observed sky manifest throughout and thus greatly simplifies our calculation. We then move on to present the calculation of the BiPoSHs and the BiPoSH power spectra. After that, we construct the minimal-variance quadratic estimator for the reduced amplitude, and calculate the cumulative signal-to-noise. We proceed to present numerical results for a local scalar-scalar-fossil bispectrum in Sec. III. They are complemented by Sec. IV, which discusses a real-space picture of the fossil-field effects on the temperature map. Finally, we make concluding remarks in Sec. V. We also comment there on a divergence in the predicted observables from the scalar-scalar-tensor correlation in single-field slow-roll inflation.", "pages": [ 1, 2 ] }, { "title": "A. Fossil parameterization", "content": "Interactions between a fossil field h and the curvature perturbation R can generate three-point functions of the type 〈RR h 〉 at horizon crossing that then convert into a primordial bispectrum involving two scalar-metricperturbation Φ modes and a fossil-field mode after the end of inflation. A realization of the h field then locally induces departures from statistical homogeneity in the scalar autocorrelation function, which then appears as a correlation between different Fourier modes of the scalar perturbation. A general parameterization, is written down in Ref. [16] to account for the possibility that the fossil field h can be scalar, vector or tensor field. Here k 1 and k 2 are wavevectors of the two scalar-perturbation modes, K is the wavevector of the fossil-field mode, and h p ( K ) is the Fourier amplitude of the fossil field. The Dirac delta function reflects the translational invariance of the underlying physics. We introduce a generic bispectrum shape function f p h ( k 1 , k 2 , K ), which is related to the primordial scalarscalar-fossil bispectrum through B ΦΦ h ( k 1 , k 2 , K ) = P p h ( K ) f p h ( k 1 , k 2 , K ) /epsilon1 p ab ( K ) ˆ k a 1 ˆ k b 2 , where P p h ( K ) is the power spectrum for the fossil field. Moreover, the symmetric three-by-three polarization tensor /epsilon1 p ab ( K ) with p = 0 , L, x, y, + , × geometrically distinguishes modulations of scalar, vector, and tensor type through the azimuthal dependence when the configuration is rotated about the direction of K . The trace tensor /epsilon1 0 ab ( K ) ∝ δ ab and the longitudinal /epsilon1 L ab ( K ) ∝ ( K a K b -K 2 δ ab / 3) K -2 describe a scalar fossil field. Two transverse-vectorial tensors /epsilon1 x,y ab ( K ) ∝ K ( a w x,y b ) satisfying K a w x,y a = 0 describe a transverse-vector fossil field. Likewise, two transversetensorial tensors /epsilon1 + , × ab ( K ) satisfying K a /epsilon1 + , × ab ( K ) = 0 describe a transverse-tensor fossil field.", "pages": [ 2 ] }, { "title": "B. The TAM formalism", "content": "Since angular observables defined on the twodimensional sky are involved, we adopt the total-angularmomentum (TAM) wave formalism, recently developed in Ref. [18], to take full advantage of the rotational symmetry of the problem from the very beginning. In the TAM formalism, scalar, vector, or tensor fields in threedimensional space are expanded in terms of a complete set of spherical waves, as opposed to the conventional expansion in terms of plane waves. These spherical waves are eigenfunctions of the Helmholtz equation with wave number K and are eigenfunctions of total angular momentum and its third component with quantum numbers J and M . Although the scalar case has been long known as the Fourier-Bessel expansion, the TAM formalism greatly simplifies calculations when vector or tensor fields have to be projected onto the two-dimensional sphere. In our case here, since three-by-three symmetric polarization tensors are used in Eq. (1), scalar, vector and tensor fossil fields can all be incorporated into a symmetric traceless tensor field h ab ( x ). As the tensor field is expanded in terms of TAM waves, the longitudinal mode h L JM ( K ) describes a scalar fossil field, the two divergencefree vectorial modes h V E JM ( K ) and h V B JM ( K ) describe a transverse-vector fossil, and the two divergence-free tensorial modes h TE JM ( K ) and h TB JM ( K ) describe a transversetensorial fossil. In terms of TAM coefficients Φ lm ( k ) and h α JM ( K ) for the scalar perturbation and the fossil field respectively, the modulation analogous to Eq. (1) reads for α = L, V E, V B, T E, T B respectively. Here Ψ k ( lm ) ( x ) and Ψ α,K ( JM ) ab ( x ) are TAM wave functions for scalar and tensor fields. The overlaps of three TAM wave functions have been worked out in Ref. [19]. Note that statistical homogeneity and isotropy require that f V E h ( k 1 , k 2 , K ) = f V B h ( k 1 , k 2 , K ) ≡ f V h ( k 1 , k 2 , K ) and f TE h ( k 1 , k 2 , K ) = f TB h ( k 1 , k 2 , K ) ≡ f T h ( k 1 , k 2 , K ).", "pages": [ 2, 3 ] }, { "title": "C. CMB BiPoSHs", "content": "Through the epochs of radiation domination and matter domination to recombination, scalar metric perturbations source temperature and E -mode polarization anisotropies of the CMB. Therefore, cross-correlations of different scalar-perturbation modes, as a consequence of modulation by the fossil field, give rise to crosscorrelations between different harmonic modes of the CMB anisotropies. For clarity, in this paper we only discuss the effect on the temperature map. Still, the inclusion of E -mode polarization is straightforward and will improve the overall sensitivity of detection, once the cross-correlations between temperature and E -mode polarization are properly taken care of. For a given fossil-field configuration, the modulation violates statistical isotropy in the anisotropies. This effect can be conveniently quantified in terms of a bipolar spherical harmonic (BiPoSH) expansion [20]. Being the two-sphere analog of the fossil-field modulation, Eq. (1), it provides a parameterization for the most general twopoint correlation function on the sky. Specifically, crosscorrelations of CMB temperature multipoles read where 〈 l 1 m 1 l 2 m 2 | l 3 m 3 〉 denotes the Clebsch-Gordan coefficient. The BiPoSH coefficients A JM l 1 l 2 , given by exist if J , l 1 and l 2 can form a triangle. The angle brackets with a subscript h denote an average over all realizations of Φ for a fixed realization of h . Here we use W l 1 l 2 l 3 m 1 m 2 m 3 as a compact notation for the usual Wigner-3 j symbol. Since the fossil is parameterized by a symmetric tensor field, it only induces BiPoSHs with J /greaterorequalslant 2. Under the TAM basis, the scalar perturbation can be described by TAM coefficients Φ lm ( k ), which source CMB temperature multipoles according to where g T l ( k ) is the scalar radiation transfer function for temperature. Combining Eqs. (2), (4), and (5), the CMB BiPoSH due to modulation of a single TAM wave of the fossil field is, where the parity P ( α ) = 0 for α = L, V E, T E and P ( α ) = 1 for α = V B, T B . The functions I α l 1 l 2 J ( k 1 , k 2 , K ) are given by where the relevant overlaps of three TAM wave-functions can be found in Eqs. (80), (81), (84), (85), and (87) of Ref. [19], for each α respectively. Note that the dependence on azimuthal quantum numbers m 1 , m 2 , and M cancels out on the right hand side, and the final result for I α l 1 l 2 J ( k 1 , k 2 , K ) is reduced to an integral over the radial coordinate. In the above result for the BiPoSH coefficients, rotational invariance is manifest in that a given A JM l 1 l 2 is only generated by TAM waves of the fossil field with the same total-angular-momentum quantum numbers J and M . Moreover, due to parity conservation, L , V E , and TE modes only induce even-parity BiPoSHs, i.e. J + l 1 + l 2 = even, while V B and TB modes only induce odd-parity BiPoSHs, i.e. J + l 1 + l 2 = odd. Therefore, vector and tensor fossils, both containing B -mode TAM waves, can be distinguished from scalar fossils from their signature in odd-parity BiPoSHs. Nevertheless, vector and tensor fossils cannot be geometrically distinguished from each other from CMB BiPoSHs, as they can with threedimensional surveys [16]. This is heuristically understood as information loss when the three-dimensional correlation function of the scalar perturbation is projected onto the sky to give two-dimensional angular correlation functions of the CMB. In addition to primordial mechanisms, late-time effects can distort a Gaussian, statistically isotropic map as well. Particularly, weak-lensing of the CMB by the foreground matter distribution also produces even-parity BiPoSHs, mimicking the effect of inflation fossils. Still, we anticipate the spectra of BiPoSHs from lensing to differ from those from modulation by fossils. The reason is that the power spectrum for lensing by the scalar potential peaks at J ∼ 60 [21], but, e.g., for a local scalar-scalar-fossil bispectrum the fossil modulation effects dominate at J /lessorsimilar 5, as our results will show later. This implies that the shape of BiPoSH power spectra can be used to break the degeneracy. Besides, weak-lensing generates B -mode polarization of the CMB, while a primordial scalar-scalar-fossil bispectrum does not.", "pages": [ 3, 4 ] }, { "title": "D. BiPoSH power spectra and estimators", "content": "A stochastic background of the fossil field is presumably generated during inflation, just like the scalar field or inflationary gravitational waves, and it is characterized by a power spectrum, Statistical homogeneity and isotropy guarantee that P V E h ( K ) = P V B h ( K ) ≡ P V h ( K ) and P TE h ( K ) = P TB h ( K ) ≡ P T h ( K ). The statistical significance with which any individual BiPoSH is detected to be nonzero is expected to be small. In light of this, we average over all realizations of the fossil field, and define bipolar auto-/cross-power spectra, to statistically measure the imprint of fossils. These correspond to four-point correlations in the temperature map. They are invariant under rotations. We emphasize that since the fossil-field power spectrum is statistically homogeneous and isotropic, statistical isotropy of the CMB is resumed after taking an ensemble average over the fossil field. To measure BiPoSHs from data, we use quadratic estimators. Estimators for BiPoSH coefficients can be constructed by Then estimators for bipolar power spectra can be written down, Note that the second term is needed to unbias the estimators with respect to a Gaussian CMB map without fossil effects. We assume a phenomenological parameterization for the fossil-field background, which is described by two parameters. One is the normalization P Z h for the power spectrum, with a fiducial-power spectrum shape ˜ P Z h ( K ). The other is the amplitude B Z h of the scalar-scalar-fossil bispectrum, with a fiducial bispectrum shape ˜ f Z h ( k 1 , k 2 , K ). Here Z can be scalar Z = L , transverse vector Z = { V E, V B } or transverse tensor Z = { TE,TB } . For vector or tensor fossils, both E modes and B modes have to exist. With these parameterization, Eq. (6) can be cast into the form where α ∈ Z and the coefficient function F J,α l 1 l 2 ( K ) can be read off from Eq. (6) as The BiPoSHs power spectra are then calculated to be where we define the reduced amplitude A Z h ≡ P Z h ( B Z h ) 2 of the fossil background. There is an observational degeneracy between the fossil-field power spectrum and scalarscalar-fossil bispectrum, so only this combination is measurable. An estimator, for the reduced amplitude A Z h can be constructed, for each possible combination of J and l i , i = 1 , . . . , 4. Taking into account the symmetry of ̂ A J,Z h,l 1 l 2 ,l 3 l 4 , without loss of generality we can fix l 1 /lessorequalslant l 2 , l 3 /lessorequalslant l 4 , and l 2 /lessorequalslant l 4 if l 1 = l 3 . We can combine these estimators to bring down the statistical error. Under the null hypothesis, it can be shown that ̂ A J,Z h,l 1 l 2 ,l 3 l 4 's, being quartic expressions in a T lm , are nearly uncorrelated with each other. This is seen through calculating the null-hypothesis correlation matrix by applying the Wick expansion. Cross-correlation elements are suppressed by either negative powers of (2 l i + 1) or a Wigner-6 j symbol relative to the autocorrelation elements. Treating ̂ A J,Z h,l 1 l 2 ,l 3 l 4 's as statistically independent estimators, we then linearly combine them, and write down the inverse-variance-weighted estimator, where ∑ ( l 1 ,l 2 ,l 3 ,l 4 ) loops over all independent combinations of multipoles. The denominator is equal to the inverse variance of ̂ A Z h , which is worked out to be Hence, a high signal-to-noise S = A Z h /σ Z A indicates a detectable BiPoSH signature in the CMB from inflation fossils.", "pages": [ 4, 5 ] }, { "title": "III. NUMERICAL RESULTS", "content": "Our discussion so far applies to the most general inflation fossil with arbitrary power spectrum P α h ( K ) and bispectrum shape f α h ( k 1 , k 2 , K ). In order to provide some illustrative numerical estimates of the detectability of the signal, we specialize to the case where the bispectrum shape is of the local type. Such a bispectrum may arise if the fossil-curvature interaction is local. This occurs, for example, for the graviton-curvature-curvature bispectrum [7] in single-field slow-roll inflation. We approximate the bispectrum in this case by the fiducial form ˜ f Z h ( k 1 , k 2 , K ) = ( k 1 k 2 ) -3 / 2 it attains in the squeezed limit k 1 , k 2 /lessmuch K where it peaks. We assume a scalefree power spectrum ˜ P Z h ( K ) = 1 /K 3 for the fossil field as should arise if no physical scale other than the Hubble scale comes into play during inflation. Specializing to those fiducial forms for the power spectrum and the bispectrum, we numerically compute sensivities σ Z A . We assume the standard flat ΛCDM cosmology with the WMAP+BAO+ H 0 best-fit cosmological parameters taken from TABLE 1 of Ref. [22], except that for consistency we assume perfect scale-invariance during inflation by setting n s = 1. The public code CAMB [23] is used to tabulate radiation transfer functions. CMB multipoles up to l max = 3000 are considered. We numerically evaluate and tabulate coefficient functions F J,α l 1 l 2 ( K ) from Eq. (15) for 5 × 10 -6 Mpc -1 /lessorequalslant K /lessorequalslant 5 × 10 -3 Mpc -1 , ranging from fossil-field modes that are well outside of the horizon today to modes that were inside of the horizon at the last-scattering surface. Examples of F J,α l 1 l 2 ( K ) are plotted in Fig. 2 to show some of the features. In the TAM-wave picture, the oscillatory nature of the F J,α l 1 l 2 ( K )'s can be understood as the value of the TAM wave function on the surface of last scattering. As K increases, spherical waves shrink inward to the origin, and hence nodes and anti-nodes alternate to pass the surface of last scattering. On the other hand, for large scale modes with very small values of K , even the first peak is well beyond the last scattering surface, and consequently BiPoSHs vanish in the infrared. Exceptions are the J = 2 even-parity modes α = L, V E, T E , whose wave functions do not vanish at the origin. Therefore, logarithmic infrared divergence arises in even-parity quadrupolar BiPoSHs due to superhorizon TAM modes. In this work, we introduce an infrared cutoff at K min = 5 × 10 -6 Mpc -1 , which does not affect the results except for the even-parity J = 2 BiPoSHs. We calculate sensitivities for A Z h at 3 σ statistical significance (Fig. 3) as a function of the largest CMB multipole l max dominated by the signal. No instrumental noise is assumed for l /lessorequalslant l max . We consider scalar, vector or tensor fossil fields Z = L, V, T respectively using two strategies. One is to fully exploit the information from both even- and odd-parity BiPoSHs. Since the signal-to-noise is dominated by even-parity BiPoSHs, this optimizes the sensitivity. In reality, however, lensing by gravitational potentials and other late-time mechanisms kick in on small angular scales l /greaterorsimilar 1000. They mimic the effect of fossil fields by generating even-parity BiPoSHs as well, making it necessary to disentangle between them. To circumvent the problem, the second strategy is to include only odd-parity BiPoSHs. They provide clean probes of fossils with non-zero spin, with moderate compromise in the overall sensitivity. Even though there is degeneracy between the reduced amplitudes A V h and A T h from measurement of the CMB BiPoSHs, one can still break it with a three-dimensional measurement from galaxyclustering [16]. The BiPoSH signal is dominated by anisotropic modulation on the largest angular scales, and J /lessorequalslant 5 almost saturates the detectability. On the other hand, modulation of small-scale anisotropies (very large l 1 and l 2 ) contributes large signal-to-noise to bring down σ Z A which is found to scale as l -2 max , in agreement with similar studies of CMB trispectra [14, 24]. Such a scaling law is justified analytically in the Sachs-Wolfe limit, valid on large scales. In that case, the radiation transfer functions are approximated by g T l ( k ) = -j l ( kr ∗ ) / 3, where r ∗ ≈ 14 Gpc is the comoving distance to the last scattering surface. For a given BiPoSH angular scale J and CMB scale l 1 , l 2 satisfying L /lessmuch l 1 ≈ l 2 ≈ l 12 and L /lessmuch l 3 ≈ l 4 ≈ l 34 , the signal squared can be shown to scale as l -3 12 l -3 34 , while the noise squared damps out more rapidly as l -4 12 l -4 34 , so that each estimator ̂ A J,Z h,l 1 l 2 ,l 3 l 4 contributes signal-to-noise squared ∝ l 12 l 34 . Given the constraints | l 1 -l 2 | < J and | l 3 -l 4 | < J , the number of such estimators scales as J ∑ l 12 /lessorequalslant l max ∑ l 34 /lessorequalslant l max . Therefore, the cumulative signalto-noise squared is S 2 ∝ J ∑ l 12 /lessorequalslant l max ∑ l 34 /lessorequalslant l max l 12 l 34 ∝ l 4 max . Nonetheless, at l max /greaterorsimilar 3000, gravitational lensing and instrumental error take over [12], cutting off the growth in sensitivity. To compare our results to the trispectrum parameter τ NL widely used in the local model of primordial non-Gaussianity [14, 25-28], we consider induced fourpoint correlations in (b) of Fig. 1 by integrating out an intermediate scalar fossil (dashed line), which mimics an intrinsic trispectrum. Since the induced trispec- rum is proportional to A L h , we find the conversion A L h = 5 . 3 × 10 -23 τ NL . In this way, we compare our results to the WMAP 5-year bound | τ NL | < 3 . 3 × 10 4 [29] and to the bound τ NL < 2800 at 2 σ from the first Planck data release [30]. Good discovery potentials for Planck with l max = 2500 can be achieved, for both even-/oddparity BiPoSHs combined and odd-parity BiPoSHs alone. The Planck satellite can improve upon WMAP's sensitivity to A Z h by a factor ∼ 25. Planck's sensitivity corresponds to that of a galaxy-clustering surveys with volume V = 60Gpc 3 h -3 and a resolution k max = 0 . 1 h Mpc -1 . Thus, Planck's sensitivity is in principle nearly two orders of magnitude better than that of the current SDSSIII BOSS survey [31]. Besides, we find good numerical agreement with the τ NL forecasts of Ref. [14].", "pages": [ 5, 6 ] }, { "title": "IV. LOCAL-DEPARTURE MODULATION IN REAL SPACE", "content": "There exists, for the local-type scalar-scalar-fossil, a simple and illustrative real-space picture of the effect of the fossil field on primordial perturbations. To see this, we write, This reproduces the squeezed limit K → 0 of Eq. (1) with the local-departure form Eq. (13). Here Φ g ( x ) is the gaussian field, and P Φ is the dimensionless normalization of the scalar-perturbation power spectrum. Note that the relation between Φ( x ) and Φ g ( x ) is local. For CMB multipoles l /lessorsimilar 100, the temperature is determined primarily by the value of the potential Φ at the surface of last scattering. Therefore, consider one TAM mode of the fossil field. The distortions to hot and cold spots induced by the fossil field are proportional to the value of the TAM wave function Ψ α,k ( JM ) ab ( x ) on the surface of last scattering. This reduces to threedimensional tensor field that lives on the surface of the two-sphere. This tensor field is a linear combination of the five tensor spherical harmonics Y β ( JM ) ( ˆ n ) , β = L, V E, V B, T E, T B , as shown in Eq. (94) of Ref. [18] where the comoving radial distance r corresponds to the comoving distance to the surface of last scattering. We clarify here that although these tensor spherical harmonics are only functions of two-dimensional ˆ n , their tensor value at each ˆ n can have components purely perpendicular ( β = TE,TB ) or parallel ( β = L ) to the line of sight, or even partially perpendicular ( β = V E, V B ). Consider a patch in the sky, which is small relative to the typical variation scale of the fossil field, but large relative to the scale of hot and cold spots. Then according to Eq. (20) the fossil field acts as a constant modulation factor across the patch. From Eq. (20), the departure is proportional to the double gradient of Φ g ( x ), but only the components purely perpendicular to the line of sight can be seen. These then distort isothermal contours on the sky. This implies that local distortion patterns are distinct for each of the five types of tensor harmonics (Fig. 4): (1) The longitudinal harmonic Y L ( JM ) ab ( ˆ n ) is purely radial, so from a projected view onto the sky, isothermal contours expand or shrink in an isotropic way. As a result, only the contrast between hot and cold spots are enhanced or reduced, but the shape and the center remain the same. (2) The vectorial harmonics Y V E,V B ( JM ) ab ( ˆ n ) establish a preferred direction at given ˆ n from a projected view onto the sky. Isothermal contours are shifted when they are perpendicular to that preferred direction, but are unchanged if parallel to it. The outcome is that spots are shifted parallel or anti-parallel to the preferred direction, and hot spots and cold spots shift in the same direction. (3) The purely transverse tensorial harmonics Y TE,TB ( JM ) ab ( ˆ n ) define two principle directions perpendicular to each other at given ˆ n from a projected view onto the sky. Isothermal contours always shift to the hot side along one principle direction, and shift to the cold side along the other principle direction. This leaves the center of the hot and cold spots unchanged, but induces ellipticity by elongating the spot along one of the principle directions and squashing it along the other. On the scale of the fossil-field modulation, the local distortion pattern varies from patch to patch (Fig. 5). Given fossil-field angular-momentum quantum numbers J and M , the full sky is divided into regions centered at zeroes of the harmonic. For modulation by the longitudinal harmonic Y L ( JM ) ab ( ˆ n ), the hot/cold contrast is enhanced in one of the two adjacent regions and is re- uced in the other. For modulation by the vectorial E -mode harmonic Y V E ( JM ) ab ( ˆ n ), rings of hot/cold spots around the center of the region either expand away from it, or sink toward it, and an adjacent region has the opposite effect. For the B -mode harmonic Y V B ( JM ) ab ( ˆ n ), rings of hot/cold spots systematically rotate around the center of the region, with opposite rotating directions in adjacent regions. Similarly, for modulation by the tensorial E -harmonic Y TE ( JM ) ab ( ˆ n ), hot/cold spots are elongated along the radial/tangential direction with respect to the center of the region. By contrast, for the B -mode harmonic Y TB ( JM ) ab ( ˆ n ) the direction of elongation is rotated by 45 · . Again, in adjacent regions the effect is the opposite. In the far-field limit, i.e. small fossil-field wavelength relative to the comoving distance to the surface of last scattering, the TAM wavefunction Ψ α,k ( JM ) ab ( x ) is asymptotically proportional to the corresponding type of tensor harmonic Y α ( JM ) ab ( ˆ n ). In that case, longitudinal TAM waves mainly modulate hot/cold contrast, divergencefree vectorial TAM waves mainly induce displacement, and divergence-free tensorial TAM waves mainly generate ellipticity. However, the interesting scales for a bispectrum of the local type are large scales, and a given TAM wavefunction typically projects to different types of tensor harmonics at last scattering and generates different types of real-space distortions simultaneously.", "pages": [ 7, 8 ] }, { "title": "V. CONCLUSION", "content": "In this paper, we have investigated the signature of inflation fossils in the anisotropies of the CMB temperature. To take into account the possibilities that the fossil can be a scalar, a vector or a tensor, we have used a generic parameterization for cross-correlations of two scalar perturbation modes as a consequence of the fossil. In harmonic space, non-zero BiPoSH coefficients are induced; in real space, the effects show up as modulations of temperature contrast and displacement and ellipticity of hot and cold spots from patch to patch across the sky. We have provided results for BiPoSH coefficients and BiPoSH power spectra for general inflation fossil scenarios. As an important example, we have numerically examined the case of a local-type modulation, featuring a scalar-scalar-fossil bispectrum dominated by the squeezed limit and a scale-free fossil power spectrum. By constructing the minimum-variance quadratic estimator, we have found that the combination A Z h = P Z h ( B Z h ) 2 can be measured. Our numerical results show that modulations on large angular scales dominate the signal, and the sensitivity improves as l -2 max . For an experiment with Planck's sensitivity l max = 2500 and with a combined analysis of both even-parity and odd-parity BiPoSHs, we find detectability at A Z h / 10 -20 = 6 . 6 , 3 . 5 , 3 . 5 for Z = L, V, T respectively, at 3 σ significance. On the other hand, odd-parity BiPoSHs alone are clean probes of vector/tensor fossils, with sensitivities A Z h / 10 -20 = 20 , 8 . 6 for Z = V, T respectively. These results are comparable to a sensitivity for the trispectrum parameter in the local model τ NL ∼ 1250. They are also significantly better than the sensitivity estimated to be available with current galaxy-clustering surveys. Even better signal-to-noise is expected if the E -mode polarization is also considered (the scalar-scalar-fossil bispectrum we discuss here does not generate or affect the B -mode polarization). In that case, the BiPoSHs arising from TT -correlation, EE -correlation and TE -correlation are correlated with each other. Therefore, a correct treatment must involve inverting the covariance matrix. We do not include the full analysis with polarization in this paper, leaving it to future work. Also, it will not be possible to geometrically distinguish between scalar-, vectorand tensor-type fossil field from the CMB. A possible solution might be to scrutinize the shape of the bipolar power spetra. The shape is found to be degenerate between vector and tensor fossils for a primordial bispectrum of local type, but it might be more discriminative for models with primordial bispectra of other shapes. In the latter case, fossil modes smaller than the scale of the last scattering surface can be important, and therefore distince BiPoSH shapes can be induced due to different polarization patterns for the scalar-, vector- and tensortype fossil TAM wavefunctions in the far-field limit, as discussed in Sec. IV. Weak lensing of the CMB by the foreground mass distribution also generates non-zero BiPoSHs and could thus constitute a background for the fossil-field signal. However, the fossil-field signal peaks (at least for a localtype coupling) at J /lessorsimilar few, while the weak-lensing BiPoSHs peak at J ∼ 100 and will have a very preciselypredictable shape. Vector and tensor fossil fields will moreover be distinguishable by the odd-parity BiPoSHs, which are not induced by lensing by density perturbations. Finally, lensing introduces a CMB-polarization B mode, which can further distinguish it from the effects of a fossil field. Before closing we comment on the nature of the infrared divergence in the prediction for the amplitude of the quadrupolar ( J = 2) BiPoSHs that arises if the fossil field has a scale-invariant spectrum and an inflatoninflaton-fossil three-point function of the local type. The predicted amplitude for the J = 2 BiPoSH then depends logarithmically on the smallest wavenumber K for the fossil field, and thus, on the onset of inflation. A similar divergence arises also in the fossil-field prediction of galaxy clustering [16]. The result seems to imply that distance scales well beyond the horizon are having significant effect on observables within the horizon. A similar logarithmic divergence has been discussed in Refs. [9, 10] where it was argued that the tensor-scalar-scalar threepoint correlation in slow-roll inflation may give rise to an power quadrupole that could be probed observationally [32, 33]. One the other hand, in Refs. [34, 35] infraredsafe correlation functions are constructed and are argued to be representative of the statistics in the local Hubble patch. We believe it is important to fully understand this effect, as it occurs from the scalar-scalar-tensor threepoint function that arises in single-field slow-roll inflation [7], even without the introduction of additional new field.", "pages": [ 8, 9 ] }, { "title": "Acknowledgments", "content": "We thank Fabian Schmidt for useful discussions. This work was supported by DoE SC-0008108 and NASA NNX12AE86G. [34] M. Gerstenlauer, A. Hebecker and G. Tasinato, JCAP 1106 , 021 (2011) [arXiv:1102.0560 [astro-ph.CO]]. [35] T. Tanaka and Y. Urakawa, JCAP 1105 , 014 (2011) [arXiv:1103.1251].", "pages": [ 9, 10 ] } ]

[ { "title": "Confronting brane inflation with Planck and prePlanck data", "content": "Yin-Zhe Ma, 1, 2, ∗ Qing-Guo Huang, 3, † and Xin Zhang 4, 5, ‡ 1 Department of Physics and Astronomy, University of British Columbia, Vancouver, V6T 1Z1, BC, Canada 2 Canadian Institute for Theoretical Astrophysics, Toront, M5S 3H8, Ontario, Canada. 3 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 4 College of Sciences, Northeastern University, Shenyang 110004, China 5 Center for High Energy Physics, Peking University, Beijing 100080, China In this paper, we compare brane inflation models with the Planck data and the prePlanck data (which combines WMAP , ACT, SPT, BAO and H 0 data). The Planck data prefer a spectral index less than unity at more than 5 σ confidence level, and a running of the spectral index at around 2 σ confidence level. We find that the KKLMMT model can survive at the level of 2 σ only if the parameter β (the conformal coupling between the Hubble parameter and the inflaton) is less than O (10 -3 ), which indicates a certain level of fine-tuning. The IR DBI model can provide a slightly larger negative running of spectral index and red tilt, but in order to be consistent with the non-Gaussianity constraints from Planck , its parameter also needs fine-tuning at some level.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The ongoing astronomical observations, such as WMAP [1], Planck [2-4], SDSS [6], ACT [7] and SPT [8], have been measuring the cosmic microwave background (CMB) and large scale structure to an unprecedented precision. This provides an excellent opportunity to probe the physics in the early Universe with the underlying fundamental theories. One of the leading candidates of generating initial fluctuations in the early Universe is inflation [9, 10]. The inflation paradigm offers a compelling explanation for many puzzles in the standard hot big-bang cosmology, such as the flatness problem, homogeneity problem and horizon problem [9]. The accelerated expansion period in the early Universe provides a nearly scale-invariant primordial power spectrum which has already been supported by the measurements of CMB anisotropy [15, 7, 8]. In spite of its phenomenological success, inflation remains a paradigm rather than a fundamental theory, which in principle can be implemented by various models from different microscopic physical constructions [11]. The fact that it is easy to construct a wide variety of inflation models does not mean that any of them will turn out to be the true mechanism. Actually, e ff ective field theory models of inflation should by definition be understood as valid only up to some energy scale that is low enough, and so the singularity problem and any 'trans-Planckian' e ff ects are out of the range of validity of the models [12, 13]. If one would like a UV completion to any e ff ective field theory ideas, one might hope that the string theory would provide such a way. Undoubtedly, inflation can be successfully realized in a string context. The string inflation model considered in this paper is the brane inflation scenario, proposed in [14, 15] originally, which o ff ers a class of observational signatures. In this scenario, the inflation is driven by the potential between the parallel dynamic brane and antibrane [16-18], and the distance between the branes in the extra compactified dimensions plays the role of the inflaton field. This inflation scenario can be realized via two viable mechanisms, namely, the slow-roll and DiracBorn-Infeld (DBI) inflations [11]. The original brane inflation model is the slow-roll inflation model [14, 16-18] where branes and antibranes are slowly moving towards each other in a flat potential. The KKLMMT model [19] provides such an example. In this model, the antibrane is fixed at the bottom of a warped throat, while the brane is mobile and experiences a small attractive force towards the antibrane [19, 20]. When the brane and antibrane collide and annihilate, the inflation ends and the hot big-bang epoch is initiated. The annihilation of the brane and antibrane makes the universe settle down to the string vacuum state that describes our Universe [19, 20]. For extensive studies on the KKLMMTmodel and other types of slow-roll brane inflation models, see Refs. [11, 20-27]. Another inflationary mechanism is the DBI inflation. In this paradigm, the speed of the rolling brane is not determined by the shape of the potential but by the speed limit of the warped spacetime [28-32]. The warped internal spaces naturally arise in the extra dimensions due to the stabilized string compactification. In order to test the inflationary paradigm and explore the dynamics of the internal space, we will scan the parameter spaces of these two types of inflation models subject only to the requirement that they provide enough e -folding number to solve the flatness, horizon and homogeneity problems. This is because solving the problems of standard cosmology is the basic motivation of the inflation paradigm and the most attractive feature of inflation models [11]. Then we will explore the observational signatures that are allowed by brane inflation dynamics and constrain the model parameters with the current observational CMB data. We will see that the current observational data are able to tighten up the parameter space of brane inflation to a great extent and the generic models need to be fine-tuned to match the current observations. Recently the Planck team just released the results from the 2 . 7 full-sky surveys [2]. For the Λ CDM model, Planck data combined with WMAP polarization data (hereafter Planck + WP) show that the index of the power spectrum satisfies [3, 4] ns = 0 . 9603 ± 0 . 0073 (1 σ CL) , at the pivot scale k 0 = 0 . 05 Mpc -1 , which rules out the exact scale invariance ( ns = 1) at more than 5 σ . If the running of spectral index α s = dns / d ln k is released as a free parameter, the spectral index becomes redder, while the running of the spectral index is not equal to zero at less than 2 σ CL, For a comparison, in [36], we combined the WMAP 9-year data [1] with ACT data [7], SPT data [8], as well as BAO data [6, 33, 34] and H 0 data [35] (hereafter, we call this combined data set the ' WMAP 9 + ' data set), and we obtained a red spectral index of power spectrum at the pivot scale k 0 = 0 . 002 Mpc -1 , But if we let the running of the spectral index be α s = dns / d ln k as a free parameter, the spectral index becomes and the running of the spectral index becomes In addition, the joint constraints on r (tensor-to-scalar ratio) and ns already become a very sensitive tool to constrain inflation models. In [1], it is found that inflation models with power-law potential φ 4 cannot provide a reasonable number of e -folds (between 50-60) in the restricted space of r -ns at around 2 σ level. Reference [36] pushes this limit further and shows that with the combination of WMAP 9, ACT, SPT, BAO and H 0 data, the inflation potential with power law form φ p can only survive if p is in the range of 0 . 9-2 . 1. Besides the above conventional parameters that have been used to constrain inflation models, Planck data are also able to constrain the non-Gaussianity of primordial fluctuations. The Planck found that the local, equilateral and orthogonal types of non-Gaussianity are These place very tight constraints on the inflation model space. Based on the WMAP 3-year and 5-year results, Refs. [23] and [20] investigated brane inflation models and showed that the KKLMMT model cannot fit WMAP + SDSS data at the level of one standard deviation and a fine-tuning (at least one part in a hundred) is needed at the level of two standard deviations. Since the CMB data have been dramatically improved over the past several years, it is meaningful to see how the status of brane inflation is a ff ected by the arrival of the new CMB data, especially the Planck and WMAP 9 + data. In this paper, we will have a close look at the constraints on the brane inflation models with the results from Planck [2-4] and prePlanck surveys [1, 6-8]. This paper is organized as follows: In Sec. II, we discuss the relationship between the e -folding number of inflation and the pivot scale of observation. In Sec. III, we discuss a simple brane inflation model neglecting the problem of dynamic stabilization. This is the simplest brane inflation model one can achieve in the multidimensional spacetime. In Sec. IV, we focus on the KKLMMT model and compare the model predictions with observational data. In Sec. V, we turn to the discussion of the infrared DBI inflation model and confront the model predictions with observational data. The conclusion is presented in the last section.", "pages": [ 1, 2 ] }, { "title": "II. NUMBEROF E -FOLDS", "content": "Before we start to constrain any inflation model, we first address an important issue in the inflation model tests: how do we compare model predictions with observational data? Inflation models are actually models of di ff erent inflation potentials, where the amplitude and shape are the features of various models. In the community of inflation theorists, people use the amplitude of potential to characterize the energy scale of inflation and a set of 'slow-roll' parameters to describe the shape of the potential. Usually the shape of the potential includes the 'slope' and 'curvature' parameters of the potential. For a given potential, the slow-roll parameters can be expressed in terms of the number of e -folds ( Ne ) which characterizes the duration of inflation. On the other hand, observations from the CMB provide constraints on the amplitude and shape of the primordial power spectrum. But since the power spectrum itself is a k -dependent quantity, the measured amplitude ( ∆ 2 R ), tilt ( ns ), tensor-to-scalar ratio ( r ) and running of spectral index ( dns / d ln k ) are referred to a particular 'pivot scale'. This indicates that for a given data set, if the pivot scale is switched to a di ff erent value, the constraints on the ( ∆ 2 R ( k 0), ns ( k 0), dns / d ln k ) can be slightly di ff erent. Therefore, to really compare model predictions with observational data, we need to associate the number of e -folds with the pivot scale of observation. Our main goal in this section is to obtain a relationship between the number of e -folds Ne and its corresponding comoving scale k . Once inflation happened, di ff erent scales (di ff erent k -modes) stretched out of the Hubble radius at di ff erent time. After inflation, the Universe experienced a short period of reheating, and then entered into radiation, matter and dark energy dominated eras. The number of e -folds is related to the processes of subsequent evolution because both the inflation and subsequent evolutionary processes contribute to the total expansion factor of the Universe (see Fig. 1 in [37]). We can therefore write [37] where we used the subscripts 'k, e, reh, eq, 0' to represent the horizon exit, end of inflation, reheating epoch, matterradiation equality epoch and present time. Number of e -folds between horizon exit and the end of inflation is Ne ( k ) = ln( ae / a k). By assuming the equation of state during the reheating era being w ( w = P /ρ ), one can reach the following equation (see also [37, 41]), where V k is the energy scale of inflation at horizon exit, and M pl ≡ 1 / (8 π G ) /similarequal 2 . 4 × 10 18 GeV is the reduced Planck mass. By defining the ratio of the energy densities between at the reheating and at the end of inflation as x ≡ ρ reh /ρ e, and regarding ρ e = V k ('slow-roll' approximation), we can rewrite Eq. (9) as To further simplify this equation, we use the requirement that the primordial perturbations have to produce the observed level of fluctuations ( Ps ( k 0) /similarequal 2 . 43 × 10 -9 ), i.e., Substituting known quantities, Eq. (10) can be greatly simplified as For a particular mode k , its corresponding Ne ( k ) relies on the equation of state w and energy scale of reheating ρ reh. Since the standard picture tells that vacuum is decayed into standard particles, ρ reh is always less than or equal to potential energy scale Vk , i.e. x ≤ 1, thus ln x is always a negative value. Therefore, if w → 0 (close to a 'matter-dominated phase'), the fourth term of Eq. (12) becomes (1 / 12) ln x , which gives a minimal number of e -folds. This means that if the equation of state is close to zero, the shape of the potential ( ∼ φ 2 ) can keep inflaton oscillating for a fairly long period of time while the Universe is expanding, therefore we need less number of e -folds to produce an observable scale of the Universe. On the other hand, if the equation of state during the reheating era is w /similarequal 1 / 3, or the reheating is instantaneous ( ρ reh = V k, i.e., ln x = 0), the fourth term vanishes, which gives the maximum number of e -folds ( ∼ φ 4 ). Since there is a great uncertainty of what energy scale reheating really happened, in the following discussion we stick to the case of instantaneous reheating, so that the number of e -folds becomes For joint WMAP 9 + SPT + ACT + BAO + H 0 data ( k 0 = 0 . 002Mpc -1 ) and Planck + WP data ( k 0 = 0 . 05Mpc -1 ), the corresponding numbers of e -folds are Typically observational predictions of slow-roll parameters (e.g. /epsilon1 v ) depend on Ne , so both sides of Eq. (14) contain Ne which could be solved simultaneously. In practice, the deviation of Ne from the typical value 60 is always small, so one can solve Eq. (14) iteratively by assuming a particular Ne and use it to calculate the potential properties, then use these to recalculate Ne , and so on. In fact, one iteration easily su ffi ces to give su ffi cient accuracy of Ne . We will illustrate this in the following sections.", "pages": [ 2, 3 ] }, { "title": "A. Model predictions", "content": "In this section, we discuss another important type of brane inflation model, namely, the infrared Dirac-Born-Infeld model (IR DBI model). The di ff erence between this model and the KKLMMTmodelisthat the rolling velocity of the brane is not determined by the shape of the potential but by the speed limit of the warped spacetime [11]. Such a warped spacetime can always emerge in the inner space of compactified Calabi-Yau manifold. s Phenomenologically, the inflaton in IR DBI model can be driven by the kinetic term, where the inflaton is not slowly rolling at all. Therefore, the sound speed of inflaton in such a model could be less than unity, providing a large tilt in the tensor power spectrum (remember nt = -r / (8 cs ) [20, 42]). Observation on large scale temperature and polarization can be used to pin down the uncertainty of the soundspeed. In addition, as shown in previous analyses [11], there are a lot of parameters that describe the structure of internal space, and we will show that some of them may be pinned down by the CMB observations (see also [4] for more detailed discussions on constraints from non-Gaussianity). In the DBI inflation, the action takes the form where V ( φ ) is the potential, X is the kinetic term, and f ( φ ) is the warp factor. For the IR DBI model, the inflaton potential is where the parameter β is in principle within a wide range 0 . 1 < β < 10 9 [11]. The scalar power spectrum of DBI inflation can be parametrized as [11] where As is the amplitude of the perturbations which depends on several parameters of the internal space, Nc is the critical number of e -folds at scale kc (critical scale where string phase transition happens), and N DBI e is the number of e -folds of inflation at relativistic rolling. The total number of e -folds is the sum of relativistic and nonrelativistic (NR) rollings, Now we can calculate the spectral index and its running, which turns out to be (see also Appendix in [11]) where x = ( N DBI e / Nc ) 8 . In addition, nontrivial sound speed cs can generate large non-Gaussianity since the inflaton is no longer slowly rolling down to the potential. The predicted equilateral and orthogonal non-Gaussianities are [43, 44] where", "pages": [ 6, 7, 8 ] }, { "title": "B. Constraints from Planck and prePlanck data", "content": "In Fig. 1, we plot the theoretical prediction of r (in terms of log 10 r ) and ns . The black and blue lines are the predictions for n = 2 and n = 4 models with µ/ M pl values being 0 . 1 and 0 . 01. The range between small and big dots corresponds to the number of e -folds within [50 , 60]. The red line across the diagram is the boundary line between the convex potential ( η v > 0) and concave potential ( η v < 0). We also plot the 1 σ and 2 σ constraints on r and ns from WMAP 9 + data and Planck + WP + BAO (hereafter Planck + ) data. From the plot, one can see that WMAP 9 + prefers a slightly lower ns comparing with Planck + data. In addition, most of the contour regions locate within η v < 0 region, indicating strong evidence of concave potential. The models with n = 2 and n = 4 lying within the contours suggests that the model prediction is consistent with the current constraints. We also plot the predicted detection limits of r from Planck polarization experiment [38] and CMBPol [39]; note that these two limits are not from actual data, but are based on the predictions of future data. One can see that even if µ/ M pl is of order 0 . 1, the model prediction is still much lower than the CMBPol detection limit. Only if µ/ M pl > 0 . 3, could the CMBPol be able to detect the tensor mode in this model. In Fig. 2 we plot the predicted α s -ns relation for the brane inflation model with the constraint results from WMAP 9 + and Planck + data. The purple contours on the left panel is the joint constraints on α s -ns from the WMAP 9 + data set with the pivot scale k 0 = 0 . 002 Mpc -1 . Therefore we use Eq. (14) to determine the number of e -folds: we substitute a fiducial number of e -folds N fid = 60 into Eq. (17) and obtain an estimate of /epsilon1 , then substitute it into Eq. (14) to obtain the corresponding number of e -folds for this model. We test that one iteration is enough for determine the specific Ne . Then with Eqs. (17)(20) we plot the α s -ns prediction with variation of the parameter µ . The red line is for the n = 2 model and the blue line is for the n = 4 model. The two lines are pretty close to each other, and they are all outside 1 σ confidence level (CL) but some range is within 2 σ CL. We then figure out which values of µ can match the results inside the 2 σ . We give a couple of trials and find that, for the n = 2 model µ/ M pl needs to be between 10 -48 and unity, and for the n = 4 model this range is [10 -30 , 1]. On the right panel of Fig. 2, we use the constraints from Planck + WP + BAO to compare with the theoretical predictions. The results are similar to the left panel, except that the range of µ/ M pl is shorten to be [10 -46 , 1] for the n = 2 model, and [10 -29 , 1] for the n = 4 model. In a word, the prototype of brane inflation with potential form (15) is consistent with the observational constraints on α s and ns . Then let us see what this implies for the energy scale of inflation in this model. The amplitude of the scalar perturbations is [21, 23, 41] which is constrained to be ∼ 2 . 2 × 10 -9 by the Planck data [2]. We substitute /epsilon1 v [Eq. (17)] into Eq. (21), and thus we obtain a relationship between the amplitude of inflation and the parameter µ ,   Then from our estimation of µ we can find that the amplitude of inflation is in the range [2 . 7 × 10 4 , 8 . 4 × 10 15 ] GeV for the n = 2 model and [1 . 3 × 10 6 , 5 . 9 × 10 15 ] GeV for the n = 4 model. These are all reasonable ranges for V , because it needs to be lower than 10 16 GeV so that we do not detect any tensor mode yet, and greater than the particle physics energy scale 10 3 GeV since the inflaton is not detected in LHC.", "pages": [ 4, 5 ] }, { "title": "IV. KKLMMTMODEL", "content": "The prototype of the brane inflation model discussed above is not a realistic model, because the distance between the brane and the antibrane would be larger than the size of the extra-dimensional space if the inflaton is slowly rolling in this scenario [20, 23]. It indicates that this model is not really reliable from the viewpoint of theory itself. The first more realistic brane inflation model which considers the effect of warped spacetime on inflaton potential is the so-called KKLMMT model [19], whose predictions are directly calculable and can be directly compared to observations. Note that, strictly speaking, the KKLMMT model is only a braneinflation-inspired model rather than a scenario with all elements of the potential computed precisely; for more complicated versions of brane inflation, see [26, 27].", "pages": [ 5 ] }, { "title": "B. Constraints from observational data", "content": "In Fig. 3, we plot the r -ns diagram similar to the structure of Fig. 1. Instead, here it is the KKLMMT model. The black solid and black dashed lines represent the trajectories for Ne = 50 and 60, respectively. Di ff erent colors of empty and filled circles mark the point where the model takes di ff erent β values. One can see how the β parameter controls the shape of the potential. If it is greater than 0 . 03, the potential turns to be convex which is not preferred by current observational data. Actually, the problem for β > 0 . 01 is that it provides a blue tilt which has already been ruled out by Planck + WP + BAO at more than 5 σ CL. In order for the model to pass this test, β value has to be much smaller than 10 -3 . In fact, since the current Planck data prefer the ns value around 0 . 96 (green contours), the models with β < 10 -3 are just about to survive since they o ff er the spectral index to be 0 . 96 but not smaller than 0 . 95 (see Fig. 4 as well). This means that as long as the CMB data prefer ns to be around 0 . 96, this model can always pass this test and survive. Nevertheless, the parameter needs to be highly fine-tuned. Finally, similar to Fig. 1, one can see that the tensor mode predicted by the KKLMMT model is really undetectably small since it is several orders of magnitude lower than the Planck polarization [38] and CMBPol limits [20, 39]. In Fig. 4 we show the comparison of the observational constraints and the model predictions on the α s -ns plane. One can see that Planck + WP + BAO prefers a slightly negative running with a very red power tilt. The tilt of the power spectrum at more than 5 σ deviates from unity (the Harrison-Zel'dovich spectrum), while the running is at less than 2 σ away from zero. On the other hand, if α s is released as a free parameter, the WMAP 9 + data set cannot tighten up ns to be less than unity. The purple contours stretch from a small negative running ( ∼ -0 . 01) with red tilt ( ∼ 0 . 96) out to a large negative running ( ∼ -0 . 04) with blue tilt ( ∼ 1 . 05) region. However, constraints from these two di ff erent data sets overlap at the small negative running and red tilt region, indicating that this is the preferable region for both data sets. In addition, we plot the model predictions for di ff erent β values, and we mark the region of model predictions in between Ne = 50 and 60 in order to have a direct vision of whether this 'physically plausible' region falls in the observational constraint contours. One can also see that the KKLMMT model cannot produce a red tilt and suitable level of negative running unless β ≤ 10 -3 at 2 σ CL. The model with β = 0 . 01 cannot fit the 2 σ joint constraints in either case. This is actually an order of magnitude tighter than the previous upper limit of β from WMAP 5-year data [20] ( β < 0 . 01 at 2 σ CL), and also much tighter than the combined constraints ( β < 6 × 10 -3 ) from WMAP 3 + SDSS [23].", "pages": [ 6 ] }, { "title": "B. Confront with current data", "content": "Since the IR DBI model has a lot of parameters that describe the internal structure of the warped space, in order to directly compare its predictions with the current observational data, we adopt the best-fit values of Nc , kc and N NR e to be 35 . 7, 10 -4 . 15 Mpc -1 and 18 . 4, respectively, according to the constraints from WMAP 5-year data [11]. In Fig. 5 we plot the predicted trajectory of the IR DBI model in the α s -ns plane. The purple and orange contours are the results from WMAP 9 + and Planck + WP + BAO as we discussed before. The red line is the trajectory corresponding to N tot e between 45 and 60, which includes a wide range of scale k . One can see that the trajectory crosses the contours of both WMAP 9 + and Planck + WP + BAO, which is quite consistent with the data. In addition, the model predictions at the two pivot scales k 0 = 0 . 002Mpc -1 and k 0 = 0 . 038Mpc -1 , which are the chosen scales of the two constraints are marked on the plot. One can see that the black dot is close to the boundary of WMAP 9 + constraints while the blue one is outside of the 2 σ contours from Planck . However, although it seems that there is a discrepancy, we remind the reader that there is some uncertainty of the subsequent evolution after inflation, so it is reasonable to allow a broader range of number of e -folds for a given pivot scale. Non-Gaussianity becomes an important tool to constrain such a non-slow-roll inflation model. The local, equilateral and orthogonal f NL parameters are given by Eq. (7), which still do not show strong signal for non-Gaussianity. However, the error-bars of local, equilateral (and orthogonal) f NL become a factor of two and four smaller than WMAP 9-year data [45]. Since the IR DBI model predicts vanishing local f NL, it is already consistent with the value given by Planck . Now we investigate the predictions of equilateral and orthogonal types of non-Gaussianity. In Fig. 6, we plot the model predictions of f eq NL and f orth NL and the current lower and upper bounds. The yellow bands in both panels are the allowed region for N tot e in between 45 and 60. Note that in Planck paper XXII [4], Ne is just allowed to be 60-90 when considering the constraints on the IR DBI model, while here we consider a more reasonable range of the number of e -folds. Given the yellow bands and the 2 σ lower bound for equilateral type of non-Gaussianity, we find that the value of β needs to be smaller than 1 . 5 in order to prevent very negative equilateral non-Gaussianity. Similarly, on the right panel, we show that β needs to be less than 2 . 5 in order to prevent large positive non-Gaussianity. These limits are consistent with the range of β < 0 . 7 as found by Planck paper XXIV [5], which uses global likelihood analysis to obtain the limit. We should notice that this is already a fine-tuning for IR DBI model, because in this model β has the lower limit 0 . 1 ( β < 0 . 01 is KKLMMT model as discussed in Sec. IV) but no real upper limit. Therefore, the current data is able to shrink the parameter space to be [0 . 1 , O (1)] is already a tight limit. Our comparison gives a intuitive understanding of why the parameter β needs to be smaller than a certain value.", "pages": [ 8, 9 ] }, { "title": "VI. CONCLUSION", "content": "In this paper, we studied brane inflation with the Planck data and the joint data set from WMAP 9-year data, SPT, ACT, BAO and H 0 data. We first discussed the relationship between the number of e -folds and the corresponding pivot scale. We clarified the case where adopting di ff erent pivot scales of the constraints, the corresponding number of e -folds could be slightly di ff erent. We then considered a toy model (prototype) of brane inflation where the problem of dynamic stabilization is neglected. Furthermore, we considered a more realistic 'slow-roll' brane inflation model (namely, the KKLMMT model) and the DBI inflation model, and examined them with the Planck and WMAP 9 + results. For the toy model, we showed that the model is consistent with the observational data at 2 σ CL, given the fact that it prefers a red tilt close to 0 . 96 and a slightly negative running. For a comparison, in our previous work [20], we found that this type of brane inflation model is consistent with the WMAP 5-year data at the level of 1 σ . The situation does not change very much when we confront the model with WMAP 9 + data and Planck data. For the KKLMMT model, we first discussed how the model parameter β a ff ects its predictions of scalar power spectrum. Then we compared the model to the WMAP 9 + data and Planck data. We found that in order for the model to provide the α s and ns allowed by the tight constraints from Planck and WMAP 9 + , the β parameter needs to be fine-tuned to be less than 10 -3 . For comparison, by using the WMAP 3-year data in [23], we found that the KKLMMT model cannot fit WMAP 3 + SDSS data at the level of 1 σ and a fine-tuning, at least eight parts in a thousand, is needed at the level of 2 σ . When the WMAP 5-year data becomes available, we found that the value of the parameter β is constrained to be less than O (10 -2 ) at the level of 2 σ [20]. Thus, we can see that the problem of fine-tuning of β becomes more severe when confronting with the recent observational data. Undoubtedly, this is not good news for the KKLMMT model. Finally, we briefly discussed the current constraints on the infrared Dirac-Born-Infeld inflation model given the current observational data. The model can predict a larger negative running ( ∼ -0 . 02) than the previous KKLMMT model. By figuring out the trajectory of the model on the α s -ns plane by varying the number of e -folds, we found that the model can predict the running of the spectral index and the tilt that are consistent with WMAP 9 + and Planck data. However, when we confronted it with the current bounds on equilateral and orthogonal non-Gaussianities, we found that in order to avoid a large non-Gaussianity the value of β which controls the shape of the potential needs to be less than 1 . 5. This limit to the IR DBI model is already a fine-tuning. To summarize, although the prototype of brane inflation can fit the data well, it is not a realistic model of the brane inflation. For the KKLMMT and IR DBI inflation models, the parameters need to be fine-tuned to satisfy the current observational requirement. The current observation of CMB from Planck is competent to place stringent limits on internal parameters of warped space.", "pages": [ 9 ] }, { "title": "Acknowledgments", "content": "We would like to thank Anthony Challinor, Xingang Chen, Gary Hinshaw and Andrew Liddle for useful discussions. Y.Z.M. is supported by a CITA National Fellowship. Part of the research is supported by the Natural Science and Engineering Research Council of Canada. Q.G.H. is supported by the Knowledge Innovation Program of the Chinese Academy of Science and by the National Natural Science Foundation of China (Grant No. 10821504). X.Z. is supported by the National Natural Science Foundation of China (Grants No. 10705041, No. 10975032 and No. 11175042) and by the National Ministry of Education of China (Grants No. NCET09-0276, No. N100505001 and No. N120505003).", "pages": [ 9 ] } ]

[ { "title": "Constraints on Neutrino Mass from Sunyaev-Zeldovich Cluster Surveys", "content": "Daisy S. Y. Mak 1, ∗ and Elena Pierpaoli 1, † 1 Physics and Astronomy Department, University of Southern California, Los Angeles, California 90089-0484, USA The presence of massive neutrinos has a characteristic impact on the growth of large scale structures such as galaxy clusters. We forecast on the capability of the number count and power spectrum measured from the ongoing and future Sunyaev-Zeldovich (SZ) cluster surveys, combined with cosmic microwave background (CMB) observation to constrain the total neutrino mass M ν in a flat ΛCDM cosmology. We adopt self-calibration for the mass-observable scaling relation, and evaluate constraints for the South Pole Telescope normal and with polarization (SPT, SPTPol), Planck, and Atacama Cosmology Telescope Polarization (ACTPol) surveys. We find that a sample of ≈ 1000 clusters obtained from the Planck cluster survey plus extra information from CMB lensing extraction could tighten the current upper bound on the sum of neutrino masses to σ M ν = 0 . 17 eV at 68% C.L. Our analysis shows that cluster number counts and power spectrum provide complementary constraints and as a result they help reducing the error bars on M ν by a factor of 4 -8 when both probes are combined. We also show that the main strength of cluster measurements in constraining M ν is when good control of cluster systematics is available. When applying a weak prior on the mass-observable relations, which can be at reach in the upcoming cluster surveys, we obtain σ M ν = 0 . 48 eV using cluster only probes and, more interestingly, σ M ν = 0 . 08 eV using cluster + CMB which corresponds to a S/N ≈ 4 detection for M ν ≥ 0 . 3 eV. We analyze and discuss the degeneracies of M ν with other parameters and investigate the sensitivity of neutrino mass constraints with various surveys specifications.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Measuring masses of neutrinos is one the major goals of particle physics and cosmology. While atmospheric and solar neutrino oscillation experiments are sensitive to neutrino flavor, mixing angle, and the mass difference among different species, cosmological data are instead more sensitive to the absolute mass scale M ν = ∑ m ν . In fact, the most stringent upper bound of the total neutrino mass is coming from CMB and large scale structures since massive neutrinos leave detectable imprints throughout the history of the universe. Most recently, [1] obtained M ν < 0 . 23 eV at 95% C.L. by combining CMB data and BAO from Sloan Digital Sky Survey (SDSS) In this work, we explore the prospects of employing ongoing and future galaxy cluster surveys detected by the SZ effect in constraining neutrino masses. Galaxy clusters are in principle a powerful tool for probing neutrino properties. Neutrino becomes nonrelativistic after the epoch of decoupling if its mass scale is smaller than O (0 . 1) eV. The relativistic behavior of neutrinos, as opposed to cold dark matter, causes the suppression of matter perturbations on small scales with respect to the case in which neutrinos are massless and all dark matter is cold. The presence of massive neutrinos affects the growth rate of perturbations in the linear regime, and, as a consequence, the shape of the matter power spectrum and cluster abundance. Current measurements from X-ray cluster surveys obtained a tight upper limit of M ν < 0 . 33 eV [2] by combining measurements of Chan- ra X-ray observations of galaxy clusters, CMB from WMAP 5 year data, BAO and type 1A supernova (both from HST Key Project). Similarly, measurement from galaxy power spectrum from the SDSS-III BAO survey + CMB + SN found M ν < 0 . 34 eV [3]. Subsequently, several works were dedicated to discuss the prospects of utilizing large future surveys of large scale structures (galaxy or galaxy clusters) in different wavelengths (e.g. [4-11]). These works showed that constraints of neutrino mass depend on assumptions of the underlying cosmology (e.g. inclusion of dark energy or flatness), cluster physics, and the use of external priors (e.g. CMB lensing extraction). Here we revisit the analysis to forecast the constraint of the total neutrino mass, in the framework of flat ΛCDM universe and, like past constraints, the standard scenario with only three neutrino species. We use cluster abundance and power spectrum as the observables that will be obtained from various SZ cluster surveys: the Planck, ACTPol, SPT, and SPTPol cluster surveys. These surveys are very promising and, in the next couple of years, will provide large samples of mass selected clusters out to high redshift. With respect to previous works [e.g 9, 10] which also employ SZ cluster surveys, we provide a more realistic survey specifications to characterize the cluster detection and include the self-calibration to characterize the uncertainties of the mass-observable relations. We also discuss the degeneracy of the neutrino mass with dark energy, which is lacking in previous studies, and compare the strength of cluster probes with CMB on the constraining power of neutrino mass. The paper is organized as follows. In Sec. II we discuss the effects of M ν on the large scale structures. In Sec. III we present the methodology which includes the description of the future SZ cluster samples and the Fisher ma- rix formalism. The main results are presented Sec. IV and discussed in Sec. V. Finally, a conclusion is presented in Sec. VI.", "pages": [ 1, 2 ] }, { "title": "II. IMPACT OF NEUTRINO MASSES ON GROWTH OF THE LARGE SCALE STRUCTURES", "content": "The presence of massive neutrinos mildly affects expansion history but significantly impacts the growth of structure through free-streaming. Fluctuations on comoving scales that enter the horizon when neutrinos are still relativistic may be reduced in amplitude because neutrinos would tend to leave the perturbation. This effect, that is neutrino mass dependent, typically occurs on length scales below the free-streaming scale: l fs = 1 /k fs = 1 / (1 . 5 √ Ω M h 2 / (1 + z ))(eV /M ν )Mpc. Thus, the growth of any structure that have scale smaller than l fs will be less efficient. A smaller neutrino mass increases the free-streaming scale, but also reduces the neutrino fraction with respect the total amount of dark matter, mitigating the overall suppression. As a result of these dependences measurements of the large scale structures such as cluster number counts and power spectrum can be used to place constraints on neutrino masses. The late-time evolution of perturbations in a ΛCDM cosmology with massive neutrinos can be accurately described by the product of a scale dependent growth function and a time dependent transfer function. For example, [12] derived a reasonable approximation to the analytical expression of the transfer function for small scales. In this work, we employ the transfer function determined numerically from CAMB [13] which provides precise estimate on the matter power spectrum and include non-linear effects at largek limit in which the analytical expressions fail to give an accurate estimate.", "pages": [ 2 ] }, { "title": "III. ANALYSIS", "content": "Our analysis closely follows the treatment of [14], here we only outline the method and refer the readers to [14] for details. For cluster abundance and clustering, we use the results of numerical simulations from [15] for the cluster mass function n ( M,z ), and [16] for the halo bias.", "pages": [ 2 ] }, { "title": "A. Cluster survey", "content": "We consider four upcoming SZ cluster surveys studied in [14]: the Planck survey, the South Pole Telescope normal and polarization survey (SPT and SPTPol respectively), the Atacama Cosmology Telescope polarization survey (ACTPol). Each of these surveys has different specifications for the selection threshold, i.e. M lim ( z ), and their properties are summarized in Tab. I. Briefly, for the Planck survey we adopt a flux limit of Y 200 ,ρ c ≥ 2 × 10 -3 arcmin 2 [17], where Y 200 ,ρ c is the integrated comptonization parameter within the radius enclosing a mean density of 200 times the critical density. This corresponds to a 5 σ detection threshold and would yield ∼ 1000 clusters. For the SPT survey, i.e. single frequency at 150 GHz, we employ the calibrated selection function of the survey by [18] and adopt a detection threshold at 5 σ . This yields ∼ 500 clusters . The SPTPol has an increased sensitivity at 150 GHz than the normal survey and we account for this, following previous work, by scaling the mass limits by a factor of 3.01/5.95. The expected number of clusters is ∼ 1000. For the ACTPol survey, we include clusters with M 200 , ˆ ρ c > 5 × 10 14 M /circledot h -1 (Sehgal 2011, private communication) which corresponds to a 90% completeness. This straight mass cut result in ∼ 500 clusters. We construct cluster sample for the Planck survey in the redshift range 0 < z < 1. We impose a lower cut z cut = 0 . 15 for the SPT, SPTPol, and ACTPol survey. Currently, the SPT team is setting a low redshift cut at z cut = 0 . 3 in their released cluster sample, due to difficulties in reliably distinguishing low-redshift clusters from CMB fluctuations in single frequency observations. Nevertheless, with upcoming multi-frequency observations, a lower cut z cut = 0 . 15 will likely be attained. We therefore apply this cut in our work.", "pages": [ 2 ] }, { "title": "B. Fisher matrix forecasts and cosmological parameters", "content": "We estimate the constraints on cosmological parameters by applying the Fisher matrix formalism to future SZ cluster surveys. This approach can best approximate the likelihood when the fiducial model is close to the true, underlying model and the likelihood is close to gaussian. Typically, the gaussian approximation is more accurate, and the use of the Fisher matrix better justified, when the likelihood is peaked and the parameter in hand has little degeneracies with other parameters. In order to achieve this goal, the use of external priors can be beneficial. For example, [19] noted that the CMB power spectra likelihood function for the neutrino mass differs from the gaussian case due to strong parameter degeneracies, particularly for models with many parameters. These authors suggested the use of CMB lensing extraction information in order to sharpen the likelihood and make it better approximated by Gaussian. We will adopt the same strategy, as described below. The Fisher matrix for the cluster number counts and power spectrum is described in detail in [14]. Similarly, for our main results, we here consider self-calibration to account for the uncertainties of the observed cluster mass. We add the Planck CMB lensing extraction (LE) that is considered to be a very promising way to constrain neutrino mass (e.g. [12, 20]). The CMB anisotropies obey Gaussian statistics in the absence of weak lensing, and therefore they are fully described by the temperature and polarization power spectrum. Weak lensing, however, introduces non-gaussianity in both the temperature and polarization anisotropies [21, 22]. Therefore, extracting the lensing information from CMB (e.g. using quadratic estimators [23-26]) would provide the lensing potential and delensed CMB anisotropies, and hence extra information to the Fisher matrix. In the following, we refer to the Fisher matrix results obtained from CMB lensing extraction as the CMB LE. As shown in [27], CMB LE is useful in providing strong neutrino mass constraints and potentially breaking of the major neutrino mass degeneracies with other parameters [19]. While very promising, the exploitation of higher order statistics may suffer from subtle ways from the effect of galactic and extragalactic contaminants. For this reason, we also consider constraints coming from the CMB power spectrum only (with lensing) when combining probes with cluster' s ones. We note that the latest Planck results were released during the preparation of this work. They derive a tight upper limit of M ν ≤ 0 . 93 eV when using CMB data alone and M ν ≤ 0 . 23 eV when further combined with BAO data. Nevertheless, these limits use information from polarization of the WMAP data and not from the Planck data itself (the Planck CMB polarization data will be employed in the next data release). Therefore instead of using these numbers as priors on M ν constraints, we derive our Planck CMB prior that takes into account the Planck polarization information which is believed to be better than that from WMAP . Thus this prior should be considered as the self-contained and improved one than the current constraint in [1]. We adopt a spatially flat ΛCDM model as the fiducial model. The set of parameters included in our analysis is (Ω b h 2 , Ω M h 2 , Ω Λ , M ν , n s , σ 8 , w 0 , w a ). The fiducial values are adopted from the best fit flat ΛCDM model from WMAP 7yr data, BAO and H 0 measurements [28]: Ω b h 2 = 0 . 0245, Ω M h 2 = 0 . 143, Ω Λ = 1 -Ω M = 0 . 73, M ν = 0 . 3 eV, n s = 0 . 963, σ 8 = 0 . 809, w 0 = -1, w a = 0. As proposed in [29, 30], we can use cluster surveys to constrain the mass observable relation by considering self-calibration, hence taking into account the systematic errors of the SZ surveys due to uncertainties in observed cluster mass. In this work, we follow [31] to introduce four nuisance parameters, B M 0 , α , σ ln M, 0 , β , that spec- the magnitude and redshift dependence of the fractional mass bias B M ( z ) = B M, 0 (1 + z ) α and the intrinsic scatter σ ln M ( z ) = σ ln M, 0 (1 + z ) β . We adopt fiducial values of B M 0 = 0, α = 0, σ ln M, 0 = 0 . 1, β = 0, hence corresponding to zero mass bias and 10% intrinsic scatter. In deriving the main results, we will not make any assumption on the four nuisance parameters and leave them free to vary. We discuss the impact of this assumption in Sec. IV C.", "pages": [ 2, 3 ] }, { "title": "A. Cluster number count and power spectrum", "content": "Tab. II summarizes the neutrino mass constraints from the Fisher matrix analysis for Planck CMB (with and without LE), cluster number counts, and power spectrum for the four cluster surveys. Constraints of M ν from cluster number counts alone are better than power spectrum ones, however, each of them is very weak when considered separately, with σ M ν > 4 eV. When combining information from both probes, the constraints are improved significantly by a factor of 4 -8. The best case is obtained from the Planck cluster survey with σ M ν = 0 . 94 eV, whereas the constraints from other surveys are a factor of two worse.", "pages": [ 3 ] }, { "title": "B. Cluster probes + CMB", "content": "Adding the Planck CMB priors breaks degeneracies (see Sec. V A) and improves the constraints (number count or power spectrum alone) further by a factor of > 4 (without LE) and > 5 (with LE). When including all the information but LE, i.e. count + power spectrum + CMB, we find the best constraint comes from the Planck and ACTPol cluster survey with σ M ν = 0 . 23 eV. This is 80% better than that obtained from Planck CMB alone ( σ M ν = 0 . 41 eV). Including CMB priors also shrinks the difference in σ M ν among different surveys in which it is now σ M ν = 0 . 23 -0 . 30 eV. Similar results are obtained when we add the CMB LE and the best constraint is σ M ν = 0 . 17 eV. This suggests that the improvements in σ M ν are mainly driven by CMB information. We note that a perfect cleaning of all the astrophysical foregrounds is assumed when computing the CMB Fisher matrix in this work. Foreground contamination dominates at small angular scales (e.g. l ≥ 1000) and would introduce extra non-gaussianity and spoil the lensing extraction process [32]. Nevertheless, [27] found that the effect of no foreground subtraction in Planck CMB (with and without LE) only degrades the M ν constraint marginally (by 9%). Therefore, our results that involve CMB information can be considered to be robust against foreground contamination. We repeat the analysis with a fiducial M ν = 0 . 1 eV instead to investigate the effect on the constraint with less massive neutrinos. The results are very close (within 15%) to those for M ν = 0 . 3 eV when using cluster probes only, and are almost the identical when CMB priors are added.", "pages": [ 3, 4 ] }, { "title": "C. Self-calibration and uncertainty of nuisance parameters", "content": "The dominant systematic errors for SZ derived constraints are the uncertainties in the mass observable relation due to structure and evolution of clusters. We can ask how much could be gained by eliminating such uncertainties. For example, we can expect some external constraints on the nuisance parameters by using detailed studies of individual clusters or combining different information from optical, weak lensing, X-ray and SZ measurements. To estimate the effect of self-calibration of systematic uncertainties on the neutrino mass constraints, we repeat the forecasts with different priors on the four nuisance parameters as summarized in Tab. III. We first discuss the results when applying a 'weak' prior, i.e. using current knowledge on the calibration on the mass proxies ∆ σ M, 0 = 0 . 1, ∆ β = 1, ∆ B M , 0 = 0 . 05, ∆ α = 1. In the case of cluster count + power spectrum, the 1 σ error reduces marginally for SPT and SPTPol, but significantly (by a factor of two) for Planck and ACTPol. This results in σ M ν = 0 . 48 eV for the Planck cluster survey which is competitive with the CMB only constraint. In the case of adding the CMB (with and without LE) priors, the 1 σ errors generally reduce by a factor of two and resulted in, for the best case as obtained by the Planck survey, σ M ν = 0 . 08 eV, which corresponds to a S/N ≈ 4 detection for M ν ≥ 0 . 3 eV Similar results are obtained when applying a 'strong prior', i.e. the four nuisance parameters are held fixed at their fiducial values, which is equivalent to assuming a perfect knowledge of cluster true masses. The constraints are improved significantly by 66 -236% in the case of cluster count + power spectrum, and a factor 2 -3 when the CMB priors are further added. The best constraint is, again with the Planck cluster survey, σ M ν = 0 . 07 eV which is a relative marginal improvement with respect to the weak prior case. While it is unrealistic to have perfect knowledge on the mass observable relations, one can achieve similar scenario by restricting the analysis to a relatively small subset of clusters for which follow up observations are available. This would ensure a sample with well calibrated mass proxies. For example, it has been shown in [2] that the ability to constrain dark energy parameters from a small sample of ≈ 50 well calibrated X-ray clusters is comparable to a larger sample of ≈ 10000 optical clusters (e.g. SDSS [33]). Unlike other parameter constraints (e.g. nonGaussianity with galaxy clusters [14]), the results of the weak prior are sufficiently close to the those from the strong prior. The prospect of achieving the weak prior conditions is promising, e.g. clusters detected in weak lensing measurements or a subsample of objects having extensive multi-wavelength follow-up. Therefore the cluster probes are good enough to provide interesting M ν constraint even without perfect knowledge of the scaling relations. As a final remark, we would like to compare our count + CMB result with [10] which similarly presented M ν constraints assuming perfect knowledge of cluster mass and used Planck cluster count + CMB. Our result ( σ M ν = 0 . 17 eV) is a factor of 2.8 worse than that obtained in [10]. We note that the discrepancy is due to the different assumption on the total number counts: ≈ 6000 in [10] and ≈ 1000 in this work for the Planck survey if a 5 σ survey detection limit is assumed. Our estimate is based on the conservative assumption that ensures high level of completeness (90%) and realistic mass limits that vary at different redshifts, while [10] assumed a constant and lower mass threshold.", "pages": [ 4 ] }, { "title": "A. Parameter Degeneracies", "content": "The dark energy equation of state w 0 and M ν is one of the major parameter degeneracies. Fig. 1 shows the 1 σ constraints on M ν and w 0 computed from cluster number counts, power spectrum, combination of the two, with and without LE of the Planck CMB. The contour for number count shows a clear diagonal alignment, and the degeneracy direction can be understood as follows: an increase in neutrino mass suppresses the growth of structure formation, this can be compensated by a larger rate of accelerated expansion (i.e. more negative w ). The constraints from power spectrum is less degenerate but show different degeneracy directions. As a result, combining information from both probes greatly improve the constraints. To see the effect of w 0 on M ν constraint, we derive σ M ν again by marginalizing over w 0 and w a . We find that, as expected, only the constraints from number count are affected (improve by a factor of > 2), while those from power spectrum are barely affected. Furthermore, only modest improvements are obtained when combining number count and power spectrum in this case. The degeneracy between curvature Ω K and neutrino mass M ν is also known to be significant and impact on both M ν and the number of neutrino species N eff , which could affect the constraints coming from CMB [e.g 34, 35]. However we note that the cluster probes used in this work are related to the growth of structures which are not sensitive to Ω K . Thus we expect that including Ω K in the Fisher matrix analysis would not impact our results. It is out of the scope of this paper to study in depth the impacts of including an extended set of parameters (e.g. Ω K , N eff ). Nevertheless it would be potentially interesting to study their effects for growth of structures and we leave it for future works.", "pages": [ 4 ] }, { "title": "B. Survey sensitivities to neutrino mass constraints", "content": "In order to better understand what aspects of SZ surveys would improve the constraints on neutrino mass, we repeat the Fisher matrix calculation that includes different range of wavenumber k and cluster mass M . One of the major dependences is the maximum k values ( k max ) as it determines the smallest scales that can be probed by a survey. The effect of massive neutrino is particularly prominent at small scales (large k values), in which free streaming of neutrinos prevent structure formation. In this work, we use the same k max = 0 . 1 h/Mpc in the power spectrum Fisher matrix for all surveys considered. We do not attempt to increase the k max beyond this value to avoid the non-linear effects at smaller scales. Furthermore, this scale is the limit that can be reached by SZ cluster surveys. Instead we study the dependence when small scale modes are lost, as shown in Fig. 2 (left). The effect of losing small scales information begins at k ≈ 0 . 06 h/Mpc, which corresponds to the free streaming scale k fs at z = 2, with M ν = 0 . 3 eV. The prospect of using smaller scale modes in constraining neutrino mass would be coming from galaxy surveys which can probe down to k ≈ 0 . 5 Mpc/h. A number of studies forecasted the neutrino mass constraints from future galaxy sur- (e.g. [36] (BOSS) and [7] (EUCLID-like)) and found that the improvement in σ M ν beyond k = 0 . 1 Mpc/h is marginal (see Fig. 6 in [36]). The other relevant dependence is the limiting cluster mass that differentiates the various SZ surveys. We show in Fig. 2 (right) the fully marginalized σ M ν from the power spectrum + Planck CMB as a function of minimum cluster mass used in the calculation. It is clear that a deeper survey that can probe down to lower mass would improve the constraints. It is also interesting to note that the SPTPol survey, despite of its small sky coverage, can perform better than the ACTPol and SPT survey because it can detect clusters down to ≈ 2 × 10 14 M /circledot . Therefore a deep survey can compensate for the sky coverage when constraining neutrino mass.", "pages": [ 5, 6 ] }, { "title": "VI. CONCLUSION", "content": "In this work, we explored the possibility of using future and upcoming SZ cluster surveys to constrain neutrino masses. We employ the Fisher Matrix analysis to forecast the sensitivities of various SZ surveys in constraining the total neutrino mass M ν in the context of flat ΛCDM cosmology. We do so by making use of the cluster number counts and power spectrum, and taking into account the self-calibration of mass-observable scaling relations. In general, we find that the M ν constraints from cluster number count and power spectrum is weak if they are considered separately, due mainly to strong parameter degeneracy between M ν and w 0 , especially in the case of cluster number count. However, such degeneracy can be broken if the two probes are combined, which helps to improve the constraints considerably. For example, a sample of ≈ 1000 clusters obtained from the Planck cluster survey gives σ M ν = 0 . 94 eV (0.43 eV with weak prior). The constraints can be further improved when combined with CMB priors. The best constraint is obtained for the Planck and ACTPol survey, with σ M ν = 0 . 23 eV (CMB) and σ M ν = 0 . 17 eV (CMB LE). This is ≈ 80(25)% im- provement with respect to the CMB (CMB LE) only constraint. The use of CMB lensing extraction can better help the cluster only constraints because it determines the neutrino's free streaming effect on the matter power spectrum and break some of the parameter degeneracies. While we find that M ν constraint is mainly driven by CMB and the addition of cluster probes, i.e. number count + power spectrum, to CMB only helps marginally, the use of clusters is still beneficial if we have good control of cluster systematics. For example, when applying a weak prior on the mass-observable relation, the 1 σ error on M ν as obtained from cluster count + power spectrum goes down to 0 . 48 eV and is competitive with CMB only constraint. If we further combine with CMB priors, σ M ν reduces to 0 . 07 eV, which corresponds to a ≈ 4 σ detection for M ν ≥ 0 . 3 eV. The prospect of achieving the weak prior conditions is promising, e.g. clusters detected in weak lensing measurements or a subsample of objects having extensive multi-wavelength follow-up. Therefore, cluster measurements are useful, as an independent probe of the M ν with respect to the CMB, in tightening the current bound on M ν . We find that a deeper cluster survey that detects smaller mass clusters, e.g. down to 2 × 10 14 M /circledot like SPTPol, improves neutrino mass constraints. This is because of the effect of free streaming of massive neutrinos that prevents structure formation to happen at small scales. Likewise, the availability of the small scale modes, i.e. the maximum k values that can be probed by a cluster survey, also helps the constraints. We show that the modes at k ≥ 0 . 06 h/Mpc are important as they help decreasing σ M ν significantly.", "pages": [ 6 ] }, { "title": "Acknowledgments", "content": "EP acknowledges support from JPL-Planck subcontract 1290790. DM acknowledges support from USC Stauffer Fellowship. EP and DM were partially supported by NASA grant NNX07AH59G. 065009 (2008), 0805.1920.", "pages": [ 6 ] } ]

[ { "title": "Holonomy Operator and Quantization Ambiguities on Spinor Space", "content": "Etera R. Livine , Johannes Tambornino 1 1 Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 All'ee d'Italie, Lyon 69007, France We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant 'generalized ladder operators' recently introduced in the U( N ) approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The recent development of spinor techniques for loop (quantum) gravity has led to various interesting applications, from a better understanding of the discrete geometries underlying spin network states to the derivation of Hamiltonian constraint operators encoding the dynamics prescribed by spinfoam models (see [1] for a review). To start with, the introduction of spinor variables allowed a compact reformulation of the loop gravity phase space [2-6], with a clear geometrical interpretation as 'twisted geometries' generalizing the discrete Regge geometries. This became particularly relevant for the construction and interpretation of spinfoam models when analyzing the hierarchy of constraints to impose on arbitrary discrete space-time geometries in order to implement a proper quantum version of general relativity [7]. Furthermore, following the generalization of these spinor variables to twistor networks allowing to describe a Lorentz connection [8, 9], these spinor techniques (or actually upgraded to twistor techniques) allowed to explore and better understand the phase space structure underlying the discrete path integral defining the spinfoam amplitudes [10-12]. Following a parallel but different line of research, this parametrization of the loop gravity phase space in terms of spinors naturally led to the definition of coherent states [13-16], which allow a slight modification and a convenient re-writing of the spinfoam amplitudes. These techniques led to some exact results for the evaluation spinfoam amplitudes [17, 18] or for the dynamics of spinfoam cosmology [19]. More generally, it is possible to describe and define the spinfoam amplitudes and 3nj symbols of the recoupling theory of spins (SU(2) representations) directly in terms of spinors and their quantization. At a semi-classical level, this allows to derive and study their asymptotical behavior (at large spins) (e.g. [20]). At the full quantum level, this spinor techniques lead to differential equations and recursion relations satisfies by the spinfoam amplitudes, which are interpreted as the Hamiltonian constraints encoding the dynamics of the spin network states for the quantum geometry. This was indeed done explicitly for the BF theory spinfoam amplitudes [21] (following the approach of [22] for 3d quantum gravity) and for the spinfoam cosmology amplitudes on the 2-vertex graph [19]. Finally, the use of spinor variables to parameterize the loop gravity phase space allowed a systematic study of the gauge invariant observables at the discrete level. This led to the identification of observables generating the basic SU(2)-invariant deformations of intertwiners. We identified in particular u ( N )-subalgebra of observables (where N is the valency of the intertwiner), which turned out powerful in the study of the intertwiner spaces [23] and the construction of appropriate coherent intertwiners [13-15] and in implementing symmetry reductions on fixed graph in order to define mini-superspace models for loop quantum gravity hopefully relevant for cosmology [19, 24]. The quantization of the spinor phase space is a priori rather straightforward since the complex variables are quantized as harmonic oscillators. The equivalence of this quantization scheme with the standard loop quantum gravity using spin network states was proved at the level of the Hilbert space on a fixed graph in [4, 5]. Nevertheless, all the relevant observables have not been consistently studied. Indeed, we have well studied the geometric observables (such as areas and angles) constructed from the triad classically and thus from the s u (2) generators at the quantum level. However the holonomy operator has not yet been explicitly constructed in this context. The goal of the present short paper is to remedy this shortcoming. We first remind the reader of the loop gravity phase space and its parametrization in terms of spinors. Then we describe the unambiguous quantization of the holonomy-flux algebra in terms of harmonic oscillators and holomorphic functions. This leads to a complete description of the grasping and holonomy operators of loop quantum gravity. In the spinorial picture these emerge both as composite operators built from some generalized ladder operators, ˆ E 's and ˆ F 's, which provide a complete set of SU(2)-invariant operators living on each vertex and acting on interwiners [4, 13, 23, 24]. This extends the work done for the classical phase space [4] to the quantum realm. Furthermore, we analyze some quantization- and operator-ordering ambiguities which are encountered in the definition of the holonomy operator on spinor space if we quantize it using the same technique as Thiemann's trick for the definition of the Hamiltonian constraint operator in loop quantum gravity [25]. We show that it leads to an anomaly and we comment on the choice of quantization scheme.", "pages": [ 1, 2 ] }, { "title": "II. LOOP GRAVITY WITH SPINORS", "content": "The Hilbert space of loop quantum gravity on a given oriented graph Γ with E edges 1 is defined as the space of L 2 functions over E copies of the SU(2) Lie group provided with the Haar measure, that is H Γ := L 2 (SU(2) E , d E g ). . This can be understood as a quantization of ( T ∗ SU(2)) E , namely one copy of the cotangent bundle T ∗ SU(2) /similarequal SU(2) × s u (2) for each edge e of the graph, which is usually parameterized with the couple ( g, J ), where g ∈ SU(2) is the holonomy of the Ashtekar-Barbero connection along the edge e and J = /vector J · /vectorσ ∈ s u (2) ∼ R 3 is related to the flux of the densitized triad through a surface dual to that edge. Furthermore, if J is assumed to live on the target vertex of e one can can use the group element g to parallel-transport it to the source vertex of e and obtain J = -g ˜ Jg -1 . The full Poisson algebra, attached to the edge e , is then given by where J and ˜ J are considered as 3-vectors and the SU(2) group element g defined in the fundamental representation as a 2 × 2 matrix with the indices A,B = 0 , 1. The next step is to impose SU(2) gauge invariance at the vertices v of the graph. This is implemented by the (first class) closure constraints, which generates SU(2) transformation on both J 's and g 's. At the quantum level, it is taken into account by going from the Hilbert space H Γ = L 2 (SU(2) E ) to the gauge-invariant Hilbert space H o Γ = L 2 (SU(2) E / SU(2) V ) where we have quotiented by the SU(2)-action at all vertices. The final step would be to implement the Hamiltonian constraints (corresponding to the invariance under space-time diffeomorphisms in the continuum theory). This issue is still-open and we will not discuss it here. We will focus on the kinematical structures of the theory. Focusing on a single edge of the graph, it was shown in [2-6] that an alternative parameterization of T ∗ SU(2) is possible, in terms of two spinors | z 〉 , | ˜ z 〉 ∈ C 2 . The spinors are interpreted as living at the source and target vertices of the edge, as illustrated on fig.1. A spinor | z 〉 is an element of C 2 and has components z A , A = 0 , 1. We denote its conjugate by 〈 z | = (¯ z 0 , ¯ z 1 ) and its dual by | z ] = /epsilon1 | ¯ z 〉 , /epsilon1 = -iσ 2 . The space C 2 has the standard positive definite inner product 〈 z | w 〉 := ¯ z 0 w 0 + ¯ z 1 w 1 . We further impose a matching constraint, M := 〈 z | z 〉 - 〈 ˜ z | ˜ z 〉 enforcing the norms of the two spinors to be equal and generating a U(1) gauge invariance (under multiplication of the two spinors by opposite phases). It can be shown that C 2 × C 2 , the space of two spinors | z 〉 and | ˜ z 〉 , reduces to T ∗ SU(2) by symplectic reduction with M (see [2, 3, 5] for details). The group and Lie-algebra variables are explicitly reconstructed as This relation obviously implies that g | ˜ z ] = | z 〉 and thus reproduce the constraint J = -g ˜ Jg -1 . Endowing C 2 with the symplectic structure { z A , ¯ z B } = -iδ AB , one easily recovers the phase space (1). In that sense, the spinors can be understood as (complex) Darboux-like coordinates for the space T ∗ SU(2) in which the symplectic structure (1) is trivial. Here, let us stress a subtlety about the orientation of the edge. Indeed the group element introduced above maps one spinor onto the other, as g | ˜ z ] = | z 〉 and g | ˜ z 〉 = -| z ], so that its inverse is defined as g -1 | z 〉 = | ˜ z ] and g -1 | z ] = -| ˜ z 〉 . This is slightly different from the change of orientation which would lead to a group element ˜ g satisfying ˜ g | z ] = | ˜ z 〉 and ˜ g | z 〉 = -| ˜ z ] as we follow the definitions given above. The difference is actually just a sign flip: This sign flip is not so important, but it is to be kept in mind. This sign ambiguity can be traced to the change from vector variables to spinor variables. Indeed, the Poisson algebra of the variables ( g, J ) is unchanged under the change g ↔-g , while the Poisson algebra of the variables ( g, z ) will be affected (by a mere sign). This is simply because a 3-vector does not see the difference between the 3d rotations (in SO(3)) induced by g and -g , while these two SU(2) transformations act differently on a spinor. This spinor parametrization provides a direct link between spin network states and discrete geometries and provides an interesting new perspective on loop quantum gravity [1, 5, 6], spinfoam models [11, 12, 14-16], quantum spinfoam cosmology [4, 19, 24], topological BF-theory [21] and group field theory [27]. Now, turning the focus to a given vertex v of the graph, one has one spinor variable z e per edge attached to v (here we do not distinguish between the z 's and ˜ z 's), as illustrated on fig.1. One can easily identify a complete set of SU(2)-invariant observables, i.e commuting with the constraints G v : These scalar products between spinors interestingly form a closed algebra [4, 13, 23] and in particular the E -observable form a u ( N ) algebra (where N is the number of edges attached to the vertex v ) which is interpreted as generating the deformation of the intertwiner for fixed boundary area [13, 23, 28]. These observables then serve as basic building blocks for all gauge-invariant observables on a given graph Γ and in particular allow to decompose at the classical level the holonomy observable into basic deformations of the spin(or) network [4]. Here we will generalize this to the quantum level.", "pages": [ 2, 3 ] }, { "title": "III. THE HOLOMORPHIC REPRESENTATION", "content": "The standard quantization scheme for T ∗ SU(2) is to consider the Hilbert space L 2 (SU(2)), with the usual orthogonal basis given by the Wigner matrices D j mn ( g ) := 〈 j, m | g | j, n 〉 and with the holonomy g acting by multiplication and the vector X acting by derivation as the s u (2) generators. Here the choice of spinor variables to parameterize the phase space leads to a different polarization, but which has been shown to be unitarily equivalent to the standard quantization (e.g. [5]). The quantization of the canonical Poisson bracket { z A , ¯ z B } = { ˜ z A , ¯ ˜ z B } = -iδ AB leads naturally to two copies of the Bargmann space F 2 := L 2 hol ( C 2 , dµ ) of holomorphic, square integrable functions with a normalized Gaussian measure, where the spinors are naturally represented as the raising- and lowering operators of harmonic oscllators. Taking into account the (area) matching constraint ˆ M = 0 one is led to the space A natural orthonormal basis of H spin is given by holomorphic polynomials in two spinors, where the spin j ∈ N / 2 gives the overall degree of the polynomial and -j ≤ . . . m, n · · · ≤ + j . These polynomials have a simple interpretation in terms of SU(2) representations: where | j, z 〉 is the SU(2) coherent state labeled by the spinor | z 〉 while | j, ˜ z ] denotes the SU(2) coherent state labeled by the dual spinor | ˜ z ] = /epsilon1 | ¯ z 〉 (see e.g. [13-15] for more details). Orthonormality and completeness read as where dµ is the Gaussian measure and I 0 ( x ) is the zeroth modified Bessel function of first kind which plays the role of the delta-distribution on H spin . As explained in [5, 6], these holomorphic polynomials P j mn are unitarily related to the standard Wigner matrices in L 2 (SU(2)), i.e. there exists a unitary map T : L 2 (SU(2)) →H spin mapping Considering the formula (3) for the holonomy g in terms of the spinors z, ˜ z , it appears that this map T extracts the holomorphic part of the group element g . The pre-factor in √ 2 j +1 then ensures that this map still conserves the norms and scalar products, i.e. that it is unitary. Details on this map and its properties can be found in [5]. The elementary operators on H spin are the ladder-operators [ a, a † ] = 1 which directly correspond to the classical spinors { z, ¯ z } = -i . Holonomies g and fluxes J then emerge as composite operators. Let us start with the basis (6) which decomposes as P j mn ( z, ˜ z ) = e j m ( z ) ⊗ ˜ e j n (˜ z ) = e j m ( z ) ⊗ e j n ( | ˜ z ]), where e j m ( z ) := ( z 0 ) j + m ( z 1 ) j -m √ ( j + m )!( j -m )! is the well known orthonormal Fock basis in F 2 . Each of the z and ˜ z spinors gets quantized into a set of two harmonic oscillator operators These act as the usual raising- and lowering-operators on F 2 respectively: At first, it might seem awkward that ¯ z be quantized as the multiplication by z while z get becomes the differentiation ∂ z . To make it more normal, one should instead consider anti-holomorphic polynomials in the spinors z and ˜ z , then ¯ z would be the multiplication by ¯ z and z the differentiation ∂ ¯ z . This detail does not truly matter. What's important is the action of the operators on the basis states e j m and how they shift the j and m . The quantization of the ˜ z -sector is carried out exactly as above for the z -sector. But since we act on slightly different wave-functions, the ˜ e j n (˜ z ) = e j n ( | ˜ z ]) instead of the e j m ( z ), we will get different pre-factors and shifts in j and n : The matching constraint is quantized as ˆ M = (ˆ a 0 ) † ˆ a 0 + (ˆ a 1 ) † ˆ a 1 -( ˆ ˜ a 0 ) † ˆ ˜ a 0 -( ˆ ˜ a 1 ) † ˆ ˜ a 1 , where we use the obvious notation a, ˜ a (corresponding to z, ˜ z ) to denote ladder-operators acting in the first and second copy of F 2 respectively. This generates a U(1)-invariant on polynomials of z and ˜ z as expected and we check that ˆ MP j mn = 0 for all basis elements of H spin . Then we require operators on the space H spin , built from these ladder-operators on F 2 , to commute with the U(1)-constraint ˆ M . Such invariant operators are the flux and holonomy operators, as we develop in the next section.", "pages": [ 3, 4, 5 ] }, { "title": "IV. QUANTIZING THE HOLONOMY-FLUX ALGEBRA", "content": "Let us start with the flux-operators, corresponding to the quantization of the 3-vectors J : The commutators are an exact representation of the classical Poisson brackets, i.e the ˆ J form an s u (2)-algebra: This is the standard Schwinger representation for the s u (2) Lie-algebra. Their action on holomorphic polynomials is easily computed and reproduces the well-known action of the s u (2) generators on basis states: Another essential operator on H spin is the area-operator ˆ E which arises as a quantization of the norm of the 3-vector | /vector J | = 1 2 〈 z | z 〉 , which is diagonalized in the standard basis (6) such that and commutes with ˆ J ± , ˆ J 3 . The geometric interpretation of this operator ˆ E is that it gives the area carried by that edge. The set { ˆ J ± , ˆ J 3 , ˆ E } forms a u (2)-algebra. The ˜ z -sector differs slightly from the formulas above. The quantization of the 3-vectors is carried out exactly the same way and the expression of the operators ˜ J in terms of the raising and lowering operators remains the same. Nevertheless the action on basis states gets a sign flip: To derive the action of the holonomy operator, we start by its action on the Wigner matrices in L 2 (SU(2)) and pull back to H spin using the unitary map T as in (8). Indeed, on L 2 (SU(2)) we know that the holonomy operators ˆ g AB (here taken as the matrix elements of the group element g in the fundamental representation of SU(2)) acts simply by multiplication, that is Nevertheless the operators ̂ ˜ J still form the expected s u (2) algebra without any sign flip (this is actually the complex conjugate representation of s u (2) compared to the z -sector). Using the SU(2)-recoupling theory, the action of the holonomy is easily computed. It is convenient to switch from the indices A,B = 0 , 1 to indices α, β = ± 1 2 . We get after the pull-back: Note that the precise pre-factor 1 2 j +1 is crucial to ensure that the classical Poisson algebra relations (1) are correctly implemented on L 2 (SU(2)). In particular, one can apply this formula to the special case when we act with the character χ 1 2 on the character χ j : as expected. Instead of using the formulas from SU(2) recoupling, we can follow our quantization rules from the classical expression (3) of the holonomy: The polynomial part is straightforwardly quantized and the only subtlety is the pre-factor 1 √ 〈 z | z 〉〈 ˜ z | ˜ z 〉 that gets regularized and quantized as (2 ˆ E + 1) -1 . This regularization, which makes the non-polynomial part a well-defined operator on all of H spin (the a priori expression (2 ˆ E ) -1 diverges on the state P 0 mn for j = 0), seems a bit ad hoc at the first glance. However, there are two reasons which justify this choice: first, this is the only regularization which ensures that ˆ g acts on L 2 (SU(2)) as required (after acting with the map T ). Second, even without knowing about the unitary mapping between L 2 (SU(2)) and H spin , one would have constructed the same operator ˆ g by starting with an arbitrary regularization 1 f ( ˆ E ) of the non-polynomial part of the group element such that it is well-defined on all states in H spin and then demand that the commutator [ˆ g AB , ˆ g CD ] = 0 is implemented with no-anomaly. This condition selects the regularization chosen here. Now that we have constructed the operators ˆ J, ˆ ˜ J and ˆ g AB , we still have to check the final commutation relations between them in order to conclude that the Poisson algebra (1) is represented correctly on H spin . The Poisson brackets between J 's and the g 's can be re-written explicitly as: which gets quantized as: It is straightforward that these commutators are satisfied by the operators as we have defined above.", "pages": [ 5, 6 ] }, { "title": "V. QUANTIZATION AMBIGUITY AND THIEMANN'S TRICK", "content": "As stated above, the ordering ambiguities in the holonomy-operator on H spin are strongly restricted by demanding that the classical Poisson algebra (1) be represented non-anomalously. A conceptually similar, but mathematically much more involved problem occurs in the definition of the Hamiltonian constraint, which generates the quantum dynamics in loop quantum gravity. The issue is that the classical Hamiltonian constraint is polynomial but for a pre-factor given by he inverse square-root of the determinant of the triad, which does not have a clear unambiguous quantization. Thiemann [25] was the first to propose a mathematically well-defined quantum operator corresponding to the classical Hamiltonian constraint. His construction relies strongly on some classical Poisson-identities, reexpressing the inverse square root of det E which appears in the classical Hamiltonian constraints as where V = ∫ Σ dσ √ | det E| is the total volume of the spatial manifold Σ. The basic variables are the Ashtekar- connection A j a and the densitized triad E a j , which is related to the (inverse) spatial metric as q ab = E a j E b j / | det E| . This Poisson-identity is used to reformulate the (Euclidean part of the) classical Hamiltonian constraint as and the corresponding quantum operator is then defined as 2 ˆ H Eucl := 1 i /planckover2pi1 T r ( ˆ F ∧ [ ˆ A, ˆ V ]). This definition, using the Poisson-identity (24) as a way to regulate a possibly diverging non-polynomial expression is rather non-standard and its physical and mathematical consistency has not been checked intensively so far. Whether the classical hypersurface deformation algebra, encoding general relativity's invariance under diffeomorphisms, is represented non-anomalously remains an open issue [29, 30]. To analyze the fate of the hypersurface deformation algebra in the full theory is rather difficult conceptually and technically. However, the spinorial formalism described in this article can be used to model a quantization based on such Poisson-identities in a much simpler setting. Indeed the holonomy g contains a similar pre-factor, given by the inverse square-root of the product of the norms of the spinors. It is possible to use a similar trick to re-absorb this pre-factor and generate it through a Poisson bracket. But in our (much) simpler framework, we know the exact quantization of the holonomy ˆ g , so we can test if such Poisson-identities lead more or less to the correct quantization or not. More precisely, we consider the Poisson-bracket of a spinor | z 〉 with the square-root of the total area E = 1 2 〈 z | z 〉 , which allows to generate an inverse square-root of the norm of z : Using the same Poisson-identities for the spinor ˜ z , we can therefore write the classical group element (3) as Similarly to the definition of the Hamilton constraint operator in loop quantum gravity we can now promote this identity to the definition of a holonomy operator by replacing the Poisson brackets by commutators, {· , ·} → -i [ · , · ]. We simply substitute the classical phase space functions √ E , √ ˜ E and | z 〉 [˜ z | - | z ] 〈 ˜ z | with the corresponding welldefined 3 operators on H spin . As a result we obtain the new definition: It is straightforward to compute its action on basis states: Note that the action of this operator is very similar to (19), but for the different pre-factors in j in front of each term. In the asymptotic limit of large spins j , we do recover that the pre-factors above give back 1 / (2 j +1) as expected at leading order. But for small j 's, the pre-factors differ, which means that the quantization of the holonomy is clearly different for small spins, i.e. close to the Planck scale. Moreover, as stated above, the exact form of the combinatorial pre-factors in (19) is important in order to obtain a non-anomalous representation of the classical Poisson-algebra (1). With the latter choice derived from quantizing the Poisson-identities, /negationslash which contradicts the fact that any two holonomy operators should Poisson-commute. In general, if we define the holonomy operator acting as: where f + ( j ) and f -( j ) are the j -dependent pre-factors corresponding to the quantization of the inverse square-root of the norms of the spinors, we can compute the resulting commutator. We only give the commutator between ̂ G ++ and ̂ G --for the sake of simplicity (to avoid a mess with the indices), but all the commutators can be computed similarly: It is fairly easy to check that this factor vanishes for our quantization ˆ g , when f -= f + = (2 j +1) -1 . On the other hand, for the quantization using the Thiemann-like trick, we get for [ g ' ++ , g ' --]: Therefore we conclude that a quantization based on the Poisson-identity (26) does not give the desired result and leads to an anomaly in the algebra at the quantum level. Defining an operator via Poisson-identities of the kind (26) amounts to a specific choice of operator ordering in the quantum theory. In the simple test case considered here, we have shown that this particular operator ordering does not lead to a proper quantum representation of the classical Poisson algebra. This is a standard with the quantization of non-polynomial observables. Having a closed algebra of observables at the classical level guides us here to choose the correct quantization and operator ordering. The quantum dynamics of loop quantum gravity (and loop quantum cosmology, which is often used as a finite dimensional toy model) relies substantially on the Poisson-identity (24) which is, at least in spirit, very similar to the one tested here. Extrapolating from our results to the case of loop quantum gravity, it would not be surprising if a similar inconsistency would show up in the quantization of the Hamiltonian constraint. To clarify this issue we think it could be helpful to study further toy models based on Poisson identities of the type (24) and check them for internal consistency.", "pages": [ 6, 7, 8 ] }, { "title": "VI. GRASPINGS AND VOLUME OPERATOR", "content": "So far we have restricted our attention to operators that were defined on a single edge of a spin network, namely the holonomy- and flux-operators (12) and (21) acting on H spin . Now we would like to discuss operators acting on the full spin network, and thus taking into account the SU(2) gauge invariance at the vertices of the graph. The Hilbert space on an arbitrary graph Γ is H Γ = L 2 (SU(2) × E / SU(2) × V ) in terms of the total number of edges E and the total number of vertices V . It can be recast in the spinor framework as [4, 5]: where i = 1 , .., N v labels the edges attached to a given vertex v . The initial definition focuses on degrees of freedom attached to the edges of the graph -the group element g e - up to the SU(2) gauge invariance at the vertices. The spinor framework allows to break the degrees of freedom on each edge into two pieces -the two spinors z e and ˜ z e - attached to its source and target vertices, up to a U(1) gauge invariance along the edge allowing to glue the two spinors (by imposing that their norm be equal). In the end, this allows to factorize the Hilbert spaces around each vertex and to re-write the full Hilbert space as encoding degrees of freedom attached to each vertex up to the U(1) gauge invariance on all edges. This is natural from the point of view that spin networks are made of intertwiners living at each vertex of the graph, or equivalently at the classical level that spinor networks can be interpreted geometrically as polyhedra dual to each vertex and glued together along the edges [2-4, 23]. This is at the heart of the U( N ) framework for spin networks [4, 13-15, 23], where the space of intertwiners at each vertex v is identified as living in an irreducible representation of the unitary group U( N v ) (which depends on the total area around the vertex v ). Indeed, now focusing on a vertex v of the graph, we have N edges attached to it (we dropped the index v off N v ), and thus N spinors z i . As we have seen earlier in section II, we have a set of SU(2)-invariant observables at the classical level given by the scalar between those spinors and their dual: where the E -matrix is Hermitian, E ij = E ji and the E -matrix anti-symmetric, F ij = -F ji At the quantum level, we have working on the Hilbert space H v := ( N ⊗ i =1 F 2 ) / SU(2). The quantization of those observables is straightforward: These operators are now the basic building blocks for all SU(2)-invariant operators acting on spin networks. The U( N ) formalism is based on the fact that the ˆ E ij operators form a closed u ( N ) algebra [23, 28]. Now we are interested in the SU(2)-invariant version of the flux and holonomy operators, studied in the previous section. We construct in the present section the grasping operators around a given vertex v , as polynomials in the operators E v and F v . In the next section, we will deal with the holonomy operators, defined around closed loops of the graph. They will involve polynomials of the operators E v and F v (for all vertices v around the loop), up to the same norm factors that appeared for the quantization of the group element on a single edge. We will pay special attention to those factors and associated operator ordering. Coming back to a single vertex, the operators ˆ E v ij and ˆ F v ij acts on the couple of edges i and j attached to the vertex v . The ˆ E -operators shift a quantum of area from one edge to another, while the ˆ F - and ( ˆ F † )-operators respectively annihilate and create a quantum of area on both edges. In this sense these operators can be regarded as generalized creation and annihilation operators. They are invariant under the action of SU(2) at the vertex. They are not however invariant under the U(1)-symmetry on the edges and thus do not qualify as operators on the fully gauge invariant Hilbert space H γ . On the other hand, the natural observables invariant under both U(1) and SU(2) symmetries are the scalar combination of the 3-vectors, i.e the scalar product /vector J i · /vector J j and higher order combinations involving vector products such as ( /vector J i ∧ /vector J j ) · /vector J k . All these observables can actually be written in terms of the E and F observables as polynomials (of the same in E,F as in the J 's). For instance, starting with the scalar product observable on a single edge, this gives the squared area carried by that edge, which is easily translated in the E -observables: This relation also holds at the quantum level, except for a correction term accounting for the quantum ordering [23, 28]: Such correction terms lead to ordering ambiguities in the area spectrum of loop quantum gravity. For instance, one can define the area directly as the operator ˆ E ii , with spectrum j the operators E ii and J 2 i Therefore selecting the particular Poisson bracket that we want to keep (without anomaly) as commutators at the quantum level would select one particular ordering over all others. The same is easily done for the scalar product observables between two different edges [23, 24]: , or as the squareroot of the SU(2) Casimir operator √ /vector ˆ J 2 i , with spectrum √ j ( j +1). Such ambiguities appear crucial in solving some of the (second class) constraints in loop gravity or spinfoam, for instance in the construction of the EPRL spinfoam model [31]. Notice nevertheless that ˆ √ /vector ˆ are different and do not have the same commutation relation with the other observables. We also look at the cubic operator ˆ U ijk ≡ -i [ /vector ˆ J i · /vector ˆ J j , /vector ˆ J i · /vector ˆ J k ] = /epsilon1 abc ˆ J a i ˆ J b j ˆ J c k . For a 3-valent vertex, this operator vanishes due to the SU(2) gauge-invariance. It is non-trivial for a 4-valent vertex and actually defines the squared volume operator (up to a numerical factor) (see e.g. [32] for the geometrical interpretation or [33, 34] for a study of this operator in loop quantum gravity and more recently [35]). The peculiarity of this operator is that it is not positive (its spectrum is real but symmetric under change of sign), so it is non-trivial to define its square-root. One can very naively take its absolute value in a basis which diagonalize it, but this seems an ad hoc definition weaken by the fact that we do not know explicitly the exact spectrum and eigenstates of the operator U . Nevertheless, it is the only well-defined proposal for a volume operator. It would seem better suited to identify the positive and negative modes of the operator but this turns out more complicated and it is not yet achieved 4 . For vertices with valency larger or equal to 5, the squared volume operator is defined by adding the operators U ijk over all (oriented) triplets of edges attached to the vertex. This operator turns out to be easily written in terms of the operators E or F . After a little algebra, we obtain: It might be interesting to study the action and spectrum of each of these combinations of E and F operators separately. One can also generalize the construction of such gauge-invariant grasping operators of higher order. The operators E and F are already SU(2) invariant, so we only have to deal with enforcing the U(1) invariance. This is achieved by requiring that matching the indices i.e requiring that each edge appear the same number of times through creation operators and annihilation operators.", "pages": [ 8, 9, 10 ] }, { "title": "VII. SPINORIAL REPRESENTATION OF THE WILSON LOOP", "content": "The second important class of operators in loop quantum gravity are the Wilson loop operators ˆ W L which, in contrast to the grasping operators, capture non-local information about the states in H Γ . In terms of holonomies the Wilson loop operators are defined as that is the trace over the oriented product of holonomies ordered along all edges e part of a (oriented) loop L (i.e. we take the inverse of a group element if the edge is oriented in the opposite direction than the loop). An expression for classical Wilson loops in terms of the classical observables E and F was given in [4]. It was however unclear how to quantize these expressions due to the ambiguity in regularizing the inverse norm factors in the holonomies at he quantum level. Having obtained an explicit form of the holonomy operator on a single edge (21) in section IV, we are now able to provide an explicit formula for the Wilson loop operators in terms of the generalized ladder operators ( ˆ E, ˆ F ). Writing the group elements g e in terms of the spinor variables as in eqn.(3), we use our operator ordering (21) and we regroup the creation and annihilation operations around the vertices along the loop as was done in [4] at the classical level. This gives: The index i labels the edge around the loop (from an arbitrary origin vertex), as illustrated on figure 2. Here we have chosen all the edges oriented in the same direction along the loop for the sake of simplicity. In the general case, we would get an overall sign for each edge oriented in the opposite direction. For given r i 's, we call the operator W { r i } L the generalized holonomy operator (following the classical nomenclature introduced in [4]). It is a polynomial operator in the operators ˆ E and ˆ F . Its action is fairly simple despite its seemingly complicated structure. It raises the spin by 1 2 on the edge i if r i = 0 and lowers it by one half if r i = 1. The non-polynomial part, given classically by the inverse norm factors, is regularized by a contribution of [2 ˆ E +1] -1 per edge, all of them ordered to the right . This is the simplest scenario, instead of the possibility of these inverse norm factors entering the generalized holonomies and messing up their structure. The slight difference from the ordering conjectured in [4], with the inverse norm factor split as a square-root on the right and one of the left, is actually due to the non-trivial factor 1 / √ 2 j +1 in the map T given in eqn.(8) between the Wigner matrices D j mn ( g ) and their holomorphic counterpart. This operator, expressed in terms of generalized ladder operators, is unitarily equivalent to the standard Wilson loops of loop quantum gravity. To understand its structure and action, let us give some examples. In the simplest case, the graph is just given by one loop with a single vertex (see figure 3). There are four sets of ladder operators, PSfrag replacements e one doublet for each end of the edge or equivalently one doublet on each leg around the single vertex. Let us denote ✶ PSfrag replacements γ these by a, b and ˜ a, ˜ b . The Wilson loop operator ˆ W for this loop can be decomposed into ˆ E - and ˆ F -operators as 5 : with The operator ˆ E gives the spin on the single edge, i.e. the area carried by that edge, while the operators ˆ F and ˆ F † act at the vertex and respectively decreases and increases the spin on the edge by one half. We can compute the action of these operators on our Hilbert space. Since we have a single loop, an orthogonal basis is given by the characters χ j = ∑ m P j mm , and we get 6 : One easily check that we indeed recover the expected action of the holonomy operator due to the factor (2 j +1) -1 , as given in eqn.(20): One can go further and check that indeed ˆ F † + ˆ F = ˆ g --+ ˆ g ++ with the group element operators given earlier in eqn.(21). Beyond this consistency check on the single loop, a more generic example is given by the following situation: consider a loop L within a graph Γ that goes through only 2 vertices (see figure 4). Computing the Wilson loop operator around the loop indicated in figure we obtain 7 : This reproduces exactly the holonomy operator derived in [24] on the 2-vertex graph using the explicit Clebsh-Gordon coefficients. In general, these generalized holonomy operators around the loop L allow to split the full holonomy operator into smaller polynomial operators which act by simple shifts on all edges of the loop. In some simple cases, they have already been used to generate recursion relations on spin network evaluations (and more particularly on the 6j-symbol of the recoupling theory of spins) and to generate the action of the Hamiltonian constraints in 2+1 Riemannian gravity [21]. We hope that this reformulation of all loop quantum gravity gauge-invariant operators in terms of the ladder operators ˆ E and ˆ F will somewhat allow a more systematic approach to the study of gauge-invariant operators entering the Hamiltonian (constraint) for loop quantum gravity in 3+1 dimensions.", "pages": [ 10, 11, 12, 13 ] }, { "title": "VIII. OUTLOOK AND CONCLUSION", "content": "In this short note, we have constructed the holonomy-flux operators in the spinor representation of loop quantum gravity. Holonomies and fluxes emerge as composite operator built from a set of harmonic oscillator operators which are considered to be the more elementary operators in this picture. Because of this compositeness, there are certain operator ordering ambiguities which need to be investigated. Here we showed that an operator ordering is selected by the requirement that the classical holonomy-flux algebra be represented non-anomalously on the spinorial Hilbert space H spin . This guarantees that this representation of the holonomy-flux algebra is unitarily equivalent to the standard one on L 2 (SU(2)). Taking SU(2)-gauge invariance at the nodes of Γ into account we constructed the familiar grasping and Wilson loop operators on H spin . They can be written in terms of the generalized ladder operators ˆ E, ˆ F introduced in the U( N )-formalism [4, 13, 14, 23, 24], capturing the gauge-invariant content of the individual intertwiner spaces. An interesting point to note is that in the spinor formalism the distinction between Wilson loops on the one hand and grasping operators on the other side becomes blurry: both are gauge invariant combinations of the same elementary ˆ E, ˆ F , the only difference being that the grasping operators are localized around a vertex of Γ whereas the Wilson loop operators contain non-local information on the spin network state. An interesting side results of this note is the observation that the spinor formalism can be used as a simple toy model to test the quantization procedure leading to the Hamiltonian constraint operator in loop quantum gravity. This quantization procedure rests on a peculiar Poisson-identity, which is used to get rid of potentially diverging operators and can be modeled by a similar (at least in spirit) Poisson-identity in the spinor formalism. Here we showed that a quantization of the holonomy operator based on that Poisson-identity leads to an anomalous representation of the holonomy flux algebra on H spin . While this calculation does not allow a direct conclusion for the full theory, we suggest that more toy models of this kind should be considered to collect (counter-?) evidence for an anomaly-free implementation of the Dirac algebra in loop quantum gravity.", "pages": [ 13 ] }, { "title": "Acknowledgments", "content": "EL and JT acknowledge support from the Programme Blanc LQG-09 from the ANR (France).", "pages": [ 13 ] } ]

[ { "title": "Binary Black Hole merger in f ( R ) theory", "content": "Zhoujian Cao, 1, ∗ Pablo Galaviz, 2, † and Li-Fang Li 3, ‡ 1 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 School of Mathematical Science, Monash University, Melbourne, VIC 3800, Australia 3 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China (Dated: October 5, 2018) In the near future, gravitational wave detection is set to become an important observational tool for astrophysics. It will provide us with an excellent means to distinguish different gravitational theories. In effective form, many gravitational theories can be cast into an f ( R ) theory. In this article, we study the dynamics and gravitational waveform of an equal-mass binary black hole system in f ( R ) theory. We reduce the equations of motion in f ( R ) theory to the Einstein-Klein-Gordon coupled equations. In this form, it is straightforward to modify our existing numerical relativistic codes to simulate binary black hole mergers in f ( R ) theory. We considered binary black holes surrounded by a shell of scalar field. We solve the initial data numerically using the Olliptic code. The evolution part is calculated using the extended AMSS-NCKU code. Both codes were updated and tested to solve the problem of binary black holes in f ( R ) theory. Our results show that the binary black hole dynamics in f ( R ) theory is more complex than in general relativity. In particular, the trajectory and merger time are strongly affected. Via the gravitational wave, it is possible to constrain the quadratic part parameter of f ( R ) theory in the range | a 2 | < 10 11 m 2 . In principle, a gravitational wave detector can distinguish between a merger of binary black hole in f ( R ) theory and the respective merger in general relativity. Moreover, it is possible to use gravitational wave detection to distinguish between f ( R ) theory and a non self-interacting scalar field model in general relativity. PACS numbers: 04.70.Bw, 05.45.Jn", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Einstein's general relativity (GR) is currently the most successful gravitational theory. It has excellent agreement with many experiments (see e.g. [1-3]). However, most of the tests involve weak gravitational fields. On the other hand, recent cosmological observations require ad-hoc explanations to fit in the framework of GR theory, for example the dark energy and dark matter problems [4-6]. In order to solve these difficulties, some alternative gravitational theories have been proposed [7, 8]. In effective form, many gravitational theories can be caste into an f ( R ) theory [9-13]. Additionally, f ( R ) theory has a relatively simple form. Therefore, it is a good alternative gravitational model. In this work, we characterize the gravitational waveform of binary black hole mergers in f ( R ) theory. In the near future, gravitational wave detection will become an observational method for astrophysics [1417]. The gravitational wave experiments can be excellent tools for testing GR in strong field regime. Moreover, it will be possible to distinguish different gravitational theories. Quantitatively, future experimental data can be used to constrain f ( R ) parameters, and to confirm or to reject alternative gravitational theories. With this in mind, we analyze the waveforms in order to quantify the differences. According to our results, it is possible to distinguish quadratic models of f ( R ) and GR with future experimental data. The quadratic form of f ( R ) is given by f ( R ) = R + a 2 R 2 . The main free parameter is the coefficient of the quadratic part a 2 . In the case a 2 = 0, f ( R ) theory reduces to GR. In linearized f ( R ) it is possible to shows that Mercury's orbit sets the value of | a 2 | ≤ 1 . 2 × 10 18 m 2 [18]. On the other hand, Eot-Wash experiments restrict the value of | a 2 | ≤ 2 × 10 -9 m 2 [19, 20]. The Laser Interferometer Space Antenna (LISA) may distinguish | a 2 | ≥ 10 17 m 2 . Binary black holes in the mass range 30 -300 M sun are expected to merge at frequencies in the most sensitive region of the Laser Interferometer Gravitational Wave Observatory (LIGO) frequency band [21]. Therefore, we focused our attention on an equal-mass binary black hole system with total mass M = m 1 + m 2 = 100 M sun . We find that the LIGO detection can distinguish | a 2 | ≥ 10 11 m 2 . The paper is organized as follows: in Sec. II, we summarize the equations of f ( R ) theory. This is followed by a description of the initial data setup in Sec. III. In Sec. IV A, we describe the numerical techniques used to solve the equations of motion. In Sec. IV B, we give some motivation and background for the configuration used in this work. The evolution of equal-mass binary black hole system is presented in Sec. IV C. Conclusions and discussions are presented in Sec. V.", "pages": [ 1 ] }, { "title": "A. Notation and units", "content": "We employ the following notation: Space-time indices take values between 0 and 3, with 0 representing the time coordinate. The first Latin indices ( a, b, c, . . . , h ) refer to four-dimensional space-time and take values between 0 and 3, while Latin indices ( i, j, k, l, . . . ) refer to threedimensional space and take values from 1 to 3. The metric signature is ( -1 , 1 , 1 , 1). Some references (e.g., [18]), use a metric signature (1 , -1 , -1 , -1). The difference is a change of sign of the scalar curvature R as well as f ( R ). We use Einstein's summation convention. The symbol a := b means that a is defined as being b. A dot over a symbol, ˙ glyph[vector]x , means the total time derivative, and partial differentiation with respect to x i is denoted by ∂ i . Differentiation with respect to the Ricci scalar R is denoted with a prime, for example f ' := df ( R ) dR . In order to simplify the calculations, we use geometric units, where the speed of light c and the gravitational constant G are normalized to 1. A variable in bold font, i.e. x , denotes physical quantities in international system units. Particularly, the values of a 2 ≈ 1 M 2 in geometric units corresponds to a 2 ≈ 10 11 m 2 for typical gravitational wave sources of binary black hole for LIGO. We use the following abbreviations: Einstein's general relativity (GR), Laser Interferometer Space Antenna (LISA), Laser Interferometer Gravitational Wave Observatory (LIGO), Einstein-Klein-Gordon (EKG), Baumgarte-Shapiro-Shibata-Nakamura (BSSN), Arnowitt-Deser-Misner (ADM) and binary black hole (BBH).", "pages": [ 2 ] }, { "title": "II. MATHEMATICAL BACKGROUND", "content": "In vacuum spacetimes, f ( R ) theory generalizes the Hilbert-Einstein action to where GR is recovered by setting f ( R ) = R . From this action, we obtain the Euler-Lagrange equations of motion Using the definition of Einstein tensor G ab := R ab -g ab R/ 2, we obtain after subtracting a Ricci tensor term Rg ab / 2 in (2), and rearranging terms, On the other hand, considering the conformal transformation ˜ g ab = e 2 ω g ab , the Ricci tensor transforms into The corresponding Ricci scalar transforms as Therefore, the Einstein tensor transformation is given by Defining ω := 1 2 ln λ , we have The substitution of (7) and (8) in (6) implies Substituting λ := f ' in (3) and the result in (9), we get Since the conformal transformation satisfies ˜ g ab = λg ab , (10) takes the form Defining φ := √ 3 16 π ln λ , we get where The right hand side of (12) has the form of the stress energy tensor of a scalar field (see e.g. [22, 23]) Therefore, in vacuum, the f ( R ) theory equations of motion are equivalent to GR equations coupled to a real scalar field The equation of motion of the scalar field is given by the trace of (2) with g ab where we have employed the conformal metric transformation. Substituting the definition of φ we get The result is the dynamical equation of a real scalar field with potential V . Therefore, the equations of motion for f ( R ) theory are equivalent to Eqs. (12) and (17), which form the EKG system of equations. Notice that the scalar field is introduced for numerical simulation convenience. Moreover, it is related to the Ricci scalar. Therefore, it does not represent a physical freedom. The equations of motion derived with the metric ˜ g ab are commonly referred to be in the Einstein frame. For physical interpretation, we need to transform them using the physical metric g ab = e -4 √ π 3 φ ˜ g ab . The equations in that form are referred to be in the Jordan frame. We use Newman-Penrose scalar Ψ 4 to analyze gravitational waveform. Therefore, it is calculated through ˜ Ψ 4 = e -4 √ π 3 φ Ψ 4 . Since the Weyl tensor is conformal invariant, the pre-factor comes from a tetrad transformation. We use 3+1 formalism to solve (12) and (17). For Einstein equations (12) we adopt the BSSN formulation as in our previous work [24]. The scalar field equations (17) can be decomposed using the 3+1 formalism as follows (see e.g., for detail about the 3+1 formalism [25, 26]): First it is useful to define an auxiliary variable ϕ := L n φ , where L n denotes the Lie derivative along the normal to the hypersurface Σ t . Expressing the Lie derivative in terms of the lapse function α and the shift vector β i , the evolution of φ is given by On the other hand, the evolution of ϕ is given by the substitution of L n φ in (17) where we used the BSSN metric conformal transformation ¯ γ ij = χγ ij and the relationships with K the trace of the extrinsic curvature, γ the determinant of the 3-metric and Γ i the contracted Christoffel ij symbol. The quantities with an upper bar are represented in the conformal metric of BSSN form. The matter densities are given by S := γ ia γ jb T For f , we consider a quadratic form f ( R ) = R + a 2 R 2 , which results in the potential This potential is analytic around φ = 0 and it can be expanded as The coefficient of φ 2 is related to the mass of the scalar field ( m = 1 / √ 6 a 2 ) and the other terms imply that the scalar field has nonlinear self-interaction. With the signature convention taken in this work, only the positive values of a 2 are physically meaningful. Therefore, we demand that a 2 ≥ 0.", "pages": [ 2, 3 ] }, { "title": "A. Formalism for numerical calculation of f ( R ) dynamics", "content": "The dynamical equations for f ( R ) theory can be written as (2), or equivalently as (12). There is a key component in BSSN formalism where ¯ Γ i are consider to be new independent functions. Similar to this, we promote φ to a new independent function. Then the evolution equation of φ is determined by (17). On the other hand, the definition of φ (15 is a constraint equation. For later reference, we summarize the equations for numerical calculation of f ( R ) dynamics as follows The constraint equation is It is interesting to note that the original dynamical equation (2) for f ( R ) theory includes 4th order derivative terms of metric. This is because f depends on R , ab which contains second derivative terms of the metric, and (2) contains second derivative terms of f . After performing a conformal transformation, we obtain the dynamical equation (12). If we look at the conformal metric ˜ g ab instead of g ab as dynamical variables, (12) involves 3rd order derivatives which come from the derivative of φ . This is because φ itself is a function of R which contains second derivative of conformal metric. In (27) and (28), we replace the 3rd order derivative terms by promoting the auxiliary variable φ as an independent variable. This treatment introduces an extra constraint equation (29) which is similar to the role of the Gamma constraint equations in BSSN numerical scheme. With this treatment, equations (27) and (28) contain at most second order derivative terms. The system of equations (27) and (28) takes the form of coupled Einstein-Klein-Gordon equations. For Einstein equation we use the BSSN formulation. We monitor the constraint equation (29) to check the consistency of our numerical solutions.", "pages": [ 3, 4 ] }, { "title": "III. INITIAL DATA", "content": "Under a 3+1 decomposition, the constraint equations read as follows: where R is the Ricci scalar, K ij is the extrinsic curvature, K the trace of the extrinsic curvature, γ ij the 3-metric, and D j the covariant derivative associated with γ ij . E and p i are the energy and momentum densities given in equations (22) and (23).", "pages": [ 4 ] }, { "title": "A. Puncture method", "content": "The constraints can be solved with the puncture method [27]. Following the conformal transversetraceless decomposition approach, we make the following assumptions for the metric and the extrinsic curvature: where ˆ A ij is trace free and ψ 0 is a conformal factor. We chose a conformally flat background metric, ˆ γ ij = δ ij , and a maximal slice condition, K = 0. The last choice decouples the constraint equations (30)-(31) to take the form where glyph[triangle] is the Laplacian operator associated with Euclidian metric. Notice that we have chosen ϕ ≡ L n φ = 0 initially. This is consistent to the quasi-equilibrium picture. So p i = 0 which results in (34). In a Cartesian coordinate system ( x i ) = ( x, y, z ), there is a non-trivial solution of (34) which is valid for any number of black holes [28] (here the index n is a label for each puncture): where r n := √ ( x -x n ) 2 +( y -y n ) 2 +( z -z n ) 2 , glyph[epsilon1] ik l is the Levi-Civita tensor associated with the flat metric, and P n and S n are the ADM linear and angular momentum of n th black hole, respectively. The Hamiltonian constraint (35) becomes an elliptic equation for the conformal factor ψ 0 . The solution splits as a sum of a singular term and a finite correction u [27], with u → 0 as r n → ∞ . The function u is determined by an elliptic equation on R 3 , which is derived from (35) by inserting (37), and u is C ∞ everywhere except at the punctures, where it is C 2 . The parameter m n is called the bare mass of the n th puncture.", "pages": [ 4 ] }, { "title": "B. Numerical Method", "content": "The Hamiltonian constraint (35) is solved numerically using the Olliptic code ([29]). Olliptic is a parallel computational code which solves three dimensional systems of nonlinear elliptic equations with a 2nd, 4th, 6th, and 8th order finite difference multigrid method [30-34]. The elliptic solver uses vertex-centered stencils and boxbased mesh refinement. The numerical domain is represented by a hierarchy of nested Cartesian grids. The hierarchy consists of L + G levels of refinement indexed by l = 0 , . . . , L + G -1. A refinement level consists of one or more Cartesian grids with constant grid-spacing h l on level l . A refinement factor of two is used such that h l = h G / 2 | l -G | . The grids are properly nested in that the coordinate extent of any grid at level l > G is completely covered by the grids at level l -1. The level l = G is the 'external box' where the physical boundary is defined. We used grids with l < G to implement the multigrid method beyond level l = G . For the outer boundary, we required an inverse power fall-off condition, where the factor B is unknown. It is possible to get an equivalent condition which does not contain B by calculating the derivative of (38) with respect to r , solving the equation for B and making a substitution in the original equation. The result is a Robin boundary condition: For the initial data, we set q = 1 and A = 0.", "pages": [ 4, 5 ] }, { "title": "1. Test problem", "content": "As a test, we set the mass parameter of the black hole to zero and consider a spherical symmetric field φ and potential V . The Hamiltonian constraint (35) reduces to a second order ordinary differential equation where the prime denotes differentiation with respect to r . In order to obtain a high-resolution reference solution, we solve (40) using Mathematica [35]. A useful transformation for the case V = 0 is ψ 1 := rψ 0 . Under this transformation, regularity at the origin implies lim r → 0 ψ 1 ( r ) = 0. The boundary condition (39) with q = 1 and A = 1 reduces to ψ ' 1 ( r max ) = 1, where r max is the radius of our numerical domain. The problem then becomes glyph[negationslash] For the case V = 0, the term r -4 produces a singularity at the origin. We cure artificially the singularity by solving the equation with a term ( r 4 + glyph[epsilon1] ) -1 instead of r -4 . For the test, the value of glyph[epsilon1] is set to 10 -12 . We considered 2 cases where in both cases r 0 = 120 M , σ = 8 M , φ 0 = 1 / 40. For case II, we set a 2 = 1. The numerical domain is a cubic box of size 4000 ( r max = 2000) and 11 refinements levels. We use the fourth order finite difference stencil since it provides a good convergence property at the boundary for large domains (see [29] for details). The convergence tests consist of a set of six solutions with grid points N i ∈ { 43 , 51 , 75 , 105 , 129 , 149 } . The comparison with the reference solution was performed along the Y axis using a 6th order Lagrangian interpolation. For each resolution, the difference E i := | u i -¯ u | gives an estimation of the error. Here u i denotes the solution produced with Olliptic , i is an index which labels the grid size, ¯ u the reference solution and | · | the absolute value (computed point by point). The functions are interpolated in a domain with grid size ∆ y = 1. The error satisfies E i ∼ Ch p i , where C is a constant, h i ∼ 1 /N i is the grid size and p the order of convergence. Using the L 1 norm of the error and performing a linear regression of ln | E i | L 1 vs ln | h i | , we estimate the convergence order p and the constant C . Figure 1 shows the result of case I. There is a good agreement between the several resolutions and the reference solution. The plot does not show noticeable differences (see Fig. 1(a) ). The solution has convergence properties, and the estimated error diminishes with increased resolution (Fig. 1(b) ). The scaled error E i /h p i also shows good convergence with convergence order p = 3 . 7 ± 0 . 2 given by linear regression (Fig. 1(c) ). The results for case II are presented in Figure 2. The solution is similar to case I, an almost constant solution between 0 and r 0 which joins a inverse power solution after Y = r 0 . However, the solution of case II is around 2 orders of magnitude larger than the solution of case I. Contrary to case I, there are noticeable differences between the reference solution and the lower-resolution ones (Fig. 2(a) ). The solution shows convergence properties and the scaled error shows convergence consistently with p = 3 . 9 ± 0 . 3 (Fig. 2(b) , (c) ).", "pages": [ 5, 6 ] }, { "title": "2. Initial data for evolution", "content": "The solution of (35) provides initial data for our evolutions. The initial parameters of the BBH are: puncture mass parameter m 1 = m 2 = 0 . 487209 (approximate apparent horizon mass equals to 0.5), initial position ( x, y, z ) = (0 , ± 5 . 5 , 0) and linear momentum ( p x , p y , p z ) = ( ∓ 0 . 0901099 , ∓ 0 . 000703975 , 0). The linear momentum parameter is tuned for non-spinning quasicircular orbits in GR. For the scalar field part, we consider that the BBH is surrounded by a shell of scalar field with initial profile with r 0 = 120 M , σ = 8 M and several values of φ 0 (see below). When a 2 goes to zero, both φ and V go to zero. Therefore, standard general relativity is recovered. On the other hand, when a 2 → ∞ , the amplitude of the scalar field tends to φ 0 while the potential vanishes. Our model provides an unified scheme to investigate standard GR ( a 2 = 0), usual f ( R ) (0 < a 2 < ∞ ) and the free EKG system in GR ( a 2 →∞ ). From the solution of the conformal factor it is possible to estimate the ADM mass through where the integration is performed in a sphere S of radius r 0 (formally the ADM mass is computed taking the limit r 0 → ∞ ). In our calculations r 0 = 1537 . 5 and the integrations is done numerically using 6th order Lagrange interpolation in the sphere and 6th order Boole's quadrature [36, 37]. The estimation of the ADM mass gives us a way to analyze the parameters φ 0 and a 2 . On one hand, it is possible to compute M ADM for the case a 2 →∞ for several values of φ 0 (see Table I). The result is a quadratic relationship (see figure 3). The quadratic behavior is consistent with the fact that the coefficient of ψ 0 in (35) for the scalar field profile (44) is quadratic in the amplitude φ 0 . On the other hand, for fixed φ 0 , we analyzed the functional behavior of M ADM as function of a 2 . Figure 4 shows the result (in this example φ 0 = 0 . 001642). For this particular value of φ 0 , the ADM mass reaches its maximum value M ADM = 1 . 16023966 when a 2 = 2 . 64353. The estimation of the value a 2 comes from the maximization of the product of the coefficients of ψ 0 and →∞ ψ 5 0 (see right hand side of (35)): where we define ˜ φ 0 := 4 √ π/ 3 φ 0 and ˜ a 2 := ˜ φ 0 a 2 2 / ( a 2 2 +1). Notice that with respect to the radial coordinate r the coefficients are evaluated in their respective maximums. We are looking for the values ( φ 0 , a 2 ) which maximize the product instead of the maximum value of C . Therefore, we can drop all the multiplicative constants. The maximization of C is performed with respect to the variable ˜ a 2 . The extrema of the function reduces to computing the roots of We computed the values numerically using Mathematica . Figure 5 shows the result. From the numerical data, it appears that ˜ a 2 is a linear function of ˜ φ 0 (see Figure 5(a) ). However, a comparison of the data with the fitted linear function showed us that a higher order polynomial is better a approximations. We choose a second order polynomial since higher order polynomials do not exhibit a significant reduction of the errors. The results for ( φ 0 , a 2 ) variables confirm that a quadratic fit is a better approximation (see Figure 5(b) ). Note that in the interval investigated a 2 ∼ 2 . 64. In international system units, it corresponds to 10 11 m 2 (considering the typical gravitational wave sources of BBH for LIGO). This value maximizes the f ( R ) effect for BBH collisions.", "pages": [ 6, 7 ] }, { "title": "A. Numerical method", "content": "The evolution of the black hole and scalar field is solved using the AMSS-NCKU code (see [24, 29, 38-40]). Although AMSS-NCKU code supports both vertex center and cell center grid style, we use the cell center style. We use finite difference approximation of 4th order. We update the code to include the dynamics of real scalar field equations (18) and (19). We use the outgoing radiation boundary condition for all variables. In addition, we update our code to support a combination of box and shell grid structures (according to [41, 42]). The numerical grid consists of a hierarchy of nested Cartesian grid boxes and a shell which includes six coordinate patches with spherical coordinates ( ρ, σ, r ). For symmetric spacetimes, the corresponding symmetric patches are dropped. Particularly, we adopt equatorial symmetry. For the nested Cartesian grid boxes, a moving box mesh refinement is used. For the outer shell part, the local coordinates of the six shell patches are related to the Cartesian coordinates by where both angles ( ρ, σ ) range over ( -π/ 4 : π/ 4). Notice that positive and negative Cartesian patches are related through the same coordinate transformation. This coordinate choice is right handed in + x , -y , + z patches and left handed in -x , + y , -z patches. Disregarding parity issues, left-handed coordinates do not bring us any inconvenience. We have applied this coordinate choice to characteristic evolutions in [43]. For an alternative approach, see [41, 42]. The coordinate radius r relates to the global Cartesian coordinate through All dynamical equations for numerical evolution are written in the global Cartesian coordinate. The local coordinates ( ρ, σ, r ) of the six shell patches are used to define the numerical grid points with which the finite difference is implemented. The derivatives involved in the dynamical equations in the Cartesian grid x i = ( x, y, z ) are related to the spherical derivatives in the shell coordinates r i = ( ρ, σ, r ) through The spherical derivatives in (52) and (53) are approximated by center finite difference. In the spherical shell two patches share a common radial coordinate and adjacent patches share the angular coordinate perpendicular to the mutual boundary. Therefore, it is not necessary to perform a full 3D interpolation between the overlapping shell ghost zones. Moreover, it is enough to perform a 1D interpolation parallel to the boundary (see [41, 44] for details). For this purpose, we use 5th order Lagrangian interpolation with the most centered possible stencil. For the interpolation between shells and the coarsest Cartesian grid box, we use a 5th order Lagrange interpolation. This is a 3D interpolation done through three directions successively. The grid structure for boxes and shells are different. There is no parallel coordinate line between the grid structures. Therefore, we have a region which is double covered. Similar to the mesh refinement interface, we also use six buffer points in the box and shell. The buffer points are re-populated at a full RungeKutta time step. For parallelization, we split the shell patches into several sub-domains in three directions. The same is done for boxes. We have tested the convergence behavior of the updated AMSS-NCKU code. Fig. 6, shows the waveform produced with three resolutions. The corresponding values of the grid size for the finest refinement level are 0.009 M , 0.0079 M and 0.007 M . From here-on, we refer to these values as the low (L), medium (M) and high (H) resolutions respectively. We shift the time in order to align the waveforms at the maximum amplitude of Ψ 4 , 22 . The results presented in sections IV B and IV C are performed with the medium resolution. The equation (15) represents a constraint equation which is introduced by reducing the 4th order derivative dynamical formulation to the 2nd order. Based on 3+1 formalism, we have Substituting L n K with the evolution equations for K ij results in Therefore, the constraint equation reads as From here-on, we will refer to (57) as the f ( R ) constraint. In Fig. 7, we show an example of the violation of this constraint during our simulations. This violation of f ( R ) constraint is much smaller than that of the Hamiltonian constraint.", "pages": [ 7, 8, 9 ] }, { "title": "B. Initial scalar field setup", "content": "One way to interpret f ( R ) theory is as an effective model of quantum gravity. In the astrophysical context, it is natural to assume that the systems are in their ground states, and correspondingly, the scalar field takes the profile of the ground state of the related quantum gravity system. We simulate the development of the scalar field from the ground state of the SchrodingerNewton system considered in [45]. Other authors model the dark matter halo [46] in the center of a galaxy with a similar profile (see e.g., [47]). Our result shows that the scalar field evolves from the ground state configuration to a shell-type profile (similar to (44)). Moreover, the shell forms in the early stages of the evolution. Fig. 8 shows two snapshots, the initial ground state profile and the final shell configuration. In our test, the initial profile of the scalar field is some general Gaussian shape, and the shell shape soon forms. Our results imply that the formation of a shell shape is generic in coupled systems of scalar field and BBH. Considering the development of a scalar field shell in the early stages of the formation of a BBH system, we starte the evolution with the profile (44). The parameters used in our simulations are listed in Table III. We divide the parameters into three groups. The first group, a 2 = 0, φ 0 = 0 corresponds to general relativity. The second group, a 2 → ∞ corresponds to the free EKG equations. In this case, the scalar field in the far zone is weak. Therefore, the waveforms in the Jordan frame are similar to the waveforms in the Einstein frame. The third group, 0 < a 2 < ∞ corresponds to general f ( R ) theory. In this case, the value a 2 is the one which maximizes M ADM for given φ 0 .", "pages": [ 9 ] }, { "title": "C. Results", "content": "In this subsection, we present the numerical simulation results for the BBH evolution in f ( R ) theory. We focus on the comparison between f ( R ) and GR evolution. We refer to the difference between them as the f ( R ) effect.", "pages": [ 9 ] }, { "title": "1. Dynamics of the scalar field induced by binary black holes", "content": "The characteristic dynamics of the scalar field in our simulations is the following. Starting from a shell shape, the scalar field collapses towards the central BBH. Then, the maximum of the scalar field reaches the black holes. At that moment in the evolution, a burst of gravitational radiation is produced . After that, the scalar field continues collapsing towards the origin of the numerical domain. The BBH excites the surrounding scalar field. The perturbations produced by the BBH collapses to the origin, thereby joining the main part of the scalar field. After reaching the origin, the scalar field is scattered in the outgoing direction. Once the scalar field moves outside of the orbit of the BBH, it is attracted by the BBH again and remains there for some time. The scalar field slowly radiates to the exterior of the numerical domain. In the process, part of the scalar field is absorbed by the black holes. In Fig. 9(a) we show the maximum of | φ | with respect to time. Since the scalar field approximates a shell shape, we only consider the radial position. The change in the amplitude of the scalar field represents the collapsing stage (increments) and the scattering stage (decrements). There are two main peaks around time = 125 M. The first peak corresponds to the initial collapse (before reaching the BBH). The second peak corresponds with the excitation of the scalar field produced by the BBH. A small third peak corresponds to the attraction produced by the BBH. Fig. 9(b) shows the radal position of max( | φ | ) with respect to time (solid line) and the radial position of black hole (dashed line). The main collapsing and scattering process is clear. There are four coincidences of the scalar field and the BBH. Three of them correspond to the peaks showed in Fig. 9(a) . We enlarge the detail of the encounters in Fig. 9(c) . As mentioned above, the collision between the scalar field and the BBH produces a burst of gravitational radiation. Fig. 10 shows the corresponding waveform of the evolution presented previously (with parameters a 2 →∞ and φ 0 = 0 . 000959). In this plot, we extract the waves at r = 200 M. After the radiation produced by the initial data configuration (so-call junk radiation), there is a peak at about time = 340 M (dashed line). This burst of radiation is not present in the BBH case (solid line). Moreover, the pattern is encoded in every even m mode of Ψ 4 . Fig. 11 shows the dependence of the amplitude of the burst as a function of φ 0 . The functional behavior is well represented by a quadratic function A + Bφ 0 + Cφ 2 0 , with A = 3 . 04 × 10 -4 ± 3 × 10 -6 , B = -0 . 08 ± 0 . 01 and C = 2273 ± 14. In the above description, we have presented the results for the free EKG system ( a 2 → ∞ ). For our representative f ( R ) case, where a 2 is finite but non-vanishing, the behavior of scalar field is qualitatively different. We compared the cases φ 0 = 0 . 000959 and a 2 = 2 . 64418 with φ 0 = 0 . 000048 and a 2 = 2 . 61877. Fig. 12 shows the results. Contrary to the free EKG case, we found only one collapsing stage without the scattering to infinity phase. In both cases, almost all of the scalar field was absorbed by the black holes. During the collapsing process, the scalar field excites the spacetime. The back reaction excites the scalar field, thereby producing several zigzags in its maximum amplitude (see Figures 12-d). After the maximum of the scalar field passes over the black hole, the dynamics of scalar field become much richer. The scalar field is constantly excited near the black hole. Fig.12-(e) shows that the scalar field is trapped in the inner region of the BBH's orbit. The black holes play the role of a semi-reflective boundary. A minor amount of scalar field escapes to infinity. In comparison with the free EKG system, the case φ 0 = 0 . 000959 and finite a 2 introduces a large amount of eccentricity to the BBH system. However, there is no burst of gravitational radiation (which corresponds to the one presented in Fig. 10).", "pages": [ 9, 10, 11 ] }, { "title": "2. Dynamics of the binary black hole induced by the scalar field", "content": "The trajectory of the BBH is strongly affected by the scalar field. When the scalar field is present, the BBH merges faster. Notice that the ADM mass is not the main cause of the fast merge. As shown in Table III, for cases φ 0 = 0 . 00048 and φ 0 = 0 . 000959, the ADM mass is larger than in the GR case. On the other hand, when φ 0 = 0 . 000048, the ADM masses for a 2 → ∞ and a 2 = 2 . 61877 are smaller and equal to the GR case respectively. However, in both cases with non-vanishing scalar field, the BBH merges faster than in the GR case (see Fig. 13). For larger values of φ 0 , for example 0 . 00048, the scalar field increases the eccentricity of the BBH's orbit in addition to making it merge faster. This extra eccentricity depends on the parameter a 2 . When a 2 is big, the resulting eccentricity is large (see Fig. 14(a) ). In addition, we observe that the f ( R ) effect makes the BBH merge faster in finite a 2 case than in the free EKG case. Previously in Sec. IV C 1, we noticed that the interaction between the scalar field and the black hole is weaker in finite a 2 case than in the free EKG case. The behavior shown in Fig. 14(a) is consistent with this conclusion. When the interaction is stronger, it introduces more eccentricity to BBHevolution. More eccentric BBH orbits produce more gravitational radiation [48]. Therefore, the mergers are faster. Although the coordinate information is gauge dependent, it is possible to verify a change in the eccentricity by looking at the gravitational waves (see Fig. 14(b) ). Notice that the amplitude of the gravitational radiation burst in finite a 2 case is smaller than in the free EKG case. In Fig. 10, we can see the change in the eccentricity for the case of φ 0 = 0 . 000959. So far, we have shown that small φ 0 for free EKG cases introduces more f ( R ) effects than finite a 2 cases. On the other hand, large φ 0 for free EKG cases introduces less f ( R ) effects than finite a 2 cases. It is possible that the nonlinear terms of the finite a 2 cases are the cause of these differences. Considering the f ( R ) effect introduced by the scalar field, we can distinguish the parameter a 2 through gravitational wave detection. LIGO's main BBH sources are black holes with several tens of solar mass. If a 2 is bigger than 10 11 m 2 , we expect to be able to distinguish between f ( R ) theory and GR, via the gravitational detection. On the other hand, LISA (or some similar spacecraft experiment) can distinguish between f ( R ) and GR if a 2 > 10 17 m 2 [18]. All together, the merger phase of BBH collisions allows distinction between the theories, as proposed by [49]. 3. Difference between f ( R ) and other Einstein-Klein-Gordon models in GR We have seen above that it is possible to distinguish between f ( R ) theory and GR via the gravitational waves. Astrophysical models often include EKG equations for the description of certain phenomena. For example, there are models of dark matter which use EKG in the weak field limit [50-53]. One example of relativistic scalar field is boson stars [54-58]. Therefore, it is interesting to ask if gravitational wave detection can be used to distinguish BBHcollisions in f ( R ) theory from another system which also contains scalar fields. In the rest of this section, we analyze the differences between the free EKG system ( a 2 → ∞ ) and the f ( R ) theory. The main difference between free EKG and f ( R ) theory is the nonlinear self interactions, present only in f ( R ) theory. If the scalar field is strong, it is easy to distinguish between free EKG and f ( R ) . If the scalar field is weak, a deeper analysis is necessary in order to distinguish between the theories. Our quantitative results support this statement. First row of Fig. 15 shows the results for φ 0 = 0 . 00048. Fig. 15(a) shows the trajectory of one of the components of the binary (the companion black hole trajectory is symmetric with respect to the X axis). We can see several crosses of the trajectories. This indicates different fluctuations on the inspiral rate. This results from the extra eccentricity introduced by the scalar field. In Sec. IV C 2 and Fig. 14, we saw that the eccentricity is larger in the free EKG system than in the representative case of f ( R ) theory. In addition, the BBH in f ( R ) theory merges faster than in the free EKG. Therefore, it is possible to distinguish between free EKG models and f ( R ) theory. The second row of Fig. 15 shows the results for φ 0 = 0 . 000048 (the value is ten times smaller). In this case, there are no noticeable differences between free EKG models and f ( R ) theory. This is consistent with our assumption that the self interaction becomes weak for small scalar field. However, the quantitative difference of the glyph[lscript] = 2, m = 2 mode of Ψ 4 is significant (see Fig. 16(a) ). Moreover, the relative difference is larger than ten percent (see Fig. 16(b) ). Once again, there is a small peak at roughly time = 240 M in Fig. 16(b) . The peak is the result of a burst of gravitational radiation produced by the free EKG model, which is absent in the f ( R ) case (see also Fig. 10). We expect that we will be able to characterize the differences using more detailed quantitative data analysis techniques. We plan to present the results in a forthcoming paper.", "pages": [ 11, 12 ] }, { "title": "V. DISCUSSION", "content": "Extending the work of [18], where the extreme mass ratio BBH systems were considered to be the gravitational wave sources for LISA, we studied an equal mass BBH system. In order to simulate BBH in f ( R ) theory with our existing numerical relativistic code, we performed transformations of the dynamical equations of f ( R ) theory from the Jordan frame to the Einstein frame. In this way, we performed full numerical relativistic simulations. The main result in [18] is that the gravitational wave detection with LISA can distinguish between f ( R ) theory and GR if the parameter | a 2 | > 10 17 m 2 . Our results im- ply that the gravitational wave detection with LIGO can do the same for | a 2 | > 10 11 m 2 . Mathematically, the dynamical equations of f ( R ) theory in the Einstein frame require a scalar field. We found an interesting dynamics between this scalar field and the BBH. For example, the BBH excites the scalar field for free EKG cases ( a 2 →∞ ) near the collision. The scalar field is constantly excited close to the BBH for finite a 2 cases. Moreover, the interaction introduces extra eccentricity to the evolution of the BBH orbit. We found that the BBH eccentricity is affected by the initial parameter of the scalar field φ 0 depending on the value of a 2 . For small φ 0 , the excitation of the BBH orbit is larger in the representative f ( R ) case in comparison with the free EKG system. On the other hand, for larger values of φ 0 the excitation of the BBH orbit is smaller in the representative f ( R ) case in comparison with the free EKG system. Using gravitational waves, it is possible to distinguish among f ( R ) theory, general relativity and a free EinsteinKlein-Gordon system. We found that the perturbation produced by the scalar field depends on the initial scalar field configuration. Specifically, the waveform exhibits a radiation burst which depends quadratically on the initial scalar field amplitude. The burst is a particular feature of the system which is useful when distinguishing between f ( R ) and GR. For an initial amplitude of scalar field φ 0 = 0 . 000048, the relative difference in the gravitational waveform between f ( R ) theory and the free EKG model is more than 10%. Therefore, gravitational wave astronomy may provide the necessary information to rule in or rule out some alternative gravitational theories. φ = 4 . 8 × 10 - 5", "pages": [ 12, 13, 14 ] }, { "title": "Acknowledgments", "content": "It is a pleasure to thank David Hilditch, Ee Ling Ng, Todd Oliynyk and Luis Torres for valuable discussions and comments on the manuscript. This work was supported in part by ARC grant DP1094582, the NSFC (No. 11005149, No. 11175019 and No. 11205226), and China Postdoctoral Science Foundation grant No. 2012M510563. raud, D. Shoemaker, J. Y. Vinet, F. Barone, L. di Fiore, L. Milano, G. Russo, J. M. Aguirregabiria, H. Bel, J. P. Duruisseau, G. Le Denmat, P. Tourrenc, M. Capozzi, M. Longo, M. Lops, I. Pinto, G. Rotoli, T. Damour, S. Bonazzola, J. A. Marck, Y. Gourghoulon, L. E. Holloway, F. Fuligni, V. Iafolla, and G. Natale, Nuclear Instruments and Methods in Physics Research A 289 , 518 (Apr. 1990)", "pages": [ 14 ] } ]

[ { "title": "Noether current from surface term, Virasoro algebra and black hole entropy in bigravity", "content": "Taishi Katsuragawa 1 and Shin'ichi Nojiri 1 , 2 We consider the static, spherically symmetric black hole solutions in the bigravity theory for a minimal model with a condition f µν = C 2 g µν and evaluate the entropy for black holes. In this condition, we show that there exists the Schwarzschild solution for C 2 = 1, which is a unique consistent solution. We examine how the massive spin-2 field contributes to and affects the Bekenstein-Hawking entropy corresponding to Einstein gravity. In order to obtain the black hole entropy, we use a recently proposed approach, which uses Virasoro algebra and central charge corresponding to the surface term in the gravitational action. PACS numbers: 04.60.-m, 04.62.+v", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "As is well known, while general relativity and the standard model based on quantum field theory are very successful in describing experiments and observations, the unification of these two frameworks still remains mysterious because of both conceptual and technical difficulties. So it is ultimate goal for modern theoretical physics to establish gravitational theory in a microscopic scale, that is, the quantum theory of gravity. One possible way to understand the underlying quantum gravity theory could be to study black hole entropy. In general relativity, black holes have concepts of temperature and entropy, and there exist thermodynamics laws of black holes [1-3]. Investigating the statistical origin of black hole entropy, we could be able to obtain quantum properties of space-time, and we may have a glimpse of aspects of quantum gravity. In fact, the black hole entropy has been calculated in string theory [4] and loop quantum gravity [5, 6]. Recently, a new approach for evaluating the black hole entropy was proposed by using the Noether current corresponding to the surface term of the action [7-9]. The basic idea of this procedure is the following. Define the Noether charges and the Lie bracket corresponding to diffeomorphism; then the Lie bracket forms the Virasoro algebra with a central extension, from which we can read off a deduced central charge and zero-mode eigenvalues of the Fourier modes of the charge. By substituting these quantities into the Cardy formulas [10-12], one can obtain the black hole entropy. This approach is the general formulation, and it has been shown that the formula of black hole entropy can be obtained not only in the Einstein gravity with the usual Einstein-Hilbert action but also in higher-curvature gravity [13, 14]. In this paper, we apply this approach to the bigravity theory. Bigravity is nonlinear massive gravity that has ghost-free construction with the dynamical metric [15-19]. This gravity model is called bigravity or bimetric gravity because the model contains two metrics and a massive spin-2 field appears in addition to the massless spin-2 field corresponding to the graviton. Therefore, by considering the black holes in bigravity, we can evaluate how the massive spin-2 field near the horizon affects the black hole entropy [20-22]. For simplicity, we consider the bigravity theory in the minimal model, and we obtain entropy of a static, spherically symmetric black hole.", "pages": [ 1 ] }, { "title": "II. BLACK HOLE SOLUTION FOR BIGRAVITY", "content": "In this section, we briefly review the construction of ghost-free bigravity [19] and consider the black hole solutions. The bimetric gravity includes two metric tensors g µν and f µν , and it contains the massless spin-2 field corresponding to the graviton and massive spin-2 field. It has been shown that the Boulware-Deser ghost does not appear in such a theory. Note that g µν and f µν do not simply correspond to the massless and massive spin-2 field individually, but they are given by linear combinations of these two fields in the linearized model; that is, when one considers perturbations of g µν and f µν by appropriately redefining two tensor fields, the Fierz-Pauli mass term is reproduced. The action of bigravity is given by Here R ( g ) and R ( f ) are the Ricci scalar for g µν and f µν , respectively. M eff is defined by and √ g -1 f is defined by the square root of g µρ f ρν , that is, For general tensor X µ ν , e n ( X )s are defined by Here, [ X ] expresses the trace of tensor X : [ X ] = X µ µ . The action for the minimal model is given by corresponding to the coefficients, By the variation (5) over g µν and f µν , we obtain [23, 24] By imposing a condition f µν = C 2 g µν with a constant C , the above two equations have the following form: Here, the cosmological constants Λ g and Λ f are defined by For the consistency, both Eqs. (9) and (10) should be identical to each other. By putting f µν = C 2 g µν , we find R µν ( f ) = R µν ( g ), R ( f ) = C -2 R ( g ). Then, we find and obtain the constraint for the constant C from Eqs. (11) and (12): Here, we define M ratio ≡ M f /M g . The solution of Eq. (14) is given by C 2 = 1. When C 2 = 1, the cosmological constants in Eqs. (11) and (12) vanish, and Eqs. (9) and (10) have the identical Schwarzschild solutions, As a result, we have shown that the bigravity theory for the minimal model has a consistent solution when f µν = g µν and cosmological constants vanish. However, for the other model, which has a different choice of β n s, it is shown that one can obtain the consistent solutions with a nonvanishing cosmological constant [25, 26]. Although we can also obtain the Kerr solutions, for simplicity, we only consider the Schwarzschild solutions in the following arguments. Note that when we interpret g µν as a physical metric, the horizon is at r = 2 M , and another tensor field f µν diverges at the same location, but the divergence can be also removed by the coordinate transformation.", "pages": [ 1, 2, 3 ] }, { "title": "III. BLACK HOLE ENTROPY FROM THE NOETHER CURRENT", "content": "In this section, we summarize the procedure by Majhi and Padmanabhan [7-9, 13]. Let us consider a general surface term as follows: Here, N a is a unit normal vector of the boundary ∂ M , g µν is the bulk metric, and σ µν is the induced boundary metric. For the Lagrangian density √ g L = √ g ∇ a ( L B N a ), the conserved Noether current corresponding to differmorphism x a → x a + ξ a is given by Here, J ab is the Noether potential, and the corresponding charge is defined as Here, d Σ ab = -d n -2 x ( N a M b -N b M a ) is the surface element of the ( n -2)-dimensional surface ∂ Σ, and h ab is the corresponding induced metric. We now choose the unit normal vectors N a and M b as spacelike and timelike, respectively. In the following disucussion, we assume Σ exists near the horizon of a black hole. Next, we define the Lie bracket for the charges as follows: which leads to the Virasoro algebra with central extension as we will see later. By using the deduced central charge and the Cardy formula, one can find black hole entropy. To derive the Noether charge and Virasoro algebra, we need to identify appropriate diffeormorphisms, that is, the related vector field ξ a . In this work, we consider static-spherical black holes, for which the metric has the following form: Here, Ω ij ( x ) is the ( n -2)-dimensional space, and h ij = r 2 Ω ij ( x ). The horizon exist at r = r h , where f ( r h ) = 0. For the metric (20), the two normal vectors N a and M a are given by Here, ρ is defined by r = ρ + h h for convenience, and in the near horizon limit, we find ρ → 0. Then, the metric has the following form: Furthermore, we introduce the Bondi-like coordinates, and the metric is transformed as We choose the vector fields ξ a so that the vector fields keep the horizon structure invariant. Then, we now solve the Killing equations for above metric: The solutions of the above equations are given by The remaining condition L ξ g uu = 0 is automatically satisfied near the horizon because the above solutions lead to L ξ g uu = O ( ρ ) and ρ → 0 at the horizon. Expressing the results in the original coordinates ( t, ρ ), we obtain Since we have the appropriate vector fields ξ a for a given T , we can calculate the charge Q . Finally, expanding T in terms of a set of basis functions T m with we obtain corresponding expansions for ξ a and Q in terms of ξ a m and Q m defined by T m . We choose T m to be the basis so that the resulting ξ a m obeys the algebra isomorphic to Diff S 1 , with { , } as the Lie bracket. Such a T m can be achieved by the choice Here, α is a constant, p is an integer, and g ( ρ ) is a function that is regular at the horizon. Note that α is arbitrary in this approach, which will not affect the final results.", "pages": [ 3, 4 ] }, { "title": "IV. ENTROPY FOR BIGRAVITY", "content": "Using the previous procedure and the black hole solution, we evaluate the black hole entropy. At first, we need to calculate surface term of the bigravity action L B and the vector field ξ a related to the diffeormophism, which leaves the horizon structure invariant. Since the interaction term does not include any derivative terms, the contribution to the surface term does not appear. Therefore, the surface term is two Gibbons-Hawking terms from Ricci scalar R ( g ) and R ( f ), with K = -∇ a N a as the trace of the extrinsic curvature of the boundary. When we consider the Schwarzschild solution, f µν = g µν and f ( r ) = 1 -2 M r with the horizon at r h = 2 M . And the metric corresponding to the coordinates ( t, ρ ) is given by The Bondi-like coordinate transformation (23) is defined as In this coordinate system, the metric is expressed as The vector fields ξ a in the original coordinates ( t, ρ ) have the following expressions: We now calculate the Noether current and the Virasoro algebra. The normal vectors for the horizon are and the Gibbons-Hawking term is given by The charge Q in the near-horizon limit ρ → 0 is given by 1 Here, κ is the surface gravity of the black hole, κ = 4 M , and the factor 2 comes from the two Gibbons-Hawking terms. In the calculation, we use the surface gravity κ , which is related to the expansion of f ( ρ + r h ) in the near-horizon limit, Now, we consider f µν = g µν , so two κ s from two tensor fields have an identical value. Finally, we calculate the central charge with the appropriate expansion of T . For T = T m , T n , the Lie bracket of the charges Q m and Q n (19) is given by We now substitute T m chosen in the previous section (30) into Eqs.(38) and (40) and implement the integration over a cross-section area A , and we obtain Therefore, we find that the central term in the algebra is given by From the central term, we can read off the central charge C and the zero mode energy Q 0 as follows: Using the Cardy formula [10-12], we eventually obtain the entropy This is twice as much as the Bekenstein-Hawking entropy in the Einstein gravity. While we obtain the entropy by using the Noether current corresponding to the surface term, it is necessary to check whether our result is appropriate in other approaches. To compare this result with other methods, we also evaluate the entropy in Wald's approach [28]. Here, because f µν = g µν , the interaction term in Eq.(5) vanishes, and the Lagrangian depends only on the metric g µν . Therefore, we may use the Wald formula: Here, R αβγδ is the Riemann tensor, and /epsilon1 αβ is the binormal for the horizon, which satisfies the condition /epsilon1 αβ = -/epsilon1 βα . For the Schwarzschild solution, we find /epsilon1 tr = 1, and the others are zero. Then, the Wald entropy is given by which is again twice as much as the Bekenstein-Hawking entropy. Generally, constructing the Noether current in Wald entropy is rather complicated when the β n s are arbitrary and f µν expresses the degrees of freedom, which is distinguished with g µν . However, the approach that we use in this paper uses only the surface term to construct the Noether current in contrast to the Wald entropy, which needs the bulk action. Therefore, we can ignore the interaction term in evaluating the black hole entropy, and the remaining terms corresponding to the Noether current are simply the two traces of the extrinsic curvature. This approach is very simple and useful for calculation.", "pages": [ 4, 5, 6 ] }, { "title": "V. CONCLUSION AND DISCUSSION", "content": "In this work, we have shown that the bigravity for the minimal model has a static, spherically symmetric black hole solution. For the minimal model, we have begun with the ansatz f µν = C 2 g µν , but, finally, we have shown that the consistency tells C 2 = 1, that is, f µν = g µν . And we have obtained the Schwarzschild solution and evaluated the entropy for it. Then, we find that the obtained entropy has a double portion of the Bekenstein-Hawking entropy in the Einstein gravity. This is because the surface terms for the two tensor fields f µν and g µν are identical with each other, and they give the same contribution to the Noether current. Of course, while we have only considered the entropy for the Schwarzschild black hole, the stationary axisymmetric solution, that is, the Kerr black hole, is also a solution. Even in this case, the entropy may have also twice as much as that of the Einstein gravity because we have f µν = g µν as well. /negationslash It is interesting that our approach may be generalized to the case of other models. For other models, we may choose different values for β n s in Eq.(1) to reproduce the Fierz-Pauli mass term [25-27]. We may have other consistent solutions with f µν = C 2 g µν but C 2 = 1 in such a model, and there could exist (anti-)de Sitter-Schwarzschild solutions and also (anti-)de Sitter-Kerr solutions if the cosmological constants do not vanish. Forthermore, if two fields are different, the surface terms could give different contributions to the Noether current, and the entropy would be changed. With the procedure using the Noether current of the surface term, we have explicitly shown that the entropy is given by sum of two entropies from two Ricci scalars in bigravity. Similar results are obtain in Refs. [21, 22], and our results do not conflict with the implication of general arguments.", "pages": [ 6 ] }, { "title": "Acknowledgements", "content": "T.K. is partially supported by the Nagoya University Program for Leading Graduate Schools funded by the Ministry of Education of the Japanese Government under the program number N01. The work by S.N. is supported by the JSPS Grant-in-Aid for Scientific Research (S) # 22224003 and (C) # 23540296.", "pages": [ 6 ] } ]

[ { "title": "T. M. HELLIWELL", "content": "Physics Department, Harvey Mudd College Claremont, California, 91711, USA [email protected]", "pages": [ 1 ] }, { "title": "D. A. KONKOWSKI", "content": "Mathematics Department, U.S. Naval Academy, 572C Holloway Road Annapolis, Maryland, 21402, USA [email protected] March 5, 2022", "pages": [ 1 ] }, { "title": "Abstract", "content": "A definition of quantum singularity for the case of static spacetimes has recently been extended to conformally static spacetimes. Here the theory behind quantum singularities in conformally static spacetimes is reviewed, and then applied to a class of spherically symmetric, conformally static spacetimes, including as special cases those studied by Roberts, by Fonarev, and by Husain, Martinez, and N'u˜nez. We use solutions of the generally coupled, massless Klein-Gordon equation as test fields. In this way we find the ranges of metric parameters and coupling coefficients for which classical timelike singularities in these spacetimes are healed quantum mechanically. PACS: 04.20.Dw, 04.62.+v", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Classical singularities, as characterized by the theorems of Hawking and Penrose, are ubiquitous in general relativistic spacetimes (see, e.g., [9]). The theorems do not necessarily indicate a divergence in the curvature, but rather geodesic incompleteness in otherwise maximal spacetimes. Spacetime geodesic incompleteness means, at least in the timelike and null cases, that classical particle paths come to an abrupt end. Classical singularities can be classified by their strengths [4, 9]. Quasiregular singularities are the mildest true singularity; they are topological in nature, basically holes in the fabric of spacetime. Conical singularities, as in idealized cosmic strings, are a good example. The other two types of singularities are stronger, curvature singularities. They are designated nonscalar or scalar depending on whether scalars in the curvature, such as the Ricci scalar and the Kretschmann scalar, diverge. Usually only C 0 scalars are considered, although there have been investigations of higher-order diverging scalar polynomial invariants (see, e.g., [21] and references therein). Nonscalar curvature singularities include those in whimper cosmologies and certain plane-wave spacetimes, whereas scalar curvature singularities are the best-known, occurring at the center of black holes or the beginning of big bang cosmologies. Naked singularities, singularities not covered by an event horizon, of any of these types are even more troublesome; they are not only mathematical curiosities but have observable gravitational effects [47, 45, 46]. The hope is that most singularities, especially naked singularities, can be 'resolved' or 'healed' in a complete quantum theory of gravity, and at least in the case of the two most prominent theories, string theory [13, 31] and loop quantum gravity [1], there is a hint that this might be the case. For example, orbifolds are erased in string theory. What can we hope for in a complete or generic theory of quantum gravity? To answer this question we have stepped back and examined the simplest quantum generalization of classical singularities, the so-called quantum singularities. Quantum singularities are generalizations of geodesic incompleteness to quantum wave packet ill-posedness. The idea originated with a paper by Wald [49] and was further developed by Horowitz and Marolf [14]. The basic idea is to study the behavior of wave packets in a spacetime, and see if they have a well-defined evolution, without placing boundary conditions at the location of the classical singularity that one hopes to heal quantum mechanically. As first proposed, the analysis was restricted to scalar (Klein-Gordon) wave packets in a static spacetime with a classical timelike singularity. That has since been extended to other fields (e.g., Maxwell and Dirac) [12] and recently the authors have proposed an extension to conformally static spacetimes [21, 22]. Here we focus on the application to the conformally static case by studying a class of spherically symmetric spacetimes that includes as special cases those studied by Roberts [40], by Fonarev [6, 27], and by Husain, Martinez, and Nunez [15].", "pages": [ 2, 3 ] }, { "title": "2 Quantum singularities in static spacetimes", "content": "A static spacetime is quantum-mechanically (QM) non singular if the evolution of a test scalar wave packet, representing the quantum particle, is uniquely determined by the initial wave packet, manifold and metric, without having to place boundary conditions at the singularity [14]. Technically, a static ST is QMsingular if the spatial portion of the relevant wave operator, here the Klein-Gordon operator, is not essentially self-adjoint [39, 38] on C ∞ 0 (Σ) in the space of square-integrable functions L 2 (Σ), where Σ is a spatial slice. Arelativistic scalar particle with mass M can be described by the positivefrequency solution [14] to the Klein-Gordon equation in a static spacetime where the spatial operator A is with K = -ξ µ ξ µ . Here ξ µ is the timelike Killing field and D i is the spatial covariant derivative on a static slice Σ. The appropriate Hilbert space is L 2 (Σ). If we initially define the domain of A to be C ∞ 0 (Σ), A is a real, positive, symmetric operator and self-adjoint extensions always exist. If there is only a single, unique extension A E , then A is essentially self-adjoint. In this case, the Klein-Gordon equation for a free scalar particle takes the form with These equations are ambiguous if A is not essentially self-adjoint, in which case the future time development of the wave function is ambiguous. This fact led Horowitz and Marolf [14] to define quantum-mechanically singular spacetimes as those in which A is not essentially self-adjoint. Note that an operator A is said to be self-adjoint if (i) A = A ∗ and (ii) Dom ( A ) = Dom ( A ∗ ), where A ∗ is the adjoint of A and Dom is short for domain [39, 38]. An operator is essentially self-adjoint if (i) is met and (ii) can be met by expanding the domain of the operator A or its adjoint A ∗ so that it is true. One way to test for essential self-adjointness is to use the von Neumann criterion of deficiency indices [48, 39], which involves studying solutions to the equation A Ψ = ± i Ψ, where A is the spatial portion of the Klein-Gordon operator, and finding the number of solutions that are square integrable ( i.e. , ∈ L 2 (Σ) on a spatial slice Σ) for each sign of i . Another approach, which we have used before(see, e.g., [21] and references therein) and will use here, has a more direct physical interpretation. A theorem of Weyl [39, 50] relates the essential self-adjointness of the Hamiltonian operator to the behavior of the 'potential' in an effective one-dimensional Schrodinger equation (made from the radial equation in a cylindrically or spherically symmetric spacetime), which in turn determines the behavior of the scalar-wave packet. The effect is determined by a limit point-limit circle criterion [39]. The technique is straightforward for static spacetimes with timelike singularities. After separating the wave equation for a static metric, with changes in both dependent and independent variables, the radial equation can be written as a one-dimensional Schrodinger equation Hu ( x ) = Eu ( x ) where the operator H = -d 2 /dx 2 + V ( x ) and E is a constant, and any singularity is assumed to be at x = 0. This form allows us to use the limit point-limit circle criteria described in Reed and Simon [39]. Definition . The potential V ( x ) is in the limit circle case at x = 0 if for some, and therefore for all E , all solutions of Hu ( x ) = Eu ( x ) are square integrable at zero. If V ( x ) is not in the limit circle case, it is in the limit point case. A similar definition pertains for x = ∞ : the potential V ( x ) is in the limit circle case at x = ∞ if all solutions of Hu ( x ) = Eu ( x ) are square integrable at infinity; otherwise V ( x ) is in the limit point case at infinity. There are of course two linearly independent solutions of the Schrodinger equation for given E . If V ( x ) is in the limit circle case at zero, both solutions are square integrable ( ∈ L 2 (Σ)) at zero, so all linear combinations are square integrable ∈ L 2 (Σ) as well. We would therefore need a boundary condition at x = 0 to establish a unique solution. If V ( x ) is in the limit point case, the L 2 (Σ) requirement eliminates one of the solutions, leaving a unique solution without the need of establishing a boundary condition at x = 0. This is the whole idea of testing for quantum singularities; there is no singularity if the solution in unique, as it is in the limit point case. The critical theorem is due to Weyl, Theorem 1 (Theorem X.7 of Reed and Simon [39, 50]). Let V ( x ) be a continuous real-valued function on (0 , ∞ ). Then H = -d 2 /dx 2 + V ( x ) is essentially self-adjoint on C ∞ 0 (0 , ∞ ) if and only if V ( x ) is in the limit point case at both zero and infinity. A useful theorem at infinity is Theorem 2 (Theorem X.8 of Reed and Simon [39]). Let V ( x ) be a continuous real-valued function on (0 , ∞ ) and suppose that there exists a positive differentiable function M ( x ) so that (iii) M ' ( x ) / ( M ( x )) 3 / 2 is bounded near ∞ . Then V ( x ) is in the limit point case (complete) at ∞ . A useful theorem near zero is Theorem 3 (Theorem X.10 of Reed and Simon [39]). Let V ( x ) be continuous and positive near zero. If V ( x ) ≥ 3 4 x -2 near zero then V ( x ) is in the limit point case. If for some /epsilon1 > 0, V ( x ) ≤ ( 3 4 -/epsilon1 ) x -2 near zero, then V ( x ) is in the limit circle case. Theorem 3 states in effect that the potential is only limit point if it is sufficiently repulsive at the origin that one of the two solutions of the onedimensional Schrodinger equation blows up so quickly that it fails to be square integrable. Another useful condition, as stated by Reed and Simon [39], which does not require that V ( x ) be positive, is that -d 2 /dx 2 + V ( x ) is limit circle at zero if V ( x ) is decreasing as x goes to zero. Horowitz and Marolf used the Hilbert space L 2 when they studied the essential self-adjointness of the spatial Klein-Gordon operator in static spacetimes with classical timelike singularities [14]. Subsequently, Ishibashi and Hosoya [16] used as the Hilbert space the 1st Sobolev space H 1 ; they then studied 'wave regularity' of Klein-Gordon waves on static spacetimes with a classical timelike singularity. Here we follow the Horowitz and Marolf definition, as it uses the usual L 2 Hilbert space of quantum mechanics. By now many spacetimes have been tested to see whether or not quantum particles heal their classical singularities. For example, we have studied quasiregular [19] and Levi-Civita spacetimes [23, 24], and used Maxwell and Dirac operators [12] as well as the Klein-Gordon operator, showing that they give comparable results. Cylindrically symmetric spacetimes were considered [20], and Blau, Frank, and Weiss [3] in addition to Helliwell and Konkowski [11] have studied two-parameter geometries whose metric coefficients are power-laws in the radius r in the limit of small r . Pitelli and Letelier have considered the global monopole [33], spherical and cylindrical topological defects [35], BTZ spacetimes [34], and have recently extended their discussions along a new path to investigate cosmological spacetimes [25, 26]. They also have a review paper [36] on the mathematical techniques of quantum singularity analysis for static spacetimes along with numerous examples. Pitelli and Saa [37] investigated quantum singularities in Horava-Lifshitz cosmology while Gurtug and co-workers examined the quantum singularity in Lovelock gravity [28], in a model of f ( R ) gravity [42], in an Einstein-Maxwell-Dilaton theory [30], and in a 2+1 dimensional magnetically charged solutions in Einstein-power-Maxwell theory [29]. Unver and Gurtug [44] studied quantum singularities in (2+1) dimensional matter coupled to black hole spacetimes. Seggev [41] studied possible extensions to stationary spacetimes. And, finally more recently Koehn [18] looked at relativistic wave packets in classically chaotic quantum billiards, a BKL-type scenario. A critical question in all of this work is: When is this use of quantum particles effective in healing classical singularities?", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Quantum singularities in conformally static spacetimes", "content": "In a previous paper we described how to extend the methods for static spacetimes to the case of conformally static spacetimes.[21] In particular, we considered conformally coupled scalar fields on conformally static spacetimes with timelike classical singularities. 1 The natural form and separability of the wave equations on conformally static spacetimes was exploited. Here we elaborate on that extension. A spacetime that admits a timelike conformal Killing vector field W is known as conformally stationary [8]. As V. Perlick [32] nicely summarizes, 'If W is complete and there are no closed timelike curves, the spacetime must be a product: M /similarequal R × ˆ M , with a Hausdorff and paracompact 3 manifold ˆ M and W parallel to the R lines. If we denote the projection from M to R by t and choose local coordinates x = ( x 1 , x 2 , x 3 ) on ˆ M , the metric takes the form with µ, ν = 1 , 2 , 3 . ' He goes on to define the more restrictive condition of conformally static, 'If ˆ φ = ∂ µ h , where h is a function of x = ( x 1 , x 2 , x 3 ), we can change the time coordinate according to t ↦→ t + h ( x ), thereby transforming ˆ φ µ dx µ to zero, i.e., making the surface t = constant orthogonal to the t -lines. This is the conformally static case.' Therefore, a conformally static spacetime g µν ( x α ) is related to a static spacetime γ µν ( x a ) by a conformal transformation C ( η ) of the metric. Here C ( η ) is the conformal factor, where η is the conformal time, related to the time t by dt = Cdη . Simply put, g µν ( x α ) = C 2 ( η ) γ µν ( x a ). Here Greek letters α, β, ... label spacetime indices that range over 0, 1, 2, 3, and Latin letters a, b, c, ... label spatial indicies that range over 1, 2, 3. The Lagrangian for a generally coupled scalar field is [2] where M is the mass of the scalar particle, R is the scalar curvature, and ξ is the coupling ( ξ = 0 for minimal coupling and ξ = 1 / 6 for conformal coupling.)[2] Varying the action S = ∫ L d 3 x gives the Klein-Gordon field equation, In the massless case with conformal coupling, this field equation is conformally invariant under a conformal transformation of the metric and field; in this case the inner product respecting the stress tensor for the field is also conformally invariant. This led Ishibashi and Hosoya [16] to state, in the context of wave regularity, that 'the calculation is as simple as that in the static case when singularities in conformally static space-times are probed with conformally coupled scalar fields.' The conformally static metric has the form where a, b, c = 1 , 2 , 3. Then as shown by Kandrup[17], mode solutions χ, ζ of a wave equation on the static portion of the metric (i.e., without the conformal factor C 2 ( η )) on a Hilbert space L 2 (Σ) have the inner product where γ is the determinant of the spatial portion of the metric. At this point we consider the radial portion alone, change variables and write the radial equation in one-dimensional Schrodinger form, Hu ( x ) = Eu ( x ), where the operator H = -d 2 /dx 2 + V ( x ) and E is a constant, with the singularity at x = 0. The inner product here is simply ∫ dx | u ( x ) | 2 and the Hilbert space is L 2 (0 , ∞ ). One can now simply apply the limit point limit circle criterion as easily as in the static case in order to determine the quantum singularity structure.", "pages": [ 7, 8 ] }, { "title": "4 A class of conformally static spacetimes", "content": "We now specialize to a class of conformally static, spherically symmetric spacetimes with metrics of the form where d Ω 2 = dθ 2 +sin 2 θdφ 2 , with the time-dependent conformal factor a 2 ( t ). We study solutions of the generally coupled, massless Klein-Gordon equation of Eq. (7). The curvature scalar is where overdot ≡ d/dt and prime ≡ d/dr . The Klein-Gordon equation separates into Φ ∼ T ( t ) F ( r ) Y /lscriptm ( θ, φ ), where the Y /lscriptm are spherical harmonics. The equation for T ( t ) then becomes while the radial equation for F ( r ) becomes where q is a separation constant. The first step in testing for quantum singularities is to convert the radial equation into a one-dimensional Schrodinger equation with normalization The conversion from r to x and F ( r ) to u ( x ) is carried out with F ( r ) = u ( x ) /S ( r ) and We are then able to write G ( x ) = E -V ( x ), where E is a separation constant and the potential is We can then explore V ( x ) near the singularity to see if a spacetime of this form is limit point (LP) or limit circle (LC), using Theorem X.10 of Reed and Simon [39]. To do so, let the singularity be located at r = r 0 , and write an arbitrary radius in the form r = r 0 + /epsilon1 . Assume that both f and S can be represented to lowest order by a power law in /epsilon1 near the singularity, namely f = C 0 /epsilon1 n and S = C 1 /epsilon1 m , so we can write /epsilon1 in terms of x by where we confine ourselves to n ≤ 1 / 2 so that /epsilon1 is real and so that the singularities occur at /epsilon1 = 0. For n < 1 / 2, the singularities are at /epsilon1 = 0 and x = 0; for n = 1 / 2, the singularities are at /epsilon1 = 0 and x = -∞ . Substituting these into V ( x ) gives, to lowest order, Consider now the n < 1 / 2 case. The singularity is at x = 0, and the potential V ( x ) is in the LP case if V ( x ) ≥ 3 / 4 x 2 near the singularity; otherwise it is LC. First, suppose that n + m < 1, so that the first term in V ( x ) dominates. We can therefore characterize the LC and LP regions in an ( m,n ) parameter plane, for any given value of the coupling constant ξ . If ξ = 0, i.e., the coupling is minimal, then the three LC regions correspond to the following, each with n < 1 / 2: (a) n -1 / 2 < m < 0, (b) 0 < m < -3 n + 3 / 2 < 3 / 4, and 3 / 4 < m < -n + 1. The two LP regions correspond to (a) m < n -1 / 2 < 0 and (b) -3 n + 3 / 2 < m < -n + 1, both with n < 1 / 2. If instead ξ = 1 / 6, i.e., the coupling is conformal, then V ( x ) is in the LP case if m ≤ (11 / 2) n -9 / 4, and in the LC case if -n + 1 > m > (11 / 2) n -9 / 4, again restricted to n < 1 / 2. The various regions can easily be worked out for other choices of coupling constant if desired. Now suppose that n + m > 1, so the second term in V ( x ) dominates. Then V ( x ) is in the LP case if /lscript ( /lscript + 1) + 2 ξ > 0, and in the LC case if /lscript ( /lscript +1)+2 ξ < 0. If /lscript ( /lscript +1)+2 ξ = 0, the second term vanishes, so the first term dominates regardless of n + m . In the n = 1 / 2 case the singularity is at x = -∞ , and it is null. Therefore the n = 1 / 2 spacetime is globally hyperbolic, the spatial Klein-Gordon operator is essentially self-adjoint, and the potential is necessarily in the LP case [14]. The use of quantum particles heals the classical singularity in this case, for all values of the other parameters.", "pages": [ 8, 9, 10, 11 ] }, { "title": "5 Special cases", "content": "As special cases we consider three important dynamical and spatially inhomogeneous exact solutions to Einstein's equations. All involve massless scalar fields coupled to gravity. These spacetimes can be taken as simple models of the collapse of matter to form or grow black holes or naked singularities, a process that normally requires numerical studies to analyze the complicated differential equations that arise. The Einstein-scalar field spacetime models in this section were constructed by Roberts [40] in 1989, Husain, Martinez, and N'u˜nez (HMN) [15] in 1994, and Fonarev [6] in 1995. The energy-momentum tensor for the minimally-coupled scalar field has in each case the form which yields the Einstein equation Here φ is the scalar field and U is its potential. The potential is zero for both the Roberts and HMN spacetimes, and U = U 0 e -√ 8 πλφ for Fonarev, where U 0 and λ are constants. Now we consider each metric in turn; all are conformally static and spherically symmetric. The spacetime is self-similar, with a massless scalar field that models collapse to a timelike naked singularity. It has a timelike scalar curvature singularity at r = 0 and a global structure that is identical to that of the negativemass Schwarzschild solution. The existence of a scalar curvature singularity requires not only that some scalar in the curvature diverge, but also that either null or timelike geodesics (or both) reach the singularity with a finite affine parameter λ . In fact, the curvature scalar R diverges as r → 0 for Roberts, so the first test is met. It is straightforward to show that inward-directed radial null geodesics in the general class of spacetimes we are considering possess the two first integrals of motion dr/dλ = -K/a 2 ( t ) and dt/dλ = K/a 2 ( t ) f 2 ( r ), where K is a positive constant. Therefore the affine length of a null geodesic starting at radius r 0 at time t 0 and ending at smaller radius r is where the notation means that the quantity t 0 + ∫ r 0 r dr/f 2 ( r ) is to be substituted for t in a ( t ). For the Roberts spacetime, with f ( r ) = 1 and a ( t ) = e t , it follows that which is finite as r → 0. Therefore there is a classical scalar curvature singularity at r = 0. Is r = 0 also quantum mechanically singular? Substituting the quantities f ( r ) and S ( r ) into the general potential V ( x ), it is straightforward to show, using Theorem 2 of Section 2, that there exists a positive constant ( M ) such that all three criteria listed in Theorem 2 criteria are satisfied, for any finite value of /lscript and ξ . Therefore the Roberts spacetime is complete at infinity. In terms of our general form, the Roberts parameters describing f ( r ) and S ( r ) near r = 0 are C 0 = 1, n = 0, C 1 = √ p , and m = 1 / 2. Therefore as x → 0, /negationslash It follows from Theorem 3 of Section 2 that the Roberts spacetime is LC (quantum mechanically singular) unless ξ ≥ 2, in which case it is LP (quantum mechanically nonsingular). In particular, the spacetime is LC for both the minimally coupled and conformally coupled cases. Quantum mechanics fails to heal the classical singularity in these cases. The time equation for the Roberts spacetime is which is a damped harmonic oscillator equation with solutions for the overdamped (6 ξ -q < 1) and underdamped (6 ξ -q > 1) cases, and for the critically damped case (6 ξ -q = 1), where A and B are constants. If we had chosen the conformal factor to be a ( t ) = constant, corresponding to a class of static spacetimes, the time equation would be simply the harmonic oscillator equation. It is interesting that the Roberts conformally static spacetime converts the time portion of the wave equation from a simple harmonic oscillator for the corresponding static spacetime, to that of a damped harmonic oscillator. In any case the time solutions do not influence whether or not a conformally static spacetime is quantum mechanically singular. (where α = ± √ 3 / 2 only.) The HMN spacetimes were constructed to model scalar field collapse. They have the conformal Killing vector v = ∂/∂t as well as the three Killing vectors associated with spherical symmetry. Since the spacetime is not asymptotically flat (but is asymptotically conformally flat), the authors interpret it to be an inhomogeneous scalar field cosmology. We are considering only the nonstatic solutions, which have scalar curvature singularities at r = 2 and t = -b/a (where a = ± 1) for both α = √ 3 / 2 and α = -√ 3 / 2. Without loss of generality b can be set equal to zero, so then the time coordinate ranges over 0 ≤ t < ∞ for a = 1 and -∞ < t ≤ 0 for a = -1. These correspond to expanding and collapsing scalar fields. HMN consider only the collapsing case, in which case the spacetime has a timelike scalar curvature singularity at r = 2 (that is, at the areal radius R = rS = 0) and a spacelike singularity at t = 0. The coordinate r ranges over 2 ≤ r < ∞ . Using f ( r ) and a ( t ) for the HMN metric in Eq. 24, one can show that radial null geodesics reach the r = 2 timelike singularity with finite affine length, so r = 2 fulfills all necessary criteria for being a classical scalar curvature singularity. Substituting the quantities f ( r ) and S ( r ) into the general potential V ( x ), it is straightforward to show, using Theorem 2 of Section 2, that there exists a positive constant ( M ) such that all three criteria listed in Theorem 2 criteria are satisfied, for any finite value of /lscript and ξ . Therefore the HMN spacetimes are also complete at infinity. In terms of our general form, the HMN parameters describing f ( r ) and S ( r ) near the timelike singularity at r = 2 are C 0 = (2) -α/ 2 , n = α/ 2, C 1 = (2) ( α +1) / 2 , and m = (1 -α ) / 2, where in each case α = ± √ 3 / 2 only. In these cases the potential becomes Regardless of the choice of signs, the 1 /x 2 terms dominate, so the HMN spacetimes are LP if ξ ≥ 2[(1 + √ 3 / 2) / (1 -√ 3 / 2)], and are otherwise LC. So in particular they are LC, and therefore quantum mechanically singular, for both the minimally coupled and conformally coupled cases. The time equation for the HMN metric is Bessel's equation /negationslash (3) Fonarev spacetimes [6, 27] have metric coefficients a ( t ) = | t/t 0 | 2 / ( λ 2 -2) , f ( r ) = (1 -2 w/r ) α/ 2 , and S 2 ( r ) = r 2 (1 -2 w/r ) 1 -α (where 0 < α 2 ≤ 3 / 4, but α 2 = 1 / 2.) Our index n is less than 1/2 in all the Fonarev solutions. The parameter λ , a so-called steepness parameter , satisfies 0 < λ 2 ≤ 6 with λ 2 = 2 for a non-negative potential; λ is related to α by α = λ/ √ λ 2 +2. If λ 2 = 6, the solution reduces to HMN. The Fonarev spacetimes are a generalization of the HMN spacetimes, to include a scalar field potential, as in Eq. [22]. Massless scalar fields with exponential potentials appear in supergravity theories [43] and in dimensionally reduced effective four-dimensional theories [7, 5], as explained by Maeda [27] in his paper on the physical interpretation of the Fonarev solution. /negationslash If the parameter w = 0, the Fonarev solution reduces to the flat FriedmannRobertson-Walker cosmology. Here we consider w = 0, which becomes the FRW spacetime as r →∞ . The coordinate r ranges 2 w < r < ∞ for w > 0 and 0 < r < ∞ for w < 0. For w = 0, there are central scalar curvature singularities at r = 0 and at r = 2 w . For 0 < λ 2 < 2, there is a null curvature singularity as t → ±∞ , while for 2 < λ 2 ≤ 6 there as a spacelike curvature singularity at t = 0. Thus the t - coordinate ranges are ∞ < t < 0 and 0 < t < ∞ , with big bang and big crunch singularities. We will study only the locally-naked timelike central singularities. 2 Inward-directed null geodesics reach the timelike singularity at r = 2 w in finite affine parameter according to Eq. 24, so r = 2 w fulfills necessary requirements to be a classical scalar curvature singularity. /negationslash /negationslash In terms of our general form, the Fonarev parameters describing f ( r ) and S ( r ) near the timelike singularity at r = 2 w are C 0 = (2 w ) -α/ 2 , n = α/ 2, C 1 = (2) ( α +1) / 2 ( w ) ( α -1) / 2 , and m = (1 -α ) / 2, where in each case 0 < α 2 ≤ 3 / 4, but α 2 = 1 / 2. Using these parameter values, it is then straightforward to show from Eq.[20] that the potential V ( x ) for these spacetimes is in the LC case if and in the LP case if /negationslash In particular, Fonarev spacetimes are LC for both the minimally and conformally coupled cases. For the Fonarev spacetime, the time equation is again Bessel's equation,", "pages": [ 11, 12, 13, 14, 15, 16 ] }, { "title": "6 Conclusions", "content": "We have confirmed that the Horowitz and Marolf definition of quantum singularities for static spacetimes can be extended to the case of conformally static spacetimes. We then tested the formalism for a class of conformally static, spherically symmetric spacetimes, a class that includes the special cases of spacetimes studied by Roberts, by Fonarev, and by Husain, Martinez, and N'u˜nez (HMN). We used as quantum fields the solutions of the generally coupled, massless Klein-Gordon equation, and Weyl's limit point limit circle criteria for judging the existence of quantum singularities. This required that we write the radial part of the Klein-Gordon equation in the form of a one-dimensional Schrodinger equation, and evaluate the behavior of the associated potential energy in the vicinity of the singularity. /negationslash In this way we found that the Roberts spacetimes are limit circle (i. e., quantum mechanically singular) for both the minimally coupled (coupling parameter ξ = 0) and conformally coupled ( ξ = 1 / 6) cases, and in general if the coupling parameter ξ < 2; they are limit point (quantum mechanically nonsingular) if ξ ≥ 2. The HMN spacetimes are limit circle if ξ < 2[(1 + √ 3 / 2) / (1 -√ 3 / 2)], and so are also limit circle for both the minimally coupled and conformally coupled cases. Similarly, Fonarev spacetimes are limit circle if ξ < 2(1 -α ) / (1 + α ), where the parameter α in these spacetimes is constrained by 0 < α 2 ≤ 3 / 4, but α 2 = 1 / 2. In particular, all Fonarev spacetimes are limit circle for both minimally coupled and conformally coupled cases. As in the studies of other metrics up to now, we have found that the use of the Horowitz and Marolf procedure is successful in healing classical singularities for some parameter values in the metrics, but not for others. One goal for the future is to understand more comprehensively and more deeply which singularities can be healed in this way, and which cannot, and why.", "pages": [ 16, 17 ] } ]

[ { "title": "The Pesky Power Asymmetry", "content": "Liang Dai, Donghui Jeong, Marc Kamionkowski and Jens Chluba Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 (Dated: March 27, 2013) Physical models for the hemispherical power asymmetry in the cosmic microwave background (CMB) reported by the Planck Collaboration must satisfy CMB constraints to the homogeneity of the Universe and quasar constraints to power asymmetries. We survey a variety of models for the power asymmetry and show that consistent models include a modulated scale-dependent isocurvature contribution to the matter power spectrum or a modulation of the reionization optical depth, gravitational-wave amplitude, or scalar spectral index. We propose further tests to distinguish between the different scenarios. PACS numbers: 98.80.-k The Planck Collaboration has reported a hemispherical asymmetry in the cosmic microwave background (CMB) fluctuations [1], thus confirming a similar power asymmetry seen in the Wilkinson Microwave Anisotropy Probe (WMAP) data [2]. The new data, with far better multifrequency component separation, make it more difficult to attribute the asymmetry to foregrounds. The asymmetry is also seen in the Planck data to extend to smaller scales, and it is thus of greater statistical significance than in the WMAP data. If the asymmetry is modeled as a dipolar modulation of an otherwise statistically isotropic CMB sky [3-5], the best-fit dipole has direction ( l, b ) = (227 , -27) and amplitude (in terms of r.m.s. temperature fluctuations on large angular scales, multipoles glyph[lscript] < 64) of A = 0 . 072 ± 0 . 022, although the asymmetry extends to higher glyph[lscript] [1, 6]. This power asymmetry is, as we explain below, extremely difficult to reconcile with inflation. Given the plenitude of impressive successes of inflation (the nearly, but not precisely, Peebles-Harrison-Zeldovich spectrum, adiabatic rather than isocurvature perturbations, the remarkable degree of Gaussianity), the result requires the deepest scrutiny. While there are some who may wave away the asymmetry as a statistical fluctuation [7], evidence is accruing that it is statistically significant. There is moreover the tantalizing prospect that it may be an artifact of some superhorizon pre-inflationary physics. Here we investigate physical explanations for the origin of the asymmetry and put forward new tests of those models. We begin by reviewing the tension between the asymmetry and inflation. We then provide a brief survey of prior hypotheses and discuss the very stringent constraints imposed by the CMB temperature quadrupole and by upper limits to hemispherical asymmetries in quasar abundances. We review some existing models for the asymmetry and then posit that the asymmetry may be due to spatial variation of standard cosmological parameters (e.g., the reionization optical depth, the scalar spectral index, and gravitational-wave amplitude) that affect CMB fluctuations without affecting the total density nor the matter power spectrum. We show how the different scenarios can be distinguished by the glyph[lscript] dependence of the asymmetry, and we discuss other possible tests of the models. The CMB power asymmetry is modeled as a dipole modulation of the power; i.e., the temperature fluctuation in direction ˆ n is (∆ T/T )(ˆ n ) = s (ˆ n )[1 + A ˆ n · ˆ p ], where s (ˆ n ) is a statistically isotropic map, A is the power-dipole amplitude, and ˆ p is its direction. However, the asymmetry cannot arise due to a preferred direction in the three-dimensional spectrum P ( k ) [8], as reality of the fractional matter-density perturbation δ ( r ) relates the Fourier component ˜ δ ( k ) for wavevector k to that, ˜ δ ( -k ) = ˜ δ ∗ ( k ), of -k . The asymmetry must therefore be attributed to a spatial modulation of threedimensional power across the observable Universe. The modulation required to explain the asymmetry can then be written in terms of a spatially-varying power spectrum, P ( k, r ) = P ( k )[1 + 2 A ˆ p · r /r ls ], where r ls is the distance to the last-scattering surface, for modes inside the comoving horizon at present ( k glyph[greaterorsimilar] H -1 0 ). Any model that modulates the power must do so without introducing a modulation in the density of the Universe. A long-wavelength isocurvature density fluctuation with an amplitude O (10%) must generates a temperature dipole two orders of magnitude greater than is observed. If the density fluctuation is adiabatic, then the intrinsic temperature dipole is cancelled by a Doppler dipole due to our infall toward the denser side. Even so, in this case, the small temperature quadrupole and octupoles constrain the density in the observed Universe to vary by no more than O (10 -3 ) [9]. These considerations make it unlikely that the power asymmetry could arise in single-clock models for inflation. In these models, the inflaton controls both the power-spectrum amplitude and the total density, thus making it difficult to introduce an O (10%) modulation in the power with a glyph[lessorsimilar] O (10 -3 ) modulation of the total density. These arguments were made precise for slow-roll inflation with a standard kinetic term in Ref. [10]. We surmise that it may be difficult to get to work also with nontrivial kinetic terms [11], especially given the increas- ingly tight constraints to such models from Planck. There are additional and very stringent constraints to power modulation-even in models that can do so without modulating the density-from the Sloan Digital Sky Survey (SDSS) quasar sample [12]. These quasars are found at distances nearly half of the comoving distance to the CMB surface of last scatter, and their abundances depend very sensitively on the amplitude of the primordial power spectrum. A detailed analysis [12] finds an upper limit A < 0 . 0153 (99% C.L.) to the amplitude of an asymmetry oriented in the direction of the CMB dipole. While this is consistent with the 3 σ lower limit A glyph[greaterorsimilar] 0 . 006 inferred from Ref. [13], it is roughly five times smaller than the central value A = 0 . 072. Of course, quasars probe the power spectrum primarily at wavenumbers k ∼ 1 Mpc -1 , while the glyph[lscript] glyph[lessorsimilar] 60 CMB power probes k glyph[lessorsimilar] 0 . 035 Mpc -1 . A model that produces a power asymmetry with a sufficiently strong scale dependence may (neglecting the possible extension of the CMB asymmetry to higher glyph[lscript] ) allow the central CMB value to be consistent with the quasar constraint. To summarize: Any mechansim that accounts for the CMB power asymmetry at glyph[lscript] glyph[lessorsimilar] 60 must (1) do so while leaving the amplitude of the long-wavelength adiabatic or isocurvature density fluctuation across the observable Universe to be glyph[lessorsimilar] 10 -3 , and (2) leave the power asymmetry at k ∼ 1 Mpc -1 small. Given that the asymmetries are seen at glyph[lscript] < 60, it is also likely that any causal mechanism must involve inflation. We now run through a number of scenarios for the power asymmetry. Ref. [10] arranged for a scale-independent power modulation, while keeping the total density fixed, by introducing a curvaton during inflation. The curvaton can then contribute appreciably to a modulation of the density-perturbation amplitude without modulating the mean density. This model is inconsistent with the quasar bound for the best-fit value A = 0 . 072. This model also predicts a non-Gaussianity parameter f (local) NL glyph[greaterorsimilar] 25( A/ 0 . 07) 2 , and would thus be ruled out for A = 0 . 072 by Planck constraints [14] to f (local) NL , even if there were no quasar constraint. Ref. [15] also presented a modified inflationary theory wherein the curvaton decays after dark matter freezes out thus giving rise to an isocurvature perturbation that is subdominant relative to the usual adiabatic perturbations from inflaton decay. A postulated superhorizon perturbation to the curvaton field then modulates the amplitude of the isocurvature contribution across the observable Universe. The model parameters can be chosen so that this spatially-varying isocurvature contribution is scale-dependent, with a CMB power spectrum that peaks around glyph[lscript] ∼ 10, falls off rapidly from glyph[lscript] ∼ 10 to glyph[lscript] ∼ 100, and is then negligibible at higher glyph[lscript] . The model predicts an isocurvature contribution to primordial perturbations that may be in tension with new Planck upper limits [16], although more analysis may be required, given the asymmetric nature of the contribution, to determine the consistency of the model with current Planck data. Ref. [17] recently postulated that the power asymmetry may arise in single-field inflation through some sort of non-Gaussianity that increases the bispectrum in the squeezed limit. In this way, the homogeneity constraint imposed by the CMB can be evaded. This model, however, gives rise to a roughly scale-independent power asymmetry and thus conflicts with the quasar constraint. It may still be possible though, to adjust the parameters to reduce the asymmetry on small scales. We now suppose that the power asymmetry may arise from a modulation of one of the cosmological parameters that affects the CMB power spectrum. There are a number of cosmological parameters-abundances of cosmic constituents, inflationary observables, fundamentalphysics parameters [18]-that affect the CMB power spectrum [19]. If there is a difference between the value of one of these cosmological parameters on one side of the sky and the value on the other side, then the CMB power spectrum one side may differ from that on the other side. For each of these parameters p , we calculate ∆ C TT glyph[lscript] = ( ∂C TT glyph[lscript] /∂p )∆ p , where the amplitude ∆ p is chosen so that it produces an asymmetry A = 0 . 072. We assume here that this asymmetry is determined from the data by weighting the asymmetry in all sphericalharmonic modes equally up to glyph[lscript] max = 64; i.e., The fractional power-spectrum differences, normalized to give A = 0 . 072 are given in Fig. 1. We then plot in Fig. 2 the fractional change ∆ P ( k ) in the matter power spectrum induced by the modulations of each of the parameters considered in Fig. 2. We do not consider parameters that only affect the recombination history (e.g., the helium abundance, fine-structure constant, etc.), as these modify CMB power only on small scales (high glyph[lscript] ), not at the lower glyph[lscript] where the asymmetry is best seen. Ref. [6] considers, among other possibilities, a modulation in Ω b , the baryon density. However, if Ω b is modulated by O (10%), while holding all other parameters fixed, it will introduce a large-scale inhomogeneity in conflict with the CMB dipole/quadrupole/octupole. These constraints can be evaded through a compensated isocurvature perturbations (CIP), wherein Ω b and Ω cdm , the dark-matter density, are both modulated in such a way that the total matter density Ω b + Ω cdm remains constant across the observable Universe [20]. Such a hypothesis results, as Fig. 2 shows, in a power asymmetry at small scales larger than allowed by the quasar constraint. It can therefore be ruled out. Ref. [4] considered a dark-energy density that varies linearly with position along the direction picked out by the power dipole. The dark-energy density is negligible at the surface of last scatter, but it affects at later times CMB fluctuations in two different ways. First of all, changes to Ω de change the ISW contribution to lowglyph[lscript] power, but this is a relatively small effect. The other consequence is a change to the angle subtended by a given comoving scale. The effect of a dipolar modulation, across the sky, of this mapping is equivalent to, and indistinguishable from, that induced by a peculiar velocity. A variation of Ω de large enough to account for the A glyph[similarequal] 0 . 072 asymmetry would thus yield a CMB temperature dipole considerably larger than that observed. This explanation can thus be learned out. More generally, a modulation of any of the parameters that affects the total density of the Universe that is large enough to account for the power asymmetry will give rise to a large-angle CMB fluctuation in gross conflict with observations. We thus now consider modulations to several parameters that affect the CMB power spectrum without changing the matter densities. We begin with a variation to the scalar spectral index n s . In considering a modulation of n s , we must specify a pivot wavenumber k 0 , at which the power on both sides of the sky is equal. Here we choose this pivot point to be k 0 = 1 Mpc -1 so that the quasar constraint is satisfied. Doing so allows us accommodate a large-scale power-asymmetry amplitude A = 0 . 072 with a value of n s glyph[similarequal] 0 . 93 on one side of the sky and n s glyph[similarequal] 0 . 99 on the other. This model then predicts that the CMB power asymmetry should decrease, but relatively slowly, with higher glyph[lscript] , as shown in Fig. 1. Along similar lines, one can vary the gravitationalwave amplitude from one side of the sky to the other. The gravitational-wave energy-density flucutation required to account for the lowglyph[lscript] power asymmetry is small enough to satisfy the homogeneity constraints, and gravitational waves contribute nothing to P ( k ). This hypothesis can thus explain the CMB power asymmetry without violating other constraints. The only difficulty with the model is that an asymmetry amplitude A = 0 . 072 requires a gravitational-wave amplitude on one side of the Universe ten times larger than the homogeneous upper limit, a magnitude that may be not only unpalatable, but also inconsistent with current data. Still, an asymmetry of smaller amplitude, say A ∼ 0 . 015 may be consistent. We finally consider a dipolar modulation of τ , the reionization optical depth. The optical depth primarily suppresses power at glyph[lscript] glyph[greaterorsimilar] 20, but there is also a small increase in power at lower glyph[lscript] from re-scattering of CMB photons. We find from Figs. 1 and 2 that a modulation of τ can account for the asymmetry in the CMB without affecting (by assumption, really) P ( k ) at quasar scales. An asymmetry A = 0 . 072 can be obtained by taking τ = 0 . 017 on one side of the Universe and τ = 0 . 21 on the other side. A value of τ glyph[similarequal] 0 . 017 implies a reionization redshift z glyph[similarequal] 3, which is lower than quasar absorption spectra allow. Still, an asymmetry A glyph[similarequal] 0 . 05 could be accommodated while preserving a reionization redshift z glyph[greaterorsimilar] 6 everywhere. We surmise, without developing a detailed microphysical model that a spatial modulation in the primordial P ( k ) at k glyph[greatermuch] Mpc -1 could give rise to such a τ modulation, or perhaps one of slightly lower amplitude, given the highly uncertain physics reionization and the possibly strong dependence to small changes in initial conditions. To summarize, we have four models that can account for the CMB asymmetry while leaving the Universe homogeneous and without affecting P ( k ) on quasar scales. There is the inflation model of Ref. [15] whose possible phenomenological weakness may be in an isocurvature contribution in tension with current upper limits, and there is modulation of n s , which is phenomenologically quite attractive. The other two models-variable gravitational-wave amplitude and variable τ -require variations in the parameters that are probably larger than are allowed. Still, we continue to consider them for illustrative purposes and in case the asymmetry amplitude is found in the future to be smaller but still nonzero. We now discuss measurements that can distinguish between the different scenarios. First of all, with Plank data we should be able to measure the difference ∆ C TT glyph[lscript] in the power spectra between the two hemispheres, as a function of glyph[lscript] . The model of Ref. [15] and a modulation of the gravitational-wave amplitude both predict little or no asymmetry at glyph[lscript] glyph[greaterorsimilar] 100, while the power asymmetry should extend to much higher glyph[lscript] if it is due to a modulation of n s or τ . There are also the temperature-polarization correlations ( C TE glyph[lscript] ) and polarization autocorrelations ( C EE glyph[lscript] ) shown in Figs. 3 and 4. The TE difference power spectrum, in particular, should help distinguish, through the sign at low glyph[lscript] of ∆ C TE glyph[lscript] , modulation of gravitational waves from modulation of τ . An asymmetry in P ( k ) can be probed at lower k than quasars probe through all-sky lensing, Comptony , and/or cosmic-infrared-background (CIB) maps, such as those recently made by Planck [21]. Asymmetries at ∼ 0 . 1 -1 Mpc -1 scales may also be probed via number counts in populations of objects other than quasars [22]. Probing asymmetries in P ( k ) at k glyph[greatermuch] Mpc -1 , as required if τ is modulated, may be more difficult in the near term, although future 21-cm maps of the neutralhydrogen distribution during the dark ages [23] and/or epoch of reionization [24] may do the trick, as may maps of the µ distortion to the CMB frequency spectrum [25]. A modulation in the gravitational-wave background may show up as an asymmetry in CMB B modes [26]. In fact, if the asymmetry is attributed to a gravitational-wave asymmetry, suborbital B-mode searches may do better to search on one side of the sky than on the other! If the asymmetry has an origin in the coupling of an inflaton to some other field, a 'fossil' field, during inflation, there may be higher-order correlation functions at smaller scales that can be sought [28]. It may also be instructive to perform a bipolar spherical harmonic (BiPoSH) [27] analysis and consider the odd-parity dipolar (i.e., L = 1, with glyph[lscript] + glyph[lscript] ' + L =odd) BiPoSH [29], which may shed light on the nature of a fossil field [30] that would give rise to the asymmetry. While a modulation in n s can account for the CMB power asymmetry, and a modulation to the gravitationalwave amplitude may do so for a smaller-amplitude asymmetry, these are both no more than phenomenological hypotheses, and there may be difficult work ahead to accommodate them within an inflationary model. Similar comments apply to variable optical depth. There may also well be a completely different explanation, like bubble collisions [31] or non-trivial topology of the Universe [32], that is not well described by the modulation models we have considered here. Any such model must still, however, satisfy the CMB constraints to homogeneity and the quasar limit to a small-scale power asymmetry. Finally, if the asymmetry does indeed signal something beyond the simplest inflationary models, then it may be possible to draw connections between it and the tension between CMB and local models of the Hubble constant, between different suborbital CMB experiments, and perhaps other anomalies in current data. This work was supported by DoE SC-0008108 and NASA NNX12AE86G.", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "CMB bounds on dark matter annihilation: Nucleon energy-losses after recombination", "content": "Christoph Weniger a , Pasquale D. Serpico b , Fabio Iocco c , Gianfranco Bertone a a GRAPPA Institute, Univ. of Amsterdam, Science Park 904, 1098 GL Amsterdam, Netherlands b LAPTh, Univ. de Savoie, CNRS, B.P.110, Annecy-le-Vieux F-74941, France and c The Oskar Klein Center for CosmoParticle Physics, Department of Physics, Stockholm University, Albanova, SE-10691 Stockholm, Sweden (Dated: October 17, 2018) We consider the propagation and energy losses of protons and anti-protons produced by dark matter annihilation at redshifts 100 < z < ∼ 2000. In the case of dark matter annihilations into quarks, gluons and weak gauge bosons, protons and anti-protons carry about 20% of the energy injected into e ± and γ 's, but their interactions are normally neglected when deriving cosmic microwave background bounds from altered recombination histories. Here, we follow numerically the energyloss history of typical protons/antiprotons in the cosmological medium. We show that about half of their energy is channeled into photons and e ± , and we present a simple prescription to estimate the corresponding strengthening of the cosmic microwave background bounds on the dark matter annihilation cross section. PACS numbers: 95.30.Cq, 95.35.+d, 98.80.Es LAPTH-007/13", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Astrophysical and cosmological observations provide compelling evidence that about 85% of all the matter in the Universe is in the form of Dark Matter (DM), an elusive substance which is currently searched for with a variety of observational and experimental channels at colliders, in underground detectors, or via indirect signals from DM annihilation or decay [1]. Cosmic microwave background (CMB) anisotropy and polarization data provide interesting constraints on the properties of DM particles [2-4]. Secondary particles injected via DM annihilation (or decay) after recombination, around redshift z ∼ O (600), would in fact inevitably affect the recombination history of the Universe and widen the surface of last scattering, which is tightly constrained by CMB observations as discussed in Refs. [5-9], and more recently in Refs. [10-16]. Possible effects on the cosmological recombination spectrum were discussed in Ref. [17]. One of the main reasons of interest for these constraints is that, in contrast with other indirect searches, they do not rely on knowledge of astrophysical DM structures, affected by the complex aspects of non-linear gravity as well as complicated feedback due to baryons [7, 8]. The CMB probe is thus as reliable as the description of the basic atomic and nuclear/particle physics processes involved is. The robustness (and astrophysical independence) of the CMB constraints motivates further efforts to assess and improve the error budget. The degree of sophistication in modeling the atomic processes down to the recombination stage is quite elevated and has also seen recent improvements, see e.g. [18]. Here we revise one aspect related to the accuracy of the nuclear/particle physics part. It is typically assumed that protons are highly penetrating and poor at transferring energy to the intergalactic medium (IGM) - a misnomer, since no galaxies have formed at such high redshifts - and their energy release to the medium is neglected [2, 6] (see however the comment in Ref. [10]). In this article, we estimate the additional energy released to the gas by the interactions of the high-energy protons and antiprotons formed among DM annihilation final states. This article is structured as follows: In Sec. II we review the basic physics substantiating the two points above. In Sec. III we describe our computational technique and present our results in Sec. IV. Finally, in Sec. V we conclude.", "pages": [ 1 ] }, { "title": "II. PHYSICAL PROCESSES", "content": "Protons and antiprotons carry a significant fraction of the overall energy emitted in DM annihilations into quarks, gluons or weak gauge bosons. Typically, this amounts to ∼ 20% of the energy channelled into e ± and γ 's, see for example Fig. 4 in [19], and a fraction of this energy will be inevitably transferred to the IGM. Neutrons and antineutrons decay very fast and behave practically like protons and antiprotons, while (anti-)deuterons and heavier nuclei are produced in negligible amounts in DM annihilations. At the epochs of interest here, which correspond to a redshift z = O (10 3 ), p/¯p's propagate in a medium which is eight or nine orders of magnitude more dense than at the present epoch, with typical proton and Helium gas densities up to O (10 2 ) particles/cm 3 . Even neglecting the interaction with photon baths of densities of O (10 11 )cm -3 , a typical p-p inelastic cross-section of 30 mb yields a collision timescale lower than the age of the Universe at decoupling. Hence, we expect that in general the probability for a nucleon to interact within a Hubble time is large, and that a significant energy deposition takes place (similar estimates can be found in Ref. [10]). More specifically, p/¯p's will undergo the following pro-", "pages": [ 1 ] }, { "title": "cesses:", "content": "In Fig. 1 we show the inelastic scattering cross-sections as a function of the p/¯p kinetic energy (top panel), and the corresponding fractional energy loss rate E -1 dE/dt times the Hubble time H -1 at redshift z = 1000 as function of the p/¯p kinetic energy (bottom panel). Values of order unity or even larger indicate potentially large effects. In case of anti-protons, we plot annihilating and non-annihilating rates separately. For nonannihilating inelastic interactions, the final energy distribution of p/¯p is taken to be constant in the physically accessible regime m p . . . E p ( E p is the energy of the primary particle, m p the p/¯p mass). Consequently, the mean energy loss during a non-annihilating collision is 〈 ∆ E 〉 /similarequal 0 . 5 T p , with T p ≡ E p -m p being the initial kinetic energy. The energy released during an annihilation event is ∆ E = E p + m p . At energies E > ∼ 10 GeV, the losses are dominated by inelastic ¯p-p and p-p scattering without annihilation; at lower energies ¯p-p annihilation and Coulomb losses become relevant and dominate for non-relativistic particles. In the next section we shall follow the evolution of a nucleon pair from a hypothetical toy annihilation process χχ → ¯ pp, computing the energy loss of the daughter nucleons down to relatively low-redshifts of a few hundreds. Obviously, only part of this energy will be absorbed by the gas. Following in detail the energy degradation down to the energy-transfer processes to the gas - heating and ionization - goes beyond our purposes (it is worth mentioning that several efforts are being put into a more realistic treatment of the related physics for the e ± and γ 's, see e.g. [23]). Rather, we will content ourselves with estimating the energy that is released into energetic electrons and photons (essentially, the fraction of the energy lost to stable particles other than neutrinos). We will express this as the electromagnetic fraction f e ± γ of the energy initially injected as nucleons, which we will precisely define below. This approach is exact as long as the CMB bounds are 'calorimetric' and the precise form of the spectrum of electromagnetic particles is irrelevant in determining the ultimate fate of the energy injected. Previous investigations suggest that this should provide a reasonable first approximation to the true result; for our purposes the accuracy is expected to be at the level of ∼ 30% (see Fig. 2 of [15], z ∼ 600). Since we are already concerned with a correction to the basic results, we shall adopt this Ansatz which greatly simplifies the problem. The only potentially problematic case is the one of Coulomb reactions of the (anti-)protons, which can produce low-energy electrons (which may behave differently from high-energy ones). However, the legitimacy of our approximation is supported by two arguments: i) this energy-loss rate has only a logarithmic dependence on the minimum kinetic energy; ii) it only matters for non-relativistic (anti-)protons, for which most of the energy coming from electroweak-scale WIMPs has already been dissipated. In fact, with the exception of very light (GeV scale) DM particles, protons are typically born relativistic or semi-relativistic.", "pages": [ 2 ] }, { "title": "III. COMPUTATION", "content": "The energy loss processes sketched above are of two distinct classes: continuous energy losses (with small en- ergy transfer per collision) when protons interact with electrons or photons; or catastrophic losses, when they undergo an inelastic collision or an annihilation (for antiprotons). For the moment we shall ignore elastic collisions, we will comment on their role at the end of Sec. IV. Continuous energy losses in the range of our interest, 100 < z < ∼ 2000, can be described via the following differential equation The last term at the RHS describes adiabatic energylosses in the expanding universe, the other two terms describe Coulomb and Thomson energy-losses. In the present work, Thomson losses will be neglected, which from Fig. 1 is clearly justified at the energies of interest. Hadronic processes are included on top of the continuous energy losses with a Monte Carlo simulation. The Monte Carlo starts with a fixed p/¯p energy E inj at some initial redshift z init and tracks the history of the particle as it moves to lower redshifts. The redshift of an individual scattering process is inferred by sampling from a survival function, which is determined by solving a differential equation whose derivate is given by the inelastic scattering rates. The amount of electromagnetic and hadronic energy lost by each particle as function of redshift z is recorded. At any given redshift z , a fraction f e ± γ ( z ) of the hadronically injected energy is actually channeled into electromagnetic form (energetic e ± and photons, see discussion above). More specifically, for annihilating DM, f e ± γ is defined as where /epsilon1 e ± γ ( z ) denotes here the energy released into electromagnetic form per comoving volume and per unit time, and n χ, 0 is today's number density of DM particles. This f e ± γ ( z ) can be derived from the integral where g e ± γ ( z, z ' ) denotes the fraction of the energy of a single DM annihilation event at redshift z ' that is released into electromagnetic energy at redshift z per unit time. We will take z max = 10000 throughout, such that /epsilon1 e ± γ ( z ) at z < 1000 is independent of z max . We split f e ± γ ( z ), and analogously the function g e ± γ ( z ), into three parts according to with the three addenda at the RHS corresponding to Coulomb losses, inelastic scattering and annihilation, respectively. These ˜ f 's denote the total (anti-)proton energy lost to secondary particles via the scatterings. All of the energy lost by Coulomb scattering, and a fraction α had of the energy lost during inelastic scattering and annihilation is channeled into electromagnetic energy and hence contributes to f e ± γ . A reasonable value is α had /similarequal 0 . 5. After a hadronic scattering, the energy eventually carried by e ± and γ contributes fully to f e ± γ , the energy carried by ν 's is completely lost, and only a fraction of the energy carried by nucleons will be transfered into e ± γ at a later stage. We will neglect this latter contribution to f e ± γ , since it is further suppressed by the small fraction of energy released into nucleons during inelastic scattering. As a proxy for the energy released in the different channels we can take the gluon-gluon (or charmed quark) channels of Fig. 4 in [19], where about 46% of the energy is found to be channelled into e ± and γ , with ∼ 27% of this amount going into nucleons. We also checked that these figures do not change appreciably as a function of M χ (M. Cirelli, private communication). This is also consistent with the expectations from nonrelativistic proton-antiproton annihilation, see e.g. Table 2 in [24]. A very simple justification of this value can be obtained in the limit where annihilation final states are just made of pions, with isospin-blind production yields (equal quantities of π 0 , π + , π -). One would expect full channeling of π 0 energy into e ± γ and about 1 / 6 of the energy stored in π ± , hence a weighted average of 44%. We thus believe that α had /similarequal 0 . 5 is a reasonable benchmark, probably affected by a 20% uncertainty. While this may be a crude description of the actual process, it is enough for providing a first estimate of the hadronic correction to CMB bounds on annihilating dark matter. On a more technical note, to obtain ˜ g Cou . ( z, z ' ), ˜ g inel . ( z, z ' ) and ˜ g ann . ( z, z ' ), we simulate particle injection at redshifts z ' = 30-10000 and record what fraction of the energy is released in certain redshift intervals z 0 . . . z 1 , divided by the corresponding time interval ∆ t ≡ t 0 -t 1 . From these ˜ g ( z, z ' )'s, we can then derive the f e ± γ ( z ) from Eqs. (3) and (4).", "pages": [ 2, 3 ] }, { "title": "IV. RESULTS AND DISCUSSION", "content": "In Fig. 2 we show the main results of this paper. In the top panel, the resulting different components of the fractional energy loss of (anti-)protons are shown as function of their injection energy at redshift z ∼ 600. Notably, the annihilation of anti-protons releases also the energy of the target proton, which yields values ˜ f ann . > 1 when considering the anti-proton channel alone. The bottom panel shows the fraction of the injection energy that is channeld into electromagnetic energy, f e ± γ , as function of redshift z and for different kinetic injection energies T p / ¯ p . For energies T p / ¯ p < 10 GeV ( T p / ¯ p > 10 GeV) about f e ± γ ∼ 50% (40%) of the energy fraction channelled into nucleons will eventually by released as electromagnetic energy at the most relevant redshifts z = 500-700. Most importantly, for DM masses below ∼ 100 GeV, the f e ± γ ( z )'s are approximately independent of redshift . This implies that the released electromagnetic energy is simply a constant correction to the prompt electromagnetic energy directly released during DM annihilation. In the current literature (see e.g. Ref. [6]), the fraction of the total injected energy finally deposited in the gas as heat, ionization and excitation energy is denoted by f ( z ), and determined neglecting annihilation channels into hadronic final states. As discussed above, CMB bounds are to a good approximation calorimetric, and the precise form of the spectrum of electromagnetic particles can be neglected at first order in determining the ultimate fate of the energy injected. 2 We can then give a simple recipe to correct these f 's: where E N ¯ N and E γ + e ± denote the prompt energy in nucleons and in electromagnetic energy, respectively, produced during an annihilation. As shown in the bottom panel of Fig. 2, we find that f e ± γ /similarequal 0 . 5 for injected p/¯p energies well below ∼ 10 GeV, i.e. for most DM masses of interest. Throughout, we neglected elastic p/¯p scattering, which is well justified for our purposes: At energies below T p < 1 GeV, elastic p-p scattering will only redistribute energy that finally is lost via Coulomb scattering. At energies above T p > 1 GeV, elastic p-p scattering is subdominant ( < ∼ 50% of the total p-p cross-section); even rare collisions with maximal energy transfer do not alter the deposition history drastically since the energy dependence shown in Fig. 2 is weak. In case of ¯p-p scattering, the elastic cross section is smaller than 40% at all energies; the main effect of collisions with large energy transfer would be a slight increase in anti-proton annihilation rate.", "pages": [ 3, 4 ] }, { "title": "V. CONCLUSIONS", "content": "It has been recently recognized that non-standard sources of heating and ionization of the IGM can be probed via CMB anisotropies. This effect has been used to derive bounds on DM annihilation cross-section and decay lifetime. To a very good approximation, these bounds do not depend on 'astrophysics' (structure formation, DM halo structure, star formation etc.). The nature of this probe, whose robustness relies essentially on the accuracy of the description of atomic and nuclear/particle physics, motivates efforts to assess and improve the error budget of bounds obtainable with this technique. In astroparticle physics, energy losses of high energy nucleons propagating in the intergalactic medium are usually neglected outside the realm of ultra-high energy cosmic rays. Energetic nucleons propagating in a highz cosmological medium over Hubble times, on the other hand, do suffer significant energy losses. A physically interesting example is provided by nucleon by-products of dark matter annihilation, which can release a significant fraction of their energy at highz ( /similarequal O (10 3 )). The aim of this paper was to provide a first estimate of this process, which goes in the direction of improving the CMB bounds for a number of channels. Under reasonable approximations, we found that the Eq. (5) is a fair description of the main effect, for a value f e ± γ /similarequal 0 . 5 over a quite large parameter space. We expect thus that for DM annihilation channels into gluons, gauge bosons and quarks for which E N ¯ N E γ + e ± /similarequal 0 . 2, this should change the CMB bounds on DM at the 10% level and should be included in forthcoming Planck data analyses.", "pages": [ 4, 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We would like to thank M. Cirelli useful comments. At LAPTh, this activity was developed coherently with the research axes supported by the labex grant ENIGMASS. Partial support of the ANR grant DMAstroLHC is also acknowledged by PS. GB acknowledges the support of the European Research Council through the ERC Starting Grant 'WIMPs Kairos'. ph.CO]].", "pages": [ 5 ] } ]

[ { "title": "Nucleosynthesis constraint on Lorentz invariance violation in the neutrino sector", "content": "Zong-Kuan Guo ∗ and Jian-Wei Hu † State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China (Dated: February 27, 2018) We investigate the nucleosynthesis constraint on Lorentz invariance violation in the neutrino sector which influences the formation of light elements by altering the energy density of the Universe and weak reaction rates prior to and during the big-bang nucleosynthesis epoch. We derive the weak reaction rates in the Lorentz-violating extension of the standard model. Using measurements of the primordial helium-4 and deuterium abundances, we give a tighter constraint on the deformed parameter than that derived from measurements of the cosmic microwave background anisotropies. PACS numbers: 14.60.St, 98.80.Es Neutrino oscillation experiments have shown that there are small but non-zero mass squared differences between three neutrino mass eigenstates (see Ref. [1] and reference therein). However, neutrino oscillations cannot provide absolute masses for neutrinos. Cosmology provides a promising way to constrain the total mass of neutrinos by the gravitational effect of massive neutrinos on the expansion history near the epoch of matter-radiation equality [2] and on the formation of large-scale structures in the Universe [3] (see also Ref. [4] for a review). Recently, a 3 σ detection of non-zero neutrino masses is reported using new measurements of the cosmic microwave background (CMB) anisotropies from the south pole telescope and Wilkinson microwave anisotropy probe (WMAP), in combination with low-redshift measurements of the Hubble constant, baryon acoustic oscillation feature and Sunyaev-Zel'dovich selected galaxy clusters [5]. These observations establish the existence of physics beyond the standard model of particle physics. Another possible signal of new physics is violation of Lorentz symmetry. The possibilities of Lorentz invariance violation were considered in string theory [6], standard model extension [7], quantum gravity [8], loop gravity [9], noncommutative field theory [10], and doubly special relativity theory [11]. Searches for Lorentz invariant violation with neutrinos have been performed with a wide range of systems [12]. Although present experiments confirm Lorentz invariance to a good precision, it can be broken in the early Universe when energies approach the Planck scale. Cosmological observations provide a possibility to test such a symmetry at high energies. Recently, measurements of the CMB power spectrum were used to probe Lorentz invariant violation in the neutrino sector [13]. Lorentz invariant violation affects not only the evolution of the cosmological background but also the behavior of the neutrino perturbations. The former alters the heights of the first and second peaks in the CMBpowerspectrum, while the latter modifies the shape of the CMB power spectrum. These two effects can be distinguished from a change in the total mass of neutrinos or in the effective number of neutrinos. The seven-year WMAP data in combination with lower-redshif measurements of the expansion rate were used to put constraints on the Lorentz-violating term. However, the resulting constraints suffer from a strong correlation between the Lorentz-violating term and the dark matter density parameter [13]. In this letter, we use current big-bang nucleosynthesis (BBN) data to constrain Lorentz invariance violence in the neutrino sector. There are two effects of Lorentz invariant violation on BBN. The first is a correction to the weak reaction rate in the Lorentz-violating standard model extension, which governs the neutron-to-proton ratio at the onset of BBN. The second is a change in the total energy density of the Universe. Since the abundances of the light elements produced during BBN depend on the competition between the expansion rate of the Universe and the nuclear and weak reaction rates, the BBN predictions depend on the Lorentz-violating term. In particular the BBN-predicted abundance of helium-4 is very sensitive to the deformation parameter. We focus on Lorentz invariance violence only in the neutrino sector and consider the following deformed dispersion relation /negationslash where E is the neutrino energy, m the neutrino mass, p = ( p i p i ) 1 / 2 the magnitude of the 3-momentum, and ξ the deformation parameter characterizing the size of Lorentz invariance violation. The dispersion relation implies that there are departures from Lorentz invariance in the neutrino sector if ξ = 0. Such a deformed dispersion relation was constructed in the framework of conventional quantum field theory [14] and derived in the Lorentz-violating extension of the standard model [15]. Here we have to point out that the dispersion relation for neutrinos given in (1) is not very general. It neglects neutrino oscillations, possible species dependence, anisotropies associated with violations of rotation sym- metry, and CPT violation. As shown recently by Kostelecky and Mewes, all of these are possible [12]. The model considered in this paper is one of many possible Lorentz-violating theories. The number density n ν and energy density ρ ν for massive neutrinos with (1) are given by [13] where g ν = 2 is the number of spin degrees of freedom. The phase space distribution for neutrinos is the FermiDirac distribution where T ν is the neutrino temperature. Thus the number and energy density can be written as n ν = (1+ ξ ) -3 / 2 n (0) ν and ρ ν = (1+ ξ ) -3 / 2 ρ (0) ν , where n (0) ν and ρ (0) ν are the standard number and energy density, respectively. Increasing ξ decreases both the number and energy density. The former leads to a reduced rate of the weak reaction prior to and during the BBN epoch since the reaction rate is proportional to the neutrino number density, while the latter results in a reduced expansion rate of the Universe. Therefore, Lorentz invariant violation affects the nucleosynthesis of light elements. We turn now our attention to the details of the computation of the weak reaction rate with Lorentz invariance violation in the neutrino sector. At early times when the temperature of the Universe was T ∼ 100 MeV, the number and energy density were dominated by relativistic particles: electrons, positrons, neutrinos, antineutrinos and photons. All of the particles were kept in thermal equilibrium by the weak reactions When the expansion rate of the Universe exceeds the reaction rate for n ↔ p processes, the baryons become uncoupled from the leptons. At the time the neutronto-proton ratio is frozen, which largely determines the primordial helium mass fraction. To estimate the neutron abundance at the onset of BBN one has to compute the reaction rate. As an example, let us consider the process ν e + n → p + e -. The differential reaction rate per incident nucleon is where |M| 2 is the squared matrix element, to be summed over initial and final state spins, m n and m p the neutron and proton mass, respectively, ( E e , p e ) the electron fourmomentum, and f e denotes the Fermi-Dirac statistical distribution for electron. The process (5) involves the gauge boson W as mediator. At tree level, one has where G F is the Fermi coupling constant, c µν are the coefficients for Lorentz violation, and C V , C A are the vector and axial coupling of the nucleon. Here the coefficients c µν are defined to be traceless and isotropic. After integrating (8) the reaction rate is where ω (0) is the standard reaction rate per incident nucleon derived in [16]. The first factor on the right-hand side of (10) arises from the neutrino propagator and the eνW coupling in the Lorentz-violating extension of the standard model [7] and the second factor from the statistical distribution for neutrinos (more general Lorentzviolating corrections involving electrons, neutrinos, neutrons and protons were discussed in [17]). At tree level, the differential reaction rates for the other five processes in (5)-(7) can be simply derived from (8) by properly changing the statistical factors and the delta function determined by the energy conservation for each reaction. Therefore, the corrections to the conversion rate of neutron into proton and its inverse are the same as in (10). From (10) we see that increasing ξ reduces the reaction rate, and thus the weak reactions freeze out at earlier time, corresponding to a higher freeze-out temperature. This leads to a larger neutron-to-baryon ratio at the onset of BBN and thus a larger abundance of primordial 4 He production. On the other hand, increasing ξ also reduces the expansion rate of the Universe due to a decrease of the energy density, which means the weak reactions freeze out at later time without corrections to the reaction rate induced by the deformed parameter. This therefore results in a lower helium-4 abundance. These two effects play opposite roles in the BBN prediction for the helium4 abundance. The abundances of the other light nuclides weakly depend on ξ by changing the neutron-to-proton ratio and the expansion rate. Considering these corrections to both the reaction rate and the expansion rate, we now estimate the freezeout temperature, T f , determined by equating the expansion rate with weak reaction rate. In the FriedmanRobertson-Walker Universe, the expansion rate obeys H 2 = 8 πGρ/ 3 where ρ ∝ T 4 at early times. Thus, we have H ∝ (1 -0 . 75 ξ ) T 2 . Since the standard reaction rate in Eq. (10) is roughly given by ω (0) ∝ T 5 [18], we have ω ∝ (1 -1 . 80 ξ ) T 5 . Setting H ∼ 4 ω since the free-neutron decay process and its inverse are neglected at the BBN epoch, one derives the freeze-out temperature where T (0) f is the standard one. For a large ξ , the weak reactions freeze out at a higher temperature. This implies that effects caused by changing the reaction rate dominate over those by changing the expansion rate due to the Lorentz invariance violation in the neutrino sector. In order to calculate the abundances of light elements produced during BBN, we modified the publicly available PArthENoPE code [19] to appropriately incorporate the Lorentz-violating term in the neutrino sector. Figure 1 shows the 4 He mass fraction and D/H abundance as a function of ξ for Ω b h 2 = 0 . 0213 (the upper panel) and 0 . 0224 (the lower panel). Both Y p and D/H increase as ξ increases since the effect of changing the reaction rate play a leading role. Moreover, the dependence of Y p on ξ is much larger, relative to its observational uncertainties, than that of D/H. Therefore, the primordial helium4 abundance can provide a sensitive probe of neutrino physics with Lorentz invariance violation. Assuming that there are three types of neutrinos with vanishing chemical potentials in the Universe, the BBNpredicted primordial abundances depend on only two parameters: Ω b h 2 and ξ . As shown in Figure 1, the abundance of deuterium is more sensitive to the baryon energy density parameter but less sensitive to the deformation parameter while that of helium-4 is more sensitive to ξ but less sensitive to Ω b h 2 . We use the observed primordial abundances of 4 He and D in combination to constrain these two parameters based on the likelihood function Here we adopt the estimate of the primordial helium mass fraction, Y p = 0 . 2565 ± 0 . 0060, derived in [20] using Monte Carlo methods to solve simultaneously for many possible systematic effects, based on 93 spectra of 86 lowmetallicity extragalactic HII regions. While some have employed a selected subset of these data for more detailed analyses, the sources and magnitudes of systematic errors have rarely been addressed. The measurement uncertainty in Y p is currently dominated by systematic errors. For the primordial deuterium abundance, we use the value of log[D / H] = -4 . 55 ± 0 . 03 obtained in [21] from measurements of the absorption lines of seven quasars at high redshifts in low-metallicity hydrogen-rich clouds with low internal velocity dispersions. Besides 4 He and D , 3 He and 7 Li are the other two nuclides predicted in measurable quantities by BBN. Since their postBBN evolutions are complicated and their measurements suffer from systematical uncertainties which are difficult to quantify (for helium-3) or are poorly understood (for lithium), the observed 3 He and 7 Li do not provide a reliable probe of BBN at present, as discussed in [22]. Thus, we do not include them in our constraints. The 4 He abundance is used to provide a constrain on the deformation parameter while the D abundance is used to provide a constrain on the baryon density parameter. Using the combination of the 4 He and D data, we find ξ = 0 . 036 ± 0 . 023 and Ω b h 2 = 0 . 0213 ± 0 . 0009 (68% confidence level). This estimated value of the deformation parameter is consistent with Lorentz invariant ξ = 0 within 95% confidence level. Compared to the results derived from the 7-year WMAP data in combination with lower-redshift measurements of the expansion rate [13], BBN gives smaller uncertainties in ξ by a factor of 4 because there is nearly no correlation between the deformation parameter and the baryon density parameter as shown in Figure 2. The estimate of Ω b h 2 is agreement with that from the CMB data [2] with errors. In summary, we have shown that the BBN puts strong constraint on the deformed parameter in the Lorentzviolating extension of the standard model, ξ = 0 . 036 ± 0 . 023. Since the BBN-predicted abundance of helium-4 is very sensitive to the deformed parameter but less sensitive to the baryon energy density parameter, there is nearly no correlation between the two parameters. Our results indicate no significant preference for departure from Lorentz symmetry in the neutrino sector in the early Universe. Compared to previous constraints on the Lorentz-violating coefficient, current BBN observation yields a weaker constraint. As listed in Table XIII of [12], the coefficient is constrained down to 10 -9 from time-of-flight measurements. Cohen and Glashow have argued that the observation of neutrinos with energies in excess of 100 TeV and a baseline of at least 500 km allows us to deduce that the Lorentz-violating parameter is less than about 10 -11 [23].", "pages": [ 1, 2, 3, 4 ] }, { "title": "Acknowledgments", "content": "We thank J. Hamann, V. A. Kostelecky, O. Pisanti and F. Wang for useful discussions. Our numerical analysis was performed on the Lenovo DeepComp 7000 supercomputer in SCCAS. This work is partially supported by the project of Knowledge Innovation Program of Chinese Academy of Science, NSFC under Grant No.11175225, and National Basic Research Program of China under Grant No.2010CB832805.", "pages": [ 4 ] } ]

[ { "title": "Gamma Rays from Bino-like Dark Matter in the MSSM", "content": "Jason Kumar 1 and Pearl Sandick 2 1 Department of Physics and Astronomy, University of Hawai'i, Honolulu, HI 96822, USA 2 Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider regions of the parameter space of the Minimal Supersymmetric Standard Model (MSSM) with bino-like neutralino dark matter in which a large fraction of the total dark matter annihilation cross section in the present era arises from annihilation to final states with monoenergetic photons. The region of interest is characterized by light sleptons and heavy squarks. We find that it is possible for the branching fraction to final states with monoenergetic photons to be comparable to that for continuum photons, but in those cases the total cross section will be so small that it is unlikely to be observable. For models where dark matter annihilation may be observable in the present era, the branching fraction to final states with monoenergetic photons is O (1 -10%). PACS numbers: 95.35.+d, 95.55.Ka", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In light of recent analyses of data from the Fermi Large Area Telescope (LAT) [1] indicating the possibility of an excess of ∼ 130 GeV photons from the Galactic Center region [2], there has been renewed interest in dark matter models that can produce monoenergetic photons through the processes χχ → γγ, γZ, and γh [3-9]. Generically, one would expect models with a monoenergetic photon line to be accompanied by a larger continuum photon spectrum. This reasoning derives from general principles: since there are tight constraints on the electromagnetic charge of dark matter, monoenergetic photon production can only arise through a millicharged coupling or through a loop diagram. These processes are expected to be suppressed relative to tree-level annihilation to other Standard Model particles. The dominant annihilation products would then produce a continuum photon flux, either from decays that produce photons or through synchrotron radiation, which is large relative to the strength of a monoenergetic photon signal. Bounds on the magnitude of the continuum spectrum arising from dark matter annihilation in the Galactic Center thus put tight constraints on dark matter models that could explain a line signal. Many new dark matter models have been proposed to evade these constraints and explain the recent Fermi data [3, 4]. Similarly, a variety of works have studied the question of whether it is possible for the 130 GeV photon excess seen by Fermi to be produced within the MSSM [5-7] or singlet extensions of the MSSM [8], while maintaining consistency with bounds on the continuum spectrum. In this work, we consider an MSSM analysis from a different perspective. We assume that the MSSM lightest supersymmetric particle (LSP) is the lightest neutralino, χ , and constitutes all of the dark matter in the universe, and we ask what we can learn about the parameters of the MSSM from the observation of a photon line signal and a small (or unobservable) continuum signal. With this goal in mind, we avoid (as far as possible) any assumptions about the MSSM parameter space based on naturalness or simplicity, instead taking as our guide only the mass of the Higgs boson (taken to be m h ∼ 126 GeV), other observational constraints, and the assumption of a large photon line signal and small continuum signal. We will find that these demands are so restrictive that they constrain the MSSM parameter space in a way comparable to (though orthogonal to) the typical assumptions that lead to scenarios like the CMSSM or mSUGRA, allowing a scan over allowed parameter space to be feasible. Alarge annihilation cross section to monoenergetic photons relative to that for continuum photons can be obtained within the MSSM if the LSP is a purely bino-like neutralino. In this case, the dominant cross section to continuum photons comes from χχ → ¯ ff , a process that is always suppressed in some way. As we demonstrate here, the suppression of the continuum cross section can be large enough that it becomes comparable to the cross section to monoenergetic photons, a scenario at odds with the intuition that tree-level annihilation to continuum photons will have a much larger cross section than loop-suppressed annihilation to final states with monoenergetic photons. Although this type of analysis is relevant to an understanding of the implications of the Fermi-LAT data, its utility extends beyond that particular observed excess, which may simply be a statistical or systematic effect [10]. Rather, one can consider the implications of the detection of a line signal with future experiments, such as GAMMA-400 [11] or C¸erenkov Telescope Array (CTA) [12], or the emergence of such a signal in the data from the Fermi satellite or ground-based gamma-ray detectors such as VERITAS [13] and HESS [14]. The ground-based CTA will have an effective area several orders of magnitude larger than the effective area of space-based telescopes, but will suffer from the background of cosmic rayinduced C¸erenkov showers. The GAMMA-400 satellite, by contrast is not expected to have a significantly larger effective area than the Fermi-LAT, but it may have much better energy resolution. Here we will assume that relevant experiments can distinguish a true line signal from a very hard continuum spectrum, as might arise from a three-body annihilation process such as χχ → f ¯ fγ . In section II, we describe the general principles that lead to the region of MSSM parameter space of interest, where the annihilation cross section to monoenergetic photon final states is comparable to that for final states that give rise to a continuum spectrum of photons. In section III we analyze how the cross section for annihilation to various final states scales with relevant model parameters. In section IV we discuss a detailed numerical scan of the annihilation cross sections in this region of parameter space and relevance of our results for more general non-MSSM scenarios. Finally, we state our conclusions in section V.", "pages": [ 2, 3 ] }, { "title": "II. GENERAL PRINCIPLES", "content": "As put forward in [7], one can parameterize the limits on dark matter annihilation obtained from gamma-ray observations in terms of two quantities, where σ ann. is the total dark matter annihilation cross section and σ ( γγ,γZ ) are the χχ → ( γγ, γZ ) annihilation cross sections, respectively. Since the neutralino LSP is a Majorana fermion, annihilation to the γh final state is p -wave suppressed, and can therefore be ignored. The rate at which dark matter annihilates to monoenergetic photons is thus proportional to σ line , and R th is the ratio of the total annihilation rate to the rate of annihilation to monoenergetic photons. If an excess of monoenergetic photons is observed, then, with some assumptions about the dark matter density profile of the source, one can estimate the quantity ( σ line v ). Assuming the spectra of continuum photons from relevant dark matter annihilation processes are roughly similar in shape, gamma-ray observations can then place an approximate upper bound on R th . Without any assumptions about the relationships between MSSM parameters, the number of free parameters is so vast that a complete scan is impractical. This consideration usually leads to assumptions motivated by, among other things, naturalness, a string theoretic origin, a putative SUSY breaking mechanism, elegance or simplicity. We will instead consider a framework where the relations assumed between parameters are motivated only by the goal of maximizing ( σ line v ) and minimizing the annihilation to continuum photons. We will find that these considerations are sufficient to drastically reduce the parameter space of the MSSM. We are guided by general considerations underlying the suppression of annihilation to continuum photons that is often seen in MSSM models. Based on these facts, the low-energy particle content of interest for minimizing R th in the limit where tree-level annihilations to gauge bosons are negligible must include a purely bino-like LSP, negligible sfermion mixing, heavy sbottoms and scharms, and heavy neutral Higgsinos. Although the production of on-shell tops is kinematically forbidden if m χ < m t , the production of off-shell tops can still provide a large contribution to ( σ ann. v ). Similarly, if m χ glyph[greatermuch] m t , then although the the χχ → ¯ tt annihilation process will be kinematically suppressed, the sensitivity of gamma-ray observations to ( σ line v ) will also be suppressed. Finally, we note that if up/down/strange-squarks are light then the χχ → ¯ uug , ¯ ddg , and ¯ ssg three-body annihilation processes are only suppressed by α s . To minimize R th , we will thus assume that all squarks are heavy. We are therefore led to consider the scenario in which the only sfermions that are possibly light are the sleptons. The charged slepton, sneutrino, and LSP masses are the only parameters that determine both σ line and R th . This scenario is consistent with constraints from the Large Hadron Collider, which has ruled out models with light first generation squarks [17].", "pages": [ 3, 4 ] }, { "title": "III. FEATURES OF THIS SCENARIO", "content": "In this section we discuss the general model features that determine the relevant quantities R th and ( σ line v ). Since the LSP is largely bino-like and the squarks and Higgsinos are very heavy, tree-level annihilation occurs largely through t -channel exchange of light sleptons and sneutrinos, as shown in Fig. 1(a). The primary two-body annihilation processes that contribute to ( σ ann. v ) in this case are χχ → ¯ ff where f = e, µ, τ, ν e,µ,τ . We parameterize the scale of the various annihilation processes in terms of α , m χ , and r , as defined above. The cross sections for these processes scale as Since v 2 ≈ 10 -6 today, and m 2 f glyph[lessmuch] m 2 χ , these processes are typically negligible. The relevant three-body annihilation processes are χχ → ¯ ll ( γ, Z ), ¯ ν l ν l Z , ¯ lν l W -, and ¯ ν l lW + , where l = e, µ, τ . These processes occur through t -channel exchange of a light sfermion, with an electroweak gauge boson emitted either from the outgoing fermions or from the internal sfermion line, as shown in Fig. 1(b). If the gauge boson is emitted from the internal line, then the diagram has two scalar propagators and scales as r -2 . There are two diagrams in which the gauge boson is emitted from the outgoing fermions, and each diagram has only one scalar propagator. But the diagrams partially cancel, and their sum also scales as r -2 [18]. The electroweak bremsstrahlung processes are further suppressed by an additional factor of α . For m χ ≈ 1 2 m Z,W , the cross section for annihilation to a threebody state with a massive vector boson will also be phase-space suppressed. Finally, the cross section for annihilation to ¯ llγ also receives an additional suppression due to angular momentum conservation; for certain regions of the three-body final state phase space, the angular momentum of the final state can only be zero if the gauge boson has helicity zero, which is not possible for a massless gauge boson. If m χ > 1 2 m Z,W , then the total three-body annihilation cross section will scale as The processes χχ → γγ , γZ proceed through box diagrams with l, ˜ l running in the loop, as shown in Fig. 1(c). The number of slepton propagators in the loop can vary between one and three, but as the sleptons become heavy, the dominant diagram will have only one slepton in the loop. We thus find The three-body annihilation process can dominate over χχ → ¯ ll only if α/r 2 > v 2 . Similarly, three-body annihilation can only dominate over χχ → ¯ bb if α/r 2 > m 2 b m 2 χ r 2 /m 4 ˜ b . Assuming that the squarks are very heavy, three-body annihilation is the dominant continuum process if r < O (10 2 ). As sleptons become heavier, however, annihilations to three-body final states become insignificant, and the two-body final states determine the annihilation cross section to continuum photons, ( σ cont. v ). So we find We thus expect to find values of R th ranging from ∼ 1 to ∼ 100. But small R th arises only when the sleptons are heavy, implying that the magnitude of ( σ line v ) will be suppressed. Maximizing the magnitude of the line signal would require R th > ∼ O (100).", "pages": [ 4, 5 ] }, { "title": "IV. PARAMETER SPACE SCAN AND ANALYSIS", "content": "We calculate the spectrum of supersymmetric particles, Higgs bosons, and dark matter observables using SuSpect [19], FeynHiggs [20], and MicrOMEGAs [21], with supplementary calculations of the annihilation cross sections to three-body final states following [15, 16]. Annihilations to 3-body final states will be most significant when sfermion mixing is negligible, so we compute this only for the first two generations. We focus on consistent models that fall at least approximately into the scenario described in section II, and consider the heavy scalars to be degenerate, with masses m 0 in the range ( m χ +3 TeV, 8 TeV). We fix the Higgs mixing parameter, µ , to be µ = m 0 , to ensure that the lightest neutralino is nearly pure bino, and allow masses for the left- and right-handed 1 ˜ e , ˜ µ , and ˜ ν e,µ in the range ( m χ , m 0 ), with ˜ τ and ˜ ν τ heavier by at least 500 GeV such that 3-body annihilations involving the third generation are subdominant to those involving the first two generations. Finally, the trilinear couplings, A , may take values in the range ( -5 m 0 , 5 m 0 ), with larger | A | preferred by the CMS [22] and ATLAS [23] measurements of m h ≈ 126 GeV. For the parameter points we consider, we require the mass of the lighter CP-even Higgs boson to lie within the range 123 GeV < m h < 129 GeV. We assume that the lightest neutralino constitutes all of the dark matter in the universe. In figure 2, we plot the cross section for production of a line signal, ( σ line v ), as a function of dark matter mass. The points are color-coded by the value of R th : R th < 5 (magenta), 5 ≤ R th < 10 (blue), and R th ≥ 10 (cyan). In the left panel, we address potential line signals observable by the Fermi satellite, for neutralino masses up to roughly 300 GeV, while in the right panel we consider neutralino masses as large as 2.5 TeV. Though HESS and VERITAS are sensitive to dark matter masses in this range, the cross sections to monoenergetic photons are all well below the current limits [14]. We note that with adequate energy resolution, the γγ and γZ line signals will be distinguishable, and R th and ( σ line v ) will not be optimal parametrizations of the signal strength. Nonetheless, any observed line signal(s) can be simply mapped to particular values of R th and ( σ line v ). From the analysis of the previous section, we expect that in the decoupling limit discussed in the previous section, models with small ( σ line v ) have large values of r , which in turn implies small R th . This expectation is borne out in figure 2: For m χ > ∼ 80 GeV, smaller R th is possible at smaller ( σ line v ). Furthermore, the smallest line cross sections (below 10 -31 cm 3 s -1 ) do indeed come from points for which all sleptons are heavy. However, R th is not necessarily small for these points unless the LSP is much heavier than the τ lepton, i.e. m 2 τ /m 2 χ < ∼ v 2 . Such large R th points do not satisfy the limit discussed in the previous section. Note also that for very light LSP's ( m χ < ∼ m Z ), R th may be small even when ( σ line v ) is large. In these cases, although annihilation to 3-body final states is the dominant mechanism for producing continuum photons, annihilation to 3-body final states with massive vector bosons is kinematically suppressed relative to the behavior approximated by Equation 3. Finally, these largest cross sections, ( σ line v ) ≈ 10 -29 cm 3 / s, are indeed obtained in the limit m ˜ l ≈ m χ , the limit discussed in Ref. [3]. GAMMA-400 may be sensitive to cross sections of this size for small enough m χ [24]. It has been shown that for m ˜ l ≈ m χ , very hard bremsstrahlung photons can mimic a gamma-ray line signal for the present Fermi-LAT energy resolution [5]. The analysis of the previous section shows that for r < ∼ 10 2 , the ratio of the annihilation cross section to 3-body final states, σ 3-body , to that to 2-body final states that contribute to the continuum photon flux, σ 2-body , is One expects that this ratio can thus be as large as ∼ 10 4 . In this limit where annihilations to 3-body final states are most significant, i.e. large R th , we expect σ 2-body : σ line : σ 3-body to be 1 : 10 2 : 10 4 . In this analysis, all electroweak bremsstrahlung cross sections are included as part of the annihilation cross section to continuum photons, despite the fact that the spectrum may be significantly different from that for annihilations to 2-body final states. Therefore one can interpret R th for the points with the largest cross sections in figure 2 as maximal, since any very hard bremsstrahlung will effectively contribute to a line signal, as perceived by Fermi-LAT, and not to the perceived continuum. The ability to distinguish the 3-body continuum spectrum, the 2-body continuum spectrum, and a line signal may indeed be very important to determining the nature of particle dark matter. In the left panel of figure 3 we focus on neutralinos with m χ ≤ 300 GeV and plot only points for which the thermal relic abundance of dark matter satisfies Ω χ h 2 ≤ 0 . 15, in rough compatibility with the observed abundance of cold dark matter [25]. We see that some of the very largest line cross sections are due to models in which the dark matter is a thermal relic. It is not surprising that we find points for which R th ∼ O (10 2 ) with ( σ line v ) ∼ 10 -29 cm 3 / s, i.e. a continuum cross section today of ( σ cont. v ) ∼ 10 -27 cm 3 / s, and possibly even a larger annihilation cross section in the early universe, allowing compatibility between the measured dark matter abundance and the thermal relic density. Indeed, there are even points for which R th < 5 that can predict a relic abundance of neutralinos compatible with the observed dark matter abundance. In fact, these models are precisely those in which the annihilation rate in the early universe is significantly enhanced due to coannihilations of neutralinos with light sleptons. This is evident in the right panel of figure 3, where we plot ( σ line v ) as a function of the mass splitting between the neutralino LSP and the lightest slepton. Green points satisfy Ω χ h 2 ≤ 0 . 15, while light gray points predict Ω χ h 2 > 0 . 15. For m χ < ∼ 80 GeV, there are models with small R th and ( σ line v ) > ∼ 10 -29 cm 3 / s that provide the best prospects for detection. Although this analysis has been presented in the framework of the MSSM, it applies more generally to any model where the dark matter candidate is a Majorana fermion that is neutral under the Standard Model gauge groups. For such a model, dark matter can annihilate at tree-level only to ¯ ff or hh , with the latter process being either p -wave suppressed or suppressed by CP-violating phases. If annihilation to ¯ ff proceeds through t -channel exchange of a scalar partner, then the analysis given here for the relation between R th and the mass of the scalar partner still holds. Note, however, that the bino-fermion-sfermion coupling is simply related to the hypercharge coupling. For a more general dark matter model, this coupling, g , is a free parameter. Since ( σ line v ) and all continuum annihilation cross sections scale as g 4 , the choice of coupling does not affect R th . However, it does affect the overall scale of the annihilation cross section; if this coupling is required to be perturbative, then σ ( γγ,γZ ) can be increased by up to 4 orders of magnitude, in agreement with the limit discussed in Ref. [3]. For models with small R th , such an enhancement could bring a line signal within the reach of GAMMA-400.", "pages": [ 6, 7, 8 ] }, { "title": "V. CONCLUSIONS", "content": "We have considered a class of MSSM models for which the ratio σ ( χχ → γγ, γZ ) /σ ( χχ → anything) is maximized. We have found that this requirement leads to a dramatic restriction in the number of relevant MSSM parameters. In particular, the region of interest studied here contains a nearly pure bino LSP, with the only light superpartners being the bino and the sleptons. Interestingly, this region of parameter space is consistent with recent bounds from the LHC, which tightly constrain models with light squarks. For such models, we find that if a photon line signal is potentially observable at current or next generation experiments, then R th can be as low as O (10 -100). Moreover, for many such models the continuum signal is dominated by three-body annihilation. Since three-body annihilation may produce a very hard photon spectrum, it is possible that such a continuum annihilation process could be mistaken for a line. Therefore our numerical results for R th represent the maximal values possible (i.e. the limit of perfect energy resolution). The improved energy resolution of next generation gamma-ray telescopes should make it possible to resolve this difference. For MSSM models with small R th , the line signal will be a few orders of magnitude too small to be observed with Fermi. However, more general models of Standard Model-neutral Majorana fermion dark matter can achieve the same R th with an enhancement in the line signal of up to four orders of magnitude, potentially bringing a pure line signal with no observable continuum spectrum within reach of observation.", "pages": [ 8, 9 ] }, { "title": "Acknowledgments", "content": "We thank the organizers of ICHEP2012 and the Center for Theoretical Underground Physics and Related Areas (CETUP* 2012) in South Dakota for their support and hospitality during the completion of this work. We are grateful to D. Marfatia and C. Kelso for useful discussions, and to K. Fukushima for comments and for assistance with the numerical code for computing the continuum annihilation cross section. We are also grateful to the anonymous referee for insightful suggestions. Support and resources from the Center for High Performance Computing at the University of Utah are also gratefully acknowledged. The work of J. K. is supported in part by Department of Energy grant DE-FG02-04ER41291.", "pages": [ 9 ] } ]

[ { "title": "Curvature perturbations of Quasi-Dilaton non-linear massive gravity", "content": "Zahra Haghani 2 , ∗ Hamid Reza Sepangi 2 , † and Shahab Shahidi 2 ‡ 2 Department of Physics, Shahid Beheshti University, G. C., Evin, Tehran 19839, Iran We study the cosmological perturbations of the recently proposed extension of non-linear massive gravity with a scalar field. The added scalar field ensures a new symmetry on the field space of the theory. The theory has the property of having a flat dS solution, in contrast to the standard dRGT massive gravity. The tensor part is the same as that of the standard dRGT and shows gravitational waves with a time dependent mass which modifies the dispersion relation. We obtain the curvature perturbation of the model on superhorizon scales for a specific choice of ω = 0 and find that the theory does not allow a constant curvature perturbation on the superhorizon scales and we have always a growing mode. The consistency of equations restricts the parameter space of the theory. PACS numbers:", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Gravity, as the oldest force known to man has also had the longest history of trials and tribulations along the road to discovering its nature. Over the course of its development, it has been witnessing a myriad of attempts to unlock its notoriously difficult and mysterious behavior from the largest to the smallest of distances. The challenge has been truly spectacular. Since the first formulation of the gravitational field by Newton and centuries later by Einstein in the form of the theory of general relativity (GR), the scenery is still cluttered with debris left from various attempts to understand its hard to grasp nature. Even today, the challenge is as fresh and as interesting as ever. Not surprisingly, building on GR, the last couple of decades have been particularly rich in new ideas and approaches which attempt to formulate the gravitational field in such a way as to pave the way to a formulation of the theory which would explain such recently observed phenomenon as the accelerated expansion of the universe, galaxy rotation curves and even the birth of the universe. One such attempt has been surfacing over the past few years in the form of what is now known as massive gravity which, as the name suggests, is a theory with a massive graviton as the building block of the gravitational field. The notion of a massive graviton has been a tempting and challenging premise in theoretical physics. One of the main motivations of having a massive graviton is that gravity could become weak at large distances, thus mimicking the effect of accelerated expanding universe. The problem is not as easy as it seems. The first attempt to build a theory for a massive spin-2 field was back in 1939 when Fierz and Pauli (FP) developed a linear theory for a massive graviton [1]. It took thirty years for physicists to find out that the theory does not reduce to the standard GR when one takes the limit m → 0 [2]. The problem was soon addressed by Vainstein [3] who proposed that adding non-linearities to the action can cure the problem and screen the effect of helicity-0 component of the massive graviton at solar system scales. The simplest possible non-linearity one can add to the FP action is by replacing the linear kinetic term for the helicity-2 field with the fully non-linear, and still ghost free, Einstein-Hilbert action. However, the resulting action have proven to have a ghost instability which was discovered by Boulware and Deser [4]. The problem arises because the lapse function is no longer a Lagrange multiplier. This new problem can then be solved if one appropriately adds interaction terms to the Lagrangian and again makes the lapse function a Lagrange multiplier order by order in non-linearities [5, 6]. In this paper, after a brief review of the theory we study the curvature perturbation around an accelerating solution and obtain the background equations. The second order Lagrangian can then be obtained using the perturbed FRW metric. It is an immediate observation from the form of the second order Lagrangian that the tensor, vector and scalar modes do not couple. The tensor mode shows a massive gravitational wave with a time-dependent mass parameter which we shall obtain in section IV A. In section IV B we consider the vector mode and show that the vector part of the action vanishes at superhorizon scales and subsequently obtain the scalar mode in section IV C. Two of the scalar modes are non-dynamical and can be integrated out immediately with three degrees of freedom remaining in the scalar sector. We show that in the superhorizon limit where one of the degrees of freedom does not play a role, the curvature perturbation can be obtained analytically and there is a vast region in the parameter space over which the curvature perturbation will grow on the superhorizon scales.", "pages": [ 1, 2 ] }, { "title": "II. A BRIEF REVIEW", "content": "Recently, a two parameter theory of massive gravity has been proposed [7]. With the aid of a new re-summation of the non-linear interaction [8] de Rham, Gabadadze and Tolley (dRGT) have constructed a theory which is free of ghost instabilities in the decoupling limit. The theory has been proven to be ghost-free in the full non-linear theory and hence a reliable effective field theory of a massive spin-2 field [9]. In all fairness, there are also criticisms towards dRGT in that superluminal shock wave solutions have been shown to appear in the theory [10]. However, subsequently it was shown that such shock waves are unstable with an arbitrary fast decaying time [11]. The Lagrangian for dRGT non-linear massive gravity can be written as where the non-linear interactions are collected in U which consists of polynomials of various traces of the matrix K µ ν ( g, φ a ) = δ µ ν -√ g µα f αν where the fiducial metric is defined as f αν = ∂ α φ a ∂ ν φ b η ab and φ a are the Stuckelberg fields responsible for the breaking of general covariance µ where the rectangular brackets denote traces, [ K ] ≡ Tr( K ) = K µ . The first term concides with the Fierz-Pauli mass term at the linear level, and the last two terms are non-linear interactions which ensure that the theory has no ghost. One of the interesting properties of the theory is that if one assumes the Stuckelberg fields to be in the unitary gauge defined as φ a = δ a µ x µ , the theory does not have a non-trivial flat FRW solution [12]. However, the theory has an open FRW solution with an additional consideration in that one must transform the field space to the open slicing of the Minkowski metric [13]. The cosmological perturbations around such a solution was also considered in [14] where the authors found that the scalar, vector and tensor modes actually decouple and as a result the vector mode does not play any role in the theory. The tensor mode describes a massive gravitational wave with a time-dependent mass. Cosmological evidences of dRGT theory is also considered in [15]. For a review on the theoretical aspect of the theory see [16]. One can also let the non-dynamical metric of the theory to have a kinetic term and hence construct a bimetric theory [17] which has been proven to be ghost free [18]. The cosmological aspects of such bimetric theory is investigated in [19]. One may continue the procedure to build a multimetric gravity theory with dRGT non-linear mass terms. In [20], the authors show that the theory is ghost free in the metric formulation only when the interactions between gravitons are not cyclic. However, in [21] the authors show that in the vielbein formulation of the theory any interactions are allowed. Also in [22] the authors show that the problem of having no flat FRW solution persists in multimetric theories. In fact if one of the metrics is assumed to be flat, all of the other metrics will become flat. One of the solutions to the problem of the non-existence of the flat FRW solution in the theory is by extending the theory in such a way that the graviton mass becomes a function of some scalar field ϕ [23]. The cosmological solutions and dynamical analysis of such theories are considered in [24]. Another way to extend the theory is to couple a scalar field to the mass Lagrangian such that the resulting new Lagrangian has an extra symmetry. In particular, one can couple the scalar field to the Lagrangian to achieve dilation invariance on the field space [25] In the Einstein frame one can write the new Lagrangian as where we define H = ˙ a/ ( aN ) and and r ≡ r ( t ) = a ( t ) /N ( t ). We will use these forms of the background equations in the second order Lagrangian. Substituting equation (10) in (11) leads to a constant Hubble parameter, showing a de Sitter solution where ˜ K µ ν is now defined as Only the pure geometric part of the above action is invariant under transformation (4), thus the acronym QuasiDilaton (QD) for the scalar field [25]. This theory has been proven to be free of ghost in the Minkowski background if ω > 6. The most interesting feature of this theory is that it admits a flat de Sitter solution even if the Stuckelberg fields are in the unitary gauge [25]. In our notation, the de Sitter solution is also stable in the decoupling limit for /negationslash", "pages": [ 2, 3 ] }, { "title": "III. THE BACKGROUND EQUATION", "content": "Let us assume that the background metric is of the form and the Stuckelberg fields take the form Note that we will work in the unitary gauge. However, in order to obtain the Stuckelberg equation from the action we assume the above form for the Stuckelberg fields, and finally set f ( t ) = t . We also assume that the QD field only depends on t . By varying the action (1) with respect to f ( t ), one obtains the constraint equation as where k 1 is an integration constant. We are interested in the set of equations for which k 1 = 0. In this case one can solve the above equation by using the ansatz in the equation which results in The solution X = 1 is not acceptable because in this case the effective cosmological constant vanishes, and the consistency of the theory does not allow one to have a flat cosmological solution [25]. Putting the above ansatz in the mass part of the Lagrangian, one can write the Friedman and Raychaudhuri equations as where From equation (12) we obtain that r should be a constant given by The scalar field equation is satisfied automatically by plugging in equations (10), (15) and (16).", "pages": [ 3, 4 ] }, { "title": "IV. SECOND ORDER LAGRANGIAN", "content": "In this section we study perturbations signified by where φ , β i and h ij are the perturbation variables of the FRW metric. The perturbation of the Stuckelberg fields in unitary gauge is and the perturbation of the dilaton field about the background solution has the form Now, we consider the infinitesimal coordinate transformation which leads to the change of the perturbed quantities as where a dot denotes the time derivation. Using the perturbations of the Stuckelberg fields one can construct the gauge invariant quantities using the perturbed Stuckelberg fields in the following manner We may decompose the gauge invariant vector and tensor parts as We note that there are four degrees of gauge freedom because of the coordinate transformation, two of which represent the scalar part and the others relate to the vector part. One may fix the gauge freedom by the choice Note that with this gauge fixing all the gauge invariant perturbation variables become equal to the original one, making our calculations simpler. Also note that the above gauge fixing is similar to the use of the unitary gauge, and so the fiducial metric takes the form The components of the f µ ν = g µρ f ρν matrix in the unitary gauge are where /epsilon1 represents a generic perturbation parameter. To compute the components of the ˜ K µ ν , we use the method presented in [14] to expand the square root in (6). To zeroth order perturbation we find where ∆ = e σ 0 /M pl . The first and second orders are where r = a N , as that in the background. The traces of ˜ K for zero and first order perturbation are given by where X = ∆ /a . To second order we obtain where From the above formulae, one may construct the mass term using equations (3). The gauge invariant second order Lagrangian can then be written as where we have defined and The definition of M i 's are given in Appendix A. We have also used equations (11) and (12) to simplify the action. The above action ensures that the scalar, vector and tensor modes do not couple to each other. Therefore we study them separately.", "pages": [ 4, 5, 6 ] }, { "title": "A. Tensor mode", "content": "Keeping only γ ij in the action (35), we obtain Variation of the above action with respect to γ ij leads to the equation of motion for tensor perturbations Fourier transforming the above equation and using the conformal time defined as one can write equation (39) as where we have dropped the indices of γ ij and define ¯ γ = a 2 γ . This equation shows that the graviton acquires a time-dependent mass in this background. This is in agreement with the result of [14] with a different mass parameter.", "pages": [ 6, 7 ] }, { "title": "B. Vector mode", "content": "We now study the vector mode of action (35). There are two vector modes S i and F i in the action. One can write the vector part of the action as One can see from the above action that the vector mode S i is an auxiliary field. Varying the action with respect to S i gives Going over to the Fourier space and substituting S i from the above equation into action (42) we obtain On the superhorizon scales one can see that the vector part of the action vanishes. Note that equation (43), expressed in Fourier space, implies S i = 0 on superhorizon scales.", "pages": [ 7 ] }, { "title": "C. Scalar mode", "content": "In this section we study the scalar perturbations of action (35). The scalar part of the second order Lagrangian can be written as As is seen from the equation above, β and Φ are non-dynamical. Transforming back to the Fourier space, one finds their equations of motion and where we have defined It is worth mentioning that the scalar perturbations of QD massive gravity has also been addressed in the decoupling limit in [25] where the authors argue that only one of the scalar modes can be captured in this limit. As we can see above, we have three scalar modes in our scalar Lagrangian. However, one combination of these scalar modes should be non-dynamical due to the ghost-free nature of the theory [25]. At this point we are interested in the behavior of the fields on the superhorizon scales where k 2 → 0. After substitution of β and Φ from equations (46) and (47) and defining the curvature perturbation on constant quasidilaton hypersurface as the Lagrangian (45) takes the form where we define For ω = 0, the field equations at the superhorizon scales are simplified to which can be analytically solved for R and ψ . Substituting R from (57) into the second equation and solving the resulting equation for ψ results in where C 1 and C 2 are integration constants and we have defined where C 3 and C 4 are some functions of C 1 , C 2 and M i . Noting that t is the conformal time, a simples analysis shows that if the condition holds, the curvature perturbation decays on superhorizon scales. On the other hand, if we have the curvature perturbation becomes constant on superhorizon scales. However, writing the above expressions in terms of α 3 and α 4 one can see that conditions (63, 64) cannot be satisfied for H 0 > 0. The other limits imply that the curvature perturbation grows on the superhorizon scales which restricts the constants α 3 and α 4 to In the case α 4 = -1 4 (1 + 3 α 3 ) only the growing mode survives and the constant α 3 admits the following range Therefore, the only possibility for the curvature perturbation in QD massive gravity is to grow on superhorizon scales.", "pages": [ 7, 8, 9 ] }, { "title": "V. CONCLUSIONS AND FINAL REMARKS", "content": "In this paper we have studied the cosmological perturbations of the Quasi-Dilaton massive gravity. This theory is the extension of the non-linear massive gravity theory recently proposed by de Rham, Gabadadze and Tolley through a scalar field. The scalar field is coupled to the mass term in such a way that the field space of the theory admits a dilatation invariance. This new symmetry of the theory enables us to obtain flat FRW solutions. If considered without matter, the theory predicts an accelerating solution which is the effect of the graviton mass. The stability of this solutions is considered in [25] where the authors find that the ω parameter has to have a positive value less than 6, and the parameter α 4 has to be less than α 2 3 / 2. The tensor mode has a different behavior as compared to that of the standard GR but similar to the gravitational waves obtained in the dRGT massive gravity theory. The gravitational waves in this theory have a non-vanishing time-dependent mass which modifies the dispersion relation of the gravitational waves. The vector mode has the property that it vanishes on the superhorizon scales. In order to find the scalar spectrum of the theory, one can use the gauge invariant variables and then integrate out two of the non-dynamical variables included in the metric perturbation. The equations of motion of the remaining three scalar perturbations can then be obtained by varying the resulting action. As a matter of fact, the procedure is so difficult that one cannot solve the equations analytically. However, as long as we are interested in studying the behavior of the theory at the superhorizon scales, we can study the lagrangian over that scales. One of the scalar perturbations does not play any role in the superhorizon scales because it always comes with a wave number. The resulting superhorizon Lagrangian can subsequently be varied with respect to ψ and the curvature perturbation R . The equations can be solved analytically if one assumes that the quasi-dilaton field has no kinetic term. We may then obtain conditions for which the curvature perturbation grows over the superhorizon scales, i. e. equation (65). However, if one considers the range of α 4 within which the solution is stable, equation (7), one may reduce the allowed parameter space to that represented by relations (65) for ω = 0. One should note that the above range for the parameter space would be different if one considered a non-zero ω parameter. It is also worth noting that relations (63,64) imply no constant curvature perturbation on superhorizon scales. It is worth mentioning that our results in this paper are in agreement with the work by Wands et al. [26] where it is proved that the comoving curvature perturbation will become constant on superhorizon scales if the energymomentum tensor of the matter is conserved. This is so since in the context of the present work, one can write the field equations of the metric as where T σ µν is the energy-momentum tensor of the dilaton field. The X µν tensor is the contribution of the graviton mass term which depends on the dilaton field, the metric and Stuckelberg fields. The covariant divergence of X µν is not zero in the full theory which implies that the energy-momentum tensor of the dilaton field from which the curvature perturbation is constructed is not constant in the full theory. So, one expects to have growing modes on superhorizon scales in QD massive gravity theory. Finally, the QD massive gravity theory has the potential of producing a reasonable inflationary scenario if one adds a potential to the action or change the graviton mass to be a function of the quasi-dilaton field. This can break the dilatation invariance, which we will study in future works. After the completion of our paper, two more works have appeared on the same subject [27, 28] where the emphasis is on the appearance of ghosts in the scalar mode on sub-horizon scales.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "We would like to thank A. E. Gumrukcuoglu for useful discussions.", "pages": [ 10 ] } ]

[ { "title": "Dissipative fields in de Sitter and black hole spacetimes: Quantum entanglement due to pair production and dissipation", "content": "Julian Adamek ∗ D´epartement de Physique Th´eorique & Center for Astroparticle Physics, Universit´e de Gen`eve, 24 Quai Ernest Ansermet, 1211 Gen`eve 4, Switzerland Xavier Busch † and Renaud Parentani ‡ Laboratoire de Physique Th'eorique, CNRS UMR 8627, Bˆat. 210, Universit'e Paris-Sud 11, 91405 Orsay CEDEX, France For free fields, pair creation in expanding universes is associated with the building up of correlations that lead to nonseparable states, i.e., quantum mechanically entangled ones. For dissipative fields, i.e., fields coupled to an environment, there is a competition between the squeezing of the state and the coupling to the external bath. We compute the final coherence level for dissipative fields that propagate in a two-dimensional de Sitter space, and we characterize the domain in parameter space where the state remains nonseparable. We then apply our analysis to (analogue) Hawking radiation by exploiting the close relationship between Lorentz violating theories propagating in de Sitter and black hole metrics. We establish the robustness of the spectrum and find that the entanglement among Hawking pairs is generally much stronger than that among pairs of quanta with opposite momenta.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The propagation of quantum fields in expanding cosmological backgrounds leads to the spontaneous creation of pairs of particles with opposite momenta [1]. For free fields, relativistic or dispersive, this pair creation (also called the dynamical Casimir effect in condensed matter physics, see e.g., Refs. [2, 3]) is associated with the building up of nonlocal correlations that lead to quantum mechanically entangled states [4, 5]. To define these states without ambiguity, we shall use the notion of nonseparability [6], see Appendix B. For dissipative fields, i.e., fields coupled to an environment, there is a competition between the squeezing of the state, which increases the strength of the correlations, and the coupling to the external bath, which reduces it [7-9]. Our principal aim is to study this competition. We shall work both in time-dependent (cosmological) settings and with stationary metrics. For simplicity and definiteness, we consider fields that propagate in a twodimensional de Sitter space and display dissipative effects above a certain momentum threshold Λ. For these fields, the final coherence level is constant and well defined. We characterize the domain in parameter space where the final state is nonseparable. The parameters are the mass of the field, the temperature of the environment, and the ratio Λ /H , where H is the Hubble constant. Since the dissipative/dispersive effects we are considering are suppressed in the infrared, our models can be conceived as providing a phenomenological approach to theories of quantum gravity, such as Hoˇrava-Lifshitz gravity [10], where Lorentz invariance is violated at high energy. In these theories, dissipative effects will necessarily appear through radiative corrections [11]. We also recall that in condensed matter, the spectrum of quasiparticles often displays dissipation above a certain threshold. Hence, our model can also be viewed as a toolbox to compute the consequences of dissipation on pair production and parametric amplification found, e.g., in the superfluid of polaritons studied in Ref. [12]. The interest in working in de Sitter space is twofold. On the one hand, the analysis of the state can be done in terms of homogeneous modes and pair creation of quanta with opposite momentum. On the other hand, the state can also be analyzed in terms of stationary modes and thermal-like effects associated with the Gibbons-Hawking temperature [1]. It is rather clear that the homogeneous representation in de Sitter can be conceived as an approximation to e.g., slow roll inflation, see Refs. [13, 14]. What is less obvious is that de Sitter also provides a reliable approximation to describe dissipative fields propagating in black hole metrics. Indeed, when the ultraviolet scale Λ is well separated from the surface gravity of the black hole, the dissipative aspects of typical Hawking quanta all occur in the near horizon region, which can be mapped into a portion of de Sitter space (when the Hubble constant is matched to the surface gravity). As a result, the state evaluated in a black hole metric can be well approximated by the corresponding one evaluated in de Sitter. In this respect, the present paper follows up on our former work [15] where we studied this correspondence for dispersive fields. The reader unfamiliar with field propagation in de Sitter space will find in that work all necessary information. This paper is organized as follows. In Sec. II we present the action which engenders dissipative effects, and we discuss the residual symmetries found in de Sitter space when considering such theories. In Sec. III, exploiting the homogeneity of de Sitter, we compute the spectral properties and the correlations of pairs with opposite momenta. In Sec. IV, exploiting the stationarity, we compute the deviations with respect to the Gibbons-Hawking temperature. We apply our model to black holes in Sec. V, and we conclude in Sec. VI. We work in units where glyph[planckover2pi1] = c = 1.", "pages": [ 1, 2 ] }, { "title": "A. Covariant settings", "content": "We study a scalar field φ that has a standard relativistic behavior at low energy but displays dispersion and dissipation at high energy, thereby violating (local) Lorentz invariance. While high-energy dispersion is rather easily introduced and has been studied in many papers both in cosmological settings [14, 16-18] and black hole metrics [19, 20], see e.g., Ref. [21] for a review, dissipation has received comparatively much less attention. When preserving unitarity and general covariance, dissipation is also technically more difficult to handle. To do so in simple terms, following [22], we introduce dissipation by coupling φ to some environmental degrees of freedom ψ , and the action of the entire system S tot = S φ + S ψ + S int is taken quadratic in φ, ψ , as in models of atomic radiation damping [23] and quantum Brownian motion [24]. Again for reasons of simplicity, we shall work in 1 + 1 dimensions. The reader interested in four-dimensional models may consult [13], where there is a phenomenological study of inflationary spectra in dissipative models. In the present work, we consider dispersion relations that contain both dispersive and dissipative effects. These relations can be parametrized by two real functions Γ , f as where Γ( P 2 ) > 0 is the damping rate, and f ( P 2 ) describes dispersive effects. To recover a relativistic behavior in the infrared, a typical behavior would be Γ ∼ P 2 and f ∼ P 4 for P 2 → 0. In Eq. (1), Ω and P 2 are, respectively, the proper frequency and the proper momentum squared as measured in the 'preferred' frame [25], i.e., the frame used to implement the dispersion relation. In condensed matter systems, it is provided by the medium. Instead, in the phenomenological approach to Lorentz violating effects we are pursuing, it should be given from the outset, either as a dynamical field endowed with an action [10, 26], or as a background field (as we shall do). To describe it in covariant terms, following Ref. [27], we introduce both the unit timelike vector field u which describes the flow of preferred observers, and the unit spacelike vector field s which is orthogonal to u . In terms of these, one has Ω = u µ p µ and P 2 = ( s µ p µ ) 2 where p µ is the momentum of the particle in an arbitrary coordinate system. In two dimensions, the metric g µν can be written as g µν = -u µ u ν + s µ s ν which expresses that u and s are orthonormal vectors. We now consider a unitary model which implements Eq. (1). This model is not unique but can be considered as the simplest one, as shall be made clear below. In covariant terms, the total action S tot = S φ + S ψ + S int is where d 2 = d 2 x √ -g ( x ) is the covariant measure. In the first line, S φ is the standard action of a massive scalar field, apart from the last term which introduces the high frequency dispersion described by f ( P 2 ). In two dimensions, the self-adjoint operator which implements P 2 is -∇ 2 s . = ∇ † s ∇ s , where ∇ s = s µ ∇ µ is an anti-self-adjoint operator (when u is a freely falling frame), ∇ † s = -∇ µ s µ its adjoint, and ∇ µ the covariant derivative. A fourdimensional version of this model can be found in [22]. glyph[negationslash] The second action, that of the ψ field, contains the extra dimensionless parameter q , which can be considered as a wave number in some extra dimension. Its role is to guarantee that the environment degrees of freedom are dense, something necessary to engender dissipative effects when coupling ψ to φ [22, 24]. The role of the frequency Λ is to set the ultraviolet scale where dissipative effects become important. The kinetic term of ψ is governed by the anti-self adjoint operator ∇ u . = -( u µ ∇ µ + ∇ µ u µ ) / 2 which implements Ω = u µ p µ . We notice that there is no spatial derivative acting on ψ . This means that the quanta of ψ are at rest in the preferred frame. This restriction can easily be removed by adding the term c 2 ψ ( ∇ s ψ ) 2 which associates to c ψ the group velocity of the low q quanta. Including this term leads to much more complicated equations because dissipative effects are then described by a nonlocal kernel, as shall be briefly discussed after Eq. (11). For reasons of simplicity, we shall work with c ψ = 0 which gives a local kernel. Moreover, in homogeneous universes c ψ = 0 also implies that the Ψ-modes are not parametrically amplified by the cosmological expansion. When working with given functions Γ( P 2 ) and f ( P 2 ), we do not expect that the complications associated with c ψ = 0 will qualitatively modify the effective behavior of φ , at least when Λ is well separated from the Hubble scale. The interaction between the two fields is given by the third action. The strength and the momentum dependence of the coupling is governed by the function γ ( P ) which has the dimension of a momentum. Its role is to engender the decay rate Γ entering Eq. (1). The last two actions possess peculiar properties which have been adopted to obtain simple equations of motion. These are The solution to the second equation is where ψ 0 q is a homogeneous solution, and where the driven solution is governed by G q ( x , x ' ), the retarded Green function of ψ q . When injecting ψ q in the rhs of the first equation, one obtains the equation of φ driven by ψ 0 q . The general solution can be written as φ = φ dec + φ dr , where the decaying part is a homogeneous solution, and where the driven part is given by In a general Gaussian φ -ψ model, the retarded Green function G ret would obey a nonlocal equation, i.e., an integro-differential equation. We have adjusted the properties of S ψ and S int precisely to avoid this. Two properties are essential. Firstly, at fixed q and along the orbits of u , Eq. (3b) reduces to that of a driven harmonic oscillator. This can be seen by introducing the coordinates ( τ, z ) defined by u µ ∂ µ = -∂ τ | z where z is a spatial coordinate which labels the orbits of u . Then, ∇ u applied on scalars is where a ( τ, z ) . = e ∫ τ dτ ' Θ( τ ' ,z ) , and where Θ . = -∇ µ u µ is the expansion of u . Hence the rescaled field obeys the equation of an oscillator of constant frequency π Λ | q | . Secondly, when summed over q , the retarded Green function of ψ obeys [22] where δ 2 ( x -x ' ) is the covariant Dirac delta, i.e., ∫ d 2 f ( x ) δ 2 ( x -x ' ) = f ( x ' ). Eq. (8) guarantees that the differential operator encoding dissipation is local. Namely, when inserting ψ q of Eq. (4) in Eq. (3a), one finds with the local differential operator One can now verify that the WKB solutions of glyph[square] diss φ = 0 are governed by a Hamilton-Jacobi action which obeys the dispersion relation of Eq. (1) with see Appendix C for more details. The reader can also verify that any modification of the actions S ψ and S int leads to the replacement of glyph[square] diss by a nonlocal operator. When considered in homogeneous and static situations, this is not problematic because one can work with Fourier modes in both space and time. However when considered in nonhomogeneous and/or nonstatic backgrounds, it becomes hopeless to solve such an equation by analytical methods. In our model, the retarded Green function thus obeys and vanishes when x is in the past of x ' , where the past is defined with respect to the foliation introduced by the u field. When canonically quantizing φ and ψ , since our action is Gaussian, the commutator G c ( x , x ' ) . =[ ˆ φ ( x ) , ˆ φ ( x ' )] is independent of ˆ ρ tot , the state of the entire system. Moreover, it is related to G ret in the usual way In this paper we only consider Gaussian states. This implies [28, 29] that the density matrix ˆ ρ tot , and all observables, are completely determined by the anticommutator of ˆ φ , that of ˆ ψ , and the mixed one containing ˆ φ and ˆ ψ . Decomposing the field operator ˆ φ = ˆ φ dec + ˆ φ dr , G ac splits into three terms. The first one involves only ˆ φ dec , the second contains both ˆ φ dec and ˆ φ dr , and the last only ˆ φ dr . When assuming that the initial conditions are imposed in the remote past, because of dissipation, only the last one is relevant. Using Eq. (5), it is given by 1 where the noise kernel is In Secs. III and IV, we compute G dr ac and extract from it pair creation probabilities and Hawking-like effects taking place in de Sitter space.", "pages": [ 2, 3 ] }, { "title": "B. Affine group in de Sitter space", "content": "The two dimensional de Sitter space possesses three Killing vector fields that generate the algebra of the Lie group SO (1 , 2). Imposing that the action is invariant under the full group precludes ultraviolet dispersive and dissipative effects such as those of Eq. (1), see Appendix A in Ref. [15] for the proof. Since we want to work with Eq. (1), we must break (at least) one of these symmetries. As in Refs. [15, 33], we preserve the invariance under a two dimensional sub-group which corresponds to the affine group. Its algebra is generated by the Killing fields K z and K t . Using the cosmological coordinates t, z of the Poincar'e patch K z = ∂ z generates translations in z and expresses the homogeneity of the sections t = cst . , whereas K t = ∂ t -Hz∂ z expresses the stationarity of de Sitter. Using X = e Ht z , this symmetry becomes manifest, Considering the two Killing fields K z and K t , there is only one unit timelike freely falling field which commutes with both of them. We call it u ff , and we call s ff the spatial unit orthogonal vector u ff · s ff = 0. Then the coordinates t, X are both invariantly defined in terms of u ff , s ff by dt = u ff µ dx µ , ∂ X | t = s µ ff ∂ µ . Imposing that the action of Eq. (9) be invariant under the affine group requires that the preferred fields u and s commute with K t and K z . This fixes u and s up to a boost, see Appendix A. For simplicity, in what follows, we work with u = u ff . In this case, the preferred frame coincides with the cosmological one, and the orbits of u are z = cst . We also impose that the states ˆ ρ tot are invariant under the affine group. This is analogous to the restriction to the so-called α -vacua which are invariant under the full de Sitter group [34, 35]. This means that G ac , G ret and N of Eq. (15) will be invariant under both K t and K z . However, because the commutator [ K z , K t ] = -HK z does not vanish, one cannot simultaneously diagonalize K z and K t . This leads to two different ways to express the two-point functions, either at fixed wave number k = -i∂ z | t , or at fixed frequency ω = i∂ t | X . Explicitly, one has where k = | k | , and where the 'any' subscript indicates that these Fourier transforms apply to any two-point function which is invariant under the affine group. (In Eq. (19a), G k only depends on k because we impose isotropy.) What is specific to this group is that the two symmetries combine in a nontrivial way, and imply that twopoint functions only depend on two quantities, and not three, as it is generally the case in homogeneous or stationary metrics. In the homogeneous representation, it implies that the product k G k any ( η, η ' ) only depends on the physical momenta P = -Hkη , P ' = -Hkη ' . Hence, in what follows, we work in the P -representation with To reach this representation when starting from the stationary G ω any ( X,X ' ) is more involved, and is explained in Appendix A.", "pages": [ 4 ] }, { "title": "A. Dissipation and nonseparability", "content": "In this section, we decompose the fields in Fourier modes of fixed k . This representation is suitable for studying the cosmological pair-creation effects induced by the expansion a ( t ) = e Ht = -1 /Hη . To express the outcome of dissipation in standard terms, we exploit the fact that Lorentz invariance is recovered in the infrared, for momenta P = ke -Ht glyph[lessmuch] Λ. In this limit, since Γ and f of Eq. (1) are negligible, the k components of ˆ φ decouple from ˆ ψ , and obey a relativistic wave equation. Hence, the k component of the (driven) field operator of Eq. (5) can be decomposed in the out basis as where the out modes obey the scalar wave equation and satisfy the standard positive frequency condition at late time. This means that the (reduced) state of ˆ φ (obtained by tracing over ˆ ψ ) can be asymptotically described in terms of conventional excitations with respect to the asymptotic out -vacuum. The out operators ˆa k , ˆ a † k obey the standard commutation rule [ˆa k , ˆ a † k ' ] = δ ( k -k ' ). For notational simplicity, we omit the δ ( k -k ' ) when writing two-point functions because it is common to all of them since we only consider homogeneous states. For instance, Tr ( ˆ ρ tot { ˆ φ † k , ˆ φ k ' } ) = δ ( k -k ' ) × G k ac . Using Eq. (21), the coefficient of the δ function is where The mean number of asymptotic outgoing particles is n k > 0, whereas the complex number c k characterizes the strength of the correlations between particles of opposite wavenumber. The relative magnitude of this number leads to the notion of nonseparability. To explain this, we recall that the correlations weighted by c k obey the following Cauchy-Schwartz inequality, see Appendix B for more details. To characterize the level of coherence, we shall use the parameter of Ref. [36] which belongs to the interval [0 , n k +1]. When δ k = 0, one has a maximally entangled squeezed state with zero entropy, and when δ k = n k + 1 one has an incoherent thermal state of maximum entropy. For homogeneous Gaussian states, one also verifies that the entropy is monotonically growing with δ k . The important and nontrivial fact is that δ k = 1 divides states that are quantum mechanically entangled from states that only possess classical correlations. To show this we recall the notion of separability. A twomode state is called separable when it can be written as a weighted sum of products of two one-mode states, where all weights are positive and can thus be interpreted as probabilities. In this case, the strength of the correlations is more restricted than Eq. (24). Indeed, one finds | c k | 2 ≤ n 2 k , see Appendix B. As a consequence, whenever a homogeneous state is nonseparable , i.e., so entangled that it cannot be represented as a classically correlated state characterized by probabilities. In terms of δ k this criterion is simply given by δ k < 1.", "pages": [ 4, 5 ] }, { "title": "B. Invariant states and P representation", "content": "Since the states we consider are invariant under the affine group, n k and c k are necessarily independent of k . We shall nevertheless keep the label k to remind the reader that we work at fixed k and not at fixed ω as in the next section. Because of the affine group, only depends on P , where ϕ k ( t ) is the (positive unit norm) out mode of Eq. (21). The norm of the mode ϕ is fixed by the Wronskian Using such ϕ and Eqs. (20) and (22), Eq. (15) can be written as (29b) In the second line, the noise kernel of Eq. (16), which is also invariant under the affine group for the set of states we are considering, has been written in the P -representation using Eq. (20). To extract n k and c k from the above equations, we need to compute G ret and N . Using Eq. (20), Eq. (12) reads The unique (retarded) solution can be expressed as with the optical depth [13], Its role is to limit the integrals over P 1 and P 2 in Eq. (29b) to low values so that I P 0 glyph[lessorsimilar] 1. All information about the state for higher values of P is erased by dissipation. In Eq. (31) we have introduced where ¯ ϕ is a homogeneous damped solution of Eq. (30). By construction, ˜ ϕ P obeys the reversible (damping free) equation 2 and is normalized by Eq. (28). Moreover, we impose that it obeys the out positive frequency condition, meaning that in the limit P → 0, it asymptotes to the out mode ϕ of Eq. (27). Hence, comparing Eq. (22) with Eqs. (29b) and (31), we find (35a) These central equations establish how the environment noise kernel N fixes the late time mean occupation number and the strength of the correlations. We now compute N . When u is freely falling, the rescaled field ˆ Ψ 0 q of Eq. (7) is a dense set of independent harmonic oscillators of constant frequency Ω q = π Λ | q | , one at each z . The frequency is constant because we set c ψ = 0 in the action for ψ , see the discussion after Eq. (2). It implies that the positive frequency mode functions are the standard e -i Ω q t / √ 2Ω q , and that the state of these oscillators remains unaffected by the expansion of the universe. Hence T ψ , the temperature of the environment, is not redshifted. We here wish to recall that for relativistic (and dispersive) fields, the vacuum state of zero temperature is the only stationary state which is Hadamard [15]. Hence, for these fields, the temperature is fixed to zero. This is not the case in our model where any temperature T ψ is acceptable. In what follows, we shall thus treat T ψ as a free parameter, and work with homogeneous thermal states. This means that the expectation value of the anticommutator of ˆ ψ 0 q is given by The factor coth(Ω q / 2 T ψ ) = 2 n Ψ q + 1 is the standard bosonic thermal distribution. The prefactor δ ( z -z ' ) / √ a ( t ) a ( t ' ) comes from the facts that ˆ Ψ 0 q of Eq. (7) is a dense set of independent oscillators, and that a ( τ, z ) reduces here to the scale factor a ( t ). To get N of Eq. (16) one should differentiate the above and integrate over q . The integration gives a distribution which should be understood as Cauchy principal value, To be able to re-express Eq. (37) in the P -representation, it is necessary to verify that it is invariant under the affine group. This is easily done using notations of the Appendix A. One verifies that the first factor simply equals δ (∆ 2 ), whereas the second line is only a function of ∆ 1 . Taking into account the derivatives of Eq. (16), in the P -representation, the noise kernel at temperature T ψ reads The symbol P . V . indicates that when evaluated in the integrals of Eq. (35), the nonsingular part should be ex- tracted using a Cauchy principal value prescription on ln( P ' /P ) = H ( t -t ' ). In the high-temperature limit, the double integrals of Eq. (35) can be evaluated analytically because N effectively acts as a Dirac delta function. Instead, when working with an environment in its ground state, or at low temperature T ψ , we are not aware of analytical techniques to evaluate these integrals. Hence, to study the impact of dissipation on coherence in (near) vacuum states, we shall numerically integrate Eqs. (35).", "pages": [ 5, 6 ] }, { "title": "C. Numerical Results", "content": "In the forthcoming numerical computations, for simplicity, we work with which contain the same ultraviolet momentum scale Λ. The dimensionless coupling g 2 controls the relative importance of dispersive and dissipative effects. In the limit g 2 → 0, we get the quartic superluminal dispersion studied in Refs. [14, 15]. The critical coupling g 2 crit = 2, greatly simplifies the calculations, since f -Γ 2 = 0 guarantees that ˜ ϕ P obeys a relativistic equation, see Eq. (34). Using a numerically stable procedure for the Cauchy principal values like in Ref. [13], we compute n k and c k of Eq. (35) in the parameter space Λ, g 2 , m 2 , and T ψ . Since all physical effects only depend on dimensionless ratios, we present the numerical results in terms of µ = m/H , λ = Λ /H , and ϑ = T ψ /H .", "pages": [ 6 ] }, { "title": "1. Massless critical case", "content": "We begin with the massless case ( µ 2 = 0) and with g = g crit . Then Eq. (34) is particularly simple since the rescaled mode ˜ ϕ of Eq. (33) reduces for all P to the out -mode ϕ P = e iP/H / √ 2 H . In this we recover the conformal invariance of the massless field in two dimensions. There usually would be no particle production when it propagates in de Sitter space, however, the conformal invariance being broken by dissipation, pair-creation will take place. In Fig. 1 we present n k and δ k when the environment is in its ground state ( T ψ = 0). For comparison, we also show n k for quartic dispersion ( g 2 = 0) which can be computed analytically in the Bunch-Davies vacuum [14]. For λ →∞ the number of particles goes to zero as 1 /λ , as is expected since conformal invariance is restored in this limit. Despite dissipation, we find that δ k < 1 for all values of λ . This indicates that the state is always nonseparable in the two-mode k basis. In addition, contrary to what might have been expected, the two-mode entanglement is stronger for smaller values of λ , i.e., stronger dissipative effects. The reason for this has to be found in the fact that λ also sets the scale where conformal invariance is broken. Let us now turn to the effects of the environment temperature T ψ . Figure 2 shows contour plots of n k and δ k for a massless field with Eq. (39), again for g = g crit . In the limit λ →∞ , we observe that n k → 0 irrespectively of the value of T ψ . This establishes that there is a robustness of the relativistic result in the limit λ → ∞ which generalizes that found for dispersive fields, see e.g., Ref. [14]. Moreover, in the high-temperature limit, Eqs.(35) can be evaluated analytically to give where erfi is the imaginary error function. We compared the corresponding contours with the numerical ones shown in Fig. 2 and found that they are practically indistinguishable for ϑ > 10. When considering the effects of T ψ , we observe two regimes. At low temperature ( ϑ glyph[lessmuch] 1), n k and δ k only depend on λ and are basically given by the zero temperature limit shown in Fig. 1. However, at large temperature ( ϑ glyph[greatermuch] 1), they depend on λ and ϑ according to Eqs. (40). As expected, the strongest signatures of quantum entanglement, δ k glyph[lessmuch] 1, are found in the region where the breaking of conformal invariance is large (and hence pair-creation is active) and when the environment temperature is small, so that the spontaneous pair-creation events are not negligible with respect to thermally induced events. On the other hand, when the temperature is large, the final state is separable since δ k glyph[greatermuch] 1. In Fig. 2 (right panel) we see that the threshold case δ k = 1 is approximatively given by ϑ ∼ λ -1 / 2 for λ glyph[lessorsimilar] 1. The hatched region for λ glyph[greaterorsimilar] 10 represents the numerical uncertainty in the region where n k is much smaller than 1.", "pages": [ 6, 7 ] }, { "title": "2. Massive fields", "content": "We note that the massless case µ 2 = 0 is an isolated point in the mass spectrum: a well-defined notion of out -quanta requires either µ 2 = 0 or µ 2 > 1 / 4. In the latter case, the asymptotic out -modes with positive frequency (see, e.g., Appendix B of Ref. [14]) are given by where ˜ µ . = √ µ 2 -1 / 4 and J denotes the Bessel function of the first kind. Figure 3 shows the contour plots of n k and δ k for a massive field with µ 2 = 5 / 4 and g = g crit , in the same parameter space ( λ , ϑ ) as in Fig. 2. The case of a Lorentzinvariant field in the Bunch-Davies state is recovered in the limit λ →∞ , ϑ → 0. Now conformal invariance is already broken by the mass term and therefore n k remains nonzero in this limit. At zero temperature, the strongest entanglement (lowest δ k ) is found at large values of λ , i.e., weak dissipation. This was expected, since dissipation reduces the strength of correlations. However, as in the massless case, the threshold of separability δ k = 1 is not crossed. When increasing the environment temperature T ψ , we see that the strength of correlation is reduced, and separable states are found. The nonseparability criterion δ k < 1 is therefore only met either when T ψ is smaller than the Gibbons-Hawking temperature T GH = H/ 2 π , or when the coupling to the environment is sufficiently weak. Notice also that the behavior at high temperature can again be obtained analytically, the integrals over the Bessel functions becoming hypergeometric functions.", "pages": [ 7 ] }, { "title": "3. Role of g in the underdamped regime", "content": "It is also interesting to consider the role of the coupling g , see Eq. (39). As g 2 approaches zero, the dissipative scale 2Λ /g 2 is moved deeper into the UV with respect to the dispersive scale which is fixed by Λ. In the limit g 2 → 0, the field becomes purely dispersive and n k , δ k can be computed analytically [14] in the Bunch-Davies vacuum. For g 2 < 2 the mode is underdamped. In this case, the solutions to Eq. (34) which correspond to asymptotic out -modes of positive frequency are given by, see Appendix B of Ref. [14], where ˜ λ . = λ/ √ 4 -g 4 and M is a Whittaker function defined in Ref. [37]. and dissipation) and δ k for a massive field in the underdamped regime. Here, we set T ψ = 0, and plot the results in the parameter space spanned by the two (dimensionless) ultraviolet scales: λ which characterizes dispersion, and 2 λ/g 2 which is the UV scale of dissipation. The latter is larger than the former in the underdamped regime. The grey areas therefore correspond to the overdamped regime which we did not study. In the weak dispersive/dissipative regime λ glyph[greaterorsimilar] 10, it is evident that δn k and δ k are both dominated by dissipative effects. For the latter, this is because dispersion alone does not lead to decoherence. For the deviation δn k , this follows from the fact that dispersion gives an exponentially small correction to the pair creation process (see Ref. [14]), while the corrections due to dissipation are only algebraically small. As a result, the hierarchy of scales does not directly fix the importance of the respective effects. On the other hand, when dispersion is strong ( λ glyph[lessorsimilar] 1) the pair creation process is basically governed by dispersive effects. The correction to the particle number due to dissipation is very small (compared to the dispersive correction). One can also observe that the degree of twomode entanglement is then basically governed by the separation between the two scales g 2 , i.e., δ k is determined by the strength of dissipation at the dispersive threshold , (Γ /P ) | P =Λ .", "pages": [ 7, 8, 9 ] }, { "title": "IV. STATIONARY PICTURE", "content": "In the absence of dispersion/dissipation, it is well known that the Bunch-Davies vacuum is a thermal (KMS) state at the Gibbons-Hawking temperature T GH = H/ 2 π [1]. It is also known that this is the temperature seen by any inertial particle detector, and that this is closely related to the Unruh effect found in Minkowski space, and to the Hawking radiation emitted by black holes [38]. In the presence of dissipation, while the stationarity of the state of φ is exactly preserved when the state of the environment is invariant under the affine group, the thermality of the state is not exactly preserved. This loss of thermality, which generalizes what was found for dispersive fields [15], questions the status of black hole thermodynamics when Lorentz invariance is violated [39-41].", "pages": [ 9 ] }, { "title": "A. Loss of thermality", "content": "To probe the stationary properties of the state, we consider the transition rates of particle detectors at rest with respect to the orbits of K t . This means that the detector is located at fixed H | X | < 1 in the coordinates of Eq. (18). In this case, the two-point functions only depend on t -t ' and can be analyzed at fixed ω = i∂ t | X , see Eq. (19b). (The above restriction on X simply expresses that the trajectory be timelike.) The transition rates are, up to an overall constant, given by Fourier transforms of the Wightman function G W [38]. The rates then determine n ω ( X ), the mean number of particles of frequency ω > 0 seen by a detector located at X , through To study the deviations with respect to the GibbonsHawking temperature T GH = H/ 2 π , we introduce the temperature function T ω ( X ) defined by It gives the effective temperature seen by the detector, and reduces to the standard notion when it is independent of ω . In the absence of dispersion and dissipation, T ω ( X ) = T GH for all values of ω , which means that the Tolman law is satisfied [15]. In the following numerical computations, for simplicity, we work at X = 0 with an inertial detector, with g = g crit , m = 0, and T Ψ = 0. Since the calculation of the commutator of φ is much faster and more reliable than that of the anticommutator, instead of using Eq. (43), GLYPH<144> n ω shall be computed with The denominator is expressed using Eq. (13). The numerator is obtained from Eqs. (69) and (38) with T ψ → 0. In addition, the principal value is replaced by a prescription for the contour of ln P/P ' = Ht to be in the upper complex plane. In this we recover the fact that when the anticommutator in the vacuum is P . V . (1 /t ), the corresponding vacuum Wightman function is 1 / ( t -iglyph[epsilon1] ). In Fig. 5, we plot the ratio T ω /T GH as a function of ω for various values of λ , and for T ψ = 0. We first observe that T ω is constant for all frequencies from zero to a few multiples of T GH . Hence, the Planckian character of the state is, to a high accuracy, preserved by dissipation, as was found in the presence of dispersion [15, 42, 43]. For higher frequencies, i.e., ω/T GH > 4, we were not able to study T ω with sufficient accuracy because of the numerical noise associated to n ω < 0 . 01. As in the dispersive case, we expect that the temperature function T ω is modified for ω glyph[greaterorsimilar] Λ. Secondly, when λ is smaller than 5, i.e., when dissipation is strong, we observe that the temperature is significantly (more than 5%) larger than T GH . These deviations are further studied in Fig. 6, where we plot the deviations of T 0 , the low-frequency effective temperature, with respect to T GH as a function of λ . We observe that the deviation due to dissipation asymptotically follows This law has been verified up to λ = 10 3 . It has to be compared with the deviation due to quartic dispersion studied in Ref. [15]. This deviation is represented by the dotted curve, and scales as T disp 0 /T GH -1 ∼ e -πλ/ 4 . In other words, the deviation due to (quadratic) dissipation decreases much slower than that due to (quartic) superluminal dispersion. The important lesson for black hole thermodynamical laws is that ultraviolet dispersion and dissipation both destroy the thermality of the state. This lends support to the claim that Lorentz invariance is somehow necessary for these laws to be satisfied.", "pages": [ 9, 10 ] }, { "title": "B. Asymptotic correlations among right movers", "content": "As explained in Sec. III A, at late time, the φ field decouples from its environment. This allows to use the relativistic out basis at fixed k to read out the state of φ . Alternatively, one can also use an out basis formed with stationary modes with fixed frequency ω . Indeed, at fixed ω , the momentum P ω ∼ | ω/X | → 0 at large | X | , and dispersive effects are negligible. Hence ˆ φ ω ( X ), the stationary component of the field operator, decouples from the environment at large | X | , and can be analyzed using relativistic modes. As we shall see, this new out basis is not trivially related to the homogeneous one used in Sec. III because it encodes thermal effects at the Gibbons-Hawking temperature. Hence the covariance matrix of the new out operators will depend on n k and c k of Eq. (35), but also on these thermal effects. At this point we need to explain why we are interested in expressing in a different basis a state which is fully characterized by n k and c k . The main reason comes from black hole physics. As shall be discussed in the next section, when certain conditions are met, the results of this section apply to the Hawking radiation emitted by dissipative fields. To compute the covariance matrix in the new basis, we recall some properties of the relativistic massless field in de Sitter. First, because of conformal invariance, the field operator splits into two sectors which do not mix, one for the right-moving U modes with k > 0, and the other for the left-moving V modes with k < 0. In addition, in de Sitter, the time-dependence of all homogeneous modes can be expressed through ϕ ( P ) of Eq. (27), which here reduces to where P > 0. This mode has a unit positive KleinGordon norm, as can be verified using the Wronskian condition of Eq. (28). We introduce an intermediate basis constructed with the stationary 'Unruh' modes ϕ ω [44]. In the P representation, they can be written as [45] They form an orthonormal and complete mode basis if ω ∈ ] - ∞ , ∞ [. The spatial behavior of the U modes is given by We now introduce the alternative out basis formed of stationary modes which are localized on either side of the horizons, henceforth called R and L modes. They behave as Rindler modes in Minkowski space. For U -modes, the horizon is located at HX = -1, and these modes are where ω > 0. The first has a positive norm, while the second has a negative one. They are easily related to the Unruh mode by computing Eq. (49). Indeed, for ω > 0, one gets where coefficients α H ω and β H ω are the standard Bogoliubov coefficients leading to the Gibbons-Hawking temperature H/ 2 π . They obey ∣ ∣ β H ω /α H ω ∣ ∣ = e -πω/H . Asymptotically in the future and in space, the U part of the field operator can thus be expressed as The V part possesses a similar decomposition, and the V modes are obtained from the U ones by replacing X → -X , and R → L . The χ V modes are thus defined on either side of HX = 1. Using the above equations, the Unruh and the Rindlerlike operators of frequency | ω | are related by We considered both U and V modes because our aim is to compute the covariance matrix of the R and L operators in terms of n k and c k of Eq. (35), where c k mixes U and V modes. To do so, we first compute the covariance matrix of the Unruh operators. When working with states that are invariant under the affine group, n k and c k of Eq. (35) are independent of k . This implies that the covariance matrix of the Unruh operators is independent of ω . Indeed, using which follows from the Fourier transforms Eqs. (52a) and (52b), one verifies that the independence of k implies that of ω . As a result, introducing V † ω = ( ˆ a † ω U , ˆ a -ω U , ˆ a † ω V , ˆ a -ω V ) , the covariance matrix of Unruh operators reads where n k and c k are given in Eq. (23). Using the matrix B ω of Eq. (53), and dropping the trivial factor of δ ( ω -ω ' ), the covariance matrix of R and L operators is where glyph[negationslash] The first two coefficients concern separately either the U , or the V -modes. They fix the spectrum and the strength of the correlations. The last two concern the U -V mode mixing, and are proportional to c k . Considering the coherence amongst pairs of U -quanta, i.e., ignoring the V -modes, as in Eq. (25), we define Using Eq. (57), we obtain We see that δ U does not depend on c k . This is to be expected since c k characterizes the correlation between modes of opposite momenta, and since there is no U -V mode mixing for two-dimensional massless fields. More importantly, Eq. (59) is valid irrespectively of the temperature of the environment T ψ . We can thus study how the separability of U -quanta is affected by T ψ . The criterion of nonseparability, δ ω U < 1, gives where n k ( T ψ ) is plotted in Fig. 2. Using this Figure, in Fig. 7 we study ln δ U ω with ω = H as a function of λ and ϑ = T ψ /H . At zero temperature T ψ = 0, we see that the pair of U -quanta with ω = H is nonseparable for λ glyph[greaterorsimilar] 0 . 2, i.e., for a rather strong dissipation since Λ = H/ 5. Using Eq. (60) we see that this is also true for all quanta with ω/H glyph[lessorsimilar] 1. More surprisingly, when λ is high enough, this pair is nonseparable even when T ψ > T GH , i.e., when the environment possesses a temperature higher than the Gibbons-Hawking temperature. Indeed, whenever T ψ glyph[lessorsimilar] √ H Λ / 2, the pair is nonseparable, as all pairs with smaller frequency ω . In other words the quantum entanglement of the lowfrequency U pairs of quanta is extremely robust when working with dissipative fields which are relativistic in the infrared. The robustness essentially follows from the kinematical character of the transformation of Eq. (53) which relates two relativistic mode bases. It is also due to the fact that n k , the number of U -V pairs created by the cosmological expansion, remains negligible in Eq. (57) as long as 1 glyph[lessmuch] Λ /H , and T ψ glyph[lessmuch] T GH (Λ /H ) 1 / 2 .", "pages": [ 10, 11, 12 ] }, { "title": "V. BLACK HOLE RADIATION", "content": "We now explain when and why the above results apply to the Hawking radiation emitted by dissipative fields. We shall be more qualitative than in the former sections because several approximations are involved in the correspondence between de Sitter and the black hole case. Our main aim is to establish that the spectrum of Hawking radiation, and the associated long distance correlations across the horizon, are both robust when dissipation occurs at sufficiently high energy with respect to the surface gravity, as was anticipated in Refs. [20, 22]. The robustness shall be established by studying the anticommutator of Eq. (A3b), and showing that its asymptotic behavior is governed by Eqs. (57a) and (57b). Firstly, being covariant, the action of Eq. (2) applies as such to any black hole metric endowed with a preferred frame described by a timelike field u . 3 Secondly, the correspondence with de Sitter becomes more precise when working with stationary settings. At the level of the background, this means that there is a Killing field K t , and that u commutes with K t . In this case, the metric can be written as As in Eq. (18), t, X are defined by dt = u ff µ dx µ , and ∂ X = s µ ff ∂ µ , where u ff is a stationary and freely falling unit timelike field. In the present case, it is no longer unique because the system is no longer translation invariant. It belongs to a one parameter family, where the parameter can be taken to be the value of v at spatial infinity [48]. When the preferred field u is freely falling (as we shall assume for simplicity), this residual invariance is lifted by working with u ff = u . By stationary settings, we also meant that the state of the environment is stationary. This implies that the noise kernel of Eq. (16) only depends on t -t ' when evaluated at X,X ' , along the orbits of the Killing field K t . When these stationary conditions are met, the (driven part of the) anticommutator of φ is (exactly) given by Eq. (A3b), where the two kernels G ω ret and N ω are now defined in the black hole metric of Eq. (61). As a result, to compare the expressions of G ω ac ( X,X ' ) evaluated in de Sitter and in Eq. (61), it is sufficient to study G ω ret and N ω . To establish the correspondence with controlled approximations, the following four conditions are necessary: The first condition is rather obvious and needs no justification. The second and the third conditions concern the metric and the field u . To characterize the near horizon region (NHR) explicitly, we shall use which possesses a future (black hole) Killing horizon at X = 0. The NHR is defined by the region | κX | glyph[lessorsimilar] D/ 2 where v is approximately linear. Hence it is a portion of de Sitter space with H = κ , see Eq. (18). It should be emphasized that the mapping also applies to the u field. In fact, when u is freely falling, the only scalar quantity which is involved in the mapping is its expansion evaluated at the horizon: Θ 0 = -∇ µ u µ = κ . Hence, in the NHR, the orbits of u coincide with those found in de Sitter. (When u is accelerating, both Θ 0 and the acceleration γ 0 must match, see Eq. (A10) and footnote 4 in Ref. [15].) Using Eq. (62), the third condition means that D cannot be too small. This condition was found in Ref. [42] when considering the spectral deviations of Hawking radiation which are due to highfrequency dispersion, see also Refs. [43, 49, 50]. For quartic dispersion, these deviations are small when D 3 / 2 glyph[greatermuch] κ/ Λ. In this case, the nontrivial dispersive effects all occur deep inside the NHR, i.e., in a portion of de Sitter space. Moreover, at fixed κ/ Λ, the spectral deviations increase when D decreases. We shall see below that these facts also apply to dissipative fields when the above four conditions are met.", "pages": [ 12, 13 ] }, { "title": "A. The stationary noise kernel", "content": "When considering the model of Eq. (2) in the metric Eq. (61) with u freely falling, the noise kernel N ω of Eq. (A3b) is where v i . = v ( X i ) and ∂ i . = ∂ X i . The stationary kernel of the last line is the Fourier transform of the anticommutator of ψ , see Eq. (36). To compute it we use the fact that the factor a ( τ, z ) of Eq. (6) is now given by (see Eq. (55) in Ref. [22] for a three-dimensional radial flow) As in Eq. (6), z labels the orbits of u . It is here completely fixed by the condition that z = X when t = 0. Since the orbits are solutions of dX/dt = v , z is implicitly given by Using the above equations to re-express the δ ( z -z ' ) of Eq. (36), one finds Its Fourier component with respect to ∆ t is trivially where ∆ t 12 = ∫ X 1 X 2 dX/v is the lapse of time from X 2 to X 1 following an orbit z = cst. which connects these two points. Since the settings are stationary, these orbits are all the same, as can be seen in Fig. 9. Using Eq. (63), the noise kernel is explicitly given by This kernel is local in that it only depends on g µν and u µ between X 1 and X 2 . Hence, when evaluated in the black hole NHR, it agrees, as an identity , with the corresponding expression evaluated in de Sitter. In conclusion, we notice that this identity follows from our choice of the action of Eq. (2). Had we used a more complicated environment, this identity would have been replaced by an approximative correspondence. In that case, the correspondence would have still been accurate if the propagation of ψ had been adiabatic. As usual, this condition is satisfied when the degrees of freedom of ψ are 'heavy', i.e., when their frequency Ω q ∼ Λ glyph[greatermuch] κ .", "pages": [ 13 ] }, { "title": "B. The stationary G ω ret", "content": "The stationary function G ω ret ( X,X 1 ) obeys Eq. (12), which is a fourth order equation in ∂ X when working with Eq. (39). Depending on the position of X and X 1 , its behavior should be analyzed using different techniques. Far away from the horizon, the propagation is well described by WKB techniques since the gradient of v is small. Close to the horizon instead, the WKB approximation fails, as in dispersive theories [49]. In this region, the P representation accurately describes the field propagation, and is essentially the same as that taking place in de Sitter. Therefore, the calculation of G ω ac ( X,X ' ) of Eq. (A3b) at large distances boils down to connecting the de Sitter-like outcome at high P to the low-momentum WKB modes. As in the case of dispersive fields, the connection entails an inverse Fourier transform from P to X space in the intermediate region II , see Fig. 9, where both descriptions are valid [20, 49, 51-53]. In the present case, these steps are performed at the level of the two-point function rather than being applied to stationary modes. In fact, we shall compute G ω ac through where the two G ω ret are expressed in a mixed X,P representation. The early configurations in interaction with the environment are described in P space, while the large distance behavior is expressed in X space. Let us give here only the essential points, more details are given in Appendix C. The validity of the whole procedure relies on a combination of the third and the fourth condition given above, namely max(1 , D -2 ) glyph[lessmuch] Λ /κ , and is limited to moderate frequencies, i.e., 0 < ω ∼ κ glyph[lessmuch] Λ. For simplicity, we consider massless fields. Then Λ /κ glyph[greatermuch] 1 guarantees that the infalling V modes essentially decouple from the outgoing U modes because the only source of U -V mixing comes from the ultraviolet sector. Hence, at leading order in κ/ Λ, it is legitimate to consider only the U modes. For massive fields with m glyph[lessmuch] Λ, the discussion is more elaborate but the main conclusion is the same: the properties of the Hawking radiation are robust. For massless fields, at fixed ω , the propagation of the U modes is governed by the effective dispersion relation, see Eq. (34), As long as P glyph[lessmuch] Λ, the U sector of G ω ret behaves as for a relativistic field, since √ F 2 -Γ 2 ∼ P (1 + O ( P/ Λ)). Instead, when P glyph[greaterorsimilar] Λ, the dispersive and dissipative terms weighted by f and Γ cannot be neglected in Eq. (12). To characterize the transition from these two regimes, we consider the optical depth of Eq. (32). When working at fixed ω , one finds where X ω ( P ) is the root of Eq. (70), as is P ω ( X ) when using X as the variable. The first expression governs G ret in the NHR where ∂ X v ∼ κ is almost constant, see Eq. (31). To leading order in Γ /P glyph[lessmuch] 1, which is satisfied everywhere but very close to the horizon, the second expression governs G ret in X space. Since v ω gr = 1 /∂ ω P is the group velocity in the rest frame, I ω = ∫ t t 1 dt ' Γ( P ω ), where the integral is evaluated along the classical outgoing trajectory. It should be noticed that, when considered in X space, I ω applies on the right and the left of the horizon. In the R region, v gr > 0, while it is negative in L, so that in both cases I ω > 0 when P 1 > P > 0, i.e., when P 1 is in the past of P . To characterize the retarded Green functions of Eq. (69), we compute I ω in the mixed representation, in the limit where P 1 is large enough so that X ω ( P 1 ) is deep inside the NHR, while X is far away from that region. For simplicity, we consider the case of Eq. (39) with g = g crit . In this case, only the dissipative effects are significant, 4 and one finds where v R ( v L ) is the asymptotic velocity on the right (left) side. From the second term, we learn that | κX | should be much smaller than Λ /κ for the Hawking quanta not to be dissipated. Since we work in the regime Λ /κ glyph[greatermuch] 1, this condition is easily satisfied. We notice that a similar type of weak damping effect of outgoing modes has been observed in experiments [54]. From the first term, we learn that I ω gives an upper bound to the domain of P which significantly contributes to Eq. (69), namely P 2 glyph[lessorsimilar] Λ κ , as in de Sitter. A lower bound of this domain is provided by the γ factors of Eq. (38). Using this equation and Eq. (A8), the integrand of Eq. (69) scales as and its behavior is represented in Fig. 8. Hence, the relevant domain of P , i.e., when T is larger than 10% of its maximum value, scales as Considered in space-time, since P ∼ e -κt , this limits the lapse of time during which the coupling to ψ occurs. Interestingly, this lapse is given by κ ∆ t ≈ 2, i.e., two e-folds, irrespective of the value of Λ /κ , and that of ω . It should be also stressed that nothing precise can be said about the domain of X , which significantly contributes because the X -WKB fails when P is so large. One can simply say that it is roughly characterized by the interval [ -X trans , X trans ], where X trans = X ω = κ ( P min ) is given by This value defines the central region III , see Fig. 8 and Fig. 9. Using the profile of Eq. (62), X trans is situated deep inside the NHR when κ/ω diss max glyph[lessmuch] 1, where the critical frequency ω diss max is given by Hence, when κ/ω diss max glyph[lessmuch] 1, the coupling between φ and ψ is accurately described in the P representation, and takes place in a portion of de Sitter. In addition, the connection between the high- and low-momentum propagation can be safely done in the intermediate region II , defined by κ | X trans | glyph[lessmuch] κ | X | glyph[lessorsimilar] D , see Fig. 9, where, on the one hand, one is still in a de Sitter-like space since v is still linear in X , and, on the other hand, the lowmomentum modes can be already well approximated by their WKB expressions. Notice finally that this reasoning only applies for frequencies ω glyph[lessmuch] ω diss max . Indeed, when ω = ω diss max , dissipation occurs around κX ∼ D , i.e. no longer in a de Sitter like background. These steps are sufficient to establish that the results of Sec. IV B apply for ω glyph[lessmuch] ω diss max . In particular, Eq. (57a) implies that the spectrum of radiation is robust (when the temperature of the environment is low enough, see Fig. 7). Namely, to leading order in κ/ Λ, the mean occupation number n ω of quanta received far away is given by the Planck distribution at the standard relativistic temperature T H = κ/ 2 π . As in dispersive settings, the real difficulty is to evaluate the spectral deviations. In this respect, we conjecture that the leading deviations due to dissipation will be suppressed by powers of κ/ω diss max . That is, they will be governed by the composite ultraviolet scale of Eq. (76) which depends on the high-energy physics, here with Γ quadratic in P , and on the extension D of the black hole NHR. This second dependence is highly relevant when D glyph[lessmuch] 1. Together with the robustness of the spectrum, one also has that of the long-distance correlations across the horizon between the Hawking quanta and their partners. These correlations are fixed by the coefficient c ω of Eq. (57b). To get the space-time properties of the pattern, one should integrate over ω , i.e., perform the inverse Fourier transform of Eq. (19b), because it is this integral that introduces the space-time coherence [20, 45, 56]. In Fig. 9, we have schematically represented the anticommutator G ac ( t -t 1 , X, X 1 ) in the t -t 1 , X plane when X 1 is taken far away from the horizon. t In brief, when κ/ω diss max glyph[lessmuch] 1 and ω/ω diss max glyph[lessmuch] 1, the nontrivial propagation only occurs deep inside the NHR which is a portion of de Sitter space. This implies that n ω and c ω are, to a good approximation, given by their de Sitter expressions of Eq. (57). Given that these (exact) expressions hardly differ from the relativistic ones when κ/ Λ glyph[lessmuch] 1, we can predict that, when computed in a black hole metric, these two observables are robust whenever the finiteness of the NHR introduces small deviations with respect to the de Sitter case. For ω/ω diss max glyph[lessmuch] 1, this is guaranteed by κ/ω diss max glyph[lessmuch] 1.", "pages": [ 14, 15, 16 ] }, { "title": "VI. CONCLUSIONS", "content": "In this paper we used (a two-dimensional reduction of) the dissipative model of Ref. [22] to compute the spectral properties and the correlations of pairs produced in an expanding de Sitter space. The terms encoding dissipation in Eq. (2) break the (local) Lorentz invariance in the ultraviolet sector. Yet, they are introduced in a covariant manner by using a unit timelike vector field u which specifies the preferred frame. In addition, the unitarity of the theory is preserved by coupling the radiation field φ to an environmental field ψ composed of a dense set of degrees of freedom taken, for simplicity, at rest with respect to the u field. Again for simplicity, the action is quadratic in φ, ψ , and the spectral density of ψ modes is such that the (exact) retarded Green function of φ obeys a local differential equation, see Eq. (10) and Eq. (12). By exploiting the homogeneous character of the settings, we expressed the final occupation number n k , and the pair-correlation amplitude c k , in terms of the noise kernel N and the retarded Green function, see Eq. (35). Rather than working with integrals over time as usually done, we used the proper momentum P = k/a ( t ) to parametrize the evolution of field configurations. Hence, Eq. (35) can be viewed as flow equations in physical momentum space. This possibility is specific to the residual symmetry group found in de Sitter space when the u field commutes with the two Killing fields K t and K z . These group theoretical aspects are explained in Appendix A. The key equations are Eq. (A6) and Eq. (A8) which show how the P representation is related to the invariant distances, to the homogeneous representation of Eq. (20), and to the stationary one. This representation is extended to Feynman rules and Schwinger-Dyson equations of (relativistic) interacting field theories in Ref. [58]. We numerically computed n k and c k in Sec. III. When considering a massless field, n k and the strength of the correlations are plotted as functions of the scale separation Λ /H , and the temperature of the environment T ψ /H , in Fig. 2. The robustness of the relativistic results is established in the limit of a large ratio Λ /H . The key result concerns the threshold values of the parameters, see the locus δ k = 1 on the right panel, for which the final state remains nonseparable, i.e., so entangled that it cannot be described by a stochastic ensemble. Various criteria of nonclassicality are compared in Appendix B. This analysis was then extended to massive fields, see Fig. 3, and to the consequences of varying the relative importance of dissipative and dispersive effects, see Fig. 4. As expected, the quantum coherence is lost at high cou- g, and when the temperature of the environment is high enough. In Sec. IV we exploited the stationarity, and we studied how the thermal distribution characterizing the GibbonsHawking effect is affected by dissipation. As in the case of dispersion [15], we found that the thermal character is, to leading order, robust. We also computed the deviations of the effective temperature with respect to the standard one T GH = H/ 2 π , see Fig. 5 and Fig. 6. In preparation for the analysis of the Hawking effect, we studied the strength of the asymptotic correlations across the Killing horizon between (right) moving quanta with opposite frequency. Quite remarkably, we found that the pairs remain entangled (the two-mode state remains nonseparable) even for an environment temperature exceeding T GH = H/ 2 π , see Fig 7. Finally, in Sec. V we extended our analysis to black hole metrics. When four conditions are met, we showed that the above analysis performed in de Sitter applies to Hawking radiation. The inequality which ensures the validity of this correspondence is κ/ω diss max glyph[lessmuch] 1, where ω diss max is the composite ultraviolet scale of Eq. (76). It depends on both the microscopic scale Λ, and D , which fixes the extension of the black hole near horizon region where the metric and the field u can be mapped into de Sitter. The validity of the correspondence in turn guarantees that, to leading order, the Hawking predictions are robust - even if the early propagation completely differs from the relativistic one, see Fig. 9. This establishes that when leaving the very high momentum P ∼ Λ (trans-Planckian) region and starting to propagate freely, the outgoing configurations are 'born' in their Unruh vacuum state [38, 59, 60]. The microscopic implementation of this state in dissipative theories is shown in Eq. (C16). As a result, as in the case of dispersive theories [42, 49], the leading deviations with respect to the relativistic expressions should be suppressed as powers of κ/ω diss max , i.e., they should be governed by the extension of the black hole NHR which is a portion of de Sitter space. In conclusion, even though our results have been derived in 1 + 1 dimensions, we believe that very similar results hold in four dimensions, at least for homogeneous cosmological metrics and for spherically symmetric ones, because a change of the dimensionality only affects the low-momentum mode propagation. Hence even if this introduces nontrivial modifications, as grey body factors in black hole metrics, they will not interfere with the high-momentum dissipative effects when the hierarchy of scales Λ /H, Λ /κ glyph[greatermuch] 1 is found. They can thus be computed separately.", "pages": [ 16, 17 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work has been supported by the FQXi Grant 'Hawking radiation in dissipative field theories' (No. FQXi-MGB-1129). J.A. wants to thank the Laboratoire de Physique Th'eorique at Orsay for hospitality and the German Research Foundation (DFG) for financial support through the Research Training Group 1147 'Theoretical Astrophysics and Particle Physics' at the University of Wurzburg, where parts of this work have been carried out. We are grateful to Ted Jacobson and Iacopo Carusotto for interesting remarks.", "pages": [ 17 ] }, { "title": "Appendix A: Affine group and P representation", "content": "We remind the reader that the affine group is the subgroup of the de Sitter isometry group which is generated by the Killing fields K z = ∂ z | t and K t = ∂ t | X , which possess the following commutator [ K z , K t ] = -HK z . The definition of the coordinates t, z, X is given in Eq. (17) and Eq. (18). In de Sitter space, there are two geometrical invariants under this group. Using the coordinates t, X , they read They are linked to the de Sitter invariant distance by The distances ∆ 1 , ∆ 2 can also be defined in a coordinate invariant manner. The interested reader will find the expressions at the end of this Appendix. When working with states that are invariant under the affine group, the n-point correlation functions only depend on ∆ 1 and ∆ 2 evaluated between the various pairs of points. Hence, any two-point functions G any ( x , x ' ) can be written as ˜ G any (∆ 1 ( x , x ' ) , ∆ 2 ( x , x ' )). However, it turns out that it is not convenient to use ∆ 1 , ∆ 2 to compute Eq. (15), and this even though the four integrals of that equation can be easily expressed in terms of two over ∆ 1 and two over ∆ 2 . The reason is that the integrals over the ∆ 2 are convolutions. Hence, it is appropriate to work with the Fourier transform with respect to ∆ 2 because, in this representation, Eq. (15) contains only two integrals. The fact that only two variables are needed is not a surprise, given the homogeneity (stationarity) of the setting. Indeed using G k ( t, t ' ) ( G ω ( X,X ' )) of Eq. (19), one immediately has To understand the relationship between these two representations, it turns out that the most convenient variables are the proper momenta P = | s µ ff p µ | and P ' = | s µ ff p ' µ | . The reasons for this are many. Firstly, P is invariantly defined; secondly, ∆ 1 is easily expressed in P, P ' space; thirdly, so is the variable conjugated to ∆ 2 ; and fourthly, P can be attributed to the field itself, so that one can easily take the even (anticommutator) and the odd part of the two-point functions. Let us explain these reasons. Once the de Sitter group is broken in a way which preserves the affine group, P is invariantly defined as the momentum associated with the orthogonal fields u ff , s ff which commute with K t and K z , and where u ff is geodesic. In our case, we work with the preferred field u = u ff , but this needs not be the case for P to be unambiguously defined as P 2 = ( s µ ff p µ ) 2 . Since P = ke -Ht , ∆ 1 is simply In addition, the momentum conjugated to ∆ 2 , defined by ¯ P . = ∂ ∆ 2 | ∆ 1 , is given by the geometrical mean The first equality follows from ∆ 2 √ ∆ 1 = X + f ( t, t ' , X ' ), and P . = ∂ X | t,t ' ,X ' . The second one follows from Eq. (A4). Hence, the Fourier transform of ˜ G any (∆ 1 , ∆ 2 ) with respect to ∆ 2 , only depends on P and P ' . Moreover, if one imposes the isotropy of the setting, ˜ G any (∆ 1 , ∆ 2 ) is even in ∆ 2 , and G any ( -P , -P ' ) = G any ( P , P ' ). Hence, in this case, all the information is contained in G any ( P, P ' ). The important point is that G any ( P, P ' ) defined by Eq. (A6) coincides with the lhs of Eq. (20). In addition, starting with the stationary representation of Eq. (19b), one can also verify that the double Fourier transform has automatically the following structure where G any ( P , P ' ) is given by Eq. (A6). Together with Eq. (20), Eq. (A6) and Eq. (A8) are the key equations of this appendix: Whenever a two-point function G any ( x , x ' ) is invariant under the affine group, its Fourier transforms G k any ( t, t ' ) and G ω any ( P , P ' ) are related to G any ( P , P ' ) of Eq. (A6) by Eq. (20) and Eq. (A8) respectively. Finally, the antisymmetry of the commutator G c is expressed as G c ( P ' , P ) = -G c ( P, P ' ) ∗ while the symmetry of G ac gives G ac ( P ' , P ) = G ac ( P, P ' ) ∗ . To conclude this Appendix, we express ∆ 1 and ∆ 2 in covariant terms. The log of ∆ 1 is given by the line integral of u ff from x to x ' , that is This is an invariant expression. Indeed, on the one hand, since u ff is geodesic, u ff µ dx µ is an exact 1-form and the above integral does not depend on the path. On the other hand, u ff is the only (timelike) unit geodesic field that commutes with K z and K t . Since, Eq. (A2) gives ∆ 2 as a combination of ∆ 1 and ∆ which are both invariantly defined, so is ∆ 2 . 7 We notice that the preferred frame fields u, s have not been used. But, if one wishes, they can be used. Indeed any couple of orthogonal fields u, s which commute with K z and K t are related to u ff , s ff by where the constant expansion is Θ = -∇ µ u µ , and where the constant acceleration is γ ν . = u µ ∇ µ u ν = γs ν .", "pages": [ 17, 18 ] }, { "title": "Appendix B: Nonseparability and Cauchy-Schwarz inequalities", "content": "In this appendix, we consider homogeneous Gaussian states. This implies that the state factorizes as glyph[negationslash] where ˆ ρ ( k ) 2 fixes the state of the two-mode system k , -k . This also implies that n k and c k of Eq. (23) only depend on k . To be general, we work with n k = n -k , which means that the state is anisotropic. Our aim is to compare three inequalities relating the norm of c k to n k and n -k which allow to distinguish quantum from classical correlations, for a recent review, see e.g., Ref [61]", "pages": [ 18 ] }, { "title": "1. CS inequality in quantum mechanics", "content": "Any quantum state (density matrix) ˆ ρ defines a (positive) scalar product on operators by: The corresponding Cauchy-Schwarz (CS) inequality implies When applied, to Eq. (B1) with ˆ A = ˆ a k and ˆ B = ˆ a † -k , one gets When n k = n -k , one obtains Eq. (24).", "pages": [ 18, 19 ] }, { "title": "2. Separability", "content": "A bi-partite state is said separable [6, 62] when it can be written as where p n ≥ 0, and where the two-mode states ˆ ρ ( k ) 2 ,n are factorized ˆ ρ ( k ) 2 ,n . = ˆ ρ ( k ) n ⊗ ˆ ρ ( -k ) n . The operators ˆ ρ ( ± k ) n are density matrices for each one-mode system at fixed k . The structure of these states defines a new scalar product. It is given by where ˆ X , ˆ Y are arbitrary operators. Considering operators that act on one sector only, i.e., ˜ A = A ⊗ 1 and ˜ B = 1 ⊗ B , one finds where the quantities with a bar are the expectation values involving only one-mode states The inequality in Eq. (B7b) comes from the positivity of Tr(ˆ ρ ( k ) n ˆ ξ † n ˆ ξ n ) applied to ˆ ξ n = ˆ A -A ( k ) n , which gives The crucial point here is that the bound is insensitive to the ordering of A and A † . Therefore, when applying the CS inequality associated with the scalar product of The only difference with Eq. (B4) is that n -k +1 has been replaced by n -k by virtue of Eq. (B9). In conclusion, the inequalities of Eq. (26) characterize the quantum states which are nonseparable.", "pages": [ 19 ] }, { "title": "3. Subfluctuant mode", "content": "glyph[negationslash] We show that nonseparable states possess a subfluctuant mode whose variance is smaller than that of the vacuum. In the isotropic case, the proof can be found in Ref. [9]. Below, we extend the proof to the anisotropic case n k = n -k . To obtain the subfluctuant mode, we diagonalize the 2 × 2 covariance matrix Tr(ˆ ρ ( k ) 2 { W † , W } ) with W = ( a -k , a † k ) by a rotation, and not by a U (1 , 1) transformation (a Bogoliubov transformation). The operators define the super- and the subfluctuant mode, and the two angles are One verifies that Tr(ˆ ρ { S k , L † k } ) = 0, and that the spread of the subfluctuant mode is Using Eq. (B10), one establishes that Tr(ˆ ρ { S k , S † k } ) < 1 implies that the state is nonseparable. QED.", "pages": [ 19 ] }, { "title": "Appendix C: Flux and long distance correlations", "content": "The expressions for the asymptotic flux and the correlation pattern are both encoded in Eq. (A3b). To obtain them, we need two things. Firstly, we need to characterize G ω ret from the asymptotic region down to the NHR. To this end, we should perform a WKB analysis of the stationary damped modes. Secondly, we need to connect the WKB modes with the high-momentum de Sitter-like physics taking place very close to the horizon.", "pages": [ 19 ] }, { "title": "1. WKB analysis", "content": "At fixed ω , using Eq. (10), glyph[square] diss φ dec = 0 implies that the decaying mode φ ω dec obeys The mode φ ω dec decays when displacing X along the direction of the group velocity. Hence, on the right of the horizon, the outgoing U -mode decays when X increases, while it decreases for decreasing X < 0 in the left region, see Fig 9. Hence, U -modes spatially decay on both sides when leaving the horizon. As in the case of dispersive fields, we look for solutions of Eq. (C1) of the form where Q ω ( X ) is expanded in powers of the gradient of v ( X ). To first order, Eq. (C1) gives where the functions Γ > 0 of Eq. (11) and F are evaluated for P = Q ω . The leading order solution, the complex momentum Q (0) ω ( X ) . = P C ω ( X ), contains no gradient, and obeys the complex Hamilton-Jacobi equation As expected, this equation gives Eq. (1) since Ω = ω -vP . To first order in the gradient, we get a total derivative Combining Eq. (C4) and Eq. (C5), we obtain the decaying WKB-mode To get this expression, we introduced v C gr = 1 /∂ ω P C ω which can be conceived as a complex group velocity. We also decomposed P C ω into its real part P ω , and its imaginary part P I ω . The oscillating exponential is the standard expression, while the decaying one is ∫ dXP I ω . The latter is equal to I ω of Eq. (71) when working to first order in Γ /P , which is here a legitimate approximation. A preliminary analysis, similar to Eq. (A12) of Ref. [49], indicates that the corrections to Eq. (C6) are bounded by O ( ω 2 Λ 2 | 1+ v | 3 + g 2 ω Λ(1+ v ) 2 ). Hence Eq. (C6) gives an accurate description everywhere but in the central region III defined by κX trans of Eq. (75). Using Eq. (C6), the U -mode contribution to the commutator is, for ω > 0, where the growing mode ϕ ω grw satisfies Eq. (C1) with the opposite sign for the last term which encodes dissipation. The expression for ω < 0 is given by G -ω c = -( G ω c ) ∗ which follows from the imaginary character of G c in t, X space. We used the sign of I ω in Eq. (C7) so that a similar expression is valid on the left of the horizon. Note also that Eq. (C7) cannot be used to estimate G ω c across the horizon because the WKB approximation fails in region III . Note finally that Eq. (C7) is valid only for Λ | X -X ' | glyph[greatermuch] 1. Having characterized in quantitative terms the impact of dissipation, we now work in conditions such that the mode damping is negligible far away from this central region. That is, we work with X,X ' obeying where the upper limit comes from the neglect of the second term in Eq. (72). Under these conditions, the anticommutator of Eq. (A3b) is, for ω > 0, given by where n ω and c ω are constant because we are far from region III , and where the R and L out modes live on one side of the horizon and have unit norm. Being undamped, they are either relativistic, or, more generally, dispersive WKB modes. In the former case, they thus behave in the regions of interest, namely I R/L and II R/L , as where v R ( L ) is the asymptotic velocity in the region R ( L , where 1 + v L < 0). As in de Sitter, the (positive unit norm) mode ϕ -ω L living in the L region has a negative Killing frequency. In Eq. (C9), n ω and c ω are unambiguously defined because the R/L modes are normalized in regions I R/L . Thus, they respectively define the spectrum emitted by the black hole, and the ω -contribution of the correlation across the horizon. To compute them, we should find the equivalent of Eq. (35). To this end, we shall use Eq. (69), and exploit the fact that their values are fixed in the domain of P given in Eq. (74).", "pages": [ 20 ] }, { "title": "2. Connection with de Sitter physics", "content": "In Eq. (69), we need (the U -mode contribution of) G ω ret ( X,P 1 ) with | X | glyph[greatermuch] X trans , since we are interested in the far away behavior of G ω ac , and with P 1 glyph[greaterorsimilar] √ κ Λ, because the integrand vanishes for lower values of P . Since P ω ( X ) glyph[lessmuch] P 1 , the retarded character of Eq. (31) is automatically implemented, which means that The commutator G ω c ( X,P 1 ), on the one hand, obeys Eq. (C1) in X , and on the other hand, behaves as in de Sitter for P 1 glyph[greaterorsimilar] √ κ Λ, when ω diss max of Eq. (76) obeys κ/ω diss max glyph[lessmuch] 1. This second condition means that the high P 1 behavior is governed by Eq. (31) and Eq. (A8). For simplicity we consider the massless case of Eq. (39), when g 2 = 2. In this model, in de Sitter, using the Unruh modes of Eq. (48), the U -mode contribution is where I P 0 is given in Eq. (32), and where we replaced its lower value P ω ( X ) glyph[lessmuch] √ κ Λ by 0 because X is taken sufficiently large. Using Eq. (51), we can reexpress Eq. (C12) in the R/L out mode basis. For ω > 0 we get In this we recover that the commutator possesses the same expression if one uses the in (Unruh) or the out mode basis. Equation (C13) applies as such to the black hole metric in the regions II R/L , κX trans glyph[lessmuch] | κX | < D/ 2, because G c , BH obeys the same equations, and its normalization is fixed by the equal time commutators. In fact, in these regions the normalized black hole modes ϕ ω R , ϕ -ω L coincide with the modes χ ω R , χ -ω L of Eq. (50). Then, the WKB character of ϕ ω R , ϕ -ω L guarantees that Eq. (C13) applies further away from the horizon, in the regions defined by Eq. (C8). Hence, in these regions, we have We kept the de Sitter modes in P space because only | P | glyph[greatermuch] κ/ Λ contribute to Eq. (69). Using Eq. (C11), inserting the above expression in Eq. (69), and comparing the resulting expression with Eq. (C9), we get These expressions are identical to those evaluated in de Sitter. Hence, n ω and c ω are respectively given by Eqs. (57a) and (57b). Therefore, to leading order in κ/ Λ, and for an environment at zero temperature, n ω and c ω retain their standard relativistic expressions. This means that the state of the outgoing modes when they leave the central region III , and propagate freely, is the Unruh vacuum [38, 59, 60]. This can be explicitly checked from Eq. (C15) by reexpressing the out modes χ ω R/L in terms of the Unruh modes of Eq. (48). In this case, one finds that the mean number of Unruh quanta n Unruh ω is given by, see Eq. (55), In other words, the role of the double integrals in Eq. (C15) and Eq. (C16), whose integrand explicitly depends on the actual 'trans-Planckian' physics governed by Λ, f ( P ), Γ( P ), is to implement the Unruh vacuum in dissipative theories. (2006), astro-ph/0505376. (2008), 0805.0424.", "pages": [ 21, 22 ] } ]

[ { "title": "Systematic approach to ∆ L = 1 processes in thermal leptogenesis", "content": "T. Frossard a , ∗ A. Kartavtsev b , † and D. Mitrouskas c ‡ a Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany b Max-Planck-Institut fur Physik, Fohringer Ring 6, 80805 Munchen, Germany c LMU Munchen, Mathematisches Institut, Theresienstr. 39, 80333 Munchen, Germany In this work we study the contribution to leptogenesis from ∆ L = 1 decay and scattering processes mediated by the Higgs with quarks in the initial and final states using the formalism of non-equilibrium quantum field theory. Starting from fundamental equations for correlators of the quantum fields we derive quantum-corrected Boltzmann and rate equations for the total lepton asymmetry improved in that they include quantum-statistical effects and medium corrections to the quasiparticle properties. To compute the collision term we take into account one- and two-loop contributions to the lepton self-energy and use the extended quasiparticle approximation for the Higgs two-point function. The resulting CP -violating and washout reaction densities are numerically compared to the conventional ones. PACS numbers: 11.10.Wx, 98.80.Cq Keywords: Leptogenesis, Kadanoff-Baym equations, Boltzmann equation", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The Standard Model (SM) of particle physics [1-3] has successfully passed the numerous experimental tests performed so far. The recent observation of the Higgs particle [4] at the LHC [5, 6] also seems to confirm the mechanism of spontaneous symmetry breaking, which is responsible for masses of the known gauge bosons and fermions. On the other hand, we know that the SM is not complete. Firstly, it does not provide a viable dark matter candidate. Secondly, it predicts that the active neutrinos are strictly massless, which contradicts the results of neutrino oscillation experiments. A simple yet elegant way to generate small but nonzero neutrino masses is to add three right-handed Majorana neutrinos to the model: where N i = N c i are the heavy Majorana fields, /lscript α are the lepton doublets and ˜ φ ≡ iσ 2 φ ∗ is the conjugate of the Higgs doublet. After the electroweak symmetry breaking the active neutrinos receive naturally small masses through the type-I seesaw mechanism. This scenario has even more far-reaching consequences as it can explain another beyond-the-SM observation, the baryon asymmetry of the universe. The Majorana mass term in (1) violates lepton number. In the early Universe a decay of the Majorana neutrino into a lepton-Higgs pair increases the total lepton number of the Universe by one unit, and a decay into the corresponding antiparticles decreases the total lepton number by one unit. If there is CP -violation then, on average, the number of leptons produced in those decays is not equal to the number of antileptons and a net lepton asymmetry is produced. It is also known that whereas the difference of the lepton and baryon numbers is conserved in the Standard Model, any other their linear combination is not [7]. This implies that the lepton asymmetry produced by the Majorana neutrinos is partially converted to the baryon asymmetry [8]. This mechanism, which is referred to as baryogenesis via leptogenesis, naturally explains the observed baryon asymmetry of the Universe. For a more detailed review of leptogenesis see e.g. [9-11]. The state-of-the-art analysis of the asymmetry generation uses Boltzmann equations with the decay and scattering amplitudes calculated in vacuum. Their applicability in the hot and expanding early universe is questionable and can be cross-checked using a first-principle approach based on the use of non-equilibrium quantum field theory. One of the most important processes for the generation of the asymmetry is the decay of the Majorana neutrino. Thermal effects enhancing CP -violation in the decay have been studied in [12-16]. The role of the flavor effects has been addressed in [17]. A first-principle analysis of the asymmetry generation in the very interesting regime of resonant leptogenesis has been presented in [18] and [19]. The effect of next-to-leading order corrections from the gauge interactions of lepton and Higgs doublets on the production and decay rate of right-handed neutrinos at finite temperature has been recently studied in [20, 21]. The asymmetry generated in the Majorana decay is partially washed out by the inverse decay and scattering processes. The latter can be classified into two categories. The first category includes ∆ L = 2 scattering processes mediated by the Majorana neutrinos. A first-principle analysis of such processes free of the notorious doublecounting problem has been presented in [22]. The second category includes ∆ L = 1 decay and scattering processes mediated by the Higgs. The latter processes are also known to play an important role in the asymmetry generation and are addressed in the present paper. The outline of the paper is as follows. In Sec. II we briefly review the canonical approach to the analysis of the ∆ L = 1 processes and derive the corresponding amplitudes and reduced cross-sections. In Sec. III we derive quantum-generalized Boltzmann equations for the lepton asymmetry, calculate the effective amplitudes of the Higgs-mediated scattering processes and compare them with the canonical ones. The obtained Boltzmann equations are used in Sec. V to derive a simple system of rate equations for the total lepton asymmetry. In section Sec. VI we present a numerical comparison of the corresponding reaction densities with the ones obtained using the canonical approach. A summary of the results is presented in Sec. VII.", "pages": [ 1, 2 ] }, { "title": "II. CONVENTIONAL APPROACH", "content": "In the scenario of thermal leptogenesis lepton asymmetry is generated in the lepton number and CP -violating decay of the heavy Majorana neutrinos. The correspond- ing CP -violating parameters receive contributions from the interference of the tree-level amplitude with the vertex [8, 12] and self-energy [13, 23-26] amplitudes, see Fig. 1. The contribution of the loop diagrams can be accounted for by effective Yukawa couplings [26]. If thermal masses of the SM particles are negligible, they are given by: where the loop-function g ij is defined as Note that this expression is valid only for on-shell final states. The first term in (3) is related to the self-energy and the second term to the vertex contribution. This expression is applicable for a mildly or strongly hierarchical mass spectrum of the Majorana neutrinos. In both cases most of the asymmetry is typically generated by the lightest Majorana neutrino, whereas the asymmetry generated by the heavier ones is almost completely washed out. For a strongly hierarchical mass spectrum, M i /lessmuch M j , the intermediate Majorana line in Figs. 1.b and 1.c contracts to a point, see Fig. 2, and the structure of the self-energy and vertex contributions is the same. In this limit: Note that in this approximation the loop integral leading to (4) depends only on the momentum of the initial state and is independent of the momenta of the final states. This implies in particular that this expression can also be used for off-shell final states. Using the effective couplings (2) we find for the decay amplitudes (squared) [22, 26]: where we have summed over flavors of the leptons in the final state as well as over the Majorana spin ( g N = 2) and the SU (2) L group ( g w = 2) degrees of freedom. Here q and p are momenta of the heavy neutrino and lepton, respectively. The decay amplitudes (5) can be traded for the total decay amplitude and CP -violating parameter: Combining (5) and (6) we then find for the (unflavored) CP -violating parameter: /negationslash The asymmetry generated by the Majorana decay is partially washed out by the inverse decay and scattering processes violating lepton number. An important role is played by the ∆ L = 2 scattering processes mediated by the heavy Majorana neutrinos [22, 26, 27]. In addition, there are ∆ L = 1 scattering process mediated by the Higgs doublet with quarks (or the gauge bosons) in the initial and final states [26, 27], see Fig. 3 and Fig. 4. The Higgs coupling to the top is considerably larger than to the other quarks of the three generations. For this reason we do not consider the latter here. The corresponding Lagrangian reads: where Q and t are the SU (2) L doublet and singlet of the third quark generation. The ∆ L = 1 processes are also CP -violating. The CP -violation is generated by the same self-energy and vertex diagrams. Strictly speaking, since the Higgs is no longer on-shell the effective couplings (2) are not applicable in this case. On the other hand, for a strongly hierarchical mass spectrum the intermediate Majorana lines in Fig. 3 and Fig. 4 again contract to a point and the momenta of the Higgs and lepton play no role. In other words, for a strongly hierarchical mass spectrum we still can use the effective couplings (2) supplemented with (4) to calculate the CP -violating scattering amplitudes. Summing over flavors and colors of the quarks and leptons in the initial and final states as well as over the corresponding SU (2) L and spin degrees of freedom we find for the amplitude of N i Q → /lscriptt scattering: where ∆ T ( k ) ≈ 1 / ( k 2 -m 2 φ ) is the Feynman (or timeordered) propagator 1 of the intermediate Higgs and we have defined Here g s = 3 is the SU (3) C factor, and p t and p Q are the momenta of the singlet and the doublet respectively. For the charge-conjugate process we find an expression similar to (9). As can be inferred from (10) in this work we neglect CP -violation in the quark sector, which is known to be small. Defining CP -violating parameter in scattering as we then obtain for /epsilon1 N i Q → /lscriptt the same expression as for the Majorana decay, see (7). In the same approximation amplitude and CP -violating parameter for N i ¯ t → /lscript ¯ Q scattering coincide with those for N i Q → /lscriptt process. Proceeding in a similar way we find for the scattering amplitude of the N i ¯ /lscript → ¯ Qt process: where Ξ φ → ¯ Qt = Ξ φQ → t because we neglect CP -violation in the quark sector. Furthermore, for a strongly hierarchical mass spectrum Ξ N i ¯ /lscript → φ = Ξ N i → /lscriptφ . The resulting expression for the CP -violating parameter then coincides with (7). If the lepton and both quarks are in the final state then instead of a scattering process we deal with a threebody Majorana decay, see Fig. 5. In complete analogy with the scattering processes we can write its amplitude in the form: Evidently, CP -violating parameter for this process coincides with that for the two-body Majorana decay. To compute the generated lepton asymmetry, the conventional approach uses the generalized Boltzmann equation for the total lepton abundance, Y L ≡ n L /s , with s being the comoving entropy density [28]. In the Friedmann-Robertson-Walker (FRW) universe the contribution of the Higgs-mediated processes to the righthand side of the the Boltzmann equation simplifies to: where we have introduced the dimensionless inverse temperature z = M 1 /T , the Hubble rate H = H | T = M 1 , and d Π p a p b ...p i p j ... ab...ij... stands for the product of the invariant phase space elements, d Π p a ≡ d 3 p/ [(2 π ) 3 2 E p ]. Note that to ensure vanishing of the asymmetry in thermal equilibrium one should also include CP -violating 2 ↔ 3 processes [10]. Since there is no need for that in the non-equilibrium QFT approach we will not consider these processes here.", "pages": [ 2, 3, 4 ] }, { "title": "III. NON-EQUILIBRIUM QFT APPROACH", "content": "The formalism of non-equilibrium quantum field theory provides a powerful tool for the description of out-ofequilibrium quantum fields and is therefore well suited for the analysis of leptogenesis. In this section we briefly review results obtained in [22] and introduce notation that will be used in the rest of the paper. As has been argued in [22], the equation of motion for the lepton asymmetry can be derived by considering the divergence of the lepton current. In the FRW Universe j µ L = ( n L , 0 ) and therefore it is related to the total lepton abundance by: Using the formalism of non-equilibrium quantum field theory one can express it through propagators and selfenergies of leptons. After a transformation to the Wigner space we obtain [22]: where d Π 4 p ≡ d 4 p/ (2 π ) 4 and the hats denote matrices in flavor space. In (16) we have taken into account that the SU (2) L symmetry is unbroken at the epoch of leptogenesis. As a consequence, the SU (2) L structure of the propagator is trivial, S αβ ab = δ ab S αβ , and summation over the SU (2) L components simply results in the overall factor g w = 2. Furthermore, in this work we restrict ourselves to the analysis of unflavored leptogenesis. Therefore, the lepton propagator can be approximated by S αβ = δ αβ S . Similar relation also holds for the lepton self-energy. Then the equation for the divergence of the lepton current takes the form: where Σ ≡ Σ αα and we have suppressed the argument ( t, p ) of the two-point functions. Note that the trace in (17) acts now in spinor space only. To convert the integration over positive and negative frequencies into the integration over positive frequencies only we have introduced in (17) CP -conjugate two-point functions and self-energies which are denoted by the bar. According to the extended quasiparticle approximation (eQP) [29-31] the Wigthmann propagators can be split into on- and off-shell parts: The off-shell parts of the lepton propagators exactly cancel out in the lepton current as they are lepton number conserving. On the other hand, as we will see later, the off-shell part of the Higgs two-point functions is crucial for a correct description of the scattering processes. The on-shell part of the Wightman propagators is related to the eQP spectral function and one-particle distribution function f /lscript by the Kadanoff-Baym (KB) ansatz: where In the limit of vanishing width the eQP spectral function ˜ S ρ approaches the Dirac delta-function [22], where we have extracted the 'scalar' part S ρ for notational convenience. In (21) we have approximately taken the gauge interactions into account in the form of effective masses of the leptons. Note that we will not attempt a fully consistent inclusion of the gauge interactions here. In the used approximation the spectral function is CP -symmetric. This implies that the spectral properties, in particular the masses, of the particles and antiparticles are the same. To evaluate the right-hand side of (17) we need to specify the form of the lepton self-energy. It can be obtained by functional differentiation of the 2PI effective action with respect to the lepton propagator. Loosely speaking, this means that the self-energies are obtained by cutting one line of the 2PI contributions to the effective action. The two- and three-loop contributions are presented in Fig. 6(a) and Fig. 6(c). The one-loop contribution to the lepton self-energy, see Fig. 6 (b), is given by [22]: where S and ∆ denote the Majorana and Higgs propagators respectively, and d Π 4 qk ≡ d Π 4 q d Π 4 k . The expression for the two-loop contribution, see Fig. 6 (d), is rather lengthy. Here we will only need a part of it: where we have introduced two functions containing loop corrections: and V nm ( t, q, k ) ≡ P Λ † nm ( t, q, k ) P to shorten the notation. Here P = γ 0 is the parity conjugation operator. The remaining terms of the two-loop self-energy can be found in [22]. As has been demonstrated in the same reference, CP -conjugates of the above self-energies can be obtained by replacing the Yukawa couplings by the complex conjugated ones and the propagators by the CP -conjugated ones. Comparing (22) and (23) we see that the two selfenergies have a very similar structure. First, the integration is over momenta of the Higgs and Majorana neutrino and the delta-function contains the same combination of the momenta. Second, they both include one Wightman propagator of the Higgs field and one Wightman propagator of the Majorana field. These can be interpreted as cut-propagators which describe on-shell particles created from or absorbed by the plasma [32]. The retarded and advanced propagators can be associated with the off-shell intermediate states. We therefore conclude that the two self-energies describe CP -violating decay of the heavy neutrino into a lepton-Higgs pair. Note that this interpretation only holds for the 'particle' part of the eQP ansatz. The inclusion of the off-shell part of the Higgs Wightman propagator gives raise to the Higgs mediated scattering processes and three-body decay, see section IV. To evaluate (22) and (23) we need to know the form of the Higgs and Majorana propagators. For the Higgs field we will adopt in this section a leading-order approximation: and a simple quasiparticle approximation for the spectral function, where m φ is the effective thermal Higgs mass. Close to thermal equilibrium the full resummed Majorana propagator is given by [22]: where ˆ Π d and ˆ Π ' denote the diagonal and off-diagonal components of the Majorana self-energy respectively, ˆ S R and ˆ S A are given by and we have introduced to shorten the notation. The first term in the square brackets of (27) describes (inverse) decay of the Majorana neutrino, whereas the remaining three terms describe two-body scattering processes mediated by the Majorana neutrino. For the 'particle' part of the eQP diagonal Wightman propagators of the Majorana neutrino one can use the KB approximation: with the spectral function given by an expression identical to (20). Substituting (28) we find in the limit of small decay width: Inserting (22) and (23) into the divergence of the lepton current (17) and integrating over the frequencies we then obtain an expression that strongly resembles the Boltzmann equation: where we have introduced with the plus (minus) sign corresponding to bosons (fermions). Note that F p a p b .. ; p i p j .. ab.. ↔ ij.. vanishes in equilibrium due to detailed balance. This implies that in accordance with the third Sakharov condition [33] no asymmetry is generated in equilibrium. In the Kadanoff-Baym formalism this result is obtained automatically and no need for the real intermediate state subtraction arises. The effective decay amplitudes Ξ are given by a sum of the tree-level, one-loop self-energy and one-loop vertex contributions. The first two: emerge from the one-loop lepton self-energy (22). The third one: is generated by the two-loop lepton self-energy (23). Substituting (34) and (35) into (6) we find to leading order in the couplings that the total decay amplitude summed over the Majorana spin degrees of freedom is given by Ξ N i = 2 g N g w ( h † h ) ii ( pq ). The self-energy CP -violating parameter reads [22]: where the 'scalar' part of the diagonal hermitian Majorana propagator is given by [22]: It describes the intermediate Majorana neutrino line in Fig. 1.b. Note that (36) has been obtained assuming a hierarchical mass spectrum of the heavy neutrinos and is not applicable for a quasidegenerate spectrum. For positive q 0 and q 2 the self-energy loop function L S is given by [22]: Simplifying (35) we find for the vertex CP -violating parameter [22]: The vertex loop function is given by: where S F = ( S > + S < ) / 2 is the 'scalar' part of the corresponding statistical propagator of the heavy neutrino. For the lepton and Higgs fields the definitions are similar. The three lines in the square brackets in (39) correspond to different cuts through two of the three internal lines of the vertex loop. The first line corresponds to cutting the propagators of the Higgs and lepton and can be simplified to [16]: The other two are cuts through the Majorana and lepton and the Majorana and Higgs lines respectively [15]. For the second cut we obtain: whereas contribution of the third cut is given by: where we have assumed M i < M j so that the (inverse) decay N i ↔ N j /lscript/lscript is kinematically forbidden. In (42) the second term vanishes for the decay process N i ↔ /lscriptφ but gives a non-zero contribution for the scattering processes, see section IV. If the intermediate Majorana neutrino is much heavier than the decaying one the last two cuts are strongly Boltzmann-suppressed. Furthermore, comparing (38) and (41) we observe that in this case pL V ≈ pL S . In the same approximation we can also neglect the 'regulator' term in the denominator of (37). The two contributions to the CP -violating parameter then have the same structure and their sum can be written in the form: In the vacuum limit L µ S = q µ and we recover (7). At finite temperatures the CP -violating parameter is moderately enhanced by the medium effects [22].", "pages": [ 4, 5, 6, 7 ] }, { "title": "IV. HIGGS MEDIATED SCATTERING", "content": "In the previous section we have approximated the full resummed Higgs propagator by leading-order expressions (25) and (26). In this section we will use a more accurate eQP approximation. As we will see, it allows one to describe Higgs-mediated ∆ L = 1 two-body scattering and three-body decay processes. Similarly to (18), the extended quasiparticle approximation for the Higgs propagator reads: Its graphic interpretation is presented in Fig. 7. For the first term on the right-hand side of (45) we can again use approximations (25) and (26). To analyze the second with and neglecting their off-shell parts, which are lepton number conserving, we can write the Higgs self-energy in the form: Substituting the one-loop lepton self-energy (22) with the Higgs propagator given by (45) into the divergence of the lepton current (17), we obtain: term we have to specify the Higgs self-energy. At oneloop level it reads: see Appendix A for more details. As is evident from (46), here we limit our analysis to contributions generated by the quarks of the third generations. Let us note that in the SM the gauge contribution to the Higgs self-energy is of the same order of magnitude and should not be dismissed in a fully consistent approximation. Using the KB ansatz for the eQP propagators of the quarks with effective thermal mass: where we have introduced a combination of the retarded and advanced propagators, which describes the intermediate Higgs line in Fig. 3 and Fig. 4. Note that in (50) the momenta are not restricted to the mass shell. In particular, the zeroth components of the momenta can have either sign. Due to the Diracdeltas in the spectral functions the frequency integration is trivial. Each of the spectral functions can be decomposed into a sum of two delta-functions, one with positive and one with negative frequency, leading to 2 4 terms. These different terms correspond to 1 ↔ 3 (inverse) decay, 2 ↔ 2 scattering and to (unphysical) 0 ↔ 4 process. An additional constraint comes from the deltafunction ensuring energy conservation. In the regime M i > m /lscript + m Q + m t only 8 terms satisfy the energy conservation. Using the relation where f ¯ a denotes the distribution function of the antiparticles, we can then recast (50) in the form: The effective scattering amplitudes in (53) correspond to different assignments for the sign of the four-momenta in (50), reflecting the usual crossing symmetry. For the tree-level and self-energy contributions to the effective scattering and decay amplitudes we obtain: and similar expressions for the CP -conjugate ones. Note that Ξ T + S N i ↔ /lscriptφ and Ξ T + S N i ¯ /lscript ↔ φ are given by the same expression since the CP -violating loop term in (34) depends only on the momentum of the Majorana neutrino. In vacuum these scattering amplitudes reduce to (9) and (12) respectively but with the Feynman propagator ∆ 2 T replaced by ∆ 2 R + A . In the latter the contribution of the real intermediate state is subtracted by construction [22]. However, in the regime m φ < m Q + m t the intermediate Higgs cannot be on-shell such that the vacuum and inmedium amplitudes become numerically equal. Since the amplitudes Ξ φ ↔ ¯ Qt and Ξ φQ ↔ t factorize in (54) and are CP -conserving, the self-energy CP -violating parameter in these processes is the same as in the Majorana decay, see (36). However, since the decay and scattering processes have different kinematics the averaged decay and scattering CP -violating parameters are not identical. Next we consider the two-loop lepton self-energy (23). Proceeding as above we find for the divergence of the lepton current an expression of the form (53) with the amplitudes given by: Since the vertex contribution to the Majorana decay amplitude depends on the momentum of the Higgs, the amplitude Ξ V N i ↔ /lscriptφ does not coincide with Ξ V N i ¯ /lscript ↔ φ and we can define two inequivalent vertex CP -violating parameters [34]. For the scattering processes N i Q ↔ /lscriptt and N i ¯ t ↔ /lscript ¯ Q as well as for the three-body decay N i ↔ /lscript ¯ Qt , the CP -violating parameter coincides with (39) with the contributions of the three possible cuts given by (41), (42) and (43) respectively. For the N i ¯ /lscript ↔ ¯ Qt process, the CP -violating parameter still has the form (39), but since the lepton is in the initial state the loop integral must be evaluated at ( q, -p ) instead of ( q, p ). For the first cut we obtain: Contributions of the second and third cuts are given by: and by respectively. As follows from (56) and (57), CP -violating parameter in the N i ¯ /lscript ↔ ¯ Qt scattering receives two vacuum contributions [34]. One is the usual cut through /lscriptφ , and the second one is given by the first term in the cut through N j /lscript . The kinematics of the second contribution corresponds to N i /lscript ↔ N j /lscript t -channel scattering and therefore requires the initial center-of-mass energy s = q + p to be greater than the final masses M j + m /lscript , meaning that contribution of this term to the reaction density is suppressed for a hierarchical mass spectrum.", "pages": [ 7, 8, 9 ] }, { "title": "V. RATE EQUATIONS", "content": "Solving a system of Boltzmann equations in general requires the use of numerical codes capable of treating large systems of stiff differential equations for the different momentum modes. This is a difficult task if one wants to study a wide range of model parameters. A commonly employed simplification is to approximate the Boltzmann equations by the corresponding system of 'rate equations' for the abundances Y a . In [35] it was shown that the two approaches, Boltzmann or the rate equations, give approximately equal results for the final asymmetry, up to ∼ 10% correction. Starting from a quantum Boltzmann equation of the type (53) we derive here the rate equation for the lepton asymmetry which includes the (usually neglected) quantum statistical factors. In our analysis we are closely following [22]. Contribution of various processes to the generation of the lepton asymmetry can be represented in the form: compare with (53), where the sum runs over each possible particle state with /lscript ∈ { j } or ¯ /lscript ∈ { a } . We assume that the SM particles are maintained in kinetic equilibrium by the fast gauge interactions. This means that their distribution function takes the form: with a time- (or temperature-) dependent chemical potential µ a = µ a ( t ). Here the upper (lower) sign corresponds to bosons (fermions). It is also useful to define the equilibrium distribution function, The fast SM interactions relate chemical potentials of the leptons, quarks and the Higgs, such that only one of them is independent. We can therefore express the chemical potential of the quarks as a function of the lepton chemical potential [36-38], Chemical potentials of the antiparticles: µ ¯ a = -µ a . The lepton chemical potential is related to the abundance by: where c /lscript depends on the thermal mass of the lepton. For m /lscript /T ≈ 0 . 2 it can be very well approximated by the zero mass limit, c /lscript ≈ 9 ξ (3) /π 2 ≈ 1 . 1. Using the identity 1 ± f a = e ( E a -µ a ) /T f a and energy conservation we can rewrite the combinations of distribution functions appearing in (59) as: where we have suppressed the momentum argument in the distribution functions for notational convenience. We can then expand (64) in the small chemical potential µ a . Assuming the Majorana neutrino to be close to equilibrium, f N i -f eq N i ∼ O ( µ a ), we see that the term in the square bracket in (64) is already of the first order in the chemical potential. We can therefore replace the distribution functions in the second line of (64) by the equilibrium ones, and expand the exponential to first order in the chemical potential. Since we assume small departure from equilibrium the Majorana distribution function that multiplies the chemical potential should also be replaced by the equilibrium one. The corresponding equation for the antiparticles is obtained from the above equation by replacing µ a → -µ a . The last step is to assume that the Majorana distribution function is proportional to its equilibrium value, Putting everything together we get the conventional form of the rate equation, where we have defined the production and washout reac- tion densities: and the numerical factor, Equation (67) must be supplemented by an equation for the heavy neutrino abundance, with the reaction density given by an expression similar to (68b): Note that these expressions are valid for two-body scattering processes with Majorana neutrino in the initial or final state as well as for Majorana (inverse) decay processes. If the quantum-statistical corrections are neglected, i.e. if the 1 ± f terms are replaced by unity and the equilibrium fermionic and bosonic distributions are approximated by the Maxwell-Boltzmann one, then (68b) and (71) are equal. For the case of a 2 ↔ 2 scattering process they read: Part of the integrations in (72) can be performed analytically and we obtain: Here s min = ( M i + m a ) 2 (assuming M i + m a > m j + m k ) and ˆ σ ( s ) is the so-called reduced cross-section: where s and t are the usual Mandelstam variables. The integration limits are given by [22]: where λ ( x, y, z ) ≡ x 2 + y 2 + z 2 -2 xy -2 xz -2 yz is the usual kinematical function. If effective thermal masses of the SM particles are neglected then the integration limits simplify to t + = 0 and t -= -( s -M 2 i ). Integrating (9) and (12) over t and neglecting the effective masses of the initial and final lepton and quarks we obtain the standard expressions (see, e.g. [39, 40]) for the reduced cross-sections of the Higgs-mediated scattering processes: where we have replaced s by x ≡ s/M 2 1 and introduced dimensionless quantities a i ≡ M 2 i /M 2 1 and a φ ≡ m 2 φ /M 2 1 . Combined with (73), expressions (76) give the conventional reaction densities of the Higgs-mediated scattering processes. Since in the conventional approach the CP -violating parameter is calculated in vacuum it is momentumindependent and therefore can be taken out of the integral. The CP -violating reaction densities are thus proportional to the washout ones: where we have again assumed a strongly hierarchical mass spectrum of the heavy neutrinos. When the medium corrections are taken into account the CP -violating parameter depends on the momenta of the initial and final states and this simple relation is violated.", "pages": [ 9, 10 ] }, { "title": "VI. NUMERICAL RESULTS", "content": "To illustrate the effect of the quantum-statistical corrections and effective thermal masses we present in this section ratios of the reaction densities to the conventional ones assuming a strongly hierarchical mass spectrum of the Majorana neutrinos. z Let us first consider the scattering processes. Ratios of the improved reaction densities to the conventional ones are presented in Fig. 8. Note that the Majorana (as well as the quark) Yukawa couplings cancel out in these ratios and for this reason we do not specify them here. The dashed lines show the ratio of the reaction density computed using (73)-(75), i.e. taking into account the effective thermal masses but neglecting the quantumstatistical corrections, to the conventional ones. For the N i ¯ /lscript ↔ ¯ Qt process (dashed red line) the effective masses decrease the available phase space and lead to a suppression of the reaction density in the whole range of temperatures. Note that the ratio does not approach unity at low temperatures. Qualitatively this behavior can be understood from (72). Let us assume for a moment that the scattering amplitude is momentum-independent. The reaction density at low temperatures can then be estimated by evaluating the distribution functions at the average momenta 〈 p i 〉 and 〈 p a 〉 ∼ 3 T . In the ratio of the reaction densities the Majorana distribution function cancels out and /negationslash /negationslash A more accurate estimate for the ratio of 〈 γ X Y 〉 MB,m =0 and 〈 γ X Y 〉 MB,m =0 is ∼ exp( -m 2 a /T 2 ). Since to a good approximation m a ∝ T we conclude that this ratio is a constant smaller than unity. In other words, despite the fact that at low temperatures the quark masses become small compared to the Majorana mass this ratio is not expected to approach unity as the temperature decreases. Note also that (in a very good agreement with the numerical cross-check) this ratio does not depend on the masses of the final states. Of course, the momentum dependence of the scattering amplitude somewhat changes the lowtemperature behavior of the reaction density. Interestingly enough, for the N i Q ↔ /lscriptt process the inclusion of the thermal masses actually enhances the reaction density at high temperatures (dashed blue line). This occurs because the induced increase of the amplitude turns out to be larger than the phase-space suppression. At low temperatures the effective masses become negligible in the scattering amplitude but still play an important role in the kinematics. As a result, the ratio becomes smaller than unity and continues to decrease as the temperature decreases. Let us note that for a (moderately) strong washout regime most of the asymmetry is typically produced at z /lessorsimilar 10 and the low-temperature behavior of the reaction densities does not affect the generation of the asymmetry. Since all particles in the initial and final states are fermions the quantum-statistical effects further suppress the reaction densities (solid blue and red lines) and render the ratio of the improved and conventional reaction densities smaller than unity for both N i Q ↔ /lscriptt and N i ¯ /lscript ↔ ¯ Qt in the whole range of temperatures. Ratios of the improved CP -violating scattering reaction densities to the conventional ones are presented in Fig. 9. For both scattering processes the improved CP -violating reaction densities are enhanced at high temperatures. This is explained by the enhancement of the CP -violation in the Majorana decay observed in [16, 22]. At the intermediate temperatures the relative enhancement of the CP -violating parameters gets smaller and is overcompen- /negationslash sated by the effective mass and Fermi-statistics induced suppression of the washout reaction densities that we have observed in Fig. 8. The low-temperature behavior is somewhat different for the two scattering processes. For the N i ¯ /lscript ↔ ¯ Qt process the effective mass and quantumstatistical effects get smaller in both the (unintegrated) CP -violating parameter and the washout reaction density, such that the ratio of the CP -violating reaction density to the conventional one slowly approaches a constant value. On the other hand, for the N i Q ↔ /lscriptt process the suppression of the washout reaction density induced by the effective masses of the initial and final states that we observed in Fig. 8 also leads to a suppression of the CP -violating reaction density that gets stronger at low temperatures. Next we consider the three-body decay. Neglecting the quantum-statistical effects and using vacuum approximation for the N i ↔ /lscript ¯ Qt decay amplitude in (68) and (71) we recover the conventional expression for the decay reaction density: Note that it is important to retain the effective thermal masses of the quarks in the calculation of Γ N i → /lscript ¯ Qt . The four-momentum of the intermediate Higgs, see Fig. 5, varies in the range ( m Q + m t ) 2 /lessorequalslant k 2 /lessorequalslant ( M i -m /lscript ) 2 . The relation m φ < m Q + m t , which is fulfilled in the SM, ensures that the intermediate Higgs remains off-shell and prevents a singularity in the Higgs propagator. The /negationslash ratio of the reaction density computed taking into account the quantum-statistical effects and effective masses to the one computed taking into account only the effective masses is presented in Fig. 10. Note that since m Q ≈ m t ≈ 0 . 4 T and m /lscript ≈ 0 . 2 T this three-body decay is kinematically allowed only at T /lessorsimilar M i . As one would expect, at high temperatures the fermionic nature of the initial and final states leads to a suppression as compared to the Boltzmann approximation. At low temperatures the quantum-statistical effects play no role and the ratio slowly approaches unity. Ratio of the CP -violating reaction density for the N i ↔ /lscript ¯ Qt process is presented in Fig. 11. At high and intermediate temperatures the medium enhancement of the (unintegrated) CP -violating parameter is overcompensated by the suppression of the washout decay reaction density that we have observed in Fig. 10. At low temperatures the effective mass and quantum-statistical effects get smaller in both the (unintegrated) CP -violating parameter and the washout reaction density, such that the CP -violating reaction density slowly approaches the conventional one. To conclude this section we present the ratio of the three-body decay and 2 ↔ 2 scattering processes to the reaction density of N i ↔ /lscriptφ process, see Fig. 12. As can be inferred from this plot, the three-body decay is subdominant in the whole range of temperatures and can be safely neglected. The inclusion of the effective masses has a very similar effect on the two-body decay and scattering reaction densities such that the ratios of the two almost do not change as compared to the one computed in the massless approximation. The inclusion of the quantumstatistical corrections has a stronger effect on the scatter- /negationslash z ing processes such that the ratio of the reaction densities is smaller than the ratio of the conventional ones. Let us also note that the scattering processes are very important at high temperatures but become subdominant at low temperatures.", "pages": [ 10, 11, 12, 13 ] }, { "title": "VII. SUMMARY", "content": "In this work we have studied ∆ L = 1 decay and scattering processes mediated by the Higgs with quarks in the initial and final states using the formalism of nonequilibrium quantum field theory. Starting from the Kadanoff-Baym equations for the lepton propagator we have derived the corresponding quantum-corrected Boltzmann and rate equations for the total lepton asymmetry. As compared to the canonical ones the latter are free of the notorious double-counting problem and ensure that the asymmetry automatically vanishes in thermal equilibrium. To compute the collision term we have taken into account one- and two-loop contributions to the lepton self-energy and used the extended quasiparticle approximation for the Higgs propagator. The impact of the SM gauge interactions on the collision term has been approximately taken into account in the form of effective thermal masses of the Higgs, lep- /negationslash /negationslash /negationslash /negationslash /negationslash tons and quarks. We find that the inclusion of the effective masses and quantum-statistical terms suppresses the washout reaction densities of the decay and scattering processes with respect to the conventional ones, where these effects are neglected, in the whole relevant range of temperatures. For the N i ¯ /lscript ↔ ¯ Qt process the ratio of the improved and conventional washout reaction densities slowly approaches a constant value close to unity at low temperatures. Interestingly enough, for the N i Q ↔ /lscriptt processes this ratio decreases even at low temperatures. Finally for N i ↔ /lscript ¯ Qt process the ratio slowly approaches unity at low temperatures. As far as the CP -violating reaction densities are concerned, we find that for the scattering processes the ratio of the improved and the conventional ones is greater than unity at high temperatures but is smaller than unity at intermediate and low temperatures because of the thermal masses and quantum-statistical effects. For the three-body decay this ratio is smaller than unity in the whole relevant range of temperatures. We expect that the effects studied here can induce a O (10%) correction to the total generated asymmetry. For a detailed phenomenological analysis it is necessary to include further phenomena such as flavour effects and process with gauge bosons in the initial and final states.", "pages": [ 13 ] }, { "title": "Acknowledgements", "content": "The work of A.K. has been supported by the German Science Foundation (DFG) under Grant KA-3274/1-1 'Systematic analysis of baryogenesis in non-equilibrium quantum field theory'. T.F. acknowledges support by the IMPRS-PTFS. We thank A. Hohenegger for useful discussions.", "pages": [ 13 ] }, { "title": "Appendix A: Higgs self-energy", "content": "The top quark contribution to the Higgs self-energy is derived from the 2PI effective action. At one-loop the contribution of the top quark is given by: where the factor g s = 3 comes from the summation over color indices and /epsilon1 = iσ 2 . In a SU (2) L symmetric state the Higgs and lepton propagators are proportional to the identity in the SU (2) L space, and so is the Higgs selfenergy, Its Wightman components are given by, Finally, performing a Wigner transform of the above equation, we find,", "pages": [ 13, 14 ] }, { "title": "Appendix B: Reaction density of 1 → 3 decay", "content": "For N i → /lscript ¯ Qt decay the general expression (71) takes the form: where we have used the explicit form of the decay amplitude (13). To reduce it to a form suitable for the numerical analysis we insert an identity: where k is the four-momentum of the intermediate Higgs. Approximating furthermore ∆ 2 R + A by ∆ 2 T we can rewrite the reaction density in the form: Note that in the regime m φ < m Q + m t realized in the considered case the Higgs is always off-shell and its width can be neglected in ∆ T . For the second line in (B3) we can use [22]: The integration limits are given by where x /lscript ≡ m 2 /lscript /M 2 i , x φ ≡ s/M i and λ ( x, y, z ) ≡ x 2 + y 2 + z 2 -2 xy -2 xz -2 yz is the usual kinematical function. For the third line we can use a similar expression with x Q = m 2 Q /s and x t = m 2 t /s . Expressed in terms of the integration variables the amplitudes take the form: Since they do not depend on the angles between the quarks and leptons the integration over ϕ is trivial and the reaction density takes the form: The three-momentum of the intermediate Higgs is given by | k | = ( E 2 k -s ) 1 2 and E k = E q -E p . Note that if we neglect the quantum-statistical factors in (B7) the reaction density takes the standard form.", "pages": [ 14 ] }, { "title": "Appendix C: Majorana spectral self-energy", "content": "We compute here the Majorana spectral self-energy. In a CP -symmetric medium it reads [22]: where we have defined the loop function L S ( q ), Using the eQP for the Higgs, see (45), one can split the function L S into a decay part, identical to the one computed in [22], where we have assumed q 0 > 0, and defined and a scattering part, Performing the frequency integration as explained above, see (53), we can rewrite (C5) as a sum of four terms, corresponding to the three scattering and one three-body decay process as well as their CP -conjugates. Assuming q 0 > 0 we obtain: In the regime m φ < m Q + m t the intermediate Higgs cannot be on-shell. Therefore one can neglect the Higgs width in ∆ 2 R + A and approximate it by ∆ 2 T . As can be inferred from the definition (C4) for the scattering terms ˜ F vanishes in vacuum, whereas for the decay term it does not. Due to Lorentz covariance in vacuum both L d S and L s S must be proportional to the four-vector q . Using (B4) and (B5) we find that the coefficient of proportionality is equal to unity for the decay contribution, i.e. L d S = q , if thermal masses of the Higgs and leptons are neglected. Using (B7) we find that for the scattering contribution the coefficient of proportionality reads: Note that since we have omitted the Higgs decay width, this expression is convergent only if m φ < m Q + m t . The vacuum result (C7) provides also a very good approximation for nonzero temperatures provided that M/T /greatermuch 1. The thermal masses of the quarks then ensure that the Higgs remains off-shell and therefore that (C7) is finite. It is important to note that due to the temperature dependence of the effective masses the coefficient (C7) is temperature-dependent as well. A numerical analysis shows that it grows as the temperature decreases. Using L S we can calculate the in-medium CP -violating parameter in N i ↔ /lscriptφ process. For a hierarchical mass spectrum [22]: where /epsilon1 vac 0 denotes the vacuum CP -violating parameter calculated neglecting contributions of the Higgs-mediated processes, i.e. using only L d S . As has been mentioned /negationslash above, if thermal masses of the Higgs and leptons are neglected then L d S = q in vacuum and we recover /epsilon1 = /epsilon1 vac 0 . Once the contribution of the Higgs-mediated processes is taken into account /epsilon1 vac = /epsilon1 vac 0 . To estimate the size of the corrections induced by (C6) we plot the ratio of thermally averaged CP -violating parameter, 〈 /epsilon1 〉 ≡ 〈 /epsilon1γ D N 〉 / 〈 γ D N 〉 , to /epsilon1 vac 0 , see Fig. 13. Note that we have neglected thermal masses of the final-state Higgs and lepton in the numerics. The blue line corresponds to the CP -violating parameter computed using (C3). In agreement with the above discussion the ratio reaches unity at low temperatures. The dashed red line corresponds to the CP -violating parameter computed using the sum of (C3) and the scattering (lines two to four) contributions to (C6). As expected, at high temperatures we observe an enhancement of the ratio, whereas at low temperatures it reaches unity. The solid red line is obtained by considering the sum of (C3) and all of the terms in (C6). Since the three-body process is kinematically suppressed at high temperatures, the dashed an solid lines overlap for z /lessorsimilar 1. At lower temperatures the quantum-statistical effects are small. However, in agreement with the discussion below (C7), the effective thermal masses of the Higgs and quarks lead to a slow rise of the ratio at low temperatures.", "pages": [ 14, 15 ] } ]

[ { "title": "Phenomenology of S 4 Flavor Symmetric extra U(1) model", "content": "Yasuhiro Daikoku ∗ , Hiroshi Okada † Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan. ∗‡ School of Physics, KIAS, Seoul 130-722, Korea † July 13, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study several phenomenologies of an E 6 inspired extra U(1) model with S 4 flavor symmetry. With the assignment of left-handed quarks and leptons to S 4 -doublet, SUSY flavor problem is softened. As the extra Higgs bosons are neutrinophilic, baryon number asymmetry in the universe is realized by leptogenesis without causing gravitino overproduction. We find that the allowed region for the lightest chargino mass is given by 100-140 GeV, if the dark matter is a singlino dominated neutralino whose mass is about 36 GeV.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Standard model (SM) is a successful theory of gauge interactions, however there are many unsolved puzzles in the Yukawa sectors. What do the Yukawa hierarchies of quarks and charged leptons mean? Why is the neutrino mass so small? Why does the generation exist? These questions give rise to the serious motivation to extend SM. Another important puzzle of SM is the existence of large hierarchy between electroweak scale M W ∼ 10 2 GeV and Planck scale M P ∼ 10 18 GeV. The elegant solution of this hierarchy problem is supersymmetry (SUSY)[1]. Recent discovery of the Higgs boson at the Large Hadron Collider (LHC) may suggest the existence of SUSY because the mass of Higgs boson; 125 -126 GeV [2], is in good agreement with the SUSY prediction. Moreover, in the supersymmetric model, more information are provided for the Yukawa sectors. In the supersymmetric model, the Yukawa interactions are introduced in the form of superpotential. Therefore, to understand the structure of the Yukawa interaction, we have to understand the structure of superpotential. In the minimal supersymmetric standard model (MSSM), as the Higgs superfields H U and H D are vector-like under the SM gauge symmetry G SM = SU (3) × SU (2) × U (1), we can introduce µ -term; in superpotential. The natural size of parameter µ is O ( M P ), however µ must be O ( M W ) to succeed in breaking electroweak gauge symmetry. This is so-called µ -problem. The elegant solution of µ -problem is to make Higgs superfields chiral under a new U (1) X gauge symmetry. Such a model is achieved based on E 6 -inspired extra U(1) model [3]. The new gauge symmetry replaces the µ -term by trilinear term; which is converted into effective µ -term when singlet S develops O (1TeV) vacuum expectation value (VEV) [4]. At the same time, the baryon and lepton number violating terms in MSSM are replaced by single G-interactions; where G and G c are new colored superfields which must be introduced to cancel gauge anomaly. These terms induce very fast proton decay. To make proton stable, we must tune these trilinear coupling constants to be very small ∼ O (10 -14 ), which gives rise to a new puzzle. The existence of small parameters in superpotential suggests that a new symmetry is hidden. As such a symmetry suppresses the Yukawa coupling of the first and the second generation of the quarks and the charged leptons, it should be flavor symmetry. We guess several properties that the flavor symmetry should have in order. At first, the flavor symmetry should be non-abelian and include triplet representations, which is the simple reason why three generations exist. At second, remembering that the quark and the charged-lepton masses are suppressed by SU (2) W gauge symmetry as the left-handed fermions are assigned to be doublet and the right-handed fermions are assigned to be singlet, the flavor symmetry should include doublets. In this case, if we assign the first and the second generation of the left-handed quarks and leptons to be doublets and the right-handed to be singlets, then suppression of Yukawa couplings is realized in the same manner as SU (2) W . At the same time, this assignment softens the SUSY-flavor problem because of the left-handed sfermion mass degeneracy. Finally, any products of the doublets should not include the triplets. In this case, we can forbid single G-interactions when we assign G and G c to be triplets and the others to be doublets or singlets. The existence of triplets G and G c compels all fermions to consist of three generations to cancel gauge anomaly. As one of the candidates of the flavor symmetries which have the nature as above, we consider S 4 [5]. In such a model, the generation structure is understood as a new system to stabilize proton [6]. In section 2, we introduce new symmetries and explain how to break them. In section 3, we discuss Higgs multiplets. In section 4, we give order-of-magnitude estimates of the mass matrices of quarks and leptons and flavor changing processes. In section 5, we discuss cosmological aspects of our model. Finally, we give conclusions in section 6.", "pages": [ 2 ] }, { "title": "2 Symmetry Breaking", "content": "At first we introduce new symmetries and explain how to break these symmetries. The charge assignments of the superfields are also defined in this section.", "pages": [ 2 ] }, { "title": "2.1 Gauge symmetry", "content": "We extend the gauge symmetry from G SM to G 32111 = G SM × U (1) X × U (1) Z , and add new superfields N c , S, G, G c which are embedded in 27 representation of E 6 with quark, lepton superfields Q,U c , D c , L, E c and Higgs superfields H U , H D . Where N c is right-handed neutrino (RHN), S is G SM singlet and G,G c are colored Higgs. The two U(1)s are linear combinations of U (1) ψ , U (1) χ where E 6 ⊃ SO (10) × U (1) ψ ⊃ SU (5) × U (1) χ × U (1) ψ , and their charges X and Z are given as follows The charge assignments of the superfields are given in Table 1. To break U (1) Z , we add new vector-like superfields Φ , Φ c where Φ c is the same representation as RHN under the G 32111 and its anti-representation Φ is originated in 27 ∗ . To discriminate between N c and Φ c , we introduce Z R 2 symmetry and assign Φ c , Φ to be odd. The invariant superpotential under these symmetries is given by where unimportant higher dimensional terms are omitted 1 . Since the interactions W S drive squared mass of S to be negative through renormalization group equations (RGEs), spontaneous U (1) X symmetry breaking is realized and U (1) X gauge boson Z ' acquires the mass where the used value g X ( M S = 1TeV) = 0 . 3641 is calculated based on the RGEs given in appendix A, and 〈 H U,D 〉 /lessmuch 〈 S 〉 is assumed based on the experimental constraint [7] m ( Z ' ) > 1 . 52TeV , (11) which imposes lower bound on VEV of S as To drive squared mass of Φ c to be negative, we introduce 4th generation superfields H U 4 , L 4 and their antirepresentations ¯ H U 4 , ¯ L 4 and add new interaction To forbid the mixing between 4th generation and three generations, we introduce 4-th generation parity Z (4) 2 and assign all 4-th generation superfields to be odd. If M Φ = 0 in W Φ , then Φ , Φ c develop large VEVs along the D-flat direction of 〈 Φ 〉 = 〈 Φ c 〉 = V , U (1) Z is broken and U (1) Z gauge boson Z '' acquires the mass where the used value g Z ( µ = M I ) = 0 . 4365 is calculated by the same way as g X . We determine the values of two gauge couplings g X , g Z by requiring three U (1) gauge coupling constants are unified at reduced Planck scale M P = 2 . 4 × 10 18 GeV as In this paper we fix the VEV as RHN obtains the mass through the quartic term in W 0 . After the gauge symmetry breaking, since the R-parity symmetry defined by remains unbroken, the lightest SUSY particle (LSP) is a promising candidate for cold dark matter. As we adopt the naming rule of superfields as the name of superfield is given by its R-parity even component, we call G,G c 'colored Higgs'. Before considering flavor symmetry, we should keep in mind following points. As the interaction W G induces too fast proton decay, they must be strongly suppressed. As the mass term M Φ ΦΦ c prevents Φ , Φ c from developing VEV and breaking U (1) Z symmetry, it must be forbidden. In W 0 , the contributions to flavor changing processes from the extra Higgs bosons must be suppressed [8]. - - - - - -", "pages": [ 3, 4 ] }, { "title": "2.2 S 4 flavor symmetry", "content": "If we introduce S 4 flavor symmetry and assign G,G c to be triplets, then W G defined in Eq.(8) is forbidden. This is because any products of doublets and singlets of S 4 do not contain triplets. The multiplication rules of representations of S 4 are given in appendix B. Note that we assume full E 6 symmetry does not realize at Planck scale, therefore there is no need to assign all superfields to the same flavor representations. In this model the generation number three is imprinted in G,G c . Therefore they may be called 'G-Higgs' (generation number imprinted colored Higgs). Since the existence of G-Higgs which has life time longer than 0.1 second spoils the success of Big Ban nucleosynthesis (BBN)[9], S 4 symmetry must be broken. Therefore we assign Φ to be triplet and Φ c to be doublet and singlet to forbid M Φ ΦΦ c . With this assignment, S 4 symmetry is broken due to the VEV of Φ and the effective trilinear terms are induced by pentatic terms The size of effective coupling constants of these terms is given by This is the marginal size to satisfy the BBN constraint [10]. This relation gives the information about the RHN mass scale if the life time of G-Higgs is measured. The assignments of the other superfields are determined based on following criterion, (1)The quark and charged lepton mass matrices reproduce observed mass hierarchies and CKM and MNS matrices. (2)The third generation Higgs H U 3 , H D 3 are specified as MSSM Higgs and the first and second generation Higgs superfields H U 1 , 2 are neutrinophilic which are needed for successful leptogenesis. To realize Yukawa hierarchies, we introduce gauge singlet and S 4 doublet flavon superfield D i and fix the VEV of D i by then the Yukawa coupling constants are expressed in the power of the parameter which is realized by Z 17 symmetry 2 . To drive the squared mass of flavon to be negative, we add 5-th and 6-th generation superfields L 5 , 6 , D c 5 , 6 as S 4 -doublets and their anti-representations ¯ L 5 , 6 , ¯ D c 5 , 6 and introduce trilinear terms as where the mass scale of these fields is given by We assign the 5th and 6-th generation superfields to be Z (5) 2 -odd. The representation of all superfields under the flavor symmetry is given in Table 2. The mass terms of 4-th generation fields are given by where which realizes gauge coupling unification at Planck scale as", "pages": [ 4, 5 ] }, { "title": "2.3 SUSY breaking", "content": "For the successful leptogenesis, the symmetry Z (2) 2 × Z N 2 must be broken softly. Therefore we assume these symmetries are broken in hidden sector and the effects are mediated to observable sectors by gravity. We introduce hidden sector superfields A,B + , B 1 , -, B 2 -, C + , C 1 , -, C 2 -, where their representations are given in Table 3. We construct O'Raifeartaigh model by these hidden sector superfields as follow [11] - - - - As the F-terms of hidden sector superfields given by supersymmetry is spontaneously broken. The flavor symmetry Z (2) 2 × Z N 2 is also broken. Since we assume the U (1) R symmetry is explicitly broken in the higher dimensional terms [12], soft SUSY breaking terms are induced by the interaction terms between observable sector and hidden sector as where the indices A,B,C runs the species of superfields and the indices a, b, c runs generation numbers. Generally, as the coefficient matrices c ab are not unit matrices, large flavor changing processes are induced by the sfermion exchange. The explicit Z (2) 2 × Z N 2 breaking terms are given by [ B ∗ + B M 2 BX = = /epsilon1m 1 - 2 P ( c ( X ( D 1 1 X ∗ 1 X ) 3 + D M + s ( X X 2 2 ) ∗ ∗ ) X X 3 3 + h.c. ) + h.c. ( X = H , H , S ) , (38)", "pages": [ 5, 6, 7 ] }, { "title": "2.4 S 3 breaking", "content": "The S 3 subgroup of S 4 is broken by the VEV of S 4 -doublet flavon D i . Here we consider the direction of VEV. For the later convenience, we define the products of D i as follows, 1 ' 3 2 (41) and the VEVs of each components of D i as Generally, the superpotential of D i is written in the form of polynomial in E 2 , E 3 , P 3 as Substituting the VEVs given in Eq.(43) to the flavon potential, we get where 2 P ] do not have the solution as Z L (2) 2 B D U D and V F is F-term contribution. As this potential is polynomial in s 3 , the stationary condition gives parameter independent solution and parameter dependent solution Which solution of two is selected for the global minimum is depends on the parameters in potential. Since the solution Eq.(48) gives wrong prediction such as massless up-quark and electron, we assume the solution Eq.(49) corresponds to the global minimum. In this paper we assume 〈 D i 〉 are real without any reason, which is important in considering CP violation in section 4. The scale of V D is determined by the minimum condition as which agrees with Eq.(22). In this paper we sometimes write SUSY breaking scalar squared mass parameters as m 2 for simplicity and assume m ∼ O (TeV).", "pages": [ 7, 8 ] }, { "title": "2.5 S 4 breaking", "content": "The superpotential of gauge non-singlet flavons Φ , Φ c is given by Since the first term in Eq.(25) drives the squared mass of Φ c 3 to be negative through RGEs, these flavons develop VEVs along the D-flat direction as follows where S 3 -symmetry is unbroken in this vacuum. The scale of V is determined by the minimum condition as which agrees with Eq.(16). In this paper we define the size of O (1) coefficient as 0 . 1 < Y X < 1 . 0. Note that there are S 3 breaking corrections in the potential of Φ , Φ c as follows the direction given in Eq.(53) is modified as follows", "pages": [ 8, 9 ] }, { "title": "3 Higgs Sector", "content": "Based on the set up given in section 2, we discuss about phenomenology of our model. In this section, we consider Higgs doublet multiplets H U a , H D a and singlet multiplets S a .", "pages": [ 9 ] }, { "title": "3.1 Higgs sector", "content": "The superpotential of Higgs sector is given by For simplicity we assume The coupling k and λ 3 drive the squared mass of S 3 to be negative. Omitting O ( /epsilon1 )-terms, Higgs potential is given by (60) where V 1-loop is 1-loop corrections from Q 3 , U c 3 , G a , G c a . The VEVs of H U 3 , H D 3 , S 3 trigger off gauge symmetry breaking at low energy scale. Z (2) 2 -breaking terms enforce S 4 -doublets developing VEVs as follows Due to the Z (2) 2 symmetry, the Yukawa couplings between H U 3 and N c are forbidden and neutrino Dirac mass is not induced. To give neutrino Dirac mass, we assume the size of VEV of H U i is given by and put the Z (2) 2 breaking parameters as follows by hand. The suppression factor O ( /epsilon1 ) may be induced by the running based on RGEs, because off diagonal elements of scalar squared mass matrix do not receive the contributions from gaugino mass parameters which tend to make scalar squared mass larger at low energy scale. We use the notation of VEVs as follows where we fix the values by In this paper, we neglect the contributions from v u,d,s except for neutrino sector. With this approximation, the potential minimum conditions are given as follows, where the 1-loop contribution is neglected in Eq.(69), which is unimportant. These equations give the boundary conditions for m 2 H U , m 2 H D , m 2 S 3 at SUSY breaking scale M S = 10 3 GeV in solving RGEs. 3 3 The mass matrices of heavy Higgs bosons are given as follows where only O (1TeV) terms are considered and notation H U H D = ( H U ) 0 ( H D ) 0 -( H U ) + ( H D ) -is used. In this approximation, generation mixing terms are negligible. The third generation mass matrices are diagonalized as follows where the zero eigenvalue in CP even Higgs bosons corresponds to lightest neutral CP even Higgs boson and the other zero eigenvalues are Nambu-Goldstone modes absorbed into gauge bosons.", "pages": [ 9, 10, 11 ] }, { "title": "3.2 Lightest neutral CP even Higgs boson", "content": "To calculate the mass of the lightest neutral CP even Higgs boson, we diagonalize 2 × 2 sub-matrix given by where O ( v EW ) terms are included. We evaluate the 1-loop contributions from top, stop, G-Higgs and G-higgsino as at the renormalization point Λ = M S [13][14], where the mass eigenvalues are given by where we neglect O ( v EW ) terms in D-term contributions. The mass matrices of stop and G-Higgs are given in following sections (see Eq.(171) and Eq.(236)). By the rotation with the O (1TeV) term is eliminated from off-diagonal element of Eq.(78) and we get If we fix the parameters at M S as given in Table 6, we get The 1-loop contribution is dominated by stop and top contributions, this is because we put k small ( k = 0 . 5) to intend the mass values of the particles in the loops are within the testable range of LHC at √ s = 14TeV as follows The value of λ 3 = 0 . 37 is tuned to realize observed Higgs mass which is mainly controlled by this parameter through ( λ 3 v EW sin β ) 2 and X T for fixed v ' s and A U 3 (= A t ).", "pages": [ 11, 12 ] }, { "title": "3.3 Chargino and neutralino", "content": "At next we consider the higgsinos and the singlinos. The mass matrix of the charged higgsinos is given by Since the (3,3) element is much larger than the other O ( /epsilon1 2 ) elements, the first and second generation higgsinos decouples and have the same mass λ 1 v ' s . With the gaugino interaction given as follows the third generation charged higgsino mixes with wino and the mass matrix is given by The mass eigenvalues of charginos are given by where χ ± 3 is almost third generation higgsino and χ ± w is almost wino. The mass matrix of the neutralinos is divided into two 3 × 3 matrices and one 6 × 6 matrix as follows The mass eigenvalues of these mass matrices are given in Table 7. The common mass of two LSPs is given by the smallest eigenvalue of 3 × 3 matrix given in Eq.(97). Note that the LEP bound for chargino [15] must be satisfied. Requiring the coupling constants λ 4 , 5 do not blow up in µ < M P , we put upper bound for them as λ 4 < 0 . 5 , λ 5 < 0 . 7, then the rough estimation of LSP mass is given by where χ 0 i, 1 is almost singlino. To realize density parameter of dark matter Ω CDM h 2 = 0 . 11, we must tune M ( χ 0 i, 1 ) ∼ 30 -35GeV to enhance annihilation cross section. This condition gives upper bound as This constraint is not consistent with the lower bound from ATLAS [16] and CMS [17] as follows Therefore we assume the lightest chargino mass is in the region in which 3-lepton emission is suppressed due to the small mass difference between chargino and neutralino compared with m Z . Note that bino-like neutralino can decay into Higgs boson and LSP through the O ( /epsilon1 2 ) mixing of Higgs bosons by the interaction", "pages": [ 12, 13 ] }, { "title": "4 Quark and Lepton Sector", "content": "In this section, we consider the quark and lepton sector and test our model by observed values given as follows, running masses of quarks and charged leptons at µ = M S = 1TeV [18] CKM matrix elements [19] and neutrino masses and MNS mixing angles [19] . 12 . 09 ) 10 (eV ) , (109) After that we estimate the flavor changing process induced by sfermion exchange. - 3 2", "pages": [ 13 ] }, { "title": "4.1 Quark sector", "content": "The superpotential of up-type quark sector is given by from which we get up-type quark mass matrix as Note that there is dangerous VEV direction such as θ = π 6 . In this direction the matrix given in Eq.(112) is given by which has zero eigenvalue. In the same way, the down type quark masses are given by from which we get down-type quark mass matrix as follows The effects of flavor violation appear not only in superpotential but also in Kahler potential as follows where dot in X · Y means inner product of two S 4 -doublets X,Y . Therefore, a superfield redefinition has to be performed in order to get canonical kinetic terms as follows [20] by which the mass matrices given above are transformed into These matrices are diagonalized as follows Therefore Yukawa hierarchies are given by On the other hand, observed values Eq.(106) give which give good agreement with Eq.(126). Where the renormalization factors √ α α u u ( M ( M S P ) ) √ α α d d ( M ( M S P ) ) and Y t ( M P ) = 0 . 28 are calculated based on RGEs given in appendix A. CKM matrix is given by = 5 . 1 , = 7 . 2 , (133) which requires accidental cancellation of two mixing matrices L u,d to reproduce the small Cabbibo angle of CKM matrix given in Eq.(107). Note that the Z (2) 2 breaking induces generation mixing in Higgs bosons then Yukawa interactions are modified as follows Since these Yukawa coupling matrices are diagonalized in the basis that the quark mass matrices are diagonalized, the extra Higgs boson exchange do not contribute to the flavor changing processes.", "pages": [ 14, 15, 16 ] }, { "title": "4.2 Lepton sector", "content": "With the straightforward calculation, the mass matrices of lepton sector are given as follows. From the superpotentials we get original mass matrices as follows Redefining the Kahler potential given by by the superfields redefinition as L = and MNS matrix is given by the modified mass matrices are given by The mixing matrices of charged leptons are given by The Yukawa hierarchy of charged leptons gives good agreement with the experimental values given by where the used value of renormalization factor is calculated based on RGEs given in appendix A. The neutrino seesaw mass matrix is given by which has one zero eigenvalue because one RHN n c 1 does not couple to left-handed leptons. Therefore mixing matrix and mass eigenvalues are given as follows  1 1 1 1 1 1  (156) With the recent experimental value of | sin θ 13 | ∼ 0 . 14 [21], MNS matrix is given by which requires accidental cancellation of two mixing matrices L e,ν to reproduce the small θ 13 .", "pages": [ 16, 17, 18 ] }, { "title": "4.3 Squark and slepton sector", "content": "Sfermion mass matrices are given as follows where m = O (10 3 GeV) and the contributions from F-terms are neglected except for top-Yukawa contributions and the contributions from D-terms are neglected except for the contributions from S 3 . After the redefinition of Kahler potential and the diagonalization of Yukawa matrices, sfermion masses are given as follows where the off diagonal parts are extracted except for stop mass matrix and δ N LL , δ E RR are omitted.", "pages": [ 18, 19 ] }, { "title": "4.4 Flavor and CP violation", "content": "The off diagonal elements of sfermion mass matrices contribute to flavor and CP violation through the sfermion exchange, on which are imposed severe constraints. Based on the estimations of the flavor and CP violations with the mass insertion approximation, the upper bounds for each elements are given in Table 4, where M Q = M (gluino) = M (squark) , M L = M (slepton) = M (photino) are assumed [22]. Note that there is another suppression factor in δ X LR as The most stringent bound for M L is given by µ → eγ as and the one for M Q is given by /epsilon1 K as Note that if Q 1 , 2 were S 4 -singlets, then ( δ U LL ) 12 would be O (1) and the most stringent bound for M Q would be given by Comparing Eq.(182) and Eq.(183), one can see that S 4 softens the SUSY flavor problem very efficiently. Before ending this section, we discuss the problem of a complex flavon VEV. If the relative phase of two VEVs 〈 D 1 〉 , 〈 D 2 〉 exists, we must include in Kahler potential, then redefinition of superfields are modified as follows Therefore the mass matrix and mixing matrix of down quark sector and off-diagonal matrix of squarks are modified as follows As the result, the most stringent bound for M Q is changed into M Q > 68TeV. This suggests new mechanism is needed to suppress CP violation. We leave this problem for future work.", "pages": [ 19, 20 ] }, { "title": "5 Cosmological Aspects", "content": "Based on our model, we consider the scenario to reproduce the cosmological parameters given as follows [19] For Ω b , we adopt leptogenesis as the mechanism to generate baryon asymmetry. For Ω CDM , we assume that dark matter consists of singlino dominated neutralino.", "pages": [ 20 ] }, { "title": "5.1 Leptogenesis", "content": "In general, leptogenesis scenario to generate baryon asymmetry causes over production of gravitino in supersymmetric model. This problem can be avoided in the case neutrino mass is generated by small VEV of neutrinophilic Higgs doublet [23]. In the diagonal RHN mass basis, superpotential of RHN is given by where we assume accidental mass hierarchy as follows ( ) ( ) Note that these particles are enough light to create in low reheating temperature such as 10 7 GeV without causing gravitino over production [9]. The interactions of right-handed sneutrinos (RHsNs) are given by where the contributions from A-terms are neglected. The Z N 2 breaking scalar squared mass terms fill in the zeros of sneurino mass matrix and gives where O ( /epsilon1 ) suppressions of Z N 2 breaking terms are assumed without any reason. Note that the O ( /epsilon1 3 ) elements are originated from small Z N 2 breaking parameters and small Y Φ as In the diagonal RHsN mass basis, the interaction terms given in Eq.(194) are rewritten by Following [24], the CP asymmetry of sneutrino N c 1 decay is calculated as follows The lightest RHN n c 1 does not receive above corrections and remains decoupled. Therefore lepton asymmetry is generated by the out of equilibrium decay of the lightest RHsN N c 1 . From the naive power counting, we obtain Using /epsilon1 1 , the B -L asymmetry generated via thermal leptogenesis is expressed as where g ∗ is the total number of relativistic degrees of freedom contributing to the energy density of the universe and dilution factor κ is defined as follows By the EW sphaleron processes, the B -L asymmetry is transferred to a B asymmetry as where N H is number of Higgs doublets which are in equilibrium through Yukawa interactions, for example N H = 1 for SM and N H = 2 for MSSM. In any way N H -dependence is not important for our rough estimation. For our parameter values, we obtain K ∼ O (1) and which is consistent with observed value Requiring the effective interaction is decoupled in order to avoid too strong wash out, we impose the condition as follows which gives upper bound for temperature as This condition is always satisfied after the decay of N c 1 starts. This scenario is different from conventional one in the point that neutrino mass is realized by the small VEV v u = O (1GeV).", "pages": [ 20, 21, 22, 23 ] }, { "title": "5.2 Dark matter", "content": "Here we calculate the relic abundance of LSP which corresponds to singlino dominated neutralino in our model [25]. The most dominant contribution to annihilation cross section of LSP is given by the interaction with Z boson. If the mass matrix given in Eq.(97) is diagonalized by the field redefinition as the interaction with Z boson is given by , G ( ν L ) = 0 . 372 , are used. The formula for the relic abundance of cold dark matter is given by m P = 1 . 22 10 19 GeV , × where m f /lessmuch m χ 0 is assumed. Substituting the values given in Eq.(218) and Eq.(219) and following values in Eq.(224) and Eq.(225), we get from which the formula is rewritten as follows where quark and lepton masses are neglected except for bottom and τ . Since the two LSPs χ 0 1 , 1 , χ 0 2 , 1 have the same mass and the same interactions, they have the same relic abundance. Therefore the required relic abundance of one LSP is Ω CDM h 2 = 0 . 055. For the allowed range given in Eq.(104), the required values for λ 4 , 5 to reproduce observed relic abundance of dark matter are given in Table 5. The allowed ranges for λ 4 , 5 are very small. Note that we should not impose LEP bound ( m χ 0 1 > 46GeV) on this LSP, because Z → χ 0 i, 1 χ 0 i, 1 is strongly suppressed by the factor ( | V a | 2 -| V b | 2 ) 2 / 2 ∼ 0 . 005 and the contribution to invisible decay width is negligible as follows [19] where phase space suppression factor ∼ 0 . 6 and the ratio of LSP number and neutrino number 2 / 3 are multiplied. where", "pages": [ 23, 24 ] }, { "title": "5.3 Constraint for long-lived massive particles", "content": "Finally we consider long-lived massive particles which are included in our model, G-Higgs, flavons and the lightest RHN. Such particles are imposed on strong constraints from cosmological observations. The superpotential of G-Higgs sector gives degenerated G-higgsino mass as which receives S 4 breaking perturbation from Kahler potential given by which solves the mass degeneracy, however generation mixing is not induced. Neglecting O ( /epsilon1 2 ) corrections and contributions from D-terms except for the contribution from S 3 , the G-Higgs mass terms are given by from which we obtain three same 2 × 2 matrices as The mass spectrum of G-Higgs and G-higgsino is given in Table 7 and the lightest particle of them is lighter G-Higgs scalar G -. The dominant contributions to the G -decay are given by the superpotential from which we obtain For simplicity, we assume G ∼ G -then decay width of G -is given by where the renormalization factors are calculated based on the RGEs given in appendix A as follows Substituting these values in Eq.(240) we obtain the life time of G -as Since the existence of a particle which has longer life time than 0.1 second spoils the success of BBN [9], we must require τ ( G -) < 0 . 1sec which impose constraint as The G-Higgs exchange may contribute to proton decay, however it seems that the suppression of power of /epsilon1 is too strong to observe proton decay [26]. The five of six flavon multiplets Φ a , Φ c a have O (1TeV) masses which are enough small to product them non-thermally through the U (1) Z gauge interaction. The lightest flavon (LF) is quasi-stable and should not produced so much in order not to dominate Ω CDM . Solving the Boltzmann equation with the boundary condition n LF ( T RH ) = 0, we get relic abundance of LF as [27] 3 Requiring the LF does not dominate dark matter as Ω LF h 2 < 0 . 01, the upper bound for reheating temperature is given by which is consistent with our leptogenesis scenario. The life time of LF is estimated as follows. The LF can decay, for example through the operator the decay width and life time are given by [27] which suggests LF does not exist in present universe. Note that three and two body decays are suppressed by small VEV v u . The lightest RHN n c 1 behaves like LF because there is no distinction between N c and Φ c under the gauge symmetry. Integrating out N c 1 and λ Z in the Lagrangian where ( ψ, Ψ) means super-multiplet and some factors are omitted for simplicity, we get from which the life time of n c 1 is given by", "pages": [ 25, 26 ] }, { "title": "6 Conclusion", "content": "In this paper we consider S 4 flavor symmetric extra U(1) model and obtain following results.", "pages": [ 27 ] }, { "title": "Acknowledgments", "content": "H.O. thanks to Dr. Yuji Kajiyama and Dr. Kei Yagyu for fruitful discussion.", "pages": [ 27 ] }, { "title": "A RGEs", "content": "O (1) coupling constants of our model consist of gauge coupling constants and trilinear coupling constants defined by from which the fine structure constants are defined as follows We define the step functions as follows The beta functions are given by t = ln µ, (262) where we include only the contributions from α 2 , 3 in 2-loop order terms. The RGEs for trilinear coupling constants are given by We define gaugino mass parameters and A-parameters as follows The RGEs for gaugino mass parameters are given by where we take account of 2-loop contributions only for M 3 . The RGEs for A-parameters are given by (2 π ) (2 π ) dA t dt dA 3 dt = 6 α + 1 2 = 3 α t α A X t t + α M A t X h + +4 α A 3 5 6 h +2 α α A Z 3 M 4 Z +2 α A θ I 4 4 , (274) A + 4 16 3 +2 α α 3 5 M A 5 3 +3 α +9 α k 2 M A k + 13 9 α M 2 Y Y RGEs for scalar squared masses are given by dm 2 H dt dm 2 H dt where Note that the relations are held. At µ = M I , we add U (1) Z D-term corrections as follows [28] We solve these RGEs using following boundary conditions. At SUSY breaking scale ( µ = M S = 1TeV), we put by hand as follows (2 π ) (2 π ) U a D a = (3 α t M 2 t + α h M 2 h +2 α 4 M 2 4 ) δ a, 3 + α 5 M 2 5 (1 - δ a, 3 ) = α h M 2 h δ a, 3 + α 4 M 2 4 (1 - δ a, 3 ) + 2 α 5 M 2 5 δ a, 3 At reduced Planck scale ( µ = M P = 2 . 4 × 10 18 GeV),we put by hand as follows Note that gauge coupling constants do not satisfy the conventional unification as The renormalization factors of first and second generation Yukawa coupling constants Y u,d,e and single G-Higgs coupling constants defined by are given by which are calculated by RGEs as follows where the contributions from Φ a , Φ c 3 , D i are neglected. The results are given in Table 6.", "pages": [ 27, 28, 29, 30 ] }, { "title": "B The multiplication rules of S 4", "content": "The representations of S 4 are 1 , 1 ' , 2 , 3 , 3 ' [29]. Their products are expanded as follows. -", "pages": [ 30, 31 ] }, { "title": "C Mass bounds of new particles", "content": "The mass bound of the lightest chargino ( χ ± 1 ) is given by 3-lepton emission through EW direct process χ ± 1 χ 0 2 → W ± Zχ 0 1 χ 0 1 . This neutralino χ 0 1 corresponds to χ 0 i 1 , in our model. Under the assumption that slepton decouples and LSP ( χ 0 1 ) is massless, excluded region of chargino mass is given by 140 < M ( χ ± 1 ) < 295GeV [16], or M ( χ ± 1 ) < 330GeV [17]. These constraints are not imposed on the chargino in the mass range In this case Z and following two lepton emissions are suppressed. Taking account of LEP bound M ( χ ± 1 ) > 100GeV [15], we consider the allowed region given by", "pages": [ 31 ] } ]

[ { "title": "Cosmological constraints on the curvaton web parameters", "content": "Edgar Bugaev ∗ and Peter Klimai † Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia We consider the mixed inflaton-curvaton scenario in which quantum fluctuations of the curvaton field during inflation lead to a relatively large curvature perturbation spectrum at small scales. We use the model of chaotic inflation with quadratic potential including supergravity corrections leading to a large positive tilt in the power spectrum of the curvaton field. The model is characterized by the strongly inhomogeneous curvaton field in the Universe and large non-Gaussianity of curvature perturbations at small scales. We obtained the constraints on the model parameters considering the process of primordial black hole (PBH) production in radiation era. PACS numbers: 98.80.-k, 04.70.-s arXiv:1212.6529 [astro-ph.CO]", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Curvaton mechanism which has been suggested ∼ 15 years ago [1-5] now is the object of intense study. It is assumed, in the standard implementation of the curvaton model, that not the inflaton field perturbations are responsible for the primordial density fluctuations and for the cosmic microwave background fluctuations, but instead the (isocurvature) perturbations of the curvaton field σ . It is assumed that this curvaton field is subdominant during inflation but in post-inflationary epoch when Hubble constant becomes small, H ∼ m (where m is the curvaton mass), curvaton starts oscillating in its potential and behaves as nonrelativistic matter. The energy density of the curvaton decreases as ∼ a -3 ( a is the scale factor) whereas the energy density of radiation produced by the inflaton decay decreases as a -4 . As a result the curvaton energy density grows relative to radiation energy density until the curvaton contribution becomes significant. If it happens before the curvaton decay one can say that curvaton mechanism is 'effective', in a sense that just the curvaton (rather than inflaton) field perturbations during inflation determine the resulting (adiabatic) curvature perturbations at cosmological scales. In scenarios with the 'effective' curvaton there is the strong constraint on a value of the curvaton mass: it must be much smaller than the Hubble constant during inflation, H i , otherwise the primordial density perturbations have too large spectral tilt. Moreover, if the ratio m 2 /H 2 i is not small, the coherent length of the curvaton field (i.e., the characteristic size of the region inside of which the field is approximately homogeneous) is also too small and, in particular, smaller than the current horizon size. In the latter case, the primordial perturbation spectrum is strongly non-Gaussian, in contradiction with observations. The condition m 2 /H 2 i /lessmuch 1 is too restrictive and prohibits an use, for a description of the curvaton, particle physics models predicting large ratios m 2 /H 2 i at inflation (e.g., some variants of supersymmetric theories). In this connection it is reasonable to consider also the mixed curvaton-inflaton scenarios [6, 7] in which the curvaton perturbations are additional to the usual perturbations produced by the inflaton. Combining two contributions, one can obtain the primordial perturbation spectrum which is in agreement with data at cosmological scales. At the same time, the prediction for smaller scales may be quite unusual: the spectrum can be, e.g., very blue (i.e., the spectral tilt is large and positive) and, besides, the perturbations can be strongly non-Gaussian. In particular, large value of the tilt arises due to nonrenormalizable and supergravity corrections to the Lagrangian of some supersymmetric theories inducing mass terms of the order H 2 [8-12]. In most curvaton scenarios it is assumed that the curvaton field in the Universe is highly homogeneous and, as a result, the non-Gaussianity is relatively small. According to the alternative hypothesis, after the long inflationary expansion, the average value of the curvaton field is close to zero, and the local value of the field has a Gaussian probability distribution, variance of which is given by the formula [13, 14] Here, m ∗ is the effective curvaton mass which differs from the true curvaton mass m [15]. The corresponding coherent length is In Eqs. (1) and (2), t σ is the spectral tilt of the perturbation spectrum of the curvaton field, t σ = d ln P σ /d ln k . The assumption that ¯ σ = 0 will have real sense if the scale of interest, /lscript R = a i /k R , will be larger than /lscript c (both scales are calculated at the end of inflation). The value of /lscript R is given by the expression Here, k end is the scale leaving the horizon at the end of inflation, a i is the scale factor at the end of inflation (and at the beginning of radiation era), N is a number of efolds after the scale k R leaves the horizon. The condition leads to the inequality N /greatermuch 1 /t σ . It means that if t σ is not small ( t σ ∼ 1), and the coherent length /lscript c is small, one anticipates the blue curvature spectrum (the curvaton contribution) and large non-Gaussianity at small scales . In this case, the data at cosmological scales are described by the inflaton fluctuations only. In the opposite case, if t σ is very small, the number of e-folds N , which is necessary for the fulfilment of the condition /lscript R /greatermuch /lscript c becomes large, N → N infl ∼ 60. In particular, if t σ ≈ 1 / 60, one has, instead of the inequality (4), Traditionally, predictions for the primordial curvature perturbation spectrum in a region of small scales are constrained with a help of primordial black holes (PBHs). PBHs are produced in the early Universe, e.g., in radiation era, due to collapses of primordial density inhomogeneities [16-22]. Experimental limits from PBH overproduction had been studied in many articles, beginning from pioneering works [23, 24]; for the latest reviews, see [25, 26]. In the concrete case of the curvaton model, the idea of PBH constraining at small scales was suggested in [27] and was considered, in more detail, in [28]. In the present work we consider the predictions of the mixed curvaton-inflaton scenario just for the case which is most relevant for the PBH constraining: we assume that i) the average value of the curvaton field in the Universe is zero, and the Eq. (1) holds, and ii) the spectral tilt t σ is relatively large ( t σ ∼ 1) and positive. In this case adiabatic perturbations at small scales are produced mostly by the curvaton, resulting in a blue curvature spectrum. Large non-Gaussianity follows in this scenario from the quadratic dependence of the curvature on the curvaton field value. In this case, the typical size of the 'curvaton domain' [29] is relatively small, it is smaller than the horizon size at the moment of the formation of PBH with a given mass. Recently, the PBH formation in a curvaton scenario was studied in [30, 31]. In contrast with the present work, authors of [30, 31] do not use the assumption about a long period of inflation happened well before the observable Universe left the horizon. They assume, instead, that the curvaton field is nearly homogeneous in the whole Universe. The possibility of an essential PBH production at small scales in such models depends on the concrete inflationary scenario used. The authors of [30] use for a curvaton field a variant of the axion model suggested in [32] which predicts extremely blue spectrum of curvature fluctuations, while the authors of [31] used the model with a convex potential [as the concrete realization of a 'hilltop curvaton' scenario (see, e.g., [33])], in which strong scale dependence of the curvature power spectrum arises due to tachyonic enhancement effects. The plan of the paper is as follows. In the next Section we derive the basic formula for the curvature perturbation spectrum used in the concrete calculations. In Sec. III, the process of PBH production in our curvaton model is considered. The last Section contains the results of the calculation and conclusions. The technical details concerning the calculation of a probability density function (PDF) of the smoothed curvature field are discussed in the Appendix.", "pages": [ 1, 2 ] }, { "title": "II. CURVATURE PERTURBATION SPECTRUM FORMULA", "content": "Calculations of primordial curvature power spectra in mixed curvaton-inflaton scenario are carried out, in most cases, using the separated universe assumption and δN -formalism [34-41]. It had been shown, in particular [41], that the nonlinear curvature perturbation on an uniform energy density hypersurface, given by the formula is conserved on superhorizon scales, for a fluid with an equation of state P = P ( ρ ). In Eq. (6), ψ is the 'nonlinear curvature perturbation' entering the expression for the locally defined scale factor In our case there are two (non-interacting) fluids, radiation from an inflaton decay and an oscillating curvaton which we consider as pressureless matter field. Assuming that the curvaton decays on an uniform total density hypersurface, one has ψ = ζ on this surface, and, from Eq. (6), one has From here, one has for the fluid densities In the sudden decay approximation [3, 42, 43], the sum of densities is, on the decay hypersurface, equal to ¯ ρ ( t dec ) (i.e., it is homogeneous quantity). It leads to the important relation [44] ∣ The second relation which is necessary for the calculation of the curvature power spectrum is the nonlinear generalization of the formula for the relative entropy perturbation. In linear theory, one has Neglecting the curvaton density compared with radiation density (at the beginning of the radiation era), one has The nonlinear extension of Eq. (14) is given by Using Eqs. (11, 15) one can connect the curvature perturbation ζ with the curvaton field value on superHubble scales during inflation. At a beginning of the curvaton oscillations, one has, in a case of the quadratic potential Here, m osc is the curvaton mass at the moment of the beginning of oscillations. For simplicity, everywhere below we neglect the change of curvaton mass after t = t osc , and put m osc ≈ m . It is convenient to study the evolution of the curvaton field (from the field value at horizon exit during inflation, σ ∗ , to the field value at the beginning of the oscillations, σ osc ) separately for the averaged value and perturbation, The equations determining the evolution are (the prime and the dot denote d dσ and d dt , respectively). Eq. (19) is written for perturbations on superhorizon scales, where the gradient term ( ∼ k 2 /a 2 ) is negligible. For a quadratic potential V , a fractional perturbation, δσ/ ¯ σ , remains constant during the evolution. As is pointed out in the Introduction, we assume that the early Universe follows the scenario considered in [13, 34] ('the Bunch-Davies case'). In this scenario, ¯ σ is close to zero. As for the δσ ∗ , one can neglect its evolution during inflation . When the curvaton field is close to a minimum of the potential, then, due to a competition between the random walk and a (slow) roll, the typical value of the field, as can be easily shown, is ∼ H 2 2 πm , which is consistent with Eq. (1). After an end of inflation, the evolution of the total curvaton field takes place (of the average value as well as of the perturbation). Following Ref. [45], we denote this evolution introducing the notation where ¯ σ e is the average value of the curvaton field at the end of inflation, (which will be put equal to zero in final formulas). In a case of the quadratic potential the evolution is linear, so and one has, finally, The following steps are straightforward (see, e.g., [46, 47]. The entropy perturbation S σr is obtained from Eq. (16), expanding left and right sides of it up to second order, Further, expanding exponents in Eq. (11) up to second order, one obtains, using the connection of S σr with ζ σ , ζ r : Here, R σ,dec is given by the formula Since, according to the definition of the R σ,dec , there is the proportionality R σ,dec ∼ ¯ ρ σ = 1 2 m ¯ σ 2 osc (the proportionality coefficient is derived below, in Sec. III-A), and since ¯ g = ¯ σ osc , it follows from Eq. (26) that only the term proportional to R σ,dec / ¯ g 2 survives in this Equation in the limit ¯ σ osc → 0. It leads to the simple formula for the curvaton-generated part of the total curvature perturbation: Everywhere below we will use for ζ ( σ ) the notation ζ σ , dropping the brackets in the index. The power spectrum of ( δσ ∗ ) 2 is expressed through the power spectrum of the curvaton field perturbation [27], and the power spectrum of the curvaton field is The spectral tilt t σ is simply connected with a value of the effective mass of the curvaton field, m ∗ : The difference N infl -N is the number of e-folds of 'relevant inflation' [27], i.e., the number of e-folds passed from the moment when the observable Universe leaves horizon up to the moment when the scale k -1 R leaves horizon. The scale k -1 R enters horizon at the radiation era, just when the curvature perturbation ζ σ is created. The value of k R determines the value of horizon mass M h and, correspondingly, the order of magnitude value of PBH mass that can be produced at this moment. Finally, we obtain for the curvature spectrum the expression For calculations using this formula, one needs the relation R σ,dec / ¯ g 2 . It is derived in the next Section, for the concrete choice of the potential [see Eq. (41)].", "pages": [ 2, 3, 4 ] }, { "title": "A. Curvaton potential", "content": "Recently, a variety of models of chaotic inflation in supergravity, in connection with the curvaton scenario and curvaton web problem, had been introduced and studied [48]. Their models and conclusions, however, can not be used in our work straightforwardly because in our curvaton scenario i) there is no degeneracy of masses of the inflaton and curvaton fields, and ii) our curvaton field is a real, single component field, rather than the radial component of a complex field, as in [48]. Both these features . i are not inconsistent with the general theory of chaotic inflation in supergravity [49, 50]: for example, the curvaton field can be imaginary part of the complex scalar field [50]. We consider the model with the simple phenomenological potential of the form The corresponding effective mass of the curvaton field is m 2 ∗ = m 2 + αH 2 and the spectral tilt is given by The evolution equation for the curvaton field δσ is given above [see Eq. (19)]. The calculation of δσ ( t ) starts at moment t = 0 corresponding to an end of inflation and the beginning of the radiation-dominated era (the reheating is assumed to be instant). The derivative g ' is calculated numerically, and the initial conditions are: In our case, because the potential (33) is quadratic, g ' = δσ osc /δσ ∗ . For the value of δσ osc , we take δσ osc ≡ δσ ( t osc ), and the moment of time when oscillations start, t osc , is determined by the condition [51] ∣ ∣ According to this condition, after an onset of the oscillation the time scale of a change of the curvaton field is smaller that the expansion time H -1 . The example of the solution of Eq. (19) for the particular set of parameters, m/H i = 0 . 1, α = 1, is shown in Fig. 1. The corresponding value of the derivative g ' is equal to 0 . 62. According to (33), the energy density of the (average) curvaton field at the moment t osc is [ H osc ≡ H ( t osc )] After the moment t = t osc , and until the curvaton's decay at t = t dec , the curvaton is assumed to behave like a pressureless matter, so a value of the curvaton density at decay time is [ a ( t osc ) ≡ a osc , a ( t dec ) ≡ a dec ]: The radiation density at the moment t dec can be related to H ( t dec ) ≡ H dec by using the Friedmann equation, (here and below we neglect ¯ ρ σ,dec compared to ¯ ρ r,dec ). From Eqs. (37, 38, 39) one obtains Now, from Eqs. (27, 40), taking into account that Ω σ,dec /lessmuch 1 and using relations a ∼ t 1 / 2 ∼ H -1 / 2 , we obtain the final formula used in our calculations, In this Equation, we used the equality 1 2 t dec = H dec = Γ σ , to obtain t dec , while t osc is calculated numerically from the condition given by Eq. (36). Note also that in a case when α = 0 and H osc = m , one obtains from Eq. (40) ( M P = m Pl / √ 8 π ), which corresponds to a well-known result (see, e.g., [3]).", "pages": [ 4, 5 ] }, { "title": "B. PDF for the curvature perturbation ζ", "content": "It is generally assumed that the perturbations of the curvaton field at Hubble exit during inflation can be well described by a Gaussian random field (correspondingly, the equation (17) for σ ∗ contains, in its right-hand side, no higher-order terms). In our curvaton model, the curvature perturbation ζ σ depends on the curvaton field quadratically. In this case, the field ζ σ is chi-squared distributed, so the probability density function for ζ σ perturbations is strongly non-Gaussian. A formula for the PDF in the case of chi-square distribution of ζ -filed perturbations, i.e., in the case when is well known [52] (in our notations, χ ≡ δσ ∗ ; in contrast with the analogous formula (28) in Sec. II, in Eq. (43) the subtraction of 〈 χ 2 〉 is performed, to provide the condition 〈 ζ σ 〉 = 0). For applications in PBH production calculations (with using the Press-Schechter formalism [53]) one must derive the PDF for the smoothed field ζ σ . This problem is thoroughly discussed in the Appendix. It is argued there that the PDF for the smoothed ζ field can be approximately written in the form Here, σ ζ ( R ) is the variance of the smoothed ζ field [it is given by Eq. (A34)] and the function p (˜ ν ) is given by Eq. (A10). Effects of the smoothing operation enter, in Eq. (44), only through the variance, while the function p (˜ ν ) is the same in smoothing and non-smoothing cases.", "pages": [ 5 ] }, { "title": "C. PBH mass spectrum and constraints", "content": "The PBH constraints are obtained using the Press and Schechter formalism generalized for a case of nonGaussian PDFs. We will follow the Refs. [54-57] working with the curvature perturbation ζ R rather than with the density contrast. The basic formula in the Press and Schechter approach is In this Equation, P is the probability that in a region of comoving size R one has ζ R > ζ c , where ζ c is the threshold value for the PBH formation in the radiation era, n ( M ) is the mass spectrum of the collapsed objects, ρ i is the initial energy density. We will use the value of ζ c = 0 . 75 corresponding to the PBH formation criterion for the density contrast, δ c = 1 / 3. The PBH mass M BH is connected with the mass of the fluctuation M by the relation [58, 59] where M h is the horizon mass corresponding to the time when the fluctuation of mass M crosses horizon in radiation era, M i is the horizon mass at the start of the radiation era, t = t i . For the constant f h we will use the value f h = (1 / 3) 1 / 2 [58, 59]. In the approximation of the fast reheating, t i coincides with the time of the end of inflation. Using Eqs. (45) and (46) one obtains the formula for the PBH number density (mass spectrum) [54]: ∣ ∣ where a i is the scale factor at the end of inflation, and a eq , t eq are scale factor and time at matter-radiation equality, respectively. The derivative ∂P/∂R is given by the expression This expression is obtained with using the formula (44) for the non-Gaussian PDF. The dependence of the PBH number density on the curvature perturbation power spectrum P ζ arises just through the derivative ∂P/∂R . If PBHs form at t = t e , one can calculate the energy density fraction of the Universe contained in PBHs at the time of formation (at this time, the horizon mass is equal to M h ( t e ) ≡ M f h [54]): Ω PBH ( M f h ) ≈ In this formula, M min BH is the minimum mass of the PBH mass spectrum, M min BH ≈ f h M f h . The PBH mass spectrum is very steep, so, with high accuracy one has where β PBH is, by definition (see, e.g., [26]), the fraction of the Universe's mass in PBHs at their formation time, Now, having Eqs. (50, 51), one can use the experimental limits on the value of β PBH [26] to constrain parameters of models used for PBH production predictions.", "pages": [ 5, 6 ] }, { "title": "IV. RESULTS AND DISCUSSION", "content": "The examples of curvaton-generated curvature perturbation power spectra are shown in Fig. 2, and some examples of the PBH mass spectra calculations are given in Fig. 3. For each curve shown in Figs. 2, 3, the model parameter Γ σ is chosen so that the predicted PBH abundance is of the same order of magnitude as the currently available limits [26] on the parameter β PBH in the corresponding PBH mass range. On the vertical axis of Fig. 3 the combination M -1 / 2 i ρ -1 i M 5 / 2 BH n BH ( M BH ) is shown; just this combination is approximately equal to β PBH , as it follows from Eq. (50). The following connection between the comoving scale k R and horizon mass M h (which is approximately equal to PBH mass) is used in Fig. 2 [60]: It is seen from Fig. 3 that for smaller values of α , the PBH mass spectra become more wide. The low mass cutoff of the curves shown is determined by the fact that no PBHs are formed before the curvaton decays at t = t dec , so the minimal PBH mass is M min BH = f h M h ( t dec ). For the constraining of the curvaton model parameters, we used the limits for β PBH ( M BH ) from the review work [26]. Demanding that PBHs are not overproduced, i.e., the value of β PBH ( M BH ) does not exceed the available limits [26], one may obtain the corresponding constraints on the parameters of the considered cosmological model. Such constraints are shown in Fig. 4 for the case of α = 0 . 4 and in Fig. 5 for α = 1. In particular, in Figs. 4a and 5a we show the limits on the combination of parameters Γ σ /m while in Figs. 4b and 5b - on the value of Γ σ itself. The prohibited (by PBH overproduction) parameter ranges lie below the corresponding lines. In the sudden decay approximation, there is a very simple approximate connection between Γ σ and the PBH mass produced. It follows from the relations Thus, constraints on Γ σ (see Figs. 4b, 5b) are at the same time constraints on the mass of PBHs that can be produced in this model [this is reflected on the vertical axis of the Figures 4b, 5b; the relation between M BH and Γ σ is given by Eq. (55)]. Deriving the constraints, we use the condition in order not to contradict with the data on the cosmological scales. One must note that the characteristic values of P ζ which determine the constraints on the model parameters shown in Figs. 4, 5 are of order of ∼ 10 -3 . 5 . This is consistent with the PBH constraints on P ζ (for non-Gaussian ζ σ -perturbations) obtained in our previous work [61] (see also [55]). In Figs. 4c, 5c we show also the number of e-folds after the scale k R leaves horizon, as a function of the constrained model parameters. It is seen that for all cases corresponding to the obtained PBH constraints, N /greatermuch 1. As pointed out in the Introduction, this is needed for the validity of the considered model. One can see from the resulting Figs. 4, 5 that, generally, PBH constraints are very weak. The forbidden region contains too small values of Γ σ /m (although the nucleosynthesis limit, Γ σ > ∼ (1MeV) 2 /M P , allows such values). The PBH constraint works only for very high values of Hubble constant during inflation, H i > ∼ 10 11 ÷ 12 . 5 GeV, and for very large values of curvaton masses, m > ∼ (10 -4 ÷ 10 -1 ) H i . For other values of parameters, the spectrum amplitude, P ζ σ , is too small and cannot be constrained. For illustrative purposes we show in Figs. 4, 5 constraints for a large interval of H i values, up to 10 15 GeV, although there is a well-known upper bound on the Hubble parameter during inflation (according to the recent results of Planck collaboration, H i /M P < 3 . 7 × 10 -5 [62]). One must note also that in the forbidden region the reheating temperatures are rather high ( T RH ∼ √ H i M P ) and, in standard supersymmetric models, gravitinos are overproduced.", "pages": [ 6, 7, 8 ] }, { "title": "Acknowledgments", "content": "The study was supported by The Ministry of education and science of Russian Federation, project 8525.", "pages": [ 8 ] }, { "title": "Appendix A: Moments of PDF of ζ -field", "content": "It follows from Eq. (28) that in our model the curvature perturbation depends on the Gaussian curvaton field δσ ∗ quadratically, In Eq. (A1) the constant term A 〈 δσ 2 ∗ 〉 is subtracted such that now 〈 ζ 〉 = 0, and ζ is the 'overcurvature'. Introducing the notation δσ ∗ ≡ χ , one has and the PDF of the χ field is PDF of the ζ field is obtained from the PDF of the χ field using the Chapman-Kolmogorov equation, Here, χ i are roots of the equation The final expression for the PDF of the ζ field is The variance of the p ζ ( ζ ) is Using this equation, the distribution (A7) can be written in the form: In this equation, the notation ν = ζ/ 〈 ζ 2 〉 1 / 2 is introduced. Note, that the product p ζ ( ζ ) dζ doesn't depend on ζ and 〈 ζ 2 〉 1 / 2 separately, i.e., The first (central) moments of the p ζ are and the first cumulants , 〈 ζ n 〉 c , are given by the relations (see, e.g., [63]) The reduced cumulants are defined by the relation (see, e.g., [64]) For the first non-trivial reduced cumulants, skewness and kurtosis, one has, respectively, The general formula for D n is remarkably simple, To find the PDF of the smoothed curvature fluctuations one must use the smoothed ζ field, Here, W ( x/R ) is the window function. In the present paper we use the Gaussian window function, defined by the equations The general expressions for the cumulants of the PDF of the smoothed ζ field have been derived in [52] using the path integral formalism. In this formalism, authors of [52] expressed cumulants through the integrals in k -space, Here, ˜ W ( kR ) is the window function in k -space, ˜ W ( kR ) = e -k 2 R 2 / 2 , P χ ( k ) is the power spectrum of the χ field, depend on the k -dependence of the power spectrum of the χ field and on the window size R . To study qualitatively the R -dependence of the cumulants it is more convenient to use the expressions for 〈 ζ n R 〉 c through the integrals in real (configuration) space [65]. The corresponding expression is As one can see from Eq. (A20), values of the cumulants Here, the connected n -point function in real space is given by the product of two-point correlation functions of χ field, In Eqs. (A23, A24) we use the notation x ij = | x i -x j | . If the power spectrum of the χ field has a form it follows from Eq. (A24) that ξ χ ( x ij ) ∼ x -t χ ij , and Integrals in Eq. (A22) converge, if 0 < t χ < 2 . 5, and scale with the window size R . Therefore, there is the proportionality 〈 ζ n R 〉 c ∼ R -nt χ , and, as a result, the reduced cumulants almost don't depend on the smoothing scale [65], The weak dependence of the reduced cumulants on R suggests that the PDF of the smoothed ζ field can be written in the form analogous to Eq. (A9) [65], ˜ ν ≡ ζ R / 〈 ζ 2 R 〉 1 / 2 . Indeed, the reduced central moments for this PDF, which are given by the relation have a form which is independent on the smoothing scale, in accordance with Eq. (A27). Evidently, the reduced cumulants which are connected with the reduced central moments by a relation analogous to (A13) also have this property. Quantitative values of D n,R are different for different values of the power spectrum index t χ (even if D n,R almost do not depend on R ). One can expect, however, that if the positive tilt of the χ -spectrum is not too large, t χ < ∼ 1, the approximate equality takes place. This problem had been studied, for the case n = 3, in [66], and, for the case n = 4, in [67]. It had been shown in [66, 67] that, really, if t χ is not small enough (e.g., if t χ = 2) the cumulants D n,R are comparatively small, D n,R /lessmuch D n , but they are close to D n in the limit t χ < ∼ 1 (just this limit is of interest for us in the present work). Assuming that Eq. (A30) holds for all n (i.e., that the reduced cumulants are the same in cases with smoothing and without smoothing), one can use for the PDF of the smoothed ζ field the expression where p (˜ ν ) is given by Eq. (A10), with a substitution ν → ˜ ν . In this approximation, the effects of the smoothing come only through the variance 〈 ζ 2 R 〉 1 / 2 while the shape of the PDF is the same as in the non-smoothing case. The variance, 〈 ζ 2 R 〉 1 / 2 ≡ σ ζ ( R ), is given by the expression followed from the general formula (A22): Note, for completeness, that moments of the PDF of the ζ field are simply connected with polyspectra of the ζ field. In particular, using the definition one can obtain from Eq. (A32) the simple formula for the variance: [50] R. Kallosh, A. Linde and T. Rube, Phys. Rev. D 83 , 043507 (2011) [arXiv:1011.5945 [hep-th]].", "pages": [ 8, 9, 10, 11 ] } ]

[ { "title": "Maximum Entropy deconvolution of Primordial Power Spectrum", "content": "Gaurav Goswami 1 and Jayanti Prasad 2 IUCAA, Post Bag 4, Ganeshkhind, Pune-411007, India", "pages": [ 1 ] }, { "title": "Abstract", "content": "It is well known that CMB temperature anisotropies and polarization can be used to probe the metric perturbations in the early universe. Presently, there exist neither any observational detection of tensor modes of primordial metric perturbations nor of primordial non-Gaussianity. In such a scenario, primoridal power spectrum of scalar metric perturbations is the only correlation function of metric perturbations (presumably generated during inflation) whose effects can be directly probed through various observations. To explore the possibility of any deviations from the simplest picture of the era of cosmic inflation in the early universe, it thus becomes extremely important to uncover the amplitude and shape of this (only available) correlation sufficiently well. In the present work, we attempt to reconstruct the primordial power spectrum of scalar metric perturbations using the binned (uncorrelated) CMB temperature anisotropies data using the Maximum Entropy Method (MEM) to solve the corresponding inverse problem. Our analysis shows that, given the current CMB data, there are no convincing reasons to believe that the primodial power spectrum of scalar metric perturbations has any significant features.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Observations of Cosmic Microwave Background (CMB) temperature anisotropies as well as polarization [1] can be used to uncover the physics of the early universe e.g. of cosmic inflation [2, 3, 4]. However, calculations of power spectra of CMB anisotropies and polarization [5, 6] involve making a number of assumptions e.g. about the reionization history of the universe, the equation of state of dark energy etc. It is also usually assumed that the primordial power spectrum of scalar metric perturbations (denoted by sPPS in this work) is a power law (with a small running). One can then use the CMB observational data to put constraints on the values of various cosmological parameters [8] including the ones specifying sPPS (usually denoted by A S , n S etc). Since this procedure leads to 'resonable' values of these parameters, it is often said that a power law sPPS is consistent with the observed data. But it is worth noticing that this is just an assumption. Cosmic inflation is the most actively investigated paradigm for explaining the origin of anisotropies in CMBsky as well as the large scale structure of the universe. The simplest versions [2, 3, 4] of inflationary models give a smooth, nearly scale-invariant (tilted red) sPPS. But there are other models which are capable of giving more complicated forms of sPPS (abnormal initial conditions, multifield models, interruptions to slow roll evolution, phase transition during inflation, see e.g. [11, 12, 13, 14, 15, 16]). Are these models ruled out by the present data? Thus, even though power law sPPS is consistent with the data, the assumption of a power law PPS (with small running) is just that: a well motivated assumption. It is worth checking, how the models in which sPPS is not just a simple power law with a small running fare against the present available data. This can be done in various ways: e.g. one could try to redo cosmological parameter estimation with the actual form of sPPS left free (see e.g. [9]). Another option is to work with inflationary models which lead to features in sPPS and redoing parameter estimation for those models (see e.g. [10, 11]). This exercise illustrates that (i) models in which sPPS is not this simple also do fit the data, (ii) very often, with these models, one can get a better fit to data than power law with small running. Given this situation, a reasonable possibility is to try to directly deconvolve sPPS from observed CMB anisotropies ( i.e. C /lscript s). Previous attempts [17] at doing so seem to suggest the existence of features in sPPS (the statistical significance of which is still being assessed [18]), e.g., a sharp infrared cutoff on the horizon scale, a bump (i.e. a localized excess just above the cut off) and a ringing (i.e. a damped oscillatory feature after the infrared break). This is consistent with many existing models of inflation and this has also motivated theorists to build models of inflation that can give large and peculiar features in primordial power spectrum (see [11, 12, 13, 14, 15, 16]). Given the fact that primoridal power spectrum of scalar metric perturbations is the only cosmological correlation whose effect is, at this stage, observable in the universe (Primordial non Gaussianity is yet to be detected in CMB data, so are B modes of polarization of CMB due to inflationary Gravitational waves), it becomes important to settle this issue of possible existence of features. In the present work, we try a new method of probing the shape of primordial power spectrum: the Maximum Entropy Method (MEM). We begin in § 2 by broadly describing the problem and its various attempted solutions. Then, in § 3, we describe in detail the algorithm that we have used. This is followed by § 4 in which we apply the algorithm to binned CMB temperature anisotropies data. We conclude in § 5 with a discussion of salient features, limitations and future prospects for the work. In the appendix § A, we present the results of applying the method on a toy problem and in the process illustrate the use of the algorithm.", "pages": [ 2, 3 ] }, { "title": "2.1 Formulation as an inverse problem", "content": "We address the issue of reconstructing the shape of the sPPS by attempting to directly solve the (noisy) integral equations giving the CMB angular power spectrum using MEM. The observed CMB TT angular power spectrum is given by (see e.g. [19]): here, /lscript is the multipole moment, k is the wave number and the quantity in the square brackets is the radiation transfer function ( η 0 denotes the value of conformal time today) and P Φ ( k ) = k 3 2 π 2 〈| Φ( k ) | 2 〉 is the power spectrum of the scalar metric perturbation in Newtonian gauge (often called Bardeen potential, Φ). Assuming a given set of values of background cosmological parameters, the radiative transport kernel can be found (see § 4), we can then formulate the problem we are dealing with as the solution of a set of integral equations i.e. as an inverse problem. The scalar primoridal power spectrum is the power spectrum of comoving curvature perturbation: here R ( k ) is the mode function 3 of the comoving curvature perturbation on super-Hubble scale (when it has become frozen). For a power law sPPS, In matter dominated universe (at the time of recombination), at linear order in perturbation theory, Φ = (3 / 5) R , so, for a power law PPS, C TT /lscript should be (in µK 2 ) We shall now replace T CMB 2 · ( 3 5 ) 2 · A S ( k k 0 ) n S -1 by a general function f ( k ) and try to find this function f ( k ). We thus have with and so the function f ( k ) shall have values of the order of magnitude of 10 3 . Given the temperature radiation transfer function (∆ T ( /lscript, k, η 0 )), the theoretical C TT /lscript can be found from Eq [1] provided, we know the sPPS. The /lscript range for which we wish to evaluate the transfer function and the corresponding C /lscript s goes from /lscript = 2 to /lscript = l max = 1500. The typical behaviour of the function is shown in the Fig (1) (with dk chosen such that the integral in the definition of C /lscript can be evaluated to a high enough accuracy). For every given /lscript , the radiation transport kernel is a highly oscillatory function of the wavenumber k . But for any /lscript , it has significant (i.e. non-negligible) values only within a small range of k values. The brightness fluctuations roughly go as j /lscript [ k ( η 0 -η ∗ )] (where j /lscript is spherical bessel function while η ∗ is the conformal time at the epoch of recombination), thus the minimum value of /lscript sets a minimum value of k at which the kernel takes up non-negligible values. This procedure tells us that since the radiation transfer function is neglible for k < k min , no matter how much power is there in sPPS at very small k -values, the CMB anisotropies cannot be used to probe the sPPS at these (very large scales). This sets the k min below which we cannot probe the sPPS. Similarly, given the fact that we have observations only till a maximum value of /lscript , this sets the maximum value of k upto which we need to sample the kernel: thus, the smallest possible angular resolution of a CMB experiment shall set the lmax that we can probe which shall set a k max , i.e. sPPS at scales smaller than this scale can not be probed by CMB experiments. Thus, /lscript = 2 determines k min while /lscript = l max determines k max . Within this range, one discretizes the k -space in such a way that the transfer function can be sampled sufficiently well and the above integral can be performed to the desired accuracy. 4 Apart from this consideration, the actual observed C /lscript s are also noisy (due to cosmic variance, instrumental noise and the effect of masking the sky). Thus Eq (1) can be written as a set of linear equations where n s is the number of bins in k -space and C N /lscript is the noise term. Thus the problem we wish to solve is: given the matrix G , the few observations ( C /lscript s), the moments of the random variables C N /lscript , how can we find the set of numbers f k ? In this paper, we shall use the binned CMB data to find sPPS. The number of (binned and hence uncorrelated) data points (WMAP) is 45 (call it n d ). To sample the kernel satisfactorily, we divide the k space into 6200 points ( n s ). Thus, we have a problem with a set of 45 noisy linear equations and 6200 unknowns to be determined.", "pages": [ 3, 4, 5 ] }, { "title": "2.2 Bayesian inversion", "content": "Recovering the primordial power spectrum f k from the observed C l can be casted as a Bayesian inversion problem in the following way. The posterior probability P ( f k | C l , G lk ) of obtaining the primordial power spectrum f k given a kernal G lk and observed C l is given by: where P ( C l | f k , G lk ) is the likelihood and P ( f k ) is the prior probability. For our case the denominator (evidence) works just a normalization and we can ignore it. For the case of Gaussian noise 5 the likelihood function can be written as where for the case when the noise covariance matrix is diagonal. Since for our problem the number of unknowns i.e., f k are far more than the number of knowns i.e., C l s therefore ordinary chi square minimization is of no use since it can make the chi square too low 6 . In order to avoid chi sqaure taking unphysical values we need some form of regularization in the form of prior. In place of maximizing the likelihood function we maximize the posterior probability. It has been a common practice to conside the following form of prior for any regularization problem where λ is the regularization parameter and S is the regularization function. There have been many form of regularization function like quadratic form etc. In the present work we use an Entropy function S ( f k ) as a regularization function which is defined in the following way where A is a parameter which parameterizes the entropy functional we use. With the regularization function the posterior probability distribution can be written as where Maximum entropy method is a particular (nonlinear) inversion method. Here the regularization function S ( f k , A ) is non-quadratic so that the equations to be dealt with to solve the optimization problem shall turn out to be non-linear. Without such a maximum entropy (ME) constraint, the inversion problem is ill-posed (since the data can be satisfied by an infinity of primordial power spectra). The condition that the entropy be a maximum selects one among these. There exist, in the literature, various arguments justifying the use of MEM over other ways of inversion (often using arguments from information theory 7 ), at this stage, we just treat it as just another nonlinear version of the general regularization scheme.", "pages": [ 5, 6 ] }, { "title": "3 The Cambridge Maximum Entropy Algorithm", "content": "So, the problem that we wish to solve involves a highly under-determined system of linear equations. As was mentioned in the last section, one way in which we can attempt to solve this problem is to formulate it as a problem involving the optimization of a non-quadratic function (which will require solving a set of non-linear equations) subject to a constraint. Since the number of unknowns is so large, we have to solve the corresponding constrained non-linear optimization problem in a very large dimensional space. Also, we have other constraints that we need to take care of e.g. the components of f are positive quantities (since f is a power spectrum), so the optimization algorithm that we use must not cause the components of f to become negative (this requirement rules out methods such as the steepest ascent). Similarly, since the the objective function is quite different from a pure quadratic form, methods such as conjugate gradient method are not very useful. Experience has shown that one of the strategies which work (despite being complicated) is the following: instead of searching for a minimum in a single search direction (e.g. in steepest ascent method), one searches in a small- (typically three-) dimensional subspace. This subspace is spanned by vectors that are calculated at each point in such a way as to avoid directions leading to negative values. The algorithm that we use is based on the one developed by Skilling and Bryan [21, 22] and is sometimes referred to as The Cambridge Maximum Entropy Algorithm. It has been extensively used in not only radio astronomy but also in other fields. Here we quickly review this algorithm for the sake of completeness.", "pages": [ 6, 7 ] }, { "title": "3.1 Entropy and χ 2", "content": "The problem to be solved involves finding a set of f k ( k = 1 , 2 , · · · , n s ) (with maximum entropy) from a dataset D /lscript ( /lscript = 1 , 2 , · · · , n d ). For any f k , let We shall use the following definition of entropy (the non-linear regularization function) here, A is a fixed number (sometimes called 'the default') that sets the normalization of f . Notice that S ( /vector 0) = 0 , S ( f k = A ) = n s · A,S ( f k = eA ) = 0. This gives, (since A is fixed), telling us that ∂ i S ( /vector 0) = ∞ , ∂ i S ( f k = A ) = 0 and ∂ i S ( f k = eA ) = -1. It is easy to see that entropy surfaces are strictly convex. Also, the expression for the various derivatives of the entropy tell us that the solution f i = A is the global maximum of entropy, this fact shall be important later. The measure of misfit that we shall use (in order to use the data) is the Chi-squared function from which we get, the gradient of C and the Hessian For a linear experiment, the surfaces of constant chi-squared are convex ellipsoids in N-dimensional space. The largest acceptable value for χ 2 at 99 percent confidence is about C aim = n d + 3 . 29 √ n d (with n d being the number of observations), see [21]. As the above equations show, quantities such as gradient of C and Hessian of C can be easily evaluated (though finding the Hessian of C is the one of the most computationally expensive tasks since the matrix G /lscriptk is 45 × 6200 and Hessian of C shall be 6200 × 6200 matrix). At every iteration, instead of searching for the maximum of S and minimum of C along a line, we search in an n dimensional subspace of the parameter space. So, instead of we shall have (with e µ being n search directions) Sufficiently near any point, every function can be approximated by a quadratic function (provided the higher order terms in the Taylor expansion can be ignored). So, within the subspace we shall model the entropy and chisquared by where s ( x ) and c ( x ) are quadratic which correspond to the first three terms in the Taylor series expansion of S ( f ) and C ( f ). The first order term in the Taylor expansion of S is which tells us what s µ should be. Similarly, c µ , g µν and h µν can be found: Thus, if we know the basis vectors, we can find the quadratic functions s ( x ) and c ( x ). Obviously, the above definitions shall not be valid to arbitrary distances from the point in question. The quadratic models are reliable only in the vicinity of the current f where cubic and higher powers can be nelgected. Thus, the step size at each iteration must be such that for some l 0 . We thus need to define the concept of distance in this abstract space. Recall that this means we need to define a metric note that the metric ¯ g ij is different from the function g µν defined by Eq. (26). Experience (see [21]) has shown that the following definition of distance works well this needs to be compared with the expression for the Hessian matrix of entropy (notice that ¯ g ij = f i δ ij ). It is straightforward to show that while choosing l 0 2 to be 1 / 5 of ∑ f works well (see [21]). The algorithm works in the following way: at every iteration, when we are at a point in the f space, one considers a distance region s.t. the quadratic model is a good approximation in that region. We now find a subspace and within this subspace, we try to find the place where", "pages": [ 7, 8, 9 ] }, { "title": "3.2 Construction of the subspace", "content": "So, how do we decide the basis vectors which span the subspace? One of our aims is to find the maximum of entropy on the surface of ellipsoid corresponding to χ 2 = C aim . So, naturally, the direction of gradient of entropy must be one of the basis vectors. Since the metric in the space of interest is not Cartesian, there shall be a distinction between contravariant and covariant components of vectors in the space. Since the 'position vector' of any point is f i , a contravariant vector, gradiant such as ∂S/∂f i is going to be a covariant vector. So the first (contravariant) basis vector is The meaning of this direction is easy to understand by recalling its equivalent in usual Cartesian space. In the usual situation, ( /vector ∇ T ) · ˆ ndr = dT (i.e. if we are at any point, and we go in the direction ˆ n by a distance of dr , the change in the value of the function is dT ). It is obvious from this expression that when ˆ n is parallel to the direction of gradient, the change df is maximum. Thus, to maximize the change in f , we shall move in the direction parallel to /vector ∇ T so that This equation should be compared with the definition of the first basis vector, Eq [35] (and since the kronecker delta is the metric in a Cartesian space, the two equations are equivalent). Thus, the first basis vector tells us the direction in which the entropy change per unit distance is maximum. Similarly, another basis vector could be since we wish to change the χ 2 at every iteration so that we eventually reach the χ 2 = C aim surface. If we find what the two search directions (defined above) become after incrementing by x 1 e 1 + x 2 e 2 , the direction e 1 shall stay within the subspace spanned by e 1 and e 2 but the direction e 2 shall go out of the subspace (see [22]). This suggests that we choose more basis vectors such as Experience has shown that a family of three or four such search directions gives quite a robust algorithm for solving the problem. In our problem, we chose the third search direction to be where, the following Eqs define the lengths L s and L c which are the gradient vectors our experience has shown that putting the factors of L s and L c in the definition of the third basis vector improves the speed of convergence of the answer.", "pages": [ 9, 10 ] }, { "title": "3.3 Optimization within the subspace", "content": "Once we have found the subspace (by finding the basis vectors in the space of all f s), we proceed as follows: we now wish to find the step, the coefficients x in Eq (23). To do this, we shall solve a corresponding constrained optimization problem in the n dimensional subspace (as was stated in the previous subsection, we worked with n = 3, but we shall continue to explain the details for a general n ). Since the functions s ( x ) and c ( x ) are quadratic, the problem in the subspace is much simpler: it is a simple problem of quadratic programming (quadratic objective function with quadratic constraint). The only additional complication is that the quadratic model is not valid to arbitrary distances from the original point, so we need to satisfy an additional distance constraint. Let us begin by recalling that both the matrices g and h are real-symmetric. Also, the matrix g is positive definite. The reason is as follows: the way we have defined the metric on the space (see Eq (33) ), where in the last step we used Eq (34). So, it is clear that g µν is a positive definite matrix (which imples that all its eigenvalues are positive). Thus if one of the eigenvalues of g µν is a small positive number, numerical errors can cause it to become negative. In our implementation of the algorithm, we choose to ignore any directions which are defined by eigenvalues which are too small. Additional simplification occurs if we simultaneously diagonalize the two matrices g and h . 8 After simultaneous diagonalization, within the subspace, the quadratic model functions ˜ S and ˜ C are given by the quantities s µ etc are now defined in terms of the new basis vectors (but the same old definitions). Since this causes the function g µν to become a Kronecker delta, the distance constraint Eq. looks like We chose the coefficient on the RHS to be 0 . 2 and we verified that the actual value of this number is unimportant. Typically, the function ˜ C is such that all its eigenvalues are (also) positive, then, the minimum value of the function ˜ C in the subspace (where the above definitions work) is Thus, no matter what the global aim C aim is, at a given iteration, within the subspace, we can not get to any values below ˜ C min . In fact, even trying to achieve ˜ C min is not a great idea since in that case we shall not use any information about ˜ S . The real challenge in the subspace is to satisfy the distance constraint. Many different elaborate tricks have been mentioned in the literature to do this. We choose to not worry about getting a quick answer, hence we do the following: in order to ensure that the distance constraint always gets satisfied (i.e. we do not go too far from the present location in just one step), we shall choose to have a ˜ C aim which is not too different from C 0 (the present value of χ 2 ). We thus choose with a chosen to be a small number (e.g. 0 . 01). This causes the algorithm to take very small 'baby steps' towards the answer. Numerical experience has shown that as far as our problem is concerned, this is good enough. Of course, the actual value of a or C aim chosen is not important as long as the distance constraint gets satisfied. The problem in the sub space is thus simplified to finding the point x a such that the function ˜ S is maximum subject to the constraint that ˜ C = ˜ C aim (and an additional constraint that the distance constraint must get satisfied). The techinique of Lagrange's undetermined multipler is useful here: we wish to find the point on the curve ˜ C = ˜ C aim where ˜ S is maximum, to find the desired point, we consider the set of points at which all the partial derivatives of the function (for an undetermined α ) vanish. For any α , such points are given by So, for every value of α , find the value of x a and then the function ˜ C : we are after that value of α which leads to ˜ C = ˜ C aim so we look for a solution of the equation ˜ C ( α ) = ˜ C aim . The function ˜ C ( α ) is a monotonically increasing function of α , see fig. (2). Since the function ˜ C ( α ) -˜ C aim often happens to be a quickly changing function of α (especially while it is changing its sign), the solution for α needs to be found to a high tolerance level. θ", "pages": [ 10, 11, 12, 13 ] }, { "title": "3.4 Stopping criterion", "content": "In solving the constrained optimization problem, one fact which becomes important is the following: at the point at which the constrained optimization problem gets solved, the gradient vectors of the two functions become parallel. Thus, if we find the unit vector in the direction of gradient of entropy and in the direction of gradient of chi squared function, the dot product of these two unit vectors (defined using the entropy metric) must become negligible as we head towards the point at which the constrained optimization problem gets solved. The unit vectors in the directions of gradients are (with L s and L c defined previously) We thus expect the angle to become too small (compared to a unit radian) as the algorithm proceeds (Fig(3)).", "pages": [ 13 ] }, { "title": "4 Recovering Primordial Power Spectrum", "content": "In this section, we shall (i) test the formalism presented in the previous section by trying to recover a featureless as well as feature-full sPPS from simulated noisy CMB data and (ii) apply the algorithm to actual WMAP 7 year binned TT angular power spectrum [1] to recover the Primordial Power Spectrum. Thus, to begin with, we shall find out the radiation transfer function for the simplest set of assumptions, inject a featureless sPPS and get noise-free C TT /lscript (which we shall refer to as theretical C /lscript s). Next we shall add noise to these pure C /lscript s.", "pages": [ 13 ] }, { "title": "4.1 The radiative transport kernel", "content": "First, we need to set the values of the various cosmological parameters and get the corresponding radiation transfer function. This can be done by making use of the codes such as CMBFAST [5], CAMB [6] or gTfast [7]. The results in this section are got from transfer function found using the code gTfast which itself is based on CMBFAST (version 4.0). It is important to notice that since in this work we shall only use the TT data, so, we only calculate the temperature radiation transfer function. To find out the transport kernel, we assume that the universe is spatially flat and dark enery is a cosmological constant (i.e. we have a spatially flat ΛCDM universe) and set the values of the cosmological parameters to their WMAP nine year values [1] (WMAP9 + bao + h0): the values of various parameters to be fed into the code gTfast are given in table 1. We also assume that there are no tensor perturbations to the metric. We use Peebles recombination (rather than using RECFAST) and assume that the Primordial flutuations are completely adiabatic. Finally ,we shall not correct the transfer function for lensing of CMB, SZ effect or other effects that cause secondary ansiotropies of CMB. This shall give us the radiation transfer function from which we can easily evaluate the matrix G /lscriptk . For the case we are dealing with, the matrix G /lscriptk shall have dimensions 1500 × 6200. Finally, we would like to state that the results one obtains and conclusions that one draws should better not depend on the exact values of these parameters.", "pages": [ 13, 14 ] }, { "title": "4.2 Recovering test spectra", "content": "We can now inject a test sPPS which is a power law with A S = 2 . 427 × 10 -9 , n S = 0 . 971 (with k 0 = 0 . 002Mpc -1 and T cmb = 2 . 72548 × 10 6 µ K) and get the corresponding theoretical C /lscript s, and add noise. The noise we add is dominated by cosmic variance at low /lscript (less than 600) values while for high /lscript values, the noise is dominated by instrumental errors. Fig (4) shows the result of using the algorithm described in the previous section to recover the sPPS in the present case. The following points are worth noting: it is clear that f = A = 54000 corresponds to P ( k ) being 2 × 10 -8 . Thus, this value of A corresponds to the situation in which P ( k ) = 2 × 10 -8 is the solution with the maximum value of entropy. 1 In practise, the process of masking the sky causes the various C /lscript s to get correlated. The simplest situation in which we can hope to recover the sPPS is the one in which the the data points corresponding to different /lscript values are uncorrelated. This happens for the binned CMB dataset (which has data only for 45 /lscript values). To make use of the binned data, we also work with a binned kernel which is defined where N is the number of /lscript values in the bin. By using this averaged kernel and applying the algorithm to simulated binned data (with the added noise equal to the noise for WMAP 7 year binned data), we get the results shown in Fig (5) (this time we show the results for many A values). We again get an answer which at scales at which the noise is large, tends to the value of the default (i.e. A ) while at scales at which the noise is relatively low, the recovery tends to fluctuate around the featuresless injected signal. For a fixed value of A , the recovery at scales at which the noise is relatively lower shall be different if we consider different realizations of the noise. This is illutrated in Fig (6): here the recovery shall be the same at scales with no data and shall be different at scales with data. The key question is whether we can recover features in the sPPS by this method. The fact that this can be done is illustrated in Fig (7): we just introduce a bumpy feature between the scales 10 -3 MPc -1 to 10 -2 MPc -1 and vary its height and see that unless the height of the bump is too small, the algorithm can recover it. Of course if we introduce a feature at a scale at which the data is not good or at which the kernel takes up negligible values, the feature shall not be recovered. Moreover, it is not surprising that the recovery is much better if the feature is more prominent.", "pages": [ 14, 15, 17 ] }, { "title": "4.3 WMAP 7 year binned CMB data", "content": "In this sub-section we apply the algorithm to actual CMB data. We use WMAP 7 year binned TT dataset and use it to recover the sPPS. The result is shown in fig 8. The details of the recovery of course depend on the chosen value of the parameter A . In the present context, the value of A represents our a priori knowledge (without using any data) of how much we think should be the scalar fluctuation in the metric in the early universe. 1 It may appear that if the conclusion depends on such an a priori knowledge, we may not get anything worthwhile. But the following fact is worth noting: it is seen that at scales at which the noise is lesser, even though the recovered P ( k ) depends on the value of A chosen, this dependence is quite weak and quite predictable (as we increase A a lot or decrease it a lot, the recovery just 'stretches' in the P ( k ) direction in the ln P -ln k plane). An interesting exercise is this: if, without using the CMB data, we still knew that the amplitude of the scalar metric perturbations is (roughly) A s , then what can this method of deconvolution tell us about the sPPS? Fig (9) illustrates how the red tilt of the PPS can be detected in such a case. One can keep on decreasing the value of A and see what happens. In this context, the case of A = 1 is very interesting since this corresponds to using another familiar definition of entropy, the recovery for this case is illustrated in Fig (10). What is interesting is that if we choose A to be too small, we begin to get an IR cut-off not very different from the one reported in the literature previously (see [17]), but, we also get an apparent UV cut-off. Moreover, such a small value of A causes the artifical features to get stretched so much that we may not consider the reconstruction to be trustworthy in this case. 1", "pages": [ 18, 19 ] }, { "title": "5 Summary and discussion", "content": "In this work, we attempted to probe the amplitude and shape of scalar primordial power spectrum (sPPS) using the CMB data. We fixed the values of various cosmological parameters (apart from the ones specifying the sPPS itself) and formulated the problem as an inverse problem. To solve the inverse problem, we use the maximum entropy method which is a non linear regularization method. There exist many possible ways to employ the maximum entropy regularization, we use a particular definition of entropy and a particular algorithm to solve the corresponding constrained non-linear optimization problem in a very large dimensional parameter space. The way we have formulated the problem, there exists a parameter (which we called A ) whose value decides the location of global maximum of entropy in the space of all P ( k )s. In the absence of any data, the algorithm shall just send every initial guess to the global maximum of entropy. Even in the presence of data, the following is worth noting", "pages": [ 19 ] }, { "title": "1. at scales where", "content": "the P ( k ) recovered by MEM depends on the value of A chosen (as P ( k ) = A is the ME solution), while at the scales where the data is good, we recover something which has comparatively lesser dependence on what A we choose. existence of any small deviations from power law behaviour can not be completely ruled out, this analysis reinforces our belief that any such possible deviations must be small. This is by no means the last word on the existence of features in sPPS, this is not even the last word on the use of MEM for this purpose.The implementation of our algorithm to this problem till now does not seem to give any reason to believe that there are any serious deviations from the power law. We would like to mention that this is not completely unexpected, even in the light of existing papers such as [17] because the error bars at scales at which the features were recovered in those works are very large: the maximum entropy method can not claim any features at scales where the error bars are so large 9 . This analysis shows that at scales at which the CMB data is trustworthy, the Primordial Power Spectrum of scalar metric perturbations is, to a very good approximation, a power law . In future, one can look at the follwing prosoects. We should be able to solve this problem of possible existence of features in sPPS without assuming the values of other cosmological parameters (i.e. without formulating this problem as a simple inversion problem). Even in the present formulation, there may be ways of combining results from different values of A to get a better recovery. One may wish to use the actual WMAP likelihood (or rather, the corresponding χ 2 eff ) as a measure of misfit, but this is not easy in the way we have attempted to solve the problem (we need to know the χ 2 and its first two derivatives). Also, we have lost a lot of information in the process of binning the kernel and working with the binned, uncorrelated data. We would like to use all that lost information. Similarly, we have only used the TT angular power spectrum of CMB, we would also like to use the polarization spectra to probe the sPPS. We may also need to post-process the recovered sPPS to get more useful information. Another interesting possibility worth exploring is the connection of Maximum Entropy deconvolution with other ways of deconvolution (e.g. Richard Lucy deconvolution).", "pages": [ 19, 20 ] }, { "title": "A Testing the method", "content": "The main text described the material necessary to employ the maximum entropy inversion in any circumstance. The following points need to be noted (these are just tried and tested facts about the algorithm, many of which are illustrated here for the case of a toy problem shown in Fig(11), whose solution is given in Fig (12)): bin constrained optimization problem in the subspace (for ˜ Q defined by Eq (48)) shall become too large. In this situation, it may be difficult to numerically find any solution for α . dimensional f -space while as we change A , we define the line f i = A as being the location of global maximum of entropy for these different values of A . This will of course mean that as we change A , the place where the entropy is maximum on the χ 2 = C aim surface shall also change. Thus, as we continuously change A , we shall get a family of recoveries (see fig (13)). So, the details of the recovered answer depends on the chosen value of this free (or adjustable) parameter. But since the global maximum of entropy is at f i = A , the value of A represents the 'background' (i.e.a priori) knowledge of how much the power in various bins is, without using any knowledge of data at all. Acknowledgment: The authors acknowledge the use of WMAP data and the use of codes such as CMBFAST, gTfast and CAMB. The authors would also like to thank Tarun Souradeep (IUCAA, Pune) for reading through the manuscript and giving useful comments. GG thanks Rajaram Nityananda (NCRA, Pune), Mihir Arjunwadkar (CMS, Pune university, Pune), Abhilash Mishra (CALTECH, Pasadena) and Ranjeev Misra (IUCAA, Pune) for discussions at various stages of the work. GG thanks Council of Scientific and Industrial Research (CSIR), India, for the research grant award No. 102(5)/2006(ii)-EU II. JP acknowledge support from the Swarnajayanti Fellowship, DST, India (awarded to Prof. Tarun Douradeep, IUCAA, Pune, India).", "pages": [ 20, 21, 22, 23 ] } ]

[ { "title": "ABSTRACT", "content": "We construct a 'Chern-Simons-like' action for N = 1 Topologically Massive Supergravity from the Chern-Simons actions of N = 1 Supergravity and Conformal Supergravity. We convert this action into Hamiltonian form and use this to demonstrate that the theory propagates a single massive ( 2 , 3 2 ) supermultiplet.", "pages": [ 1 ] }, { "title": "Alasdair Routh 1", "content": "1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K. email: [email protected]", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Theories of gravity in spacetimes of three dimensions (3D) have been studied extensively over the past few decades, both as tools for understanding gravity in four or more dimensions and for their own intrinsic interest. It is well-known that in 3D, massless spin-2 particles have no local degrees of freedom, and correspondingly 3D General Relativity (GR) is 'trivial'. One thing this suggests is that studying 3D GR and its quantisation could be a helpful tool on the way to understanding the quantisation of 4D GR, see for example [1, 2]. It also suggests that in order to find a 'non-trivial' 3D gravity theory, one should look at models of interacting massive spin-2 particles. Massive gravity is of general interest [3] and understanding the situation in 3D could help the development of more complicated 4D models. In 3D a massive spin-2 particle has one local degree of freedom, and the first theory of such a particle to be discovered was Topologically Massive Gravity (TMG) [4]. This theory breaks parity, as a parity invariant theory must have two massive spin-2 particles of opposite helicities. Recently, this has been realised by another massive gravity model, New Massive Gravity (NMG) [5,6], which propagates two modes of the same mass and opposite helicities and is parity invariant. The combination of the two models is called Generalised Massive Gravity (GMG) which propagates two modes of different masses and opposite helicities, and TMG and NMG can be realised as limits of this more general model. The above discussion generalises to supergravity. The 3D massless super-multiplet containing a spin-2 particle as its highest spin state also has no local degrees of freedom, and so like its bosonic counterpart, 3D supergravity is 'trivial'. Supersymmetric counterparts of TMG, NMG and GMG have been found [7-9], and in this paper we will be particularly interested in the N = 1 supersymmetric extension of TMG, called N = 1 Topologically Massive Supergravity (TMSG) first described in [10]. 3D General Relativity [1, 11] and Conformal Gravity [12] can both be formulated as Chern-Simons theories of the 3D Poincar'e Group (or A/dS Group if a cosmological constant is included) and Conformal Group respectively. In this form, their actions are integrals of 3-form Lagrangians constructed from exterior products of 1-forms and exterior derivatives. These 1-forms are the dreibein, spin-connection and in the case of Conformal Gravity extra fields corresponding to special conformal transformations and dilatations. TMG was originally formulated by adding together the action of 3D GR and an alternative gauge-fixed second-order action for Conformal Gravity, but it was recently noticed that one could get the same theory by combining the two Chern-Simons actions [13], putting TMG into 'Chern-Simons-like' form. By this we mean that it is similar to a Chern-Simons theory in that it is described by an action which is the integral of a 3-form Lagrangian constructed from 1-form fields and exterior derivatives. However we do not require that the action arises from a group structure as in an actual Chern-Simons theory. Such 'Chern-Simons-like' actions are worth considering for their interesting and useful properties. They are constructed without the use of a metric, they are relatively simple to work with, and importantly they are first-order. This last point relates to the fact that we treat the spin connection as an independent variable, to be determined by the field equations. Being first-order makes these actions very easy to put into Hamiltonian form. As any term can only have one time-derivative, after a time/space decomposition the actions are automatically in the form ' P · ˙ X -λ i C i '. The spacelike components of the 1-form fields can be interpreted as canonical variables X,P while the time components of the fields are always non-dynamical and act as Lagrange multipliers λ i imposing various constraints C i . We can use the Hamiltonian analysis described by Dirac in [14] to find any additional constraints which must be included. The ' P · ˙ X ' term defines Poisson brackets between the dynamical variables, and using these elementary Poisson brackets, the Poisson brackets of the constraints can be computed. We can then distinguish socalled first-class and second-class constraints and hence determine the number of physical modes the theory propagates. In [13], 'Chern-Simons-like' actions for TMG, NMG and GMG were found and their constraint structures analysed as described above, agreeing with previous results in [1521]. The aim of this paper is to extend that work to three different N = 1 supergravity theories. We will use a component approach, but superspace methods for Conformal Supergravity and TMSG have recently been studied in [22]. We will first review N = 1 Supergravity and Conformal Supergravity in their ChernSimons forms. The Supergravity action is well-known, and while the first-order ChernSimons version of the Conformal Supergravity action has almost been constructed several times in the literature [23-25], the explicit expression given here is perhaps novel. We will then use these theories to construct a 'Chern-Simons-like' version of TMSG and finally transfer each supergravity theory into Hamiltonian form and analyse the constraint structure to determine the number of degrees of freedom each has.", "pages": [ 2, 3 ] }, { "title": "2 The Supergravity Theories", "content": "We will begin our analysis by presenting the Chern-Simons forms of the 3D Supergravity and Conformal Supergravity actions from which we construct TMSG, which will allow us to introduce our conventions. We will then discuss TMSG itself. For a more thorough discussion of the corresponding bosonic theories using the same formulation and conventions, see [13], of which this paper is an extension 1 .", "pages": [ 3 ] }, { "title": "2.1 Supergravity", "content": "Let us first recall the Einstein-Cartan formulation of 3D gravity. This model uses the dreibein e a = e µ a dx µ , a Lorentz-vector valued one-form which gives rise to the metric g µν = η ab e µ a e ν b , and the spin connection ω µ a = 1 2 /epsilon1 abc ω µbc , another 1-form which in three dimensions can be dualised as shown. We also dualise the Riemann tensor R a = 1 2 /epsilon1 abc R bc = dω a + 1 2 /epsilon1 abc ω b ω c , and define the covariant derivative of a Lorentz-vector valued one-form, Dh a = dh a + /epsilon1 abc ω b h c . We implicitly take the wedge product of adjacent forms. In this notation the standard Einstein-Hilbert action becomes the integral of the Lagrangian three-form This is a 'first-order' formulation, we consider the spin connection to be an independent variable rather than a fixed function of the dreibein. Varying both of the forms, we get the following Euler-Lagrange equations corresponding to ( e, ω ) respectively De a is the torsion which the second equation sets to be zero. This determines the spin connection to be the usual one, which we will call ω ( e ). The first equation is then equivalent to the Einstein equation, G µν +Λ g µν = 0. Note that this form of the action for GR is first order in time derivatives, which will be important when we look at the Hamiltonian form later. We can extend this action to a supergravity action by introducing a gravitino, an anti-commuting Majorana spinor valued 1-form, ψ µ α . Define the covariant derivative of a spinor valued one-form, Dχ = dχ + 1 2 ω a γ a χ where the γ a satisfy { γ a , γ b } = 2 η ab with the ( -++) convention, and whenever a representation of the γ a is called for we will use a real representation defined in terms of the Pauli matrices Our supergravity action is The cosmological constant is Λ = -λ 2 , non-positive in supergravity. Again vary each form to get the Euler-Lagrange equations corresponding to ( e, ψ, ω ) The third equation is the usual supergravity torsion condition, which defines the spin connection in terms of the dreibein and the gravitino, we will denote this spin connection ω ( e, ψ ). The first and second equations are then the standard trivial equations of motion for the graviton and gravitino in 3D.", "pages": [ 3, 4 ] }, { "title": "2.2 Conformal Supergravity", "content": "Conformal Supergravity can be constructed as the Chern-Simons theory of the 3D N = 1 superconformal algebra, Osp (1 | 4) [23-25], just as Conformal Gravity has been constructed as the Chern-Simons theory of the conformal group [12]. As a Chern-Simons theory, Conformal Supergravity has a field corresponding to each transformation generator in the N = 1 superconformal algebra. The translations and Lorentz rotations give rise to e a and ω a respectively, and the supersymmetry transformations correspond to ψ . There are also special conformal transformations, conformal supersymmetry transformations and dilatations, so we must introduce new 1-form fields f a , φ (an anti-commuting Majorana spinor) and b associated to these. To construct the action, we have followed the working of [24], and explicitly written out their expression (3.1) in terms of the basic forms - - To find the equations of motion, first eliminate every variable except e a and ψ using their equations of motion Using the Bianchi identity DDe a = /epsilon1 abc R b e c , one can see that the ω and φ equations imply the b equation. As with Conformal Gravity, this is because b can be gauged away by special conformal transformations, so we can set it to be zero as it must drop out of the final equations of motion. The φ equation determines φ ( e, ψ ) and then the ω equation defines f a ( e, ψ ) where e is the determinant of the dreibein, R µ = /epsilon1 µνρ D ν ψ ρ and S µν ( e ) ≡ R µν -1 4 Rg µν is the 3D Schouten tensor. The e a and ψ equations then give the equations of motion for the bosonic and fermionic modes. This action is of course invariant under the superconformal algebra, the action of which is shown in the table above.", "pages": [ 4, 5, 6 ] }, { "title": "2.3 Topologically Massive Supergravity", "content": "Just as the TMG Lagrangian can be constructed as -1 × the Einstein-Cartan Lagrangian + 1 µ × the Conformal Gravity Lagrangian [13], TMSG can be constructed as -1 × the Supergravity Lagrangian + 1 µ × the Conformal Supergravity Lagrangian. This 'ChernSimons-like' action, which we will demonstrate is equivalent to the TMSG action given in [10], is Now consider the relation between the Euler-Lagrange equations of this theory and those of Chern-Simons Conformal Supergravity above. The TMSG equations are which differ from the corresponding Conformal Supergravity equations only by the extra term on the right hand side of the ω equation. Notice that the ω equation which determines f a ( e, ψ, φ, b ) is linear in f a , so write f a = f a 0 + f a 1 , where f a 0 is the solution of the Conformal Supergravity equations above. Then the ω and b equations become The first equation is equivalent to while the second equation says that the left hand side of this equation vanishes, implying b = 0. The first equation then sets f a 1 = 0. This implies that f a is the same as in Conformal Supergravity, and we can determine that φ is as well. Looking ahead to the Hamiltonian formulation, after choosing a time co-ordinate on our space time, the b i will be dynamical variables, and the equation b = 0 indicates the presence of constraints on them. As in [13], it will make later calculations simpler if we have as few constraints as possible, so we can consider setting b µ dx µ = b 0 dx 0 in the Lagrangian. Although b i = 0 is a consequence of the equations of motion, this change may in principle alter the theory as it eliminates the two equations of motion of b i , so we must check that the equations of motion are unaltered. Our previous analysis holds until the step where we determine f a 1 = 0 and b = 0. The single b 0 equation becomes and the ω equation becomes which implies that b 0 = 0 and thus f a 1 = 0. This was the only place we used the b equation, so the final equations of motion do not change, and we may set b i = 0 in the Lagrangian without changing the theory. Now we have expressions for all variables in terms of e a and ψ , the Euler-Lagrange equations for these two variables then give us the equations of motion for Topologically Massive Supergravity, the former being the Topologically Massive Gravity equation with extra fermion terms and the latter an equation for ψ . If we substitute the expressions for φ and f a into the Lagrangian to make it depend on e a and ψ only, it becomes where ω µ a = ω µ a ( e, ψ ) implicitly. This is the same as the usual result [10,26,27] with a cosmological constant 2 .", "pages": [ 6, 7 ] }, { "title": "3 Hamiltonian Formulation", "content": "Now that we have first order Lagrangians for our theories, we can easily convert them into Hamiltonian form and then perform an analysis of the Poisson brackets structure of the constraints to determine the number of degrees of freedom of each theory [14]. We will introduce the necessary concepts by working through the simpler theories first. The analysis for the purely bosonic theories was performed in [13].", "pages": [ 7 ] }, { "title": "3.1 Supergravity", "content": "Consider the theories given by integrating each Lagrangian 3-form we have discussed over a 3-manifold with a Cauchy hypersurface. We will assume the spacetime can be foliated by spacelike surfaces indexed by a time t such that we can decompose our forms as, for example, e µ a dx µ = e 0 a dt + e i a dξ i . We transfer our Lagrangian 3-forms into the usual Lagrangians by L = L dx 0 dx 1 dx 2 , getting expressions which are first order in time derivatives for each of our theories. Our first example, Supergravity, has Lagrangian To make the necessary calculations easier, rescale the fields as Decompose the spacetime directions into time and spacelike directions The fields e 0 a , ω 0 a and ψ 0 can be seen to be non-dynamical, and act as Lagrange multipliers imposing the constraints C a e , C a ω and C ψ respectively, defined as This Lagrangian is now in Hamiltonian form, by which we mean that it is a symplectic term minus a Hamiltonian. The Hamiltonian of a time reparametrisation invariant theory like all those we are working with must vanish, so takes the form of a collection of constraints. The symplectic term tells us the Poisson brackets of the theory Note that the Poisson brackets of commuting variables are anti-symmetric while the Poisson brackets of anti-commuting variables are symmetric. To work out the number of degrees of freedom the theory has, we must calculated the matrix of Poisson brackets of constraints, where x, y label the fields, e, ω, ... and A, B a Lorentz index for e, ω, f a spinor index for ψ, φ and is absent for b . We evaluate this matrix on the constraint surface, where all constraints are set to 0. The rank of this matrix will then be the number of second class constraints in the theory, the rest being first class. We use the formula 'dimension of physical phase space = dimension of phase space - (2 × number of first class constraints) - (1 × number of second class constraints)' to work out the number of physical modes, which is half the dimension of the physical phase space as in [13]. To compute the Poisson brackets of the constraints, we integrate them against test functions, arbitrary smooth functions with compact support so all surface terms vanish and find The algebra of Poisson brackets closes, equivalently all Poisson brackets are zero when evaluated on the constraint surface, so all the constraints of the theory are first class. The dimension of the phase space is 16, as e i a and ω i a have 6 components each and ψ i has 4, and there are 8 constraints as e a 0 and ω a 0 have 3 components each and ψ 0 has 2. The physical phase space therefore has dimension 16 -(2 × 8) -(1 × 0) = 0; as expected there are no propagating modes.", "pages": [ 8, 9 ] }, { "title": "3.2 Conformal Supergravity", "content": "Now we do the same thing for Conformal Supergravity, which has Lagrangian It will again be convenient to rescale some of the fields to simplify the calculation; change The Lagrangian then decomposes into where the constraints are We can compute the Poisson brackets as well, which have been conveniently normalised by the rescaling done earlier Again use the test function method to calculate the Poisson brackets of the constraints The algebra closes again, so all constraints are first class and Conformal Supergravity has 28 -(2 × 14) -(1 × 0) = 0 propagating modes, as expected.", "pages": [ 9, 10, 11 ] }, { "title": "3.3 Topologically Massive Supergravity", "content": "The Topologically Massive Supergravity action, setting b i = 0 as we have earlier checked does not alter the theory, is The Poisson bracket structure is not affected by an overall factor in the Lagrangian so consider the Lagrangian µ L , the case µ = 0 is Conformal Supergravity which we have already dealt with. Redefine fields and then drop the tildes we have introduced for convenience. After rewriting everything in terms of these new variables and decomposing the spacetime directions, the Lagrangian becomes where the constraints are, writing l = λ µ The complicated field redefinitions earlier were designed to simplify the Poisson brackets as much as possible We can now work out the Poisson brackets of the constraints. As we are interested in whether the constraints are first or second class, for brevity we shall ignore all multiples of constraints which appear on the right hand side of these equations, equivalently we shall work out the Poisson brackets evaluated on the constraint surface. σα j a ¯ ψ i γ χ j , a We want to work out the rank of the 14 × 14 matrix of Poisson brackets P AB xy . Compared to a purely bosonic theory, there is potentially an additional complication as this matrix is a supermatrix. We are trying to find the number of independent second class constraints, which is the number of linearly independent rows or columns of the matrix. A set of vectors { v i } is linearly independent if λ i v i = 0 ⇒ λ i = 0. If we allow the { v i } to be Grassmann odd, we can use the same definition as long as we also allow the { λ i } to be Grassmann odd, and then everything procedes as in the bosonic case. To minimise the amount of calculation necessary, first notice that since the three C a Ω commute with everything, as in [13] we can separate these conditions out, using them to pick the local frame e 1 a = (0 , 1 , 0), e 2 a = (0 , 0 , 1) and reducing the problem to finding the rank of the 11 × 11 submatrix, ˆ P formed by removing the Ω rows and columns. Next, we compute and that all other Poisson brackets with C 0 k vanish. The b column is therefore linearly independent of the other columns, and similarly the b row is linearly independent of the other rows. Let the 10 × 10 submatrix formed by removing the b row and column of ˆ P be denoted Q . Reorder the rows and columns so Now recall that the rank of a matrix, the dimension of its column space col ( M ) and the dimension of its row space row ( M ) are all equal. By linear independence of the b row and column The problem therefore reduces to calculating the rank of Q . We have already fixed e i a , and Ω i a and b do not appear in Q , so the matrix is composed of the 6 elements of k i a and the 4 elements of each of ψ i α and χ i α . After writing the matrix out in terms of these variables, the row space of the matrix can be seen to be spanned by the k 1 0 , k 2 0 , ψ 1 0 and ψ 2 0 rows. More explicitly, first subtract appropriate combinations of the k 1 0 and k 2 0 rows from all the others in order to remove all elements of k i a from them, the resulting matrix then has rank 4 by inspection. Therefore, Q has rank 4, ˆ P has rank 6 and the original matrix, P , also has rank 6. TMSG therefore has a 26 -(2 × 8) -(1 × 6) = 4 dimensional physical phase space, or equivalently 2 propagating modes. We know that TMSG is supersymmetric [10] so this is a massive spin ( 2 , 3 2 ) supermultiplet. Note that this is independent of the value of λ .", "pages": [ 11, 12, 13, 14, 15 ] }, { "title": "4 Discussion", "content": "We have constructed a first-order Chern-Simons action for Conformal Supergravity, and then combined this with the Chern-Simons Supergravity action to get a 'Chern-Simonslike' action for TMSG which is equivalent to the existing formulation. This new action has a number of nice properties, being first-order, metric independent and convenient to work with. We slightly modified this action by setting b i = 0 which we showed does not change the dynamics, a step performed for all dynamically non-trivial 'Chern-Simonslike' actions so far studied here and in [13]. Using this action we have then shown that TMSG has two propagating modes, which must be a spin ( 2 , 3 2 ) supermultiplet, while the Chern-Simons theories it is constructed from of course have no local degrees of freedom. It would be interesting to extend these results to other similar models of massive 3D supergravity [7-9]. It was not obvious that a 'Chern-Simons-like' action for TMSG had to exist, and the existence of 'Chern-Simons-like' actions for other massive 3D supergravity theories is similarly unclear. One complication is that supersymmetry acts differently on each of the Chern-Simons actions composing TMSG. That the final action is supersymmetric is shown by equivalence to an existing supersymmetric model. One possible extension would be to N = 1 supersymmetric versions of NMG and GMG, a supersymmetrisation of the bosonic 'Chern-Simons-like' models presented in [13]. Such a theory would involve an extra Lorentz vector 1-form, k a , and one would have to supersymmetrise a term One could begin by defining a superpartner χ to h a , then the first term could be supersymmetrised as the usual Einstein-Hilbert term is by adding 2¯ χDχ . However beyond this the large number of possible extra terms, as well as the existence in the bosonic version of an additional Lorentz scalar 1-form [13], and the problem of establishing that the final action is supersymmetric make this a complicated task. Another interesting extension would be to N = 2 TMSG. The Supergravity action would then contain a one-form A corresponding to R-Symmetry and perhaps a one-form C corresponding to a central charge, and the Conformal Supergravity action would gain a term AdA [28, 29]. The theory would be expected to propagate a spin ( 2 , 3 2 , 3 2 , 1 ) supermultiplet, the spin-1 mode being formed from A or C , but it is difficult to see how any 3-form terms of the type needed for the action to be 'Chern-Simons-like' could give rise to the required Maxwell action for A or C . Acknowledgements I would like to thank Paul K. Townsend, at whose suggestion and under whose supervision this work was carried out. I am supported by the STFC.", "pages": [ 15, 16 ] }, { "title": "References", "content": "[29] P. S. Howe, J. M. Izquierdo, G. Papadopoulos and P. K. Townsend, 'New supergravities with central charges and Killing spinors in 2+1 dimensions,' Nucl. Phys. B 467 (1996) 183", "pages": [ 18 ] } ]

[ { "title": "Causal loop in the theory of relative locality", "content": "Lin-Qing Chen Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada We find that relative locality, a recently proposed Planck-scale deformation of special relativity, suffers from the existence of causal loops. A simple and general construction of such on-shell loop processes is studied. We then show that even in one of the weakest deformations of the Poincar'e group in relative locality, causality can be violated. The search for quantum gravity has led to the idea that special relativity should be modified at high energies in such a way that (modified) Lorentz transformations leave the Planck scale invariant. Doubly special relativity (DSR) has been proposed as an embodiment of this idea [1, 2]. However, it was shown in [3] that this theory has observer dependent non-localities for distant interactions. Relative locality is a reincarnation of DSR that tries to clarify the issue by proposing a radically different way of thinking about physics. The birth of relative locality came from the insight that we never directly observe the spacetime we postulate. Our usual picture of spacetime is constructed operationally from the measurements of energies, momenta and times of events[4, 5]. The notions that everything shares a universal spacetime and that locality has absolute observer-independent meaning could be unwarranted assumptions. As a theory describing quantum gravity induced modifications to relativistic dynamics of particles, relative locality was proposed at the regime where glyph[planckover2pi1] → 0 , G → 0, while their ratio m P = √ glyph[planckover2pi1] /G is held fixed for every observer[4, 5]. So effects due to the presence of the Planck mass are expected. Some RL phenomology has been studied in [4-7] Relative locality takes momentum space P as primary and formulates classical dynamics on the phase space T ∗ ( P )[4]. The geometry of momentum space is not preassumed to be that of a linear space but could have curvature, torsion and non-metricity in general, and should be tested by experiments. There is no global projection that gives a description of processes in a universal spacetime. The notion of absolute locality and universal spacetime is equivalent with the assumption that the conservation law of momenta is linear. [4]. We start with giving a brief review of the classical dynamics and phase space structure of relative locality. After defining the causal structure in RL, we go on to show that the theory has solutions that are causal loops. The general conditions allowing for such loops to happen are then studied. We illustrate this construction when the geometry of momentum space is taken to be that of κ -Poincar'e, which is the most well studied and the simplest non-trivial geometry of momentum space in RL [7, 8]. The appearance of the causal loops is a result of the phase space structure of the theory and the causal loops vanish in the limit of special relativity.The existence of causal loops implies that it would be non-trivial to construct a quantum theory of RL with unitarity, a worry which should be addressed in future research. If RL were shown to be a well-tested theory in the future, causal loops will challenge our understanding of the fundamental role of causality. More probably, this result implies that RL is incomplete or wrong.", "pages": [ 1 ] }, { "title": "I. CLASSICAL PARTICLE DYNAMICS AND PHASE SPACE IN RELATIVE LOCALITY", "content": "Momentum space is assumed to be a manifold P with a metric g ab and a connection Γ ab c . The geodesic distance D ( p ) from the origin to a point p ∈ P is interpreted as the rest mass of a particle with momentum p . The nonlinear addition rule of momenta p ⊕ q , which should be found experimentally, determines the connection: Introduce an inverse operation glyph[circleminus] satisfying ( glyph[circleminus] p ) ⊕ p = 0 which turns incoming momenta into outgoing momenta. Now we can write the conservation law associated with any interaction vertex as a non-linear equation K a ( p I ) ≡ 0. For example, a three vertex can be written as ( p ⊕ q ) glyph[circleminus] k = 0. The order of addition is important, since it corresponds to micro-causal structure of interactions[4]. The torsion of momentum space is a measure of the noncommutativity and the curvature is a measure of nonassociativity of addition rule of momenta.[4, 6, 8] Dynamics of point particles is defined by the action: where s is an affine (time) parameter along the trajectory of the particle and an interaction labeled by i happens at s i for each particle; x J are Hamiltonian spacetime coordinates which are defined as being canonically conjugate to p J a : { x a I , p J b } = δ a b δ J I and x a J ∈ T ∗ p J . The mass shell condition C J ( p ) ≡ D 2 ( p J ) -m 2 J is imposed by the Lagrange multiplier N J . The interaction part of the action is a Lagrange multiplier times the conservation of momenta K a ( p 1 , p 2 ... ) ≡ 0. By varying the action, we get the equations of motion, the two of which we will be concerned with: ± indicates an incoming/outgoing particle respectively. Equation (3) tells us how free particles propagate on one cotangent space ('Hamiltonian spacetime' T ∗ p ) of their phase space; and equation (4) describes how cotangent spaces T ∗ p of different particles are connected by interaction events z . Physics should be invariant under momentum space diffeomorphisms, i.e. redefinition of coordinates on the momentum manifold. Such a transformation is given by: p a → ˜ p a = f a ( p ) s.t. the geodesic distance between the origin and p is preserved. Under this transformation, K ( p ) → ˜ K = K ( f ( p )), x a transforms like a covector: x a → ˜ x a = x b ( ∂f a ( p ) /∂p b ) -1 ∈ T ∗ p , z a → ˜ z a = z c ( δ ˜ K a /δ K c ) -1 . By applying these transformations to our action we see that it is unaltered, as desired.", "pages": [ 1, 2 ] }, { "title": "II. CAUSAL LOOP PROCESS", "content": "In relativity the causal relationships between events are expressed as geometrical relationships between points on the spacetime manifold, while in RL we do not have a universal spacetime. We thus have to come back to the most fundamental notion of causality between events. Define event B to be in the causal past of event A if in the process that is being considered, there exists a sequence of events B,B 1 , ...B n , A, n ≥ 0 s.t. from each event there exists an outgoing free-propagating particle coming to the next event. We write it as B ≺ A . Vice versa we can define causal future C glyph[follows] A . The above definition of causal relationship is actually a strict partial order. In analogy with the notion in general relativity, one type of causality violation is if ∃ events A,B s.t. A ≺ B,B ≺ A i.e. causal loop. Let us check the existence of the simplest causal loop. Assume two bi-particle collision events A and B , defined by conservation laws K A ≡ 0 and K B ≡ 0. Particle 0 with momentum p 0 is created from event A and then collides with another particle at event B . The twist is now to consider particle 1 with momentum p 1 created at event B and then colliding with another particle at event A , which creates the particle 0. We use x 0 A and x 0 B to label the starting point and ending point of particle p 0 's worldline which lives on T ∗ p 0 ; and similarly x 1 B and x 1 A for particle p 1 , as in Fig[1]. T P T P Particle 0 freely proporgates from event A to B for proper time τ 0 on T ∗ p 0 ; the end of its worldline is related with the starting point of particle 1's worldline by interaction B: Similarly, the other part of the loop is described by where u 0 , u 1 are given by equation of motion (3). Thus the existence of a causal loop process is equivalent to the following equation having a physical solution for x µ 0 A and proper times τ 0 , τ 1 : The above is a general condition. Once we specify the geometry of momentum space and write down the conservation laws of two vertices, the above condition (7) will then give a system of four linear equations with six free unknowns: x µ 0 A , τ 0 and τ 1 . Based on the Cramer's rule, if the matrix ( M A -M B ) has rank four, i.e. the determinant is non-zero, any physical choice of τ 0 and τ 1 yields a unique solution for x µ 0 A . If the matrix ( M A -M B ) has rank less than four, we can fix a ratio τ 1 = βτ 2 , β ∈ R + first and then check the number of remaining unknowns of the system of the homogeneous equations. Assuming the number of unknowns is d ( d ≤ 5 now), if ∃ β s.t. the new matrix of coefficients of the system of equations has rank < d , then we can always get at least one physical solution for τ 1 > 0. When for ∀ β ∈ R + the new coefficients matrix has a rank ≥ d , only then the equation does not have physical solutions, which means the causal loop (corresponding to these specific interaction vertices) does not occur. The solution will be invariant under the momentum space diffeomorphsim, because the equation (7) is just based on the equations of motion, which are invariant. In the special relativity limit of RL, beacuse of the linearity of momentum space, which also implies trivial isomorphism between cotangent spaces, the transport operators M A , M B are all identity matrices. This makes the above equation (7) immediately degenerate to τ 0 u a 0 + τ 1 u a 1 = 0, which doesn't have physical solution. It means that this kind of causal loop will not occur in special relativity. The causal loop formed by two interaction events is just the simplest case. Similar processes can be constructed by other forms of vertices and more events: A ≺ B , B ≺ ...N , N ≺ A , which conditions enjoy the similar form with the simplest case: These compose a system of four linear equations with 4+ n unknowns, where n is the number of interactions that form the causal loop. Compared with eq. (7), the above conditions are even easier to be satisfied, since there are more unknowns in the same number of equations. More general causal loops that contain branching out can be always decomposed into a few simple loops described by the above process.", "pages": [ 2, 3 ] }, { "title": "III. TWO-EVENT CAUSAL LOOP IN κ -POINCAR ' E MOMENTUM SPACE", "content": "In this section we will illustrate the existence of causal loops in a specific geometry of momentum space. κ -Poincar'e Hopf algebra, a dimensionful deformation of the Poincar'e group, describes a momentum space with de Sitter metric, torsion and nonmetricity, which is the first well-studied example of the non-trivial geometry of momentum space in relative locality [9], [8], [7]. It is also one of the weakest deformation of Poincar'e group as a Hopf algebra[9]. The line element of the κ -Poincar'e momentum space in comoving coordinates is given by: where κ is a large energy scale close to Planck energy. The mass-shell condition is given by the geodesic distance from point p to the origin of momentum space[8]: from which we can get e.o.m.(3). The addition rule of momenta on κ -Poincar'e momentum space is as follows[8]: As an example of the simplest causal loop, we look at two events AB , which non-linear conservation of momenta are K A = ( k ⊕ p 1 ) glyph[circleminus] ( p 0 ⊕ l ) ≡ 0 , K B = ( p 0 ⊕ q ) glyph[circleminus] ( r ⊕ p 1 ) ≡ 0, see fig. 2 . The conservation of momenta can be writen out explicitly as (12). Now, we have all the necessary elements to calculate the condition (7) for this specific example. It turns out that the matrix ( M A -M B ) has rank three and an unknown x 0 0 A drops out, which is due to the linear addition rule of the zero component of momenta, in (12). For simplicity, assume that the particles with momentum p 0 , p 1 have same rest mass m . Using τ 0 to rescale the other unknowns and solve the above equation for τ 1 /τ 0 and x i /τ 0 , i = 1 , 2 , 3, we get the solution, in which the ratio of two proper times is where F = e k 0 /κ -e r 0 /κ just for shorthand. Thus as long as there exists physical values of momenta such that the solution for x i 0 A /τ 0 is finite and τ 1 /τ 0 is positive, this specific causal loop can occur. We check it as follows. For simplicity and without lose of generality, we set the last two spatial components for all the momenta to be zero. Because the energy of single particle has to be smaller than κ , energies p 0 0 , p 1 0 , k 0 , r 0 ∈ (0 , κ ). The requirement of timelike on-shell momenta m ( p ) > 0 constrains the spatial components, We can then plot the region in terms of ( p 1 0 , p 1 1 ) that allow the equation to have physical solution for x and τ 1 , τ 2 under a specific relationship among other momenta. See Fig.3 k Surprisingly, for any low energy scale of momenta choice (as long as κ is not infinity), there always exist solutions for positive τ s and finite x 0 A . At low energy, the solution of x oA in eq. (7) is proportional to κ/m : thus when the scale κ is much larger than the scale of momenta, x 0 A has quite large scale compared with τ 0 . At high energy, the scales of solutions are reasonable. Two random examples taken from two different energy scales are shown in Table (I). If the current form of relative locality were to be correct, then we would expect causal loops to be common at high energy.", "pages": [ 3 ] }, { "title": "IV. DISCUSSION AND CONCLUSIONS", "content": "We have shown that relative locality allows causal loops solutions, and it is surprisingly simple to construct them. This is a generic feature arising from the phase space structure of RL: there is no universal spacetime, and the configuration information lives on the different cotangent spaces of the momentum manifold. The nonlinear conservations of momenta at interaction events determine how the cotangent spaces T ∗ p of different particles are connected by the interaction, Eq.(4). Due to the nontrivial relations between cotangent spaces, a particle can come back to the event that causes its own creation as one of the incoming participants of that interaction. In the last section, the causal loop's existence depends on specific choices of points on the cotangent space T ∗ p (let us call it'x dependence'). In general, after fixing the proper times, the solution set of x µ oA (if it exists) is a lower dimensional subspace. In the example we calculated, it was a line on T ∗ p 0 . The same unusual feature of 'x dependence' is also present in many of the usual loop processes without causal issues in relative locality, e.g.[10]. Some nets of vertices lead to the dependence on 'where' the events are on the cotangent space, while some do not. We do not yet have fundamental reasons for choosing some forms of vertices rather than others just for the sake of 'x independence'. This necessitates future work in understanding the physical meaning of the Hamiltonian spacetimes and the choices of vertices. In [7] authors enforced the translation invariance on T ∗ p , which is a more strict symmetry than 'x independence'. However, the approach there requires the use of very non-local interaction vertices to achieve the symme- try, which is not physical. Future work should address the following questions. Is 'x dependence' of many loops a generic feature? Should translation invariance on T ∗ p be a fundamental symmetry of relative locality? How do we achieve 'x independence' or translation invariance on T ∗ p in a physical way? Can we remove the causal loops by enforcing those symmetries in the theory? Does the existence of causal loops imply that there is an inconsistency in the theory? In general relativity, Einstein equations have closed-timelike-curve (CTC) solutions. One way out of the grandfather paradox is the Novikov self-consistency principle, which states that events in CTC influence each other in a self-adjusted, cyclical way and the only solutions can occur locally are those which are globally self-consistent [11]. However, it has been shown that at the quantum level, unitarity fails for interacting fields in CTC and the subjective probabilities of events can be different for different observers [12-14]. It probably points towards that spacetimes with CTC are unphysical. In relative locality, a well-defined quantum theory has not been built yet, but there is ongoing research into constructing it. The lesson from quantum field theory in curved spacetime shows that it would be non-trivial to have causal loops in a consistent unitary theory. It is thus essential to study whether it is even possible to construct a unitary quantum field theory for relative locality. If the current form of relative locality were to be experimentally established in the future, we need to rethink causality as a fundamental property of nature. It would influence some approaches to quantum gravity which take discrete causal structure as basic assumption. Another possibility is that relative locality is an incomplete or a wrong theory in its current form, and the revision of it should exclude causal loops. Acknowledgements: I am very grateful to L. Smolin, L. Freidel, G. Amelino-Camelia, A. Banburski, T. Rempel, S. Hossenfelder, J. Kowalski-Glikman, F. Mercati, J. Hnybida, T. Wang and J. Ziprick for helpful discussions and suggestions. 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. The work was supported by NSERC and FQXi.", "pages": [ 4 ] } ]

[ { "title": "Embedding of two de-Sitter branes in a generalized Randall Sundrum scenario.", "content": "Rodrigo Aros 1, ∗ and Milko Estrada 1, 2, † 1 Departamento de Ciencias F'ısicas, Universidad Andr'es Bello, Av. Rep'ublica 252, Santiago,Chile 2 Universidad Internacional SEK, Fernando Manterola 0789, Providencia, Santiago, Chile (Dated: August 6, 2018) In this work it is studied a generalization of Randall Sundrum model. It is obtained new behaviors of warp factor , constraints in that the strong brane has tension of positive sign and, new relations between the coupling constants κ 4 , κ 5 and between the Higgs and Planck masses. Keywords: Randall Sundrum, brane world, de Sitter, hierarchy problem", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The lack of a theory that unifies the four fundamental interactions is nowadays one of the most important issue of physics. In particular it is necessary to explain the huge difference between the values of the Higgs mass, m H ≈ 1 TeV , and the Planck mass m P ≈ 10 19 GeV . This is (called) the hierarchy problem . To address this problem in 1999 Lisa Randall and Raman Sundrum [4] proposed a model (RS) including two statics 3-branes imbedded in AdS 5 space with tensions of equal magnitude but opposite signs and our universe corresponding to the positive tension brane. This space has also Z 2 symmetry. Using coordinates x M = ( t, x, y, z, φ ), with φ ∈ [ -π, π ], our universe is located at φ = 0 and the secondary brane, which is called the strong brane, at φ = π = -π . In this model it is also proposed that Planck scales in four and five dimensions are of the same order, i.e. , M 4 ≈ M 5 . The line element considered in this model is given by where e -2 kr | φ | is called warp factor. In this way the value of mass in the two different branes is related by m φ =0 = e -krπ m φ = π . This allows that Planck mass at strong brane to be of same order of Higgs mass at our universe provided e krπ ≈ 10 15 . RS model however is not adequate to describe the current observations showing that our universe is under an accelerate expansion. Because of that in this work we study a model that consider dS 4 brane-worlds imbedded in (A)dS 5 . The line element is given in this case by where L is the radius dS 4 . In the next sections Einstein equations are solved yielding warp factors, tension of both branes and new relations between the coupling constants κ 4 and κ 5 and between Higgs and Planck masses. It will be analyzed the cases Λ 5 D = ± 6 l 2 and Λ 5 D → 0.", "pages": [ 1 ] }, { "title": "II. EMBEDDING OF TWO BRANES dS 4 IN A SPACE-TIME ( A ) dS 5 .", "content": "Let us consider the EH action coupled with two branes [2], i.e. , where κ 5 , g , R 5 D and Λ 5 D are the coupling constant, the metric determinant, Ricci scalar and the cosmological constant 5 D respectively. Here T 1 , T 2 , h 0 y h π are the tensions and the induced metrics on the weak and strong branes respectively. The equation of motion are The ( φ, φ ) component of Eq.(4), takes values ∓ for Λ 5 D = ± 6 l 2 . The ( u, v ) component is where g uv ( σ u ) = L 2 t 2 η uv . Replacing Eq.(5) in Eq.(6), for the values of φ = 0 = π , yields /negationslash /negationslash The last condition is satisfied for the space dS 4 . Therefore a factor A ( φ ) which is solution of the ( φ, φ ) Einstein equation, is also also a solution of the ( u, v ) component /negationslash /negationslash for values φ = 0 = π . The induced metrics on the weak and strong branes are respectively Replacing the Eqs. (5), (7) and (8) in Eqs.(6) yields This yields the tensions ∣ The tension of a brane acts as a cosmological constant, which on our brane universe is associated with a energy density of positive sign and a negative pressure. Therefore, solutions that yield T 1 < 0 are not to be considered.", "pages": [ 1, 2 ] }, { "title": "III. FOUR DIMENSIONAL EFFECTIVE THEORY", "content": "Since the extra dimension, said φ , can not be detected by current experiments it is interesting to consider the form adopted by EH action in five dimensions. This is given by By comparing this equation with the four dimensional EH action it is obtained the relation where the coupling constants κ 4 and κ 5 are related with the Planck scales M 4 and M 5 through the relation To proceed is necessary to fix some parameters. Recalling [4] it is convenient to fix the values of κ 4 and κ 5 such that M 4 ≈ M 5 . To propose new relations between the Higgs and Planck mass it will be proceed ' a la Randall and Sundrum by considering the action for a Higgs field on the strong brane. This is where H stands for the Higgs field. Here v π is the Higgs vacuum expectation value, λ is the coupling constant and D u is the gauge covariant derivative. By normalizing H → e A ( π ) H equation (8) implies where the vacuum expectation value is suppressed. This yields v ≡ e -A ( π ) v π . Any mass parameter at weak brane correspond to the physical mass defined by In the rest of paper we will study in different scenarios the values of r that produce a mass of order of Higgs in our brane universe considering a mass of order of Planck at the strong brane. The line element is where it can be noticed that there is not an exponential dependence. Therefore, the scaling differs from Randall Sundrum model along the fifth dimension. By replacing the factor A ( φ ) of equation (17) in equations (10) and (11), the tensions are where the constant C > 0 and therefore T 1 > 0. From equation (13) is obtained where it can be observed that the r dependence is no longer weak and thus differs from Randall Sundrum model. On the other hand, from equation (16) arises the relation between the mass m φ =0 y m φ = π given by ∣ ∣ First at all, we will study the case where C = E r . In this case the line element (17) represents a Minkowski space at φ = 0 and at a time of order of the current age of our universe, t = E ≈ 10 61 t p ≈ 10 61 l p (where l p and t p are the lenght and time of Planck, respectively) [11]. With this value of C , the tensions are: It is worth to notice that T 1 > 0 but T 2 < 0 or T 2 > 0 provided r < E π or r > E π . If E ≈ 10 61 l p the last constraint is uninteresting, since, large values for r would imply deviations from Newtonian gravity at solar system scales [3]. It is interesting to observe that at the limit r → 0 the Randall and Sundrum model is reproduced, i.e. T 1 = -T 2 . By substituting C = E r in equation (20), we obtain the relation between κ 4 and κ 5 : where, using M 5 ≈ M 4 ≈ M Planck = 10 19 GeV yields, κ 2 4 ≈ 10 -38 GeV -2 and κ 2 5 ≈ 10 -57 GeV -3 . On the other hand, considering Λ 4 D = 3 L 2 ≈ 10 -122 l -2 p it turns out that L ≈ 10 61 l p . Finally, using a value comparable to the age of universe E = 2 . 820947918 · 10 60 l p , we find a radius of order of 2 l p . With these values for E and L in the equation (21) implies a separation between branes πr of approximately 10 45 l p + E to establish a mass of order of Higgs on our brane universe. This, unfortunately, is not a reasonable value from physical point of view. Because of that one can consider that C is not of the same order of E r . For instance, if C is a constant of value C ≈ 5 · 10 44 then a mass of order of Higgs arises on our brane universe for values of L ≈ 10 61 l p and r ≈ 2 l p . Furthermore, the tensions T 1 and T 2 have positive and negative signs, respectively. Equation (20) gives in this cases that 10 -10 M 5 ≈ M 4 . Despite the fact that in this last case we do not obtain that M 4 ≈ M 5 , a null five dimensional cosmological constant could be of physical interesting, since at the above paragraph we obtain a mass of order of Higgs on our brane universe, which has tension > 0.", "pages": [ 2, 3 ] }, { "title": "B. Case Λ 5 D < 0", "content": "In this case is necessary to consider the line element which for r = 1 is similar to reference [5]. In order to recover a Minkowski space at a time t = E and φ = 0 it is necessary to fix C = sinh -1 ( E l ). In this case the tensions are given by It is worth to notice that T 1 > 0 but T 2 could be positive or negative depending on the r . For l ≈ l p [12] and E ≈ 10 61 l p , T 2 < 0 for r < 44 . 91 l p and T 2 > 0 for r > 44 . 91 l p . Therefore a value of r near and greater than 44 . 91 l p , which is ≈ 1 . 5 times bigger than the radio of Randall and Sundrum model could be of physical interest. In this last case the model has two branes of positive sign. Unfortunately the relation between constants κ 4 and κ 5 in this case is highly non linear. This is given by In this relation, as in Eq. (24), it is observed a dependence on r, L and E . In this equation is difficult to adjust all constants, due to the number of terms and the sensibility of equation under small changes on the values of the constants. The mass relation is ∣ ∣ where, again using l ≈ l p and E ≈ 10 61 l p , it is required a radius r = 78 . 133 l p to produce a Higgs mass on our brane universe. This radius is ≈ 2 . 6 times bigger than Randall and Sundrum model one. It is remarkable that with this value of C it can be reproduced a Higgs mass on our brane universe.", "pages": [ 3 ] }, { "title": "C. Case Λ 5 D > 0", "content": "In this case is worth considering a line element of the form The tension of branes are respectively The relation between κ 4 and κ 5 is given by Finally, this yields a relation between the masses For C = sin -1 ( E l ) the space at the brane is Minkowski at φ = 0 and t = E . Since there is not constraints for the value of cosmological constant in five dimensions it will be consider | E l | ≤ 1. Furthermore, due to that E and l are positive constants, C is located in the first two quadrants. By taking E ≈ 10 61 l p , l ≈ √ 2 E and r ≈ l p , the constant C can be determined depending on the ± in Eq.(33). For C = C + = π 4 and C = C -= 3 4 π . Furthermore, and independently of the sign ± , T 1 > 0 and T 2 < 0 respectively. Considering the values of E , l and r , the relation (33), it is interesting to notice that it can be imposed that ∓ l ± l cos 2 ( πr l ) ≈ 0. In this case where the constants are adjusted such that M 4 ≈ M 5 and L ≈ 10 61 l p . Finally for the masses in relation (34), it is not obtained a mass of order of Higgs. Now, using a constant C = ∓ πr l ∓ sin -1 ( 10 45 l p l ) on equation (34), we can obtain a mass of order of Higgs with the values r ≈ 6 l p , l ≈ 10 61 l p and L ≈ 10 61 l p . With this values the equation (33) yields the relation M 4 ≈ 3 . 42 M 5 , where M 4 = M planck and, it is obtained a model where, at the equations (31) and (32), T 1 is positive and T 2 = -T 1 .", "pages": [ 3, 4 ] }, { "title": "IV. CONCLUSIONS", "content": "It have been established new constraints for the values of compactification radius r at the scenarios Λ 5 D ± 6 l 2 and Λ 5 D → 0. These constraints allow to consider models where the tension of our brane is positive, the Planck constants 4 D and 5 D are approximately of same order, and a mass of order of Higgs on our brane universe is recovered. A very interesting result is obtained at the case Λ 5 D > 0 with C = ∓ πr l ∓ sin -1 ( 10 45 l p l ) , , since all assumptions described above are derived . Other interesting result is obtained for Λ 5 D < 0 where for a reasonable radius of compactification one recovers a mass of order of Higgs with E of order of age of universe and two branes have positive tensions.", "pages": [ 4 ] } ]

[ { "title": "Ionization of hydrogen by neutrino magnetic moment, relativistic muon, and WIMP", "content": "Jiunn-Wei Chen, 1, 2 C.-P. Liu, 3 Chien-Fu Liu, 1 and Chih-Liang Wu 1 1 Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan 2 National Center for Theoretical Sciences and Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan 3 Department of Physics, National Dong Hwa University, Shoufeng, Hualien 97401, Taiwan", "pages": [ 1 ] }, { "title": "Abstract", "content": "We studied the ionization of hydrogen by scattering of neutrino magnetic moment, relativistic muon, and weakly-interacting massive particle with a QED-like interaction. Analytic results were obtained and compared with several approximation schemes often used in atomic physics. As current searches for neutrino magnetic moment and dark matter have lowered the detector threshold down to the subkeV regime, we tried to deduce from this simple case study the influence of atomic structure on the the cross sections and the applicabilities of various approximations. The general features being found will be useful for cases where practical detector atoms are considered.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The electromagnetic (EM) properties of neutrinos, in particular the magnetic dipole moments, µ ν , are of fundamental importance not only in particle physics but also astrophysics and cosmology (for reviews, see, e.g., Refs. [1, 2]). In the Standard Model with massive neutrinos, a non-vanishing µ ν arises as a result of one-loop electroweak radiative correction; for Dirac neutrinos, ∗ it is given by µ ν = 3 . 20 × 10 -19 ( m ν eV ) µ B , where the Bohr magneton µ B = e/ (2 m e ) with e and m e being the magnitude of charge and mass of electron. † From the current mass upper limit set on the electron neutrino in the tritium β decay [3], m ν e < 2 eV , one can estimate that µ ν e /lessorsimilar 10 -18 µ B is indeed very tiny in the Standard Model. The best direct limits on µ ν so far are extracted mostly from neutrino-electron ( νe ) scattering: with the reactor antineutrinos, µ ¯ ν e < 2 . 9 × 10 -11 µ B by the GEMMA collaboration [4] and µ ¯ ν e < 7 . 4 × 10 -11 µ B by the TEXONO collaboration [5]; with the solar neutrinos, µ ν /circledot < 5 . 4 × 10 -11 µ B by the Borexino collaboration [6]. Many stronger, but indirect, limits ranging from 10 -11 to 10 -13 were inferred from astrophysical or cosmological constraints, however, they are subject to model dependence and theoretical uncertainty. Because the current limits, whether direct or indirect, are orders of magnitude away from the Standard Model prediction, it makes the search of µ ν a powerful probe of new physics. The cross section of neutrino scattering off a free electron through the EM interaction with µ ν is [7] ∣ where α is the fine structure constant, E ν the neutrino incident energy, and T the neutrino energy deposition. The 1 /T feature indicates a way of improving the limit on µ ν by lowering the detector threshold of T . Currently the thresholds can be as low as a few keV (e.g., the Germanium semiconductor detectors deployed by both the GEMMA and TEXONO collaborations), and the next-generation detectors are geared up to extend down to the subKeV regime [8, 9]. While one expects improved limits from such experimental upgrades, a theoretical issue regarding how the electronic structure of detectors affects the simple free νe scattering formula naturally arises, as the associated energy scale is comparable to the atomic scale. Recently there have been discussions about whether atomic structure can possibly enhance an atomic ionization (AI) cross section [10, 11], and the robustness of an free electron approximation in low energy transfer [12, 13]. With experiments keep pushing down the detector threshold, the need for more reliable cross section formulae will certainly grow. Another type of experiments where AI can be relevant is the search for dark matter (DM), as it shares many similar detection techniques as for µ ν . Most current search focus on the weakly-interacting massive particles (WIMPs) with masses about GeV to TeV scales favoured for astrophysical reasons - with nuclear recoil in targets being the main observable. Recently the subGeV DM candidates, generically classified as light dark matter (LDM), start to get attention [14], and the associated AI processes in targets can be used to constrain the interaction of LDM candidates with electrons and their masses [15]. Given the importance of understanding the detectors' response, in particular in low energy regime, our study starts by considering the simplest atom - hydrogen. By treating the electrons as non-relativistic particles and including the one photon exchange together with the Coulomb interaction, the problem is solved analytically with O ( v 2 e ) and O ( α 2 ) errors, where v e is the electron velocity. We then compare our result against various widely-used approximation schemes for the AI through µ ν or DM scattering, we try to draw useful information about the applicabilities of these approximation schemes under various kinematic conditions. This knowledge serves as a precursor to our currently-ongoing projects with realistic atomic species. The article is organized as follows: In Sec. II, we lay down the general formalism for AI cross sections through EM interactions. The analytic results for the atomic response functions of hydrogen-like atoms are given explicitly, and approximation schemes including the free electron approximation (FEA), equivalent photon approximation (EPA), longitudinal photon approximation (LPA), and the one of Kouzakov, Studenikin, and Voloshin (KSV) [13] are introduced. The case of AI by µ ν is studied in Sec. III, with particular attention to the issue whether atomic structure enhances or suppresses the cross sections while scattering occurs at atomic scales. In Sec. IV, the well-known AI process by relativistic muon is revisited. A detailed account of why EPA works for this case but not for µ ν is given. Finally we extend the above formalism to a QED-like gauge model for the DM interaction with normal matter, and study the hydrogenic response under various DM kinematics in Sec. V. A brief summary is in Sec. VI.", "pages": [ 2, 3, 4 ] }, { "title": "II. FORMALISM", "content": "Consider the ionization of a hydrogen-like atom H by a lepton l , through one photon exchange, as shown in Fig. 1. We will treat the electron as a nonrelativistic particle and include all its Coulomb interactions in the initial and final states. This problem can be solved analytically. The results will be referred as the 'full' ones in comparison to various approximations to be discussed later on - and have errors on the order of O ( v 2 e , α 2 ) . The unpolarized differential cross section in the laboratory frame, i.e., the velocity of the incident lepton /vectorv 1 = 0 and the velocity of the atomic target /vectorv H = 0 , is expressed as ‡ /negationslash where the four momenta k 1 = ( ω 1 , /vector k 1 ) and k 2 = ( ω 2 , /vector k 2 ) are of the initial and final leptons, p H = ( M H , /vector 0) of the initial atom, p R = ( E R , /vectorp R ) and p r = ( E r , /vectorp r ) of the final H + + e -state in the center-of-mass and relative coordinates, and q µ = k µ 1 -k µ 2 = ( T, /vectorq ) of the virtual photon; respectively; and Q 2 = q µ q µ . The leptonic tensor is obtained by a sum of the final spin state s 2 and an average of the initial spin state s 1 of the leptonic electromagnetic (EM) current, j l , matrix elements; and similarly the atomic tensor involves a sum of the final angular momentum state m j f and an average of the initial angular momentum state m j i of the atomic EM current, j A , matrix elements, where | i 〉 and | f 〉 refer to atomic initial and final states, respectively. In this work, we use the relativistic form for j µ l The Dirac and Pauli form factors, F ( l ) 1 and F ( l ) 2 , which describe the helicity-preserving and helicity-changing EM couplings, are constant for elementary leptons: F ( l ) 1 is the charge e l (in units of e ) and F ( l ) 2 the anomalous magnetic dipole moment κ l (in units of µ B ) . Since we are only interested in the case that the energy deposition by the incident particle is small enough such that electrons can be treated as non-relativistic particles, the charge and spatial current densities in momentum space are ρ ( A ) is leading in the 1 /m e expansion while /vector j ( A ) is subleading. The proton contribution can be neglected because its contribution to /vector j ( A ) is O (1 /m p ) and is smaller than the electron contribution by a factor of m e /m p . Its contribution to ρ ( A ) is smaller than the electron contribution by at least one power of m e /m p in the multiple expansion, because the size of the proton wave function is smaller than that of the electron wave function by a m e /m p factor. After performing the spin sum, contraction of the leptonic and atomic tensors, and implementing the current conservation condition to relate the longitudinal spatial current to the charge density where q ≡ | /vectorq | , the cross section can be cast into the following form through defining the longitudinal and transverse response functions, R L and R T , and the corresponding kinematic factors, V L and V T . The kinematic factors, which depend on the energy transfer T and momentum transfer q , are for couplings with the F ( l ) 1 and F ( l ) 2 form factors, respectively. § The response functions, which are also functions of ( T, q ) but independent of the form of leptonic coupling, are where B is the binding energy of the H atom, M = m e + m p ≈ m p , and µ red = m e m p / ( m e + m p ) ≈ m e . Note that the center-of-mass degrees of freedom in the final state have been integrated out by the momentum conservation, which yield /vector p R = /vectorq ; and the resulting energy conservation delta function properly takes care the nuclear recoil effect. Consider now the ionization of a hydrogen-like atom from its ground state, i.e., the 1 s orbit, the relevant atomic spatial wave functions for the initial ( i ) and final ( f ) states are and in atomic units (so the barred quantities are ¯ r = r m e α , ¯ p r = p/ ( m e α ) , etc.), where Γ( z ) and 1 F 1 ( a, b, z ) are the Gamma and confluent hypergeometric functions, respectively. The evaluations of R L and R T can be done analytically by the Nordsieck integration [16-19] and yield: The first term in R T is the contribution from the convection current, and the second one from the spin current. The overall α 2 factor appearing in both terms reflects the 1 /m 2 e order in the non-relativistic expansion. In comparison, R L is O ( α 0 ) . The single differential cross section with respect to the energy transfer can then be computed by integration over the lepton scattering angle θ with a constrained range of cos θ : For latter discussion, we note that for a fixed energy transfer, the square of four momentum transfer: only depends on cos θ , therefore, the integration over cos θ is equivalent of integrating over Q 2 (or q 2 ). While it is straightforward to obtain complete and analytic results for ionizations of the hydrogen atom to the order outlined above, ¶ we shall discuss several approximation schemes often employed in atomic calculations, and compare them with the full calculations for this case study in the following sections.", "pages": [ 4, 5, 6, 7 ] }, { "title": "A. Free Electron Approximation (FEA)", "content": "The FEA is expected be a good approximation if the photon wavelength is much smaller than the size of the atom (or the typical distance between electrons in a multi-electron system) such that the atomic effect is no longer important. Thus, a necessary (but not sufficient) condition for this approximation to be valid is that the scattering energy needs to be high (compared with the typical scale of the problem). In this approximation, the electron before and after ionization is treated as a free particle. The free electron cross section of Eq.(1) is multiplied by the step function θ ( T -B ) to incorporate the binding effect: Energy and momentum conservation fixes Q 2 = -2 m e T in this two-body phase space.", "pages": [ 8 ] }, { "title": "B. Equivalent Photon Approximation (EPA)", "content": "The equivalent photon approximation [20, 21] treats the virtual photon as a real (and thus transversely polarized) photon. It could be a good approximation for low energy processes where the photon is soft such that Q 2 ≈ 0 because q µ ≈ 0 for every component of µ . At high energies, besides soft photon emissions, when the initial and final state electrons are highly relativistic and almost collinear, the emitted 'collinear' photon also has Q 2 ≈ 0 . While the soft photon emission is likely to dominate the phase space of low energy scattering, whether the soft and collinear photon emission will dominate the high energy scattering depends on the transition matrix elements. The total cross section σ γ for the photoionization process γ +H → H + + e -is where the photon energy E γ = T and the superscript ' 0 ' denotes that the photon is 'onshell', i.e., T 2 = q 2 . Then EPA relates σ γ to a corresponding lepto-ionization process (involving a virtual photon) by the following two steps: (i) ignoring the longitudinal response function R L and (ii) substituting the off-shell response function R T by the on-shell R 0 T extracted from the photo-ionization process, i.e., with the energy spectrum of equivalent photon N ( T ) defined by where the integration range of cos θ is the same as Eq. (22). Because it directly feeds the photo-ionization cross sections (experimental accessible) to the corresponding leptoionization cross sections, a lot of theoretical work and uncertainties can be saved when it works properly. At this point, we should make an important remark as regards the approximation scheme adopted in Ref. [10]: Even though it is in the spirit of the EPA, however, it makes a stronger assumption that the integration leading to energy spectrum of equivalent photon is also dominated by the Q 2 ≈ 0 region (or staying constant), i.e., To distinguish this stronger version of the EPA from the conventional one, we shall denote it as the EPA ∗ scheme.", "pages": [ 8, 9 ] }, { "title": "C. Longitudinal Photon Approximation (LPA)", "content": "The longitudinal photon contribution is leading order in the 1 /m e expansion while the transverse photon contribution is subleading. Thus, it might be a good approximation for non-relativistic systems: The difference of this approximation to the full calculation is a measure of how importantly the transverse current contributes to the process.", "pages": [ 9 ] }, { "title": "D. Approximation Scheme of Kouzakov, Studenikin, and Voloshin (KSV)", "content": "The KSV scheme includes the longitudinal photon contribution which is leading order in the 1 /m e expansion and approximates the subleading transverse photon contribution by a relation only strictly suitable for the electric dipole ( E 1 ) transition in the long wavelength limit, i.e., q → 0 : This relation can be derived from the Siegert theorem [22] for E 1 , which is based on current conservation. It can also be explicitly checked by taking the same limit to Eqs. (19,20). In general R T is not dominated by E 1 in the processes that we are considering, which requires q r A /lessmuch 1 where r A is the size of the atom. But Eq. 30 can still be a good approximation to the cross section calculations as long as R T remains subleading to R L . In Refs. [12, 13], the authors adopted the above relation so the cross section were calculated without need to evaluate the transverse response function which is harder to compute:", "pages": [ 10 ] }, { "title": "III. IONIZATION BY NEUTRINO MAGNETIC MOMENT", "content": "In case the incident lepton is a neutrino ( ν ) or antineutrino ( ¯ ν ), as e ν = 0 , the EM breakup process is thus sensitive to the neutrino magnetic moment µ ν = κ ν µ B (which is purely anomalous). The energy spectrum for reactor antineutrinos typically peaks around few tens of keV to MeV (see, e.g., Ref. [5]); setting ω 1 = 1MeV , a plot of the single differential cross section with energy loss up to 1 keV is given in Fig. 2. Not shown in these figures are the results of the approximation schemes KSV and LPA. They both agree with the full calculation to good extents: within 10 -5 for the former and 10 -3 for the latter in the entire range. In other words, this atomic bound-to-free transition is dominated by the atomic charge operator, while the transverse current operator is negligible, which implies the inadequacy of the EPA scheme. The dominance of the charge operator over the transverse current can be roughly understood by a comparison of their corresponding kinematic factors V ( F 2 ) L and V ( F 2 ) T . For (a) ω 1 = 1 MeV LPA (both not shown) are in excellent agreement with the full calculation. neutrino scattering where x ≡ cos θ , they are As T 2 /ω 2 1 /lessmuch 1 in our consideration, both functions peak near x = 1 , and they have similar maximum values: V ( F 2 ) L /Q 4 | max = ω 2 1 /T 2 and V ( F 2 ) T /Q 4 | max = 2 ω 2 1 /T 2 , and widths. Since the transverse response function does not get enhancement from the kinematic factor V ( F 2 ) T over V ( F 2 ) L , its contribution to the cross section is suppressed by the usual non-relativistic order α 2 . The good agreements with the FEA scheme (Fig. 2a) is not really a surprise: the energetic neutrino emits a virtual photon with wavelength smaller than the atomic size so that the binding effect does not manifest in a short distance. The only exception is near the ionization threshold (Fig. 2b) where the virtual photon wavelength is larger than the atomic size, and the binding effect suppresses the cross section in comparison to FEA. Also shown in these figures is the result of the EPA ∗ scheme. Note that the curve in Fig. 2a has to be scaled down by a factor of 100 in order to be cast on the same plot as other calculations; in other words, the EPA ∗ hugely overestimates the cross section by several (a) ω 1 = 1 MeV , T = 20 eV orders of magnitude. There is only a tiny region near the ionization threshold (see Fig. 2b) where the EPA ∗ does work; that is where the virtual photon approaches the real photon limit ( T = q ). The origin of such an overestimate can be clearly seen in Fig. 3a, where the double differential cross section dσ/ ( dT dx ) is plotted as a function of x with a fixed energy loss T = 20eV . Due to the kinematic constraint, the maximum scattering angle θ max = 6 . 28 · . However, even within this small range of peripheral scattering angle, the differential cross section decreases dramatically by 12 orders of magnitude from the forward angle as a combined result of the kinematic factors V L,T and the response functions R L,T . Therefore, the flatness of dσ/ ( dT dx ) required by the EPA ∗ is severely violated and results in this overestimation. On the other hand, one does see the EPA ∗ start to work when the incident neutrino energy ω 1 drops below the binding momentum of the hydrogen-like atom ∼ Z m e α . For hydrogen, the scale is about 3 . 73 keV , and Fig. 4 shows that varying ω 1 from 3 keV , 2 keV , to 1 keV , the EPA ∗ result becomes reasonably good. As evidenced from Fig. 3b, the double differential cross section for ω 1 = 1 keV and T = 20 eV , even though not looking completely flat, does vary only modestly with increasing θ , and the agreement is getting better when ω 1 is further decreased. In the meanwhile, the FEA is no longer a good approximation since the de Broglie wavelength of the incident neutrino is on the order of atomic size so the binding effect is not negligible.", "pages": [ 10, 11, 12 ] }, { "title": "IV. IONIZATION BY MUON", "content": "Replacing the incident lepton from a neutrino to a muon ( µ -) , as e µ -= -1 , the EM breakup process is instead dominated by the F 1 coupling, while the F 2 coupling can be ignored for the smallness of muon g -2 ≈ 0 . 001 [1]. Consider relativistic muons with 10 0 , 1 , 2 , 3 GeV energies, the differential cross sections are plotted in Fig. 5. (Because the muon is relativistic while the electron non-relativistic, the final state interaction between the muon and electron can be ignored, see, e.g., Ref. [23].) The noticeable differences in comparison to what have been drawn in the previous neutrino case are: (1) The differential cross section falls off more quickly as T increases, i.e., the recoil electrons tend to have relatively smaller energies. (2) The FEA results are insensitive to ω 1 and largely underestimates in all cases. (3) There are substantial contributions from the transverse current, despite its built-in O ( Z 2 α 2 ) suppression due to the non-relativistic kinematics of atomic electrons. In fact, when ω 1 becomes big enough, the interaction with the atomic transverse current dominates over the one with the charge and longitudinal current, as indicated by the competition between the EPA and the LPA curves in Fig. 5, and one expects the larger ω 1 increases, the better the EPA works. /negationslash The reason for such differences is primarily due to the associated kinematic factors. At the Q 2 → 0 limit, they behave like V ( F 1 ) L ∝ Q 4 and V ( F 1 ) T ∝ Q 0 for muon ( m µ = 0 ), and V ( F 2 ) L ∝ Q 6 and V ( F 2) T ∝ Q 4 for neutrino ( m ν ≈ 0 ) ionization, respectively. As the differen- (a) ω 1 = 1 GeV (c) ω 1 = 100 GeV tial cross section dσ/dT involves an 1 /Q 4 weighted integration over Q 2 , only the transverse part in muon ionization receives a strong weight at peripheral scattering angles (where Q 2 ≈ 0 ).This explains the importance of the transverse current in relativistic muon ionization and its insignificance in neutrino ionization. Also, with the allowed scattering angles become closer to the exact forward direction as the muon incident energy increases (with T fixed), the kinematics becomes real-photon-like and eventually the huge enhancement by the 1 /Q 4 weight is able to overcome the non-relativistic suppression in the transverse response function. The failure of the FEA in relativistic muon ionization can also be understood in a similar way: In the FEA scheme, the differential cross section dσ/dT is determined from a specific kinematics Q 2 FEA = -2 m e T by energy-momentum conservation; on the other hand, the full calculation with two-body kinematics involves an integration over allowed | Q 2 | ranging from ≈ 0 to some maximum value determined by the maximum scattering angle. Because the 1 /Q 4 factor that enhances the contributions from the Q 2 ≈ 0 region, the Q 2 FEA = -2 m e T ceases to be a good representative point. A semi-quantitative understanding could be obtained by the following approximate forms of V ( F 1 ) L and V ( F 1 ) T . For a relativistic muon they are One sees that unlike the previous case for which V ( F 2 ) T and V ( F 2 ) L are comparable in most range of x , V ( F 1 ) T /Q 4 is comparable to V ( F 1 ) L /Q 4 only for 1 -x /greaterorsimilar T 2 2 ω 2 1 . As the scattering angle further decreases, V ( F 1 ) T /Q 4 starts to dominate over V ( F 1 ) L /Q 4 , and when 1 -x /lessorsimilar m 2 µ ω 2 1 T 2 2 ω 2 1 , it overwhelms by a factor ω 2 1 m 2 µ /greatermuch 1 . Also because of this huge weight on extremely small angles, the FEA scheme with | Q 2 FEA | = 2 m e T overestimates the averaged | Q 2 | for the realistic situation and leads to an underestimation. Though one sees that the EPA serves as a better approximation than the FEA in the relativistic muon ionization, however, as shown in Fig. 5, even at ω 1 = 1000 GeV , it can still not be taken as a good approximation to the full result. The main reason is its non-zero mass which limits the lowest | Q 2 | to be reached If m µ is adjusted to smaller values /lessorsimilar 1 eV , then indeed the EPA accounts for /greaterorsimilar 80% of the cross section for ω 1 on the orders of GeV -TeV , as shown in Fig. 6. In other words, in case one seeks a better description of relativistic muon ionization or other processes alike beyond the EPA, the contribution from charge and longitudinal current should be included.", "pages": [ 13, 14, 15 ] }, { "title": "V. IONIZATION BY WIMP", "content": "Instead of a relativistic muon, consider now the atomic ionization by some non-relativistic, weakly interacting massive particle, χ , which could be a dark matter (DM) candidate. Suppose this particle is of galactic origin with a mean velocity v χ ∼ 220 / (3 × 10 5 ) , its kinetic energy ≈ 1 2 m χ v 2 χ = 270 ( m χ GeV ) eV ; therefore, in order to ionize a hydrogen, m χ /greaterorsimilar 60 MeV . To make use of the general formalism developed in Sec. II, we postulate a QED-like fermionic DM-electron ( χe ) interaction in which the new U (1) gauge boson has mass m b and the interaction strength α χe ≡ g χe α . Fig. 7 shows the differential cross sections for m χ = 100 MeV and 1 GeV , with either a massless gauge boson m b = 0 , which corresponds to an infinitelyranged interaction, or a very massive one m b = 125 GeV , which leads to a extremely shortranged interaction. †† Not shown in Fig. 7(a)-(d) are the results of the KSV and LPA, as they are in excellent agreement with the full calculations in all cases illustrated. Accordingly, the failure of the EPA scheme is anticipated. On the other hand, although it is expected that the binding effect should suppress the FEA results, the several orders of magnitude overestimation by the FEA scheme in the entire range of energy transfer, evidenced in panels (a)-(d), indicates the inadequacy of the FEA scheme in such a kinematic regime. Figs. 7(e) and (f) show that the differential cross section becomes 'saturated' when m χ becomes much bigger than 1 GeV , which is about the mass of the hydrogen target. This can be understood by transforming the laboratory frame, where the hydrogen target is stationary, to the DM rest frame which coincides the center-of-mass frame for m χ /greatermuch m p : the kinematics only depends on v χ and the reduced mass ≈ m p . Also by comparing the case with m b = 0 , Figs. 7(a,b,e), and m b = 125 GeV , Figs. 7(b,d,e), the differential cross sections, apart from some overall scale factors, show a slower decreasing with energy transfer T as the range of the χe interaction decreases. If, on other hand, the neutral fermionic dark matter has a non-zero (anomalous) magnetic moment, or its coupling to the U (1) gauge boson is via the Dirac bilinear ¯ χσ µν q ν χ/ (2 m e ) , a different constant α χe = κ χe α is assigned to characterize the interaction strength. This anomalous-magnetic-moment-like interaction yields quite different results, as shown in Fig. 8, from the previous case with the same kinematics. The most noticeable difference seen in Figs. 8(a)-(d) is that the KSV and LPA no longer work, and in fact, largely underestimate. This implies not only substantial contributions from the transverse response but also the breakdown of long wavelength approximation, which has been good for all cases previously discussed. However, the EPA does not work either: it yields a huge overestimate which implies the transverse kinematics is not dominated by the photon-like, Q 2 ≈ 0 , region. Therefore, one encounters a very subtle kinematic regime where none of the approximation schemes work and requires a full calculation. While Figs. 8(e) and (f) show a similar cross section saturation for m χ /greatermuch m p and the range effect on the differential cross section as previously found, a comparison of Fig. 8 and Fig. 7 shows that the large energy transfer regime is more suppressed in the QED-like interaction than the anomalous-magnetic-moment-like interaction. The general trend observed above for the χe cross sections that the atomic charge operator dominates in the QED-like interaction while the transverse current operator in the anomalous-magnetic-moment-like interaction is opposite to what have been concluded for the ionizations by relativistic muons ( F 1 coupling) and neutrinos ( F 2 coupling). This difference is also partially due to the corresponding kinematic factors: With m 2 χ /greatermuch | Q 2 | , q 2 , T 2 , they are and The square of four momentum transfer in DM scattering is where r E is the fraction of the DM kinetic energy transfer to the atom, i.e., 2 T/ ( m χ v 2 χ ) . For most range of x (which is less restricted unless m χ /greatermuch m p ) and r E (which can never be zero for ionization), one can estimate | Q 2 | ∼ m 2 χ v 2 χ . Because T 2 = ( r 2 v 2 χ / 4) m 2 χ v 2 χ , q 2 = T 2 + | Q 2 | ∼ m 2 χ v 2 χ . Using these estimates, the leading orders in v χ are O (1) for V ( F 1 ) L , O ( v 2 χ ) for V ( F 1 ) L ; and O ( v 4 χ ) for V ( F 2 ) L , O ( v 2 χ ) for V ( F 2 ) T , respectively. Therefore, in ratio to the charge operator, the transverse current is suppressed by O ( v 2 χ ) /O (1) in the F 1 -type coupling, while it is enhanced by O ( v 2 χ ) /O ( v 4 χ ) in the F 2 -type coupling due to the kinematic factors.", "pages": [ 16, 18, 20 ] }, { "title": "VI. CONCLUSION", "content": "We studied the ionization of hydrogen by scattering of neutrino magnetic moment, relativistic muon, and weakly-interacting massive particle with a QED-like interaction. Analytic results were obtained and compared with several approximation schemes often used in atomic physics. It is found that for the case of neutrino magnetic moment, the atomic charge operator dominates the process, and for typical reactor neutrino energies about tens of keV to a few MeV , the atomic binding effect is negligible. For relativistic muon scattering, on the other hand, the transverse current operator becomes dominant with increasing incident muon energy. In this case, the equivalent photon approximation yields a reasonable result, however, for further improvement, the contribution from the charge operator needs to be taken into account. Also, due to the special weight by kinematics, the free electron approximation largely underestimates the result. The WIMP scattering is the most kinematics-sensitive case, and the free electron approximation fails badly. Depending on the coupling to the dark matter particle, the cross section is dominated by the charge operator for the F 1 -coupling, and the transverse current operator for the F 2 -coupling. While the longitudinal photon approximation works for the former, none of the approximations under study work for the latter.", "pages": [ 20 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank Henry T. Wong for stimulating discussions and comments. The work is supported in part by the NSC of ROC under grants 99-2112-M-002-010-MY3, 102-2112-M-002013-MY3 (JWC, CFL, CLW) and 98-2112-M-259-004-MY3, 101-2112-M-259-001 (CPL).", "pages": [ 21 ] } ]

[ { "title": "Light Dirac right-handed sneutrino dark matter", "content": "Ki-Young Choi Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Republic of Korea and Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Republic of Korea", "pages": [ 1 ] }, { "title": "Osamu Seto", "content": "Department of Life Science and Technology, Hokkai-Gakuen University, Sapporo 062-8605, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "We show that mostly right-handed Dirac sneutrinos are a viable supersymmetric light dark matter candidate. While the Dirac sneutrino scatters with nuclei dominantly through the Z -boson exchange and is stringently constrained by the invisible decay width of the Z boson, it is possible to realize a large enough cross section with the nucleon to account for possible signals observed at direct dark matter searches, such as CDMS II(Si) or CoGeNT. Even if the XENON100 limit is taken into account, a small part of the signal region for CDMS II(Si) events remains outside the region excluded by XENON100. PACS numbers:", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Light weakly interacting massive particles (WIMPs) with masses around 10 GeV have received a lot of attention, motivated by the results of some direct dark matter (DM) detection experiments. DAMA/LIBRA has claimed detection of the annual modulation signal by WIMPs [1]. CoGeNT has found an irreducible excess [2] and annual modulation [3]. CRESST has observed more events than expected backgrounds can account for [4, 5]. The CDMS II Collaboration has just announced [6] that their silicon detectors have detected three events and its possible signal region overlaps with the possible CoGeNT signal region analyzed by Kelso et al. [7]. However, these observations are challenged by the null results obtained by other experimental collaborations, such as CDMS II [8, 9], XENON10 [10], XENON100 [11, 12] and SIMPLE [13]. Recently, Frandsen et al. [14] have pointed out that the XENON10 exclusion limit in Ref. [10] might be overconstraining. It has been stressed that the signal region due to low-energy signals in CDMS II(Si) extends outside the XENON exclusion limit [15]. The Fermi-LAT collaboration has derived stringent constraints on the s -wave annihilation cross section of WIMPs by analyzing the gamma-ray flux from dwarf satellite galaxies [16]. In particular, in the light-mass region below O (10) GeV, the annihilation cross section times relative velocity 〈 σv 〉 of O (10 -26 )cm 3 / s, which corresponds to the correct thermal relic abundance Ω h 2 /similarequal 0 . 1, has been excluded. Light WIMPs have been investigated as a dark matter interpretation of this positive data. In fact, very light neutralinos in the minimal supersymmetric Standard Model (MSSM) [17, 18] and the next-to-MSSM (NMSSM) [19, 20] or very light right-handed (RH) sneutrinos in the NMSSM [21-23] have been regarded as such candidates. However, these candidates hardly avoid the above Fermi-LAT constraint. 1 In this paper, we show that mostly right-handed Dirac sneutrinos are viable supersymmetric light DM candidates and have a large enough cross section with nucleons to account for possible signals observed at direct DM searches. Dirac sneutrinos scatter off nuclei dominantly via the Z -boson exchange process through the suppressed coupling and mostly with neutrons rather than protons. Although this Z -boson-mediated scattering does not relax the tension among direct DM search experiments and its availability is limited by the invisible decay width of the Z boson, a part of the signal region for CDMS II(Si) events [6] remains outside the excluded region by XENON100 [12]. We examine the cosmic dark matter abundance as well as the constraints from indirect dark matter searches for a viable model of Dirac sneutrino dark matter. The paper is organized as follows. In Sec. II, we estimate the DM-nucleon scattering cross section through the Z -boson exchange process and show the experimental bounds and signal regions for this case. We impose the bound from the Z boson invisible decay width too. In Sec. III, after a brief description of the model, we examine other cosmological, astrophysical, and phenomenological constraints. We then summarize our results in Sec. IV.", "pages": [ 2, 3 ] }, { "title": "A. Invisible Z -boson decay", "content": "We are going to consider light Dirac sneutrino DM scattering with nuclei through the Z -boson exchange process in the direct detection experiments. Since the property of the Z boson is well understood, the possibility of a light sneutrino has been stringently constrained from the invisible decay width of the Z boson. First, we briefly summarize the bound. The Z -boson invisible decay is (20 . 00 ± 0 . 06)% for the total decay width of the Z -boson decay Γ Z = 2 . 4952 ± 0 . 0023 GeV [27]. This gives a constraint on the neutrino number which couples to the Z boson, given by [27] The LEP bound on the extra invisible decay width is given as [28] If there is a light sneutrino which couples to the Z boson, the Z boson can decay into light sneutrinos. The spin-averaged amplitude is Here, C eff parametrizes the suppression in the sneutrino-sneutrinoZ boson coupling as shown in Fig. 1. For pure left-handed sneutrinos, C eff = 1. The decay width of the Z boson into light sneutrino DM is given by and we impose the upper bound (2) on this. This bound corresponds to for a few GeV dark matter particle. The contour plot of the invisible decay width is also shown in Fig. 3.", "pages": [ 3, 4 ] }, { "title": "B. Direct detection", "content": "Dirac sneutrino DM can have elastic scattering with nuclei in the direct detection experiments. The most relevant process is due to the Z -boson exchange as in the left diagram in Fig. 2. The Z -boson exchange cross section with nuclei A Z N is given by where M DM and m N denote the dark matter mass and nucleus mass, respectively, A N and Z N are the mass number and proton number of the nucleus, and G F is the Fermi constant [29]. Here µ X is the reduced mass defined by and m n stands for the neutron mass. In the expression (7), σ Z χn denotes the DM scattering cross section with a neutron, and we have used the fact that the Z boson dominantly couples with a neutron (as opposed to a proton) as (1 -4 sin 2 θ W ) /similarequal 0 . 076, and hence we have neglected the contribution from scattering with a proton. Usually the bound or signal of the direct detection experiments is given to the WIMPnucleon scattering cross section, assuming the isospin-conserving case. This is true for a conventional WIMP such as a neutralino, where Higgs boson-exchange processes are dominant. For the Z -boson-mediated case, the DM interacts dominantly with a neutron, and thus the bound should be modified according to this. Using Eq. (7), the corresponding WIMP-neutron cross section, σ ( Z ) n , for the Z -boson-mediated case is related to the isospinconserving (IC) WIMP-nucleon scattering cross section, σ (IC) n , by For Xenon A /similarequal 130 , Z = 54, and for Si in CDMS II A = 28 , Z = 14. These factors give enhancement on the cross section by factors 4 and 3, respectively. In Fig. 3, we show the contour of the Z -boson extra invisible decay width and the WIMPneutron scattering cross section in the plane of C eff and the dark matter mass M DM . The contours of the predicted scattering cross section with a neutron (blue) are given in units of 10 -40 cm 2 with those of the extra Z -boson invisible decay width (red). The red region is disallowed by the LEP bound on the Z -boson extra invisible decay given in Eq. (2). In Fig. 4, we show the WIMP-neutron scattering cross section versus dark matter mass. We show the constraint from XENON100 [12], and the signals measured by CDMSII-Si [6] and CoGeNT [7] with the contour of the Z -boson extra invisible decay width. Following Ref. [14], we do not include the XENON10 limit in this paper to keep our discussion conservative. We find that a still barely compatible region exists for a dark matter mass around 6 GeV and the WIMP-nucleon cross section σ ( Z ) n /similarequal 10 -40 cm 2 .", "pages": [ 4, 5, 6 ] }, { "title": "III. OTHER CONSTRAINTS", "content": "The discussion and conclusion in the previous section are model independent and were made applicable for any scalar DM scattering with a nucleon dominantly through Z -boson exchange by introducing the coefficient C eff . In this section, we discuss other DM phenomenologies and experimental constraints. To do this, we need to specify the particle model for Dirac sneutrino dark matter. One model has been constructed with nonconventional supersymmetry (SUSY)-breaking mediation [30]. Light sneutrino DM has been studied in Refs. [31, 32] and has unfortunately turned out to be hardly compatible with LHC data, mainly due to the SM-like Higgs boson invisible decay width [32]. There is another available model proposed by us [33] in the context of the neutrinophilic Higgs doublet model [34-37]. Therefore in the rest of this section, as an example, we discuss other DM phenomenologies based on this model.", "pages": [ 6, 7 ] }, { "title": "A. Brief description of the model in Ref. [33]", "content": "The neutrinophilic Higgs model is based on the concept that the smallness of the neutrino mass might not come from a small Yukawa coupling but rather from a small vacuum expectation value (VEV) of the neutrinophilic Higgs field H ν . As a result, neutrino Yukawa couplings can be as large as of the order of unity for a small enough VEV of H ν . Other aspects-for instance, collider penomenology [38-40], astrophysical and cosmological consequences [33, 41-43], vacuum structure [44], and variant models [45-47], -have also been studied. The supersymmetric neutrinophilic Higgs model has a pair of neutrinophilic Higgs doublets H ν and H ν ' in addition to up- and down-type two-Higgs doublets H u and H d in the MSSM [41]. A discrete Z 2 parity is also introduced to discriminate H u ( H d ) from H ν ( H ν ' ), and the corresponding charges are assigned in Table I. Under this discrete symmetry, the superpotential is given by where we omit generation indices and dots represent the SU(2) antisymmetric product. The Z 2 parity plays a crucial role in suppressing tree-level flavor-changing neutral currents and is assumed to be softly broken by tiny parameters of ρ and ρ ' ( /lessmuch µ, µ ' ). Here, we do not introduce lepton-number-violating Majorana mass for the RH neutrino N to realize a Dirac (s)neutrino. By solving the stationary conditions for the Higgs fields, one finds that tiny soft Z 2 -breaking parameters ρ, ρ ' generate a large hierarchy of v u,d ( ≡ 〈 H u,d 〉 ) /greatermuch v ν,ν ' ( ≡ 〈 H ν,ν ' 〉 ) expressed as It is easy to see that neutrino Yukawa couplings y ν can be large for small v ν using the relation of the Dirac neutrino mass m ν = y ν v ν . For v ν ∼ 0 . 1 eV, it gives y ν ∼ 1. At the vacuum of v ν,ν ' /lessmuch v u,d , physical Higgs bosons originating from H u,d are almost decoupled from those from H ν,ν ' , except for a tiny mixing of the order of O ( ρ/M SUSY , ρ ' /M SUSY ), where M SUSY ( ∼ 1 TeV) denotes the scale of soft SUSY-breaking parameters. The former H u,d doublets almost constitute Higgs bosons in the MSSM - two CP -even Higgs bosons h and H , one CP -odd Higgs boson A , and a charged Higgs boson H ± - while the latter, H ν,ν ' , constitutes two CP -even Higgs bosons H 2 , 3 , two CP -odd bosons A 2 , 3 , and two charged Higgs bosons H ± 2 , 3 . Thus, our model does not suffer from a large invisible decay width of SM-like an Higgs boson h even for a large y ν and a light lightest-supersymmetric-particle (LSP) dark matter. At the vacuum, the mixing between left- and right-handed sneutrinos is estimated as We find that the RH sneutrino ˜ N has very suppressed interactions with the SM-like Higgs boson or Z boson at tree level, since they are proportional to the mixing of left-handed and RH neutrinos' sin θ ˜ ν in Eq. (12). However, radiative corrections induce a sizable coupling between RH sneutrinos and the Z boson. We have parametrized the effective interaction between the RH sneutrino DM and Z boson by C eff ; then, the vertex induced by the scalar ( H ν -like Higgs boson and ˜ ν L ) loop 2 is given as with where k µ 1 and k µ 2 are the ingoing and outgoing momenta of the RH sneutrino and for simplicity we take equal masses for particles in the loop, M = M H ν = M ˜ ν L . By comparing Fig. 4 and Eq. (7) with Eq. (14), we find the parameter set can explain the CDMS II Si result.", "pages": [ 7, 8, 9 ] }, { "title": "B. Annihilation cross section", "content": "The dominant tree-level annihilation mode of ˜ N in the early Universe is the annihilation into a lepton pair ˜ N ˜ N ∗ → ¯ f 1 f 2 mediated by the heavy H ν -like Higgsinos as described in Fig. 5. The final states f 1 and f 2 are charged leptons for the t -channel ˜ H ν -like charged Higgsino ( ˜ H ± ν ) exchange, while thy are neutrinos for the t -channel ˜ H ν -like neutral Higgsino ( ˜ H 0 ν ) exchange. The thermal averaged annihilation cross section for this mode in the early Universe when using the partial wave expansion method is given by [48] where we used 〈 v 2 rel 〉 = 6 T/M DM with v rel being the relative velocity of annihilating dark matter particles, m f is the mass of the fermion f , and M ˜ H ν /similarequal µ ' denotes the mass of the ˜ H ν -like Higgsino. For simplicity we have assumed that Yukawa couplings are universal for each flavor. Since the s -wave contribution of the first term on the right-hand side is helicity suppressed, the p -wave annihilation cross section of the second term is relevant for the dark matter relic density at the freeze-out epoch. In the neutrinophilic Higgs model, the sneutrino has - in addition to the tree-level processes - a sizable annihilation cross section into two photons through a one-loop diagram, which has been pointed out in Ref. [33]. The charged components of the H ν scalar doublet and charged scalar fermions make the triangle or box loop diagram, and the two photons can be emitted from the internal charged particles. For the mass spectrum we are interested in now, M H ν , M ˜ l /greatermuch M ˜ N , we obtain the annihilation cross section to two photons via one loop as where we have used M H ν = M H ' ν = M ˜ l ≡ M ch for simplicity. Therefore for the total annihilation cross section of RH sneutrino DM, we obtain Now if we attempt to reproduce the latest CDMS II-Si data by taking a parameter set given by Eq. (15), we find that two-photon production via one loop is dominant and thus for the given parameters in Eq. (15). This loop-induced annihilation does not only dominate the tree-level annihilation but also exceeds the standard value 〈 σv 〉 /similarequal 10 -9 GeV -2 . This DM appears to not have the correct thermal relic abundance if the relic density is determined from its thermal freeze-out.", "pages": [ 9, 10, 11 ] }, { "title": "C. Dark matter relic abundance and indirect DM search constraints", "content": "As stated above, from Eq. (19) we see that the standard thermal relic density of ˜ N with zero chemical potential leads to a too small value for Ω h 2 /lessmuch 0 . 1. However, we know that our Universe is baryon asymmetric. Hence, we expect that lepton asymmetry is also nonvanishing. In fact, the sphaleron process, which interchanges baryons and leptons, plays an important role in many baryogenesis mechanism and leaves a similar amount of baryon asymmetry and lepton asymmetry. Because our model is supersymmetric, a promising mechanism would be Affelck-Dine (AD) baryo(lepto)genesis [49]. Candidates for a promissing AD field φ are, e.g., ¯ u ¯ d ¯ d or LL ¯ e directions with the nonrenormalizable superpotential ∆ W = φ 6 /M 3 , where M is a high cutoff scale for this operator. The generated baryon ( q = B ) or lepton ( q = L ) asymmetry for those directions have been studied by many authors and evaluated as [50-55] for a relatively low reheating temperature after inflation T R in gravity-mediated SUSYbreaking models, where m φ and A φ are soft SUSY-breaking mass and A term for the AD field, δ is an effective CP phase, M P is the reduced Planck mass, and M is taken to be around the grand unification scale. Then, the charge of Q -balls, even if they are formed, is small enough for a Q -ball to evapolate quickly [56] and to not affect the dark matter density. 3 To be precise, this generated B -L asymmetry is related to the baryon and lepton asymmetry generated by the sphaleron process [57], Since a Dirac sneutrino carries a lepton number and has a large annihilation cross section [as in Eq. (19)], our sneutrino is one of the natural realizations of the so-called asymmmetric dark matter (ADM) [58-64], and in our model only ˜ N remains after annihilation with ˜ N ∗ . Thus, the relic abundance is actually determined by its asymmetry and the mass. For a novanishing sneutrino asymmetry similar to the baryon asymmetry, and a mass of about 5 -6 GeV, the correct relic density for dark matter Ω ˜ N h 2 /similarequal 0 . 1 is obtained. Finally, we note that our model is free from any indirect search for DM annihilation; in other words, DM annihilation cannot produce any signal because of the ADM property, namely, the absence of anti-DM particles in our Universe.", "pages": [ 11, 12 ] }, { "title": "IV. CONCLUSION", "content": "We have shown that mostly right-handed Dirac sneutrinos are a viable supersymmetric light DM candidate and have a large enough cross section with nucleons to account for possible signals observed at direct DM searches. The Z -boson-mediated scattering does not relax the tension among direct DM search experiments and is constrained by the invisible decay width of the Z boson. Nevetherless, we have found that a part of the signal region for CDMS II(Si) events remains outside the excluded region by XENON100. As an example of specific particle models, we have shown that a Dirac right-handed sneutrino with neutrinophilic Higgs doublet fields is a viable light dark matter candidate.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "K.-Y.C. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology Grant No. 2011-0011083. K.-Y.C. acknowledges the Max Planck Society (MPG), the Korea Ministry of Education, Science and Technology (MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics (APCTP).", "pages": [ 12, 13 ] } ]

[ { "title": "Detectability of gravitational effects of supernova neutrino emission through pulsar timing", "content": "Ken D. Olum, 1 Evan Pierce, 1 and Xavier Siemens 2 1 Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA 2 Center for Gravitation and Cosmology, Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin, 53201", "pages": [ 1 ] }, { "title": "Abstract", "content": "Core-collapse supernovae emit on the order of 3 × 10 53 ergs in high-energy neutrinos over a time of order 10 seconds, and so decrease their mass by about 0.2 M /circledot . If the explosion is nearly spherically symmetric, there will be little gravitational wave emission. Nevertheless, the sudden decrease of mass of the progenitor may cause a change in the gravitational time delay of signals from a nearby pulsar. We calculate the change in arrival times as successive pulses pass through the neutrino shell at different times, and find that the effect may be detectable in ideal circumstances. PACS numbers: 95.30.Sf 97.60.Gb 97.60.Bw 04.30.Db", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Core-collapse supernovae emit most of their energy in the form of neutrinos, with on the order of 3 × 10 53 ergs emitted. Such neutrinos were detected from SN1987A over the course of 13 seconds. If the emission is anisotropic, these neutrinos could be the source of a gravitational wave signal, but that will not be our interest here, nor will we be concerned with any non-gravitational interactions of the emitted neutrinos. Instead we consider gravitational time-delay effects resulting from the sudden expulsion of mass of order 0.2 M /circledot into a shell that moves outward at the speed of light. We propose to detect these effects via pulsar timing in the case where there is a stable pulsar located near the supernova. The magnitude of the effect can be estimated as follows. Suppose a pulsar and a core-collapse supernova are are equidistant from Earth and separated from each other by distance b . Before the supernova, signals from the pulsar are gravitationally redshifted by an amount z i = GM i / ( c 3 b ), where M i is the mass of the star before the supernova, G Newton's constant, and c the speed of light. We have ignored effects of higher order in GM/ ( bc 3 ). Signals emitted from the pulsar once time b has elapsed after the supernova are redshifted by the smaller amount z f = GM f / ( c 3 b ), where M f is the mass of the star excluding energy emitted at the speed of light, which is primarily in the form of neutrinos. Later signals thus are blueshifted relative to those emitted before the supernova by where m is the mass emitted in neutrinos. If we have an observation period which is also equal to b , the blueshift z will cause an advance of the time of arrival by Gm/c 3 over the period of observation, with no dependence on b . With m ∼ 0 . 2 M /circledot , Gm/c 3 ∼ 1 µs , and since pulsar timing accuracy is at the level of tens of nanoseconds, such an effect could easily be detected. In the rest of this paper, we analyze this effect more carefully. In Sec. II we give the effect of passing through the shell on the frequency of a photon and the motion of an observer. In Sec. III we compute the resulting blueshift of the pulsar signals as observed on Earth, in Sec. IV we give the resulting advancement of pulse arrival times, and in Sec. V we discuss the prospects for observation.", "pages": [ 2 ] }, { "title": "II. NEUTRINO SHELL METRIC", "content": "We will now compute the metric of the spacetime containing the expanding neutrino shell. We will not include the mass that stays behind in the remnant, because it was present before and after the supernova and thus does not contribute at first order to the change in time delay. We also ignore the ordinary matter ejected during the explosion, because this moves at much less than the speed of light and so will not reach distances comparable to b during a reasonable period of observation. So we will take the spacetime inside the expanding shell to be flat and spacetime outside to be given by the Schwarzschild metric with mass m . One could join these two regions by using the Israel junction conditions [1], but it seems to be somewhat easier just to use the Vaidya metric [2] directly. So let u be a null coordinate constant on outgoing rays and let m ( u ) be the mass inside coordinate u . Let u = 0 be the coordinate of the neutrino shell, so that m ( u ) decreases from m to 0 in a narrow range of u around 0. We then have the metric in spherical coordinates [2], where d Ω 2 = dθ 2 +sin 2 θdφ 2 , and we work in units where c = G = 1. We can convert to a Schwarzschild-like time coordinate t by taking so Inside the shell we have flat space, and outside we have the Schwarzschild metric, Let V a out = ( V t out , V r out , V θ out , V φ out ) be some vector outside the shell. If we parallel transport V a along any path x ( λ ) that crosses the shell, we find We can take a very short path across the shell, so that the connection will only be important if it has components depending on dm/du. There is just one such component, Γ r uu = -( dm/du ) /r . So the only term in the parallel transport equation is Thus the integral does not depend on the path, and crossing the shell from outside ( m ( u ) = m ) to inside ( m ( u ) = 0) decreases V r by ( m/r ) V u , so and otherwise V a in = V a out . We can convert to ( t, r ) coordinates using to find We will ignore terms of order m 2 throughout, so we can write", "pages": [ 2, 3, 4 ] }, { "title": "III. BLUESHIFT FROM SHELL", "content": "Now consider a photon that is emitted outside the shell but observed on Earth after the shell has passed. The mass of the shell leads to a gradual deflection of the photon path while the photon is outside the shell, and crossing through the shell leads to a sudden deflection. But the size of these deflections is linear in m and so their effects on frequency and arrival time will be proportional to m 2 , and can be ignored. The important effect is the difference in frequency of photons that pass through the shell at different places. Let ω 0 be the photon frequency measured by a stationary observer at infinity. If the photon crosses the shell at radius r , it will have immediately before the crossing the frequency 4-vector with ( K r ) 2 + fr 2 [( K θ ) 2 +sin 2 θ ( K φ ) 2 ] = ω 0 . Inside the shell, the photon frequency becomes If the photon makes angle α to the radial direction at the time of crossing, we thus have This corresponds to the photon having received an inward boost with velocity m/r . If the photon is traveling radially inward, then ω in = ω 0 f -1 R . For photons traveling nearly radially outward, and thus barely overtaken by the shell, ω in = ω 0 . One might think that the sudden frequency change on crossing the shell would lead to a discontinuous frequency shift of observed signals, and that such an effect could be observed even at great distances from the supernova. But this is not the case. Consider an observer who is stationary at fixed radius r outside the shell. His 4-velocity in ( t, r, θ, φ ) coordinates is If he measures the photon immediately before crossing the shell, he finds frequency ω 0 f -1 / 2 , which represents the blueshift due to the mass of the shell. If the observer moves inertially as the shell passes, his 4-velocity afterwards is The photon crosses at the same place, and the observer's measurement of the photon frequency is just as before. The shell gives equal boosts to the observer and the photon, so the observed frequency does not change. The only measurable effect is thus a smooth change in observed frequencies depending on where and when different photons crossed through the shell. We expect the distance between Earth and the supernova to be so large that the gravitational effect of the neutrino shell passing us is negligible, so we will take the observed frequency to be just ω in . When we observe a photon at time t , we find a frequency where r ( t ) is the radial position at which the shell and the photon cross, and α ( t ) is the angle between the radial direction and the photon direction at that time. We compute these quantities in the next section.", "pages": [ 4, 5 ] }, { "title": "IV. TIME-SHIFT OF PULSAR DATA FROM SHELL", "content": "We will first consider a pulsar located very far behind the supernova. We let b be the distance of closest approach between the supernova center and the photon trajectory, as shown in Fig. 1, and let t = 0 denote the time when the neutrino shell reaches Earth. We take the distance to Earth to be much larger than b and we continue to work at first order in m . If a photon crosses the shell at radial distance r , it will reach Earth at time Using b = r sin α , we can solve for r in terms of t , From Eqs. (19)-(21), the observed frequency of a photon arriving at time t is thus Correspondingly if the period between pulses observed before the supernova is T 0 , the period between pulses observed at time t is The period begins to decrease at t = 0, but only at third order. It reaches a minimum at t = √ 3 b , and then returns asymptotically to T 0 . The time of arrival advancement at time t is where τ ≡ t/b . Reinserting units, we can write the prefactor Thus the total magnitude of this effect is easily large enough to be detected. Unfortunately, the timescale is quite long, being given by the impact parameter b . Thus observations are only possible when the supernova and the pulsar (or the line of sight to the pulsar) are quite nearby. We discuss this in the next section. Equation (25) is plotted in Fig. 2. The time advancement increases very slowly at the beginning, so little effect can be seen until about t = b . There is an inflection point at t = √ 3 b , and eventually the increase is only logarithmic, but in fact the curve is quite flat from t ∼ b to t ∼ 3 b . Now consider the case where the pulsar is located near the supernova, rather than far behind. In this case we will see the time-of-arrival curve exactly as for a distant pulsar, until the shell has passed the pulsar. After that, the pulse interval, and thus the slope of the ∆ t vs. t curve, will remain the same. This helps observation if it happens when the slope is large, and hinders it if the slope is small. If the pulse emitted when the shell passes the pulsar is received at time t , then the pulsar is further from Earth than the supernova by distance x = ( t 2 -b 2 ) / (2 t ). The best possible geometry has this happen at t = √ 3 b , when the slope is greatest, which gives x = b/ √ 3. But we can see from Fig. 2 that the effect of the finite distance to the pulsar is unimportant as long as the pulsar is not too far in front of the supernova.", "pages": [ 5, 6 ] }, { "title": "V. PROSPECTS FOR OBSERVATION", "content": "Timing of pulsars requires modeling the unknown distance to the source, pulse interval, and spin-down rate. (One must also subtract the known motion of the observing station relative to the sun, and the periodic motion of the pulsar if it is in a binary.) Thus one must make a quadratic fit to observed data, and only deviations from this fit can be detected. In Fig. 3 we show potential data from a scenario with b = 10 light-years. In this case the effect is clearly observable, even after the best quadratic fit has been subtracted. The effect comes primarily from the time-of-arrival advancement when t ∼ b . Thus if the the period of observation after the supernova were significantly smaller than b , the effect would not be observable. A chance alignment of of a background pulsar with a nearby supernova is quite unlikely. Even if we were to be lucky enough to see a supernova in our own galaxy, the typical distance would be of order the distance to the galactic center, about 8kpc. A disk of radius 10 lightyears (3pc) at this distance would subtend solid angle only 4 × 10 -7 steradians, and thus the chance that the line of sight to any given pulsar would pass through this disk is only 3 × 10 -8 . Abetter possibility is that the supernova and the pulsar are both part of the same compact star-forming region. For example, consider the R136 region in the Large Magellanic Cloud. This region produced a supernova seen by John Herschel in 1836, and also contains a pulsar, PSR J0537-6910, which was formed in a supernova about 4000 years ago. The angular distance between these objects is 21 '' . Taking a distance of 50kpc to R136 gives a transverse distance of only 16 light-years between the supernova and the pulsar. SN 1987A occurred at the edge of the Tarantula Nebula, about 700 light years from PSR J0537-6910. The time-of-arrival advancement due to this supernova is thus still in the initial quartic segment and would not be detectable within a reasonable period of observation. A supernova in a binary star system that already contains a pulsar would be an even better but far rarer target. Signals from the pulsar that pass through the neutrino shell reach Earth before signals emitted after the shell has reached the pulsar. Thus the effect could in principle be observed before any disturbance of the pulsar due to its proximity to the supernova.", "pages": [ 6, 7, 8 ] }, { "title": "Acknowledgments", "content": "We would like to thank Ben Shlaer and Carrie Thomas for useful conversations. This work was supported in part by the National Science Foundation under grant numbers 0855447 and 1213888.", "pages": [ 8 ] } ]

[ { "title": "A Clustering Analysis of the Morphology of the 130 GeV Gamma-Ray Feature", "content": "Eric Carlson 1 , Tim Linden 1 , Stefano Profumo 1 , 2 and Christoph Weniger 3 1 Department of Physics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064 2 Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064 and 3 GRAPPA Institute, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands Recent observations indicating the existence of a monochromatic γ -ray line with energy ∼ 130 GeV in the Fermi-LAT data have attracted great interest due to the possibility that the line feature stems from the annihilation of dark matter particles. Many studies examining the robustness of the putative line-signal have concentrated on its spectral attributes. Here, we study the morphological features of the γ -ray line photons, which can be used to differentiate a putative dark matter signal from astrophysical backgrounds or instrumental artifacts. Photons stemming from dark matter annihilation will produce events tracing a specific morphology, with a statistical clustering that can be calculated based on models of the dark matter density profile in the inner Galaxy. We apply the DBSCAN clustering algorithm to Fermi γ -ray data, and show that we can rule out the possibility that 1 (2, 4) or fewer point-like sources produce the observed morphology for the line photons at a 99% (95% , 90%) confidence level. Our study strongly disfavors the main astrophysical background envisioned to produce a line feature at energies above 100 GeV: cold pulsar winds. It is highly unlikely that 4 or more such objects have exactly the same monochromatic cosmic-ray energy needed to produce a γ -ray line, to within instrumental energy resolution. Furthermore, we show that the larger photon statistics expected with Air Cherenkov Telescopes such as H.E.S.S.-II will allow for extraordinarily stringent morphological tests of the origin of the 'line photons'. PACS numbers: 98.70.Rz, 95.55.Ka, 95.35.+d, 97.60.Gb", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The launch of the Fermi Large Area Telescope (LAT) in 2008 has allowed for a greatly expanded view of the γ -ray sky, including a significantly enhanced energy and angular resolution, compared to previous missions [1]. These characteristics have allowed, in particular, for a thorough investigation of the extremely dense population of high-energy γ -ray sources in the Galactic center (GC) region [2]. The GC region is known to host such diverse γ -ray sources as supernova remnants [3], highly ionized gas [4], dense molecular clouds [5], massive O/B stars [6], both young and recycled pulsar populations [7], as well as being the densest region of dark matter in the Galaxy [8]. Notably, no other location in the sky is expected to provide a signal from dark matter annihilation which is as bright as the GC (for a recent general review of gamma-ray searches for signals from dark matter annihilation see Ref. [9]). While this makes the GC region an extremely interesting location for a multitude of scientific studies, it also means that additional information, such as characteristic spectra for each source class, must be carefully considered in order to separate the desired signal from the bright background. Recently, Bringmann et al. [10] and Weniger [11] found indications for a unique spectral signature in observations of the region surrounding the GC, which is consistent with dark matter annihilation. Specifically, in sky regions optimized for large signal-to-noise ratios for various dark matter density profiles, they observed an excess of photons with an energy spectrum resembling a 130 GeV γ -ray line, smeared by the finite energy resolution of the Fermi-LAT telescope. The significance of this excess was found to be 3 . 2 σ globally [11] - enough to make the feature interesting on statistical grounds. The feature is strongest when using regions of interest that have been optimized for dark matter density distributions following: (1) a Navarro-Frenk-White (NFW) profile [12], (2) an Einasto profile [13], and (3) a generalized NFW profile, with a radial slope governing the dark matter density profile (r -α ) set to α = 1 . 15, similar to what could result from adiabatic contraction [14, 15]. The reported feature is much weaker for profiles where the dark matter density is cored near the GC. A monochromatic γ -ray line has been long considered the 'Holy Grail' for dark matter indirect detection, given the difficulty of producing a high-energy monochromatic signal with ordinary astrophysical processes. Thus, this observation prompted a number of follow-ups. Profumo and Linden [16] noted that the observation of a monochromatic γ -ray signal could be qualitatively mimicked by an additional power-law component which breaks strongly at an energy of 130 GeV, and posited the Fermi bubbles [17] as a possible source for this excess (although rather strong breaks are required to fit the data, see Ref. [9]). Most notably, Su and Finkbeiner [18] localized the emission to within approximately 5 · of the GC, finding a 6.5 σ (5.0 σ after including a trials factor) preference for a line signal following an off-center Einasto profile compared to the assumption of a simple power-law background (see also [19]). This finding disputes the implication of the Fermi bubbles as a source for the excess, as the latter are observed to extend over a much larger emission region. The off-centered nature of the dark matter profile is at the moment only marginally statistically significant [20, 21]. However, if confirmed, an off-center peak would pose a challenge to traditional models of the Galactic dark matter density which assume the largest dark matter density to fall on top of the peak of the baryonic mass density 1 . However, a recent analysis by Kuhlen et al. [22] found that the peak in the dark matter density may, in fact, be displaced by hundreds of parsecs from the dynamical center of the Galaxy - although this scenario might be incompatible with the assumption of a cusped profile. More data are needed to clarify whether the 130 GeV feature in the data persists with larger statistics or if it is only a statistical fluke. Since the observation of a γ -ray line is extremely sensitive to inaccuracies in the energyreconstruction of γ -rays observed by Fermi-LAT, a great deal of interest has also focused on searching for possible instrumental abnormalities affecting the photons belonging to the γ -ray line. The most notable characteristic of any such instrumental effect would be the observation of line activity either across the entire sky, or across a certain region of instrumental phase space. Interestingly, some early results found an excess of 130 GeV events in observations of the Earth-limb [18, 23, 24]. This is troubling, as the vast majority of limb photons are known to result from the di-photon π 0 decay spectrum created as cosmic-ray protons interact with the upper layers of the Earth's atmosphere. There is no conceivable model in which dark matter annihilation could create a γ -ray line in this region [18]. However, this line activity was not detected along the Galactic plane, which provides significantly more photons than the GC, and inhabits similar regions of instrumental phase space. A comprehensive study by Finkbeiner et al. [23] did not find any significant evidence for systematic features in the energy reconstruction of the Fermi-LAT, which would be able to artificially produce a γ -ray line (see however Refs. [25-27]). Follow up analyses by the Fermi-LAT team are currently ongoing, but have revealed two noteworthy results. At the time of the initial discovery of the γ -ray line, efforts were already ongoing to improve the energy normalization of the Fermi-LAT data, accounting for a decrease in the calculated energy of γ -rays over time due to radiation damage to the calorimeter. This effect linearly increased the reconstructed energy of high energy photons, moving the line signal from 130 GeV up to approximately 135 GeV. However, this reprocessing did not greatly affect any other signature of the posited line analysis. We note that throughout the rest of this paper, we refer to the '130 GeV line', as we are using a version of the Fermi-LAT data which has not been reprocessed. However, all results shown here are very nearly applica- e to an analysis of the 135 GeV line observed in the reprocessed data. Additionally, the Fermi-LAT analysis did uncover one troubling aspect of the posited γ -ray line. Employing a parameter CTBBestEnergyProb (which is not publicly available), they investigated the confidence they had in the energy reconstruction of each photon belonging to the γ -ray line. In the case where a true γ -ray line feature is present in the data, this should increase the statistical significance of the observation, as the proceedure adds additional statistical weight to the line photons which are most likely to have a correctly measured energy. However, when this analysis was applied to the observed photon data, the statistical significance of the line feature was found to decrease moderately. This signals that the line feature has photons with a somewhat poorer energy resolution than would generally be expected[28]. However, further inquiry of these systematic issues is required, as none of the systematics can clearly account for the entire statistical strength of the line feature. If interpreted as a signal of particle dark matter annihilation or decay, the large observed intensity of the 130 GeV γ -ray line (along with strong constraints on the total continuum emission from additional hadronic states [29]) has proved a difficult, though by no means intractable, particle physics problem. Numerous models have already been posited to 'brighten' the γ -ray line [30-59]. Summarizing the myriad particle physics details of these models lies beyond the scope of the present paper. Although often characterized as a 'smoking gun' signature for the annihilation or decay of particle dark matter, tentative observations of the 130 GeV line have spurred the question of whether any traditional astrophysical mechanism might mimic a line in the relevant energy range. Aharonian et al. [60] argued that the only plausible mechanism for the creation of an astrophysical γ -ray line is through inverse Compton scattering of ambient photons by a jet of nearly monoenergetic electrons and/or positrons, occurring in the deep Klein-Nishina regime. If the latter kinematic regime holds, the photon acquires nearly the entire energy of the incoming lepton, allowing for a nearly monoenergetic lepton spectrum to efficiently transfer into a sharply peaked γ -ray feature. One possible class of astrophysical objects that possesses the potential to host the needed leptonic monochromatic 'jet', as well as the ambient photons in the needed energy range (here, a few eV) is cold ultrarelativistic pulsar (PSR) winds [60]. It is important to note that this scenario presents a potential difficulty in explaining the observed spread of 130 GeV photons beyond a single point source, as different PSRs would be expected to exhibit γ -ray lines at different energies. One way to test this one astrophysical background is therefore to study whether the morphology of the observed 130 GeV photons can be reproduced with a small number of point sources or not. This is the key objective of the present study. Several other approaches have tested the dark matter nature of the 130 GeV photons. For example, any dark matter interpretation of the 130 GeV line implies additional regions of interest for follow up searches, where the so-called J -factor (the line-of-sight integral of the dark matter density squared, smeared over the instrumental point-spread function) is expected to be largest. Most importantly, dwarf spheroidal galaxies and galaxy clusters have been singled out as promising regions to search for a dark matter signal. Observations of Milky Way dwarfs have not uncovered any evidence of a 130 GeV signal [61]: this is not unexpected, as the estimated annihilation cross-section to γγ implied from GC observations predicts less than one photon to arrive from the population of dwarf spheroidal galaxies. Interestingly, Hektor et al. [62] argued for an observation of a 130 GeV line in a population of nearby galaxy clusters. However, it should be noted that this signature is only significant when very large ROIs of ∼ 8 · are considered, which is much larger than the expected angular size of the galaxy clusters under investigation. Using a similar method, Su and Finkbeiner [63] investigated the population of unassociated Fermi-LAT point sources - that is, point sources detected by the FermiLAT instrument which have not been identified at other wavelengths. They found a statistically significant detection for a double 130 GeV and 111 GeV line, with 14 unassociated sources showing evidence of a line photon. Furthermore, they found no significant detection of γ -ray line emission in the control sample of Fermi-LAT point sources that have already been associated with various astrophysical phenomena. This caused them to conclude that some portion of the unassociated point-source sample may contain previously unknown dark matter substructures. However, Hooper and Linden [64] argue against this conclusion, noting that each unassociated source is identified primarily based on its continuum emission between energies of 100 MeV-10 GeV, rather than based on the detection of a single line photon. The intensity and spectrum of this continuum emission can then be compared to the signal from dark matter annihilation to any final state producing a γ -ray continuum, and is expected in order to produce the thermal relic abundance of dark matter. They find that for at least 12 of the 14 indicated unassociated point sources, the continuum emission is not compatible with any dark matter annihilation pathway. Furthermore, they argue that the latitude distribution of the identified sources is not consistent with that expected from any model of dark matter subhalo formation. A second analysis by Mirabal [65] argued that while these 14 sources remain unidentified, at least 12 of the sources (not identical to those from [64]) are spectrally strongly consistent with AGNs. In addition to considering the energy signature of the 130 GeV line, understanding the morphology of the photons belonging to the source class producing the observed feature will be key to elucidate the physics behind the line phenomenon. Notably, the point-spread function of front-converting events at energies near 100 GeV approaches 0.1 · [1], which is significantly smaller than the ∼ 5 · region of interest implicated by [18], allowing for the actual morphology of the line emission to be closely mirrored by Fermi-LAT observations. To first order, observations indicating a morphology consistent with widely accepted dark matter density profiles would provide additional evidence for a dark matter interpretation (the data indeed points to that direction, see e.g. Fig. 3 of Ref. [9]), while measurements consistent with either a population of point sources, or a significantly flattened profile may point to other astrophysical or instrumental interpretations. In this paper , we examine the morphology of photons belonging to the γ -ray line more closely, analyzing with a sound statistical approach the distribution of the arrival direction of photons by employing a clustering algorithm to pinpoint the correlation between the arrival directions of photons putatively belonging to the line feature. We then compare these results against simulated models where the line is produced by dark matter annihilation or by an astrophysical process associated with a few point sources (for example a handful of PSRs). The key result of our study is that current data disfavor a scenario where the line photons stem from 4 or fewer point sources . Given how unlikely it is that 4 or more pulsars produce a gamma-ray line at exactly the same energy (within the LAT energy resolution), our study disfavors the PSR scenario over a truly diffuse and un-clustered origin for the photons. In Section II we describe the data employed in our observations of the γ -ray line feature, the specifics of the algorithms used to determine the photon morphology, our models for both the annihilation of dark matter and emission from PSRs, and the diffuse background in the region. In Section III we present the results of our study both for current Fermi observations, and for projections for upcoming observations of the GC with the Atmospheric Cherenkov Telescope (ACT) H.E.S.S.-II. Finally, in Section IV we discuss the interpretation of the results, and present our conclusions.", "pages": [ 1, 2, 3 ] }, { "title": "A. Photon Selection", "content": "In order to analyze the population of photons stemming from the putative γ -ray line emission, we must make a photon selection which isolates the line photons from those correlating to background events. We follow here the same photon selection employed in Ref. [66], which provides the location of observed Fermi-LAT photons in three energy bands, 70-110 GeV, 120-140 GeV, and 150300 GeV over a 10 · square window centered on the GC. Since the Fermi-LAT energy resolution is approximately 10% at 130 GeV, we assume photons in the 120-140 GeV band to encompass the photons related to the γ -ray line observation, while photons in the low and high energy bands correspond to background events not associated with the γ -ray line observation. Below, a power-law fit to the sidebands will be used to fix the background rate in our 120-140 GeV simulations while the remainder of the photon excess will make up the signal. We note that there is some evidence for a second line at energies of around 111 GeV [18], however the weak significance of that feature makes its impact on the population of 70110 GeV γ -rays negligible. In comparing the photon morphology from the line region against the 'side-band' photons, we assume that the background morphology remains approximately invariant throughout the 70-300 GeV energy range. This assumption is warranted in light of observations indicating that the primary component of diffuse emission through this region stems from π 0 -emission tracing the Galactic gas [29, 67]. While some unresolved point-sources may also be present, it is unlikely that any given source contributes multiple photons to the observed high-energy γ -ray emission, making the spectral features of each individual source irrelevant. Before proceeding with the description of the clustering algorithm we employ in this analysis, we describe in the following sections the simulated data sets we use to validate our analysis. Section II B details the simulated line events from both dark matter annihilation scenarios with different dark matter density profiles, and scenarios with one or more point sources. Section II C details the simulated background events. Finally, sec. II D describes the clustering algorithm we employ in the present study.", "pages": [ 3, 4 ] }, { "title": "B. Dark Matter and Pulsars Models", "content": "In order to establish a quantitative measure for the clustering properties of γ -rays due to either dark matter or one or more point-sources in the GC region, we produce Monte Carlo simulations of the expected positions of photons stemming from each model. In the case of PSRs, we examine models featuring between 1 and 6 point sources to explain the excess 130 GeV emission. We randomly pick the distribution of each point source following a surface density distribution ρ (r) ∝ r -1 . 2 , as motivated by the observed density distribution of O/B stars in the inner Galaxy [68] and we produce an excess number of photons which are distributed randomly (assuming equal brightness) between the simulated pulsars. In the case of dark matter annihilation, we predict the annihilation signal to follow the integral over the line of sight of the square of the dark matter density. We choose two independent dark matter density profiles motivated by models of dark matter structure formation. We first examine a generalized Navarro-Frenk-White [12] profile, with a density profile In our standard analysis we choose α =1 and r s =22 kpc fitting the best numerical results from the Aquarius simulation [13]. In order to evaluate the effect of changing the dark matter density profile, we also consider an Einasto profile with a density distribution [13]: assuming, here, that α = 0.17 [13]. In each case, we assume that the annihilation rate is proportional to ρ 2 (r), and then integrate over the line of sight from the solar position R glyph[circledot] = 8.3 kpc [69] in order to generate the dark matter morphology that would be observed by the FermiLAT. We additionally consider two alternative profiles. First, the case of decaying dark matter following the NFW profile given in Eq. (1), with the decay rate now proportional to ρ (r). Second, the case of isotropic emission, i.e. a uniform surface profile. Since the clustering properties of the source are highly dependent on the number of observed photons, we calculate for Fermi-LAT (H.E.S.S.-II), 10 5 (2000) realizations of 48 (5000) photons following the distribution assumed in each of these cases over a 10 · (4 · ) square window. For H.E.S.S.-II observations, we estimate the number of photons from a relatively short exposure time, on the order of a 6.25 hours, using an effective area given for the H.E.S.S.-II telescope with the flux in the 130 GeV energy range measured by the Fermi-LAT. In each case, we must also consider the smearing of target photons based on the point-spread function of the Fermi-LAT telescope. In order to accomplish this accurately for observations at the GC, we employ the Fermi tools to estimate the point-spread function for photons entering both the front and back of the instrument at different θ -angles. Specifically, we employ the gtpsf tool developed by the Fermi-LAT collaboration in order to calculate the effective PSF given the total exposure of the GC region from all locations in the Fermi-LAT instrumental phase space (i.e. how much was the GC viewed from different spacecraft orientations). For the selected observation period, the average (68%, 95%) containment radius over the observation is (0.124 · ,0.529 · ) for front-converting events ( ∼ 56% of exposure area) and (0.258 · ,0.907 · ) for rear converting events ( ∼ 44% of exposure area). The resulting PSF for each photon (signal and background) is randomly chosen based on this weighted average of instrument coordinates and the incoming photon is smeared based on the given PSF. In the case of Atmospheric Cherenkov Telescope (ACT) simulations, the PSF depends somewhat sensitively on the angle of incidence of the incoming photons. Following Aharonian et al [70] we approximate the H.E.S.S. point-spread function as an energy independent, two-component Gaussian with the probability density of an event smearing to radius θ given by, Where σ 1 = 0 . 046, σ 2 = 0 . 12, and A rel = 0 . 15 and an overall normalization A .", "pages": [ 4, 5 ] }, { "title": "C. Background Models", "content": "In order to characterize the morphology of the expected diffuse background we follow the detailed PASS 7 Galactic Diffuse Model, which contains both spectral and morphological information generated by observations of both HI and CO line surveys, which constrain the distribution of interstellar gas. The γ -ray morphology and spectrum are then generated by convolving these maps with the modeled cosmic-ray densities utilizing the Galprop code [71], and calculating the expected γ -ray emission from processes including π 0 -decay, bremsstrahlung emission, and inverse-Compton scattering. Utilizing these simulations, we then generate a Monte Carlo population of background γ -rays following a morphology compatible with observations across the γ -ray spectrum. In the case of the Fermi-LAT telescope, we assume zero cosmic-ray contamination, since we are considering only the GC region, which is very bright in γ -rays. Given the calculated intensity of the γ -ray line, we calculate an average of 12 signal, and 36 background photons between an energy of 120-140 GeV. In each simulation of the Fermi-LAT data, we allow the strength of this signal to float using Poisson statistics, setting the mean intensity to be 12 signal photons. For Fermi-LAT observations we model a 10 · square window around the GC. In the case of ACT observations, we note that cosmicray contamination is a much larger issue, since hadronic showers dominate the data collected by these telescopes. Extrapolating the cosmic-ray and γ -ray signals from recent low-energy H.E.S.S. observations at 300 GeV [72] and using the best estimates for the H.E.S.S.-II instrumental characteristics [73], we find that 86% of the total background signal will stem from cosmic-ray backgrounds. Thus, for ACT observations we create a simulation composed of 4.35% signal photons, 13.15% diffuse background photons (following the Fermi PASS-7 Galactic diffuse model) and 82.5% isotropic background photons. As in the case of Fermi-LAT observations, we allow the total number of signal photons to float using Poisson statistics, and maintain a background which is 86% isotropic, and 14% diffuse. For H.E.S.S.-II observations we model a 4 · square window around the GC, consistent with the smaller field of view of ACT instruments. The background model described above is also used to estimate the average background count N b during the computation of the cluster significances. N b is calculated by integrating the background template, normalized to the correct background count over the 95% containment area of the cluster members (100% containment in cases with fewer than 20 cluster members). This allows for a statistical measure which traces the local morphology of the background and is thus minimally dependent on its anisotropic structure, reducing the significance of 'hotspots' in the background which may be falsely identified as true clusters.", "pages": [ 5 ] }, { "title": "D. Clustering Algorithm", "content": "In order to classify the spatial morphology of photons in a statistically robust way, we employ the Density Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm [74], which is capable of both distinguishing cluster points from noise and constraining the maximum connectivity size based on the instrumental point-spread function. DBSCAN possesses two input parameters, corresponding to the assumed radius ( glyph[epsilon1] ) of each cluster neighborhood and the number of points (N min ) which must be contained within a neighborhood to form a new cluster or add to an existing cluster. Our implementation of DBSCAN, modified from the Scikit-Learn python package [75], works as follows: Traditional DBSCAN implementations also define the notion of 'density reachable' to indicate points which are not themselves density-connected, but which lie within a core point's glyph[epsilon1] -neighborhood. This property is, however, not symmetric, and cluster assignment in general depends on the input ordering of the data. Our algorithm ignores density-reachable points, thus ensuring deterministic results. While we assume, in this analysis, that each profile is centered at the position of the GC, our results are independent of this assumption as the DBSCAN algorithm focuses on the relative position between photons, oblivious to any zero point of the profile. We note that a slightly modified DBSCAN algorithm has already been employed on Fermi-LAT data in the past [76]. To exemplify the use of the DBSCAN algorithm, Figure 1 shows the DBSCAN analysis of the Fermi-LAT photon events measured with an energy between 120 140 GeV (left), and of a simulated model containing two point sources near the galactic center (right). In each case, we show the DBSCAN glyph[epsilon1] -neighborhoods for each core point of each detected cluster. For the Fermi results, DBSCAN finds only one cluster (interestingly centered on the actual Galactic center location!), while in the 3 pulsar simulation case, the algorithm correctly identifies three clusters, at the positions corresponding to where the pulsar photons were generated. The following discussion explains in detail the procedure we employ to apply DBSCAN to γ -ray data. To quantitatively compare our models against Fermi data, we follow Tramacere and Vecchio [76] and employ the likelihood ratio proposed by Li and Ma [77] to calculate the cluster significance, s , in terms of the number of cluster photons N s and background photons N b : Here, N b represents the expected background counts, determined by integrating a diffuse background model (discussed in Subsection II C), while N s is based on the total photon count contained in the cluster; we effectively adopt α = 1 in the notation of Ref. [77]. According to Ref. [77], as long as N s and N b are not too sparse, one can equate a cluster with significance s to an ' s -standard deviation observation'. Thus a cluster significance s = 2 implies the cluster is a 2 σ fluctuation above the mean background as computed in Subsection II C. We will use this nomenclature in our analysis. With the individual cluster significance in hand, we now define the 'global' significance S as the mean significance of each detected cluster weighted by the number of photons in that cluster. We then optimize the choices of the DBSCAN parame- rs glyph[epsilon1] and N min for ACT simulations by maximizing the global significance for the clustering results from our pulsar simulations. Finally we explain why this optimization procedure does not work with the limited Fermi photon count at 130 GeV and choose appropriate DBSCAN parameters based on the Fermi spatial point spread function. The value of glyph[epsilon1] must be large enough that true cluster elements are not excluded, but small enough that noise is not included. The variable glyph[epsilon1] , as a result, is closely tied to the physical size of the instrumental PSF. One must additionally choose a value N min large enough such that the background does not easily fluctuate above this number, but low enough that one has a high efficiency of finding real clusters. To this end, we use simulations of 1, 2, and 3 pulsar models for Fermi simulations, and 2, 4, and 6 pulsar models for ACT observations to determine a region of ( glyph[epsilon1], N min ) parameter space which simultaneously optimizes the significance and detection efficiency. Displayed in the top row of Figure 2 is the global clustering significance (shown in filled contours) and number of detected clusters with s > 1 . 29 (inset, labeled contours) for 48 photon Fermi simulations of 1, 2, and 3 pulsars as a function of the DBSCAN parameters glyph[epsilon1] and N min . In the bottom row, we again plot the global clustering significance and number of detected clusters with s > 2 for 5000 photon ACT simulations with columns from left to right corresponding to 2, 4, and 6 pulsar models. We note that we apply a firm cut that N min must be at least 3, as this represents the lowest possible non-trivial clustering which may be analyzed by the DBSCAN algorithm. Inspection of the all three columns for both Fermi and ACT simulations reveals that the clustering algorithm detects clusters at high significance over large coincident regions of DBSCAN parameter space. In the case of ACT observations where the clusters are better differentiated from the background, we also see that these regions also detect the correct number of clusters until the number of pulsars becomes to large to reliably detect all true clusters. This indicates that the results are robust for most reasonable choices of scan parameters while in the case of 6 pulsars, the number of detected clusters is somewhat more sensitive to parameter choices. For these ACT simulations, we see that choosing N min too small, or glyph[epsilon1] too large can lead to the identification of extra (false) clusters which lowers the overall significance. We choose our scan parameters based on the 6 pulsar simulations (bottom right) which have the lowest signal to noise ratio among the models we consider here. These considerations motivate a choice of glyph[epsilon1] = 0 . 35 · , N min = 3 for Fermi simulations and glyph[epsilon1] = 0 . 05 · , N min = 8 for ACT projections as a balance between preserving significance and detecting most of the clusters at s > 2. We note that detailed studies on the behavior of DBSCAN settings applied to Fermi-LAT data at lower energies have found qualitatively comparable optimization regions for DBSCAN parameters [76]. In summary, for ACT observations we expect ∼ 5000 photons for a 6h exposure, and use our significance measure balanced against the number of detected clusters to optimize the DBSCAN parameters. We find glyph[epsilon1] = 0 . 05 · and N min = 8. For Fermi observations, we choose glyph[epsilon1] = 0 . 35 · and N min = 3, which represents the lowest level of non-trivial clustering. We note that there are only 48 photons in our sample, and thus we do not expect to be able to identify more than a few clusters corresponding to point sources with our analysis technique.", "pages": [ 5, 6, 7 ] }, { "title": "III. RESULTS", "content": "In order to compare our models of the expected 130 GeV line signal produced by both dark matter and pulsars, we first calculate the clustering properties of the actual Fermi dataset in the energy range of 120-140 GeV using the DBSCAN algorithm. We find only one detected cluster with a significance of s = 1 . 29, an angular scale of 0.22 · (defined as the mean pairwise distance of each pair of cluster members), and 3 member photons (see Fig. 1, left). We first define the parameters useful for differentiating different emission classes and then compare to current | | Fermi data and projections for upcoming H.E.S.S.-II observations. In addition to the global significance, S , we define three quantities in the space of clusters with significance s > 1.29 for Fermi and s > 2 for ACT observations. The quantities we employ are chosen to best capture the results of each simulation based on the output of DBSCAN, and provide useful information on the clustering properties of each source model. Specifically, we use: In Figure 3 we show the results of the DBSCAN algorithm applied to both the Fermi data (vertical dasheddotted line), compared to results from Monte Carlo simulations of dark matter and point source emission from pulsars for 48 photons (left; resembling Fermi-LAT observations) and 5000 photons (right; resembling ACT observations). The dashed lines correspond to diffuse dark matter annihilation models - NFW (blue dashed), Einasto (red dashed), NFW decay model (cyan dashed) and a flat distribution (green dashed). Pulsar models are represented by solid lines - 1 (magenta), 2 (yellow), 3 (black), 4 (blue), 5 (green), and 6 (red). Each histogram is normalized and if no clusters are found, the significance defaults to zero. We thus see that it is much less likely for diffuse models to produce clusters dense enough to be picked up by DBSCAN, indicating that our combination of glyph[epsilon1] and N min are reasonably efficient at rejecting spurious background clusters, especially in the ACT case. The global cluster significance S (top row of Figure 3) provides the strongest metric for differentiating pointsource from diffuse emission. Diffuse models should possess virtually identical clustering properties as the extent of any true structure is much larger than the instrumental point spread function. Thus, diffuse sources should only occasionally produce low significance, loosely grouped clusters due to background fluctuations. One possible exception for dark matter models is the identification of a single point source at the GC where the dark matter annihilation rate can be very large. It is notable that the single cluster found in the Fermi data lies precisely on the galactic center. We see in the case of Fermi simulations (top left panel) that there is a fairly sharp cutoff in the fraction of models with global significance S < ∼ 1. This lower bound is set by DBSCAN's minimum cluster detection requirements, as well as the location of the cluster with respect to the background template, which determines the number of background photons in that region. Because the cluster detected in the Fermi data is loose, we expect the single detected Fermi cluster to be close to the effective cutoff ( S = 1 . 29 for the detected cluster). The second row of Figure 3 shows the mean clustering radius . For ACT observations there is a clear division between the point source and smaller diffuse emission scales. However, the Fermi-LAT simulations do not offer any useful discrimination between models with such low photon counts. For point sources, this distribution is governed by the average value of the PSF and possesses an asymmetric tail at larger scales due to the inclusion of background photons and events whose true position is determined by the long PSF tails. For diffuse models, the distribution is governed dominantly by the glyph[epsilon1] DBSCAN parameter until the background density becomes dominant. Displayed in the third row is the distribution of the total number of clusters , N clusters , found with significance s > 1.29 (s > 2). In the case of ACT observations we also require clusters to have at least 10 core points to be included whereas we required only 3 for Fermi. We expect to be able to discern at most N sig /N min point sources if the signal photons happen to distribute themselves evenly between sources. Even in this case, these true clusters may still lie below the significance threshold. It is realistic to identify only 2-3 true clusters with 12 signal photons (48 total) if they in fact have point source progenitors. This is reflected here. For diffuse sources, only a fraction the detected clusters pass the significance cut. As the number of events is increased, this significance cut could be increased to maintain a high acceptance/rejection ratio, though it is clear that for diffuse models, we typically obtain either zero clusters or one cluster per simulation with our current significance cuts, while still efficiently detecting several clusters in the case of pulsar models. Finally, the fourth row contains the distribution of the mean number of cluster members , N members . For a random distribution of events between pulsars, we expect a Poissonian distribution with a mean of approximately N sig /N pulsars (contributions from the background should typically be < 1 photon for Fermi, but the number of signal photons can fluctuate significantly during Poisson sampling). For Fermi-LAT observations we expect 12 ± 3 . 5 signal photons distributed between N pulsars . In the case of ACT observations, we expect 232 ± 15 photons. Occasional spurious clusters will force this distribution downwards, although this effect is reduced by the significance weighting and because we only consider 'core points' to be cluster members, thus rejecting those lying on the cluster boundaries. In order to quantify in how confidently point-source or diffuse models for the 130 GeV excess can be rejected, we count the fraction of simulations which are incompatible with Fermi-LAT data for each tentative source class. A simulation is deemed incompatible if at least 1 cluster is detected at significance s > 1 . 29 corresponding to the maximum number and significance of clusters detected in the Fermi-LAT data. The first column of Table I shows the fraction of simulations for each model with at least one cluster ( n = 1) with s i > 1 . 29. In column 3 we show the fraction of simulations which have two clusters detected with a significance of s i > 1 . 29 in Fermi-LAT simulations, which further demonstrates the vast statistical separations between the clustering properties of diffuse and point-source models. In the subsequent columns we show similar data for the H.E.S.S.-II telescope, which also requires a cluster to have at least 10 core points, and clearly provides an ever greater ability to differentiate between source classes producing the γ -ray line. Our statistical approach shows that Fermi-LAT data already rule out models where the 130 GeV γ -ray line is produced by 1, (2, 4) or fewer pulsars at the 99% (95%, 90%) confidence level (CL). Specifically, only one cluster was detected in the Fermi-LAT data with a significance s = 1 . 29, while a cluster with a larger significance is observed in more than 90% of simulations with any ensemble of less than 4 point sources. Due to the greatly increased effective area of the H.E.S.S.-II telescope, we find an even greater statistical separation between our models of diffuse and point source emission. This indicates that H.E.S.S.-II will be able to conclusively differentiate models of the 130 GeV line using purely statistical properties.", "pages": [ 7, 8, 9, 10 ] }, { "title": "IV. DISCUSSION AND CONCLUSIONS", "content": "If confirmed, the tentative detection of a γ -ray line in the Fermi data might potentially turn into one of the most important breakthroughs on physics beyond the Standard Model, pointing towards the mass of the dark matter particle. Key future developments include the analysis of Fermi γ -ray events with the forthcoming Pass 8 version of the Fermi-LAT analysis software, which will include a major overhaul of the energy reconstruction algorithm [78]. It will also be crucial to identify whether the excess events at 130 GeV from the Earth's limb are indeed a statistical fluke. This question will be answered by increased exposure, which will increase the current statistical sample [18, 28]. At present, barring instrumental effects, a 130 GeV line could be ascribed to either new physics, presumably a dark matter particle decaying or pair-annihilating into a 2 γ (or γ Z, γ h etc.) final state, or to one or more pulsars featuring a cold wind with electrons with an energy at, or very close to, 130 GeV. This latter possibility, it is argued in Aharonian et al. [60], is the only 'traditional' astrophysical process envisioned thus far that could produce a sharp gamma-ray line in the energy regime of interest. It is therefore of the utmost importance to discriminate between a cold pulsar wind scenario and a dark matter scenario, if indeed the line is resilient to future tests and observations. Discriminating dark matter and pulsar interpretations of the 130 GeV line may not be possible based solely on the spectral characteristics of the Fermi-LAT data. In the present study we sought to use morphological information, i.e. the 130 GeV events' arrival direction, to establish whether the signal is likely due to multiple point sources as opposed to a truly diffuse origin. This is a meaningful question, since the signal region is much larger than the instrumental angular resolution, and it should be thus possible to discriminate a finite number of point sources, as expected in the pulsar case, from a distribution that follows a diffuse morphology, such as what expected from dark matter annihilation or decay. To quantitatively approach the issue of discriminating pulsars versus dark matter on a morphological basis, we employed the DBSCAN algorithm which distinguishes clusters from background noise based on the local photon density. We defined a statistical significance measure, and we optimized the algorithm's two physically well-constrained parameters in order to reconstruct as accurately as possible the potential 'clusters' producing the observed γ -ray events. As a result of our analysis of the available Fermi-LAT data, we concluded that at the 99%, 95%, 90% confidence level the data need at least 2, 3, 5 or more point sources, respectively, while the events' morphology is perfectly consistent with various dark matter density profiles. We conclude that the data strongly disfavor the hypothesis of a small number of pulsars as the origin of the signal. If the pulsar scenario is indeed the culprit for the 130 GeV events, it is necessary to postulate a relatively large population of pulsars (likely at least 4) with cold winds featuring electrons with exactly, to within the instrumental energy resolution, the same energy. This appears, to say the least, quite problematic. A diffuse origin seems therefore the most likely scenario for the 130 GeV photons. Our clustering algorithm approach, clearly, is not optimized to discriminate between different diffuse morphologies: in fact, on the basis of our results, we find that we cannot discriminate between different diffuse morphologies (like dark matter annihliation vs. decay). Present and future observatories have the potential to shed additional light on the presence and characteristics of the 130 GeV line [79]. Improvements to the H.E.S.S. telescope (H.E.S.S.-II) have reduced the γ -ray threshold to around 50 GeV, allowing for the independent determination of a line signal from the GC region. This is especially important, as the 10 4 m 2 collecting area of the H.E.S.S. telescope will quickly alleviate the low-statistics issues involved in Fermi-LAT studies [60]. Furthermore, future instruments such as Gamma-400 [80] and CTA [81] are likely to provide the necessary effective area and energy-resolution to definitively and conclusively test the existence and nature the 130 GeV line feature. However, in the near future, the most important contribution is expected to come from Fermi-LAT itself: Additional data taken since last year, and the continuous accumulation of more data over the next years, will show whether the signature persists or is a rare statistical fluke. Simultaneously, the availability of pass 8 events, based on a set of completely rewritten event reconstruction algorithms for the LAT, will allow a fresh look on possible instrumental systematics.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "This work is partly supported by NASA grant NNX11AQ10G. SP also acknowledges partial support from the Department of Energy under contract DEFG02-04ER41286. 1205.3276.", "pages": [ 11, 12 ] } ]

[ { "title": "Possible High-Energy Neutrino and Photon Signals from Gravitational Wave Bursts due to Double Neutron Star Mergers", "content": "He Gao 1 , Bing Zhang 1 , 2 , Xue-Feng Wu 3 and Zi-Gao Dai 4 1 Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA, [email protected];[email protected] 2 Kavli Institute for Astronomy and Astrophysics and Department of Astronomy, Peking University, Beijing 100871, China 3 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China 4 School of Astronomy and Space Science,Nanjing University, Nanjing 210093, China As the technology of gravitational-wave and neutrino detectors becomes increasingly mature, a multi-messenger era of astronomy is ushered in. Advanced gravitational wave detectors are close to making a ground-breaking discovery of gravitational wave bursts (GWBs) associated with mergers of double neutron stars (NS-NS). It is essential to study the possible electromagnetic (EM) and neutrino emission counterparts of these GWBs. Recent observations and numerical simulations suggest that at least a fraction of NS-NS mergers may leave behind a massive millisecond magnetar as the merger product. Here we show that protons accelerated in the forward shock powered by a magnetar wind pushing the ejecta launched during the merger process would interact with photons generated in the dissipating magnetar wind and emit high energy neutrinos and photons. We estimate the typical energy and fluence of the neutrinos from such a scenario. We find that ∼ PeV neutrinos could be emitted from the shock front as long as the ejecta could be accelerated to a relativistic speed. The diffuse neutrino flux from these events, even under the most optimistic scenarios, is too low to account for the two events announced by the IceCube Collaboration, but it is only slightly lower than the diffuse flux of GRBs, making it an important candidate for the diffuse background of ∼ PeV neutrinos. The neutron-pion decay of these events make them a moderate contributor to the sub-TeV gamma-ray diffuse background. PACS numbers: 95.55.Vj; 95.85.Ry; 98.70.Rz I. Introduction. The next-generation gravitational-wave (GW) detectors, such as the advanced LIGO, advanced VIRGO and KAGRA interferometers [1], are expected to detect GW signals from mergers of two compact objects. One of the top candidates of these gravitational wave bursts (GWBs) is the merger of two neutron stars (i.e. NS-NS mergers) [2]. The study of the electromagnetic (EM) counterpart of such GWBs is of great interest [3]. Numerical simulations show that mergers of binary neutron stars would leave two remnants, a postmerger compact object and a mildly anisotropic ejecta with a typical velocity of ∼ 0 . 1 -0 . 3 c (where c is the speed of light) and typical mass of ∼ 10 -4 -10 -2 M /circledot [4]. Even though a black hole is usually taken as the post-merger product, observational data and numerical simulations suggest that for a stiff equation of state of nuclear matter and a small enough total mass of the two neutron stars, the postmerger product could be a stable hypermassive, millisecond magneter [5-9]. Recently, Ref. [7, 8] have systematically studied the EM signals for the NS-NS scenario with a stable millisecond magnetar post-merger product. Zhang [7] proposed that the proto-magnetar would eject a near-isotropic Poyntingflux-dominated outflow, the dissipation of which would power a bright early X-ray afterglow for essentially every GWB of NS-NS merger with a magnetar central engine. Gao et al. [8] proposed that after the dissipation, within the framework of an energy injection scenario [10], a significant fraction of the wind energy would be used to push the ejecta launched during the merger, which would ac- rate the ejecta to mildly or even highly relativistic speed, making a strong external shock upon interaction with the ambient medium. Electrons are accelerated in the shocked region, giving rise to broad band afterglow through synchrotron emission [8]. Protons are also expected to be accelerated in these shocks, serving as efficient high-energy cosmic ray accelerators. On the other hand, as propagating to us, photons emitted via magnetic dissipation at a smaller radius from the engine [7] would first pass through the external shock front, and have a good chance to interact with the accelerated protons. Strong photo-meson interactions happen at the ∆-resonance, when the proton energy E p and photon energy E γ satisfy the threshold condition where Γ is the bulk Lorentz factor, m ∆ = 1 . 232 GeV and m p = 0 . 938 GeV are the rest masses of ∆ + and proton, respectively. The ∆ + particle decays into two channels. The charged pion channel gives ∆ + → nπ + → ne + ν e ¯ ν µ ν µ , with a typical neutrino energy E ν /similarequal 0 . 05 E p . The neutron pion channel gives the ∆ + → pπ 0 → pγγ . Note that the broad-band photons produced in the shocked region could also serve as the seed photons for pγ interaction. However, since their peak flux in the X-ray band [8] is much lower than that of the internal dissipation photons [7], we do not consider their contribution. With the multi-messenger era of astronomy ushered in, studying multi-messenger signals in astrophysical sources is of the great interest (e.g. [11]). The high-energy neutrino detectors such as IceCube have reached the sensitivity to detect high energy neutrinos from astrophysical objects for the first time. Gamma-ray bursts (GRBs) have been proposed to be one of the top candidates of PeV neutrinos [12]. However, a dedicated search of high energy neutrinos coincident with GRBs have so far led to null results [13, 14], which already places a meaningful constraint on GRB models [14-16]. Very recently, the IceCube collaboration announced their detections of two neutrino events with an energy approximately 1-2 PeV[17, 18], which could potentially represent the first detections of high-energy neutrinos from astrophysical sources. Among the proposed sources of such cosmic rays, GRBs stand out as particularly capable of generating PeV neutrinos at this level [18, 19]. However, the absence of associated GRBs for these two events calls for alternative cosmological PeV neutrino sources. Here we investigate the possible neutrino signals associated with NS-NS mergers with a millisecond magnetar central engine using the photomeson interaction mechanism delineated above. II. General picture. First of all, we adopt the ansatz that NS-NS merger events leave behind a massive millisecond magnetar and an essentially isotropic ejecta with mass ∼ (10 -4 -10 -2 )M /circledot . Shortly after the merger, the neutron star is able to cool down quickly so that a Poynting-flux-dominated outflow can be launched [20, 21]. Since the postmerger magnetar would be initially rotating near the break-up angular velocity, its total spin energy E rot = (1 / 2) I Ω 2 0 /similarequal 2 × 10 52 I 45 P -2 0 , -3 ergs (with I 45 ∼ 1 . 5 for a massive neutron star) may be universal. Here P 0 ∼ 1 ms is the initial spin period of the magnetar. Throughout the paper, the convention Q = 10 n Q n is used in cgs units, except for the ejecta mass M ej , which is in units of solar mass M /circledot . Given nearly the same total energy, the spindown luminosity and the characteristic spindown time scale critically depend on the dipole magnetic field strength B p , i.e. L sd = L sd , 0 / (1 + t/T sd ) 2 , where L sd , 0 /similarequal 10 49 erg s -1 B 2 p, 15 R 6 6 P -4 0 , -3 , and the spindown time scale T sd /similarequal 2 × 10 3 s I 45 B -2 p, 15 P 2 0 , -3 R -6 6 , where R = 10 6 R 6 cm is the stellar radius. Here we take the spindown luminosity L sd , 0 as the total luminosity of the Poynting-flux-dominated outflow and the spindown time scale T sd as its duration. For simplicity, we neglect the possible gravitational wave spin down of the new-born magnetar [22]. Note that both dipole magnetic field strength and spindown timescale could have a relatively large parameter space, which would add uncertainties to the following results. Initially, the heavy ejecta launched during the merger is not far away from the magnetar, so that in a large solid angle range, the magnetar wind would hit the ejecta before self-dissipation of the magnetar wind happens. In this case, a good fraction ( η ) of the magnetic energy may be rapidly discharged upon interaction between the wind and the ejecta. The Thomson optical depth for a photon to pass through the ejecta shell is τ th ∼ σ T M ej / (4 πR 2 m p ). By setting the optical depth equals to unity, we define a photosphere radius R ph = 2 . 5 × 10 14 M 1 / 2 ej , -3 cm for the ejecta. When R < R ph , the spectrum of the dissipated wind is likely quasi-thermal due to the large optical depth of photon scattering. The typical photon energy can be estimated as E ph , t ∼ k ( L sd , 0 η/ 4 πR 2 σ SB ) 1 / 4 /τ th ∼ 27 eV L 1 / 4 sd , 0 , 47 η 1 / 4 -1 M -1 ej , -4 R 3 / 2 14 , where σ SB is the StefanBoltzmann constant. Alternatively, when R > R ph , the typical synchrotron energy could be estimated as E γ,t /similarequal 1 . 8 × 10 4 keV L 1 / 2 sd , 0 , 47 R -1 15 η 3 / 2 -1 σ 2 4 , where σ is the magnetization parameter of the Poynting flow when the magnetar wind catches the ejecta [23]. In order to estimate the value of σ , we assume that the protomagnetar has σ 0 ∼ 10 7 at the central engine and the magnetized flow is quickly accelerated to Γ ∼ σ 1 / 3 0 at R 0 ∼ 10 7 cm, where σ ∼ σ 2 / 3 0 [24]. After this phase, the flow may still accelerate as Γ ∝ R 1 / 3 , with σ falling as ∝ R -1 / 3 [25]. Consequently, we have E γ,t /similarequal 1 . 8 keV L 1 / 2 sd , 0 , 47 η 3 / 2 -1 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 15 . As it is pushed forward by the magnetar wind, at a late time the ejecta is far away enough from the central engine, so that before hitting the ejecta, the magnetar wind already starts to undergo strong self-dissipation, for instance, through internal-collision-induced magnetic reconnection and turbulence (ICMART) process [23]. In this case, the typical synchrotron frequency can be still estimated as above, except that the emission radius is set to the self-dissipation radius, which we parameterize as the ICMART radius R i = 10 15 R i , 15 , rather than the blastwave radius [7, 23], i.e. E γ,t /similarequal 1 . 8 keV L 1 / 2 sd , 0 , 47 η 3 / 2 -1 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 i , 15 . Notice that for a substantial range of M ej , we have R ph < R i . Overall, the seed photon energy for pγ interaction can be summarized as In the mean time, the magnetar-wind-powered ejecta would interact with the ambient medium, forming a blastwave similar to GRB afterglow. Depending on the unknown parameters such as M ej , B p (and hence L sd , 0 ) [8], the blastwave could be accelerated to a mildly or even highly relativistic speed, due to the continuous energy injection from the magnetar wind. Protons are accelerated from the forward shock front along with electrons via the first-order Fermi acceleration process. Consequently, when the seed photons due to magnetar wind dissipation (Eq.2) pass through the shocked region, significant neutrino production due to pγ interaction through ∆-resonance would happen, as long as the condition R ≡ Γ γ M m p c 2 E p , t > 1 is satisfied. Here, E p , t = 0 . 147 GeV 2 Γ 2 /E γ,t is the corresponding proton energy for the typical seed photon at ∆-resonance, and γ M is the maximum proton Lorentz factor. It can be estimated by balancing the acceleration time scale and the dynamical time scale, which gives γ M ∼ Γ teB ' ζm p c , where ζ is a parameter of order unity that describes the details of acceleration and B ' is the comoving magnetic field strength. Once pγ interaction happens, significant neutrinos with energy /epsilon1 ν ∼ 0 . 05 E p , t would be released, the neutrino emission fluence may be estimated as where E tot ∼ 4 πR 3 n Γ(Γ -1) m p c 2 / 3 is the total energy of all the protons, f γ p , t ≡ E γ p , t E tot is the energy fraction of the relevant protons, and f π is the fraction of the proton energy that goes to pion production. Assuming a power-law distribution of the shock accelerated protons: N ( E p ) dE p ∝ E -p p dE p (hereafter assuming p > 2), one can obtain f γ p , t = ( γ p , t γ m ) 2 -p , where γ m = (Γ -1) p -2 p -1 +1 is the minimum proton Lorentz factor. The fraction of the proton energy that goes to pion production could be estimated as f π ≡ 1 2 (1 -(1 -< χ p → π > ) τ pγ ), where τ pγ is the pγ optical depth and < χ p → π > /similarequal 0 . 2 is the average fraction of energy transferred to pion. Notice that f π is roughly proportional to τ pγ when τ pγ < 3 [16]. III. Neutrino energy and fluence. The dynamics of the blastwave is defined by energy conservation [8] where L 0 = ξL sd , 0 is the magnetar injection luminosity into the blastwave, and M sw = (4 π/ 3) R 3 nm p is the swept-up mass from the interstellar medium. Initially, one has ( γ -1) M ej c 2 /greatermuch ( γ 2 -1) M sw c 2 , so that the kinetic energy of the ejecta would increase linearly with time until R = min( R sd , R dec ), where the deceleration radius R dec is defined by the condition ( γ -1) M ej c 2 = ( γ 2 -1) M sw c 2 . By setting R dec ∼ R sd , we can derive a critical ejecta mass which separate regimes with different blastwave dynamics [8]: Case I: M ej < M ej , c , 1 or R sd > R dec . In such case, the ejecta can be accelerated linearly until the deceleration radius R dec ∼ 3 . 9 × 10 17 M 2 / 5 ej , -4 L -1 / 10 sd , 0 , 47 n -3 / 10 0 , where bulk Lorentz factor of the blastwave is Γ dec ∼ 12 . 2 L 3 / 10 sd , 0 , 47 M -1 / 5 ej , -4 n -1 / 10 0 . After that, the blastwave decelerates, but still with continuous energy injection until R sd ∼ 1 . 0 × 10 18 ξ 1 / 2 L -1 / 4 sd , 0 , 47 n -1 / 4 0 , where Γ sd ∼ 7 . 5 ξ -1 / 4 L 3 / 8 sd , 0 , 47 n -1 / 8 0 . During the acceleration phase, the blastwave passes the non-relativistic to relativistic transition line Γ -1 = 1 at radius R N ∼ 2 . 2 × 10 14 M ej , -4 L -1 sd , 0 , 47 . For the different radius range of the typical photon energy shown in Eq. 2, we can investigate whether pγ interaction at ∆-resonance can occur, and if so, the typical energy and fluence of neutrino emission. We first assume that the blastwave is always nonrelativistic when R < = R ph , since R N is comparable with R ph with a high probability. In this range, we have R = 0 . 1 η 1 / 4 -1 L -5 / 12 sd , 0 , 47 M -1 / 3 ej , -4 n 1 / 2 0 R 11 / 6 14 < 1, implying that p γ interaction at ∆-resonance could hardly happen. Second, at R ph < R < R i , we have R = 26 . 0 η 3 / 2 -1 L -1 / 6 sd , 0 , 47 M 2 / 3 ej , -4 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -4 / 3 15 > 1, so that pγ interaction would happen at ∆-resonance. The typical neutrino energy and fluence could be estimated as /epsilon1 ν = 1 . 1 × 10 -2 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 47 M -2 / 3 ej , -4 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 7 / 3 15 , and f ν = 1 . 6 × 10 -12 η -0 . 05 -1 L 0 . 65 sd , 0 , 47 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 3 . 2 15 . Next, similar to the previous stage, at R i < R < R dec , we have R = 120 . 7 η 3 / 2 -1 L -1 / 6 sd , 0 , 47 M 2 / 3 ej , -4 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 i , 15 R 1 / 3 17 > 1, and the typical neutrino energy and fluence are /epsilon1 ν = 0 . 21 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 47 M -2 / 3 ej , -4 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 5 / 3 i , 15 R 2 / 3 17 and f ν = 1 . 6 × 10 -8 η -0 . 05 -1 L 0 . 65 sd , 0 , 47 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 1 . 17 i , 15 R 2 17 , respectively. Finally, when approaching the spindown radius, i.e., R dec < R < R sd , one has R = 1 . 2 × 10 3 η 3 / 2 -1 n 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 i , 15 R 2 18 > 1, and the typical neutrino energy and fluence are /epsilon1 ν = 0 . 24 PeV η -3 / 2 -1 n -1 / 2 0 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 5 / 3 i , 15 R -1 18 and f ν = 1 . 6 × 10 -6 η -0 . 05 -1 L 0 . 65 sd , 0 , 47 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 1 . 17 i , 15 R 2 18 , respectively. For better illustration, we take L sd, 0 = 10 47 and M ej = 10 -4 M /circledot as an example and plot the evolution of /epsilon1 ν and f ν for this dynamical case in Figure 1. Case II: M ej ∼ M ej , c , 1 or R sd ∼ R dec . In this case, the ejecta would be continuously accelerated until R sd = 1 . 2 × 10 18 ξ 3 L -1 sd , 0 , 49 M -2 ej , -4 , where the bulk Lorentz factor reaches Γ sd = 83 . 3 ξM -1 ej , -4 . Similar to case I, for R < = R ph , we do not expect significant pγ interaction since R = 0 . 01 η 1 / 4 -1 L -5 / 12 sd , 0 , 49 M -1 / 3 ej , -4 n 1 / 2 0 R 11 / 6 14 < 1. In the next stage R ph < R < R i , one has R = 12 . 0 η 3 / 2 -1 L -1 / 6 sd , 0 , 49 M 2 / 3 ej , -4 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -4 / 3 15 > 1. The expected neutrino energy and fluence are /epsilon1 ν = 0 . 02 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 49 M -2 / 3 ej , -4 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 7 / 3 15 , and f ν = 3 . 2 × 10 -11 η -0 . 05 -1 L 0 . 65 sd , 0 , 49 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 3 . 2 15 , respectively. Finally, at R i < R < R sd , one has R = 55 . 9 η 3 / 2 -1 L -1 / 6 sd , 0 , 49 M 2 / 3 ej , -4 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 i , 15 R 1 / 3 17 > 1, and /epsilon1 ν = 0 . 5 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 49 M -2 / 3 ej , -4 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 5 / 3 i , 15 R 2 / 3 17 , f ν = 3 . 2 × 10 -7 η -0 . 05 -1 L 0 . 65 sd , 0 , 49 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 1 . 17 i , 15 R 2 17 , respectively. In this case, we take L sd, 0 = 10 49 and M ej = 10 -4 M /circledot , and plot the evolution of /epsilon1 ν and f ν in Figure 1. Case III: M ej > M ej , c , 1 or R sd < R dec . Similar to Case II, the ejecta would be accelerated to a relativistic speed of Γ sd = 16 . 7 ξM -1 ej , -3 until R sd = 5 . 0 × 10 16 ξ 3 L -1 sd , 0 , 49 M -2 ej , -3 . Similarly, when R ≤ R ph , one has R = 0 . 004 η 1 / 4 -1 L -5 / 12 sd , 0 , 49 M -1 / 3 ej , -3 n 1 / 2 0 R 11 / 6 14 < 1, and hence, no significant neutrino emission. At R ph < R < R i , one has R = 35 . 1 η 3 / 2 -1 L -1 / 6 sd , 0 , 49 M 2 / 3 ej , -3 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -4 / 3 15 > 1, and ν /epsilon1 ν = 8 . 4 × 10 -3 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 49 M -2 / 3 ej , -3 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 7 / 3 15 , f ν = 3 . 2 × 10 -11 η -0 . 05 -1 L 0 . 65 sd , 0 , 49 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 3 . 2 15 . At R i < R < R sd , one has R = 163 . 1 η 3 / 2 -1 L -1 / 6 sd , 0 , 49 M 2 / 3 ej , -3 n 1 / 2 0 σ 4 / 3 0 , 7 R 2 / 3 0 , 7 R -5 / 3 i , 15 R 1 / 3 17 > 1, and /epsilon1 ν = 0 . 2 PeV η -3 / 2 -1 L 1 / 6 sd , 0 , 49 M -2 / 3 ej , -3 σ -4 / 3 0 , 7 R -2 / 3 0 , 7 R 5 / 3 i , 15 R 2 / 3 17 , f ν = 3 . 2 × 10 -7 η -0 . 05 -1 L 0 . 65 sd , 0 , 49 n 0 σ -0 . 93 0 , 7 R -0 . 47 0 , 7 R 1 . 17 i , 15 R 2 17 . For this case, we take L sd, 0 = 10 47 and M ej = 10 -3 M /circledot and plot the evolution of /epsilon1 ν and f ν in Figure 1. Note that there is another critical ejecta mass M ej , c , 2 ∼ 6 × 10 -3 M /circledot I 45 P -2 0 , -3 ξ (defined by setting E rot ξ = 2( γ -1) M ej , c , 2 c 2 ), above which the blast wave would never reach a relativistic speed [8]. The dynamics is similar to Case III, with the coasting regime in the non-relativistic phase. In this case, we always have R < 1, therefore no significant neutrino flux is expected. IV. Detection prospect. From the above calculation, one can see that when the post-merger product is a millisecond magnetar and the outgoing ejecta could be accelerated to a relativistic speed, ∼ PeV neutrinos could indeed be emitted from NS-NS mergers scenario. These neutrinos are well suited for detection with IceCube[26]. As shown in Figure 1, for different initial conditions, i.e., different combinations of M ej and L sd , the maximum neutrino fluence is always reached at the spin-down time scale. We therefore take the neutrino energy and fluence at this epoch as the typical values for each specific NS-NS merger event. For the events happening at 300 Mpc, the optimistical typical neutrino fluence could be as large as 10 -6 -10 -5 GeV cm -2 (corresponding to σ 0 = 10 7 , 10 6 respectively), one or two orders of magnitude lower than the typical fluence of GRBs[15]. Given the typical neutrino energy ∼ PeV and the IceCube effective area ∼ several 10 6 cm 2 [26, 27], optimistically only several 10 -6 -10 -5 neutrinos are expected to be detected by IceCube for a single event. In any case, these events would contribute to the ∼ PeV neutrino background. The NS-NS merger event rate is rather uncertain, i.e., (10 -5 × 10 4 ) Gpc -3 yr -1 [28]. Considering that only a fraction of NS-NS merger event would leave behind a massive neutron star rather than a black hole, and that only a sub-fraction of these mergers have the right M ej and L sd , 0 to make relativistic blastwaves, the event rate of NS-NS mergers that generate PeV neutrinos may be at least one order of magnitude lower, i.e. ∼ (1 -5 × 10 3 ) Gpc -3 yr -1 . Even with the most optimistic estimate, the ∼ PeV diffuse back ground is ∼ 10 -10 GeV cm -2 s -1 sr -1 . It takes tens of years to get two events. So these systems are not likely the origin of the two reported PeV events announced by the Icecube collaboration [17]. Nevertheless, compared with the GRB event rate 1 Gpc -3 yr -1 [30], this scenario may gain the event rate by 1-2 orders of magnitude than GRBs. Noticing that a typical GRB has a fluence 1-2 orders of magnitude higher than a magnetar-wind-powered NSNS merger remnant, our scenario could contribute to the ∼ PeV neutrino diffuse background, which is comparable or slightly lower than that of GRBs. V. High energy photon emission. Besides high-energy neutrino emission, the decay of π 0 produced in pγ interactions would lead to the production of high energy gamma-ray photons. Assuming that half of the ∆ + decays go to the π + channel (neutrino production), while the other half go to the π 0 channel ( γ -ray production), the typical gamma-ray photon energy and fluence values would be comparable to the neutrinos we studied in section III . However, such high-energy photons may interact with the synchrotron emission photons in the shock [8] to produce electron/positron pairs, γγ → e ± , and initiate an electromagnetic cascade: the pairs would emit photons via synchrotron and inverse Compton, which would be converted back to pairs, and the pairs would emit photons again, etc. Photons can escape only when the γγ optical depth becomes lower than unity [29]. Following the calculation shown in Ref.[8], we find that the γγ optical depth exceeds unity for photon energy above /epsilon1 γγ ∼ 100GeV. For simplicity, we assume that the total energy of the π 0 -decay photons would finally show up around 100GeV through an EM cascade. These photons are within the energy windows of the Fermi/LAT. In the most optimistic situation, the photon flux for an event at 300 Mpc could be as high as 10 -11 erg cm -2 s -1 , which is essentially 10 -10 photons cm -2 s -1 . The effective area of LAT for 100GeV photons is around 9000 cm 2 [31], suggesting that even for T sd ∼ 10 5 , one single NS-NS merger event could not trigger LAT. Nevertheless, the total diffuse flux from these events could reach ∼ several 10 -7 MeV cm -2 s -1 sr -1 optimistically, giving a moder- ub-TeV γ -ray background, i.e., 4 × 10 -4 MeV cm -2 s -1 sr -1 , according to Fermi/LAT observation[32]. VI. Acknowledge. We thank stimulative discussions with Zhuo Li and Qiang Yuan. We acknowledge the National Basic Research Program ('973' Program) of China (Grant No. 2009CB824800 and 2013CB834900), National Science Foundation (AST-0908362), and National Natural Science Foundation of China (grant No. 11033002 & 10921063). XFW acknowledges support by the One-Hundred-Talents Program and the Youth Innovation Promotion Association of Chinese Academy of Sciences.", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "Weyl-Cartan-Weitzenbock gravity through Lagrange multiplier", "content": "Zahra Haghani 1 , ∗ Tiberiu Harko 2 , † Hamid Reza Sepangi 1 , ‡ and Shahab Shahidi 1 § 1 Department of Physics, Shahid Beheshti University, G. C., Evin,Tehran 19839, Iran and 2 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom We consider an extension of the Weyl-Cartan-Weitzenbock (WCW) and teleparallel gravity, in which the Weitzenbock condition of the exact cancellation of curvature and torsion in a Weyl-Cartan geometry is inserted into the gravitational action via a Lagrange multiplier. In the standard metric formulation of the WCW model, the flatness of the space-time is removed by imposing the Weitzenbock condition in the Weyl-Cartan geometry, where the dynamical variables are the spacetime metric, the Weyl vector and the torsion tensor, respectively. However, once the Weitzenbock condition is imposed on the Weyl-Cartan space-time, the metric is not dynamical, and the gravitational dynamics and evolution is completely determined by the torsion tensor. We show how to resolve this difficulty, and generalize the WCW model, by imposing the Weitzenbock condition on the action of the gravitational field through a Lagrange multiplier. The gravitational field equations are obtained from the variational principle, and they explicitly depend on the Lagrange multiplier. As a particular model we consider the case of the Riemann-Cartan space-times with zero non-metricity, which mimics the teleparallel theory of gravity. The Newtonian limit of the model is investigated, and a generalized Poisson equation is obtained, with the weak field gravitational potential explicitly depending on the Lagrange multiplier and on the Weyl vector. The cosmological implications of the theory are also studied, and three classes of exact cosmological models are considered. PACS numbers: 04.20.Cv, 04.50.Kd, 98.80.Jk, 98.80.Es", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "General relativity (GR) is considered to be the most successful theory of gravity ever proposed. Its classic predictions on the perihelion advance of Mercury, on the deflection of light by the Sun, gravitational redshift, or radar echo delay have been confirmed at an unprecedented level of observational accuracy. Moreover, predictions such as the orbital decay of the Hulse-Taylor binary pulsar, due to gravitational - wave damping, have also fully confirmed the observationally weak-field validity of the theory. The detection of the gravitational waves will allow the testing of the predictions of GR in the strong gravitational field limit, such as, for example, the final stage of binary black hole coalescence (for a recent review on the experimental tests of GR see [1]). Despite these important achievements, recent observations of supernovae [2] and of the Cosmic Microwave Background radiation [3] have suggested that on cosmological scales GR may not be the ultimate theory to describe the Universe. If GR is correct, in order to explain the accelerating expansion of the Universe, we require that the Universe is filled with some component of unknown nature, called dark energy, having some unusual physical properties. To find an alternative to dark energy to explain cosmological observations, in the past decade many modified theories of gravity, which deviate from the standard GR on cosmological scales have been proposed (see [4] for a recent review on modified gravity and cosmology). On the other hand, because of its prediction of space-time singularities in the Big Bang and inside black holes GR could be considered as an incomplete physical model. In order to solve the singularity problem it is generally believed that a consistent extension of GR into the quantum domain is needed. Since GR is essentially a geometric theory, formulated in the Riemann space, looking for more general geometric structures adapted for the description of the gravitational field may be one of the most promising ways for the explanation of the behavior at large cosmological scales of the matter in the Universe, whose structure and dynamics may be described by more general geometries than the Riemannian one, valid at the Solar System level. The first attempt to create a more general geometry is due to Weyl [5], who proposed a geometrized unification of gravitation and electromagnetism. Weyl abandoned the metric-compatible Levi-Civita connection as a fundamental concept, since it allowed the distant comparison of lengths. Substituting the metric field by the class of all conformally equivalent metrics, Weyl introduced a connection that would not carry any information about the length of a vector on parallel transport. Instead the latter task was assigned to an extra connection, a socalled length connection that would, in turn, not carry any information about the direction of a vector on parallel transport, but that would only fix, or gauge, the conformal factor. Weyl identified the length connection with the electromagnetic potential. A generalization of Weyls theory was introduced by Dirac [6], who proposed the existence of two metrics, one unmeasurable metric ds E , affected by transformations in the standards of length, and a second measurable one, the conformally invariant atomic metric ds A . In the development of the generalized geometric theories of gravity a very different evolution took place due to the work of Cartan [7], who proposed an extension of general relativity, which is known today as the EinsteinCartan theory [8]. The new geometric element of the theory, the torsion field, is usually associated from a physical point of view to a spin density [8]. The Weyl geometry can be immediately generalized to include the torsion. This geomety is called the Weyl-Cartan space-time, and it was extensively studied from both mathematical and physical points of view [9]. To build up an action integral from which one can obtain a gauge covariant (in the Weyl sense) general relativistic massive electrodynamics, torsion was included in the geometric framework of the Weyl-Dirac theory in [10] . For a recent review of the geometric properties and of the physical applications of the Riemann-Cartan and Weyl-Cartan space-times see [11]. /negationslash A third independent mathematical development took place in the work of Weitzenbock [12], who introduced the so-called Weitzenbock spaces. A Weitzenbock manifold has the properties ∇ µ g σλ = 0, T µ σλ = 0, and R µ νσλ = 0, where g σλ , T µ σλ and R µ νσλ are the metric, the torsion, and the curvature tensors of the manifold, respectively. When T µ σλ = 0, the manifold is reduced to a Euclidean manifold. The torsion tensor possesses different values on different parts of the Weitzenbock manifold. Therefore, since their Riemann curvature tensor is zero, Weitzenbock spaces possess the property of distant parallelism, also known as absolute, or teleparallelism. Weitzenbock type geometries were first used in physics by Einstein, who proposed a unified teleparallel theory of gravity and electromagnetism [13]. The basic idea of the teleparallel approach is to substitute, as a basic physical variable, the metric g µν of the space-time by a set of tetrad vectors e i µ . In this approach the torsion, generated by the tetrad fields, can be used to describe general relativity entirely, with the curvature eliminated in favor of torsion. This is the so-called teleparallel equivalent of General Relativity (TEGR), which was introduced in [14], and is also known as the f ( T ) gravity model. Therefore, in teleparallel, or f(T) gravity, torsion exactly compensates curvature, and the space-time becomes flat. Unlike in f ( R ) gravity, which in the metric approach is a fourth order theory, in the f ( T ) gravity models the field equations are of second order. f ( T ) gravity models have been extensively applied to cosmology, and in particular to explain the late-time accelerating expansion of the Universe, without the need of dark energy [15]. An extension of the teleparallel gravity models, called WCW gravity, was introduced recently in [16]. In this approach, the Weitzenboock condition of the vanishing of the sum of the curvature and torsion scalar is imposed in a background Weyl-Cartan type space-time. In contrast to the standard teleparallel theories, the model is formulated in a four-dimensional curved space-time, and not in a flat Euclidian geometry. The properties of the gravitational field are described by the torsion tensor and the Weyl vector fields, defined in a four-dimensional curved space-time manifold. In the gravitational action a kinetic term for the torsion is also included. The field equations of the model, obtained from a Hilbert-Einstein type variational principle, allow a complete description of the gravitational field in terms of two vector fields, the Weyl vector and torsion, respectively, defined in a curved background. The Newtonian limit of the model was also considered, and it was shown that in the weak gravitational field approximation the standard Poisson equation can be recovered. For a particular choice of the free parameters, in which the torsion vector is proportional to the Weyl vector, the cosmological applications of the model were investigated. A large variety of dynamical evolutions can be obtained in the WCW gravity model, ranging from inflationary/accelerated expansions to non-inflationary behaviors. The nature of the cosmological evolution is determined by the numerical values of the parameters of the cosmological model. In particular a de Sitter type late time evolution can be naturally obtained from the field equations of the model. Therefore the WCW gravity model leads to the possibility of a purely geometrical description of dark energy where the late time acceleration of the Universe is determined by the intrinsic nature of the space-time. Recently, the use of Lagrange multipliers in the formulation of dynamical gravity models has attracted considerable attention. The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality nonholonomic constraints, which are capable of reducing the dynamics [17]. The extension of f ( R ) gravity models via the addition of a Lagrange multiplier constraint has been proposed in [18]. This model can be considered as a new version of f ( R ) modified gravity since dynamics, and the cosmological solutions, are different from the standard version of f ( R ) gravity without such constraint. Cosmological models with Lagrange multipliers have been considered from different points of view in [19]. It is the purpose of the present paper to investigate a class of generalized WCW type gravity models, in which the Weitzenbock condition of the exact compensation of torsion and curvature is introduced into the gravitational action via a Lagrange multiplier approach. We start our analysis by considering the general action for a gravitational field in a Weyl-Cartan space-time, and we explicitly introduce the Weitzenbock condition into the action via a Lagrange multiplier. By taking the Weyl vector as being identically zero, we obtain the field equations of this gravity model in a Riemann-Cartan space time, with the Weitzenbock condition being described by a proportionality relation between the scalar curvature and torsion scalar, included in the gravitational action via a Lagrange multiplier method which mimics the teleparallel gravity. The weak field limit of the general theory is also investigated, and a generalized Poisson equation, explicitly depending on the Lagrange multiplier and the Weyl vector is obtained. The cosmological implications of the model are investigated for three classes of models. The solutions obtained describe both accelerating and decelerating expansionary phases of the Universe, and they may prove useful for modeling the early and late phases of cosmological evolution. The paper is organized as follows. The gravitational action of the WCW theory with Lagrange multiplier is introduced in Section II. The gravitational field equations are derived in Section III. Some particular cases are also considered in detail. The field equations for the case of the zero Weyl vector are presented in Section IV. The weak field limit of the theory is investigated in Section V, and the generalized Poisson equation is obtained. The cosmological implications of the theory are investigated in Section VI, and several cosmological models are presented. We discuss and conclude our results in Section VII. Some aspects of the Weyl invariance of the theory are considered in the Appendix.", "pages": [ 1, 2, 3 ] }, { "title": "II. WCW GRAVITY MODEL WITH LAGRANGE MULTIPLIER", "content": "In this Section we formulate the action of the gravitational field in the WCW gravity with a Lagrange multiplier. A Weyl - Cartan space CW 4 is a four-dimensional connected, oriented, and differentiable manifold, having a metric with a Lorentzian signature chosen as ( -+++), curvature, torsion, and a connection which can be determined from the Weyl nonmetricity condition. Hence the Weyl-Cartan geometry has the properties that the connection is no longer symmetric, and the metric compatibility condition does not hold. The Weyl non-metricity condition is defined as where w µ is the Weyl vector. Expanding the covariant derivative, we obtain the connection in the Weyl-Cartan geometry where the first term in the LHS is the Christoffel symbol constructed out of the metric and the contorsion tensor C λ µν is defined as with torsion tensor T λ µν given by One can then obtain the curvature tensor of the WeylCartan space-time as Using equation (2) and contracting the curvature tensor with the metric, we obtain the curvature scalar where R is the curvature scalar constructed from the Christoffel symbols and we have defined T β = T α βα . Also all covariant derivatives are with respect to the Riemannian connection described by the Christoffel symbols constructed out of the metric g µν . We also introduce two tensor fields W µν and T µν , constructed from the Weyl vector and the torsion vector, respectively where T = T µ T µ . The most general action for a gravitational theory in the Weyl-Cartan space-time can then be formulated as where L m is the matter Lagrangian which depends only on the matter fields and the metric, and is independent on the torsion tensor and the Weyl vector. We have also added a kinetic term for the Weyl vector and two possible kinetic terms for the torsion tensor. In equation (9), ˆ α and ˆ β are arbitrary numerical constants, and κ 2 = 16 πG . Substituting definition of the curvature scalar from equation (6), the action for the gravitational field becomes where we have defined α = κ 2 ˆ α and β = κ 2 β . The Weitzenbock condition requires that the sum of the scalar curvature and torsion be zero. In order to impose this condition on the gravitational field equations of the theory, we add it to the action by using a Lagrange multiplier λ . The gravitational action then becomes We note that for λ = -1 one gets the original WCW action [16].", "pages": [ 3 ] }, { "title": "III. THE GRAVITATIONAL FIELD EQUATIONS OF THE WCW GRAVITY MODEL WITH A LAGRANGE MULTIPLIER", "content": "Let us now derive the field equations of the WCW gravity with the Lagrange multiplier. By considering the irreducible decomposition of the torsion tensor and imposing a condition on the terms of the decomposition, we obtain an explicit representation of the Weitzenbock condition. The field equations of a simplified model in which the constant α = 0 are also obtained explicitly.", "pages": [ 4 ] }, { "title": "A. The gravitational field equations and the effective energy-momentum tensor", "content": "Variation of the action (12) with respect to the Weyl vector and the torsion tensor results in the equations of and respectively. Variation of the action with respect to the Lagrange multiplier λ gives the Weitzenbock condition (11). Now, varying the action with respect to the metric and using the condition (11), we obtain the dynamical equation for the metric as where T m µν is the energy-momentum of the ordinary-matter. The generalized Einstein field equation (15) can be written as where we have defined the effective energy-momentum tensor as and use has been made of equation (11).", "pages": [ 4 ] }, { "title": "B. The decomposition of the torsion tensor", "content": "The torsion tensor can be decomposed irreducibly into where Q ν and S ρ are two vectors, and the tensor t µνρ is symmetric under the change of the first two indices, and satisfies the following conditions By contracting equation (18) over µ and ρ we obtain Q µ = T µ . Assuming that t µνρ ≡ 0 [16], one may formu- motion late the Weitzenbock condition as Now in equations (14) and (15) the terms with coefficient (1 + λ ) can be simplified to and respectively. Taking the trace of equation (14) over indices β and µ , we have Now, substituting the LHS of the above equation into (14) we obtain /negationslash If one assumes λ = -1 then S µ = 0. We note that from equation (21) one has R = 8 / 3 T which implies that the vector T µ should be space-like for the accelerating Universe with R = 6( ˙ H +2 H 2 ), where H is the Hubble parameter.", "pages": [ 4, 5 ] }, { "title": "C. The case α = 0", "content": "In order to further simplify the gravitational field equations of the WCW model with a Lagrange multiplier, let us assume that α = 0, as in [16]. In this case from equation (24) we find /negationslash provided that T = 0. Substituting (24) into (14) we obtain which implies that T µ = Aw µ , where A is a constant. In order to obtain the value of the constant A , we take the covariant divergence of equation (13), with the result The above equation implies that A = 3 / 2, so we conclude that Substituting the above equation into (13) we obtain the dynamical field equation of the Weyl vector Now, using (29), we write equation (14) as which implies where w 2 = w α w α . Substituting T µ and λ from equations (29) and (32) into the metric field equation we obtain the effective energy-momentum tensor of the WCW model with the Lagrange multiplier, equation (17), as In summary, one may obtain the Weyl vector from equation (30) and then the Lagrange multiplier λ from equation (32). The field equation (16), together with equation (33) can then be used to obtain the evolution of the metric. Hence a complete solution of the gravitational field equations in the WCW model with a Lagrange multiplier can be constructed, once the thermodynamic parameters of the matter (energy density and pressure) are known. conserved due to the Bianchi identity. One can easily prove this statement in the case α = 0. Using equation (33), one may write equation (16) as It is worth mentioning that because of the general covariance, the matter energy-momentum tensor should be Taking the divergence of the above equation one obtains Now, using the identity and considering the Weitzenbock condition which reads R = 6 w 2 , where R is the Ricci scalar, one easily finds ∇ µ T m µν = 0.", "pages": [ 5, 6 ] }, { "title": "IV. THE LIMITING CASE w µ = 0 AND THE TELEPARALLEL GRAVITY", "content": "In this Section we consider the limiting case in which the Weyl vector becomes zero. We also assume α = 0 for with The variation of the action with respect to the Lagrange multiplier gives the Weitzenbock condition (11). Now consider the decomposition of the torsion tensor, given by equation (18), with t µνρ = 0. One can again obtain S µ = 0 by the same trick as in Section III. We then obtain the Weitzenbock condition in the form From equation (38) one can isolate the Lagrange multiplier By substituting equation (42) one can check that equation (38) is automatically satisfied. The metric then simplicity. The action of the theory becomes One may then obtain the field equations for the torsion tensor and the metric as and equation becomes", "pages": [ 6, 7 ] }, { "title": "A. The case β = 0", "content": "For β = 0 the torsion has no kinetic term. Putting β = 0 in equation (38) and using equation (22), we obtain T ρ = 0. The trace of equation (38) then gives S µ = 0. Therefore, from the field equations we obtain T µ ρν = 0 and the theory reduces to a Brans-Dicke type theory, with equations of motion and respectively, where T m is the trace of the energymomentum tensor. We have used the Weitzenbock condition R = 0 to obtain equation (45).", "pages": [ 7 ] }, { "title": "V. THE NEWTONIAN LIMIT AND THE GENERALIZED POISSON EQUATION", "content": "In this Section, we will obtain the generalized Poisson equation describing the weak field limit of the WCW theory with Lagrange multiplier. Taking the trace of equation (15), using the Weitzenbock condition (21), and noting that S µ = 0 in our setup, we obtain Now, using equation (24) to eliminate the /square T term, we find In the limit of the weak gravitational fields the (00) component of the metric tensor takes the form g 00 = -(1 + 2 φ ), where φ is the Newtonian potential. In this limit we have R = -∇ 2 φ , and obtain the generalized Poisson equation as In obtaining the above equation we have assumed that the matter content of the Universe is pressureless dust, and we have used the Weitzenbock equation to keep terms up to first order in φ . In the particular case α = 0, from equation (32) we find that the Lagrange multiplier is of the order of φ . Using equation (29) we obtain the generalized Poisson equation as For w = 0, we recover the standard Poisson equation of Newtonian gravity.", "pages": [ 7 ] }, { "title": "VI. COSMOLOGICAL SOLUTIONS", "content": "In this Section we consider the cosmological solutions and implications of the WCW model with Lagrange multiplier. We assume that the metric of the space-time has the form of the flat Friedmann-Robertson-Walker (FRW) metric, Also in the following we suppose that the tensor t µνρ vanishes, t µνρ = 0. As we have mentioned in the previous Section, S µ = 0 and T α should be space-like in order to obtain an accelerating solution. We consider only models in which the Universe is filled with a perfect fluid, with the energy-momentum tensor given in a comoving frame by where ρ and p are the thermodynamic energy density and pressure, respectively. The Hubble parameter is defined as H = ˙ a/a . As an indicator of the accelerated expansion we will consider the deceleration parameter q , defined as If q < 0, the Universes experiences an accelerated expansion while q > 0 corresponds to a decelerating dynamics.", "pages": [ 7 ] }, { "title": "A. The case α = 0", "content": "In this case the cosmological dynamics is described by equation (30) which represents the dynamical equation for the Weyl vector together with equations (32) and (16) which determine the Lagrange multiplier and the scale factor, respectively. The Weitzenbock condition is Let us assume that the Weyl vector is of the form The Weitzenbock equation reduces to and the Lagrange multiplier can be obtained as where we have defined Ψ = ˙ ψ/ψ . The dynamical equation for the Weyl vector is The off diagonal elements of the metric field equation gives One can then check that the Weyl equation (57) is automatically satisfied. By substituting H from (58) to the diagonal metric equations one obtains and We note that in this case we have four equations, (55), (58), (59) and (60) for four unknowns a , ψ , ρ and p . The Lagrange multiplier can then be obtain from equation (56). Equation (58) can be immediately integrated to give /negationslash where C 0 is an arbitrary constant of integration. With the use of ψ ( t ) = C 0 /a ( t ), the Weitzenbock condition, equation (55), becomes or equivalently which immediately leads to where C 1 and C 2 are arbitrary constants of integration. By assuming the initial conditions a (0) = a 0 and H (0) = H 0 , respectively, we obtain C 2 = a 2 0 , and C 1 = 2 a 2 0 H 0 . Thus for the Hubble parameter we obtain The energy density of the Universe can be obtained from equation (60) as The thermodynamic pressure is found in the form For t = 0 we obtain the initial values of the density and pressure as and respectively. Once the initial conditions ( a 0 , H 0 , ρ 0 , p 0 ) are known, from Eqs. (68) and (69) the values of the integration constants can be determined. The deceleration parameter can be obtained as If the initial values of the scale factor and Hubble parameter satisfy the condition a 0 H 0 < √ 3 C 0 , q < 0 for all times then the Universe is in an accelerated expansionary phase. If a 0 H 0 = √ 3 C 0 , q ( t ) ≡ 0 then the Universe is in a marginally inflating state. Finally, the Lagrange multiplier for this model can be obtained as /negationslash", "pages": [ 7, 8, 9 ] }, { "title": "B. The case α = 0", "content": "We assume that the Weyl vector is space-like, mimicking the proportionality of the torsion and the Weyl vector as in the case α = 0. Let us assume that By substituting these forms of the torsion and Weyl vector into equation (14) we obtain, after some algebra, the Lagrange multiplier where we have defined By using equation (73), the field equation (13) becomes The Weitzenbock equation takes the form Substituting equations (73) and (76) into (15), one obtains and Eqs. (75), (76), (77), (78) and (79) form a closed system of differential equations for five unknowns ψ , φ , H , p and ρ . Equation (73) can then be used to determine the Lagrange multiplier. In the following we will look only for a de Sitter type solution of the field equations (75)-(79) with H = H 0 = constant and a = exp( H 0 t ), respectively. Then the Weitzenbock condition (76) immediately gives and Φ ≡ 0, respectively. Equation (75) takes the form with the general solution given by where c 1 and c 2 are arbitrary constants of integration. The simplest case corresponds to the choice c 1 = 0, c 2 = 0, giving By substituting this form of ψ into equation (79) we obtain the value of α as For the energy density of the Universe we obtain One can see that the energy density and the pressure is positive if H 0 ≥ 1 /κ 2 √ 3 / 2.", "pages": [ 9, 10 ] }, { "title": "C. Cosmological models with w µ = 0", "content": "Finally, we consider the cosmological implications of the WCW model with Lagrange multiplier with w µ = 0. Assuming the following form for the torsion, the Weitzenbock condition is formulated as The Lagrange multiplier can be obtained in the form where we have defined Φ = ˙ φ/φ . The metric field equations take the form and respectively. Let us consider the case a ( t ) = t s . In this case one obtains and the energy density and pressure take the form In order to have a consistent solution, φ should be real and ρ and p must be positive. This restricts the range of s to For the deceleration parameter we obtain In the case a ( t ) = e H 0 t we have with the matter energy density and the pressure becoming exactly zero", "pages": [ 10, 11 ] }, { "title": "VII. CONCLUSION", "content": "In this paper we have considered an extension of the Weitzenbock type gravity models formulated in a WeylCartan space time. The basic difference between the present and the previous investigations is the way in which the Weitzenbock condition which in a RiemannCartan space time requires the exact cancellation of the Ricci scalar and the torsion scalar, is implemented. By starting with a general geometric framework, corresponding to a CW 4 space - time described by a metric tensor, torsion tensor and Weyl vector, we formulated the action of the gravitational field by including the Weitzenbock condition via a scalar Lagrange multiplier. With the use of this action the gravitational field equations have been explicitly obtained. They show the explicit presence in the field equations of a new degree of freedom, represented by the Lagrange multiplier λ . The field equations must be consistently solved together with the Weitzenbock condition which allows the unique determination of the Lagrange multiplier λ . The weak field limit of the model was also investigated and it was shown that the Newtonian approximation leads to a generalization of the Poisson equation where besides the matter energydensity, the weak field gravitational potential also explicitly depends on the Lagrange multiplier and the square of the Weyl vector. An interesting particular case is represented by the zero Weyl vector case. For this choice of the geometry the covariant divergence of the metric tensor is zero and the Weitzenbock condition takes the form of a proportionality relation between the Ricci scalar and the torsion scalar, respectively. When one neglects the kinetic term associated to the torsion, the model reduces to a BransDicke type theory where the role of the scalar field is played by the Lagrange multiplier. /negationslash The cosmological implications of the theory have also been investigated by considering a flat FRW background type cosmological metric. We have considered three particular models, corresponding to the zero and non-zero values of the coupling constant α , and to the zero Weyl vector respectively. For α = 0 the field equations can be solved exactly, leading to a scale factor of the form a ( t ) = √ 3 c 2 0 t 2 +2 H 0 a 2 0 + a 2 0 . The energy density and the pressure are monotonically decreasing functions of time and are both non-singular at the beginning of the cosmological evolution. The nature of the cosmological expansion - acceleration or deceleration - is determined by the values of the constants ( C 0 , a 0 , H 0 ) and three regimes are possible: accelerating, decelerating, or marginally inflating. In the case α = 0, we have considered only a de Sitter type solution of the field equations. Such a solution does exist if the matter energy density and pressure are constants, or, more exactly, the decrease in the matter energy density and pressure due to the expansion of the Universe is exactly compensated by the variation in the energy and pressure due to the geometric terms in the energy-momentum tensor. In the case of the cosmological models with vanishing Weyl vector we have investigated two particular models corresponding to a power law and exponential expansion, respectively. In the case of the power law expansion, the energy density and pressure satisfy a barotropic equation of state, so that p ∼ ρ where both the energy and pressure decay as t -6 . Depending on the value of the parameter s , both decelerating and accelerating models can be obtained. On the other hand, for a vanishing Weyl vector, the de Sitter type solutions require a vanishing matter energy density and pressure and hence the accelerated expansion of the Universe is determined by the geometric terms associated with torsion which play the role of an effective cosmological constant. In the present paper we have introduced a theoretical model for gravity, defined in a Weyl-Cartan space-time, in which the Weitzenbock geometric condition has been included in the action via a Lagrange multiplier method. The field equations of the model have been derived by using variational methods, and some cosmological implications of the model have been explored. Further astrophysical and cosmological implications of this theory will be considered elsewhere.", "pages": [ 11, 12 ] }, { "title": "Appendix A: Note on Weyl gauge invariance", "content": "Suppose that length of a vector at point x is l . In the Weyl geometry, the length of the vector under parallel transportation to the nearby point x ' is l ' = ξl . On the other hand, the change in the length of the vector can be written as So, the change in the Weyl vector is From the above relations, one obtains the change in the metric tensor [10] M. Israelit, Gen. Rel. Grav. 29 , 1411 (1997); M. Israelit, The torsion tensor is invariant under the above gauge transformation, i.e., We note that the curvature tensor (6) is covariant with the power -2, which means and the metric determinant has power 4. Naturally, one demands to make the Lagrangian (9) gauge-invariant. In order to do so one can add a scalar field β or a Dirac field with power -1 and write the first term in equation (9) as √ -gβ 2 K to make it gauge-invariant. However, the Weitzenbock condition (11) is neither gauge invariant nor covariant. In fact, one may write where ∇ is the metric covariant derivative and we have defined k α = ∂ α log ξ . In order to make the Weitzenbock condition gauge-covariant, one should add to W some terms containing the torsion tensor and the Weyl vector. This generalization of the Weitzenbock condition by adding torsion and Weyl tensors will be considered in our future work [20]. Gen. Rel. Grav. 29 , 1597 (1997); M. Israelit, Found. Phys. 28 , 205 (1998); M. Israelit, Hadronic J. 21 , 75 (1998); M. Israelit, Proceedings of the 8-th Marcel Grossmann Meeting, Pirani and Ruffini Editors, World Scientic, Singapore, p. 653 (1999).", "pages": [ 12 ] } ]

[ { "title": "Geometric Origin of Stokes Phenomenon for de Sitter Radiation", "content": "Sang Pyo Kim ∗ Department of Physics, Kunsan National University, Kunsan 573-701, Korea † and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 18, 2018) We propose a geometric interpretation for the Stokes phenomenon in de Sitter spacetime that particles are produced in even dimensions but not in odd dimensions. The scattering amplitude square for a quantum field between the in-vacuum and the transported one along a closed path in the complex-time plane gives the particle-production rate that explains not only the Boltzmann factor from the simple pole at infinity, corresponding to the cosmological horizon, but also the sinusoidal behavior from simple poles at the north and south poles of the Euclidean geometry. The Stokes phenomenon is a consequence of interference among four independent closed paths in the complex plane. PACS numbers: 04.60.-m, 04.62.+v, 03.65.Vf, 03.65.Sq", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Nature in some circumstances distinguishes the dimensionality of spacetime through the underlying theory. The fundamental solution to the wave equation has a delta-function support in even dimensions while it has a step-function support in odd dimensions. Another interesting feature of dimensionality is that a de Sitter (dS) spacetime produces particles in even dimensions while it does not produce any particles in odd dimensions [1, 2]. Polyakov interpreted the reflectionless scattering of a quantum field in an odd-dimensional dS spacetime as a soliton of the Korteweg-deVries (KdV) equation [3, 4]. In fact, the quantum field in all odd-dimensional dS spacetimes has the Posch-Teller potential [5], whose asymptotically reflectionless scattering implies the solitonic nature of produced particles [6]. In the phase-integral method [7] each mode of a quantum field in time-dependent gauge fields or curved spacetimes has at least one pair of complex turning points in the complex-time plane, for which the Hamilton-Jacobi action between each pair determines the particle-production rate for that channel. Remarkably the actions from more than one pair of complex turning points may contribute constructively and destructively to the particle production, known as the Stokes phenomenon. The Stokes phenomenon in Schwinger mechanism discovered by Dumlu and Dunne explains the substructure of produced particles for some time-dependent electric fields [8-10]. In a dS spacetime a quantum field has one pair of complex turning points in the planar coordinates and two pairs in the global coordinates. The actions along Stokes lines connecting two anti-Stokes lines have both the imaginary part determining the dS radiation and the real part resulting in constructive or destructive interference for dS radiation in the global coordinates [11]. In this paper we propose a geometric interpretation in the complex-time plane of the Stokes phenomenon for dS radiation. Instead of tunneling paths and their actions in the phase-integral method, we study the quantum evolution operator for the field and calculate the geometric contributions to the transported in-vacuum in the complex-time plane. It is shown that each harmonics of the field in the global coordinates of dS spacetime obtains not only a geometric factor for dS radiation which originates from the simple pole at infinity corresponding to the cosmological horizon but also interfering terms from finite simple poles at the north and south poles of the Euclidean geometry which explain the sinusoidal behavior responsible for the presence or absence of particle production in even and odd dimensions. The geometric transition of the time-dependent Hamiltonian in the complex plane leads to an exponential decay of the initial state through level crossings [12, 13]. The geometric transition has been formulated to include the higher corrections in Ref. [14]. Recently it has been shown that the in-vacuum of a time-dependent oscillator transported along a closed path in the complex plane may gain a geometric contribution from the simple pole at infinity and that the geometric factor explains Schwinger pair production in a constant electric field and dS radiation in the planar coordinates [15]. It has been further argued that the scattering amplitude between the transported in-vacuum and the in-vacuum determines multiple pair production, depending on the winding number of the closed path in the complex plane. However, these models have only one pair of complex turning points and rule out the Stokes phenomenon, whereas the global coordinates of dS spacetime provide two pairs of complex turning points for each harmonics of quantum field and lead to the Stokes phenomenon. The organization of this paper is as follows. In Sec. II the real-time evolution of a quantum field is formulated in the functional Schrodinger picture. In Sec. III the scattering amplitudes between the in-vacuum and the transported one along closed paths in the complex-time plane are computed and the particle-production rate is given by summing the scattering amplitude squares for all independent paths of winding number one. It is shown that the Stokes phenomenon is a consequence of the interference among different paths, which has a geometric origin. In Sec. IV we compare the result of this paper with other methods and discuss the physical implications.", "pages": [ 1, 2 ] }, { "title": "II. EVOLUTION OPERATOR IN REAL TIME", "content": "For the sake of simple harmonics decomposition we consider a complex scalar in the global coordinates of a (d+1)dimensional dS spacetime (in units of c = /planckover2pi1 = 1) The field equation for the complex scalar field with mass m may be derived from the Lagrangian 1 where /square = (1 / √ -g ) ∂ µ ( √ -gg µν ∂ ν ). Decomposing ψ and ψ ∗ by the spherical harmonics on S d , ∇ 2 u κ ( x ) = -κ 2 u κ ( x ) with κ 2 = l ( l + d -1) , ( l = 0 , 1 , · · · ) [17] and symmetrizing them, we obtain the Hamiltonian where π κ = ˙ ψ ∗ κ +(˙ g/ 4 g ) ψ ∗ κ and π ∗ κ = ˙ ψ κ +(˙ g/ 4 g ) ψ κ , and where for a massive scalar ( m>dH/ 2) In the functional Schrodinger picture, the evolution operator for the field obeys the time-dependent Schrodinger equation Each Hamiltonian is diagonalized by the time-dependent annihilation and creation operators as Then the evolution operator is expressed by the spectral resolution [15] where H κD ( t ) = ω κ ( t ) diag (1 / 2 , · · · , n +1 / 2 , · · · ) is the diagonal matrix and Φ T ( t ) = ( | 0 κ , t 〉 , · · · , | n κ , t 〉 , · · · ) is the row vector of the number states for (7), and A κ ( t ) is the induced vector potential from the time-dependent number states with entries Here T denotes the transpose of the matrix or vector. Note that ω κ ( t ) > 0 and A κ ( t ) does not have any singularity, so ˆ U κ ( t 0 , t 0 ) = I for any path along the real-time axis and the in-in formalism thus becomes trivial. In the real-time dynamics the in-out formalism carries all physical information through the scattering matrix between the out-vacuum and the in-vacuum. Hence, to implement the in-in formalism for particle production, the real-time dynamics should be extended to the complex-time plane, as will be shown in the next section.", "pages": [ 2, 3 ] }, { "title": "III. GEOMETRIC INTERPRETATION OF STOKES PHENOMENON", "content": "It has been known for long that the quantum evolution of a time-dependent Hamiltonian system exhibits a rich structure in the complex-time plane, such as geometric phases and nonadiabatic evolutions [12, 13, 18]. Now we extend the quantum evolution (6) to a complex plane. For that purpose we assume that the geometry (1) and the Hamiltonian (3) have an analytical continuation in the whole complex plane, which is realized by the conformal mapping Being interested in the quantum evolution along a path z ( t ) in the complex plane, we analytically continue Eq. (6) to where ˆ H κ ( z ) := ∂t ∂z ˆ H κ ( t ( z )). The quantum field theory in analytically continued geometries has also been discussed in Ref. [19]. Hence the spectrally resolved evolution operator (8) is analytically continued to a closed path C ( z ) in the complex plane provided that 〈 m,z | n, z 〉 holds along the path. Then the lowest order of the Magnus expansion [20, 21] gives the scattering amplitude between the in-vacuum and the transported one along a path C ( n ) of winding number n with the base point t 0 [15] In the in-out formalism the vacuum persistence, which is the magnitude of the square of the scattering amplitude between the out-vacuum and the in-vacuum, is the probability for the out-vacuum to remain in the in-vacuum. The decay of the vacuum persistence results from one-pair and multipair production in bosonic theory [22]. Similarly, the magnitude of the scattering amplitude square (12) is the rate for multiparticle production and depends only on the information of simple poles included in the path. The pair-production rate in time-dependent electric fields has been proposed of the form | e -i ∮ C (1) ω ( z ) dz | in the phase-integral method [23]. The dynamical phase has an extension to the complex plane along C ( n ) ( t 0 ) as where the branch points are Cutting the branch points z + , z -and their conjugates z ∗ + , z ∗ -as shown in Fig. 1, the integrand in Eq. (14) is an analytic function. The integrand (14) has a simple pole at z = ∞ , which corresponds to the cosmological horizon, and which is located outside the path and can be obtained by the large z expansion [24]. The geometric contribution from the pole at infinity is universal for all paths of nonzero winding numbers. Further, there are two finite simple poles at z i = i and z i = -i , which correspond to the north and south poles of the Euclidean geometry of dS space (1). The simple poles at z = ± i classify four independent paths of winding number 1 with the base point t 0 in the z plane: the first class C (1) I does not include any pole at z = ± i as shown in Fig. 1, the second class C (1) II and the third class C (1) III include only one pole at z = ± i as shown in the left panel of Fig. 2, and the fourth class C (1) IV includes both poles at z = ± i as shown in the right panel of Fig. 2. The scattering amplitude between the in-vacuum and the transported one along a path of each class always receives a geometric contribution -2 iπ Res ω ( ∞ ) from the simple pole at z = ∞ , which is located outside the path. The particle-production rate is the magnitude of the sum of the scattering amplitude square for each class path Here the first term in the parenthesis comes from C (1) I ( t 0 ), the second term from C (1) II ( t 0 ) and C (1) III ( t 0 ), and the last term from C (1) IV ( t 0 ). It should be noted that the magnitude is taken after summing over all independent paths of winding number 1. In the limit of large action ( | ∮ ω κ | /greatermuch 1 and l /greatermuch 1), we approximately have λ ≈ l + d/ 2 -1 / 2 and obtain the particle-production rate Hence, in odd dimensions ( d even) the particle-production rate vanishes while in even dimensions it is the leading Boltzmann factor of the exact formula [2] Finally, we compare the result of this paper with the Stokes phenomenon in the phase-integral method [11]. In the complex plane { z ∗ -, z -} and { z + , z ∗ + } constitute two pairs of complex turning points and each pair gives the Hamilton-Jacobi action for the scattering over barrier where J denotes the class II or III . The imaginary and real parts of the actions determine the exponential and oscillatory behaviors for the particle-production rate, respectively [11]. Thus the Stokes phenomenon for dS radiation originates from the interference among four independent paths involving two simple poles at the north and south poles of the Euclidean geometry.", "pages": [ 3, 4 ] }, { "title": "IV. CONCLUSION", "content": "We have shown that the Stokes phenomenon for dS radiation, constructive interference in even dimensions and destructive interference in odd dimensions, has a geometric interpretation in the complex-time plane. In contrast to the trivial real-time dynamics in the in-in formalism, the transported in-vacuum of a quantum field along a closed path in the complex-time plane may gain geometric contributions from possible simple poles and the magnitude of the scattering amplitude square between the in-vacuum and the transported one gives the particle-production rate for that path. The global coordinates of a dS spacetime have two finite simple poles corresponding to the north and south poles of the Euclidean geometry as well as the simple pole at infinity, corresponding to the cosmological horizon. It is shown that the four classes of paths, either including or not including the finite simple poles, provide each channel for dS radiation, which explains not only the leading Boltzmann factor from the simple pole at infinity but also the sinusoidal behavior from finite simple poles. Thus the Stokes phenomenon for dS radiation has a geometric interpretation in the complex-time plane. We now compare the geometric interpretation of this paper with other tunneling approaches to dS radiation. In the tunneling interpretation of dS radiation the cosmological horizon in the static coordinates plays an essential role in emitting particles from vacuum fluctuations near the horizon [25-32]. On the other hand, the geometric interpretation relies on the nonstationary nature of the time-dependent Hamiltonian of a quantum field in dS spacetime. The quantum evolution in the complex-time plane provides the geometric factor when the in-vacuum is transported along a closed path and returns to the initial time. In fact, the magnitude of the scattering amplitude square between the in-vacuum and the transported one along an independent path gives a channel for particle production. Further, the two simple poles from the north and south poles of the Euclidean geometry result in the interference among independent paths, constructive in even dimensions and destructive in odd dimensions. It would be interesting to investigate physics behind two methods by comparing different coordinates for the embedding spacetime.", "pages": [ 4, 5, 6 ] }, { "title": "Acknowledgments", "content": "The author would like to thank Misao Sasaki for useful discussions on de Sitter spacetimes. He also thanks Eunju Kang for drawing figures. This paper was initiated and completed at Yukawa Institute for Theoretical Physics, Kyoto University. This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2012R1A1B3002852). [13] A. Joye, H. Kunz, and Ch.-Ed Pfister, 'Exponential Decay and Geometric Aspect of Transition Probabilities in the Adiabatic Limit,' Ann. Phys. 208 , 299 (1991).", "pages": [ 6 ] } ]

[ { "title": "What wormhole is traversable?", "content": "- A case of a wormhole supported by a spherical thin shell - 1 , 2 Ken-ichi Nakao ∗ , 2 Tatsuya Uno † and 3 Shunichiro Kinoshita 1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2 Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan 3 Osaka City University Advanced Mathematical Institute, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan (Dated: June 12, 2021)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We analytically explore the effect of falling matter on a spherically symmetric wormhole supported by a spherical shell composed of exotic matter located at its throat. The falling matter is assumed to be also a thin spherical shell concentric with the shell supporting the wormhole, and its self-gravity is completely taken into account. We treat these spherical thin shells by Israel's formalism of metric junction. When the falling spherical shell goes through the wormhole, it necessarily collides with the shell supporting the wormhole. To treat this collision, we assume the interaction between these shells is only gravity. We show the conditions on the parameters that characterize this model in which the wormhole persists after the spherical shell goes through it. PACS numbers: 04.20.-q, 04.20.Jb, 04.70.Bw ‡", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The wormhole is a tunnel-like spacetime structure by which a shortcut or travel to disconnected world is possible. Active theoretical studies of this fantastic subject began by an influential paper written by Morris, Thorne and Yurtsever[1] and Morris and Thorne[2]. The earlier works are shown in the book written by Visser[3] and review paper by Lobo[4]. However, it is not trivial what is the mathematically rigorous and physically reasonable definition of wormhole in general situation, although we may find a wormhole structure in each individual case. Hayward gave an elegant definition of wormhole by using trapping horizon and showed that the violation of the null energy condition is a necessary condition for the existence of the wormhole in the framework of general relativity, where the null energy condition is T µν k µ k ν ≥ 0 for any null vector k µ [5, 6]. The exotic matter is necessary to make a wormhole, but where is an exotic matter? In Refs.[1] and [2], the authors discussed possibilities of quantum effects. Alternatively, such an exotic matter is often discussed in the context of cosmology. The phantom energy, whose equation of state is p = wρ with w < -1 and positive energy density ρ > 0, does not satisfy the null energy condition, and a few researches showed the possibility of the wormhole supported phantom-like matter[7-9]. Recently, theoretical studies from observational point of view on a compact object made of the exotic matter, possibly wormholes, have also reported[10, 11]. It is very important to study the stability of wormhole model in order to know whether it is traversable. The stability against linear perturbations is a necessary condition for the traversable wormhole, but it is insufficient. The investigation of non-linear dynamical situation is necessary, and there are a few studies in this direction[12-14]. In this paper, we study the condition that a wormhole persists even if it experiences non-linear disturbances. In our model, the wormhole is assumed to be supported by a spherical thin shell composed of the exotic matter, and hence the wormhole itself is also assumed to be spherically symmetric. The largest merit of a spherical thin shell wormhole is the finite number of its dynamical degrees of freedom. By virtue of this merit, we can analyze this model analytically even in highly dynamical cases. The thin shell wormhole was first devised by Visser[15], and then its stability against linear perturbations was investigated by Poisson and Visser[16]. Recently, the linear stability of the thin shell wormhole in more general situation has been investigated by Garcia, Lobo and Visser[17]. In this paper, we consider a situation in which a spherical thin shell concentric with a wormhole supported by another thin shell enters the wormhole. These spherical shells are treated by Israel's formulation of metric junction[18]. When the shell goes through the wormhole, it necessarily collides with the shell supporting the wormhole. The collision between thin shells has already studied by several researchers[19-21], and we follow them. Then, we show the condition that the wormhole persists after a spherical shell passes the wormhole. This paper is organized as follows. In Sec. II, we derive the equations of motion for the spherical shell supporting the wormhole and the other spherical shell falling into the wormhole, in accordance with Israel's formalism of metric junction. In Sec. III, we derive a static solution of thin shell wormhole which is the initial condition. In Sec. IV, we reveal the condition that a shell falls from infinity and goes through the wormhole. In Sec. V, we study the motion of the shells and the change in the gravitational mass of the wormhole after collision. In Sec. VI, we show the condition that the wormhole persists after the shell goes through it. Sec. VII is devoted to summary and discussion. In this paper, we adopt the geometrized unit in which the speed of light and Newton's gravitational constant are one.", "pages": [ 2, 3 ] }, { "title": "II. EQUATION OF MOTIONS FOR SPHERICAL SHELLS", "content": "We consider two concentric spherical shells which are infinitesimally thin. The trajectories of these shells in the spacetime are timelike hypersurfaces: The inner hypersurface is denoted by Σ 1 , and the outer hypersurface is denoted by Σ 2 . These hypersurfaces divide a domain of the spacetime into three domains: The innermost domain is denoted by D 1 , the middle one is denoted by D 2 , and the outermost one is denoted by D 3 . We also call Σ 1 and Σ 2 the shell-1 and the shell-2, respectively. This configuration is depicted in Fig. 1. By the symmetry of this system, the geometry of the domain D i ( i = 1 , 2 , 3) is described by the Schwarzschild solution whose line element is given by with where M i is the mass parameter. We should note that the coordinate t i is not continuous across the shells, whereas r , θ and φ are continuous across the shells. The location of the horizon is given by a solution of the equation f i ( r ) = 0 as The positive root exists if and only if M i is positive. Since finite energy and finite momentum concentrate on the infinitesimally thin domains, the stress-energy tensor diverges on these shells. This means that these shells are categorized into the so-called curvature polynomial singularity through the Einstein equations. Even though Σ A ( A = 1 , 2) are spacetime singularities, we can derive the equation of motion for each spherical shell which is consistent with the Einstein equations by so-called Israel's formalism. Let us cover the neighborhood of one singular hypersurface Σ A by a Gaussian normal coordinate λ , where ∂/∂λ is a unit vector normal to Σ A and directs from D A to D A +1 . Then, the sufficient condition to apply Israel's formalism is that the stress-energy tensor is written in the form where Σ A is located at λ = λ A , δ ( x ) is Dirac's delta function, and S µν is the surface stressenergy tensor on Σ A . The junction condition of the metric tensor is given as follows. We impose that the metric tensor g µν is continuous across Σ A . Hereafter, we denote the unit normal vector of Σ A by n µ instead of ∂/∂λ . The intrinsic metric of Σ A is given by and the extrinsic curvature is defined by where ∇ ( i ) α is the covariant derivative with respect to the metric in the domain D i . This extrinsic curvature describes how Σ A is embedded into the domain D i . In accordance with Israel's formalism, the Einstein equations lead to where tr S is the trace of S µν . Equation (7) gives the condition of the metric junction. By the spherical symmetry, the surface stress-energy tensors of the shells should be the perfect fluid type; where σ and P are the energy per unit area and the pressure on Σ A , respectively, and u µ is the 4-velocity. By the spherical symmetry, the motion of the shellA is described in the form of t i = T A,i ( τ ) and r = R A ( τ ), where i = A or i = A + 1, that is to say, i represents one of two domains divided by the shellA , and τ is the proper time of the shell. The 4-velocity is given by where a dot means the derivative with respect to τ . Then, n µ is given by Together with u µ and n µ , the following unit vectors form an orthonormal frame; The extrinsic curvature is obtained as and the other components vanish, where a prime means a derivative with respect to its argument. By the normalization condition u µ u µ = -1, we have Substituting the above equation into Eq. (14), we have From the u -u component of Eq. (7), we obtain the following relations. Here, we assume the following equation of state where w A is constant. Substituting Eq. (18) into Eq. (17), we obtain", "pages": [ 3, 4, 5, 6 ] }, { "title": "A. The shell-1: Initially inner shell", "content": "We assume that the domains D 1 and D 2 form a wormhole structure by the shell-1. This means that n a ∂ a ln r | D 1 < 0 and n a ∂ a ln r | D 2 > 0 (see Fig. 2), and we have Here, note that Eq. (20) implies ˙ T 1 , 1 is negative, whereas ˙ T 1 , 2 is positive. Hence, the direction of the time coordinate basis vector in D 1 is opposite with that in D 2 . From θ -θ component of Eq. (7), we obtain the following relations. Equation (21) is satisfied only if σ 1 is negative, and hence we assume so. Here, we introduce a new positive variable defined by From Eq. (19), we have where µ is a positive constant, and, for notational simplicity, hereafter we denote w 1 by w . Let us rewrite Eq. (21) into the form of the energy equation for the shell-1. First, we write it in the form and then take a square of the both sides of the above equation: Furthermore, we rewrite the above equation in the form By taking a square of the both sides of the above equation, we have where where - ( m - 1 ) - + M r - ( m 2 r ) Equation (27) is regarded as the energy equation for the shell-1. The function V 1 corresponds to the effective potential. In the allowed domain for the motion of the shell-1, an inequality V 1 ≤ 0 should hold. But, this inequality is not a sufficient condition of the allowed region. The left hand side of Eq. (24) is non-negative, and hence the right hand side of it should also be non-negative. Then, substituting Eq. (26) into the right hand side of Eq. (24), we have Further manipulation leads to By the similar argument, we obtain Then, by the similar procedure, we have Hence, we have Finally, we obtain the following constraint; or In order to find the allowed domain for the motion of the shell-1, we need to take into account the constraint (35) or (36) in addition to the condition V 1 ≤ 0. = 1 M 2 M 1 2 M 1 2 1 2", "pages": [ 6, 7, 8 ] }, { "title": "B. The shell-2: Initially outer shell", "content": "For simplicity, we assume that the outer shell (shell-2) is composed of dust, i.e., w 2 = 0. The proper mass of the shell-2 is defined by By Eq. (19), we find that m 2 is constant. We assume that σ 2 takes any value except for the trivial case σ 2 = 0, and hence m 2 can take any value except for the trivial case m 2 = 0. We assume the wormhole structure does not exist around the shell-2. Hence, the extrinsic curvature of the shell-2 is given by By using the above result, the θ -θ component of the junction condition leads to In the case of m 2 > 0, we find from the above equation that f 2 ( R 2 ) > f 3 ( R 2 ), or equivalently, M 3 > M 2 . From the above equation, we have Since the left hand side of the above equation is non-negative, the following inequality should be satisfied. By taking the square of the both sides of Eq. (40), we have Substituting the above result into the left hand side of Eq. (41), we have In the case of m 2 < 0, we find from Eq. (39) that f 2 ( R 2 ) < f 3 ( R 2 ), or equivalently, M 3 < M 2 . From Eq. (39), we have Since the left hand side of the above equation is non-negative, the following inequality should be satisfied. By taking the square of the both sides of Eq. (44), we have Substituting the above result into the left hand side of Eq. (45), we have From Eqs. (43) and (47), we have By taking a square of both sides of Eq. (42), we obtain an energy equation for the shell-2, where with Note that E is a constant which corresponds to the square of the specific energy of the shell-2. The allowed domain for the motion of the shell-2 satisfies V 2 ≤ 0 and Eq. (48) as long as R b ≥ 2 M 3 . 1", "pages": [ 9, 10 ] }, { "title": "III. STATIC WORMHOLE SOLUTION", "content": "We consider a situation in which the wormhole supported by the shell-1 is initially static. For simplicity, we assume the symmetric wormhole, i.e., E = 0 or equivalently, M 1 = M 2 = M wh . In this case, the analysis becomes very simple. In order that the wormhole structure is static, the areal radius R 1 = a of the shell-1 should satisfy V 1 ( a ) = 0 = V ' 1 ( a ). Furthermore, in order that this structure is stable, V '' 1 ( a ) > 0 should be satisfied. These conditions lead to From Eq. (53), we have and Since a should be positive, we find that both M wh > 0 and 2 w +1 < 0 should be satisfied, or both M wh < 0 and 2 w + 1 > 0 should be satisfied. Substituting Eq. (55) into the left hand side of Eq. (54), we have The above inequality implies -1 / 2 < w < -1 / 4. Hence, the static symmetric wormhole is stable, only if Substituting Eq. (55) into Eq. (52), we have Substituting the above equation into Eq. (55), we have and hence where The factor F ( w ) is monotonically increasing in the domain -1 / 2 < w < -1 / 4, and F ( w ) → -∞ for w →-1 / 2, whereas F ( w ) → 0 for w →-1 / 4.", "pages": [ 10, 11 ] }, { "title": "IV. THE CONDITION OF THE ENTRANCE TO THE WORMHOLE", "content": "We show the condition that the shell-2 enters the wormhole supported by the shell-1. The allowed domain for the motion of the shell-2 is determined by the conditions (48) and V 2 ≤ 0. The shell-2 is assumed to come from the spatial infinity. By this assumption, E > 1 or E = 1 with M d ≥ 0 should hold from the condition that V 2 ( r ) < 0 for sufficiently large r . If M d ≥ 0, we easily see that V 2 ( r ) < 0 always holds for E ≥ 1, and hence the shell-2 can enter the wormhole (see Fig. 3). In the case of M d < 0 (hence E should be larger than unity), the equation V ' 2 ( r ) = 0 has a positive root where a prime represents a derivative with respect to the argument. Furthermore, if V 2 ( r m ) > 0, or equivalently, the shell-2 falling from infinity may be eventually prevented from the entrance to the wormhole by the potential barrier. If the inequality in Eq. (63) holds, the equation V 2 ( r ) = 0 has two positive roots given by If the equality of Eq. (63) holds, R + agrees with R -. The shell-2 cannot enter the domain R 2 < R + as long as Eq. (63) is satisfied. Hence, if Eq. (63) is satisfied, R + < a should be satisfied so that the shell-2 enters the wormhole. Let us investigate V 2 ( R b ), where R b is defined in Eq. (48). We have  Let us consider two cases M 3 ≤ 0 and M 3 > 0, separately. In the former case, V 2 ( R b ) ≥ 0 for M 3 ≤ 0 since M 2 = M wh < 0 is assumed. Since Eq. (63) is satisfied, the domain of V 2 ≤ 0 is R 2 ≥ R + and 0 ≤ R 2 ≤ R -, and furthermore, we find R -≤ R b ≤ R + by the inequality V 2 ( R b ) ≥ 0. The constraint (48) implies that the only domain of R 2 ≥ R + is allowed for the motion of the shell-2. In the latter case, since M 3 is necessarily larger than M 2 , we have Hence, if R b is larger than or equal to 2 M 3 , V 2 ( R b ) > 0 and hence the situation is similar to the former case: The allowed domain for the shell-2 is R 2 ≥ R + . As mentioned in the footnote 1, if R b is smaller than 2 M 3 , the allowed domain for the shell-2 is determined by the only condition V 2 ≤ 0. To summarize, one of the following three conditions should be satisfied so that the shell2 falling from infinity enters the wormhole. By using the relation M 2 = M wh and M 3 = M wh + m 2 √ E ,", "pages": [ 12, 13 ] }, { "title": "V. COLLISION BETWEEN THE SHELLS", "content": "Let us consider a process in which the shell-2 shrinks and collides the shell-1 which supports the wormhole. The situation may be recognized by Fig. 4. The collision occurs at r = a . Then, in this section, we show how the mass parameter in the domain between the shells changes by the collision. We assume that the interaction between these shells is gravity only. Thus, after the collision, these shells merely go through each other: The 4-velocities u α i ( i = 1 , 2) of the shells are continuous at the collision event, respectively. We assume that the proper mass m A of each shell does not change. In the domain D 2 , we have two tetrad basis ( u α i , n α i , e α ( θ ) , e α ( φ ) ), where i = 1 , 2. We can express the 4-velocity u α 1 of the shell-1 by using the tetrad basis ( u α 2 , n α 2 , e α ( θ ) , e α ( φ ) ), and converse is also possible; The components of u α i and n α i with respect to the coordinate basis in D 2 , i.e., ( t 2 , r, θ, φ ), are given by where f 2 = f 2 ( a ). Hence, we have", "pages": [ 13, 14, 15 ] }, { "title": "A. Shell-1 after the collision", "content": "The tetrad basis ( u α 2 , n α 2 , e α ( θ ) , e α ( φ ) ) is available also in the domain D 3 . The components of u α 2 and n α 2 with respect to the coordinate basis in D 3 are given by where f 3 = f 3 ( a ). By using the above equations, we obtain the components of u α 1 with respect to the coordinate basis in D 3 as The above components are regarded as those of the 4-velocity of the shell-1 in the domain D 3 just after the collision event. By using Eqs. (39) and (79), we have By taking the square of Eq. (39) and using Eq. (49), we have The above equation implies Then, we have We can check that the normalization condition -f 3 ( u t 3 1 ) 2 + f -1 3 ( u r 1 ) 2 = -1 is satisfied. The above result implies that after the collision, the derivative of the areal radius of the shell-1 with respect to its proper time becomes Since the shell-2 falls into the wormhole, ˙ R 2 is negative. This fact implies that the shell-1 or equivalently the radius of the wormhole throat begin shrinking just after the shell-1 collides with the shell-2 if m 2 is positive. By contrast, if m 2 is negative, the shell-1 start to expand after the shell-2 goes through the wormhole. This result implies that m 2 plays a role of not only the proper mass of the shell-2 but also the active gravitational mass of it. The domain between the shell-1 and the shell-2 after the collision is called D 4 . By the symmetry, D 4 is also described by the Schwarzschild geometry with the mass parameter M 4 . From the junction condition between D 4 and D 3 , the shell-1 obeys the following equation just after the collision; From the above equation and Eq. (85), we obtain", "pages": [ 15, 16 ] }, { "title": "B. Shell-2 after the collision", "content": "Since the tetrad basis ( u α 1 , n α 1 , e α ( θ ) , e α ( φ ) ) is available also in the domain D 1 . By using Eqs. (14), (15) and (20), the components of u α 1 and n α 1 with respect to the coordinate basis in D 1 are given by where f 1 = f 1 ( a ). As already noted just below Eq. (20), the time component of u α 1 with respect to the coordinate basis in D 1 is negative. By using the above equations, we obtain the components of u α 2 with respect to the coordinate basis in D 1 as where we have used the symmetric condition f 1 ( a ) = f 2 ( a ). Since ˙ R 2 is negative, the shell-2 begins expanding after the collision. This is a reasonable result because of the wormhole structure. From the junction condition between D 1 and D 4 , we have From Eq. (91), since ˙ R 2 2 is unchanged by the collision, we have", "pages": [ 17 ] }, { "title": "C. The mass parameter M 4 in D 4", "content": "Equations (87) and (94) impose the following constrains on one unknown parameter M 4 ; and by the definition of V 2 , i.e., Eq. (50), we have Since M 1 = M 2 = M wh , Eqs. (96) and (97) lead to By solving the above equation with respect to M 4 , we obtain two roots, M 4 = M 3 and M 4 = 2 M 1 -M 3 -m 2 2 /a . Since the shell-1 is static before the collision, V 1 ( a ) should vanish, and this condition leads to By using the above condition and Eq. (95), we find that is a solution, where we have used M 1 = M 2 = M wh and M 3 = M wh + m 2 √ E . Hence, after the collision, the wormhole becomes asymmetric. Asymmetric wormhole is necessarily metastable, and hence the wormhole might collapse.", "pages": [ 17, 18 ] }, { "title": "VI. THE CONDITION THAT THE WORMHOLE PERSISTS", "content": "In this section, we consider the condition that the wormhole stably exists after the entrance of the shell-2. First of all, a > 2 M 3 should hold. If it is not the case, the wormhole is enclosed by an event horizon after the shell-2 enters a domain of R 2 ≤ 2 M 3 . Second, the effective potential of the shell-1 should have a negative minimum between positive potential domains. From Eq. (86), we see that the effective potential of the shell-1 after the collision ¯ V 1 is given by where By the condition (57), ¯ V 1 ( r ) → -∞ for r → 0, whereas ¯ V 1 ( r ) → 1 for r → ∞ . Hence, the effective potential ¯ V 1 should have at least two extremums (see Fig. 5). The equation ¯ V ' 1 ( r ) = 0 is rewritten in the form where x = r 4 w +1 . Since, as mentioned, the effective potential ¯ V 1 should have two extremum, the discriminant of the above quadratic equation should be positive, i.e., The two real roots of ¯ V ' 1 ( r ) = 0 is given by The maximum of the effective potential ¯ V 1 is at r = R ex+ , and the condition ¯ V 1 ( R ex+ ) > 0 should hold. Furthermore, the radius of the wormhole at the moment of the collision should be larger than R ex+ . If so, ¯ V 1 ( R ex -) is necessarily negative, and hence we need not impose ¯ V 1 ( R ex -) < 0 in addition to a > R ex+ . To summarize, all of the following four conditions should be satisfied so that the wormhole persists after the shell-2 goes through it. P1) a > 2 M 3 . P3) a > R ex+ . There are three independent parameters in this model. The initial static wormhole is characterized by the constant of proportionality in the equation of state, w , and its gravitational mass, M wh : Note that µ and a are the functions of w and M wh by Eqs. (58) and (60). However since M wh may be regarded as a unit, the remaining parameter is only w , whereas the dust shell is characterized by two parameters, its proper mass m 2 and the square of conserved specific energy E . In Figs. 6-8, we depict the domains that satisfy the conditions of the entrance of the shell-2 to the wormhole, E1)-E3), and the conditions of the persistence of the wormhole, P1)-P4), in ( E,m 2 )-plane in three cases of w = -7 / 16 , -3 / 8 , -5 / 16, respectively. In the domain shaded by straight lines, at least one of E1), E2) and E3) is satisfied. The domain shaded by dots is the intersection of the domains each of which P1)-P4) are satisfied. If the parameters E and m 2 take values in the intersection of the domains shaded by straight lines and dots, the wormhole does not collapse but merely oscillates after the shell-2 goes through the wormhole.", "pages": [ 18, 19, 20 ] }, { "title": "VII. SUMMARY AND DISCUSSION", "content": "We analytically studied the dynamical process in which a spherical thin shell of dust goes through a wormhole supported by a spherical thin shell composed of the matter whose tangential pressure is proportional to its surface energy density with a constant of proportionality w . We treated these thin shells by Israel's formalism of metric junction. The negativ surface energy density of the shell is necessary to form the wormhole structure. This result is consistent with the known fact that the wormhole structure needs the violation of the null energy condition. We considered the situation in which the wormhole is initially static and has Z 2 symmetry with respect to the spherical thin shell supporting it, and found that the gravitational mass of the static wormhole should be negative, and the constant of proportionality in the equation of state should satisfies -1 / 2 < w < -1 / 4, in order that the wormhole is stable against linear perturbations. Then we studied the condition that the wormhole persists after a spherical thin dust shell concentric with it goes through it. We assumed that the interaction between the wormhole shell and the dust shell is only gravity, or in other words, the 4-velocities of these shells are assumed to be continuous at the collision event. In this model, there are three free parameters: The constant of proportionality, w , which characterizes the wormhole shell, the square of conserved specific energy E and the proper mass m 2 , which characterize the dust shell, in the unit that the initial gravitational mass of the wormhole is one. Then, we showed that there is a domain of the non-zero measure in ( m 2 , E )-plane for three values of w , in which the wormhole persists after the dust shell goes through it. In this paper, we investigated the case of the only linear equation of state for the shell supporting the wormhole. We need to investigate whether the present result strongly depends on the equation of state. This will be discussed elsewhere.", "pages": [ 20, 22, 23 ] }, { "title": "Acknowledgments", "content": "KN thanks the participants of 'workshop on theories and possibilities of observations of wormholes' held at Rikkyo university in October 2012 for useful discussions.", "pages": [ 23 ] } ]

[ { "title": "Avenues for Analytic exploration in Axisymmetric Spacetimes. Foundations and the Triad Formalism.", "content": "Jeandrew Brink, 1, 2 Aaron Zimmerman, 1, 3 and Tanja Hinderer 1, 4 1 National Institute for Theoretical Physics (NITheP), Western Cape, South Africa 2 Physics Department, Stellenbosch University, Bag X1 Matieland, 7602, South Africa 3 Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA 4 Maryland Center for Fundamental Physics & Joint Space-Science Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA (Dated: July 27, 2018) Axially symmetric spacetimes are the only vacuum models for isolated systems with continuous symmetries that also include dynamics. For such systems, we review the reduction of the vacuum Einstein field equations to their most concise form by dimensionally reducing to the threedimensional space of orbits of the Killing vector, followed by a conformal rescaling. The resulting field equations can be written as a problem in three-dimensional gravity with a complex scalar field as source. This scalar field, the Ernst potential is constructed from the norm and twist of the spacelike Killing field. In the case where the axial Killing vector is twist-free, we discuss the properties of the axis and simplify the field equations using a triad formalism. We study two physically motivated triad choices that further reduce the complexity of the equations and exhibit their hierarchical structure. The first choice is adapted to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a simplification of the three-dimensional Ricci tensor and the boundary conditions on the axis. We illustrate its properties by explicitly solving the field equations in the case of static axisymmetric spacetimes. The other choice of triad is based on geodesic null coordinates adapted to null infinity as in the Bondi formalism. We then explore the solution space of the twist-free axisymmetric vacuum field equations, identifying the known (unphysical) solutions together with the assumptions made in each case. This singles out the necessary conditions for obtaining physical solutions to the equations. PACS numbers: 04.20.-q, 04.20.Cv, 04.20.Jb", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Numerical relativity has revolutionized our understanding of General Relativity (GR) in the last decade, allowing us to study situations of high curvature and strongly nonlinear dynamics (see [1] for a comprehensive review). In particular, numerical relativity has allowed for the solution of the two-body problem in GR, giving a description of the interaction and merger of compact objects. Successful numerical simulations of merging black holes have shown that these events can be well described by Post-Newtonian theory up until the black holes are quite near merger, and after merger black hole perturbation theory accurately describes the ring-down. Where perturbation theory fails, a simple transition between the regimes of the 'chirp' waveform associated with PostNewtonian theory and the exponential decay to a stationary black hole is observed. A primary focus of current research is to combine these computationally expensive simulations with analytical approximations to create full inspiral-merger-ringdown gravitational waveforms [2-8]. Such waveforms will serve as templates for the matchedfiltering based signal detection methods that will be used in ground-based gravitational-wave detectors coming into operation within the next few years [9-12]. Despite the success of perturbation and numerical methods in modeling binary merger waveforms, a detailed understanding of the nonlinear regime of a binary merger remains an open problem. It is in this stage of the merger that the black hole binary emits most of its radiated energy (see [13] and the references therein) and experiences a possibly strong kick due to beamed emission of radiation [14-17]. As such, deeper analytic understanding of nonlinear dynamics in GR, including better insights into the two-body problem, gravitational wave generation, and black hole formation, remains a primary research goal. The purpose of this paper is to review and expand on the analytic techniques involved in the study of the field equations in axisymmetry. Along the way we will collect many known and useful results, placing them into a unified context and notation. We intend this comprehensive overview of the state of knowledge in the field to serve as a launching point for future analytic investigations and searches for physically relevant, exact dynamical solutions in an era where a wealth of numerical data from simulations is available to guide our intuition. We will largely restrict the discussion to the simplest case where there is no rotation about the axis of symmetry (so that the Killing vector of the symmetry is 'twistfree'). While specializing to such a great degree does limit the scope of our discussion, at least two interesting scenarios are still included in the spacetimes under consideration. The first is the case of a head-on merger of two non-rotating black holes, the simplest instance of the two body problem in GR. The second is the critical collapse of axially symmetric gravitational waves, which gives insights into the formation of black holes (for a review see [18]). Our approach to exploring the Einstein field equations in this context closely follows the methods developed by Hoenselaers, Geroch, and Xanthopolous [19-24]. The basic idea is to reduce the number of equations to a minimum by applying a dimensional reduction and conformal rescaling to the axisymmetric field equations. The resulting equations are then expressed on a null basis, in the manner of the Newman-Penrose (NP) formalism [25] but in only three dimensions. This triad formalism imposes an additional structure on the equations to be solved, which can lead to valuable physical insights as in the NP formalism. As will be illustrated in the text, the resulting system of equations is simple enough to allow us to keep track of the assumptions made in trying to obtain a solution and to analyze the properties of a given solution. This approach may have the potential to make dynamical spacetime problems analytically tractable and to provide a consistent framework for systematically characterizing the results of axisymmetric numerical simulations. The formulation given here also has a close connection to that used to find solutions to the well-studied stationary axisymmetric vacuum (SAV) equations. To place our work in a broader context and to motivate the approach to the field equations advocated here, we now briefly review the development both of the field of exact solutions as well as aspects of the subsequent development of numerical relativity. Symmetry has often played a primary role in arriving at a solution to the field equations (Ref. [26] contains a comprehensive review). Famous solutions such as the Schwarzschild black hole, de Sitter, Anti de Sitter, and Friedmann-Robertson-Walker cosmological solutions all possess large numbers of symmetries. Relaxing the degree of symmetries present, but still imposing sufficient symmetry to make headway in solving the field equations, leads to the study of stationary, axisymmetric vacuum (SAV) spacetimes (equivalently, spacetimes with two commuting Killing vectors), which has been completely solved [19-21, 27-36]. It was shown that the task of solving the SAV equations can be reduced to seeking a solutions of Ernst's equation [27, 37] on a flat manifold. Various techniques to generate new solutions from known ones were developed in [20, 21, 28, 29], based on examining the integral extension (prolongation structure) of the SAV field equations. Over the ensuing decade a variety of additional techniques were explored, including the use of harmonic maps [19, 30], Backland transformations [31, 32], soliton and inverse scattering techniques [38], and the use of generating functions to exponentiate the infinitesimal Hoenselaers-Kinnersley-Xanthopoulos (HKX) transformations [33-36], to name a few. These techniques are all interrelated [39], and each has in turn taught us about the structure and properties of the SAV field equations. They allow, for example, the generation of an SAV spacetime with any desired asymptotic mass- and current-multipole moments [40]. Unfortunately, by their nature, SAV solutions cannot include gravitational radiation and tell us little about the dynamics of spacetime. Building on the progress made in studying in spacetimes with two Killing vectors, in the early and mid1970's triad methods were developed for spacetimes with a single symmetry, and applied to stationary spacetimes [41] and to dynamical, axisymmetric spacetimes [22-24]. At this time, however, the availability of increasingly powerful computers offered a promising new approach to obtaining solutions of the Einstein field equations for fully generic spacetimes by numerical means. In the relativity community at large, the major focus of research on solving the field equations shifted from systematically exploring the analytic structure to attempting their solution numerically. However, the numerical integration of the field equations proved to be unexpectedly difficult, especially in the axisymmetric case. The advent of strongly hyperbolic and stable formulations of the field equations (e.g. the commonly used BSSN [42, 43] and generalized harmonic [44] formulations) made the long term simulations of binary black hole simulations an exciting reality. With the steady progress since the breakthrough by Pretorius [45], the merger of compact objects has become routine [3, 46], although still computationally limited in duration and mass ratio. The insights afforded by these successes can now serve to guide research efforts aimed at obtaining an analytical understanding of dynamical solutions to the field equations. The relative simplicity of the gravitational waveforms and other observables generated during the highly nonlinear phase of a binary coalescence indicate that even this phase of merger could potentially be amenable to analytic techniques. A renewed interest in analytic investigations of axisymmetric spacetimes [47] has already led to interesting results such as the discovery and use of geometric inequalities [48-55], studies of the radiation in a head-on collision [56-58], models for understanding gravitational recoil [59, 60] and geometrical insights on gravitational radiation [61-63]. Initially, symmetries played an important role in the development of numerical relativity because of the great reduction in computational cost in axisymmetry compared to a fully 4D simulation. Some of the first successful work in numerical relativity was done in axisymmetry [64-67], following the initial attempt of [68]. Coordinate singularities at the axis of symmetry [69-73], and growing constraint violations, even when using strongly hyperbolic formulations of the field equations [74], presented computational challenges in fully axisymmetric codes. Because of these difficulties, successful codes capable of long-term evolutions of axisymmetric systems have only recently been developed [67, 75-77]. The continued interest in axisymmetric simulations is driven mainly by the desire to understand the critical collapse of gravitational waves [78, 79] and in the higher accuracy and lower computational cost of these simulations. In addition, similar dimensional reductions as in the axisymmetric case are also used in numerical simu- lations of spacetimes in theories with higher dimensions; see e.g. the mergers studied in [80, 81]. In addition to the simplicity of the nonlinear dynamics observed in numerical simulations of the merger event, there are other tantalizing indications that the axisymmetric problem could be solvable analytically. The field equations of axisymmetric spacetimes can be written in terms of a generalized Ernst potential on a curved background, given the appropriate dimensional reduction [19], discussed in detail in Sec. II. In this formulation some of the techniques used to find solutions for the SAV field equations, such as harmonic maps, have a straightforward generalization to the dynamic axisymmetric case. Viewing the numerical simulations in the context of the analytic techniques employed in the past may help provide new insight into questions such as the nature of initial junk radiation in numerical simulations, the reasons for the robustness of certain approaches such as the puncture method [82], the nature of singularity formation during a collapse process, and the possible distinction between which features of initial data contribute to the the mass of the final black hole and which components are ultimately radiated away (similar to the way in which poles and scattering data can be differentiated in the nonlinear solution of the KdV equations [83]). The intent of this work is to provide a framework that could be used in future work to explore and interact with the results of axisymmetric simulations, drawing on the accumulated analytic and numerical results available for these spacetimes to date. We now briefly outline the structure and contents of the paper.", "pages": [ 1, 2, 3 ] }, { "title": "A. Overview of this paper", "content": "We will begin our discussion in full generality, explicitly carrying out in Sec. II the series of reductions that ends in the field equations for vacuum, twist-free, axisymmetric spacetimes. Here we largely follow the discussion of [20], although in Appendix A we present the derivation in the familiar notation of the 3+1 decomposition used in numerical relativity. The resulting set of equations is equivalent to 3D GR coupled to a complex scalar potential E which obeys the Ernst equation. The manifold S on which these fields are defined is obtained by conformally rescaling the metric on the quotient space ¯ S with the norm of the axial KV. The space ¯ S should be thought of as the physical 4D manifold M modulo the orbits of the Killing vector (KV) ξ µ , or ¯ S = M /ξ µ . We then specialize to the case of non-rotating spacetimes, where the field equations are equivalent to 3D GR with a real harmonic scalar field source that obeys the Klein-Gordon equation. In Sec. II F we discuss general considerations regarding the existence of an axis, and note that the problem of divergences at the axis is in principal easily handled analytically. In Sec. III we express the 3D field equations in terms of a triad formulation that was first developed by Hoense- laers [22-24], but seemingly not used by other authors (it should be compared to a similar triad formulation presented by Perj'es [41] and used in the case of stationary spacetimes). This formulation is derived by the projection of the dimensionally-reduced field equations onto a 3D null basis, composed of two null and one orthonormal spatial vector. The field equations and Bianchi identities are then written out in full, in terms of the 3D rotation coefficients. Considering the success of the NP equations and the valuable insights they provide, this formulation of the axisymmetric field equations merits a more thorough investigation than it appears to have received. In Sec. IV, we relate the 3D rotation coefficients and curvatures quantities to the familiar NP quantities on M , thus providing a dictionary between NP quantities and the quantities that arise in the triad formulation. This facilitates a connection to known results and an interpretation of the physical content of the triad equations. In the triad formulation, we have the freedom to specialize our choice of basis vectors. In Sec. V, we present two useful choices of triad vectors and accompanying coordinates which serve to simplify the field equations. The first choice is, to our knowledge, new, and is analogous to the use of Lagrangian coordinates in fluid mechanics. In this triad choice the spatial triad leg is adapted to the gradient of the scalar field that encodes the dynamical degree of freedom in the twist-free axisymmetric spacetime. By virtue of the field equations, this scalar field is a harmonic coordinate which asymptotically becomes a cylindrical radius. This first coordinate choice is well-suited for analyzing the behavior of the metric functions and rotation coefficients on and near the axis. The second triad choice is inspired by the tetrad commonly used in the NP formalism, where one null vector is taken to be geodesic and orthogonal to null hypersurfaces. This choice is useful in that it connects directly to many known solutions of the field equations, and to the dynamics at asymptotic null infinity, where the peeling property [84-88] holds. Our purpose in Sec. VI is twofold. The first is to catalog known axisymmetric vacuum solutions, together with the assumptions that lead to each solution in terms of the triad formalism. This isolates the conditions required for the spacetime to represent a physically relevant solution. Secondly, we provide two example derivations of (known) spacetimes in the context of the triad equations, to illustrate typical techniques used to find solutions in this formulation. While we generally do not say much about the extensively-studied SAV spacetimes (see e.g. [26]), in Sec. VII we discuss the equations governing SAV spacetimes in the context of our new coordinate choice from Sec. V A. We conclude in Sec. VIII. Additional useful results are collected in a series of appendices. Throughout this paper, we use geometrized units with G = c = 1. We use Einstein summation conventions, with Greek indices indicating 4D coordinate indices (in practice these can be taken as abstract tensor indices). Latin indices from the middle of the alphabet ( i, j, k, . . . ) run over 3D coordinates (two spatial coordinates and one time coordinate), and Latin indices from the beginning of the alphabet ( a, b, c, . . . ) run over 3D triad indices. Indices with a hat correspond to tetrad components of a tensor in the physical manifold M and run over 1 , 2 , 3 , 4. Indices preceded by a comma either indicate a partial derivative with respect to the coordinates, as in f ,i , or the directional derivative of a scalar quantity with respect to the members of a null basis, as in f ,a . Similarly, we use semicolons to denote covariant differentiation in the coordinate basis, as in V µ ; ν . Indices preceded by a bar as in v a | b indicate the intrinsic derivative on the triad basis. Symmetrization of indices is denoted by enclosing them in parenthesis, and anti-symmetrization by using square brackets. We use a spacetime signature of ( -+++) on the 4D spacetime, and ( -++) on the 3D quotient space. Note that this modern convention differs from the signature used by many authors in the literature referenced here. An asterisk denotes complex conjugation.", "pages": [ 3, 4 ] }, { "title": "II. REDUCTION OF THE AXISYMMETRIC FIELD EQUATIONS", "content": "In this section, we review a formalism for expressing the full four dimensional Einstein field equations in a simpler three dimensional form when there is a single continuous symmetry present in the spacetime. We then specialize the resulting equations to the vacuum case, and then to spacetimes that admit a twist-free Killing vector. The formalism for the dimensional reduction was presented by Geroch [20] for a single symmetry, and extended by him to the case of two commuting symmetries [21] in order to study SAV spacetimes. This reduction has been extensively used, especially in the investigation of stationary spacetimes [26]. We closely follow Geroch's derivation and notation in what follows. We also compare the dimensional reduction to the familiar 3 + 1 decomposition used in numerical relativity, for which [89, 90] provide excellent references. Finally, Dain's review of axisymmetric spacetimes [47] complements the discussion provided here and throughout this paper. The reduction in complexity when one studies the field equations for a 3D Lorentzian metric as opposed to a 4D metric becomes immediately apparent by counting the number of independent components of the Weyl tensor, given by N ( N +1)( N +2)( N -3) / 12 in N dimensions. That is, zero independent components in 3D, and ten in 4D. In the reduction to axisymmetric, vacuum spacetimes discussed in greater depth in Sec II A, all of the gravitational field's dynamical degrees of freedom enter as two scalar functions whose gradients serve as sources for the 3D Ricci curvature. In the twist-free case, one of these scalars vanishes. The fact that the gravitational field is determined by a single remaining scalar demonstrates the tremendous simplification over the full 4D case with no symmetries present. The reduction proceeds in three steps. The first step is to derive the equations on the three manifold. Presented in Sec. II A, this process is similar to the 3+1 spacetime split familiar to numerical relativists. As a second step, we specialize to vacuum spacetimes. The last step of the reduction is a conformal rescaling, discussed in Sec. II D, which simplifies the 3D field equations further and makes apparent the existence of a generalized Ernst potential.", "pages": [ 4 ] }, { "title": "A. The space of orbits and the general reduction of the field equations", "content": "We begin by considering a 4D manifold M that admits a metric g µν and a Killing vector (KV) field ξ µ . Throughout this paper, we will consider ξ µ to be spacelike; however, the same formalism is easily extended to the case of a timelike symmetry [20, 26]. The KV field represents a continuous symmetry, and it defines a set of integral curves called the orbits of ξ µ . Motion along these orbits leaves the spacetime invariant and preserves the metric. This means that tensor fields on M have vanishing Lie derivative along ξ µ . For the case of the metric tensor L ξ g µν = 0 leads to the Killing equation, Intuitively, we see that one of the dimensions of M is redundant, and so we would like to reduce the study of this spacetime to the study of some 3D space. Naively, one would think of considering dynamics in M only on surfaces to which ξ µ is orthogonal. In practice however, ξ µ is only orthogonal to such a foliation of submanifolds of M if its twist ω µ , given by vanishes. When ω µ = 0, the KV ξ µ points in the same direction as the gradient of some scalar function φ on M , but this is not true in general [91]. Instead of considering some hypersurface in M , we consider a new space, which we call ¯ S following Geroch [20]. The space ¯ S is defined as the collection of orbits of ξ µ in M ; it is a 3D space that can be shown to posses all the properties of a manifold. The space ¯ S can be represented as a surface in M only if ω µ = 0. Figure 1 provides an illustration of the case of a twist-free symmetry with closed orbits. We denote with an over-bar tensor fields on ¯ S . These fields are orthogonal to the KV on all their indices, e.g. ¯ T α β ξ β = ¯ T α β ξ α = 0. A metric ¯ h µν on ¯ S can be defined by 'subtracting' the exterior product of two unit vectors pointing in the direction of the KV from the metric g µν . The resultant metric on ¯ S is Note that ξ µ ¯ h µν = 0 and the Lie derivative of ¯ h µν along ξ µ vanishes. The function λ that appears in Eq. (2.3) is the norm of the spacelike KV, and will play a key role in the reduction that follows. By raising an index on ¯ h µν using g µν , we can define a projection operator ¯ h α ν which projects 4D fields onto ¯ S . Arbitrary tensor fields can be projected into ¯ S by contracting all of their indices onto the projector, and similarly for tensors of arbitrary rank. We also define the operator ¯ D α by contracting the usual 4D covariant derivative of a tensor field with the projector on all its indices, It can be shown that the operator ¯ D α obeys all the usual axioms associated with the unique covariant derivative operator on a manifold with metric ¯ h µν [20]. Given the metric ¯ h µν on ¯ S and a compatible covariant derivative, we can compute the Riemann tensor on ¯ S , and relate it to the 4D Riemann tensor and the KV ξ µ . In doing so, the 4D field equations will be expressed entirely in terms of quantities on ¯ S . This projection of the 4D field equations is achieved by writing out GaussCodazzi equations generalized to the case of a timelike quotient space. This calculation, although computationally intensive, is only a slight modification of the standard techniques of the 3 + 1 split often used in numerical relativity and is detailed in Appendix A. Here, we summarize the key results that will be used later in the text. The contracted Gauss equation expresses the 3D Ricci curvature ¯ R αβ on ¯ S in terms of the Ricci tensor R µν on the manifold M , derivatives of the norm λ of the KV and its twist ω µ as Since ¯ S is a 3D manifold all the curvature information on ¯ S is contained in the Ricci tensor ¯ R αβ associated with ¯ h µν , with the remaining geometric content of M given by the magnitude λ and twist ω µ of ξ µ . Note that Eq. (2.7) has the same form as the Einstein field equations on the three manifold S with additional source terms on the right hand side; in the case where there are matter fields, we would re-express R µν in terms of the stress energy tensor T µν . We are primarily interested in the vacuum field equations, in which case R µν = 0 and the geometry on the three manifold is entirely sourced by λ and ω µ . As such, we need equations governing the evolution of λ and ω µ in order to complete our reduction of the field equations. This second set of equations is analogous to the Codazzi equations [89], since they are derived by applying the Ricci identity to the unit vector tangent to the KV. They are detailed in Appendix A 2. The resulting equation governing λ is where the 3D wave operator is defined using ¯ D 2 ≡ ¯ D α ¯ D α . The twist ω µ obeys the equations Together, Eqs. (2.7)-(2.10) can be solved on ¯ S for ¯ h αβ , λ and ω µ . We can then find an expression for the KV ξ µ using the identity, derived in Appendix A 2, together with the fact that ξ µ ¯ h µν = 0. With the KV and ¯ h µν , we can finally reconstruct the full 4D metric g µν on M , completing the solution of the field equations. The field equations on ¯ S are greatly simplified compared to the full Einstein field equations, but they are still formidable. As such, we will make a series of further specializations with the aim of rendering them tractable. In the past, the assumption of a second, timelike symmetry has resulted in the SAV equations and their solution. We will briefly discuss the SAV equations in Sec. VII, in the context of a convenient coordinate system we introduce in Sec. V A. Since our purpose is to pursue new solutions, outside of Sec. VII we will not assume any further symmetries. Instead, we give the reductions of the field equations in the case of vacuum, and then twist-free spacetimes in the sections that follow.", "pages": [ 4, 5, 6 ] }, { "title": "B. Coordinates adapted to the symmetry", "content": "In this section we detail the consequences of using a coordinate system adapted to the Killing symmetry. For a spacetime admitting a KV there exists coordinates x µ = ( x i , φ ) on M such that ξ µ = δ µ φ , where φ is a coordinate that does not appear in the metric, L ξ g µν = ∂g µν /∂φ = 0 [26, 92]. To find the form of the metric g µν in coordinates adapted to an axial KV, we first note that which also implies that ξ φ = λ . We denote the remaining covariant components of ξ µ by B i , so that ξ µ = ( B i , λ ). Since fully projected quantities on ¯ S are orthogonal to ξ µ , e.g. ¯ V α ξ α = ¯ V φ = 0, the φ components of projected tensors vanish, and the remaining components of ¯ h µν are the 3 × 3 block of components ¯ h ij . Using this in Eq. (2.3), the metric g µν takes a simple form Denoting the inverse of ¯ h ij by ¯ h ij and using it to raise and lower 3D indices, we can define B i = ¯ h ij B j and B 2 = ¯ h ij B i B j . This allows us to write the inverse of the metric (2.13) as The determinant of g µν can be expressed as Finally, in this basis the relationship between twist of the KV and B i can be found by defining the projected antisymmetric tensor ¯ /epsilon1 αβγ = /epsilon1 αβγµ ξ µ / √ λ . Using the definition of the twist (2.2) and projecting onto ¯ S we have from which we can see that if B i vanishes, so does the twist. This decomposition of the 4D metric and its inverse in terms of the 3D metric and the KV should be compared to the analogous decompositions of the 4D metric into a spatial metric, lapse, and shift vector in a 3 + 1 split, e.g. as found in [89]. For the remainder of this text, we will use coordinates adapted to the Killing symmetry, so that the decompositions (2.13) and (2.14) hold. The most useful consequence of this choice is that all of the information contained in quantities projected onto ¯ S is contained in the components on the coordinate basis x i . As such, we will write projected four dimensional indices α, β, . . . as Latin three dimensional indices such as i, j, k, . . . which run over coordinates on ¯ S .", "pages": [ 6 ] }, { "title": "C. The vacuum field equations", "content": "We now consider the case of vacuum 4D spacetimes. This sets the 4D Ricci tensor to zero in the equations derived in Sec. II A. Importantly, we see from Eq. (2.10) that the curl of the twist vector vanishes. We can thus define a twist potential ω such that From Eqs. (2.8), (2.9), and (2.7), recalling that we may use 3D indices for quantities projected onto ¯ S , we have as our field equations", "pages": [ 6 ] }, { "title": "D. The conformally rescaled equations and the Ernst potential", "content": "A further simplification to the reduced field equations (2.18) can be obtained by conformally rescaling the metric ¯ h ij . We define h µν to be and investigate the conformally rescaled 3D manifold which we will call S . The vacuum field equations (2.18) can now be rewritten in terms of h ij , bearing in mind that the Christoffel symbols associated with the two metrics are related by The wave operator, D 2 associated with h µν is related to ¯ D 2 as and further, ¯ D i f ¯ D i f = λD i fD i f . Substituting these identities into Eqs. (2.18), the field equations can be expressed using the metric h ij [20, 47] The symbol R 3D ij denotes the Ricci curvature of the rescaled three manifold with metric h ij . There is additional structure in these equations which can be made more apparent by introducing the complex Ernst potential E = λ + iω [27]. In terms of this potential, Eqs. (2.22) become It is important to note that the Ernst potential usually discussed in the context of stationary spacetimes is based on the norm and twist of a timelike KV, rather than the spacelike KV as discussed in this section. This results in some sign differences in various definitions, c.f. the relevant chapters of [26]. The relationship between the Ernst potential defined here and the Ernst potential used in conjunction with SAV spacetimes is explained further in Sec. VII.", "pages": [ 6, 7 ] }, { "title": "E. Reduction to the case of twist-free Killing vectors", "content": "The axisymmetric field equations (2.23), though much simplified from their full 4D form, remain intractable. For the remainder of this paper, we restrict our exploration to the situation depicted in Fig. 1, where ξ µ is hypersurface orthogonal, so that ω = 0. In doing so we eliminate the possibility of the study of rotating axisymmetric spacetimes, but we benefit from further simplifications to the field equations. A number of physically interesting dynamical spacetime solutions are twist free, these including the head-on collision of black holes and non-spinning, axisymmetric critical collapse. The twist-free assumption reduces the problem of finding solutions to the field equations to the study of a harmonic scalar ψ on the three manifold S , where we define ψ via The field equations (2.23) become and the Ricci scalar associated with the three metric, which we denote as R , is given by the contraction of (2.25), where we used semicolons in place of D i to condense the notation for the covariant derivatives. The scalar R is the only nonzero eigenvalue of R 3D ij and corresponds to the eigenvector ψ ; i . Some general properties of gravity in 3D are discussed in [93]. In particular, since the 3D gravitational field has no dynamics, due to the vanishing of the Weyl tensor, the only dynamical degree of freedom in the problem is the scalar ψ . This reduced number of variables drastically simplifies the calculations. In the sections that follow we present a systematic way of analyzing Eqs. (2.25) using a triad formalism, without immediately specializing to any given coordinate system. The fact that ψ is harmonic makes it a convenient choice of coordinate on S that in addition greatly simplifies the components of the Ricci tensor R 3D ij . The full implication of choosing ψ as a coordinate, as well as another gauge choice adapted to geodesic null coordinates on S , are discussed in the sections that follow.", "pages": [ 7 ] }, { "title": "F. The axis", "content": "All of the previous results in this section hold for KVs with generic orbits. Here, we review some additional results which apply if the orbits are closed, as in the case of axisymmetry. Motion along the orbits of a KV maps the spacetime onto itself, and by the definition of the KV this map preserves the metric. This map may have fixed points, where it is simply the identity operator, and these fixed points comprise the axis of the spacetime. Much is known about the axis of an axisymmetric spacetime, see e.g. [94-96]. A key result due to Carter [94] is that any vacuum spacetime with a KV that has closed orbits and is asymptotically flat admits fixed points and therefore isolated systems which possess an axial KV ξ µ will have an axis. This axis is 2D and timelike [94], and will be denoted W 2 . On the axis the magnitude of the axial KV vanishes, Note that the derivative of the KV, ξ µ ; ν cannot vanish on the axis, or else ξ µ would vanish everywhere (see e.g. [91] for further discussion). When the axis is free of singularities, a condition known as elementary flatness holds in a neighborhood of the axis. This condition expresses the fact that in the local Lorentz frame of a small neighborhood about a point on W 2 , we can make a loop around the axis, and the circumference of this loop must be equal to 2 π times its radius. If this is not true, then there is a conical singularity in this small neighborhood, and traversing the circle around them results in a deficit (or surplus) angle. One way to express elementary flatness is to find a set of coordinates in which the line element has the form ds 2 0 = g ρρ dρ 2 + λdφ 2 near the axis, holding the third spatial coordinate fixed. Dividing the proper length around a circle by 2 π times the proper distance to the axis yields a constant K D which, if different from unity, gives a mea- the deficit angle [97], Acoordinate invariant form of this same condition that is more useful from our perspective was given by Mars and Senovilla [95], We derive this result using a specific coordinate system in Sec. V A. Expressing Eq. (2.29) in terms of ψ and the conformal three metric h ij we have Equation (2.30) provides explicit boundary conditions for quantities on S as the axis λ = 0 is approached, if we wish our axis to be free of conical singularities.", "pages": [ 7, 8 ] }, { "title": "III. THE TWIST-FREE FIELD EQUATIONS EXPRESSED USING A TRIAD FORMALISM", "content": "To explore the field equations on the 3D manifold S , we employ a triad formalism in which we choose a basis for the tangent bundle before further selecting coordinates on the manifold. In this section we follow Hoensaelers [22-24] in writing out the 3D field equations (2.25) and Bianchi identities on S in a manner similar to the Newman-Penrose (NP) equations [25, 98]. This form of the equations is particularly convenient for the study of the exact solutions of the field equations, since it makes manifest what the various possible assumptions and simplifications might be for special and physically interesting cases. The procedure is to define a null (or orthonormal) triad and write out in full the field equations expressed in this basis (we note that a similar formalism was developed by Perj'es in [41] in the context of stationary spacetimes, using a complex triad). Our approach largely follows the conventions for the tetrad formalism used in Chandrasekhar's text [98], which also gives general background on the technique. We begin by selecting a triad basis so that the metric expressed on this triad basis contains only constant coefficients. A null triad choice that is particularly useful is one for which and the non-zero metric components are η 12 = η 21 = -1 and η 33 = 1. The orientation of the triad is fixed by the equations Given the normalization in Eq. (3.3), the metric on the coordinate basis is expressed in terms of the triad vectors as The fundamental variables in a triad formalism are the Ricci rotation coefficients γ abc , which record how the basis vectors change as we traverse the manifold. They are defined by The rotation coefficients are antisymmetric in the first two indices γ abc = γ [ ab ] c (note our ordering of indices induces a sign change from Chandrasekhar's definition [98]). In 3D there are nine independent real rotation coefficients, as opposed to the 24 real rotation coefficients that exist in 4D. We adopt the following naming convention first introduced in [22], The projection of the 3D Ricci tensor R ij (we drop the superscript 3D from here on) onto this basis gives us six curvature scalars, which we denote The Ricci scalar is given by 1 The field equations describe how the rotation coefficients listed in Eqs. (3.7) change in a particular basis direction to ensure that Eqs. (2.25) are satisfied. The change along a basis direction is a directional derivative given by The basis-dependent directional derivative can be related to the intrinsic covariant derivative of a tensor projected onto the triad basis, by taking into account the manner in which the basis itself changes. Using Eqs. (3.6), (3.10), and (3.11) one can show that the relationship between the directional and intrinsic derivatives is Recall that triad indices are raised using the constant metric η ab defined by η ab η bc = δ a c , which has the same component form as η ab . It is important to note that in a triad formalism, the directional derivatives of a scalar function do not commute, while intrinsic derivatives do. The commutation relations for directional derivatives are Using the relationship between intrinsic and directional derivatives, we now express the field equations on the triad basis in terms of the rotation coefficients [98]. The Ricci tensor obeys Writing out the field equations (3.14) in full leads to In the twist-free case, the curvature scalars φ i appearing in the above expressions are obtained from the Ricci tensor R ab computed by projecting Eqs. (2.25) onto the triad, Of the above set of nine field equations there are six equations that contain Ricci curvature components and three that do not. These three equations constitute the 3D version of the eliminant relations (see Chandrasekhar [98]). As expected, there are fewer equations on S than in the 4D case (9 here versus 36 equations in 4D). The three, 3D Bianchi identities are written as 2 In terms of the rotation coefficients, the commutation relations (3.13) are and must be used whenever interchanging the order of directional derivatives. Finally, it is useful to note that the operator D 2 f = f ,a | a can be expressed as This concludes the general triad formulation using the rotation coefficients as fundamental variables. The equations given are valid for both twisting and twist-free spacetimes, with the only difference being the complexity of the 3D Ricci tensor. These equations can be simplified to a great degree by a judicious choice of triad. We explore two especially useful triad choices in Sec. VI, where we also specialize to the case of twist-free spacetimes.", "pages": [ 8, 9, 10 ] }, { "title": "IV. RELATING PHYSICAL 4D QUANTITIES TO COMPUTED 3D QUANTITIES", "content": "In this section we provide the explicit correspondence between NP quantities on the physical 4D spacetime M and the computationally concise quantities on the conformal manifold S . Knowledge of this correspondence is useful for various reasons: (i) Initial conditions for integrating the much simpler 3D field equations (3.15)-(3.23) are most readily specified on M . (ii) The boundary conditions on the axis, discussed in Sec. II F and Appendix D, require information about the smoothness of the physical quantities on M , since the 3D conformal metric h ij is singular on the axis. (iii) Searching for solutions to the field equations involves making choices of the triad and the gauge, and having a direct translation of the assumptions made in 3D to the implications for the physical quantities is advantageous. This relationship between specializations in 3D and 4D also identifies the conditions on the 3D quantities corresponding to known solutions. To exhibit the correspondence, we first note that in the twist free case, the metric decomposition of Sec. II B simplifies to the case where B i = 0 and the 4D metric g µν can thus be expressed as The metric ˜ g µν , which is conformal to the physical metric, provides a useful intermediate step for the calculations that follow.", "pages": [ 10 ] }, { "title": "A. Spin coefficients", "content": "We now define the relationship between the NP spin coefficients on M and the rotation coefficients defined in Eqs. (3.7). There is some freedom in the choice of tetrad as we go between the 4D and 3D manifolds, which we fix by choosing the tetrad so that the directions of all the null basis vectors coincide, and so that the parametrization of the out-going null vectors are the same. In order to avoid confusion, all quantities on M such as spin coefficients, ˆ κ , ˆ /epsilon1 and Weyl scalars, ˆ Ψ i are given with a hat (ˆ . ). Quantities associated with the conformally rescaled 4-metric ˜ g µν are all indicated with a tilde (˜ . ), and 3D quantities will remain unadorned. The standard complex null tetrad on M is with the non-zero metric components being ˆ η ln = ˆ η nl = -ˆ η mm ∗ = -ˆ η m ∗ m = -1. Now consider another tetrad constructed by augmenting the triad (3.1) with the vector d µ = e -2 ψ δ µ φ which has the same direction as the KV ξ µ , to yield the tetrad where we have omitted the tilde's to emphasize that this tetrad is built from the same triad vectors that we use on S (although strictly speaking they are the lift of these vectors onto a conformal 4D space). It can be verified directly using Eqs. (3.5) and (4.1) that the conformal metric ˜ g µν can be expressed as where the covector d µ is d µ = e 2 ψ δ φ µ . To find the relationship between the NP spin coefficients on M and the rotation coefficients on S , we first calculate the rotation coefficients associated with the conformally related metric ˜ g µν on the basis in Eq. (4.3). To do this expediently we introduce the quantities λ abc that are defined as [98] and are antisymmetric in the first and third indices. The major advantage of working with the quantities λ abc is that they can be computed using coordinate derivatives rather than covariant derivatives. This property makes the comparison between quantities defined on different metrics given the same coordinate choice easy. Given a set of λ abc 's the rotation coefficients can be constructed using the relation The 24 rotation coefficients associated with the conformal metric ˜ g µν can be related to the nine rotation coefficients associated with h ab by noting that ˜ λ abc = λ abc when a, b, c run over 1 , 2 , 3. The remaining 15 rotation coefficients can be subdivided into nine coefficients of the form ˜ γ a 4 b , three ˜ γ ab 4 coefficients, and three ˜ γ a 44 coefficients . From the definitions in Eq. (4.5) and the vector d µ , it is straightforward to verify that ˜ λ ab 4 = ˜ λ a 4 b = 0, and so the 12 coefficients ˜ γ a 4 b and ˜ γ ab 4 vanish. There are then only three non-zero rotation coefficients, in addition to those in Eq. (3.7). Given the rotation coefficients associated with the augmented tetrad in Eq. (4.3), the spin coefficients associated with the physical space tetrad in Eq. (4.2) can be obtained from a transformation of the form Specifically, P b ˆ a = e -2 ψ Q b ˆ a and the nonzero components of Q b ˆ a are Note that the fact that Q 1 1 = 1 ensures that the parametrization of outgoing null vector l i on the three manifold coincides with the associated vector on the 4D spacetime. The vectors on M are then given in terms of the tetrad (4.3) by By repeatedly using the definition (4.5) on the different tetrads, we find that the λ abc functions associated with the physical tetrad (4.2) [and thus the rotation coefficients via Eq. (4.6)] are related to those on the augmented tetrad given in Eq. (4.3) by where the constant metric ˜ η ab has the non-zero components, ˜ η 12 = ˜ η 21 = -1 and ˜ η 33 = ˜ η 44 = 1. Since the P a ˆ b 's are functions only of ψ all the physical rotation coefficients reconstructed using Eq. (4.6) given Eq. (4.11) can be written in terms of the nine rotation coefficients on the triad basis, the three directional derivatives of the scalar function ψ , and functions of ψ itself. It can also be observed that all the physical rotation coefficients expressed on the basis in Eq. (4.10) are real. The physical spin coefficients using the NP naming convention [98], when expressed in terms of the rotation coefficients defined on S are The identifications in Eqs. (4.12) gives us the benefit of all the usual intuition regarding the spin coefficients in the 4D spacetime when computing quantities on the manifold S . We will explore these relationships and their physical implications more fully in Sec. VI when we review the exact solutions to the field equations.", "pages": [ 10, 11 ] }, { "title": "B. Curvature and Weyl scalars", "content": "The second set of quantities that are useful for exploring the physical content of spacetime, such as gravitational radiation, are the Weyl scalars. In this section we will show that they have a particularly simple representation in terms of the 3D rotation coefficients and directional derivatives of ψ . The fact that the Weyl tensor is conformally invariant implies that on the coordinate basis ˆ C α βγδ = ˜ C α βγδ . Lowering the index α , expressing the tensor on the tetrad basis in Eq. (4.10), and subsequently using Eq. (4.8) to express it on the augmented basis in Eq. (4.3), we obtain an expression for the physical Weyl tensor in terms of the Weyl tensor on the augmented basis, The quantity ˜ C abcd is readily computed in terms of the rotation coefficients and directional derivatives of ψ on S from the standard expression for the Riemann tensor [98], which in vacuum is identical to the Weyl tensor: Writing out Eqs. (4.14) in full, making use of the definitions of φ i given in Eqs. (3.8) and (3.24), and substituting in the field equations (3.15)-(3.23) wherever necessary yields the following expressions for the Weyl scalars on the physical manifold: It is important to note that the assumption of twist-free axisymmetry greatly decreases the number of independent functions to be considered: the NP spin coefficients and Weyl scalars which in general are complex are all real in the twist-free case, effectively cutting the problem of finding solutions in half. Further simplifications can be achieved with specific gauge and tetrad choices.", "pages": [ 11, 12 ] }, { "title": "V. TWO TRIAD CHOICES", "content": "In this section we discuss the implications of two physically-motivated triad choices which further simplify Eqs. (3.15)-(3.23) and the Bianchi identities (3.26)(3.28). The first choice is to use ψ as a coordinate and to associate the triad direction c a with its gradient. This choice greatly simplifies the Ricci tensor on the three manifold and is suited to applying the boundary condition on the axis. The second is to use geodesic null coordinates. This allows us to make direct contact with the Bondi formalism and thus the emitted radiation reaching future null infinity I + in asymptotically flat spacetimes.", "pages": [ 12 ] }, { "title": "A. Choosing ψ as a coordinate", "content": "The field equations (2.25) describe a gravitational field on a three manifold sourced by a harmonic scalar field ψ which obeys D 2 ψ = 0. In 4D gravity, harmonic coordinates have been successfully employed, e.g. for proving the well-posedness of the Cauchy problem for the Einstein equations [91, 99]. The usefulness of harmonic coordinates in 4D, together with the fact that the 3D Ricci tensor greatly simplifies if ψ is chosen as a coordinate leads us to investigate this gauge choice further. We now specialize our triad so that c a points in the same direction as the gradient of ψ . The normalization condition c a c a = 1 implies that where R = 2 ψ ,a ψ ,a is the 3D Ricci scalar defined in (2.26). Note that the sign of R determines whether ψ ,a is timelike, spacelike, or null. For Schwarzschild, R > 0, and so we might expect this to be true of a physically reasonable spacetime, especially one that settles down to Schwarzschild after some dynamical evolution, and as such we will assume that ψ ,a is spacelike. Given the definition of c a in Eq. (5.1) we can express the Ricci tensor (2.25) as R ij = Rc i c j , and so the six curvature scalars defined in Eqs. (3.8) are φ 5 = φ 4 = φ 3 = φ 2 = φ 1 = 0 and φ 0 = R . This greatly simplifies the Bianchi identities, which are and which gives the rotation coefficients appearing in Eq. (5.2) the interpretation of being proportional to the rate of change of ln R in a particular direction. Because R is a scalar, the curl of its gradient, /epsilon1 abc R | bc = 0, must vanish. Equivalently, the commutator equations (3.29) with f = R must hold. This augments the field equations with the following three equations, The fact that c a points along the gradient of a scalar places additional conditions on the rotation coefficients. To see this, we compute the intrinsic derivative of c a and express the result on the triad basis to obtain   Now, noting that c a | b + 1 2 c a (ln R ) ,b = √ 2 /Rψ | ab , and using the directional derivatives of R computed in (5.2), we have   However since ψ ,a is a gradient, this matrix should be symmetric. Thus γ = -/epsilon1 . Further, we note that D 2 ψ = 0 is automatically satisfied. The Bianchi identities (5.2), in addition to the field equations (3.15)-(3.23), allow us to find a particularly simple expression for the wave operator of ln R , It is interesting to note that if ψ is chosen as a coordinate and the tetrad leg c a is fixed using (5.1), then the directional derivatives of ψ that enter into the 4D expressions for the NP scalars become particularly simple. Explicitly ψ , 1 = ψ , 2 = 0 and ψ , 3 = √ R/ 2. This implies that the expressions for the Weyl scalars (4.15) become The rotation coefficients that enter these expressions are the same rotation coefficients that appear in the second derivative of ψ expressed on the triad basis, Eq. (5.5). This underscores the fact that the scalar ψ sources the gravitational field. Another important consequence of Eqs. (5.7) is that for this tetrad choice, if l a is geodesic, i.e. β = 0, then the geodesic is a principal null geodesic of the spacetime, ˆ Ψ 0 = 0. Thus far the other triad vectors are unspecified, except that they are null and orthogonal to c a . With c a fixed, we still have freedom to boost along l a . The equivalent of the Lorentz transformations for the 3D triad are discussed fully in Appendix B. Here we consider the effect of a boost of the form Using the definitions in Eqs. (3.7), we find that under such a boost, six of the coefficients are simply multiplied by factors of A , while three have nontrivial transforms, The full transforms are given in Eq. (B2); interestingly, the above coefficients with a nontrivial transform do not enter into the expressions for the Weyl scalars in Eqs. (5.7). We can always use our boost freedom to set at least one of ˜ α, ˜ η, or ˜ δ to zero. Note that if a boost exists that can set ˜ α = ˜ η = ˜ δ = 0, then it can be shown that R = 0 and that the resulting spacetime is flat. Also, if one triad leg c a is chosen according to Eq. (5.1), it is not possible to apply a boost to render the null vector l a geodesic, or equivalently to set the coefficient β to zero. An example which illustrates this fact is in the asymptotic region of a radiating spacetime, where our choice of c a would point along a cylindrical radius; meanwhile, the outgoing null geodesics define a radial direction, and it is clear that these two directions are not orthogonal. Rather, we would need to locally choose some other null direction to define l a . We now ask whether it is possible to find a coordinate t whose gradient is timelike and orthogonal to c a , i.e. that t ,a c a = 0. The first step is to define a timelike unit vector T a as From the normalization conditions (3.3) it is straightforward to verify that T a T a = -1. We would like to determine if T a is hypersurface orthogonal, so that it can be written as T a = -/rho1t ,a . This is possible if and only if T a is twist-free, T [ a D b T c ] = 0. In 3D, this is equivalent to the vanishing of the scalar For a general l a and n a this will not be true, but we can choose a boost A that will transform η such that W = 0. By Eq. (5.9), we see we must choose We have so far fixed our triad, and selected the harmonic coordinate ψ and the coordinate t whose gradient lies parallel to T a . Let us call the third coordinate s . On the coordinate basis ( t, s, ψ ) the assumptions thus far imply that in all generality the we can express the covariant components of the triad as where l t , n t , h s and h ψ are free functions of ( t, s, ψ ). The factor /rho1 in the definition of T a is /rho1 = ( l t + n t ) / √ 2. The metric on the coordinate basis is constructed using Eq. (3.5). To see if any further metric functions can be set to zero, consider a coordinate transformation that leaves the coordinates t and ψ unchanged but chooses a new coordinate s ' , such that s = f ( t, s ' , ψ ). We find that the metric can be expressed in the same form except with the functions l t , n t , h s , h ψ transformed as It is thus always possible to choose a gauge in which h ' ψ = 0. Dropping the primes, the resulting metric on the coordinate basis is For the rest of this section we make this coordinate choice. The covariant components of the triad vectors are The choice of ψ as a coordinate is an unfamiliar one, and to help build some intuition we present the Minkowski metric, triad, and rotation coefficients in this coordinate system in Appendix C. The rotation coefficients in general axisymmetric spacetimes can be expressed in terms of the functions entering Eqs. (5.13) and (5.16), and are listed in Appendix D. The expression in (D1) for the coefficient /epsilon1 , can be integrated using the Bianchi identity (5.2), Combining these equations shows that the metric functions obey [ ln R ( h s /rho1 ) 2 ] ,ψ = 0, which after integration provides There is still some residual coordinate freedom in Eq. (5.15) in that we can apply a coordinate transformation to the s and t coordinates without changing the form of the metric. In particular by using the coordinate transformation s = f 2 ( t, s ' 2 ) and using a restricted version of Eq. (5.14) it is possible to choose a gauge in which g ( t, s ) = 1 so that we have ( h s /rho1 ) 2 R = 1. We will not necessarily make this specialization in the rest of the text.", "pages": [ 12, 13, 14 ] }, { "title": "1. Field equations adapted to the ψ coordinate choice", "content": "In this subsection we specialize the field equations (3.15)-(3.23) to the case where we use ψ as a coordinate and where c a = √ 2 /Rψ ,a . Recall that this choice implies that γ = -/epsilon1 , ψ , 1 = ψ , 2 = 0 and ψ , 3 = √ R/ 2. With this specialization, we re-order the general field equations (3.15)-(3.23) augmented by the commutation relations (5.3). One of the field equations is redundant with one of the commutation relations, while the remaining 11 equations can be split into a subset of four equations that contain directional derivatives in the l a and n a directions only, and a group of seven equations that fix the directional derivatives of certain rotation coefficients in the c a direction, We showed that in the coordinate basis ( t, s, ψ ) the metric can be written in the form (5.15). With the choice of l a and n a in (5.16), all the equations (5.20) contain only derivatives with respect t and s , and effectively constitute a set of constraint equations that have to be satisfied for every constant ψ surface. As can be seen from the above set of equations, choosing ψ as a coordinate does not greatly simplify the field equations. For this coordinate and triad choice the major simplifications occur in the Bianchi identities (5.2), the form of the metric (5.15), and the simple form of the corresponding Weyl scalars. An additional advantage of this coordinate and triad choice that will be discussed in the next section is the easy identification of the axis.", "pages": [ 14 ] }, { "title": "2. Axis conditions as λ → 0", "content": "On the axis, which for the three metric is denoted by the boundary conditions ψ → -∞ or λ → 0, we now explore the conditions on the triad quantities required for the elementary flatness condition to hold. The first step is to observe that working in a coordinate system where ψ is a coordinate makes it easy to prove the equivalence of the two forms of the axis conditions, K D = 1 in Eq. (2.28) and the coordinate invariant expression in Eq. (2.29). Assuming the metric h ij can be written in the form (5.15), the metric on the space orthogonal to the axis W 2 is merely ds 2 0 = g ψψ dψ 2 + λdφ 2 , where the 4D metric component g ψψ = 2 / ( λR ). The elementary flatness condition (2.28) now reads where use has been made of the fact that λ is not a function of φ . Applying l'Hˆopital's rule and differentiating above and below the line with respect to ψ , the elementary flatness condition becomes or equivalently R → 2 e -4 ψ . By definition, R = 2 ψ ,a ψ ,a , showing that the covariant expression (2.30) and thus Eq. (2.29) are equivalent to the elementary flatness condition. Note that the elementary flatness condition, in conjunction with the condition found when examining the rotation coefficient /epsilon1 , Eq. (5.19), implies that the determinant of metric on the subspace normal to the axis also remains finite as we approach the axis. To see this explicitly, observe that det[ h ˜ i ˜ j ] = -2 h 2 s /rho1 2 , where ˜ i, ˜ j ∈ { s, t } . By the condition found in Eq. (5.19) in the gauge where g ( s, t ) = 1 we have det[ h ˜ i ˜ j ] = -2 /R . The determinant associated with the corresponding part of the four metric becomes det[ g ˜ i ˜ j ] = -2 / ( Re 4 ψ ), which by the elementary flatness condition approaches the value -1 on the axis as expected. Symmetry dictates that a null vector on the axis remains on the axis when it is sent out to infinity or toward the origin. Thus on the axis ˆ l µ and ˆ n µ are geodesic, provided they are chosen to lie along the ingoing and out-going directions. In terms of the NP scalars (4.12), this translates into ˆ κ = βe -ψ / √ 2 → 0, and ˆ ν = ζe 3 ψ / √ 2 → 0. In Appendix D, explicit formulas for the expansions of the metric quantities about the axis are given and discussed. The special case of the static Schwarzschild black hole is examined in Sec. VII C where the scaling of the solution, the 3-curvature R and all the rotation coefficients are explicitly computed.", "pages": [ 14, 15 ] }, { "title": "B. Geodesic null coordinates", "content": "We now examine the equations in a coordinate system adapted to asymptotic null infinity, where the concept of emitted radiation is well defined. Akin to the standard methods used in the NP formulation (see e.g. [25, 85, 100]), this coordinate system and triad choice is tied to the tangent vectors of null geodesics. We begin with a family of null hypersurfaces in S , and we label these by a coordinate u , so that h ij u ,i u ,j = 0. We then choose the covariant representation of one null triad vector to be the gradient of the coordinate u , setting l i = -u ,i . Since l i is the gradient of a coordinate, it has vanishing curl. The intrinsic derivative of l i on the triad basis is The fact that l a | b is symmetric immediately sets β = α = 0 and η = -/epsilon1 . Note that β = α = 0 implies that l a is geodesic and affinely parametrized on S , by Eqs. (3.7). Also, recall that null geodesics are conformally invariant, and we can verify that here Eqs. (4.12) imply that if β = 0 then ˆ κ = 0 in M . Thus if l a is the generator of a geodesic null congruence on S , the corresponding null congruence in the physical manifold is also geodesic. The above conditions on the rotation coefficients further imply that the field equation (3.22) is trivially satisfied. If we choose as another coordinate the affine parameter p along the geodesic that l a is tangent to, we have l i f ; i = f , 1 = ∂ p f . Lastly, we label our third coordinate χ . Expressing the null vectors on the ( u, p, χ ) coordinate system, we have The normalization conditions (3.3) allow us to restrict some of the components of the remaining triad vectors, giving n u = 1 and c u = 0. Using the expression for the metric in terms of the triad vectors (3.5), we can see that h uu = h uχ = 0 and h up = -1 follows. Three more metric functions fully determine h ij . We parametrize these remaining metric components following the convention of [101, 102] so that the contravariant form of the metric becomes where v i are free functions of the coordinates. The covariant form of the metric on S is then given by This metric holds for any null foliation of the manifold S , where constant u surfaces denote the null hypersurfaces, the affine parameter p serves as a coordinate along a particular geodesic and the coordinate χ , usually associated with an angular coordinate, labels the geodesics within the hypersurface. We further need to fix the triad legs n i and c i . One such choice that satisfies the normalization condition (3.3) and gives the correct form of the metric (5.26) is The corresponding covariant vectors are On this triad, the directional derivatives applied to a function f are If the chosen coordinates ( u, p, χ ) are to be valid, they must satisfy the commutation relations given in Eq. (3.29). Applying the commutation relations to each successive coordinate provides a simple way of relating the rotation coefficients to derivatives of the metric functions of Eq. (5.27). The commutators acting on χ yield the coefficients Applying the commutation relations to u reiterates that α = β = 0 and η = -/epsilon1 . Finally, applying the commutation relations to p fix 1. Field equations adapted to the geodesic null coordinate choice When working with geodesic null coordinates, where γ = η = -/epsilon1 and α = β = 0, the field equations (3.15)(3.23) can be expressed in simplified form in terms of the five remaining rotation coefficients as One of the equations is trivially solved and has been omitted. The remaining equations have been reordered, and some are linear combinations of the original set. These combinations are (5.37) = (3.21)/2 -(3.15)/2 -(3.19) describing the derivative δ , 1 ; the combination (5.38)=(3.21)/2 -(3.15)/2, yielding an expression for the combination θ , 2 -/epsilon1 , 3 ; and finally the combination (5.40) = (3.23) -(3.16) to obtain an expression for ζ , 1 + /epsilon1 , 2 . The remaining equations are simplified analogues of their counterparts in Eqs. (3.15)-(3.23). The reordering makes apparent the fact that a hierarchy exists in the reduced system of equations, which in turn makes it possible to formally integrate the field equations in a systematic way. Suppose we begin on a null hypersurface of constant u on which the directional derivatives of the function ψ are given, so that φ i , i = 0 . . . 5 are known. Equations (5.35) and (5.31) can be integrated with respect to p to obtain the rotation coefficient θ , and subsequently the metric function v 3 . In a similar fashion Eqs. (5.36) and (5.34) yield /epsilon1 and v 2 , and subsequently (5.37) and (5.32) give δ and v 1 . The metric functions v 3 , v 2 and v 1 are thus determined within the null hypersurface up to boundary terms. The requirement that D 2 ψ = 0 determines ι = v 3 , 2 using Eq. (3.30). Thus the manner in which the metric function v 3 changes away from the initial null hypersurface is known. Equation (5.38) then serves as a consistency condition which restricts some of the six integration constants that arise while integrating Eqs. (5.35)-(5.37). The other integration constants are determined by boundary conditions that will be discussed more fully in Sec. V C. Equation (5.40) implicitly determines v 2 , 2 . Equation (5.41), in conjunction with the condition D 2 ψ = 0 provides an evolution equation of ψ . The remaining two equations, (5.39) and (5.42), are eliminant relations that are trivially solved when the metric functions are substituted into the field equations. The hierarchy of field equations that arise when they are expressed on a coordinate system adapted to a null hypersurface has been extensively studied in the four dimensional context. It is known, for instance, that the equivalent equations on M are formally integrable on a constant u surface [85, 101, 103, 104]. The asymptotic behavior of the metric in geodesic null coordinates, and the associated boundary conditions are further discussed in Sec. V C, where the relationship to the Bondi formalism is explored. In Appendix F we give an explicit example of how the field equations are systematically integrated in an asymptotic region far from a gravitating system, although there l a is affinely parametrized with respect to M . The gauge and triad choice discussed in this section has the advantage of eliminating four of the nine rotation coefficients. This reduction in complexity makes apparent a hierarchy in the field equations that hints at the possibility of finding an analytic solution to the problem. In Sec. VI A we carry out an example calculation in which the field equations (5.35)-(5.42) are systematically solved in a special case. It should be noted however that this simplicity comes at a cost. Unlike the case where the triad was adapted to the coordinate ψ , and the Ricci tensor only had one non-zero component, the Ricci tensor on this triad is constructed from the three independent quantities ψ ,a . It has two degenerate eigenvectors with zero eigenvalues, and a single normalized eigenvector with nonzero eigenvalue 2 ψ ,a ψ ,a . /negationslash The analysis performed in this section assumes that l i is affinely parametrized in S . If we adjust the parameter along each geodesic, p → p ' ( u, p, χ ), this results in h up ' = -1, but otherwise preserves the form of the metric. A physically motivated alternative to affine parametrization in S is to boost l i so that it is affinely parametrized in the physical 4D spacetime, and then to use the affine parameter τ as the second coordinate instead of the parameter p . The field equations that result form this choice of parametrization are detailed in Appendix E. Some gauge freedom remains when l i is affine in S , and p is used as a coordinate. Shifting the origin of the affine parameter along each geodesic separately, p ' = p + f ( u, χ ), transforms the metric function of (5.27) according to Relabeling the individual geodesics within a spatial slice, χ ' = g ( u, χ ), transforms the metric functions of (5.27) to [105] Finally, it is also possible to relabel the null hypersurfaces, setting u ' = h ( u ), p ' = p/h ,u . The metric components transform as", "pages": [ 15, 16 ] }, { "title": "C. Asymptotic flatness and the peeling property", "content": "We will complete our discussion of useful coordinate systems on S by discussing the asymptotic limit of the metric far from an isolated, gravitating system. We will consider only spacetimes that are asymptotically flat and therefore admit the peeling property [84-88]. According to the peeling property, the Weyl scalars expressed on an affinely parametrized out-going null geodesic tetrad admit a power series expansion at future null infinity (denoted I + ) of the form where τ is the affine parameter along the out-going null geodesics in M and ˆ Ψ ( n ) i are constant along an out-going geodesic, i.e. ˆ Ψ ( n ) i ( u, χ ). The work of Bondi, van der Burg and Metzner [101], as well as Tamburino and Winicour's approach [100], indicate that the the metric functions also admit a power series expansion if expressed in terms of geodesic null coordinates. Appendix F details a triad-based derivation of the asymptotic series expansions of the metric functions, in the restricted context of axisymmetric spacetimes. In this derivation, the 'Bondi news function' is identified with the derivative of the dominant coefficient in the expansion of the shear of the out-going null tetrad leg. The calculation is performed assuming that the out-going null geodesic is affinely parametrized in M , and the corresponding field equations given in Appendix E are used. The results obtained in Appendix F for the asymptotic expansion of the metric can be summarized as follows. In terms of affinely parametrized null coordinates, the 4D line element can be expressed as where, according to Eqs. (F2), (F4), (F11), (F15), (F18), and (F21), the metric functions admit the following asymptotic expansion as the affine parameter τ →∞ : Note that the coordinate χ is chosen here to be χ = cos θ , and θ is the usual polar angle. The axis occurs as χ →± 1. The free functions that enter into the metric are σ (0) ( u, χ ) and the dominant terms associated with the Weyl scalars ˆ Ψ (0) i ( u, χ ), i ∈ { 0 , 1 , 2 } . The dominant terms of ˆ Ψ (0) 3 and ˆ Ψ (0) 4 are fixed by these free functions through Eqs. (F25) and (F26) or equivalently The field equations determine the evolution of ˆ Ψ (0) i ( u, χ ), i ∈ { 0 , 1 , 2 } from one null hypersurface to another via Eqs. (F22), (F27), and (F28). As can be observed from (5.49), the free function σ (0) ,u carries the gravitational wave content of the spacetime and is often referred to as the 'Bondi news function.' A solution that settles down to a Schwarzschild black hole in its final state requires that in the limit u → ∞ the scalars behave as where the constant M is the mass of the final black hole. For u < ∞ , ˆ Ψ (0) i , i ∈ { 0 , 1 , 2 } are then determined by these final conditions, provided that σ (0) ( u, χ ) is given, using the evolution equations (F22), (F27), and (F28). For easy reference these equations are repeated here : We now examine how the metric and Weyl scalars behave on the axis in the limit of large distance from the isolated source. As noted in Sec. II F, and explored further in Appendix D, the metric functions have a power series expansion in λ = e 2 ψ near the axis of symmetry. In addition, these expansions are such that the metric functions vanish sufficiently quickly in the approach to the axis so that there are no 'kinks' at the axis [101]. Note that from (5.48), the coefficient of the dudχ term in the metric, namely e w 3 w 2 , is only regular on the axis if both σ (0) and ˆ Ψ (0) 1 vanish on the axis, and respectively scale like where ˜ σ (0) and ˜ Ψ (0) 1 need not vanish on the axis. Substituting these scalings into the evolution equations for the dominant expansion terms for the Weyl scalars, Eqs. (5.51)-(5.53) and (5.49), shows that on the axis indicating that the spacetime is Type D on the axis, and there is no radiation to infinity along the axis. This is to be expected, since spin-2 transverse radiation cannot propagate along the axis and still obey axisymmetry. The only nonzero Weyl scalar is ˆ Ψ 2 , and the dominant coefficient can depend only on u , The metric functions in the near-axis, large τ limit are with the g χχ term becoming singular at the poles simply due to our coordinate choice. Changing from χ to the coordinate θ gives g θθ = τ 2 + O ( τ ) while fixing g uθ = 0 on the axis. Asymptotically, the only dynamics present are the variation of the multipole moments with changing u , where M ( u ) clearly gives a monopole mass moment. The results given thus far are for a metric whose τ coordinate coincides with the affine parameter of the geodesic null vector ˆ l µ on the physical manifold M . In order to convert to affine geodesic null coordinates on the manifold S , and so read off the asymptotic behavior of the metric (5.27), we need to consider the effect of the transformation between ( u, p, χ ) and ( u, τ, χ ) coordinates, where p = p ( u, τ, χ ). Expanding dp in (5.27) in terms of du , dτ and dχ and equating the result with (E3) yields the following relationship between the metric functions and the derivatives of the affine parameter p , Integrating the first equation of (5.58) with respect to τ yields Inverting the series (5.59) to obtain an explicit expression for τ in terms of p is complicated by the logarithmic term. The leading order expression can however easily be found and is By working out the series expansions of Eq. (5.58) in terms of τ and then substituting Eq. (5.60) into the result, the asymptotic behavior of the metric (5.27) can be found to be", "pages": [ 16, 17, 18 ] }, { "title": "VI. SOLUTIONS TO THE TWIST-FREE AXISYMMETRIC VACUUM FIELD EQUATIONS", "content": "In this section we will characterize the properties of known twist-free solutions to the axisymmetric vacuum field equations within the framework that was established in the previous sections. The aim is to identify existing solutions and to catalog the assumptions made in finding them. With the exception of the Schwarzschild solution, none of the existing asymptotically flat solutions have physical significance. The hope is that this characterization will help establish the necessary properties a new dynamical solution, such as the head-on collision, must posses. A number of insights that can be gleaned by relating the four dimensional physical quantities to the three dimensional rotation coefficients are discussed in Sec. III. This section thus relies heavily on Sec. IV, and in particular Eqs. (4.12) and Eqs. (4.15), which give the 4D NP spin coefficients and associated Weyl tensor in terms of the 3D rotation coefficients discussed in Sec. III. Wherever possible we will also express the properties of the known solutions in terms of the two geometrically motivated triad and coordinate choices of Sec. V A and Sec. V B. We begin the discussion of analytic solutions with an example of the systematic solution of the field equations mentioned in Sec. V B. We consider the special case where the spacetime admits a coordinate choice in which we can simultaneously choose ψ as a coordinate with c a = √ 2 /Rψ ,a and find a null coordinate u such that the geodesic null vector l a = -u ,a is orthogonal to c a . This example has the benefit that it draws on our general results for both coordinate choices discussed in Sec. V A and Sec. V B. Having found one solution, we then place it in context with known solutions using a classification scheme based on the optical properties of the geodesic null congruence that l a is tangent to. It should be noted that the scope of many of the known solutions discussed in this section often extends beyond the restricted arena of twist-free axisymmetry, but we will restrict our discussion to this realm.", "pages": [ 18 ] }, { "title": "A. Special case: spacetimes that admit the coordinate choice ( u, p, ψ )", "content": "Any three metric can be expressed on an affinely parametrized geodesic null coordinate basis ( u, p, χ ) as in Eq. (5.27). In this section we will consider the special case where the third coordinate χ = ψ . Making the triad choice defined in Eqs. (5.25) and (5.29) we thus require that ψ ,i be spacelike and orthogonal to l i . From the results in Sec. V B on the geodesic null coordinate choice we have that α = β = 0, /epsilon1 = -η = -γ and that the simplified field equations presented in Eqs. (5.35)-(5.42) hold. As discussed in Sec. V A, the choice of ψ as a coordinate naturally sets ψ , 1 = ψ , 2 = 0 and the metric function e -2 v 3 = R/ 2. Furthermore the only non-zero curvature scalar is φ 0 = R which also simplifies Eqs. (5.35)-(5.42). We now proceed to solve this set of field equations. First, we note that Eq. (5.35), θ ,p = θ 2 can be solved by setting θ = -[ p + f ( u, ψ )] -1 . We use the coordinate freedom discussed in Sec. V B to relabel the origin of the affine parameter p by an arbitrary function of f ( u, ψ ), to give Performing one more integration using the commutation relation (5.31), v 3 ,p = p -1 , allows us to obtain the metric function v 3 ; equivalently, the scalar curvature R = 2 e -2 v 3 is The next field equation (5.36), /epsilon1 ,p = 0 indicates that /epsilon1 = /epsilon1 ( u, ψ ) only. Once more a commutation relation can be integrated to obtain the metric function v 2 . In this case Eq. (5.33), ( v 2 p ) ,p = -2 /epsilon1p implies that Before proceeding, let us use the fact that ψ has been chosen as a coordinate and examine the simplified Bianchi identities (5.2). The first equation is trivially satisfied, while the third equation (ln R ) , 3 = 4 /epsilon1 places restrictions on the integration constants already obtained. Writing out the directional derivative in terms of coordinate derivatives and substituting in the solutions for R , v 3 , /epsilon1 and v 2 we have This expression must vanish for all powers of p , which implies that /epsilon1 = c 2 = c 1 ,ψ = 0. A consequence of this result is that v 2 = 0, and in addition η = γ = /epsilon1 = 0, and c 1 = c 1 ( u ) is a function of u only. The final Bianchi identity gives the coefficient ι , Substituting the results obtained thus far into the third field equation, (5.37) we obtain δ ,p = -R/ 2, and thus the rotation coefficient δ = c 1 ( u ) / (2 p ) + c δ ( u, ψ ). The integration constant c δ ( u, ψ ) is fixed to the value c δ = c 1 ,u / (2 c 1 ) by evaluating the field equation (5.38). The commutation equation (5.32), v 1 ,p = -δ , yields an expression for the final metric function v 1 , Since v 2 = 0, the final commutation equation (5.34) sets ζ = c 3 ,ψ √ c 1 / 2 p -1 but the field equation (5.40) implies that ζ ,p = 0. Thus we have that c 3 = c 3 ( u ) is a function of u only and ζ = 0. All metric functions now depend on the variables p and u only. It is useful to observe that we still have the freedom to relabel the null hypersurfaces of constant u as discussed in Eq. (5.44). If we transform to a new set of coordinates ( u ' , p ' , ψ ) such that u ' = ∫ du √ c 1 ( u ) / 2 and p ' = p √ 2 /c 1 ( u ), the metric function v ' 1 expressed on the new coordinate basis can be written in the form v ' 1 = -ln p ' + c ' 3 ( u ' ). It is always possible to choose a coordinate u that labels the null hypersurfaces in a manner such that c 1 ( u ) = 2. For the rest of the section we make this choice (omitting the primes). The final field equation (5.41) reduces to ι , 2 = -ι ( ι -δ ). Upon substituting in ι = -v 1 p -1 , δ = p -1 and v 1 = -ln p + c 3 we find that c 3 ,u = 0, so that if we define ln A = c 3 , A is merely a constant. In summary, when ψ ,i is orthogonal to a geodesic null vector l i = u ,i the metric functions are and the rotation coefficients take on the values Having successfully solved the 3D field equations in this special case, let us examine some of the implications the solution has for the 4D spacetime associated with the original axisymmetric problem. The 4D NP spin coefficients for the solution found in this section are easily obtain from Eqs. (4.12) These expressions for the spin coefficients show the corresponding 4D null congruence is also geodesic and affinely parametrized. By writing down the Weyl scalars using Eq. (5.7) we observe that ˆ l µ is a principal null direction. A general classification scheme using the spin coefficients will be discussed more fully in Sec. VI C. For now it is useful to observe that the fact that ˆ ρ = ˆ σ and that ˆ l µ is a geodesic principal null vector indicates that this spacetime is a cylindrical-type Newman-Tamburino solution [106]. These spacetimes do not depend on any free functions of u . In fact, the metric of Eq. (6.7) corresponds to the particular case of a cylindrical-type Newman-Tamburino spacetime with one of the two arbitrary constants that parametrize these solution set to zero [26]. The axis conditions offer no additional constraints to this solution. As we approach the axis, we have e 2 ψ → 0. In order for the axis to be free of singularities, the elementary flatness condition, Eq. (2.29) must hold. In this particular case, we would have h ij ψ ,i ψ ,j = p -2 , and so elementary flatness would require as we approach the axis. However, we then have that p → 0 as we approach the axis, and we can see from the Weyl scalars (6.10) that the spacetime is singular as p → 0. We can conclude that this solution possesses a curvature singularity along the axis.", "pages": [ 18, 19, 20 ] }, { "title": "B. Spacetimes with special optical properties", "content": "In the next subsection we review known, special solutions to the axisymmetric, vacuum field equations, classifying them according to their optical properties. The classification will be made according to the properties of the null congruence that the tetrad vector ˆ l µ is tangent to in M , and by extension the congruence that the triad vector l i is tangent to in S . One of the benefits of the NP formalism is that the spin coefficients are directly related to the optical properties of a given spacetime. By seeking solutions with specified optical properties, many simplifications become possible, and the assumptions made are physically transparent. As detailed in Sec IV, by choosing to work in twist-free axisymmetry we have at least halved the complexity of the problem of solving the 4D field equations. The 4D spin coefficients computed from the 3D rotation coefficients are all real, which has immediate implications for the null congruence they describe in 4D, and we will discuss these implications here. Consider a general axisymmetric spacetime whose axial KV is twist-free and explore the behavior of the congruence of null curves that ˆ l µ is tangent to in M . The expansion and twist of this congruence are described by the real and imaginary parts of the spin coefficient ˆ ρ , respectively, and constitute the first two optical scalars. From Eqs. (4.12) we know that and is manifestly real, and this shows that a spacetime with a twist-free, axial KV admits a twist-free null congruence (c.f. [26, 107]). Furthermore, a twist-free congruence is hypersurface orthogonal, and so the fact that ˆ ρ is real implies that ˆ l u is proportional to the gradient of some potential function u . As in Sec. V B, in this case ˆ l µ is geodesic, since l ν l µ ; ν = u ,ν u ; µν = ( u ,ν u ,ν ) ,µ / 2 = 0, and we have The final optical scalar that characterizes the geometrical properties of the null congruence is the shear, which measures the distortion of the congruence and is given by An interesting property that arises from restricting the discussion to twist-free axisymmetric spacetimes is that, if the spacetime further has a null direction along which the derivative of ψ vanishes, the associated congruence has ˆ ρ = ˆ σ . The vanishing of any of the other directional derivatives of ψ gives analogous reductions to the 3D rotation coefficients, as can be seen by studying Eqs. (4.12). A number of solutions to the field equations have been found which admit a geodesic, hypersurface orthogonal null congruence (where /Ifractur [ ˆ ρ ] = 0). We will discuss these spacetimes and their relation to the form of the field equations developed in this paper in the next subsection. Our focus will be on asymptotically flat solutions, which can represent isolated systems, and we will reserve our discussion of stationary, axisymmetric spacetimes until Section VII. Figure 2 gives a summary of the solutions we will be considering, along with the reductions and assumptions employed to yield the known results.", "pages": [ 20 ] }, { "title": "C. Principal null geodesic congruences", "content": "We have showed that in any twist-free, axisymmetric spacetime there exists a geodesic, hypersurfaceorthogonal null congruence. If in addition, the tangent to this congruence is also assumed to be a principal null direction, so that ˆ Ψ 0 = 0, the field equations can be solved and the exact metric expressions are known. To see this, it is easiest to work in a triad where l a is affinely parametrized with respect to the physical manifold M . In this case, we have α = -2 ψ , 1 on S , ˆ /epsilon1 = 0 on M , and Eqs. (4.15) and (3.20) become Note that the directional derivative in these equations can be interpreted as a derivative with respect to the affine parameter τ in M and expressed as f , 1 = f ,τ . For Eq. (6.15) the two solutions are Using Eq. (6.15), Eq. (6.16) can be rewritten as ( θ + ψ , 1 ) , 1 = ( θ + ψ , 1 ) 2 . The two solutions to this equation are /negationslash /negationslash By substituting the solutions (6.18) and (6.17) into the 4D spin coefficients, Eqs. (4.12), several distinct conditions on the optical scalars ˆ ρ and ˆ σ can be identified. The expansion free case, where θ = ψ , 1 = 0, implies that ˆ ρ = ˆ σ = 0 and is known as the Kundt solution. On the other hand if ψ , 1 = 0, or if θ = -ψ , 1 , we have that ˆ ρ = ± ˆ σ = 0 which characterizes a cylindrical-type Newman-Tamburino spacetime. In the case that ψ , 1 = 0 and ψ , 1 + θ = 0 the transformation τ → τ ' + f ( u, χ ) can always be used to set c 1 = -c 2 . The optical scalars thus become ˆ ρ = -τ/ ( τ 2 -c 2 1 ) and ˆ σ = c 1 / ( τ 2 -c 2 1 ). This case can be split into two distinct scenarios: If c 1 = 0, ˆ σ = 0 and the spacetime can be classified as a RobinsonTrautman spacetime. If on the other hand c 1 = 0, then ˆ σ is nonzero and the result would again be a NewmanTamburino spacetime, but of spherical type. However, in the case of axisymmetry, we can solve the field equations explicitly for a nonzero c 1 , by integrating the hierarchy of field equations and matching powers (and transcendental functions) of τ at each step; the resulting spacetime has vanishing curvature and so is actually flat. This conforms to the known fact that the (non-trivial) spherical Newman-Tamburino solution can have at most only a single ignorable coordinate, namely the parameter labeling the null hypersurfaces u [26, 108]; in other words, the spherical-type solutions are incompatible with axisymmetry. /negationslash In the subsequent subsections we examine the properties of the each of the spacetimes mentioned here in greater depth. /negationslash", "pages": [ 21 ] }, { "title": "1. Newman-Tamburino spacetimes", "content": "Newman-Tamburino spacetimes are characterized by the properties /negationslash /negationslash The metric for these solutions can be found explicitly [105, 106] and except for special cases, the spacetimes are of the generic Petrov Type I. The Newman-Tamburino solutions are divided into two classes, 'spherical type' and 'cylindrical type' solutions. The spherical type is the more general, requiring ˆ ρ 2 = ˆ σ ˆ σ ∗ . The cylindrical type requires ˆ ρ 2 = ˆ σ ˆ σ ∗ . Only the cylindrical type solutions admit a spatial KV [108], and so are the case of interest for our study. Since all of the 4D spin coefficients are real in twist-free axisymmetry with our tetrad choice, these solutions require that the more restrictive condition ˆ σ = ± ˆ ρ holds, or equivalently in terms of the 3D quantities, In the case with ψ , 1 = 0, the proper circumference of the orbits of the axial KV is unchanging along the geodesic null congruence. Therefore, these solutions represent a spacetime that expands in the direction of the congruence. Note that the congruence is not simply frozen at a constant parameter τ , since the expansion ˆ ρ is nonzero for these solutions. The Newman-Tamburino solutions do not correspond to spacetimes of physical interest. In addition, the metric functions have a simple polynomial dependence on the coordinate u , which shows that the dynamics of these spacetimes are very simple. Since the properties of these solutions are well understood [105, 106] and the general derivation of the metric functions is lengthy, we do discuss these spacetimes further. Instead, recall that the solution found Sec. VI A is a special case of the cylindrical type Newman-Tamburino solutions, where in addition to ψ , 1 = 0 we assumed that ψ , 2 = 0. The solution found in Sec. VI A is parametrized by one constant A , while the general cylindrical-type metric contains two arbitrary constants [26, 105, 106].", "pages": [ 21 ] }, { "title": "2. Robinson-Trautman spacetimes", "content": "/negationslash The second class of solutions where the congruence is geodesic, principal null, shear-free, (ˆ κ = ˆ σ = ˆ Ψ 0 = 0) and expanding (ˆ ρ = 0) is known as the Robinson-Trautman spacetimes [109, 110]. The solutions to the field equations in such a case can be reduced to a single nonlinear partial differential equation and have been well studied, see e.g. [111] and the references therein. While these equations have been used to study radiating sources in an exact, strong field setting, they do not represent physical systems, such as a stage of head-on collision of black holes. In terms of the 3D rotation coefficients, the conditions for the Robinson Trautman solutions are /negationslash Note that since these vacuum spacetimes admit a shearfree geodesic null congruence, the Goldberg-Sachs theorem [25, 26, 112] states that they are algebraically special, so that ˆ Ψ 0 = ˆ Ψ 1 = 0. We can verify this directly from our the 3D equations. To do so, we use an affine parametrization with respect to the physical manifold M , as discussed in Appendix E. This sets α = -2 ψ , 1 . Substituting θ = -2 ψ , 1 into the field Eq. (6.16) immediately gives ˆ Ψ 0 = 0. Showing that ˆ Ψ 1 = 0 can also be made to vanish requires more finesse. For this, we use Eq. (4.15) with β = 0 to obtain an expression for ˆ Ψ 1 , In Eq. (6.22) the commutation relation (3.29) has been used to interchange to order of differentiation on ψ . The field equation (3.18), specialized to the case where θ = α = -2 ψ , 1 and β = 0, can be written as Observe that with the simplifications so far (ˆ κ = ˆ σ = ˆ /epsilon1 = ˆ Ψ 0 = 0), the two 4D Bianchi identities [26, 98] that govern only the directional derivatives of ˆ Ψ 0 and ˆ Ψ 1 can be expressed in our notation as The corresponding expressions in terms of the 3D quantities associated with the triad discussed in Appendix E, where /epsilon1 = -γ = -η -2 ψ , 3 , is If we apply the directional derivative l i ∂ i to the second of Eqs. (6.25), use the commutation relations (3.29) to switch the directional derivatives, and repeatedly use Eqs. (6.25), we obtain the equation η , 1 = -2 ηψ , 1 . Substitution of this into Eq. (6.23) implies that ψ , 1 ( ψ , 3 + η ) = -ψ , 13 , and thus ˆ Ψ 1 = 0. To complete the discussion of the Goldberg-Sachs theorem for twist-free axisymmetric spacetimes, note that substituting the condition ˆ Ψ 0 = ˆ Ψ 1 = 0 into the 4D Bianchi identities immediately implies that ˆ κ = ˆ σ = 0, and thus the existence of a geodesic shear-fee null congruence.", "pages": [ 21, 22 ] }, { "title": "3. Expansion-free spacetimes", "content": "We now consider the last case in our class of spacetimes that admit a geodesic principal null congruence, namely spacetimes that are both shear-free and expansion-free (ˆ κ = ˆ σ = ˆ ρ = ˆ Ψ 0 = 0). These metrics are Kundt solutions and have been extensively studied [26, 113, 114]. In terms of the 3D quantities, the Kundt metrics have the properties β = θ = ψ , 1 = 0. Since ψ , 1 = 0, setting α = 0 implies that the geodesics can be affinely parametrized in both M and S simultaneously. The fact that ψ , 1 = 0 also greatly simplifies the 3D curvature scalars, setting φ 5 = φ 4 = φ 3 = 0. Observe that choosing l i = -u ,i and setting θ = 0 in addition to α = β = 0 and η = -/epsilon1 implies that D 2 u = 0, so that u is a harmonic coordinate. This can be seen by taking the trace of Eq. (5.24). In the Kundt metrics, many of the rotation coefficients and metric functions are independent of the affine parameter p , which simplifies the calculations. The solution of the field equations in this case provides another simple and illustrative example of the integration of the 3D field equations. We now proceed to solve the hierarchy of field equations (5.35)-(5.42) in Sec. V B, in conjunction with the commutation relations applied to coordinates. In addition to ψ , 1 = 0, which implies that ψ is independent of the affine parameter, Eqs. (5.36) and (5.39) give /epsilon1 , 1 = ι , 1 = 0. Further, the commutation relation (5.31) yields v 3 , 1 = 0. Thus v 3 , ψ , /epsilon1 and ι are functions of only two coordinates u and χ . Note that it is always possible to use the coordinate freedom to define a new coordinate χ ' = g ( u, χ ) such that v ' 3 = 0 in the new coordinate system, by choosing g ,χ = e v 3 . Since ι = v 3 , 2 by Eq. (5.31), it is possible to set The next metric function, v 2 , can be found using the commutation relation (5.32), v 2 , p = -2 /epsilon1 . Since /epsilon1 = /epsilon1 ( u, χ ) only we can integrate the equation to yield v 2 = -2 /epsilon1p + c 1 ( u, χ ). The ability to shift the origin of the affine parameter by a function of u and χ by defining a new parameter p ' = p + g ( u, χ ) allows us to set c 1 = 0, and thus In obtaining the expressions for v 3 and v 2 we have used up most of the coordinate freedom with respect to the χ and p coordinates, except for transformations of the type χ ' = χ + f 1 ( u ) and p ' = p + f 2 ( u ) which do not spoil any of the simplifications so far. In order to learn more about the function /epsilon1 consider the harmonic condition on ψ , Eq. (3.30). With the reductions employed thus far, Eq. (3.30) reduces to ψ , 33 = 2 /epsilon1ψ , 3 . Furthermore the field equation (5.38) simplifies to the expression /epsilon1 , 3 = /epsilon1 2 + ψ 2 , 3 . Taking the sum and the difference of these two equations we obtain For functions of the form f = f ( u, χ ) the fact that v 3 = 0, implies that the directional derivative f , 3 becomes the coordinate derivative f ,χ , for these functions it is further true that f , 2 = f ,u . Before solving Eq. (6.28), note that Eq. (5.37) can be reduced to δ , 1 = /epsilon1 2 -ψ 2 , 3 , indicating that δ , 1 is independent of p and allowing us to integrate the equation to find an explicit expression for δ , where w δ is an integration constant. In addition, δ has to satisfy the difference of Eqs. (5.42) and (5.40), δ , 3 + /epsilon1 , 2 = -2 ψ , 2 ψ , 3 . Substituting in Eq. (6.29) and evaluating, we obtain the following constraint on w δ : Integrating the commutation relation (5.32), δ = -v 1 , 1 , the metric function v 1 can found to have the form where w 1 is a new integration constant. To see what additional constraints are to be imposed on the integration constants by the field equations, we express the only remaining coefficient ζ in terms of the metric functions using Eq. (5.34) with v 3 = 0, Substituting this expression for ζ into Eq. (5.41) and expressing the result using coordinate derivatives yields the constraint This is the final condition that has to be satisfied. All that remains now is to explicitly integrate Eq. (6.28). There are three possible cases: where the functions c i ( u ) are arbitrary functions of u only. For the remainder of the discussion we shall concentrate on the most generic case, Eqs. (6.36). Taking the difference of the Eqs. (6.36) and integrating one more time gives an expression for ψ , ∣ ∣ The Ricci scalar R = 2 ψ 2 ,χ and the coefficient /epsilon1 are It is straightforward to verify that the solution given here matches that presented in the original derivation of Kramer and Neugebauer [107], who also use the conformal 2+1 decomposition. The solution provided by Hoenselaers using the triad method [24] excludes the twist-free case, but McIntosh and Arianrhod [114] show (after making some corrections) that it matches the above form. We now investigate the condition of elementary flatness to see if it provides any additional constraints on the free functions. The axis is located at those points where which occurs when χ → c 3 . The condition for elementary flatness, Eq (2.29) or equivalently e 4 ψ R → 2, can be expressed as The requirement that the axis be free of conical singularities does thus provide a further constraint on the free functions, relating c 4 to c 3 -c 2 . Let us further explore the properties of the spacetime by computing the Weyl scalars. Since the Kundt spacetime is geodesic and shear free, it is algebraically special, with ˆ Ψ 0 = ˆ Ψ 1 = 0. The other Weyl scalars can be obtained in terms of the triad variables by making use of Eqs. (4.15), This shows that the metric is in general of Petrov Type II. In the case where c 2 = c 3 , we can see from our solution in Eq. (6.37) that ψ = c 4 ( u ), and so using Eq. (6.42) we have that ˆ Ψ 2 = 0. In this case the metric is of Type III. In case 2 mentioned in Eq. (6.35), /epsilon1 = -ψ χ also implies ˆ Ψ 2 = 0 and that the spacetime is also of Type III. Finally, we consider the special case 1 of Eq. (6.34) where /epsilon1 = ψ ,χ = 0, for which the vanishing of the Weyl scalars ˆ Ψ 2 = ˆ Ψ 3 = 0 implies that the metric is Type N. This completes the full classification of axisymmetric spacetimes that admit a geodesic principal null congruence.", "pages": [ 22, 23 ] }, { "title": "D. Other axisymmetric, twist-free spacetimes", "content": "Given the complete classification of spacetimes with special optical properties, it is clear that dynamical spacetimes of physical interest are not represented in this class of solutions. We must necessarily consider spacetimes whose geodesic, hypersurface-orthogonal congruence is not a principal null congruence, and Ψ 0 = 0. Known exact solutions that do not admit a geodesic principal null congruence include the twist-free solutions with the restriction (ˆ ρ/ ˆ σ ) ; µ ˆ l µ = 0 which have been found by Bilge [115]. However, Bilge and Gurses also showed that this class of spacetimes, though generally of Type I, is not asymptotically flat [116]. The class of twist-free spacetimes that obey (ˆ ρ/ ˆ σ ) ; µ ˆ l µ = 0 includes the vacuum Generalized Kerr-Schild (GKS) metrics, when they have twist-free congruences. GKS metrics are of the form /negationslash where g µν and ˜ g µν are both solutions to the vacuum field equations; ˆ l µ is a geodesic null vector with respect to both metrics (it forms the twist-free congruence); and H is some function on spacetime [115]. Gergely and Perj'es showed that the GKS spacetimes which admit twist-free congruences are the homogeneous, anisotropic Kasner solutions [117] (although two of the three constants that classify the Kasner spacetime must be set equal in order for there to be a single rotational symmetry and so an axial KV). Another method to find solutions with one spatial KV, which we mention for completeness, is to use a different triad choice than those discussed here. The idea, as developed by Perj'es in the study of stationary spacetimes [41] where the KV is timelike and the conformal 3D quotient space of the Killing orbits is spacelike, is to orient one of the triad legs along an eigenray . The eigenray is a curve defined such that if its spatial tangent vector is geodesic in the 3D quotient space, it is the projection of a null ray in the full 4D space. The triad formalism of Perj'es adapts naturally to the case of a spatial KV rather than a timelike KV, in which case the eigenray is timelike and the triad is chosen such that If the eigenray vector is, in fact, assumed to be geodesic, then solutions to the field equations can be found. In stationary spacetimes these were found in [41, 118], and in the case of a spacetime with a spatial KV they were found by Luk'acs [119]. When the geodesic eigenray is shearing, the solutions are Kasner (as in the case of the twist-free GKS spacetimes), and when it is not shearing they are Type D, and so very restricted. Thus, future studies which aim to extract physical information about isolated dynamical, axisymmetric spacetimes will have to focus on general spacetimes, where none of the principal null directions are geodesic, and which do not fall within Bilge's class of metrics.", "pages": [ 23, 24 ] }, { "title": "VII. STATIONARY AXISYMMETRIC VACUUM SPACETIMES", "content": "As discussed in Sec. I, the SAV field equations have been completely solved, in the sense that techniques exist that can generate any SAV solution. In this section, we briefly discuss the case of SAV spacetimes in the context of the conformal 3D metric and Ernst equation of Sec. II D. We then specifically focus on the twist-free case, and catalog the simplifications to the triad formalism discussed in Sec. III when static spacetimes are considered. We recast the static metric in the form discussed in Sec. V A with ψ chosen as a coordinate. Finally, as an illustrative example, we explore the properties of the Schwarzschild metric re-expressed using ψ as a coordinate, and we derive the scaling of the rotation coefficients as the axis is approached. We compare the results to the general expansions derived in the time-dependent case in Appendix D.", "pages": [ 24 ] }, { "title": "A. SAV spacetimes with twist", "content": "A stationary, axisymmetric spacetime possesses two KVs. One is spacelike, with closed orbits, which we denote ξ µ as in previous sections. The other is timelike, and we call it η µ . The ignorable coordinate associated with η µ we denote t , and we will later work in a gauge were η i = δ i t . The KV η µ places additional restrictions on the spacetimes considered up until now. Carter showed that for axisymmetric spacetimes which are asymptotically flat, the two KVs commute [94]. In addition, in vacuum, the pair of KVs are surface forming, which allows us to use coordinates t and φ in order to separate the metric into a pair of two dimensional blocks. Since any two metric is conformally flat, we can further choose coordinates ρ and z such that these coordinates are isotropic on their block [91], and express the metric in the Weyl canonical form, where the functions U, A, k are functions of ρ and z only. This form of the metric is extensively used for exploring the SAV field equations. Associated with the metric functions is an Ernst potential, which is often used in constructing solutions to the field equations. It should be noted that the Ernst potential usually employed in the discussion of SAV spacetimes in the literature E ( η ) is associated with the timelike KV η µ , and is not the Ernst potential E associated with the axial KV ξ µ introduced in Eq. (2.23). In the SAV context, E ( η ) = e 2 U + iϕ where e 2 U = -η µ η µ and ϕ ,µ = /epsilon1 µνσρ η ν η ρ ; σ . In order to make a direct connection with the notation used in this paper, we redefine the metric functions and cast the metric of Eq. (7.1) in the form with e 2 ψ = λ = ξ µ ξ µ as before. The metric functions of the two forms are related by Using the metric (7.2), the conformal 3-metric is given by It turns out that for the line element (7.2), the Ernst equation for E = λ + iω and field equations for ψ, γ, and B can be solved in an identical manner to the more usual SAV case, where U, k, and A are sought and when E ( η ) = f + iϕ . As derived previously, the field equations reduce to a single nonlinear equation for E , namely the first equation in Eqs. (2.23). Written out in the coordinates associated with metric (7.2), the equation for E becomes Here we have defined ∇ 2 as the usual flat space Laplace operator, in cylindrical coordinates. When working with SAV spacetimes it is often useful to introduce the complex coordinate ζ = ( ρ + iz ) / √ 2 to express the equations more compactly on the complex plane. In these coordinates ∂ ζ = ( ∂ ρ -i∂ z ) / √ 2. Once E is known the metric functions for γ and B can be found by making use of the line integrals", "pages": [ 24, 25 ] }, { "title": "B. Twist-free SAV spacetimes", "content": "We now specialize to the case of static axisymmetric spacetimes, which are twist-free. The KV ξ µ is hypersurface orthogonal, and thus ω = A = B = 0. The Ernst equation (7.5) reduces to the 3D cylindrical Laplacian applied to ψ , The two metric functions γ and ψ are related to the metric functions in Weyl canonical metric (7.1) by Since ln ρ is a homogeneous solution to the cylindrical Laplacian, we see that if U obeys the cylindrical Laplace equation, then ψ does also, and vice versa. All homogeneous solutions to the cylindrical Laplace equation are known (for example in terms of Legendre polynomials). The only difficulty is in specifying boundary conditions whose corresponding solution gives a spacetime with the desired physical interpretation. We will examine this issue using specific examples. In the context of the line element (7.1), a single Schwarzschild black hole is generated by a line charge of length 2 M placed on the axis. With this as the boundary condition, the Laplace equation for U can be solved; in Weyl coordinates, the solution is where r 2 ± = ρ 2 +( z ± M ) 2 and e 2 ψ = ρ 2 e -2 U . The function ψ for the Schwarzschild metric in ( ρ, z ) coordinates is plotted in Fig. 3. The black hole lies on the axis between z/M = ± 1, where the equipotential lines of ψ meet the axis at almost right angles. Further from the axis the surfaces of constant ψ rapidly approach surfaces of constant cylindrical radius. For the single black hole, as we will discuss in the next section, the appropriate axis conditions guaranteeing elementary flatness are satisfied at ρ = 0 outside the line charge. Multiple static black holes solutions can be found by placing multiple line charges along the axis [120]. Mathematically this corresponds to the superposition of multi- ple U i potentials given in Eq. (7.10), centered at positions z i and with masses M i . The corresponding ψ potential is then constructed using ψ = ln ρ -∑ m i =1 U i . Given the ψ potential of a superimposed set of black holes, we can always construct a solution for the metric function γ satisfying Eq. (7.6), which for static spacetimes becomes /negationslash Despite the fact that a solution can be found, it is not possible to find a solution for which elementary flatness holds along every connected component of the axis. From Eq. (7.1) we see that if k = 0 along a component of the axis, then elementary flatness does not hold there. Similarly, comparison of the line element (7.2) with Minkowski space written with ψ as a coordinate [see Eq. (C1) in Appendix C], shows that elementary flatness requires γ = 2 ψ -ln ρ along the axis. The conical singularities that result when the elementary flatness condition is not met are interpreted as massless strings or struts which hold the black holes apart, keeping them stationary. The fact that these singularities always appear in static, axisymmetric black hole solutions is in line with our intuition that black holes should attract each other, and can never remain stationary at a fixed separation. Approximate models based on using the tension on the strut to evolve binaries in a head-on collision scenario can be found e.g. in [121-123]. In the time-dependent case the harmonic equation governing ψ once again suggests that a generalized superposition principal could hold, at least on the initial time slice. In the time-dependent case, however the counterpart of equations (7.12) defining the gradient of the potential γ do not exist. Instead the second order elliptic equations for γ associated with the initial value problem must be solved. This allows the freedom to impose the boundary conditions guaranteeing elementary flatness. We thus expect that in dynamical spacetimes the conical singularities can be removed. We have yet to fully explore the time-dependent equations in the framework provided by this paper.", "pages": [ 25, 26 ] }, { "title": "C. Static Axisymmetric spacetimes with ψ chosen as a coordinate", "content": "In order to explore SAV spacetimes within the framework of more general axisymmetric spacetimes, we transform to a coordinate and triad system adapted to the scalar field ψ . To do this, consider the coordinate system ( t, s, ψ ), where t remains the ignorable coordinate associated with the timelike KV, ψ is a potential that obeys Eq. (7.8) and the coordinate s is orthogonal to ψ and defined by The integrability condition for s is guaranteed by the vanishing of the Laplace equation (7.8) for ψ . Changing coordinates from ( t, ρ, z ) to ( t, s, ψ ) yields a metric in the form where the functions S and R are the normalization factors defined by S = 2 s ; i s ; i and R = 2 ψ ; i ψ ; i . Note that from the definitions in Eq. (7.13) and the metric (7.4), it follows that S = ρ 2 R . This is consistent with the result (5.19) obtained by integrating the equations for the coefficient /epsilon1 , in the special case where g ( t, s ) = 1. This means that once R and ρ ( s, ψ ) are known the entire metric in ( ψ, s ) coordinates is known. In terms of the Weyl canonical coordinates ( ρ, z ), we have that R = 2( ψ 2 ,ρ + ψ 2 ,z ) e -2 γ , and in addition from the static field equations for γ , Eq. (7.12), we also have the identity R 2 = 4 e -4 γ ( γ 2 ,ρ + γ 2 ,z ) /ρ 2 . Let us now take a closer look at the rotation coefficients associated with the metric (7.14). When we consider the more general metric in Eq. (5.15), make the triad choice (5.16), and substitute both that l t = n t and that the metric functions are independent of time into Eqs. (D1), we find Furthermore note that for the choice of triad vectors (5.16), the directional derivatives obey the relation f , 2 = -f , 1 for any time-independent function f . Noting that h 2 s = 1 / ( ρ 2 R ) and that /rho1 = ρ in the static case, the remaining independent rotation coefficients can be expressed in terms of only the functions R , ρ and their derivatives as follows, To further illustrate the implications of choosing ψ as a coordinate, we now turn to the concrete case of the Schwarzschild metric. Computationally it is useful to introduce prolate spheroidal coordinates ( x, y ) related to the Weyl coordinates by the transformation In spheroidal coordinates, D 2 ψ = 0 is equivalent to requiring that In terms of the ( x, y ) coordinates, the norms of the Schwarzschild azimuthal and timelike KVs are Note that the upper and lower segments of the axis are identified by the coordinate values y = 1 and y = -1, respectively. The event horizon of the black hole is indicated by x = 1. By direct substitution it is easy to verify that both ψ and U satisfy Eq. (7.18). It is further possible to verify that the potential s = ( x -1) y has a gradient orthogonal to the gradient of ψ and obeys Eq. (7.13). It can thus be used as the second spatial coordinate. The lines of constant ψ and s coordinates are plotted in the ( ρ, z ) plane in Fig. 3. Since in Weyl coordinates the ( ρ, z ) plane is conformally flat, the fact that the curves intersect orthogonally in Fig. 3 indicates that their gradients are orthogonal to each other. The strong warping influence of the black hole at ρ = 0, -1 ≤ z ≤ 1 on constant ψ surfaces in these isotropic coordinates is clearly visible. This behavior can be ascribed in large part to the coordinates, which compress the black hole horizon onto the axis. As e ψ → 0 away from the hole, contours of constant ψ approach the axis, but on the black hole ψ acts as an angular coordinate, changing as the surface is traversed. The 3D Ricci curvature scalar R for the Schwarzschild metric is and obeys the axis condition (2.30). Also note that Eq. (7.19) is written is the same form as the more general series expansion of R about the axis given in Eq. (D8). In Schwarzschild case, the series truncates after the first order. To facilitate compact notation later on, let us define a function R 0 that is finite on the axis by The nontrivial rotation coefficients for the Schwarzschild metric are From the above expressions it is clear that on the axis, e 2 ψ = 0 ( y = ± 1), we have that β = θ = 0 while /epsilon1 and α diverge. The fact that β = 0 is an indication that the null vector l a is geodesic on the axis, as is expected from symmetry. The divergence of α is an indication that this geodesic has been poorly parametrized. A better choice would be to boost the null vector so that it is affinely parametrized. Note that in the boosted frame the vector ( ˜ l a + ˜ n a ) / √ 2 = ( Al a + A -1 n a ) / √ 2 is no longer hypersurface orthogonal. Choosing A so that the component of the l a vector along the t coordinate direction corresponds to that in the Kinnersley frame [124] results in an affine parametrization on the axis. This boost transformation is achieved by setting A = ρ ( x +1) / [ √ 2( x -1)]. The transformation of the rotation coefficients into the boosted, affinely parametrized frame are given in Eq. (B2), and is straightforward to compute. In this frame, the Weyl scalars computed from Eq. (5.7) are On the axis, the only nonzero Weyl scalar is ˆ Ψ 2 = -( x +1) -3 , where x + 1 can be associated with the standard Schwarzschild radius along the axis. As we move off the axis the other Weyl scalars take on non-zero values. This is expected because our choice of triad is adapted to the gradient of ψ rather than being a geodesic null triad. In fact, anywhere off the axis in the Schwarzschild spacetime, a triad adapted to the ψ coordinate can never have its null vector l a be geodesic, as can be verified using Eq. (7.21). This provides an explicit example of the general arguments regarding the boost transforms in (5.9). When the Schwarzschild metric is recast into a form with ψ as a coordinate, many of the equations appear to be unwieldy and offer little additional insight, but they do allow for an explicit verification that a frame exists in which the near-axis scaling of the rotation coefficients and metric functions computed in Appendix D hold. The main motivation for choosing a triad adapted to the ψ coordinate is that the results obtained in this analysis of the SAV case generalize to dynamic spacetimes. The Weyl scalars and rotation coefficients should give some indication of the expected behavior of their counterparts in dynamical spacetimes, near the axis and near black holes. It should be noted that many of the features that make the standard SAV analysis elegant are due to the availability of isotropic ( ρ, z ) coordinates, and the ability to linearly superimpose multiple solutions in the static case. These properties do not generalize to time-dependent spacetimes. One feature that is common to both the SAV and the time-dependent analysis is that ψ is a harmonic function that plays a crucial role in determining the configuration of the spacetime. It is hoped that a generalized superposition principal by which a new solutions for ψ can be formed by the 'sum' of two or more existing solutions can be found. Such a superposition of solutions, although straightforward to achieve on an initial value slice, will subsequently be complicated by the fact that the potential ψ influences the evolution of the metric functions that determine its own evolution in a possibly nonlinear way, making a 'sum' of two solutions nontrivial.", "pages": [ 26, 27, 28 ] }, { "title": "VIII. CONCLUSIONS", "content": "In this paper we have reviewed the reduction of the Einstein field equations in the case where the spacetime admits an axial Killing vector. The problem of finding solutions to the field equations in 4D then reduces to a problem of finding solutions to the field equations in 3D with an additional scalar field source term. We specialized to the case of vacuum spacetimes, and then to twist-free spacetimes, where the field equations become especially simple, but are still capable of describing spacetimes of physical relevance. Of particular interest is the case of a head-on collision of non-spinning black holes. In order to recast the equations into a form that seems especially amenable to investigation and intuition, we have presented a triad-based formulation of the equations due to Hoenselaers. We have expanded upon the original work of Hoenselaers, linking this formalism to the Newman-Penrose formalism and discussing two triad and coordinate choices that help to simplify the equations. We have also reviewed the known twist-free axisymmetric solutions which are not necessarily captured by the SAV equations, classifying them according to their optical properties, which correspond to certain simplifications in the field equations. We have introduced and explored the use of a harmonic coordinate ψ on the 3 manifold S . Recall that S is conformally related to the manifold ¯ S of orbits of the KV. The function ψ corresponds to a scalar field source for the Einstein field equations on S , and its use as a coordinate simplifies the curvature on S considerably. There is a natural expansion about the axis of symmetry in terms of λ = e 2 ψ , which provides inner boundary conditions for the field equations on S , and we have provided these series expansions and their connection to the elementary flatness condition. We have further revisited the case of static, axisymmetric spacetimes using ψ as a coordinate, in order to concretely illustrate the near-axis behavior of the triad and curvature quantities. Meanwhile, the assumption of asymptotic flatness provides the usual outer boundary behavior for the metric functions and curvature quantities, in terms of a Bondi expansion in geodesic null coordinates far from the sources. In order to develop a unified notation, we have given explicit expressions for the Bondi expansion in the triad formalism, eventually arriving at series expansions for the metric h ij on S . We intend this work to serve as a comprehensive and usable reference for the challenging goal of arriving at new solutions to the field equations. One immediate application of this work is to investigate the behavior of the scalar ψ in numerical axisymmetric simulations, such as the head-on collision of black holes. The observed behavior of ψ and the norm of its gradient R in simula- ons may give new insights. For example, by tracking these quantities, one could clearly quantify how the nonlinear collision differs from simple linear super-position of black holes. Numerical computation of the other triad quantities can also help to succinctly quantify these simulations, once an appropriate triad is fixed. This method of recasting a numerical simulation has the advantage of identifying the variables commonly employed in most solution generation techniques as well as highlighting the role of the generalized Ernst potential ( e 2 ψ ). Similarly, the investigation of ψ and R for the numerical studies of the critical collapse of gravitational waves in axisymmetry [18, 79], may lead to further understanding. In this case, the spacetime likely has additional symmetries which can help guide further analytic and numerical investigation of this solution. In the triad formalism reviewed in this work, we have presented the asymptotic expansion of the field equations and a similar expansion near the axis of symmetry. These expansions give the boundary conditions and dynamics which would be needed in any sort of axisymmetric evolution. It is natural to consider the connection between the triad formalism and the initial data for such an evolution. This data may be in the form of quantities on an initial spatial or null slice. Previous work on null initial data [125-127] immediately carries over to the triad quantities. Future work can examine the relationship between ψ and R and the momentum and Hamiltonian constraints on an initial spatial slice as well as the subsequent evolution of these fields. Note that while the Hamiltonian and momentum constraints do not constrain ψ and R , it may be possible to identify a preferred choice for the ψ associated with two black holes where the high frequency content is minimal. For example, the form that the scalars take in known initial data formulations for head-on collisions such as those in [128] and the associated emitted junk radiation is of interest. It is clear, though, that new techniques will still be needed in order to make analytic progress in the headon collision. In the past, analytic methods have allowed for an exploration of curvature quantities on the horizons of the holes in a head-on merger [129, 130]. With this and the expansions of the field equations near the boundaries, it would seem that an integration of the field quantities along null surfaces would be possible. One barrier to such an integration is the expectation that the null surfaces will caustic and become singular, especially near merger. For example, the horizon data of [129, 130] could only be numerically evolved on null slices contacting a merged horizon, due to the lack of a null foliation for a bifurcated horizon [131] (actually [131] used the initial data in the context of the evolution of the fission of a white hole). Ideally, a non-singular set of null coordinates could be found to cover the entire region of spacetime of interest, as illustrated schematically in Fig. 1. In such a coordinate system the triad formulation proves to be a powerful tool. We leave the search for such a set of null surfaces as the subject of future work.", "pages": [ 28 ] }, { "title": "Acknowledgments", "content": "We thank Yanbei Chen and Anıl Zengino˘glu for valuable discussions. JB would like to thank Y. Chen and C. Ott for their hospitality while at Caltech. TH and AZ would like to thank NITheP of South Africa for their hospitality during much of this work. AZ is supported by NSF Grant PHY-1068881, CAREER Grant PHY0956189, and the David and Barbara Groce Startup fund at Caltech. TH acknowledges support from NSF Grants No. PHY-0903631 and No. PHY-1208881, and the Maryland Center for Fundamental Physics.", "pages": [ 29 ] }, { "title": "Appendix A: Dimensional reduction of the 4D field equations", "content": "In this appendix we review some of the results pertaining to the curvature of the three dimensional manifold ¯ S whose induced metric ¯ h µν is related to the metric on the four dimensional manifold M by Eq. (2.3). In this subsection ξ µ is not necessarily a KV, but merely assumed to be timelike, i.e. ξ µ ξ µ = λ > 0. Just as in Sec. II, Eq. (2.6), the covariant derivative operator ¯ D α is defined by the full contraction of the 4D derivative operator with the projector ¯ h α ν = δ α ν -λ -1 ξ ν ξ α . For convenience, the definition of ¯ D α , Eq. (2.6) is repeated below,", "pages": [ 29 ] }, { "title": "1. Generalized Gauss-Codazzi equations for a timelike projected manifold", "content": "As in the case of the 3+1 split in numerical relativity [89], the Gauss-Codazzi equations describe the relationship between the 3D and 4D curvature tensors associated with the metrics ¯ h µν and g µν respectively. The Gauss equation can be derived by considering the 3D Ricci identity, which defines the contraction of the 3D Riemann tensor with an arbitrary covector ¯ V α on ¯ S , The derivation proceeds by writing out the derivative operators on the right hand side in terms of the 4D quantities g µν , ∇ µ and ξ µ using Eqs. (2.3) and (A1). As an intermediate step we define the quantity ¯ K βα to be and expand the double covariant derivative operator applied to ¯ V γ as To derive the second line of Eq. (A4) we repeatedly use the definition of ¯ K αβ given in Eq. (A3), the fact that ¯ h µ α ¯ h α β = ¯ h µ β , and the identity ¯ V τ ξ τ = 0 . Substituting Eq. (A4) into Eq. (A2) results in an expression relating the Riemann tensor on ¯ S to the Riemann tensor on M , This result holds for any vector ξ µ with norm λ . The top line of Eq. (A5) resembles the usual Gauss equation often encountered in a 3+1 split of spacetime if the tensor ¯ K αβ is identified with the extrinsic curvature of the embedded hypersurface (there is a relative sign change in front of the terms containing quadratic products in the tensor ¯ K βα that results from the fact that we are considering a timelike rather than spacelike 3 manifold). The tensor ¯ K αβ defined in Eq. (A3) can be identified with the extrinsic curvature of a hypersurface embedded in M only if ξ µ is hypersurface orthogonal. The second line of Eq. (A5) contains a term with the prefactor ¯ K [ δγ ] . In general if ξ µ has twist, this term is non-vanishing and must be retained. For all vectors ¯ V α whose Lie derivative with respect to ξ µ vanishes, L ξ ¯ V α = 0, we can simplify the second term in Eq. (A5) using ¯ h τ α λ -1 / 2 ξ σ ∇ σ ¯ V τ = ¯ K βα ¯ V β to yield Since the vector ¯ V β is arbitrary provided L ξ ¯ V α = 0 and ¯ V µ ξ µ = 0, it can be dropped from Eq. (A6) to give an expression for the curvature on the 3 manifold in terms of projected quantities. An important consequence of the 3D Riemann tensor so obtained is that in order for it to have the correct symmetries the tensor ¯ K βα must be antisymmetric, ¯ K βα = ¯ K [ βα ] . The condition ¯ K ( αβ ) = 0 is the same as requiring that the projection of the 4D Killing equation hold. A concise way of expressing the generalized Gauss equation Eq. (A6) in terms of the vector ξ µ in the case where K ( αβ ) = 0 is Note that the since the curvature tensor ¯ R αβγδ is defined on a 3D manifold which has a vanishing Weyl tensor, ¯ R αβγδ can be constructed solely from the Ricci tensor ¯ R αβ . As a result only the contracted Gauss equation needs to be considered. The components of the 4D curvature tensor where one index has been projected onto n µ , the unit normal in the ξ µ direction (and not to be confused with the null triad or tetrad vectors used elsewhere in this text), are related to quantities defined on the 3 manifold ¯ S via the Codazzi equations. These equations can be derived by applying the 4D Ricci identity to the unit vector n µ = λ -1 / 2 ξ µ and projecting the result onto the 3D manifold ¯ S When expanding the right hand side of Eq. (A9) in terms of 3D quantities, it is useful to observe that ¯ K βα can be expressed as the gradient of the unit vector n µ which has been twice contracted with the the projection operator. Expanding this relation and using the fact that n µ n µ = 1 yields the identity where ¯ a β = n ν ∇ ν n β is a measure of how the unit vector n ν is changing when parallel propagated. Note that ¯ h µ α ¯ a µ = ¯ a α since ¯ a µ n µ = 0. Substituting Eq. (A10) into Eq. (A9) yields a generalized Codazzi equation The last term once again vanishes in the case where ξ µ is hypersurface orthogonal, but has to be retained if we consider vectors ξ µ with twist. Contracting Eq. (A11) on the indices α , γ with the metric ¯ h αγ yields the contracted Codazzi relation where ¯ K = ¯ K α α . There is one more nonzero contraction of the 4D Riemann tensor with the projector ¯ h µ α and the unit vector n µ , which is computed by contracting the second and third indices of the Riemann tensor with n µ and the remaining indices with the projection operators ¯ h µ α . Equations (A6), (A11) and (A13) express the 4D curvature tensor in terms of projected quantities for a projection operator based on an arbitrary spacelike vector ξ µ . When ξ µ is hypersurface orthogonal, these expressions reduce to the usual Gauss-Codazzi equations. The case where ξ µ is a KV is addressed in the next section.", "pages": [ 29, 30 ] }, { "title": "2. Field Equations expressed on the 3D quotient manifold in the case where ξ µ is a Killing Vector", "content": "We now specialize the results of the Gauss-Codazzi equations derived in the previous section to the case where ξ µ is a KV obeying Eq. (2.1). When ξ µ is a KV the derivation and results presented here are equivalent to that found in [20]. The Killing equation (2.1) implies that the tensor ¯ K αβ defined in Eq. (A3) is antisymmetric and thus the generalized Gauss equation (A7) holds. The Killing equations along with the identity ξ µ λ ,µ = 0 can be used to express the covector ¯ a β defined below Eq. (A10) as This result, in conjunction with Eq. (A10), then allows us to write the gradient of the KV as Substituting Eq. (A15) into the definition of the twist, Eq. (2.2), we obtain the expression which can be inverted using the identity /epsilon1 µνρσ /epsilon1 µτχ/epsilon1 = -6 δ τ [ ν δ χ ρ δ /epsilon1 σ ] and the fact that ξ µ ¯ K µν = 0, to yield ¯ K βα in terms of the twist Finally substituting Eq. (A17) back into Eq. (A15) yields the identity Given the expressions for ¯ K αβ , ¯ a β and ∇ µ ξ ν in terms of the twist and norm of the KV, we can begin to evaluate the Gauss-Codazzi equations. An expression for Ricci tensor on the three manifold can be found by substituting the double contraction of the KV with the 4D Riemann tensor, Eq. (A13), into the contracted Gauss equation (A8) and using the relations given in this section to arrive at Since ¯ K αβ is antisymmetric, ¯ K = 0. Substituting Eq. (A17) and subsequently Eq. (A18) into the contracted Codazzi equation (A12) we obtain This equation used extensively in Sec. II A where it is referred to as Eq. (2.7). The first term on the right hand side of (A13) vanishes because of the symmetry in the indices α , δ on the left hand side of (A13) and the antisymmetry of ¯ K µα . which can be rewritten to yield which relates the twist of ω µ to the 4D Ricci curvature. This equations is reference as Eq. (2.10) in Sec. II A . The divergence of ω α is found by considering the totally antisymmetric part of the generalized Codazzi equation, or equivalently contracting ξ ν /epsilon1 ναγδ with Eq. (A11), which becomes The first Bianchi identity, R µνρσ + R ρµνσ -R σµνρ = 0 sets the term on the left hand side of Eq. (A20) to zero. The right hand side of Eq. (A20) can be evaluated using the following expression for the derivative of the extrinsic curvature, The resulting expression for the divergence of the twist vector quoted in Eq. (2.9) is An equation governing the harmonic operator applied to λ can be obtained by contracting the final projection of the Riemann tensor in Eq. (A13) with the three metric ¯ h αδ , and making use of the antisymmetry of ¯ K µσ and the expressions (A17) for ¯ K αβ and (A14) for ¯ a β . The result is quoted in Eq. (2.8) and given below, This completes the derivation of the reduced field equations on ¯ S , used in Sec. II to discuss axisymmetric spacetimes.", "pages": [ 30, 31 ] }, { "title": "Appendix B: Lorentz transforms of the 3D tetrad", "content": "Here we discuss the effect of Lorentz transforms of the triad. As usual, these come in three types: boosts along the null vector l i , and rotations about each of the two null vectors l i and n i . First we discuss boosts. Let Then ˜ l i ; j = Al i ; j + A ,j l i and ˜ n i ; j = n i ; j /A -A ,j n i /A 2 . The nine rotation coefficients become where the directional derivatives are with respect to the original triad. The six curvature scalars transform as Next, let us consider rotations about the null vectors. The usual rotations about the null vectors in a null tetrad in the 4D are restricted to those that do not mix the axial KV ξ µ with the three vectors that span the 3D hypersurfaces which correspond to S . These transforms must leave the difference and sum of the complex NP spatial vectors ˆ m µ and ˆ m ∗ µ invariant (since these correspond to the normalized KV ˆ d µ and the spatial vector ˆ c µ , respectively). The usual rotations by complex parameters (see e.g. [26]) are reduced to rotations by real parameters, a and b . For a rotation about l i , we have For a rotation about n i we have in complete analogy We are primarily interested in a fixed null direction l i , so let us consider rotations about this vector. We have the following transforms for the rotation coefficients, In these expressions, the directional derivatives are with respect to the original triad vectors. The six curvature scalars transform as Finally, we consider rotations about n i . We first note that when interchanging l i and n i , the rotation coefficients exchange identities as Thus, the effect of a rotation around n i by a factor b on the rotation coefficients can be derived from the expressions given for a rotation around l i by first applying the above relations to those transforms, and then taking a → b and swapping directional derivatives in the l i and n i directions, f , 1 → f , 2 and vice versa. The rotation coefficients thus transform as We can write similar transformations for the curvature scalars φ i by noting that, under the exchange of l i and n i the curvature scalars transform as { φ 5 , φ 4 , φ 3 , φ 2 , φ 1 , φ 0 } → { φ 2 , φ 4 , φ 1 , φ 5 , φ 3 , φ 0 } and applying these transforms and a → b to Eqs. (B7).", "pages": [ 31, 32 ] }, { "title": "Appendix C: Minkowski spacetime with ψ as a coordinate", "content": "To gain a better intuition into the choice of ψ as a coordinate, let us consider Minkowski space in cylindrical coordinates ( t, z, ρ, φ ). In these coordinates ρ = e ψ and so dρ = e ψ dψ . Inserting this gives the line element Next, consider the metric on the conformal space S . We have We can see immediately that R = 2 e 4 ψ , which trivially obeys the axis condition in S as ψ → -∞ . Looking at how the metric functions enter the triad in Eqs. (5.15) and (5.16), we see that an appropriate choice for a null triad adapted to the timelike gradient T a is Using this triad, we can compute the rotation coefficients and begin to get a sense of the way each coefficient should behave as we approach the axis. However, first let us note that the triad chosen above has some troubling features. Comparing these to the corresponding tetrad vectors in M , as given by Eq. (4.10), we see that Near the axis, we see that our chosen ˆ l µ and ˆ n µ vectors are poorly behaved; ˆ l µ blows up on the axis, and ˆ n µ vanishes. We must boost the triad vectors by a factor of A = e ψ in order for the corresponding physical tetrad to the be well behaved on the axis. Computing the rotation coefficients on S with the boosted triad legs l i = (1 , 1 , 0) / √ 2 and n i = e -2 ψ (1 , -1 , 0) / √ 2 leads to with all others vanishing. Note that under a null boost, γ and /epsilon1 do not change; we cannot prevent the pathological behavior of these coefficients on S as ψ → -∞ . Meanwhile, η does transform, and using a boost A = e mψ we find ˜ η = η -me -2 ψ , which can be used in this case to make ˜ η = 0 with the choice m = -1. This infinite boost at the axis has no effect on the vanishing of the other coefficients, and returns us to the triad we originally considered in Eq. (C3).", "pages": [ 32 ] }, { "title": "Appendix D: Rotation coefficients associated with a triad adapted to the ψ coordinate", "content": "The rotation coefficients associated with the triads in Eqs. (5.13) (with h ψ = 0) and (5.16) and the corresponding metric Eq. (5.15) are now computed using equations (4.5) and (4.6). Recall that for the case under consideration η = 1 2 ( β -γ + ζ -/epsilon1 ) and γ = -/epsilon1 . The remaining coefficients are From Eq. (D1) the dominant scaling of the rotation coefficients near the axis can now be obtained by requiring that the physical metric expressed in terms of ( t, s, λ, φ ) coordinates is regular as the axis is approached, λ → 0. To examine the behavior of the metric components as we near the axis, we will quote a result of Rinne and Stewart [74]. Consider a local Lorentz frame in a neighborhood near a point on the axis, p ∈ W 2 , and let us use Cartesian coordinates ( x, y ) on the space orthogonal to W 2 , so that the KV can be represented as If we insist that scalar quantities have a regular expansion in ( x, y ) about the axis, and that their Lie derivative with respect to ξ µ vanishes, then it can be shown that the expansion must in fact be of the form noting that λ = x 2 + y 2 near the axis if our coordinates are appropriately normalized. By applying the same Lie derivative argument to the metric, it can be shown [74] that the ( t, s ) block of the metric admits expansions in λ as if the metric functions are scalar quantities, It is always possible to choose s and t coordinates on the axis to be orthogonal to each other. This choice sets the first term in the off-diagonal metric function g (0) ts to zero. The series expansions about the axis in Eq. (D5) also set the series expansion for the functions entering into the metric h ij in Eq. (5.15). Explicitly we have that the functions h s , n t and l t admit the following expansions near the symmetry axis: where the coefficients /rho1 ( n ) = ( l ( n ) t + n ( n ) t ) / √ 2. If the metric is chosen to be diagonal on the axis, we further have that l (0) t = n (0) t . In Eq. (5.19) we integrate the equations describing the rotation coefficient /epsilon1 by means of one of the Bianchi identities to yield ( h s /rho1 ) 2 = g ( t, s ) /R . This result remains valid in the neighborhood of the axis and allows us to find an expression for the function g ( t, s ) in terms of the expansion coefficients h (0) s and /rho1 (0) , namely This result, together with the expansions given in Eq. (D6) and the series expansion of R implied by the axis condition, allows us to determine that on the axis the rotation coefficients scale as While β and ζ have O ( λ 0 ) terms, we know that null rays which remain on the axis must be geodesic, and thus these terms must vanish. We then have Recall that when working with a triad adapted to ψ as a coordinate, Eq. (5.7) states that the Weyl scalar ˆ Ψ 0 = √ R/ 2 β , and as a result (D10) implies that the geodesics along the axis are principal null. It also implies that if the metric is chosen to be diagonal on the axis so that l (0) t = n (0) t , then this property persists to order O ( λ 2 ), since l (1) t = n (1) t . We now substitute the expansions into the field equations (5.20) and (5.21), and begin to solve them order by order in λ . We start by looking at the subset of equations that have directional derivatives only in the l a , n a directions, namely Eqs. (5.20), and choose to set the metric diagonal on the axis. The dominant terms that arise from the sum and difference of the first two equations in Eqs. (5.20) give respectively. These imply that with k 1 a constant. Together Eqs. (D9), (D10) and (D11) give The dominant λ -1 term in the fourth equation of (5.20) and the first equation in (5.21) governing /epsilon1 , 3 can only vanish if both k 1 = 0 and the equation holds. With this, the equations in (5.21) that govern δ , 3 , α , 3 , θ , 3 , and ι , 3 are satisfied to O ( √ λ ) and the third equation in (5.20) to O ( λ ). Setting the O ( λ 0 ) term to zero in the equation governing /epsilon1 , 3 and the O ( λ 0 ) term to zero in those for ζ , 3 , β , 3 in (5.21), and furthermore setting the O ( λ 0 ) term to zero in the fourth equation of (5.20) fixes the R (2) , n (2) t , l (2) t and h (2) s coefficients in terms of /rho1 (0) , h (0) s , R (1) , and their derivatives. These expressions are lengthy, and we will only give the off diagonal term here, Equation (D14), indicates that for a time-dependent metric the off-diagonal term l t -n t is O ( λ 3 / 2 ). It is also clear that when the spacetime is dynamic, the diagonalization of the ( t, s ) block cannot be maintained off the axis. This off-diagonal term can be interpreted as a shift governing the motion of the coordinates along constant ψ slices as time progresses. Continuing on, it appears that the expansion coefficients of the metric functions are fixed at each higher order by the lowest order terms /rho1 (0) , h (0) s , and R (1) . The same behavior occurs far from gravitating sources in asymptotically flat spacetimes, where the expansion is in orders of inverse affine distance. This is the asymptotic expansion of the Bondi formalism, which we discuss in Sec. V C and Appendix F.", "pages": [ 32, 33, 34 ] }, { "title": "Appendix E: Geodesic null coordinates affinely parametrized with respect to the 4D manifold", "content": "In this appendix we write down the field equations adapted to a geodesic null coordinate system that is affinely parametrized with respect to the physical manifold M . The results obtained in this section can be directly derived from Sec. V B by applying a boost A = e 2 ψ using Eqs. (B2), and then selecting a new parameter and Substituting the expressions for the coefficients γ , α , β , and η found from these relations into the field equations (3.15)-(3.23) and reorganizing gives As expected, the hierarchy present in Eqs. (5.35)-(5.42) (where l a is affine in S ) persists, which allows us to formally integrate the field equations. We conclude this appendix by detailing how the metric functions in Eq. (E3) transform with the remaining along the geodesic. However, because this choice of coordinates is used in our discussion of the Bondi expansion, we will give the equations in full here. Let us once again choose a null coordinate u such that h ij u ,i u ,j = 0, but this time we choose the null vector l i = -e 2 ψ u ,i . The symmetry of u ,a | b = -( e -2 ψ l a ) | b , when expressed on the triad basis, leads to the following conditions on the rotation coefficients, Making use of Eqs. (4.12), we immediately see that this choice for l i yields ˆ κ = ˆ /epsilon1 = 0, so we see that ˆ l µ is geodesic and affinely parametrized in the physical space. Let τ denote the affine parameter in M , and choose the triad to be Then the three metric on S becomes Applying the commutation relations to χ , u , and τ respectively yield coordinate freedom [105]. These transformations include shifting the the origin of the affine parameter along each geodesic separately, τ ' = τ + f ( u, χ ), which transforms the metric function of (5.27) according to Relabeling the individual geodesics within a spatial slice, χ ' = g ( u, χ ) transforms the metric functions of (E3) to And finally, by relabeling the null hypersurfaces, setting u ' = h ( u ), τ ' = τ/h ,u ,the metric components transform as", "pages": [ 34, 35 ] }, { "title": "Appendix F: Derivation of the Asymptotic expansion of an affinely parametrized metric in null coordinates", "content": "In this appendix we systematically solve the field equations given in Appendix E in the asymptotic regime far from an isolated, gravitating system. Our method of solution further illustrates the integration of the hierarchy of the field equations that results when they are expressed on a null slicing. We focus only on spacetimes that admit the peeling property [84-88]. In this case the Weyl scalars have a power series expansion at future null infinity of the form ˆ Ψ i = τ i -5 ∑ n =0 τ -n ˆ Ψ ( n ) i , where ˆ Ψ ( n ) i are constant along an out-going null geodesic, i.e. ˆ Ψ ( n ) i ( u, χ ), and τ is the affine parameter along the geodesic. Using Eq. (5.46) as a starting point, we derive the power series expansion of the metric functions and the rotation coefficients in terms of a series in 1 /τ . This information makes apparent the boundary conditions that have to be imposed when solving the complete set of equations, and provides explicit information about the fall-off of all rotation coefficients at null infinity . Consider first the properties of the null tetrad vector ˆ l µ . We will take ˆ l µ to be the tangent to out-going, null geodesics far from the isolated system, so that β = 0, and also affinely parametrized in M , which sets α = -2 ψ , 1 . Directional derivatives in the l i direction on S can thus be expressed as f , 1 = f ,τ . The simplified field equations and form of the metric for this choice of parametrization is given in Appendix E. Specifying the series expansion for ˆ Ψ 0 on a null hypersurface of constant u gives almost all of the data required to continue the spacetime off the hypersurface. During the calculation that follows, we quantify how this information is transmitted to the metric functions. From Eq. (4.15) and (5.46) we have that the leading order terms of the function ψ are related to the coefficient ˆ Ψ (0) 0 via ψ , 11 +( ψ , 1 ) 2 = τ -5 ˆ Ψ (0) 0 + O ( τ -6 ). Solving this equation term by term, we find that Here, σ (0) = σ (0) ( u, χ ) is a function whose properties have yet to be defined. We will see that σ (0) corresponds to the dominant term in the series expansion of the shear of ˆ l µ . Its labeling corresponds to the choice made in [132], whose derivation we initially follow closely when working out the expansion properties of the optical scalars. In the series expansions that follow we keep terms of sufficiently high order to indicate where dominant terms of the expansions of the Weyl scalars enter into the metric functions. Integrating Eq. (F1) with respect to τ adds an additional integration constant ψ (0) ( u, χ ), and allows us to express ψ as A series expansion for θ can be found by substituting a power series ansatz for θ into Eq. (E6) and using Eq. (F1) to define the series expansion for ψ , 1 . It can be shown that the leading order behavior of the solution admits only two possibilities, namely θ = -τ -1 + O ( τ -2 ) or θ = -2 τ -1 + O ( τ -2 ). The former corresponds to a cylindrical type spacetime that is not astrophysically relevant. We will only consider the latter case and further discuss the justification for this choice after Eq. (F4). Using our coordinate freedom to relabel the origin of the affine parameter, τ ' = τ + f ( u, χ ), it is always possible to set the next coefficient in the expansion to zero [132]. Examining the remaining coefficients in Eq. (E6) term by term leads to the series expansion With this, we can integrate Eq. (E4), w 3 , 1 = -θ , to arrive at an expression for the metric function w 3 , which adds the integration constant w (0) 3 ( u, χ ) to our list of undetermined expansion coefficients. Note that the leading order term of the metric coefficient g χχ = e 2 w 3 -2 ψ in M is proportional to τ 2 , which is typical for a surface of constant τ that is asymptotically spherical. Had we made the selection θ = -τ -1 + O ( τ -2 ) above, the leading order term would have been independent of τ . The next set of variables to be considered in the integration hierarchy are /epsilon1 , ˆ Ψ 1 , and the metric function w 2 . For our chosen triad, determining /epsilon1 also fixes two other rotation coefficients; from Eqs. (E1) and (E4) we have that η = -2 ψ , 3 -/epsilon1 and γ = -/epsilon1 . The peeling property of the Weyl scalars (5.46) in conjunction with the expression for ˆ Ψ 1 given in Eq. (4.15) and the field equation (E7) yields which can be systematically solved to yield a series solution for /epsilon1 , Next, using Eq. (E7), we find a series expansion for ψ , 3 of the form The higher order coefficients in the expansion for /epsilon1 are fixed in terms of existing quantities as follows, and so far the coefficients /epsilon1 (0) and /epsilon1 (1) are unconstrained. It turns out that the leading coefficient /epsilon1 (0) can be set to zero using a gauge transform. To see this, note that we can obtain an expansion for the metric function w 2 using Eq. (E5). The resulting expansion is It is then possible to make use of the gauge transformation χ ' = g ( u, χ ) to relabel the geodesics such that /epsilon1 (0) = 0, using Eq. (E15). The resulting expansions for /epsilon1 , ψ , 3 and w 2 reduce to The next set of variables obtained via the systematic integration of the field equations includes δ , the Weyl scalar ˆ Ψ 2 , the metric function w 1 , and the derivative ψ , 2 . The coefficient δ can be obtained by integrating the field equation (E8), where the commutation relations, Eq. (3.29), the definition (4.15) of ˆ Ψ 2 and the harmonic equation (3.30) for ψ , have been used to replace directional derivatives in the n i direction with known series expansions. The resulting equation implies that δ must admit the series expansion The expansion for δ can now be used to obtain the expansions for the metric function w 1 using Eq. (E5), δ = -2 w 1 ψ , 1 -w 1 , 1 . The first two terms in the resulting expression are If the metric g µν is to be asymptotically flat, the metric function e 2 ψ w 1 must be finite as τ → ∞ , and thus the integration constant δ (1) = 0. The resulting expansion for the metric function w 1 becomes A series for the directional derivative ψ , 2 can be obtained from the field equation (E8) using the commutation relation to switch the order of differentiation on ψ . The resulting expression becomes which implies that ψ , 2 has the series expansion Since the series expansions for ψ and the three metric functions w 1 , w 2 , w 3 are given, the directional derivatives of any function expressed as a series in 1 /τ can also be expanded as a series. We begin by examining the directional derivatives of ψ to determine what restrictions the resulting expressions place on the existing expansion coefficients. By considering the directional derivative ψ , 3 and the expansion (F10) we obtain the result Examining the directional derivative ψ , 2 and the expansion (F17), we obtain δ (1) = -e -2 ψ (0) ψ (0) ,u = 0 which implies that ψ (0) = ψ (0) ( χ ) is independent of u . We also have that Finally, we examine the remaining field equations to obtain further restrictions on the expansion coefficients. Substituting the expansions and directional derivatives obtained thus far into Eq. (E9) yields the condition w (0) 3 ,u = 2 ψ (0) ,u = 0 , at order τ -3 , which implies that Note however that the coordinate transformation χ ' = g ( χ ) transforms the metric function as e w ' 3 = e w 3 /g ,χ and allows us to set f (0) ( χ ) to any arbitrary function of our choosing. We can understand the meaning of a choice of f (0) by insisting that as τ → ∞ , the ( χ, φ ) block of the physical metric on M has the geometry of a sphere. In other words, g AB → τ 2 Ω AB , where Ω AB is a metric on the unit 2-sphere, and { A,B } ∈ ( χ, φ ). With this requirement on the angular geometry of out-going null surfaces, a particular choice of f (0) allows us to fix both ψ (0) and to identify the particular angular coordinate χ corresponding to this choice of f (0) . For example, setting f (0) ( χ ) = -2 ψ (0) corresponds to the choice where the metric on the unit 2-sphere Ω AB has unit determinant. In this case, we find that e 2 ψ (0) = 1 -χ 2 , where χ = cos θ , θ is the usual angular coordinate, and the axis is located at χ = ± 1. For comparison, note that setting f (0) ( χ ) = 0 corresponds to the Fubini study metric representation of the sphere used in [132]. Henceforth we will set f (0) ( χ ) = -2 ψ (0) , which means that we have selected χ = cos θ and that w (0) 3 = 0. Evaluating Eq. (E9) at order τ -4 yields the additional condition Next, at order τ -5 we obtain an equation that evolves ˆ Ψ (0) 0 from one hypersurface to the next hypersurface Using equations (E4) and (E5), the expansions for ι and ζ can be shown to have the form where the expansion coefficients are related to those functions already defined by Using Eq. (4.15) the dominant terms in the remaining Weyl scalars can be shown to be where The evolution equations that propagate the coefficients ˆ Ψ (0) 1 and ˆ Ψ (0) 2 from one null hypersurface to the next can be obtained by examining (E11) and (E12) respectively at O ( τ -6 ), yielding the expressions The field equations (E10) and (E13) yield no additional constraints and vanish to O ( τ -8 ). Also note that the harmonic equation for ψ has been satisfied. The results obtained thus far are now briefly summarized. The 4D line element can be expressed as where, according to Eqs. (F2), (F4), (F11), (F15), (F18), and (F21), the metric functions admit the following asymptotic expansion as the affine parameter τ →∞ : To fully specify the solution we must make a choice for the functions ˆ Ψ (0) i ( u, χ ), i ∈ { 0 , 1 , 2 } on a hypersurface u 0 and specify the function σ (0) ( u, χ ) for all u and χ . A natural hypersurface to choose is u →∞ and to specify the functions to correspond to the Schwarzschild solution. In this case The constant M is the mass of the final black hole and C is an arbitrary constant. The fact that the shear of the null bundle must be regular on the axis sets C = 0. For u < ∞ , ˆ Ψ (0) i , can now be determined provided σ (0) ( u, χ ) is given. The results given this section are further discussed in Sec. V C. The expansion of the 4D spin coefficients can be obtained from the results in this appendix using Eqs. (4.12). The series expansion of the shear ˆ σ is found to be confirming that σ (0) is indeed the dominant term in the expansion of the shear. It should be noted that our expansion for the spin coefficient ˆ τ is and that the prefactor of 1 / 3 in front of the τ -3 term differs from the result obtained in [132].", "pages": [ 35, 36, 37, 38 ] } ]

[ { "title": "Note on black hole no hair theorems for massive forms and spin1 2 fields", "content": "Sourav Bhattacharya ∗ Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, INDIA. January 24, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We give a proof of the non-perturbative no hair theorems for a massive 2-form field with 3-form field strength for general stationary axisymmetric and static (anti)-de Sitter or asymptotically flat black hole spacetimes with some suitable geometrical properties. The generalization of this result for higher form fields is discussed. Next, we discuss the perturbative no hair theorems for massive spin1 2 fields for general static backgrounds with electric or magnetic charge. Some generalization of this result for stationary axisymmetric spacetimes are also discussed. All calculations are done in arbitrary spacetime dimensions. Keywords: Stationary axisymmetric black holes, no hair theorems, forms, spin1 2", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The black hole no hair theorems state that any realistic gravitational collapse must come to a final stationary state characterized only by parameters like mass, angular momentum, and charges corresponding to long range gauge fields (see e.g. [1, 2, 3, 4, 5, 6], and references therein). The proof of the no hair theorem for a given matter field for a given black hole spacetime thus essentially involves the proof of vanishing of that matter field in the exterior of that spacetime. Considerable effort has been given so far to investigate no hair theorems for various matter fields, such as scalars with or without non-minimal couplings, massive 1-form and spin-2 fields [5, 6, 7, 8, 9, 10, 11, 12]. We refer our reader to [13, 14, 15, 16] for some exception to this theorem and to e.g. [9] for a more detailed review on no hair theorems. We also refer our reader to [17, 18, 19] for an account of possible observational consequences related to the black hole no hair theorems. Since the concern of this paper is to discuss the no hair theorems associated with massive forms and spin1 2 fields, let us take a brief account of progress on this topic now. In [20], the perturbative no hair theorems for massless p -forms with ( p +1)-form field strengths in arbitrary dimensional static spherically symmetric spacetimes were addressed, by choosing a suitable gauge. Interestingly, when one considers massive 2-form in the context of a topologically massive gauge theory, a black hole may have a topological charge detectable via Aharonov-Bohm like effects [21]. The no hair properties for spin1 2 fields corresponding to various static spherically symmetric black hole spacetimes has been discussed in [22, 23, 24, 25] via time dependent perturbation techniques, including the presence of a cosmic string [26], which gives topology other than S 2 . Interestingly, it was indicated in [25] using the bosonisation scheme techniques that an asymptotically flat black hole spacetime carrying a non-Abelian charge may face instability if perturbed by a Dirac fermion. A demonstration of the Price's theorem [27] for Schwarzschild-de Sitter spacetime for massless spinor zero modes can be found in [28]. A proof of non-existence of time-periodic Dirac hair in asymptotically flat static or stationary axisymmetric spacetimes of dimension four using the variable separated Dirac equation [29] and the properties of the self adjoint Dirac operator can be found in [30, 31, 32, 33, 34, 35, 36, 37, 38]. These proofs have also been generalized for the de Sitter black hole spacetime [39] (see also [9]). The main concern of this work is higher dimensional general spacetimes satisfying Einstein's equations with or without a cosmological constant Λ, for which uniqueness properties including the topology are not yet very well known. The uniqueness properties of black hole spacetimes in higher dimensions does not seem to have trivial generalizations of what is obtained in dimension four (like the Birkhoff or Robinson-Carter theorems, [29]). Consequently, the statement of such uniquenesses may be quite different and may contain qualitative new features in higher spacetime dimensions. Therefore, an essential step in this direction involves the study of matter fields in such spacetimes, i.e. to check the validity of the no hair theorems. In higher dimensions, there may exist solutions where variable separation for the equation of motion may be quite complicated. The situation obviously gets much more involved when we include backreaction of the matter field. Most importantly, one cannot rule out the existence of more than one solution with the same geometrical properties. All these clearly indicate that in higher dimensions we should attempt the problem in a more unified way, rather than making a case by case study. Consequently, proofs of these theorems in such spacetimes should involve a general coordinate independent set up, which we describe in the next Section for both stationary axisymmetric and static spacetimes. For static spacetimes we shall not assume any spatial symmetry (spherical symmetry, for example). We shall not also assume any particular topology. The proofs will be coordinate independent and will mainly be based on the symmetry and suitable geometrical properties of the spacetime and hence matter fields, some reasonable energy conditions and Killing identities. We note that the proof of a no hair theorem usually involves the demonstration of vanishing of a particular matter field by forming vanishing integrals of sum of positive definites, and this chief characteristic of all such proofs are the same. Though, it may be non-trivial to construct such integrals depending upon the nature of the matter field and the spacetime, including its dimensionality. An explicit example of this will be encountered in Section 4, where we shall discuss fermions in charged black hole backgrounds with arbitrary dimensions. There is a Ricci scalar term in the squared Dirac equation, and is related to the trace of the energy-momentum tensor of the Maxwell field. This trace is vanishing in four spacetime dimensions, but not in higher ones. In particular, we cannot assign a definite sign with it for dyonic black holes. Consequently, we have to manipulate the calculations by using Killing identity to give it a suitable form. Based on the set up described in the next section, we give a proof for no hair theorems for massive 2- and higher forms in stationary axisymmetric spacetimes (with or without Λ) with two commuting Killing vector fields and without ignoring backreaction in Section 3. As a corollary, a proof for general static spacetimes is also given. We note that the study of massive form fields can be particularly interesting and motivating in the context of dark matters [40, 41]. The no hair properties for massive spin1 2 fields without backreaction will be discussed in Section 4. First we shall discuss fermion zero mode solutions in general static black hole spacetimes with charge. This result is further generalized to general static electrically charged spacetimes with Λ ≤ 0 for fermions with real frequencies. Next we discuss the case of stationary axisymmetric (anti)-de Sitter spacetimes with arbitrary number of commuting Killing fields. The discussions for spin1 2 fields will be an extension of [9], where vacuum or Λ-vacuum stationary axisymmetric spacetimes with two commuting Killing fields are addressed. We shall use mostly positive signature for the metric ( -, + , + , + , . . . ), and set 8 πG = c = ¯ h = 1.", "pages": [ 1, 2 ] }, { "title": "2 The geometrical constructions and assumptions", "content": "Let us start with an outline of the geometry we shall work in and derive some useful expressions. We assume that the spacetime is an n -dimensional smooth manifold with a Lorentzian metric g ab , and satisfies Einstein's equations, and there is no naked curvature singularity anywhere in our region of interest. This means that invariants constructed from the curvature and energy-momentum tensors are bounded everywhere in our region of interest. We assume that the spacetime connection is torsion-free. We assume that any backreacting classical matter energy-momentum tensor satisfies the weak and null energy conditions, i.e. for any two timelike and null vector fields t a and n a , we have T ab t a t b ≥ 0 and T ab n a n b ≥ 0. We also assume that for any two future directed timelike vector fields t a 1 and t a 2 , the quantity T ab t a 1 t b 2 ≥ 0 [42]. This means that the energy density measured by any future directed timelike observer corresponding to any future directed energy current must be positive definite. Interestingly, such energy condition implies that 'sufficiently' small bodies in general relativity move along timelike geodesics (see [42] and references therein). It is also easy to see that this energy condition can in fact be related to the dominant energy condition : for any future directed timelike vector field t a 1 , -T ab t a 1 is non-spacelike. For static spacetimes in arbitrary dimensions, there exists by definition a timelike Killing vector field orthogonal to a family of spacelike hypersurfaces, Σ. We do not need to assume any spatial symmetry for Σ. The case of stationary axisymmetric spacetimes is more complicated which we will describe below. The degree of complications depend on the number of axisymmetric Killing fields nonorthogonal to the timelike Killing field ξ a . Let us start with an n -dimensional stationary axisymmetric spacetime having three commuting Killing fields ( ξ a , φ a , φ 1 a ), respectively generating stationarity and axisymmetries, and non-orthogonal to each other. Any spatial isometry orthogonal to ξ a may be present, but will not complicate the calculations. The generalization to higher number of non-orthogonal Killing fields will be clear from the following discussions. It will be a generalization of [43] (see also references therein), in which details for such spacetimes with two commuting Killing fields can be found. We have We assume that the ( n -3)-dimensional spacelike surfaces orthogonal to ( ξ a , φ a , φ (1) a ) form integral submanifolds, which implies [44], For convenience, we construct a set of basis vectors ( χ a , φ a , φ a ) as so that χ a , φ a and ˜ φ a are orthogonal to each other everywhere. This requirement fixes the functions α i ( x ) and λ ( x ). Let the norms of ( ξ a , φ a , ˜ φ a ) be ( -λ ' 2 , f 2 , ˜ f 2 ) respectively. Using the second of the above equations into the first, we rewrite the basis as ˜ ˜ ˜ so that α = -ξ · φ f 2 , ˜ α = -ξ · ˜ φ ˜ f 2 , and λ = -φ 1 · φ f 2 . We find so that χ a is timelike when β 2 ≥ 0. The price we have paid doing this orthogonalization is that, ˜ φ a and χ a are not Killing fields, It is easy to see using the commutativity of the Killing fields that and ˜ In terms of our new basis the integrability conditions (2) become which permit solutions of the form Contracting the first of Eq.s (10) by χ a φ b , using Eq.s (7), the orthogonality of χ a , φ a , and ˜ φ a , keeping in mind that µ ia 's are orthogonal to ( χ a , φ a , ˜ φ a ), and the Killing equation for φ a gives ν 1 ( x ) = 0. Similarly, contraction with χ a ˜ φ b and φ a ˜ φ b and use of Eq.s (6), (7), (8) and the orthogonalities give ν 2 ( x ) = 0 = ν 3 ( x ) respectively. Similarly we find that all the other ν i ( x )'s vanish identically. where µ ia are 1-forms orthogonal to χ a , φ a and ˜ φ a and ν i ( x ) are functions which we have to determine for our purpose. Let us now determine the 1-forms µ ia . Contracting the first of Eq.s (10) by χ a , using Eq.s (6), (7), and the orthogonality between χ a , φ a , ˜ φ a , we find µ 1 b = 2 β -1 ∇ b β . Contracting the equation with φ b gives by the second of Eq.s (8). Next we contract the equation with φ b to find by the last of Eq.s (8). Putting these all in together we find which implies χ [ a ∇ b χ c ] = 0, and hence χ a is orthogonal to the family of ( n -1)-dimensional spacelike hypersurfaces, say Σ, which contain φ a and ˜ φ a . Eq. (13) and the last of Eq.s (6) give an useful expression, Similarly we can solve for ∇ a φ b and ∇ a ˜ φ b , but we do not need their explicit expressions for our present purpose. Let us consider a 1-form µ a on Σ, On any β 2 = 0 hypersurface H where κ is a function on H . The above equation follows from Eq. (13), It is clear from Eq. (15) that µ a coincides with χ a and becomes null on H . It is easy to see using the torsion-free condition that £ χ κ = 0. A Killing or true horizon of this spacetime is any β 2 = 0 null hypersurface H . This requires a proof, which is the following. Let us write χ a in terms of the Killing fields, χ a = ξ a + α 1 φ a + α 2 φ 1 a (Eq. (3)). Let τ be the parameter along χ a , i.e. χ a ∇ a τ := 1. Let c be a constant along χ a and we define a 1-form k a = e -cτ χ a . We compute using χ a φ a = 0 = χ a φ 1 a , the fact that £ χ α 1 = 0 = £ χ α 2 (follow from the commutativity of Killig fields), We further compute Now let ˆ h ab be the induced metric on the (n-2)-dimensional hypersurface orthogonal to both µ a and χ a , where f -2 1 is the norm of φ 1 a , and f 2 12 = φ a φ 1 a , and ˆ h ' ab is the induced metric on the remaining (n-4)-dimensional spacelike surfaces, orthogonal to both φ a and φ 1 a . We contract Eq. (19) with ˆ h bc , Using ∇ ( b χ c ) = φ ( b ∇ c ) α 1 + φ 1 ( b ∇ c ) α 2 , and the fact that £ φ α (1 , 2) = 0 = £ φ 1 α (1 , 2) , we get /negationslash Next we contract Eq. (19) with the combination : ζ [ bc ] = ( φ [ b φ 1 c ] + ∑ n -4 i =1 φ [ b X c ] i + ∑ n -4 i =1 φ [1 b X c ] i + ∑ n -4 i,j =1 ,i = j X [ b i X c ] j ) , where X a i | n -4 i =1 are basis vectors of ˆ h ' ab in Eq. (20), we find using Eq. (14) Next we contract Eq. (19) with ζ ( bc ) = ( φ ( b φ 1 c ) + ∑ n -4 i =1 φ ( b X c ) i + ∑ n -4 i =1 φ (1 b X c ) i + ∑ n -4 i,j =1 X ( b i X c ) j ) to find Let us now consider the β 2 = 0 surface H . Following [44], we shall now construct a null geodesic congruence on H . If we choose c = κ on H , Eq.s (16), (18) show that the vector field k a is a null geodesic on H . The Raychaudhuri equation for the null geodesic congruence k a reads [44] where s is an affine parameter, and θ , σ ab and ω ab are respectively the expansion, shear and rotation of the congruence given by where all the derivatives are taken on the spacelike ( n -2)-plane orthogonal to χ a or µ a on H . Eq.s (22), (23) show θ = 0 = ω ab on H for the null geodesic congruence k a . Then using Eq.s (24), (26), the Einstein equations R ab -1 2 -n [ T -2Λ] g ab = T ab into Eq. (25), we find since we have assumed that any backreacting matter energy-momentum tensor satisfies the null energy condition. The left hand side is a spacelike inner product and hence must be positive definite. Therefore the left hand side must vanish on H to avoid any contradiction. We also note that on H , χ a coincides with ∇ a β 2 , and £ χ α 1 = 0 = £ χ α 2 . All these suggest that α 1 and α 2 are constants on any β 2 = 0 hypersurface, so that χ a becomes a null Killing field there and hence any null hypersurface H is a Killing horizon of the stationary axisymmetric geometry we are considering. Then following similar steps as in four spacetime dimensions [44], we can show that κ is a constant on H . For spin1 2 fields we shall take µ a = ∇ a β 2 to be one of the basis vectors on Σ. It is clear from this choice (Eq.s (15), (16)) that our calculations for such fields will be valid for non-extremal or near-extremal solutions ( κ = 0), but not for the strictly extremal κ = 0 case. /negationslash The projector h a b which projects tensors onto the spacelike hypersurfaces Σ is given by Let D a be the spacelike induced derivative : D a ≡ h a b ∇ b . We have [44] where T is tangent to Σ, T a 1 a 2 ··· b 1 b 2 ··· := h a 1 c 1 · · · h b 1 d 1 · · · T c 1 c 2 ··· d 1 d 2 ··· . Now it is clear that we can generalize the above calculations by adding more commuting Killing fields non-orthogonal to ξ a . For example, for four commuting non-orthogonal Killing fields ( ξ a , φ a , φ 1 a , φ 2 a ), we will have χ a = ξ a + αφ a + α 2 φ 1 a + α 3 φ 2 a . Next we can orthogonalize the axisymmetric Killing fields to write the analogous form of Eq.s (4). The integrability conditions (2) or (9) now involves four vector fields and we can solve them as earlier. Thus the process goes on for higher number of Killing fields. For two commuting Killing vector fields ξ a , and φ a , we have χ a = ξ a + αφ a , with α = -ξ · φ φ · φ . Eq. (14) in this case becomes [43] We shall also require the following expression for two commuting Killing fields: We shall also require the projector in this case onto the integral ( n -2)-planes (say Σ) orthogonal to both χ a , φ a We shall denote the induced connection on Σ by D , defined similarly as what we did for Σ For the cosmological constant to be vanishing or negative, we assume the spacetime to be respectively asymptotically flat or anti-de Sitter. For Λ > 0, we shall assume the existence of a de Sitter Killing horizon (with β 2 = 0) surrounding the black hole horizon. Apart from the existence of the cosmological horizon as an outer boundary and regularity, no precise asymptotics on spacetime or matter fields will be imposed for the de Sitter case. We assume that any physical matter field, or any observable concerning the matter field also obeys the symmetries of the spacetime, be it continuous or discrete [5, 6, 45]. Thus if X is a physical matter field or a component of it, or an observable quantity associated with it, we must have its Lie derivative vanishing along a Killing field. Likewise, if the spacetime has any discrete symmetry, we shall assume any physical matter field obeys the symmetry. For static spacetimes we have a time reversal symmetry ξ a →-ξ a , whereas for stationary axisymmetric spacetimes with two commuting Killing fields have symmetry under the simultaneous reflections ξ a →-ξ a and φ a →-φ a . As we have seen above that the classical energy conditions play crucial role in constructing the geometry, unlike form fields, we shall ignore backreaction of the spinors on the spacetime since spinors do not obey any classical energy condition [29, 46]. We shall also assume for the spin1 2 case following [5, 6] that the Compton wavelength of the massive field is much small compared to the length scale of the black hole horizon. We note that this is not a strong assumption, since if we have a spinor having Compton wavelength comparable to the black hole horizon size, the assumption of negligible backreaction may be invalidated. This completes the necessary geometrical set up and clarifies all assumptions, and we shall now go into the proofs.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3 Massive forms", "content": "We shall start with a free theory of massive 2-form field B ab with 3-form field strength H abc , The equation of motion for the B field reads We shall consider this theory in a stationary axisymmetric spacetime with two commuting nonorthogonal Killing fields ξ a and φ a . An explicit example with Λ = 0 of such an n -dimensional spacetime can be found in [47]. We have by symmetry requirement which gives where the hypersurface orthogonal timelike vector field χ a is defined in the previous section. The discrete symmetry of the spacetime under simultaneous reflections ξ a → -ξ a and φ a → -φ a should also be obeyed by any physical matter field. Since the above simultaneous reflections imply χ a →-χ a , we shall set any cross component of B ab along χ [ a X b ] or φ [ a X b ] , for any X a orthogonal to both χ a and φ a , to zero. For static spacetimes this statement will concern only the time-space cross components, as there is in general only time reversal symmetry. We start with the component Ψ = ( βf ) -1 χ a φ b B ab . Contracting Eq. (34) with χ b φ c , using Eq.s (30), (31) we find where we have defined e a = ( βf ) -1 χ b φ c H abc . It is clear that e a χ a = 0 = e a φ a . This, along with the symmetry requirement and the commutativity of the Killing fields give Since ∇ a β and ∇ a f are orthogonal to both χ a and φ a , and so is e a , we shall write the above equation on the spacelike ( n -2)-submanifolds, Σ using Eq. (32). We have, where D is the induced connection on Σ. This equation can be simplified using orthogonalities between e a , χ a and φ a , and the Lie derivatives. We find after some calculations a very simple looking equation, We also find using Eq. (36), Using £ φ α = 0 = £ φ ( χ b B bc ) , the above equation further simplifies to We now multiply Eq. (40) with βf Ψ and use the above equation to get which we integrate in the exterior of the black hole horizon. The total divergence can be converted to a surface integral at the boundaries. For Λ ≤ 0, the boundaries are black hole horizon ( β = 0) and the spatial infinity, where we impose sufficient fall-off condition on the matter field, whereas for the de Sitter case the outer boundary is the de Sitter or cosmological horizon ( β = 0). In any case, the surface integrals go away and we are left with a vanishing integral of positive definites, which shows Ψ = 0 = e a . We shall use this result along with the symmetry arguments to show that the remaining components are vanishing too. By the requirement of discrete symmetry, there is no other component of B ab which can be directed along χ a : B ab χ a = 0, and hence we have to deal with only purely spatial part of B ab . We note that for purely spatial B ab , By antisymmetry, the quantity χ a H abc is purely spacelike. Therefore, since B ab is antisymmetric the free index in φ a B ab must be purely spatial and orthogonal to φ b . But these components are ruled out by the discrete symmetry. Thus χ a H abc = 0 and hence H abc is purely spatial. We now project Eq. (34) with the help of Eq. (28) onto Σ. We find after some algebra, which we contract with B bc and rewrite as We integrate this equation as before, and get that all the spatial part of B ab and H abc are vanishing. This completes the no hair proof for massive 2-form fields for stationary axisymmetric spacetimes endowed with two commuting Killing fields. We shall now generalize this result for higher form fields in an analogous manner. Let us consider a free massive 3-form B abc with 4-form field strength H abcd , with equation of motion with the totally antisymmetric definition and the symmetry conditions The requirements from the discrete symmetry applies as well. Contracting Eq. (47) with χ c φ d , and using Eq.s (30), (31), we get where we have defined βfF ab = H abcd χ c φ d , and βfe a = B abc χ b φ c . Antisymmetries guarantee that F ab and e a are orthogonal to both χ a and φ a . The above is the analogue of Eq. (37). Since all the tensors appearing in Eq. (50) are tangent to Σ, we shall project it as earlier to get We also have, using Eq.s (48), (49), where in the last equality we have used orthogonalities e a χ a = 0 = e a φ a as well. Contracting Eq. (51) with βfe b we get which we integrate as earlier to get e a = 0 = F ab throughout. By the discrete symmetry, the remaining components of B abc must be purely spatial. Then we can show as earlier that H abcd is purely spatial. Then we may project Eq. (47) onto Σ to get which we contract with B bcd and integrate by parts to find all the remaining purely spatial components of B and H to be vanishing. The process goes on for higher free massive form fields and hence it proves the desired no hair result for general stationary axisymmetric spacetimes with two commuting Killing fields. For static spacetimes of arbitrary dimensions, the hypersurface orthogonal timelike vector field χ a coincides with the Killing field ξ a . In this case the various Lie derivatives of the matter fields involve ξ a only. Since there is a time reversal symmetry, we set all the space-time cross components to be zero, i.e. a massive p -form B ab... is purely spatial. This, along with £ ξ B abc... = 0 implies the ( p +1)-form field strength H = dB is also purely spatial. This leads to equation like (54) in this case from which the no hair result follows. We note that we do not need to use any symmetry other than ξ a . Hence for static spacetimes, this result is valid irrespective of any spatial symmetry. We were unable to generalize the forgoing results for more than two commuting Killing fields, as we could not handle the resulting equations to put them in nice forms from which something meaningful can be extracted.", "pages": [ 7, 8, 9 ] }, { "title": "4 Massive spin1 2 fields", "content": "Let us now come to the massive spin1 2 case. We shall assume that the probability density Ψ † Ψ associated with a spinor Ψ and its derivative is bounded on the horizon (or horizons for de Sitter). We also assume that the norm of the conserved current j a = Ψ γ a Ψ is bounded there. Since we are working with mostly positive metric signature, the anti-commutation for γ -matrices is where g ab is the spacetime metric with mostly positive signature. The matrix γ 0 is Hermitian, whereas all the spatial γ 's are anti-Hermitian. Let us first consider static black hole spacetimes endowed with a magnetic charge and purely magnetic field. In this case we shall investigate only the so called zero-energy solutions. We work in a gauge in which the gauge field A b is purely spatial. We note that this is in general not possible for stationary axisymmetric spacetimes. The equation of motion is iγ a ̂ ∇ a Ψ -m Ψ = 0 , i ̂ ∇ a Ψ γ a + m Ψ = 0 , (56) where ' ̂ ∇ ' is the gauge-spin covariant derivative : ̂ ∇ a Ψ = ∇ a Ψ -ieA a Ψ, and ' ∇ ' is the usual spin covariant derivative. The constant ' e ' is the charge of the spinor. 'Squaring' the first of Eq.s (56) we have where F ab is the electromagnetic field strength, and R is the Ricci scalar. Taking the Hermitian conjugate of the above equation, we compute where i, j denote purely spatial indices. We shall now write the above equation in terms of the spacelike derivative operator D a , using Eq. (28). By computing h ab ̂ ∇ a ̂ ∇ b Ψ and using Eq. (58), and noting that the hypersurface orthogonal vector field χ a coincides with the Killing field ξ a in this case, we find after some algebra where 'H.c.' denotes Hermitian conjugate, and we have used the fact that when a derivative acts on Ψ † Ψ, the gauge connection vanishes, and A a is purely spatial : A a ξ a = 0. We shall now simplify Eq. (59) using the Lie derivative of spinors [48]. Since in this case we are only investigating zero modes, the spinor Ψ has no explicit dependence on the parameter along ξ a , which means [9, 48] which gives the expression for ξ a ∇ a Ψ. Using this in Eq. (59), and using Eq. (14) (with ˜ α = 0 = α ) or Eq. (30) (with α = 0), and Eq. (55) we find after some algebra which we multiply with β to write as We note that the quantity Ψ † Ψ is not tangent to Σ, but is the timelike component of the vector Ψ γ a Ψ. Also, there can be summation on timelike index in the spin connection ω abc γ b γ c associated with D a (although ' a ' is spacelike). Clearly, unlike tensors, now there is no natural way to project the entire derivative onto Σ. Therefore, the derivative operator ' D ' appearing in the above equations should be interpreted as spacelike directional spin-covariant derivative associated with the full spacetime metric, as D a acts on quantities not necessarily tangent to Σ. Accordingly, when we integrate, we shall use the full invariant volume measure [ dX ]. We next consider the Killing identity for ξ b , which we contract with ξ b , use Eq. (14) (with α = 0 = α ) or Eq. (30) (with α = 0), to find which we multiply with Ψ † Ψ and rewrite as We now integrate the above equation using full spacetime volume element [ dX ]. Since the 1-form ∇ a β 2 satisfies Frobenius condition and hence hypersurface orthogonal, the total divergence can be converted into surface integrals on the horizon and infinity or on the two horizons (for the de Sitter), all of which are β 2 = constant hypersurfaces. We recall from Section 2 that one of the basis tangent to Σ is µ a = ∇ a β 2 , whose norm vanishes on the horizon(s) as O ( β 2 ), Eq.s (15), (16). Now the surface integral at horizon(s) looks like ∫ H Ψ † Ψ ∇ a β 2 ds a , where ds a in the (n-1)-dimensional volume element, with the unit normal directing along µ a . Since the volume element on the horizon contains a β , and we have assumed the quantity Ψ † Ψ is bounded on the horizons, the above surface integral is bounded there, and it contains µ a ∇ a β 2 . But this is vanishing on the horizon(s), so the integral on horizon(s) vanish. For Λ ≤ 0, we impose sufficiently rapid fall-off on Ψ at infinity, so that it vanishes there too, leaving us with the vanishing volume integral of the right hand side of Eq. (65). We now combine this with the integral of Eq. (62), recalling ∇ a β = D a β , we find after some rearrangement and using Einstein's equations R ab -1 2 Rg ab +Λ g ab = T ab , we have Since the index ' a ' is spatial in the above equation, the first and the last terms are positive definite, whereas the third term is positive definite since the Maxwell field obeys weak and null energy conditions and fermion's backreaction has been ignored. Let us now examine the second term which contains the so called (Hermitian) anomalous correction to the fermion mass. Since the black hole is the source of the magnetic charge, the magnetic field should decrease with distance from the black hole horizon. If Q is the magnetic charge of the black hole, we must have in our units the quantity eQ to obey some certain smallness conditions, otherwise since A a contains a Q , the term corresponding to e Ψ γ a Ψ A a would backreact into the energy-momentum tensor. Also as we have discussed at the end of Section 2, for backreactionless massive fields the Compton wavelength ( ∼ m -1 ) of the field is small compared to the black hole horizon size. Since the anomalous term has dimensions ∼ length -2 , it is clear that the mass term should dominate it everywhere outside the black hole horizon. For Λ < 0, the mass term would dominate the Λ term too, since the AdS length scale should obviously be larger than the black hole length scale. Putting these all in together, we find that a magnetic static black hole spacetime of arbitrary dimension cannot support fermion zero modes in its exterior, i.e. Ψ = 0, provided we can ignore fermion's beackreation. An immediate corollary of the above result for neutral fermions is obtained by setting e = 0, for black holes with electric and/or magnetic (non-)Abelian charge(s). We note that unless we used the Killing identity for ξ b , we could not have obtained this conclusion. We shall now consider electrically charged static black holes, assuming there is only electric field. From now on we do not need to confine to the zero modes only. Following [9], we define a 2-form S ab from the conserved current 1-form j a = Ψ γ a Ψ, so that using ∇ a j a = 0. After using Einstein's equations in n -dimensions, each component λ of the above equation becomes where T ab does not contain the fermion contribution as earlier. We integrate the above equation between the black hole horizon and infinity (or the cosmological horizon for de Sitter). The total divergences can be converted into surface integrals and on the horizon(s) have the form where as before the direction ' a ' corresponds to a unit vector along the basis µ a = ∇ a β 2 , which becomes null and coincides with ξ a on H . For Λ ≤ 0, imposing suitable fall-off at infinity makes the surface integral vanishing there. We have by the symmetry requirement, which gives after using Eq. (30) (with α = 0 for static case) and set a = 0. The first term goes away (since ξ a ∇ a β = 0 , everywhere), and we are left with the second term only, and ξ 0 can be taken as -β 2 . Now we can evaluate the directional covariant derivative in the second of Eq. (71) on the horizon(s), where µ a = ∇ a β 2 coincides with ξ a and its norm vanishes as O ( β 2 ) there. We recall our assumption that both Ψ † Ψ and j a j a are bounded on the horizon(s), which implies β -1 j a ∇ a β 2 is also bounded on the horizon(s). Thus it is clear that ξ b ∇ b j λ | λ =0 is vanishing as at least O ( β ) on the horizon(s). Thus the second integral in Eq. (71) is vanishing. Another way to see this is to integrate the conservation equation ∇ a j a = 0, and convert it to surface integrals on the horizon and infinity (or on horizons for de Sitter), and since the surface integrand is j a µ a , the result follows from comparison with Eq. (73) with a = 0. On the other hand, since S ab is antisymmetric in its indices, and µ a coincides with ξ a on the horizon(s), we have S 0 bµ b = 0 there. Putting these all in together, we see that setting λ = 0 in Eq. (70) means We next note that j a can never be spacelike. Although this is obvious intuitively, but can be proven as the following. If possible we assume that j a is spacelike in some region of the spacetime. We erect a local Lorentz frame at some point P in this region and rotate it to one of the spatial axis of this frame to coincide with j a . But this will mean the 'time' component of j a to be vanishing identically, which means Ψ = 0. This is clearly a contradiction, and hence j a must be non-spacelike and future directed. In particular, since we are dealing with massive fields, it must be timelike. Then the energy condition discussed at the beginning of Section 2 guarantees that the first term in Eq. (74) is positive. The third term is positive or zero if Λ ≤ 0, whereas for Maxwell field T = ( 1 -n 4 ) F ab F ab ≥ 0 if there are only electric fields. This shows that for Λ ≤ 0, there can be no Dirac hair for static electrically charged spacetimes endowed only with electric fields, provided we can ignore backreaction of spinors and Eq. (72) is satisfied. This of course include real frequency solutions, when ξ a is a coordinate vector field. What happens if we are working in four spacetime dimensions with Λ = 0? Eq. (74) then only contains the first term, which is positive definite. Hence we must have T a 0 j a = 0 throughout. We next decompose T a 0 along j a and orthogonal to it. Since F ab is non-vanishing, we must have j a j a = 0 throughout. But j a is future directed timelike, so this is a contradiction. Therefore we must have j a = 0, which means Ψ = 0 throughout. We were unable to find an analogous proof for Λ > 0. Perhaps there are some additional conditions or identities which should be used (as we used Killing identity in the previous part), but we were unable to find any. We shall conclude this Section by noting the following for Λ-vacuum (positive or negative) stationary axisymmetric spacetimes in arbitrary dimensions with arbitrary number of commuting Killing fields. Let us first consider a stationary axisymmetric spacetime with three commuting Killing fields as discussed in Section 2. The symmetry requirement in this case becomes The hypersurface orthogonal vector field χ a is given by Eq. (3). Then using Eq.s (75), (14) we find in place of Eq. (73) which is formally the same as the static case. Consequently by our choice of basis we arrive at Eq. (74) with only the third term. This guarantees Ψ = 0 throughout. Since it is clear that for an arbitrary stationary axisymmetric spacetime with commuting but non-orthogonal Killing fields { ξ, φ, φ 1 , φ 2 , . . . } with integral spacelike submanifolds orthogonal to these Killing fields, the hypersurface orthogonal timelike vector field χ a can be constructed from their linear combinations and the Killing horizon(s) can be specified, we conclude that for stationary axisymmetric (anti)-de Sitter spacetimes falling into the category we discussed in Section 2, there can be no backreactionless Dirac hair with real phases.", "pages": [ 9, 10, 11, 12, 13 ] }, { "title": "5 Discussions", "content": "It is time to summarize the various results we obtained in this paper. Using the necessary geometrical set up and assumptions described in Section 2, we investigated no hair properties of general stationary axisymmetric and static black hole spacetimes. In Section 3, we demonstrated the no hair proof for massive 2- and higher forms. In the next Section we discussed the case of massive spin1 2 fields. Apart from symmetries, energy conditions, and regularities, we have not used any particular functional form of the metric or matter fields, the reason is the so far not very well understood uniqueness nature of black hole spacetimes in higher dimensions, with or without Λ. We did not have to perform any complicated variable separations, which even in four spacetime dimensions is a formidable task. We note that our results are also valid even if we are not dealing with an exact solution of Einstein's equations. As long as the spacetime falls into the category described in Section 2, our calculations apply. An example of this would be the axisymmetric static spacetime constructed in [29]. An exact solution of stationary axisymmetric black hole spacetime with three commuting Killing vector fields can be found in [49]. Our analysis is valid for spacetimes with Killing horizons like black string spacetimes (see e.g. [50, 51]), for black holes with toroidal topology [52], and as well as for the black rings [53]. Our result is also valid for multi-black hole spacetimes described in e.g. [29], which is not spherically symmetric, and not necessarily be axisymmetric as well, or other multi horizon black hole spacetimes in higher dimensions (see [54] for a vast review and list of references), only we have to replace the inner boundary integral with the sum of integrals on all the black hole horizons, since we have considered the horizon(s) in a purely geometric way as β 2 = 0 null hypersurface(s). It remains as an interesting task to further generalize the massive spin1 2 result for arbitrary stationary axisymmetric spacetimes carrying electric or magnetic charge.", "pages": [ 13, 14 ] }, { "title": "Acknowledgment", "content": "I thank Amitabha Lahiri for useful discussions.", "pages": [ 14 ] } ]

[ { "title": "Finite size source effects and the correlation of neutrino transition probabilities through supernova turbulence", "content": "Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA James P. Kneller ∗ and Alex W. Mauney † (Dated: October 8, 2018) The transition probabilities describing the evolution of a neutrino with a given energy along some ray through a turbulent supernova are random variates unique to each ray. If the source of the neutrinos were a point then all neutrinos of a given energy and emitted at the same time which were detected in some far off location would have seen the same turbulent profile therefore their transition probabilities would be exactly correlated and would not form a representative sample of the underlying parent transition probability distributions. But if the source has a finite size then the profiles seen by neutrinos emitted from different points at the source will have seen different turbulence and the correlation of the transition probabilities will be reduced. In this paper we study the correlation of the neutrino transition probabilities through turbulent supernova profiles as a function of the separation δx between the emission points using an isotropic and an anisotropic power spectrum for the random field used to model the turbulence. We find that if we use an isotropic power spectrum for the random field, the correlation of the high (H) density resonance mixing channel transition probability is significant, greater than 0.5, for emission separations of δx = 10 km, typical of proto neutron star radii, only when the turbulence amplitude is less than C /star ∼ 10%; at larger amplitudes the correlation in this channel drops close to zero for this same separation of δx = 10 km. In contrast, there is significant correlation in the low (L) density resonant and non-resonant channels even for turbulence amplitudes as high as 50%. Switching to anisotropic spectra requires the introduction of an 'isotropy' parameter k I whose inverse defines the scale below which the field is isotropic. We find the correlation of all transition probabilities, especially the H resonance channel, strongly depends upon the choice of k I relative to the long wavelength radial cutoff k /star . The spectral features in the H resonance mixing channel of the next Galactic supernova neutrino burst may be strongly obscured by large amplitude turbulence when it enters the signal due to the finite size of the source while the presence of features in the L and non resonant mixing channels may persist, the exact amount depending upon the degree of anisotropy of the turbulence. PACS numbers: 47.27.-i,14.60.Pq,97.60.Bw", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The neutrino signal from the next core-collapse supernova in our Galaxy will give us an unprecedented opportunity to peer into the heart of an exploding star and confront our current paradigm of how these stars explode with observations. But decoding the message will be no easy task because the neutrino signal will have experienced so many flavor-changing events on the trip from proto-neutron star to our detectors that scramble the information, see for example Kneller, McLaughlin & Brockman [1] and Lund & Kneller [2]. The first flavor changing effect the signal experiences is due to neutrino self interactions / collective effects in the region up to ∼ 1000 km above the proto neutron star [3-19] followed by the Mikheyev, Smirnov & Wolfenstein (MSW) [20, 21] effect which is complicated in supernovae by the impression of the shockwave racing through the stellar mantle [1, 2, 22-27]. Turbulence in the mantle, seeded during the earlier neutrino heating/Standing Accretion Shock Instability phase [28-35], also needs to be included usually by modelling [12, 36-39]. Finally, there is the possibility of Earth matter effects leaving an imprint in the signal though a recent study expects this effect to be minimal [40]. What makes decoding the signal even more of a challenge is that the neutrinos we receive at a given instant and with a given energy will not have experienced the same flavor evolutionary history. The neutrinos arriving at a detector will have been emitted from different locations at the source and both the neutrino collective and the MSW+turbulence effects will vary from trajectory to trajectory. Starting with Duan et al. [5], the self interaction effects in calculations where the neutrino emission over the source is assumed to be spherically symmetric have been seen to be 'angle dependent' in the sense that a neutrino following a pure radial trajectory differs from one emitted at an angle relative to the normal. Presumably allowing for aspherical source emission would only make the trajectory dependence even stronger. Similarly the MSW plus turbulence effects are also trajectory dependent. If we temporarily cast aside the turbulence and focus on the gross structure of the explosion i.e. the lowest angular multipole moments, an aspherical passage of the shock through the star, by itself, leads to a line-ofsight dependence. But, one must recall that we will not observe the neutrinos from a supernova at widely differ- ent lines of sight, all our detectors are here on Earth. The size of the source is of order the proto-neutron star radius, i.e. ∼ 10 km while the shock effects show up in the signal when the shock has propagated out to r ∼ 10 4 km. As long as the curvature of the shock is over a lengthscale greater than the source size the neutrinos which appear in our detectors will all have seen essentially the same profile. When one re-inserts the turbulence into the profile, one realizes this approximation may no longer be valid because turbulence extends to much smaller lengthscales even when the shock is far out in the stellar mantle. The density profile along two, essentially parallel lines of sight to a distant detector separated by ∼ 10 km will no longer be negligibly dissimilar and one must consider how the dissimilarity of the profiles propagates to the neutrinos. Any correlation will lead to a potential new feature of the neutrino signal. It has been shown that the transition probabilities for a single neutrino - the set of probabilities that relates the initial state to the state after passing through the supernova - is not unique when turbulence is inserted into a profile: it will depend upon the exact turbulence pattern seen by the neutrino as it travelled through the supernova [38, 39, 41]. Those transition probabilities are drawn from distributions whose properties will depend upon the stage of the explosion, the character of the turbulence, and the neutrino energy and mixing parameters. If the coherence of two neutrinos emitted at the same time and with the same energy but from different locations is small then the final states are uncorrelated and one would expect that the flux at a detector would just be the mean of whatever distribution describes the transition probabilities multiplied by the initial spectra. But if the coherence is high then all the neutrinos will have the same set of transition probabilities which one might expect to 'scintillate' together as the turbulence evolves. Of course, this ignores the issue of energy resolution and temporal binning of the signal that becomes necessary because of the limited statistics. The purpose of this paper is to consider the issue of finite source size and the correlation of the neutrino transition probabilities along parallel trajectories through turbulent supernova profiles. Our calculations expand upon the work of Kneller & Volpe [39] and Kneller & Mauney [41] upon which we rely heavily for the techniques used to calculate the turbulence effects and as context for our results. We begin by describing the calculations we undertook paying particular attention to the construction of the random fields used to model the turbulence. The basic approach to determining the effects of turbulence are then demonstrated, followed by the computation of the transition probability correlation as a function of the separation between the emission points. We finish by summarizing our findings and discuss the implications for the Galactic neutrino burst signal.", "pages": [ 1, 2 ] }, { "title": "II. DESCRIPTION OF THE CALCULATIONS", "content": "The neutrino transition probabilities are the set of probabilities of measuring some neutrino state ν i given an initial neutrino state ν j i.e P ( ν j → ν i ) = P ij . We shall denote antineutrino transition probabilities by ¯ P ij . If the S -matrix relating the initial and final wavefunctions is known then these probabilities are just the square amplitudes of the elements of S . The S -matrix is calculated from the Schrodinger equation where H is the Hamiltonian. The Hamiltonian is the sum of the vacuum contribution H 0 and the MSW potential V which describes the effect of matter. The vacuum Hamiltonian is diagonal in what is known as the 'mass' basis and in this basis H 0 is defined by two mass squared differences δm 2 ij = m 2 i -m 2 j and the neutrino energy E . The mass basis is related to the flavor basis by the MakiNakagawa-Sakata-Pontecorvo [42, 43] unitary matrix U . The most common parametrization of U is in terms of three mixing angles, θ 12 , θ 13 and θ 23 , a CP phase and two Majoranna phases. The MSW potential V is diagonal in the flavor basis because matter interacts with neutrinos based on their flavor. The neutral current interaction leads to a contribution to V which is common to all flavors. This may be omitted because it leads only to a global phase which is unobservable. The charged current potential only affects the electron flavor neutrino and antineutrinos and is given by √ 2 G F n e ( r ) where G F is the Fermi constant and n e ( r ) the electron density. In matter the two contributions to H means neither the mass nor the flavor states diagonalize the matrix. But there is a basis known as the matter basis which does diagonalize H i.e. for a given value of the electron density there is a matrix ˜ U such that ˜ U † H ˜ U = K where K is the diagonal matrix of eigenvalues. When the MSW potential vanishes the matter basis becomes the mass basis up to arbitrary phases. The matter basis is the most useful for studying the evolution of neutrinos through matter because it removes the trivial adiabatic MSW transition and it will be the basis we use to report our results in this paper. We refer the reader to Kneller & McLaughlin [44] and Galais, Kneller & Volpe [14] for a more detailed description of the matter basis. We now turn our attention to the turbulent density profiles through which we shall send our neutrinos. As usual, we shall model the turbulence by multiplying a turbulence free density profile by a Gaussian random field. Since the spatial extent of the neutrino emission, of order 10 km, is much smaller than the radial location of the turbulence, of order r ∼ 10 4 -10 5 km, we shall ignore any curvature of the density profile features and use a plane-parallel model for the supernova. The z axis of our Cartesian co-ordinate system is aligned with the radial direction of the profile. The profile we adopt is from a one-dimensional hydrodynamical simulation of a supernova taken from Kneller, McLaughlin & Brockman [1]. This profile is shown in figure (1) and is the same one used in Kneller & Mauney [41]. The figure shows the presence of two shocks: the forward shock at r s and the reverse shock at r r . In multi-dimensional simulations of supernova both these shock fronts are aspherical and fluid flow through the distorted shocks leads to strong turbulence in the region between them. Our selection of this profile also determines the neutrino energy we shall use since we wish the neutrinos to have an H resonance density that does not intersect the shocks. Therefore we pick 25 MeV for the neutrino energy and the reader may observe that the two-flavor resonance density for a 25 MeV, shown in the figure, does not intersect the shocks as required. The turbulence is inserted by multiplying the profile in the region between the reverse and forward shocks by a factor 1 + F ( r ) where F ( r ) is a three-dimensional Gaussian random field with zero mean. The random field is represented by a Fourier series, that is In this equation the parameter C /star sets the amplitude of the fluctuations while the two tanh functions are included to suppress fluctuations close to the shocks and prevent discontinuities. The parameter λ is a damping scale which we set to λ = 100 km. The random part of F appears in the the second half of equation (2) because the set of co-efficients { A } and { B } are independent standard Gaussian random variates with zero mean. The k n are a set of wavenumbers and, finally, the paramaters V n are k-space volume co-efficients. The method of fixing the N k k 's, V 's, A 's and B 's for a realization of F is 'variant C' of the Randomization Method described in Kramer, Kurbanmuradov, & Sabelfeld [45] which we have generalized to three dimensions. The Randomization Method in general partitions the space of wavenumbers into N k regions and from each we select a random wavevector using the power-spectrum, E ( k ), as a probability distribution. The volume paramaters V n are the integrals of the power spectrum over each partition if the power spectrum is normalized to unity. Variant C of the Randomization Method divides the k-space so that the number of partitions per decade is uniform over N d decades starting from a cutoff scale k /star . Throughtout this paper we shall use a wavenumber cutoff k /star set to twice the distance between the shocks i.e. k /star = π/ ( r s -r r ). The logarithmic distribution of the modes increases the efficiency of the algorithm in the sense that we can use a 'small' value of N k and also the agreement between the exact statistical behavior of the field and that of an ensemble of realizations is uniform over some range of lengthscales i.e. it is scale invariant. This feature is important for our study because the oscillation wavelength of the neutrinos is constantly changing as the density evolves. The minimum lengthscale we need to cover has been shown by Friedland & Gruzinov [38] and Kneller, McLaughlin & Patton [46] to be the reduced oscillation wavelengths for the neutrinos and antineutrinos i.e. λ ij = 1 / | δk ij | and ¯ λ ij = 1 / | δ ¯ k ij | - where δk ij and δ ¯ k ij are the differences between the eigenvalues i and j of the neutrinos and antineutrinos respectively. Kneller & Mauney [41] showed the wavelengths in the turbulence region were of order 1 km or greater which is approximately four orders of magnitude smaller than the shock separation. This means we need to pick N d ≥ 4 to cover the necessary decades in k-space.", "pages": [ 2, 3 ] }, { "title": "A. The power spectrum", "content": "The final component of our calculations we have yet to discuss is the power spectrum E ( k ). In this paper we shall consider two power spectra and our first choice, due to its simplicity, is a normalized three-dimensional, isotropic inverse power-law spectrum given by for | k | ≥ k /star where | k | is the magnitude of the wavevector k . Throughout this paper we shall adopt the Kolmogorov spectrum where α = 5 / 3. The one dimensional power spectrum for the k z component of the wavevector is which differs from the one dimensional power spectrum used by Kneller & Mauney [41] because for | k z | ≥ k /star the power is suppressed by the factor 1 /α and the one dimensional spectrum is non-zero for | k z | ≤ k /star . The twopoint correlation function B ( δ r ) for this choice of a power spectrum depends only the magnitude of the separation, δr , and may be calculated analytically to be where Γ( n, x ) is the incomplete Gamma function. There is one last quantity to determine: the number of N k of elements in the sets of random wavenumbers, coefficients and volumes. To find this quantity we compare the statistical properties of an ensemble of random field realizations with the exact expressions as a function of the ratio N k /N d for a given N d . The statistical property we compute is the second order structure function G 2 ( δ r ) which is given by where δ r is the separation between two points. The function G 2 ( δ r ) is related to the two-point correlation function B ( δ r ) via G 2 ( δ r ) / 2 = 1 -B ( δ r ). For the isotropic power spectrum both G 2 and B are only functions of the magnitude of δ r and the correlation function is given above. In figure (2) we show the ratio R ( δr ) of the numerically calculated structure function for the isotropic random field to the exact solution as a function of the scale k /star δr when we use either N k = 50 wavenumbers spread over N d = 5 decades or N k = 90 wavenumbers over N d = 9 decades. The numerical calculation is the aver- 1 * age of 30 , 000 realizations of the turbulence and the error bar on each point is the error on the sample mean. The figure indicates that the method we use to generate random field realizations reproduces the analytic results for the structure function very well and with high efficiency because good agreement between the statistics of the ensemble and the exact result requires just N k /N d = 10. In fact we find even N k /N d ratios of just N k /N d ∼ 2 -3 are sufficient to give acceptable agreement but we re-assure the reader we shall stick with N k /N d = 10. But isotropic and homogeneous three-dimensional turbulence is perhaps not a realistic scenario for supernova because the gravitational potential and the general fluid flow are in the radial direction. Only on sufficiently small scales should the turbulence become isotropic. This division into large and small lengthscales indicates we should partion the power-spectrum so that for | k z | ≥ k I the spectrum is isotropic, where k I is the isotropy scale, between k /star ≤ | k z | ≤ k I the spectrum is anisotropic and then below the cutoff scale, | k z | ≤ k /star , the power spectrum should be set to zero since there should be no modes on scales larger than 1 /k /star . For | k z | ≥ k I where the spectrum is isotropic we use a power spectrum resembling equation (3) Note the additional factor of α in the numerator. For k /star ≤ | k z | ≤ k I we write the spectrum as the product E ( k x , k y , k z ) = E ( k x , k y ) × E ( k z ). The spectrum E ( k z ) is chosen to be a continuation of the inverse power-law given above while the spectrum in the xy directions, E ( k x , k y ), is the spectrum of the isotropic/homogeneous region in these directions fixed at | k z | = k I . The spectrum for k /star ≤ | k z | ≤ k I is thus The reader may verify the power spectrum defined by equations (7) and (8) is normalized. This anisotropic three-dimensional power spectrum yields a one-dimensional spectrum along the z direction given by power spectrum but if we consider the two-point structure function of the random field for points orientated along the z direction then we can compute that in this direction Compared to the isotropic spectrum above, this anisotropic spectrum differs in important ways. First, even if we set k I = k /star we observe that the lack of power in the region | k z | ≤ k /star means we have to compensate by increasing the structure / decreasing the correlation by the factor α . This increase is the reason for the appearance of the extra factor α in equation (7). Next, as we increase the ratio f I = k I /k /star , we push more and more of the structure of the field in the xy direction to ever smaller scales reducing even further the correlation of the field at some fixed non-radial separation δx compared to the isotropic case. This extra power at small scales can be seen in figure (3) which is a plot of the ratio of the one-dimensional two-point structure function in the x direction relative to the structure function along the z direction at the same separation scale for three values of f I . As promised, when f I = 1 there is an equal amount of structure in the field along both radial and * non-radial directions but as f I increases we push more and more of the structure of the field in the xy direction to smaller scales. The anisotropic power spectrum we have constructed means the turbulence along different parallel rays is less correlated than the turbulence along two rays at the same separation when the power spectrum is isotropic. If that's the case then the transition probabilities for the neutrinos travelling along those two rays should also be less correlated and below we quantify the decrease.", "pages": [ 3, 4, 5 ] }, { "title": "III. RESULTS", "content": "Now that we have the random fields to model the turbulence we are all set to generate turbulent profiles and send neutrinos and antineutrinos through them. To achieve higher efficiency we follow six neutrinos and six antineutrinos simultaneously through every realization of the turbulence with one neutrino and one antineutrino emitted at x ∈ { 0 , 10 4 , 10 5 , 10 6 , 10 7 , 10 8 } cm. Each time we generate a new realization we end up with a different set of transition probabilities so by repeating the calculation many times - in our case a minimum of one thousand times but often much larger - we can create an ensemble of transition probabilities of size N from each emission point. Once we have our ensemble we can then go ahead and compute means 〈 P ij ( x ) 〉 , variances V ij ( x ), and, of course, correlations The correlation of the antineutrino transition probabilities will be denoted as ¯ ρ ij . In the large N limit the error on the correlation is expected to be σ ρ = (1 -ρ 2 ) / √ N -1. Combining the results from the six emission points we can form fifteen separations δx so fifteen correlations but two points must be remembered: first, groups of them will cluster e.g. we will have a value for the correlation at δx = 10 km but also two more at δx = 9 km and δx = 9 . 9 km and second, these groups of transition probability correlations are themselves correlated - half the data in each correlation value is the same for all members of the cluster. Nevertheless these clusters are useful because they serve as a consistency check 1 - we should expect the results to be similar within each cluster - and also they give us an indication if the error in the results are comparable to the expected, large-N error σ ρ given above. We also need to specify the neutrino mixing parameters we have used. The hierarchy will be set to normal and we shall comment on how our results translate to the inverted hierarchy. As discussed, the neutrino energy will be fixed at E = 25 MeV, typical of supernova neutrino energies and we shall set the neutrino mixing parameters to be δm 2 12 = 8 × 10 -5 eV 2 , δm 2 23 = 3 × 10 -3 eV 2 , sin 2 2 θ 12 = 0 . 83, and sin 2 2 θ 23 = 1. The recent measurements of the last mixing angle θ 13 by T2K [47], Double Chooz [48], RENO [49] and Daya Bay [50] are all in the region of θ 13 ≈ 9 · . We shall adopt this value for the majority of this paper but this result is sufficiently new that we shall show on occasion results with multiple values of θ 13 in order to put this result in context. Finally, the turbulence amplitude C /star will be allowed to vary but we shall focus upon larger values. With the measurement of a large value of θ 13 the turbulence effects are negligible for amplitudes of order C /star ∼ 1% [41].", "pages": [ 5, 6 ] }, { "title": "A. The point source statistics", "content": "Before we show our results for the correlation of the transition probabilities as a function of the emission separation, we consider first the statistical properties of the ensembles for each emission point. In addition to being interesting in their own right and useful as a reference, these calculations allow us to test that our 3D random field generator is working properly because the ensembles for each point of emission should be consistent and independent of x . In figure (4) we show the frequency distribution of P 23 for the six emission locations x using the mixing paramaters given above, C /star = 30%, N k = 50, N d = 5 and sin 2 (2 θ 13 ) = 0 . 1 and the isotropic power spectrum. The sample size is N = 3265 for each emission point. In each panel of the figure the reader will observe that the transition probability is almost uniformly distributed - there is a slight decrease in the frequency of higher values of P 23 - but, more importantly, there is no observed trend with x . A closer inspection of figure (4) also hints at some correlation: the bottom few panels of the figure are very similar. We have reproduced this calculation for other choices of the N k and N d paramaters. The results are shown in figure (5) where we plot the mean values and standard deviations of P 11 , P 23 and P 31 for ensembles of neutrinos emitted at x = 0 as a function of the parameter N k keeping the ratio N k /N d fixed at N k /N d = 10. There is no discernable trend with N k and we march on confident that setting N k = 50 and N d = 5 does not bias our results. We now allow the values of C /star and θ 13 to float and consider both the isotropic and anisotropic power spectrum. The evolution of the transition probability means as a function of C /star for the two power spectra and two choices * * of θ 13 are shown in figure (6). There are many interesting trends discussed in detail in Kneller & Mauney [41]. Large amplitude turbulence works its way through to affect every mixing channel, not just the H resonance channel P 23 , as promised so that by C /star = 0 . 5 we observe 〈 P 12 〉 ∼ 20%, 〈 P 13 〉 ∼ 10%, 〈 P 23 〉 ∼ 50%, and 〈 ¯ P 12 〉 ∼ 20%, 〈 ¯ P 13 〉 ∼ 5%, 〈 ¯ P 23 〉 ∼ 1%. To put this in context, in the absence of turbulence all these transition probabilities are zero when θ 13 = 9 · . The only neutrino mixing channel with reasonable sensitivity to θ 13 is the H resonance channel P 23 and even then the disparity in 〈 P 23 〉 at C /star ∼ 0 . 1 disappears by C /star ∼ 0 . 3. In contrast the antineutrinos are very sensitive to θ 13 even at large turbulence amplitudes: the expectation value for P 13 varies by a factor of ∼ 2 when θ 13 is changed from sin 2 2 θ 13 = 4 × 10 -4 to sin 2 2 θ 13 = 0 . 1, ¯ P 13 and ¯ P 23 on the other hand change by ∼ 1 -2 orders of magnitude between the same limits. While these trends are interesting, the purpose of figure (6) is to compare the use of the isotropic and anisotropic power spectra. Except for the H resonance mixing channel P 23 , the isotropic power spectrum gives values of 〈 P ij 〉 which are smaller than the anisotropic spectrum. The neutrinos are more sensitive to the turbulence when the power spectrum is anisotropic because the neutrinos are sensitive to the amplitude of the turbu- e modes of order the neutrino oscillation wavelength [38, 46] which is typically in the range of ∼ 10 km in the H resonance region. The anisotropic spectrum removed all power for the fluctuations in the radial direction at the long wavelengths above 1 /k /star - which is of order 10 4 km in our calculation - and to compensate we needed to increase the power on the smaller wavelengths which means and effective increase of their amplitude. In fact we already know the exact amount the amplitude is effectively increased because we pointed out the 1 /α factor that appears in the one dimensional power spectrum in the isotropic case compared to the one-dimensional spectrum derived from the anisotropic turbulence. Thankfully, our expectations are confirmed by figure (6) because the increase of all the mixing channels except P 23 is on the expected scale of α . The isotropy scale paramater k I , which sets the scale in the radial direction below which the turbulence is isotropic, does not play a role for these point source statistics. The one-dimensional power spectrum along the radial direction is independent of the isotropy scale k I which can be seen when comparing equations (4) and (9). So if the one-dimensional power spectrum is independent of k I then the effect of switching the power spectrum from isotropic to anisotropic is solely due to the removal of radial long-wavelength fluctuations. The transition probability P 23 behaves slightly differently but is 8 entirely consistent with the understanding of the effects in the other channels. At smaller amplitudes and the smaller value of θ 13 there is no effect of the power spectrum switch upon 〈 P 23 〉 because the depolarization limit has been reached. At the larger mixing angle depolarization has not achieved and switching the power spectrum leads to the effects as seen in P 12 and P 13 . The two-flavor depolarization limit is reached for the sin 2 2 θ 13 = 0 . 1 case when C /star ∼ 30%. At around this same turbulence amplitude there begins the shift to three-flavor depolarization where 〈 P 23 〉 = 1 / 3. Whatever the mixing angle used, we see that the mean value 〈 P 23 〉 as a function of C /star using the anisotropic spectrum begins the transition at smaller C /star than the same calculation using the isotropic spectrum because of the increased amplitude of the small scale fluctuations in the former case.", "pages": [ 6, 7, 8, 9 ] }, { "title": "B. The correlation through isotropic turbulence", "content": "We now turn to the correlation of the transition probabilities as a function of the distance between the emission points and consider first the case of the isotropic power spectrum. Our result for the correlation of the transition probabilities, except ¯ P 23 , as a function of the separation δx at various values of θ 13 and turbulence amplitudes C /star is shown in figure (7). ¯ P 23 is excluded is because it is difficult to calculate its correlation reliably. What one notices immediately about the results are that ρ 12 , ρ 13 , ¯ ρ 12 and ¯ ρ 13 all show little sensitivity to either θ 13 or C /star - which is in contrast to figure (6). The reason for the lack of sensitivity of these correlations to θ 13 and C /star is explained by the exponential distributions these transition probabilities possess. Both the turbulence amplitude and the mixing angle simply 'rescale' the ensemble of transition probabilities and, as equation (11) shows, this rescaling cannot alter the correlation. One also sees that the correlation of all these transition probabilities is high, /greaterorsimilar 0 . 5, for all separations δx /lessorsimilar 100 km. In contrast the correlation of P 23 is sensitive to both θ 13 and C /star . When C /star is of order C /star ∼ 10% the sensitivity to θ 13 arises because the distributions of P 23 at the different mixing angle choices are very different: for sin 2 2 θ 13 = 4 × 10 -4 the distribution is uniform, for sin 2 2 θ 13 = 0 . 1 it is strongly skewed to small values of P 23 . As C /star increases the sensitivity disappears because the distributions at each value of θ 13 become similar: this is the same behavior seen in figure (6). Finally, for C /star = 10% the currently preferred value of θ 13 gives greater correlation at a given seperation than smaller values of θ 13 . The correlation ρ 23 is high for δx /lessorsimilar 10 km, a scale of order the proto-neutron star diameter, at for C /star = 10% and decreases rapidly as C /star increases. For C /star /greaterorsimilar 0 . 3 the transition probability P 23 of two neutrinos emitted from points on the proto-neutron star separated by a distance greater than δx /greaterorsimilar 1 km are essentially independent. ρ ρ ρ", "pages": [ 9 ] }, { "title": "C. The correlation through anisotropic turbulence", "content": "The change to the mean point source transition probabilities when switching to an anisotropic power spectrum is both understandable and measurable but, overall, the effects are small and of the order of factors of α i.e. the amplitude by which the small scale fluctuations in the anisotropic spectrum increased in amplitude compared to the isotropic spectrum. That insensitivity no longer holds when we examine the correlations of the transition probabilities because these quantities are functions of the isotropy scale paramater k I . The correlations of the transition probabilities as a function of the separation between the emission points is strongly sensitive to the amount of turbulence power in the perpendicular directions and increasing k I relative to the fixed scale k /star shifts the power from long wavelength, small k x and k y , 8 to much shorter wavelengths, as shown in figure (3). The effects of introducing and varying f I = k I /k /star are shown in figure (8) for the case C /star = 0 . 3 and sin 2 2 θ 13 = 0 . 1. For f I = 1 the difference between the isotropic and anisotropic power spectra are minimal but at larger ratios of the two scales the correlation at some given separation δx drops noticeably in all three channels though the reduction in the correlation of P 12 and P 13 is not as severe as that for the transition probability P 23 . It is still the case that the correlation of P 12 and P 13 at typical proto-neutron star radii of δx ∼ 10 km is larger than 0.5 if f I /lessorsimilar 10 for this particular mixing angle choice and turbulence amplitude. Pushing even more power to smaller scales would lead to minimal correlation of these two transition probabilities. In the case of P 23 , the correlation at δx ∼ 10 km is already small for this mixing angle and turbulence amplitude even in the isotropic and f I = 1 cases so pushing more power of the fluctuations in perpendicular directions to smaller wavelengths completely removes the correlation of P 23 over the protoneutron star radial scale. If we now vary the turbulence amplitude we generate figure (9) which is, again, for a mixing angle of sin 2 2 θ 13 = 0 . 1. Examining the results we quickly observe the same general trends with changes in f I as seen in figure (8): increasing f I reduces the correlation with ρ 23 affected to a greater degree than ρ 12 , ρ 13 , ¯ ρ 12 and ¯ ρ 13 . Likewise, the trends seen in figure (7) for changes in C /star are also reproduced.", "pages": [ 9, 10, 11 ] }, { "title": "IV. SUMMARY AND CONCLUSIONS", "content": "Supernova turbulence and its effects upon both the flavor composition of neutrinos that pass through it and their correlations depend upon many numerous parameters one needs to introduce to describe the turbulence. All affect the result and here we try to succinctly summarize our results. For a neutrino energy of 25 MeV and using a supernova density profile taken from a simulation 4 . 5 s post-bounce, we find in a normal hierarchy that the correlation of the H resonance mixing channel transition probability P 23 as a function of the emission separation δx drops considerably as C /star increases for both the cases of isotropic and anisotropic turbulence. If the turbulence amplitude is of order C /star ∼ 0 . 1 then the correlation of the transition probability P 23 for neutrinos emitted from opposite sides of the proto-neutron star, i.e. separated by ∼ 10 km, is marginal for the isotropic spectrum and for the anisotropic only when f I /lessorsimilar 10. For C /star /greaterorsimilar 0 . 3 there is essentially no correlation of the H resonance transition probabilities for neutrinos emitted from opposite sides of the proto-neutron star. At these amplitudes the turbulence along parallel trajectories separated by ∼ 10 km is just too different to permit any correlation of this transition probability in the supernova neutrino burst signal. If we switch to an inverted hierarchy then it will be the transition probability ¯ P 13 which behaves this way. In contrast, the correlation of the transition probabilities P 12 , P 13 , ¯ P 12 and ¯ P 13 in a normal hierarchy as a function of emission separation is largely independent of C /star and θ 13 . The correlation decreases as the ratio f I = k I /k /star increases but remains significant for separations of order the proto-neutron star radius even for f I ∼ 100. When switching to an inverted hierarchy the mixing channels which behave this way are P 12 , P 23 , ¯ P 12 and ¯ P 23 . These mixing channels, particularly P 12 and ¯ P 12 , are the most promising for observing flavor scintillation assuming the energy resolution of our neutrino detectors does not wash out the effect and the temporal correlation remains high.", "pages": [ 11 ] }, { "title": "Acknowledgments", "content": "This work was supported by DOE grant de-sc0006417, the Topical Collaboration in Nuclear Science 'Neutrinos and Nucleosynthesis in Hot and Dense Matter', DOE grant number de-sc0004786, and an Undergraduate Research Grant from NC State University.", "pages": [ 11 ] } ]

[ { "title": "Superradiance and statistical entropy of hairy black hole in three dimensions", "content": "Myungseok Eune, 1, ∗ Yongwan Gim, 2, † and Wontae Kim 1, 2, 3, ‡ 1 Research Institute for Basic Science, Sogang University, Seoul, 121-742, Republic of Korea 2 Department of Physics, Sogang University, Seoul 121-742, Republic of Korea 3 Center for Quantum Spacetime, Sogang University, Seoul 121-742, Republic of Korea (Dated: February 23, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We calculate the statistical entropy of a rotating hairy black hole by taking into account superradiant modes in the brick wall method. The UV cutoff is independent of the gravitational hair, which gives the well-defined area law of the entropy. It can be shown that the angular momentum and the energy of matter field depend on the gravitational hair. For the vanishing gravitational hair, it turns out that the energy for matter is related to both the black hole mass and the black hole angular momentum whereas the angular momentum for matter field is directly proportional to the angular momentum of the black hole. PACS numbers: 04.70.Dy, 04.50.Kd Keywords: Black Hole, Thermodynamics, Modified Gravity", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "There has been much attention to three-dimensional topological massive gravity [1, 2] because it has rich structures even though it is lower dimensional gravity. Recently, new massive gravity [3, 4] has been also intensively studied, in particular, it can be shown that there is a new type of a rotating black hole solution apart from the rotating BanadosTeitelboim-Zanelli (BTZ) black hole [5]. A new rotating black hole has three hairs: two of them are mass and angular momentum and the other corresponds to the gravitational hair [6-9]. On the other hand, the statistical origin of the entropy for black holes has been studied in terms of the brick wall method [10]. Subsequently, there have been extensive applications of the brick wall method to various black holes [11-13]. In connection with the rotating hairy black hole, there exists a superradiant mode so that the brick wall method is nontrivial, and it should be treated carefully [14]. Moreover, a thin-layer method has been introduced [15, 16], where it is in thermal equilibrium locally and the divergent term due to a box with infinite size does not appear anymore. For a thin layer near the horizon, this method is valid if the proper thickness is taken keeping the local equilibrium state, since the degree of freedom is dominant near the horizon. In this paper, we would like to calculate the entropy of the rotating hairy black hole [6] from new massive gravity [3, 4] by using the brick wall method [10] in thin-layer approximations [15, 16], where the angular speed of a particle near the horizon can be approximately fixed to a constant. So we will show how to take into account the superradiant mode in the entropy calculation of the rotating hairy black hole which has complicated metric components. The entropy of the hairy black hole in connection with the superradiant mode for the rotating case deserves to be studied. In Sec. II, a new hairy rotating black hole and its thermodynamic quantities are introduced. Additionally, one can write down the explicit form of the horizons and the radius of the ergosphere analytically. In Sec. III, we calculate the thermodynamic quantities such as the entropy, the angular momentum, and the energy by considering the superradiant and the nonsuperradiant modes simultaneously. A summary is given in Sec. IV.", "pages": [ 2 ] }, { "title": "II. ROTATING HAIRY BLACK HOLE", "content": "We consider the rotating black hole described by [6-9] where N ( r ) = 1+ b/lscript 2 (1 -η ) / (4Σ), f ( r ) = (Σ 2 /r 2 ) [Σ 2 //lscript 2 + b (1 + η )Σ / 2 + b 2 /lscript 2 (1 -η ) 2 / 16 -µη ], Ω 0 ( r ) = a ( µ -b Σ) / (2 r 2 ), and Σ( r ) 2 = r 2 -µ/lscript 2 (1 -η ) / 2 -b 2 /lscript 4 (1 -η ) 2 / 16. a , b , and µ are integration constants and Λ and η have been defined as Λ = -1 / (2 /lscript 2 ) and η ≡ √ 1 -a 2 //lscript 2 . For the limit of b = 0, the metric functions reduce to N = 1, f = -µ + r 2 //lscript 2 + µ 2 a 2 / (4 r 2 ), and Ω 0 = µa/ (2 r 2 ), and Eq. (1) simply describes the rotating BTZ black hole [5]. Note that the range of the rotating parameter a is given by -/lscript ≤ a ≤ /lscript . Then the Arnowitt-Deser-Misner (ADM) mass and the ADM angular momentum can be obtained as [8] respectively. G is a Newton constant in three dimensions, which is set to G = 1 for convenience. The entropy has been also obtained as S = π/lscript √ 2 M (1 + η ) [6, 7, 9]. Now, the condition of f ( r ) = 0 gives which yields the horizon r + . Equation (1) describes the rotating BTZ black hole, which has two horizons of r ± = /lscript √ µ (1 + η ) / 2 and r 0 = /lscript √ µ (1 -η ) / 2. Especially for a = 0, the black hole has two horizons r ± = 2 /lscript ( √ M±| b | /lscript/ 4) with r 0 = 0. Next, the Hawking temperature of the black hole can be obtained from the surface gravity, Consequently, the first law of thermodynamics d M = T H d S +Ω H d J is satisfied. The maximum Ω + and the minimum Ω -of the angular velocity for a particle are given by Note that the angular velocity of a particle on the event horizon becomes and the radius r e of the ergosphere is explicitly written as which can be calculated from Ω -( r e ) = 0.", "pages": [ 3, 4 ] }, { "title": "III. STATISTICAL ENTROPY", "content": "Now, we consider a scalar field in a thin layer between r + + h to r + + h + δ with h /lessmuch r + and δ /lessmuch r + , where h is a cutoff parameter and δ is a small constant related to the thickness of the thin layer. It satisfies the massless Klein-Gordon equation, /square Φ( t, r, φ ) = 0. Assuming Φ( t, r, φ ) = Ψ ωm ( r ) e -iωt + imφ , we obtain rN∂ r ( rNf∂ r Ψ ωm ) + r 2 N 2 f 2 k 2 Ψ ωm = 0, where k ( r ; ω, m ) = N -1 f -1 √ ( ω -Ω + m )( ω -Ω -m ). In the WKB approximation with Ψ ∼ e iS ( r ) , k is the radial momentum defined by k = ∂S/∂r . Therefore, the number of states less than the energy ω and the angular momentum m is given by where' k '( r ; ω, m ) = k ( r ; ω, m ) if k 2 > 0 and ' k '( r ; ω, m ) = 0 if k 2 < 0. The free energy of a rotating black hole should be written as [14] where where the 'NS' and 'SR' denote the nonsuperradiant mode with ω -m Ω H > 0 and superradiant mode with ω -m Ω H < 0, respectively, λ is the set of ( ω, m ), and the density of the number of states is given by g ( ω, m ) = dn/dω for the NS mode and g ( ω, m ) = -dn/dω for the SR mode. Substituting Eq. (9) into Eqs. (11) and (12), we obtain where ω max ( m ) and ω min ( m ) denote the maximum and the minimum of ω for a given m in each mode, respectively. For convenience, Eq. (13) can be rewritten as From Eq. (14), the free energy of the SR mode is written as Then, the total free energy which consists of the nonsuperradiant and superradiant modes (10) becomes which leads to where in the leading order of the cutoff and the thickness. Note that the second term in the free energy for the positive mode in Eq. (17) and the second term in the free energy for the superradiant mode in (18) canceled out. Thus the entropy can be simplified as where β H is defined as the inverse of the Hawking temperature T H . The proper lengths for the UV cutoff parameter and the thickness are defined by ¯ h ≡ ∫ r + + h r + dr √ g rr /similarequal 2 √ h/ √ f ' ( r + ) and ¯ δ ≡ ∫ r + + h + δ r + + h dr √ g rr /similarequal 2( √ h + δ -√ h ) / √ f ' ( r + ). Then, the entropy is written as S = 3 ζ (3) r + ¯ δ/ [4 π 2 ¯ h ( ¯ δ + ¯ h )]. Recovering dimensions, the entropy becomes where A ≡ 2 πr + and /lscript P ≡ /planckover2pi1 G/c 3 are the area of the event horizon and the three-dimensional Plank length, respectively. If the cutoff is chosen as ¯ h ( ¯ h + ¯ δ ) / ¯ δ = [3 ζ (3) / (2 π 3 )] /lscript P , the entropy (22) agrees with the Bekenstein-Hawking entropy S BH = c 3 A/ (4 G /planckover2pi1 ). Finally, let us calculate angular momentum of matter, which becomes and the internal energy of the system is written as Note that the angular momentum (23) and the energy (24) of matter have well-defined limits and they are compatible with the results in Ref. [14] for b = 0. On the other hand, it would be interesting to note that a partition function from free energy (20) can be compared with the result for the partition function of the corresponding two-dimensional conformal field theory (CFT) on the boundary of three-dimensional anti-de Sitter (AdS) spacetime [17]. For this purpose, we write down the free energy (20) by using Eqs. (5) and (7) along with the proper lengths ¯ h and ¯ δ as where we restricted to the case of b = 0 and identified β with β H . If one chooses the cutoff as ¯ h ( ¯ δ + ¯ h ) / ¯ δ = [3 ζ (3) /π 3 ] /lscript P , then the free energy (25) is simplified as In order to write down the free energy in terms of dimensionless quantities, we rescale the free energy, the inverse Hawking temperature, and the angular velocity at the horizon by /lscript P F → F , β H //lscript → β , and /lscript Ω H → Ω, respectively. Then, the free energy becomes F = -π 2 / [6 β 2 (1 -Ω 2 )]. Since the relation between the partition function Z and the free energy is given by βF = -ln Z , we can obtain which agrees with the result given in Ref. [17]. However, it may depend on the cutoff within our brick wall formulation so that the coefficient can be adjusted. As a result, the degrees of freedom near the horizon can be described by the boundary degrees of freedom. In fact, the bulk degrees of freedom can be read off from the boundary degrees of freedom from the AdS/CFT while the bulk degrees of freedom can be also described by the degrees of freedom near the horizon based on the brick wall formalism. Combining these two notions, the boundary degrees of freedom at both ends can be connected.", "pages": [ 4, 5, 6, 7 ] }, { "title": "IV. SUMMARY", "content": "In the course of calculations, the second term of the free energy for the positive mode in Eq. (17) and the second term of the free energy for the superradiant mode in (18) canceled out so that from the simplified resulting free energy we have obtained the statistical entropy satisfying the area law by determining the UV cutoff which is independent of the hairs of the black hole, and additionally derived the angular momentum and the energy of matter field. The energy E is always positive and it depends on the mass of the black hole, the angular momentum of the black hole, and the gravitational hair b . For the limit of b = 0, the angular momentum can be reduced to J = 1 2 J and E = 1 2 M + 1 6 √ M 2 -J 2 //lscript 2 . It means that the angular momentum of the matter is directly proportional to that of the black hole while the energy is related to the mass and angular momentum of the black hole simultaneously.", "pages": [ 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work was supported by the Sogang University Research Grant 201310022 (2013).", "pages": [ 8 ] } ]

[ { "title": "Electroweak bremsstrahlung in bino-like dark matter annihilations", "content": "Kenta Shudo ∗ and Takeshi Nihei † Department of Physics, College of Science and Technology, Nihon University, 1-8-14, Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8308, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate the effects of electroweak bremsstrahlung on bino-like neutralino dark matter pair annihilations in the minimal supersymmetric standard model (MSSM). We calculate the nonrelativistic pair annihilation cross sections via W -strahlung from leptonic final states, χχ → W/lscript ¯ ν , and compare them with the contributions of the relevant two-body final states. We explore the case that sleptons lie below the TeV scale, while squarks are extremely heavy. It is found that the electroweak bremsstrahlung can give a dominant contribution to the cross section for some parameter regions which include slepton coannihilation regions with the observed relic abundance. We also evaluate the neutrino spectra at injection in the Sun. It is shown that energetic neutrinos via weak bremsstrahlung processes can be dominant over contributions of the two-body final states.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Clarifying the nature of cold Dark Matter (DM) is one of the key issues in recent astrophysics and cosmology [1]. The relic abundance of cold DM in the present Universe is determined by recent astronomical observations with great precision as Ω χ h 2 = 0.1199 ± 0.0027 [2]. 1 Among the diverse candidates, the lightest superparticle (LSP) in supersymmetric models is one of the most attractive ones for the DM particle [3, 4]. In the minimal supersymmetric standard model (MSSM) [5], the LSP is typically the lightest neutralino given by a linear combination of neutral gauginos and higgsinos where ˜ B is the U (1) Y gaugino (bino), ˜ W 3 is the neutral SU (2) L gaugino (wino), and ˜ H 0 1 and ˜ H 0 2 are the two neutral higgsinos with opposite hypercharges. The coefficients N 1 i ( i = 1 , 2 , 3 , 4) are the elements of the 4 × 4 unitary matrix N which diagonalizes the neutralino mass matrix [5, 6]. Assuming the GUT relation for the gaugino masses, a bino-like LSP is realized for relatively light gauginos. Cosmic rays produced by DM annihilations in the galactic halo provide a way of indirect detection of DM. For a bino-like LSP, the dominant annihilation channel is fermion pair production χχ → f ¯ f . The neutralino pair annihilation cross section in the nonrelativistic limit is helicity suppressed ( ∝ m 2 f /m 2 χ ) for light fermions due to the Majorana nature of the neutralino [7]. It is known, however, that gauge boson emissions can lift the helicity suppression, since the emitted gauge boson carries unit angular momentum. Indeed, it has been shown that the bremsstrahlung χχ → f ¯ fγ can potentially give characteristic signals of DM in gamma-ray observations [8, 9]. In recent years, the significance of electroweak bremsstrahlung emitting W / Z bosons has been recognized in the literature [10-22]. In particular, the weak bremsstrahlung is expected to be more important in evaluating neutrino flux than the usual bremsstrahlung emitting photons, since the former emits primary neutrinos. The weak bremsstrahlung in a leptophillic dark matter model has been examined in Refs. [10, 11] where χχ → W/lscriptν can give a dominant contribution over χχ → /lscript + /lscript -γ . The effect of the W/lscriptν final states in this model is maximized in the limit where the dark matter mass is nearly degenerate with the mass of the SU (2) L doublet bosons /negationslash ( η 0 and η ± ) which mediate the annihilation process similarly to sleptons. Gammaray signals in this model were investigated in Ref. [12]. Various signals, including positrons, were studied in Ref. [13]. Neutrino spectra including neutrino oscillation effects were explored in Ref. [14]. Helicity dependent effects on the neutrino spectra were studied for SU (2) L singlet Majorana fermion dark matter [15]. For a wino-like dark matter, initial state W / Z radiations were found to be important [16]. Gamma rays from bino-like dark matter annihilations in the MSSM were examined including the three-body final state with weak bremsstrahlung [17]. Neutrino signals from weak bremsstrahlung in the MSSM were studied for a bino-like TeV dark matter scenario [18]. Antiproton constraints have been studied in Ref. [19] where effects of a longitudinal W -boson emission are examined in the presence of SU (2) L breaking effects ( m η 0 = m η ± ), including the case of the constrained MSSM. The massive threebody final state, Wtb , was considered in Ref. [20]. The role of weak bremsstrahlung for the relic density of DM was analyzed in Ref. [21]. In this paper, we investigate effects of electroweak bremsstrahlung on bino-like neutralino dark matter pair annihilations in the phenomenological MSSM where various SUSY parameters are chosen freely. We calculate the nonrelativistic pair annihilation cross sections via W -strahlung, χχ → W/lscript ¯ ν , and compare them with the contributions of relevant two-body final states. We consider the case that squarks are extremely heavy ( /greaterorsimilar 10 TeV), while sleptons are much lighter than squarks. In this case, the weak bremsstrahlungs with quarks, χχ → Wud , Wcs , Wtb via t - and u -channel squark exchanges are suppressed. Then we discuss leptonic processes with primary neutrinos This paper is organized as follows. In Sec. II, we describe the relevant MSSM interactions. In Sec. III, we calculate the cross sections and neutrino spectra for the weak bremsstrahlung. In Sec. IV, we discuss the narrow width approximation for the massive weak bremsstrahlung process χχ → Wtb . In Sec. V, we present our numerical results. Finally concluding remarks are given in Sec. VI. A simplified expression for the cross section in the unbroken SU (2) L limit in the slepton sector is provided in Appendix A. The contributions of two-body final states are listed in Appendix B.", "pages": [ 2, 3 ] }, { "title": "II. RELEVANT MSSM INTERACTIONS", "content": "The relevant interaction Lagrangian of the neutralino χ with leptons and sleptons can be written as where P L = 1 -γ 5 2 and P R = 1+ γ 5 2 . The fields ˜ /lscript I ( I = 1, 2) denote the charged slepton mass eigenstates, and ˜ ν /lscript is the sneutrino ( /lscript = e , µ , τ ). Flavor mixings and CP violations are neglected. The coupling constants in Eq. (3) are given by where g ' and g denote the gauge coupling constant for the U (1) Y and SU (2) L , h /lscript = gm /lscript / ( √ 2 m W cos β ) is the Yukawa coupling constant for the lepton, and N ij is the element of the unitary matrix to diagonalize the neutralino mass matrix [6, 23]. The vacuum angle β is given by tan β = v 2 /v 1 , where v 1 and v 2 are the vacuum expectation values of the two neutral Higgs bosons. The unitary matrix ˜ V /lscript diagonalizes the charged slepton mass squared matrix as ˜ V /lscript M 2 ˜ /lscript ˜ V † /lscript = diag ( m 2 ˜ /lscript 1 , m 2 ˜ /lscript 2 ). The mass squared matrix, neglecting the m 2 /lscript terms, is approximately given by [6] where m 2 ˜ /lscriptL and m 2 ˜ /lscriptR are the soft supersymmetry (SUSY) breaking mass parameters for the left- and right-handed sleptons, respectively, s W = sin θ W , and A /lscript is the trilinear scalar coupling constant for the slepton. The mass eigenstates are related with the chiral bases ˜ /lscript L and ˜ /lscript R as ˜ /lscript I = ( ˜ V /lscript ) I 1 ˜ /lscript L + ( ˜ V /lscript ) I 2 ˜ /lscript R . The sneutrino mass squared is given by m 2 ˜ ν = m 2 ˜ /lscriptL + 1 2 m 2 Z cos 2 β . The W -boson emission from the slepton involves the following interaction: We follow the convention of Ref. [6] for the MSSM parameters. We assume the GUT relation for the gaugino masses: M 1 = 5 3 M 2 tan 2 θ W where M 2 and M 1 are the gaugino masses for the SU (2) L and U (1) Y gauginos. Numerically, this implies M 1 ∼ M 2 / 2. The neutralino in Eq. (1) is bino-like for M 1 /lessmuch | µ | , where µ is the Higgsino mass parameter.", "pages": [ 4, 5 ] }, { "title": "III. CROSS SECTIONS FOR WEAK BREMSSTRAHLUNG", "content": "In this section, we present the cross sections for the weak bremsstrahlung process (2) in the nonrelativistic limit v → 0, where v is the relative velocity between the two neutralinos in the center of mass frame. The relevant Feynman diagrams for χχ → W + /lscript ¯ ν /lscript via t - and u -channel slepton exchange are shown in Fig. 1. The diagram A (B) proceeds via pair production χχ → ν ∗ ¯ ν ( χχ → /lscript ¯ /lscript ∗ ) followed by the W -boson emission from the neutrino (charged lepton). In the diagram C, the W boson is emitted by the virtual sleptons. There exist other diagrams which can contribute to the process (2) in principle. However, in the parameter region we consider, they give only negligible effects. Contributions via s -channel Z -boson and pseudoscalar Higgs ( A ) exchange χχ → Z ∗ / A ∗ → /lscript ¯ /lscript ∗ → W/lscript ¯ ν give no significant effects, since the neutralino coupling to the Z boson or the pseudoscalar Higgs boson is highly suppressed for a bino-like LSP. 2 Initial state radiation, in which an initial neutralino emits the W boson, is negligible for a bino-like LSP, since a pure bino does not couple to a W boson. Contributions via W -boson pair production followed by the W -boson decay to the leptonic pair, χχ → WW ∗ → W/lscript ¯ ν , are negligible due to the suppressed coupling of the bino-like LSP to the W boson. 3 The 4-momentum of each particle in Fig. 1 is assigned as follows: We define the sum of the t - and u -channel diagram for each diagram A, B and C in Fig. 1 as M A = M A t + M A u , M B = M B t + M B u and M C = M C t + M C u , respectively. After the Fierz rearrangements, these matrix elements can be written as /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash where q 1 = p 1 + p 3 , q 2 = p 2 + p 3 , and ε 3 = ε ( p 3 ) is the polarization vector of the W boson. The spinors u and v are denoted as u χ = u ( k 1 ), v χ = v ( k 2 ), u /lscript = u ( p 1 ), v ν = v ( p 2 ), where the spin indices are suppressed. The Lorentz invariants t 1 and t 2 are defined as Note that k 1 = k 2 in the nonrelativistic limit v → 0. In evaluating the matrix elements, we neglect the lepton masses compared with m W and m χ , while we take into account the lepton masses in slepton mass matrices to keep potentially large left-right mixings for the sleptons. 4 The Gordon decomposition for the s-wave limit, m χ v χ γ α γ 5 u χ = -k α 1 v χ γ 5 u χ , can be used to further simplify Eqs. (8)-(10). The differential cross section for the process χχ → W + /lscript ¯ ν /lscript is given by where E W and E ν are the energy of the W boson and the neutrino at the center of mass frame. To include the charge-conjugated process as in Eq. (2), the differential cross section in Eq. (12) must be doubled. The helicity sum of the matrix element squared for diagrams A, B and C are given as follows: ∑ spins 2 ∑ 4 m 2 2 W t 1 1 - m 2 ˜ /lscriptI t 1 1 - m 2 ˜ /lscriptJ g |M B | = I,J The auxiliary functions F 1 , F 2 , · · · , F 5 are given by where the Lorentz invariants x , y and z are defined by Note that x + y + z = 4 m 2 χ + m 2 W , where m χ and m W denote the mass of the neutralino and W boson, respectively. The Lorentz invariants in Eqs. (11) and (20) can be written in terms of the energies as x = 4 m χ ( m χ -E ν ), y = 4 m χ ( m χ -E W ) + m 2 W , z = 4 m χ ( m χ -E /lscript ), t 1 = z 2 -m 2 χ , t 2 = x 2 -m 2 χ , where E /lscript is the energy of the charged lepton at the center of mass frame of the initial neutralinos, and the lepton mass is neglected. Let us discuss the behavior in the heavy slepton limit: m ˜ /lscriptI = m ˜ ν ≡ ˜ m → ∞ . The slepton mass dependence of each amplitude is M A , M B ∼ 1 / ˜ m 2 , and M C ∼ 1 / ˜ m 4 . However, by summing up Eqs. (13), (14) and (16), the leading 1 / ˜ m 2 terms cancel out between diagrams A and B, resulting in the total amplitude suppressed as ∼ 1 / ˜ m 4 [11]. On the other hand, the amplitude for the leptonic two-body process, χχ → τ + τ -, is suppressed only by ∼ 1 / ˜ m 2 . Therefore, the ratio ( σv ) Wτν / ( σv ) τ + τ -falls down for heavy sleptons. We also evaluate neutrino spectra at injection from the center of the Sun. The primary neutrino spectrum via χχ → W/lscript ¯ ν is obtained by integrating Eq. (12) over E W as where The cross section is obtained by integrating Eq. (21) over E ν in the range 0 < E ν < m χ ( 1 -m 2 W 4 m 2 χ ) . This integration can be done analytically, although the expressions are lengthy. The secondary neutrino spectra from decay of the W boson and the tau lepton are evaluated as follows. The neutrino spectrum via W -boson decay is written as where ( dN ν dE ν ) W ( E W , E ν ) is the neutrino energy distribution per W -boson decay with energy E W , and d ( σv ) W/lscript ¯ ν /dE W is the W -boson spectrum obtained by integrating Eq. (12) over E ν . The secondary neutrino spectrum from tau decay is obtained in a similar fashion using the neutrino distribution per tau decay, ( dN ν dE ν ) τ ( E τ , E ν ). In evaluating the contributions via relevant two-body processes χχ → τ + τ -, t ¯ t , b ¯ b and W + W -, we further need the neutrino distributions from the top and bottom quarks. Neutrino distribution ( dN ν dE ν ) i ( E i , E ν ) from the parent particle ( i = W , τ , t , b ) with energy E i is affected by matter effects in the Sun. We neglect decay of the light quarks and the muon, since they stop before decay in the Sun. We also neglect the contribution of the charm quark, since it is subdominant compared with that of the bottom quark. For the energy distributions ( dN ν dE ν ) i in the Sun, we use the result of Ref. [24] where the distributions are obtained with the Monte Carlo code PYTHIA [25]. For the energy of the parent particles not tabulated in the reference, we simply adopt linear interpolations.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "IV. NARROW WIDTH APPROXIMATION FOR χχ → Wtb", "content": "In this section, we discuss the relation between the massive weak bremsstrahlung process χχ → Wtb and the corresponding two-body process χχ → t ¯ t using a narrow width approximation. The cross section for Wtb can be obtained from M A , M B and M C by replacing ν and /lscript with t and b , respectively, and taking both left- and right-handed stops into account. When m χ > m t , the top quark pair production χχ → t ¯ t opens up. Then, the three-body cross section for χχ → Wtb calculated with only M A reduces to the cross section of the two-body process χχ → t ¯ t evaluated with only t - and u -channel diagrams. In this sense, the massive weak bremsstrahlung χχ → W + ¯ tb is included in the two-body process χχ → t ¯ t followed by the on-shell top quark decay for m χ > m t [20]. On the other hand, the leptonic W -strahlung in Eq. (2) can never be included in any two-body process, since a W -boson emission from on-shell lepton is kinematically forbidden. We have calculated the cross section for the Wtb final state in the same way as the leptonic case. Integrating the differential cross section d 2 ( σv ) ( ˜ t ) W ¯ tb / ( dE W dE t ) over E W , the top quark energy distribution via only stop exchange can be obtained as where ∆ t = x -m 2 t = 4 m χ ( m χ -E t ), Γ t is the decay width for the top quark, N c = 3 is the color factor, and The coupling constants C ( t ) LI and C ( t ) RI can be defined in a similar fashion as C ( /lscript ) LI and C ( /lscript ) RI in Eq. (4). The constant D ( t ) LI is defined by Integration of Eq. (24) over E t gives the cross section ( σv ) ( ˜ t ) W ¯ tb . If the integration range includes the pole E t = m χ ( x = m 2 t ), the narrow width approximation can be justified where the propagator of the top quark can be replaced with the delta function as Under this approximation, the integration of Eq. (24) over x (equivalently over E t ) indeed reduces to the two-body s-wave cross section via t - and u -channel stop exchange [3] (see Appendix B) We neglect the s-channel contributions for the three-body processes for simplicity. This is a good approximation for the leptonic process χχ → W/lscriptν . For the Wtb final state, however, this is not a good approximation, since squarks are extremely heavy in the present analysis. In our numerical calculation, we use the total two-body expression for χχ → t ¯ t in Eq. (B1) rather than in Eq. (28). Therefore, our result for the Wtb final state below the t ¯ t threshold m χ /lessorsimilar m t is not smoothly connected to that for the t ¯ t final state. This does not affect our conclusion, since the Wtb contribution below the threshold is subdominant in the parameter range we consider in the present analysis.", "pages": [ 9, 10, 11 ] }, { "title": "V. NUMERICAL RESULTS", "content": "/negationslash In this section, we present our numerical results. In this analysis, we examine the phenomenological MSSM, where various SUSY parameters are chosen freely. In order to find the parameter ranges where the weak bremsstrahlung is important, we consider the scenario that the sleptons lie below the TeV scale, while the squarks are extremely heavy. Throughout the analyses, squark mass parameters are taken as m ˜ q /greaterorsimilar 10 TeV for all the squarks to be consistent with the null results of superparticle searches at the LHC [27]. The pseudoscalar Higgs boson mass is fixed at m A = 2 TeV in every figure. For the slepton mass parameters, we assume a common value m ˜ e = m ˜ µ for the left- and right-handed soft SUSY breaking masses for the selectrons and smuons, while the left- and right-handed stau mass parameters, m ˜ τL and m ˜ τR , are chosen independently. For the trilinear scalar couplings, we vary A t , A b and A τ for the stop, sbottom and stau, respectively. All the others are set to zero: A q = 0 ( q = t , b ) for the other squarks, and A e = A µ = 0. The s-wave cross section times the relative velocity, σv , for the weak bremsstrahlung processes χχ → W/lscriptν , including both W + /lscript -¯ ν /lscript and W -/lscript + ν /lscript , are plotted in Fig. 2 as a function of m χ together with the contributions of the relevant two-body processes. In Fig. 2 (a), the MSSM parameters are chosen as tan β = 2, µ =1TeV, m ˜ q =14 TeV, and m ˜ e = m ˜ µ = m ˜ τL = m ˜ τR =240 GeV. The trilinear coupling A t is chosen to satisfy the Higgs mass constraint m h ∼ 125 GeV [28, 29], although the trilinear coupling for the squarks are irrelevant in the present analysis where squark exchange diagrams are sufficiently suppressed by the heavy squark masses. The bold solid line corresponds to the sum of all the leptonic processes, ∑ /lscript W/lscript ¯ ν /lscript + h.c. The contributions of Weν e and Wµν µ are identical, ( σv ) Weν e = ( σv ) Wµν µ , which are essentially described by the simplified expression in Appendix A. On the other hand, the result for Wτν τ is slightly larger ( σv ) Wτν τ ∼ 1 . 2 × ( σv ) Weν e , where the difference originates from the left-right mixing for the staus. The contributions of the relevant two-body processes are shown with thin lines. 5 The solid, dashed, dot-dashed and dotted lines correspond to τ + τ -, t ¯ t , b ¯ b and W + W -, respectively. For a relatively large value of m χ /lessorsimilar m ˜ /lscript , the weak bremsstrahlung dominates over the two-body contributions. The range of m χ filled with gray corresponds to the cosmologically allowed region where the relic abundance constraint 0.11 < Ω χ h 2 < 0.13 is satisfied. The relic abundance is obtained using DarkSUSY [30]. In the present scenario with a bino-like LSP, the relic density is typically too large. The allowed region in Fig. 2 (a) appears with the help of slepton coannihilations which lead to an effective enhancement of the pair annihilation cross section for m χ ≈ m ˜ /lscript [31-35]. In Fig. 2 (b), the soft mass parameter for the right-handed stau is taken to be larger than the other slepton mass parameters as m ˜ τR = 480 GeV. In this case, the contributions of Weν e and Wµν µ are the same as in Fig. 2 (a), while that of τ + τ -gets suppressed due to the larger m ˜ τR . The contribution of Wτν τ becomes identical to those of Weν e and Wµν µ , since the effect of left-right mixing for the staus, ∼ m τ ( A τ -µ tan β ) / ( m 2 ˜ τL -m 2 ˜ τR + δ τ ), is reduced by taking different values for m ˜ τR and m ˜ τL , where δ τ ∼ -0 . 04 m 2 Z cos 2 β . Thus the relative magnitude of weak bremsstrahlung is increased compared with the result in Fig. 2 (a). The weak bremsstrahlung is dominant for all values of m χ . In Fig. 2 (c), the soft mass parameters for both the left- and right-handed stau are taken to be larger than that for the selectron and smuon as m ˜ τL = m ˜ τR = 480 GeV. The bold dashed line represents the result for only Wτν τ . The contribution of Wτν τ is reduced due to the larger m ˜ τL . Figure 2 (d) is the result for tan β = 10, µ = 3 TeV, and m ˜ q = 12 TeV, where the mass parameters for staus are taken to be larger as m ˜ τL = m ˜ τR = 2 TeV. For a large tan β , the contribution of the τ + τ -final state is enhanced due to the larger left-right mixing, while Weν e and Wµν µ remain unchanged. However, by taking much larger masses for staus than in Figs. 2 (a)-(c), the weak bremsstrahlung can be dominant over the two-body processes. The process Wτν τ is suppressed as ∼ 1 /m 8 ˜ τL . The bold dotted line represents the contribution of the Wtb final state below the t ¯ t threshold m χ < m t . This proceeds via off-shell top quark effect χχ → t ∗ ¯ t followed by the decay t ∗ → Wb [20]. In the present analysis, we include only t - and u -channel diagrams for the Wtb final state as explained in Sec. IV. If we had included the s -channel diagrams for Wtb , the result below the threshold would be smoothly connected to the two-body t ¯ t result. We define the ratio R = ( σv ) 3 W / ( σv ) 2 , where ( σv ) 3 W is the total leptonic W -strahlung contribution, and ( σv ) 2 = ∑ f = t,b,τ ( σv ) f ¯ f + ( σv ) WW is the sum of all the relevant two-body contributions included in the present analysis. 6 The contours of the ratio R are plotted in the ( µ , M 2 ) plane in Fig. 3 with bold lines. The relavant parameters for Figs. 3 (a)-(d) are taken to be the same as in Figs. 2 (a)-(d), respectively. The contours of ( σv ) 3 W are shown with thin lines. As M 2 gets across 2 m t ∼ 350 GeV, the ratio becomes small suddenly due to opening of the t ¯ t channel. The strip filled with gray corresponds to the cosmologically allowed region with the correct Ω χ h 2 . In Fig. 3 (a), the W -strahlung cross section ( σv ) 3 W becomes larger as M 2 increases. In the cosmologically allowed range, the weak bremsstrahlung can be comparable with the two-body processes for 800 GeV /lessorsimilar µ /lessorsimilar 1100 GeV. The area filled with light gray is excluded where the lighter stau is lighter than the neutralino. The figure includes a small µ range where the LSP is higgsino-like. In this region, the WW final state becomes dominant, though cosmologically allowed regions exist for higgsino-like LSP. Therefore, the weak bremsstrahlung is negligible for a higgsino-like LSP in the present analysis. We do not show the result for µ < 0, since the behavior is similar to that for µ > 0. In Figs. 3 (b) and (c), where the mass parameters for staus are larger, the contribution of W/lscriptν /lscript is similar to that in Fig. 3 (a). However, the ratio can be significantly larger than in Fig. 3 (a), since the relevant two-body final state, τ + τ -, gets smaller than in Fig. 3 (a). In Fig. 3 (d), the ratio is further enhanced by taking a large value for stau mass parameters as m ˜ τL = m ˜ τR = 2 TeV for tan β = 10. Finally, neutrino spectra at injection in the Sun are shown in Fig. 4. The panel (a) is the result for tan β = 2, M 2 = 450 GeV, µ = 1 TeV, m ˜ q = 14 TeV, and m ˜ e = m ˜ µ = m ˜ τL = m ˜ τR = 240 GeV. The bold solid line corresponds to the primary neutrino spectrum of weak bremsstrahlung including all the flavors and the charge conjugated states. The bold dashed and dotted lines are the results for the secondary neutrinos from the tau lepton and the W boson, respectively, produced via weak bremsstrahlung. The contributions of two-body processes are drawn with thin lines. The neutralino mass in this case is m χ ≈ 221.9 GeV. One can see that the primary neutrino from weak bremsstrahlung can give a significant contribution particularly in the high energy range 0 . 8 m χ /lessorsimilar E ν /lessorsimilar m χ . The result for m ˜ τL = m ˜ τR = 480 GeV is shown in the panel (b). In this case, the weak bremsstrahlung is dominant for the wide range of E ν , since the τ + τ -final state is suppressed by the large m ˜ τL and m ˜ τR . It is notable that primary neutrino contributions of weak bremsstrahlung are nearly flavor independent for a common slepton mass, while the largest contribution among the two-body process, τ + τ -, produces mainly tau neutrinos in the Sun. Hence W/lscriptν can strongly affect the flavor contents of energetic neutrinos [14]. Even when the three-body final state W/lscriptν is not the dominant channel in the total cross section, the neutrino spectrum at high energies, E ν /lessorsimilar m χ , can still be dominated by W/lscriptν . The energetic neutrinos mainly originate from the internal bremsstrahlung, diagram C in Fig. 1. Indeed, in Fig. 4(a), the total cross section of the three-body process is slightly smaller than the sum of the two-body results. Nevertheless, the three-body process is dominant in the high energy region in Fig. 4(a). It must be kept in mind that one should take into account the usual bremsstrahlung effect ( σv ) 3 γ as well to evaluate the total three-body cross section. However, as far as a neutrino flux is concerned, the weak bremsstrahlung is expected to give the major contribution. Also, Z -boson strahlung processes, e.g., χχ → Z/lscript + /lscript -, should be included to discuss the total contributions from weak bremsstrahlung.", "pages": [ 11, 12, 13, 14 ] }, { "title": "VI. CONCLUSIONS", "content": "We have examined the effects of electroweak bremsstrahlung on the bino-like neutralino dark matter pair annihilations in the MSSM. We have calculated the nonrelativistic pair annihilation cross sections and neutrino spectra via W -strahlung, χχ → W/lscript ¯ ν . It has been shown that the electroweak bremsstrahlung can give a dominant contribution to the cross section for some parameter regions which include cosmologically allowed ranges with the observed relic abundance. It has been found that the weak bremsstrahlung processes can give characteristic signals in the neutrino spectrum at injection in the Sun. In the present analyses, we considered extremely heavy squarks. When the squark masses are comparable with the slepton masses, the weak bremsstrahlung typically gives only a subdominant contribution for m χ > m t due to the unsuppressed t ¯ t final state. Below the t ¯ t threshold, the Wtb final state can be relevant as shown in Ref. [20].", "pages": [ 15 ] }, { "title": "Acknowledgments", "content": "This work was supported in part by a CST grant for Gakujyutsusyo, Nihon University. The authors are very grateful to S. Naka, S. Deguchi and A. Miwa for useful discussions and comments.", "pages": [ 15 ] }, { "title": "Appendix A: Unbroken SU(2) limit in the slepton sector", "content": "/negationslash In this section, we provide a simple expression for the unbroken SU (2) L limit in the slepton sector, taking the common slepton masses m ˜ ν = m ˜ /lscriptI ≡ ˜ m , and the common left-handed slepton coupling constants C ( ν ) L = C ( /lscript ) L 1 ≡ C L with C ( /lscript ) L 2 = C ( /lscript ) RI = 0, and keeping m W = 0. This corresponds to the choices N 12 = 0, h /lscript = 0, ( ˜ V /lscript ) 11 = 1 and ( ˜ V /lscript ) 12 = 0 in Eq. (4). In this limit, the neutralino is a pure bino, and only the lefthanded sleptons contribute. Writing the matrix element M = M A + M B + M C as M = ε µ 3 M µ , it is confirmed that the Ward identity p µ 3 M µ = 0 is satisfied using p 1 u /lscript = p 2 v ν = 0 [11]. This implies that the longitudinal polarization of the W boson does not contribute to the s-wave amplitude in this limit. The differential cross section for χχ → W/lscript ¯ ν can be greatly simplified as /negationslash /negationslash Integrating over dE ν and dE W analytically, Eq. (A1) reduces to the cross section found in Ref. [11]. Note that the cross section is severely suppressed as ∼ 1 / ˜ m 8 in the heavy slepton limit. /negationslash In the numerical analyses in Sec. V, the SU (2) L breaking effects lead to m 2 ˜ ν -m 2 ˜ /lscript 1 ∼ m 2 Z cos 2 β cos 2 θ W without the left-right mixing for the sleptons. In the presence of the mass splitting m ˜ ν -m ˜ /lscript 1 = 0, a longitudinal W -boson emission χχ → W L /lscript ¯ ν /lscript can enhance the cross section [19]. 7", "pages": [ 15, 16 ] }, { "title": "Appendix B: Two-body processes", "content": "In this appendix, the s-wave cross sections for the relevant two-body processes are summarized for convenience [3]. The s-wave cross section for the fermion pair production χχ → f ¯ f is given by where The quantities F A , F Z and F ˜ fI represent the amplitude of s -channel pseudoscalar Higgs boson ( A ) exchange, s -channel Z -boson exchange, and t - and u -channel sfermion exchange, respectively. The constant N c is the color factor: N c = 3 for quark pairs, and N c = 1 for leptons. The coupling constants C ffA P and C χχA P describe the interaction of the pseudoscalar Higgs A with bilinears ¯ fiγ 5 f and ¯ χiγ 5 χ , respectively. The coupling constants C ffZ A and C χχZ A determine the axial vector interaction of the Z boson with bilinears ¯ fγ µ γ 5 f and ¯ χγ µ γ 5 χ , respectively. The expressions for these coupling constants can be found in Ref. [23]. The decay widths for the Z boson and the pseudoscalar Higgs, Γ Z and Γ A , are taken into account. Note that the contribution of the sfermion exchange, F ˜ fI , includes a factor of the fermion mass m f , since C ( f ) RI ( C ( f ) LI ) is proportional to m f for I = 1 ( I = 2). The s-wave cross section for the W -boson pair production χχ → W + W -via chargino exchange diagrams is given by ∣ ∣ where m χ + p denotes the chargino mass ( p = 1, 2). The coupling constants C χ + p χW -V and C χ + p χW -A defined in Ref. [23] describe the neutralino-chargino-W vector/axial-vector interactions. Neutrino spectra for the two-body processes can be obtained from Eq. (23) by replacing the primary spectrum with the delta function distribution. For instance, the neutrino spectrum via χχ → W + W -can be written as by replacing d ( σv ) W/lscript ¯ ν /dE W in Eq. (23) with ( σv ) WW δ ( E W -m χ ).", "pages": [ 16, 17 ] } ]

[ { "title": "Isospin violating dark matter being asymmetric", "content": "Nobuchika Okada ∗ Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487, USA Osamu Seto † Department of Life Science and Technology, Hokkai-Gakuen University, Sapporo 062-8605, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "The isospin violating dark matter (IVDM) scenario offers an interesting possibility to reconcile conflicting results among direct dark matter search experiments for a mass range around 10 GeV. We consider two simple renormalizable IVDM models with a complex scalar dark matter and a Dirac fermion dark matter, respectively, whose stability is ensured by the conservation of 'dark matter number.' Although both models successfully work as the IVDM scenario with destructive interference between effective couplings to proton and neutron, the dark matter annihilation cross section is found to exceed the cosmological/astrophysical upper bounds. Then, we propose a simple scenario to reconcile the IVDM scenario with the cosmological/astrophysical bounds, namely, the IVDM being asymmetric. Assuming a suitable amount of dark matter asymmetry has been generated in the early Universe, the annihilation cross section beyond the cosmological/astrophysical upper bound nicely works to dramatically reduce the antidark matter relic density and as a result, the constraints from dark matter indirect searches are avoided. We also discuss collider experimental constraints on the models and an implication to Higgs boson physics.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Light weakly interacting massive particles (WIMPs) with a mass around 10 GeV have been currently a subject of interest, motivated by some recent results in direct dark matter (DM) detection experiments. DAMA/LIBRA has claimed detections of the annual modulation signal by WIMPs [1]. CoGeNT has found an irreducible excess [2] and annual modulation [3]. CRESST has observed many events that expected backgrounds are not enough to account for [4, 5]. However, these observations are challenged to the null results obtained by other experimental collaborations, CDMS [6], XENON10 [7], XENON100 [8, 9], and SIMPLE [10]. Light WIMPs have been investigated for a dark matter interpretation of those data. For instance, very light neutralino in the minimal supersymmetric standard model (MSSM) [11, 12] and the next-to-MSSM (NMSSM) [13, 14] or very light right-handed sneutrino [15, 16] in the NMSSM. On the other hand, the Fermi-LAT Collaboration has derived constraints on an s -wave annihilation cross section of a WIMP based on the analysis of gamma ray flux [17]. Annihilation modes of a light WIMP is now severely constrained. The isospin violating dark matter (IVDM) [18] has been proposed as a way to reconcile the tension between inconsistent results among the direct DM detection experiment, since different nuclei for target material have been used in the detector of each experiments. The possible consistency between DAMA, CoGeNT [2], and XENON [7, 8] was pointed out [18], while the discrepancy between CoGeNT and CDMS cannot be resolved by IVDM because both of them use germanium as the target. However, recently it was reported [19] that CDMS-II Si have observed three events and its possible signal region overlaps with the possible CoGeNT signal region analyzed by Kelso et al. [20]. The fitting data with IVDM have been examined by several groups [20-28], and constraints from indirect [29] and direct [30] DM detection experiments also have been derived. In this paper, we consider two simple IVDM models with a complex scalar DM and a Dirac fermion DM, respectively. In most of the previous works, the IVDM models have been proposed by introducing a new U (1) gauge symmetry with Z ' boson [31-33] or an extension of the Higgs sector [32, 34, 35]. In contrast to those models, to realize the different cross sections with respect to up quarks and down quarks, we introduce fourth generation quarks in the scalar DM model and scalar quarks in the fermion DM model, respectively. Our models are similar to a model briefly mentioned in Ref. [18]. The paper is organized as follows. In the next section, we describe our models of scalar and fermion DMs. In Sec. III, we identify the allowed region of the mass and couplings of the mediator quarks or scalar quarks by imposing the condition of the isospin violating elastic scattering cross section with nuclei. In Sec. IV, we calculate the annihilation cross section of the IVDM to examine the resultant thermal relic density as well as the constraint from Fermi-LAT data for the parameter region found in Sec. III. Constraints from collider experiments are discussed in Sec. V. Section VI is devoted to conclusions.", "pages": [ 2, 3 ] }, { "title": "A. Model of scalar dark matter with fermion mediators (model S)", "content": "First, we consider a simple model with a complex scalar dark matter, whose particle contents are given in Table I. In addition to the Standard Model (SM) particle contents, we have introduced the SM SU (2) singlet Dirac fermions ( U and D ) whose representations are the same as SU (2) singlet up and down quarks, a complex scalar DM ( φ ), and a real scalar S , with a global U (1) G symmetry. The stability of φ is ensured by the global U (1) G symmetry assumed to be conserved. All the SM particles are neutral under the global symmetry. The gauge and global symmetric Lagrangian relevant to our discussion is given by where H is the SM Higgs doublet, u R ( d R ) is the SM right-handed up (down) quark singlet, and V is a scalar potential for H , φ , and S . We assume a suitable scalar potential for our discussion: not only the Higgs doublet but also the scalar S develop vacuum expectation values and we expand these scalar fields as with the vacuum expectation values, v = 246 GeV and v s . After the electroweak symmetry breaking, the SM singlet scalar and the Higgs boson have a mass mixing such that where h 1 and h 2 are the mass eigenstates with masses m h 1 ≤ m h 2 , respectively. The existence of a light scalar particle mixed with the SM Higgs boson is constrained by the LEP experiments [36, 37]. We consider a small mixing, for example, sin α < 0 . 1, so that the mass eigenstate h 1 ( h 2 ) is almost the SM singlet scalar (the SM Higgs boson). For such a small mixing, the lower mass bound on h 1 disappears, and in the following analysis we consider m h 1 < 10 GeV. Terms in the scalar potential relevant to our analysis below are triple scalar couplings parametrized as with dimensionless couplings λ 1 , 2 , 3 . Since the SM-like Higgs boson h 2 can decay to the lighter scalars, h 1 and φ ( h 1 subsequently decays to lighter SM particles), the couplings λ 2 , 3 should be small in order not to significantly alter the Higgs boson branching ratio from the SM prediction. To simplify our analysis, we assume λ 2 /greatermuch λ 3 and further parametrize λ 1 , 2 as λ 1 = λ cos α and λ 2 = λ sin α with λ = √ λ 2 1 + λ 2 2 . We will discuss a phenomenological constraint on these parameters from the invisible decay branching ratio of the SM Higgs boson in Sec. V B.", "pages": [ 3, 4 ] }, { "title": "B. Model of fermion dark matter with scalar mediators (model F)", "content": "Next, we consider a simple model with a Dirac fermion DM, whose particle contents are given in Table II. In addition to the SM particle contents, we introduce color triplet scalars ( ˜ Q L , ˜ U R , and ˜ D R ) that are analogous to the scalar quarks in the MSSM, and a Dirac fermion DM ( ψ ). Similarly to the model S, a global U (1) G symmetry has been introduced to ensure the stability of the Dirac fermion DM. All the SM fields are neutral under the global symmetry. The relevant part of the Lagrangian is given by where ˜ H = iσ 2 H ∗ , ˜ Q L = ( ˜ U L ˜ D L ) T , q L = ( u L d L ) T is the SM doublet quark of the first generation, and A U,D are parameters with mass-dimension one. After the electroweak symmetry breaking, the mass eigenstates of ˜ U are obtained as with a mixing angle θ u . Similarly, ˜ D 1 and ˜ D 2 are obtained with an angle θ d . With the mass eigenstates, the Yukawa interactions between the dark matter fermion and the SM quarks in Eq. (5) are rewritten as", "pages": [ 4, 5 ] }, { "title": "III. DARK MATTER ELASTIC SCATTERING WITH NUCLEI", "content": "The dark matter scattering cross section with nucleus ( N ) made of Z protons ( p ) and A -Z neutrons ( n ) is given by for a scalar dark matter, while for a Dirac fermion dark matter The effective coupling with a proton f p and a neutron f n is expressed, by use of the hadronic matrix element, as where α q is an effective coupling of the DM particle with a q -flavor quark defined in the operators  with its mass m q , f ( i ) Tq , and f ( i ) TG where i = p, n are constants. In our analysis, we use the following values: f ( p ) Tu = 0 . 0290, f ( p ) Td = 0 . 0352, f ( n ) Tu = 0 . 0195, f ( n ) Td = 0 . 0525, f ( i ) Ts = 0, and f ( i ) TG = 1 -∑ q = u,d,s f ( i ) Tq . Those f ( i ) Tu and f ( i ) Td are quoted from Ref. [38], while we set f ( i ) Ts = 0 because recent studies of the lattice simulation [39] as well as chiral perturbation theory [40] imply negligible strange quark content. It has been pointed out [18] that the results of XENON100, CoGeNT and CRESST can be compatible, if the following relations are satisfied: /negationslash Note that f n = f p , and therefore the dark matter particle has isospin violating interactions with quarks.", "pages": [ 6 ] }, { "title": "1. Model S", "content": "For the model S, there are two contributions to the effective coupling α q . One is from the exchange of the scalars h 1 and h 2 , for which we find where we have assumed m 2 h 1 /lessmuch m 2 h 2 , with m h 2 being the SM(-like) Higgs boson mass. Note that α q /m q is independent of q , so that this contribution conserves the isospin. The other contribution is from the exchange of the Dirac fermions, U and D : 2 2 . Clearly this contribution violates the isospin symmetry. where we have assumed m φ /lessmuch M U,D For simplicity, let us assume f D /lessmuch f U , and the total contribution is given by Note that the existence of the two terms is crucial to realizing the opposite signs between f p and f n , because the heavy quark U (and also D ) always positively contributes to α q . Figure 1 shows the contours for various values of σ p SI , along with the (red) straight line corresponding to the condition f n /f p = -0 . 7. The two conditions in Eq. (12) are satisfied for Here we have fixed the dark matter mass as m φ = 8 GeV.", "pages": [ 6, 7 ] }, { "title": "2. Model F", "content": "For the model F, the effective coupling α q is given by Here, for simplicity, we have taken a limit, M 2 ˜ U 1 /lessmuch M 2 ˜ U 2 and M 2 ˜ D 1 /lessmuch M 2 ˜ D 2 . This effective coupling violates the isospin symmetry and f n /f p < 0 can be realized when the relative signs between sin 2 θ u f R u and sin 2 θ d f R d are opposite. We further simplify the system by setting 1", "pages": [ 7, 8 ] }, { "title": "IV. DARK MATTER ANNIHILATION CROSS SECTION", "content": "In this section, we estimate the annihilation cross section of the scalar/fermion dark matter particles for the parameters identified in the previous section to satisfy the conditions for the IVDM. We will see that the s -wave annihilation cross section of the dark matter is too large to reproduce the observed relic abundance. In order to achieve the correct relic abundance, one may consider a nonthermal dark matter scenario. However, this scenario cannot be viable, because the s -wave annihilation cross section already exceeds the upper bound obtained by the Fermi-LAT observations [17]. In the last part of this section, we will propose a simple scenario to realize the IVDM being consistent with the Fermi-LAT observations.", "pages": [ 9 ] }, { "title": "A. Model S", "content": "The dominant dark matter annihilation process is found to be φ † φ → b ¯ b mediated by the scalars, h 1 and h 2 , in the s channel. Assuming m 2 h 1 /lessorsimilar m 2 φ /lessmuch m 2 h 2 , the s -wave annihilation cross section is evaluated as where m b = 4 . 2 GeV is the bottom quark mass. Using the values in Eq. (16), we find, for example, 〈 σv 〉 /similarequal 6 . 12 pb for m φ = 8 GeV and m h 1 = 2 . 9 GeV. This cross section is roughly one order of magnitude larger than the typical dark matter annihilation cross section 〈 σv 〉 /similarequal 1 pb to achieve the observed relic density. Thus, in this case, the resultant dark matter abundance becomes too small. In order to realize the observed relic density, we may assume a nonthermal production of dark matter particles in the early Universe. However, this cannot be a phenomenologically viable scenario, because the dark matter annihilation cross section to the bottom quarks is constrained by the Fermi-LAT data as 〈 σv 〉 /lessorsimilar 0 . 5 pb [17]. In this case, the dark matter is overabundant and the relic density should be diluted by some mechanism in the history of the Universe. Since such a scenario is quite ambiguous, we do not consider it in this paper.", "pages": [ 10 ] }, { "title": "B. Model F", "content": "The s -wave annihilation modes are given by t -channel ˜ U/ ˜ D exchange with u ¯ u/d ¯ d final states. In a limit m 2 ψ /lessmuch M 2 ˜ U 1 , 2 , ˜ D 1 , 2 , the cross sections is found to be with where we have used Eq. (18) and the limit M 2 ˜ U 1 /lessmuch M 2 ˜ U 2 and M 2 ˜ D 1 /lessmuch M 2 ˜ D 2 . Using the values in Eqs. (20), we find the annihilation cross section as which is too large to reproduce the correct thermal relic density of the dark matter particle in the present Universe. In order to make the relic abundance right, we may consider a nonthermal production of the dark matter particles in the early Universe. However, as in the model S, such a scenario is not viable by the Fermi-LAT observations [17]. The upper bound on the cosmic antiproton flux obtained by the Fermi-LAT observations is interpreted to a cross section upper bound of DM annihilations to up and down quarks as [29]", "pages": [ 10, 11 ] }, { "title": "C. Solution to too large annihilation cross section", "content": "As we have seen, for a given parameter set to realize a large enough isospin violating scattering cross section with nuclei, the resultant annihilation cross section is too large to satisfy cosmological and astrophysical constraints. For relic density, one may assume a nonthermal dark matter production. However, as we have seen, such an idea cannot work because of the severe upper bound on the dark matter annihilation cross section from the Fermi-LAT observations. In order to avoid the Fermi-LAT constraints, we propose an extension of our model to the so-called 'asymmetric dark matter' scenario [41-47]. This scenario is suitable to our model, because the global U (1) G symmetry introduced in our model leads to the conservation of the dark matter number. Once a suitable DM-antiDM asymmetry is created in the early Universe, the too large annihilation cross section nicely works to leave only the dark matter in the present Universe with the observed relic abundance. Since the relic abundance of antidark matter particles in the present Universe is much smaller than the dark matter one, a cosmic ray flux produced by DM and antiDM annihilations becomes much smaller and hence the constraint from the Fermi-LAT observations can be avoided. A relic density of the dark matter particles in the presence of dark matter asymmetry (chemical potential) has been analyzed in detail by solving the Boltzmann equations [48]. For example, with a suitable initial dark matter asymmetry, the observed relic abundance of the dark matter particle can be obtained by the s -wave annihilation cross section 〈 σv 〉 = O (1) pb, while the relic abundance of antidark matter particle is found to be 2 orders of magnitude smaller than the dark matter one. As annihilation cross sections become larger, the relic abundance of anti-DM particle becomes exponentially smaller. This result is almost independent of WIMP dark matter mass. We apply the result to our scenario, so that the cosmic ray flux from DM-anti-DM pair annihilations is significantly suppressed and the constraint from the Fermi-LAT observations is avoided.", "pages": [ 11, 12 ] }, { "title": "A. Constraints on the mediator (s)quarks from LHC", "content": "Our model includes heavy (s)quarks, which can be produced at the Large Hadron Collider (LHC) mainly through the gluon fusion process. The heavy (s)quarks, once produced, decay to the SM quarks and the dark matter particles, and this process is observed as the hadronic final states with transverse missing energy. Searches for such events have been performed at the LHC experiments, and the null result, so far, sets the lower bound on heavy (s)quark masses as /greaterorsimilar 800 GeV [49]. This bound is obtained for the so-called simplified MSSM, where scalar quarks of the first two generations are produced at the LHC and decay to quarks and the lightest superpartner neutralino. Since we only introduced one generation of heavy (s)quarks, the mass bound on the mediator (s)quarks should be a little milder, but let us apply the bound for conservative discussion. From Eqs. (16) and (20), we can see that this mass bound is satisfied with the couplings being in a perturbative regime, f 2 U / (4 π ) , f 2 ˜ U, ˜ D / (4 π ) /lessmuch 1.", "pages": [ 12 ] }, { "title": "B. Constraint from Higgs boson invisible decay", "content": "In model S, the scalar mass eigenstate h 2 is approximately identified as the SM Higgs boson. Through the mass mixing with the singlet scalar s , the SM Higgs boson decays to a pair of the dark matter particles. 1 This decay width is given by The current ATLAS [51] and CMS [52] data for the Higgs boson production and its various decay modes are mostly consistent with the SM expectations, and the branching ratio of an invisibly decaying Higgs boson is constrained (at 3 σ ) as [53] where Γ SM = 4 . 07 MeV [54] is the SM prediction of the total decay width of a Higgs boson with a 125 GeV mass. Using the result in Eq. (16), we can give the annihilation cross section of Eq. (21) and the Higgs invisible decay rate of Eq. (28) as a function of only m h 1 , with a fixed dark matter mass m φ = 8 GeV. The correlation between these two quantities is shown in Fig. 3 by varying m h 1 in the range of 1 GeV ≤ m h 1 ≤ 7 . 0 GeV. Here the vertical line denotes the upper bound, BR( h 2 → invisible) = 0 . 35 at 3 σ [53] while the horizontal line corresponds to a typical value ( 〈 σv 〉 = 1 pb) of the WIMP dark matter annihilation cross section for reproducing the observed relic abundance. The upper bound BR( h 2 → invisible) = 0 . 35 is obtained by m h 1 /similarequal 2 . 9 GeV, for which we find the annihilation cross section 〈 σv 〉 /similarequal 6 . 1 pb. Note that the asymmetric IVDM scenario we have proposed in the previous section can be consistent with the constraint on the Higgs invisible decay rate. In order for the asymmetric dark matter to be consistent with the observed relic abundance, we have a lower bound on the annihilation cross section as 〈 σv 〉 /greaterorsimilar 1 pb [48]. Applying this bound, we read BR( h 2 → invisible) /greaterorsimilar 8% from Fig. 3. Precision measurements of Higgs decay width at future collider experiments such as the international linear collider, photon collider and muon collider can reveal the existence of the dark matter.", "pages": [ 12, 13 ] }, { "title": "VI. CONCLUSIONS", "content": "The IVDM scenario with destructive interference between the dark matter scatterings with a proton and a neutron offers an interesting possibility to reconcile conflicting results among direct dark matter search experiments for a light WIMP with mass around 10 GeV. In this paper, we have considered two simple IVDM models and investigate various phenomenological aspects of the models, such as realization of the IVDM scenario, the constraints on dark matter annihilation cross sections from the dark matter relic abundance as well as an indirect search for dark matter, and collider experimental constraints on the extra particles introduced in our models. One model introduces a complex scalar as a dark matter particle along with heavy extra quarks and a SM singlet real scalar, through which the dark matter particle couples with the SM up and down quarks. Isospin violating effective couplings are realized by the interference between processes mediated by the heavy quarks and the scalar. In the other model, we have introduced a Dirac fermion as a dark matter particle along with heavy colored scalars analogous to squarks in the MSSM, through which the dark matter particle couples with the SM quarks. The interference between two processes mediated by up-type squarks and down-type squarks realizes the isospin violating effective couplings. For both models, we have identified a parameter region suitable for the IVDM scenario. With the parameter regions, we have also calculated the relic abundance of the dark matter which is found to be too large to reproduce the observed relic abundance. We have noticed that for both models, the calculated dark matter annihilation cross sections exceed the upper bound obtained by the Fermi-LAT observations too, and therefore the parameter regions for realizing the IVDM scenario are excluded. We have proposed a simple scenario to reconcile the IVDM scenario with the Fermi-LAT observations, namely, the IVDM being asymmetric. In our models, a global U (1) G symmetry has been introduced whose conservation ensures the stability of a dark matter particle. At the same time, this global symmetry leads to the conservation of the dark matter number and this structure is suitable for the asymmetric dark matter scenario. As discussed above, we have found that the dark matter annihilation cross section is too large to satisfy cosmological and astrophysical constraints simultaneously. In fact, when a suitable asymmetry between dark matter-antidark matter is generated in the early Universe, the large cross section nicely works to leave only the dark matter in the present Universe. Thus, the relic abundance of the antidark matter particle is much less than the dark matter relic abundance; as a result, the flux of cosmic rays created by annihilations of the dark matter and antidark matter particles is dramatically suppressed and the constraint by the Fermi-LAT observations is avoided. Since a variety of models to account for generating the dark matter asymmetry has been proposed (for an incomplete list, see e.g., [55-65]), we do not propose a specific model for it in this paper. However, we should note that some 'dark matter number violating' operator, in other words the global U (1) G breaking terms, is necessary to generate the dark matter asymmetry in the Universe and such an operator might induce a dark matter number violating mass term at low energies, which must be sufficiently suppressed [66] not to spoil the asymmetric dark matter scenario. Concretely speaking, in model F, for instance, we may introduce the following scenario by means of a scalar condensate, which is analogous to the Affleck-Dine baryogenesis [67]. Although none of scalar fields carrying U (1) G charges develop vacuum expectation values at the present Universe, we may add the global U (1) G as well as the CP violating potential, which is given as a function of the gauge invariant product, ˜ U R ˜ D R ˜ D R 2 , in the scalar potential. During the time that the Universe undergoes a false vacuum with nonvanishing expectation value 3 of 〈 ˜ U R ˜ D R ˜ D R 〉 , dark matter asymmetry can be dynamically generated through the evolution of the coherent scalar in the similar way as the Affleck-Dine baryogenesis [67]. Note that although the global U (1) G symmetry is explicitly broken by terms with ˜ U R ˜ D R ˜ D R , the model still possesses a residual Z 3 symmetry under which we may assign charges as ˜ Q L : ω , ˜ U R : ω , ˜ D R : ω , ψ : ω 2 , where ω = e i 2 π/ 3 . This Z 3 symmetry forbids a Majorana mass term for the dark matter. As above, in order not to induce the dark matter number violating mass term, the U (1) G breaking should arise via operators that respect a Z N subgroup of U (1) G , with N ≥ 3, independently of what mechanism actually generates the asymmetry. Then, this Z N symmetry forbids a dark matter number violating mass term for the dark matter. We have also considered collider experimental constraints on our model. Colored fermions and scalars introduced in our models can be produced at the LHC and their decays to the SM quarks and dark matter particles yield the signal events with jets and missing transverse energy. We have confirmed that our IVDM scenario is realized consistently with the current LHC bound on the mass of the colored particles. In the model S, the SM Higgs boson invisibly decays to a pair of dark matter particles and the upper bound on the invisible decay rate is given by the LHC data. We have found a parameter region in which the IVDM scenario is consistent with the LHC bound on the Higgs boson invisible decay rate. Interestingly, our successful asymmetric IVDM scenario leads to a lower bound on the invisible decay rate about 8%, so that precision measurements of the Higgs decay width at future collider experiments can test our scenario. Observable effects of the asymmetric dark matter scenario in neutron stars have been investigated [68, 69]. Since the dark matter particles do not self-annihilate, once captured in neutron stars, dark matter particles are continuously accumulating and neutron stars eventually collapse into black holes. Observations of old neutron stars provide constraints on parameters of the asymmetric dark matter scenario. In particular, such constraints are more severe for the case with a scalar dark matter because of the absence of Fermi degeneracy pressure. However, since the resultant constraints highly depend on the strength of dark matter self-interactions [69], we do not consider the constraints from the black formation in our scenario.", "pages": [ 13, 14, 15, 16, 17 ] }, { "title": "Acknowledgments", "content": "This work was supported in part by the DOE Grant No. DE-FG02-10ER41714 (N.O.), and by the scientific research grants from Hokkai-Gakuen (O.S). O.S would like to thank the Department of Physics and Astronomy at the University of Alabama for their warm hospitality where this work was initiated.", "pages": [ 17 ] } ]

[ { "title": "Quentin Wallemacq ∗", "content": "January 27, 2018 IFPA, Dép. AGO, Université de Liège, Sart Tilman, 4000 Liège, Belgium", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present a dark matter model reproducing well the results from DAMA/LIBRA and CoGeNT and having no contradiction with the negative results from XENON100 and CDMS-II/Ge. Two new species of fermions F and G form hydrogen-like atoms with standard atomic size through a dark U (1) gauge interaction carried out by a dark massless photon. A Yukawa coupling between the nuclei F and neutral scalar particles S induces an attractive shorter-range interaction. This dark sector interacts with our standard particles because of the presence of two mixings, a kinetic photon - dark photon mixing, and a mass σ -S mixing. The dark atoms from the halo diffuse elastically in terrestrial matter until they thermalize and then reach underground detectors with thermal energies, where they form bound states with nuclei by radiative capture. This causes the emission of photons that produce the signals observed by direct-search experiments.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Direct searches for dark matter have been accumulating results in recent years, starting with the DAMA/NaI experiment that observed a significant signal since the late 90's. Its successor, DAMA/LIBRA, has further confirmed the signal and improved its statistical significance to a current value of 8 . 9 σ [1]. Some other experiments such as CoGeNT [2], CRESST-II [3], and very recently CDMS-II/Si [4], are going in the same direction and report observations of events in their underground detectors, while others, such as XENON100 [5], or CDMS-II/Ge [6] continue to rule out any detection. These experiments challenge the usual interpretation of dark matter as being made only of Weakly Interacting Massive Particles (WIMPs). Because of the motion of the solar system in the galactic dark-matter halo, incident WIMPs would hit underground detectors where they could produce nuclear recoils, which would then be the source of the observed signals. However, this interpretation of the data induces strong contradictions between experiments with positive and negative results as well as tensions between experiments with positive results [5]. In this context, alternatives have been proposed to reconcile the experiments. Among them, mirror matter [7] and millicharged atomic dark matter [8] provide explanations respectively in terms of Coulomb scattering of millicharged mirror nuclei on nuclei in the detectors or in terms of hyperfine transitions of millicharged dark atoms analogous to hydrogen colliding on nuclei. In these scenarios, millicharged dark species are obtained by a kinetic mixing between standard photons and photons from the dark sector. Mirror matter in the presence of kinetic photon mirror photon mixing gives a rich phenomenology that can reproduce the signals of most of the experiments, but some tensions remain with experiments such as XENON100 or EDELWEISS. Millicharged atomic dark matter can explain the excess of events reported by CoGeNT but keeps the contradictions with the others. Another scenario has been proposed by Khlopov et al. [9, 10], in which new negatively charged particles (O --) are bound to primordial helium (He ++ ) in neutral O-helium dark atoms (OHe). The approach here is quite different because the interactions of these OHe with terrestrial matter are determined by the nuclear interactions of the helium component. Therefore, instead of producing nuclear recoils, these dark atoms would thermalize in the Earth by elastic collisions and reach underground detectors with thermal energies, where they form bound states with nuclei by radiative capture, the emitted photon being the source of the signal. Therefore, the observation of a signal depends on the existence of a bound state in the OHe - nucleus system and can provide a natural explanation to the negative results experiments, in case of the absence of bound states with the constituent nuclei. However, a careful analysis of the interactions of OHe atoms with nuclei [11] has ruled out the model. Nevertheless, the scenario presented here keeps many of the features of the OHe, but avoids its problems. Our model aims at solving the discrepancies between experiments with positive results, as well as to reconcile them with those without any signal. It presents common features with the ones mentioned above [7, 8, 9]. It contains dark fermions that possess electric millicharges due to the same kind of photon - dark photon mixing as in the mirror and atomic-dark-matter scenarios, but also another mixing between σ mesons and new dark-scalar particles creating an attractive interaction with nucleons, which couple to σ mesons in the framework of an effective Yukawa theory. The dark matter will be in the form of hydrogenoid atoms with standard atomic sizes that interact sufficiently with terrestrial matter to thermalize before reaching underground detectors. There, dark and standard nuclei will form bound states by radiative capture through the attractive exchange between dark fermions and nuclei. An important feature of such a model is that it presents a self-interacting dark matter, on which constraints exist from the Bullet cluster or from halo shapes [12]. According to [13], these can be avoided if the self-interacting candidate is reduced to at most 5% of the dark matter mass content of the galaxy, the rest being constituted by conventional collisionless particles. In the following, the dark sector will therefore be a subdominant part of dark matter. In Section 2, the ingredients and the effective lagrangian of the model are described. Constraints from vector-meson disintegrations are considered and the interaction potentials between dark and standard sectors are derived in Section 3, from the lagrangian of Section 2. The thermalization of the dark atoms in terrestrial matter is studied in Section 4 and constraints on model parameters are obtained, to thermalize between the surface and an underground detector. The radiative-capture process within a detector is described in Section 5, where the capture cross section and the event rate are derived. Section 6 gives an overview of the reproduction of the experimental results.", "pages": [ 1, 2 ] }, { "title": "2 The model", "content": "We postulate that a dark, hidden, sector exists, consisting of two kinds of new fermions, denoted by F and G , respectively coupled to dark photons Γ with opposite couplings + e ' and -e ' , while only F is coupled to neutral dark scalars S with a positive coupling g ' . This dark sector is governed by the lagrangian where the free and interaction lagrangians L dark 0 and L dark int have the forms and Here, ψ F ( G ) , A ' and φ S are respectively the fermionic, vectorial and real scalar fields of the dark fermion F ( G ) , dark photon Γ and dark scalar S , while m F ( G ) and m S are the masses of the F ( G ) and S particles. F ' stands for the electromagnetic-field-strength tensor of the massless dark photon Γ . Moreover, we assume that the dark photons Γ and the dark scalars S are mixed respectively with the standard photons γ and neutral mesons σ through the mixing lagrangian where m σ = 600 MeV [14] is the mass of σ and ˜ /epsilon1 and ˜ η are the dimensionless parameters of kinetic γ -Γ and mass σ -S mixings. These are supposed to be small compared with unity. The model therefore contains 7 free parameters, m F , m G , m S , e ' , g ' , ˜ /epsilon1 and ˜ η , and the total lagrangian of the combined standard and dark sectors is where L SM stands for the lagrangian of the standard model. The F and G fermions will form dark hydrogenoid atoms in which F will play the role of a dark nucleus binding to nuclei in underground detectors, while G acts as a dark electron. F has then to be heavy enough to form bound states and we will seek masses of F between 10 GeV and 10 TeV, while requiring m G /lessmuch m F . Due to the mass mixing term in (4), F will interact with nucleons through the exchange of S and this attractive interaction will be responsible for the binding. It cannot be too long-ranged but it must allow the existence of nucleus F bound states of at least the size of the nucleus. Because the range of the interaction is of the order of m -1 S , this leads us to consider values of the mass of S between 100 keV and 10 MeV. The other 4 parameters will not be directly constrained by the direct-search experiments, but only the products ˜ /epsilon1e ' and ˜ ηg ' . However, a reasonable choice seems to be ˜ /epsilon1, ˜ η /lessmuch 1 together with e ' /similarequal e and g ' /similarequal g , where e is the charge of the proton and g = 14 . 4 [15] is the Yukawa coupling of the nucleon to the σ meson. In summary, we will consider :", "pages": [ 2, 3 ] }, { "title": "3 Dark-standard interactions", "content": "The mixings described by (4) induce interactions [7, 8] between dark fermions F and G and our standard particles. It is well known that, to first order in ˜ /epsilon1 , a kinetic mixing such as the one present in (4) will make the dark particles F and G acquire small effective couplings ± ˜ /epsilon1e ' to the standard photons. One can define the kinetic mixing parameter in terms of the electric charge of the proton e through /epsilon1e ≡ ˜ /epsilon1e ' , which means that the particles F and G will interact electromagnetically with any charged particle of the standard model with millicharges ± /epsilon1e . The mass mixing from (4) characterized by ˜ η induces an interaction between F and σ , through the exchange of S , and hence an interaction between F and any standard particle coupled to σ , e.g. the proton and the neutron in the framework of an effective Yukawa theory. Since ˜ η is small, the interaction is dominated by one σ + S - exchange and the amplitude of the process has to be determined before passing to the non-relativistic limit in order to obtain the corresponding interaction potential. As for /epsilon1 introduced above, one defines η by ηg = ˜ ηg ' . In the following, except in Section 3.1, /epsilon1 and η will be used instead of ˜ /epsilon1 and ˜ η. In a similar way as in [8], the dark fermions F and G will bind to form neutral dark hydrogenoid atoms of Bohr radius a ' 0 = 1 µα ' , where µ is the reduced mass of the F -G system and α ' = e ' 4 π . In principle, the galactic dark matter halo could be populated by these neutral dark atoms as well as by a fraction of dark ions F and G , but ref. [16] shows that supernovae shock waves will evacuate millicharged dark ions from the disk and that galactic magnetic fields will prevent them from re-entering unless /epsilon1 < 9 × 10 -12 ( m F,G / GeV), which is far below the values that we will be interested in in the following to explain the signals of the direct-dark-mattersearch experiments. Therefore, the signals will only be induced by the interactions of the dark atoms with matter in the detectors.", "pages": [ 3, 4 ] }, { "title": "3.1 Constraints from Υ and J/ψ disintegrations", "content": "A direct consequence of the mass mixing term in (4) is that a certain fraction of σ 's can convert into S scalars and then evade in the dark sector. This can be seen in the disintegrations of quarkonium states such as the J/ψ meson and the 1 S and 3 S resonances of the Υ meson. The studied and unseen processes are generically represented by where Q ¯ Q = Υ(1 S ) , Υ(3 S ) or J/ψ (1 S ) . Because of the partity -1 of these states, the disintegration in two particles of parity +1 is forbidden, and one hence avoids the constraints from the first process. From [17], [18] and [19], the 90% C.L. upper limits on the branching ratios of the second process are respectively In the limit where the momenta of the constituent quarks are nul ( p = ( M Q ¯ Q / 2 , /vector 0) , where M Q ¯ Q is the mass of the Q ¯ Q meson), we get where B ( Q ¯ Q → e + e -) is the branching ratio of the disintegration of Q ¯ Q into a positron-electron pair, α = e 2 4 π = 1 137 is the fine structure constant, β = g 2 4 π = 16 . 5 , and m e is the mass of the electron. B ( Q ¯ Q → e + e -) = (2 . 38 ± 0 . 11)% , (2 . 03 ± 0 . 20)% and (5 . 94 ± 0 . 06) % [14], respectively for Q ¯ Q = Υ(1 S ) , Υ(3 S ) and J/ψ (1 S ) . Putting together (7) and (8), one gets allowed regions for parameters ˜ η and m S from processes (7). But for the rather small values of m S considered here, expression (8) turns out to be independent of the mass of the scalar particle and the most stringent constraint comes from the disintegration of J/ψ (1 S ) :", "pages": [ 4 ] }, { "title": "3.2 Interactions of F and G fermions with nucleons and electrons", "content": "The kinetic and mass mixings introduced in the lagrangian of the model give rise, in the nonrelativistic limit, to interaction potentials between the particles F and G and standard protons, neutrons and electrons. The kinetic γ -Γ mixing induces a Coulomb interaction with protons or electrons with a potential given by where the plus sign is for the proton -F and electron -G couplings, and the minus sign for the electron -F and proton -G interactions. The σ -S mass mixing gives rise, in the non-relativistic limit, to the one σ + S - exchange potential between F and a nucleon Note that in the limit m S → m σ , expression (11) becomes V M ( r ) = -ηm σ β 2 e -m σ r , although this particular case won't be considered in the following.", "pages": [ 4, 5 ] }, { "title": "4 Thermalization of dark FG atoms in terrestrial matter", "content": "Because of the motion of the Earth (and of the Sun) through the galactic dark matter halo, an effective wind of dark atoms hits the surface of our planet. These dark atoms penetrate the surface and undergo elastic collisions with terrestrial atoms, and lose part of their energy at each collision. If the number of collisions and the elastic-diffusion cross section are sufficiently large, then the dark atoms can deposit all their energy in the terrestrial matter before going out on the other side of the Earth, or even thermalize between the surface and an underground detector. The diffusions can be of two types : electromagnetic (atom - dark atom) and σ + S -exchange (nucleus F ), from potentials (10) and (11). In the following, we shall consider the terrestrial surface as made of 'average' atoms of silicon, with atomic and mass numbers Z m = 14 and A m = 28 and mass m m = A m m p , where m p is the mass of the proton. The nuclear radius will be neglected here, since it is much smaller than the wavelength of the incident particles at these energies, and has therefore no influence on the elastic cross section.", "pages": [ 5 ] }, { "title": "4.1 Interaction of dark FG atoms with terrestrial atoms", "content": "We assume that m F /greatermuch m G , and hence that m FG /similarequal m F , where m FG is the mass of an FG dark atom, so that in the dark bound state FG , F plays the role of a dark nucleus while G is spherically distributed around it. In this context, the dark FG atoms, as well as the terrestrial ones, are assimilated to uniformly charged spheres of charges -/epsilon1e and -Z m e and radii a ' 0 and a 0 , representing the respective electronic clouds, with opposite point-like charges at their centers, corresponding to the respective F and silicon nuclei. Because the elastic interaction cross section of a dark atom with a terrestrial atom has to be large enough to allow thermalization before reaching an underground detector, the atomic size of a dark atom will be of the same order as a standard one. We take 1 Å as a reference for the atomic size and set a ' 0 = 1 m G α ' = a 0 = 1 Å. In view of the suggestion e ' /similarequal e of Section 2 , this gives m G /similarequal m e . We then obtain the atom - dark atom electrostatic interaction potential as : r being the distance between both nuclei and ' at ' standing for ' atomic ' . The shape of V at is represented in Figure 1 for a silicon atom and for the best fit value of the kinetic mixing parameter /epsilon1 = 6 . 7 × 10 -5 , discussed in Section 6. It shows a very shallow potential well at r /similarequal a 0 . Its depth, of the order of 10 -3 eV, doesn't allow to create atom dark atom bound states, as they would be destroyed by thermal excitation in the Earth, where T ∼ 300 K corresponds to thermal energies of the order of 10 -2 eV. At smaller distance, when r /lessorsimilar 0 . 6 Å, the Coulomb repulsion between nuclei starts to dominate. Thus no atomic bound state can form with elements between the surface and an underground detector. In addition to this atom-dark atom interaction, both nuclei interact through σ + S -exchange, corresponding to the potential (11) multiplied by the number of nucleons in a silicon nucleus: where ' nucl ' stands for ' nuclear ' . Because m σ /greatermuch m S , this potential is very similar to a pure Yukawa potential ∼ -1 r e -mr . Although it creates a deeper attractive well at short distance (of the order of m -1 S /similarequal 100 fm), this narrower potential will neither admit stable bound states with the relatively light nuclei present in terrestrial matter. Therefore, the interactions of FG dark atoms in the Earth can be considered as purely elastic.", "pages": [ 5 ] }, { "title": "4.2 Elastic diffusion cross section", "content": "The elastic differential cross sections corresponding to the potentials (12) and (13) can be obtained by evaluating the square of the modulus of the diffusion amplitude in the framework of the Born approximation in the center of mass frame of the nucleus F system : with and where K = 2 k sin θ/ 2 and k = √ 2 µE are the transferred and initial momenta. θ is the deflection angle with respect to the collisional axis and µ = m F m m m F + m m is the reduced mass of the nucleus F system. The total differential cross section corresponding to V at + V nucl is finally given by the sum of (14) and (15) without forgetting the interference term :", "pages": [ 6 ] }, { "title": "4.3 Energy loss per unit path length : dE dx", "content": "At each collision with an atom of the terrestrial surface, a dark atom loses an energy /triangle K = p 2 (cos θ -1) m m in the frame of the Earth, where p is the relative momentum. The energy loss per unit length in the frame of the Earth is then obtained by integrating over all diffusion angles where n m is the numerical density of terrestrial atoms. Of course, the linear path approximation is valid only when m F /greatermuch m m , but it gives in the other cases an upper limit on the penetration length of the dark atoms through the Earth, which is of interest here. To obtain it, one just needs to integrate the inverse of (17) from the initial energy of the dark atoms E 0 to the thermal energy of the medium E th = 3 2 T m , where T m is the temperature", "pages": [ 6, 7 ] }, { "title": "4.4 Penetration at a depth of 1 km", "content": "Figure 2 shows the region (in blue) of mixing parameters /epsilon1 and η where x ≤ 1 km , 1 km being the typical depth at which underground detectors are located, for the best fit values m F = 650 GeV and m S = 0 . 426 MeV obtained in Section 6. In the blue region, thermalization occurs before reaching 1 km, while outside the dark atoms hit the detector with non-thermal energies and can cause nuclear recoils. The best-fit model, characterized by m F = 650 GeV, m S = 0 . 426 MeV, /epsilon1 = 6 . 7 × 10 -5 and η = 2 . 2 × 10 -7 clearly satisfies the condition with x /similarequal 40 m. Some interesting features are present in Figure 2 . At low η ( η /lessorsimilar 10 -9 ), thermalization is realized entirely by the electromagnetic atom - dark atom interaction V at , for sufficiently large /epsilon1 ( /epsilon1 /greaterorsimilar 10 -4 ) . When η increases ( 10 -9 /lessorsimilar η /lessorsimilar 3 × 10 -8 ), the limit on /epsilon1 slightly increases. This conter-intuitive behavior is due to the negative interference term present in the total elastic cross section (16) that increases with η . For a certain range of η ( 3 × 10 -8 /lessorsimilar η /lessorsimilar 6 × 10 -8 ), 3 regimes are visible : the first at low /epsilon1 , where thermalization is mostly ensured by the nuclear interaction; the second at intermediate /epsilon1 , where thermalization before 1 km is not possibe because the interference term partly compensates ( dσ d Ω ) at and ( dσ d Ω ) nucl in (16); the third at higher /epsilon1 , where thermalization is dominated by V at . Finally, at higher η ( η /greaterorsimilar 6 × 10 -8 ), all values of /epsilon1 are possible, meaning that nuclear interaction alone would be sufficient to thermalize.", "pages": [ 7 ] }, { "title": "5 Interactions in underground detectors", "content": "The dark atoms thermalize by elastic collisions in terrestrial matter between the surface and the underground detector. Once they reach thermal energies, they start drifting towards the center of the earth until they reach the detector, where they undergo collisions with the atoms of the active medium. Because of the Coulomb barrier due to the repulsion between nuclei (seen in Figure 1 at r /lessorsimilar 0 . 6 Å), most of these collisions are elastic but sometimes tunneling through the barrier can occur and bring a dark nucleus F into the region of the potential well present at smaller distance, due to the exchange of σ and S between F and the nuclei of the detector. There, E1 transitions produce de-excitation of the system to low-energy bound states by emission of photons that can be detected, causing the observed signal. In the following, only the part of the potential that is relevant for the capture process is considered, i.e. the region 0 < r /lessorsimilar 0 . 6 Å, where the interaction is dominated by the exchanges between F and the nucleus. The long-range part of the potential, 10 3 to 10 4 times smaller, does not affect the initial diffusion eigenstate and the final bound state of the process and is therefore neglected, and the dilute electronic and G distributions, mostly transparent to each other, follow passively their respective nuclei.", "pages": [ 7, 8 ] }, { "title": "5.1 Interactions of fermions F with nuclei", "content": "Because of their interactions with nucleons, the dark particles F interact with nuclei. If a nucleus N of mass number A and atomic number Z is seen as a uniformly charged sphere of radius R = r 0 A 1 / 3 , the integration of expressions (10) and (11) over its electric and nuclear charge distributions gives for the Coulomb potential, and ( ) ( ( )) Figure 3 shows the shape of the total potential V N = V N C + V N M for light, intermediate and heavy nuclei, all involved in underground detectors : Sodium (DAMA/LIBRA), Germanium (CoGeNT, CDMS-II), Iodine (DAMA/LIBRA) and Xenon (XENON100). All these potentials exhibit a Coulomb barrier, then an attractive well at shorter distance. The height of the barrier as well as the depth and the width of the well are determined by the values of the parameters /epsilon1 , η and m S , taken here equal to the prefered values of Section 6, but also depend on the nucleus. Typically, the depth of the well is of several keV and the Coulomb barrier goes up to several eV with a maximum being localized at about 2000 fm. for the one σ + S - exchange potential between F and a nucleus. In expression (20), V 0 = 3 η m 2 σ + m 2 S β/ 2 r 3 0 m 2 S -m 2 σ , where r 0 = 1 . 2 fm.", "pages": [ 8 ] }, { "title": "5.2 Bound-state formation mechanism", "content": "At thermal energies, to order v/c , only the partial s -wave of an incident plane wave on an attractive center is affected by the potential. Considering the center-of-mass frame of the nucleus -F system, this means that the largest contribution to tunneling corresponds to tunneling through the Coulomb barrier at zero relative angular momentum l . Due to selection rules, E1 transitions to final bound states at l = 0 are forbidden. It can also be shown that M1 and E2 transitions to such final levels are not present [20], leaving only the possibility of captures of the particles F in two steps, i.e. first to levels at l = 1 after tunneling and then to levels at l = 0 , each one corresponding to an E1 transition. The radiative capture of thermal particles F therefore requires the existence of bound states at least up to l = 1 in the potential wells of Figure 3. ) V e k ( N V 0 - - 5 10 - 15 - 20 - 25 0 . 1 1 r 10 (fm) The transition probability per unit time for an electric multipole radiation of order q is given by [20] where m = -q, ..., q , ω is the angular frequency of the emitted radiation and the matrix element Q qm = e ∑ N j =1 ' r q j Y m ∗ q ( θ j , ϕ j ) ψ ∗ f ψ i d -→ r . The sum is over all the electric charges e j of the system and the spherical harmonics Y m q are evaluated at the positions of each of them. ψ i and ψ f are respectively the initial and final states of the transition. In the framework of this model, one has for the E1 capture from an s - state in the continuum to a bound p - state, expressed in the center-of-mass frame of the nucleus F system in terms of relative coordinates -→ r = -→ r F --→ r N : where m is the mass of the nucleus. The term in Q 1 m due to the millicharged dark ion F has been neglected with respect to the term of the nucleus because of the factor /epsilon1 , that brings a factor /epsilon1 2 in the transition probability. The initial and final states are expressed as R and R f being respectively the radial parts of the eigenfunctions of the system at relative angular momenta l = 0 and l = 1 , corresponding to energies E (positive, incident) and E f (negative, lowest bound energy level at l = 1 ) in the center-of-mass frame. k = √ 2 µE , where µ is the reduced mass of the nucleus F system, is the momentum of the incident plane wave. The factor 1 k comes from the decomposition of a plane wave into partial waves. The link between the transition probability λ (1 , m ) and the capture cross section σ capt (1 , m ) is made via the relation λ (1 , m ) = nσ capt (1 , m ) v , where n is the number density of incident particles and v = | -→ v F --→ v N | is the relative velocity. ψ i is normalized in such a way that there is one incident particle per unit volume ( n = 1 ), by numerically solving the radial Schrodinger equation at l = 0 for the positive energy E and matching the function R ( r ) with the asymptotically free amplitude. The total E1 capture cross section σ capt is then obtained by summing the cross sections corresponding to the three possible values of m and one finally gets Sodium Germanium Iodine Xenon 100 1000 where D = ' ∞ 0 rR f ( r ) R ( r ) r 2 dr and µ = m F m m F + m is the reduced mass of the F - nucleus system. R f and E f are obtained by solving the radial Schrodinger equation at l = 1 with the WKB approximation and R f is normalized by demanding that ' R 2 f ( r ) r 2 dr = 1 .", "pages": [ 8, 9, 10 ] }, { "title": "5.3 Event counting rate", "content": "In the active medium of a detector made of nuclei N at temperature T , both F and N have velocity distributions P F ( -→ v F lab ) and P N ( -→ v N lab ) , where ' lab ' stands for 'laboraty frame' . We take them of the same Maxwellian form The event counting rate R per unit volume of the detector is given by where n F and n N are the numerical densities of F and N in the detector -and < σ capt v > is the thermally averaged capture cross section times the relative velocity Passing to center-of-mass and relative velocities -→ v CM and -→ v , using (24), (25), (26), (27) and performing the integration over the center-of-mass variables, we get where E = 1 2 µv 2 is the total energy in the center-of-mass frame. Considering the annual modulation scenario and requiring that the density of particles F in the detector is determined by the equilibrium between the incoming flux at the terrestrial surface and the down-drifting thermalized flux, driven by gravity, one can write down the numerical density n F within the detector as a function modulated in time : where ω = 2 π T orb is the angular frequency of the orbital motion of the Earth around the Sun and t 0 /similarequal June 2 is the period of the year when the Earth and Sun orbital velocities are aligned. The constant part is given by while the annual modulation of the concentration is characterized by the amplitude V h = 220 × 10 5 cm/s is the orbital velocity of the Sun around the galactic center, V E = 29 . 5 × 10 5 cm/s is the Earth orbital velocity around the sun, γ /similarequal 60 · is the inclination angle of the Earth orbital plane with respect to the galactic plane, n 0 = 3 × 10 -4 S 3 cm -3 is the local density of the dark atoms, n /similarequal 5 × 10 22 cm -3 is the numerical density of atoms in the terrestrial crust, g = 980 cm/s 2 is the acceleration of gravity and n 〈 σ at v 〉 is the rate of elastic collisions between a thermalized dark atom FG and terrestrial atoms. σ at is obtained by integrating the differential cross section (14) from section 4.2 over all diffusion angles in the case of a silicon atom and v is the relative velocity between a dark atom and a terrestrial atom. Note that σ at dominates over σ nucl at low energies, so there is no need to consider σ tot here. - × Expression (29) may be inserted into (28) to get an annually modulated counting rate per unit volume of the detector In counts per day and per kilogram (cpd/kg) of detector, the constant and modulated parts of the signal will respectively be given by with where Q = 1000 g, t = 86400 s, N Av = 6 . 022 × 10 23 and M mol is the molar mass of the active medium of the detector in g/mol.", "pages": [ 10, 11 ] }, { "title": "6 Results", "content": "The presented model intends to reproduce the positive results of direct dark matter searches experiments, such as DAMA/LIBRA and CoGeNT, without contradicting the negative results of some others, such as XENON100 or CDMS-II/Ge. The DAMA/LIBRA experiment observes an integrated modulation amplitude ˜ R m DAMA = (0 . 0464 ± 0 . 0052) cpd/kg in the energy interval (2 -6) keV [1], while the temporal analysis of CoGeNT has given ˜ R m CoGeNT = (1 . 66 ± 0 . 38) cpd/kg in the interval (0 . 5 -2 . 5) keV [21]. Here, in a first approximation and for simplicity, the signal is supposed to be made of one monochromatic line of energy ∆ E DAMA , ∆ E CoGeNT . It would be very interesting to reproduce the observed energy distributions of the rates by taking into account the possible transitions to the different s - states, but this is postponed to another paper. One first solves the Schrodinger equation independent on time with potential V N = V N C + V N M in cases of Iodine ( 127 I component of DAMA/LIBA detector), Germanium ( 74 Ge component of CoGeNT detector) and Xenon ( 132 Xe component of XENON100 detector) with the WKB approximation. This gives good estimates of the eigenvalues and eigenfunctions of the respective two-body bound state problems. The bound eigenfunctions are normalized numercially before computing the constant or modulated number density of F particles (30) or (31). The constant or modulated part of the event rate is finally computed for each nucleus from (33) with the expression (24) of the capture cross section, at the operating temperatures of the different detectors, i.e. T = 300 , 73 and 173 K for DAMA/LIBRA, CoGeNT and XENON100 respectively. One set of parameters that reproduces the data well and the corresponding transitions energies ( ∆ E ), lowest levels at l = 1 ( E l =1 ) and rates ( R 0 and R m ) are given in Table 1. The energies of the signals and the event rates are well reproduced for the DAMA and CoGeNT experiments. The lowest levels at l = 1 give rise to E1 captures that emit photons at threshold ( 2 keV for DAMA) or below threshold ( 0 . 5 keV for CoGeNT) and only the photon emitted during the second E1 transition from a p - state to an s - state is observed, making the captures look like single-hit events. The low predicted rate for XENON100 corresponds, over the total exposure of the experiment [5], to /similarequal 0 . 6 events. Therefore, no dark matter event should have occured within the XENON100 detector, which is consistent with observations. Also, if we set g ' = g , so that η = ˜ η, the best fit value of η is well below the limit (9) obtained from vector meson disintegrations. Computing the penetration length (18) with the parameters of Table 1, one finds that the dark atoms thermalize after /similarequal 40 m, so that they reach the detectors at thermal energies, as required by the model and already announced in Subsection 4.4. In a cooled detector, the dark atoms also have to thermalize when they pass from the laboratory room to the active medium, i.e. at the edge of the detector or over a distance smaller than its size. One can roughly estimate the penetration in a detector with the same formula (18), by setting E 0 = 3 2 T room and E th = 3 2 T , even if here the motions of the atoms in the thermalizer should be taken into account and the straight-line-path approximation is more questionable. This gives, for CoGeNT and XENON100, penetration lengths /similarequal 1 Å, which is clearly much smaller than the size of the detectors and corresponds to thermalizations directly at the edges. This model predicts an event rate consistent with zero in any cryogenic detector ( T /similarequal 1 mK), due to the Coulomb barrier of the nucleus F potential that prevents particles with very small energies to be captured in the well. This is in agreement with the negative results of the cryogenic CDMS-II/Ge (Germanium) experiment, in which thermalization when entering the detector is realized after /similarequal 1 µ m. In the same manner, we predict no events in the cryogenic CDMS-II/Si (Silicon) and CRESST-II detectors, in contradiction with the three events recently observed by the former and the signal of the latter. However, the penetration length in a cryogenic detector made of Silicon as CDMS-II/Si is /similarequal 1 mm, i.e. 3 orders of magnitude larger than its equivalent in Germanium. This is essentially due to the smaller electric charge of a Silicon nucleus, giving a weaker stopping power. In this case, more collisions happen near the edge of the detector, while the dark atoms are still at room temperature and hence more likely to cross the Coulomb barrier. These peripheral collisions should therefore be studied in detail to explain the events of some cryogenic detectors. In this analysis, attention has been paid to the Iodine component of the DAMA detector, while it is constituted by a crystal of NaI, and hence also of Sodium. Some part of the signal could come from this other component, but it turns out that the only bound state with 23 Na is very shallow ( -61 eV) and is at l = 0 . There is therefore no p - state on which the capture can happen, and the signal of DAMA is due only to its Iodine component. One can try to reproduce data directly with the Sodium component, but in that case the levels obtained afterward with Iodine are much too low (because the potential well is lower, as seen in Figure 3) and give rise to a signal out of the detection interval of DAMA. The fact that DAMA data are reproduced with the heavy component, Iodine, and not with the light one, Sodium, is in fact an advantage of the model, since in this situation, light isotopes do not have any bound states with dark atoms. The first element presenting an s bound state is Oxygen ( Z = 8 ) while the first one having at least one p bound state is Phosphorus ( Z = 15 ). Binding is therefore impossible for very light nuclei with Z ≤ 7 , preventing the formation of anomalous isotopes during BBN, while heavy isotopes cannot form on Earth with nuclei Z ≤ 14 , representing the majority of terrestrial elements.", "pages": [ 11, 12 ] }, { "title": "7 Conclusion", "content": "We have presented a model in which a fraction of the dark matter density ( 5% or less) is realized by two new species of fermions F and G , forming hydrogenoid atoms with standard atomic size through a dark U (1) gauge interaction carried out by a dark massless photon. Dark scalar particles S are exchanged by the nuclei F because of a Yukawa coupling between F and S . A kinetic photon - dark photon mixing and a mass σ -S mixing, respectively characterized by small dimensionless mixing parameters /epsilon1 and η , induce interactions between the dark sector and the ordinary one. The dark atoms interact elastically in terrestrial matter until they thermalize, in such a way that they reach underground detectors with thermal energies. There, they form bound states with nuclei by radiative capture, causing the emission of photons that create the observed signals. The model reproduces well the positive results from DAMA/LIBRA and CoGeNT, without contradicting the negative results from XENON100 with the following parameters : m F = 650 GeV, m S = 0 . 426 MeV, /epsilon1 = 6 . 7 × 10 -5 and η = 2 . 2 × 10 -7 . It naturally prevents any signal in a cryogenic detector ( T ∼ 1 mK), which is consistent with CDMS-II/Ge. Further studies have to be performed to explain the presence of a signal in CRESST-II, and possibly in CDMS-II/Si, especially by considering the collisions of the dark atoms at the edge of the detector, when they are still at room temperature while the detector is colder.", "pages": [ 12, 13 ] }, { "title": "Acknowledgments", "content": "I am grateful to my advisor, J.R. Cudell, for key reading suggestions and many discussions concerning this work. My thanks go to M. Khlopov for inspiring ideas and discussions and to M. Tytgat for useful comments. I thank the Belgian Fund F.R.S.-FNRS, by which I am supported as a Research Fellow.", "pages": [ 13 ] } ]

[ { "title": "Nonexistence of the final first integral in the Zipoy-Voorhees space-time", "content": "Andrzej J. Maciejewski ∗ J. Kepler Institute of Astronomy, University of Zielona G'ora, Licealna 9, PL-65-417 Zielona G'ora, Poland.", "pages": [ 1 ] }, { "title": "Maria Przybylska †", "content": "Institute of Physics, University of Zielona G'ora, Licealna 9, 65-417 Zielona G'ora, Poland", "pages": [ 1 ] }, { "title": "Tomasz Stachowiak ‡", "content": "Center for Theoretical Physics PAS, Al. Lotnikow 32/46, 02-668 Warsaw, Poland We show that the geodesic motion in the Zipoy-Voorhees space-time is not Liouville integrable, in that there does not exist an additional first integral meromorphic in the phase-space variables.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The question of the integrability of the test particle motion in the Zipoy-Voorhees metric has recently attracted some attention, with both numerical [1, 2] and analytical investigations [3]. The authors of [3] were able to exclude the existence of some polynomial first integrals, but they argue that some weaker form of integrability might take place taking into account the results of [1]. On the other hand, the results of [2] indicate chaotic behaviour of the system, but the region where that happens is very small when compared to the phase space dominated by invariant tori, and the integration was performed with the Runge-Kutta method of the 5th order only. Since it is known [4, 5] that integrable systems can exhibit numerical chaos (particularly for the R-K method), the results of [2] should be taken cautiously. Our own numerical integration produced a Poincar'e section visibly shifted from the one in [2] (see the end of section IV ), and since we used a more accurate method, it poses the question of whether the picture would be further deformed as the precision was increased. In other words, to decide on the integrability of the problem, a rigorous mathematical analysis is required rather than numerical simulations. The physical problem and its significance are the same as in the classical paper by Carter [6] - that the existence of an additional first integral in the Kerr spacetime makes the problem completely integrable. Carter's integral is not generated by a Killing vector, so it is not a usual symmetry of the manifold, but it is quadratic in momenta which has important consequences. Such integrals translate into the separability of the HamiltonJacobi equation and d'Alembertian [7], which in turn appears in the Teukolsky [8] equation. That is to say, both the classical problem of particle motion in this space-time, the linear perturbation equations governing the gravitational waves and potentially quantum equations in that background become considerably easier to solve. This fact is also used in numerical approaches, when trying to determine possible spectra of gravitational radiation in anticipation of the observed data [9]. It is then natural to analyze other space-times which could serve as models of compact objects, and the stationary axisymmetric ones are one direction to explore. However, despite some numerical evidence [1] we find that the particular Zipoy-Voorhees metric with the parameter δ = 2 is not integrable. To be more precise, we consider the motion of a test particle as a Hamiltonian system with n degrees of freedom and ask for the existence of an additional constant of motion I n that would yield Liouvillian integrability with respect to the canonical Poisson bracket {· , ·} . That is, for all first integrals I k we would have { I k , I l } = 0, where the Hamiltonian is included as H = I 1 , and I 2 , . . . , I n -1 are also already known. It turns out, that no such first integral can be found in the class of meromorphic functions, and we will use the differential Galois theory to prove that. Recall that a function is called meromorphic when its singularities (if it has any) are just poles; so by allowing first integrals that are potentially singular at some points of the phase space we are considering a fairly wide class of functions. The reason for using this particular theory is that it gives the strongest known necessary conditions for the integrability of dynamical systems. It was used for proving the nonintegrability of the hardest problems of classical mechanics, like the three-body problem [10, 11], which had been open for centuries. For an accessible overview of applications, see [12].", "pages": [ 1 ] }, { "title": "II. FORMULATION OF THE PROBLEM", "content": "The Zipoy-Voorhees metric under consideration is given by where x , y and φ form the prolate spheroidal coordinates. Instead of working directly with the geodesic equations we take the Hamiltonian approach with where the canonical coordinates are The equations then read with i = 0 , 1 , 2 , 3, and the normalization of four-velocities gives the value of the (conserved) Hamiltonian to be H = -1 2 µ 2 . The new time parameter is the rescaled proper time µτ = s , which allows us to include the zero geodesics for photons without introducing another affine parameter but with simply µ = 0. Since the metric has two Killing vector fields ∂ t and ∂ φ , the two momenta p 0 and p 3 are conserved. Together with the Hamiltonian they provide three first integrals. The question then is whether there exists one more first integral that would make the system Liouville integrable. To answer this question we employ the differential Galois approach to integrability. More specifically, we use the main theorem of the Morales-Ramis theory [15]. Theorem 1 If a complex Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the differential Galois group of the variational and the normal variational equations along any nonconstant particular solution of this system is Abelian.", "pages": [ 1, 2 ] }, { "title": "III. THEORETICAL SETTING", "content": "Let us try to explain the involved mathematics somewhat. For detailed exposition of the differential Galois theory the reader is referred to books [13, 14]. The Morales-Ramis theory is exposed in [15, 16], and a short introduction with application to another relativistic system can be found in [17]. To describe the differential Galois approach to the integrability we consider a general system of differential equations We assume that the right-hand sides are meromorphic in the considered domain. Let ϕ ( τ ) be a nonequilibrium solution of this system. Then the variational equation (VE) along this solution have the form It is not difficult to prove that if the original system has an analytic first integral I ( u ), then the variational equation have a time-dependent first integral I · ( τ, ξ ) which is polynomial in ξ . Similarly, one can show that if I ( u ) is a meromorphic first integral, then the variational equations (6) have a first integral I · ( τ, ξ ) which is rational in ξ . The Ziglin lemma, see p. 64 in [16], says that if the system (5) has 1 ≤ k < m functionally independent first integrals I j ( u ), j = 1 , . . . , k , then the variational equations (6) have the same number functionally independent first integrals I · j ( τ, ξ ) which are rational functions of ξ . In the considered theory, time is assumed to be a complex variable, and for complex τ ∈ C , the solution ϕ ( τ ) can have singularities. Assume that τ 0 ∈ C is not a singular point of ϕ ( τ ). Then in a neighborhood of τ 0 there exist m linearly independent solutions of the variational equations (6). They are the columns of the fundamental matrix Ξ( τ ) of the system (6). This matrix can be analytically continued along an arbitrary path σ on the complex plane avoiding the singularities of the solution ϕ ( τ ). Assume that σ is such a closed path, or loop, with the base point τ 0 . Let ̂ Ξ( τ ) be a continuation of Ξ( τ ). Solutions of a system of n linear equations form a linear n -dimensional space. Thus, in a neighborhood of τ 0 , each column of ̂ Ξ( τ ) is a linear combination of columns of Ξ( τ ). We can write this fact in the form ̂ Ξ( τ ) = Ξ( τ ) M σ , where M σ is a complex nonsingular matrix, i.e., M σ ∈ GL( m, C ). In fact, the matrix M σ depends only on the homotopy class [ σ ] of the loop. Taking all loops with the base point τ 0 we obtain an a group of matrices M⊂ GL( m, C ) which is called the monodromy group of the equation (6). One can show that if I · ( τ, ξ ) is a first integral of (6), then I · ( τ, ξ ) = I · ( τ, Mξ ) for an arbitrary M ∈ M , and for an arbitrary τ from a neighborhood of τ 0 . In other words, if the original system has a meromorphic first integral, then the monodromy group has a rational invariant. Hence, if the system possesses a big number of first integrals, then the monodromy group of variational equations cannot be too big because it has a large number of independent rational invariants. This observation can be transformed into an effective tool if we restrict our attention to Hamiltonian systems and the integrability in the Liouville sense (complete integrability). The above facts are the basic ideas of the elegant Ziglin theory [18, 19]. The problem in applying this theory is that the monodromy group is known for a very limited number of equations. At the end of the previous century the Ziglin theory found a nice generalization. It was developed by Baider, Churchill, Morales, Ramis, Rod, Sim'o and Singer, see [15, 16, 20] and references therein. In the context of Hamiltonian systems it is called the Morales-Ramis theory, and in some sense, it is an algebraic version of the Ziglin theory. It formulates the necessary conditions for the integrability in terms of the differential Galois group G ⊂ GL( m, C ) of the variational equations. It is known that it is a linear algebraic group and that it contains the the monodromy group. By definition it is a subgroup of GL( n, C ) which preserves all polynomial relations between solutions of the considered linear system, see [21]; and for a wide class of equations it is generated by M . The differential Galois group can serve for a study of integrability problems on the same footing as the monodromy group. Namely, first integrals of (6) give rational invariants of G . If the considered system is Hamiltonian then necessarily m = 2 n , and groups M and G are subgroups of the symplectic group Sp(2 n, C ). It can also be shown that the differential Galois group G is a Lie group. If the system is completely integrable with n meromorphic first integrals, then G has n commuting rational invariants. The key lemma, see p. 72 in [16], states that if the above is the case, then the Lie algebra of G is Abelian. This means exactly that the identity component of G is Abelian. Determination of the differential Galois group is a difficult task. Fortunately, in the context of integrability, we need to know only if its identity component is Abelian. If it is not Abelian then the system is nonintegrable. If we find that a subsystem of VE has a non-Abelian identity component of the differential Galois group, then conclusions are the same. This is why, in practice, we always try to distinguish a subsystem of VE. It is easy to notice that ψ ( t ) = f ( ϕ ( t )) is a solution of (6). Using it we can reduce the dimension of VE by one. If the system (5) is Hamiltonian, then first we restrict it to the energy level of the particular solution. In effect, in Hamiltonian context we can easily distinguish a subsystem of variational equations of dimension 2( n -1), which are called the normal variational equations (NVE). The difficulty of investigation of the differential Galois group of NVE depends, among other things, on the form of its matrix of coefficients, and so also on the functional form of particular solution. Quite often, by an introduction of a new independent variable z = z ( τ ) we can transform NVE to a system with rational coefficients The set of rational functions C ( z ) is a field, and equipped with the usual differentiation it becomes a differential field. Solutions of a system with rational coefficients are typically not rational. The smallest differential field containing all solutions of (7) is called the PicardVessiot extension of C ( z ). The differential Galois group G of (7) tells us how complicated its solutions are, i.e., if the equations are solvable. Here solvability means that all solutions can be obtained from a rational function by a finite number of integrations, exponentiation and algebraic operations [13]. This category of functions, called Liouvillian, includes all elementary functions, as well as some transcendental, such as the logarithm or elliptic integrals, and is commonly referred to as 'closed-form' or 'explicit' solutions. The following classical result connects the structure of G with the form of the solutions. Theorem 2 System (7) is solvable, i.e., all its solutions are Liouvillian, if and only if the the identity component of its differential Galois group is solvable, The connection of this theorem with integrability is the following. If it is possible to show that either NVE, or a subsystem of NVE are not solvable, then the identity component of their differential Galois group is not solvable, so, in particular is not Abelian. Thus, by Theorem 1, the system is not integrable. The question of whether a given system with rational coefficient is solvable can be resolved completely for a system of two equations (or one equation of second order). In this case there is an effective algorithm by Kovacic for finding the Liouvillian solutions [22]. This algorithm gives a definite answer, and if Liouvillian solutions exist it provides their analytical form. There exist a similar, almost complete algorithm for systems of three equations and some partial results for systems of four equations.", "pages": [ 2, 3 ] }, { "title": "IV. PROOF OF NONINTEGRABILITY", "content": "The plan of attack is thus to look for particular solutions for which the NVE has a block structure so that a two-dimensional subsystem can be separated. We then rewrite it as a second-order linear differential equation with rational coefficients and apply the Kovacic algorithm to see if it has any Liouvillian solutions. Note that the system has no external parameters, and only the values of particular first integrals enter as internal parameters. They are synonymous with initial conditions, so that if we manage to find just one solution, for particular values of µ , p 0 and p 3 , such that the respective NVE is unsolvable, we will have proven that there cannot exist another first integral over the whole phase space. It might so happen, unlike in the Carter case, that the system exhibits some particular invariant set on which there exists an additional integral. For example, one could have ˙ I 4 = H , which would mean that I 4 is conserved on the zero-energy hypersurface µ = 0, which is clearly a physically distinguished case. We will then have to look for particular solutions on those sets to make the results even more restrictive than just the lack of a global first integral. The obvious particular solution to look at is a particle moving along a straight line, through the center in the equatorial plane, which in prolate coordinates means y = 0 and p 3 = 0. The nontrivial equations then read: Or, upon rescaling time by we have where the dot denotes differentiation with respect to u , and we have omitted the first equation, as the other two do not depend on t . This two-dimensional subsystem defines the particular solution around which we will construct the NVE as mentioned before. The conservation of the Hamiltonian now reads which together with the equation for ˙ x yields so that x ( u ) is expressible by the Jacobi elliptic functions. This fact is important, as we will change the independent variable from u to x which is permissible (does not change the identity component of the Galois group) only if the function x ( u ) defines a finite cover of the complex plane [14]. The variational equations along this solution separate so that the NVE read where the variations ξ correspond to the perturbations of variables y and p 2 . This is another step of the reduction mentioned in the previous section - the particular solution only have x and p 1 components, and the NVE only has components in the orthogonal directions of y and p 2 . Introducing a new dependent variable /negationslash and taking x as the new independent variable, the NVE can be brought to the standard form of with the rational coefficient r where R is the following polynomial /negationslash Since for all physical particles we have p 0 = 0 all the others parameters can be rescaled by it which we use in what follows. As is customary, we will use the same notation as in Kovacic's paper, adhering exactly to the steps and cases of the algorithm [22]. We note that a linear equation like (15) has local solutions in some neighborhood of a singularity x /star of r ( x ), which take the form /negationslash where g is analytic at zero, g (0) = 0, and α is called the characteristic exponent. The algorithm checks if it is possible to construct a global solution, which, in the simplest case, is of the form for a polynomial P ( x ) and rational ω ( x ). The degree of P is then linked with the exponents and that provides preliminary restrictions on the parameters' values and integrability. The application of the algorithm itself is straightforward, and the only complication is that the singularities and exponents might depend on parameters. Fortunately there are only several special values of µ that influence the outcome, and we outline the general steps in the two subsections below. For details, the reader is referred to [22], and another version of the algorithm, as applied to the dynamical system of the Bianchi VIII cosmology, can be found in [23].", "pages": [ 3, 4 ] }, { "title": "A. General r ( x )", "content": "The poles of r ( x ) are /negationslash and for all of them to be different we must have µ 2 = 1 and µ = 0. All are of order 2, and the order at infinity is 4, so that we need to check all the cases of the algorithm. In case 1, the characteristic exponents α ± c of (15) form the following set where the first pair corresponds to ∞ , and the combinations give only nine non-negative integers (not all distinct) as possible degrees of the appropriate polynomial P , which enters into the solution of (15). However, the respective test solutions of the form as in (20) require that µ = 0, and have to be discarded so that this case cannot hold. In case 2, the families of exponents E c are which in turn give 131 possible integer degrees for the appropriate polynomial. Checking them one by one, we find that they require µ = ± 1 in order to form a solution, so that this case can be discarded as well under the current assumptions. In case 3, the families E c contain 6 × 13 = 78 numbers, which make 4826809 combinations for d out of which 230856 are non-negative integers. We thus first resort to checking for the presence of logarithms in the solutions, which would prevent this case [14]. The only poles with integer difference in the exponents are 0 and ∞ . Using the Frobenius method [24], we get the two independent solutions around zero /negationslash As can be seen, the logarithm is present when µ 2 = 5, and since the solutions around ∞ do not have logarithms at all, the only possibility for case 3 left here is with µ 2 = 5.", "pages": [ 4, 5 ] }, { "title": "B. Special subcases", "content": "/negationslash In order to exclude the special energy hypersurfaces µ = 0, µ 2 = 1 and µ 2 = 5, we have to resort to a more general particular solution, namely one with p 3 = 0. As already mentioned, it is enough to find one solution for each such surface, and that means we can take a specific value of p 3 . The corresponding NVE will only have numeric coefficients, and checking for its Liouvillian solutions is much easier, for it suffices to use one of available implemented routines, for example the 'kovacicsols' of the symbolic system Maple. The solution will also be expressible by (hyper)elliptic function as defined by the Hamiltonian constraint and the counterparts of the NVE given in (13) will read We then proceed exactly as above, taking x as the independent variable and reducing the system to one equation of the form ξ '' 2 = rξ 2 . For each hypersurface in question, the value of p 3 = 1 leads to NVE that are not solvable with Liouvillian functions. This finishes the proof for all possible levels of the Hamiltonian. To further illustrate the complexity of this system, we have also obtained a Poincar'e section for the cross plane y = 0 shown in Fig. 1. The numerical integrator was based on the Bulirsch-Stoer modified midpoint scheme with Richardson extrapolation. We note that the special solution defined by (12) lies entirely in the plane y = 0 and is a trajectory beginning and ending at the singularity so it does not contribute to the section. It also lies outside the visible chaotic region, which is confined to a very small subset of the phase space, as mentioned in [2].", "pages": [ 5 ] }, { "title": "V. GENERAL METRIC", "content": "The above results carry, to some extent, to the general Zipoy-Voorhees metric given by where δ ∈ R . The main problem that arises for arbitrary δ is that the special solution might no longer be a (hyper)elliptic function, because the Hamiltonian now gives so the right-hand side is not necessarily a polynomial or rational function. Accordingly, the rationalization of the NVE might not preserve the identity component of the differential Galois group. However, when δ is rational we can still proceed by taking a new dependent variable to be as this leads to the normal form (15) which involves only integral powers of w and w δ . Assuming then that δ = p/q , we can make the NVE rational by taking w 1 /q as the new variable if need be. Unfortunately, the number of poles (and their values) now depends on p and q , so the Kovacic algorithm has to be applied to each δ separately, but for each of them it is as straightforward as above to use the Maple package, once suitable numeric values of the parameters have been chosen. For example, we have verified that δ = 1 / 2 is also nonintegrable, confirming the numerical evidence of [2] that for both δ > 1 and δ < 1, the general metric does not admit additional first integrals.", "pages": [ 5, 6 ] }, { "title": "VI. CONCLUSIONS", "content": "Our main result can be stated as Theorem 3 There does not exist an additional, meromorphic first integral of the geodesic motion in the Zipoy- Voorhees metric (1) , i.e., the system is not Liouville integrable. This confirms the previous considerations of [3] and goes much further than excluding first integrals polynomial in momenta up to a certain fixed small degree. Meromorphic functions include not only the analytic functions of both momenta and coordinates, but also rational and transcendental ones as long as their singularities are just poles. In particular, it follows that even if a conserved quantity exists, it cannot be expressed by an explicit formula of the above type. This result thus strongly reduces the possibility of using constants of motion expansion in solving the equations of geodesic motion or gravitational waves because the decomposition in terms of normal frequencies requires one to calculate their values directly from the initial conditions of the coordinates and momenta [9]. Of course, further techniques can be used to better understand and describe the motion, especially in the region where the dynamics is regular, but the fundamental physical property of this space-time is that no additional conservation law holds.", "pages": [ 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This research has been supported by grant No. DEC2011/02/A/ST1/00208 of National Science Centre of Poland. 1 p", "pages": [ 6, 7 ] } ]

[ { "title": "Dirk Puetzfeld ∗", "content": "ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany", "pages": [ 1 ] }, { "title": "Yuri N. Obukhov †", "content": "Theoretical Physics Laboratory, Nuclear Safety Institute, Russian Academy of Sciences, B.Tulskaya 52, 115191 Moscow, Russia (Dated: September 10, 2018) We derive multipolar equations of motion for gravitational theories with general nonminimal coupling in spacetimes admitting torsion. Our very general findings allow for the systematic testing of whole classes of theories by means of extended test bodies. One peculiar feature of certain subclasses of nonminimal theories turns out to be their sensitivity to post-Riemannian spacetime structures even in experiments without microstructured test matter. PACS numbers: 04.20.Fy; 04.50.Kd; 04.20.Cv Keywords: Equations of motion; Conservation laws; Approximation methods", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In a recent work [1] we derived the conservation laws for the most general class of nonminimally coupled gravity theories. Here we are going to work out the equations of motion for this whole class of theories by using Synge's expansion technique [2] in combination with a multipolar framework 'a la Dixon [3]. The framework does not only cover the metric case, but it is also general enough to cope with theories which go beyond the usual Riemannian framework [4]. In particular it allows for a generalized discussion of microstructured media. The results obtained here extend the ones in [5-13]. In particular, they offer a new perspective on placing possible observational constraints on new geometric features like torsion. Our notations and conventions are those of [4]. In particular, the basic geometrical quantities such as the curvature, torsion, etc., are defined as in [4], and we use the Latin alphabet to label the spacetime coordinate indices. Furthermore, the metric has the signature (+ , -, -, -). The structure of the paper is as follows: In section II we briefly discuss the class of theories under consideration. In particular we provide the conservation laws, which in turn are crucial for the subsequent derivation of the multipolar equations of motion in section III. Apart from providing the general form of these equations, we study the pole-dipole equations of motion in detail, and thereby find an analogue to the classical Mathisson-Papapetrou [14, 15] equations for the whole class of nonminimal coupling theories under consideration. Furthermore, we discuss the case of test matter without microstructure and its peculiar type of coupling to post-Riemannian spacetime features. Our final conclusions and an outlook on open problems is given in section IV. Appendices A and B contain a brief overview of our conventions and some frequently used formulas.", "pages": [ 1 ] }, { "title": "II. GENERAL NONMINIMAL GRAVITY", "content": "In order to be as general as possible, we consider matter with microstructure, namely, with spin. An appropriate gravitational model is then the Poincar'e gauge theory in which the metric tensor g ij is accompanied by the connection Γ ki j that is metric-compatible but not necessarily symmetric. The gravitational field strengths are the Riemann-Cartan curvature and the torsion: In [1], we worked out the conservation laws for a general nonminimal gravity model in which the interaction Lagrangian reads The coupling function F ( g ij , R kli j , T kl i ) depends arbitrarily on its arguments. In technical terms, F is a function of independent scalar invariants constructed in all possible ways from the components of the curvature and torsion tensors. The matter Lagrangian has the usual form L mat = L mat ( ψ A , ∇ i ψ A , g ij ). A Lagrange-Noether analysis, see [1], yields the following conservations laws: Here we made use of the following abbreviations, i.e. for the canonical energy-momentum tensor, for the canonical spin tensor, and for the metrical energy-momentum tensor. Furthermore, we made use of the so-called modified covariant derivative, which is defined as usual by Lowering the index in (4) and antisymmetrizing, we derive the conservation law for the spin This is a generalization of the usual conservation law of the total angular momentum for the case of nonminimal coupling.", "pages": [ 1, 2 ] }, { "title": "A. Purely Riemannian theory", "content": "Our results contain the Riemannian theory as a special case. Suppose the torsion is absent T ij k = 0. Then for usual matter without microstructure (spinless matter with τ m n i = 0) the canonical and the metrical energymomentum tensors coincide, Σ k i = t k i . As a result, the conservation law (5) reduces to", "pages": [ 2 ] }, { "title": "B. Further generalization: Matter with intrinsic moments", "content": "Our formalism allows one to consider also the case when matter couples to the gravitational field strengths not just through an F -factor in front of the Lagrangian but directly via Pauli-type interaction terms in L mat : In Maxwell's electrodynamics similar terms describe the interaction of the electromagnetic field to the anomalous magnetic and/or electric dipole moments. For Dirac spinor matter [16, 17], the Pauli-type quantities I klm n ( ψ A , g ij ) and J kl n ( ψ A , g ij ) are interpreted as the (Lorentz and translational, respectively) 'gravitational moments' that arise from the Gordon decomposition of the dynamical currents. The on-shell conservation laws are then given by: The skew-symmetric part of (13) describes the generalized conservation of the angular momentum: /negationslash For the Riemann-Cartan curvature tensor the pairs of indices do not commute, R ijkl = R klij , and one cannot reduce the two terms in the second line of (15). However, in the purely Riemannian case of General Relativity, the torsion vanishes and the curvature tensor has more symmetries (in particular, the pairs of indices do commute). Then the system (14) and (15) reduces to the familiar Mathisson-Papapetrou form The symmetric part of equation (16) describes the relation between the metrical and canonical energymomentum tensors. When deriving (16), we took into account that in view of the contraction in (12), we have the symmetry properties The form of the system of conservation laws (16)-(17) is very close to Dixon's equations describing the dynamics of material body with the dipole and quadrupole moments. However, it is important to stress that in contrast to Dixon's integrated moments of usual structureless matter, τ [ ik ] n and I ilnm are the intrinsic spin and quadrupole moments of matter with microstructure. The above conservation laws can also be viewed as a direct generalization of the ones for spinning particles and polarized media given in [18]. It is worthwhile to note that in the Riemann-Cartan spacetime the conservations laws (14) and (15) contain two types of intrinsic quadrupole moments. We identify I ijkl with the rotational (Lorentz) quadrupole moment, whereas J kl i is naturally interpreted as the translational quadrupole moment. These quantities are coupled to the corresponding rotational and translational gravitational field strengths, i.e., to the curvature R ijkl and the torsion T kl i , respectively.", "pages": [ 2 ] }, { "title": "III. EQUATIONS OF MOTION", "content": "The conservation equations (4) and (5) form the basis for a general multipolar analysis. In the following we are going to derive the equations of motion for test bodies by utilizing the expansion technique of Synge [2]. Since we are now working in a spacetime which allows for more structure, we now also have - apart from the metric g ab - the torsion T ab c . This leads to an additional degree of freedom regarding the transport operations in the underlying multipolar formalism. We can proceed in two ways: (i) extend Synge's technique to non-Riemannian spacetimes - thereby switching to a new type of (non-geodesic) reference curve; or (ii) use the standard Riemannian approach and treat torsion as an additional variable. Here we follow the latter strategy.", "pages": [ 2, 3 ] }, { "title": "A. Rewriting conservation laws", "content": "The Riemann-Cartan connection can be decomposed into the Riemannian (Christoffel) connection plus the post-Riemannian piece: Here the contortion tensor reads We use the hat to denote objects and operators (such as the curvature, covariant derivatives, etc) defined by the Riemannian connection (19). Using the decomposition (20), we rewrite the conservation laws (4)-(5) as We can develop the usual Riemannian world-function based multipole expansion starting from (22) and (23). Defining auxiliary variables like in [13], i.e. A ( g ij , R ijk l , T ij k ) := log F , A i := ∇ i A , A ij := ̂ ∇ j ∇ i A etc., we rewrite (22) and (23) as follows: Here we introduced the shortcut Ξ ij := g ij L mat .", "pages": [ 3 ] }, { "title": "B. Multipolar approximation", "content": "We will now derive the equations of motion of a test body by utilizing the covariant expansion method of Synge [2]. For this we need the following auxiliary formula for the absolute derivative of the integral of an arbitrary bitensor density ˜ B x 1 y 1 = ˜ B x 1 y 1 ( x, y ) (the latter is a tensorial function of two spacetime points): Here v y 1 := dx y 1 /ds , s is the proper time, D ds = v i ̂ ∇ i , and the integral is performed over a spatial hypersurface. Note that in our notation the point to which the index of a bitensor belongs can be directly read from the index itself; e.g., y n denotes indices at the point y . Furthermore, we will now associate the point y with the world-line of the test body under consideration. Denote We start by integrating (24) and (25) using (26): Here the derivatives are straightforwardly evaluated: where z stands either for x or for y . We now introduce integrated moments 'a la Dixon in [3], i.e. Then (29) and (30) take the form", "pages": [ 3, 4 ] }, { "title": "C. Vanishing spin current", "content": "For the special case of vanishing spin current τ abc = 0, we infer from (24) that the canonical energy-momentum tensor is symmetric Σ [ ij ] = 0, and that it coincides with the metrical energy-momentum tensor in view of (4). Furthermore, we have as a starting point for the derivation of the equations of motion Due to the antisymmetry of the contortion, the contraction in the first term with the symmetric t moment - in the case of an absent spin current - vanishes identically. Hence we are left with structurally the same equation as in [13], the only 1 difference being that here A ( g ij , R ijk l , T ij k ) is a function of the curvature and the torsion. For the vanishing spin all the corresponding multipole moments (36) and (37) vanish, too: s y 2 ...y n +1 y 0 y 1 = 0 and q y 3 ...y n +1 y 0 y 1 y 2 = 0 for any n . In addition, the multipole moments t y 2 ...y n +1 y 0 y 1 are symmetric in the last two indices.", "pages": [ 4, 5 ] }, { "title": "1. Monopole order ( τ ab c = 0 )", "content": "At the monopole order we have Substituting (42) into (41), we recover the equation of motion [13] As we see, the nonminimal coupling is manifest in the nongeodetic motion of the monopole test particle.", "pages": [ 5 ] }, { "title": "2. Pole-dipole order without spin ( τ ab c = 0 )", "content": "At the pole-dipole order we obtain Note that we did not make any simplifying assumptions about the spacetime which still has the general RiemannCartan geometric structure with nontrivial torsion. Nevertheless, neither torsion nor contortion contributes to the equations of motion (45) and (46).", "pages": [ 5 ] }, { "title": "D. General pole-dipole equations of motion", "content": "Let us consider the general case when the extended body consists of material elements with microstructure, i.e., with spin. In the pole-dipole approximation, the relevant moments 2 are p a , p ab , t ab , t abc , ξ ab , ξ abc , s ab , q abc , and we neglect all higher multipole moments. Then for n = 1 and n = 0, eq. (38) yields whereas (39) for n = 2, n = 1, and n = 0 yields p a = K cd t + K cd ; b t Combining (47) with (49), we derive Furthermore, we can substitute (47) into (52) and thus express t [ a | c | b ] in terms of the p -, q -, and s -moments: Antisymmetrizing (50), we find Combining this equation with (48), we eliminate t [ ab ] and using (47) derive Next, substituting (47), (48), and (53) into (51), we obtain after some algebra We now introduce the integrated orbital angular momentum and the integrated spin angular momentum of an extended body as respectively. Then, after a straightforward but rather lengthy computation, we can recast (56) and (57) into the final form a cd a bcd D ds Here we defined the total energy-momentum vector and the total angular momentum tensor by In addition, we introduced a redefined moment By construction, Q bc a = -Q cb a . In the derivation of (59) and (60) we made use of (47), (54) and took into account the geometrical identity ̂ The equations of motion (59) and (60) generalize the results obtained in [13] to the case when extended bodies are built of matter with microstructure and move in a Riemann-Cartan spacetime with nontrivial torsion. The latter can be proved by substituting the decomposition of the Riemann-Cartan connection (20) into the curvature definition (1). Furthermore, it is helpful to notice that q cd [ a K b ] cd +2 q c [ a | d | K dc b ] ≡ Q cd [ a T cd b ] +2 Q [ a cd T b ] cd and q cdb K bcd ; a ≡ Q bc d ∇ a T bc d .", "pages": [ 5, 6 ] }, { "title": "1. Minimal coupling", "content": "When the coupling function is constant, F = 1, that is for the minimal coupling case, we obtain and the equations of motion Comparing these equations to the conservation laws (14) and (15), it is remarkable that the redefined dipole spin moment (63) actually took over the role of the translational quadrupole moment. That is, up to a factor ( -2), conventionally introduced in (58), we can identify Q bc a with J bc a . This interesting feature was not reported before.", "pages": [ 6 ] }, { "title": "2. Nonminimal coupling: a loophole to detect torsion?", "content": "It is satisfying to see that the structure of the equations of motion (66)-(67) is in agreement with the earlier results of Yasskin and Stoeger [6]. Therefore, we confirm once again that spacetime torsion couples only to the integrated spin S ab , which arises from the intrinsic spin of matter, and the higher moment q abc . Hence, usual matter without microstructure cannot detect torsion and, in particular, experiments with macroscopically rotating bodies such as gyroscopes in the Gravity Probe B mission do not place any limits on torsion [19]. However, this conclusion is apparently violated for the nonminimal coupling case. As we see from (59) and (60), test bodies of structureless matter could be affected by torsion via the derivatives of the coupling function F ( g ij , R kli j , T kl i ). On the other hand, this possibility is qualitatively different from the ad hoc assumption that structureless particles move along auto-parallel curves in the Riemann-Cartan spacetime made in [2023]; see the critical assessment in [19]. The trajectory of a monopole particle, described by (43), is neither geodesic nor auto-parallel. The same is true for the dipole case when the nonminimal coupling force is combined with the Mathisson-Papapetrou force.", "pages": [ 6 ] }, { "title": "IV. CONCLUSION", "content": "We have obtained equations of motion for material bodies with microstructure, thus generalizing the previous works [5, 6, 8, 9, 18] to the general framework with nonminimal coupling. The master equations (38) and (39) describe the dynamics of an extended body up to an arbitrary multipole order. It turns out that, despite a rather complicated general structure of the equations of motion, most of the terms in (38) and (39) show up only at the quadrupole order or higher orders. In the special case of minimal coupling (which is recovered when F = 1), our results can be viewed as the covariant generalization of the ones in [5, 6], as well as the parts concerning Poincar'e gauge theory of [8]. A somewhat surprising result in the present nonminimal context with torsion, is the - indirect - appearance of the torsion through the coupling function F even in the lowest order equations of motion for matter without intrinsic spin - see eqs. (41)-(42). This clearly is a distinctive feature of theories which exhibit nonminimal coupling, which sets them apart from other gauge theoretical approaches to gravity. As we have shown in [6, 8, 9], and as it is also discussed at length in the recent review [19], in the minimally coupled case only microstructured matter couples to the post-Riemannian spacetime features - in particular, in the minimally coupled case one needs matter with intrinsic spin to detect the possible torsion of spacetime. As we have shown in the current work, this is no longer the case in the nonminimally coupled context. In other words, supposing that one can come up with a sensible background model for spacetime including torsion, it could be somewhat constrained through standard test bodies - i.e. made from regular matter - through the derived equations of motion, in particular through (41)-(42) in the monopolar case. Despite the progress made here, we would also like to point out some open questions and directions for future investigations. (i) In a post-Riemannian context, there is naturally more freedom regarding the possible geometry of spacetime. This additional freedom could also be used for an extension and modification of the multipolar framework itself in general spacetimes encompassing, besides the curvature, also new quantities like torsion. In particular, one could carry out the derivations in the present work with a modified world-function formalism, i.e. one which is no longer based on the geodesic structure of the spacetime - see also [24-26] for some generalizations in this direction. While such a modification remains a possibility, which is somewhat linked to the discussion of which types of curves are 'natural' in specific spacetimes, one should also be clear that one would loose comparability with almost all of the previous works on equations of motion. (ii) Another generalization concerns the generalization to the metric-affine case, i.e. including, apart from the torsion, also the nonmetricity of spacetime. The results in this paper already hint into this direction. In general non-Riemannian spacetimes, one can expect a direct coupling term, not only through the function F , on the level of the equations of motion. This will eventually lead to more 'fine grained' possible tests of post-Riemannian geometric structures.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant LA-905/8-1/2 (D.P.).", "pages": [ 7 ] }, { "title": "Appendix A: Conventions & Symbols", "content": "In the following we summarize our conventions, and collect some frequently used formulas. A directory of symbols used throughout the text can be found in table I. For an arbitrary k -tensor T a 1 ...a k , the symmetrization and antisymmetrization are defined by where the sum is taken over all possible permutations (symbolically denoted by π I { a 1 . . . a k } ) of its k indices. As is well-known, the number of such permutations is equal to k !. The sign factor depends on whether a permutation is even ( | π | = 0) or odd ( | π | = 1). The number of independent components of the totally symmetric tensor T ( a 1 ...a k ) of rank k in n dimensions is equal to the binomial coefficient ( n -1+ k k ) = ( n -1+ k )! / [ k !( n -1)!], whereas", "pages": [ 7 ] }, { "title": "Auxiliary quantities", "content": "̂ the number of independent components of the totally antisymmetric tensor T [ a 1 ...a k ] of rank k in n dimensions is equal to the binomial coefficient ( n k ) = n ! / [ k !( n -k )!]. For example, for a second rank tensor T ab the symmetrization yields a tensor T ( ab ) = 1 2 ( T ab + T ba ) with 10 independent components, and the antisymmetrization yields another tensor T [ ab ] = 1 2 ( T ab -T ba ) with 6 independent components. The covariant derivative defined by the Riemannian connection (19) is conventionally denoted by the nabla or by the semicolon: ∇ a = ' ; a '. ̂ Our conventions for the Riemann curvature are as follows: The Ricci tensor is introduced by ̂ R ij := ̂ R kij k , and the curvature scalar is ̂ R := g ij ̂ R ij . The signature of the spacetime metric is assumed to be (+1 , -1 , -1 , -1). In the following, we summarize some of the frequently used formulas in the context of the bitensor formalism (in particular for the world-function σ ( x, y )), see, e.g., [2, 27, 28] for the corresponding derivations. Note that our curvature conventions differ from those in [2, 28]. Indices attached to the world-function always denote covariant derivatives, at the given point, i.e. σ y := ∇ y σ , hence we do not make explicit use of the semicolon in case of the world-function. We start by stating, without proof, the following useful rule for a bitensor B with arbitrary indices at different points (here just denoted by dots): Here a coincidence limit of a bitensor B ... ( x, y ) is a tensor determined at y . Furthermore, we collect the following useful identities: (A11)", "pages": [ 7, 8 ] }, { "title": "Appendix B: Covariant expansions", "content": "Here we briefly summarize the covariant expansions of the second derivative of the world-function, and the derivative of the parallel propagator: The coefficients α, β, γ in these expansions are polynomials constructed from the Riemann curvature tensor and its covariant derivatives. The first coefficients read as follows: In addition, we also need the covariant expansion of a usual vector: 34:317, 1964.", "pages": [ 8, 9 ] } ]

[ { "title": "Universality and thermalization in the Unruh Effect", "content": "W. G. Brenna, 1, ∗ Eric G. Brown, 1, † Robert B. Mann, 1, ‡ and Eduardo Mart'ın-Mart'ınez 2, 3, 4, § 1 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 2 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 3 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 4 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada We explore the effects of different boundary conditions and coupling schemes on the response of a particle detector undergoing uniform acceleration in optical cavities. We analyze the thermalization properties of the accelerated detector via non-perturbative calculations. We prove nonperturbatively that if the switching process is smooth enough, the detector thermalizes to the Unruh temperature regardless of the boundary conditions and the form of the coupling considered.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The interaction between matter and the gravitational field has evaded a complete quantum description since the first attempts to formulate a quantum theory of gravity more than 60 years ago. Lacking a satisfactory quantum description of the gravitational interaction, quantum field theory in curved spacetimes (which links general relativity with quantum field theory) is thus far the most satisfactory framework to describe the interaction of quantum fields with the space-time curvature. As of today none of its predictions has been experimentally confirmed beyond analogue gravity [1], and bringing those effects within experimental reach is a matter of great interest [2-5]. One of the chief predictions of quantum field theory (QFT) in curved spacetimes is the well-known Unruh effect [6]. It dictates that a detector with constant acceleration a in free space, in which the field is in the Minkowski vacuum, will experience a response equivalent to its submersion into a heat bath with a temperature proportional to its acceleration. This phenomenon is intrinsically related to the so-called Hawking effect [7, 8], and understanding it is essential in order to investigate more complex phenomena such as black hole dynamics and possible quantum corrections to relativistic gravity. The first derivations of the Unruh effect (based on the characterization of the Minkowski vacuum in a unitarily inequivalent Rindler quantization scheme via Bogoliubov transformations) are not above criticism. A number of very strong assumptions have to be made in order to justify the observation of a thermal bath by an accelerated observer in the Minkowski vacuum. For example, the infinite amount of energy required to sustain the eternal Rindler trajectory. It has also been argued that difficulties arise in defining a Minkowski vacuum when boundary conditions are specified on the scalar field on a manifold [9]. Despite these criticisms, later results in the context of axiomatic quantum field theory [10] were used provide a model-independent derivation of the Unruh effect. Furthermore, derivations based on accelerated particle detectors [11] that appeared soon after the original derivation reassert the importance of the Unruh effect. The standard approach in grappling with the difficult computational problem of an arbitrary system interacting with a scalar field on a curved spacetime manifold is to use an Unruh-DeWitt detector [11, 12]. This simplified detector model considers a two-level system coupled to a scalar field with a monopole interaction of the form where ˆ m is the monopole-moment of the detector, φ [ x ( τ ) , t ( τ )] is the field operator evaluated along the worldline of the detector, and λ ( τ ) is a numbervalued function that represents the strength and timedependence of the coupling. It has been shown that, although simple, this Hamiltonian is a good model of the light-matter interaction when no exchange of angular momentum is involved [13]. Typically the Unruh-DeWitt model is used within the framework of perturbation theory, and very often restricted to lowest-order calculations (see, for example, [12, 14, 15]). However, perturbation theory is not always applicable and breaks down when analyzing scenarios involving strong coupling, large energy, or large time scales. While it is relatively easy to perturbatively show that the response of an accelerated detector to the vacuum state is Planckian [12], perturbation theory is not the most appropriate approach to study the thermalization properties of the detector. In practice this is mainly because higher orders of perturbation theory would be required, increasing the calculational complexity even beyond those of non-perturbative methods. More importantly, thermalization is an equilibrium result achieved over the course of long time scales. In general, such time scales will not be accessible to perturbation theory since the perturbative parameter, i.e. |〈 H 〉 ∆ T | becomes larger as time increases. Since thermalization is, in general, an equilibrium process that requires analysis in the limit ∆ T → ∞ , reasonable criticism may be raised about a perturbative claim of thermalization. Indeed, to check whether or not the detector evolves to an exactly thermal state, we will consider long time-scale evolution combined with adiabatic switching (to perturb the system the least when the interaction is switched on). Concretely, one should not only check that the probability of excitation of the detector has a Planckian response, one should also check to what extent the state of the detector becomes thermal if the detector is carefully switched on and if the interaction lasts for long enough times. This requires a complete calculation of the detector's density matrix; it is a common misconception that a detector's Planckian response implies that the detector thermalizes. For instance, the detector could evolve to a squeezed thermal state which may exhibit the same probability of excitation as some thermal state, but which is not actually a thermal state. By the use of nonperturbative methods we can make sure that thermalization is achieved and that it is not an artifact of the use of perturbation theory in regimes beyond its applicability. Such non-perturbative methods were recently developed and applied to examine the response of a detector within a cavity containing a scalar field [16]. The cavity was a wave-guide with periodic boundary conditions in which the detector was allowed to entirely cycle several times during its evolution. Whereas this is physically reasonable for the case of periodic boundary conditions, it is not the correct setting to compare with more general boundary conditions. We consider in this paper the thermality of accelerated detectors in optical cavities with different boundary conditions. Extending the work of Brown et al. [16] (see also [17]), we will demonstrate non-perturbatively that an accelerated Unruh-DeWitt detector coupled to the vacuum state of a scalar field thermalizes to a temperature proportional to its acceleration, regardless of the boundary conditions imposed. The scenarios we consider here also differ from previous work in the way that the detector trajectories are defined with respect to the cavity. For example, we modify the cavity length such that the detector remains inside a single cavity during its interaction with the field, which is of capital importance for physicality in the case of non-periodic cavities. We do note that there has been an effort to understand how imposing different boundary conditions modifies the response of detectors in non-inertial scenarios in free space. For example, work has been done in a very different context to examine the continuum Rindler case [18], and a number of boundary conditions in HartleHawking vacua have been studied [19]. However, these studies do not tell us if the boundary effects of a cavity will prevent a uniformly accelerating particle detector from thermalizing due to the Unruh effect. To our knowledge the only work that addresses this issue is the aforementioned paper by Brown et al. [16] in the periodic cavity case. In addition to the study of the universality of the Unruh effect and the existence of thermalization for different sets of boundary conditions, we shall also briefly com- ment on the effect of different methods of coupling the detector to the field. Typically, we couple the detector locally to the field through the detector's monopole moment, ˆ µ M = ( ˆ a d +ˆ a † d ) . Here, we will also explore the effects of a different form of coupling, namely the coupling of the detector's monopole moment to the momentum of the field. Our findings indicate that in all of the scenarios under consideration, the Unruh effect occurs. We observe that the detector achieves thermalization with temperature proportional to acceleration. Thus, not only does the Unruh effect occur inside a cavity (which imposes an IR-cutoff on the field and, furthermore, isolates the field in the cavity from the rest of the spacetime), it appears to occur independently of the details of this IR cutoff and of the spatial distribution of the cavity modes. This demonstrates that the Unruh effect, which many have argued relies on idealized details and thus cannot lead to thermalization [20], is in fact a very general and universal phenomenon and that thermalization of particle detectors can be computed non-perturbatively. Not only is this a remarkable result from a fundamental point of view, it also gives hope to the possibility of an experimental realization of the Unruh effect in quantum optical settings, where it has been shown that general relativistic scenarios like the one we study here can already be simulated [21]. Our paper is organized as follows. In Sect. II we discuss the physical setup of our system, elaborating on the differences between the various scenarios and boundary conditions considered. In Sect. III we explain the oscillator-detector model that we will be using in our study, as presented in [16], and go on to discuss how we solved for the evolution of the detector-field system. In Sect. IV we present our results on the thermalization of the accelerating detector along with the linear dependence of its temperature on acceleration. Furthermore, we demonstrate that these results are largely independent of the boundary conditions imposed on the field. In Sect. V we finish with some concluding remarks.", "pages": [ 1, 2 ] }, { "title": "II. THE SETTING", "content": "We will consider a uniformly accelerated point-like detector in its ground state going through a cavity prepared in the vacuum state. The trajectory will be such that the detector starts moving inside the cavity with a given initial speed, with a constant acceleration in a direction opposite to its initial motion. Hence the atom will be decelerated while crossing the cavity. The detector reaches the center of the cavity exactly when it reaches zero speed, and then travels back to the initial point with increasing speed until it reaches the position in which it started, having the same speed as when it entered the cavity but in the opposite direction. This trajectory of the atom within the cavity as a function of the detector's proper time τ is shown in Fig. 1. For larger values of the acceleration the detector will exit the cavity. In principle this is an issue as we mirror the field modes outside of the cavity. However, over the range of accelerations that we will consider the coupling decays so quickly past the edges that these tails will not contribute significantly to the observed final state. In addition, we find that the linearity of the temperature plots is preserved even when the detector escapes from the cavity. To have a clean signature one must be careful with the way in which the detector is switched on [16, 22] since a sudden switching stimulates strong quantum fluctuations that may overcome the Unruh effect. In order to reduce switching noise we apply the same approach as [16]: the interaction is smoothly switched on following a Gaussian time profile so that switching quantum noise effects are reduced. In particular, the switching function that we use has the form We prepare the ground state of the detector and the vacuum of the field at a time τ = -T where the interaction is switched on following the Gaussian profile above, the atom has some initial speed and starts decelerating until it reaches the centre of the cavity at time τ = 0. The atom continues accelerating until the time τ = T when it reaches the initial point again. In the settings that we shall analyze, we will consider as parameters T = 4, δ = 8 / 7 and λ 0 = 0 . 01. With these parameters we find that the switching is more than smooth enough for our purpose; namely the detector's response from the switching noise is negligible compared with the response from acceleration. We are going to consider different scenarios that correspond to different cavity field settings. Let us rewrite the interaction Hamiltonian (1) in the interaction picture in the following general form where we have expanded the field operator in terms of an orthonormal set of field mode functions u n [ x, t ] which will depend on the boundary conditions that are imposed upon the field. The normalization of the field modes is performed with respect to the Klein-Gordon inner product [12]. If we choose a cavity of length L and we impose the Dirichlet boundary conditions φ [ L, t ] = φ [0 , t ] = 0 (the two walls of the cavity are ideal mirrors) we find that the mode functions are the stationary waves where n ∈ Z + and ω n = k n c = nπc/L . In the case of a periodic cavity of length L (which would correspond to physical settings such as closed optical fibres or microwave guides or any other setting with a torus topology), the modes are the set of right and left-moving waves where n is an integer, k n = 2 nπ/L (negative (positive) n corresponds to left-moving (right-moving) modes) and ω n = | k n | c . In the case of a Neumann cavity of length L (this is to say, ∂ x φ [ L, t ] = ∂ x φ [0 , t ] = 0), the modes become where n ∈ Z + and ω n = k n c = nπc/L . Finally, as we described above, the worldline of the detector inside the cavity parametrized in terms of its proper time will be given by and the interaction will be smoothly switched on following the curve (2) from time -T to T with suitable values of L and T such that the atom always remains in the cavity while the interaction is 'on' (see Fig. 1).", "pages": [ 2, 3 ] }, { "title": "III. GAUSSIAN FORMALISM", "content": "We will introduce the non-perturbative oscillatordetector model. Replacing the usual two-level system in the Unruh-DeWitt model with a harmonic oscillator is somewhat common in the literature [3, 23-27]. However, in almost all of these cases, a perturbative approach was used, and not many practical non-perturbative results have been obtained in the past. Here we will use the powerful non-perturbative Gaussian formalism developed in [16, 17] to analyze the thermalization properties of the detector. The only restrictions that apply to this formalism in a cavity scenario are that the initial state of the system is a Gaussian state (such as the vacuum, a coherent or squeezed state or a thermal state) and that the interaction Hamiltonian is quadratic in the quadrature operators (in order to preserve the state's Gaussianity through time evolution). Restricting our consideration to quadratic Hamiltonians is quite reasonable since the interaction between matter and light is of this nature [28]. In addition, as the Unruh effect has almost solely been studied in the context of free space, this provides an excellent excuse for us to tread into unknown waters by considering different cavity settings. Furthermore, any experimental verification of the Unruh effect is likely to be more easily implementable in the context of optical cavities, so it is important to understand the phenomenon in such a scenario. Let us summarize the tools that we are going to employ in this paper to obtain, non-perturbatively, the response of a particle detector to the field vacuum. Instead of the interaction picture, it will be more convenient to work in the Heisenberg picture, following [16]. Let us form a vector from the detector's and field's annihilation and creation operators in the Heisenberg picture: where the subscript d corresponds to the detector and the others correspond to the modes of the field. The commutators of the components of this vector generate a symplectic form Similarly, we can form a vector of quadrature operators of the form These operators are related to the creation and annihilation operators of each mode by In order to ensure that Gaussian states will remain Gaussian over the course of their evolution, we also need to evolve by a quadratic Hamiltonian: where ˆ H I will in our case be given by (1) in the Heisenberg picture. Here Ω d is the frequency of the oscillatordetector and ω n is the frequency of the n th field mode. In our notation we use τ to denote the proper time of the detector (which is generally moving with respect to the cavity) and t to denote the lab time, which is the time with respect to which the field evolves. When computing the system's evolution as generated by some Hamiltonian we must be careful to choose a specific time parameter and construct the Hamiltonian accordingly. In the above Hamiltonian we choose to evolve with respect to the detector's proper time τ . Since the field evolves with respect to t , this means that we must include a 'blue-shift factor' on the free field's Hamiltonian. A more complete and rigorous explanation of this can be found in [16]. Since it is Gaussian, the state of the detector-field system can be completely described by a covariance matrix consisting of the first and second moments of the quadrature operators. In our scenario we need not consider states with first moments other than zero because of the absence of Hamiltonian terms linear in the quadrature operators, and so we determine the state by a covariance matrix of the form where ˆ x is the vector formed by the dimensionless position and momentum operators from equation (6). Thus the state of our single detector can be completely described using the 2 × 2 covariance matrix of the form The time evolution of the entire covariance matrix, including both the detector and the field, is governed by the equation of unitary evolution [16] where S is a symplectic matrix: S Ω S T = S T Ω S = Ω . In addition, the symplectic matrix generated by a (generally time-dependent) Hamiltonian ˆ H ( τ ) satisfies the equation with initial condition S (0) = I . Here F sym = F + F T , where F is a phase-space matrix encoding the form of the Hamiltonian via Keeping in mind that we will continue working in the Heisenberg picture (our operators are fully time dependent), we will make use of some computational techniques beyond what was indicated in [16] that are inspired by the principle of the interaction picture. Here, we make use of an exact solution of the free symplectic time-evolution matrix to speed up the computation. Namely, we split the evolution matrix into an exactly solvable part (we could call it the free part) and a nonexact part (that we could call the interaction part). From equation (12), we take where K 0 is exactly solvable. For our circumstances, we choose Here F sym d and F sym f are the symmetrized matrices corresponding to the free Hamiltonians of the detector and field, respectively. That is, their non-symmetrized versions satisfy Ω d ˆ a † d ˆ a d = ˆ x T F d ˆ x and ∑ n ω n ˆ a † n ˆ a n = ˆ x T F f ˆ x . Although, strictly speaking, K 0 is time-dependent due to the dt/dτ factor, the fact that this is a total derivative and that we are integrating over τ to solve the dynamics means that we are still able to solve for the free evolution exactly, so that if K 1 ( t ) = 0, equation (12) has the exact solution Applying this interaction-picture-like approach, we define It is then easily seen that (12) becomes The evaluation of S I ( τ ) can then be accomplished by standard numerical techniques. The full Heisenberg evolution matrix is then simply S ( τ ) = S 0 ( τ ) S I ( τ ), and the evolved state of the detector-field system is given by σ ( τ ) = S ( τ ) σ 0 S ( τ ) T .", "pages": [ 3, 4, 5 ] }, { "title": "IV. UNRUH TEMPERATURE AND THERMALIZATION", "content": "Here we present the results obtained by applying the formalism of Sect. III to the scenario outlined in Sect. II. Our goal is to test the universality of the Unruh effect with respect to a change of boundary conditions in the cavity. We begin by considering an oscillator detector uniformly accelerating through a cavity field that is initially in the vacuum state. It should be noted that when considering accelerating detectors it is necessary to include many modes in the field expansion. This is due to the fact that the detector will experience an exponentially changing time dilation with respect to the cavity frame which translates into a modulating blueshift of the mode frequencies as seen by the detector. Thus, the field modes with which the detector is resonant will rapidly change and, if the acceleration or time of evolution is large enough, very high mode numbers can make significant contributions to the evolution of the detector. In our work we have been vigilant to ensure that enough field modes were included such that further additions do not modify the results obtained for the detector. Because the non-perturbative Gaussian formalism presented in Sect. III slows down considerably for a large number of field modes, we use the standard perturbative formalism up to the first order to ensure complete convergence with respect to the number of field modes. For example, in the periodic case this method involves evaluating the integral so, for periodic boundary conditions we therefore need to evaluate which is the contribution to the excitation probability amplitude for a given field mode, where glyph[epsilon1] is used to sum over left- and right-moving modes and the last term in the exponential is the Gaussian switching function (see below for more details). We can use them (after normalization) in a partial sum over the number of field modes. That is, This gives the first-order perturbation theory result for the probability of transition of the detector from the ground state to the first excited state. Now, when beginning its evolution the detector is initially in its ground state, but through its interaction with the field it will generally become excited. After the evolution of the detector-field system is complete we will examine the state of the detector, which will be fully specified by its 2 × 2 covariance matrix σ d . In order to conclude that the detector has experienced a thermal Unruh bath during its acceleration, we look for two things: first, that the detector has evolved to a thermal state and, second, that the corresponding temperature grows linearly with the acceleration experienced by the detector.", "pages": [ 5 ] }, { "title": "A. Thermality", "content": "The non-perturbative approach from [16] is very well suited for testing the thermality of our detector. Not only is thermality easily tested, but because we have the exact (non-perturbative) state of the detector we are able to make a definitive statement about thermalization. In order to make conclusions about thermalization using perturbation theory, one would at the least need to expand to higher orders, and at the worst it would be impossible due to the long time scales typically required for thermalization to occur where perturbation theory may break down. Once the detector has completed its evolution it will be, up to phase-space rotation (i.e. free evolution), in a squeezed thermal state of the form where ν is the covariance matrix's symplectic eigenvalue and r is its squeezing parameter. These diagonal entries are just the eigenvalues of σ d , λ ± = νe ± r , from which the symplectic eigenvalue and squeezing parameter follow as ν = √ λ + λ -and e 2 r = λ + /λ -. We must now determine whether the amount of thermality introduced by ν is a much greater contributor to the energy of the detector's state compared to the amount of squeezing. If so, then the state can be said to be nearly thermal. If they are comparable, or if squeezing has the greater contribution, then we can not claim that the detector thermalizes. To compare these two effects we study how they contribute to the free energy of the detector. For small squeezing (which is satisfied in our scenario) the energy above the ground state energy, to leading order in r , a power series expansion straightforwardly yields where Ω d is the detector frequency. Since ν is of order unity (and is in fact remains very close to unity for our situation) a good test for thermality is If this inequality is satisfied then the detector can be said to be very nearly thermal. Equivalently, if is very small, the detector is said to be thermal. We find that the detector thermalizes very well in all of the three boundary conditions considered: periodic, Dirichlet, and Neumann. We find numerically that for the parameters given in Sect. II, δ is on the order of 10 -6 in all three cases. That is, the squeezing experienced by the oscillator is extremely minute compared to its thermality, and thus the detector can be said to be very nearly thermal. The first of our two conditions to verifying the Unruh effect (thermality and temperature proportional to acceleration) is satisfied for all three boundary conditions.", "pages": [ 5, 6 ] }, { "title": "B. Unruh temperature", "content": "We are now in a position to compute the temperature of the evolved detector σ d . For a single oscillator of frequency Ω d the form of an exactly thermal state is σ therm d = diag( ν, ν ), for which the temperature is [29] Since in our scenario we have already confirmed that our detector thermalizes to an excellent approximation we are able to use this equation to compute the temperature of our detector with negligible error, where ν = √ λ + λ -as above. For each of the three boundary conditions (periodic, Dirichlet, and Neumann) we compute this temperature for various values of acceleration. These results are displayed in figure (2). Notice that our least-squares fit is performed on first-order perturbative results. For small coupling strength the perturbative result for the transition probability of the accelerated detector is in close agreement with the result obtained by the nonperturbative approach. Specifically, the probability computed up to leading order (as explained above) is of O ( λ 2 ) and it is easily shown that the next relevant order, and thus the difference between the perturbative and non-perturbative answers, is O ( λ 4 ). For small temperatures, therefore, one need not use the non-perturbative approach to estimate the temperature of the oscillator detector, assuming that the detector is in a thermal state. When including a very large number of field modes it is computationally more convenient to plot the first order perturbative results with three objectives in mind: 1) from the probability (20) and assuming thermality, compute the temperature using the standard Boltzmann distribution and check that it varies linearly with acceleration, 2) check that the non-perturbative results (using (24)) are computed with enough numerical accuracy to reproduce the perturbative plot up to O ( λ 4 ), ensuring that had we used an infinite number of modes both methods would converge, and 3) use the non-perturbative results to ensure thermality of the system. Note that this last step cannot be easily carried out with a perturbative calculation (this was done and discussed in the previous section). From the plot of the trajectory (Fig. 1), notice that very high acceleration data points involve the atoms exiting the cavity for a small part of their trajectories. To prevent switching noise from damaging the results, we keep the field continuous beyond the cavity, but because the switching function is effectively zero when the detector crosses the boundary, the interaction will be negligible. Notice that the high acceleration results are not qualitatively different from lower acceleration results, justifying our negligibility assumption. Remarkably we find that for all three boundary conditions the temperature grows linearly with acceleration, demonstrating that the qualitative features expected of the Unruh effect are very much independent of the details of the cavity. This settles any doubt regarding the existence of the Unruh effect when an IR-cutoff for the field is introduced (i.e. when inside a cavity). There has been some skepticism [30-32] stemming from the large number of technical assumptions that go into the canonical derivation of the Unruh effect [6] and how the presence of a cavity might alter or even eliminate its existence. We have demonstrated not only thermality and the existence of the effect in a cavity (first shown in [16] and reaffirmed here), but also that the boundary conditions ascribed to this cavity are all but irrelevant (see Fig. 3). Indeed the numerical similarity between the different cases is striking. We note that the slope of the detector temperature with respect to acceleration is not equal to the value of 1 / 2 π predicted by the canonical free-space derivation. This is not overly surprising since we are working in a cavity setting rather than free space; significant border effects should be expected when studying such phenomena. Our results demonstrate, however, that the inclusion of an IR cutoff does not destroy the Unruh effect understood as the thermal response of a particle detector with a temperature proportional to the acceleration, and what is more, that the detector actually thermalizes to that particular temperature. If we were to take our cavity to the continuum limit we would expect (at least in the case of periodic boundary conditions) the slope to converge to the usual value of 1 / 2 π . We leave such a study for future work.", "pages": [ 6, 7, 8 ] }, { "title": "C. XP Coupling", "content": "In this portion of the paper we shall briefly discuss the effect of varying the form of coupling between the field and the detector. In particular, the question we ask is: what happens when the monopole coupling of equation (1) is modified such that the detector couples to the conjugate momentum of the field instead of the field itself? In this form, we can no longer maintain the pointlike coupling assumption for the detector-field coupling. After a Fourier transform, it is apparent that a pointlike X -P coupling is akin to coupling the detector's monopole moment to the field in an extremely delocalized way, such that the detector couples to the field everywhere. Thus this coupling is ill-defined and, without the introduction of spatial smearing, will yield divergences. We will therefore regularize the interaction assuming that the detector's coupling strength to the canonical momentum of the field varies with the field frequency. A frequency-dependent effective coupling appears naturally when considering spatially smeared detectors [13, 33], although in our case we would make the simpler assumption that the coupling strength is inversely proportional to the frequency of the mode. In this way we can analyze the following coupling for periodic boundary conditions: where λ n → λ ( τ ) /ω n so that the energy density falls off with high energy modes at the same rate as in the X -X coupling case. We note here that in first order perturbation theory, the negative sign does not contribute to the probability of transition and this scenario is exactly the same as the X -P coupling. In the non-perturbative case, we produced the same plot as the periodic curve of figure 2 with the modified coupling; all data points were exactly the same. This tells us that the periodic X -P coupling sign change is still a symmetry of the non-perturbative case. We believe that this should be observable when examining the equations (3) and (25) but we do not prove this here.", "pages": [ 8 ] }, { "title": "V. CONCLUSIONS", "content": "In this work we have non-perturbatively solved for the evolution of an oscillator detector undergoing uniform acceleration through a cavity field. We have confirmed recent previous work [16] demonstrating that the Unruh effect does indeed occur inside a cavity, and furthermore we have demonstrated that this result is independent of the boundary conditions applied to the field, implying that the Unruh effect is a very universal phenomenon. Specifically, we have considered vacuum cavity fields with periodic, Dirichlet, and Neumann boundary conditions. In all three cases we have observed that an accelerating oscillator detector evolves to a thermal state and that the temperature obtained by the detector increases linearly with its acceleration. Furthermore the results between the three cases are numerically very similar. This indicates that not only is the phenomenon qualitatively universal but, with respect to the case of boundary conditions, also quantitatively universal. We have also made some conclusions regarding the use of different detectorfield couplings that further strengthens these claims. Moreover, our use of the non-perturbative oscillator model has allowed us to make significantly stronger claims regarding the thermality experienced by the detector than can be made using the standard perturbative framework typically employed in the literature. That is, we have concluded that in our scenario an accelerating detector in fact evolves to a thermal state, rather than merely exhibiting a thermal response function. Questions of thermalization cannot be made in perturbation theory without resorting to higher order expansions, and in some scenarios may actually be impossible due to the large time scales often required for thermalization where perturbation theory breaks down. More generally, the results of this paper suggest that the Unruh effect and similar phenomena such as Hawking radiation may be largely independent of the details of the system [34]. In addition to theoretical interest, such universality bodes well for an eventual experiment where the Unruh effect could be measured.", "pages": [ 8 ] }, { "title": "VI. ACKNOWLEDGMENTS", "content": "The authors would like to thank very much William Donnelly for his helpful comments and insight on the interaction picture approach to the Gaussian formalism. This work was supported in part by the National Sciences and Engineering Research Council of Canada. E. M-M. was partially funded by the Banting Postdoctoral Fellowship Programme. W. B. was funded by the Vanier CGS Award.", "pages": [ 8 ] } ]

[ { "title": "Stabilizing the Semilocal String with a Dilatonic Coupling", "content": "Leandros Perivolaropoulos ∗ and Nikos Platis † Department of Physics, University of Ioannina, Greece (Dated: August 7, 2018) We demonstrate that the stability of the semilocal vortex can be significantly improved by the presence of a dilatonic coupling of the form e q | Φ | 2 η 2 F µν F µν with q > 0 where η is the scale of symmetry breaking that gives rise to the vortex. For q = 0 we obtain the usual embedded (semilocal) NielsenOlesen vortex. We find the stability region of the parameter β ≡ ( m Φ m A ) 2 ( m Φ and m A are the masses of the scalar and gauge fields respectively). We show that the stability region of β is 0 < β < β max ( q ) where β max ( q = 0) = 1 (as expected) and β max ( q ) is an increasing function of q . This result may have significant implications for the stability of the electroweak vortex in the presence of a dilatonic coupling (dilatonic electroweak vortex). The Nielsen-Olesen (NO) vortex [1, 2] is a topologically stable static solution of the Abelian-Higgs model. The Lagrangian density of this model is of the form where Φ is a complex scalar field, V (Φ) = λ 4 ( | Φ | 2 -η 2 ) 2 , D µ = ∂ µ -iA µ and F µν = ∂ µ A ν -∂ ν A µ . 1 The NO vortex ansatz is of the form Variation of the Lagrangian (1) leads to the field equations for f ( r ) and a ( r ) as where u ≡ a ( r ) r and we have implemented the following rescaling: The NO boundary conditions to be imposed on (4) and (5) are f (0) = u (0) = 0, f ( r →∞ ) = 1 and u ( r →∞ ) = m . Clearly, the NO solution for f ( r ), u ( r ) depends on a single parameter β ≡ λ 2 e 2 which is the squared ratio of the scalar field mass m Φ = √ λη √ 2 over the gauge field mass m A = eη . The energy density of the NO vortex is of the form The NO vortex solution can also be embedded in generalizations of the Abelian-Higgs model. For example the semilocal Lagrangian [3] is obtained by promoting the U (1) gauge symmetry of the Abeian-Higgs model to an SU (2) global × U (1) gauge symmetry. This is achieved by replacing the complex scalr Φ by a complex doublet The embedded NO vortex ansatz (semilocal vortex) is of the form while for the gauge field eq. (3) remains unchanged. By varying the semilocal Lagrangian it is easy to show that the field equations obeyed by f ( r ) and a ( r ) (or u ( r ) ≡ a ( r ) r ) are identical to the NO equations (4) and (5). Thus the NO vortex solution is embedded in the generalized semilocal Lagrangian. However, due to the S 3 topology of the semilocal vacuum, the stability of the embedded vortex is not topological. It is only dynamical and is valid for a finite range of the parameter β . It may be shown [4-7] that this range of stability is 0 < β < 1. The NO vortex can be embedded in several other generalizations of the Abelian-Higgs model which involve broken U(1) symmetries. For example it can be embedded in the bosonic sector of standard Glashow-SalamWeinberg (GSW) electroweak model [8] with SU (2) L × U (1) Y symmetry. One type of such embedded vortices is also known as the electroweak Z -vortex [9-11]. There is a parameter region of dynamical stability of the electroweak Z -vortex. It is determined by two parameters: the squared ratio β of the Higgs mass m H over the Z µ mass m Z ( β ≡ ( m H m Z ) 2 ) and the Weinberg angle θ w [11]. Thus, the stability range of the embedded electroweak Z -vortex is of the form 0 < β < β max ( θ w ). For θ w = π 2 the bosonic sector of the GSW Lagrangian reduces to the semilocal Lagrangian and therefore β max ( θ w = π 2 ) = 1. For θ w < π 2 we have β max ( θ w ) < 1 and therefore the stability range decreases. The experimental values β = ( m H m Z ) 2 = ( 91 . 2 GeV 125 GeV ) 2 /similarequal 0 . 53 and sinθ w = 0 . 23 are outside of the stability region [11]. There has been a wide range of studies aiming at constructing models where the stability of the semilocal and/or electroweak vortex is improved. These studies have attempted to improve the stability using thermal effects [12], extra scalar fields [13, 14], external magnetic fields [15], fermions [16, 17] or spinning scalar field in a charged background [18]. Most of these studies have either lead to models where the vortex stability is worse than the usual semilocal Lagrangian or to particularly contrived models requiring external backgrounds. Even for the classical stability region of semilocal and electroweak strings, there are instabilities at the quantum level [19]. In this study we attempt to increase the stability region of the semilocal vortex by considering a simple and generic generalization of the Abelian-Higgs model Lagrangian: the dilatonic Abelian-Higgs model defined to be of the form where B ( | Φ | 2 ) = e q | Φ | 2 η 2 is a dilatonic coupling that allows for dynamics of the effective gauge coupling e √ B ( | Φ | 2 ) . In the limit B ( | Φ | 2 ) → 1 ( q → 0) we obtain the usual Abelian-Higgs model with the NO vortex solution. Considering now the NO ansatz in the dilatonic Abelian- Higgs model we obtain the rescaled field equations for the dilatonic gauge vortex where as usual β = λ 2 e 2 . The corresponding energy density is of the form Using the NO boundary conditions, it is straightforward to obtain the dilatonic vortex solution of equations (13), (14) for various values of the parameters q and β (Fig 1). A simple mathematica code for the derivation of this solution, based on the minimization of the energy density (15), is provided in the Appendix. In order to investigate the stability of the embedded dilatonic vortex we generalize the dilatonic AbelianHiggs Lagrangian to a dilatonic semilocal Lagrangian with SU (2) global × U (1) gauge symmetry. We then consider the perturbed fields Φ and A µ as The energy perturbations due to δ Φ 2 and δA µ decouple and can only lead to increase of the embedded dilatonic vortex energy. The corresponding energy perturbation is identical to the energy perturbation of the topologically stable NO vortex and therefore it is positive definite. Thus the stability of the dilatonic embedded vortex is determined by the energy perturbation due to δ Φ 1 ≡ g . The energy of the perturbed vortex is of the form where we have included only perturbations due to g and E 0 is the unperturbed energy of the embedded dilatonic vortex. The energy perturbation due to g may be written in the form where ˆ O is a Schrodinger-like Hermitian operator of the form and we have only kept terms up to second order in g . The Schrodinger potential corresponding to ˆ O is For values of the parameters q and β for which ˆ O has no negative eigenvalues we have δE g ≥ 0 and therefore no instability develops. In order to determine if ˆ O has negative eigenvalues we may solve ˆ Og ( r ) = 0 with boundary conditions g (0) = 1, g ' (0) = 0 and check if the solution crosses the g = 0 line and goes to -∞ asymptotically. If it does then it is easy to show that there must exist at least one bound state (negative eigenvalue). The param- eter region in the q -β space where ˆ O has no negative eigenvalues is shown in Fig. 2 (sector I). In order to construct this plot, for each value of q > 0 we find a stability region 0 < β < β max ( q ). As expected β max ( q = 0) = 1. Interestingly β max ( q > 0) > 1 and the stability region increases as we increase q . The improvement of stability is due to the fact that the effective Schrodinger potential (22) corresponding to the operator ˆ O becomes shallower as we increase q (Fig. 3). Thus ˆ O becomes less receptive to negative eigenvalues.The new repulsive term in the Schrodinger potential (22) (proportional to q ) originates from the dilatonic term in the Lagrangian (16). This term favors energetically a lower value for the field Φ at the origin and therefore it makes the perturbation g more costly energetically at r = 0. This leads to improved stability for the dilatonic embedded gauge vortex. We have also investigated the dilatonic embedded global vortex in the presence of an external magnetic field. This solution is obtained by replacing the covariant derivatives in the Lagrangian (12) by regular ones. In the absence of a dilatonic coupling this vortex is unstable and there is no free parameter in the Lagrangian. However, in the presence of a dilatonic coupling and a gaussian external magnetic field we have shown that the embedded global vortex gets stabilized in the region where the magnetic field is present. These results will be presented elsewhere. The existence of dilatonic gauge and global vortices may also have implications as a new class of models predicting spatial variation of the fine structure constant α on cosmological scales. Such models have been discussed in Ref. [22-26] and are motivated from recent quasar absorption spectra observations that may hint towards possible spatial variation of α on cosmological Hubble scales [27]. The dilatonic semilocal model is perhaps the simplest generalization of the semilocal model that can lead to dramatic improvement of the semilocal vortex stability. The existence of such a dilatonic coupling in the realistic GSWmodel is therefore also expected to lead to improvement of the stability of the Z -string [20, 21] and perhaps create a stability region for the W -string (an alternative embedding of the NO vortex in the GSW model) [20]. The analysis of the stability of the dilatonic electroweak vortices constitutes and interesting extension of the present study. We thank Tanmay Vachaspati for useful comments and John Rizos for his help on some numerical aspects of this work. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program Edu- cation and Lifelong Learning of the National Strategic Reference Framework (NSRF) - Research Funding Program: ARISTEIA. Investing in the society of knowledge through the European Social Fund.", "pages": [ 1, 2, 3, 4 ] }, { "title": "APPENDIX", "content": "In Fig. 4 we show the Mathematica code used to find numerically the dilatonic vortex solutions by minimization of the energy functional (15). The algorithm is particularly simple and has a wide range of applications including the numerical derivation of the NO vortex solution. Note that due to stiffness of the ODE system (13)-(14), Mathematica is unable to solve it using the NDSolve routine. A similar code can be used to investigate the stability of the embedded dilatonic vortex by minimizing the energy functional (19) with respect to the three functions f , u and g . For parameter values leading to a non-zero g at the defect core we have instability. Even though this approach is more involved and subject to some numerical uncertainties, in most cases it leads to consistent results with the more accurate and simple perturbative method based on finding if the operator ˆ O has negative eigenvalues.", "pages": [ 4 ] } ]

[ { "title": "Phenomenology of Supersymmetric Models with a Symmetry-Breaking Seesaw Mechanism", "content": "Lauren Pearce, 1 Alexander Kusenko, 1, 2 and R. D. Peccei 1 1 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA 2 Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8568, Japan We explore phenomenological implications of the minimal supersymmetric standard model (MSSM) with a strong supersymmetry breaking trilinear term. Supersymmetry breaking can trigger electroweak symmetry breaking via a symmetry-breaking seesaw mechanism, which can lead to a low-energy theory with multiple composite Higgs bosons. In this model, the electroweak phase transition can be first-order for some generic values of parameters. Furthermore, there are additional sources of CP violation in the Higgs sector. This opens the possibility of electroweak baryogenesis in the strongly coupled MSSM. The extended Higgs dynamics can be discovered at Large Hadron Collider or at a future linear collider.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The minimal supersymmetric standard model (MSSM) provides an appealing framework for physics beyond the Standard Model. However, the lack of direct experimental evidence for superpartners, as well as the recent discovery of a Higgs boson with a mass that is heavier than one would naively expect in the MSSM, challenges the viability of low-energy supersymmetry in its simplest realizations. At the same time, there exists a possibility for a strongly coupled form of MSSM, which is associated with a large value of the trilinear supersymmetry breaking coupling [1-3]. In this strongly coupled regime, the squarks form bound states via the exchange of Higgs bosons, creating some additional composite states, which can have some non-zero vacuum expectation values (VEVs). This class of models has several intriguing features. First, the particle content of the low-energy effective theory is different from what one expects in the MSSM: in particular, the model predicts more Higgs bosons and fewer superpartners to be observed at the electroweak scale. Second, the bound state quartic coupling is not related to the gauge coupling, which relaxes the upper bound on the mass of the lightest Higgs boson [3]. Third, the model contains gauge singlet states, whose presence allows for the electroweak transition to be first order; this model also has additional sources of CP violation. The latter creates sufficient conditions for electroweak baryogenesis, as we discuss below. In this paper we will explore the ramifications of strongly coupled MSSM for electroweak baryogenesis, as well as collider phenomenology.", "pages": [ 1 ] }, { "title": "II. SYMMETRY BREAKING SEESAW", "content": "Let us briefly review the strongly coupled phase of the MSSM [1-3]. Supersymmetry breaking introduces the following trilinear terms in the Lagrangian: and a similar term in the down sector. If the coupling A t is sufficiently large, the stop squarks present in the theory can form bound states. One of these bound states is an SU L (2) doublet with the same quantum numbers as the fundamental Higgs boson H u . The kernel of this state is shown in Fig. 1. Although it is a crossed kernel, as opposed to the more familiar ladder diagram, the mass of the bound state can be found numerically as a function of A t , and sufficiently large couplings can cause the mass squared of the bound state, m 2 BS , to become negative [3]. If this happens, electroweak symmetry breaking occurs. It is an interesting feature of the model that electroweak symmetry breaking is triggered by supersymmetry breaking in a way that is fundamentally different from the weakly coupled MSSM. The discussion in Refs. [1, 2] and, in part, in Ref. [3], considered such large values of A t for which m 2 BS < 0. However, as was pointed out in Ref. [3], this is a sufficient, but not necessary condition. It is possible that supersymmetry breaking would trigger electroweak symmetry breaking even for some much lower values of A t . The bound-state composite Higgs doublet generically mixes with the fundamental Higgs doublet, due to the same A t H u ˜ t ∗ L ˜ t R coupling. Thus the mass matrix, in the basis of H u and the bound state, has the form where α is a numerical constant determined by strong dynamics, m h is the mass term of the fundamental Higgs doublet, and m BS is the mass of the bound state doublet. A negative eigenvalue in this mass matrix signals that electroweak symmetry is broken, which can happen when both diagonal elements are positive. One of the eigenvalues is negative when which is possible for any positive values of m BS and A t , provided that m h is a small enough (positive) number. The possibility of breaking electroweak symmetry for a relatively small A t , smaller than the value required to drive the bound state mass to zero, opens a broad range of possibilities in the MSSM, which were not considered in Refs. [1, 2]. This type of symmetry breaking, which relies on the mixing and the mass matrix similar to the seesaw neutrino mass matrix, was dubbed a symmetrybreaking seesaw mechanism [3]. An additional benefit of the symmetry-breaking seesaw mechanism is that it automatically preserves SU C (3). The two squarks in the bound state carry color, and the bound states of the form shown in Fig. 1 include an SU C (3) octet along with the color singlet. If the octet acquires a non-zero VEV, it would break SU C (3) making the model unacceptable. However, to acquire a VEV, the bound state must mix with the fundamental Higgs boson, which is only possible for an SU C (3) singlet state. The quantum numbers of the fundamental Higgs boson of the MSSM dictate the pattern of symmetry breaking by selecting the only state that is phenomenologically acceptable, among the numerous possibilities.", "pages": [ 1, 2 ] }, { "title": "III. DESCRIPTION OF BOUND STATES", "content": "In Ref. [3], the primary focus was the possibility of spontaneous symmetry breaking; in this current work, we are interested in phenomenology. Let us begin with a discussion of the bound states present in this model. In addition to the SU L (2) doublet mentioned above, there are SU L (2) singlets and SU L (2) triplets; an example of the relevant kernels are shown in Fig. 2. The vertices in these bound states are all proportional | A t | ; because Yukawa interactions are attractive in all channels, all of these states exist if the doublet bound state mentioned above exists. We will assume that only A t (and possibly A b ) are large enough to produce bound states; these bound states appear in the up and down Higgs sectors respectively. We have summarized the possible bound states, along with their quantum numbers, in Table I. Let us summarize the states present in the model and their salient features. We observe that rows 1 and 2 in Table I are hermitian conjugates of each other; consequently, these rows may be combined to form complex representations. Rows 3 and 4 may also be similarly combined. Thus, the first two rows in the table describe a complex SU L (2) triplet and a complex SU L (2) singlet, while the next two rows describe another complex SU L (2) singlet. All of these states carry color charge, and they all also have fractional electric charge. The implications of these states will be discussed further in Section VIII on collider phenomenology. Similarly, the next two rows (5 and 6) are also hermitian conjugates of each other, which may be combined into a complex SU L (2) doublet. This is the doublet discussed in Section II above. As mentioned, there are additionally 8 colored states which do not generally acquire vacuum expectation values. The last two rows (7 and 8) describe two real singlets and one real triplet; these come in both colored and colorless versions. As we will show in Section V, the electroweak phase transition is generally first order due to these singlets. Thus, we see that our model of strongly interacting supersymmetry has a rather extended Higgs sector. With so many degrees of freedom, it is important to ensure that no colored states acquire a vacuum expectation value. We note that there are no terms which involve only a single colored field, because the Lagrangian must be invariant under SU C (3). Thus, even if all uncolored states acquire non-zero vacuum expectation values, there is no term linear in a colored field; such a term would necessarily induce spontaneous symmetry breaking of SU C (3). These colored fields generally acquire corrections to their mass values during electroweak symmetry breaking; we will assume that their masses sufficiently large that these corrections do not drive any of the mass-squared values negative. The full model, including both up and down sectors, is rather complicated. Therefore, we will make the simplifying assumption that the sectors are relatively decoupled, and we will only consider the up sector. It is also possible that the bound states form only in the up sector. We expect our results to hold in the more general model. Next we will proceed to discuss the phenomenology of our model. First, we will discuss flavor-changingneutral-currents; this is a concern in any model in which extra Higgs doublets are present. Second, we will consider the properties of the electroweak phase transition in this model. Then we will discuss CP violation and baryogenesis, and, finally, we will make some remarks regarding collider phenomenology.", "pages": [ 2, 3 ] }, { "title": "IV. FLAVOR CHANGING NEUTRAL CURRENTS", "content": "Any model which introduces additional Higgs doublets must address the issue of flavor-changing-neutralcurrents (FCNCs), which are highly constrained experimentally and generically large when additional SU L (2) doublets are introduced. One well-known method of suppressing FCNCs is to have the doublets couple to different types of quarks; for example, in the MSSM the Higgs doublet H u only couples to up-type quarks and the doublet H d only couples to down-like quarks. This suppresses FCNCs provided that the mixing between the two doublets (after supersymmetry is spontaneously broken) is not too large; this assumption that the sectors are relatively decoupled is made both by the MSSM and our model. However, even if the up-sector and down-sector are decoupled, our model potentially has large FCNCs because we have two doublets within each sector; in particular, both the fundamental doublet H u and the bound state doublet Φ u couple to up-type quarks. Therefore, we consider a second way of suppressing FCNCs: if diagonalizing the quark matrix with respect to interactions with one of the doublets also approximately diagonalizes the quark matrix with respect to interactions with the other doublet, then FCNCs are small, because they are proportional to the off-diagonal elements. Equivalently, FCNCs are suppressed if the Yukawa couplings between the quarks and the first doublet are be approximately proportional to the Yukawa couplings between the quarks and the second doublet. This approach is typically disfavored because it frequently requires fine-tuning, but we will demonstrate that this condition is naturally satisfied by our model. We recall that the bound state doublet is comprised of up squarks exchanging H u bosons. We note that there is no tree-level coupling between up-squarks and quarks; however, there is a tree-level coupling between H u and the quark: the Yukawa coupling y q . Therefore, to lowest order, an up-type quark sees only the H u contribution in the bound state, and so the coupling between the quark and the bound state is y ' q = βy q . The above argument is shown diagrammatically in Fig. 3; the lowest order contribution to the Yukawa coupling between a quark and the bound state Higgs doublet comes through the exchange of a true Higgs boson H u . Therefore, it is proportional to the Yukawa coupling y t , and the proportionality constant β describes the mixing between H u and the bound state Φ u . This mixing β is clearly independent of the quark involved on the right hand side of the diagram. Therefore, the Yukawa couplings satisfy y ' q = βy q , and hence FCNCs are indeed suppressed naturally, without fine-tunings. We may be concerned about corrections from higher order diagrams; particularly in regards to the up-quark Yukawa coupling, which is quite small. The next order corrections will come from diagrams like those shown in Fig. 4, in which particle X is a gaugino. However, if the incoming quarks are up quarks, then the gaugino must convert an up quark into a stop squark, and the only gauginos which can change flavor is winos. However, these are forbidden; winos can change up quarks only into down, strange, or bottom squarks. Thus, there are no contributions from diagrams of this form to the up quark and charm quark Yukawa couplings. Such diagrams do contribute to the top quark Yukawa coupling; for example, in this case the exchanged gaugino could be a gluino, Zino, photino, or Higgsino. However, these are all smaller than the first order contribution discussed above. For future reference, let us relate y q and y ' q to the Yukawa couplings between the quark and the mass eigenstates; this will be relevant in our discussion of baryogenesis in Section VI below. Let us assume that the mass eigenstates are related to H u and Φ u by Then the relevant Yukawa couplings are given by In particular we note that and we expect β ∼ sin( θ ) due to its close relation to the mixing.", "pages": [ 3, 4 ] }, { "title": "V. ELECTROWEAK PHASE TRANSITION", "content": "Let us briefly discuss the evolution of this model with temperature. At sufficiently high temperatures, the model behaves as the standard (weakly interacting) MSSM. At lower temperatures, the model undergoes a phase transition to a strongly interacting phase, and electroweak symmetry breaking takes place. One can make an analogy with QCD, which is described by quarks and gluons at high temperature, but at some lower temperatures baryons and mesons become the appropriate degrees of freedom. Likewise, in our model one should use the fundamental degrees of freedom of MSSM for temperatures well above a TeV, but one should consider bound states as new degrees of freedom at low temperatures. In the strongly coupled phase, the model should be described by an effective Lagrangian written in terms of low-energy degrees of freedom, which include the bound states. Ideally, one would like to calculate all the parameters in terms of the parameters of MSSM, which is the ultraviolet completion of the theory. However, because the theory is strongly coupled, it is not feasible to calculate these parameters explicitly. A calculation on the lattice may be possible [4], but no detailed results are available at present. In the absence of such a calculation, one can only parametrize the low-energy couplings using generic values consistent with symmetries. (This is analogous to the approach that one took to strong interactions before QCD was discovered and understood.) Obviously, this approach is limited in what can be predicted. However, since the values of 'fundamental' MSSM parameters in the high-energy Lagrangian are unknown and are not strongly constrained, the low-energy effective approach appears to be well justified. We will now discuss the electroweak phase transition, which can be generically first-order in this model. We note that the colorless SU L (2) gauge singlets are not associated with any symmetry breaking; therefore, they may acquire vacuum expectation values in the strongly coupled phase. Such vacuum expectation values have no physical meaning and may be removed with a field redefinition that makes the tadpole diagrams vanish orderby order in perturbation theory. We will assume that this has been done in writing the effective potential. We have already shown that the colored fields do not acquire nonzero VEVs, and we will neglect them for the remainder of this section. The effective potential for the colorless Higgs fields is written in Appendix A; the cubic terms given in Equation (A3) are particularly important for the first-order phase transition. The notation used in the Appendix and this section is as follows: the complex SU L (2) doublet mass eigenstates are Ψ 1 and Ψ 2 , where Ψ 1 has a negative mass-squared eigenvalue due to the seesaw symmetry breaking mechanism, the real SU L (2) singlet mass eigenstates are S 1 and S 2 , and the real SU L (2) triplet field is V . Due to the negative mass-squared of Ψ 1 , the origin of the potential is not a local minimum and at least the neutral component of doublet Ψ 1 acquires a nonzero vacuum expectation value. The terms A S 1 S 1 Ψ † 1 Ψ 1 and ˜ A S 1 S 2 Ψ † 1 Ψ 1 produce terms linear in S 1 and S 2 respectively, and so consequently 〈 S 1 〉 = 0 nor 〈 S 2 〉 = 0 can be a local minimum. Hence, once Ψ 1 acquires a nonzero vacuum expectation value, the singlets also acquire a nonzero vacuum expectation value. When both Ψ 1 and the singlets have acquired non-zero VEVs, the neutral component of the other doublet, Ψ 2 , must also acquire a nonzero vacuum expectation value due to terms such as S 1 Ψ † 1 Ψ 2 ; the charged component does not acquire a nonzero vacuum expectation value. Next let us consider the triplet; we parametrize it as V a = ( V 1 , V 2 , V 3 ). The cubic terms in Equation (A3) include where we have identified the charge states. When the neutral component of Ψ 1 acquires a nonzero vacuum expectation value, the above equation produces a term linear in V 0 ; consequently, this field also acquires a vacuum expectation value. The consequences of this for the ρ 0 parameter and neutrino masses is discussed in Section VII. The presence of gauge singlet Higgs states and the existence of tree-level cubic couplings generically make the phase transition strongly first-order. This is in contrast with the Standard Model and MSSM, in which the cubic terms are forbidden by the SU L (2) symmetry. In the Standard Model, the transition is not first-order for the allowed range of the Higgs masses (more specifically, for the Higgs mass above 45 GeV). In the MSSM, the transition is weakly first order, and only for such parameters for which the two-loop corrections generate a sufficient barrier in the potential [5-8]. In our case, finite temperature corrections, calculated in the same manner as in the Standard Model [9], [10], produce terms proportional to T 2 M 2 , where M 2 are the mass eigenvalues of the shifted fields, as functions of the vacuum expectation values. When the fields are shifted, the cubic terms produce terms in the mass eigenvalues linear in the vacuum expectation values. Such linear terms produce a barrier which results in a strongly first order phase transition. This is identical to that manner in which singlets produce a barrier in the Next-to-Minimal Supersymmetric Model (NMSSM) [11-13]. The potential written in Appendix A has many parameters describing the cubic and quartic terms; this makes a comprehensive study of the temperature evolution of the potential impractical. However, given the large number of parameters we expect there to be relatively large region of parameter space in which the phase transition occurs at a temperature of the order of the electroweak scale and v u = √ | 〈 Ψ 1 〉 | 2 + | 〈 Ψ 2 〉 | 2 < ∼ 246 GeV, with the difference to be made up in the down sector.", "pages": [ 4, 5 ] }, { "title": "VI. BARYOGENESIS", "content": "The generic possibility of a strongly first-order phase transition reopens the possibility of baryogenesis at the electroweak scale [14, 15]. In addition to the first-order phase transition, one needs CP violation for a successful baryogenesis. The full potential written in Appendix A has 8 complex parameters; two of these may be eliminated by rotating the complex doublets Ψ 1 and Ψ 2 . This leaves 6 physical phases; therefore this model can accommodate additional CP violation beyond that present in the Standard Model. If one also includes the down sector, and allows the two sectors to mix, there are many more physical CP-violating phases. This CP-violation in the Higgs sector must be communicated to the matter sector for successful baryogenesis. This can be accomplished through interactions in the bubble wall with the top quark; only the top quark Yukawa coupling is sufficiently large for the interactions to be thermal equilibrium during the phase transition. We note that the analysis in this section is similar to [16], which considered a simpler model with two Higgs doublets and a complex singlet, of which only one doublet and the singlet acquired a nonzero vacuum expectation value. In general, both mass eigenstates Ψ 1 and Ψ 2 couple to up-type fermions, and thus the effective Lagrangian contains terms of the form where T L = ( t L , b L ) is the doublet which includes the left-handed top and bottom quarks, and a, b are SU L (2) indices. We recall that y q and y ' q are proportional to each other, as described by equation (6), which suppresses FCNCs. If we write the vacuum expectation values of the doublets after spontaneous symmetry breaking as then these terms become 〈 〉 When the various fields acquire nonzero vacuum expectation values, this top quark Yukawa coupling is modified, and these modifications can introduce a nonzero physical phase into the coupling. An example of one such contribution is shown in Fig. 5. The tree-level corrections to the top-quark Yukawa coupling after spontaneous symmetry breaking are given by which gives the standard top quark mass term; the potentially nonzero phase is absorbed in a rotation of the top quark. To simplify our analysis, we will assume that ξ 1 /greatermuch ξ 2 ; this is particularly reasonable since the mass term of Ψ 1 in the effective potential is negative but the mass term of Ψ 2 is positive. Therefore, the dominant contribution to the top quark's mass is from the Yukawa coupling between Ψ 1 and the top quark, and thus we take y t ≈ √ 2 m t /v = . 996. We will define the other vacuum expectation values to be 〈 S 1 〉 = ξ 3 , 〈 S 2 〉 = ξ 4 , and 〈 V 3 〉 = 2 V 0 = ξ 5 . where ˜ y tf summarizes the contributions from diagrams that do not contribute a net phase. (We have used the freedom to rotate Ψ 1 and Ψ 2 to make the parameters λ S 12 and ˜ λ S 12 real; we also remind the reader that we have set up our effective Lagrangian such that the singlets have zero vev before spontaneous symmetry breaking.) Let us assume that the corrections are small with respect to y t ; then this corrected Yukawa coupling may be written as where This change of phase in the Yukawa coupling can be transformed according to the standard techniques [17], [18]; the quarks are rotated by an amount proportional to their hypercharge to eliminate the phase, which introduces a new kinetic term for the top quark which violates CP. The phase φ t can be approximated as space independent, although time dependent, because the mean free path of the top quarks and gauge bosons is small compared to the scale on which φ t varies (which is approximately the thickness of the electroweak bubble walls). Then this additional term in the Lagrangian has the form of a chemical potential for baryon number. Consequently, during the transition the free energy is minimized for nonzero baryon number. If the system could evolve to the minimum of free energy, it would reach the baryon density where α is a constant of order 1; for a simple two-doublet model, it is 72 / 111 [17]. During the phase transition, the sphaleron-induced B + L violation processes drive the system toward this equilibrium value [19]: but the minimum of the free energy is not reached because the transition takes place too quickly. The sphaleron transition rate is [14, 15, 18] where κ , σ , and γ are dimensionless constants. For the Standard Model, κ is expected to be between .1 and 1 [20], while σ is expected to be between 2 and 7 [19]. Integrating the above rate gives the baryon asymmetry produced during the phase transition where ∆ φ t is the change in the phase of the top quark Yukawa coupling during the phase transition; this is not the same as φ f because the sphaleron B + L -violating interactions may go out of thermal equilibrium before the phase transition is complete. The entropy density is and so baryon-to-entropy ratio after the phase transition is To match the observed value of n B /s ∼ 10 -10 , the change in phase must be of order 10 -2 (assuming g S ( T ew ) ∼ 100). This is a reasonable number; given the form of equation (14), we expect this to be satisfied for a relatively large region of parameter space. Thus, we conclude that the electroweak phase transition in the strongly coupled MSSM can account for the observed matter asymmetry.", "pages": [ 5, 6 ] }, { "title": "VII. IMPLICATIONS OF THE TRIPLET VACUUM EXPECTATION VALUE", "content": "We have noted in Section V that it is an unavoidable consequence of this model that the neutral component of the hypercharge Y = 0 Higgs triplet acquires a nonzero vacuum expectation value. In this section, we discuss the phenomenological consequences of this, both in regards to the ρ 0 parameter and neutrino masses. Models in which a single Y = 0 Higgs triplet acquire a vacuum expectation value have been considered [21-25]; the low energy behavior of this theory was described in detail in Ref. [26]. Such models are quite constrained by precision measurements of the ρ 0 parameter, which is experimentally measured to be ρ 0 = 1 . 0004 + . 0003 -. 0004 [27]. A triplet nonzero vacuum expectation values modifies ρ 0 by [26] We recall that we must have v u ≤ 246 GeV; this means that we must have | 〈 V 0 〉 | = | 〈 V 3 〉 | / 2 < ∼ 2 . 5 GeV, or equivalently, If v u ≈ |〈 Ψ 1 〉 | , we expect | 〈 V 3 〉 | ≈ A V 1 v 2 u /m 2 V ; the above condition becomes We expect A V 1 and m V , like the other parameters in the effective Lagrangian, to be near the electroweak scale. The exact values of A V 1 and m V are determined from the high energy (MSSM) Lagrangian through strong dynamics, and it is infeasible to estimate them. It may be that a lattice calculation shows that triplet states are less strongly bound than the singlet states, and thus have larger masses, or it may be that the our model requires some fine-tuning to satisfy this condition. We note that if A V 1 and v u are both on the 100 GeV scale, then we only require m V ∼ O (TeV). Another concern with the triplet acquiring a nonzero vacuum expectation value is that generically such vacuum expectation values may produce large neutrino masses [28-31]. However, our triplet, like the rest of our bound states, is comprised of squarks and true Higgs bosons, and to lowest order interacts with the neutrino only through the exchange of Higgs doublet bosons. We observe, however, that this is just the closure of the usual seesaw mass diagram; both of these are shown in Fig. 6. This is suppressed for the same reason the regular seesaw diagram is; the contribution is y 2 v 2 u /m R , which is suppressed by the large mass value of the right-handed neutrino. Thus, although our model may require fine-tuning to satisfy current experimental constraints, the requisite fine-tuning is rather small, and the triplet vev does not generate unacceptably large neutrino masses.", "pages": [ 6, 7 ] }, { "title": "VIII. COLLIDER PHENOMENOLOGY", "content": "Finally, we make some qualitative remarks regarding the collider phenomenology of this model. As we have shown in Section III, this model has a rather extended Higgs sector, with numerous states. As a result, it will be difficult to discern individual states at an experiment such as the Large Hadron Collider. However, there may still be detectable consequences. The gauge singlet states can be detected via deviations of the Higgs decay branching ratios from the predictions of the Standard Modal [32, 33]. Many of the Higgs states present in this model carry color charge; this is in contrast to the Standard Model and the weakly interacting MSSM, in which the Higgs sector contains only colorless states. Again, due to the large number of such states they may be difficult to discern individually; however, these states may influence the number and structure of jets observed in high-energy scattering processes. Secondly, we have noted in Section III the presence of a Y = 4 / 3 triplet which carries color charge. Since SU C (3) symmetry is preserved, these states must form a colorless combinations by joining with other colored particles; most frequently by pulling quarks from the quantum vacuum. This process produces jets with integer charge. However, some of these jets will carry charge ± 2; for example, if a ˜ t L ˜ t L bound state combines with an up quark. Additionally, this model predicts numerous singly charged states; these arise from the extra doublet as well as the triplets. The Standard Model, in contrast, only has an electrically neutral Higgs boson, while the MSSM has one set of ± 1 charged Higgs bosons. Furthermore, some of the singly charged states carry color charge, again in contrast to the MSSM. Therefore, searches for charged scalar bosons may produce evidence for our model.", "pages": [ 7 ] }, { "title": "IX. CONCLUSION", "content": "We have considered phenomenological implications of a strongly coupled realization of MSSM [1-3]. The possibility that supersymmetry breaking could trigger electroweak phase transition leading to a low-energy effective theory with composite Higgs-like states is intriguing. Such a strongly coupled realization of MSSM could reconcile supersymmetry with the relatively high value of the Higgs boson mass measured at LHC. The model predicts the existence of additional Higgs bosons, which can be discovered at LHC or at a proposed future linear collider. The pattern of electroweak symmetry breaking is constrained so that the color SU C (3) symmetry is preserved. A generic prediction is the existence of a gauge-singlet Higgs boson, which gives rise to tree-level cubic terms in the effective potential. This makes the electroweak phase transition strongly first-order for generic values of parameters. The multi-state Higgs sector, which includes composite Higgs-like states, has a number of CP-violating phases. The combination of a first-order phase transition and new sources of CP violation opens a possibility for a successful electroweak baryogenesis.", "pages": [ 7 ] }, { "title": "X. ACKNOWLEDGMENTS", "content": "We thank J. M. Cornwall for very helpful, stimulating discussions. This work was supported by DOE Grant DE-FG03-91ER40662 and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.", "pages": [ 7 ] }, { "title": "Appendix A: Effective Potential", "content": "In Sections V and VI, we made reference to the effective potential which describes our strongly interacting model before electroweak symmetry breaking. In this appendix, we give the full potential; this is important in showing that we have sufficient freedom to set the sym- metry breaking parameters as desired, and so in determining the tree-level contribution to equation (14). Let us call the mass eigenstates of the doublets Ψ 1 and Ψ 2 , and the mass eigenstates of the singlets S 1 and S 2 . We recall that Ψ 1 and Ψ 2 are complex fields, while the others are real. Then the potential can be written as where the mass terms are One of the doublet mass eigenvalues is negative due to the seesaw symmetry breaking mechanism, and we emphasize that the triplet, V , is real. The cubic terms are Finally, the possible quartic terms are V 4 a (Ψ 1 , Ψ 2 ) = λ 1 (Ψ † 1 Ψ 1 ) 2 + λ 2 (Ψ † 2 Ψ 2 ) 2 + λ 12 (Ψ † 1 Ψ 1 )(Ψ † 2 Ψ 2 ) + λ ' 12 (Ψ † 1 Ψ 2 ) 2 + λ '' 12 (Ψ † 1 Ψ 2 )(Ψ † 2 Ψ 1 ) + h.c. + λ 1 (Ψ 1 τ Ψ 1 ) · (Ψ 1 τ Ψ 1 ) + λ 2 (Ψ 2 τ Ψ 2 ) · (Ψ 2 τ Ψ 2 ) + λ 3 (Ψ 1 τ Ψ 1 ) · (Ψ 2 τ Ψ 2 ) + λ 4 (Ψ 1 τ Ψ 2 ) · (Ψ 1 τ Ψ 2 ) , V 4 b (Ψ 1 , Ψ 2 , S 1 ) = λ S 1 S 2 1 Ψ † 1 Ψ 1 + λ S 2 S 2 1 Ψ † 2 Ψ 2 + ˜ λ S 1 S 2 2 Ψ † 1 Ψ 1 + ˜ λ S 2 S 2 2 Ψ † 2 Ψ 2 + λ S 12 S 2 1 Ψ † 1 Ψ 2 + ˜ λ S 12 S 2 2 Ψ † 1 Ψ 2 + λ ' S 12 S 1 S 2 Ψ † 1 Ψ 2 + h.c., V 4 c (Ψ 1 , Ψ 2 , V ) = λ V 1 Ψ † 1 Ψ 1 V T V + λ V 2 Ψ † 2 Ψ 2 V T V + λ V 12 Ψ † 1 Ψ 2 V T V + h.c., V 4 d ( S 1 , S 2 , V ) = λ SV S 2 1 V T V + ˜ λ SV S 2 2 V T V + λ S 12 S 1 S 2 V T V + λ V ( V T V ) 2 , V 4 e ( S 1 , S 2 ) = λ S S 4 + λ S S 4 + λ ' S 3 S 2 + λ ' S 1 S 3 + λ SS S 2 S 2 We note that the following 8 parameters are generally complex: A S 12 , ˜ A S 12 , A V 12 , λ ' 12 , λ S 12 , ˜ λ S 12 , λ ' S 12 , and λ V 12 . In principle, all of the many parameters that appear in this Lagrangian are determined by the high energy theory through strong dynamics; however, it is not feasible to calculate them from first principles, although some advanced lattice techniques could make it possible. We expect the generic values of these parameters to lie between the supersymmetry-breaking scale and the electroweak scale. Given the large number of parameters, we expect there to be region of parameter space in which the phase transition occurs at temperatures on the electroweak scale (which is closely related to the barrier height), and that the doublet vacuum expectation values satisfy v u = √ | 〈 Ψ 1 〉 | 2 + | 〈 Ψ 2 〉 | 2 < ∼ 246 GeV.", "pages": [ 8 ] } ]

[ { "title": "On the Potential for General Relativity and its Geometry", "content": "Gregory Gabadadze, 1, ∗ Kurt Hinterbichler, 2, † David Pirtskhalava, 3, ‡ and Yanwen Shang 4, § 1 Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY, 10003, USA 2 Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada 3 4 Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada Department of Physics, University of California, San Diego, La Jolla, CA 92093", "pages": [ 1 ] }, { "title": "Abstract", "content": "The unique ghost-free mass and nonlinear potential terms for general relativity are presented in a diffeomorphism and local Lorentz invariant vierbein formalism. This construction requires an additional two-index Stuckelberg field, beyond the four scalar fields used in the metric formulation, and unveils a new local SL(4) symmetry group of the mass and potential terms, not shared by the Einstein-Hilbert term. The new field is auxiliary but transforms as a vector under two different Lorentz groups, one of them the group of local Lorentz transformations, the other an additional global group. This formulation enables a geometric interpretation of the mass and potential terms for gravity in terms of certain volume forms. Furthermore, we find that the decoupling limit is much simpler to extract in this approach; in particular, we are able to derive expressions for the interactions of the vector modes. We also note that it is possible to extend the theory by promoting the two-index auxiliary field into a Nambu-Goldstone boson nonlinearly realizing a certain spacetime symmetry, and show how it is 'eaten up' by the antisymmetric part of the vierbein.", "pages": [ 1 ] }, { "title": "1. INTRODUCTION AND SUMMARY", "content": "Einstein's gravity is the theory that describes the two degrees of freedom of the massless helicity-2 representation of the Poincar'e group, and their two derivative self-interactions. One may ask whether it is possible to alter the interactions of the graviton beyond those dictated by the Einstein - Hilbert (EH) action. At the lowest, zero-derivative level, such a deformation would correspond to adding a potential for the metric perturbation. An obvious example is the potential described by the cosmological constant (CC) term, L 0 ∼ √ -g Λ. This changes neither the number of propagating degrees of freedom of general relativity (GR), nor the consistency of the theory, but necessarily alters the background spacetime. The CC is the only such term - other potentials inevitably change the number of degrees of freedom. The Fierz-Pauli term [1] is the unique consistent quadratic potential that gives rise to 5 degrees of freedom, as required by the massive spin-2 representation of the Poincar'e group. Adding a generic potential to the EH action however leads to the loss of all four Hamiltonian constraints of GR, and thus a total of six propagating degrees of freedom, one of which is necessarily a ghost [2]. Nevertheless, there exists a special class of mass and potential terms (the often-called dRGT terms [3, 4], see [5] for a review) that make the graviton massive, while retaining one of the four Hamiltonian constraints. This remaining constraint projects out the ghostly sixth degree of freedom [6, 7], see also [8-10]. In addition to the CC term, the dRGT construction allows for 3 free parameters. One combination is the graviton mass, m , and the other two independent combinations, α 3 and α 4 , set the strength of the nonlinear potential. The theory can be formulated by using four spurious diffeomorphism scalars, φ ¯ a - first introduced in an earlier proposal for massive gravity [11] - to allow for a manifestly diffeomorphism-invariant description. Adopting these four scalars, and following [4], one can define a matrix with components K µ ν = δ µ ν -√ g µα ∂ α φ ¯ a ∂ ν φ ¯ b η ¯ a ¯ b , that can be used to build invariants supplementing the EH action by the graviton mass as well as zero-derivative interactions that guarantee 5 degrees of freedom on an arbitrary background. One such term is given by [4] The remaining two possible terms L 3 , 4 , cubic and quartic in K respectively, can be obtained by the higher order generalization of (1) 1 , In addition to being invariant under the global Poincar'e subgroup, ISO (3 , 1) GCT , of the group of general coordinate transformations (GCT), the theory is invariant under an additional, global internal Poincar'e group, ISO (3 , 1) INT , realized on the 'flavor' indices of the scalars, as first pointed out by Siegel in an earlier context [11] Generation of the graviton mass occurs in the phase defined by the vacuum expectation value (VEV) of the order parameter 〈 ∂ µ φ ¯ a 〉 = δ ¯ a µ . This results in the spontaneous symmetry breaking pattern of the global symmetry group The unbroken ISO (3 , 1) ST group guarantees that the resulting theory is invariant under the ordinary spacetime (ST) Poincar'e transformations. Three of the four auxiliary scalars φ ¯ a are 'eaten' by the graviton to form a massive spin-2 representation of the latter group, while the fourth, potentially ghostly scalar is made non-dynamical by the single remaining Hamiltonian constraint of massive GR, originating from the specific structure of the dRGT terms L 2 , 3 , 4 . The dRGT theory gets rid of the sixth ghostly mode, and also guarantees that the remaining 5 are unitary degrees of freedom at low energies and on nearly-Minkowski backgrounds (i.e., the backgrounds with typical curvature smaller than the graviton mass square). However, the theory does not guarantee that for more general backgrounds the 5 physical modes are healthy. In fact, some of their kinetic terms may change signs around certain cosmological backgrounds. Moreover, for a large region of the α 2 , α 3 parameter space, the potential is known to violate the null energy condition and one often gets kinetic and gradient terms that give rise to superluminal group and phase velocities. Most of the above issues stem from one and the same source: the dRGT theory is strongly coupled at the energy/momentum scale Λ 3 ≡ ( M Pl m 2 ) 1 / 3 [3, 4]. As a result, a typical curvature of order m 2 produces order 1 corrections to the kinetic terms for fluctuations, often giving rise to vanishing or negative kinetic terms, or superluminal group and phase velocities (for brief comments on the current state of affairs on all these issues, see Section 6). As for any strongly coupled theory, an extension above the scale Λ 3 is desirable 2 . However, it is hard to think of such an extension since the Lagrangian contains square roots of the longitudinal modes (represented by the φ ¯ a 's). This inconvenience might be mitigated by using the vierbeins, which are square roots of the metric. The goal of the present work is to rewrite the theory in terms of the vierbeins in a GCT and local Lorentz transformation (LLT) invariant form. The hope is that this form of the theory might make it easier to find a weakly coupled completion. Also, irrespectively of that, the vierbein formulation itself merits a separate consideration. A vierbein reformulation of the theory was given by one of us and R. A. Rosen 3 [9]. That work focused on a unitary gauge description, which for a single massive graviton is not GCT or LLT invariant. In the present work, we give a GCT and LLT invariant action for a massive graviton. We find that such a formulation requires a new two-index Stuckelberg field, λ a ¯ a , in addition to the four scalar fields φ ¯ a used in the metric description. The new field is auxiliary and enters the action algebraically. To recover dRGT, this field should transforms as a vector under two different Lorentz groups, λ a ¯ a → Q a b ( x ) λ b ¯ a , and λ a ¯ a → L ¯ b ¯ a λ a ¯ b , where Q ( x ) belongs to SO (3 , 1) LLT , while the constant matrix L belongs to the global group SO (3 , 1) INT . Moreover, we note that the mass and potential terms - once written in the GCT and LLT invariant form - are amenable to an extension with λ ∈ SL (4) and unveil a new local symmetry w.r.t. simultaneous transformations, e a µ → Q a b ( x ) e b µ and λ a ¯ a → Q a b ( x ) λ b ¯ a , where Q ( x ) ∈ SL (4). Thus, the enhanced symmetry group of the mass and potential terms, SL (4) × G GCT , is larger than the symmetry group of the EH action. This observation suggests an extension of the theory by additional fields (see Section 2, and Section 3 for a GL (4) symmetric extension). As we will discuss in Section 3, the vierbein formulation enables one to give a geometric interpretation to the mass and potential terms - they can be expressed in terms of certain volume forms. There are other benefits as well: we find that the decoupling limit is much simpler to extract in this approach. The original results of [3] can be obtained with significantly less effort. Moreover, it is straightforward to derive closed-form expressions for the vector modes, which have not been obtained in complete generality before. We also note that the field λ a ¯ a ∈ SO (3 , 1) can be represented as λ a ¯ a = exp( v a ¯ a /f ), where v is an antisymmetric field (once indices are lowered with η ) and f is some dimensionful constant. Then, v can be promoted into a dynamical Nambu-Goldstone field parametrizing a coset ( SO (3 , 1) GCT × SO (3 , 1) INT ) /SO (3 , 1) Diag . We show that these six bosons are 'eaten up' by the antisymmetric part of the vierbein. This extends ghost-free massive gravity to a theory where the six antisymmetric components of the vierbein become dynamical.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2. VIERBEIN FORMULATION", "content": "The formulation of massive GR, as well as its extensions, is significantly simplified in the vierbein formalism [9]. Introducing the vierbein field e a µ , g µν = e a µ e b ν η ab with η ab = diag ( -1 , 1 , 1 , 1), the cosmological constant term can be written as d 4 x L 0 ∼ d 4 x √ -g Λ ∼ Λ ε abcd e a ∧ e b ∧ e c ∧ e d , where the one form e a is defined as e a ≡ e a µ d x µ . The ghost-free interactions of the vierbein perturbations can be represented in a similar fashion; e.g. in the unitary gauge, one such term is given by where 1 a ≡ δ a µ d x µ represents a unit vierbein. The two contributions L 0 , 2 to the potential can be supplemented by the two other independent terms L 3 , 4 , involving respectively three and four powers of ( e -1 ), contracted with the ε symbol in a similar fashion 4 , The above terms together with the Einstein-Hilbert term define an action for the 16 vari- ables in the vierbein which is neither GCT nor LLT invariant, whereas the metric formulation is an action for 10 metric variables (plus four scalars in the Stuckelberg formulation). Nevertheless, both formulations are dynamically equivalent. Following [9], we first show that the vierbein action is dynamically equivalent to the same action only with the additional constraint that the vierbein is symmetric (with respect to the Minkowski metric). In matrix notation, We parametrize the general vierbein as a constrained vierbein ˆ e satisfying (8), times a Lorentz transformation, parametrized as the exponential of a matrix ˆ B (which is antisymmetric with respect to η ) 5 , The ˆ B 's do not enter the Einstein-Hilbert term, since this term is invariant under local Lorentz transformations. Thus, the 6 variables in ˆ B appear only in the mass and potential terms (which in the metric formulation depend on the inverse metric g -1 through the matrix K = 1 -√ g -1 ∂φ∂φ ). These fields therefore appear without derivatives - they are auxiliary fields. We now vary with respect to ˆ B and look at the equations of motion, in powers of ˆ B . The lowest order terms contain no powers of ˆ B (other than the variation δ ˆ B ). Therefore, the only terms that appear at lowest order are the ones containing traces of one power of δ ˆ B along with powers of ˆ e -1 . Because ˆ e -1 is symmetric and δ ˆ B antisymmetric, and because δ ˆ B appears only linearly, the terms in the equations of motion linear in ˆ B all vanish. This means that the equations of motion of ˆ B start linearly in ˆ B , and are solved by ˆ B = 0. Plugging this solution back into the action, we see that the action with unconstrained vierbeins is dynamically equivalent to the action with symmetric vierbeins. To relate the potential with symmetric vierbeins to the potential in the metric formulation we use the matrix representation g = e η e T , Using the parametrization (9) and the symmetry property of ˆ e -1 : Thus in the unitary gauge, ∂ µ φ ¯ a = δ ¯ a µ , we can write, Due to the presence of the unit vierbein, in the form presented above the first order theory lacks invariance under both the GCT and LLT, characteristic of general relativity. Both of the symmetries however can be restored via corresponding Stuckelberg fields. For this, one introduces the auxiliary scalars φ ¯ a , analogous to those of the metric description of massive GR, as well as the 'link' field λ a ¯ a . The latter transforms as a contravariant vector under the local Lorentz group, λ a ¯ a → Q a b ( x ) λ b ¯ a , where Q ( x ) ∈ SO (3 , 1) LLT , and as a covariant vector under the global group SO (3 , 1) INT , λ a ¯ a → L ¯ b ¯ a λ a ¯ b . Using these fields, the mass and potential terms can be rewritten in a manifestly GCT × LLT-invariant form via the 'k-vierbein', k a µ ≡ e a µ -λ a ¯ a ∂ µ φ ¯ a , (As before, the glyph[epsilon1] 's here are the epsilon symbols, i.e. there are no factors of √ -g .) In the unitary gauge defined by λ ¯ a a = δ ¯ a a and ∂ µ φ ¯ a = δ ¯ a µ , one recovers the L 2 , 3 , 4 of (1). Away from this gauge, the theory acquires invariance under GCT, as well as under LLT, realized on the vierbein and the link fields as follows The transformations (16) with Q ( x ) ∈ SO (3 , 1) LLT represent a symmetry of the entire action, the potentials (13) -(15) and the Einstein-Hibert term. However, the potential terms themselves, (13)-(15), without the EH term, can have a larger symmetry. To see this, we first note that these potentials are invariant under the formal field redefinition (16) with Q ( x ) ∈ SL (4). Now, we defined λ to be a SO (3 , 1) matrix and therefore, such transformations with SL (4) matrices would take them outside of SO (3 , 1). This observation suggests that in the theory where the EH term is absent, the λ can be promoted to a SL (4) valued field. The resulting terms, (13)-(15), will have a local SL (4) symmetry, in addition to being invariant under GCT's. This extended local SL (4) symmetry is the defining property of the mass and potential terms. However, the EH term does not respect the SL (4). Therefore, there are two ways to combine the EH term with the potentials (13)-(15): (1) To define a theory where λ is an SO (3 , 1) valued field; (2) Alternatively, to define a theory with λ ∈ SL (4). In this paper we chose the former case because that is the theory of a single massive graviton. The latter choice gives a theory with 9 = dim SL(4) -dim SO(3 , 1) additional fields, and might be an interesting model to look at in the future. Thus, for λ ∈ SO (3 , 1), in the unitary gauge, λ = 1 , with φ ¯ a kept unfixed, one recovers the GCT-invariant but LLT non-invariant formulation of massive GR 6 . The relevant symmetry breaking pattern, corresponding to this case, involves six broken generators, while the remaining six correspond to the diagonal part of LLT and internal Lorentz groups. The equation of motion for λ , evaluated in the unitary gauge, gives precisely the constraint (8), needed for the theory to reduce to massive GR. As already remarked above, it is useful to represent the vierbein as e a µ = exp( ˆ B a b )ˆ e b µ and the λ field as λ a ¯ a = exp( v a ¯ a /f ), where both B and v are antisymmetric fields (once indices are lowered by η ). Under LLT, both of these fields shift by a coordinate dependent gauge function, so one or the other of them may be gauged away, but not both. One linear combination of ˆ B and v is invariant under LLT. This combination has no kinetic term in our construction, and it is algebraically determined by classical equations of motion guaranteeing, by the same arguments given earlier. Only the five helicities of the graviton are propagating degrees of freedom in the theory. An interesting alternative is to give dynamics to the gauge-invariant combination by regarding it as a Nambu-Goldstone field parametrizing the coset corresponding to the sym- metry breaking pattern (17). The kinetic term for this field also breaks the local SL (4) of the potential down to the group of LLT's. We will discuss this possibility in Section 5. Before then we will stay in the framework of massive gravity and the gauge invariant part of the λ field will be regarded as non-dynamical.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "3. GEOMETRIC INTERPRETATION AND GENERALIZATIONS", "content": "In this Section, we will give this formulation of the theory a geometric interpretation. Let us consider two manifolds of the same dimension 7 , and a smooth mapping between them φ : M → E . When a set of coordinates is given, the mapping φ consists of 4 smooth functions which we denote by φ a ( x ). (We ignore any possible topological obstructions at the moment. Such a smooth mapping always exists locally within certain patches of both M and E .) We denote, at each point of x ∈ M and φ ( x ) ∈ E , the cotangent spaces T ∗ M ( x ) and T ∗ E ( φ ) respectively. A set of vierbeins e a = d x µ e a µ is defined for every T ∗ M ( x ) which endow M with a metric g µν ≡ e a µ e b ν η ab . Usually, for the mapping φ between M and E to be compatible with their Riemannian structures, one must assume that the metric on M coincides with the metric pulled back from E through the functions φ a ( x ), i.e. both manifolds share identical Riemannian geometries and the mapping φ represents nothing other than a simple coordinate transformation. Physically, the two are indistinguishable. If, on the other hand, we insist that the manifold E should stay flat, we may choose to define the vierbeins in each T ∗ E ( φ ( x )) as θ a = d φ a . Together with the torsion-free condition, such a choice guarantees that the curvature tensor on E vanishes. But, such a construction of θ a does not respect the local Lorentz symmetry of T ∗ E ( φ ( x )), leaving only the global version intact 8 . Now that the two manifolds M and E are endowed with totally different Riemannian structures, there is no natural way to mix the cotangent vectors living in T ∗ M ( x ) and those living in T ∗ E ( φ ( x )). Indeed, if we just write terms such as e a -d φ a , they violate invariance w.r.t. the LLT's. The link fields λ a ¯ a ( x ) are introduced to remedy this. Due to the specific transformation of λ a ¯ a under the two Lorentz groups, we are able to map the forms in one cotangent space to the other and introduce mixing via the 'k-vierbein' e a -λ a ¯ a d φ ¯ a , where d φ ¯ a = ∂ µ φ ¯ a dx µ . We write the mass and potential in terms of the forms As discussed in the previous section, these expressions manifestly respect the local Lorentz symmetry on M , defined by at the cost of introducing the Stuckelberg fields λ a ¯ a . Notice that we can equally well write terms by multiplying λ ¯ b a - which we define to be the inverse matrix of λ a ¯ b - onto e a instead of d φ ¯ a . So we could write, as an example, This formulation however is equivalent to the one in (18). In the new form, the invariance under LLT is manifestly visible since d φ ¯ a are invariant, and the LLT transformations of e a are simply compensated by the opposite rotation for λ ¯ a a so the combination λ ¯ a a e a remains invariant automatically. Note that in this latter formulation one can directly extend λ to a GL (4)-valued field, and then have the mass and potential terms invariant under local GL (4), instead of SL (4) discussed in Section 2. The GL (4) invariant form can also be achieved in the original formulation, if the mass and potential terms (13)-(15) are multiplied by det( λ -1 ), with λ ∈ GL (4). The terms in (18) are quite reminiscent of the CC term in GR - these terms strongly resemble some sort of volume forms. In particular, one linear combination of the four terms gives the CC term (up to a total derivative). If, for the moment, we imagine that the fields φ ¯ a are the embedding coordinates of the manifold M into a higher dimensional flat manifold (so that ¯ a takes the values of 1 , 2 , . . . , D where D > 4), a term L ∼ λ a ¯ a λ b ¯ b λ c ¯ c λ d ¯ d d φ ¯ a ∧ d φ ¯ b ∧ d φ ¯ c ∧ d φ ¯ d ε abcd , with a fixed matrix λ a ¯ a that projects the D -dimensional tangent vectors down to the tangent space of M , is the volume form for the surface M as embedded in E . Here, in our formulation, there are two major differences. First of all, we are dealing with the mixing terms among the vierbeins of two different manifolds, M and E , with different geometries but an identical dimensionality. Secondly, we must integrate w.r.t. all possible embeddings parameterized by λ a ¯ b to make a comparison between different volume forms meaningful. Both differences complicate the geometrical identification of these mixing terms. However, for any fixed λ a ¯ b , each term in (18) can be given a geometric interpretation in terms of a difference between certain volume forms of the two different manifolds. Consider the simplest example, d 4 x L 1 ∼ ( e a -λ a ¯ a d φ ¯ a ) ∧ e b ∧ e c ∧ e d glyph[epsilon1] abcd . Apart from the volume form of M , it contains the term λ a ¯ a d φ ¯ a ∧ e b ∧ e c ∧ e d glyph[epsilon1] abcd . If we choose the gauge λ a ¯ a = δ a ¯ a , and focus only on the term ( a, b, c, d ) = (1 , 2 , 3 , 4), we recognize this as the volume form of M 3 × R , where M 3 denotes a 3-dimensional submanifold spanned out by the cotangent vectors e 2 , e 3 , and e 4 , and R denotes the 'flat dimension' parameterized by φ 1 ( x ). So, L 1 gives a difference between the two types of volume forms: the one of M and another from those of M 3 × R , with M 3 now representing a 3-dimensional submanifold of M spanned by any three of the four vierbeins e a . Individually, each such term depends on the arbitrary choice of e a , φ ¯ a , as well as the embedding matrix λ a ¯ a , but when all the indices are contracted and the fields are integrated over, we obtain a well-defined notion of a relative volume forms of the two manifolds. Fig. 1 gives an illustration to this. The left figure represents the original volume form of M , with the 4-th dimension suppressed, and the right one depicts the volume form obtained when the direction along that of e 1 is 'straightened'. The difference between the two volume forms is d 4 x L 1 . Likewise, we may interpret terms λ a ¯ a λ b ¯ b d φ ¯ a ∧ d φ ¯ b ∧ e c ∧ e d glyph[epsilon1] abcd as the volume form for various different M 2 × R 2 , where M 2 denotes the 2-dimensional submanifolds spanned by an arbitrary pair of e a and e b . A linear combination of all the four terms in (18) - that is a most general potential for GR that includes the CC term - can be thought as linear combination of all possible departures of the volume forms of M from those of M 4 -n × R n , with n = 1 , 2 , 3 , 4 denoting the number of dimensions that have been 'straightened out'. The principles outlined here allows one to consider various generalizations. For example, the dimensionality of E does not have to coincide with the dimensionality of M . If the dimensionality of E is D , the index ¯ a takes values from 1 to D in the vector representation of SO ( D -1 , 1), while the auxiliary fields λ a ¯ b transforms as bi-vector of SO (3 , 1) LLT and SO ( D -1 , 1) INT respectively. If D > 4, the extra coordinates will correspond to extra physical scalar fields with a Galileon-like symmetry. The construction remains consistent, in the sense that a Boulware-Deser like ghost will not be introduced. Such an extension of dRGT was already considered in [16]. Its vierbein formulation was given in [17] and was used to show ghost-freedom. The present formalism provides the LLT invariant vierbein formulation of this theory. In the extreme case where D = 1, φ ¯ a reduces to a single scalar φ (the index ¯ a takes only one value) and λ a ¯ b reduces to a single Lorentz vector v a ( x ) subjected to the condition v 2 = 1. Following the discussion given above, one finds that one of the natural interaction terms to consider is which, after integrating out v a , gives rise to an action of the Cuscuton type [18] Last but not least, one may consider an even more general class of theories where the internal global symmetry does not have to be the Lorentz symmetry but is instead described by an arbitrary Lie group G . As long as φ ¯ a is in some representation R of G and the field λ a ¯ b is in the bi-representation of R and the Lorentz group, we may consider interactions of φ ¯ a with gravity described by the Lagrangians given in (18). If one further gauges this internal symmetry, one arrives at a broader class of theories, which includes the bi-gravity theories considered in [19].", "pages": [ 9, 10, 11, 12, 13 ] }, { "title": "4. THE DECOUPLING LIMIT IN THE FIRST ORDER FORMULATION", "content": "In this Section, we will illustrate the advantages of the first-order formalism for the analysis of the decoupling limit (DL) of massive GR. In addition to reproducing very easily the already well known scalar-tensor interactions that arise in this limit, we will derive an allorders expression for the DL interactions involving the vector helicity of the massive graviton. To the best of our knowledge, the vector interactions have previously been unknown in closed form, though partial results are available [20-22]. We start by decomposing the vierbein field as before where ˆ B a b ≡ B a b /M 1 / 2 Pl is an antisymmetric generator of LLT, ˆ B ab = η ac ˆ B c b = -ˆ B ba , while ˆ e is the vierbein, symmetric on its lower indices, ˆ e µν ≡ ˆ e b µ η bν = ˆ e νµ . The symmetric vierbein and the auxiliary scalars are decomposed into background values and their perturbations as where π ¯ a = η ¯ aµ ( ∂ µ π/ Λ 3 3 + mA µ / Λ 3 3 ) and Λ 3 ≡ ( M Pl m 2 ) 1 / 3 . The scalings for various perturbation fields have been chosen so as to recover the correct quadratic terms in the decoupling limit of the theory. In the ghost-free theories at hand, this limit is m → 0, M Pl →∞ , with Λ 3 held finite [3, 23, 24]. Concentrating first on the S -π interactions that result from the Lagrangian (13), one can easily see that only terms with a single S and a certain number of π 's survive in the decoupling limit where the indices on the ε symbols are contracted with the help of the unit vierbein. At linear order, the vierbein and metric perturbations are related as 2 S a µ η aν = h µν , therefore the above scalar-tensor interactions are nothing but the well-known ghost-free DL interactions of the helicity - 0 and helicity - 2 gravitons in massive GR [3]. Including the independent interactions L 3 , 4 with three and four powers of k , one equally easily reproduces the remaining h ( ∂ 2 π ) 3 interaction of the decoupling limit of massive GR. As a next step, we use the above formalism to derive a closed-form expression for the vector-scalar interactions in the DL. To illustrate, we will start with the case when the two free parameters of dRGT are chosen so that all the scalar-tensor nonlinear interaction at the scale Λ 3 identically vanish [3]. For this parameter choice a linear combination of L 2 , L 3 and L 4 can be expressed, up to a total derivative, in terms of L 1 and a CC term with a tuned value [25]; the resulting theory was dubbed 'the minimal model'. In the GCT and LLT invariant vierbein formalism the minimal model takes the form: where the one form k is defined in the usual way k d = dx β k d β using the 'k-vierbein' k a µ ≡ e a µ -λ a ¯ a ∂ µ φ ¯ a . In spite of the absence of the nonlinear helicity-0 interactions with helicity-2 at the scale Λ 3 , the minimal model has nonlinear interaction terms of the vector mode with the helicity-0 at the scale 9 Λ 3 . As can be straightforwardly checked, the potentially diverging contributions, e.g. of the form εεB∂ 2 π , in fact vanish due to the symmetry properties of the B field (this is precisely what allows to consistently set the scaling of the field B to be ( M Pl ) -1 / 2 ). Keeping all finite terms involving B in the decoupling limit and expanding the wedge product in (23), one obtains 10 This is the simplest all-orders expression. It involves the auxiliary field B . We may, if we like, integrate it out to obtain an expression involving only the physical fields π and A , at the cost of generating an infinite number of terms. In matrix notation (all indices are understood to be contracted with the help of the flat metric), the equation of motion for B yields, where F µν = ∂ µ A ν -∂ ν A µ denotes the field strength for the vector mode, and P is a tensor of the schematic form ( ηη + η∂∂π ) appropriately antisymmetrized 11 . When substituted back into the action, the last equation gives the closed-form expression for the vector-scalar interactions in the decoupling limit of the 'minimal' massive GR The lowest - order term in the expansion of the latter Lagrangian in powers of ∂∂π yields the (correct-sign) kinetic term for the vector, while higher order terms give its interactions with the scalar helicity. Moving away from the minimal model, for the most general form of the potential the Lagrangian has the following schematic form in the decoupling limit 12 Varying w.r.t. the non-dynamical field B yields an expression for it in terms of π and A , that can be substituted back into the action, recovering the complete decoupling limit form of the vector-scalar interactions. These interactions are derived in [26]. The resulting expressions can be readily used for studying dynamics of the given sector of the theory on various background solutions.", "pages": [ 13, 14, 15 ] }, { "title": "5. DYNAMICAL ANTISYMMETRIC FIELD", "content": "While in pure massive gravity the link fields λ a ¯ a are non-dynamical, one can go further and consider a generalization with dynamical link fields, nonlinearly realizing the symmetry breaking pattern (17). Given the symmetries at hand, the most general Lagrangian at low energy can be written as a function of the fields with definite transformation properties under GCT × LLT × ISO (3 , 1) INT , The covariant derivative D µ acts on the LLT indices through the standard expression D µ λ a ¯ a = ∂ µ λ a ¯ a + ω a µ b λ b ¯ a , where the spin connection ω a µ b can be expressed in terms of the vierbein and its derivatives in a torsion-free theory. Being a Lorentz matrix-valued field, λ is most conveniently expressed in terms of the antisymmetric generator, λ = exp( v/f ), where f denotes the 'decay constant' of v . The decay constant f is an adjustable parameter of the theory. The lowest-order non-trivial invariant that one can form from these fields can be written as follows and includes the kinetic term for the six degrees of freedom present in v a ¯ a . Note that the kinetic term for the λ field, alongside with the EH term, breaks the local SL (4) symmetry of the mass and potential terms if we were to promote the λ to a SL (4) valued field. We will write f ∼ ˆ f ( M Pl Λ 3 ) 1 / 2 , where ˆ f is dimensionless. The Λ 3 decoupling limit remains intact as long as ˆ f remains fixed in this limit, i.e. does not depend parametrically on any other scales. Supplementing the action by the ghost-free potential terms, for example Eq. (13), one obtains a set of interactions of v with the rest of the fields present in the theory (for the moment, we choose the LLT gauge defined by B = 0.) At the linearized order, (13) yields the mass term, as well as a mixing with the vector mode in the decoupling limit (we disregard the distinction between the LLT and spacetime indices for notational simplicity) A shift in the v field, v µν → ˆ v µν -F µν Λ 3 ˆ f ( 2+ ˆ f 2 glyph[square] Λ 2 3 ) diagonalizes the action, bringing it to the following form Another peculiar feature of the above action is that ˆ v acquires a mass | m 2 v | ∼ Λ 2 3 / ˆ f 2 . This is below the cutoff of the effective theory to the extent that ˆ f glyph[greatermuch] 1. Note that, in order to reproduce the correct sign of the vector kinetic term at low energies, m 2 v has to be tachyonic; however, one could expect higher powers of v (e.g. v 4 ) to also be present, and these could stabilize the v potential. Likewise, the kinetic term of the vector acquires a modification. In the regime, ˆ f 2 glyph[square] / Λ 2 3 glyph[lessmuch] 1, the modification is irrelevant and A µ propagates the usual two vector polarizations of the massive graviton. Note that the residue of the vector particle propagator vanishes at the position of the pole of the v field. One can give the above generation of the mass m v an Anderson mechanism - like interpretation. Indeed, both of the antisymmetric fields, B and v , nonlinearly realize the local Lorentz invariance. One can always choose a gauge in which either of the two, e.g. B , is frozen to be zero, however one combination of these is gauge invariant (at the linear level, the invariant combination is simply ˆ f Λ 1 / 2 3 B -v ). Then, the gauge-invariant combination (which reduces to v in the B = 0 gauge) acquires a mass due to the spontaneous breaking of LLT. Finally, we comment on ghost-freedom of the interactions of the antisymmetric field v with the rest of the modes, present in the decoupling limit Lagrangian. The object k µ a is decomposed (excluding the symmetric vierbein perturbation) in the B = 0 gauge as follows, Most of the terms, that follow from the expansion of (13) are easily checked to be safe from more that two derivatives acting on fields in the resulting equations of motion - either on the basis of antisymmetry of v , or due to the presence of the ε symbols in the corresponding expressions. The only two interactions for which this property is not apparent are of the vv∂∂π∂∂π -type. The first of these is ε µν ·· ε ab ·· v a ρ ∂ µ ∂ ρ πv b σ ∂ ν ∂ σ π . The only potentially dangerous, three-derivative term arises in the equation of motion for π (all other similar terms vanish by antisymmetrization), and has the following form, Now, antisymmetrization in the a and b indices tells us that the object in the parentheses is antisymmetric in the ( ρ, σ ) pair. Contracted with ∂ ρ ∂ σ on the scalar, the term at hand vanishes. Likewise, a potentially dangerous term in the Lagrangian ε µν ·· ε ab ·· ∂ µ ∂ a πv b ρ v ρ σ ∂ ν ∂ σ π yields an apparently ghostly contribution to the π -equation of motion However, the object in the square parentheses in this expression is manifestly symmetric under µ → ν . Contracted with the antisymmetric ε µν ·· , this again yields zero. Of course, although this is a nice consistency-check, such a vanishing of the three-derivative terms in the equations of motion is by no means surprising and follows automatically from the inherent ghost-freedom of the potential (13).", "pages": [ 15, 16, 17, 18 ] }, { "title": "6. BRIEF COMMENTS ON THE LITERATURE", "content": "In this section, we briefly discuss the status of massive gravity as applied to the real world. In this approach, the graviton mass is taken to be of the order of the present day Hubble parameter, m ∼ H 0 ∼ 10 -33 eV (for phenomenological bounds on the graviton mass see [27]). Although this is a very small parameter as compared to the Planck scale, such smallness is robust - the mass parameter does not get renormalized by large quantum corrections [23, 28]; this is unlike the cosmological constant which does receive large renormalizations. Therefore, it is appealing to describe the observed cosmic acceleration as an effect due to a nonzero graviton mass. Massive gravitons can produce a state with the stress-tensor mimicking dark energy (the so-called self-accelerated solutions [29-34]). Massive gravity dark energy is expected to have a slightly different predictions from those of CC based cosmology, and the differences may be tested observationally. These solutions produce dark energy with the equations of state identical to that of CC, but different fluctuations. Unfortunately, certain fluctuations about these solutions are problematic - some of the physical 5 degrees of freedom have vanishing kinetic terms, destabilizing the background [35]. Extensions of dRGT by additional scalars [31, 36] or bi and multi-gravity [9, 19], or further extensions [37, 38], also exhibit selfaccelerated solutions. Recently, an extension by scalars has been proposed by De Felice and Mukohyama [39] and shown to have a self-accelerated solution with stable fluctuations - a first example of this kind. Spherically symmetric solutions and black holes in massive GR have been studied in [30, 40]. A general issue in dRGT is that it is a strongly coupled theory at the distance scale (Λ 3 ) -1 , which for the above value of the graviton mass is ∼ 1000 km. This scale is background dependent, and decreases for realistic backgrounds [41], but never enough for one to feel comfortable with it. The higher dimensional operators - that best manifest themselves in the decoupling limit - are suppressed by this scale. Moreover, on realistic backgrounds these operators give rise to order 1 or larger classical renormalization of the kinetic terms of fluctuations. That is how some of these kinetic terms vanish or flip their signs on the self-accelerated backgrounds. Therefore, dRGT needs an extension beyond the strong coupling scale in order for it to be potentially applicable to the real world. This extension is unknown at present, but for it to work it should introduce new states at or below the scale Λ 3 . Therefore, many properties of the backgrounds and fluctuations sensitive to scales above Λ 3 can get modified in an extended theory 13 . Furthermore, in the decoupling limit dRGT gets related [3] to the Galileons [43]. The latter are known to exhibit superluminal propagation on nontrivial backgrounds. So does dRGT for a large portion of the α 3 , α 4 parameter space. For theories satisfying the Froissart bound, this has been argued [44] to preclude a standard UV completion by a local, Lorentz invariant field or string theory, however, theories with long-rage fields do not necessarily obey this bound; moreover, there is no claim to rule out a possible Lorentz-violating, nonlocal or intrinsically higher dimensional completion. Furthermore, there is an exception for some special values of α 3 , α 4 , where subluminality for a spherically symmetric solution is achieved at the expense of not having an asymptotically flat background 14 [41, 45]. The question of whether superluminality can lead to prohibitive acausality is entangled with the strong coupling issue [49]. The conclusion of acausality of massive gravity [50, 51] that has been reached by constructing superluminal shock waves and characteristics is, in the context of a low energy theory, not warranted without a further nuanced study. A well known counterexample is the following: quantum electrodynamics (QED) in an external gravitational field, at energies below the electron mass, gives rise to dimension 6 operators, one of which yields superluminal characteristics for a photon propagating in a given nontrivial gravitational background [52]. However, this superluminality - which appears within the effective theory - does not mean that QED supplemented by GR is an acausal theory. In spite of a large body of literature on the issue of superluminality vs. acausality, some with split views, we believe that the low energy effective field theory understanding of systematic criteria for potential harms, or their absence, of superluminal low energy group and phase velocities is still to be precisely formulated [53]. Notes added: Ref. [26] has studied the decoupling limit of dRGT using the vierbein formalism. This work, even though it appeared later than v1 of the present work, should be considered as concurrent on the main idea of studying the decoupling limit in this formalism; moreover the results of [26] on the decoupling limit are superior to ours in their completeness. The remarkable work [54], appearing in 2006, introduced almost all of the ingredients of massive gravity, including the Stuckelbergs for the LLT's (but not the φ a fields). Unfortunately, Ref. [54] adopts an incorrect conclusion regarding the existence of the Boulware-Deser ghost. We thank Andrew Tolley for bringing this to our attention. Acknowledgments: The authors benefited from communications with Nikita Nekrasov, Warren Siegel, Andrew Tolley, and Arkady Vainshtein, on topics related to the subject of the present work. GG is supported by the NSF grant PHY-0758032 and NASA grant NNX12AF86G S06. 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 and Innovation. The work of KH was made possible in part through the support of a grant from the John Templeton Foundation. The work of YS is supported by the John Templeton Foundation through Professor John Moffat. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. DP has been supported by the U.S. Department of Energy under contract No. DE-SC0009919. GG would like to thank the Perimeter Institute for hospitality. KH and DP would like to thank the Center for Particle Physics and Cosmology at New York University for their hospitality. th/0210184].", "pages": [ 18, 19, 20, 21, 22 ] } ]

[ { "title": "On the Mutual Information in Hawking Radiation", "content": "Norihiro Iizuka 1, ∗ and Daniel Kabat 2, † 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JAPAN 2 Department of Physics and Astronomy, Lehman College, City University of New York, Bronx NY 10468, USA We compute the mutual information of two Hawking particles emitted consecutively by an evaporating black hole. Following Page, we find that the mutual information is of order e -S where S is the entropy of the black hole. We speculate on implications for black hole unitarity, in particular on a possible failure of locality at large distances. Hawking's discovery that black holes emit thermal radiation [1] is one of the few tangible results in quantum gravity, and the resulting conflict with unitarity [2] has driven much of the research in the field. See [3] for a review. The goal of the present paper is to obtain new insight into this issue, from a computation of the mutual information carried by successive Hawking particles. Consider two successive Hawking particles emitted by an evaporating black hole, as shown in Fig. 1. Motivated by the AdS/CFT correspondence we assume that a conventional quantum mechanical description of this process is available. In particular we assume there is an underlying Hilbert space with unitary time evolution that describes the microscopic degrees of freedom. In this fine-grained description there is no tension with unitarity: the two Hawking particles are correlated due to their shared history, in which they both interacted with the microscopic black hole degrees of freedom. In this fine-grained description, Hawking radiation from a black hole is no different from the blackbody radiation emitted by any other hot macroscopic object. However for a black hole we would like to consider a coarse-grained description, in which the black hole is characterized just by its macroscopic thermodynamic properties such as energy and and entropy. It seems reasonable that this coarse-graining gives rise to the usual notion of a semiclassical spacetime. That is, only in a coarse-grained description could one hope to describe the black hole using the usual Schwarzschild metric, and could one hope to describe Hawking radiation using effective field theory on the Schwarzschild background. In support of this view, note that the usual black hole metric only captures macroscopic properties such as mass or charge. Also Hawking's calculation, carried out in this context, shows that a black hole emits uncorrelated thermal radiation. This behavior is expected in a coarsegrained description of blackbody radiation, since such radiation is completely characterized by a macroscopic quantity, namely the temperature of the black hole. In this setting, to understand unitarity, the main challenge is identifying which properties of the coarse-grained description deviate most significantly from the underlying microscopic description. Given some underlying microscopic theory, what modification to the coarse-grained description is most appropriate for restoring unitarity? To sharpen our discussion we consider the correlation between the two successive Hawking particles a and b shown in Fig. 1. We have in mind two particles that are emitted almost simultaneously from well-separated points on the horizon, so that the separation between a and b is large and spacelike. The correlation can be measured by the mutual information where S a is the entropy of particle a , S b is the entropy of particle b , and S ab is the entropy of both. According to Hawking's calculation a and b are uncorrelated and the mutual information vanishes. Under seemingly reasonable assumptions this will remain true even in the presence of interactions [3]. But if the entire system (including the black hole) is in a random pure state, the true correlation between a and b can be obtained from the fundamental work of Page [4]. Page considers a Hilbert space of dimension m entangled with another Hilbert space of dimension n ≥ m , and shows that in a random pure state the average entropy is For large n the sum can be estimated using the EulerMaclaurin formula, which gives To apply this to the situation at hand, let N a be the dimension of the Hilbert space of particle a , let N b be the dimension of the Hilbert space of particle b , and let N bh = e S be the dimension of the Hilbert space of the black hole. For particle a , for example, we have a Hilbert space of dimension N a entangled with a Hilbert space of dimension N b N bh . Thus Using (1) and (3) we find that for large N bh , the mutual information in the Hawking particles a and b is This is our main result. It shows that the mutual information carried by two successive Hawking particles is of order e -S . For example if each Hawking particle could carry one bit of information then N a = N b = 2 and I ab ≈ 9 8 e -S , while if each Hawking particle could carry a large amount of information then I ab ≈ 1 2 N a N b e -S . As we discussed above, the usual semiclassical picture of gravity must be modified in order to reproduce these correlations. Roughly speaking the possible modifications fall into three categories.", "pages": [ 1, 2 ] }, { "title": "1. Modify the interior", "content": "It could be that microscopic quantum gravity effects become important at or inside the (stretched) horizon of the black hole, invalidating the use of the classical Schwarzschild geometry in this region and generating correlations between outgoing Hawking particles. However outside the horizon semiclassical gravity and effective field theory could be valid. Proposals of this type include fuzzballs [5, 6] and firewalls [7, 8].", "pages": [ 2 ] }, { "title": "2. Modify the exterior", "content": "It could be that effective field theory is not trustworthy, even at macroscopic distances outside the black hole. For example, it could be that the underlying theory of quantum gravity leads to violations of locality over large distances, in a way that generates correlations and restores unitarity. Some models with non-locality have been discussed in [911].", "pages": [ 2 ] }, { "title": "3. Modify both", "content": "Perhaps both the interior region of the black hole and the rules of effective field theory outside the black hole receive important corrections due to microscopic quantum gravity effects. Unfortunately, just from considerations of unitarity, there is no clear way to decide between these possibilities. But since most models discussed in the literature take other approaches, let us indulge in a little speculation about the possibility of non-locality outside the horizon. A key principle in local field theory is microcausality, that is, the property that field operators commute at spacelike separation. If we are prepared to give up on locality outside the horizon, it could be that spacelike separated field operators no longer commute. We have in mind that the resulting non-locality extends over macroscopic distances, and would thus fall into the category of modifying the exterior of the black hole. But we must admit that in order to restore unitarity, non-locality which extends to the stretched horizon could do the job. In fact, AdS/CFT may provide some motivation for the radical idea of non-locality over macroscopic distances. Order-by-order in the 1 /N expansion of the CFT one can construct CFT operators which mimic local field operators in the bulk [12, 13]. The algorithm involves starting from a single primary field and adding an infinite tower of higher dimension operators. In the 1 /N expansion one can show that the resulting CFT operators commute whenever the bulk points are spacelike separated. 1 But at finite N it seems unlikely that the higher dimension operators required for bulk locality could exist. Instead it's more likely that bulk observables will fail to commute at spacelike separation, even over macroscopic distances, by an amount which is nonperturbatively small in the 1 /N expansion. As a toy model for this idea, consider a pair of independent harmonic oscillators characterized by with all other commutators vanishing. We think of these oscillators as representing two independent degrees of freedom in some underlying microscopic description ('the boundary'). Suppose these boundary operators can be mapped to bulk operators which describe the Hawking particles shown in Fig. 1. We assume a boundary-tobulk map depending on two parameters θ and φ , explicitly given by This map can be thought of as a Bogoliubov transformation followed by setting The Bogoliubov transformation (11), (12) preserves the canonical commutation relations. But this is not true of the transformation (13), (14), due to the relative -sign which appears in (14). Rather the combined map leads to bulk commutators with all other commutators vanishing. Note that θ drops out of the commutation relations. This is expected since θ in (11), (12) parametrizes a Bogoliubov transformation between the bulk and boundary degrees of freedom, which by definition is a transformation that preserves the commutators. Having non-zero φ , on the other hand, leads to a non-zero commutator between ˆ a and ˆ b . We think of this as representing a bulk commutator which is non-zero at spacelike separation. This is a drastic modification to local field theory - a risky game to play - and it's not clear whether a consistent theory can be constructed along these lines. But let's proceed, and explore the connection between non-commutativity and entanglement. One can start from the microscopic vacuum and build a Fock space If one only acts on the vacuum with operators of type a one never notices the non-commutativity (likewise for type b ). But suppose the Hawking particles shown in Fig. 1 correspond to a two-particle state (which we haven't bothered to normalize) In general this state is entangled. To see this we split the Hilbert space into α and β oscillators, H = H α × H β . The choice of splitting is somewhat arbitrary, and leads to a freedom that we discuss in more detail below. Given the splitting, we construct the density matrix ˆ ρ = | ψ 〉〈 ψ | and trace over H β to obtain the reduced density matrix for particle a . 2 Properly normalized, this procedure gives where ξ + is defined by ξ + = tanh( θ + φ ). The associated entropy is The mutual information between a and b is I ab = S a + S b -S ab . But S b = S a , while the combined system is in a pure state with S ab = 0. So the mutual information is simply twice the result (22), Of course this result depends on how we decide to split the Hilbert space. In other words, it depends on what we decide to trace over in constructing ˆ ρ a . But the freedom to choose a splitting can be absorbed into a shift of the Bogoliubov parameter θ . More precisely θ parametrizes the freedom to split the Hilbert space into H α ' × H β ' , where α ' and β ' are the independent oscillators defined in (11), (12). 3 One can use this freedom to set θ + φ = 0, which makes the mutual information in the state (19) vanish. But even if the mutual information in this particular state vanishes, there will still be other states that carry mutual information. For example the state has mutual information which can be obtained from (23) by replacing ξ + → ξ -≡ tanh( θ -φ ), namely In attempting to make both (23) and (25) small the best one can do is set θ = 0. Then (23) and (25) are equal, and at leading order for small φ the mutual information in either of the states ˆ a † ˆ b † | 0 〉 or ˆ b † ˆ a † | 0 〉 is This toy model suggests that two operators which have a commutator that is O ( φ ) as in (16), typically produce entangled states with a mutual information that is O ( φ 2 ) as in (26). Since we know how much mutual information is present in Hawking radiation, we can estimate how big commutators must be at spacelike separation. Our results suggest that two field operators should have a commutator of order e -S/ 2 in the presence of a black hole, in order to account for the mutual information I ab ∼ e -S carried by two successive Hawking particles. (More precisely, we have in mind that matrix elements of the commutator in a typical state of the black hole plus Hawking radiation should be of order e -S/ 2 .) In the context of AdS/CFT black hole entropy is O ( N 2 ), so this effect is non-perturbatively small in the 1 /N expansion of the CFT.", "pages": [ 2, 3, 4 ] }, { "title": "Acknowledgements", "content": "Weare grateful to Gilad Lifschytz for valuable discussions and for comments on the manuscript. DK thanks the YITP for hospitality during this work. NI is supported in part by JSPS KAKENHI Grant Number 25800143. DK is supported by U.S. National Science Foundation grant PHY-1125915 and by grants from PSC-CUNY.", "pages": [ 4 ] } ]

[ { "title": "Emergence of space and the general dynamic equation of Friedmann-Robertson-Walker universe", "content": "Wen-Yuan Ai, Xian-Ru Hu, Hua Chen, and Jian-Bo Deng ∗ Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P. R. China (Dated: October 16, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "A novel idea that the cosmic acceleration can be understood from the perspective that spacetime dynamics is an emergent phenomenon was proposed by T. Padmanabhan. The Friedmann equations of a Friedmann-Robertson-Walker universe can be derived from different gravity theories with different modifications of Padmanabhan's proposal. In this paper, a unified formula is proposed, in which those modifications can be treated as special cases when the formula is applied to different universe radii. Furthermore, the dynamic equations of a FRW universe in the f ( R ) theory and deformed Hoˇrava-Lifshitz theory are investigated. The results show the validity of our proposed formula. PACS numbers: 04.50.-h, 04.60.-m, 04.70.Dy", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Black hole physics has provided strong hints of a deep and fundamental relationship between gravitation, thermodynamics and quantum theory. At the heart of this relationship is black hole thermodynamics [1, 2], which claims that a black hole can be regarded as a thermodynamical system. But why would such a geometric object have thermodynamical characteristics? With the deep study of black holes proceeding, now it is generally believed that our spacetime is emergent. In 1995, Jacobson [3] first derived the Einstein equations by applying the Clausius relation δQ = TδS on a local Rindler causal horizon. Here δS is the change in the entropy, while δQ and T , respectively, represent the energy flux across the horizon and the Unruh temperature seen by an accelerating observer just inside the horizon. Verlinde [4], by claiming that gravity may not be a fundamental interaction but should be interpreted as an entropic force caused by changes of entropy associated with information on the holographic screen, put forward the next great step towards understanding the nature of gravity. Further, Newton's law of gravitation, the Possion equation, and in the relativistic regime the Einstein field equations, were derived with the holographic principle and the equipartition law of energy assumed. Earlier, Padmanabhan [5] observed that the equipartition law of energy for horizon degrees of freedom (DOF), combined with the thermodynamic relation S = E/ 2 T , leads to Newton's law of gravity. See Refs. [6, 7] for a review. Nonetheless, in most investigations, only the gravitational field equations are treated as the equations of emergent phenomena, with the preexisting background geometric manifold assumed. If gravity can be treated as an emergent phenomenon, spacetime itself should be an emergent structure. It is obviously very hard to think that the time used to describe the evolution of dynamical variables is emergent from some pregeometric variables and the space around finite gravitational systems is emergent. Surprisingly, the conceptual difficulties disappear when one considers the emergence of spacetime in cosmology. Recently, Padmanabhan [8] argued that our universe provides a natural setup to implement the issue that the cosmic space is emergent as cosmic time progress . He argued that the expansion of the universe is due to the difference between the surface DOF and the bulk DOF in a region of emerged space and successfully derived the Friedmann equation of a flat FriedmannRobertson-Walker (FRW) universe. A simple equation dV/dt = L 2 p ∆ N was proposed in Ref. [8], where V is the Hubble volume and t is the cosmic time. ∆ N = N sur -N bulk with N sur being the number of DOF on the boundary and N bulk the number in the bulk. Following Ref. [8], Cai generalized the derivation process to the higher ( n + 1)-dimensional spacetime. He also obtained the corresponding Friedmann equations of the flat FRW universe in Gauss-Bonnet and more general Lovelock cosmology by properly modifying the effective volume and the number of DOF on the holographic surface from the entropy formulas of static spherically symmetric black holes [9]. The authors of [10] took another viewpoint and derived the Friedmann equations with generalized holographic equipartition. They assumed that ( dV/dt ) is proportional to a function f (∆ N ). Note that the authors of Ref. [9, 10] only derived the Friedmann equations of the spatially flat FRW universe. In Ref. [11], by modifying the original proposal of Padmanabhan, Sheykhi derived the Friedmann equations of the FRWuniverse with any spatial curvature. He gave a new equation dV/dt = L 2 p (˜ r A /H -1 )∆ N , where ˜ r A is the apparent horizon radius and H is the Hubble constant. Here V is the volume of the bulk inside the apparent horizon and ∆ N is the difference between the number of DOF on the apparent horizon and in the bulk. However, different modifications relate to different gravity theories are necessary in order to obtain Friedmann equations. There are even disparate formulas for obtaining Friedmann equations of a flat FRW universe and for obtaining the equations of one with any spatial curvature. We believe that if the idea suggested by Padmanabhan is right, there should be a unified formula describing the emergence. In this paper, we propose a general formula and argue that the modifications made by Cai and Sheykhi can be treated as special cases with the formula being applied to different universe radii when the general relationship between the number of DOF on the surface and the entropy of the horizon [12] is adopted. With our proposed formula, further study of the dynamic equations of a FRW universe in the f ( R ) theory and the deformed Hoˇrava-Lifshitz (HL) theory can be carried out. The paper is organized as follows: In Sec. III, we present our formula and show in what cases the formula would be reduced to those of Padmanabhan, Cai and Sheykhi. In Sec. III, the f ( R ) theory and deformed HL theory are investigated. Section IV is for conclusions and discussions. We take k B = 1 through all this paper.", "pages": [ 2, 3 ] }, { "title": "II. GENERAL FORMULA FOR THE EMERGENCE OF COSMIC SPACE", "content": "First, let us recall Padmanabhan and the others' work [8, 9, 11, 12]. Padmanabhan notices that for a pure de Sitter universe with Hubble constant H, the holographic principle can be expressed in terms of the form [8] where N sur denotes the number of DOF on the spherical surface of Hubble radius H -1 , N sur = 4 πH -2 /L 2 p , with L p being the Planck length, while the bulk DOF N bulk = | E | / (1 / 2) T . Here | E | = | ρ +3 p | V , is the Komar energy and the horizon temperature T = H/ 2 π . For the pure de Sitter universe, by substituting ρ = -p into Eq. (1), the standard result H 2 = 8 πL 2 p ρ/ 3 is obtained. One can get | E | = (1 / 2) N sur T from the equality (1), which is the standard equipartition law. Since it relates the effective DOF residing in the bulk to the DOF on the boundary surface, Padmanabhan called it holographic equipartition . However, as shown by lots of astronomical observations, our real universe is not a pure de Sitter but asymptotically de Sitter universe. Padmanabhan further considers that, for our real universe, the expansion is caused by the emergence of space and relates to the difference ∆ N = N sur -N bulk . He proposes a simple equation as [8] This suggests that the expansion of the universe is being driven towards holographic equipartition. It can be treated as the thermodynamical process on the horizon and therefore comes to be an emergent phenomenon. Putting the above definition of each term, one obtains This is the standard dynamical equation for a FRW universe in general relativity. With the help of the continuity equation . ρ +3 H ( ρ + p ) = 0, one gets the standard Friedmann equation where k is an integration constant which can be interpreted as the spatial curvature of the FRW universe. Here the cosmological constant term has already been taken into account in the Komar energy. In fact, in order to have the asymptotic holographic equipartition, Padmanabhan takes ( ρ + 3 p ) < 0. This implies the existence of dark energy is necessary. Put another way, the existence of a cosmological constant in the universe is required for asymptotic holographic equipartition [8]. Then Cai generalized the above derivation process to the higher (n+1)-dimensional case with n > 3. In this case, the number of DOF on the holographic surface is given by [4] Here A = n Ω n /H n -1 and α = ( n -1) / 2( n -2) with Ω n being the volume of an n-sphere with unit radius. In this case, Cai made a minor modification for the proposal (2) as [9] where the volume V = Ω n /H n . Here the bulk DOF remains the same i.e. N bulk = -2 E/T when we only consider the accelerating phase with ( n -2) ρ + np < 0 and the bulk Komar energy is [13] With the help of the continuity condition, after some simple algebra, the Friemann equation can be obtained finally. Moreover, for Gauss-Bonnet gravity and Lovelock gravity, the volume increase was modified as follows [9] where ˜ A/L n -1 p = 4 S , S is the entropy formula of black holes modified in Gauss-Bonnet gravity and Lovelock gravity. Further, the number of DOF on the surface must be taken special forms until the Friedmann equations of the spatially flat FRW universe can be obtained [9]. To get the Friedmann equations of the FRW universe with any spatial curvature in Gauss-Bonnet and Lovelock gravities, Sheykhi proposed [11] where ˜ r A = 1 / √ H 2 + k/a 2 is the apparent horizon radius, H is the Hubble constant and V is respect to the volume of the bulk inside the apparent horizon, N sur and N bulk are respectively the number of DOF on the apparent horizon and in the bulk. Even though we can derive Friedmann equations from special forms of N sur , it is suspect that N sur must be chosen in such complicated forms. Compared with this, the general relationship between the number of DOF on the surface and the entropy of the horizon proposed by the authors of Ref. [12] is more natural, i.e. With this relationship adopted, we propose a more general equation When n = 3, we have α = 1, S = A/ 4 L 2 p , dN sur /dt = dA/ ( L 2 p dt ) and dV/dt = dA/ 2 Hdt , thus Eq.(1) proposed by Padmanabhan is recovered immediately. By substituting N sur = 4 S = ˜ A/L n -1 p , d ˜ V /dt = d ˜ A/ ( n -1) Hdt into Eq. (11) and applying Eq. (11) to the Hubble radius, we can obtain Eq. (6), which is the modification given by Cai. Similarly, if we apply Eq. (11) to the apparent horizon radius, in this case, d ˜ V /dt = ˜ r A d ˜ A/ ( n -1) dt , substituting it into Eq. (11), Eq. (9) proposed by Sheykhi is obtained immediately. We have shown the universality of Eq. (11). When we consider the (3+1)-dimensional Einstein gravity, it gets back to Padmanabhan's original equation. However, Eq. (11) can also describe the evolution of a universe in higher (n+1)-dimensional Einstein gravity, GaussBonnet gravity or more general Lovelock gravity. Our proposed equation is the general form that obtains Eq. (6) when applied to the Hubble radius and Eq. (9) when applied to the apparent radius.", "pages": [ 4, 5, 6 ] }, { "title": "III. THEDYNAMICEQUATIONSOFAFRWUNIVERSEINTHE f ( R ) THEORY AND DEFORMED HL THEORY", "content": "Up to now, we have argued that Eq. (11) is the general formula which can be used to investigate the emergence of spacetime in cosmology. Now we will use it to push further study on the dynamic equations of a FRW universe in the f ( R ) theory and the deformed HL theory. For simplicity, we only consider the case when Eq. (11) is applied to the Hubble sphere. Let us first consider the emergence of space in f ( R ) theory. f ( R ) gravity was first proposed in 1970 by H. A. Buchdahl [14], as a type of modified gravity theory that generalizes Einstein's general relativity. It is actually a family of theories, each one defined by a different function of the Ricci scalar. When f ( R ) = R , it goes back to general relativity. In f ( R ) theory, We take the Komar energy generally as [15] ∣ ∣ where T µν is the total energy-momentum tensors which contains two parts, the energymomentum tensor of matter and gravity. U µ , U ν are the components of U a , which is the tangent vector of isotropic observers' wordline in a FRW universe. One can obtain the total energy density ρ and total pressure p through T µν . The black hole entropy is given by [16] From relation (10), we get where f ' denotes f ( R ) derivatives taken with respect to R . Substituting V = Ω n /H n into Eq. (7) and using N bulk = -2 E/T , we have According to Eq. (11), we obtain the dynamic equation This is the formal dynamic equation of the FRW universe we got from the emergence of space in the (n+1)-dimensional f ( R ) theory. When n = 3, we have We know that the total energy-momentum tensor T µν cannot be determined if we do not give the specific form of the theory. Now, we will determine that in the FRW model. By assuming the background spacetime is spatially homogeneous and isotropic, one can find the FRW metric where d Ω 2 n -2 denotes the line element of an (n-1)-dimensional unit sphere and the spatial curvature constants k = 0 , 1 , -1 correspond to a flat, closed, and open universe, respectively. Based on the cosmological principle, we take the components of T µν as where i, j run over 1 , 2 , ..., n -1. As in Ref. [12], we can use the Einstein-Hilbert (EH) action to determine the total energy-momentum tensor. In the f ( R ) theory the EH action takes the form where κ 2 = 8 πG . L m is the Lagrangian density of matter in this theory. After using the variational principle δS = 0, one can obtain where T ( m ) µν is the energy-momentum tensor of the matter. Adopting the Einstein tensor G µν = R µν -Rg µν / 2, one has With the energy-momentum tensor of the gravity caused by higher-order derivatives defined as one can arrive at With the matter to be perfect fluid assumed, according to the form Eq. (19) and one can get in a FRW universe, where ˙ R denotes f ( R ) derivatives taken with respect to t . By substituting Eqs. (26) and (27) into Eq. (16), we finally obtain the dynamic equation of a FRW universe from the idea of the emergence of space. When f ( R ) = R , it has f ' = 1, ρ g = p g = 0. Thus, from Eq. (17) the standard dynamic equation of the FRW universe in general relativity is obtained, showing consistency. We regard Eq. (16) or Eq. (17) as the modified dynamic equation of a FRW universe with a global correction from the idea of the emergence of space. Next, we will consider the emergence of space in deformed HL gravity. Motivated by Lifshitz theory in solid state physics, Hoˇrava proposed a new gravity theory at a Lifshitz point [17-19]. The theory has manifest 3-dimensional spatial general covariance and time reparametrization invariance. This is a non-relativistic renormalizable theory of gravity and recovers the four dimensional general covariance only in an infrared limit. Thus, it may be regarded as a UV complete candidate for general relativity. In the deformed HL gravity, we also take the form of the Komar energy as in the f(R) theory. The entropy has the form [20] where the parameter ω = 16 µ 2 /κ 2 . The entropy/area relation has a logarithmic term, which is a characteristic of HL gravity theory. However, as the parameter ω →∞ , it goes back to the one in Einstein gravity. Using the relation in Eq. (10), one gets N bulk is as same as Eq. (15). By substituting every term into Eq. (11), we get When n = 3, then Ω = 4 π/ 3. Thus, we have Now, we have proved that the standard dynamic equation of the FRW universe in general relativity can be recovered when the parameter ω →∞ . We define the total energy and pressure of the universe in HL theory as [21, 22] where the matter is assumed to be a perfect fluid. The cosmological constant term, the curvature term, and the dark radiation term are [22] At last, with the speed of light and Newton's constant in the IR limit given by [20] one can get the specific form of entropy Eq. (29). From this, with Eqs. (31) and (33), one can finally get the complete dynamic equation of a FRW universe. Here, as in the case of f ( R ) theory, Eq. (31) or Eq. (32) is the modified dynamic equation in HL theory from the global viewpoint of emergence. So far, we have shown that Eq. (11) can be used to derive dynamic equations of a FRW universe in the f ( R ) gravity and deformed HL gravity. Now, we would like to say that our equation can be treated as the unified dynamic equation of a FRW universe in all the gravity theories involved in this paper. Note that the dynamic equations for a FRW universe in f ( R ) and the deformed HL theory obtained here are different from those obtained in Ref. [12]. Since we have argued that Eq. (11) is a more general formula of describing the cosmic emergence, we believe that it is better to use Eq. (11) than Eq. (2) to derive the dynamic equation of a FRW universe while applying the idea of Padmanabhan to the gravity theories beyond general relativity.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "IV. DISCUSSION AND CONCLUSION", "content": "In this paper, we investigated the novel idea proposed by Padmanabhan [8], which states that the emergence of space and the universe's expansion can be understood by calculating the difference between the number of DOF on the surface and in the bulk. We also summarized the work done by Cai, Sheykhi and others [9, 11, 12]. Since different modifications of Padmanabhan's proposal relate to different gravity theories until finally the Friedmann equations are obtained, which seems unreasonable, we proposed a general formula Eq.(11) which can be reduced to the different modified ones in different cases. When we consider the (3+1)-dimensional Einstein gravity, it gets back to Padmanabhan's original Eq. (2). We argued that the equations proposed by Cai and Sheykhi are, respectively, special cases of our proposed equation being applied to the Hubble radius and apparent radius. We also pushed for further research on the dynamic equations of a FRW universe in f ( R ) theory and deformed HL theory using the proposed formula. With the idea proposed by Padmanabhan, we finally obtained the modified dynamic equations from the perspective of emergent phenomena. The resulting equations show a strong consistency with the standard dynamic equations of the FRW universe in general relativity when n = 3, f ( R ) = R and ω →∞ . These results imply that our formula is not only valid for general relativity, higher dimensional Einstein gravity, Gauss-Bonnet gravity, and Lovelock gravity, but is also valid for f ( R ) gravity and deformed HL gravity. Therefore, Eq. (11) can be treated as the unified formula governing the emergence of cosmic space and can be widely used to study the dynamic equations of FRW universes. Even though we have shown the universality of Eq. (11), we have not found the origin of our proposed formula. However, since Eq. (11) is valid for all the gravity theories mentioned here, we still believe it has a deep one. This, surely, is worthy of further investigation.", "pages": [ 10, 11 ] } ]

[ { "title": "Black Holes or Firewalls: A Theory of Horizons", "content": "Yasunori Nomura, Jaime Varela, and Sean J. Weinberg Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, CA 94720, USA Theoretical Physics Group, Lawrence Berkeley National Laboratory, CA 94720, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present a quantum theory of black hole (and other) horizons, in which the standard assumptions of complementarity are preserved without contradicting information theoretic considerations. After the scrambling time, the quantum mechanical structure of a black hole becomes that of an eternal black hole at the microscopic level. In particular, the stretched horizon degrees of freedom and the states entangled with them can be mapped into the nearhorizon modes in the two exterior regions of an eternal black hole, whose mass is taken to be that of the evolving black hole at each moment. Salient features arising from this picture include: (i) the number of degrees of freedom needed to describe a black hole is e A / 2 l 2 P , where A is the area of the horizon; (ii) black hole states having smooth horizons, however, span only an e A / 4 l 2 P -dimensional subspace of the relevant e A / 2 l 2 P -dimensional Hilbert space; (iii) internal dynamics of the horizon is such that an infalling observer finds a smooth horizon with a probability of 1 if a state stays in this subspace. We identify the structure of local operators responsible for describing semi-classical physics in the exterior and interior spacetime regions, and show that this structure avoids the arguments for firewalls-the horizon can keep being smooth throughout the evolution. We discuss the fate of infalling observers under various circumstances, especially when the observers manipulate degrees of freedom before entering the horizon, and we find that an observer can never see a firewall by making a measurement on early Hawking radiation. We also consider the presented framework from the viewpoint of an infalling reference frame, and argue that Minkowski-like vacua are not unique. In particular, the number of true Minkowski vacua is infinite, although the label discriminating these vacua cannot be accessed in usual non-gravitational quantum field theory. An application of the framework to de Sitter horizons is also discussed.", "pages": [ 1 ] }, { "title": "1 Introduction and Summary", "content": "General relativity and quantum mechanics are two pillars in contemporary fundamental physics. The relation between the two, however, is not clear. On one hand, one can build quantum field theory on a fixed curved background, calculating quantum properties of matter in the existence of gravity. On the other hand, a naive application of such a semi-classical procedure often leads to puzzles that signal the incompleteness of the picture. A well-known example is the overcounting of degrees of freedom that arises when the interior spacetime and outgoing Hawking radiation of a black hole are treated as independent objects on a certain equal-time hypersurface (called a nice slice) [1]. It is clearly an important and nontrivial task to understand how the world as described by general relativity emerges in a consistent theory of quantum gravity. An elegant way to address the overcounting problem described above was put forward in Refs. [2, 3] under the name of black hole complementarity. This hypothesis asserts that With these assumptions, the issue of overcounting can be solved-the distant picture having Hawking radiation and the infalling picture with the interior spacetime are two different descriptions of the same physics; in particular, they are related by a unitary transformation associated with the reference frame change [5]. This complementarity picture, however, has recently been challenged in Refs. [6-9], which assert that the smoothness of horizon as implied by general relativity, (iv), is incompatible with the other assumptions, (i) - (iii). If true, this would have profound implications for fundamental physics; in particular, it would force us to abandon one of the standard assumptions in contemporary physics-unitary quantum mechanics, locality at long distances, or the equivalence principle. The authors of Refs. [6-8] argue that the simplest option is to abandon the equivalence principle-an observer falling into a black hole hits a 'firewall' of high energy quanta at the horizon. This would be a dramatic deviation from the prediction of general relativity. In this paper, we present a quantum theory of black hole (and other) horizons in which the standard assumptions of complementarity, (i) - (iv), are preserved. Our construction builds on earlier observations in Refs. [10-14]. In Refs. [10,11], two of the authors suggested that there are exponentially many black hole vacuum states corresponding to the same semi-classical black hole, and that there can be a (semi-)classical world built on each of them, all of which look identical at the level of general relativity but are represented differently at the microscopic level. It was argued that this structure can evade the firewall argument with appropriate internal dynamics for the horizon. In Ref. [12], the same picture was considered in an infalling reference frame in which the manifestation of the exponentially many microscopic states in this reference frame was discussed. More recently, Verlinde and Verlinde considered a similar picture in which the Hilbert space structure for the relevant degrees of freedom was identified more explicitly and in which a concrete qubit model demonstrating the basic dynamics of black hole evaporation was presented [13, 14]. In this paper we develop these observations further, identifying how the distant and infalling descriptions as suggested by general relativity emerge dynamically from a full quantum state obeying the unitary evolution law of quantum mechanics. In particular, we identify the structure of operators responsible for describing the exterior and interior regions of the black hole, which allows us to address explicitly the arguments made in Refs. [6-8]. The basic hypothesis of our framework is: The quantum mechanical structure of a black hole after the horizon is stabilized to a generic state (after the scrambling time [15]) is the same as that of an eternal black hole of the same mass at the microscopic level. In particular, the degrees of freedom associated with the stretched horizon and the outside states entangled with them can be mapped to the nearhorizon states of the eternal black hole in one and the other external regions, respectively. (These near-horizon states are described in a distant reference frame, using an equal-time hypersurface determined by the outside timelike Killing vector.) The precise mapping is such that the outside states entangled with the stretched horizon and the near-horizon states in one side of the eternal black hole respond in the same way to local operators representing physics in the exterior of the black hole. It is important that this identification mapping is made in each instant of time; for example, the mass of the corresponding eternal black hole must be taken as that of the evolving black hole at each moment. Note also that the identification with the eternal black hole is made only for the stretched horizon degrees of freedom and the outside states entangled with them; the structure of the other modes need not follow that of the eternal black hole. We can summarize these concepts by saying that an eternal black hole (of a fixed mass) provides a model for an evolving black hole for a timescale much shorter than that of the evolution. A schematic picture for this mapping is depicted in Fig. 1. Key elements to understand physics of black holes (and firewalls) arising from the picture described above are is thus e A / 4 l 2 P × e A / 4 l 2 P = e A / 2 l 2 P . The actual black hole states, however, occupy only a tiny e A / 4 l 2 P -dimensional subspace of the e A / 2 l 2 P -dimensional Hilbert space relevant for these degrees of freedom [13], as suggested by black hole thermodynamics. All the other states represent 'firewall states,' which do not allow for a semi-classical interpretation of the interior region. spacetime region act nontrivially both on the stretched horizon and the outside entangled modes. This 'asymmetry' arises because the stretched horizon degrees of freedom represent the exterior modes outside the horizon in the other side of the eternal black hole under the identification map described above. We find that these elements elegantly address the questions raised by the firewall argument. Representative results include We note that the framework presented here and resulting physical predictions also apply to other horizons, including de Sitter and Rindler horizons, with straightforward adaptations. We will discuss these cases toward the end of the paper. The organization of the rest of this paper is as follows. In Section 2, we describe the microscopic structure of the black hole vacuum states. In Section 3, we see how these states are embedded in the larger Hilbert space relevant for the stretched horizon degrees of freedom and the states entangled with them. We discuss how the vacuum and non-vacuum black hole states as well as the firewall states arise in this large Hilbert space, and identify the form of local operators responsible for describing the exterior and interior spacetime regions. We argue that the dynamics of quantum gravity can be such that a black hole stays as a black hole state under time evolution (not becoming a firewall state), and that an infalling observer interacting with such a state will see a smooth horizon with a probability of 1 because of the properties of the internal dynamics of the horizon. In Section 4, we discuss the fate of infalling observers under various circumstances, especially when the observers manipulate degrees of freedom before entering the horizon. We also describe how the present framework is realized in an infalling reference frame. We argue that locally (and global) Minkowski vacuum states are not unique at the microscopic level, although the same semi-classical physics can be built on any one of them, so that this degeneracy need not be taken into account explicitly in usual applications of quantum field theory, e.g. to the problem of scattering. In Section 5, we discuss how our framework is applied to de Sitter horizons.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Microscopic Structure of Black Holes", "content": "Here we discuss the microscopic structure of black holes, following Refs. [10-14]. Suppose we describe a system with a black hole, which for simplicity we take to be a Schwarzschild black hole in 4-dimensional spacetime, from a distant reference frame. We assume that, for any fixed black hole mass M , the entire system is decomposed into three subsystems: 1 Among all the possible quantum states for the C degrees of freedom, some are strongly entangled with the states representing ˜ B . 2 We call the set of these quantum states B : Following the locality hypothesis, we consider that systems C and R , more precisely operators acting only on C or R , are responsible for physics outside the stretched horizon, which is well described by local quantum field theory at length scales larger than the fundamental (string) length l ∗ . On the other hand, the interior spacetime for an infalling observer, as we will argue, is represented by operators acting on the combined ˜ BB system (on both ˜ B and B states). In our analysis below, we ignore the center-of-mass drift and spontaneous spin-up of black holes [17], which give only minor effects on the dynamics. Suppose, as usual, we quantize the system in such a way that the Hamiltonian near (and outside) the horizon takes locally the Rindler form. Then, a black hole vacuum state is described by one in which some of the states for the C degrees of freedom, i.e. B , are (nearly) maximally entangled with the states for ˜ B [18]. The basic idea of Refs. [10,11] is that there are exponentially many ( ≈ e A / 4 l 2 P where A = 16 πM 2 l 4 P is the horizon area 3 ) black hole vacuum states | ψ i 〉 which correspond to the same semi-classical black hole, and that there can be a (semi-)classical world built on each of them, all of which look identical to general relativity but are represented differently at the microscopic level (consistently with the no-hair theorem). More specifically, the states | ψ i 〉 , 3 Here and below, similar expressions are valid at the leading order in expansion in powers of l 2 P / A , in the exponent for the number of states (or in entropies). With this understanding, we will use the equal sign below, instead of the approximate sign. which live in the combined ˜ BB system, can be written as [11,13] Here, α ( i ) j are coefficients that satisfy the orthonormality condition and the condition for each | ψ i 〉 being maximally entangled and | ˜ b j 〉 and | b k 〉 ( j, k = 1 , · · · , e A / 4 l 2 P ) are elements of H ˜ B and H B with [13] where H ˜ B and H B are the Hilbert space factors that contain all the possible states for ˜ B and B , respectively. To be more precise, the black hole vacuum states | ψ i 〉 are written as instead of Eq. (1). Here, | α ( i ) j | 2 = 1 / ∑ j ' max j ' =1 e -β j ' E j ' , and E j and β j are the energy of the state | b j 〉 and the reciprocal of the temperature relevant for it (i.e. the effective blue-shifted local Hawking temperature relevant for the state). The expressions in Eqs. (1 - 3) are the ones in which the Boltzmann factors, e -β j E j / 2 , are ignored and j max is replaced by the effective Hilbert space dimension for the | b j 〉 states, which we identify as the Hilbert space dimension for the | ˜ b j 〉 states. The conditions in Eq. (2), therefore, must be regarded as approximate ones. /negationslash For simplicity, below we will use the expressions in Eqs. (1, 2) for | ψ i 〉 's, which is a good approximation for our purposes. The more precise expression of Eq. (4), however, suggests why the number of independent black hole vacuum states | ψ i 〉 is only e A / 4 l 2 P , despite the fact that the dimension of the Hilbert space for the combined ˜ BB system is much larger, dim H ˜ BB = e A / 2 l 2 P . If maximal entanglement between ˜ B and B were the only condition for a smooth horizon, then we would have e A / 2 l 2 P smooth horizon black hole states. In order for the horizon to be smooth, however, the ˜ B and B states must be entangled in a particular Boltzmann weighted way; in particular, | b j 〉 having energy E j must be multiplied by | ˜ b j 〉 having exactly the opposite energy -E j , not by some | ˜ b k 〉 with E k = -E j . Here, the concept of energy for the ˜ B states arises through identification of these states as the modes outside the horizon in the other side of an eternal black hole; see Section 3.3. Assuming that there are no B states exactly degenerate in energy, this only leaves a room to put phase factors in front of various | ˜ b j 〉| b j 〉 terms, leading to only dim H B (not dim H ˜ BB ) independent states as shown in Eq. (4), where dim H B is the effective Hilbert space dimension for the | b j 〉 states. Taking the number of independent black hole states to be e A / 4 l 2 P as implied by the standard thermodynamic argument, the dimensions of H ˜ B and H B are fixed as in Eq. (3). As emphasized in Refs. [13,14], this implies that space spanned by the states | ψ i 〉 comprises only a tiny, e A / 4 l 2 P -dimensional, subspace of the Hilbert space representing the combined ˜ BB system: dim H ˜ BB = e A / 2 l 2 P /greatermuch e A / 4 l 2 P . 4 In general, a black hole vacuum state can be represented by an arbitrary density matrix defined in space spanned by the | ψ i 〉 's. In the case where entanglement between the black hole and the rest may be ignored, the entire system can be written as where | r 〉 is an element of H R , the Hilbert space factor comprising all the possible states for subsystem R . If the black hole is formed by a collapse of matter that has not been entangled with its environment, then the state of the system is well approximated by Eq. (5) until later times (see below). With such a formation, the number of possible black hole microstates is expected to be much smaller than e A / 4 l 2 P (presumably of order e c A 3 / 4 /l 3 / 2 P where c is an O (1) coefficient [21]); but after the scrambling time t sc ∼ M 0 l 2 P ln( M 0 l P ) [15], all these states are expected to evolve into generic states of the form in Eq. (5): As time passes, the black hole becomes more and more entangled with the rest in the sense that the ratio of the entanglement entropy between ˜ BB and R , S ˜ BB = S R , to the BekensteinHawking entropy at that time, S BH = 4 πM 2 l 2 P , keeps growing, which saturates the maximum value S ˜ BB /S BH = 1 after the Page time t Page ∼ M 3 0 l 4 P , where M 0 is the initial mass of the black hole [16]. Therefore, the state of the system at late times must be written more explicitly as [10-12] where | r i 〉 's are elements of H R . In other words, at these late times the logarithm of the dimension of space spanned by | r i 〉 's is of order S BH (and equal to S BH after the Page time), while at much earlier times it is negligible compared with S BH . The state at early times, therefore, can be well approximated by Eq. (5) for the purpose of discussing internal properties of the black hole. As we will see, the structure of the black hole states described above, together with dynamical assumptions discussed in Section 3, elegantly addresses questions raised by the firewall argument. Before turning to these issues, however, we make a comment on the structure of Hilbert space to avoid possible confusion. As described at the beginning of this section, we have divided the system with a black hole of mass M into three subsystems ˜ B , C , and R ; this division, therefore, implicitly depends on the mass M . Since the black hole mass varies with time, the Hilbert space in which the state of the entire system evolves actually takes the form where we have explicitly shown the M dependence of ˜ B , B , C , and R , and dim H ˜ B ( M ) = dim H B ( M ) = e 4 πM 2 l 2 P as seen in Eq. (3). H C ( M ) -B ( M ) is the Hilbert space spanned by the states for the C degrees of freedom orthogonal to B (i.e. not entangled with ˜ B ), and we define H 0 to be the Hilbert space for the system without a black hole. As the black hole evolves, the state of the system moves between different H M 's; for example, a state that is an element of H M 1 with some M 1 will later be an element of H M 2 with M 2 < M 1 . 5 To help understand the meaning of Eq. (8), let us consider a system in which a black hole was formed at time t 0 with the initial mass M 0 : | Ψ( t 0 ) 〉 ∈ H M 0 . Suppose at some time t with t -t 0 /lessmuch t Page , the black hole mass is M ( < M 0 ). Then, the state of the system at that time, | Ψ( t ) 〉 , is given (approximately) by an element of H M in the form of Eq. (5). Now, at a later time t ' with t ' -t 0 > ∼ t Page , the mass of the black hole becomes smaller, M ' ( < M ). The state of the system | Ψ( t ' ) 〉 is then given by an element of H M ' that takes the form of Eq. (7) with (almost all) | r i 〉 's linearly independent. Finally, after the black hole evaporates, the system is described by an element of H 0 (which is generally time-dependent, representing the propagation of Hawking quanta).", "pages": [ 6, 7, 8, 9 ] }, { "title": "3 Black Hole Interior vs Firewalls", "content": "In this section, we discuss the structure of elements in the Hilbert space factor H ˜ B ⊗ H B and operators acting on it, assuming that there is no extra matter near and outside the stretched horizon. (If there is extra matter, it simply changes the identification of the B states in the Hilbert space for the C degrees of freedom.) For simplicity, we focus our discussion mostly on these entities for a fixed M . The evolution of a black hole, which leads to a variation of M , is discussed in Section 3.4 only to the extent needed.", "pages": [ 9, 10 ] }, { "title": "3.1 Black hole states and firewall states", "content": "Let the Hilbert space spanned by the black hole vacuum states | ψ i 〉 be H ψ : At the leading order, the dimension of H ψ is e A / 4 l 2 P , i.e. where A = 16 πM 2 l 4 P . (Here and below we use the approximate symbol, ≈ , to indicate that an expression is valid at the leading order in expansion in inverse powers of A /l 2 P .) The basic idea of Refs. [10, 11] is that a semi-classical world can be constructed on each of | ψ i 〉 's, and that all of these worlds look identical to general relativity. In the language here, this implies that an operator ˆ O that can be used to describe a semi-classical world for an infalling observer may be written in the block-diagonal form in the H ˜ B ⊗H B space if an appropriate basis is chosen. Moreover, by taking an appropriate basis in each block, all the operators in the diagonal blocks ˆ o i , which we call branch world operators (and are represented here by e ≈A / 4 l 2 P × e ≈A / 4 l 2 P matrices), may be brought into an identical form The resulting basis states can be arranged in the form where we have listed the e ≈A / 2 l 2 P basis states in H ˜ B ⊗H B in the form of a column vector; i.e., each block contains one of the black hole vacuum states | ψ i 〉 's, and in each block the | ψ i 〉 can be put at the bottom of the column vector of e ≈A / 4 l 2 P dimensions. In the basis of Eq. (13), the outgoing creation/annihilation operators for an infalling observer (˜ a ˜ ω or ˆ a ω in Ref. [18] or a ω in Ref. [6]) take the form of Eq. (11) with all the branch world operators taking the same form, which we denote by ˆ a ω . 6 (For simplicity, below we only consider spherically symmetric modes to keep the shape of the horizon, but the extension to other cases is straightforward.) By acting (a finite number of) ˆ a † ω 's on one of the | ψ i 〉 's, one can construct a state in which matter exists in the interior of the black hole as viewed from an infalling observer. How many such states can we construct from a vacuum state | ψ i 〉 , keeping the classical spacetime picture in the interior? We expect that the number of these states (for each | ψ i 〉 ) is of order e ≈ c A n /l 2 n P with n < 1 [21]. This implies that the number of all the states in H ˜ B ⊗H B that allow semi-classical interpretation in the black hole interior is Namely, the Hilbert space factor H cl ( ⊃ H ψ ) spanned by all these semi-classical-i.e. not necessarily vacuum-black hole states satisfies consistent with the counting expected by the holographic bound [21, 22]. As we will see more explicitly in the next subsection, an arbitrary superposition of elements of H cl (or an arbitrary density matrix in H cl ⊗H ∗ cl ) represents a black hole state in which an infalling observer sees smooth horizon. In particular, this implies that in order for the infalling observer not to find any drama, the black hole state need not take the maximally entangled form in Eq. (1)-it can even be in a separable form of | ˜ b j 〉| b j 〉 (without summation in j ), since these states can be obtained as a superposition of (maximally entangled) | ψ i 〉 's. Once again, the condition for an infalling observer to see smooth horizon is not that a black hole (vacuum) state has a maximally entangled form, but that it stays in the e ≈A / 4 l 2 P -dimensional subspace H cl (or H ψ ) in the e ≈A / 2 l 2 P -dimensional space H ˜ B ⊗ H B (called the balanced form in Ref. [13] for the vacuum states). This is because as long as the black hole state stays in H cl , the dynamics of the horizon makes the observer see the state in the basis determined by | ψ i 〉 , as will be discussed more explicitly in the next subsection. This therefore replaces/refines (in a sense) the maximally-entangled condition of Ref. [23] for the existence of smooth classical spacetime (in the present context, beyond the horizon). The vast majority of the states in H ˜ B ⊗ H B that do not belong to H cl are 'firewall states.' They do not admit the smooth classical spacetime picture in the interior of the horizon; in particular, they include states in which a diverging number of a ω quanta, including high energy modes, are excited on | ψ i 〉 . It may be possible to view these states as representing the situation in which singularities of general relativity exist near the horizon, not just at the center (see Ref. [24]), so that there is no classical spacetime in the interior region.", "pages": [ 10, 11, 12 ] }, { "title": "3.2 No firewall for black hole states-dynamical selection of the basis", "content": "We now argue that if the state stays in subspace H cl in the Hilbert space factor H ˜ B ⊗H B , then an infalling observer does not see firewalls. We begin by discussing why operators responsible for describing (semi-)classical worlds for an infalling observer take the special block-diagonal form of Eqs. (11, 12). Why can't general Hermitian operators acting on the e ≈A / 2 l 2 P -dimensional space H ˜ B ⊗H B be observables in these worlds? As discussed in Refs. [5, 25], observables in classical worlds-which emerge dynamically from the full quantum dynamics-correspond, in general, to only a tiny subset of all the possible quantum operators acting on the microscopic state of the system. These observables represent the information that can be amplified in a single term in a quantum state (i.e. the information that can be shared and compared by multiple physical 'observers' in the system), and are selected as a result of the dynamics of the system (the selection of the measurement basis). The statement that operators used to describe a semi-classical world for an infalling observer take the form of Eqs. (11, 12), therefore, comprises an assumption on the internal dynamics of the ˜ BB system, i.e. the microscopic Hamiltonian acting on the ˜ B and B degrees of freedom, which determines the form of operators representing observables in a semi-classical world [10]. In fact, this is precisely the physical content of the complementarity hypothesis, which has to do with how classical spacetime emerges in a full quantum theory of gravity. In the distant description, an object falling into the horizon will interact strongly with surrounding highly blue-shifted Hawking quanta, making it (re-)entangled with the basis determined by the | ψ i 〉 's. Here we take this dynamical assumption for granted, which one might hope to eventually derive from the microscopic theory of the ˜ B and B degrees of freedom. With this interpretation of semi-classical observables, it is now easy to see that an arbitrary state in H cl , or more generally an arbitrary density matrix in H cl ⊗H ∗ cl , does not lead to a firewall for an infalling observer. Consider, for simplicity, that the ˜ B and B degrees of freedom are in a pure state 2 where c i 's are arbitrary coefficients with ∑ e ≈A / 4 l P i =1 | c i | 2 = 1. If the infalling observer interacts with this state, then he/she will 'measure,' or 'feel,' it in the basis determined by ˆ O 's in Eqs. (11, 12); i.e. he/she will find that the black hole is in a particular state | ψ i 〉 with probability | c i | 2 . Since all the | ψ i 〉 states represent the same semi-classical black hole with smooth horizon (at the level of general relativity), this implies that the observer will find that the horizon is smooth with a probability of 1-the observer does not see a firewall. 7 In fact, one can obtain the same conclusion by calculating the average number of high energy a ω quanta (i.e. a ω quanta with ω /greatermuch 1 /Ml 2 P ) for the states in H ψ . In the basis of Eq. (13), the number operators for a ω modes take the form for all ω /greatermuch 1 /Ml 2 P , since | ψ i 〉 's are black hole vacuum states. The average number of high energy a ω quanta is then for any ω /greatermuch 1 /Ml 2 P , where the traces are taken over arbitrary basis states in H ψ . (Note that ¯ N a ω is independent of the basis chosen.) Since an expectation value of a number operator, ˆ N a ω , is positive semi-definite, this implies that typical states in H ψ do not have firewalls. By the definition of H cl , the same argument also applies to the states in H cl . In Ref. [8], a similar calculation was performed with the conclusion that typical black hole states do have firewalls. A crucial element in the calculation of Ref. [8] was the statement/assumption that the eigenstates of the number operator ˆ b † ˆ b provide a complete basis for unentangled black hole states (see also [26]), where ˆ b is the annihilation operator for a Killing mode that is located outside the stretched horizon. If this were true, then we could calculate the average number of high energy quanta for the black hole states ¯ N a ω , defined analogously to Eq. (18), by going to the basis spanned by the ˆ b † ˆ b eigenstates. Now, the semi-classical relation where β ( ω ) and γ ( ω ) are some functions, implies that the expectation value of an a ω -number operator in a ˆ b † ˆ b eigenstate is O (1) for any ω /greatermuch 1 /Ml 2 P . This would, therefore, give ¯ N a ω ≈ O (1), implying that typical black hole states must have firewalls. Why has our calculation led to the opposite conclusion? The key point is that with the structure of the Hilbert space discussed here, the traces over H ψ in Eq. (18) cannot be taken as those over ˆ b † ˆ b eigenstates. Below, we examine this point more closely. While doing so, we also discuss the structure of quantum operators that can be used to describe the exterior and interior spacetime regions of the black hole.", "pages": [ 12, 13, 14 ] }, { "title": "3.3 Exterior and interior operators", "content": "As we have seen in Section 3.1, the creation/annihilation operators for quanta on | ψ i 〉 's take the form of Eqs. (11, 12) (e.g. with ˆ o = ˆ a ω and ˆ a † ω for outgoing modes), which generically act nontrivially both on the ˜ B and B degrees of freedom. In general, one may take a linear combination of these operators to construct operators that represent a mode localized in the exterior or interior of the horizon (with the latter viewed from an infalling observer). What form would such operators take? Let us consider the annihilation operator ˆ b for an outgoing mode that is localized outside the stretched horizon. We consider a mode in B , i.e. a mode (significantly) entangled with stretched horizon degrees of freedom ˜ B . 8 This is the mode used in the argument of Ref. [8]. The operator ˆ b can be constructed by taking a linear combination of creation/annihilation operators ˆ O 's with ˆ o = ˆ a ω and ˆ a † ω . Because of the assumption that low energy physics outside the stretched horizon is well described by local quantum field theory built on C ( ⊃ B ) and R degrees of freedom, this operator must take the form i.e. act only on the B degrees of freedom, where ˆ b and ˆ b B are operators defined in e ≈A / 2 l 2 P -dimensional and e ≈A / 4 l 2 P -dimensional Hilbert spaces, respectively. The complementarity hypothesis asserts that semi-classical physics-in particular physics responsible for the Hawking radiation process-persists, implying that the relation in Eq. (19) must be preserved (with ˆ a ω and ˆ a † ω interpreted as the corresponding operators in the full e ≈A / 2 l 2 P -dimensional Hilbert space). This implies that the action of the particular linear combination of ˆ a ω 's and ˆ a † ω 's appearing in the right-hand side of Eq. (19) on ˜ B must be trivial, so that ˆ b takes the form of Eq. (20). What about operators representing an interior mode? One might naively think that those operators, collectively written as ˆ d , take the form ˆ d = ˆ d ˜ B ⊗ 1 , analogous to Eq. (20). However, the structure of the black hole vacuum states in Eq. (1) (or Eq. (4)) and that of the Hilbert space in Eq. (3) suggest that this is not the case. These structures are exactly those of near-horizon modes of an eternal black hole with the same mass M . We therefore postulate that The quantum mechanical structure of a black hole after the scrambling time (when formed by a collapse) is the same as that of an eternal black hole (even) at the microscopic level . In particular, the relevant Hilbert space describing a black hole of mass M has dimension e ≈A / 2 l 2 P (= e ≈ 8 πM 2 l 2 P ), not e ≈A / 4 l 2 P , although the dynamics will make the state sweep only e ≈A / 4 l 2 P -dimensional subspace of it as the time passes or as the initial condition for the collapse is scanned. (This sweeping will be ergodic in the subspace after a sufficient coarse-graining.) This is, obviously, consistent with the semi-classical expectation that a black hole formed by a collapse looks like an eternal black hole when it is probed late enough. Here we require that it is also the case quantum mechanically at the microscopic level, including the form of operators representing various excitations. More precisely, we consider that the B and ˜ B degrees of freedom for a black hole of mass M correspond, respectively, to the near-horizon degrees of freedom in one and the other external regions-often called regions I and III-of an eternal black hole with the same mass M as viewed from a distant reference frame. (See Fig. 1 for a schematic depiction.) Here, the near-horizon modes are defined such that the reactions of the modes in region I to the exterior operators of the form in Eq. (20) are the same as those of B ; for example, these modes have energies of the order of, or smaller than, local Hawking temperatures. The near-horizon modes in region III can then be defined through entanglement with those in region I. We assume that the Hilbert space structure of the B and ˜ B states for a collapse-formed black hole (often called a one-sided black hole) is the same as that of the states in the two exterior regions of an eternal black hole (two-sided black hole) with the quantization hypersurface taken as an equal-time hypersurface determined by the outside timelike Killing vector. This suggests that operators responsible for describing the interior spacetime region take the form The structure of operators in Eq. (21) can be motivated by the fact that an equal-time hypersurface determined by the outside timelike Killing vector forms a Cauchy surface, so that all the local operators in the interior spacetime can be obtained by evolving local operators of the form ˆ o region III ⊗ 1 + 1 ⊗ ˆ o region I on the initial equal-time hypersurface along a set of hypersurfaces depicted by dotted lines in the right panel of Fig. 1. (Note that this evolution preserves this particular form of operators, since it can be represented by acting some unitary operator U and its inverse U -1 from left and right, respectively.) We assume that the interior region of the black hole under consideration can be constructed in this way with ˆ o region I and ˆ o region III acting only on the near-horizon modes in the respective regions, leading to Eq. (21). We emphasize again that the 'identification' of the B and ˜ B states with the eternal black hole states is made at each instant of time (or in a sufficiently short time period compared with the timescale for the evolution of the black hole); in particular, the mass of the eternal black hole must be taken as that of the evolving black hole at each moment M ( t ), not the initial mass M 0 . This implies that an infalling object passes through the horizon of the eternal black hole at the center of the Penrose diagram depicted in the right panel of Fig. 1. where ˆ d is defined in the full e ≈A / 2 l 2 P -dimensional Hilbert space H ˜ B ⊗ H B , while ˆ d ˜ B and ˆ d B are defined in e ≈A / 4 l 2 P -dimensional Hilbert spaces H ˜ B and H B , respectively. We are now at the position of discussing why our analysis in Section 3.2 has led to the opposite conclusion as that in Ref. [8]. The key point is that in our present framework, the black hole vacuum states | ψ i 〉 provide a complete basis of the ( e ≈A / 4 l 2 P -dimensional) Hilbert space H ψ ( ⊂ H ˜ B ⊗H B ), while the ˆ b † ˆ b eigenstates provide that of a different (again, e ≈A / 4 l 2 P -dimensional) Hilbert space H B . This implies, in particular, that the black hole vacuum states | ψ i 〉 can not all be made ˆ b † ˆ b eigenstates by performing a unitary rotation in the space spanned by | ψ i 〉 (i.e. H ψ ) in which the traces in Eq. (18) are taken. This can be seen more explicitly as follows. Let us write | ψ i 〉 's in the form of Eq. (1). Since the ˆ b † ˆ b eigenstates span a basis in H B , we may write | b j 〉 as /negationslash where | e k 〉 's are the ˆ b † ˆ b eigenstates. (Note that these eigenstates may be degenerate; i.e., | e k 〉 and | e k ' 〉 with k = k ' need not have different eigenvalues under ˆ b † ˆ b .) Substituting Eq. (22) into Eq. (1), we obtain where An important point is that the element in H ˜ B that is entangled with | e k 〉 , i.e. | ˜ e ( i ) k 〉 , depends on the vacuum state, i.e. on the index i . This prevents us from finding a basis change in H ψ that makes all the | ψ i 〉 's ˆ b † ˆ b eigenstates. (Otherwise, the rotation represented by the matrix ( f -1 ) k ( i ) would do the job.) The traces in Eq. (18), therefore, cannot be taken over ˆ b † ˆ b eigenstates, avoiding the conclusion in Ref. [8].", "pages": [ 14, 15, 16, 17 ] }, { "title": "3.4 Dynamical evolution", "content": "We have seen that the ˜ B and B degrees of freedom can represent very different objects-black holes and firewalls-depending on their quantum mechanical states. In particular, if they are in a state that is an element of e ≈A / 4 l 2 P -dimensional Hilbert space H cl( M ) (or represented by a density matrix that is an element of H cl( M ) ⊗H ∗ cl( M ) ≡ L cl( M ) ), then an infalling observer interacting with these degrees of freedom will find that the horizon is smooth and see the interior spacetime; if not, he/she will see a firewall. (In general, if the ˜ BB state involves a component in H cl( M ) , the infalling observer will see the interior spacetime with the corresponding probability.) Here, we have restored the index M to remind us that H cl( M ) is, in fact, a component of H M ; see Eq. (8). We do not know a priori the answer to these questions. We may, however, interpret the success of general relativity with the global spacetime picture to mean that the dynamical evolution keeps the ˜ BB degrees of freedom to stay in subspace H cl( M ) (or L cl( M ) ), i.e. in a state that allows for the smooth classical spacetime interpretation in the interior of the horizon (at least, in the absence of a certain special manipulation of the degrees of freedom by an infalling observer; see Section 4). This interpretation can be made particularly plausible by considering the extension of the framework to (meta-stable) de Sitter space, where the validity of the global spacetime picture beyond the de Sitter horizon is strongly supported by the successful prediction for density perturbations in the inflationary universe; see Section 5. We therefore postulate that the dynamics of quantum A natural interpretation of these black hole and firewall states-i.e. the elements of H M -is that they both represent objects that lead to the Schwarzschild spacetime of mass M in the region outside the (stretched) horizon, because they both use the same C ( ⊃ B ) and R degrees of freedom and local quantum field theory is supposed to be valid in this region. A crucial question then is: what does the dynamics of the (entire) system tell us about the properties of an object under consideration? In particular, if the object is formed by a collapse of matter with the initial mass M 0 , are the corresponding ˜ B and B degrees of freedom in a state in H cl( M 0 ) (or L cl( M 0 ) )? And if so, do they stay in H cl( M ) (or L cl( M ) ) when the mass is reduced to M ( < M 0 ) by time evolution? /negationslash gravity is such that it keeps an element of H cl( M ) (or L cl( M ) ) to stay in H cl( M ' ) (or L cl( M ' ) ) under time evolution, where M ' = M in general. In the context of black hole physics, this implies that the state of the system takes the form of Eq. (5) when the black hole is formed by (isolated) matter, which evolves into Eq. (7) as time passes. This evolution is indeed consistent with the standard dynamics for the black hole evaporation process, e.g. the generalized second law of thermodynamics. An explicit qubit model representing this process was described in Ref. [14]. As discussed in Ref. [10], the evolution described above is also consistent with the standard analysis of the information flow in black hole evaporation in a unitary theory of quantum gravity [16], avoiding the paradox raised by Ref. [6]. (In fact, this was how the structure of the black hole states discussed in this paper was first found.) In Ref. [6], it was argued that for an old black hole, the conditions for unitarity and smooth horizon, which were respectively given by were mutually incompatible, where S X represents the von Neumann entropy of subsystem X . The structure of the black hole states discussed here, however, elegantly avoids this conclusion, keeping the standard assumptions of black hole complementarity. For a state with an old black hole given in Eq. (7), the conditions for unitarity and smooth horizon are given, respectively, by [10] where ˜ S ( i ) X are branch world entropies , the von Neumann entropy of subsystem X calculated using the state representing the (semi-)classical world i : | Ψ ( i ) 〉 = | ψ i 〉| r i 〉 (without summation in the right-hand side). The point is that the relations in Eq. (26) are not incompatible with each other; in fact, they are all satisfied for a generic state of the form of Eq. (7) after the Page time. The reason for this success can be put as follows: What is responsible for unitarity of the evolution of the black hole state is not an entanglement between early radiation and modes in B as imagined in Ref. [6], but an entanglement between early radiation and the way B and ˜ B degrees of freedom are entangled . In other words, the information of the black hole is contained in the coefficients d i in Eq. (7), i.e. how the black hole is made out of the e ≈A / 4 l 2 P vacuum states.", "pages": [ 17, 18 ] }, { "title": "3.5 Gauge/gravity duality", "content": "Here we comment on how a black hole (or a firewall) may be realized in the gauge theory side of gauge/gravity duality. As an example, we might consider a setup in Ref. [7] in which an evaporating black hole is modeled by a conformal field theory (CFT) coupled to a large external system. How can the structure discussed in this paper be realized in such a setup? One possibility is that the whole structure of the Hilbert space described so far, in particular the Hilbert space of e ≈A / 2 l 2 P dimensions for the horizon and the entangled degrees of freedom, exists at the microscopic level in the corresponding gauge theory. The whole e ≈A / 2 l 2 P degrees of freedom are not visible in standard thermodynamic considerations in the gauge theory side, since the dynamics populates only the e ≈A / 4 l 2 P subspace ( H cl ) of the whole Hilbert space ( H ˜ B ⊗ H B ) relevant for these degrees of freedom. The local operators responsible for describing the exterior and interior regions are still given in the form of Eqs. (20) and (21), respectively. We note that a similar construction of operators in the exterior and interior regions were discussed in Ref. [27], in which the ˜ B degrees of freedom were thought to arise effectively after coarse-graining the outside degrees of freedom. Our picture here is different-we consider that both the stretched horizon ( ˜ B ) and the outside ( B ) degrees of freedom exist independently at the microscopic level. It is simply that standard black hole thermodynamics does not probe all the degrees of freedom because of the properties of the dynamics. An alternative possibility is that the gauge theory only contains states in H cl for the horizon and the entangled degrees of freedom. 9 In this case, the gauge theory cannot describe a process in which the state for these degrees of freedom is made outside H cl (i.e. creation of a firewall), or it may simply be that such a process does not exist in the gravity side as well (see a related discussion in Section 4.1).", "pages": [ 18, 19 ] }, { "title": "4 Infalling Observer", "content": "In this section, we discuss what our framework predicts for the fate of infalling observers under various circumstances. We also discuss how the physics can be described in an infalling reference frame, rather than in a distant reference frame as we have been considering so far.", "pages": [ 19 ] }, { "title": "4.1 Physics of an infalling observer: a black hole or firewall?", "content": "As we have seen, as long as the black hole state is represented by a density matrix in L cl = H cl ⊗H ∗ cl , an infalling observer interacting with this state sees a smooth horizon with a probability of 1. Here we ask what happens if the observer manipulates the relevant degrees of freedom, e.g. measures some of the B or R modes, before entering the horizon. This amounts to asking what black hole state such an observer encounters when he/she falls into the horizon. We first see that the observer finds a smooth horizon with probability 1 no matter what measurement he/she performs on Hawking radiation emitted earlier from the black hole. This statement is obvious if the object the observer measures is not entangled with the black hole, e.g. as in Eq. (5), so we consider the case in which the observer measures an object that is entangled with the black hole. To be specific, we consider the case in which the observer measures the | r i 〉 degrees of freedom in Eq. (7) before he/she enters the black hole. Without loss of generality, we assume that the outcome of the measurement was ∑ k U jk | r k 〉 , where U jk is an arbitrary unitary matrix. The state of the black hole the observer encounters is then ∑ k U † kj d k | ψ k 〉 . Since this is a superposition of the black hole states | ψ i 〉 's, the observer finds a smooth horizon with a probability of 1, as discussed in Section 3.2. We conclude that it is not possible to create a firewall by making a measurement on early Hawking radiation. On the other hand, if an infalling observer can directly access the B states entangled with the stretched horizon modes, then he/she may be able to see a firewall. Consider that the black hole state | Ψ 〉 is given by Eq. (7). We may then expand the B states in terms of the ˆ b † ˆ b eigenstates as in Eq. (22), leading to where f ( i ) k and | ˜ e ( i ) k 〉 are given in Eqs. (23, 24). Suppose the observer measures the B degrees of freedom are in a ˆ b † ˆ b eigenstate | e k 〉 . Then the relevant state is without summation in k . Now, the state | r i 〉 is in general a superposition of decohered classical states | r cl j 〉 , | r i 〉 = ∑ j g ( i ) j | r cl j 〉 , 10 so the observer finds the ˜ BB state is in one of the | ˜ ψ j 〉 's, where This state is not in general a superposition of the smooth horizon states | ψ i 〉 's; in other words, | ˜ ψ j 〉 is not an element of H cl . Thus, if the infalling observer directly measures B to be in a ˆ b † ˆ b eigenstate (which will require the detector to be finely tuned) and enters the horizon right after (i.e. before the state of the black hole changes), then he/she will see a firewall with an O (1) probability. 11 Here we have assumed that such a measurement can be performed, although it is possible that there is some dynamical (or perhaps computational [28]) obstacle to it. We note that a similar argument to the one here applies to any surface in a low curvature region, not just the black hole horizon. This way of seeing a firewall, therefore, does not violate the equivalence principle. We finally mention that even if an observer finds B to be in a ˆ b † ˆ b eigenstate, if he/she enters the horizon long after that, then he/she may see a smooth horizon rather than a firewall. Suppose that the observer measured a ˆ b † ˆ b eigenstate when the black hole had mass M . This implies that he/she was entangled with the ˆ b † ˆ b eigenstate in B ( M ), where we have explicitly shown the M dependence of the decomposition. Now, at a much later time, the black hole has a smaller mass M ' ( /lessmuch M ). The mode which was in B ( M ) may then be in R ( M ' ). If this is the case, then the observer is no longer entangled with a B mode necessary to see a firewall, i.e. B ( M ' ); instead, he/she is simply entangled with the environment (early radiation) of the black hole, R ( M ' ). This observer will therefore see a smooth horizon when he/she enters the black hole, as discussed at the beginning of this subsection. The fact that an observer was once entangled with a B mode is not enough to see a firewall; he/she must be entangled with a B mode of the black hole at the time of entering the horizon in order to see a firewall.", "pages": [ 19, 20, 21 ] }, { "title": "4.2 Description in an infalling reference frame", "content": "We now consider how the physics for an infalling object is described in an infalling reference frame, rather than in a distant frame as has been considered so far. Following Ref. [5], we consider that such an infalling description is obtained by performing a unitary transformation on the distant description, corresponding to a change of the clock degrees of freedom in full quantum gravity. According to the complementarity hypothesis, the Hamiltonian-the generator of time evolutionafter the transformation takes the form local in infalling field operators, including the interior operators discussed in Section 3.3. Since the complementarity transformation is supposed to be unitary, the e ≈A / 4 l 2 P states | ψ i 〉 must be transformed into e ≈A / 4 l 2 P different states which must all look like locally Minkowski vacuum states. In particular, this implies that in the limit that the black hole is large A → ∞ , i.e. in the limit that the horizon under consideration is a Rindler horizon, there are infinitely many Minkowski vacuum states labeled by i = 1 , · · · , e ≈A / 4 l 2 P = ∞ . This seems to contradict our experience that we can do physics without knowing which of the Minkowski vacua we live in. Isn't the Minkowski vacuum unique, e.g., in QED? Otherwise, we do not seem to be able to do any physics without having the (infinite amount of) information on the Minkowski vacua. The structure of operators discussed in Section 3.1, however, provides the answer. After the complementarity transformation, the form of operators is preserved; in particular, all the local operators responsible for describing semi-classical physics take the block-diagonal form, Eq. (11), with all the (branch world) operators in the diagonal blocks taking the identical form, Eq. (12). This implies that no matter which superposition of Minkowski vacua we live in, we always find the same semi-classical physics (which is everything in the non-gravitational limit). More precisely, if we live in a vacuum represented by state c i | ψ i 〉 ( ∑ i | c i | 2 = 1), then we find ourselves to be in a particular vacuum | ψ i 〉 with probability | c i | 2 (i.e. the measurement basis is | ψ i 〉 ), but all of these vacua lead to the same semi-classical physics. In order to discriminate different vacua, we need to consider operators beyond local field operators, e.g. those of Eq. (11) with ˆ o i taking different forms for different i in the basis of Eq. (13). Such operators will be either highly nonlocal or act on the boundary/horizon of the infalling/locally Minkowski description of spacetime. In the true Minkowski space, the boundary is located only at spatial infinity. In the case of an infalling description of a black hole vacuum, however, spacetime is Minkowski vacuum-like only locally, and the nonzero curvature effect can lead to a horizon (as viewed from the infalling frame, not the original one as viewed from a distant frame) at a finite spatial distance, e.g. along the lines of Ref. [12]. Probing microscopic degrees of freedom on such a horizon, therefore, might allow us to access the information on which | ψ i 〉 vacuum the observer is in.", "pages": [ 21, 22 ] }, { "title": "5 de Sitter Space", "content": "The quantum theory of horizons described in this paper is applicable to horizons other than those of black holes with straightforward adaptations. Here we discuss an application to de Sitter horizons. The analysis is parallel with the case of black hole horizons. The de Sitter horizon is located at r = 1 /H in de Sitter space, where r is the radial coordinate of the static coordinate system ( t, r, θ, φ ) and H the Hubble parameter. The stretched horizon is located where the local Hawking temperature becomes of order the fundamental/string scale M ∗ = 1 /l ∗ : T ( r ∗ ) = M ∗ / 2 π (the factor of 2 π is for convenience), i.e. Following the analysis of the black hole case, we consider that the dimensions of the Hilbert spaces for the stretched horizon degrees of freedom, ˜ B , and the states entangled with them, B , are given by where A = 4 π/H 2 is the area of the (stretched) de Sitter horizon. We may expect that at the leading order in l 2 P / A , the dimension of the Hilbert space spanned by all the possible states inside the de Sitter horizon ( r < r ∗ ) is the same as that of H B , which we assume to be the case. (As can be seen from the distribution of thermal entropy ∝ T ( r ) 3 , most of these states are localized near the horizon.) The dimension of the total Hilbert space needed to describe de Sitter space is then at the leading order in l 2 P / A . As in the case of black holes, the states which allow for the interpretation of classical spacetime outside the de Sitter horizon span only a tiny subspace of the entire Hilbert space In fact, the Hilbert space spanned by the vacuum states already has the same logarithmic dimension at the leading order in l 2 P / A : with the basis states for measurement taking the maximally entangled form as in Eq. (1) (ignoring Boltzmann factors). As long as a state stays in H cl (or is represented by a density matrix in H cl ⊗H ∗ cl ), an object that hits the horizon can be thought of going to space outside the horizon; otherwise, it hits a firewall/singularity. The information about the object that goes outside will be stored in the ˜ BB degrees of freedom, which may later be recovered, for example if the system evolves into Minkowski space (or another de Sitter space with a smaller vacuum energy). As in the black hole case, we expect that the dynamics of quantum gravity is such that a state in H cl keeps staying in H cl . In fact, this is observationally indicated by the success of the prediction for density perturbations in the inflationary universe, which is based on the global spacetime picture of general relativity [29]. Finally, we consider the limit H → 0 in which the de Sitter space approaches Minkowski space. In this limit, the number of vacuum states becomes infinity each differing in the way ˜ B and B states are entangled; see Eq. (1). Since most of the B states are localized near the horizon, which is located at spatial infinity in this limit, probing the structure of (infinitely many) Minkowski vacua will require access to the boundary at infinity. This is the same picture we have arrived at in Section 4.2 by taking the large mass limit of a black hole horizon. The fact that various horizons, especially black hole and de Sitter horizons, can be treated on an equal footing is an important ingredient for the quantum mechanical treatment of the eternally inflating multiverse advocated in Refs. [5,30], in which the eternally inflating multiverse and the many worlds interpretation of quantum mechanics are unified as the same concept. It would be interesting to see if there are other implications of the present framework beyond what have been discussed in this paper.", "pages": [ 22, 23 ] }, { "title": "Acknowledgments", "content": "We thank Juan Maldacena for useful conversations. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC02-05CH11231, and in part by the National Science Foundation under grant PHY-1214644.", "pages": [ 24 ] } ]

[ { "title": "Limits on anisotropic inflation from the Planck data ∗", "content": "Jaiseung Kim † Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany and Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark", "pages": [ 1 ] }, { "title": "Eiichiro Komatsu", "content": "Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany and Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI) (Dated: October 6, 2018) Temperature anisotropy of the cosmic microwave background offers a test of the fundamental symmetry of spacetime during cosmic inflation. Violation of rotational symmetry yields a distinct signature in the power spectrum of primordial fluctuations as P ( k ) = P 0 ( k )[1 + g ∗ ( ˆ k · ˆ E cl ) 2 ], where ˆ E cl is a preferred direction in space and g ∗ is an amplitude. Using the Planck 2013 temperature maps, we find no evidence for violation of rotational symmetry, g ∗ = 0 . 002 ± 0 . 016 (68% CL), once the known effects of asymmetry of the Planck beams and Galactic foreground emission are removed. PACS numbers: 98.70.Vc, 98.80.Cq, 98.80.-k Cosmic inflation [1-5], an indispensable building-block of the standard model of the universe, is described by nearly de Sitter spacetime. The metric charted by flat coordinates is given by ds 2 = -dt 2 + e 2 Ht d x 2 , where H is the expansion rate of the universe during inflation. This spacetime admits ten isometries: three spatial translations; three spatial rotations; one time translation accompanied by spatial dilation ( t → t -λ/H and x → e λ x with a constant λ ); and three additional isometries which reduce to special conformal transformations in t →∞ . The necessary time-dependence of the expansion rate, Ht → ∫ H ( t ' ) dt ' , breaks the time translation symmetry hence the spatial dilation symmetry, yielding the two-point correlation function of primordial fluctuations that is nearly, but not exactly, invariant under x → e λ x [6]. The magnitude of the deviation from dilation invariance is limited by that of the time-dependence of H , i.e., -˙ H/H 2 = O (10 -2 ). In the usual model of inflation, six out of ten isometries remain unbroken: translations and rotations. Why must they remain unbroken while the others are broken? In this paper, we shall test rotational symmetry during inflation, using the two-point correlation function of primordial perturbations to spatial curvature, ζ , generated during inflation. This is defined as a perturbation to the exponent in the spatial metric, ∫ H ( t ' ) dt ' → ∫ H ( t ' ) dt ' + ζ ( x , t ). In Fourier space, we write the two-point function as 〈 ζ k ζ ∗ k ' 〉 = (2 π ) 3 δ (3) ( k -k ' ) P ( k ), and P ( k ) is the power spectrum. Translation invariance, which is kept in this paper, gives the delta function, while rotation invariance, which is not kept, would give P ( k ) → P ( k ) with k ≡ | k | . Dilation invariance would give k 3 P ( k ) = const . , whereas a small deviation, k 3 P ( k ) ∝ k -0 . 04 , has been detected from the CMB data with more than 5σ significance [7, 8]. Following Ref. [9], we write the power spectrum as P ( k ) = P 0 ( k ) [ 1 + g ∗ ( k ) ( ˆ k · ˆ E cl ) 2 ] , where ˆ E cl is a preferred direction in space, g ∗ is a parameter characterizing the amplitude of violation of rotational symmetry, and P 0 ( k ) is an isotropic power spectrum which depends only on the magnitude of the wavenumber, k . This form is generic, as it is the leading-order anisotropic correction that remains invariant under parity flip, k → -k . 'Anisotropic inflation' models, in which a scalar field is coupled to a vector field (see Ref. [10-12] and references therein) can produce this form. 1 A very longwavelength perturbation on super-horizon scales can also produce this form via a three-point function [17]. A preinflationary universe was probably chaotic and highly anisotropic, and thus a remnant of the pre-inflationary anisotropy may still be detectable [18]. We shall ignore a potential k dependence of g ∗ in this paper. We expand g ∗ ( ˆ k · ˆ E cl ) 2 using spherical harmonics: We then write the power spectrum as where we have absorbed g ∗ / 3 into the normalization of the isotropic part, ˜ P 0 ( k ) ≡ P 0 ( k )(1 + g ∗ / 3), and defined g 2 M ≡ 8 π 15 g ∗ 1+ g ∗ / 3 Y ∗ 2 M ( ˆ E cl ) with g 2 M for M < 0 given by g 2 , -M = ( -1) M g ∗ 2 ,M . There are 5 parameters to be determined from the data. We denote the parameter vector as h ≡ { g 20 , Re[ g 21 ] , Im[ g 21 ] , Re[ g 22 ] , Im[ g 22 ] } . We search for h in the covariance matrix of the spherical harmonics coefficients of CMB temperature maps, C l 1 m 1 ,l 2 m 2 ≡ 〈 a l 1 m 1 a ∗ l 2 m 2 〉 , where a lm = ∫ d 2 ˆ n T (ˆ n ) Y ∗ lm (ˆ n ). The anisotropic power spectrum of Eq. 2 gives [19] where the matrices denote the Wigner 3j symbols, and D l 1 l 2 ≡ 2 π ∫ k 2 dk ˜ P 0 ( k ) g Tl 1 ( k ) g Tl 2 ( k ) with g Tl ( k ) the temperature radiation transfer function. In the limit of weak anisotropy, the likelihood of the CMB data given a model may be expanded as The first and second derivatives are given by where H i ≡ 1 2 [ C -1 a ] † ∂ C ∂h i [ C -1 a ] , and a denotes a lm measured from the data and C ≡ 〈 aa † 〉 , both of which include noise and the other data-specific terms. We obtain an estimator for h by maximizing the likelihood with respect to h [20] The covariance matrix, C , is neither diagonal in pixel nor harmonic space. In order to reduce the computational cost, we shall approximate it as diagonal in harmonic space. While this approximation makes our estimator sub-optimal, it remains un-biased. The new estimator is where ˜ a lm ≡ ∫ d 2 ˆ n T (ˆ n ) M (ˆ n ) Y ∗ lm (ˆ n ) is the spherical harmonic coefficients computed from a masked temperature map ( M (ˆ n ) = 0 in the masked pixels, and 1 otherwise), and C l and N l are the signal and noise power spectra, respectively. The matrix F is defined by with f sky ≡ ∫ d 2 ˆ n 4 π M (ˆ n ) the fraction of unmasked pixels. Here, 〈 ˜ a ∗ l 1 m 1 ˜ a l 2 m 2 〉 h =0 in Eq. 9 is the 'mean field,' which is non-zero even when g ∗ = 0. Data-specific issues such as an incomplete sky coverage, inhomogeneous noise, and asymmetric beams generate the mean field. From ˆ h i , we need to estimate g ∗ and ˆ E cl . As the estimator ˆ h i consists of the sum of many pairs of coefficients a lm , we expect the estimated value to follow a Gaussian distribution (the central limit theorem). Therefore, the likelihood of g ∗ and ˆ E cl is where G is the covariance matrix of ˆ h , which we compute from 1000 Monte Carlo simulations. Since h ( g ∗ , ˆ E cl ) has nonlinear dependence on g ∗ and ˆ E cl , we obtained the posterior distribution of g ∗ and ˆ E cl by evaluating Eq. 11 with the Markov Chain Monte Carlo sampling [21]. 2 We use the Planck 2013 temperature maps at N side = 2048, which are available at the Planck Legacy Archive [23-25]. (We upgrade the low-frequency maps, which are originally at N side = 1024, to N side = 2048.) We use the map at 143 GHz as the main 'CMB channel', and use the other frequencies as 'foreground templates'. We reduce the diffuse Galactic foreground emission by fitting templates to, and removing them from, the 143 GHz map. This is similar to the method called SEVEM by the Planck collaboration [26]. We derive the templates by taking a difference between two maps at neighboring frequencies. This procedure ensures the absence of CMB in the derived templates, producing five templates: (30 -44), (44 -70), (353 -217), (545 -353), and (857 -545) [GHz]. To create these difference maps, we first smooth a pair of maps to the common resolution. We smooth the lowfrequency maps at 30-70 GHz as a ( ν ) lm → a ( ν ) lm b G l /b ( ν ) l , where b ( ν ) l is the beam transfer function at a frequency ν [27] and b G l is a Gaussian beam of 33 ' (FWHM). We smooth the high-frequency maps at 217-857 GHz as a ( ν ) lm → a ( ν ) lm b (143) l /b ( ν ) l , where b (143) l is the beam transfer function at 143 GHz [28]. After the smoothing, we mask the locations of point sources and the brightest region near the Galactic center (3% of the sky) following SEVEM [26]. As the smoothed sources occupy more pixels, we enlarge the original pointsource mask as follows: we create a map having 1 at the source locations and 0 otherwise, and smooth it. We then mask the pixels whose values exceed e -2 . We fit the templates to the 143 GHz map on the unmasked pixels (86% of the sky). The left and middle panels of Figure 1 show the original and foreground-reduced maps at 143 GHz, respectively. We still find significant foreground emission on the Galactic plane. We thus mask the regions contaminated by the residual foreground emission, combining the masks of various foreground-reduced maps produced by the Planck collaboration ( NILC , Ruler , SEVEM , and SMICA [26]), and the point-source mask. We show the combined mask in the right panel of Figure 1, which leaves 71% of the sky unmasked, and is similar to the 'union mask' of the Planck collaboration, except for a slightly enlarged point-source mask due to smoothing. We use Eqs. 9 and 11 to compute g LM from the masked foreground-reduced map. We restrict our analysis to the multipole range of 2 ≤ /lscript ≤ 2000. We compute the mean field from 1000 Monte-Carlo realizations of signal and noise. The signal map is T S (ˆ n ) = ∑ lm √ C l x lm b ( ν ) l p l Y lm (ˆ n ), where C l is the bestfit 'Planck+WP' power spectrum [8], p l the pixel window function, and x lm a Gaussian random variable with unit variance. The noise map is T N (ˆ n ) = √ N (ˆ n ) y (ˆ n ), where N (ˆ n ) is the noise variance map provided by the Planck collaboration, and y (ˆ n ) a Gaussian random variable with unit variance. We create high-frequency maps at N side = 2048, while we create low-frequency maps at N side = 1024 and upgrade to N side = 2048. We also compute g LM from the signal plus noise simulations, and compute the covariance matrix, G , in Eq. 11. Finally, we compute the posterior distribution of g ∗ and ˆ E cl by evaluating Eq. 11 using the CosmoMC sampler [21]. The left panel of Figure 2 shows the log-likelihood of locations of a preferred direction, ln L ( ˆ E cl ), given the Planck data. We find a significant detection of g ∗ = -0 . 111 ± 0 . 013 (68% CL) with ˆ E cl pointing to ( l, b ) = (94 · . 0 +3 · . 9 -4 · . 0 , 23 · . 3 ± 4 · . 1) in Galactic coordinates. This direction lies close to the Ecliptic pole at ( l, b ) = (96 · . 4 , 29 · . 8). This is essentially the same result as found from the WMAP data. Following the first detection reported in Ref. [29], the subsequent analysis finds g ∗ = 0 . 29 ± 0 . 031 with ( l, b ) = (94 · , 26 · ) ± 4 · from the WMAP 5-year map at 94 GHz in the multipole range of 2 ≤ /lscript ≤ 400 [30] (also see [20]). They find a negative value at 41 GHz, g ∗ = -0 . 18 ± 0 . 04. These signals, however, have been explained entirely by the effect of WMAP 's asymmetric beams coupled with the scan pattern [31, 32]. To confirm their results, we use the foreground-reduced WMAP 9year maps [32], finding g ∗ = -0 . 484 +0 . 021 -0 . 023 , 0 . 105 +0 . 036 -0 . 028 , and 0 . 355 +0 . 038 -0 . 037 at 41, 61, and 94 GHz, respectively, in the multipole range of 2 ≤ /lscript ≤ 1000. The directions lie close to the Ecliptic pole. We find g ∗ < 0 from the Planck 143 GHz map. This is because the orientations of the semi-major axes of 143 GHz beams are nearly parallel to Planck 's scan direction [28], which lies approximately along the Ecliptic longitudes. As the beams are fatter along the Ecliptic longitudes, the Planck measures less power along the Ecliptic north-south direction than the east-west direction, yielding a quadrupolar power modulation with g ∗ < 0. 3 We quantify and remove the effect of beam asymmetry by computing g LM from 1000 signal plus noise simulations, in which the signal is convolved with Planck 's asymmetric beams and scans. We have used the EffConv code, which is developed by the Planck collaboration and publicly available 4 with the Planck effective beam data files [28, 34]. The middle panel of Figure 2 shows ln L ( ˆ E cl ) given the simulation data. We reproduce what we find from the real data: g ∗ = -0 . 101 ± 0 . 0004 with (96 · . 1 ± 0 · . 1 , 25 · . 9 ± 0 · . 1) (the error bars are for the average of simulations). Using this result as the mean field (i.e., 〈 ˜ a ∗ l 1 m 1 ˜ a l 2 m 2 〉 h =0 in Eq. 9), we recompute ln L ( g ∗ , ˆ E cl ), finding no evidence for g ∗ (see also the right panel of Figure 2, which shows no preferred direction). Our best limit is g ∗ = 0 . 002 ± 0 . 016 (68% CL), 0 . 002 +0 . 031 -0 . 032 (95% CL) and 0 . 002 +0 . 047 -0 . 048 (99.7% CL). We have also analyzed the foreground-reduced 100 GHz map, which has less foreground emission than the 143 GHz map. We find 28- and 7σ detections of g ∗ in the Ecliptic-pole directions before and after the beam asymmetry correction, respectively. The 100 GHz beam is much less symmetric than the 143 GHz one [28]; thus, the beam simulation needs to be more precise for removing the asymmetry to the sufficient level. We find g ∗ = -0 . 308 ± 0 . 011 before the beam asymmetry correction, which is consistent with the 100 GHz beams being more elongated along Planck 's scan direction. Finally, we study the effect of Galactic foreground emission. Using the raw 143 GHz without cleaning, we find significant anisotropy: g ∗ = 0 . 340 and 0 . 328 ± 0 . 018 before and after the beam asymmetry correction, respectively. The directions lie close to the Galactic pole; [1] A. A. Starobinsky, Phys.Lett. B91 , 99 (1980). thus, the foreground reduction plays an important role in nulling artificial anisotropy in the data. We summarize our finding in Table I. After removing the effects of Planck 's asymmetric beams and Galactic foreground emission, we find no evidence for g ∗ . Our limit, about 2% in g ∗ , provides the most stringent test of rotational symmetry during inflation. JK would like to thank Belen Barreiro, Carlo Baccigalupi, Jacques Delabrouille, Sanjit Mitra, Anthony Lewis and Niels Oppermann for helpful discussions. We acknowledge the use of the Planck Legacy Archive (PLA). The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php? project=planck&page=Planck_Collaboration . We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science Division at the NASA Goddard Space Flight Center. We also acknowledge the use of the EffConv [34], HEALPix [35], CAMB [36], and CosmoMC packages [21]. [2] K. Sato, Mon.Not.Roy.Astron.Soc. 195 , 467 (1981). orientations of the 41 GHz beams are nearly parallel to WMAP 's scan direction, whereas the 61 and 94 GHz maps give g ∗ > 0, as the orientations are nearly perpendicular to the scan direction [33]. This explanation is due to Ref. [31].", "pages": [ 1, 2, 3, 4 ] } ]

[ { "title": "Chaotic brane inflation", "content": "Benjamin Shlaer Institute of Cosmology, Department of Physics and Astronomy Tufts University, Medford, MA 02155, USA We illustrate a framework for constructing models of chaotic inflation where the inflaton is the position of a D3-brane along the universal cover of a string compactification. In our scenario, a brane rolls many times around a nontrivial one-cycle, thereby unwinding a Ramond-Ramond flux. These 'flux monodromies' are similar in spirit to the monodromies of Silverstein, Westphal, and McAllister, and their four-dimensional description is that of Kaloper and Sorbo. Assuming moduli stabilization is rigid enough, the large-field inflationary potential is protected from radiative corrections by a discrete shift symmetry.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Perhaps the simplest phenomenological model of inflation [1-3] is due to Linde's monomial potential [4], which undergoes what is called 'chaotic inflation' due to its expected behavior on large scales. Chaotic inflationary models are not obviously natural in the context of effective field theory precisely because of the requirement that the potential be sufficiently flat over superPlanckian field distances. In the quadratic model, the inflaton mass must be of order 10 -5 M P , and higher-order terms in the potential must remain subdominant for field values as large as 10 M P , which requires a functional finetuning from an effective field theory point of view. An elegant solution to this problem was presented in Refs. [5, 6], whereby an axion 'eats' a three-form potential, and so acquires a mass [7]. The axion potential is purely quadratic, being protected from radiative corrections by the underlying shift symmetry. Aside from the simplicity of the model, chaotic inflation is interesting because of its distinct phenomenological predictions; it is capable of sourcing significant primordial tensor perturbations, which may be detectible in the cosmic microwave background. Here we will find a stringy realization of large-field inflation. As in brane inflation [8-10], the inflaton represents the position of a D3-brane in a six-dimensional compactification manifold, assumed to be sufficiently stable. The potential felt by the D3-brane due to the five-form field strength F 5 gives rise to the four-dimensional effective potential of the inflaton. Crucially, this field strength depends not just on the location of the D3-brane(s), but also on their history, i.e., the number of times they have traversed any nontrivial one-cycles of the compactification manifold. This 'flux wrapping' will allow for the possibility of large-field brane inflation, which cannot otherwise exist [11, 12] because increasing the field range typically requires increasing the compactification volume, which in turn increases the four dimensional Planck mass. It should be pointed out that there are already a few stringy realizations of large-field inflation, e.g. Refs. [1315], as well as Refs. [16-18]. After this paper was completed, we learned of related work on unwinding fluxes [19], and their application to inflation [20, 21]. In the probe approximation, the potential felt by a D3 must be exactly periodic, just as for an axion. Furthermore, the five-form flux takes quantized values over the five-cycle which is dual to the one-cycle. Assuming rigid moduli stabilization, the flux potential is exactly quadratic in the discrete flux winding ∮ M 5 F 5 . By turning on the coupling of the D3 to the background fiveform flux, the periodicity is lifted, and the discrete flux becomes a continuous parameter, contributing an exactly quadratic term to the potential. We now illustrate this with a simple example.", "pages": [ 1 ] }, { "title": "II. CHARGES IN COMPACT SPACES WITH NONTRIVIAL FIRST HOMOLOGY", "content": "As a warm-up example, let us consider a single electron and positron in the compact space S 2 × S 1 . The action and equations of motion are given by where M 1 is the oriented world lines of the charges. Integration and differentiation of the equations of motion require that the point-particle current δ 3 ( M 1 ) satisfy or equivalently (see Appendix), respectively. We abuse the notation ∩ to mean both the intersection and the winding number of the intersection, so the left-hand side of Eq. (4) should be read as stating that there are equally many positive points as negative points in the total intersection. These equations simply state that no net charge can occupy a compact space, and electric current is conserved. We can either ensure that M 1 has no net time-like winding (as we have done), or add a diffuse background 'jel- lium' charge to the action. A homogenous jellium contribution is just proportional to the spatial volume form, Let us imagine that M 1 represents a single positive charge and a single negative charge. We can compute the potential between them by finding the Green's function on this space. We expect the usual Coulomb interaction to be modified by two effects: with n = ( S 2 × S 1 ∩ M 1 ) . The uniform charge density cancels the tadpole. It is the latter effect which we find useful here, as it enables one to change the electric flux on the S 1 . It is straightforward to calculate the difference in flux caused by transporting one of the two charges around the S 1 . The transport of one of the particles around the onecycle means that M 1 acquires winding number equal to one. The flux on the S 1 is measured by choosing a fixed time and z coordinate, and then integrating the dual field strength ˜ F 2 = ∗ d A 1 over the S 2 . To calculate the change in flux caused by a single winding of a particle, let us define a 3-manifold (with boundary) M 3 which spans an interval in time [ t i , t f ] times the full S 2 cycle. Then and so Thus the electric field in the z direction changes by one unit each time a particle is transported around the circle in the z direction. A simple interpretation of this is that the charge drags the field lines around the cycle. We can immediately write down the homological piece of the potential. If the metric is given by d s 2 = -d t 2 + R 2 ( d θ 2 +sin 2 θ d φ 2 ) +d z 2 , with z ≡ z + L , then where ∆ z is the z separation of the two charges as measured on the universal cover. The flux part of the electron potential is thus where V ⊥ = ∫ vol 3 ( S 2 × S 1 ) = 4 πR 2 L . The flux potential cancels the jellium [22, 23] contribution. We can think of the jellium term in the potential as arising due to the finite compactification volume. A jellium term is required in the potential felt by a probe charge, since the field strength is then single-valued. But transport of a physical charge around a nontrivial cycle does not leave the field strength invariant, and so the probe charge is an inadequate description. In a sense, one can say that the configuration space of charges and flux is not simply the product of the compact manifold and its first homology, but rather is a nontrivial fibration: one can change the flux by transporting charges around the one-cycles associated with them. Although the potential is exactly quadratic classically (and even in perturbation theory), there are nonperturbative corrections. Pair production will eventually discharge any potential exceeding twice the electron mass. This is such a slow enough process that we can safely ignore it. Furthermore, adiabatic motion of a charge will never be able to wind more than one unit of flux because of avoided level crossing [6]. This is not a problem except on timescales long compared to m -1 exp( mL ), where m is the electron mass.", "pages": [ 1, 2 ] }, { "title": "III. INGREDIENTS FOR CHAOTIC BRANE INFLATION", "content": "The ingredients we will need is an F-theory compactification of Type IIB string theory which contains at least one mobile D3-brane. Furthermore, the six-dimensional transverse space must have a nontrivial first homology, i.e., H 1 ( M 6 ) = /CI or /CI N . Because the D3 moduli in the direction of the nontrivial one-cycles are lifted at tree level, these models may lack supersymmetry. All closed string moduli must be sufficiently stabilized, in order that inflation may take place in this background. It is further necessary that the periodic portion of the potential be flat enough that the full potential has only a single minimum. In the probe approximation, the D3 has a discreet 'shift symmetry' associated with transport about the one-cycle, but this may not be sufficient to guarantee local flatness. As illustrated before, the inflationary potential exists due to nontrivial winding of the five-form flux about the homological one-cycle. The D3-brane moves classically through this cycle to unwind the flux. We will assume that the moduli stabilization is rigid enough to ignore the backreaction of the dynamical flux. This assumption is generically false in known warped flux compactifications [24-26], but such effects may actually flatten the potential [15]. The potential induced by brane monodromy is where L 6 is the volume of the compact space, M 10 , P is the ten-dimensional reduced Planck mass, and µ D3 = M 4 10 , P is the D3 charge. We assume the string coupling to be of order unity. In terms of the four-dimensional reduced Planck mass, M 2 P = M 8 10 , P L 6 , and for a canonically normalized inflaton φ = √ µ D3 z , we find the potential To achieve reasonable density perturbations, the quadratic model needs the inflaton mass to obey which requires the compactification scale to be In terms of the inflaton, this scale corresponds to a field distance Hence, successful large-field inflation will require the brane to undergo of order a few thousand revolutions, so any model must have first Homology large enough to permit this, i.e. with N /greaterorsimilar 2 × 10 3 . This rather large number can be relaxed by no more than two orders of magnitude by allowing the size of the one-cycle to be much larger than the natural scale 6 √ L 6 . In the /CI case, the quadratic approximation for V ( φ ) must eventually break down, due to backreaction. If we assume the modulus L is very heavy, it can be written as L ( φ ), which grows as more flux is wound on the transverse space. If this is the only effect of backreaction, the inflaton potential is flattened at large flux values, and reaches a maximum if d L ( φ ) / d φ exceeds 2 3 L ( φ ) /φ . However, a steepening [15] of the inflaton potential could instead occur due to kinetic coupling between the inflaton and L ( φ ), say of the form for sufficiently negative n . We will assume that the kinetic coupling is subdominant, and hence that backreaction leads to a flattening of the inflaton potential at sufficiently large field values φ ∼ Nf φ . Qualitatively speaking, a potential which is quadratic at small field values, and flat at large field values can be thought of as approximately sinusoidal over the range | φ | /lessorsimilar Nf φ . The /CI N case has extended periodicity z ≡ z + NL , and so the homological part of the potential in each of the above cases is approximately given by with where N represents either the backreaction scale or the size of the homology group. This scenario could be called natural brane inflation, following Refs. [27, 28], although it avoids the problems associated with large axion decay constants f φ by the appearance of the large factor N in the potential of Eq. (16), allowing f φ /lessmuch M P /lessmuch Nf φ . Alternative approaches to this problem can be found in Refs. [29, 30]. We additionally require the single-valued part of the potential to be relatively flat, meaning λ /lessorsimilar 1 /L . To achieve 60 e -folds of slow-roll inflation, we must arrange the scalar field φ to initially have a super-Planckian vacuum expectation value, φ /greaterorsimilar 15 M P . The Hubble scale during inflation is then H ≈ 10 -5 M P , which is almost two orders of magnitude below the Kaluza-Klein scale 2 π/L . However, the four-dimensional potential will be of order V ≈ 10 -10 M 4 P , which exceeds the tension of a D3brane by two orders of magnitude, opening the possibility for brane tunneling [31] or nucleation [32]. Because these are slow processes, our description remains valid. Indeed, brane nucleation 1 could give rise to the mobile inflaton, although inflation will then end with brane-antibrane annihilation, but unlike Ref. [33], the bubble need not self-annihilate until after many laps are completed. The final D3-D3 annihilation will result in the formation of a cosmic-string network [34].", "pages": [ 2, 3 ] }, { "title": "IV. DISCUSSION", "content": "We have provided a simple framework for large-field brane inflation. To construct realistic models, a number of hurdles must first be addressed, the most significant of which is moduli stabilization. However, because our framework relies on a nontrivial first homology group, much of the progress made on moduli stabilization of warped compactifications does not apply here. Another potential difficulty may arise in obtaining a flat enough periodic portion of the brane potential. If the modulation of the quadratic piece is too large, there may not be a long enough slow-roll trajectory. On the other hand, a periodic modulation of the inflaton potential can lead to detectable non-Gaussanity in the cosmic microwave background [35]. Finally, it is unlikely that supersymmetry can be unbroken in the models considered here, since the D3 moduli receive an explicit mass, rather than from a spontaneous uplifting, say by the introduction of antibranes. Nevertheless, a number of intriguing features arise here, foremost being a UV description of large-field inflation. The monodromies of this framework are extremely easy to visualize, being simply the motion of a (pointlike) brane around a one-cycle. By traversing the cycle (perhaps several thousand times) a Ramond-Ramond flux is unwound, realizing either chaotic or natural inflation, both of which predict significant tensor modes in the cosmic microwave background.", "pages": [ 3, 4 ] }, { "title": "Acknowledgments", "content": "We thank Daniel Baumann, Xingang Chen, Liam McAllister, Enrico Pajer, Lorenzo Sorbo, Alexander Westphal, and Xi Dong for helpful conversations. Funding was provided through NSF grant PHY-1213888.", "pages": [ 4 ] }, { "title": "Appendix A: The de Rham delta function", "content": "Here we review a simple notation [36] appropriate for calculating the effects of localized sources coupled to gauge potentials. The new object is a singular differential form which we call the 'de Rham delta function.'", "pages": [ 4 ] }, { "title": "1. Definition", "content": "On a D -dimensional oriented manifold M D with M r ⊆ M D an oriented submanifold of dimension r , we define the de Rham delta function δ D -r ( M r ) as follows: where the pullback is implicit on the rhs. The subscripts denote the order for differential forms, and superscripts denote the dimension for manifolds. Stokes' theorem then implies and so where we have used the fact that ∂ ( M r ∩M s ) = ( ∂ M r ∩ M s ) ∪ ( -1) D -r ( M r ∩ ∂ M s ). Here ∪ is essentially the group sum of r -chains in M D . This definition of ∪ is equivalent to Following the definition we also find which leads to the relation This identity illuminates some generic features of submanifolds.", "pages": [ 4 ] }, { "title": "2. Coordinate representation", "content": "The coordinate representation of δ D -r ( M r ) is straightforward in coordinates where the submanifold is defined by the D -r constraint equations, with i = 1 ...D -r , via where δ ( D -r ) is the usual ( D -r )-dimensional Dirac delta function. The well-known transformation properties of the Dirac delta function make this object automatically a differential form. (Thus the only meaningful zeros of the λ i are transversal zeros, i.e., those where λ i changes sign in any neighborhood of the zero.) If a submanifold is D -dimensional, then the corresponding de Rham delta function is simply the characteristic function, δ 0 ( M ' D ) = χ M ' D , with One may describe submanifolds with boundary by multiplication with this scalar de Rham delta function using Eq. (A4). As an example, if M 1 is the positive x axis in /CA 3 , then where the characteristic function is the Heaviside function Θ( x ). Notice that the orientation of this submanifold has been chosen to be along the + x direction, consistent with Eq. (A2) and the fact that its boundary is minus the point at the origin. The λ i do not need to be well defined on the entire manifold, and in fact they only need to be defined at all in a neighborhood of M r . Thus despite its appearance in Eq. (A6), δ D -r ( M r ) is not necessarily a total derivative. If all of the λ i are well defined everywhere, then M r is an algebraic variety. By Eq. (A2) and Eq. (A6) it can be seen that all algebraic varieties can be written globally as boundaries. It may be that the λ i are well defined only in a neighborhood of M r , in which case M r is a submanifold. Near points not on ∂ M r we may think of M r locally as a boundary, just as we may think of a closed differential form as locally exact. Nonorientable submanifolds will correspond to constraints that may be double valued, that is λ i may return to minus itself upon translation around the submanifold. We considered such cases in Ref. [36]. Another important case occurs when M r is only an immersion, i.e., it intersects itself. The λ i are path dependent here, as well. Consider the immersion S 1 ⊂ /CA 2 defined by the constraint λ = 2 arcsin( y ) -arcsin( x ) = 0. This looks like a figure-eight centered on the origin of the plane. Clearly λ is multivalued, and to get a complete figure-eight requires summing over two branches of λ . We suppress the sum in Eq. (A6). Notice that the figure-eight immersion satisfies The self-intersection of this immersion is twice the point at the origin, but since the orientation (sign) of the intersection is negative for exactly one of the two points of intersection, the total self-intersection vanishes. For evencodimemsion immersions, the sum over branches allows for nonzero self-intersection from cross terms. By the antisymmetry of the wedge product, one may show that the self-intersection of an immersion of odd codimension will always vanish, assuming M D is orientable. One may also compute the n th self-intersection of an immersion via", "pages": [ 4, 5 ] }, { "title": "3. Geometry", "content": "If we introduce a metric on our manifold, we can measure the volume of our submanifolds with the volume element from the pull-back metric. We use ∗ r to denote the Hodge star on M r , so the induced volume element is ∗ r 1. Looking at the coordinate definition of δ D -r ( M r ) given in Eq. (A5), we see that each of the λ i is constant along the submanifold, and so d λ i is orthogonal to the submanifold. Thus, a good candidate for a volume element on M r is ∗ (d λ 1 ∧ ... ∧ d λ D -r ). To remove the rescaling redundancy in the λ i 's we may divide by || ∧ i d λ i || , where the norm of a differential form C r is defined by the scalar Hence, we write where the pull-back is implicit on the rhs. Despite its appearance, ∗ δ D -r ( M r ) || δ D -r ( M r ) || is an r -form living in all D dimensions, although it is only well defined near M r . This is because in Eq. (A9), the Dirac delta function coefficients cancel, leaving dependence only on the smooth λ i s. Because of this, we may define d ∗ δ D -r ( M r ) || δ D -r ( M r ) || using the full D -dimensional exterior derivative. This is well defined on M r , and when restricting to points on the submanifold, From Eq. (A9) it is evident that More generally, the action by this Hodge star can be rewritten as where F s is an s -form living on M r and the pull-back is implicit on the rhs.", "pages": [ 5 ] }, { "title": "4. Topology", "content": "One feature of the de Rham delta function is that Poincar'e duality becomes manifest. Here we assume M D is orientable and compact with no boundary. Then Eq. (A2) tells us that δ D -r ( M r ) is closed if and only if M r is a cycle, and δ D -r ( M r ) is exact if M r is a boundary. To complete this correspondence between the r th homology and the ( D -r )th de Rham cohomology of M D , we need to show that But this is not true when torsion is present. Consider the manifold /CA P 3 = SO(3) which has a single nontrivial one-cycle ζ 1 , i.e. H 1 ( /CA P 3 ; /CI ) = /CI 2 . Since the group sum of two of these cycles is trivial, they must form a boundary, ζ 1 + ζ 1 = ∂ M 2 and so 2 δ 2 ( ζ 1 ) = d δ 1 ( M 2 ) which means Since ζ 1 is not a boundary, it is false to claim that all exact de Rham delta functions are Poincar'e dual to boundaries. Only by using real coefficients can we make the statement that De Rham's theorem gives us the isomorphism between homology and cohomology and Poincar'e duality asserts that The de Rham delta function provides the isomorphism In fact, if the cohomology basis is chosen such that then where ˜ M r ( i ) is the Poincar'e dual of M D -r ( i ) , and together they satisfy i.e., their net intersection is a single positive point if i = j , and is empty otherwise.", "pages": [ 5, 6 ] } ]

[ { "title": "V. A. Belinski", "content": "ICRANet, 65122 Pescara, Italy; Phys. Dept., Rome University 'La Sapienza', 00185 Rome, Italy; IHES, F-91440 Bures-sur-Yvette, France.", "pages": [ 1 ] }, { "title": "Abstract", "content": "The window is found in the space of the free parameters of the theory of viscoelastic matter for which the Friedmann singularity is stable. Under stability we mean that in the presence of the shear stresses the generic solution of the equations of relativistic gravity possessing the isotropic, homogeneous and thermally equilibrated cosmological singularity exists.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Observations show that the early Universe was isotropic, homogeneous and thermally balanced. A number of authors [1-3] expressed the point of view that also the initial cosmological singularity should be in conformity with these properties, that is should be of the Friedmann type. But it is well known that the Friedmann singularity for the conventional types of matter is unstable which means that space-time cannot start isotropic expansion unless an artificial fine tuning of unknown origin. This instability is due to the sharp anisotropy which develops unavoidably near the generic cosmological singularity. However, an intuitive understanding suggests that anisotropy can be damped down by the shear viscosity which being taking into account might results in the generic solution with isotropic Big Bang. To search an analytical realization of such a possibility there would be inappropriate to use just the Eckart [4] or LandauLifschitz [5] approaches to the relativistic hydrodynamics with dissipative processes. These theories are valid provided the characteristic times of the macroscopic motions of the matter are much bigger than the time of relaxation of the medium to the equilibrium state. It might happen that this is not so near the cosmological singularity since all characteristic macroscopic times in this region tend to zero in which case one need a theory which takes into account the Maxwell's relaxation times on the same footing as all other transport coefficients. In a literal sense such a theory does not exists, however, it can be constructed in an approximate form for the cases when a medium do not deviates too much from equilibrium and relaxation times do not exceed noticeably the characteristic macroscopic times . It is reasonable to expect that these conditions will be satisfied automatically for a generic solution (if it exists) near isotropic singularity describing the beginning of the thermally balanced Friedmann Universe accompanying by the arbitrary infinitesimally small corrections. The main target of the efforts of many authors (starting from the first idea of Cattaneo [6] up to the final formulation of the generalized relativistic theory by Israel and Stewart [7, 8]) was to bring the theory into line with relativistic causality, that is to eliminate the supraluminal propagation of the thermal and viscous excitations. The existence of such supraluminal effects was the main stumbling-block for the Eckart's and Landau-Lifschitz's descriptions of dissipative fluids. One of the first applications of the Israel-Stewart theory to the problems of cosmological singularity was undertaken in the article [9]. Already in this paper the stability of the Friedmann models under the influence of the shear viscosity has been investigated and it was found that relativistic causality and stability of the Friedmann singularity are in contradiction to each other. Then the final conclusion was: ' ...relativistic causality precludes the stability of isotropic collapse. The isotropic singularity cannot be the typical initial or final state. ' However, in the present paper it will be shown that this 'no go' conclusion was too hasty since it was the result of too restricted range for the dependence of the shear viscosity coefficient on the energy density. As usual, in the vicinity to the singularity where the energy density ε diverges we approximate the coefficient of viscosity η by the power law asymptotics η ∼ ε ν with some exponent ν. In the article [9] (due to some more or less plausible thoughts) we choose the values of this exponent from the region ν > 1 / 2 . For these values of ν the negative result of paper [9] remains correct, but recently it made known that the boundary value ν = 1 / 2 leads to the dramatic change of the state of affairs. It turns out that for this case there exists a window in the space of the free parameters of the theory in which the Friedmann singularity becomes stable and at the same time no supraluminal signals exist in its vicinity. This possibility was overlooked in [9]. It is worth to add that also the case ν < 1 / 2 is analyzed in the present article but it is of no interest since it leads to the strong instability of the isotropic singularity independently of the question of relativistic causality. Also it is necessary to stress that here as well as in the old paper [9] only the standard models for a physical fluid is considered for which the pressure is non-negative and is less than the energy density. To make the present paper self-contained we will reproduce below the principal (although updated) points on which the analysis of the work [9] was based. Then to read the present paper there is no necessity to turn to our old publication.", "pages": [ 1 ] }, { "title": "II. BASIC EQUATIONS IN THE PRESENCE OF THE SHEAR STRESSES", "content": "Shear stresses generate an addend S ik to the standard energy-momentum tensor of a fluid 1 : and this additional term has to satisfy the following constraints [5]: Besides we have the usual normalization condition for the 4-velocity: If the Maxwell's relaxation time τ of the stresses is not zero then do not exists any closed expression for S ik in terms of the viscosity coefficient η and 4-gradients of the 4-velocity. Instead the stresses S ik should be defined from the following differential equations [7]: which due to the normalization condition for velocity is compatible with constraints (2). In case τ = 0 expression for S ik , following from this equation, coincides with that one introduced by Landau and Lifschitz [5]. If the equations of state p = p ( ε ) , η = η ( ε ) , τ = τ ( ε ) are fixed then the Einstein equations together with equation (4) for the stresses gives the closed system where from all quantities of interest, that is g ik , u i , ε, S ik can be found. Since we are interesting in behaviour of the system in the vicinity to the cosmological singularity where ε →∞ the viscosity coefficient η in this asymptotic domain can be approximated by the power law asymptotics with some constant exponent ν. Beforehand the value of this exponent is unknown then we need to investigate its entire range -∞ < ν < ∞ . As for the relaxation time τ the choice is more definite. It is well known that η/ετ represents a measure of velocity of propagation of the shear excitations. Then we will model this ratio by a positive constant f (in a more accurate theory f can be a slow varying function on time but in any case this function should be bounded in order to exclude the appearance of the supraluminal signals). Consequently we choose the following model for the relation between relaxation time and viscosity coefficient: For the dependence p = p ( ε ) we follow the standard approximation with constant parameter γ : Now the system (1)-(8) is closed and we can search the asymptotic behaviour of its solution in the vicinity to the cosmological singularity. It is convenient to work in the synchronous reference system where the interval is Our task is to take the standard Friedmann solution in this system as background and to find the asymptotic (near singularity) solution of the equations (1)-(8) for the linear perturbations around this background in the same synchronous system. The background solution is 2 : where t > 0 and t c is some arbitrary positive constant (it is worth to remark that in the comoving and at the same time synchronous system the right hand side of the equation (4) is identically zero then the background value S ik = 0 indeed satisfies this equation). We have to deal with the following linear perturbations (as usual any quantity X we write as X = X (0) + δX where X (0) represents the background value of X ): In the linearized version of the system (1)-(9) around the Friedmann solution (10)-(11) will appear only these variations. The variations δu 0 and δS 0 k can not be of the first (linear) order because of the exact relations u i u i = -1 and u i S ik = 0 and properties (11) of the background. The variations δτ and δη of the relaxation time and viscosity coefficient, although exist as the first order quantities, will disappear from the linear approximation since they reveal itself only as factors in front of the terms vanishing for the isotropic Friedmann seed. Let's introduce for the quantities (12) the following notations: (Here and in the sequel we use two different entries δ αβ and δ αβ for one and the same 3-dimensional Kronecker delta). In terms of these quantities the linearized version of the equations (1)-(5) in synchronous reference system over the Friedmann space-time (10)-(11) becomes: where H = δ αβ H αβ . In these formulas R is the background scale factor given in (10) and ε, τ, η are the background values of these quantities defined by the the relations (6), (7) and (11) (in principle they should be written as ε (0) , τ (0) , η (0) but we omit the index (0) to simplify the writing). To find the general solution of these equations we apply the technique invented by Lifschitz [10] (see also [11]) and used by him to analyze the stability of the Friedmann solution for the perfect liquid. Since all coefficients in the differential equations (14)-(17) do not depend on spatial coordinates we can represent all quantities of interest in the form of the 3-dimensional Fourier integrals to reduce these equations to the system of the ordinary differential equations in time for the corresponding Fourier coefficients. First of all we substitute the expression for E from equation (16) to the right hand side of equation (14) and expression for V α from (15) to the right hand side of (17). This gives the closed system of equations for tensorial perturbations H αβ and K αβ and corresponding system of ordinary differential equations in time for their Fourier coefficients ˜ H αβ and ˜ K αβ (any space-time field Φ ( t, x 1 , x 2 , x 3 ) we represent as Φ ( t, x 1 , x 2 , x 3 ) = ∫ ˜ Φ( t, k 1 , k 2 , k 3 ) e ik α x α d 3 k ). Any symmetric tensorial Fourier coefficient containing six independent components can be expended in the Lifshitz basis which consists of the six basic elements. These elements can be constructed from an orthonormal triad ( l (1) α , l (2) α , l (3) α ) in the euclidean k -space where that is l (1) α is the unit directional vector of k -space and l (2) α , l (3) α are another two unit vectors normal to k α and to each other. The aforementioned basic elements are: Then ˜ H αβ and ˜ K αβ can be expended in the following way: (here we introduced the new index J which takes only two values 2 and 3), where the amplitudes λ, µ, σ ( J ) , ω ( J ) , A, B ( J ) , D ( J ) depend on time (and on the components of the wave vector). Only Q αβ has non-zero contraction δ αβ Q αβ , that's why in the expansion (23) for the shear stresses this component is absent (remember that the second condition of (2) calls δ αβ K αβ = 0). The reason why the Lifshitz basis is better than any other lies in the fact that in this basis the system of the differential equations (14)-(17) rewritten in terms of the λ, µ, σ ( J ) , ω ( J ) , A, B ( J ) , D ( J ) splits in the three separate and independent subsets: the first for λ, µ, A, the second for σ ( J ) , B ( J ) and the third for ω ( J ) , D ( J ) . The equations for λ, µ, A are: For the four amplitudes σ ( J ) , B ( J ) we have: and equations for two pairs ω ( J ) , D ( J ) are: If we know functions λ, µ and σ ( J ) the Fourier components ˜ V α , ˜ E of perturbations of velocity and energy density, as follows from the equations (15) and (16) (also making use the equation (24) to eliminate the second derivative ¨ µ ), can be expressed in terms of these functions by the relations:", "pages": [ 2, 3, 4 ] }, { "title": "III. ON THE PROPAGATION OF THE SHORT WAVELENGTH PULSES", "content": "To study the waves of the short wavelength (formally k → ∞ ) it is convenient to pass to the conformally flat version of the Friedman metric -ds 2 = R 2 ( T ) [ -dT 2 + ( dx 1 ) 2 + ( dx 2 ) 2 + ( dx 3 ) 2 ] , introducing the new time variable T by the relation dT = dt/R. Then in the limit when k dominates in the equations (24)-(30) this system has the following set of solutions: with large phases and slow varying amplitudes (index sva ). Substituting these expressions into the equations (24)-(30) and keeping only the terms of highest order with respect to the large quantity k one get the velocities of propagation of perturbations: This result have been obtained in [9] and it shows that gravitational waves (perturbation ω ( J ) , D ( J ) ) propagate with velocity of light but in order to exclude the supraluminal signals for two other types of perturbations it is necessary to demand υ 2 1 < 1 and υ 2 2 < 1 . Both of these conditions in the region 1 /lessorequalslant γ < 2 will be satisfied if", "pages": [ 4 ] }, { "title": "IV. EXTREME VICINITY TO THE SINGULARITY", "content": "The useful property of the equations (24)-(30) is an essential simplification and unification of their mathematical forms near singularity. Indeed near the singular point t → 0 in the limit when t is much less than everything else (including t /lessmuch k -1 ) we can neglect in these equations by all terms containing the factor k 2 R -2 which are much smaller than all other terms 3 . Consequently the asymptotic form of the equations (24)-(30) in the vicinity to singularity is: In the solution µ = C ( -1) µ t -1 + C (0) µ of the first equation both arbitrary constants C ( -1) µ and C (0) µ can be removed by the coordinate transformations which still exist in the synchronous system [11], that is µ in this approximation represents a pure gauge (non physical) excitation. We can take without loss of generality but keeping in mind that also constant C (0) µ can be put to zero 4 . The other pairs of perturbations are described by the identical equations so it is enough to consider only one such pair, for example ( λ, A ) . As analysis shows there are three principally different characters of behaviour of perturbations for the three different ranges of values of the index ν in formula (6) for the viscosity coefficient, namely ν < 1 / 2 , ν > 1 / 2 and ν = 1 / 2 . For the first two ranges it is convenient to represent relations (6) and (7) (taking into account expression (11) for ε ) in the following form: with some arbitrary positive constant t τ . Then from the equations (39) follow that ˙ λ and A can be expressed in term of an auxiliary function F ( t ) as: after which equations (39) reduce to one ordinary equation of second order for F : If instead of t and F ( t ) we introduce the new time variable y and new function W ( y ) by the relations: where then (45) gives the Whittaker equation [12]: where constants L and M are: It is easy to check that due to the condition of causality (37) the quantity 3 ( γ +2) 2 -32 f under the square root in expression for M can never be negative. Then M is real and without loss of generality we can choose its positive branch M > 0 . For the boundary value ν = 1 / 2 the representation (43) and (45)-(49) does not works and this special case we will consider separately (see below).", "pages": [ 4, 5 ] }, { "title": "A. Case ν < 1 / 2 .", "content": "In this case, as follows from (46), near singularity ( t → 0) the variable y →∞ . Then the asymptotic behaviour of the function W ( y ) at infinity is characterized by the superposition of two terms y -L e y/ 2 and y L e -y/ 2 (this can be seen directly from the equation (48) without necessity to go to a reference book for the asymptotic properties of the two Whittaker fundamental solutions W L,M ( y ) and W -L,M ( -y )). Then relations (46) and (44) show that perturbations contain the strongly divergent mode for which λ, A ∼ exp [ 1 2(1 -2 ν ) ( t t τ ) 2 ν -1 ] . This mode de- stroys the background regime. Consequently the values ν < 1 / 2 are of no interest for us since in this case does not exists a general solution of the gravitational equations with the Friedmann singularity.", "pages": [ 5 ] }, { "title": "B. Case ν > 1 / 2 .", "content": "For ν > 1 / 2 the singularity t → 0 corresponds to y → 0 . In this asymptotic region the superposition of two modes W ± ∼ y 1 2 ± M forms the general solution for the Whittaker equation 5 . For the function F ( t ) this corresponds to the modes F ± ∼ t (2 ν -1) ( α + 1 2 ± M ) which gives the following asymptotic behaviour for two time-dependent perturbation modes for metric: λ ± ∼ t (2 ν -1) ( α + 1 2 ± M ) +1 . With definitions (47) and (49) for constants α and M we have: For stability of the Friedmann solution it is necessary for both exponents s ± to be positive ( λ ± must disappear in the limit t → 0). Because the first term in s ± in the region 1 /lessorequalslant γ < 2 is positive and the square root is positive we need to provide only the inequality s -> 0 and it is easy to show that this is equivalent to the restriction f > 3 4 γ (2 -γ ) for the constant f. However, this restriction is exactly opposite to the causality condition (37). Consequently, also for ν > 1 / 2 , assuming the absence of the supraluminal excitations, there is no way to provide stability of the Friedmann solution near singularity. This result has been obtained already in [9]. It is worth to remark that this state of affairs is in conformity with the general statement, already expressed in the literature, on the connection between the existence of the supraluminal signals and instability of the equilibrium states [13, 14].", "pages": [ 5, 6 ] }, { "title": "C. Case ν = 1 / 2 .", "content": "For ν = 1 / 2 instead of (43) we have to write: where β is some dimensionless positive arbitrary constant. Relations (44) are the same and equation for the auxiliary function F ( t ), which follows from (39) and (51) becomes: This equation has exact solutions in the form of two power law modes with power exponents following from the corresponding quadratic equation. Using the relation ˙ λ = 2 F it is easy to show that the final result for the perturbation λ is: where C (0) λ , C (1) λ , C (2) λ are three arbitrary constants (depending on the wave vector) and where the sign plus corresponds to s 1 and minus to s 2 and square root we take to be positive in case when it is real. In order to have λ /lessmuch 1 at t → 0 both exponents s 1 and s 2 should be either positive or they should have the positive real part. At the same time in both cases we have to satisfy the relativistic causality condition (37). Then we have two possibilities. Either in which case s 1 and s 2 are real and positive, or which corresponds to the complex conjugated s 1 and s 2 but with positive real part. It is easy to show that in the space of parameters f, β, γ there are two regions, exposed on the Fig. 1, in which either the first or the second of these sets of requirements is satisfied. The set of inequalities (55) is satisfied in the triangle ABD and the set (56) is valid in the triangle BCD . The caption to this figure contains all necessary information on the admissible domains for the values of the parameters f, β, γ . Fom the first of the equations (39) follows amplitude A : where Now one can make asymptotics for λ, µ and A more precise taking into account those terms in equations (24)(26) containing the factor k 2 R -2 , that is the terms which have been neglected in the first approximation. Analysis shows that their influence consists in generation the small time-dependent corrections to the arbitrary constants appeared in the first approximation. These corrections are completely expressible in terms of parameters of the first approximation, that is they do not bring any new arbitrariness in the solution. The fact is that the exact general solution of equations (24)- (26) for the functions λ, µ and t 2 A has the form F (0) ( t )+ t s 1 F (1) ( t )+ t s 2 F (2) ( t ) with the same exponents s 1 and s 2 given by the formula (54) and with functions F (0) , F (1) , F (2) each of which can be expressed in the form of the Taylor series in the small parameter ζ : which parameter tends to zero in the limit t → 0 because ( kt/R ) 2 = k 2 t 4 / 3 γ c t 2(3 γ -2) / 3 γ and γ /greaterorequalslant 1. The first time-independent terms in these power series are just the arbitrary constants figured in the formulas (42),(53) and E (57). The general structure of the exact solution for λ, µ and t 2 A is: where all α -coefficients in front of the powers of parameter ζ are constant quantities which depend on the four arbitrary constants C (0) µ , C (0) λ , C (1) λ , C (2) λ and external numbers f, β, γ . There is no big sense in taking into account corrections containing the powers of ζ in the factors in front of the powers t s 1 and t s 2 since this would give the small unimportant addends to the asymptotics. The same is true for the corrections of the orders ζ 2 and higher in terms which do not contain powers t s 1 and t s 2 . However, to keep the terms α (0) λ 1 ζ , α (0) µ 1 ζ and α (0) A 1 ζ in the asymptotics is necessary because in general they, although small, play a role in the behaviour of the solution and the first non-vanishing term in the asymptotic expression for the energy density depends on them. Calculations gives the following result for the coefficients α (0) λ 1 , α (0) µ 1 and α (0) A 1 : Then the final sufficient asymptotics for the amplitudes λ, µ and A is: where The exact coincidence of the forms of equations (39)(41) means that in the main approximation the other pairs of amplitudes σ ( J ) , B ( J ) and ω ( J ) , D ( J ) are described by the same formulas (53)-(54) and (57)-(58) with only difference that instead of C (0) λ , C (1) λ , C (2) λ one should take the new arbitrary constants C (0) σ ( J ) , C (1) σ ( J ) , C (2) σ ( J ) and C (0) ω ( J ) , C (1) ω ( J ) , C (2) ω ( J ) respectively. After that one can calculate corrections to this main approximation taking into account the influence of the terms in equations (27)-(30) containing the factor k 2 R -2 . These calculations are analogous to those we made for the amplitudes λ, µ, t 2 A and the final results are: where the coefficients α (0) ω ( J ) 1 and α (0) D ( J ) 1 are: Due to specific structure of equations (27) and (28) the solutions for perturbations σ ( J ) , t 2 B ( J ) do not contain corrections of the order t s 3 . The two arbitrary constants C (0) σ ( J ) can be removed by the coordinate transformations which still remain in the synchronous system (in addition to those by which we already eliminated constant C ( -1) µ and can eliminate constant C (0) µ in function µ ). Consequently the total number of the arbitrary physical constants in the Fourier coefficients (which generate the arbitrary 3-dimensional physical function in the real x -space) of the solution is 13, these are C (0) λ , C (1) λ , C (2) λ , C (1) σ ( J ) , C (2) σ ( J ) , C (0) ω ( J ) , C (1) ω ( J ) , C (2) ω ( J ) . This is exactly the number of arbitrary independent physical degrees of freedom of the system under consideration, that is 4 for the gravitational field, 1 for the energy density, 3 for the velocity and 5 for the shear stresses (five because the six components S αβ follows from the six differential equations of the first order in time with one additional condition δ αβ S αβ = 0). Then the solution we constructed is generic. The asymptotic solutions for the Fourier coefficients for perturbations of the velocity and energy density follow from (31)-(32): It is evident that the asymptotic behaviour of all perturbations satisfy the basic requirement to be small in relative sense. This condition means that variations (12) must be small with respect to the corresponding background values, that is the quantities δg αβ R 2 , δε ε , δu α and δS αβ εR 2 in the limit t → 0 should be much less than unity (the necessity to be small for the last ratio follows from the condition δS αβ /lessmuch T (0) αβ = pg (0) αβ ∼ εR 2 ). In terms of the Fourier amplitudes these requirements are ˜ H αβ /lessmuch 1 , t 2 ˜ E /lessmuch 1 , ˜ V α /lessmuch 1 , t 2 ˜ K αβ /lessmuch 1 and all of them are satisfied since all time-dependent terms in the left hand sides of these inequalities are going to die away as t → 0 and the six arbitrary constants in the metric perturbations ˜ H αβ we are free to take to be infinitesimally small. The interpretation of these constants is well known: their appearance simply indicates that the isotropic part of the perturbed metric g αβ in the x -space instead of the seed value R 2 δ αβ acquires the more general form R 2 a αβ ( x 1 , x 2 , x 3 ) where a αβ in perturbative solution should be closed to δ αβ but in the nonperturbative context (see below) becomes an arbitrary symmetric 3-dimensional tensor. All this means that in the real x -space a generic non perturbative solution exists with the following asymptotics for the metric near singularity: where R = ( t/t c ) 2 / 3 γ and exponents s 1 , s 2 and s 3 are defined by the relation (54) and (69). The additional terms denoted by the triple dots are small corrections which contain the terms of the orders t 2 s 3 , t s 1 + s 3 , t s 2 + s 3 as well as all their powers and cross products. The main addend a αβ represents six arbitrary 3-dimensional functions (in the linearized version they are generated by the arbitrary constants C (0) λ , C (0) µ , C (0) σ ( J ) , C (0) ω ( J ) in the Fourier coefficients). Each tensor b (1) αβ and b (2) αβ consists of the six 3-dimensional functions subjected to the restrictions a αβ b (1) αβ = 0 and a αβ b (2) αβ = 0 (here a αβ is inverse to a αβ ), consequently b (1) αβ and b (2) αβ contain another ten arbitrary 3-dimensional functions (in the linearized version they are generated by the ten arbitrary constants C (1) λ , C (2) λ , C (1) σ ( J ) , C (2) σ ( J ) , C (1) ω ( J ) , C (2) ω ( J ) in the Fourier coefficients). In case of complex conjugated s 1 and s 2 the components b (1) αβ and b (2) αβ are complex but in the way to provide reality of the metric tensor. The last term b (3) αβ and all corrections denoted by the triple dots in the expansion (79) are expressible in terms of the a αβ , b (1) αβ , b (2) αβ and their derivatives then they do not contain any new arbitrariness. The shear stresses, velocity and energy density follows from the exact Einstein equations in terms of the metric tensor (79) and its derivatives and all these quantities also do not contain any new arbitrary parameters. In result the solution contains 16 arbitrary 3-dimensional functions the three of which represent the gauge freedom due to the possibility of the arbitrary 3-dimensional coordinate transformations. Then the physical freedom in the solution corresponds to 13 arbitrary functions as it should be. This result is the generalization of the so-called quasiisotropic solution constructed in [15] (see also [11]) for the perfect liquid. However, in case of perfect liquid the isotropic singularity is unstable and asymptotics found in [15] corresponds to the narrow class of particular solutions containing only 3 arbitrary physical 3-dimensional parameters.", "pages": [ 6, 7, 8, 9 ] }, { "title": "V. CONCLUDING REMARKS", "content": "1. The results presented show that the viscoelastic material with shear viscosity coefficient η ∼ √ ε can stabilize the Friedmann cosmological singularity and the corresponding generic solution of the Einstein equations for the viscous fluid possessing the isotropic Big Bang (or Big Crunch) exists . Depending on the free parameters f, β, γ of the theory such solution can be either of smooth power law asymptotics near singularity (when both power exponents s 1 and s 2 are real and positive) or it can have the character of damping (in the limit t → 0) oscillations (when s 1 and s 2 have the positive real part and an imaginary part). The last possibility reveals itself as a weak trace of the chaotic oscillatory regime which is characteristic for the most general asymptotics near the cosmological singularity and which can not be described in closed analytical form (for the short simplified review on the oscillatory regime see [16]). The present case show that the shear viscosity can smooth such chaotic behaviour up to the quiet oscillations which have simple asymptotic expressions in terms of the elementary functions of the type t Re s sin [(Im s ) ln t ] and t Re s cos [(Im s ) ln t ] . 2. In the generic isotropic Big Bang described here some part of perturbations are presented already at the initial singularity t = 0 which are the three physical components of the arbitrary 3-dimensional tensor a αβ ( x 1 , x 2 , x 3 ) in formula (79). Another ten arbitrary physical degrees of freedom are contained in the components of two tensors b (1) αβ and b (2) αβ in this formula and they come to the action in the process of expansion. This picture has no that shortage of the classical Lifshitz approach when one is forced to introduce some unexplainable segment between singularity t = 0 and initial time t = t 0 when perturbations arise in such a way that inside this segment it is necessary to postulate without reasons the validity of the exact Friedmann solution free of any perturbations. 4. In our analysis the case of stiff matter ( γ = 2) have been excluded. This peculiar possibility should be investigated separately. It is known that for the perfect liquid with stiff matter equation of state a generic solution with isotropic singularity is impossible (see [16] and references therein). The asymptotic of the general solution for this case have essentially anisotropic structure although of the smooth (non-oscillatory) power law character. It might be that viscosity will be able to isotropize such evolution, however, it is not yet clear how the viscous stiff matter should be treated mathematically. The simple way to take γ = 2 in our previous study does not works. where σ and T is the entropy density and temperature of a (perturbed) fluid. Here we used the fact that in our model chemical potential vanish (that is γε = Tσ ) and that principal assumption of the Israel-Stewart theory that the Gibbs relation (in our case dε = Tdσ ) is universal in the sense that it is valid for the arbitrary displacements of the thermodynamical parameters, that is not only between neighbouring equilibrium states. Equation (4) for stresses being multiplied by S ik gives: Substituting this into the previous formula we obtain: The 4-vector in the brackets in the left hand side of the last equation represents the generalization of the LandauLifshitz entropy flux for the case when relaxation time τ of the shear stresses is not zero. This expression for the entropy flux is the same that have been proposed by the Israel-Stewart theory [7, 8]. If the background solution is an real equilibrium state in the literal sense then the action of the operator u k ∂ k on the background values of quantities τ, η, T gives zero and also u k ; k = 0 for the background values of the 4-velocity. Then the factor ( τu k / 4 ηT ) ; k in front of S mn S mn in the last term of the equation (82) disappears in the first approximation. Then this last term belongs to the third approximation since also S mn vanish for the background solution. Consequently up to the second order in the deviation from the equilibrium the equation (82) provides correct result, that is for any future directed evolution the entropy increases because the quantity S mn S mn is always positive due to the properties (2) of the stresses. However, the Friedmann background is not an equilibrium state in the aforementioned literal sense. This solution describes the quasi-stationary evolution in which the Universe passes the continuous sequence of equilibrium states with different equilibrium parameters but with one and the same conserved entropy. Due to this evolution the background value of the factor ( τu k / 4 ηT ) ; k in equation (82) is not zero, moreover, it is not small with respect to the factor 1 / 2 ηT in the first term in the right hand side of the equation (82). It is easy to get ( τu k / 4 ηT ) ; k from formulas (10)-(11) and (51) using expression T = γε c ( ε/ε c ) ( γ -1) /γ for the background temperature ( ε c is an arbitrary constant). In result the entropy production equation (82) for our model take the form and one can see that constant β -2 is negative. Indeed, the first inequality in both sets of stability conditions (55) and (56) is β < (3 γ -2) /γ but for any value of parameter γ from the interval 1 /lessorequalslant γ < 2 the quantity (3 γ -2) /γ is less than 2. It might be thought that the negativity of the right hand side of equation (83) means that the second law of thermodynamics precludes the physical realization of the generic isotropic Big Bang. However, it can happen that such conclusion again would be too hasty because, as we already said, the entropy of gravitational field might normalize the situation. As of now no concrete calculation can be made inasmuch no theory of the gravitational entropy exists. Nevertheless in the model under investigation it looks plausible that gravitational entropy, being proportional to some invariants of the Weyl tensor [2], indeed would be able to change the state of affairs because for the background Friedmann solution this tensor is identically zero and it will start to increase in the course of expansion. Then increasing of the gravitational entropy would compensate the decreasing of the matter entropy. For those who believe that the Universe began by an isotropic expansion the negativity of the right hand side of the equation (83) stands as a hint that gravitational entropy indeed exists. By the way it is worth to remark that practically in all publications (including [7, 8]) dedicated to the extended thermodynamics in the framework of the General Relativity the condition σ k ; k /greaterorequalslant 0 for the entropy flux of the matter is accepted from the beginning as one's due. Moreover, namely from this condition follows the structure of the additional dissipative terms in the energymomentum tensor and particle flux. Such strategy is undoubtedly correct not only for the 'everyday life' but also for the majority of the astrophysical problems where the gravitational fields are relatively weak. However the cases with extremely strong gravity as in vicinity to the cosmological singularity need more precise definition of what we should understand under the total entropy of the system.", "pages": [ 9, 10 ] }, { "title": "VI. ACKNOWLEDGEMENTS", "content": "It is a pleasure to thank G.Bisnovatiy-Kogan for the useful critics and stimulating discussions and E.Vladimirova for the help which accelerates the creation of the final version of the manuscript. (1992).", "pages": [ 10, 11 ] } ]

[ { "title": "Weighing neutrinos in f ( R ) gravity", "content": "Jian-hua He 1, ∗ 1 INAF-Osservatorio Astronomico, di Brera, Via Emilio Bianchi, 46, I-23807, Merate (LC), Italy We constrain the neutrino properties in f ( R ) gravity using the latest observations from cosmic microwave background(CMB) and baryon acoustic oscillation(BAO) measurements. We first constrain separately the total mass of neutrinos ∑ m ν and the effective number of neutrino species N eff . Then we constrain N eff and ∑ m ν simultaneously. We find ∑ m ν < 0 . 462eV at a 95% confidence level for the combination of Planck CMB data, WMAP CMB polarization data, BAO data and highl data from the Atacama Cosmology Telescope and the South Pole Telescope. We also find N eff = 3 . 32 +0 . 54 -0 . 51 at a 95% confidence level for the same data set. When constraining N eff and ∑ m ν simultaneously, we find N eff = 3 . 58 +0 . 72 -0 . 69 and ∑ m ν < 0 . 860eV at a 95% confidence level, respectively. PACS numbers: 98.80.-k,04.50.Kd", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The determination of the neutrino mass is an important issue in fundamental physics. The Standard Model of particle physics had assumed that all three families of neutrinos: electron neutrinos ν e , muon neutrinos ν µ and tau neutrinos ν τ are massless, and that the neutrino cannot change its flavor from one to another. However, the results from solar and atmospheric experiments [1] showed that the flavour of neutrinos could oscillate. The mixing and oscillating of flavors implies nonzero differences between the neutrino masses, which in turn indicates that the neutrinos have absolute mass. If the neutrino does have absolute mass, it will be the lowest-energy particle in the extensions of the Standard Model of particle physics. However, such observations of flavor oscillations can only show that the neutrinos have mass, and cannot exactly pin down the absolute mass scale of neutrinos. Particle physics experiments are able to place lower limits on the effective neutrino mass, which, however, depends on the hierarchy of the neutrino mass spectra[2](also see Ref.[3] for reviews). On the other hand, cosmological constraints on neutrino properties are highly complementary to particle physics. Massive neutrinos, if above 1eV , will become nonrelativistic before recombination[4], leaving an impact on the first acoustic peak in the cosmic microwave background(CMB) temperature angular power spectrum due to the early-time integrated Sachs-Wolfe (ISW) effect; neutrinos with mass below 1eV will become nonrelativistic after recombination, altering the matter-radiation equality; the massive neutrino will also suppress the matter power spectrum on small scales, since neutrinos cannot cluster below the free-streaming scales [5](see[6] for reviews). Combining various cosmological observations can put rather tight constraints on the sum of the neutrino mass. The most recent measurements from the Planck satellite[7] on the CMB in combination with the baryon acoustic oscillation(BAO)[8-11], WMAP polarization(WP) and the highl data on the CMB from the Atacama Cosmology Telescope(ACT)[12] and the South Pole Telescope(SPT)[13] give an upper limit for the sum of the neutrino mass as ∑ m ν < 0 . 23eV(95%C . L . ) in the spatially flat ΛCDM model with the effective number of neutrino species as N eff = 3 . 04 . It is even more promising that with the upcoming ESA Euclid mission[14] in the near future, the neutrino mass can be constrained up to an unprecedented accuracy simply by cosmological observations[15]. The allowed neutrino mass window could be closed by forthcoming cosmological observations. Nevertheless, it is important to recall that the constraints on neutrino properties are usually found within the context of a Λ CDM model or within the context of a dark energy model[16]. Considering different cosmological models, degeneracies may arise among neutrinos and other cosmological parameters. Cosmological constraints on neutrino properties are highly model dependent. References[15, 17] have investigated this issue in the framework of a dark energy model with varying total neutrino mass and number of relativistic species. The aim of this paper is, however, to extend such investigations to modified gravity models. For simplicity, we consider the f ( R ) gravity [18] and particularly focus on a specific family of f ( R ) models that can exactly reproduce the Λ CDM background expansion history of the Universe. This family of f ( R ) models has only one more parameter than the Λ CDM model, which can be characterized by which is approximately the squared Compton wavelengths in units of the Hubble scale [19]. Cosmological constraints on these models without taking into account neutrino mass have already been presented in the literature. On linear scales, the WMAP nine-year data in combination with the matter power spectra of LRG from SDSS DR7 data can only put weak constraints on these models: B 0 < 3 . 86(95%C . L . ) [20, 21]. Tighter constraints can be obtained from the galaxy-ISW correlation data, which puts the constraint up to B 0 < 0 . 376(95%C . L . ) [20, 22]. Using the data of cluster abundance, the constraints are dramatically improved up to B 0 < 1 . 1 × 10 -3 (95%C . L . ) [22, 23]. However, the tightest constraints so far come from the astrophysical tests[24] which place the upper bound for B 0 as B 0 < 2 . 5 × 10 -6 . On the other hand, the cosmological constraints on f ( R ) mod- s taking into account neutrino mass have also already been presented in the literature[25, 26]. However, these works are done within the framework of parameterized gravities. We still need to get more accurate results by solving the full linear perturbation equations in the f ( R ) gravity. In this paper, we will explore the neutrino properties in f ( R ) gravity based on our modified version of CAMB code [27], which solves the full linear perturbation equations in the f ( R ) gravity [20]. We will conduct the Markov chain Monte Carlo(MCMC) analysis on our model based on the COSMOMC package[28] and constrain the cosmological parameters using the latest observational data. Besides examining the total mass of active neutrinos ∑ m ν , we will also investigate the effective number of neutrino species N eff since a detection of N eff > 3 . 04 will imply additional relativistic relics or nonstandard neutrino properties[29]. This paper is organized as follows: in section II, we will briefly outline the details of the basic equations in f ( R ) cosmological models. In section III, we will discuss about how the f ( R ) gravity impacts on the neutrino constraints. In Sec. IV, we will list the observational data used in this work. In Sec. V, we will present the details of our numerical results. In Sec. VI, we will summarize and conclude this work.", "pages": [ 1, 2 ] }, { "title": "II. f ( R ) GRAVITY", "content": "In f ( R ) gravity, the Einstein-Hilbert action is given by where κ 2 = 8 πG and L ( m ) is the matter Lagrangian. With variation with respect to g µν , we obtain the modified Einstein equation where F = ∂f ∂R . If we consider a homogeneous and isotropic background universe described by the flat Friedmann-Robertson-Walker(FRW) metric the modified Friedmann equation in f ( R ) gravity is given by[30] Taking the derivative of the above equation, we obtain where the dot denotes the time derivative with respect to the cosmic time t , and ρ is the total energy density of the matter which consists of the cold dark matter, baryon, photon, and neutrinos. p is the total pressure in the Universe. If we convert the derivatives in Eq.(6) from the cosmic time t to x = ln a , Eq.(6) can be written as where E ≡ H 2 H 2 0 and dρ dx = -3( ρ + p ) . For convenience, in the above equation, the energy density ρ is in units of H 2 0 , and we set κ 2 = 1 in our analysis. In order to mimic the Λ CDM background expansion history, we can parameterize E ( x ) as [31] which includes the effect of neutrinos. Ω 0 c and Ω 0 b represent present-day cold dark matter and baryon density, respectively. Ω 0 d is the effective dark energy density which is a constant. T ν 0 = (4 / 11) 1 / 3 T cmb = 1 . 945K is the present-day neutrino temperature and Ω 0 r = 2 . 469 × 10 -5 h -2 for T cmb = 2 . 725K . m ν represents the neutrino mass and we assume that all massive neutrino species have the equal mass. The function f ( y ) in the above expression is defined by After fixing the background expansion, Eq.(7), governing the behavior of the scale field F ( x ) in f ( R ) gravity, can be solved numerically, given the initial condition in the deep-matterdominated epoch[20]: where the index is defined by p + = 5+ √ 73 12 . The above initial conditions are still applied here, because the relativistic neutrinos are far less than the total amount of nonrelativistic species(including baryons, cold dark matter and nonrelativistic neutrino)in the Universe at this moment. Equation (7) has analytical solutions[32] if we ignore the relativistic species in the Universe. Noting the fact that p + > 0 , our model only has growing modes in the solutions of Eq.(7), which satisfy and our model thus can go back to the Λ CDM model at high redshift. This family of f ( R ) models has only one more parameter than the Λ CDM model, which can be characterized either by D or by the Compton wavelengths B 0 . In this work, we will sample D directly in our MCMC analysis and treat B 0 as a derived parameter. In order to avoid the instabilities in the high-curvature region[33], we need to set D < 0 , which keeps the Compton wavelength B always positive during the past expansion of the Universe B > 0 . We set the initial conditions for the background in Eq.(6) roughly at the point a i ∼ 0 . 03 around which the value of the , scalar field F ( x ) obtained by solving Eq.(6)rather weakly depends on the exact choice of a i , given Eq(10) as the initial conditions. For the perturbed spacetime, we solve the full linear perturbation equations in the f ( R ) gravity based on our modified version of the CAMB code [20]. In our code, we plug in the f ( R ) gravity perturbation at a = 0 . 03 , before which we set the perturbation as δF = 0 , ˙ δF = 0 such that the equations completely go back to the standard equations in the Λ CDMmodel.", "pages": [ 2, 3 ] }, { "title": "III. THE INTEGRATED SACHS -WOLFE EFFECT AND THE CMB LENSING", "content": "Before going further to present our MCMC analysis, we will discuss in this section about how the f ( R ) gravity impacts the neutrino constraints. The f ( R ) model studied in this paper actually has rather weak impacts on the early Universe. It only has late-time effects and impacts mainly on the late-time integrated Sachs -Wolfe(ISW) effect and the CMB lensing. For the ISW effect, the f ( R ) gravity will suppress the power of the ISW quadrupole as the parameter B 0 which characterizes the f ( R ) gravity is relatively small [19]. As B 0 increases, the suppression will reach its maximum and then become reduced. Further increasing B 0 , there is a turnaround point above which the suppression will turn into excess, which increases the power of the ISW quadrupole as well as the total quadrupole. In order to better understand this phenomenon, in Fig.1 we plot the total temperature angular power spectra and the ISW spectra as well, which are calculated by where P χ is the primordial power spectrum, j l ( x ) is the spherical Bessel function, and ε is the optical depth between η and the present. The potential Φ -which accounts for the ISW effect, is defined by and its derivative with respect to the conformal time η is given by where Φ and Ψ in the above equation are the Bardeen potentials[34] and σ , Z , η T are the perturbation quantities in the synchronous gauge. We present the equivalent expressions in the synchronous gauge for Φ -here because the CAMB code is based on the synchronous gauge. For illustrative purposes, we take the cosmological parameters for the fiducial model as the best-fitted values of the Λ CDM model as reported by the Planck team Ω 0 b = 0 . 049 , Ω c = 0 . 267 , Ω Λ = 0 . 684 , h = 0 . 6711 , n s = 0 . 962 , 10 9 A s = 2 . 215 , τ = 0 . 0925 [7]. From Fig.1, we can see that the suppression of the ISW power spectra reaches its maximal around D ∼ -0 . 25( B 0 ∼ 0 . 92) then the power turns to grow from its minimal as further increasing the value of | D | . Around D ∼ -0 . 45( B 0 ∼ 1 . 94) , the power spectra of the f ( R ) model go back to being similar to that of the Λ CDM model. The f ( R ) gravity and the Λ CDM model give almost the same temperature angular power spectrum at this point. However, the value of B 0 for this point depends on the cosmological parameters of the fiducial model. To show this, in Fig.2, we plot the power spectra of the model with different values of Ω m = 0 . 24 and h = 0 . 73 which are the same as those used in Ref. [19] and keep the other cosmological parameters unchanged. We find that around D ∼ -0 . 37( B 0 ∼ 1 . 5) , the suppression reaches its maximal and around D ∼ -0 . 60( B 0 ∼ 3) , the power spectra go back to being similar to that of the Λ CDMmodel. Our results are actually well consistent with Ref. [19], if we take the same values of the cosmological parameters. In Fig. 3, we show the angular power spectra of the lensing potential ψ ≡ -Φ -for a few representative values of D . From Fig. 3, we can see that, contrary to the ISW effect, f ( R ) gravity always enhances the power of the lensing potential. The larger the value of | D | , or equivalently, of B 0 , the more enhancement in the power spectrum of the potential. We should remark here that in the original CAMB code, ψ is calculated by using an approximation. However, this approximation does not apply to the f ( R ) gravity. We need to use the exact expression of Eq.(14) to calculate ψ instead. Then we follow the standard routine in the CAMB code to calculate C ψ l . The detailed derivations of C ψ l can be found in Ref. [35]. The phenomenon as described above of the impact of the f ( R ) gravity on the ISW effect (e.g. Fig.1,Fig.2) and the CMBlensing (e.g. Fig.3) can be explained by the evolution of the metric potential Φ -[19]. In Fig. 4, we show the value of Φ -/ Φ -i with respect to the scale factor a . We choose the wave number as k = 3 × 10 -3 h Mpc -1 from which the power of the quadrupole mainly arises[19]. Φ -is calculated by Eq.(14) and Φ -i is the value of the potential in the Λ CDM model at a = 0 . 03 . As is well-known, the ISW effect is driven by the evolution of the potential Φ -, which depends on the relative difference of the potential Φ -at the initial time Φ -i and the present time Φ -0 . From Fig. 4, we can see that the gravitational potential Φ -always decays in the Λ CDM model at late times of the Universe. However, in the f ( R ) gravity, the potential will be enhanced against such decay due to the existence of the extra scalar field δF . Φ -in the f ( R ) gravity will decay less than that in the Λ CDM model when the value of B 0 is relatively small (e.g. B 0 = 0 . 161 ). Then, for a certain value of B 0 (e.g. B 0 ∼ 0 . 920 ), Φ -0 at present will be comparable to Φ -i at early times. The ISW effect is canceled out at this point. For large enough B 0 (e.g. B 0 = 1 . 938 ), the potential at present will overwhelm the potential at early times Φ -0 > Φ -i and the ISW effect in the f ( R ) gravity will change its sign. However, the amplitude of the ISW effect in- 0 becomes much larger. This explains what we observed in Fig. 1 and Fig. 2. For the CMB lensing, we can find that, contrary to the ISW effect, the angular power spectrum of the lensing potential C ψ l [35] depends on the absolute value of the amplitude of the potential Φ -, which increases monotonously with B 0 as shown in Fig. 4. It is the case, therefore, that the larger the value of B 0 , the larger the power of C ψ l . On the other hand, neutrinos with mass heavier than a few eV will become nonrelativistic before the recombination, triggering significant impact on the CMB anisotropy spectrum. However, this situation is strongly disfavoured by current observational bounds even in the case of f ( R ) gravity as we shall see later. Therefore, we will not discuss this case here. Neutrinos with a mass ranging from 10 -3 eV to 1eV will be relativistic at the time of matter-radiation equality and will be nonrelativistic today, which can potentially impact the CMB in three ways(see [6] for reviews). The massive neutrino can shift the redshift of equality which affects the position and am- plitude of the peaks; it can also change the angular diameter distance to the last scattering surface which affects the overall position of CMB spectrum features; the massive neutrino can affect the late time ISW effect as well. We will focus on the ISW effect in this work. In Fig. 5, we plot the total angular power spectrum and ISW effect for a few representative values of the density of the massive neutrinos Ω ν . We can see that the massive neutrinos will suppress the power of the ISW effect and the power of the total power spectrum. We also plot the impact of the massive neutrinos on the angular power spectrum of the lensing potential in Fig. 6. We can see that the massive neutrinos will always enhance the power of the lensing potentials. From the above analysis, we can see that with the cos- mological parameters of the fiducial model around Ω m ∼ 0 . 32 , h ∼ 0 . 67 , which is favored by the Planck results [7], if B 0 < 0 . 92 , the impact of f ( R ) gravity on the ISW effect and the CMB lensing is degenerate with the impact of the massive neutrinos. Moreover, for f ( R ) models with B 0 > 0 . 92 , the impact of f ( R ) gravity on the ISW effect could partially compensate the effect of massive neutrinos since f ( R ) gravity enhances the power as B 0 grows if B 0 > 0 . 92 . This compensation would further boost the degeneracy between B 0 and ∑ m ν as we shall see later.", "pages": [ 3, 4, 5 ] }, { "title": "IV. CURRENT OBSERVATIONAL DATA", "content": "In this work, we adopt the CMB data from the Planck satellite[7], as well as the highl data from the Atacama Cosmology Telescope(ACT)[12] and the South Pole Telescope(SPT)[13]. For the Planck data, we use the likelihood code provided by the Planck team, which includes the high-multipoles l > 50 likelihood following the CamSpec methodology and the low-multipoles ( 2 < l < 49 ) likelihood based on a Blackwell-Rao estimator applied to Gibbs samples computed by the Commander algorithm. For the highl data, we include the ACT 148 × 148 spectra for l ≥ 1000 , and the ACT 148 × 218 and 218 × 218 spectra for l ≥ 1500 . For SPT data, we only use the high multipoles with l > 2000 . In our analysis, the WMAP polarization data will be used along with Planck temperature data. For comparison, we also present the results obtained from WMAP nine-year data in this work. The likelihood code[36] contains both temperature and polarization data. The temperature data include the CMB anisotropies on scales 2 ≤ l ≤ 1200 ;the polarization data contain TE/EE/BB power spectra on scales (2 ≤ l ≤ 23) and TE power spectra on scales (24 ≤ l ≤ 800) . In addition to the CMB data, we also add the measurement on the distance indicator from the baryon acoustic oscillations(BAO) surveys. BAO surveys measure the distance ratio between r s ( z drag ) and D v ( z ) where r s ( z drag ) is the comoving sound horizon at the baryon drag epoch, which is defined by where η is the conformal time and R ≡ 3 ρ b / (4 ρ r ) . The drag redshift z drag indicates the epoch for which the Compton drag balances the gravitational force, which happens at g d ∼ 1 , where with ˙ g = -an e σ T (where n e is the density of free electrons and σ T is the Thomson cross section). z drag is defined by g d ( η ( z drag )) = 1 . The quantity D v ( z ) is a combination of the angular diameter distance D A ( z ) and the Hubble parameter H ( z ) . Although the f ( R ) model studied in this work exhibits strong scale-dependent growth history even in the linear regime(see Fig 7), which changes not only the amplitude but also the shape of the matter power spectra, in real space the scale of the BAO peak in the two-point correlation function of the density z field does not change for this family of f ( R ) models: =0 From Fig 8, we can see that the BAO scales do not shift in this family of f ( R ) models. The locations of the BAO peaks in the f ( R ) models relative to that in the Λ CDM model shift no more than ± 1 . 5Mpc / h , which is mainly subject to the numerical errors. In this paper, we therefore can safely adopt the BAOdata. We follow the Planck analysis [7] and use the BAO measurements from four different redshift surveys: z = 0 . 57 from the BOSS DR9 measurement [8]; z = 0 . 1 from the 6dF Galaxy Survey measurement [9]; z = 0 . 44 , 0 . 60 and 0 . 73 from the WiggleZ measurement[10]; z = 0 . 2 and z = 0 . 35 from the SDSS DR7 measurement[11].", "pages": [ 5, 6 ] }, { "title": "V. NUMERICAL RESULTS", "content": "In this section, we explore the cosmological parameter space in our f ( R ) model using the Markov chain Monte Carlo analysis. Our analysis is based on the public available code COSMOMC [28] as well as a modified version of the CAMB code which solves the full linear perturbation equations in the f ( R ) gravity [20]. The parameter space of our model is where Ω b h 2 and Ω c h 2 are the physical baryon and cold dark matter energy densities respectively, 100 θ MC is the angular size of the acoustic horizon, A s is the amplitude of the primordial curvature perturbation, n s is the scalar spectrum powerlaw index, τ is the optical depth due to reionization, ∑ m ν is the sum of the neutrino mass in eV, N eff is the effective number of neutrinolike relativistic degrees of freedom and D is the parameter which characterizes the f ( R ) gravity. We will sample the parameter D directly in our work and treat B 0 as a derived parameter. The priors for the cosmological parameters are listed in Table I. In this work, we will pay particular attention to the neutrino properties. We will fix N eff = 3 . 046 to constrain the total mass of neutrinos ∑ m ν and, in turn, fix ∑ m ν = 0 . 06[eV] to constrain the effective number of neutrino species N eff . Finally, we will constrain N eff and ∑ m ν simultaneously.", "pages": [ 6 ] }, { "title": "A. Constraints on the total mass of active neutrinos", "content": "In this subsection, we report the constraints on the total mass of active neutrinos ∑ m ν assuming N eff = 3 . 046 . The numerical results are shown in Table II. In Fig.10, we show the one-dimensional marginalized likelihood for the total neutrino mass ∑ m ν as well as other cosmological parameters D, n s , Ω c h 2 , 100 θ MC , H 0 . We start by presenting the results obtained from the data combinations associated with WMAP nine-year data . From TableII, we can find that WMAP nine-year data along place very poor constraints on ,", "pages": [ 6 ] }, { "title": "TABLE I. Uniform priors for the cosmological parameters", "content": "∑ m ν , Ω c h 2 and H 0 . ∑ m ν remains almost unconstrained and the 2 σ (95%C . L . ) range of marginalized likelihood for ∑ m ν almost spans the whole range as our priors listed in Table I. However, if we add the BAO data, the constraint can be improved significantly, because the BAO data can improve the constraint on H 0 and breaks the degeneracy between H 0 and ∑ m ν . The combination of WMAP+BAO gives However, adding the BAO data does not improve the constraint on f ( R ) gravity. We find D < 0 . 542( B 0 < 2 . 54)(95%C . L . ; WMAP + BAO) which is even slightly larger than the constraints obtained from WMAP data alone D < 0 . 518( B 0 < 2 . 37)(95%C . L . ; WMAP) . Adding the highl measurement from the CMB can further improve the constraint on ∑ m ν because the WMAP data do not have enough accuracy on the highl angular power spectra. The combination of WMAP9+BAO+highL places the constraint at Compared with the constraints associated with WMAP data, Planck data show more robust constraints on ∑ m ν as well as the f ( R ) gravity. Although the Planck data alone in combination with WMAP polarization(WP) data only place very weak constraints on the total neutrino mass, they put tighter constraints on the f ( R ) gravity D < 0 . 346( B 0 < 1 . 36) (95%C.L.) due to fact that f ( R ) gravity produces the quadrupole suppression on the temperature angular power spectra[19] and the Planck data have a more accurate measurement on the large-scale ( 2 < l < 50 ) temperature angular power spectra than that of the WMAP data. The data combination Planck+WP, however, can not put a tight constraint on H 0 , as shown in Fig.10. Planck+WP therefore gives very poor constraint on ∑ m ν due to the degeneracy between H 0 and ∑ m ν . Therefore, it can be expected that adding BAO data can improve the constraints significantly. We find with | D | < 0 . 379( B 0 < 1 . 54) (95%C.L.). The constraint on ∑ m ν has been improved by almost 50% by adding the BAO data. On the other hand, we find that the highl data do not show a significant improvement on the constraint of ∑ m ν but slightly improve on the constraint of f ( R ) gravity due to the tighter constraint on Ω c h 2 (see Table II). We find and | D | < 0 . 298( B 0 < 1 . 14) (95%C.L.). In order to show the degeneracy between B 0 and ∑ m ν . We plot the Marginalized two-dimensional likelihood ( 1 , 2 σ contours) constraints on B 0 and ∑ m ν in Fig 9. We can see that when B 0 > 1 , there are tails in the contours, which means the degeneracy sharpens here. This is because the impact of f ( R ) gravity on the ISW effect could partially be compensated by the massive neutrinos if B 0 > 1 as discussed previously.", "pages": [ 7 ] }, { "title": "B. Constraints on N eff", "content": "In this subsection, we consider the constraints on the effective number of neutrino species, N eff , assuming the total . c mass of active neutrinos as ∑ m ν = 0 . 06eV . The numerical results are shown in Table III. In Fig.11, we show the onedimensional marginalized likelihood on the effective number of neutrino species N eff as well as other cosmological parameters D, n s , Ω c h 2 , 100 θ MC , H 0 . WMAP nine-year data along place rather weak constraints on the effective number of neutrino species at the 95% C.L. However, the constraints on N eff as well as other cosmological parameters are improved significantly when the BAO data are added. The combination of the WMAP+BAO data set improve the constraint on N eff up to We find that after adding the highl data, the constraints can be further improved. The error bars have shrunk almost by 50% compared to the case without the highl data. The other cosmological parameters are also better constrained after adding the highl data(see Table III). Particularly, Ω c h 2 is constrained up to 0 . 1151 +0 . 0048 -0 . 0048 where the error bars have reduced by almost 75% . For the WMAP data set, we can find that the results are compatible with the standard value N eff = 3 . 046 within the 1 σ range. Compared with the results obtained from the combination of WMAP data, Planck data show robust constraints on N eff as well as the f ( R ) gravity. Planck data alone in combina- tion with WMAP polarization(WP) data (Planck+WP) give the constraints as The best-fit value strongly favors N eff > 3 . 046 , which indicates the existence of extra species of relativistic neutrinos. The standard value N eff = 3 . 046 is only on the edge of the 1 σ range (see Table III) but is still compatible within the 2 σ range. Adding the BAO data can improve the constraints significantly. The combination of Planck+WP+BAO data set gives However, we find that further adding the highl data does not show a significant improvement on the constraint of N eff . The combination of Planck+WP+BAO+highL data sets only give which is almost the same as the result in the Λ CDM model as reported by Planck team N eff = 3 . 30 +0 . 54 -0 . 51 (95%C . L . ) [7]. This result is expected because the f ( R ) models investigated in this work only change the late-time growth history of the Universe and do not change the matter-radiation equality. If the parameter Ω c in the f ( R ) gravity model is tightly constrained, the constraints on N eff , in this case, should be quite close to that in the Λ CDMmodel.", "pages": [ 7, 8, 9 ] }, { "title": "C. Simultaneous constraints on N eff and ∑ m ν", "content": "In this subsection, we report the joint constraints on the total mass of active neutrinos ∑ m ν and the effective number of species N eff . In this work, we assume three active neutrinos share a mass m ν = ∑ m ν / 3 . The extra species of neutrinos δN eff = N eff -3 . 046 are relativistic and massless. When N eff < 3 . 046 , the temperature of the three active neutrinos is reduced accordingly, and no additional relativistic species are assumed. Based on these assumptions, we conduct the MCMC analysis and the numerical results are shown in table IV. In Fig.12, we show the one-dimensional marginalized likelihood on ∑ m ν , N eff and other cosmological parameters D, n s , Ω c h 2 , 100 θ MC . We first present the results obtained from the data combination associated with WMAP data. WMAP data along yields very poor constraints on both ∑ m ν and N eff The ∑ m ν remains almost unconstrained and the error bars on N eff are quite large. However, these bounds can be significantly tightened by adding BAO data. We find However, ∑ m ν still remains almost unconstrained. After adding the highl data, we find the constraints are improved significantly. (24) Similar to previous sections, the Planck data again show robust constraint on both N eff and ∑ m ν . We find However, compared with the results in previous section where ∑ m ν is fixed, the constraint on N eff , in this section, is clearly weakened if ∑ m ν can vary. This point is quite different from the case in the Λ CDM model as reported by the Planck team[7], where the joint constraints do not differ very much from the bounds obtained when introducing these parameters separately. This is because ∑ m ν is degenerate with f ( R ) gravity and looses the constraint on Ω m h 2 = Ω ν h 2 +Ω c h 2 + Ω b h 2 and so does the matter-radiation equality. The constraint on N eff is, therefore, weakened as well. After adding the BAO data, the constraints are improved up to (26) However, we find that adding the highl data does not show significant improvement on the constraints.", "pages": [ 9 ] }, { "title": "VI. CONCLUSIONS", "content": "In this work, we have analyzed the performance of constraints on neutrino properties from the latest cosmological observations in the framework of f ( R ) gravity using massive MCMC analysis. We have analyzed the constraints on the total mass of neutrinos ∑ m ν assuming N eff = 3 . 046 ; we have also analyzed the constraints on the effective number of neutrino species N eff assuming ∑ m ν = 0 . 06[eV] ;finally,we have analyzed the constraints on N eff and ∑ m ν simultaneously. To conclude, we summarize our main results with the tightest error bars in TableV and also compare them with the results obtained by the Planck team[7] within the context of the Λ CDM model. We can find that the constraints on ∑ m ν when fixing N eff = 3 . 046 in f ( R ) gravity are a factor of 2 larger than those of the Λ CDMmodel. When fixing ∑ m ν = 0 . 06eV , the constraint on N eff in f ( R ) gravity is almost the same as that in the Λ CDM model. However, when running ∑ m ν and N eff simultaneously, the constraints on N eff and ∑ m ν in the f ( R ) model are both significantly weaker than that in the Λ CDM model due to the degeneracy between the late time growth history in f ( R ) gravity and ∑ m ν . c In summary, constraints on neutrino properties from cosmological observations are highly model dependent. Tighter constraints on the neutrino properties can only be achieved when the modified gravity models are also well constrained. Stringent constraints on the f ( R ) model can be obtained on nonlinear scales using the data from cluster abundance[37]. However, the chameleon mechanism[38, 39] plays an important role on nonlinear scales. At early times, since the background curvature is very high, the nonliner perturbation for the f ( R ) models which can go back to the Λ CDM model at high curvature regime lim R → + ∞ F ( R ) = 1 generally follows the \"high-curvature solution\" [40], where the effective Newtonian constant in overdensity regions is extremely close to that of the standard gravity G eff ∼ G [41] and the chameleon mechanism works very efficiently in this period. If the \"highcurvature solution\" in high density regions could persist until present day, the thin-shell structure can be formed natu- n high density regions for the galaxies in the Universe. If the galaxies are sufficiently self-screened, the stars inside a galaxy can naturally be self-screened as well. The model thus can evade the stringent local tests of gravity. However, in high density regions, the \"high-curvature solutions\" are not always achieved for f ( R ) models at late times in the Universe. For the family of f ( R ) models studied in this work, neglecting the effects of massive neutrinos, we do not find any \"high-curvature solutions\" or \"thin-shell\" structures in the dense region for the models with | f R 0 = 1 -F | > 10 -4 and there is a factor of 1 / 3 enhancement in the strength of Newtonian gravity[41]. This means that these models could be ruled out by local tests of gravity and, conservatively speaking, the viable f ( R ) models should be with | f R 0 = 1 -F | < = 10 -4 ( B 0 < 5 . 5 × 10 -4 ) [42]. From the tightest astrophysical constraints B 0 < 2 . 5 × 10 -6 [24] which is in the bound of B 0 < 5 . 5 × 10 -4 , we can learn that, for viable f ( R ) mod-els, at ast, the chameleon screening mechanism should work very efficiently. However, this estimation is only based on our simulations in the case without taking account of massive neutrinos. There are no N-body simulations available at the moment, to our best knowledge, that have been calibrated with neutrinos in any forms of f ( R ) models. To calibrate neutrinos in f ( R ) simulations is an urgent object of our future work. Acknowledgment: J.H.He acknowledges the Financial support of MIUR through PRIN 2008 and ASI through contract Euclid-NIS I/039/10/0. We thank B. R. Granett for carefully reading the manuscript.", "pages": [ 9, 10, 11, 12 ] } ]

[ { "title": "Intermediate inflation from rainbow gravity", "content": "John D. Barrow 1 and Jo˜ao Magueijo 2, 3 1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge, CB3 0WA,UK 2 Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom 3 Dipartimento di Fisica, Universita La Sapienza and Sezione Roma1 INFN, P.le A. Moro 2, 00185 Roma, Italia (Dated: July 16, 2018) It is possible to dualize theories based on deformed dispersion relations and Einstein gravity so as to map them into theories with trivial dispersion relations and rainbow gravity. This often leads to 'dual inflation' without the usual breaking of the strong energy condition. We identify the dispersion relations in the original frame which map into 'intermediate' inflationary models. These turn out to be particularly simple: power-laws modulated by powers of a logarithm. The fluctuations predicted by these scenarios are near, but not exactly scale-invariant, with a red running spectral index. These dispersion relations deserve further study within the context of quantum gravity and the phenomenon of dimensional reduction in the ultraviolet.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In recent work [1-3] we have investigated the viability of cosmological scenarios based on modified dispersion relations (MDRs), when combined with Einstein gravity. Notably, it was found that the MDR linking the energy E and momentum p , with constant γ and characteristic running scale λ , is associated with exactly scale-invariant fluctuations when the constant parameter γ = 2, with suitable modifications leading to small deviations from exact scale-invariance [1]. This is particularly interesting, since these dispersion relations appear to model well the phenomenon of dimensional reduction in the ultra-violet (UV), for which there is growing evidence in numerous approaches to quantum gravity [4-17]. The mechanism producing fluctuations is analogous to that of varying speed of light/sound models [18-20] and, at face-value, dispenses with the need for inflation to perform this role. Nonetheless, it is noteworthy that it is possible to change units, or 'frame', so as to render the dispersion relations trivial. The non-trivial phenomenology of the theory is then shifted elsewhere, specifically to the theory of gravity. This operation was performed in [2, 3] and it is equivalent to what is done with Brans-Dicke theory when one conformally changes from the Einstein frame, with a varying G , to the Jordan frame, with a constant G but modified gravity. Specifically in [2] we showed that the new frame, where the speed of light, c, is constant, is the frame of 'rainbow gravity' [21], and that typically one has inflation in this frame, even if the strong energy conditions are not broken. A similar phenomenon was found before in [22]. This is by no means always the case. Indeed, the very topical case γ = 2 (associated with UV spectral dimension 2) does not lead to inflation in the dual frame from the point of view of fluctuations; instead, it leads to the switching off of gravity altogether. Also, whenever we do have inflation in the dual frame, it is not standard infla- ion. It is inflation driven by the gravity theory, rather than by the matter content (or an 'inflaton field'). Also, (near) scale-invariant fluctuations can be obtained under very different conditions to standard inflation: for example, we do not need to be near de Sitter. For this reason in [2] these models of inflation were labelled 'esoteric inflation'. The specific models derived from (1) lead to powerlaw or de Sitter inflation. The purpose of this paper is to obtain more general MDRs associated with intermediate inflation [23-26]. This form of inflation is a generalisation of de Sitter inflation in which the expansion scale factor evolves with /negationslash with A > 0 and 0 < n ≤ 1 constants. Subject to Einstein gravity it creates scale-invariant fluctuations when n = 2 / 3 as well as n = 1 (which is de Sitter). It has been found to arise in a wide class of scalar-tensor gravity theories [27] and in general relativistic cosmologies where there is an effective equation of state, linking the density ρ and the pressure P, of the form ρ + P = Γ ρ B . For Γ = 0 and B = 1 2 or 1, a zero-curvature FRW universe has an exact solution of the form (2) with [26] /negationslash This is equivalent to a family of exact solutions containing a single scalar field φ with a particular self-interaction potential, V ( φ ) [26].", "pages": [ 1 ] }, { "title": "II. RAINBOW INFLATION REVISITED", "content": "It was proved in [2] that MDRs of the form (1) combined with Einstein gravity can be mapped into a rainbow frame with trivial dispersion relations but a modified theory of gravity. In general, this modified gravity theory is very different, but it was shown in [2] that for background solutions with no curvature (i.e. FRW models with K = 0) this amounts to keeping Einstein gravity and adopting an 'effective' equation of state in the rainbow frame with where w = P/ρ is the linear equation of state factor in the Einstein frame. We stress that the modified gravity theory is more complex in general, in particular for the perturbations around these solutions. Here we present an alternative derivation of this result which is not only particularly simple, but will mimic the method used for finding intermediate inflationary solutions in the next Section. If (1) is valid then at high energies (in the 'UV' limit) we have: Here p is the physical momentum, so if we focus on a comoving mode labelled by k , then: and this property is valid so long as there is spatial translational invariance. Therefore, in the Einstein frame, c is both energy and time dependent, due to the expansion. We may define the rainbow frame (in which c is constant) directly in terms of proper time, in a procedure that is equivalent to that used in [2]. First define a disformal transformation by keeping the spatial coordinates and a unchanged but replacing t by Since in Einstein gravity for K = 0 Friedmann expansion we have, we must also have so that By comparing with (9) we can read off that in this context (i.e. for FRW, K = 0 solutions) the new gravity theory is equivalent to Einstein gravity, but with the matter content modified according to (5). Notice, however, that the Hubble constant is now k -dependent, something that will affect indirectly the perturbations (in addition to the direct effects of modified gravity). It turns out that this is the most general situation for which: Specifically, we have power-law inflation in the rainbow frame if and de Sitter inflation if we saturate the second identity: It is curious that γ = 2 (associated with running to spectral dimension d S = 2 in the UV) combined with radiation ( w = 1 / 3) in the Einstein frame, produces de Sitter inflation in the rainbow frame. In this case and so (where the proportionality constant could be k -dependent). We see that the Hubble constant is now k -dependent one of the many esoteric properties of these inflationary models, and a common feature in rainbow gravity.", "pages": [ 1, 2 ] }, { "title": "III. INTERMEDIATE INFLATION", "content": "We can obtain more general inflationary solutions if we allow for more general MDR, specifically those that in the high-energy (UV) λp /greatermuch 1 limit take the form: where the function g need not be a power-law. The speed of light is now given by c = E/p ≈ g , with a general profile. Just as with power-law inflation, we can obtain intermediate inflation solutions but the multiplicative constants will be k -dependent. We can reverse engineer the MDRs associated with intermediate inflation by adapting the argument in Section II. For simplicity, let us first illustrate the argument by the case where we start with radiation in the Einstein frame, so that and seek to obtain the well-known special case of intermediate inflation with scale-invariant inhomogeneity spectrum where n = 2 / 3 , so where we explicitly allow for k -dependence in the multiplicative factors. From (19), we conclude that we must have where f ( k ) and g ( k ) are still to be specified. By changing variables, we can rewrite (8) as where we have used (18) to conclude that t ∝ a 2 , specific to the radiation case. By comparing (21) and (20), we can just read off the consistency condition: where we have started fixing some of the free functions in the initial ansatz. We therefore arrive at the UV-limit expression: where D is an arbitrary ( k -independent) constant. We can now check directly that this MDR results in the intermediate inflationary solution in the dual frame: This argument may be generalized to express any equation of state w in the Einstein frame as an intermediate inflationary solution in the rainbow frame of the form log a ∝ ˜ t n . Performing the calculation we find that we should impose: where for completeness we have linked the IR limit with the UV solution required. In the UV: and the exponents in the MDRs are related to the solutions in the Einstein and rainbow frame by The solution in the rainbow frame can be given more completely by: This reduces to our illustrative solution (24) for n = 2 / 3 (i.e. β = 1 / 2) and w = 1 / 3 (i.e. γ = 2). It also reduces to the de Sitter case (15) for n = 1 (i.e. β = 0) and w = 1 / 3 (i.e. γ = 2), when the integration constants are all adjusted to be equivalent. We note that the parameter D has to be chosen so that g remains positive for a range of p . At face value we should conclude that D imposes a maximal momentum, since for we finally get g = 0. However we could also take the modulus and extend the MDRs up to infinite momentum.", "pages": [ 2, 3 ] }, { "title": "IV. FLUCTUATIONS IN RAINBOW INTERMEDIATE INFLATION", "content": "The conditions for viable fluctuations in models based on MDRs are entirely different from those based on standard inflation. Specifically, for γ = 2 and β = 0 the model is known to lead to exact scale invariance regardless of the value of w , within a certain range [1]. This corresponds to any inflationary ˜ w in the rainbow frame, in contrast with standard theory (which requires near de Sitter inflation). Furthermore, within the standard theory, only intermediate inflation with the specific value n = 2 / 3 leads to exact scale-invariance. Rainbow intermediate inflation may therefore be expected to predict departures from exact scale-invariance. This is confirmed by calculation. It is easiest to perform the calculation in the Einstein frame. Then, the equation for the cosmological perturbations is: where the prime denotes derivative with respect to conformal time η defined with respect to the Einstein frame time (i.e. dt = adη ). In terms of the variable v the (comoving gauge) curvature perturbation is given by ζ = -v/a . The speed of light/sound is given in the UV limit by: We want to study dual intermediate inflationary models associated with near-scale invariance. This dual requirement constrains us to select γ ≈ 2 and w ≈ 1 / 3. Then a ∝ η and the suitably normalized solution describing vacuum fluctuations inside the 'horizon' is given by This should be glued to v ∼ F ( k ) a when ckη ∼ 1 in order to find the spectrum left frozen outside the horizon (a procedure explained in more detail in [1]). To leading order the glueing point satisfies k ( λk ) 2 ∝ η ∝ a , so this finally translates into: or where E is a constant. We see that intermediate inflation in the rainbow frame is near scale-invariant, with a red running spectral index. No longer is the n = 2 / 3 case special: near-scale invariance is valid for all rainbow intermediate inflation models, with their n simply controling the power of the logarithmic modulation.", "pages": [ 3, 4 ] }, { "title": "V. CONCLUSIONS", "content": "The models just presented could be very interesting observationally. The Planck satellite results have put pressure on model builders to predict departures from strict scale-invariance [28]. Whilst these can be easily accommodated within standard inflation, the issue arises as to how natural those departures are (or seen in another way, how predictive with respect to them the theory actually is). Intermediate inflation has long been seen as an interesting direction to explore regarding this issue [29, 30]. In this paper we obtained intermediate inflation by postulating the appropriate MDRs in the Einstein frame capable of transforming into it in the rainbow frame. The required MDRs are of the general form (25), which is the central result of this paper. The fact that we naturally obtained departures from exact scale-invariance (see Eq. (35)) without fine-tuning of parameters is very interesting, and will be explored in a future publication. As pointed out before (e.g. in [31]) Occam's razor sometimes may dismiss the best-fit model. It remains to understand better what MDRs of the type (25) mean within the context of quantum gravity and the phenomenon of dimensional reduction in the ultraviolet. The logarithmic factor certainly has a 'renormalization' flavour. We are currently working out the spectral dimension running function d S ( s ) for these MDRs, as well as an array of related implications, with a view to clarifying their more fundamental meaning.", "pages": [ 4 ] }, { "title": "Acknowledgments", "content": "We thank G. Amelino-Camelia, M. Arzano, G. Gubitosi, M. Lagos and the anonymous referee of [2] for the discussions and comments leading to this paper. JM was supported by an International Exchange Grant from the Royal Society; JDB and JM both acknowledge STFC consolidated grant support.", "pages": [ 4 ] } ]

[ { "title": "Instability of a Kerr black hole in f(R) gravity", "content": "Yun Soo Myung a Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Korea", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the stability of a rotating (Kerr) black hole in the viable f ( R ) gravity. The linearized-Ricci scalar equation shows the superradiant instability, leading to the instability of the Kerr black hole in f ( R ) gravity. PACS numbers: 04.60.Kz, 04.20.Fy Keywords: Kerr black hole; superradiant instability; modified gravity a [email protected]", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "One of modified gravity theories, f ( R ) gravity [1, 2, 3, 4] has much attentions as a strong candidate for explaining the current and future accelerating phases in the evolution of universe [5, 6]. On the other hand, the Schwarzschild-de Sitter black hole was firstly obtained for a constant curvature scalar from f ( R ) gravity [7]. A Schwarzschild-anti de Sitter black hole solution was obtained from f ( R ) gravity by requiring a negatively constant curvature scalar [8]. The trace of stress-energy tensor should be zero to obtain a constant curvature black hole when f ( R ) gravity couples with matter of the Maxwell field [8], the Yang-Mills field [9], and a nonlinear Maxwell field [10]. Most of astrophysical black holes including supermassive black holes are considered to be a rotating black hole. A rotating black hole solution [11] should be stable against the external perturbations because it stands as a realistic object in the sky [12]. The stability analysis of the Kerr black hole is not as straightforward as one has performed the stability analysis of a spherically symmetric Schwarzschild black hole [13, 14, 15] because it is an axis-symmetric black hole. Here we would like to mention that the stability analysis is based on the linearized equations and thus, it does not guarantee the stability of black holes at the nonlinear level. The Kerr black hole has been proven to be stable against a massless graviton [16, 17, 18] and a massless scalar [19]. However, there exist the superradiant instability (the black-hole bomb) when one chooses a massive scalar [20, 21, 22, 23, 24, 25] and a massive vector [26]. For example of f ( R ) = R + hR 2 , the Kerr black hole is unstable because it could be transformed into a massive scalar-tensor theory [27]. It is known that the Kerr solution could be obtained from a limited form (6) of f ( R ) gravity [28]. Interestingly, it was shown that a perturbed Kerr black hole could distinguish Einstein gravity from f ( R ) gravity [29]. However, the stability analysis of f ( R )-rotating black hole is a formidable task because f ( R ) gravity contains fourth-order derivative terms in the linearized equation. Transforming the limited form of f ( R ) gravity into the scalartensor theory might resolve difficulty, which leads to the fact that the f ( R )-rotating black hole is unstable against a massive scalar perturbation when one used the black-hole bomb idea [30]. In this work, we examine the stability of a rotating black hole in the viable f ( R ) gravity. We consider the linearized Ricci scalar as a truly massive spin-0 graviton propagating on the Kerr black hole spacetimes. Solving its linearized equation shows a superradiant instability, which dictates the instability of the Kerr black hole in f ( R ) gravity. This will be compared to the Gregory- Laflamme instability of the massive spin-2 graviton in the dRGT massive gravity [31, 32] and the fourth-order gravity [33].", "pages": [ 2, 3 ] }, { "title": "2 f ( R ) -rotating black holes", "content": "We start with the f ( R ) gravity action with κ 2 = 8 πG . The Einstein equation takes the form where the prime ( ' ) denotes the differentiation with respect to its argument. It is well-known that Eq. (2) provides a solution with constant curvature scalar R = ¯ R . In this case, Eq. (2) reduces to and thus, the trace of (3) determines the constant curvature scalar to be with Λ f the cosmological constant. The subscript ' f ' denotes that the Λ f arose from the f ( R ) gravity. Substituting this expression into (3) leads to the Ricci tensor /negationslash To find the Kerr black hole solution with Λ f = 0 ( ¯ R µν = ¯ R = 0), one requires f (0) = 0 with f ' (0) = 0. To this end, one has to choose a specific form of f ( R ) as [28] /negationslash In Table 1, we list viable models of f ( R ) gravity which provide the form (6). Hence, a model of f ( R ) = R -µ 4 /R could not provide a rotating black hole [35, 36] because f (0) → -∞ and f ' (0) → ∞ . Also, the form of f ( R ) = α √ R + β [37, 38] is excluded because f (0) = α √ β = 0. By the same token, the two models of f ( R ) = R p e q/R and f ( R ) = R p (ln[ αR ]) q [39] are not suitable for seeking the Kerr black hole solution. - - In this work, we use the Boyer-Lindquist coordinates to represent an axissymmetric Kerr black hole solution with mass M and angular momentum J [11] with Here we use Planck units of G = c = /planckover2pi1 = 1 and thus, the mass M has a length scale. In the nonrotating limit of a → 0, (7) recovers the Schwarzschild black hole, while the limit of a → 1 corresponds to the extremal Kerr black hole. From the condition of ∆ = 0( g rr = 0), we determine two horizons which are located at The angular velocity at the event horizon takes the form In general, one introduces the metric perturbation around the Kerr black hole to study the stability of the black hole Hereafter we denote the background quantities with the 'overbar' ( ¯ R µν = 0 , ¯ R = 0). The Taylor expansions around the zero curvature scalar background is employed to define the linearized Ricci scalar [34] as The linearized equation to (2) is given by where the linearized Ricci tensor and scalar could be expressed in terms of h µν as Considering (15) and (16), the linearized equation (14) is a fourth-order differential equation with respect to the metric perturbation h µν , which is not a tractable equation to be solved. Choosing the Lorentz gauge of ¯ ∇ ν h µν = ¯ ∇ µ h/ 2 and using the trace-reversed perturbation of ˜ h µν = h µν -h ¯ g µν / 2, equation (14) takes a relatively simple from [29] where the mass squared m 2 f is defined by /negationslash In case of Einstein gravity [ f ( R ) = R, f ' (0) = 1 , f '' (0) = 0], Eq. (17) leads to a well-known second-order equation for ˜ h µν since the last fourth-order term is decoupled from (17). However, we could not solve (17) directly for m 2 f = ∞ because it is a coupled fourth-order equation for ˜ h µν and ˜ h .", "pages": [ 3, 4, 5 ] }, { "title": "3 Superradiant instability of Ricci scalar", "content": "It is well-known that f ( R ) gravity has 3 degrees of freedom (DOF) without ghost in Minkowski spacetimes: 2 DOF for a massless spin-2 graviton and 1 DOF for a massive spin-0 graviton. The massive spin-0 graviton is usually described by the trace h of h µν , but it could be represented by the linearized Ricci scalar δR because δR = -¯ ∇ 2 h/ 2 = ¯ ∇ 2 ˜ h/ 2 under the Lorentz gauge. For this purpose, we may take the trace of (14) with ¯ g µν . Then, we have a massive equation for δR which is considered as a second-order equation that describes the linearized Ricci scalar propagating on the background of Kerr black hole. In the previous work [30], we have replaced δR by a scalaron δA which could be interpreted to be a massive scalar in the scalar-tensor theory. This result is meaningful only if the scalaron approach (the scalar-tensor theory) represents f ( R ) gravity truly. However, it is noted that the linearized Ricci scalar by itself is regarded as a physically propagating scalar because the f ( R ) gravity includes a massive scalar graviton with single DOF. In Table 1, we list m 2 f for viable f ( R ) models. In order to not have a tachyonic scalar, it should be positive ( m 2 f > 0) which implies that f ' (0) > 0 and f '' (0) > 0. Thus, one requires either λ < 0 or n < 0 for the Starobinsky model ( f S ) [40]. Also, 0 < c 1 < 1 is required for the n = 1 Hu-Sawiciki model ( f n =1 HS ) and c 1 < 0 for the n = 2 Hu-Sawiciki model ( f n =2 HS ) [41]. However, these are not mandatory to explain the accelerating universe when one uses viable f ( R ) gravity [3]. Reminding the axis-symmetric background (7), it is convenient to separate the linearized Ricci scalar into [43] where S m l ( θ ) are spheroidal angular functions with l the spheroidal harmonic index and m the azimuthal harmonic number. Also, we choose a positive frequency ω of the mode here. Plugging (20) into the linearized massive equation (19), one has the angular and radial equations for S m l ( θ ) and u ( r ) where A lm is the separation constant whose form is given by [44, 22] for ω /similarequal m f . The radial Teukolsky equation takes the Schrodinger form where the tortoise coordinate y is defined by dy = r 2 + a 2 ∆ dr and a ω -dependent potential V ω ( r ) is given by Its asymptotic form is given by and its form near the event horizon is Here we impose the two boundary conditions of purely ingoing waves near the horizon and a decaying (bounded) solution at spatial infinity. These are known to be boundary conditions for quasibound states [20]. Near the horizon and at the spatial infinity, the linearized Ricci scalar takes the form Then, we may choose an ingoing mode near the horizon From (29), a bound state of exponentially decaying mode at spatial infinity is characterized by the condition The three boundary conditions (28)-(30) imply a discrete set of resonances { ω n } which corresponds to bound states of the linearized Ricci scalar. In addition, let us consider a wave of e -iωt e imφ with m > 0 and real ω which is propagating into a rotating black hole with angular velocity Ω. If the frequency of the incident wave satisfies the condition [16] then the scattered wave is amplified. This is called the superradiance condition for a bosonic field [44]. The existence of superradiant modes can be converted into an instability of the black hole background if a mechanism to trap these modes in a vicinity of the black hole is provided. There are two mechanisms to achieve it. If one surrounds the black hole by putting a reflecting mirror, the wave will bounce back and forth between black hole and mirror, amplifying itself each time and eventually producing a nonnegligible backreaction on the black hole background. This yields an exponentially growing mode which can be no longer considered as a perturbation, demonstrating the instability of the black hole. Secondly, the nature may provide its own mirror when one introduces a massive scalar. Press and Teukolsky have suggested to use this mechanism to define the black-hole bomb [16] by introducing a massive scalar with mass M propagating around the Kerr black hole with mass M . For ω < M ( ω 2 < M 2 ), the mass term works as a mirror effectively. The maximum growth rate for the instability is associated with modes with ω = ω R + iω I . The sign of ω I usually determines whether the solution is decaying ( ω I < 0) or growing ( ω I > 0) in the time evolution. It was shown that ω I M ∼ 6 × 10 -5 for mirrorlike boundary conditions [21] and ω I M ∼ 1 . 72 × 10 -7 for massive scalars [25]. Here the growth time scale is given by τ = 1 /ω I . More explicitly, according to the Hod's argument [45], two ingredients are necessary to trigger the instability of the Kerr black hole when one uses a massive scalar perturbation: 1) The existence of an ergoregion where superradiant amplification of the waves takes place. 2) The existence of a trapping potential well ( ∼ ) for quasibound states is between the potential barrier from ergoregion and potential barrier from the mass (see Fig. 15 of Ref.[12] and Fig. 7 of Ref.[46]). The first ingredient is usually implemented by the superradiance condition (31). The second ingredient is supplied by the condition of the bound states for modes in the regime Combining (31) with (32), one finds a restricted regime for the mass which implies an inequality between mass m f of the Ricci scalar and the angular velocity Ω of the rotating black hole which is the main result of our work. The bound (34) is reminiscent of the Gregory-Laflamme s -mode instability [47] for a massive spin-2 graviton with mass µ propagating on the spherically symmetric Schwarzschild black hole spacetimes (mass 2 M S = r 0 ). Choosing the transverse-traceless gauge of ¯ ∇ µ h µν = 0 and h = 0, its linearized equation takes the form which describes 5 DOF of a massive spin-2 propagating on the Schwarzschild black hole spacetimes. To this end, we would like to mention that the stability of the Schwarzschild black hole in four-dimensional massive gravity is determined by using the Gregory-Laflamme instability of a five-dimensional black string. It turned out that the small Schwarzschild black holes in the dRGT massive gravity [31, 32] and fourth-order gravity [33] are unstable against the metric and Ricci tensor perturbations because the inequality is satisfied as On the other hand, the dynamics of Ricci scalar with mass m f is expected to be stable in the complementary regime Similarly, the massive graviton is stable if it propagates around the large Schwarzschild black hole which satisfies the bound [48]", "pages": [ 6, 7, 8, 9 ] }, { "title": "4 Discussions", "content": "We have investigated the stability of a rotating black hole in the viable f ( R ) gravity explicitly. Even though viable f ( R ) gravity is promising to describe the current accelerating universe, it does not have a room to accommodate a rotating black hole because the Kerr black is unstable against the Ricci scalar perturbation. This superradiant instability (the black-hole bomb) arose from the nature of f ( R ) gravity which provides a massive scalar graviton with single DOF, in addition to a massless spin-2 graviton with 2 DOF. This implies strongly that the Kerr black holes do not exist in f ( R ) gravity and/or they do not form in the process of the f ( R ) gravitational collapse [31]. On the other hand, we expect from the scalar-tensor theory [34] that the Schwarzschild black hole is stable against the Ricci scalar perturbation in viable f ( R ) gravity because it is a nonrotating black hole. We summarize the type of black hole instabilities found in the dRGT massive gravity, fourth-order gravity, and f ( R ) gravity in Table 2. Let us compare the instability condition of Kerr black hole in f ( R ) gravity with the instability condition of Schwarzschild black hole in dRGT massive gravity and fourth-order gravity. The instability of the Schwarzschild black hole in fourdimensional massive gravity is determined by using the Gregory-Laflamme instability of a five-dimensional black string. The two conditions of µ ≤ 0 . 438 M S and m 2 ≤ 1 2 M S imply that the small Schwarzschild black holes in the dRGT massive gravity [31, 32] and fourth-order gravity [33] are unstable against the s -mode metric and Ricci tensor perturbations. These instabilities arose from the massiveness of s -mode spin-2 graviton propagating on the nonrotating small black hole with mass M S . On the other hand, the condition of m f < √ 2 m Ω arose from the massiveness of spin-0 graviton with azimuthal number m propagating on the rotating black hole with angular velocity Ω. Even though the massiveness is a common factor for both instabilities, the phenomena of the instability are different: GL black string instability and black hole bomb. Finally, we conclude that the massive graviton instabilities are quite different from the Regge-Wheeler-Zerilli stability for a massless graviton [13, 14, 17]. It suggests that a massive gravity is hard to possess the black hole.", "pages": [ 10 ] }, { "title": "Acknowledgement", "content": "This work was supported supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.2012- ≤ ≤ R1A1A2A10040499).", "pages": [ 10, 11 ] } ]

[ { "title": "Large scale cosmic perturbation from evaporation of primordial black holes", "content": "Tomohiro Fujita, 1 Keisuke Harigaya, 1 and Masahiro Kawasaki 2,1 We present a novel mechanism to generate the cosmic perturbation from evaporation of primordial black holes. A mass of a field is fluctuated if it is given by a vacuum expectation value of a light scalar field because of the quantum fluctuation during inflation. The fluctuated mass causes variations of the evaporation time of the primordial black holes. Therefore provided the primordial black holes dominate the Universe when they evaporate, primordial cosmic perturbations are generated. We find that the amplitude of the large scale curvature perturbation generated in this scenario can be consistent with the observed value. Interestingly, our mechanism works even if all fields that are responsible for inflation and the generation of the cosmic perturbation are decoupled from the visible sector except for the gravitational interaction. An implication to the running spectral index is also discussed.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Recent observations of the cosmic microwave background radiation determine the cosmological parameters with increased accuracy and hence we know the amplitude and the tilt of the initial cosmological perturbations on the large scale [1]. However, the mechanism which generates such perturbations is still unknown. Even if one assumes that inflation occurs and it stretches the quantum fluctuations of light scalar fields into the cosmological ones [2], what actually produces the observed perturbations is quite obscure. For example, in curvaton [3] or modulated reheating [4] scenarios, the scalar field which is responsible for the large scale cosmic perturbations is different from the inflaton itself. In addition, all known scenarios, such as single field inflation, curvaton and modulated reheating, are not necessarily the only possibilities to be considered. Thus it is important to investigate another feasible mechanism generating the perturbation, in order to understand what actually happened in the early Universe. Primordial black holes (PBHs) are black holes which formed in the early Universe. After the pioneer work by Zel'dovich and Novikov [5] in 1966, PBHs have attracted much attention for a long time. Now it is known that PBHs can be produced by large density perturbations due to inflation or preheating [6] as well as a sudden reduction in the pressure [7], bubble collisions [8], collapses of cosmic strings [9], and so on [10]. PBHs with small masses, M BH /lessorsimilar 10 15 g, evaporate until today by the Hawking radiation [11] and lead to rich phenomenologies including entropy productions, baryogenesis [12] and neutrino radiations [13]. On the other hand, PBHs with large masses, M BH /greaterorsimilar 10 15 g, survive today and they can be the candidate of cold dark matter and might affect the large scale structure [14], possibly seeding the supermassive black holes [15]. However, although a number of earlier works investigate the possible roles of PBHs, PBHs have never been studied as the origin of the cosmic perturbation. Before showing detailed calculations, let us briefly explain the basic idea of our mechanism where the fluctuation of the PBH evaporation time generates cosmic perturbations: First of all, we assume that one field ( ψ ) acquires its mass m ψ from another field (light scalar field φ ) by the Higgs mechanism. Then the mass of ψ , m ψ , is fluctuated if the field value of the light scalar field φ is also fluctuated by inflation. Second, a lifetime of a PBH depends on m ψ if m ψ is larger than the initial Hawking temperature T BH of the PBH. It can be understood step by step. Since the Hawking temperature T BH is inversely proportional to the PBH mass M BH , T BH increases as M BH decreases due to the Hawking radiation. Next, the PBH can emit particles whose masses are smaller than T BH . Thus the number of particle species in the Hawking radiation rises as T BH increases. Moreover the energy loss rate of the PBH is proportional to the degree of freedom of the radiated particles. Therefore the energy loss of the PBH accelerates when T BH exceeds m ψ . As a result, the variation of m ψ causes the variation of the time of the PBH evaporation. Finally, if PBHs dominate the Universe, the fluctuation of the PBH evaporation time is nothing but the fluctuation of the (second) reheating time. 1 Thus cosmic perturbations are generated via the PBH evaporation. This novel mechanism has several interesting features. First, the generation of perturbations by this mechanism is general because it is natural to expect that an unknown particle acquires its mass by the Higgs mechanism, if some symmetry forbids the mass term. Second, this mechanism is still viable even if all fields which are relevant to inflation and the generation of the cosmic perturbations are decoupled from the visible sector except for the gravitational interaction. It is because particles in the visible sector are emitted by the Hawking radiation even if PBHs are formed from an invisible sector. Third, the PBH dominated universe leads to the rich phenomenologies as we have mentioned. It would be interesting to explore these scenarios in combination with our mechanism. In this article, however, we focus on the generation mechanism of cosmic perturbations via PBHs. This article is organized as follows. First, we explain the generation mechanism in detail and show that the observed magnitude of the cosmic perturbations can be realized. Next, we make a prediction of the running spectral index when the PBHs are produced due to a bluetilted perturbation generated by the inflaton. We show that the negatively large running is predicted. The final section is devoted to conclusions and discussion.", "pages": [ 1, 2 ] }, { "title": "II. PERTURBATION OF PBH LIFETIME", "content": "In this section, we compute the curvature perturbation produced by the PBH evaporation when a mass of a field is fluctuated by the Higgs mechanism. A PBH loses its mass due to the Hawking radiation [11] which is characterized by the Hawking temperature, where M BH is the mass of the PBH and M Pl ≈ 2 . 43 × 10 18 GeV is the reduced Planck mass. The mass loss rate is given by where g ∗ is the effective degrees of freedom of the radiated particles and r s ≡ M BH / 4 πM 2 Pl is the Schwarzschild radius. 2 Let us consider a case where a field ψ is added to the standard model. We define m ψ as the mass of ψ and g ψ as its effective degrees of freedom. Then g ∗ varies approximately as where g sm = 106 . 75 is the total degrees of freedom in the standard model. One can find the solution of Eq. (2) under Eq. (3) as where m ψ is assumed to be larger than the initial Hawking temperature T 0 . Otherwise, ψ is emitted from the beginning and then τ does not depend on m ψ . On the other hand, ψ makes a small difference if m ψ /greatermuch T 0 because a black hole evaporates rapidly at t /similarequal τ . Next, we introduce a scalar field φ and assume that ψ is a fermion field. We suppose that the interaction between φ and ψ is given by the Yukawa coupling and it gives ψ the mass m ψ , where y is a Yukawa coupling constant. Provided that ψ does not acquire any significant mass except for Eq. (5), the fluctuation of φ which is generated during inflation perturbs the mass of ψ . Then it causes the perturbation of the PBH lifetime τ through Eq. (4). One can find that the perturbation of τ , where δφ is the fluctuation of φ and the contribution of φ to g ∗ is ignored. Note that in order for δφ to survive until the PBH evaporation, φ should not gain a large thermal mass by interactions with the radiation which is originated from the inflaton. Let us evaluate the resultant curvature perturbation. Here we assume that the PBHs dominate the Universe before their evaporation. The Universe is in the matter dominated era before the PBH evaporation and enters the radiation dominated era after that. Then the curvature perturbation ζ generated at the evaporation of the PBHs can be calculated in the same way as the modulated reheating cases in which ζ is given by ζ MR = -δ Γ / (6Γ), where Γ is the decay rate of the inflaton [4]. We derive this formula in the Appendix. Since in the PBH evaporation case Γ corresponds to the inverse of the PBH lifetime τ -1 , ζ generated by the PBH lifetime perturbation is given by where τ sm /greatermuch δτ is assumed. Provided that φ is light during inflation and the power spectrum of its fluctuation is P δφ = H inf / 2 π , the power spectrum of the curvature perturbation is given by where H inf is the Hubble parameter during inflation. Although several generation mechanisms of PBHs are proposed, we simply assume that the PBHs form right after the end of inflation without specifying a concrete model. In that case, the typical mass of the PBHs is evaluated as where M horizon denotes the energy density in the horizon volume at the end of inflation and γ ≈ 0 . 2 is the numerical factor representing the effect that the pressure of radiations prevents the structure formation [16]. The initial Hawking temperature is Substituting Eq. (10) into Eq. (8), we obtain When g ψ /lessmuch g sm , Eq. (11) is evaluated as Let us discuss whether the observed curvature perturbation, P 1 / 2 ζ ∼ 10 -5 , can be realized by the evaporation of the PBHs. To obtain the observed curvature perturbation, m ψ must be close to H inf . Otherwise, the Yukawa coupling y must be larger than the unitarity bound ∼ 4 π . However, the required coincidence of m ψ and H inf is not a fine-tuned one; it is just within a factor of few. The coincidence can be realized more naturally in the following way. If φ couples to several fields, as is the case for the standard model Higgs, it is not unnatural that one of these fields has a mass close to the Hubble scale within a factor of a few. Note also that the perturbation depends on the degree of freedom of the fermions, g ψ . If φ couples to several fermions with the same Yukawa coupling, which can be guaranteed by assuming a flavor symmetry among the fermions, the required closeness of the mass scales is relaxed. Before closing this section, let us note the condition in which the PBHs dominate the Universe prior to their evaporation. Provided that the PBHs form right after the end of inflation and the inflaton oscillation phase is negligible (instant reheating), the density parameter of the PBHs at their formation epoch, β ≡ Ω PBH ( t form ), is constrained as where Eq. (9) is used. If β is smaller than the lower bound in Eq. (13), the resultant curvature power spectrum P ζ decreases by a factor of (3Ω PBH ( τ ) / (4 -Ω PBH ( τ ))) 2 .", "pages": [ 2, 3 ] }, { "title": "IV. IMPLICATION TO THE RUNNING SPECTRAL INDEX", "content": "In this section, we briefly discuss the prediction of the running spectral index when the PBHs are produced due to a blue-tilted perturbation from the inflaton. To be concrete, we discuss this based on the hybrid inflation [17], in which a blue-tilted spectral is easily obtained. 3 We assume the following standard hybrid inflaton potential: where s is the inflaton and · · · includes the interactions with the waterfall sector. With a simple calculation, we obtain the relation where P inf ζ , η and N are the curvature perturbation generated by the inflaton, the second slow-roll parameter and the number of e-foldings, respectively. The indices ∗ and e denote that the value is evaluated at the horizon exit of the scale of the interest and the end of the inflation, respectively. Note that the curvature perturbation should be large enough at the small scale in order to produce the PBHs, and should be small at the large scale, Therefore we obtain the lower bound for η , Such large η is natural in the supergravity theory [18]. On the other hand, the spectral index n s of the perturbation generated by the PBH evaporation is given by n s = 1 -2 /epsilon1 ∗ , since the perturbation is originated from a light field φ other than the inflaton. If the mass of φ is so large that it affects the spectral index, φ would begin an oscillation before the evaporation of the PBHs. To obtain the value consistent with the Planck results [1], n s = 0 . 9607 ± 0 . 0063 (95% C.L.), /epsilon1 ∗ = 0 . 020 ± 0 . 003 is required. In the end, we obtain the prediction of the running of the spectral index, Thus, a large running is easily obtained.", "pages": [ 3 ] }, { "title": "IV. CONCLUSIONS AND DISCUSSION", "content": "In this article, we have proposed a new generation mechanism of cosmic perturbations from the evaporation of PBHs. It has been shown that the mechanism is compatible with the observed magnitude of the curvature perturbation. The implication to the running spectral index has also been discussed. As has been mentioned in the Introduction, the generation mechanism of cosmic perturbations from the evaporation of PBHs has an interesting feature. Even if all fields responsible for inflation and cosmic perturbations are decoupled from the visible sector, this mechanism is viable.", "pages": [ 4 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "This work is supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 25400248 (M.K.), No. 21111006 (M.K.) and also by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. T.F. and K.H. acknowledge the support by JSPS Research Fellowship for Young Scientists.", "pages": [ 4 ] }, { "title": "APPENDIX: CURVATURE PERTURBATION AND THE EVAPORATION RATE", "content": "In this appendix, we derive the formula for the curvature perturbation when the evaporation rate of the PBHs, Γ, fluctuates, with the aid of the so-called δN formula [19]. We assume that the PBHs dominate the Universe before their evaporation. We take a flat time slice t i well before the PBHs evaporate but well after the PBHs dominate the Universe as the initial time slice. We also take a uniform density time slice t f well after the PBHs evaporate as the final time slice. Note that the Universe is in the matter dominated era before the PBH evaporation and enters the radiation dominated era after that. Therefore, the number of e-foldings between the two slices is given by where τ is the lifetime of the PBHs, τ = Γ -1 . By taking a variation of Eq.(19), we obtain which is used in Eq.(7)", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "We extend our recent work on the quasilocal formulation of conserved charges to a theory of gravity containing a gravitational Chern-Simons term. As an application of our formulation, we compute the off-shell potential and quasilocal conserved charges of some black holes in threedimensional topologically massive gravity. Our formulation for conserved charges reproduces very effectively the well-known expressions on conserved charges and the entropy expression of black holes in the topologically massive gravity.", "pages": [ 1 ] }, { "title": "Quasilocal Conserved Charges with a Gravitational Chern-Simons Term", "content": "Wontae Kim ab ∗ , Shailesh Kulkarni ac † and Sang-Heon Yi a ‡", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Conserved charges in general relativity is very important and rather subtle concept. The main obstacle is related to the construction of the completely generally covariant energy-momentum tensor for gravitational field. Many people, including Einstein himself, have unsuccessfully tried to find such a tensor or to construct its alternatives, for instance, energy momentum pseudotensor [1]. Now there is a general consensus that such a construction does not exist, since the local conservation law for general relativity turns out to be meaningless. However, several approaches have been suggested to construct total conserved charges in general relativity, one of which is the formalism accomplished by Arnowit-Deser-Misner(ADM) [2]. This approach uses a linearization of metric around the asymptotically flat spacetime and becomes cumbersome for the gravity actions which contain higher curvature or higher derivative terms. An extension of ADM formalism to higher curvature theories of gravity - known as the Abbott-Deser-Tekin(ADT) formalism - was provided in [3, 4]. Unlike the ADM formalism, the ADT method is covariant and also applicable to the asymptotically AdS geometry. There also exist other approaches to conserved charges, which are based on quasilocal concepts (for review, see [5]). One of such formulations is the Brown-York formalism [6] which needs to be improved for asymptotic AdS space by introducing the appropriate counter terms [7]. This formulation has been especially useful in the context of the AdS/CFT correspondence. Another such a formulation is known as the Komar integrals [8] which is not known to be completely consistent with the results in the existing literatures. For instance, the mass and angular momentum calculated via Komar integrals contain the well-known factor two discrepancy when compared to ADM formalism. In the covariant phase space approach, initiated by Wald the conserved charges were computed by using the Noether potential [9, 10, 11, 12]. Wald's formulation has a distinct advantage in that it holds for any generally covariant theory of gravity and captures the entropy of black holes which can be regarded as the natural extension of Beckestein-Hawking area law [13]. Furthermore, this method established the first law of black hole thermodynamics in any covariant theory of gravity. There exists an interesting connection between the on-shell ADT potential and the linearized on-shell Noether potential. Indeed, it was observed that at the asymptotic boundary, the linearized Noether potential around the on-shell background (which solves the Einstein equations), when combined with the surface term, produces the known expression for the ADT potential [14, 15, 16, 17]. This relation, although very interesting, is indirect and shown to hold in Einstein gravity only. In our recent work [18], we have provided a non-trivial generalization of the above connection to any covariant theory of gravity. This was achieved by elevating the ADT potential to the off-shell level. Then, by using the corresponding off-shell expression for the linearized Noether potential supplemented with the surface term, we were able to show the desired connection directly. Integrating the resultant expression for the ADT potential along the one parameter path in the solution space, we finally obtained the expression for the quasilocal conserved charges which is identical with the one given by Wald's covariant phase space approach. At the on-shell level, this result establishes that our construction through the quasilocal extension of the ADT formalism is completely equivalent to the covariant phase space formalism which encompasses the black hole entropy. There are certain aspects of our formalism which we would like to highlight at this stage. We may recall that in the conventional analysis of ADT potentials/charges one has to use the equations of motion for the background. As a result, the procedures become highly complicated when the higher curvature or higher derivative terms are present in the Lagrangian. On the other hand, our formalism uses the off-shell (or background independent) expression for ADT and Noether potentials which are shown to be related in a one to one fashion. One can exploit this correspondence to obtain the conserved ADT charges for any covariant theory of gravity in a more efficient way. The off-shell Noether potential has already been used in the literatures in somewhat different ways. For instance, the entropy for black holes was computed from the off-shell Noether potential [19, 20, 21]. In another work [22], we have used the off-shell Noether potential to compute the entropy for rotating extremal as well as non-extremal BTZ black holes in new massive gravity coupled to a scalar field. We have also computed the angular momentum of hairy AdS black holes and shown its invariance along the radial direction. This fact was used to verify the holographic c-theorem for hairy AdS black holes. Most of the studies on constructing the Noether potential and the corresponding conserved charges have been limited to covariant theories of gravity. There are some attempts to generalize the Wald's formalism to the apparently non-covariant Lagrangians which often include gravitational Chern-Simons terms. Gravitational Chern-Simons terms are closely related to anomaly and appear frequently in the string theory context. Moreover, it has important implications in the AdS/CFT correspondence. One of such a theory with a gravitational Chern-Simons term is the three-dimensional topologically massive gravity(TMG) proposed by Deser, Jackiw and Templeton [23]. The extension of the Wald's procedure to the TMG, especially for the black hole entropy, was provided by Tachikawa in [24]. The entropy computed from this approach matches exactly with the one obtained by indirect ways [25, 26, 27, 28, 29, 30]. This on-shell approach was extended in conjunction with the covariance of black hole entropy [31]. On the other hand, the mass and angular momentum for the non-covariant theories like TMG have been obtained independently of the entropy, for instance, by using the ADT formalism [32, 33], by the canonical method [34] or by using the direct codified computer implementation [35] of the formalism given in [14]. Another interesting aspect of TMG is the existence of warped AdS black hole solutions. These solutions with central charge expressions were a starting point for the warped AdS/CFT correspondence [36] and the Kerr/CFT correspondence [37], which may be extended to the dS/CFT case [38, 39, 40]. Interestingly, the exact relationship among the ADT charges and the Noether potential is still missing for TMG. At first glance, the Noether potential introduced in our previous paper [18] becomes non-covariant for an apparently non-covariant Lagrangian like TMG. On the contrary, the ADT potential is completely covariant since its construction is based on the covariant equations of motion. Therefore, it is not clear that the formalism given in our previous paper can be extended to this case. In the present work we would expedite this apparently non-covariant case and show that the formalism works well even in this case. As a specific example, we will take the topologically massive gravity to elaborate our formalism. This paper is organized as follows. In the next section we propose a general framework for calculating quasilocal conserved potentials and charges for a theory of gravity of the apparently non-covariant Lagrangian. We then implement this procedure to TMG and show that our formalism matches completely with the covariant phase space approach to TMG. The quasilocal mass and angular momentum for the rotating Banados-Teitelboim-Zanelli(BTZ) [41] and warped AdS black holes [42] are computed in section 3. Finally, we summarize our findings in section 4.", "pages": [ 2, 3, 4 ] }, { "title": "2.1 Formalism", "content": "In this section we extend our formulation of quasilocal conserved charges developed in [18]. First, we obtain a generic expression for the off-shell Noether current. This current, apart from the usual covariant terms, involves a non-covariant piece. This means that the off-shell Noether current itself is not a covariant quantity and so does not have a good physical interpretation just like Levi-Civita connection. However, it turns out that its linearized expression is related to the off-shell covariant ADT potential and so it can be used for the computation of conserved charges through the one-parameter path on the on-shell solution space. Let us consider an action which contains the apparently non-covariant term. The variation of the action with respect to g µν will be taken by where E µν = 0 denotes the equations of motion (EOM) for the metric and Θ denotes the surface term. Note that the surface term Θ becomes non-covariant since we are considering the apparently non-covariant Lagrangian, though E µν is a covariant expression. Under the diffeomorphism denoted by the parameter ζ , the Lagrangian transforms as where Σ µ term denotes an additional non-covariant term when the Lagrangian contains a noncovariant term like the gravitational Chern-Simons term. In this case, the identically conserved current can be introduced as Unlike the covariant Lagrangian case, this off-shell Noether current J µ , and the potential K µν are not warranted to be covariant. This is naturally expected, since the Lagrangian L , the surface term Θ and the boundary term Σ, all take the non-covariant forms. Just as in the covariant case, there are some ambiguities in the form of the Noether potential K , which turn out not to affect the final expression for quasilocal conserved charges. In contrast, the ADT potential [3, 4, 43, 44, 45] is introduced in a completely covariant way. The on-shell ADT current is introduced for a Killing vector ξ µ as J µ = δ E µν ξ ν , which can be shown to be conserved by using EOM, Bianchi identity and the Killing property of ξ . Then, the ADT potential Q is introduced by J µ = ∇ ν Q µν . Since these on-shell current and potential, which use the EOM, are highly involved for a higher curvature or derivative theory of gravity, the background independent ADT current and potential were used for TMG [33] and new massive gravity [46]. In Ref. [18], we have realized the importance of the identically conserved ADT current and extended its use to a generic case. Explicitly, the off-shell ADT current and its potential for a Killing ξ can be defined by which can be shown to be conserved identically by using the Bianchi identity and the Killing property of ξ without using EOM. Since this off-shell ADT potential is based on the covariant EOM, it takes the covariant form even for the apparently non-covariant Lagrangian. This covariant nature of the ADT potential may lead some worries about the inapplicability of our formalism to the apparently non-covariant case. However as we shall see below, the formalism can be extended successfully even to such a case. For matching the linearized off-shell Noether potential and the off-shell ADT potential, the diffeomorphism parameter ζ will be taken as a Killing vector ξ in the following. To extend the formalism in Ref. [18] to this case, let us introduce the formal Lie derivative for non-covariant Θ term as Note that this Lie derivative satisfies the Leibniz rule. This Lie derivative of a non-covariant quantity is different from its diffeomorphism variation, which is not the case for a covariant one. Let us denote the difference between the Lie derivative and the diffeomorphism variation of (non-covariant) Θ-term as By using the property of the Θ-term [11, 47] for a Killing vector ξ and introducing Ξ µν as one can see that This is the extension of the quasilocal formula for conserved charges in the covariant Lagrangian case to the apparently non-covariant one. The left hand side of the above equation is covariant by construction (see Eq. (4)), while each term in the right hand side is not warranted generically to be covariant. One may note that the additional term Ξ µν is responsible for the covariantization of the right hand side. We would like to emphasize again that this quasilocal ADT potential is defined only up to the total derivative of a certain antisymmetric tensor U µνρ just in the covariant case. This ambiguity does not affect the final expression for the conserved charges since it is a total derivative under the integral over aclosed subspace. By using the above quasilocal ADT potential and using the one-parameter path in the solution space, just like the covariant case, one can introduce conserved charges for the Killing vector ξ as We would like to emphasize that the background and the variation are on-shell in the end, since we have taken the path in the solution space. The on-shell conservation of Q µν ADT has been used for construction of conserved charges. Using Eq. (9), the conserved charge Q ( ξ ) can be obtained through the Noether potential and surface terms as where ∆ K denotes the finite difference of K -values between two end points of the one-parameter path in the solution space. The right hand side in Eq. (11) can be regarded as the extension of the covariant phase space expression to the apparently non-covariant Lagrangian case, which was done at the on-shell level in [24]. On the bifurcate Killing horizon H , the second term in the right hand side would vanish and the final expression gives us the well-known Wald's entropy as ( κ/ 2 π ) S = Q H . Our construction shows that the conventional ADT charges should agree exactly with those from the covariant phase space approach. Conversely speaking, the quasilocal extension of the ADT formalism can reproduce the Wald's entropy for black holes. One of the lessons in this formulation is that the ADT charges and Wald's entropy do not need to be computed independently. Rather, they are directly related in our formulation and should be consistent with the first law of black hole thermodynamics by construction, as was shown by Wald [9, 11].", "pages": [ 4, 5, 6, 7 ] }, { "title": "2.2 Gravitational Chern-Simons term", "content": "In this section we apply our formulation of quasilocal conserved charges to a specific example: TMG in three dimensions. It turns out that the ADT potential can be obtained in a very concise form and consistent with the previously known results. Let us take the action for TMG in three dimensions [23] as The last term for the gravitational Chern-Simons term is given by The equations of motion for TMG are given by where /epsilon1 -tensor is defined such that √ -g /epsilon1 012 = 1. Our convention for the curvature tensor is taken as [ ∇ µ , ∇ ν ] V ρ = R ρ σµν V σ and the mostly plus metric signature is employed. where G µν denotes Einstein tensor and C µν denotes the Cotton tensor defined by The above Cotton tensor is traceless, symmetric, and divergence-free, which is the threedimensional analog of the Weyl tensor. One may note that it can also be written as C µν = /epsilon1 αβ ( µ ∇ α R ν ) β . In the following, we use h µν for the linearized metric interchangeably with δg µν and all the indices are raised and lowered by the background metric g . The quasilocal ADT potential for the Ricci scalar part has been known to be given by which can also be derived from the quasilocal ADT formalism given in [18]. Since the construction has already been done for the covariant terms, let us focus on the gravitational Chern-Simons term in the following. The surface term for the gravitational Chern-Simons term under a generic variation turns out to be Note that the surface Θ-term is non-covariant though the EOM is covariant. Under a diffeomorphism with a parameter ζ , Christoffel symbol transforms as where L ζ denotes the Lie derivative defined in the same way with the Θ-term as Then, one can see that L CS transforms under diffeomorphism as where the additional boundary term Σ CS is given by The surface term for this diffeomorphism is given by One may note that the above Σ-term and the Θ-term have some ambiguities. Nevertheless, those do not affect our essential steps and so the above explicit expressions are taken for definiteness. According to the generic formulation given in Eq. (3), the off-shell current and Noether potential for a gravitational Chern-Simons term are introduced by By using three-dimensional identities given in the Appendix, one can obtain the off-shell Noether potential in the form of 1 The additional term Ξ µν CS can be shown to be given by Collecting the above results, one can obtain the contribution of the gravitational Chern-Simons term to conserved charges and the entropy of black holes. First, let us consider the contribution of the Chern-Simons term to the entropy of black holes. By using Eq. (9) with the on-shell background metric and taking ζ as the Killing vector ξ H for the Killing horizon H , one can show that the contribution of the Chern-Simons term is given by where we have used the property of the bifurcate Killing horizon such that ξ vanishes on H . This expression is completely covariant and can be integrated into a finite form which is consistent with the one obtained in the covariant phase space approach [24, 31]. Now, by using our relation given in Eq.(9), one can obtain the quasilocal ADT potential for the three-dimensional gravitational Chern-Simons term as 2 Note that this expression is completely covariant as was shown generically to be the case in the previous section. We would like to compare our results to the previously known expressions of the ADT potential for the gravitational Chern-Simons term. To achieve this goal, let us introduce the totally antisymmetric tensor U µνρ as Using the Killing property of ξ and the identities given in the Appendix, one can show that U µνρ satisfies As a result, one can verify that the above expression of the ADT potential for the gravitational Chern-Simons term Q µν CS can be rewritten as 3 where η is defined by η µ ≡ 1 2 /epsilon1 µαβ ∇ α ξ β . This computation shows us explicitly the equivalence of our expression of the background independent or off-shell ADT potential to the one given in [33]. Though our expression of the off-shell ADT potential is more succinct and illuminating, we would like to emphasize that we can use Eq. (11) for the computation of conserved charges instead of the explicit expression of the ADT potential. Using Eq. (11), one can also obtain the entropy of black holes in TMG at one stroke. In order to apply Eq. (11) to black hole solutions in TMG in the next section, let us summarize what we have computed. In TMG, the off-shell Noether potential, Θ-term, and Ξ-term are given by It is interesting to note that each of the above expressions are non-covariant, as expected.", "pages": [ 7, 8, 9, 10 ] }, { "title": "3 Black holes and their charges", "content": "In this section, we compute the mass and angular momentum of some black holes in TMG as the simplest example of our formulation. Since our formulation was shown to give us the background independent ADT potential which is equivalent to the previously known expression in [33], the mass and angular momentum for black holes 4 in TMG are assured to be given by the same expression. However, it is illuminating and fruitful to reproduce these results by using the expression of conserved charges given in Eq.(11). In all the given examples, upper index components of relevant Killing vectors are taken to be constant and so Ξ-term contribution vanishes. In our convention the metric of the BTZ black hole [41] is taken in the following form of The Killing vectors for the time-translational and rotational symmetry will be chosen as ξ = ∂ L∂t , ∂ ∂θ , respectively. To utilize the formula given in Eq.(11), take an infinitesimal parametrization of a one-parameter path in the solution space as follows By expanding the above BTZ metric in terms of dr ± and keeping terms up to the relevant order, one can obtain the infinitesimal expression of the Θ-term. And then, one can integrate this expression to obtain conserved charges. Let us consider the quasilocal angular momentum of the BTZ black hole at first. After a bit of computation, one can see that, just like the covariant case, the quasilocal angular momentum for the rotational Killing vector ξ R = ∂ ∂θ comes entirely from the ∆ K -term, of which the relevant component is As a result, the angular momentum of the BTZ black hole is given by By noting that the nonvanishing components of the infinitesimal Θ-term and the ∆ K -term for a Killing vector ξ T = 1 L ∂ ∂t are one can show that the mass of the BTZ black hole in TMG is given in the form of These expressions match completely with the known results. Note that our convention is such that the first law of black hole thermodynamics holds in the form of dM = T H d S BH -Ω dJ . Now, let us consider the warped AdS black hole in TMG, of which expressions for the mass and angular momentum are rather involved. The metric of the warped AdS black hole may be taken as [42] where Z 2 ≡ ρ 2 +2 ωρ +(1 -β 2 ) ω 2 + β 2 ρ 2 0 / (1 -β 2 ). Two of the four parameters in the above metric, β and ζ , are related to the Lagrangian parameter 1 /L 2 and 1 /µ as follows The other two parameters ω and ρ 0 are related to the mass and angular momentum of this black hole. In this case one can choose the infinitesimal one-parameter path in the solution space as 5 As in the case of the BTZ black hole, it is sufficient to keep various terms up to linear parts of the variations. Then, the quasilocal conserved angular momentum for the rotational Killing vector ξ R = ∂ ∂θ can be shown to come entirely from the ∆ K µν -term, while the quasilocal mass for the timelike Killing vector ξ T = ∂ ∂t has another contribution from the Θ-term. Let us consider the angular momentum of the warped AdS black hole at first. By using the relevant component of the ∆ K -term for the rotational Killing vector ξ R one can obtain the quasilocal angular momentum of the warped AdS 3 black hole as Now, let us turn to the mass of the black hole for the timelike Killing vector ξ T . In this case the nonvanishing component of the infinitesimal Θ-term for the above chosen path turns out to be By combining this with the ∆ K contribution one can see that mass is given by Note that the above expressions for the mass and angular momentum match completely with those given in [33] up to sign convention for angular momentum. (See [36, 50, 51] for a dual CFT interpretation for these black holes.)", "pages": [ 10, 11, 12 ] }, { "title": "4 Conclusion", "content": "In this paper we have extended our previous formalism for quasilocal conserved charges to a theory of gravity with a gravitational Chern-Simons term. This formulation turns out to be very effective to obtain the ADT potential and quasilocal charges. In fact, we have shown that this quasilocal extension of the ADT method even to an apparently non-covariant Lagrangian is completely equivalent to the covariant phase space approach. We have explicitly verified that this formulation reproduces the known background independent ADT potential for TMG up to the irrelevant total derivative of a totally antisymmetric tensor. Furthermore, quasilocal conserved charges for the BTZ black holes and the warped AdS black holes are reproduced, which are completely consistent with the previously known results. It would be very interesting to develop this formulation further to encompass the asymptotic Killing vectors, which is relevant to the construction of the asymptotic Virasoro algebra in the context of the AdS/CFT, Kerr/CFT and dS/CFT correspondence. This would allow us to extract the information of the central charge and eventually the black hole entropy. Another interesting direction would be to use the off-shell Noether potential, K µν (see Eq. (29)) in the stretched horizon approach developed by Carlip [52]. This will lead to the near horizon Virasoro algebra and the entropy of black holes.", "pages": [ 12, 13 ] }, { "title": "Acknowledgments", "content": "We would like to thank B. Tekin for a useful correspondence. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) through the CQUeST of Sogang University with grant number 2005-0049409. W. Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MOE) (2010-0008359). S.-H.Yi was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No. 2012R1A1A2004410). S. Kulkarni was also supported by the INSPIRE faculty scheme (IFA-13 PH-56) by the Department of Science and Technology (DST), India.", "pages": [ 13 ] }, { "title": "Appendix", "content": "Here we shall give some identities and formulae which are useful in the text, especially in Section 2.2. In three dimensions we have the following identities We have used the following convention for /epsilon1 tensor and the integration measure A Killing vector ξ satisfies For a Killing vector ξ , let us introduce another vector field η formed by contracting the covariant derivative of ξ with the /epsilon1 -tensor Such a vector field η obeys Here, h µν and h represents the linearized metric and its trace, respectively. Another useful identity for the ∇ ρ U µνρ computation is", "pages": [ 14, 15 ] } ]

[ { "title": "Axisymmetric Dirac-Nambu-Goto branes on Myers-Perry black hole backgrounds", "content": "Viktor G. Czinner ∗ Centro de Matem´atica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal and HAS Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49, Hungary Stationary, D -dimensional test branes, interacting with N -dimensional Myers-Perry bulk black holes, are investigated in arbitrary brane and bulk dimensions. The branes are asymptotically flat and axisymmetric around the rotation axis of the black hole with a single angular momentum. They are also spherically symmetric in all other dimensions allowing a total of O (1) × O ( D -2) group of symmetry. It is shown that even though this setup is the most natural extension of the spherical symmetric problem to the simplest rotating case in higher dimensions, the obtained solutions are not compatible with the spherical solutions in the sense that the latter ones are not recovered in the non-rotating limit. The brane configurations are qualitatively different from the spherical problem, except in the special case of a 3-dimensional brane. Furthermore, a quasi-static phase transition between the topologically different solutions cannot be studied here, due to the lack of a general, stationary, equatorial solution. PACS numbers: 04.70.Bw, 04.20.Jb, 04.50.-h, 11.25.-w.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Possible interactions between branes and black holes in higher dimensions are interesting and important problems in many fields of modern theoretical physics. One direction, which has been recently introduced by Frolov [1], is a spherically symmetric black hole interacting with Dirac-Nambu-Goto (DNG) test branes [2] in arbitrary brane and bulk dimensions. This brane - black hole system, beyond the interest of its own, has also proven to be very useful as a toy model for various other problems. For example, it posses striking similarities in its properties to the problem of topology changing and merger transitions between higher dimensional black solutions [3-5], and also shows a self-similar behavior, very similar to the Choptuik critical collapse phenomenon [6]. Furthermore, it also turned out to be a relevant model in investigating holographic phase transitions in strongly coupled gauge theories [7, 8], via the gauge/gravity correspondence [9]. Generalizations to the system, by considering small thickness corrections to the branes, have also been studied lately by Frolov and Gorbonos [10], and more extensively (also within a more general framework) by us [1113]. The motivation for this extension was to consider higher order, curvature corrections to the thin brane action, which, in the holographic dual picture, correspond to finite 't Hooft coupling corrections, and provide a more realistic description of the phase transition [8]. In the present paper, as a sequel to our previous works on the subject matter [11-13], we provide another generalization of the problem into a different direction. We investigate the brane - black hole system in the rotating case by considering a Myers-Perry black hole in the background with a single angular momentum. The motivation of this work is also clear, we would like to understand the role that rotational effects play when a quasistatic, topology changing transition is considered in the system. The problem is interesting not only from the geometrical point of view, but also because it may provide further insights to other topology changing- or merger transition problems in higher dimensional, classical general relativity, or to certain holographic phase transitions in the gauge/gravity dual picture. In constructing the model, we follow the method of [1] as closely as possible, and define the DNG-branes with the highest possible symmetry properties that the background allows. By this construction the branes posses a total of O (1) × O ( D -2) group of symmetry, and just as in the spherical case, the brane action simplifies radically, resulting the problem of an ordinary differential equation (highly non-linear though) for the brane configurations. We present and analyze the general solution of this problem, first analytically in far distances, and later numerically in the near horizon region. As a result, we obtain that due to the coordinate parametrization of the Myers-Perry metric, this rotating problem is not compatible with the spherical results of [1], in the sense that the latter ones are not recovered in the non-rotating limit. Although the ideal situation would be to provide a rotating solution which is the 'corresponding' one to the spherical problem in the above sense, nevertheless we could not find an appropriate coordinate system in which this could be done in a natural way as presented by Frolov in [1], and as we also do it here. Consequently, we conclude that while the construction of the problem is the closest possible to the spherical case, the obtained results are qualitatively different, except in the special case of a 3-dimensional brane. Furthermore, we also find that stationary equa- torial solutions do not generally exist for arbitrary brane dimensions, except again for the case of a 3-dimensional brane, and as a result, we cannot study the quasi-static phase transition in the geometric setup as we did in the thickness corrected spherical case [11, 12] by following the method of Flachi et al. [14]. The plan of the paper is as follows. In section II. we define the rotating brane - black hole system analogous to the spherical case. In section III. we obtain the brane equation and discuss its incompatibility with the results of [1]. In section IV., first we discuss the analytic properties of the solutions in the near horizon region and derive unique boundary conditions for the topologically different solutions from regularity requirements. Then, we obtain the far distance solution in an analytic form, and analyze its properties. In section V. we present and illustrate the numerical results in the near horizon region, and finally in section VI. we draw our conclusions. In addition, we discuss the problem of the coordinate systems in an appendix section.", "pages": [ 1, 2 ] }, { "title": "II. THE BRANE - BLACK HOLE MODEL", "content": "The metric of the N -dimensional Myers-Perry solution [15] in Boyer-Lindquist coordinates with a single angular momentum is given by where and d Ω 2 N -4 is the line element on an ( N -4)-dimensional unit sphere. The parameters µ and a are related to the total mass, M , and angular momentum, J , of the black hole as where is the area of an ( N -2)-dimensional unit sphere S N -2 . For simplicity, without any loss of generality, we can fix the value of the mass parameter µ to 1. Test brane configurations in an external gravitational field can be obtained by solving the Euler-Lagrange equation derived from the Dirac-Nambu-Goto action [2] where is the induced metric on the brane and ζ µ ( µ = 0 , . . . , D -1) are coordinates on the brane world sheet. The brane tension does not enter into the brane equations, thus for simplicity it can also be put equal to 1. We introduce coordinates in the bulk as and it is assumed that the brane is stationary, spherically symmetric in the ϑ i ( i = 1 , . . . , n = D -3) dimensions, rotationally symmetric in the ϕ coordinate, and, if D < N -1, its surface is chosen to obey the equations With the above properties the brane world sheet allows an O (1) × O ( D -2) group of symmetry, and can be completely defined by the single function θ = θ ( r ). We shall use coordinates ζ µ on the brane as where n = D -3. With this parametrization the induced metric on the brane surface is given by where, and throughout the paper, over-dot denotes the derivative with respect to the radial coordinate, r . The Dirac-Nambu-Goto action (7) reduces to where ∆ t is an arbitrary interval of time, A n is the area of the unit sphere S n , the 2 π factor is obtained from the integration with respect to ϕ , and the Lagrangian takes the form", "pages": [ 2 ] }, { "title": "III. THE BRANE EQUATION", "content": "Test brane configurations are solutions of the EulerLagrange equation which for the Lagrangian (12) reads as where α and β are The horizon of the black hole is defined as the largest solution of ∆ = 0, and one can consider the non-rotating problem by taking the a → 0 limit. In the case of the non-rotating limit however, one notices that the Euler-Lagrange equation, obtained from (14), is not identical to the one that has been obtained and analyzed by Frolov in [1], and what we also investigated in the presence of thickness corrections in the spherically symmetric case [11-13]. After some analysis one can show that the difference stems from the different coordinate systems used by the Myers-Perry and Schwarzschild-Tangherlini solutions [16], and which disappears in standard 4-dimensions in the a → 0 limit, but remains present in higher dimensions, even after taking the non-rotating limit. The detailed calculation to show this is a bit lengthy, therefore we present it as an Appendix at the end of the paper. As a consequence, it is very important to emphasize that the DNG-brane, defined as θ ( r ) in the previous section, is not the 'corresponding' brane to the one that we investigated in the Schwarzschild-Tangherlini case, in the sense, that it does not reproduce the spherical results of [1] in the non-rotating limit. This is because the angular coordinate θ , through which the brane is defined in the Myers-Perry metric, is different from the one (denoted with the same letter θ ) in the Schwarzschild-Tangherlini solution, even after taking the non-rotating limit. They correspond trivially only in 4-dimensions, where we are accustomed to obtain the Schwarzschild coordinates in the non-rotating limit of the Kerr solution. It may also worth to mention that we've been trying to find an appropriate coordinate system for the rotating case, where those 'corresponding' branes, which would reproduce the solutions of [1] as their non-rotating limit, could be defined naturally. The problem, however, turns out to be very difficult, because in those systems where the limit in the bulk is automatic, either the definition of the rotationally symmetric brane is problematic, or the coordinate transformations involve angles from the extra dimensions of the metric, which cannot be integrated out from the action in the simple way as we did in (11). Although we believe that the problem should ultimately be resolved in one way or another, nevertheless, we were not able to obtain a satisfactory resolution so far. Accordingly, in the present paper we are focusing on those DNG-branes which are defined in section II, and are the solutions of the Euler-Lagrange equation (14). The problem is, of course, interesting in its own right, being the most naturally defined DNG-brane problem on a rotating black hole background in arbitrary dimensions, and also the most natural extension of the spherical problem to the simplest rotating case. However, we have to keep in mind that it is essentially different from the one, that would provide back the Schwarzschild-Tangherlini solution of [1] in the non-rotating limit.", "pages": [ 2, 3 ] }, { "title": "IV. ASYMPTOTIC AND REGULARITY ANALYSIS", "content": "In this section we present the near horizon- and far distance asymptotic solutions of the brane equation. From regularity requirements in the near horizon region, we obtain unique boundary conditions for the problem which will be used for the numerical solution in the following section.", "pages": [ 3 ] }, { "title": "A. Near horizon behavior", "content": "For a brane crossing the horizon (black hole embedding case or supercritical branch in Frolov's terminology [1]), (14) has a regular singular point on the horizon, r = r 0 . A regular solution at this point has the following expansion near the horizon where the regularity requirement impose the condition ∣ Consequently, supercritical solutions are all uniquely determined by their boundary value θ 0 . In the Minkowski embedding (subcritical) case, the brane does not cross the horizon, and its surface reaches its minimal distance from the black hole at r 1 > r 0 , which, for symmetry reasons, occurs at θ = 0. A regular (but not smooth or even differentiable, see [11-13]) solution of (14) near this point has the asymptotic behavior where the regularity requirement on the axis of rotation impose the conditions and with Hence, all subcritical solutions are also uniquely determined by the single parameter, r 1 .", "pages": [ 3, 4 ] }, { "title": "B. Far distance solution", "content": "Since the Myers-Perry solution is asymptotically flat, the brane function θ ( r ) has to converge to a constant value, θ ∞ , as r → ∞ . The explicit value of θ ∞ is not known for the moment (in contrast with the spherical case where it was π/ 2 for all dimensions), rather it can be obtained by the following consideration. The far distance solution of (14) can be searched in a perturbative form where ν ( r ) is a first-order small function compared to θ ∞ , and we require that We shall only keep the linear terms of ν in (14) which yields the asymptotic equation The general solution of (24) reads as with Before running into the analysis of the complex powers in the solution, we notice that the first term of (25) is a constant. Thus (25) can only be a good solution of (24) if B = 0, due to the requirement (23). This implies the asymptotic constraint and yields the asymptotic value According to this result, we can conclude that for each brane dimension, n , the solutions have different asymptotic behavior, and the asymptotic value, θ ∞ , coincides with the Schwarzschild-value, π/ 2, only in the case of n = 0, that is, when the brane is 3-dimensional. It is interesting to note here that 3-dimensional branes tend to behave differently from their higher dimensional counterparts in other aspects too. In our previous works [12, 13], we also found that 3-dimensional branes had exceptional analytic properties in the near horizon region when thickness corrections had been considered in the non-rotating case. Another interesting feature to note is that the asymptotic value does not depend on the rotation parameter of the black hole, it is determined solely by the number of inner dimensions of the brane in which it is spherically symmetric. After deriving the value for the asymptotic constants, we can obtain the corresponding asymptotic solutions by plugging back θ ∞ into (24), which results the asymptotic equation or plugging it directly into (25), and take a bit of time with the power analysis. Either case, the asymptotic solution takes the form It may seem, for the first sight, that branes with n ≤ 4 dimensions have different far distance asymptotics than the ones with dimensions n ≥ 5. The real change occurs, however, at n = 3, as we can see it from the following analysis. In the n = 0 case, the rotation of the black hole does not seem to affect the asymptotic behavior directly, and, as we mentioned earlier, this is the exceptional case of the 3-dimensional brane, when the asymptotic value, θ ∞ , is π/ 2, just as in the Schwarzschild problem. When n = 1, the first term dominates the solution, because the second one, which is controlled by the rotation parameter, decays faster. In the case of n = 2, both terms decay essentially as r -2 . In all other cases, starting from n ≥ 3, the first terms in the solution decay much faster than the second one, which results that all the branes with D = 6 or more dimensions have an almost uniform convergence to the asymptotic value in the far distance region, and this is controlled by the rotation parameter of the black hole. The coefficients p and p ' in the solutions are continuous functions of the θ 0 or r 1 boundary parameters that we obtained previously from regularity requirements in the near horizon region. On the other hand, because of the complicated asymptotic behavior, the interpretation of p or p ' is not so clear as it was in the Schwarzschild case (being the distance of the brane from the asymptotic value at infinity [1]).", "pages": [ 4, 5 ] }, { "title": "V. NUMERICAL RESULTS", "content": "After obtaining the far distance solution of the problem in analytic form and also deriving boundary conditions from regularity requirements in the near horizon region, we can consider the numerical solution of (14). As it was shown earlier, the boundary value θ 0 , or the radial coordinate r 1 , uniquely determines the corresponding superor subcritical solutions, respectively. The numerical solution itself does not require very advanced techniques, we have performed it by using the Mathematica /circleR NDSolve function. On FIG. 1 we are plotting a sequence of D = 3 ( n = 0) brane solutions from both topologies in the near horizon region. The asymptotic constant in this special case is π/ 2 and we have chosen the value 0 . 4 for the rotation parameter, a . As a result (just as we expect from the far distance analysis) the brane configurations are very similar to what we had before in the spherical case [1, 11]. By increasing the brane dimension from D = 3 ( n = 0) to D = 4 ( n = 1), and keeping the bulk dimension fixed ( N = 6), we can see the interesting new result on the asymptotic behavior. We plotted this situation on FIG. 2. In this case, the asymptotic value, θ ∞ , is π/ 4, and it can be seen that all solution tend asymptotically to this value (in good agreement with the far distance analysis) independently of the near horizon boundary values. By increasing the number of brane dimensions, n , the value of the asymptotic constant, θ ∞ , changes according to (26), but the qualitative picture of the solutions remains essentially similar to what we see on FIG. 2. For the sake of illustration, on FIG. 3, we also plot the D = 5 ( n = 2) dimensional case with N = 6. By changing the value of the rotation parameter, the near horizon configurations are also changing together with the asymptotic convergence to θ ∞ , that we discussed earlier in the far distance solution. In order to illustrate this change, we define the function and compute the ∆ θ ( r ) functions for a sequence of brane solutions with different boundary values, θ 0 , equally distributed around the θ ∞ = π/ 4 value in the θ 0 ∈ (0 , π/ 2) region, just as on FIG. 2 and FIG. 3. The corresponding curves are plotted on FIG. 4. for two different rotation parameter values, a = 0 . 1 (left picture) and a = 0 . 9 (right picture). In the case of slow rotation ( a = 0 . 1), the ∆ θ ( r ) functions have an almost 'mirror symmetric' amplitude distribution around the θ ∞ = π/ 4 value (right picture on FIG. 4.), while in the case of a large rotation parameter ( a = 0 . 9), the picture becomes very asymmetric. The branes with boundary values θ 0 ∈ ( π/ 4 , π/ 2) deviate strongly from θ ∞ in the near horizon region, while the branes with θ 0 ∈ (0 , π/ 4) approach the asymptotic value very quickly. In our previous works [11, 12], we also investigated the problem of a quasi-static evolution of a brane from the equatorial plane in black hole embeddings, to a Minkowski embedding topology, through a topology change transition. The question was very natural there, following the method developed in [14], because the equatorial configuration was a general solution in every dimensions of the spherical problem. In the present rotating case however, as we saw above, the equatorial configuration is a solution of the problem only in the exceptional case of the 3-dimensional brane, and we cannot use this method for a general discussion. Although we could analyze the topological phase transition in this special case, nevertheless we believe that it would be misleading since the relevant problems are usually obtained from higher dimensions, like the case of the holographic dual phase transition. As a consequence, the question remains open in the present rotating case.", "pages": [ 5, 6 ] }, { "title": "VI. CONCLUSIONS", "content": "In the present work, we studied the problem of rotationally symmetric, stationary, Dirac-Nambu-Goto branes on the background of a Myers-Perry black hole with a single angular momentum. In defining the interacting brane - black hole system, we strongly followed the spherical problem given by Frolov [1]. Although this model is the most natural extension of the spherical setup to the simplest rotating case, we found that due to the non-equivalent coordinate parametrization, the obtained solutions are not compatible with the spherical solutions in the sense that the latter ones are not recovered in the non-rotating limit. Our efforts, to find an appropriate coordinate system in which the rotating problem could be formulated naturally, in a way that the spherical case could also be reproduced in the a → 0 limit, has not succeeded so far. It is an open question whether it can be done at all. After clarifying the above situation, we analyzed the properties of the obtained problem, and presented its solution both analytically, at far distances, and numerically, in the near horizon region. In the latter case, we found that the analytic properties of the test brane solutions, both on the axis of rotation and on the horizon, are very similar to what we saw in the spherical case. From regularity requirements we could obtain unique numerical solutions for each, freely chosen, boundary value in both topologies. By analyzing the far distance solutions, we obtained a new interesting result that the asymptotic behavior of the rotating solutions are qualitatively different from the spherical problem, except in the special case of a 3-dimensional brane. This difference change the entire structure of the brane configurations in the near horizon region too, because all solutions are attracted asymptotically to the same constant value, independently of the near horizon boundary conditions. Furthermore, the asymptotic value is different for every brane dimensions, and it is controlled solely by the dimension parameter of the brane. Another interesting result is that the rotation of the black hole has a direct effect only on how the solutions tend to the asymptotic value, and we illustrated this phenomenon in the cases of a small and a large rotation parameter. One of the motivations of this work was to understand the role that rotational effects may play in a quasi-static, topology changing phase transition of the system. As a negative result, we obtained that the problem cannot be studied here in the geometrical way that we applied in the thickness corrected spherical problem [11, 12], due to the lack of a general, stationary, equatorial solution for arbitrary dimensions. Consequently, we could not obtain general results on the phase transition in this paper, so the question remains open for the rotating case. The lack of the equatorial solution has another consequence which is connected to the stability of the rotating brane - black hole system. It has been shown by Hioki et al. [17] that equatorial solutions are stable against small perturbations in the spherical case. Stability is an important issue in higher dimensions, and it would be also important to know whether similar results may hold for the present axisymmetric case too. Unfortunately, because of the lack of the equatorial solution, the question of stability can not be studied here by using the method of Hioki et al. for the general case. As another stability issue, in this paper we have not considered the cases of extremal and ultra-spinning black holes. The reason for this is the fact that ultra-spinning black holes are expected to be unstable [18], and the instability limit occurs at a surprisingly low value of the angular momentum, i.e. not far in the ultra-spinning regime. In fact, the magnitude of the critical rotation parameter, a , has been estimated by Emparan and Myers for several dimensions [18], and it turned out that a typical value is around a ≈ 1 . 3. According to this, in the present paper we constrained ourselves to keep the value of the rotation parameter small enough to stay away from the presumably unstable regime.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I am grateful for valuable discussions with Bal'azs Mik'oczi and Alfonso Garc'ıa Parrado G'omez-Lobo. Most calculations, especially the numerical parts, have been performed and checked by the computer algebra package MATHEMATICA 9 . The research leading to this result has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under the grant agreement No. PCOFUND-GA-2009-246542 and from the Foundation for Science and Technology of Portugal.", "pages": [ 7 ] }, { "title": "Appendix: The Schwarzschild-Tangherlini limit of the Myers-Perry solution", "content": "The N -dimensional Myers-Perry metric with a single angular momentum in Boyer-Lindquist coordinates is given in (1), while the Schwarzschild-Tangherlini (ST) solution of the same dimension is given by where In both formulas d Ω 2 k is the metric of a k -dimensional unit sphere S k , which is parametrized with the polar coordinates, ξ k , defined by the following recursive relation By taking the limit of a → 0 in the MP metric, the ST solution has to be reproduced. In order to check this, after taking the limit in the coefficient functions, one arrives to the following equation for the metric on the ( N -2)-dimensional unit sphere, Applying the recursive relation given in (A.3) we can rewrite (A.4) into the form From (A.5) it is clear that if N > 4, the angular parametrization of the 2-sphere in question is different from that of the ST metric of the same dimension. This difference however disappears in standard 4-dimensions since the last terms are zero on both sides yielding the equivalence In order to see the ST limit of the MP metric for N > 4, one needs to verify that the angular parametrization given in (1) is equivalent with the Schwarzschild parametrization. To show this, we need to find the transformation laws from the spherical coordinates defined by the polar angles ξ 1 and ξ 2 , to the coordinate system defined by the angles θ and ϕ . The transformation rules are the solution of the following system of equations obtained from (A.5), where θ = θ ( ξ 1 , ξ 2 ) and ϕ = ϕ ( ξ 1 , ξ 2 ). This system can be integrated in a closed form with the solution or equivalently the inverse transformations are To see how the angles θ and ϕ parametrize the unit 2sphere let us utilize the standard Cartesian coordinates x, y, z given by Here the polar angle ξ 1 ∈ [0 , π ] is measured from the positive z -direction, and the azimuthal angle ξ 2 ∈ [0 , 2 π ] runs in the x -y plane measured from the positive x -direction. Expressing now x , y and z as functions of θ and ϕ through the transformation formulas (A.13) and (A.14) we get It is thus clear that θ and ϕ are also spherical polar coordinates of the unit 2-sphere in a way that the polar angle θ ∈ [0 , π ] is measured from the positive y -direction, and the azimuthal angle ϕ ∈ [0 , 2 π ] runs in the z -x plane measured from the positive z -direction. According to this, we observe that in standard 4 dimensions the axis of rotation of the Kerr black hole is or- thogonal to the x -y plane (corresponding to the labeling of the standard Cartesian coordinates above), however in dimensions N > 4, the axis of rotation 'switches' to be orthogonal to the z -x plane instead. One has to be thus very careful in taking the Schwarzschild-Tangherlini limit of the Myers-Perry solution in higher dimensions, because the angular parametrization of the two solutions remains different even in the non-rotating limit. (2009).", "pages": [ 7, 8 ] } ]

[ { "title": "The impact of the Hall effect on high energy density plasma jets", "content": "P.-A. Gourdain, C. E. Seyler", "pages": [ 1 ] }, { "title": "Abstract", "content": "Using a 1-MA, 100ns-rise-time pulsed power generator, radial foil configurations can produce strongly collimated plasma jets. The resulting jets have electron densities on the order of 10 20 cm -3 , temperatures above 50 eV and plasma velocities on the order of 100 km/s, giving Reynolds numbers of the order of 10 3 , magnetic Reynolds and Péclet numbers on the order of 1. While Hall physics does not dominate jet dynamics due to the large particle density and flow inside, it strongly impacts flows in the jet periphery where plasma density is low. As a result, Hall physics affects indirectly the geometrical shape of the jet and its density profile. The comparison between experiments and numerical simulations demonstrates that the Hall term enhances the jet density when the plasma current flows away from the jet compared to the case where the plasma current flows towards it. Pacs numbers : 52.38.Ph, 52.72.+v, 95.30.Qd, 52.65.-y, 52.30.Cv, 52.65.Kj, 52.30.-q, 52.77.Fv High energy density (HED) plasmas are an exciting research medium to study extreme states of matter. With kinetic pressures larger than one megabar, they open a new range of opportunities in understanding the properties of warm dense matter and the kinematics of flows at large Reynolds (Re = ρ vL/ µ ), magnetic Reynolds (ReM = vAL µ0 / η ) and Péclet (Pe = vL/ χ ), numbers. In computing Re, ReM, and Pe it is found that the flows in accretion disks surrounding black holes, proto stars and active galaxies are likely to be turbulently advective with significant magnetic fields. When using the mass density ρ , speed v, dynamic viscosity µ , magnetic diffusivity η/µ0 , the scale length of the flow L and heat diffusivity χ of plasmas, these numbers scale as: Re ∝ vLZ 4 A 1/2 niT -5/2 , ReM ∝ vLT 3/2 Z -1 and Pe ∝ vLZ(Z+1)niT -5/2 . Here ni is the plasma ion number density, Z the ionization number, A the atomic mass and T the plasma temperature. So, even if scale lengths are small in laboratory experiments, HED plasmas are extremely dense and these numbers can reach large values, placing them at the forefront of laboratory astrophysics 1 . Ultimately only numerical codes can possibly encompass the sizes of most astrophysical objects and the art of numerical simulations does reside in finding the models which can best represent the phenomena observed by astrophysicists. In particular, laboratory experiments can help to validate these numerical codes. While kinetic effects cannot be ignored in astrophysical plasmas, today's large scale simulation efforts focus on the less computationally intensive fluid models. However the magnetohydrodynamics (MHD) model may not capture correctly celestial mechanics. For instance the Hall effect, which is absent from the MHD model, was shown to play a critical role in space dynamics, for instance in the magnetic polarity of galactic jets 2 . This letter wishes to highlight how the Hall term can alter the properties of a strongly collimated HED plasma jet produced by the ablation of a thin metallic foil and it will discuss how these results can be scaled back to astrophysical plasmas. While the jet absolute parameters fall short of astrophysical jets, dimensionless parameters, such as the ratio of the jet radius to its length or the ratio of jet density to the background plasma density, are similar to that of astrophysical jets. While Re, ReM or Pe are smaller, this letter shows that the importance of the Hall term is actually linked to the ion inertial length, the scale length of the system and the Alvén Mach number rather than Re or Pe. However the impact of large Re and Pe flows on the Hall electric field may reduce its impact. In the extended magnetohydrodynamics (XMHD) framework, the Hall term generates an electric field perpendicular to the resistive electric field when electrical currents flows across magnetic fields. This Hall electric field can be easily included in Ohm's law to give a simplified version of the generalized Ohm's law (GOL), All bold quantities here represent vectors. E is the total plasma electric field, v the flow velocity, η the plasma resistivity, n e is the electron number density, J the electrical current, B the magnetic field and e is the electron charge. In this equation we have ignored the contribution of the electron pressure to the plasma electric field. The Hall term, J x B , can be neglected in particular situations that we explain now by rewriting the GOL in its dimensionless form, i.e. Eq. (2) was obtained dividing Eq. (1) by the local plasma characteristic velocity of the flow v 0 and the local magnetic field B . It is important to note that all terms in Eq. (2) are now dimensionless except for the characteristic length scale of the jet L and the ion inertial length δ i . Their ratio is dimensionless. In general, the ion inertial length δ i =( m i / µ 0 e 2 Z 2 n i ) 1/2 is the distance below which ion motion decouples from electron motion, which is still frozen in the magnetic field 3 . We assume here electrical quasi-neutrality, i.e. n i = Zn e . To keep scaling parameters consistent across all the XMHD equations, the characteristic flow velocity v 0 has to be the local Alfvén speed u A , i.e. B /( m i n i µ 0 ) 1/2 . As a result, the dimensionless Eq. (2) uses the Alfvén Mach vector M A , which is the ratio of the plasma local velocity vector to the local Alfvén speed. In this case the ReM is also the Lundquist number S . Hall physics introduces a great numerical challenge in computing plasma flows on time scales far below the characteristic electron frequencies. MHD codes assume that δ i/L is small and simply drop the Hall term from Eq. (2) thereby greatly reducing the total computational time. This letter makes the point that the systematic dismissal of Hall physics based on the presumed smallness of δ i/L is shown to be ill-advised in flows with low Alfvén Mach numbers (<<10). In recent years, we have explored an HED experiment using the plasma produced by a thin metallic foil to test the impact of Hall physics on HED plasma jets. In this experimental setup, the foil is stretched on the anode of a pulsed power generator and connects to the cathode via a hollow metallic pin placed under the foil along the foil geometrical axis. Published research conducted at Cornell University 4,5,6 and Imperial College 7,8,9 presents in greater details the properties and potential applications of such configurations, henceforward called radial foil configurations. The basic idea is that plasma currents converge towards the central pin and JxB forces lift the foil upwards. During this process a small portion of the total plasma current (~ 5 to 10%) flows above the foil where Ohmic resistance heats the plasma. The ablating plasma expands into the vacuum and drags electrical current away from the foil surface. Most of the ablation and plasma motion occurs near the pin, where the JxB forces are intense due to radial convergence. The ablated plasma is forced onto the geometrical foil axis by magnetic pressure and forms a dense, vertical jet with axial velocity on the order of 80 km/s as measured in Ref. 4. The ablated plasma and the base of the jet are visible on the experimental laser Schlieren images presented in Figure 1. Plasma Schlieren imaging 10 records only the light rays which have been diffracted away from the optical focus of the collection optics by electron density gradients. Such regions appear dark in the figure due to publication imperatives. As time progresses, a plasma bubble forms, then expands into the low density plasma above the foil. Kink instabilities 11 disrupt the column at the center of the bubble which quickly breaks apart. While reproducibility of the plasma bubble phase is not guaranteed due to instabilities, the plasma jet phase reproduces nicely from shot to shot as long as current drive waveforms are similar. Using experimental plasma properties previously published 4,5,6 for jet and ablated plasmas, we can estimate the following dimensionless plasma parameters for the jet : Re~10 3 , ReM ~ 1, Pe ~ 1 and MA ~ 10; and in the ablated plasma: Re~10 4 , ReM~10 , Pe~10 and MA~ 2. To highlight the Hall effect experimentally when cathode and anode shapes are different, the current direction has to be reversed. While the plasma velocity stays the same, all electromagnetic terms on the right hand side of Eq. (2) change signs but the Hall term (i.e. J x B ). A dozen shots were done with standard (radially inward) and reverse (radially outward) electrical currents. We present herein the discharges which highlight best the impact of the Hall effect on plasma dynamics. Figure 1 shows the differences of the ablated plasma for standard (left) and reverse (right) currents. The plasma instabilities responsible for the elongated diffraction patterns (direction highlighted by the arrows) caused by inhomogeneity in ablation plasma density 5 point away from the axis for standard currents (SC), whereas they point towards the axis for a reverse currents (RC). Since the ablation differs for both plasmas it seems reasonable to assume that the jet density will be affected by the current direction. Indeed laser interferometry shows substantial differences in the jet density profiles. Using a fringe-counting algorithm, such as the IDEA code 12 , it is relatively straightforward to obtain the areal electron number density of both jets and their surroundings from laser interferometry (150 ps pulse length at 532 nm). At this wavelength, one fringe shift corresponds to an electron areal number density of 3.72x10 17 cm -2 . Since the plasma dynamics is quasi-axisymmetric during the early stages of the plasma discharge, it is possible to obtain the local electron number density using a robust Abel inversion technique 13 . Figure 2 shows the experimental local electron density for standard (left panel) and reverse (right panel) currents. Gray masks hide the location where densities could not be computed properly due to the absence of or an inaccuracy in counting interference fringes. Overall, both jets have similar radii, on the order of 400 µm. However, the RC jets are taller. The local electron density varies from 10 20 cm -3 at the base of both jets, to 5x10 19 cm -3 at mid height and 1.5x10 19 cm -3 at the top of the jets. Additionally, the RC jet has a larger electron density for a given height as compared to the SC jet. It is important to note that the RC jet interferogram was taken 4 ns earlier than the SC jet. Overall the reverse current case seems to confine the plasma better on axis, thus enhancing the jet density. This effect is easily seen on the plasma electron density profiles presented in Figure 3 at 1.5 mm and 2 mm above the foil. At each height, the plasma densities for both cases were renormalized to highlight profile dissimilarities instead of local density differences. The SC jet profiles are broader than RC profiles. Further SC profiles have a tendency to be flat or slightly hollow near the jet axis. RC profiles are systematically peaked, indicating that the plasma is pushed on axis with greater strength. Both jets are also visible on data captured by XUV four frame pinhole camera. Due to diffraction caused by the 50 micron pinholes, photon energies below 40 eV are cut-off and hardly reach the photocathode of the quadrant camera, giving a lower bound on the electron temperature of the plasma jet. This energy corresponds to an ionization number of the aluminum plasma on the order of 3. As a result, the ion inertial length δ i is on the order of 10 µm, 100 µm and 150 µm at the base, mid height and top of both jets respectively. If we take the 200 µm jet radius as the characteristic length L, the ratio δ i / L is smaller than 1 inside the plasma jet and much smaller at the base of the jet. A better characteristic scale length L is the magnetic field scale length which reduces to for axi-symmetric systems. Even with this length, the Hall effect is weak in the jet. However experimental evidence shows noticeable differences inside the jet and further investigation is required to understand the dissimilarities between the standard and reverse current cases. To fully explain the experimental data presented herein we use the PERSEUS 14 code (Plasma as an Extended-mhd Relaxation System using an Efficient Upwind Scheme) that can simulate HED plasmas generally and radial foil dynamics in particular. This code includes the Hall, electron inertia and electron pressure terms, and runs as fast as a standard MHD code by computing the Hall term in a local semiimplicit manner. The electron pressure was 'turned off' in these simulations to focus only on the Hall term. The simulations are two-dimensional in r-z cylindrical coordinates. The plasma ionization Z and gas constant γ were assumed constant throughout the computational domain, 3 and 1.15 respectively. Despite these restrictions, simulations confirm the trends observed in both experiments. The ion density for both standard (left) and reverse (right) currents, using a log10 scale in Figure 4-a, shows that the jet with reverse current is taller and denser than the jet with standard currents. The code also reproduces correctly the plasma instabilities visible in in the ablated plasma of Figure 1-a and b. Since the choice of the plasma scale length is rather arbitrary, the ion inertial length criterion of Eq. (2) does not define well the plasma regions where the Hall term dominates. However the simulation gives access to a wealth of plasma parameters and we can compare precisely the Hall electric field with the dynamo electric field to understand the circumstances in which one dominates over the other. We therefore find more judicious to use the Hall-Dynamo Criterion (CHD), given by: The CHD compares the strength of the Hall electric field to the strength of the electric field generated by dynamo. Where the ion inertial length criterion requires only the measurements of ni and Z, CHD also requires the measurements of B, J and v, making it more difficult to determine experimentally. As Figure 4-b shows, Hall electric fields dominate over the dynamo electric field in most of the ablated (outer) plasma. The Hall effect plays a minor role in the remainder of the plasma volume, especially in the plasma jet. This result supports the experimental ion inertial length argument discussed previously. In fact, when no axial magnetic field is present, currents and flows are always perpendicular to the magnetic field in axi-symmetric systems. As a result We can actually connect both criteria if we use the magnetic field scale length as our characteristic scale length L Since CHD measures the absolute strength of the Hall electric field compared to the dynamo electric field, Eq. (6) shows that the ion inertial length criterion δ i/L can overestimate or underestimate the impact of the Hall effect depending if the flow is super-Alfvénic or sub-Alfvénic respectively. Figure 4-c shows that the ion inertial length criterion artificially enhances the impact of the Hall term near the jet axis, where MA is larger since B is small there, especially in the top section of the jet where most of the volume is devoid of current. It artificially reduces the impact of the Hall effect under the foil, where MA is smaller due to large B in this region. As a result, the following criterion is better apt at judging the importance of the Hall term of plasma dynamics. In conclusion, experimental data show the Hall term affects the dynamics of strongly collimated plasma jets produced by radial foils, particularly the jet geometry and its density profile. However the Hall criterion δ i/LMA shows that Hall physics dominates only the low-density plasma region surrounding the plasma jet. Surprisingly this effect is strong enough to alter the jet dynamics. The plasma flow stream lines, plotted in Figure 4-a, congregates closer to the axis for reverse electrical currents. This increase in radial inward flow is responsible for the denser, taller jets observed in reverse current cases and it is consistent with the density profiles presented in Figure 3. One can reconcile the impact of the Hall term onto the jet by seeing the ablated plasma surrounding the jet as a virtual electrode where the electric field is dominated by Hall physics. Consequently, Hall-dominated currents and flows in the region surrounding the jet act as boundary conditions to dynamo or Ohmic-dominated currents and flows in the jet region. It is rather evident that HED jets which have δ i/LMA >> 1 and S >> 1 will be strongly influenced by Hall physics. What is more remarkable is that if the ratio of jet density to background density is on the order of 10, a HED jet can have δ i/LMA << 1 and still be influenced by Hall physics when the background plasma has δ i/LMA >> 1 and S >> 1. Since S is large enough in our experiments to allow the expression of Hall physics, the major obstacle to extend our conclusions to astrophysical jets is in the low Re and Pe of our experimental jets. While they are low compared to astrophysical jets, Re or Pe do not enter directly the GOL scaling of the electric field, only δ i/LMA and S (i.e. ReM) do. As a result, we believe that our experimental and numerical results can be scaled to astrophysical jets when the density ratio of the astrophysical jet density to the stellar background density is smaller than 10. If δ i/LMA >> 1 and S >> 1 in this background plasma, then the electric field surrounding the astrophysical jet will be strongly dominated by Hall physics and the jet will be altered indirectly by Hall physics. If the plasma density of the astrophysical jet is much larger than the background plasma (100 or more) then the conclusion presented herein may not apply. Large Re and Pe can alter the properties of the jet in such ways that the external Hall electric field will not penetrate deep enough in the jet to alter its density profile and experiments working at larger Re and Pe are necessary..", "pages": [ 1, 2, 3, 4, 5, 6 ] }, { "title": "Acknowledgments", "content": "Research supported by the NSF Grant # PHY-1102471 and the NNSA/DOE Grant Cooperative Agreement # DE-FC52-06NA 00057. Figure 1. Negative black and white Schlieren shadowgraphy for shot # 02580 (standard current) 51 ns into the plasma discharge for shot # 02579 in (reverse current) 53 ns into the plasma discharge. Both shadowgraphs were obtained using a ND:YAG pulsed laser in the green (532 nm). The initial foil location is indicated by the dashed line under which we sketched the 1-mm diameter pin. The direction of plasma instabilities is indicated with arrows. Figure 2. Local electron density of a) shot # 02178 (standard current) 76 ns into the current pulse and b) shot # 02173 (reverse current) 72 ns into the current pulse. Figure 3. Electron density profiles for both standard and reverse currents 1.5 and 2 mm above the foil, normalized to 3 and 2, respectively. Figure 4. a) Plasma ion density, b) Hall-Dynamo Criterion (C HD ) and c) ion inertial length criterion on the logarithmic scale. All values have been clipped to the minima and maxima of both scales. The white lines in panel a) are plasma flow stream lines.", "pages": [ 6 ] } ]

[ { "title": "Potential Vorticity Formulation of Compressible Magnetohydrodynamics", "content": "Wayne Arter EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, UK. OX14 3DB (Dated: May 20, 2018) Compressible ideal magnetohydrodynamics (MHD) is formulated in terms of the time evolution of potential vorticity and magnetic flux per unit mass using a compact Lie bracket notation. It is demonstrated that this simplifies analytic solution in at least one very important situation relevant to magnetic fusion experiments. Potentially important implications for analytic and numerical modelling of both laboratory and astrophysical plasmas are also discussed. PACS numbers: 52.30.Cv, 52.55.Fa, 96.60.Q-", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "Ideal MHD is a model for magnetised plasma where the collisionality is low, so that dissipative effects can be neglected, yet where the charged particles still interact sufficiently strongly via the electromagnetic field they can be treated as a single fluid. The ideal MHD model is applied to a wide range of laboratory and astrophysical situations, where there are long periods of relative quiescence in which Maxwellian particle distributions can be approached, interrupted by often violent transients. Ideal MHD instabilities are thought to be implicated in the triggering of the sawtooth crash phenomenon in tokamak magnetic fusion experiments and flaring in the solar and stellar context, see textbooks such as [1]. The former is important as it limits the performance of devices ultimately intended to generate nuclear power, and the latter is implicated in the generation of solar magnetic storms which can disrupt terrestrial power grids, navigation and communication systems. Both these topics are presently the subject of intensive investigation, magnetic fusion as the multi-billion dollar ITER tokamak enters the construction phase, whereas multiple satellite missions are collecting data on solar and stellar magnetic fields. It is often mathematically convenient when employing ideal MHD, to assume that the plasma fluid is incompressible , but the reality in the above-mentioned situations is that the plasma density varies by one or more orders of magnitude over the region of interest. This work presents what is believed to be a novel, mathematically convenient formulation of compressible MHD. The equations of ideal MHD as usually formulated are well-known and are to be found in many textbooks, see eg. [1, § 4.3]. As explained there, the problem admits a variational formulation which is of great utility for practical stability analysis, and a functional Hamiltonian formulation in terms of Lie derivatives [2], of great theoretical importance for understanding stability and evolution. More direct approaches to ideal MHD stability are also now used [1, § 6], and the results presently to be described are more relevant to the latter school. The potential vorticity is the ordinary vorticity ω of the plasma (the curl of the mean flow U of ions and electrons), divided by the mass density ρ , ie. ˜ ω = ω /ρ . The possibility of combining the equation for the time evolution of vorticity with that for density evolution to give a simple equation for the rate of change of potential vorticity, was first realised for a classical fluid by Helmholtz as described by [3, § 146] in the mid-19th Century. In the mid-20th Century, Wal'en, according to [4, § 4-2] was the first to realise that a mathematically identical relation governed the evolution of the magnetic flux per unit mass ˜ B = B /ρ where B is the magnetic field. For incompressible plasma, Arnold & Khesin [5, § I.10.C] combined these results in late-20th Century to give an elegant formulation of ideal MHD in terms of Lie brackets of vector fields. The Lie bracket is here the generalisation to arbitrary vector fields of the 'flux-freezing' operator, ie. the operator which determines the advection of divergencefree (solenoidal) fields B and ω [6, § 3.8]. The novelty of the present work is to extend this formalism to compressible MHDandexplore the implications. In particular, the peculiar, coordinate invariant nature of the Lie bracket makes it easy to generalise solutions to arbitrary geometry in some cases, both analytically and numerically. The next section contains a detailed mathematical derivation of the key formula. A discussion of the implications for analytic and numerical solution follows, and finally some important possible applications are summarised.", "pages": [ 1 ] }, { "title": "MATHEMATICS", "content": "In terms of the operators of Classical Vector Mechanics, the Lie derivative of a vector can be defined as: which will help explain the equivalence with the vector advection operator, the first term on the right. Indeed, Wal'en's result for magnetic induction in a perfectly conducting medium is Introducing component notation for vectors in general non-orthogonal coordinate systems, as described in many textbooks e.g. [7], it turns out that the Jacobians thereby introduced (of the co-ordinate transformation from Cartesians), cancel among the terms in Eq. (1), so that where u k , v k are the contravariant components of the 3vectors u , v respectively, and the summation convention is implied. It follows that where [ · , · ] denotes the Lie bracket of Schutz [8]. It will be now be proved that the equation for the evolution of potential vorticity in compressible ideal MHD may be written where the potential current ˜ J = ∇× B /ρ . The customary vorticity equation in ideal MHD is where vorticity ω = ρ ˜ ω = ∇× U , and current J = ρ ˜ J = ∇× B = ∇× ( ρ ˜ B ). When proceeding further, it is convenient and often physically justifiable, by a barotropic or isentropic assumption, to neglect the term in the pressure p , and if not, the resulting additional term is easily representable in general geometry. It follows that to establish the equivalence of Eqs (5) and (6), it is necessary to show that ∆ = 0 , where Now, Eq. (7) is a vector equation, so validity in any coordinate frame implies validity in all, hence it is sufficient to establish the result in Cartesian coordinates, where The curl term may be expanded using the identity where R = ∇ ρ/ρ . Setting v = ˜ B × ˜ J , and expanding the resulting curl-cross operation, there is cancellation of the two terms from the Lie derivative, leaving Since ∇· J = 0, it follows that and likewise since ∇· B = 0, Substituting Eq. (11) and Eq. (12) in Eq. (10), and expanding the last term as dot products, shows that, as required ∆ = 0 . The set of evolution equations is completed by mass conservation This does not involve a vector Lie derivative, but, using the standard expression for the divergence operator in general curvilinear coordinates, it may be written where √ g is the Jacobian and the g ik is the metric tensor, which upon introducing ˜ ρ = ρ √ g may be written provided that √ g does not change with time. Like the neglect of the pressure term above, this latter inessential assumption is often physically reasonable. Unfortunately, the ideal MHD equations are here completed by the two definitions of potential vorticity and potential current, which do explicitly contain metric information, viz. and In the above, e ikl = e ikl is the alternating symbol, taking values 1, -1 or 0, depending whether ( ikl ) is an even, odd or non-permutation of (123). Finally, note that Eq. (2) and Eq. (15) together ensure that ∇· B = 0, only if initially", "pages": [ 1, 2 ] }, { "title": "SOLVING THE NEW SYSTEM", "content": "The new model system for ideal barotropic compressible MHD evolution consists of Eq. (5), Eq. (2), Eq. (15), Eq. (16) and Eq. (17). The simplification of the first three has been gained at the expense of complicating the last two 'static' relations. Nonetheless, evolution equations are harder to treat numerically, because any errors in the discretisation tend to combine over time. Moreover, it will be evident that problems solved in Cartesian geometry will test all aspects of the coding of the evolutionary equations. Thus, there is considerable computational advantage to be gained. There is obviously the concern that the magnetic field computed may not be accurately solenoidal, but this is an issue for many other discretisations also. The main difficulty is in the inversion of Eq. (16) to give the velocity field U corresponding to a freshly evolved potential vorticity (since ˜ B itself is evolved, Eq. (17) does not need to be inverted). However, this inversion, together with the computation of the irrotational part of U , is a classical hydrodynamical problem, and a variety of strategies may be found in the literature. On present machine architectures, introducing the vector potential for velocity then solving the coupled system Eq. (15) and Eq. (16) by a pseudo-timestepping algorithm is probably to be preferred. Similar numerical solution strategies were successfully employed in electromagnetics by the current author and collaborators [9, 10]. Vorticity formulations are common in plasma modelling as they are helpful in several physically relevant limits, and in particular, a vorticity formulation has been used successfully in nonlinear, compressible MHD [11]. Turning to analytic results, first consider MHD equilibrium solutions with no time dependence and U = 0 , implying L ˜ B ( ˜ J ) = 0. In the case of force-free fields, meaning J ∝ B , substituting ˜ J = λ ˜ B in the Lie derivative in component form show this is a solution provided B · ∇ λ = 0, i.e. exactly the same constraint on λ that follows from the solenoidal constraint on B and J when seeking the solution J = λ B . Hydromagnetic force-free solutions, with the additional constraint that U = λ 2 B , now cease to exist however, because U is not solenoidal unless the flow is incompressible. Moving now to time dependent solutions, interest attaches to the 'flux compression' solution [12, § 4.6], which is postulated on purely kinematic grounds (i.e. from Eq. (2)) and which may be written for a compressible flow U with density ρ provided that U = ( U x ( x, y, t ) , U y ( x, y, t ) , 0). Here, c is an arbitrary constant and ( x, y, t ) are the usual Cartesian coordinates. This solution is of practical importance for fusion experiments, where external magnets are used to generate a time dependent flux designed so as to compress plasma 'frozen' to it. It is easy to establish that if B = cρ ˆ z then J × B /c 2 = ( ∇ ρ × ˆ z ) × ρ ˆ z = -∇ ( 1 2 ρ 2 ), and so there are compressible MHD solutions of the form Eq. (19), for 2-D solutions of compressible hydrodynamics compatible with an additional pressure gradient of this form. One possibility is illustrated in simple geometry in Figure 1. /negationslash The simple form of the new evolution equations enables a generalisation of the flux-compression solution to general curvilinear coordinates. It is important to emphasise that the following is not simply re-expressing B = ρ ˆ z in different coordinate systems, nor is there a loss of generality in choosing units for density such that c = 1. The obvious generalisation is to take ˜ B 3 = 1 ( ˜ B 1 = ˜ B 2 = 0), implying a 2-D density to ensure a solenoidal B , since Eq. (18) requires ∂ ˜ ρ/∂x 3 = 0. The next step is to ensure that L ˜ B ( ˜ J ) = 0, which as may be seen using the coordinate form Eq. (3), simply requires ∂ ˜ J j /∂x 3 = 0. Similarly L U ( ˜ B ) = 0 may be satisfied by a flow with ∂U j /∂x 3 = 0 (note that U 3 = 0 is therefore allowed). From the 'static' relations, it will be seen that a solution with ˜ J j independent of x 3 is possible provided ∂g ik /∂x 3 = 0. Put in the language of differential geometry [8, § 3.11], if ˜ B is a Killing vector, there is a flux-compression solution. Further to explore the implications of this, introduce generalised toroidal coordinates ( /rho1, s, w ) (cf. ( r, θ, φ ) as commonly employed in plasma physics [7]) so that where It will be seen that ψ c ( /rho1, s, w ) = const. as /rho1 varies form a set of nested toroidal surfaces with major axis R 0 . Introduce helical coordinates ( u, v ) on each surface, so that s = u -v/q ( ψ ), w = v + u/q ( ψ ), and write ψ ( /rho1, u, v ) = ψ c ( /rho1, s, w ). Suppose that ψ is rotationally symmetric about the z -axis and satisfies the GradShafranov equation, ie. ψ is a flux function for an equilibrium magnetic field, then the curves of Eq. (20) as v varies at constant u and ψ are equivalent to lines of the equilibrium field with helical twist q ( ψ ). (Note that u and v need only be suitably periodic functions of the regular toroidal angles θ and φ . To define an equilibrium fully requires defining these functions, but this is inessential for what follows.) The metric tensor in a coordinate system ( x 1 , x 2 , x 3 ) is given by Taking ( x 1 , x 2 , x 3 ) = ( /rho1, u, v ) and using suffix , i to denote differentiation with respect to x i , the components of g ik are straightforwardly calculated as and when q is constant, s ,i = (0 , 1 , -1 /q ), w ,i = (0 , 1 /q, 1). This x k coordinate system has been chosen so that the equilibrium field expected in the tokamak confinement device may be expressed as ˜ B 3 = 1 ( ˜ B 1 = ˜ B 2 = 0), but it will be seen that in general, the metric tensor does depend on x 3 = v through s = u -v/q . By inspection, however, in the limit when q is large, g ik depends only on x 1 and x 2 . Hence a purely toroidal field, ie. one tangent to circles about the major axis x = y = 0 of the torus, allows for flux freezing solutions. Further, when ψ/R 0 is small, the s -dependence of g ik is weak, so a helical field in a torus with relatively large major radius is also in this category. The preceding limits illustrate two of the Killing vector solution symmetries [6, § 5.2.4] (the third is simply invariance in a Cartesian coordinate). Other possibilities for new analytic solutions outside of ˜ B 3 = 1 are opened up when it is realised that the helical field considered above is just one example of the use of Clebsch variables [7, § 5] to represent a solenoidal vector field as a single contravariant component. Alternatively, the vector potential may be introduced, leading to an interesting calculus involving R , consistent with the fact that exponentially varying density profiles (implying constant R ) are often studied analytically.", "pages": [ 2, 3, 4 ] }, { "title": "APPLICATIONS", "content": "For magnetic fusion physics, the above, new analytic flux-compression solutions represent a possible nonlinear development of interchange modes [13, § 12.1.2]. They would seem to represent an efficient and rapid means whereby mass (and hence heat) might escape from a discharge, hence might be implicated in situations where there is rapid transient cooling, such as the sawtooth crash in the centre of the tokamak discharge ( ψ small), and ELMs (Edge Localised Modes) in divertor discharges (large q limit). The preceding section has also speculated that the new formalism could be used efficiently to simulate ideal MHD evolution of discharges in generalised coordinates, say defined by an arbitrary MHD equilibrium. In astrophysics, observed magnetic fields usually exhibit a significant degree of disorder, so it is unclear how important the new flux-compression solutions might be, as they rely on at least a degree of coordinate invariance. It is speculated that, in stars with a strong internal toroidal field (such as the Sun is believed to possess), the rotationally symmetric solution might help model the convection pattern, accounting for the largely latitudinal variation of the solar differential rotation. Regardless, it should be helpful that, in the new equations, the field geometry appears only in the state equations. It will for example, be simpler to generate more realistic solutions from symmetric ones by varying g ik starting with the unit tensor. This could be useful, say, for modelling sunspot penumbrae both analytically and computationally, since there the magnetic field is predominantly directed radially outwards in the horizontal direction.", "pages": [ 4 ] }, { "title": "Acknowledgement", "content": "I thank Anthony J. Webster and John M. Stewart for their various helpful inputs. This work was funded by the RCUK Energy Programme under grant EP/I501045 and the European Communities under the contract of Association between EURATOM and CCFE. The views and opinions expressed herein do not necessarily reflect those of the European Commission.", "pages": [ 4 ] } ]

[ { "title": "Three-body forces and proton-rich nuclei", "content": "J. D. Holt, 1, 2 J. Men'endez, 3, 4 and A. Schwenk 4, 3 1 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA 2 Physics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, USA 3 Institut fur Kernphysik, Technische Universitat Darmstadt, 64289 Darmstadt, Germany 4 ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum fur Schwerionenforschung GmbH, 64291 Darmstadt, Germany We present the first study of three-nucleon (3N) forces for proton-rich nuclei along the N = 8 and N = 20 isotones. Our results for the ground-state energies and proton separation energies are in very good agreement with experiment where available, and with the empirical isobaric multiplet mass equation. We predict the spectra for all N = 8 and N = 20 isotones to the proton dripline, which agree well with experiment for 18 Ne, 19 Na, 20 Mg and 42 Ti. In all other cases, we provide first predictions based on nuclear forces. Our results are also very promising for studying isospin symmetry breaking in medium-mass nuclei based on chiral effective field theory. PACS numbers: 21.60.Cs, 21.10.-k, 23.50.+z, 21.30.-x Exotic nuclei with extreme ratios of neutrons to protons can become increasingly sensitive to new aspects of nuclear forces. This has been shown in shell model studies with three-body forces for the neutron-rich oxygen [1, 2] and calcium [3] isotopes, which present key regions for exploring the evolution to the neutron dripline and for understanding the formation of shell structure. Calculations with three-nucleon forces predicted an increase in binding of the neutron-rich 51 , 52 Ca isotopes compared to existing experimental values, which was recently confirmed by high-precision Penning-trap mass measurements [4]. The pivotal role of 3N forces has also been highlighted in large-space coupled-cluster calculations [5, 6]. Proton-rich nuclei provide complementary insights to strong interactions, exhibit new forms of radioactivity, and are key for nucleosynthesis processes in astrophysics, such as the rapid-proton-capture process that powers X-ray bursts [7, 8]. Although the proton dripline is significantly better constrained experimentally than the neutron dripline, nuclear forces remain unexplored in medium-mass proton-rich nuclei. Because the proton dripline is closer to the line of stability, it has also been mapped out empirically by calculating the energies of proton-rich systems from known neutron-rich nuclei using the isobaric multiplet mass equation (IMME) [9, 10] or Coulomb displacement energies [11]. This suggests that proton-rich nuclei provide an important testing ground for nuclear forces including known Coulomb and isospin-symmetry-breaking effects. In this Letter, we present the first study of 3N forces for proton-rich nuclei. The couplings of 3N forces are fit to few-nucleon systems only, and we provide predictions for the ground-state energies (Figs. 1 and 3) and spectra (Figs. 2 and 4) along the chains of N = 8 and N = 20 isotones to the proton dripline. For the interactions studied here, 3N forces provide repulsive contributions as protons are added, similar to the neutron-rich case. This is expected due to the Pauli principle combined with the leading two-pion-exchange 3N forces [1]. Our results suggest a two-proton-decay candidate 22 Si, whose Q value is very sensitive to the calculation; within theoretical uncertainties it could also be loosely bound. For the N = 20 isotones, we predict the dripline at 46 Fe and the twoproton emitter 48 Ni [12, 13]. Furthermore, we find good agreement with experimental spectra of 18 Ne, 19 Na, 20 Mg and 42 Ti and provide predictions for the isotones where excited states have not been measured. We consider a shell model description of the N = 8 and N = 20 isotones and determine the interactions among valence protons, on top of a 16 O and 40 Ca core, based on nuclear forces from chiral effective field theory (EFT) [14]. At the NN level, we take the chiral N 3 LO potential of Ref. [15] and evolve to a lowmomentum interaction V low k with cutoff Λ = 2 . 0 fm -1 using renormalization-group methods, which improve the many-body convergence [16]. Three-nucleon forces are included at the N 2 LO level. These consist of the long-range two-pion-exchange part, as well as onepion-exchange and short-range contact terms [14]. The shorter-range 3N couplings c D and c E are determined by fits to the 3 H binding energy and the 4 He radius for Λ 3N = Λ = 2 . 0 fm -1 [17], without further adjustments in the many-body calculations presented here. Note that 3N forces depend on the NN interaction used, so that the contributions from 3N forces differ depending on the cutoff in chiral NN potentials, and when used with bare chiral interactions (see, e.g., Ref. [6]) versus with renormalization-group-evolved interactions. Excitations outside the valence space are included to third order in many-body perturbation theory (MBPT) [18, 19] in a space of 13 major harmonicoscillator shells. We have checked that the matrix elements are converged in terms of intermediate-state excitations. For the N = 8 isotones, we consider both the standard sd -shell and an extended sdf 7 / 2 p 3 / 2 valence space with /planckover2pi1 ω = 13 . 53 MeV, and for N = 20, the pf and pfg 9 / 2 spaces with /planckover2pi1 ω = 11 . 48 MeV. The extended valence spaces proved important in this framework for oxygen and calcium isotopes [2-4]. In addition to the NN-force contributions, we include the normal-ordered (with respect to the core) one- and two-body parts of 3N forces in 5 major shells [2, 4]. The normal-ordered parts dominate over the contributions from residual three-body interactions [6, 20]. The latter are expected to be weaker in normal Fermi systems due to phase-space limitations in the valence shell compared to the core [21]. For the valence proton single-particle energies (SPEs) in 17 F and 41 Sc, we solve the Dyson equation selfconsistently including the contributions from NN and 3N forces in the same spaces and to the same order in MBPT. Our calculated SPEs are given in Table I, in comparison with empirical SPEs taken from the experimental spectra of 17 F and 41 Sc. The MBPT SPEs are similar to the empirical values, but the s 1 / 2 and p 3 / 2 orbitals are at higher energy in both spaces. This finding is similar to the calculated neutron SPEs in oxygen and calcium isotopes [2, 3], which are more bound and differ due to Coulomb and isospin-symmetry-breaking interactions. All calculations based on NN forces are performed with the empirical SPEs in the standard sd -and pf -shells (NN forces only lead to poor SPEs), while those involving 3N forces use MBPT SPEs in both standard and extended valence spaces. In this work, we focus on 3N forces, whose contributions are of the order of a few MeV, but an explicit inclusion of the continuum naturally becomes important for weakly bound states and can lead to very interesting contributions, typically of several hundred keV [5, 22]. Therefore, we only show spectra to 22 Si and to the two-proton emitter 48 Ni. Note that in the case of weakly bound or unbound states, additional attractive contributions from the continuum are expected [22]. We first consider the ground-state energies of the N = 8 isotones from 18 Ne to 24 S, which we compare with experiment when available. As data is limited, we also employ the IMME [9]. This relates the energies in an isospin multiplet (of states with the same quantum numbers in different isobars A ) by a quadratic dependence in isospin projection T z = ( Z -N ) / 2, The energies of proton-rich nuclei can thus be obtained from their -T z isobaric analogues via E ( A,T,T z ) = E ( A,T, -T z )+2 b ( A,T ) T z , using a standard fit of the empirical b -coefficient [9], b = (0 . 7068 A 2 / 3 -0 . 9133) MeV, with the atomic mass evaluation (AME2011) [23] for known neutron-rich nuclei. Moreover, we include for comparison the extrapolated values of AME2011, although the IMME is considered to be more accurate. Figure 1 shows the calculated ground-state energies, obtained from exact diagonalization in the valence spaces with NN-only and NN+3N forces, compared with the AME2011 experimental values and extrapolation, and with the IMME. As expected, the IMME reproduces well experimental data. It finds 22 Si to be bound, though only by 10 keV, with respect to 20 Mg. In the calculations based on NN forces only, we see a systematic overbinding throughout the isotone chain, which becomes more pronounced for larger mass number. Three-nucleon forces provide key repulsive contributions to ground-state energies and good agreement with experiment is obtained in both valence spaces. The extended-space predictions become more bound beyond 20 Mg, the last measured isotone. For both valence spaces, the proton dripline is predicted at 20 Mg, though 22 Si is unbound with respect to 20 Mg by only 0 . 1 MeV in the extended space, compared with 1 . 6 MeV in the sd -shell. This makes a prediction of the dripline difficult, and an experimental measurement of the 22 Si ground-state energy would present a decisive constraint for 3N forces. All calculations find a sharp decrease in binding energy past 22 Si, clearly indicating the dripline has been reached. Amore detailed picture can be developed from the oneand two-proton separation energies given in Table II. S p and S 2 p are key quantities for determining two-proton emission candidates. In general, we find good agreement between our calculations and the experimental (and IMME) values. While the sd -shell energies are slightly closer to experiment for lighter isotones, the extendedspace calculations agree best with the IMME beyond A = 20, approximately the same point, 21 O, at which the added valence-space orbitals become important in the oxygen isotopes in this framework [2]. Spectroscopic data in the N = 8 isotones exists to 20 Mg. In Fig. 2, we compare the experimental low-lying states in 18 Ne, 19 Na, and 20 Mg to those calculated with NN+3N forces in the sdf 7 / 2 p 3 / 2 valence space. Calculations with 3N forces in the sd -shell give very similar spectra up to 19 Na, while for 20 Mg, 21 Al, and 22 Si they are more compressed than in Fig. 2. In 18 Ne we find good agreement for the first excited 2 + and 4 + states. The ground state and first two excited states in 19 Na have been measured with tentative spin and parity assignments [25, 26]. The ordering of the first two states in our calculation disagrees with the tentative assignments, but the spacing between them is only 0 . 1 MeV. The 1 / 2 + state is predicted in our calculation close to the 1 / 2 + in the mirror 19 O, but 0 . 9 MeV above experiment. This 19 O19 Na 1 / 2 + difference is a clear example of the Thomas-Ehrman effect [25, 27]. Since in our 19 Na calculation the s 1 / 2 orbital is unbound, continuum coupling is expected to reduce the 1 / 2 + energy. In 20 Mg, only information on the first excited state has been published [28], but a second excited state has been measured recently [29]. While a tentative assignment of 4 + was given to this state, we predict two close-lying states (2 + and 4 + ) at very similar energy. In our predictions for 21 Al and 22 Si, we note the high 2 + state in 22 Si as a possible indication of a subshell closure analogous to 22 O [2]. For all cases, the differences of excitation energies between these protonrich nuclei and the corresponding mirror oxygen isotopes are less than 0 . 8 MeV. Next, we show in Fig. 3 the ground-state energies of the N = 20 isotones from 42 Ti to 48 Ni, where the IMME also reproduces well the limited experimental data [30]. Calculations with NN forces already lead to a reasonable description of experiment, with energies only modestly overbound (within 1 MeV) beyond 45 Mn. When 3N forces are included, the additional repulsion systematically improves the agreement with data. The extended-space calculations agree very well with the IMME throughout the isotone chain, while the pf -shell results deviate for 47 Co and 48 Ni. In all calculations the proton dripline is robustly predicted at 46 Fe. The one- and two-proton separation energies are given in Table III. The experimental and IMME values generally fall within the NN+3N calculations in the pf and pfg 9 / 2 valence spaces. The difference in S p and S 2 p between the two calculations only becomes larger than 0 . 7 MeV for 46 Fe, 47 Co and 48 Ni. This indicates that, in our framework, the g 9 / 2 orbital becomes relevant around A = 47 and provides extra binding, similar to the calcium isotopes [3, 4]. Indeed, our pfg 9 / 2 result for S 2 p of 48 Ni is only 0 . 3 MeV larger than recent experiment [12, 13]. Spectroscopic data is only available for 42 Ti in the N = 20 isotones. We show the predicted spectra based on NN+3N calculations in the pfg 9 / 2 valence space in comparison with experiment in Fig. 4. The energies of the first 2 + , 4 + and 6 + are well reproduced. There are two observed states between 2 + 1 and 4 + 1 that do not appear in our calculation. We attribute these to neutron (4p2h) excitations, expected around 40 Ca [31]. For the remaining isotones, we show our predictions for the energies of the first five excited states below 5 MeV. Similar to 22 Si, we note the high energy of the 2 + state in 48 Ni as a tentative closed subshell signature. The excitation energy difference with respect to the mirror calcium isotopes is smaller than 0 . 3 MeV, in agreement with the experimental knowledge in this region [32]. The calculated spectra in the pf -shell are similar, though modestly compressed, up to 44 Cr, and more compressed beyond. In summary, we have presented the first study of 3N forces in proton-rich medium-mass nuclei. Our results for ground- and excited-state energies are in very good agreement with experiment, including the prediction of a recently discovered state in 20 Mg [29]. A future measurement of the ground-state energy of 22 Si will provide an important constraint for 3N forces. We make predictions for the unexplored spectra of the N = 8 and N = 20 isotones. Our extended-space calculations for the ground-state energies are of similar quality as empirical IMME predictions, which is very promising for studying isospin symmetry breaking in medium-mass nuclei based on chiral EFT interactions. Our work presents a bridge to future studies, based on nuclear forces, of exotic nuclei with proton and neutron valence degrees of freedom. We thank M. Pfutzner, B. Blank, I. Mukha and A. Poves for helpful discussions, and the TU Darmstadt for hospitality. This work was supported by the US DOE Grant DE-FC02-07ER41457 (UNEDF SciDAC Collaboration), DE-FG02-96ER40963 (UT), the Helmholtz Alliance Program of the Helmholtz Association, contract HA216/EMMI 'Extremes of Density and Temperature: Cosmic Matter in the Laboratory', the BMBF under Contract No. 06DA70471, and the DFG Grant SFB 634. Part of the calculations were performed on Kraken at the National Institute for Computational Sciences. Physics, 1990, Vol. 364, p. 1; M. Hjorth-Jensen, T. T. S. Kuo and E. Osnes, Phys. Rept. 261 , 125 (1995).", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "Cosmology of a Friedmann-Lamaˆıtre-Robertson-Walker 3-brane, Late-Time Cosmic Acceleration, and the Cosmic Coincidence", "content": "Ciaran Doolin 1 and Ishwaree P. Neupane 1, 2, 3 1 Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch 8041, New Zealand 2 Centre for Cosmology and Theoretical Physics (CCTP), Tribhuvan University, Kathmandu 44618, Nepal 3 CERN, Theory Department, CH-1211 Geneva 23, Switzerland Alate epoch cosmic acceleration may be naturally entangled with cosmic coincidence - the observation that at the onset of acceleration the vacuum energy density fraction nearly coincides with the matter density fraction. In this Letter we show that this is indeed the case with the cosmology of a Friedmann-Lamaˆıtre-Robertson-Walker (FLRW) 3-brane in a five-dimensional anti-de Sitter spacetime. We derive the four-dimensional effective action on a FLRW 3-brane, from which we obtain a mass-reduction formula, namely, M 2 P = ρ b / | Λ 5 | , where M P is the effective (normalized) Planck mass, Λ 5 is the five-dimensional cosmological constant, and ρ b is the sum of the 3-brane tension V and the matter density ρ . Although the range of variation in ρ b is strongly constrained, the big bang nucleosynthesis bound on the time variation of the effective Newton constant G N = (8 πM 2 P ) -1 is satisfied when the ratio V/ρ /greaterorsimilar O (10 2 ) on cosmological scales. The same bound leads to an effective equation of state close to -1 at late epochs in accordance with astrophysical and cosmological observations. PACS numbers: 98.80.Cq, 04.65.+e, 11.25.Mj Introduction .- The paradigm that the observable Universe is a branelike four-dimensional hypersurface embedded in a five- and higher-dimensional spacetime [1] is fascinating as it provides new understanding of the feasibility of confining standard-model fields to a D(irichlet)3-brane [2]. This revolutionary idea, known as brane-world proposal [3-6], is supported by fundamental theories that attempt to reconcile general relativity and quantum field theory, such as string theory and M theory [7]. In string theory or M theory, gravity is a truly higher-dimensional theory, becoming effectively four-dimensional at lower energies. This behavior is seen in five-dimensional brane-world models in which the extra spatial dimension is strongly curved (or 'warped') due to the presence of a bulk cosmological constant in five dimensions. Warped spacetime models offer attractive theoretical insights into some of the significant questions in particle physics and cosmology, such as why there exists a large hierarchy between the 4D Planck mass and electroweak scale [3] and why our late-time low-energy world appears to be fourdimensional [8, 9]. For viability of the brane-world scenario, the model must provide explanations to key questions of the concurrent cosmology, including - (i) why the expansion rate of the Universe is accelerating and (ii) why the density of the cosmological vacuum energy (dark energy) is comparable to the matter density - the so-called cosmic coincidence problem. In this Letter, we show that the cosmology of a Friedmann-Lamaˆıtre-Robertson-Walker (FLRW) 3-brane in a five-dimensional anti-de Sitter (AdS) spacetime can address these two key questions as a single, unified cosmological problem. Our results are based on the exact cosmological solutions and the four-dimensional effective action obtained from dimensional reduction of a five-dimensional bulk theory. Model .- A 5D action that helps explore various features of low-energy gravitational interactions is given by where M 5 is the fundamental 5D Planck mass, L b m is the brane-matter Lagrangian, and V is the brane tension. The bulk cosmological constant Λ 5 has the dimension of ( length ) -2 , similar to that of the Ricci scalar R 5 . As we are interested in cosmological implications of a warped spacetime model, we shall write the 5D metric ansatz in the following form where k ∈ {-1 , 0 , 1 } is a constant which parametrizes the 3D spatial curvature and d Ω 2 is the metric of a 2-sphere. The equations of motion are given by with ρ b ≡ ρ + V and p b ≡ p -V , where ρ and p are the density and the pressure of matter on a FLRW 3-brane. The parameter h that appeared in Eq. (1) is the determinant of four-dimensional components of the bulk metric, i.e., h µν ( x µ ) = g µν ( x µ , y = 0) . Bulk Solution .- Using the restriction G 0 5 = 0 and choosing the gauge n 0 ≡ n ( t, y = 0) = 1 , in which case t is the proper time on the brane, one finds that the warp factor a ( t, y ) that solves Einstein's equations in the 5D bulk and equations on a FLRW 3-brane is given by [10] where a 0 ≡ a ( t, y = 0) and ¯ ρ b = ρ b /M 3 5 . The form of n ( t, y ) is obtained using n = ˙ a/ ˙ a 0 . The integration constant C enters into the brane analogue to the first Friedmann equation where H ≡ ˙ a 0 /a 0 is the Hubble expansion parameter. The brane analogue to the second Friedmann equation is where w b ≡ p b /ρ b is the effective equation of state on a FLRW 3-brane. The brane evolution equations are quite different from Friedmann equations of standard cosmology: the distinguishing features are (i) the appearance of the brane energy density in a quadratic form, (ii) the dependence of H 2 on Λ 5 , and (iii) the appearance of the bulk radiation term C /a 4 0 . If the radiation energy from bulk to brane (or vice versa) is negligibly small, then it would be reasonable to set C = 0 . In the following we assume that C = 0 unless explicitly shown. Cosmic acceleration .- With V = const, the brane energyconservation equation, ˙ ρ b +3 H ( ρ b + p b ) = 0 , reduces to where w = p/ρ is the EOS of matter on the 3-brane and ρ ∗ is a constant. With Eq. (7), Eq. (5) takes the following form: where Λ 4 ≡ Λ 5 2 + ¯ V 2 12 , ¯ ρ ∗ = ρ ∗ M -3 , and ¯ V ≡ V/M 3 5 . This admits an exact solution when k = 0 , which is given by where H 0 ≡ √ Λ 4 3 and ν ≡ ¯ V 6 H 0 . From this we find that the Hubble expansion parameter is given by The deceleration parameter changes sign from positive to negative when γH 0 t ∼ 1 . 1 (cf. Fig. 1). This implies a transition from decelerating to accelerating expansion. The onset time of acceleration depends on ν but only modestly; generically, we expect that ν = ¯ V / 6 H 0 = √ ¯ V 2 / 12Λ 4 /greaterorsimilar O (1) . In the Randall-Sundrum (RS) limit ( Λ 4 = 0 ), we find that a 0 ( t ) ∝ ( 2 t + γνH 0 t 2 ) 1 /γ , which shows that the scale factor scales as t 1 /γ at early epochs and as t 2 /γ as late epochs. The crossover takes place when H 0 t ∼ 2 / ( γν ) . In the generic case with Λ 4 > 0 , the scale factor grows in the beginning as t 1 /γ (as in the Λ 4 = 0 case), but at late epochs it grows almost exponentially, a 0 ( t ) ∝ [ e γH 0 t -2 ν/ (1 + ν ) ] 1 /γ . Cosmic coincidence : Consider the Friedmann constraint /negationslash where As shown in Fig. 1, Ω ¯ ρ 2 starts out as the largest fraction around H 0 t /greaterorsimilar 0 , but Ω ¯ ρ quickly overtakes it when H 0 t /greaterorsimilar 0 . 15 . Gradually, Ω 4 , which measures the bare vacuum energy density fraction, surpasses these two components. Notice that Ω Λ + Ω ¯ ρ /similarequal 1 when H 0 t /greaterorsimilar 0 . 5 . We can see, for ν /similarequal 2 , that Ω m /similarequal 0 . 26 and Ω Λ /similarequal 0 . 74 when H 0 t /similarequal 0 . 75 . The crossover time between the quantities Ω m ≡ Ω ¯ ρ + Ω ¯ ρ 2 and Ω 4 depends modestly on ν . This provides strong theoretical evidence that dark energy may be the dominant component of the energy density of the Universe at late epochs, and it is consistent with results from astrophysical observations [11, 12]. Unlike some other explanations of cosmic coincidence, such as quintessence in the form of a scalar field slowly rolling down a potential [13], the explanation here of cosmic coincidence does not require that the ratio Ω m / Ω Λ be set to a specific value in the early Universe. Because of the modification of the Friedmann equation at very high energy, namely, H ∝ ρ , new effects are expected in the earlier epochs and that could help to address the challenges that the Λ CDM cosmology faces at small (subgalaxy) scales [12]. Effective Equation of State .- Eq. (6) can be written as As H 0 t →∞ , H → H 0 , q →-1 , and when ¯ V /greatermuch ¯ ρ , which generally holds on large cosmological scales, we obtain This is consistent with the result inferred from WMAP7 data: w b = -0 . 980 ± 0 . 053 ( Ω k = 0 ) and w b = -0 . 999 +0 . 057 -0 . 056 ( Ω k = 0 ) [12]. In the earlier epochs with γH 0 t /lessorsimilar 1 . 2 , we have w b > -1 / 3 , showing that a transition from matter to dark-energy dominance is naturally realized in the model. In Fig. 2 we exhibit the parameter space for { ν, H 0 t } with a specific value of w b at present. If any two of the variables { ν, H 0 t, w } are known, then the remaining one can be calculated. Typically, if ν /similarequal 2 and H 0 t /similarequal 0 . 75 , then w b /similarequal -0 . 985 . In particular, the effective equation of state w b is given by where ζ ≡ V/ρ . For brevity, suppose that the brane is populated mostly with ordinary (baryonic) matter plus cold dark matter, so w /similarequal 0 ( γ /similarequal 3 ). In this case, cosmic acceleration occurs when ζ > 1 / 2 (or w b < -1 / 3 ). This result is consistent with the behavior of the 4D effective potential. Dimensionally reduced action .- The gravitational part of the action (1) is where the prime (dot) denotes a derivative with respect to y ( t ). In order to derive from this a dimensionally reduced 4D effective action, we may separate a '' and n '' into nondistributional (bulk) and distributional (brane) terms Using n = ˙ a/ ˙ a 0 and the solution (4), the nondistributional part of the action (17) is evaluated to be-∗ where R 4 = 6 ( a 0 /a 0 + ˙ a 2 0 /a 2 0 + k/a 2 0 ) . In the above we have employed the background solution (4) and integrated out the y -dependent part of the 4D metric. The distributional part of the action (17) is evaluated to be The sum of I 1 and I 2 gives a dimensionally reduced action with the effective potential given by The finiteness of Newton constant is required at low-energy scale where one ignores the effects of ordinary matter field on the brane. In this limit, the extra dimensional volume is finite in the same way as in canonical Randall-Sundrum models. In the presence of matter fields, we must consider a normalized Planck mass which generically depends on 4D coordinate time, since ρ b is time dependent. From Eq. (21) we read off the normalized Planck mass In the limit that Λ 4 = 0 and V /greatermuch ρ , Eq. (23) reduces to the formula or identification 8 πG (0) N /similarequal V/ (6 M 3 5 ) used in [10], where G (0) N is the bare Newton's constant identified in the low-energy limit (or when the matter density is much lower than the brane tension). The mass reduction formula for RS flat-brane models [4], M 2 P = M 3 5 √ -6 / Λ , is obtained as a special limit of our result, namely, M 3 5 Λ 5 ≡ Λ , ρ = 0 , and Λ 4 = 0 . We make a remark here in regard to the scenario with Λ 5 = 0 . The Dvali-Gabadadze-Porrati model [14] corresponds to a flat 5D bulk. In their model, it is argued that R 4 is generated from loop-level coupling of brane matter to the 4D graviton. At least at a classical level, R 4 is not generated in the dimensional reduction of the 5D action if Λ 5 = 0 , and this is exactly what we found. AdS/FLRW-cosmology correspondence .- In the limit ζ ≡ V/ρ /greatermuch 1 , the 4D effective potential is approximated by Note that Λ 4 → 3 4 Λ e ff as ζ →∞ . This result, which relates the bare cosmological constant to the 4D effective potential in the limit ρ → 0 , is a direct manifestation of AdS/FLRWcosmology correspondence. In a general case with finite x , /negationslash for any value of 3D curvature constant k . The boundary action (20) is crucial to correctly reproduce the RS limit, i.e., Λ e ff → 0 as ρ → 0 and Λ 4 → 0 . If ρ > 0 , then Λ e ff = 0 even if Λ 4 = 0 . This shows that the vacuum energy on the brane or brane tension need not be directly tied to the effective cosmological constant on a FLRW 3-brane. With Λ 4 = 0 , there is no accelerated expansion of the Universe, at least in a late epoch. To quantify this, take γ = 4 . We then find Λ eff = -¯ ρ 2 / 18(1 + ζ ) < 0 , implying a decelerating Universe. If γ = 3 , then Λ eff > 0 in the range 0 . 177 /lessorsimilar ζ < 2 . 822 , but in this range w b > -1 / 3 . A small deviation from RS fine-tuning can naturally lead to accelerated expansion; the onset time of this acceleration primarily depends on the ratio Λ 4 / ¯ V 2 . The larger deviations from RS fine-tuning imply an earlier onset of cosmic acceleration. This can be seen by plotting the 4D effective potential (cf. Fig. 3) or analyzing the solution given by Eq. (9). For Λ 4 /greaterorsimilar 0 , the model correctly predicts the existence of a decelerating epoch which is generally required to allow cosmic structures to form. Constraints from big bang nucleosynthesis (BBN) .- In the limit V /greatermuch ρ , so ρ b /similarequal V , Eq. (23) is approximated as Cosmological observations, especially BBN constraints, impose the lower limit on V , namely, From Eq. (23) we find the time variation of the effective Newton constant (or 4D gravitational coupling) From Eq. (9) we can see, particularly at late epochs or when H 0 t > 0 . 5 , that ˙ ρ/ρ →-γH 0 . The BBN bound, namely which is also comparable to current constraints from lunarlaser ranging [15], translates to the condition that ζ > 10 2 (and -1 < w b ∼ -0 . 99 ) on large cosmological scales. Further constraints .- Next we consider perturbations about the background metric given by Eq. (2), along with the solution (4). The transverse traceless part of graviton fluctuations δg ij = h ij ( x µ , y ) = ∑ ϕ m ( t ) f m ( y ) e i ˜ k · x leads to a complicated differential equation for the spatial and temporal functions, which take remarkably simple forms at y = 0 + , namely, where m 2 is a separation constant. Equation (29b) is equivalent to a standard time-dependent equation for a massive scalar field in 4D de Sitter spacetime. The masses of KaluzaKlein excitations are bounded by m 2 > 4 H 2 / 9 , in which case the amplitudes of massive KK excitations rapidly decay away from the brane. This, along with a more stringent bound coming from Eq. (29a), implies that H < 3 V/ (8 M 3 5 ) . This is similar to the bound coming from the background solution, namely Λ 4 < ¯ V 2 / 12 or H 0 < V/ (6 M 3 5 ) . Conclusion .- Brane-world cosmology with a small deviation from RS fine-tuning ( Λ 4 = 0 ) is able to produce a latetime cosmic acceleration. The model puts the constraints The smaller the deviation from the RS fine-tuning the larger the duration of cosmic deceleration, prior to the late-epoch acceleration. For the background solution (9), the brane tension is not fine-tuned but only bounded from below. However, once the ratio Λ 4 /M 2 P is fixed in accordance with the observational bound Λ 4 /M 2 P ∼ 10 -120 , the ratio ¯ V / 6 H 0 also gets fixed, in which case there is a fine-tuning between the bulk cosmological constant and brane tension. The method of dimensional reduction gave a simple formula, M 2 P = ρ b / | Λ 5 | , which relates the normalized Planck mass M P to the matter-energy density on the brane and the bulk cosmological constant. As ρ b is time varying, this suggests that a mass normalized gravitational constant is timedependent. This is acceptable since analysis of primordial nucleosynthesis has shown that G N can vary, although the range of variation is strongly constrained. The BBN bound is satisfied when the ratio V/ρ is larger than O (10 2 ) or when the effective equation of state -1 < w b ∼ -0 . 99 . At cosmological scales the background evolution of a FLRW 3-brane becomes increasingly similar to Λ CDM but the model is essentially different from Λ CDM at earlier epochs. With precise determination of the present deceleration parameter or the effects of a time varying equation of state, we can hope to explore the late-time role of high-energy field theories in the form of brane worlds and many new physical ideas. We wish to acknowledge useful conversations with Naresh Dadhich, Radouane Gannouji, M. Sami, Misao Sasaki, Tetsuya Shiromizu, Shinji Tsujikawa and David Wiltshire. I.P.N. is supported by the Marsden Fund of the Royal Society of New Zealand (RSNZ/M1125). (2010).", "pages": [ 1, 2, 3, 4, 5 ] } ]

[ { "title": "Action and entanglement in gravity and field theory", "content": "Yasha Neiman 1, ∗ 1 Institute for Gravitation & the Cosmos and Physics Department, Penn State, University Park, PA 16802, USA (Dated: June 8, 2021) In non-gravitational quantum field theory, the entanglement entropy across a surface depends on the short-distance regularization. Quantum gravity should not require such regularization, and it's been conjectured that the entanglement entropy there is always given by the black hole entropy formula evaluated on the entangling surface. We show that these statements have precise classical counterparts at the level of the action. Specifically, we point out that the action can have a nonadditive imaginary part. In gravity, the latter is fixed by the black hole entropy formula, while in non-gravitating theories, it is arbitrary. From these classical facts, the entanglement entropy conjecture follows by heuristically applying the relation between actions and wavefunctions. PACS numbers: 03.65.Ud,04.20.Fy,04.70.Dy", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "The Bekenstein-Hawking formula for black hole entropy [1-3] is an important clue for any attempt at quantum gravity. In Planck units with c = /planckover2pi1 = 8 πG = 1, it relates the entropy σ H of a black hole with horizon H to the horizon's area A as: In classical diff-invariant theories of gravity other than General Relativity (GR), the entropy formula (1) is modified according to Wald's prescription [4, 5]. Despite much progress, the precise physical meaning and the range of applicability of the black hole entropy formula remains unclear. One possibility was recently articulated by Bianchi and Myers [6]. They conjecture that eq. (1) (generalized a-la Wald) is a universal formula for the entanglement entropy across a surface H , whenever the relevant states admit a semiclassical spacetime interpretation. Similar statements were made previously by many authors, as reviewed in [6]. This conjecture is in sharp contrast with the situation in nongravitational quantum field theory. There, the entanglement entropy between adjoining regions is UV-divergent. Given a short-distance regulator, it will depend on the cutoff scale, as well as on the specific field theory. Thus, the Bianchi-Myers conjecture implies that gravity provides a universal short-distance regulator (at the Planck scale) for all possible sets of matter fields. Strictly speaking, entanglement entropy is only defined when the total state of the system is pure. This will not be the case in general, especially if the total state is itself defined in a bounded region of space. We therefore consider a slight generalization of the Bianchi-Myers conjecture. Let there be a semiclassical state in a spatial region AB , separated into subregions A and B by a surface H . We can then state the conjecture, along with the UVsensitivity of entanglement in non-gravitating theories, as: Here, σ A , σ B and σ AB are the von Neumann entropies of the state in the corresponding regions, and σ H is the black hole entropy formula evaluated on H . The LHS of (2) is known as the mutual information. When the overall state is pure, we have σ AB = 0, while σ A and σ B both equal the entanglement entropy. This is the reason for the factor of 2 on the RHS.", "pages": [ 1 ] }, { "title": "SUMMARY OF RESULTS", "content": "In this paper, we point out that eq. (2) has a classical counterpart in the actions of adjoining spacetime processes. Consider a spacetime region of the type depicted in figure 1(a). It describes an evolution between initial and final states on spacelike hypersurfaces, joined together along an 'entangling surface' H . This can be viewed in two different ways, illustrated in figures 1(b,c). In one view, there are two separate causally closed processes, taking place in each of the spatial regions A and B , with respective actions S A and S B . In the other view, there is a single process taking place in the larger region AB with action S AB , such that the spacelike hypersurfaces housing the initial and final states happen to intersect at the surface H . Our central statement is as follows. In a classical theory of gravity that is second-order in time derivatives, the actions S A , S B and S AB satisfy the relation: where σ H is again the black hole entropy formula evaluated on H . Eq. (3) encompasses several properties of the action that will be surprising for many readers. First, as noticed by Brill and Hayward [7], the gravitational action is non-additive. Second, as noticed by the author [8-10], it has an imaginary part that is closely related to the black hole entropy formula. Third, in nongravitational field theory, these properties do not necessarily disappear, but instead their magnitude becomes undetermined. The above features are all related to the corner contributions to the action's boundary term. The necessary details will be reviewed and developed in the sections that follow. The similarity between eqs. (2) and (3) seems more than just superficial. In fact, on a heuristic level, the formula (3) for the action's non-additivity implies the mutual information conjecture (2). The reasoning is as follows. First, action differences exponentiate into wavefunction ratios: These square into ratios of density matrix eigenvalues: Finally, the logarithms of density matrix eigenvalues give entropies, providing the desired link from (3) to (2): These considerations suggest a new role for the classical action: through its non-additive imaginary part, the action knows about the entanglement entropy of semiclassical states. However, as stated in (3) and will be shown below, this can only be put to good use in gravitational theories: otherwise, there is too much freedom in the action's definition.", "pages": [ 1, 2 ] }, { "title": "NON-ADDITIVITY OF THE GRAVITY ACTION", "content": "In this section, we review the relevant facts from [7] on the non-additivity of the GR action due to corner contributions. We postpone the discussion of the action's imaginary part by considering first the Euclidean theory, where the action is real. While we consider GR for simplicity, the results extend straightforwardly to Lovelock gravity [11], using the corner contributions from [8] and their behavior under 2 π rotations from [10, 12]. The action of GR in a spacetime region Ω with boundary ∂ Ω is given by: Here, the ± signs correspond to Euclidean and Lorentzian spacetime, respectively. g µν is the spacetime metric (with mostly-minus signature in the Lorentzian), g and R are its determinant and Ricci scalar, Λ is the cosmological constant, and L M is the (minimally coupled) matter Lagrangian. In the boundary term [13, 14], h ab is the intrinsic metric, h is its determinant, n µ is the boundary normal oriented such that n µ v µ > 0 for outgoing vectors v µ , and K = ∇ a n a is the trace of the extrinsic curvature. C is an arbitrary functional of h ab and matter fields that doesn't contain normal derivatives. Consider now a region shaped as in figure 1, but in Euclidean spacetime. At the surface H , the boundary has a corner: it is non-differentiable, and turns suddenly by a finite angle. At such surfaces, the extrinsic curvature has a delta-function singularity. When the K term in (7) is integrated through this singularity, it picks up a 'corner contribution' of A ( θ -π ) [15, 16], where A is the corner's area, and θ is the dihedral angle between the intersecting hypersurfaces. The crucial point is that the πA pieces in the corner contributions are non-additive under the gluing together of spacetime regions. In our setup, this plays out as follows. Let θ A,B be the dihedral angles indicated in figure 1(a). Then the corner contributions at H in figure 1(b) read: After the gluing, i.e. in figure 1(c), the relevant dihedral angles are instead 2 π -α and θ A + θ B + α , with α as indicated in figure 1(a). The corner contributions at H become: The other terms in the action (7) are additive. Though the C term may have singularities at H due to discontinuities in the derivatives of h ab , the transition from figure 1(b) to 1(c) only changes the order in which these are integrated. Thus, the non-additivity of the action comes entirely from the extrinsic curvature term, and reads: In higher-curvature Lovelock gravity, the corner contributions [8] are more complicated, involving integrals over the angle of n µ . The difference S A + S B -S AB can then be expressed as an integral over a full turn. From the analysis in [10, 12], one again obtains the relation (10) with the black hole entropy formula σ H (but not with the surface area A ).", "pages": [ 2, 3 ] }, { "title": "IMAGINARY PART OF THE GRAVITY ACTION", "content": "We now return to Lorentzian GR. For rotation angles around timelike corner surfaces, the situation is the same as in the Euclidean, up to signs. However, in figure 1, the surface H is spacelike, while the dihedral angles around it are boosts in a Lorentzian plane. Normally, one considers boost angles within a single quadrant of the plane, where they span the entire real range ( -∞ , ∞ ). However, this is not enough for the corners in figure 1(b): the boundary normal there goes from past-pointing timelike on the initial hypersurface to future-pointing timelike on the final one. This requires crossing two quadrant boundaries in the Lorentzian plane. To evaluate such corner contributions, we must assign boost angles to the entire plane, rather than just to a single quadrant. This assignment is illustrated in figure 2. It was found in [17] by analytically continuing the inverse trigonometric functions, and was rederived in [8] using a contour integral. The crucial point is that in order to span the entire plane, the angle must become complex. Specifically, the angle picks up an imaginary contribution πi/ 2 per quadrant crossing, i.e. per signature flip of the rotating vector. The angle for a full turn is then 2 πi , in analogy with the Euclidean 2 π . In the contour-integral approach, the πi/ 2 jumps come from bypassing poles in a dz/z integral, after decomposing the rotating vector as n µ ∼ L µ + z/lscript µ in a null basis ( L µ , /lscript µ ). The imaginary parts of corner angles plug into the action's corner contributions, making the action complex. While the sign of the angles' real part is determined by the sign of the boundary term in (7), one must make a separate choice for the sign of the imaginary part. This reflects the two separate P, T transformations that are present in Lorentzian signature. The choice argued for in [8] is the one that makes Im S positive. This leads to amplitudes e iS that are exponentially damped rather than exploding. In analogy with (8), we now have the corner contributions from H in figure 1(b) as: where θ A,B are the (real, positive) dihedral angles from figure 1(a). For figure 1(c), the relevant dihedral angles are now 2 πi -α and θ A + θ B + α , with Im α = π . The corresponding corner contributions to the action read: Thus, the action's non-additivity takes the form: which establishes the upper line in eq. (3). The analogous result in Lovelock gravity again follows similarly, using the results of [8, 10] for the imaginary parts of the corner integrals. We expect the result to also hold for arbitrary two-time-derivative matter couplings. For example, a conformally coupled scalar is handled straightforwardly, as discussed in [10].", "pages": [ 3 ] }, { "title": "THE AMBIGUITY IN NON-GRAVITATIONAL ACTIONS", "content": "It remains to establish the lower line in eq. (3), i.e. that non-gravitational field theory actions have an arbitrary non-additive imaginary part. This is surprising at first, since the matter Lagrangian L M in (7) has no such property. The crucial point is that when the metric is non-dynamical, we have a lot of freedom in redefining the action without disturbing its variational principle. For example, we can add to the Lagrangian any functional of the metric. More to the point, we can add to the boundary term any functional of the metric h ab , the extrinsic curvature K ab and the matter fields (but not the matter fields' normal derivatives). In particular, we can add any multiple of the GR boundary term or its Lovelockgravity generalizations [18]. In this way, we can write different actions that vary identically under variations of the matter fields, while having arbitrary non-additive imaginary parts. In gravitational theories, this freedom is not present. There, the extrinsic curvature term is fixed by the Lagrangian, so as to satisfy a variational principle where h ab but not K ab is held fixed on ∂ Ω. The ambiguity in the non-gravitational action is not just an esoteric loophole. It's manifested whenever we take the limit from a gravitating to a non-gravitating theory by sending the coupling G to zero. Consider again the GR action (7), setting Λ = C = 0 and restoring the units of G : As we send G to zero, this action does not reduce to the L M term! By the Einstein equations, the bulk EinsteinHilbert term has a finite limit: where T ν µ is the matter stress-energy tensor. As for the York-Gibbons-Hawking boundary term, it diverges: the boundary's extrinsic curvature does not become small as G is sent to zero. In particular, the non-additive imaginary part (13) diverges as: This is consistent with the expectation that the mutual information diverges as the short-distance cutoff G is removed.", "pages": [ 4 ] }, { "title": "DISCUSSION", "content": "We have shown that the action in gravitational and non-gravitational field theories has a property (3) that is directly analogous to the mutual information conjecture (2). The gravitational part of the result is demonstrated for GR with minimally coupled matter and for Lovelock gravity, and probably holds for all two-time-derivative theories. The restriction to two time derivatives is necessary to ensure a standard variational principle, and thus a well-motivated boundary term. Put differently, it implies that the variational principle makes sense at arbitrarily short distance scales. This requirement is consistent with the fact that both the mutual information and the action's corner contributions are short-distance effects. We've seen that the conjecture (2) can be heuristically derived from the action's property (3). This suggests that the Lorentzian classical action plays a role in encoding state statistics, in addition to its role in the variational principle (as discussed in [8], the variational principle is unaffected by the action's imaginary part). These issues should be explored further. A possible avenue may be the general-boundary approach to mixed and entangled states, along the lines of [19].", "pages": [ 4 ] }, { "title": "Acknowledgements", "content": "I am grateful to Eugenio Bianchi, Norbert Bodendorfer and Rob Myers for discussions. This work is supported in part by the NSF grant PHY-1205388 and the Eberly Research Funds of Penn State.", "pages": [ 4 ] } ]

[ { "title": "Spacetime Transformations from a Uniformly Accelerated Frame ∗", "content": "Yaakov Friedman and Tzvi Scarr Jerusalem College of Technology Departments of Mathematics and Physics P.O.B. 16031 Jerusalem 91160, Israel e-mail: [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We use Generalized Fermi-Walker transport to construct a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K ' to an inertial frame K . The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz . We compute the metric at an arbitrary point of a uniformly accelerated frame. PACS : 03.30.+p ; 02.90.+p ; 95.30.Sf ; 98.80.Jk. Keywords : Uniform acceleration; Lorentz transformations; Spacetime transformations; Fermi-Walker transport; Weak Hypothesis of Locality", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The accepted physical definition of uniformly accelerated motion is motion whose acceleration is constant in the comoving frame . This definition is found widely in the literature, as early as [1], again in [2], and as recently as [3] and [4]. The best-known example of uniformly accelerated motion is one-dimensional hyperbolic motion . Such motion is exemplified by a particle freely falling in a homogeneous gravitational field. Fermi-Walker transport attaches an instantaneously comoving frame to the particle, and one easily checks that the particle's acceleration is constant in this frame (see [5], pages 166-170, or section 5.2 below). In [6], we showed that 1D hyperbolic motion is not Lorentz invariant. These motions are, however, contained in a Lorentz-invariant set of motions which we call translation acceleration . Moreover, we introduced three new Lorentz-invariant classes of uniformly accelerated motion. For null acceleration, the worldline of the motion is cubic in the time. Rotational acceleration covariantly extends pure rotational motion. General acceleration is obtained when the translational component of the acceleration is parallel to the axis of rotation. A review of these explicit solutions appears in section 2. In this paper, we establish that all four types (null, translational, rotational and general) do, in fact, represent uniformly accelerated motion by showing that they have constant acceleration in the instantaneously comoving frame. Fermi-Walker transport will no longer be adequate here to define the comoving frame because we must now deal with rotating frames. Instead, we will use generalized Fermi-Walker transport . Similar constructions appear in [7] and [5]. Our construction begins in section 3.1, where we define the notion of a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. As mentioned above, the construction uses Generalized Fermi-Walker transport. This leads us, in section 3.2, to the definition of a uniformly accelerated frame . Here, we explain the connection between our approach and that of Mashhoon [7]. It is also here that we show that the four types of acceleration all have constant acceleration in the comoving frame. We also show here that if K ' and K '' are two uniformly accelerated frames with a common acceleration, then the spacetime transformations between K ' and K '' are Lorentz , despite the fact that neither K ' nor K '' is inertial. The main results appear in section 4. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K ' to an inertial frame K . We show that these transformations extend the Lorentz transformations between inertial systems. We also compute the metric at an arbitrary point of a uniformly accelerated frame. Section 5 is devoted to examples of uniformly accelerated frames and the corresponding spacetime transformations. We summarize our results in section 6.", "pages": [ 1, 2 ] }, { "title": "2 Four Lorentz-invariant Types of Uniformly Accelerated Motion", "content": "In [6], uniformly accelerated motion is defined as a motion whose four-velocity u ( τ ) in an inertial frame is a solution to the initial value problem where A µν is a rank 2 antisymmetric tensor. Equation (1) is Lorentz covariant and extends the 3D relativistic dynamics equation F = d p dt . The solutions to equation (1) are divided into four Lorentz-invariant classes: null acceleration, translational acceleration, rotational acceleration, and general acceleration. The translational class is a covariant extension of 1D hyperbolic motion and contains the motion of an object in a homogeneous gravitational field . In [6], we computed explicit worldlines for each of the four types of uniformly accelerated motion. We think of these worldlines as those of a uniformly accelerated observer . Recall that in 1 + 3 decomposition of Minkowski space, the acceleration tensor A of equation (1) has the form where g is a 3D vector with physical dimension of acceleration, ω is a 3D vector with physical dimension 1 / time, the superscript T denotes matrix transposition, and, for any 3D vector ω = ( ω 1 , ω 2 , ω 3 ), where ε ijk is the Levi-Civita tensor. The factor c in A provides the necessary physical dimension of acceleration. The 3D vectors g and ω are related to the translational acceleration and the angular velocity, respectively, of a uniformly accelerated motion. We raise and lower indices using the Minkowski metric η µν = diag(1 , -1 , -1 , -1). Thus, A µν = η µα A α ν , so Using the fact that the unique solution to (1) is given by the exponential function we found in [6] that the general solution to (1) is where ± α and ± iβ are the eigenvalues of A , and the D µ are appropriate constant fourvectors which depend on A and can be computed explicitly. By integrating u ( τ ), one obtains the worldline of a uniformly accelerated observer. The four classes indicated in (5) (null, translational, rotational and general) are Lorentzinvariant . The translational class is a covariant extension of 1D hyperbolic motion. The null, rotational, and general classes were previously unknown.", "pages": [ 2, 3, 4 ] }, { "title": "3 Uniformly Accelerated Frame", "content": "In this section, we use Generalized Fermi-Walker transport to define the notion of the comoving frame of a uniformly accelerated observer . We then show that in this comoving frame, all of our solutions to equation (1) have constant acceleration . We show that our definition of the comoving frame is equivalent to that of Mashhoon [7]. We also show that if two uniformly accelerated frames have a common acceleration tensor A , then the spacetime transformations between them are Lorentz , despite the fact that neither frame is inertial.", "pages": [ 4 ] }, { "title": "3.1 One-Parameter Family of Inertial Frames", "content": "First, we define the notion of a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. The coordinates in this family of comoving frames will be used as a bridge between the observer's coordinates and the coordinates in the lab frame K . The family of frames is constructed by Generalized Fermi-Walker transport of the initial frame K 0 along the worldline of the observer. In the case of 1D hyperbolic motion, this construction reduces to Fermi-Walker transport [8, 9]. In fact, Fermi-Walker transport may only be used in the case of 1D hyperbolic motion. This is because Fermi-Walker transport uses only a part of the Lorentz group - the boosts. This subset of the group, however, is not a subgroup , since the combination of two boosts entails a rotation. Generalized Fermi-Walker transport, on the other hand, uses the full homogeneous Lorentz group, and can be used for all four types of uniform acceleration: null, linear, rotational, and general. The construction of the one-parameter family { K τ : τ ≥ 0 } is according to the following definition. Definition 1. Let ̂ x ( τ ) be the worldline of a uniformly accelerated observer whose motion is determined by the acceleration tensor A , the initial four-velocity u (0) , and the initial position ̂ x (0) . We first define the initial frame K 0 . The origin of K 0 at time τ = 0 is ̂ x (0) . For the basis of K 0 , choose any orthonormal basis ̂ λ = { u (0) , ̂ λ (1) , ̂ λ (2) , ̂ λ (3) } . Next, we define the one-parameter family { K τ ( A, ̂ x (0) , ̂ λ ) } of inertial frames generated by the uniformly accelerated observer. For each τ > 0 , define K τ as follows. The origin of K τ at time τ is set as ̂ x ( τ ) . The basis of K τ is defined to be the unique solution λ ( τ ) = { λ ( κ ) ( τ ) : κ = 0 , 1 , 2 , 3 } , to the initial value problem We remark that the choice of the initial four-velocity u (0) for ̂ λ (0) is deliberate and required by Generalized Fermi-Walker transport. Claim 1. For all τ , we have λ (0) ( τ ) = u ( τ ) . This follows immediately from (1). Claim 2. The unique solution to (6) is This follows immediately from (4). Claim 3. For all τ , the columns of λ ( τ ) are an orthonormal basis. To prove this claim, it is enough to note that A is antisymmetric. Therefore, exp( Aτ/c ) is an isometry . Analogously to (5), the general solution to (6) is This claim follows from the fact that the type of acceleration is based solely on the eigenvalues of A . Claim 5. Let A denote the acceleration tensor as computed in the lab frame K , and let ˜ A ( τ ) denote the tensor as computed in the frame K τ . Then ˜ A ( τ ) is constant for all τ . To prove this claim, first note that λ ( τ ) is the change of matrix basis from K to K τ . Hence, using claim 2 and the fact that A and exp( Aτ/c ) commute, we have Claim 6. For all τ , we have λ ( τ ) ˜ A ( τ ) = Aλ ( τ ) . This follows from the first equality in (9).", "pages": [ 4, 5, 6 ] }, { "title": "3.2 Uniformly Accelerated Frame", "content": "Two frames are said to be comoving at time τ if at this time, the origins and the axes of the two frames coincide, and they have the same four-velocity. We now define the notion of a uniformly accelerated frame . Definition 2. A frame K ' is uniformly accelerated if there exists a one-parameter family { K τ ( A, ̂ x (0) , ̂ λ ) } of inertial frames generated by a uniformly accelerated observer such that at every time τ , the frame K τ is comoving to K ' . In light of this definition, we may regard our uniformly accelerated observer as positioned at the spatial origin of a uniformly accelerated frame. This approach is motivated by the following statement of Brillouin [10]: a frame of reference is a 'heavy laboratory, built on a rigid body of tremendous mass, as compared to the masses in motion.' Our construction of a uniformly accelerated frame should be contrasted with Mashhoon's approach [7], which is well suited to curved spacetime, or a manifold setting. There, the orthonormal basis is defined by where ˜ A is a constant antisymmetric tensor. Notice that the derivative of each of Mashhoon's basis vectors depends on all of the basis vectors, whereas the derivative of each of our basis vectors depends only on its own components. In particular, Mashhoon's observer's four-acceleration depends on both his four-velocity λ (0) and on the spatial vectors of his basis, while our observers's four-acceleration depends only on his four-velocity. This seems to be the more natural physical model: is there any a priori reason why the fouracceleration of the observer should depend on his spatial basis? We show now, however, that the two approaches are, in fact, equivalent. The two approaches are equivalent if we identify Mashhoon's tensor ˜ A with our own tensor ˜ A (0): ˜ A = ˜ A ( τ ) = ˜ A (0). Then, by equation (6) and claim 6, we have which is (10). We now show that all of our solutions of equation (1) have constant acceleration in the comoving frame. Let A be as in (2). Denote ˜ u = (1 , 0 , 0 , 0) T . By claim 1 and 2 we have Thus, the acceleration of the observer in the comoving frame is constant and equals ˜ g from the decomposition (2) for ˜ A . We end this section by showing that if K ' and K '' are two uniformly accelerated frames with a common acceleration tensor A , then the spacetime transformations between K ' and K '' are Lorentz , despite the fact that neither K ' nor K '' is inertial. Let K ' and K '' be two uniformly accelerated frames with a common acceleration tensor A . Without loss of generality, let the lab frame K be the initial comoving frame K 0 of K ' , so that the initial orthonormal basis of K ' is the identity I . Let ̂ λ be the initial orthonormal basis of K '' . Then the basis of K ' at time τ is λ ' ( τ ) = exp( Aτ/c ), while the basis of K '' at time τ is λ '' ( τ ) = exp( Aτ/c ) ̂ λ = λ ' ( τ ) ̂ λ . Thus, the change of basis from K ' to K '' is accomplished by the Lorentz transformation with matrix representation ̂ λ . This implies, in particular, that there is a Lorentz transformation from a lab frame on Earth to an airplane flying at constant velocity, since they are both subject to the same gravitational field.", "pages": [ 6, 7, 8 ] }, { "title": "4 Spacetime Transformations from a uniformly accelerated frame to the lab frame", "content": "In this section, we construct the spacetime transformations from a uniformly accelerated frame K ' to the lab frame K . This will be done in two steps.", "pages": [ 8 ] }, { "title": "Step 1: From K τ to K", "content": "First, we will derive the spacetime transformations from K τ to K . The idea here is as follows. Fix an event X with coordinates x µ in K . Find the time τ for which ̂ x ( τ ) is simultaneous to X in the comoving frame K τ . Define the 0-coordinate in K τ to be y (0) = cτ . Use the basis λ ( τ ) of K τ to write the relative spatial displacement of the event X with respect to the observer as y i λ ( i ) ( τ ) , i = 1 , 2 , 3. The spacetime transformation from K τ to K is then defined to be Transformations of the form (12) have a natural physical interpretation and were also used in [7]. Moreover, they extend the Lorentz transformations . To see this, let K ' be an inertial frame (simply set A = 0). We will show that the Lorentz transformations K ' → K can be written as in (12). Suppose that K ' moves with 3D velocity v = ( v, 0 , 0) with respect to K . Assume, as usual, that the observer located at the spatial origin of K ' was at the origin of K at time t = 0. Let x µ = ( x 0 = ct, x i ) denote the coordinates of an event in K , and let y ( µ ) denote the event's coordinates in K ' . In K , the observer has constant four-velocity ̂ u = γ (1 , v/c, 0 , 0), and the observer's worldline in K is ̂ x ( τ ) = cτ ̂ u = ̂ uy (0) . In this case, the comoving frame of K ' is The Lorentz transformations K ' → K are usually written as In this form, the transformations correspond to Figure 1(a), in which the event A is written as a linear combination of unit vectors along the x ' and t ' axes. However, we can write these transformations equivalently as which is exactly In this form, the transformations correspond to Figure 1(b), in which the event A is written as the vector sum of the worldline of the observer located at the origin of K ' and the event's spatial coordinates in this observer's comoving frame. It is worthwhile noting the properties of the transformations (14) when K ' is inertial ( A = 0). First of all, only when K ' is inertial are the transformations (14) linear, since only in this case does the observer's position depend linearly on y (0) . Next, note that when K ' is inertial, the transformations (14) are well defined on all of Minkowski space. For each value of τ , let X τ be the 3D spacelike hyperplane consisting of all events simultaneous (in K τ ) to ̂ x ( τ ). These hyperplanes are parallel and, therefore, pairwise disjoint. Now, let X be an event with coordinates y (0) , y (1) , y (2) , y (3) in K ' . This event is simultaneous to the event ̂ x ( τ 0 ) = ( y (0) , 0 , 0 , 0), which corresponds to the observer at time τ 0 = y (0) /c . The vector X -̂ x ( τ 0 ) belongs to the hyperplane X τ 0 and may therefore be decomposed as X -̂ x ( τ 0 ) = y ( i ) λ ( i ) ( τ 0 ) in K τ 0 . Since the X τ are pairwise disjoint, the vector X -̂ x ( τ 0 ) does not belong to any other X τ . Hence, the transformations are well defined everywhere. glyph[negationslash] Returning to the general case ( A = 0), we are now ready to show that the spacetime transformations from K τ to K have the form (14). Let K ' be the uniformly accelerated frame determined by A , ̂ x (0), and ̂ λ . The worldline ̂ x ( τ ) of the observer is obtained by integrating his four-velocity u ( τ ), and the comoving frame matrix λ ( τ ) is given by (8). In order to use (14), it remains only to establish well-defined spatial coordinates y ( µ ) in K τ . Since K ' is accelerated, the hyperplanes X τ are no longer pairwise disjoint. Nevertheless, since X τ is perpendicular to u ( τ ), there exist a neighborhood of τ and a spatial neighborhood of the observer in which the X τ are pairwise disjoint. Thus, spacetime can be locally split into disjoint 3D spacelike hyperplanes. This insures that, at least locally, the same event does not occur at two different times. Hence, within the locality restriction, we may uniquely define coordinates for the observer. This implies that, at least locally , the spacetime transformations from K τ to K are given by (14). A similar construction can be found in [11], in which the authors use radar 4coordinates .", "pages": [ 8, 9, 10 ] }, { "title": "Step 2: From K ' to K τ", "content": "At this point, we invoke a weaker form of the Hypothesis of Locality introduced by Mashhoon [12, 13]. This Weak Hypothesis of Locality is an extension of the Clock Hypothesis. The Weak Hypothesis of Locality Let K ' be a uniformly accelerated frame, with an accelerated observer with worldline ̂ x ( τ ) . For any time τ 0 , the rates of the clock of the accelerated observer and the clock at the origin of the comoving frame K τ 0 are the same, and, for events simultaneous to ̂ x ( τ 0 ) in the comoving frame K τ 0 , the comoving and the accelerated observers measure the same spatial components. Consider an event with K coordinates x µ . By step 1, we have, for a unique τ 0 , x = ̂ x ( τ 0 ) + y ( i ) λ ( i ) ( τ 0 ). Hence, the K τ 0 coordinates of the event x are y (0) = cτ 0 and y ( i ) . Since x and ̂ x ( τ 0 ) are simultaneous in K τ 0 , the Weak Hypothesis of Locality implies that the spatial coordinates y ( i ) coincide with the spatial coordinates in K ' . We have thus proven the following: Let K ' be a uniformly accelerated frame attached to an observer with worldline ̂ x ( τ ). Let { K τ ( A, ̂ x (0) , ̂ λ ) } be the corresponding one-parameter family of inertial frames. Then the the spacetime transformations from K ' to K are Unless specifically mentioned otherwise, we will always choose the lab frame K to be the initial comoving frame K 0 . This implies that ̂ λ = I and A = ˜ A . We end this section by calculating the metric at the point y of K ' . First, we calculate the differential of the transformation (15). Differentiating (15), we have Define ¯ y = (0 , y ). Using (10) (but writing A for ˜ A , as is our convention), this becomes Finally, since we obtain Therefore, the metric at the point ¯ y is This formula was also obtained by Mashhoon [7]. We point out that the metric is dependent only on the position in the accelerated frame and not on time .", "pages": [ 10, 11 ] }, { "title": "5 Examples of Spacetime Transformations from a Uniformly Accelerated frame", "content": "In this section, we consider examples of uniformly accelerated frames and the corresponding spacetime transformations.", "pages": [ 11 ] }, { "title": "5.1 Null Acceleration ( α = 0 , β = 0 )", "content": "Since, in this case, | g | = | c ω | and g · ω = 0, we may choose g = ( g, 0 , 0) and c ω = (0 , 0 , g ). From (2), we have Then Thus, from (8), we have The observer's four-velocity is, therefore, His four-acceleration is which shows that the acceleration is constant in the comoving frame. Integrating (23), we have Using (22) and y (0) = cτ , the spacetime transformations (15) are", "pages": [ 12, 13 ] }, { "title": "5.2 Linear Acceleration ( α > 0 , β = 0 )", "content": "Without loss of generality, we may choose where g > cω > 0. In order to simplify the calculation of the exponent of A , we perform a Lorentz boost to the drift frame corresponding to the velocity Since g > cω > 0, we have | v | ≤ c . In the drift frame, the acceleration tensor A becomes and leads to 1D hyperbolic motion. Hence, If ω = 0, we recover the usual hyperbolic motion of a frame. Thus, the previous formula is a covariant extension of hyperbolic motion. From the first column of (29), the observer's four-velocity is Hence, the observer's four-acceleration is which shows that the acceleration is constant in the comoving frame. Note that our definition of linear acceleration is more general than the usual d u dt = g . From formula (30), we have Since dτ dt = γ -1 , and γ is the zero component of u ( τ ), we have which is not constant unless ω = 0. This provides a proof of the fact mentioned in part I that the equation d u dt = g is limited to the particular case ω = 0. Integrating (30), we have Using (29) and y (0) = cτ , the spacetime transformations (15) are", "pages": [ 13, 14, 15 ] }, { "title": "5.3 Rotational Acceleration ( α = 0 , β > 0 )", "content": "Without loss of generality, we may choose where cω > g > 0. In order to simplify the calculation of the exponent of A , we perform a Lorentz boost to the drift frame corresponding to the velocity Since cω > g > 0, we have | v | ≤ c . In the drift frame, the acceleration tensor A becomes and leads to pure rotational motion. Hence, If g = 0, we recover the usual rotation of the basis about the z axis. Thus, the previous formula is a covariant extension of rotational motion. From the first column of (35), the observer's four-velocity is Hence, the observer's four-acceleration is which shows that the acceleration is constant in the comoving frame. Integrating (36), we have Using (35) and y (0) = cτ , the spacetime transformations (15) are", "pages": [ 15, 16 ] }, { "title": "6 Summary", "content": "Definition 1 introduces a new method of constructing a family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. Our construction uses generalized Fermi-Walker transport, and we have shown that our approach is equivalent to that of Mashhoon (equation (10)). Thus, we may use the two approaches interchangeably. Mashhoon's approach is better suited to curved spacetime, that is, a manifold setting. Our approach, on the other hand, leads to a decoupled system of differential equations, and is, therefore, easier to solve for explicit solutions. Moreover, all of our solutions (5) for uniformly accelerated motion have constant acceleration in the comoving frame (see equation (11)). In fact, the value of this constant acceleration is g , the linear acceleration component of the acceleration tensor A . We have also shown at the end of section 3 that the spacetime transformations between two frames K ' and K '' are Lorentz not only when K and K ' are inertial, but also when K and K ' are two uniformly accelerated frames, provided that each frame experiences the same acceleration. In section 4, we used the Weak Hypothesis of Locality to obtain local spacetime transformations (formula (15)) from a uniformly accelerated frame K ' to an inertial frame K . These transformations extend the Lorentz transformations. We have also computed (equation (19)) the metric at an arbitrary point of K ' . The metric depends only on position, and not on time . In the process of solving the examples of section 5, we used the 'drift frame.' What is the physical meaning of this frame? What is the physical significance of the drift velocity? In an upcoming paper, we will obtain velocity and acceleration transformations from K ' to K . We will also derive the general formula for the time dilation between clocks located at different positions in K ' . It turns out that this time dilation depends on the state of the clock, that is, on its position and velocity. In computing the spacetime transformations from a uniformly accelerated frame, we used the Weak Hypothesis of Locality, which is an extension of Einstein's Clock Hypothesis. Not all physicists agree with this hypothesis. L. Brillouin ([10], p.66) wrote that 'we do not know and should not guess what may happen to an accelerated clock.' It is shown in [14] that if the Clock Hypothesis does not hold, then there is a universal limitation a max on the magnitude of the 3D acceleration g . In [6], we showed that the 3D acceleration must be replaced by an antisymmetric tensor A in order to achieve covariance. Thus, we expect that if the Clock Hypothesis is not valid, then the maximal acceleration will put a bound on the admissible acceleration tensors. In this case, the set of admissible acceleration tensors will form a bounded symmetric domain known as a JC ∗ -triple (see [15]). In [16], the first author shows how to modify the 3D Relativistic Dynamics Equation to a 3D Extended Relativistic Dynamics Equation in order to preserve the bound on accelerations. In order to make this extended equation covariant, one needs to apply a similar procedure to that used in [6] to make the 3D Relativistic Dynamics Equation covariant. In this way, we hope to obtain the spacetime transformations from a uniformly accelerated frame in case the Clock Hypothesis is not valid. We would like to thank B. Mashhoon, F. Hehl, Y. Itin, S. Lyle, and Ø. Grøn for challenging remarks which have helped to clarify some of the ideas presented here.", "pages": [ 17, 18 ] } ]

[ { "title": "ABSTRACT", "content": "R esearchin A stronomyand A strophysics Key words: stars: variables : other - galaxies: individual (LMC)", "pages": [ 1 ] }, { "title": "Analysis of a selected sample of RR Lyrae stars in LMC from OGLE III", "content": "B.-Q. Chen 1 , 2 , B.-W. Jiang 1 and M. Yang 1 P.R.China; [email protected] , [email protected] , [email protected] Abstract Asystematic study of RR Lyrae stars is performed based on a selected sample of 655 objects in the Large Magellanic Cloud with observation of long span and numerous measurements by the Optical Gravitational Lensing Experiment III project. The Phase Dispersion Method and linear superposition of the harmonic oscillations are used to derive the pulsation frequency and variation properties. It is found that there exists an Oo I and Oo II dichotomy in the LMC RR Lyrae stars. Due to our strict criteria to identify a frequency, a lower limit of the incidence rate of Blazhko modulation in LMC is estimated in various subclasses of RR Lyrae stars. For fundamental-mode RR Lyrae stars, the rate 7.5 % is smaller than previous result. In the case of the first-overtone RR Lyr variables, the rate 9.1 % is relatively high. In addition to the Blazhko variables, fifteen objects are identified to pulsate in the fundamental/first-overtone double mode. Furthermore, four objects show a period ratio around 0.6 which makes them very likely the rare pulsators in the fundamental/second-overtone double-mode.", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "RR Lyrae stars (RRLS) are pulsating variables on the horizontal branch in the H-R diagram. They have short periods of 0.2 to 1 day and low metal abundances Z of 0.00001 to 0.01. Usually they can be easily identified by their light curves and color-color diagrams (Li et al., 2011). RRLS is famous for the 'Blazhko effect' (Blaˇzko, 1907), a periodic modulation of the amplitude and phase in the light curves, which is still a mystery in theory today. The main photometric feature of the Blazhko effect is that the frequency spectra of the light curves are usually strongly dominated by a symmetric pattern around the main pulsation frequency f 0 , i.e., kf 0 and kf 0 ± f BL where f BL is the modulation frequency and k is the harmonic number (Kov'acs, 2009). Moreover, the higher-order multiplets such as quintuplets and higher, i.e. multiplets kf 0 ± lf BL , are now also attributed to the Blazhko effect (Benk\"o et al., 2011). On the other hand, the amplitudes of modulation components are usually asymmetric so that one side could be under the detection limit in highly asymmetric cases, which may lead to the asymmetric appearance of the frequency spectra. Several models are proposed to explain this effect, including the non-radial resonant rotator/pulsator (Goupil & Buchler, 1994), the magnetic oblique rotator/pulsator (Shibahashi, 2000), 2:1 resonance model (Borkowski, 1980), resonance between radial and non-radial mode (Dziembowski & Mizerski, 2004), 9:2 resonance model (Buchler & Koll'ath, 2011) and convective cycles model (Stothers, 2006). However, none of them is able to interpret all the observational phenomena about the Blazhko effect. A systematic study of RRLS helps to understand their nature, such as the incidence rate of various pulsation modes, the distribution of modulation frequency and amplitude and the dependence of the variation properties on environment. This has been performed on the basis of some datasets with large amount of data for variables. The early micro-lensing projects, MACHO (Alcock et al., 2003) and OGLE (Soszynski et al., 2008) that surveyed mainly LMC, SMC and the Galactic bulge, and the variables-oriented all-sky survey project, ASAS (Szczygieł & Fabrycky, 2007), have given particularly great support to such study. Using the MACHO data, Alcock et al. (2000) and Nagy & Kov'acs (2006) analyzed the frequency of 1300 first overtone RRLS in LMC with an incidence rate of the Blazhko variables of 7.5%. Meanwhile, Alcock et al. (2003) made a frequency analysis of 6391 fundamental mode RRLS in LMC that resulted in an incidence rate of the Blazhko variables of 11.9%. With the OGLE-I data, Moskalik & Poretti (2003) searched for multiperiodic pulsators among 38 RRLS in the Galactic Bulge. Mizerski (2003) made a complete search for multi-period RRLS from the OGLE-II database, while Collinge et al. (2006) presented a catalogue of 1888 fundamental mode RRLS in the Galactic bulge from the same database. Recently, Moskalik & Olech (2008) conducted a systematic search for multiperiod RRLS in ω Centauri, a globular cluster, and found the incidence rate of Blazhko modulation pretty high, about 24% and 38% for the fundamental and first-overtone RRLS respectively. Jurcsik et al. (2009) got a 47% incidence rate in the dedicated Konkoly survey sample of 30 fundamental-mode RRLS in the Galactic field. Kolenberg et al. (2010) also claimed at least 40% RRLS in the Kepler space mission sample of 28 objects exhibiting the modulation phenomenon. The OGLE-III database released in 2009 (Soszy'nski et al., 2009) contained 24906 light curves that are preliminarily classified as RRLS in the Large Magellanic Cloud, the ever largest sample of RRLS. This database covers a time span of about 10 years that makes a large-scale analysis of the RRLS variation possible. Soszy'nski et al. (2009) analyzed the basic statistical features of RRLS in the LMC and divided them into 4 subtypes: RRab, RRc, RRd and RRe. With this sample of RRLS, the study of the structure of LMC was developed. Pejcha & Stanek (2009) investigated the structure of the LMC stellar halo; Subramaniam & Subramanian (2009) found that RRLS in the inner LMC trace the disk and probably the inner halo; Feast et al. (2010) established a small but significant radial gradient in the mean periods of Large Magellanic Cloud (LMC) RR Lyrae variables. The data of RRLS in SMC and the Galactic bulge are also released (Soszy˜nski et al., 2010; Soszy'nski et al., 2011) and some work is also done with those data (Pietrukowicz et al., 2011). It is worth to note that these work mainly deals with the structure of LMC other than the RR Lyr variables. In this paper, we focus the study on the RR Lyr variables themselves based on the released OGLE-III database. With the PDM and Fourier fitting methods, we make a precise systematic frequency analysis of carefully selected 655 RRLS in LMC, and a detailed classification of the RRLS in the LMC based on which to discuss the incidence rate of the Blazhko modulation in various pulsation modes. The data and the sample are illustrated in section 2, the method is introduced in section 3, the detailed classification of RRLS in LMC in section 4 and discussion in section 5.", "pages": [ 1, 2 ] }, { "title": "2 THE SAMPLE", "content": "Soszy'nski et al. (2009) presented a catalog of 24,906 RR Lyrae stars discovered in LMC based on the OGLE-III observations and classified them into 17,693 fundamental-mode, 4958 first-overtone, 986 double-mode and 1269 suspected second-overtone RRLS. Thanks to their generosity, all the data are released. This catalog has three columns recording the observational Julian Date, magnitude in the I or V band and error of the magnitude. Since the number of measurements in the V band is much fewer than in the I band, our analysis mainly makes use of the I-band data. For over 20,000 objects, precise analysis of the frequency for all of them seems an improbable task. Fortunately, the statistical properties can be reflected by a much smaller sample. Thus, we concentrate our study on a sample of RRLS that were measured as many times as possible with high precision. As the amplitude of some RRLS is rather small as about 0.1 mag, the measurements with assigned photometric error bigger than 0.1 mag are dropped. In the released database, the number of measure- ments is usually not as numerous as claimed in the OGLE-III catalog web. Most of them have fewer than 400 measurements, as can be seen in Fig. 1 that displays the distribution of the number of measurements for all the RRLS. In our sample, only the objects with more than 1000 measurements are kept. To exclude the foreground stars, the criterion ¯ I ≥ 18 mag is added. In the I/V-I diagram (Fig. 1), it can be seen that this brightness cutoff constrains the sources in the major RRLS area and excludes some sparse sources seemingly to be giant or foreground stars. With the limitation of brightness and number of measurements, our sample consists of 655 RRLS. Through the color-magnitude diagram of the 655 RRLS in comparison with the complete group of all the 24906 RRLS sources identified by Soszy'nski et al. (2009) in Fig. 1 (top panel), we can see that the sample agrees with the majority of RRLS. The miss of the faint sources (I > 19.5) is mainly due to our request of high photometry quality. In the same time, some red RRLS with V-I bigger than about 0.8 are neither included. The faint and red RRLS may be caused by extinction as they coincide pretty well with the A V =1 trend (the extinction law is taken from Mathis (1990)) in Fig. 1. Thus the missed faint and red stars should have intrinsically similar brightness and color with the majority, and their absence in our sample shall not influence the statistical variation properties of RRLS. This sample of 655 RRLS has three advantages for the frequency analysis. Firstly, more than six-hundred stars are already big enough to obtain the statistical parameters objectively and to understand the common properties of RRLS in LMC. Secondly, the sample is not too big, so that we can not only make the frequency analysis accurately, but also check carefully for individual object, to make sure every result is reliable. Thirdly, the sample we compose contains the highest-quality data for RRLS in the OGLE-III project, and the derived variability properties should be highly reliable. As mentioned earlier, all stars in our sample have more than 1000 measurements. The time span is about 4000 days (i.e. close to 11 years), and the interval of two adjacent measurements is mostly shorter than 20 days. Theoretically, based on the effect of random and uncorrected noise, using the least square fit of a sinusoidal signal, we can roughly estimate the error of the frequency 10 -7 , and the error of amplitude 10 -3 (Montgomery & Odonoghue, 1999). The objects in our sample are listed in Table 1, with the OGLE name, position, number of measurements both in original OGLE catalog and in our calculation, the magnitude in the I and V bands that are the average over all the measurements , the subtype of variation which will be discussed later, and MACHO ID if available. It should be noticed that the I magnitudes in the tables afterwards are the average value of the Fourier fitting.", "pages": [ 2, 3 ] }, { "title": "3 FREQUENCY ANALYSIS", "content": "Most studies of the RRLS frequencies (e.g. Kolenberg et al. 2010) make use of the Period04 software that analyze the Fourier spectrum of the observed light curve. The advantage of Period04 is its high precision and intuitive power spectrum with both the frequency and amplitude shown, appropriate for analyzing multi-period light-curves. But, the Fourier analysis fits a sinusoidal signal, while for RRLS in the fundamental mode, their light-curves are very asymmetric. Besides, RRLS often have several harmonics, so that more than one period would be found for a single period RRLS. An alternative N method to determine the frequency of light variation is the Phase Dispersion Method (PDM, Stellingwerf 1978) independent on the shape of the light curve or irregular distribution of measurements in the time domain. Although PDM also brings about a strong signal of the harmonics, this can be looked over in the folded phase curve, which needs a careful eye-check. The mediate volume of our sample makes it possible to check the phase curve one by one. Moreover, the result from PDM provides a mutual check between the two main-stream methods in the study of variable stars. The PDM method looks for the right period of light variation in a range of trial periods by fixing the period of the minimum phase dispersion for the folded phase curve. The phase dispersion Θ PDM is defined as the ratio between the summed phase dispersion in all the phase bins to the phase dispersion of all the measurements. A perfect periodical light curve would produce Θ PDM = 0 . The period of light variation of the RRLS sample is searched in two steps. In the first step, the frequency range is set to the whole range for RRLS variation, i.e. from 1 c/d to 5 c/d, with a step of 10 -5 and 50 bins in the phase space [0,1.0]. The PDM analysis yields the first guessed frequency ( f PDM ) at the minimum Θ PDM . With this frequency, the folded phase light curve is plotted and checked by eyes to exclude the harmonics, often the double or triple, which results in a crude estimation of the main frequency, f est = f PDM ∗ n (n=1,2,3... according to the order of the harmonics in the phased light-curves). In the second step, the frequency resolution is increased to 10 -7 and the range of frequency is shrunk to [ f est -0.05, f est +0.05]. Then a PDM analysis is performed once more to yield the minimum Θ PDM that tells the frequency with higher accuracy. Thanks to the long time span and numerous measurements of RRLS in the sample, such high precision is achievable in determining the frequency. This frequency is the main pulsation frequency, f 0 or corresponding period P 0 in following text, to distinguish among the fundamental, first overtone and second overtone modes. Once the main frequency is determined, the light curve is fitted by a linear superposition of its harmonic oscillations: where n is the highest degree of the harmonics, M ( t ) the measured magnitude in the I or V band, M 0 the mean magnitude, and ω = 2 π/P 0 the circular frequency. In fact, it's the phased light-curve instead of the light-curve itself that's fitted for the sake of higher significance: where Φ t = ( t -t 0 ) /P - | ( t -t 0 ) /P | , t is the time of observation and t 0 is the epoch of maximum brightness. Eq. (2) can be re-written as: where A k = √ a 2 k + b 2 k and φ k = arctan( b k /a k ) . The parameters A k and φ k can be transformed to the Fourier parameters R ij = A j /A i and φ ij = jφ i -iφ j ( i , j refers to different k , (Simon & Lee, 1981)), both of which are widely used in expressing the features of the light-curves, and even to derive the physical parameters of the variables such as the metallicity (e.g. Jurcsik & Kovacs 1996). In principle, the degree of the harmonics can be arbitrarily high, however, the highest degree in practice is set to 5 or 6, being able to reflect the essential shape of the light curve. Among all the 655 stars, only 64 stars need the sixth harmonics and all the others with n ≤ 5. Moreover, it avoids over-fitting, even for very complex light curves such as shown in Fig. 2. The main pulsation frequency we derived by this method is almost the same as that of Soszy'nski et al. (2009), with the difference ≤ 0.0001. The upper four panels in Fig. 2 show the process of determining the primary frequency, from the estimation of the frequency, through the accurate measurement of the frequency and the folded phase curve to the final fitting of the phase curve. The secondary period is searched in the residual after subtracting the variation in the main frequency with its harmonics. The method is the same as for the principle period. The difference lies in the amplitude of the secondary oscillation being much smaller than the principle one. This procedure is carried repeatedly until an assigned threshold, which is set to guarantee the significance of the derived period. The phase dispersion parameter Θ is related to the significance of the period, the smaller the better. Another related parameter is the probability P ( F ( P/ 2 , N -1 , ∑ ( n j ) -M ) = 1 / Θ) (Stellingwerf, 1978). However, Θ and P depends on the number and quality of measurements so they do not necessarily have the same cut-off value in different cases and become ambiguous in marginal cases. We developed an independent parameter, S/N ≡ ¯ Θ -min(Θ) σ Θ , which reflects the S/N of the minimum Θ in the Θ distribution. Combining all the three parameters, the specific thresholds are set to Θ PDM < 0 . 63 -0 . 85 , P < 0 . 01 -0 . 05 or S/N > 10 -15 for a reliable period determination. Once the period is determined, a fitting is performed as for the principle period, but the highest degree of the harmonics is taken to be 3 instead of 6 in the first run. The lower four panels in Fig. 2 show the procedure for determining a secondary period, including the first guess and final determination of the frequency and the fitting of the phased light curve.", "pages": [ 3, 4, 5, 7 ] }, { "title": "4 CLASSIFICATION", "content": "RRLS are considered as the pure radial pulsator basically. According to the pulsation modes, it was classically divided into four types: RRab pulsating in the fundamental (FU) mode, RRc in the first overtone (FO) mode, RRe in the second overtone mode (SO) and RRd in the double (FU and FO) modes. Alcock et al. (2000) introduced a new system of notation with a digit to mark the primary pulsation mode to replace the letters, i.e. RR0, RR1, RR2 and RR01 instead of RRab, RRc, RRe and RRd, more intuitive to mnemonics. When the modulation of period and amplitude is considered, Nagy & Kov'acs (2006) adopted additional letters to classify RRLS, PC for period change, BL for the Blazhko effect and MCfor closely spaced multiple frequency components. To make a complete view of the variation type, we combine both notations into a more detailed designation to make the phenomenological classification of RRLS by following Nagy & Kov'acs (2006). The identification of the main pulsation mode is clear for separating the RR0 and RR1 classes, as proved in many previous studies. The period is longer and the amplitude is mostly larger in RR0 RRLS than in RR1, and the shape of light curve of RR0-type RRLS is more asymmetric than the RR1type, as shown in the period-amplitude and period-skewness diagrams in Fig. 3, where the skewness is calculated from the phased light curve. The definition of skewness is E ( x -µ ) 3 /σ 3 , where x is the observed magnitude, µ the mean of x, σ the standard deviation of x, and E ( t ) represents the expected value of the quantity t . Examples of phased light-curves in our sample are shown in Fig. 5. The gap between RR0 and RR1 is also clearly shown in the period-Fourier coefficients diagram in Fig. 4. The puzzle comes with the identification of the RR2-type RRLS, those pulsating in the secondovertone mode. The RR2 stars are believed to have even shorter period, slightly smaller amplitude and more symmetric light curve. In the period-amplitude diagram, they should locate on the left of RR1 stars, e.g. the magenta points in Fig.2 of Soszy'nski et al. (2009). In the period-amplitude diagram of our sample (Fig. 3, top), no clear gap is found in the shorter-period group of stars. There is neither apparent peak in the period distribution of all single-mode RRLS shown in Fig. 6, which is very similar to that of all the RRLS in LMC (Soszy'nski et al., 2009). Concerning the shape of the light curves, their skewness is calculated. From the appearance in the bottom panel of Fig. 3, the separation between RR0 and RR1 is again apparent with RR1 being systematically more symmetric, while no more subgroup can be further distinguished in the relatively symmetric group. Thus, if a RR2 group is to be assigned, the borderline would be very arbitrary both in the period-amplitude and period-skewness diagrams. Because there is no systematic features, we are conserved in identifying a group of RR2 stars. In fact, it's also possible that the stars with shorter period, smaller amplitude and more sinusoidal light curve may be metal-rich RR1 stars (Bono et al., 1997a).", "pages": [ 7 ] }, { "title": "4.1 Single period RRLS", "content": "The notation 'RR-SG' refers to the single period RRLS, i.e. no more frequency is found in the residual after removing the primary frequency and its harmonics. In our sample, 556 stars were found to be RR-SG stars, 84.9% of all the 655 sample RRLS. Out of these RR-SG stars, 424 (76%) are RR0-SG in the FU mode and 132 (24%) are RR1-SG RRLS in the FO mode. The RR0-SG RRLS are three times as many as RR1-SG RRLS. In Table A.1 we listed their period of light variation, minimum phase dispersion Θ , amplitude and mean magnitude in the I band, and subtype. The histogram of the RRLS periods (Fig. 6) displays two classical prominent peaks at 0.58 d and 0.34 d for RR0 and RR1 stars respectively. The average period of our sample for RR0 stars ¯ P RR0 -SG is 0.587 d and for RR1 stars ¯ P RR1 -SG is 0.328 d . We compare our division of the subtype into RR0 and RR1 with that of Soszy'nski et al. (2009), regardless of RR2 stars. One star (ID: 303, OGLE ID: OGLE-LMC-RRLYR-14697) is found to be discrepant, which is classified as RR1 in our work and RR0 in their work. In Fig. 3 that is the main criteria for classification, this star is specially denoted by a pentagon. In either the period-amplitude or the period-skewness diagram, this star locates in the central area of RR1 stars. When compared with the RR0 and RR1 stars in the phased light curve (Fig. 5), its shape is apparently asymmetric, but not the same as RR0 stars (not so steep as RR0 stars). On the other hand, its small amplitude and short period bring it to the RR1 group. This example also tells that the skewness of the light curve of one class covers a wide range, and makes the separation of RR2 stars hard. The distribution of RR-SG in Fig. 3 is composed of two typical parts, one sequence of RR0-SG and the other shape of bell for RR1-SG. The distribution of RR0-SG seems to be composed of two parts, one densely clumped on the left forms a line shape, the other loosely distributed on the right. This shape of distribution reminds one of the dichotomy into Oosterhoff I and II classes of Galactic RRLS due to different evolving phases. Such dichotomy was found in the Galactic fundamental-mode RRLS (Szczygieł et al., 2009). Fig. 7 compares the Bailey diagram of RR0-SG stars in LMC with that in the Galactic bulge (Collinge et al., 2006) and the Galactic field (Szczygieł et al., 2009), where the contours are for the density of RR0-SG stars, the solid and dashed lines are from Equation 2, 3, 4 and 5 of Szczygieł et al. (2009) for their fitting to the Oo I (left), Oo II (right) groups and their borderlines respectively in the Galactic field RRLS. It can be seen that the RR0 stars in LMC can be divided into the Oo I and Oo II groups as well as in the Galactic field and bulge. Although the two groups are not fitted, the solid and dash lines from Szczygieł et al. (2009) generally agree with the distribution. Moreover, the Oo I group RR0 stars are clearly dominating over the Oo II stars. This is consistent with the fact that the average period ¯ P RR0 -SG =0.587 d is more approximate to 0.549 d of the average Oo I clusters than to 0.647 d of the average Oo II clusters (Clement & Rowe, 2000; van Agt & Oosterhoff, 1959). According to the contour diagram, the Oo I group RR0 in LMC stars have a slightly longer period ( /triangle lg P ∼ 0 . 02 ) than those in the bulge. If the bulge and field appearance in the contour diagram is taken into account, the Oo I distribution forms a series from the bulge to the field and then to the LMC from left to right, i.e. the period from short to long. This sequence coincides with that of the average metallicity in these three environments. Since metallicity influences the opacity and the mechanism for the light variation of RRLS is the κ mechanism, the shift of the Oo I group may be caused by the metallicity difference. Indeed, the metal abundance also influences the distribution of OoI and Oo II RR0 stars in the Galactic field (Szczygieł et al., 2009).", "pages": [ 8, 10 ] }, { "title": "4.2 Multiple period RRLS", "content": "There are 99 (15.1%) RRLS with variation detected in the residual of the light curve after removing the principle frequency and its harmonics. They are further classified into several subclasses according to the number of additional frequencies and their locations relative to the main frequency, including the RR01, RR-BL, RR-MC, RR-PC and miscellaneous subtypes.", "pages": [ 10 ] }, { "title": "4.2.1 RR01 stars", "content": "RR01 refers to the RR Lyrae stars pulsating in double radial modes, one is the FU mode and the other is the FO mode, classically RRd stars. In our sample, 15 ( 2 . 3% in the sample) stars are classified as RR01 stars. Seven of them have two frequencies detected. Other eight stars have three frequencies detected, but the third frequency is either the sum or the difference of the first and second frequency and thus dependent. All these 15 stars are listed in Table A.2, with the period and amplitude in the FU mode P 0 and A 0 , the minimum phase dispersion Θ 0 in deriving the FU mode, the mean magnitude in the I band, the period and amplitude ratios between FO and FU modes, and the minimum phase dispersion Θ 1 in deriving the FO frequency. In the period-magnitude diagram (Fig. 3, top), both the FU and FO periods of these double mode RRLS are shown by black open triangles. The periods are either at the long side of the single FO mode or the short side of the FU RRLS. Their amplitudes are small, all smaller than 0.3 mag, which is comparable to that of the RR1 stars. The dominance of FO mode in RR01 can explain this small amplitude. It can be seen that the amplitude of the FU mode of RR01 stars is much smaller than those single mode FU stars at corresponding short period while comparable to those single mode FO stars. In fact, RR01 stars distinguish themselves from the other single-mode FU stars by their small amplitude in the period-amplitude diagram. Meanwhile, their light curves appear differently by a positive skewness from the FO RRLS most of which have a negative skewness although both are more symmetric than the FU RRLS. The lower panel of Fig. 3 shows such difference. The period ratio P 1 /P 0 ranges from 0.7422 to 0.7465, with an average of 0.7436. The amplitude ratio A 1 /A 0 is significantly larger than one in 13 stars, and two smaller than one (but bigger than 0.8), with an average of 1.533. It can be concluded that the RR01 stars mainly pulsate in the FO mode. In the Petersen diagram for RRLS (Fig. 8), the period and amplitude ratios are plotted versus the FU period, and compared with the results of Alcock et al. (2000) from the MACHO data. The increase of both ratios with the period is clear and consistent with Alcock et al. (2000). The distribution of the period ratio overlaps completely with that of Alcock et al. (2000). The ratio of the amplitude does not rise so high as the Alcock et al. (2000) result, although the rising tendency with period is the same. Moreover, the RR01 stars whose dominating mode is fundamental (i.e. A 1 /A 0 < 1 ) have smaller P 1 /P 0 than the average, specifically, their P 1 /P 0 is all smaller than 0.743, even when including those from Alcock et al. (2000).", "pages": [ 10, 11, 12 ] }, { "title": "4.2.2 RR-BL stars", "content": "RR-BL stars refer to the Blazhko stars. As mentioned in Introduction, the early identification of RR-BL stars was the symmetric appearance of the frequency spectrum. With the development of the study of the Balzhko effect, some asymmetric patterns are considered to be its variation. Thus, in the frequency pattern, they may appear as: (1) two frequencies which have one closely spaced frequency around the principle frequency (RR-BL1), (2) three frequencies which have two side frequencies closely and symmetrically distributed around the main frequency (RR-BL2), and (3) more than three frequencies which have multiple components at closely spaced frequencies (RR-MC). All of them are the consequence of the modulation of the amplitude and/or phase, which can be explained by the modulation of a single (sinusoidal or non-sinusoidal) oscillation (Szeidl & Jurcsik, 2009; Benk\"o et al., 2011). RR-BL1 stars RR-BL1 stars have one frequency close to the main frequency, bigger or smaller. Alcock et al. (2000) marked them as ν 1, and here we use the definition of Nagy & Kov'acs (2006) to mark them as BL1. There are 41 (6.3% of all the 655 stars in the sample) such RR-BL1 stars, forming a much larger group than other multi-period RRLS. Taking into account the main pulsation mode, they are classified into 32 RR0-BL1 stars in the FU mode and 9 RR1-BL1 stars in the FO mode. The main pulsation period of RR0-BL1 stars ranges from 0.38 d to 0.76 d and of RR1-BL1 from 0.26 d to 0.37 d. The main pulsation amplitude of RR0-BL1 stars ranges from 0.106 mag to 0.747 mag and of RR1-BL1 from 0.073 mag to 0.352 mag. In Fig. 3 and Fig. 4, these RR-BL1 stars are mixed homogeneously with other single-mode RRLS, which means they have normal main period and amplitude of light variation as those single-mode stars. The sum of all the differences between every side frequency and main frequency δf = ∑ i ∆ f i , where ∆ f i = f i -f 0 and i = 1 , 2 ... for frequency at the first, second overtone and so on , is usually used to characterize the asymmetry of the frequency distribution. For RR-BL1 stars, there is only one side frequency, resulting δf = f 1 -f 0 . About 75% (24 out of 32) of these RR0-BL1 stars and 67 % (6 out of 9) of these RR1-BL1 stars have δf positive. This is very different from the 37% proportion for RR1-BL1 stars derived from the MACHO data by Alcock et al. (2000). But it agrees well with the percentage 80% for RR0-BL1 stars from the study of the Blazhko variables by Kov'acs (2002). For RR0-BL1 stars, the Blazhko periods vary between about 23 days and over 1500 days with an average of about 180 days and rms of 348 days. For RR1-BL1 stars, the Blazhko periods vary from 6.4 days to about 3000 days with an average of 745 days and rms of 1404. The shortest modulation period (6.4 d) is comparable to that found for RR1-BL in LMC by Nagy & Kov'acs (2006) and also consistent with those in the Galactic field for RR0-BL stars (Jurcsik et al., 2005b), i.e. around 6 days which is about 20 times of the main pulsation period. On the other end, the longest modulation period ∼ 3000 d is comparable to the time span of the data available, it is at least partly limited by the observational time coverage. With the continuation of the OGLE project, longer modulation period can be expected. The distribution of the Blazhko periods of RR-BL1 stars is shown in Fig. 10. The RR0-BL1 stars exhibit a normal distribution with the peak around 60-70 days. The RR1-BL1 stars shows a quite scattering distribution in a much wider range, although the group of only 9 RR1-BL1 objects makes this statistical significance less convincible. It seems that there is a preferred range of Blazhko period for RR0-BL1 from 32 d to 200 d ( lg P from 1.5 to 2.3), but no such preferred range for RR1-BL1 , which agrees well with the result of Nagy & Kov'acs (2006). We listed the main frequency and other variation parameters of RR-BL1 stars in Table A.3. The amplitude ratio ( A 1 /A 0 ) of RR0-BL1 varies from 0.119 to 0.382 with an average of 0.237 and of RR1BL1 varies from 0.324 to 1.081 with an average of 0.574. RR1-BL1 have a larger amplitude ratio than RR0-BL1, with one star (ID: 126) even larger than one but its main amplitude is small, only 0.087 mag. The asymmetric frequency can be regarded as the extreme case of the amplitude asymmetry in the Blazhko effect when the invisible symmetric component is completely submersed in the noise. In Fig. 9 is shown an example of such situation (star ID: 78, OGLE ID: OGLE-LMC-RRLYR-09295). The upper two figures are the frequencyΘ PDM diagram and the phased light-curves of the first-loop period searching for the main frequency 1.7927. The lower left figure shows the successful search for the secondary frequency at 1.8108, and the lower right figure shows that no more reliable frequency can be derived since all the three parameters at f =1.7746 ( Θ PDM =0.86, sig.=0.015 and S/N=8.7) are below our threshold. On the other hand, it may be expected that this frequency would be detected given a higher sensitivity of observation or a lower threshold. This example further supports that the missing of another frequency component in RR1 stars be caused by the asymmetry of the modulated amplitude. RR-BL2 Stars For RR-BL2 stars, the secondary frequencies indicate the modulation of the amplitude and phase. There are 11 (1.7%) RR-BL2 stars and they can be divided into two subtypes, 4 RR0-BL2 and 7 RR1-BL2 based on the main pulsation mode. This percentage (1.7%) is much smaller than the RR-BL1 stars (6.3%). It is also low in comparison with the results of previous studies, which will be discussed later. We think such low percentage is mainly due to our very strict criteria to identify a frequency so that some third frequencies have been dropped like the case of Star 78 shown in Fig. 9. This 1.7% percentage should be taken as the lower limit of the percentage of RR-BL2 stars. The differences between the side and main frequencies are shown in Table A.4, i.e. /triangle f + and /triangle f -. They are both smaller than 0.1. In addition, the difference between /triangle f + and /triangle f -is all smaller than 0.0003. It's these two features that bring them into the RR-BL2 class. On the ratio of the two amplitudes A + and A -( A + /A -), it changes from 0.76 to 1.60. This range of ratio means the two components have pulsation amplitudes at the same order, or we are only sensitive to such situation. This is understandable since a large ratio of the two amplitudes would surely make the weak component invisible and move the star into the RR-BL1 group. Such bias can only be alleviated by a very-high-sensitivity observation. This fact can also account for the low percentage of the BL2 stars. As shown in Fig. 3 by the symbol asterisks, the RR-BL2 stars have ordinary period and amplitude in the principle pulsation mode. The amplitude ranges from 0.283 mag to 0.567 mag and the period from 0.465 d to 0.647 d, with the average A 0 = 0.44 mag and P 0 =0.58 d for RR0-BL2. For RR1-BL2 stars, the amplitude ranges from 0.158 mag to 0.265 mag and the period from 0.270 d to 0.489 d, with the average A 0 = 0.21 mag and P 0 =0.33 d. According to /triangle f + and /triangle f -, the modulation period varies from about 43 to over 2700 days with an average of 1349 days for RR0-BL2; and from 12 to 2902 days with an average of 1288.5 days for RR1-BL2. The distribution of the modulation frequencies are shown in Fig. 10. Because the volume of the RR-BL2 stars is small, the distribution does not exhibit any outstanding feature. However, the situation becomes clearer when the RR-BL1 stars are included, which is reasonable since BL1 stars can be regarded as the extreme case of RR2 and both are Blazhko variables. Consequently, our sample of 655 RRLS contains 52 Blazhko stars. In Fig. 10, the period distribution of all the RR0-BL and RR1-BL stars is shown. Because of the dominance of RR-BL1 stars, the distribution of RR-BL stars is similar to that of RR-BL1 stars, i.e. with a preferred range of period from a few tens to a couple of hundred days. In regards to the modulation amplitude, a correlation is found with the main pulsation amplitude. As shown in Fig. 10 (bottom), a linear fitting results in that A i = 0 . 106 ∗ A 0 +0 . 057 and the correlation coefficient is 0.605 which means significant correlation. The error here we adopted is the maximum of the photometric error assigned in the catalog which is apparently bigger than the error in the fitting. The order of the error is mostly around 0.1 mag. This correlation was not found before and neither predicted in any models for the Blazhko effect. But it indicates that the Blazhko modulation is related to the main pulsation mode and it should be taken into account in models. On the contrary, Jurcsik et al. (2005a) found that the possible largest value of the modulation amplitude, defined as the sum of the Fourier amplitudes of the first four modulation frequency components, increases towards shorter period variables. RR-MC stars We have got 4 RR-MC in our sample. Actually all of them have three side frequency components. One RR-MC star is in the FU mode and three are in the FO mode. According to the structure of the side frequencies, they show three patterns, similar to the RR-MC stars described in Nagy & Kov'acs (2006): (a) two of the three frequencies are symmetric to f 0 ; (b) none is symmetric to the others, but three frequencies are at both sides to f 0 ; (c) three side frequencies are all at one side to f 0 . In Table A.5, the four RR-MC stars are shown with their patterns in the column 'Notes'. These stars with multiplets may also be Blazhko stars (Benk\"o et al., 2011).", "pages": [ 12, 13, 14, 15, 16, 17, 18 ] }, { "title": "4.2.3 RR-PC stars", "content": "RR-PC refers to period-changing stars. It's difficult to distinguish PC stars from MC stars or BL stars since they all have closely spaced side frequencies. Nagy & Kov'acs (2006) defined the PC stars as that they have close components which can not be eliminated within three prewhitening cycles or their separation from the main pulsation component is ≤∼ 1/T, where T is total time span. The definition is followed in this work. In our sample, we find that no star has any frequency detected after four prewhitening loops, i.e. the number of frequencies are not bigger than 4. The two stars which have a fourth frequency component both have at least one frequency with the separation from the main pulsation component ≤∼ 1/T. So they are classified as RR-PC stars with no doubt. In addition, there are some RRLS which have fewer than four frequency components but with some frequency whose separation from the main component is ≤∼ 1/T, the number of such stars is 18. Altogether, there are 20 RR-PC stars, that is about 3 percent of the sample. Their variation properties are listed in Table A.6. Our attempt to analyze the period variation is hampered by the large interval between adjacent measurements which ranges from 0.003 d to 300 d with a mean value of about 3.6 d, that is to say, the interval is several periods long.", "pages": [ 18 ] }, { "title": "4.2.4 Other RRLS", "content": "Eight (1.2%) multi-period RRLS in our sample cannot be classified into the above subtypes. They have more than one pulsation frequency, but their frequencies are not closely spaced from the main frequency. Table 2 shows their frequencies and other variation parameters. Four of them have one period around 0.3 d and the other around 0.5 d, yielding a period ratio around 0.6 which is the canonical period ratio between the SO and FU modes (Bono et al., 1997b). So we suspect that although we can't find single SO mode pulsating RRLS, but they can co-exist with the FU mode. The four RR02 candidates are shown by red triangles in Fig. 3 where P 1st and P 2nd refer to the primary and secondary period respectively. As same as the RR01 stars, the interaction between the two modes have led their amplitude and period to be on the short/long end of the FU/SO mode, which bring them together in the period-amplitude diagram (Fig. 3). Moreover, Star 228 seems to have a long modulation period for its FU mode from the fact that a third frequency is found to be closely spaced to its FU frequency. Only four stars were clearly claimed to be RR02 stars in the recent two years. All of them were discovered in space mission. They are V350 Lyr (Benk\"o et al., 2010) and KIC 7021124 (Nemec et al., 2011) from the Kepler mission, CoRoT 101128793 (Poretti et al., 2010) and V1127 Aql (Chadid et al., 2010) from the CoRoT mission. We found that star MACHO 18.2717.787 unidentified by Nagy & Kov'acs (2006) from MACHO dataset could also be such double-mode RR02 star with the period ratio of 0.5810. Poretti et al. (2010) computed a grid of linear RRLS models in a large stellar parameter space which delineated a rough range of the RR02 stars in the Petersen diagram. A similar work was presented by Nemec et al. (2011) who used the Warsaw pulsation hydrocode including turbulent convection. In Fig. 11, the range of RR02 in the Petersen diagram defined by the two models are delimited by solid and dash lines respectively. It can be seen that the two models agree with each other generally but also disagree in particular at the short fundamental periods. In this diagram, Star MACHO 18.2717.787 denoted by a blue dot is definitely inside the model range. The four RR02 candidates from our sample are shown by red dots with other five such stars by dots in other colors. Star 159 and 535 are undoubtedly inside the range by both models. Star 228 is just outside the upper border of the models, but can not be excluded since the models surely have some uncertainty. The only star which apparently deviates from the models is Star 34 although it is not too far. Puzzlingly, all the nine stars take a trend that the period ratio increases with FU period, which is opposite to the models. Interestingly, such dis- epancy between observation and model occurs exactly the same in RR01, the other double-mode stars (see Fig.5 of Alcock et al. 2000). Two stars have a secondary period around 1 day. This frequency is not taken as the alias because the phased light-curves at this frequency show reliable periodicity. We hereby denote them as RR0D1, following Alcock et al. (2000). The other three stars can not be classified into any of the classes described above. We just mark them by '?' in Table 2.", "pages": [ 18, 19 ] }, { "title": "5 DISCUSSION AND SUMMARY", "content": "The incidence rates of each subclass are shown in Table 3. The majority, 85%, is single-mode pulsators. The RRLS exhibiting the Blazhko effect (sum of RR-BL1 and RR-BL2) is the second most numerous group. With 52 RR-BL stars, they consist 7.9% of the sample. The incidence rates of Blazhko stars are compared with previous results in Table 4. As our identification of a frequency is quite strict, the percentages in our sample should be taken as the lower limit of the Blazhko incidence rate. For RRLS in LMC, the Blazhko variables (sum of RR-BL2 and RR-BL1 stars) occur less frequently in RR0 (7.5%) than in RR1 stars (9.1%). For RR1 stars, we can see an increasing trend of the incidence rates from 2.0% and 7.5% in previous work to 9.1% in present work, this can be explained by the longer time span and more precise data. But this trend does not show in RR0 stars although the data has been improved as for RR1 stars. To further analyze the reason, it is found that the incidence rate of RR-BL1 is comparable to previous work, the rate is 6.7% for RR0-BL1 and 5.1% for RR1-BL1, compared to 6.5% (Alcock et al., 2003) and 3.5% (Nagy & Kov'acs, 2006). But in regarding to the RR-BL2 stars, the incidence rate of RR1-BL2 is comparable to the work before, with 5.1% to 3.5%. Meanwhile the incidence rate of RR0-BL2 is abnormally low, with only 0.8% compared to 5.4% in Alcock et al. (2003). As mentioned in previous section, the reason may lie on our very strict criteria to accept a frequency which can move a BL2 star into a BL1 star in the case of high asymmetry of the amplitude in the two side frequencies. This explanation finds support in the fact that the RR0-BL2 stars have the amplitude ratio A + /A -not far from unity. Another possible reason is that, among RR0-BL1 stars, 75% (24 out of 32) have f + , and all the four RR0-BL2 stars have A + > A -. In the work of Kov'acs (2002), 80% of RR0-BL1 stars were found to have A + component. We suspect that for RR0-BL stars, there maybe some unknown effect to make A + much larger than A -, which made a lot of A -components missing. This effect does not appear in RR1 stars, for example, Alcock et al. (2000) found 37% RR1-BL1 stars have A -components, and in our sample only 43% RR1-BL2 stars have A + > A -. Based on Eq.(45) from Benk\"o et al. (2011), most of the RR0 stars have π < φ m < 2 π ; while for those RR1 stars, φ m is evenly distributed between zero and 2 π . People used to believe that the incident rates of the Blazhko variables are lower in LMC than in the Galaxy. But this only suits for RR0 stars, may not be true for RR1, as the work of Nagy & Kov'acs (2006) already suggested. From our work, with the long time span of observation, the Blazhko incidence rate for RR1 stars is larger in LMC than in the Galaxy bulge, as the rates are 5.1% and 4.0% for RR1-BL1 and RR1-BL2 respectively in our LMC sample in comparison with the 3.1% and 1.5% from Moskalik & Poretti (2003) or 2.9% and 3.9% from Mizerski (2003) for the bulge RRLS sample. On the other hand, both the improvement of the observational precision and the extension of observational 0 span increase the possibility to detect the Blazhko effect. Based on the data from the Kepler mission Kolenberg et al. (2010) and the Konkoly Blazhko Survey (Jurcsik et al., 2009), the incidence rate can exceed forty percent, but they are small samples with no more than 30 objects. Thus such comparison may not be conclusive as the observation and the analysis techniques are not uniform, and the samples are very different. According to these observations of LMC, SMC, bulge and ω Cen, there is no clear relation between the incidence rate and metallicity. What causes the difference in the Blazhko incidence rate in different environments is unclear, which could be part of the difficulty in understanding the mechanism for the Blazhko effect. Acknowledgements We sincerely thank the OGLE team for their continuing efforts and generosity in sharing data. We also thank the referee for his very helpful suggestions. This work is supported by the NSFC grant No. 10973004.", "pages": [ 19, 20, 21 ] }, { "title": "References", "content": "Alcock, C., Allsman, R., Alves, D. R., et al. 2000, ApJ, 542, 257 Alcock, C., Alves, D. R., Becker, A., et al. 2003, ApJ, 598, 597 Benk˝o, J. M., Kolenberg, K., Szab´o, R., et al. 2010, MNRAS, 409, 1585 Benk˝o, J. 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