Conformal Symmetry and Cosmological Entropy Production

Introducing an effective refraction index of an isotropic cosmic medium, we investigate the cosmological fluid dynamics which is consistent with a conformal, timelike symmetry of a corresponding ``optical'' metric. We demonstrate that this kind of symmetry is compatible with the existence of a negative viscous pressure and, consequently, with cosmological entropy production. We establish an exactly solvable model according to which the viscous pressure is a consequence of a self-interacting one-particle force which is self-consistently exerted on the microscopic particles of a relativistic gas. Furthermore, we show that a sufficiently high decay rate of the refraction index of an ultrarelativistic cosmic medium results in an inflationary expansion of the universe.


Introduction
Standard relativistic cosmology relies on the cosmological principle according to which the Universe is spatially homogeneous and isotropic on sufficiently large scales. This symmetry requirement largely fixes the spacetime metric which may be written in the Robertson-Walker form (see, e.g., [1]) ds 2 = −dt 2 + a 2 (t) dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 . (1) The quantity k can take the values k = 0, ±1 and characterizes the three possible cases of the spatial curvature. Only one unknown function, the scale factor a, has to be determined by the gravitational field equations. For many purposes the cosmic substratum may be modeled as a fluid. So called fundamental observers are observers which are at rest with respect to the mean motion of the cosmic fluid. The symmetry requirements of the cosmological principle restrict the corresponding effective energy momentum tensor T ik (eff ) to be of the structure T ik (eff ) = ρu i u k + P h ik .
Here, ρ is the energy density measured by an observer comoving with the fluid four-velocity u i which is normalized by u a u a = −1. The quantity h ik ≡ g ik + u i u k is the spatial projection tensor with h ik u k = 0. The total pressure P is the sum P = p + Π of an equilibrium part p > 0 and a non-equilibrium part Π ≤ 0 which is connected with entropy production. A perfect fluid is characterized by Π = 0. A scalar, viscous pressure is the only entropy producing phenomenon which is compatible with the symmetry requirements of the cosmological principle. Neither heat flows nor shear stresses are reconcilable with the assumption of spatial homogeneity and isotropy. This demonstrates that symmetries are generally connected with conditions on the properties and on the dynamics of the matter which generates this spacetime. Spacetime symmetries are invariantly characterized by symmetry groups. Isometries of a spacetime are described by Killing vectors (KVs). To characterize spacetimes that admit a conformal group one uses conformal Killing vectors (CKVs). A CKV ξ a of the metric g ik is defined by where £ ξ is the Lie derivative along the vector field ξ i . The semicolon denotes the covariant derivative with respect to g ik . For the special case φ = 0 the vector ξ is a KV. Maximally symmetric spaces have the maximally possible number of KVs. The Minkowski space, e.g., has 10 KVs corresponding to its well-known 10 symmetries and conservation laws. According to the cosmological principle, the four-dimensional spacetime contains maximally symmetric three-dimensional subspaces of constant curvature, corresponding to the spatial symmetries of homogeneity and isotropy, characterized by spacelike KVs. Timelike symmetries are not generally expected to exist in the expanding universe since a timelike KV characterizes a stationary spacetime. However, it is known [2] that under certain circumstances a conformal, timelike symmetry is possible in a Friedmann-Lemaître-Robertson-Walker (FLRW) universe. The existence of timelike or conformal, timelike KVs is closely related to "global" equilibrium properties of perfect fluids. For a gaseous fluid system of particles with mass m > 0, the global equilibrium condition can only be satisfied if the quantity ξ a ≡ u a /T , where T is the fluid temperature, is a (timelike) KV, equivalent to a stationary spacetime. For massless particles on the other hand, the condition for global equilibrium requires u a /T to be a CKV. A CKV is compatible with the cosmological expansion. Macroscopically, a gas of massless particles obeys the equation of state for radiation, p = ρ/3. Only for this perfect fluid equation of state a global equilibrium is possible in the expanding universe. (The fact that there exists an equilibrium also in the non-relativistic limit is not relevant in this paper.) Any deviation from p = ρ/3 will destroy the conformal symmetry. In other words, the conformal symmetry singles out perfect fluids with the equation of state for radiation. Apparently, this also implies that a conformal symmetry is incompatible with a non-vanishing entropy production. The purpose of the present paper is to show that, contrary to this expectation, entropy production due to a scalar viscous pressure may well be consistent with a conformal, timelike symmetry, albeit not a symmetry of the spacetime metric itself but the symmetry of an associated "optical" metric, characterized by a time dependent refraction index. The introduction of an effective refraction index of the medium as a new scalar parameter to characterize the cosmological evolution was shown to be useful in the context of cosmological particle production [3]. Here we explore the role of this parameter in more detail and for a different kind of non-equilibrium states for which the (not necessarily small) deviations from equilibrium are due to an increase in the entropy per particle, whereas the number of particles is preserved. In particular, we demonstrate that a time variation of the refraction index may take into account such kind of entropy producing deviations from the perfect fluid behavior. The standard global equilibrium for p = ρ/3 is then recovered as the limiting case for a refraction index unity. In this limit the optical metric coincides with the spacetime metric. Moreover, we show that a time dependent refraction index may be regarded as a consequence of the action of self-interacting forces on the microscopic constituents of an ultrarelativistic cosmic medium. Although entropy is produced, the microscopic particles are governed by an equilibrium distribution function. The deviations from standard equilibrium are mapped onto a non-zero, time varying chemical potential which is related to the refraction index in a simple way. Furthermore, we investigate the conditions under which the corresponding viscous pressure is sufficiently large to violate the strong energy condition and discuss simple examples for an inflationary evolution of the early universe.
The paper is organized as follows. In section 2 we summarize basic thermodynamic relations for a bulk viscous fluid. Section 3 presents the conformal symmetry concept of an optical metric and clarifies its relation to the production of entropy within the medium. Section 4 is devoted to a microscopic derivation of the fluid dynamics from relativistic gas theory in terms of a selfinteracting one-particle force. Cosmological applications, in particular the conditions for power-law inflation, are discussed in section 5, while section 6 sums up our conclusions on the relation between cosmological entropy production and the conformal symmetry of an optical metric. Units have been chosen so that c = k B = h = 1.

Basic thermodynamics
We assume the cosmic substratum to be modeled by a homogeneous and isotropic fluid with the energy-momentum tensor (2) and a particle number flow vector N i , given by where n is the particle number density. The conservation laws for particle number and energy are and respectively, where H ≡ȧ/a is the Hubble rate andṅ ≡ n ,i u i etc. In the following we recall basic relations for the evolution of the relevant thermodynamical quantities. With the help of the Gibbs equation (see, e.g., [4]) where s is the entropy per particle and T is the equilibrium temperature, we obtain nTṡ =ρ − (ρ + p)ṅ n .
Using here the conservation laws (6) and (7) yields Since n, T and H are positive, the entropy per particle increases for a negative viscous pressure Π. From the Gibbs-Duhem relation (see, e.g., [4]) where µ is the chemical potential, it follows that We assume equations of state in the general form i.e., particle number density and temperature are taken as the independent thermodynamical variables. Differentiating the latter relation and using the balances (6) and (7) provides us with an evolution law for the temperature, where and Moreover, we have used the general relation which follows from the fact that the entropy is a state function, i.e., ∂ 2 s/∂n∂T = ∂ 2 s/∂T ∂n. For the special case of a perfect fluid, i.e.,ṡ = 0, and for ultrarelativistic matter, i.e., p = ρ/3, the temperature law (14) specifies tȯ which is the well-known behavior T ∝ a −1 for the radiation temperature in an expanding universe.
With the help of Eqs. (13), (6), (9) and (14), the time evolution of the pressure may generally be written aṡ where is the square of the adiabatic sound velocity c s [see, e.g. [5]]. Using Eq. (17) in Eq. (12) yields The relations so far are completely general. Below we shall use them in connection with a relativistic Maxwell-Boltzmann gas with internal self-interactions. It is expedient to emphasize that a viscous pressure Π according to (19) influences the time behavior of the quantity µ/T . Lateron we shall be interested in a special case in which this non-equilibrium term is the only "source" for (µ/T ) · .

Conformal symmetry of optical metrics
Given a spacetime metric g ik and an isotropic medium with a four-velocity u a and a refraction index n r , one introduces the "optical" metricḡ ik by [6] Optical metrics are known to be helpful in simplifying the equations of light propagation in isotropic, refractive media. With respect to a metricḡ ik light propagates as in vacuum. Here we are interested in the role of optical metrics in relativistic gas dynamics. Guided by the connection between spacetime symmetries and equilibrium properties, outlined in the introduction, our starting point is to assume ξ a ≡ u a /T to be a conformal Killing vector of the optical metric, i.e., The main objective of this paper is to explore the fluid dynamics which is compatible with the latter requirement. In particular, we are interested in the question to what extent the CKV condition (21) admits the production of entropy in a cosmological context. In a first step, using the relations the last one being valid for any scalar S, the left-hand side of Eq. (21) is written as In a second step we find an explicit expression for the first term on the righthand side of Eq. (22). As any symmetric tensor, the Lie derivative of g ik may be decomposed into contributions parallel and perpendicular to the four-velocity. For homogeneous and isotropic media such a split amounts to where we have used that in a FLRW universe u i ;k = Hh i k ⇒ u i ;i = 3ȧ/a. Combining Eqs. (22) and (23) we obtain for the left-hand side of Eq. (21). Together with the definition (20) one finds Tȧ a for the conformal factor in (21) anḋ for the change rate of the refraction index. The last relation implies i.e., all deviations from aT = const (pure radiation, p = ρ/3, cf. Eq. (16)) have been mapped onto the time-varying refraction index n r . This demonstrates explicitly, that n r is indeed a useful quantity to characterize a fluid dynamics with a temperature evolution that is different from that forṡ = 0 and p = ρ/3. Inserting now the general temperature law (14) into Eq. (25), we obtaiṅ The variable Σ depends onṡ according to Eq. (15). Consequently, Eq. (26) establishes a relation between the rate of change of the refraction index and the expansion rate for a viscous cosmological fluid under the condition of a conformal symmetry of the optical metricḡ ik . This implies the statement that a conformal symmetry (21) does not contradict a production of entropy. To obtain a better understanding of this feature, we present a derivation for such kind of fluid dynamics for a relativistic gas in the following section.

Kinetic theory
The particles of a relativistic gas are assumed to move under the influence of a four-force F i in between elastic binary collisions, described by Boltzmann's collision integral C [f, f ]. The equations of motion of the gas particles are where γ is a parameter along the particle worldline which for massive particles may be related to the proper time τ by γ = τ /m. Since the particle fourmomenta are normalized according to p i p i = −m 2 , the force F i has to satisfy the relation p i F i = 0. The corresponding equation for the invariant one-particle distribution function f = f (x (τ ) , p (τ )) may be written as (cf. [4,7,8,9]) where is the Liouville operator. It will turn out that a suitably chosen effective oneparticle force F i may give rise to a time-dependent refraction index on the macroscopic level. The particle number flow 4-vector N i and the energy momentum tensor T ik are defined in a standard way (see, e.g., [7,8]) as Since we shall identify the particle number flow in (30) with the corresponding quantity in Eq. (5), we have used here the same symbol, although these expressions may not coincide in the general case. The integrals in the definitions (30) and in the following are integrals over the entire mass shell p i p i = −m 2 . The entropy flow vector S a is given by [7,8] where we have restricted ourselves to the case of classical Maxwell-Boltzmann particles. Using the general relationship [9] p a1 ....p an p b f dP we find for the balance equations and In particular, the energy balance becomes The entropy production density is a sum of two terms: Here, is the familiar contribution of Boltzmann's collision integral, while takes into account an entropy production due to the action of the force F i . Since Boltzmann's H theorem guarantees σ c ≥ 0, we have We restrict ourselves to the class of forces which admit solutions of Eq. (28) that are of the type of Jüttner's distribution function where α = α (x) and β a (x) is timelike. For f → f 0 the collision integral vanishes, i.e., C f 0 , f 0 = 0. With f replaced by f 0 in the definitions (30) and (31), the quantities N a , T ab and S a may be split with respect to the unique four-velocity u a according to Here we have identified the general fluid quantities of the previous section with those following from the dynamics of a Maxwell-Boltzmann gas. Note that the energy-momentum tensor T ik in Eqs. (30) and (41) does not coincide with the tensor T ik (eff ) introduced in Eq. (2). The exact integral expressions for n, ρ, p are given, e.g., in [4]). The entropy per particle s is For f → f 0 the energy balance (35) takes the forṁ The entropy production density becomes With the identification β i = u i /T we may write where we have introduced the quantity Π by Notice that we have identified this kinetic theory based viscous pressure with the corresponding phenomenological quantity of the previous sections. Consequently, the energy balance (43) takes the form (7). This implies the conclusion that a non-vanishing viscous pressure may generally be compatible with an equilibrium-type distribution function, provided, the gas particles move in a suitable force field. Conventionally, a bulk viscous pressure arises as a consequence of deviations from an equilibrium distribution. Wheras these deviations have to be small in standard non-equilibrium theories, a corresponding restriction is absent in the present case. Substituting the distribution (40) into Eq. (28) we obtain To make further progress, we have to chose specific force types. The most general force compatible with the cosmological principle is [10] where E ≡ −u i p i is the particle energy measured by a comoving observer. Generally, F may still depend on E. With the condition (47) then specifies to Now we restrict ourselves to the case ∂F/∂E = 0 and consider massless particles p i p i = −m 2 = 0, which macroscopically corresponds to an equation of state p = ρ/3. As a consequence we find α ,a = −3F u a ⇒α = 3F (51) and β (a;b) ≡ 1 2 Combining the relations (51) and (52), we obtain Using the equation of state p = nT = ρ/3 in Eq. (12) and identifying α = µ/T , we have µ T In the standard case (without a force contribution) one has α = const (= 0) and recovers the familiar cooling rateṪ /T = −ȧ/a for a relativistic fluid. We conclude that the force field manifests itself in the existence of a non-vanishing, time dependent chemical potential. From (54) and (53) we recover which shows the consistency of our approach. Comparing (53) with (23) we obtainṪ i.e., with the identificationṅ r /n r =α/3 we recover the conformal symmetry relation (25). Moreover, we have traced back the time variation of the refraction index to a self-consistent interaction within the system. From the moment equation (34) it follows that Since ρ = 3nT we reproduce Eq. (7) with This is consistent with (46) and (48). The quantity F depends on macroscopic fluid quantities. According to (27) and (48), these macroscopic quantities determine the motion of the individual microscopic particles which themselves are the constituents of the macroscopic medium. Consequently, the force F i describes a self-interaction of the fluid. This self-interaction is accompanied by a time variation of the chemical potential, equivalent to a time dependent refraction index. From the entropy production density (44) we find consistent with (45) and the phenomenological relation (10). The ratioṡ/ (3H) = −Π/ (nT ) which characterizes the quantity Σ is just the ratio between nonequilibrium pressure Π and equilibrium pressure p = nT . Via Eqs. (51) and (58), relation (60) also implies With the first equation in (61) we have recovered the special case of Eq. (19) for p = nT , ρ = 3nT , c 2 s = ∂p/∂ρ = 1/3. Furthermore, we have precised, how a decaying refraction index generates a negative contribution to the pressure, equivalent to positive entropy production. Integrating the part (µ/T ) · = 3ṅ r /n r of Eq. (61) by assuming µ/T → 0 for n r → 1 (standard radiation with µ = 0 and n r = 1), we find exp µ T = n 3 r .
Consequently, the distribution function (40) can be written as i.e., the dissipative force gives rise to a time varying factor n 3 r in the equilibriumtype distribution function.
In order to check the consistency of the entire framework, we may now use the force (48) with (58) in the microscopic equations of motion in Eq. (27). Contracting the latter with the macroscopic four-velocity results in where we have used that Du i dγ = u i ;n p n .
It follows that the distribution function (63) is indeed maintained for particles under the action of the force F i , which proves the consistency of our approach. While the second factor in the expression (63) for f 0 remains constant, the first one changes according to Eq. (61).

Conformal symmetric universe
Now we combine the non-equilibrium fluid dynamics of the previous section with the gravitational field equations. In a homogeneous and isotropic, spatially flat universe Einstein's equations reduce to The relation between Π andṅ r in Eq. (61) may be solved with respect to the non-equilibrium pressure, For ρ + 3P we obtain The case ρ + 3P = 0 is equivalent to Consequently, for a decaying refraction index with |ṅ r /n r | ≥ 2ȧ/a the strong energy condition ρ + 3P > 0 is violated and we have accelerated expansion. This may be explicitly demonstrated for a simple example. With Π from (68) the energy balance (7) may be writteṅ The simplest case is to assume a decay rate which is proportional to the Hubble rate:ṅ The constant factor β in the range 0 ≤ β < 1 has been chosen for convenience. Of course, β can be assumed constant at most piecewise. Integration of Friedmann's equation yields a ∝ t Obviously, there is accelerated expansionä ≥ 0 for β ≥ 1/2, which is consistent with (70). The limit β = 0 corresponds to the standard radiation dominated universe. If β approaches β ≈ 1, we have ρ ≈ const and, according to Eq. (67), an approximately constant Hubble rate H, i.e., exponential expansion. Any β in the range 0 ≤ β < 1 is connected with entropy production. Combining (68) and (72) we find and, with (60),ṡ = 12βȧ The entropy per particle grows logarithmically with the cosmic time. According to (58), (74), and (48), the corresponding force is The action of this force on the microscopic constituents of the medium realizes a power-law evolution (73) of the universe, which is a manifestation of a conformal symmetry (21) of the optical metric (20) and, moreover, implies a positive entropy production according to (75). This represents an exactly solvable model, both macroscopically and microscopically, of a specific non-equilibrium configuration of a self-interacting cosmic gas.

Conclusions
We have shown that the introduction of a refraction index n r of the cosmic medium as a new scalar parameter allows us to understand early phases of the cosmological evolution as a specific non-equilibrium configuration of a selfinteracting relativistic gas. The latter is characterized by an equilibrium-type distribution function of the microscopic particles. Deviations from equilibrium are mapped onto a time-varying chemical potential which is related to n r in a simple way. The refraction index defines an optical metricḡ ik = g ik − 1 − n −2 r u i u k which coincides with the spacetime metric g ik for n r = 1. A decaying refraction index gives rise to a negative contribution to the scalar pressure of the cosmic substratum. We have established an exactly solvable, self-consistent model for this type of dynamics which is connected with entropy production, but at the same time realizes a conformal, timelike symmetry of the optical metric. In other words, for any deviation from the behavior of an ultrarelativistic perfect fluid which is due to a scalar, viscous pressure, one may find a refraction index such that the evolution of the universe appears as the manifestation of a conformal, timelike symmetry of the corresponding optical metric. For a decay rate of the refraction index which is of the order of the Hubble expansion rate, the associated negative pressures may lead to a violation of the strong energy condition ρ + 3P > 0. Consequently, the resulting dynamics is inflationary.