de Sitter local thermodynamics in f ( R ) gravity

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2][3][4][5][6][7] It was used to construct an inflationary model of the early Universe -the Starobinsky inflation, which is controlled by the R 2 contribution to the effective action.This class of models, f (R) ∝ R − R 2 /M 2 , was also reproduced in the so-called q-theory, 8 where q is the 4-form field introduced by Hawking 9 for the phenomenological description of the physics of the deep (ultraviolet) vacuum (here the sign convention for R is opposite to that in Ref. 2 ).The Starobinsky model is in good agreement with the observations.However, despite the observational success, the theory of Starobinsky inflation is still phenomenological.][12] In this paper we do not discuss the problem of the UV-completion.We consider the de Sitter stage of the expansion of the Universe, and use the f (R) gravity for the general consideration of the local thermodynamics of the de Sitter state.The term local means that we consider the de Sitter vacuum as the thermal state, which is characterized by the local temperature.This consideration is based on observation, that matter immersed in the de Sitter vacuum feels this vacuum as the substance with the local temperature T = H/π, where H is the Hubble parameter.This temperature is twice larger than the Gibbons-Hawking one, and it has no relation to the cosmological horizon.The existence of the local temperature, suggests the existence of the other local thermodynamic quantities, which participate in the local thermodynamics of the de Sitter state.In addition to the the local entropy density s and local vacuum energy density ǫ, there are also the local thermodynamic variables related to the gravitational degrees of freedom.
The f (R) theory demonstrates that the effective gravitational coupling K (it is the inverse Newton constant, K = 1/16πG) and the scalar curvature R are connected by equation K = df /dR.This suggests that K and R are the thermodynamically conjugate variables. 13,14This pair is similar to the pair of electrodynamic variables, electric field E and electric induction D, which participate in the thermodynamics of dielectrics.
Example of the influence of the de Sitter vacuum to the matter immersed into this vacuum is provided by an atom in the de Sitter environment.As distinct from the atom in the flat space, the atom in the de Sitter vacuum has a certain probability of ionization.The rate of ionization is similar to the rate of ionization in the presence of the thermal bath with temperature T = H/π. 15,168][19] .That is why it is natural to consider the temperature T = H/π as the local temperature of the de Sitter vacuum.Although the local temperature is twice larger than the Gibbons-Hawking temperature assigned to the horizon, T GH = H/2π, there is the certain connection between the local thermodynamics and the thermodynamics of the event horizon.It appears that the total entropy of the volume V H bounded by the cosmological horizon coincides with the Gibbons-Hawking entropy, S = sV H = A/4G.This demonstrates that the local thermodynamics in the (3 + 1) de Sitter is consistent with the global thermodynamics assigned to the cosmological horizon, although the origin of such bulk surface correspondence is not very clear.
Here we extended the thermodynamic consideration to the f (R) gravity.Using the local thermodynamics with T = H/π, we obtained the general result for the total entropy inside the horizon, S = sV H = 4πKA, where K = df /dR is the effective gravitational coupling.This is in agreement with the global thermodynamics of de Sitter cosmological horizon, which provides the further support for the local thermodynamics with the local temperature T = H/π in the de Sitter vacuum in the (3 + 1)-dimensional spacetime.

A. Local temperature and local entropy
We consider the de Sitter thermodynamics using the Painlevé-Gullstrand (PG) form, 20,21 where the metric in the de Sitter expansion is ( Here the shift velocity is v(r) = Hr.This metric is stationary, i.e. does not depend on time, and it does not have the unphysical singularity at the cosmological horizon.That is why it is appropriate for consideration of the local thermodynamics both inside and outside the horizon.Since the vacuum serves as the thermal bath for matter, it is not excluded that the quantum vacuum may have its own temperature and entropy. 22If so, then the quasi-equilibrium states of the expanding Universe may have different temperatures: the temperature of the gravitational vacuum and the temperature of the matter degrees of freedom. 23n this section we discuss the vacuum without the excited matter ignore the thermal activated creation of matter from the vacuum.The excitation and thermalization of matter by the vacuum thermal bath will be discussed in Sec.IV.
If the vacuum thermodynamics is determined by the local the activation temperature T = H/π, then in the Einstein gravity with cosmological constant the vacuum energy density is quadratic in temperature: This leads to the free energy density of the de Sitter vacuum, F = ǫ vac − T dǫ vac /dT , which is also quadratic in T , and thus the entropy density s vac in the de Sitter vacuum is linear in T : The temperature T and the entropy density s vac are the local quantities which can be measured by the local static observer.

B. Gibbs-Duhem relation
The T 2 dependence of vacuum energy on temperature suggests the modification of the thermodynamic Gibbs-Duhem relation for quantum vacuum and to the reformulation of the vacuum pressure.The conventional vacuum pressure P vac obeys the equation of state w = −1 and enters the energy momentum tensor of the vacuum medium in the form: In the de Sitter state the vacuum pressure is negative, P vac = −ǫ vac < 0. This pressure P vac does not satisfy the standard thermodynamic Gibbs-Duhem relation, T s vac = ǫ vac +P vac , because the right hand side of this equation is zero.The reason for that is that in this equation we did not take into account the gravitational degrees of freedom of quantum vacuum.Earlier it was shown, that gravity contributes with the pair of the thermodynamically conjugate variables: the gravitational coupling K = 1 16πG and the scalar Riemann curvature R, see Refs. 8,24,25.][28][29] The quantities K and R can be considered as the local thermodynamic variables, which are similar to temperature, pressure, chemical potential, number density, spin density, etc., in condensed matter physics.Indeed, since the de Sitter spacetime is maximally symmetric, its local structure is characterized by the scalar curvature alone, while all the other components of the Riemann curvature tensor are expressed via R: That is why the scalar Riemann curvature as the covariant quantity naturally serves as one of the thermodynamical characteristics of the macroscopic matter. 30,31On the other hand, the gravitational coupling K = df /dR serves as the analog of the chemical potential, which is constant in the full equilibrium.The new thermodynamic variables, which come from the gravity, and Eq. ( 3) for the entropy density allow us to introduce the corresponding Gibbs-Duhem relation for de Sitter vacuum, which has the conventional form: This equation is obeyed, since ǫ vac + P vac = 0; R = −12H 2 ; and T s vac = 12π 2 KT 2 = 12KH 2 , which supports the earlier proposal that K and R can be considered as the thermodynamically conjugate variables. 24,25he Eq.( 6) can be also written using the effective vacuum pressure, which absorbs the gravitational degrees of freedom: Then the conventional Gibbs-Duhem relation is satisfied: The equation ( 8) is just another form of writing the Gibbs-Duhem relation (6).But it allows to make different interpretation of the de Sitter vacuum state.The introduced effective de Sitter pressure P is positive, P = ǫ vac > 0, and satisfies equation of state w = 1, which is similar to matter with the same equation of state.As a result, due to the gravitational degrees of freedom, the de Sitter state has many common properties with the non-relativistic Fermi liquid, where the thermal energy is proportional to T 2 , and also with the relativistic stiff matter with w = 1 introduced by Zel'dovich. 32

C. Hubble volume entropy vs entropy of the cosmological horizon
Using the entropy density in Eq.( 3), one may find the total entropy of the Hubble volume V H -the volume surrounded by the cosmological horizon with radius R = 1/H: where A is the horizon area.This Hubble-volume entropy coincides with the Gibbons-Hawking entropy of the cosmological horizon.However, here it is the thermodynamic entropy coming from the local entropy of the de Sitter quantum vacuum, rather than the entropy of the horizon degrees of freedom.Anyway, the relation between the bulk and surface entropies in the local vacuum thermodynamics suggests some holographic origin.Although such bulk-surface correspondence is valid only in the (3 + 1)-dimension. 33

D. Hubble volume vs the volume of Universe
It is not excluded that our Universe is finite.Its volume V might be comparatively small, not much larger than the currently observed Hubble volume V H . 34 If the Universe is finite and if the de Sitter state represents the excited thermal state of the quantum vacuum, the thermal fluctuations of the deep quantum vacuum may become important.According to Landau-Lifshitz, 35 the thermal fluctuations are determined by the compressibility of the system and by its volume.In case of the Universe with the volume V , the fluctuations of the vacuum energy density are given by: 36 Using temperature T = H/π, vacuum compressibility 36 χ vac ∼ 1/M 4 Pl , and the energy density of the de Sitter state < ǫ vac >∼ M 2 Pl H 2 , one can estimate the relative magnitude of fluctuations: The volume of the present Universe exceeds the Hubble volume, V > V H , and thus the thermal fluctuations of the vacuum energy density are still small.

III. THERMODYNAMICS OF DE SITTER STATE IN f (R) GRAVITY
A. Gibbs-Duhem relation in f (R) gravity Let us show that equation S hor = 4πKA remains valid also in the f (R) gravity, but with the gravitational coupling determined as the thermodynamic conjugate to the curvature.In the f (R) gravity the action is: The generalization of the modified Gibbs-Duhem relation for the de Sitter states (i.e. for the states with constant four-dimensional curvature) in the f (R) gravity is: In the equilibrium de Sitter state the curvature is determined by equation: Here K is the natural definition of the variable, which is thermodynamically conjugate to the curvature R, while ǫ vac serves as the corresponding thermodynamic potential.

B. Entropy of cosmological horizon in terms of effective gravitational coupling
The local entropy of the de Sitter state follows from Eq.( 13), assuming that the local temperature of the equilibrium dS states is T = H/π.Then the total entropy of the Hubble volume V H is given by the same Eq.(9): But now K is the effective gravitational coupling in Eq.( 14).8][39] .But here it was obtained using the local thermodynamics of the de Sitter vacuum.This demonstrates that the local thermodynamics of the de Sitter vacuum is valid also for the f (R) gravity.The effective gravitational coupling K serves as one of the thermodynamic variable of the local thermodynamics.This quantity plays the role of the chemical potential, which is thermodynamically conjugate to the curvature R, and it is constant in the thermodynamic equilibrium state of de Sitter spacetime.
For illustration, we consider an example of the modification of the gravitational coupling K in the de Sitter environment.In the conventional Einstein gravity, where f (R) = K 0 R + Λ, the de Sitter state has the equilibrium value of the curvature, R 0 = −2Λ/K 0 = −12H 2 .Let us add the quadratic term to the Einstein action: 8,37 Then one obtains the following equations for the equilibrium value of the curvature R 0 , the entropy of the Hubble volume S hor and the equilibrium value of the effective coupling K: The equilibrium curvature in the de Sitter space R 0 is obtained from Eq.( 15).It is the same as in Einstein gravity, because the quadratic terms in Eq.( 15) are cancelled.The local entropy s vac , which follows from Eq.( 13), is determined by the modified gravitational coupling K.As a result, the entropy of the Hubble volume in Eq.( 19), which we identify with the entropy of the horizon S hor , is also determined by the modified coupling K.The latter is given by Eq. (20).The local entropy s vac changes sign for K < 0, while the cosmological expansion is still described by the de Sitter metric.However, the negative K requires the negative parameter p < 0, which marks the instability of such de Sitter vacuum. 37

IV. LOCAL TEMPERATURE AND DE SITTER DECAY
The extension of the thermodynamics to the f (R) gravity supports the idea that the de Sitter vacuum is the thermal state with the local temperature T = H/π.On the other hand the nonzero local temperature of the vacuum suggests that the de Sitter vacuum is locally unstable towards the creation of thermal matter from the vacuum by thermal activation.This is distinct from the mechanism of creation of the pairs of particles by Hawking radiation from the cosmological horizon, which may or may not lead to the decay of the vacuum energy.There are still controversies concerning the stability of the de Sitter vacuum caused by Hawking radiation, see e.g. 40,41and references therein.
To describe the decay of the vacuum due to activation and thermalization of matter, the extension of the Starobinsky analysis of the vacuum decay [42][43][44][45] is needed.The thermal exchange between the vacuum and the excited matter generates the thermal relativistic gas with temperature approaching the vacuum temperature T = H/π.Its energy density in thermal equilibrium must approach ǫ M ∼ T 4 , or ǫ M = bH 4 , where the dimensionless parameter b depends on the number of the massless relativistic fields.That is why during the heating in the vacuum bath the matter energy density tends to the local thermal equilibrium.The energy exchange between the vacuum heat bath and matter can be described by the following dynamical modification of the Friedmann equations, 46 where the dissipative Hubble friction equation ∂ This equation describes the tendency of matter to approach the local temperature of the vacuum, T = H/π.The extra gain of the matter energy, 4bH 5 , must be compensated by the corresponding loss of the vacuum energy: Here we use for simplicity the conventional general relativity with ǫ vac = KR.This phenomenological description of the energy exchange between vacuum and matter does not depend on the details of the microscopic (UV) theory, and requires only the condition for slow variation of the Hubble parameter, Ḣ ≪ H 2 .Since the vacuum energy density is ǫ vac ∝ KH 2 , one obtains from Eq.( 22) the following time dependence of the Hubble parameter and of energy densities: Here M Pl is the Planck mass, M 2 Pl = K, and t Pl = 1/M Pl is Planck time.We assume that t 0 ≫ t Pl , and thus Ḣ ≪ H 2 .8][49][50][51] .9][50] the parameter t 0 is related to the initial value of the Hubble parameter at the beginning of inflation at t = 0: This H(t = 0) corresponds to the scaleron mass M in Starobinsky inflation.The time t 0 ∼ E 2 Pl /H 3 t=0 is called the quantum breaking time of space-times with positive cosmological constant. 52,53 CONCLUSION The local thermodynamics of the de Sitter state in the Einstein gravity gives rise to the Gibbons-Hawking area law for the total entropy inside the cosmological horizon.Here we extended the consideration of the local thermodynamics to the f (R) gravity.We obtained the same area law, but with the modified gravitational coupling K = df /dR, which is in agreement with the global thermodynamics.This supports the suggestion that the de Sitter vacuum is the thermal state with the local temperature T = H/π, and that the local thermodynamics is based on the thermodynamically conjugate gravitational variables K and R. The variable K plays the role of the chemical potential, which is constant in the thermal equilibrium.
The local temperature T = H/π has the definite physical meaning.It is temperature, which is experienced by the external object in the de Sitter environment.In particular, this temperature determines the local activation processes, such as the process of ionization of an atom in the de Sitter environment.

I. Introduction 1 II.
Thermodynamics of the de Sitter state 2 A. Local temperature and local entropy 2 B. Gibbs-Duhem relation 3 C. Hubble volume entropy vs entropy of the cosmological horizon 4 D. Hubble volume vs the volume of Universe 4 III.Thermodynamics of de Sitter state in f (R) gravity 4 A. Gibbs-Duhem relation in f (R) gravity 4 B. Entropy of cosmological horizon in terms of effective gravitational coupling 5 IV.Local temperature and de Sitter decay 5