1. Introduction

The accelerated cosmic expansion indicates the presence of a negative-pressure component to the total energy density of the universe today. This component is the source of the accelerated expansion and, either material fluid or geometry, is known as dark energy. The observational evidence and theoretical consistency of dark energy is well supported; for a review see Refs. [¹-³] and references therein. Dark energy nowadays is a fundamental element of the standard cosmological model, however the physical mechanism behind it still a mystery. The simplest and best-known model of dark energy is the energy of the vacuum represented as the cosmological constant added to the Einstein field equations. The main feature of vacuum energy is that its energy density is constant in time and is spatially smooth. The cosmological constant with cold dark matter are the key elements of the standard cosmological model or ΛCDM.

In cosmology, a perfect fluid description is adequate to model the known cosmic material components (e.g. photons, baryons, neutrinos) and also dark matter (even though we don’t have a consolidated microscopic theory of dark matter, when modeled as dust, a pressureless perfect fluid, there is theoretical and observational consistency). Dark energy, like the other main cosmological components, as a first approach is modeled in the framework of perfect fluids in a homogeneous and isotropic expanding universe. The perfect fluid approach applied to describe the cosmic components fit very well the cosmological observations at background and linear perturbation level; and also for some components has an underlying microscopic theory that support it.

To obtain an accelerated cosmic expansion the material content of the universe must violate the strong energy condition, ∑ _i (ρ _i + 3p _i ) < 0, where ρ _i and p _i are the energy density and the pressure of each component respectively. The total pressure must be negative. Since baryonic and dark matter are pressureless and radiation pressure is ρ _r /3 > 0, then there must be a source of negative pressure, this source is the dark energy. In the standard approach, the dark energy is considered as a fluid with negative pressure and barotropic equation of state, p = wρ (with w constant), and value w = −1. The dark energy equation of state p = −ρ implies that the energy density of dark energy is a constant, i.e., has no dynamics.

Recent observational reconstructions of the dark energy equation of state shows that the parameter w could be dynamical, i.e., depends on time (scale factor) [⁴, ⁵] allowing to cross the phantom divide line (w < −1), and there are even studies that suggest a slight preference to the cosmological phantom regime w < −1 [⁶-⁸], but it is not clear if it corresponds properly to a phantom fluid or to an additional physical effect which gives an effective phantom dark energy equation of state.

Dynamical dark energy, w = w(α), together with the assumption of dark energy as a perfect fluid brings up some thermodynamic problems such as the positiveness of the entropy, temperature, and chemical potential implies that w ≥ −1, which is in direct conflict with phantom dark energy [⁹,¹⁰].

One way to avoid some thermodynamic problems is to suppose that the dark energy is a fluid with bulk viscosity breaking the perfect fluid hypothesis. A fluid with bulk viscosity means that dissipative processes occur, which in cosmic fluids has the advantage of allowing the violation of the dominant energy condition (ρ + p < 0) without the dark energy necessarily being phantom [¹¹]. In this case we have an effective pressure given by p _eff = p + Π, where p = wρ is the barotropic pressure and Π < 0 is the viscous pressure. The possibility of explaining the accelerated expansion of the universe at the late future as an effect of the effective negative pressure due to bulk viscosity in the cosmic fluids was first considered in [¹²,¹³]. Viscous matter or radiation cosmologies can be mapped into the phantom dark energy scenario [¹⁴,¹⁵] and also a viscous fluid is able to produce a Little Rip cosmology as a purely viscous effect [¹⁶].

A perfect fluid in equilibrium generates no entropy and no frictional heating because its dynamic is reversible and without dissipation. We know that real fluids behave irreversibly, and if we want to model dissipative processes we require a relativistic theory of dissipative fluids. A classical irreversible thermodynamics was first extended from Newtonian to relativistic fluids by Eckart in 1940 [¹⁷]. In the Eckart theory, the effective pressure of the cosmic fluid is modeled as Π = −3ξH, where ξ is a function and H the Hubble parameter. The Eckart theory has the problem that dissipative perturbations propagate at infinite speeds; this noncausal feature is its main limitation and therefore this approach could be useful to find insight from toy models but not as a realistic theory. Nevertheless, Eckart theory has been used widely to model bulk viscosity in dark matter and dark energy models, for example, interacting viscous dark matter and dark energy [¹⁸, ¹⁹], the possibility of crossing the phantom divide line [²⁰], the magnitude of the viscosity to achieve this crossing using cosmological data [²¹], Big Rip singularities for various forms of the equation of state parameter and the bulk viscosity [²²], and unified dark fluid cosmologies [²³-²⁶]. The causal extension of the Eckart theory is the so-called Israel-Stewart theory [²⁷,²⁸]. This approach presents a better description than Eckart theory, including a casual description of dissipative processes associated to small deviations of equilibrium. The Israel-Stewart theory converges to the Eckart theory, when the collision time-scale in the transport equation of fluid is zero, i.e., when the bulk viscous model is noncausal and unstable. An interesting review about viscous cosmology can be found in [²⁹] and for a dynamical analysis for a bulk viscosity dark matter model in the full Israel-Stewart formalism see [³⁰]. A pioneer work about dissipative processes in cosmology is [³¹] and a highly recommended summary about relativistic fluid dynamics, dissipative relativistic fluids, applications to cosmology and astrophysics and bulk viscous perturbations is [³²]. Bulk viscous dark energy has been studied in several contexts: as phantom dark energy [¹⁵,³³,³⁴]; dark energy with bulk viscosity observational constraints were studied in Refs. [³⁵-³⁸]; the present acceleration of the universe as effect of bulk viscosity of self-interacting scalar field [³⁹];

An important problem in cosmology is that most of the dark energy models are able to adjust fairly well the observational data, this degeneracy hinders the tests and selection of more well-grounded dark energy models. The data shows only an accelerated cosmic expansion, but does not reveal the intrinsic nature of the source that causes this acceleration. Despite unknown, dark energy should be consistent with the known laws of physics; for that reason the dark energy when modeled as an exotic fluid must satisfy the bounds imposed by the laws of thermodynamics. In this vein, dark energy thermodynamics has been studied by several authors: a remarkable theoretical treatment in the context of perfect fluids can be found in Ref. [⁹]; thermodynamic properties of dark energy with varying equation of state parameter w(α) [¹⁰, ⁴⁰, ⁴¹]; and, thermodynamic properties of dark energy as a self-interacting complex scalar field [⁴²]. Studies about the thermodynamics of dark energy with viscosity: the validity of the generalized second law of thermodynamics in a non-flat universe in the presence of viscous dark energy [⁴³]; some thermodynamic aspects of an alternative to the standard expression for bulk viscosity [⁴⁴]; the thermodynamic stability analysis of non-interacting diffusive cosmic fluids with barotropic equation of state [⁴⁵]; and, thermodynamics of viscous dark energy in the braneworld context [⁴⁶,⁴⁷]. Following this path, in this work we investigate some general thermodynamic aspects of dark energy modeled as a perfect fluid with constant equation of state parameter w = w ₀, dynamical dark energy w = w(α), and bulk viscosity dark energy in the Eckart and Israel-Stewart theories.

This paper is organized as follows: in Sec. 2 is presented the general cosmological and thermodynamic considerations used all along the paper. The paper is divided into two parts, the first one, dedicated to dark energy as a perfect fluid, first, Sec. 3 summarizes the main results of the thermodynamics of dark energy with constant equation of state parameter, the same treatment but with dynamical dark energy is presented in Sec. 4. In second part, first in Sec. 5 are established the main ideas of a dissipative fluid in a flat, homogeneous and isotropic universe used in this work, then in Sec. 6 is presented the dark energy thermodynamic analysis in the Eckart framework to then in Sec. 7 in the Israel-Stewart. Finally, in Sec. 8 we presented the conclusions.

2. General considerations

In this section we present the general cosmological and thermodynamic considerations used all along the paper. Let us consider a Friedman-Lemaître-Robertson-Walker universe (FLRW), i.e., a homogeneous, isotropic spacetime, and the flat universe case. We are considering the late future time universe in which the dominant component is the dark energy, ∑ _i ρ _i ≈ ρ. In what follows we assume only one fluid as the main component of the universe, which experiments a dissipative process during cosmic evolution. With these considerations the Friedmann equations are

where the dot denotes derivatives with respect to the cosmic time, H=a˙/a is the Hubble parameter and we use natural units 8πG = c = 1.

In a FLRW cosmology the energy density conservation equation is

where ρ is the energy density and p is the pressure.

The thermodynamic assumptions are: the physical three dimensional volume of the universe at a given time can be expressed in terms of the scale factor V = V ₀ α ³(t) (where V ₀ is the three dimensional volume at the present time), the internal energy of the a cosmological fluid is U = ρV , and the first law of thermodynamics is expressed like

where T, S, µ and N are the temperature, entropy, chemical potential and the number of particles respectively.

The temperature will be assumed as a function of the number of particle density, n = N/V , and the energy density, ρ, therefore T = T(n,ρ). The last assumption gives the following useful relation [³¹,³²]

These are the main general work hypothesis of this work.

3. Dark energy as a perfect fluid with w = constant

In this section we will present the main thermodynamic properties of dark energy as a barotropic perfect fluid with constant equation of state parameter. This section is a summary of the previous work [⁹].

The main considerations in this section are: a barotropic equation of state p = wρ (with the condition w < −1/3 to ensure the accelerated expansion) and the conservation of the number of particle current, n ^α = nu ^α (u ^α is the 4-velocity), which for a perfect fluid this is a conserved quantity n;αα=0:

where w is a constant. The last two Eqs. (6) and (7) in the energy density conservation Eq. (3) and the first law of thermodynamics (4) gives:

Now we can calculate some relevant thermodynamic quantities.

4. Dark energy as a perfect fluid with w = w(α)

In this section we will study dynamical dark energy as a perfect fluid with the same background hypothesis of the previous section; the conservation of particle Eq. (7), the perfect fluid energy density conservation Eq. (8) and the first law of thermodynamics (9), hold in this case. But with the difference that now the equation of state has the form

The core idea of dynamical dark energy is that the equation of state parameter is a variable quantity, w(α), since it is constructed from two variable quantities.

5. Dissipative dark energy

A common feature of many of dark energy models is the assumption that it can be modeled as a perfect fluid. As stated before, a perfect fluid generates no entropy and no frictional heating because its dynamics is reversible and without dissipation. This approach works quite well in standard cosmology, but the observational reconstructions of the dark energy equation of state suggest the possibility of phantom dark energy which is incompatible with the perfect fluid hypothesis at thermodynamic level. Real fluids behave irreversibly, for this reason, if we want to maintain the fluid hypothesis to dark energy and ensure its thermodynamic compatibility, it is interesting to consider dissipative processes like bulk viscosity. Qualitatively, the bulk viscosity can be interpreted as a macroscopic consequence coming from the frictional effects within the fluid.

For a dissipative fluid, the particle 4-current will be taken to be of the same form as n;αα=0, this corresponds to choosing an average 4-velocity in which there is no particle flux, known as the particle frame. At any event in spacetime, it is considered that the thermodynamic state of the fluid is close to a fictitious equilibrium state at that event. The local equilibrium scalars are denoted with a bar: n-, ρ-, p-, S-, T- and the local equilibrium 4-velocity is u-μ. In the particle frame, it is possible to choose u-μ such that the number and energy densities coincide with the local equilibrium values, while the pressure in general deviates from the local equilibrium pressure:

where Π is the bulk viscous pressure. We are considering in the last Eqs. (32) that the thermodynamical system is in the near equilibrium regime, if the fluid is out of equilibrium as a result of dissipative effects then there is no unique average 4-velocity spoiling the formalism [³¹]. In the presented case, the near equilibrium condition occurs when the local equilibrium pressure p is dominant over the viscous pressure (which encodes deviations from equilibrium), i.e., when the condition

is satisfied. This condition is the same for both non causal and causal approaches. For notation, from now we will drop the bar of the thermodynamic variables and write the pressure as p _eff = p+Π, where p is the usual barotropic pressure p = wρ. The energy density conservation equation in this case is

From the first law of thermodynamics (4) and remembering that for the energy U = ρV , volume V = V ₀ α ³ and number of particle density n = N/V , can be derived an expression for the entropy in terms of the bulk viscous pressure

The above equation sets a strong general constraint in this framework: due to n and T are positive, and H in an expanding universe is positive too to ensure a non-negative entropy production (second law of thermodynamics) the bulk viscous pressure necessarily must be Π ≤ 0.

6. Eckart bulk viscosity dark energy model

The simplest approach to treat bulk viscosity in cosmology is the Eckart theory [¹⁷]. This approach has some important limitations, the main one is that it is a noncausal approach to dissipative processes.

In this framework the bulk viscosity pressure is given by

where the bulk viscosity depends on the function ξ and the Hubble parameter. This formalism has been widely used at background level. Parameter constraints from observational data of the Eckart bulk viscosity dark energy can be found in Refs. [³⁵, ³⁶]; in both works the main results are that the Bayesian evidence show that the Eckart viscous dark parameters are small and statistically indistinguishable from ΛCDM with the current data and the effective EoS is slightly phantom. First, in Ref. [³⁵] the authors study three Eckart viscous dark energy models: p~de=-ρde-3ηH2 and p~de=ωρde-3ηH2, both combined with presureless matter and curvature. Note that ξ=ηH and when it is only considered dark energy, it is equal to model 2 of this work. Using combined CMB +SNIa +BAO +Cosmic chronometer +gravitational lensing data they found the 2σ upper bounds of the parameter η<0.003, also for the ω DE model they get ω=-1.001-0.011+0.012. In [³⁶] the authors make a Bayesian analysis using CMB +SNIa +BAO +Cosmic chronometer data for three models: p~de=ωρde-3ξ0H2 model I (when it is only considered dark energy, it is equal to model 2 of this work), p~de=-ρde-3ξ0ρde model II, and p~de=ωρde-3ξ0H model III (when it is only considered dark energy, it is equal to model 1 of this work). They define the parameter ξ~=(8πG/H0)ξ0 and the general parameter constraints are: ξ~=-0.0097±0.013 for model I, ξ~=-0.012±0.019 for model II and ξ~=-0.002±0.008 for model III. Draws attention the negative sign of ξ~, this implies Π>0 which is forbidden, however the error statistically allows the results. Finally, they found ω∼-1.05, slightly phantom too.

From (34), the energy density conservation equation is

First, we can sketch some general features without determining the functional form of ξ. From the Friedmann Eqs. (1) and (2) we have

Note that for the case without viscosity ξ = 0 and w = −1 the last equation boils down into H˙ = 0 which is the de Sitter case. Now, assuming w = const., integrating the last equation calculate H as a function of the bulk viscosity we have [⁴⁸].

where C is an integration constant. Integrating again, for w ≠ −1, we obtain an expression for the scale factor:

where D is a another integration constant.

7. Israel-Stewart bulk viscosity dark energy model

The Israel-Stewart theory provides a better description than the Eckart theory. It is a causal and stable theory of thermal phenomena in the presence of gravitational fields. This theory besides solving the non-causal problem of the Eckart theory, enrich the framework including new features like the entropy has terms of second order in the dissipative variables and incorporates transient phenomena on the scale of the mean free path/time, outside the quasi-stationary regime of the classical theory. In the Israel-Stewart theory we have the same Friedmann equations with an equation for the causal evolution of the bulk viscous pressure given by

The last equation is known as the transport equation of the viscous pressure Π. Where τ is the relaxation time for bulk viscous effects (in the limit τ = 0 the theory is noncausal), ξ the bulk viscosity coefficient and T the temperature.

It is clear that the Israel-Stewart theory is much more complex than the Eckart theory. In addition to having to propose or infer a function for ξ and for the relaxation time τ, we must know the temperature T and solve a differential equation involving these physical quantities. Let’s see how we can make a proposal and look under which conditions an analytical solution of Eq. (58) can be obtained.

First for the temperature, following [¹⁵, ³¹, ³²], we assume that the temperature depends only on the particle number an energy density, T = T(n,ρ), with this

The conservation equations of energy density and particle density in the Israel-Stewart theory are

substituting both expressions in Eq. (60) and using (5) we have

If w is not a function of ρ, then (∂p/∂ρ) _n = w (which w is not necessarily a constant). Taking this into account this, it is easy to prove that the expression for the temperature

is solution of (63). Which is the same as for the perfect fluid case and Eckart bulk viscosity.

For ξ, we assume a power-law for the bulk viscosity in terms of the energy density of the fluid [⁴⁹],

where s is a constant arbitrary parameter and ξ ₀ a positive constant.

And finally we follow [⁴⁸] in which a simple relation between τ and ξ is proposed

Equations (65) and (66) are the two main hypotheses assumed in order to find a solution for the transport equation of the viscous pressure (58).

Combining the Friedmann Eqs. (1), (2), with the pressure in this context p _eff = p + Π = wρ + Π, it is easy to obtain

With the Eq. (67), a differential equation for the Hubble parameter can be constructed substituting (64), (65) and (66) into the transport Eq. (58), then differentiate (67) and equal both expressions considering the energy density conservation Eq. (61). The resulting equation for the Hubble parameter after these considerations is [⁴⁸]:

There is not a general analytical solution to the last equation, it may be solved numerically for specific cases of the parameters of the model. In Ref. [¹⁵] the authors solved this equation for several values of s numerically. The authors found that for s ≠ 1/2 (or s ≤ −1/2) there is no phantom solution. However, for the particular case of s = 1/2, i.e. ξ = ξ ₀ √ρ, the Eq. (68) has a particular analytical solution. Under this assumption the equation reduces to

Rewriting the last equation we have

Notice that clearly if the bulk viscosity parameter is zero ξ ₀ = 0, then (from the transport Eq. (58)) the bulk viscosity pressure is null Π = 0, and we recover the standard Hubble equation

For the cosmological constant case, w = −1, Eq. (70) boils down into the well-known de Sitter case H˙=0. Also notice that under all the hypothesis considered to derive the evolution Eq. (70), for the standard case w = −1, the evolution equation gives a solution in which the bulk viscosity does not appear regardless of how ξ, τ and T are chosen.

Henceforth we will assume ξ ₀ ≠ 0 and w + 1 ≠ 0 to to exclude the standard case. For mathematical convenience, we define

then the Eq. (70) transforms into

To sketch a solution, the following Ansatz is proposed for the Hubble function in the late future universe [¹⁵,⁴⁸,⁵⁰,⁵¹]:

where t _s is a finite time in the future at which the Big Rip occurs and A is a positive constant in order to describe an expanding universe. For s ≠ 1/2 it can be seen by direct inspection that a more general Ansatz with the form, H(t) = A(t _s − t) ^p with p < 0, does not reduce the differential equation to an algebraic polynomial in A [¹⁵].

Substituting the Ansatz (75) into Eq. (74) it is obtained a quadratic equation for the constant A

whose solution is

Integrating (75) we obtain the scale factor as a function of time, [¹⁵,⁵⁰]

where t ₀ is the present time. With this it can be easily calculated the number of particles

It is clear that at the time t = t _s the size of the universe becomes infinite and the number density of particles goes to zero.

Finally, with the Ansatz (75) we calculate the bulk viscous pressure:

Notice that the bulk viscous pressure is divergent at the Big Rip time.

8. Conclusions

We have discussed in the present work a general treatment for dark energy thermodynamics in several scenarios for the late future time: by considering it as a perfect fluid with constant and variable equation of state parameter w, and as a dissipative fluid with bulk viscosity both in the Eckark and Israel-Stewart frameworks. General equations have been derived for the dark energy temperature, energy density and entropy in a flat, homogeneous and isotropic universe. From the results of the entropy evolution, some theoretical thermodynamic constraints are imposed for the w parameter in order to satisfy a positive the entropy and grow rate.

We first recall some main results of dark energy as a perfect fluid with constant w; previously presented in a clear way in the work [9]. In this context the temperature is directly proportional to the internal energy T ∝ U, the energy density is a power-law of the temperature ρ ∝ T ^(1+w)/w , the entropy is constant in time (adiabatic) S˙=0, and if the chemical potential is null µ = 0 then the phantom regime is forbidden (w ≥ −1). Then, we apply the same treatment for the dynamical dark energy case w(α). The reader should note two interesting points: one is that despite the dynamical character of w there is no entropy production S = constant; and second, the chemical potential plays an important role, if it is null then necessarily w(α) = w = const. ≥ −1, i.e, is not dynamical, if it is positive the phantom regime is avoided, and if it is negative the dark energy could be either quintessence-like (w > −1) or phantom-like (w < −1).

In the second part of this work, the dark energy is modeled as a dissipative fluid with bulk viscosity where the (effective) pressure is assumed as the sum of the barotropic and the bulk viscous pressures p _eff = p + Π. This is the core of this work. Within the Eckart framework the viscous pressure is given by Π = −3Hξ(t) and two cases were studied: ξ(t) = ξ ₀ = const. and ξ = (β/√3)ρ ^1/2 . Both cases have the relation T ∝ ρ ^w/(1+w) . In the first case, the entropy grows exponentially and puts the condition w > −1 to preserve the entropy and temperature positive, or, if w > −1 then T < 0. Although the possibility of negative temperature for phantom energy has been studied [⁴⁰], it seems a cumbersome hypothesis and therefore quintessence is favored. In the second case, the entropy grows as a power-law, and quite an interesting fact in this case is that the phantom regime is allowed under not so strict conditions. The next step is to use the causal Israel-Stewart theory. We have solved the bulk viscous pressure transport equation assuming ξ = ξ ₀ ρ ^1/2 and τ = ξ/ρ which for the late future time universe gives rise to Big Rip cosmological solutions, H = A(t _s −t)⁻¹ [¹⁵,⁴⁸,⁵⁰]. Under these particular assumptions the temperature evolves as T = T ₀ α ^2w/A(w+1) , that is, in the quintessence regime the temperature decreases as the universe expands, whereas the temperature increases in the phantom regime. This is a characteristic of phantom and Big Rip solutions, here the energy density increases with the cosmic expansion as well the internal energy and consequently the temperature too. Calculating the entropy we obtain a power-law in time and imposing the conditions A > 0 and S > 0, an important constraint 0 < w < 1/2 is derived. The solution implies a cosmological scenario of a barotropic fluid with 0 < w < 1/2 which behaves like a dark fluid (even phantom) driven by the bulk viscosity. So, in the Israel-Stewart formalism, some solutions behave as dark energy and even allow to cross the phantom divide line with an effective equation of state lesser than −1 and constant in time without evoking an exotic barotropic fluid with w < −1. An important extension to the study done is to obtain other cosmological solutions additional to H = A(t _s − t)⁻¹ and constrain the model parameters with observational data to track the most plausible model.

Despite its intrinsic nature is not well understood yet, reconstructions of the dark energy equation of state from observational data seem to suggest a time-dependent w and even the crossing of the phantom divide line w = −1 from above to below is not only possible but could indeed be a condition for a successful description of observations. For this reason it is important to examine the phantom regime carefully. It has important problems and if modeled as a perfect fluid, it unfeasible from the point of view of classical thermodynamics. Hence, new perspectives beyond the perfect fluid approach in standard cosmology have to be considered in order to have a better understanding of the phantom regime, such as fluids with bulk viscosity that reduce the kinetic pressure of the cosmological fluid and provide a richer thermodynamic framework. In fact, one could actually dive deeper and seek for the foundations of dark energy thermodynamics from statistical physics (and possibly quantum mechanics) to establish basic general principles for a dark energy theory, or if it is the case, give solid arguments in favor of alternatives like modified gravity avoiding the hypothesis of dark energy as a substance; all this important but long beyond the scope of this work.

Finally, dark energy thermodynamics and bulk viscous dark energy are topics that have been studied from several different approaches for years, in this work we summarize particular results previously obtained by other authors on both subjects to make a broader and unified study, extending its scope in order to search for insights about the nature of the dark energy from a general theory that is classical thermodynamics and more general fluids than the perfect fluid.