Emerging the dark sector from thermodynamics of cosmological systems with constant pressure

Abstract

We investigate the thermodynamics of general fluids that have the constriction that their pressure is constant. We first consider the more general thermodynamic properties of this class of fluids finding the important result that for them adiabatic and isothermal processes should coincide. We therefore study their behaviors in curved space-times where local thermal equilibrium can be appealed. Thus, we show that this dark fluid degenerates with the dark sector of the $\Lambda$CDM model only in the case of adiabatic evolution. We demonstrate that, adding dissipative processes, a phantom behavior can occur and finally we further highlight that an arbitrary decomposition of the dark sector, into ad hoc dark matter and dark energy terms, may give rise to phantom dark energy, whereas the whole dark sector remains non-phantom.

Appendix: Exact solutions in Eckart’s theory

Exact cosmological solutions in Eckart’s theory can be obtained for the dark fluid when the viscosity has a power-law dependence upon the energy density of this fluid

$$\begin{aligned} \xi =\xi _{0} \rho ^{m}, \end{aligned}$$

(110)

where $\xi _{0}>0$ and $m$ are constant parameters. This type of behavior for the viscosity has been widely investigated in the literature, albeit there is no fundamental complete approaches for choosing it, see for example Ref. [74]. We will assume this type of behavior which allows us to obtain suitable cosmological solutions and compare with other results present in the literature. Neglecting all contributions to the total energy momentum tensor, except the dark fluid, we can write down

$$\begin{aligned} \pi (\rho ) = -3 H \xi _{0} \rho ^{m} = - \sqrt{3} \sqrt{8\pi G} \xi _{0} \rho ^{m + 1/2}. \end{aligned}$$

(111)

Only for reasons of mathematical simplicity the case $m=1/2$ is mostly considered. As a first glance to the study of the behavior of this fluid when dissipation is taken into account, it is reasonable to explore this simple case. Thus, in the following

$$\begin{aligned} \pi (\rho ) = -\sqrt{3} \tilde{\xi }_0 \rho , \end{aligned}$$

(112)

where we define $\tilde{\xi }_0 \equiv \sqrt{8\pi G} \xi _{0}$. The continuity equation becomes

$$\begin{aligned} \dot{\rho } + 3 H (\alpha \rho + P_0) = 0, \end{aligned}$$

(113)

where $\alpha \equiv 1-\sqrt{3}\tilde{\xi }_{0}$. This can be integrated to get the energy density as a function of the scale factor:

$$\begin{aligned} \rho = - \frac{P_0}{\alpha } + \left( \rho _0 + \frac{P_0}{\alpha } \right) a^{-3\alpha }. \end{aligned}$$

(114)

Using Eq. (112) we obtain $\pi$ as a function of the scale factor

$$\begin{aligned} \pi (a) = -\sqrt{3} \tilde{\xi }_0 \left[ - \frac{P_0}{\alpha } + \left( \rho _0 + \frac{P_0}{\alpha } \right) a^{-3\alpha } \right] \end{aligned}$$

(115)

We notice that Eq. (83), and hence Eq. (39), is recovered by

$$\begin{aligned} h(a_*) = \left( \rho _0 + \frac{P_0}{\alpha } \right) a_*^{3(1-\alpha )} - \frac{\sqrt{3} \tilde{\xi }_0 P_0}{1-\sqrt{3}\tilde{\xi }_0} a^3_* \end{aligned}$$

To study this solution in more detail, we can combine the Friedmann and continuity equations to obtain

$$\begin{aligned} 2\dot{H}+ 3 \alpha H^{2} + 8 \pi G P_{0} =0. \end{aligned}$$

(116)

The case $\alpha >0$ can be integrated to give

$$\begin{aligned} H(t)=\sqrt{-\frac{8 \pi G P_{0}}{3\alpha }}\left[ \frac{e^{ \sqrt{- 24 \pi G P_{0}\alpha }\,t} - A}{ e^{\sqrt{- 24 \pi G P_{0}\alpha }\,t} + A } \right] , \end{aligned}$$

(117)

where $A$ is defined by

$$\begin{aligned} A \equiv e^{ \sqrt{-24 \pi G P_{0}\alpha }\,t_0}\frac{\sqrt{\frac{- 8\pi G P_{0}}{3\alpha }}-H_{0}}{\sqrt{\frac{-8\pi G P_{0}}{3\alpha }}+H_{0}}. \end{aligned}$$

(118)

On the contrary, the case $\alpha < 0$ is obtained by analytic continuation of Eq. (117), which turns out to be a real function of $t$. We notice that the solution behaviors are strongly dependent upon the sign of the parameter $\alpha$.