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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.

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Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    We will clarify later that standard dark energy fluids turn out to be unstable when dissipative effects are involved.

  2. 2.

    In [4] the barotropic condition is not considered and the EoS of the dark fluid in a FRW universe is imposed from the beginning.

  3. 3.

    See [5] for an explicit demonstration.

  4. 4.

    Equation (9) is not exactly the first law of thermodynamics. Several authors consider it as a sort of combination between the first and second laws of thermodynamics. Often the expression is known in the literature as Gibbs equation.

  5. 5.

    In this section, we will clump the notation by explicitly showing the dependence of every considered function.

  6. 6.

    Defining the arbitrary function \(L(U,V,N)= (\partial S/\partial V)_{U,N} - c (\partial S/\partial U)_{V,N}\) with \(c\) a constant, the necessary condition is provided by the relation \(L(U,V,N) = \alpha (\partial S/\partial U)_{V,N}\), where \(\alpha \) is a constant. This condition reduces to Eq. (11) with \(P_0=c+\alpha \).

  7. 7.

    The local equilibrium is not strictly necessary if the dark fluid is the only energy component. A necessary and sufficient condition for a corresponding thermodynamical description is that the differential form \(Td\sigma \) is integrable. In flat space, it is easily satisfied (and for some authors the temperature is defined as an integration factor), but in curved space–times, it is not necessarily true, as studied in [62, 63]. Nevertheless, for the dark fluid Eq. (36) can be rewritten as \(Td\sigma = d(\rho /n) + P d(1/n) = d((\rho + P_0)/n)\) and therefore it is integrable. However, we do not follow this approach, since we will consider dissipative processes in Sect. 5.

  8. 8.

    This result can also be derived from thermodynamical arguments, where \(P_{b}=nkT\) and \(\rho _{b} = n ( m + 3kT/2)\) are good approximations when \(k_B T\ll m\), then it follows from Eq. (44) that \(T_{b} \propto a^{-2}\).

  9. 9.

    The correspondence between this approach and Eckart’s theory can be seen as follows: Assume the solution \(\rho = \rho (a)\) has an inverse \(a=a(\rho )\) (if not, consider the inverse piecewise). From the continuity equation (81) we obtain

    $$\begin{aligned} \pi (\rho ) = -\frac{1}{3} \left( \frac{d \ln a(\rho ) }{d\rho } \right) ^{-1} - \rho - P_0. \end{aligned}$$

    In Eckart’s theory \(\pi (\rho ,H) = -3 H \zeta (\rho )\), with the aid of Friedmann equation in the case in which other fluids than the dark fluid can be neglected we obtain

    $$\begin{aligned} \zeta (\rho ) = \sqrt{\frac{3}{8\pi G}} \rho ^{-1/2} \left( \frac{1}{9} \left( \frac{d \ln a(\rho ) }{d\rho } \right) ^{-1} + \frac{1}{3}(\rho + P_0) \right) . \end{aligned}$$

    In the appendix of this article, we work out an exact Big Rip solution in Eckart’s theory and we explicitly show the correspondence to our formalism.

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Acknowledgments

A. A. and O. L. want to thank prof. S. Capozziello and G. Carmona for useful discussions. A. A. acknowledges the hospitality of the Departamento de Física, Universidad de Santiago de Chile, where part of this work was done. A. A. and J. K. are financially supported by the project CONACyT-EDOMEX-2011-C01-165873 (ABACUS-CINVESTAV). N. C. acknowledges the support to this research by CONICYT through Grant Nos. 1140238. O. L. is financially supported by the European PONa3 00038F1 KM3NeT (INFN) Project.

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Correspondence to Orlando Luongo.

Appendix: Exact solutions in Eckart’s theory

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 \(\alpha \).

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Aviles, A., Cruz, N., Klapp, J. et al. Emerging the dark sector from thermodynamics of cosmological systems with constant pressure. Gen Relativ Gravit 47, 63 (2015). https://doi.org/10.1007/s10714-015-1904-6

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Keywords

  • Dark sector
  • Thermodynamics of dark energy
  • Dark degeneracy