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


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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  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.


  1. 1.

    Cervantes-Cota, J.L., Smoot, G.: Cosmology today—a brief review. AIP Conf. Proc. 1396, 28–52 (2011). arXiv:1107.1789

  2. 2.

    Copeland, E.J., Sami, M., Tsujikawa, S.: Dynamics of dark energy. Int. J. Mod. Phys D15, 1753–1936 (2006). hep-th/0603057

  3. 3.

    Kunz, M.: The dark degeneracy: on the number and nature of dark components. Phys. Rev D80, 123001 (2009). astro-ph/0702615

  4. 4.

    Hu, W., Eisenstein, D.J.: The structure of structure formation theories. Phys. Rev. D59, 083509 (1999). astro-ph/9809368

  5. 5.

    Aviles, A., Cervantes-Cota, J.L.: The dark degeneracy and interacting cosmic components. Phys. Rev. D84, 083515 (2011). arXiv:1108.2457

  6. 6.

    Luongo, O., Quevedo, H.: A unified dark energy model from a vanishing speed of sound with emergent cosmological constant. Int. J. Mod. Phys. D23, 1450012 (2014)

  7. 7.

    Xu, L., Wang, Y., Noh, H.: Unified dark fluid with constant adiabatic sound speed and cosmic constraints. Phys. Rev. D85, 043003 (2012). arXiv:1112.3701

  8. 8.

    Bielefeld, J., Caldwell, R.R., Linder, E.V.: Dark energy scaling from dark matter to acceleration. Phys. Rev. D90, 043015 (2014). arXiv:1404.2273

  9. 9.

    Ballesteros, G., Lesgourgues, J.: Dark energy with non-adiabatic sound speed: initial conditions and detectability. JCAP 014, 1010 (2010). arXiv:1004.5509

  10. 10.

    Piattella, O.F., Fabris, J.C., Bilic, N.: Note on the thermodynamics and the speed of sound of a scalar field. Class. Quant. Gravit. 31, 055006 (2014). arXiv:1309.4282

  11. 11.

    Kamenshchik, A.Y., Moschella, U., Pasquier, V.: An alternative to quintessence. Phys. Lett. B511, 265–268 (2001). gr-qc/0103004

  12. 12.

    Bento, M., Bertolami, O., Sen, A.: Generalized Chaplygin gas, accelerated expansion and dark energy matter unification. Phys. Rev. D66, 043507 (2002). gr-qc/0202064

  13. 13.

    Bilic, N., Tupper, G.B., Viollier, R.D.: Unification of dark matter and dark energy: the inhomogeneous Chaplygin gas. Phys. Lett. B535, 17–21 (2002). astro-ph/0111325

  14. 14.

    Fabris, J.C., Goncalves, S., de Sa Ribeiro, R.: Bulk viscosity driving the acceleration of the Universe. Gen. Relativ. Gravit. 38, 495–506 (2006). astro-ph/0503362

  15. 15.

    Zimdahl, W., Schwarz, D.J., Balakin, A.B., Pavon, D.: Cosmic anti-friction and accelerated expansion. Phys. Rev. D64, 063501 (2001). astro-ph/0009353

  16. 16.

    Avelino, A., Nucamendi, U.: Can a matter-dominated model with constant bulk viscosity drive the accelerated expansion of the universe? JCAP 0904, 006 (2009). arXiv:0811.3253

  17. 17.

    Li, B., Barrow, J.D.: Does bulk viscosity create a viable unified dark matter model? Phys. Rev. D79, 103521 (2009). arXiv:0902.3163

  18. 18.

    Avelino, A., Nucamendi, U.: Exploring a matter-dominated model with bulk viscosity to drive the accelerated expansion of the Universe. JCAP 1008, 009 (2010). arXiv:1002.3605

  19. 19.

    Piattella, O.F., Fabris, J.C., Zimdahl, W.: Bulk viscous cosmology with causal transport theory. JCAP 1105, 029 (2011). arXiv:1103.1328

  20. 20.

    Eckart, C.: The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid. Phys. Rev. 58, 919–924 (1940)

  21. 21.

    Muller, I.: Zum paradoxon der Warmeleitungstheorie. Z. Phys. 198, 329 (1967)

  22. 22.

    Israel, W.: Nonstationary irreversible thermodynamics: a causal relativistic theory. Ann. Phys. 100, 310–331 (1976)

  23. 23.

    Israel, W., Stewart, J.: Transient relativistic thermodynamics and kinetic theory. Ann. Phys. 118, 341–372 (1979)

  24. 24.

    Pavon, D., Jou, D., Casas-Vazquez, J.: On a covariant formulation of dissipative phenomena. Ann. l’inst. Henri Poincar (A) 36, 79 (1982)

  25. 25.

    Hiscock, W., Lindblom, L.: Stability and causality in dissipative relativistic fluids. Ann. Phys. 151, 466–496 (1983)

  26. 26.

    Lima, J., Alcaniz, J.S.: Thermodynamics and spectral distribution of dark energy. Phys. Lett. B600, 191 (2004). astro-ph/0402265

  27. 27.

    Gonzalez-Diaz, P.F., Siguenza, C.L.: Phantom thermodynamics. Nucl. Phys. B697, 363–386 (2004). astro-ph/0407421

  28. 28.

    Pereira, S., Lima, J.: On phantom thermodynamics. Phys. Lett. B669, 266–270 (2008). arXiv:0806.0682

  29. 29.

    Myung, Y.S.: On phantom thermodynamics with negative temperature. Phys. Lett. B671, 216–218 (2009). arXiv:0810.4385

  30. 30.

    Silva, R., Goncalves, R., Alcaniz, J., Silva, H.: Thermodynamics and dark energy. Astron. Astrophys. 537, A11 (2012). arXiv:1104.1628

  31. 31.

    Aviles, A., Bastarrachea-Almodovar, A., Campuzano, L., Quevedo, H.: Extending the generalized Chaplygin gas model by using geometrothermodynamics. Phys. Rev. D86, 063508 (2012). arXiv:1203.4637

  32. 32.

    Luongo, O., Quevedo, H.: Cosmographic study of the universe’s specific heat: a landscape for cosmology? Gen. Relativ. Gravit. 46, 1649 (2014). arXiv:1211.0626

  33. 33.

    Silva, H., Silva, R., Gonalves, R., Zhu, Z.-H., Alcaniz, J.: General treatment for dark energy thermodynamics. Phys. Rev. D88, 127302 (2013). arXiv:1312.3216

  34. 34.

    WMAP Collaboration, Komatsu, E., et al.: Seven-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological interpretation. Astrophys. J. Suppl. 192, 18 (2011). arXiv:1001.4538

  35. 35.

    Planck Collaboration, Ade, P., et al.: Planck 2013 results. XVI. Cosmological parameters. arXiv:1303.5076

  36. 36.

    Chuang, C.-H., Prada, F., Beutler, F., Eisenstein, D.J., Escoffier, S., et al.: The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: single-probe measurements from CMASS and LOWZ anisotropic galaxy clustering. arXiv:1312.4889

  37. 37.

    Caldwell, R.: A phantom menace? Phys. Lett. B545, 23–29 (2002). astro-ph/9908168

  38. 38.

    Caldwell, R.R., Kamionkowski, M., Weinberg, N.N.: Phantom energy and cosmic doomsday. Phys. Rev. Lett. 91, 071301 (2003). astro-ph/0302506

  39. 39.

    Cheng, C., Huang, Q.G.: The dark side of the Universe after Planck. Phys. Rev. D 89, 043003 (2014). astro-ph/1306.4091

  40. 40.

    Rest, A., et al.: Cosmological Constraints from Measurements of Type Ia Supernovae discovered during the first 1.5 years of the Pan-STARRS1 Survey. Astrophys. J. 795, 1, 44. astro-ph/1310.3828

  41. 41.

    Shafer, D.L., Huterer, D.: Chasing the phantom: a closer look at Type Ia supernovae and the dark energy equation of state. Phys. Rev. D 89, 063510 (2014). astro-ph/1312.1688

  42. 42.

    Xia, J.-Q., Li, H., Zhang, X.: Dark energy constraints after Planck. Phys. Rev. D 88, 063501 (2013). hep-lat/0602003

  43. 43.

    Carroll, S.M., Hoffman, M., Trodden, M.: Can the dark energy equation-of-state parameter w be less than \({-}1\)? Phys. Rev. D68, 023509 (2003). astro-ph/0301273

  44. 44.

    Barrow, J.D.: Sudden future singularities. Class. Quant. Gravit. 21, L79–L82 (2004). gr-qc/0403084

  45. 45.

    Cataldo, M., Cruz, N., Lepe, S.: Viscous dark energy and phantom evolution. Phys. Lett. B619, 5–10 (2005). hep-th/0506153

  46. 46.

    Cruz, N., Lepe, S., Pena, F.: Dissipative generalized Chaplygin gas as phantom dark energy. Phys. Lett. B646, 177–182 (2007). gr-qc/0609013

  47. 47.

    Barranco, J., Bernal, A., Nunez, D.: Dark matter equation of state from rotational curves of galaxies. arXiv:1301.6785

  48. 48.

    Weinberg, S.: The cosmological constant problem. Rev. Mod. Phys. 61, 1–23 (1989)

  49. 49.

    Liddle, A.R., Urena-Lopez, L.A.: Inflation, dark matter and dark energy in the string landscape. Phys. Rev. Lett. 97, 161301 (2006). astro-ph/0605205

  50. 50.

    Reyes, L.M., Aguilar, J.E.M., Urena-Lopez, L.A.: Cosmological dark fluid from five-dimensional vacuum. Phys. Rev. D84, 027503 (2011). arXiv:1107.0345

  51. 51.

    Grande, J., Pelinson, A., Sola, J.: Dark energy perturbations and cosmic coincidence. Phys. Rev. D79, 043006 (2009). arXiv:0809.3462

  52. 52.

    Linder, E.V., Scherrer, R.J.: Aetherizing lambda: barotropic fluids as dark energy. Phys. Rev. D80, 023008 (2009). arXiv:0811.2797

  53. 53.

    Novosyadlyj, B., Sergijenko, O., Apunevych, S., Pelykh, V.: Properties and uncertainties of scalar field models of dark energy with barotropic equation of state. Phys. Rev. D82, 103008 (2010). arXiv:1008.1943

  54. 54.

    Novosyadlyj, B., Sergijenko, O., Durrer, R., Pelykhc, V.: Constraining the dynamical dark energy parameters: Planck 2013 vs WMAP9. J. Cosmol. Astropart. Phys. 05, 030 (2014). arXiv:1312.6579

  55. 55.

    Sergijenko, O., Novosyadlyj, B.: Perturbed dark energy: classical scalar field versus tachyon. Phys. Rev. D 80, 083007 (2009). arXiv:0904.1583

  56. 56.

    Xu, L., Wang, Y., Noh, Hyerim: Unified dark fluid with constant adiabatic sound speed and cosmic constraints. Phys. Rev. D 85, 043003 (2012). arXiv:1112.3701

  57. 57.

    Wheeler, T.D., Stroock, A.D.: The transpiration of water at negative pressures in a synthetic tree. Nature 455(7210), 208 (2008)

  58. 58.

    Caupin, F., Arvengas, A., Davitt, K., Azouzi, M.E.M., Shmulovich, K.I., Ramboz, C., Sessoms, D.A., Stroock, A.D.: Exploring water and other liquids at negative pressure. J. Phys. Condens. Matter 24, 284110 (2012)

  59. 59.

    Stanley, H., Barbosa, M., Mossa, S., Netz, P., Sciortino, F., Starr, F., Yamada, M.: Statistical physics and liquid water at negative pressures. Phys. A Stat. Mech. Appl. 315, 281 (2002)

  60. 60.

    Quevedo, H.: Geometrothermodynamics. J. Math. Phys. 48, 013506 (2007). physics/0604164

  61. 61.

    Vazquez, A., Quevedo, H., Sanchez, A.: Thermodynamic systems as extremal hypersurfaces. J. Geom. Phys. 60, 1942–1949 (2010). arXiv:1101.3359

  62. 62.

    Krasinski, A., Quevedo, H., Sussman, R.: On the thermodynamical interpretation of perfect fluid solutions of the Einstein equations with no symmetry. J. Math. Phys. 38, 2602–2610 (1997)

  63. 63.

    Hernandez, F.J., Quevedo, H.: Entropy and anisotropy. Gen. Relativ. Gravit. 39, 1297–1309 (2007). gr-qc/0701125

  64. 64.

    Weinberg, S.: Entropy generation and the survival of protogalaxies in an expanding universe. Astrophys. J. 168, 175 (1971)

  65. 65.

    Maartens, R.: Causal thermodynamics in relativity. astro-ph/9609119

  66. 66.

    Misner, C., Thorne, K., Wheeler, J.: Gravitation. W.H. Freeman and Company, London (1973)

  67. 67.

    Israel, W., Stewart, J.: Thermodynamics of nonstationary and transient effects in a relativistic gas. Phys. Lett. A 58(4), 213–215 (1976)

  68. 68.

    Astashenok, A.V., Odintsov, S.D.: Confronting dark energy models mimicking \(\Lambda \)CDM epoch with observational constraints: future cosmological perturbations decay or future Rip? Phys. Lett. B718, 1194–1202 (2013). arXiv:1211.1888

  69. 69.

    Pavon, D., Zimdahl, W.: A thermodynamic characterization of future singularities? Phys. Lett. B708, 217–220 (2012). arXiv:1201.6144

  70. 70.

    Frampton, P.H., Ludwick, K.J., Scherrer, R.J.: The little rip. Phys. Rev. D84, 063003 (2011). arXiv:1106.4996

  71. 71.

    Brevik, I., Elizalde, E., Nojiri, S., Odintsov, S.: Viscous little rip cosmology. Phys. Rev. D84, 103508 (2011). arXiv:1107.4642

  72. 72.

    Frampton, P.H., Ludwick, K.J., Nojiri, S., Odintsov, S.D., Scherrer, R.J.: Models for little rip dark energy. Phys. Lett. B708, 204–211 (2012). arXiv:1108.0067

  73. 73.

    Astashenok, A.V., Nojiri, S., Odintsov, S.D., Yurov, A.V.: Phantom cosmology without big rip singularity. Phys. Lett. B709, 396–403 (2012). arXiv:1201.4056

  74. 74.

    Maartens, R.: Dissipative cosmology. Class. Quant. Gravit. 12, 1455–1465 (1995)

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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

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

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

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

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

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

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

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

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:

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

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

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

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

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

The case \(\alpha >0\) can be integrated to give

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

where \(A\) is defined by

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

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).

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  • Dark sector
  • Thermodynamics of dark energy
  • Dark degeneracy