Viscous Cosmology and Thermodynamics of Apparent Horizon

It is shown that the differential form of Friedmann equations of FRW universe can be recast as a similar form of the first law ,TAdSA = dE + WdV , of thermodynamics at the apparent horizon of FRW universe filled with the viscous fluid. It is also shown that the generalized second law of thermodynamics holds at the apparent horizon of FRW universe and preserves dominant energy condition. PACS numbers: 04.70.Dy, 97.60.Lf Email address: [email protected] Email address: [email protected] Introduction: In the cosmological setting, one can associate Hawking temperature and entropy with the apparent horizon analogous to the hawking temperature and entropy associated with the black hole horizon [1, 2, 3]. In the case of de Sitter space, the event horizon matches with the apparent horizon of FRW universe with k = 0 however for more general cosmological models, the event horizon may not exist but the apparent horizon associated with the Hawking temperature and entropy always exists. The thermodynamic properties associated with the apparent horizon of FRW universe filled with the perfect fluid has been studied by many authors (see for examples [1, 2, 4, 6, 5, 7, 8]). The extension of this connection between thermodynamics and gravity has also been carried out in the braneworld cosmology [9, 10, 11, 12]. For a general static spherically symmetric and stationary axisymmetric spacetimes, It was shown that the Einstein field equations can be rewritten [18, 19, 20] as a first law of thermodynamics. More recently, by considering a masslike function, it has been shown [13] that the equilibrium thermodynamics should exist in the extended theories of gravity and the Friedman equations in various theories of gravity can be rewritten as a first law TdS = dE at the apparent horizon of FRW universe. More recently, Cai et al [5] has shown that by employing Clausius relation, δQ = TdS, to the apparent horizon of a FRW universe, they are able to derive the modified Friedmann equation by using quantum corrected entropy-area relation. The cosmological models with perfect cosmic fluid has been studied widely in literature however viscous cosmic fluid came much later in the study of the universe [17]. It has been revealed by Barrow [14](also see [15]) that the thermodynamic entropy associated with bulk viscosity violates the dominant energy condition and can decrease very well. This fact was resumed [16] in the framework of the solutions given in reference [15]. By taking into account the entropy of the sources only, it has been deduced that the entropy of the bulk viscous sources cannot decrease since it would violate the second law of thermodynamics for large cosmic time values. However, in order to state the generalized second law, one has to include both the entropy connected with the sources and the entropy associated with background geometry. The most of the study of the second law in the cosmological context has been carried out at the event horizon (see for examples [15, 16]) however the validity of second law within the apparent horizon of FRW universe is of great important to investigate. In this letter we will also investigate this issue. The cosmological models with bulk viscosity have been studied in the literature from various point of view (see for examples [15, 21, 22]). The purpose of this letter is twofold. The first is to show that the Friedmann equations of FRW universe pertaining cosmic bulk viscosity can be rewritten as a similar form of the first law of thermodynamics. The other is to discuss the generalized second law within the apparent horizon of FRW universe. Let us start with a FRW universe of metric ds = −dt + a(t)( dr 2 1 − kr2 + r dΩ), (1)