Kaluza–Klein Bulk Viscous Fluid Cosmological Models and the Validity of the Second Law of Thermodynamics in f(R, T) Gravity

Abstract

The authors considered the bulk viscous fluid in f (R, T ) gravity within the framework of Kaluza-Klein space time. The bulk viscous coefficient (ξ) expressed as ξ = ξ 0 + ξ 1ȧ a + ξ 2ä a , where ξ 0 , ξ 1 and ξ 2 are positive constants. We take p = (γ − 1)ρ, where 0 ≤ γ ≤ 2 as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of f (R, T ) = R + 2f (T ), where f (T ) = λT , λ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient (ξ). The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to ξ 2ä a , hence the second law of thermodynamics is not valid, however, it is valid with the generalized second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when, the bulk viscous coefficient is proportional to ξ = ξ 1ȧ a , ξ = ξ 1ȧ a + ξ 2ä a and ξ = ξ 0 + ξ 1ȧ a + ξ 2ä a , so the second law of thermodynamics and the generalized second law of thermodynamics is satisfied throughout the evolution. We calculate statefinder parameters of the model and observed that, it is different from the ∧CDM model. Finally, some physical and geometrical properties of the models are discussed.where h.c. means Hermitian conjugate, S E is the Einstein action describing gravity, E = det(−g µν ) 1 2 , α is the gravitational coupling constant and R is the Ricci scalar.The original Kaluza-Klein theory derive with one extra spatial dimension. The appropriate metric tensor for five dimensional space time iŝThe fifth dimension is postulated to be comfactified, rolled-up in a small circle, which provides us the explanation for the un-observability of the extra dimension. Hence the topology of the five dimensional space time is M 4 × S 1 , where M 4 is the standard four-dimensional Minkowski spacetime and S 1 is a circle with very small radius. The simplest way to imagine space with one extra dimension is to imagine a small circle at every point of 3-dimensional space.Inflation is an important idea in cosmology. There are two scenarios proposed in Kaluza-Klein cosmology. The first scenario [3] is: the scale of standard 3-dimensional space expands when the scale of internal space changes slowly with time. The second scenario ([4], [5]) is: inflation occurs near the singularity a(t) → ∞, b(t) → 0 (t → t 0 ).Expansion of our universe is in an accelerating way which is suggested by type Ia supernova observational data ([6], [7]). Myrzakulov [8] constructed several concrete models describing the trefoil and figure-eight knot universes from Bianchi-type I cosmology and examined the cosmological features and properties in detail. Yesmakhanova et al [9] constructed a cosmological model by assuming the periodic forms for pressure and en...

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