FLRW-cosmology in generic gravity theories
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Regular Article - Theoretical Physics
FLRW-cosmology in generic gravity theories Metin Gürsesa , Yaghoub Heydarzadeb Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey
Received: 30 September 2020 / Accepted: 4 November 2020 © The Author(s) 2020
Abstract We prove that for the Friedmann–Lemaitre– Robertson–Walker metric, the field equations of any generic gravity theory in arbitrary dimensions are of the perfect fluid type. The cases of general Lovelock and F(R, G) theories are given as examples.
1 Introduction The Friedmann–Lemaitre–Robertson–Walker (FLRW) metric is the most known and most studied metric in General Relativity (GR). FLRW metric is mainly used to describe the universe as a homogeneous isotropic fluid distribution [1–5]. For inhomogeneous cosmological solutions, see for example [6–8]. On the other hand, current cosmological observations indicate that our universe is undergoing an accelerating expansion phase. The origin of this accelerating expansion still remains an open question in cosmology. Several approaches for explaining the current accelerated expanding phase have been proposed in the literature such as introducing cosmological constant [9], dynamical dark energy models and modified theories of gravity [10–13]. Amongst the latter, higher order curvature corrections to Einstein’s field equations have been considered by several authors [14–17]. In the context of modified theories, some attempts for a geometric interpretation of the dark side of the universe as a perfect fluid have been done [18–23] but the picture is not complete yet. In this work, we put one step forward to prove that the perfect fluid from of the dark component of the Universe is true for any generic modified theory of gravity. A generic gravity theory derivable from a variational principle
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can be given by the action 1 D √ I = d x −g (R − 2) κ
+F(g, Riem, ∇Riem, ∇∇Riem, · · · ) + L M ,
(1)
where g, Riem, ∇Riem, ∇∇Riem, etc in F denote the spacetime metric, Riemann tensor and its covariant derivatives at any order, respectively, and L M is the Lagrangian of the matter fields. The function F(g, Riem, ∇Riem, ∇∇Riem, · · · ) is the part of the Lagrange function corresponding to higher order couplings, constructed from the metric, the Riemann tensor and its covariant derivatives. The corresponding field equations are 1 G μν + gμν + Eμν = Tμν . κ
(2)
Here Eμν is a symmetric divergent free tensor obtained from the variation of F(g, Riem, ∇Riem, ∇∇Riem, · · · ) with respect to the spacetime metric gμν . Our treatment, in this work, is to consider this tensor, Eμν , as any second rank tensor obtained from the Riemann tensor and its covariant derivatives at any order. Since the Ricci tensor Rμν and Ricci scalar R are obtainable from the Riemann tensor we did not consider the function F depending on explicitly on the Ricci tensor and Ricci scalar. There are so
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