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Almost-Pseudo-Ricci Symmetric FRW Universe with a Dynamic Cosmological Term and Equation of State

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Almost-Pseudo-Ricci Symmetric FRW Universe with a Dynamic Cosmological Term and Equation of State universe Article Almost-Pseudo-Ricci Symmetric FRW Universe with a Dynamic Cosmological Term and Equation of State Sanjay Mandal 1 , Avik De 2 , Tee-How Loo 3 and Pradyumn Kumar Sahoo 1,* 1 Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad 500078, India; [email protected] 2 Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Jalan Sungai Long, Cheras 43000, Malaysia; [email protected] 3 Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia; [email protected] * Correspondence: [email protected] Abstract: The objective of the present paper is to investigate an almost-pseudo-Ricci symmetric FRW spacetime with a constant Ricci scalar in a dynamic cosmological term L(t) and equation of state (EoS) w(t) scenario. Several cosmological parameters are calculated in this setting and thoroughly studied, which shows that the model satisfies the late-time accelerating expansion of the universe. We also examine all of the energy conditions to check our model’s self-stability. Keywords: FRW universe; EoS parameter; cosmological parameters; energy conditions 1. Introduction Citation: Mandal, S.; De, A.; Loo, T.-H.; Sahoo, P.K. Ricci calculus in semi-Riemannian manifolds has a very rich history in the literature, Almost-Pseudo-Ricci Symmetric FRW which might have started from the concept of local symmetry, where the curvature tensor Universe with a Dynamic is parallel in the sense that rr Rijkl = 0, and later extended to conformally symmetric mani- Cosmological Term and Equation of folds [1], recurrent manifolds [2], conformally recurrent manifolds [3], pseudo-symmetric State. Universe 2021, 7, 205. manifolds [4], weakly symmetric manifolds [5], weakly Ricci symmetric manifolds [6], https://doi.org/10.3390/ pseudo-Ricci symmetric manifolds [7], etc., as well as their interactions with general rela- universe7070205 tivity. It should be noted that the standard theory of gravity governed by Einstein’s field equations (EFEs), which were introduced in 1915, is a very complicated system of partial Academic Editor: Alberto Vecchiato differential equations and that different types of symmetries are commonly used to find the exact solutions; these kinds of restrictions on Riemannian or Ricci curvature tensors Received: 15 May 2021 or on their covariant derivatives are very popular for such uses. In the present paper, we Accepted: 9 June 2021 will consider an almost-pseudo-Ricci symmetric manifold (APRS)n, which was defined by Published: 22 June 2021 Chaki and Kawaguchi [8] as a non-flat semi-Riemannian manifold whose Ricci curvature tensor Rij is not identically zero and satisfies the condition Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in riRjk = (Ai + Bi)Rjk + AjRki + AkRij, (1) published maps and institutional affil- iations. where Ai and Bi are the associated covariant tensors. The geometrical aspects of (APRS)n can be found in [9,10] and the references therein. This concept was introduced as a more generalized version of the pseudo-Ricci symmetric (PRS)n manifold [7], which is defined as a non-flat semi-Riemannian manifold whose Ricci curvature tensor Rij is not identically Copyright: © 2021 by the authors. zero and satisfies the condition Licensee MDPI, Basel, Switzerland. This article is an open access article riRjk = 2AiRjk + AjRki + AkRij. (2) distributed under the terms and conditions of the Creative Commons While studying curvature restrictions on a certain kind of conformally flat space of Attribution (CC BY) license (https:// class one, the authors [11] obtained the expression of the covariant derivative of the Ricci creativecommons.org/licenses/by/ curvature, which motivated the introduction of the (PRS)n. When Ai = Bi in (1), (APRS)n 4.0/). Universe 2021, 7, 205. https://doi.org/10.3390/universe7070205 https://www.mdpi.com/journal/universe Universe 2021, 7, 205 2 of 12 reduces to (PRS)n. A vital result in (PRS)n is that if its Ricci scalar is constant, then it is considered Ricci scalar flat [7]. If the Ricci tensor Rij of a Lorentzian manifold satisfies (1), then it is said to be an almost-pseudo-Ricci symmetric spacetime. In [9], the authors proved the existence of a conformally flat (APRS)n with a non-zero and non-constant Ricci scalar with a non-trivial example. Some geometric properties of (APRS)n were studied in [10], and a sufficient condition for such a spacetime to be perfect fluid was obtained; a conformally flat case was also briefly considered, and non-trivial examples were constructed. A perfect fluid (APRS)4 spacetime solution of Einstein’s field equations without a cosmological constant was investigated in [12]. The authors of [13] showed that a conformally flat (APRS)4 reduced to a Robertson–Walker spacetime, and in Einstein’s field equations with a cosmo- logical constant, the stress–energy tensor was a perfect fluid with non-constant pressure and energy density when the underlying metric was (APRS)4. (APRS)4 spacetimes were very recently studied with respect to modified gravity theories in [14]. It was shown that under certain conditions that were imposed on its scale factor, a Robertson–Walker spacetime was (APRS)4 and vice-versa. Investigations of various energy conditions were carried out in some popular models of F(R)-gravity in (APR)4 spacetime. The original field equations of general relativity introduced in 1915 by Einstein were amended by him after a couple of years in order to abolish the provision of a non-static universe in his theory by introducing the so-called cosmological constant L. Einstein’s field equations with the cosmological constant are expressed as R R − g = T − Lg , (3) ij 2 ij ij ij where R is the trace of the Ricci curvature tensor Rij and Tij is the stress–energy tensor describing the matter content, which is assumed to be a perfect fluid and is given by Tij = pgij + (r + p)uiuj, where r and p are the energy density and the isotropic pressure, respectively, and the timelike ui is the velocity vector field of the fluid. Moreover, at one point in the history of cosmological research, the cosmological constant turned into “the greatest blunder," which was pointed out by Einstein after the emergence of observational evidence of an expanding universe in the 1930s. Several studies were performed to explain this late-time inflation, but without any concrete conclusions. More recently, the present generation of cosmologists proposed the existence of an as-yet undetected dark energy (DE) that drives the evolution of the late-time universe, and the cosmological constant regained the limelight as the simplest contributing factor of the DE. However, currently, cosmologists believe that the accelerated expansion of the universe is driven by the unknown form of energy called “dark energy”, but the actual properties of dark energy are still unknown to us. The cosmological constant is the only candidate on the fundamental level of gravity that describes the accelerated expansion of the universe—i.e., in GR—as an alternative to dark energy. Moreover, “What is the value of the cosmological constant?” is a long-debatable question. Nevertheless, if string theory is the ultimate quantum gravity theory, then according to the second of the swampland conjectures, there is evidence that exact de Sitter solutions with a positive cosmological constant cannot account for the fate of the late-time universe [15,16]. Therefore, the above arguments motivate us to study the cosmological models with a time-dependent cosmological constant. Therefore, so-called constant L is now considered as a time-varying dynamic variable in order to remove some fundamental problems that it has faced as the central character of DE. There are innumerable dynamic L(t) models in the literature; one of the oldest and most popular is the model L ∼ H2, which was introduced by Carvalho et al. [17] and Waga [18]. In addition, Z. Yousaf investigated some interesting cosmological models based on L [19,20]. In addition, p and r in the stress–energy tensor are assumed to be related by an equation of state of the form p = wr. However, the choice of the constant EoS parameter is too restrictive [21], and thus, amongst a variety of possibilities, researchers considered linear and periodic functional forms of the EoS as alternatives [22–24]. Universe 2021, 7, 205 3 of 12 The present paper is organized as follows: In Section2, we construct the cosmological model and calculate the Hubble parameters (H). In Section3, we discuss the cosmological parameters, such as the Hubble (H), deceleration (q), jerk (j), snap (s), and lerk (l) param- eters, and present their profiles for two different cases of time-varying EoS models. In Section4, we examine the energy conditions to check the self-consistency of our models. Finally, we discuss our results in Section5. 2. The Field Equations In an (APRS)4, the covariant derivative of the Ricci tensor satisfies (1). Hence, we have riRjk − rkRij = BiRjk − BkRij, which, on contraction over j and k, gives us j riR = 2RBi − 2RijB . (4) For a constant Ricci scalar R, we get from (4): l RBi = Ril B . (5) Inserting (5) into (3) and considering the four velocity vectors of the fluid ui = Bi, we obtain R + r + L(t) = 0. (6) 2 Again, the trace of (3) gives − R = 3p − r − 4L(t). (7) Combining these, we conclude that R R r = − − L(t), p = − + L(t) (8) 2 2 Using the EoS parameter w(t),(8) gives R L(t)(1 + w(t)) = (1 − w(t)). (9) 2 In an FRW spacetime with a decay law L(t) = bH2, the above equation ultimately gives us H˙ w(t)(b + 6) + (b − 6) = .
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