Dynamics of a parametrized dark energy model in gravity
Introduction
Observational data indicating high redshift supernovae emphasizes the rising belief of late-time cosmic acceleration. This phenomenon also gets its support from the observations related to weak lensing, microwave background and large scale structure. A current problem of modern cosmogenesis is knowing what produces the repulsion during cosmic expansion. The phenomena can theoretically be explained by either adding an unusual matter component with substantial negative pressure to the energy–momentum tensor [1], [2], [3], [4], [5], [6], [7] or changing gravity itself.
The most straightforward dark energy candidate is the cosmological constant. The cosmological constant is equivalent to a fluid having an equation of state , where is the ratio of pressure that dark energy puts on the Universe to the energy per unit volume. Traditionally, Einstein added the cosmological constant into the field equation for gravity for the reason that it allows for a closed, static, finite Universe in which the geometry is determined by the matter’s energy density [8]. Einstein thought that ordinary matter would be required to bend geometry, a necessity that, according to him, was strongly tied to the Mach’s principle. This optimism was quickly dashed when de Sitter discovered a solution to Einstein’s equations with a cosmological constant including no matter [9]. Despite two seminal studies by Friedmann and one by Lemaitre [10], [11], [12], most workers were resistant to the concept of an expanding cosmos. Lemaitre’s work helped the community accept the idea of expanding the entire Universe. The cosmological constant has a tumultuous history and has frequently been embraced or dismissed for erroneous or inadequate reasons. The steady-state model was the first cosmological model that used the cosmological constant as a fundamental quantity [13], [14], [15]. It took advantage of the fact that a Universe with a cosmological constant has the property of time translational invariance in a given coordinate system. Unlike current cosmology, which readily conjures negative energies or pressure, largely abandoned steady-state cosmology was with the discovery of CMBR.
A remarkable result in modern cosmology, that is, the presence of cosmological constant fuelling the Universe’s current acceleration, has been growing steadily, as demonstrated in [16], [17], [18]. It was recently discovered that the enigmatic dark energy governs the late-time dynamics of the current accelerating cosmos. According to the interpretation of the astrophysical findings, such dark energy fluid (assuming it is fluid) is distinguished by negative pressure, and its equation of state parameter is approaching [19]. Mamon et al. have discussed a unified model of dark matter and dark energy [20]. Numerous studies cover various areas related to gravity and accompanying cosmic dynamics. Alternative gravitational theories represented by Lagrangians based on distinct general functions of the Ricci scalar have been shown to create lucid theoretical models to describe the experimental evidence of the Universe’s acceleration. Allemandi et al. continued this examination of cosmological possibilities of alternative gravity theories that rely on (other) curvature invariants [21]. A comprehensive investigation of the dynamics of gravity cosmological models is discussed in [22]. Viable options other than dark energy for explaining late-time cosmic acceleration can be found in modified gravitational theories. Also, few instances of discussions on low-curvature Einstein–Hilbert action corrections are noticed. Such frameworks feature unstable de Sitter solutions in general and, based on the theory’s characteristics, can have late-time accelerating attractor solutions [23]. Higher-order gravity theories have recently gained much attention as potential options for explaining observable cosmic acceleration without the requirement for a scalar field.
The function of the Ricci scalar curvature entering the gravitational Lagrangian and regulating the evolution of the Universe is a crucial component [24]. The fundamental goal of the work in [25] was to examine the existence of compact spherical systems that represent anisotropic matter distributions in the context of alternative gravity theories, notably the gravity theory. The goal of this study in [26] is to revamp a particular class of gravity models in which the Einstein–Hilbert action is supported by an arbitrary function of the energy–momentum-tensor’s trace without losing its generality. Rosa introduced an alternative scalar-tensor representation in gravity [27]. Gonclaves et al. used reconstruction methods to obtain cosmological solutions in the scalar-tensor representation of gravity [28]. Quite a number of researchers have already executed impressive work in gravity [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. Mishra et al. have reconstructed cosmological models using hybrid scale factor [42], [43].
General relativity’s generalization can be used as a feasible explanation of the early-time inflation with late-time cosmic acceleration. The various representations of a multitude of modified theories, such as traditional and Hoava-Lifshitz gravity, scalar-tensor theory, Gauss–Bonnet theory etc., as well as relationships between them, are studied in [44]. Nojiri & Odintsov [45] propose that the expansion of the cosmos may result in the dominance of dark energy over standard matter. The effective quintessence aptly describes the existing cosmic speed-up. , and , are a few standard modified gravity theories. The objective of [46] is to highlight all the relevant details on inflation, dark energy and bouncing cosmologies through multiple modified gravity models. The open irreversible thermodynamic interpretation of a simple cosmological model is described in detail for the gravity theory [47]. In [48], the cosmological implications of the non-minimal coupling matter-geometry coupling were thoroughly explored. In modified gravity models, the type of the coupling entirely and distinctively determines both the matter Lagrangian and the energy–momentum tensor.
The paper is organized as follows. Section 2 provides a succinct treatment of theory. We get highly non-linear field equations by taking the as a blend of a -dependent part having terms up to the quadratic power of and a linear -dependent part. We apply an ansatz for the scale factor to find the solution of the field equations and to investigate the behaviour of the geometrical parameters such as the Hubble parameter and the deceleration parameter . The numerical results obtained for the energy density , fluid pressure and the ratio of pressure that dark energy puts on the universe to the energy per unit volume for the proposed model are depicted for their interpretations. The energy conditions are examined to analyse and interpret the resulting solution. In Section 3, we exhibit the jerk parameter, snap parameter, the lerk parameter, Om diagnostic, the velocity of sound and statefinder diagnostic tools graphically to test the correctness of our model. Section 4 covers various cosmological tests for determining distances in cosmology using the parametrization above. Section 5 contains concluding remarks on the works presented in the manuscript.
Section snippets
FLRW space–time in -gravity
The Einstein field equations are given by where is the Ricci tensor, the Ricci scalar, the covariant metric tensor of order 2, the gravitational constant, the speed of the light, and the energy–momentum-tensor. The energy–momentum-tensor on the right-hand side of Einstein’s equations represents the contribution of the Universe’s matter component, whereas the left-hand side symbolizes pure geometry. There are two ways to cause an accelerated expansion:
Cosmographical parameters
The cosmographic analysis of the geometrical parameters is extended by the incorporation of dimensionless higher derivative components of the scale factor , namely jerk, snap and lerk parameters [57], [58]. Jerk is also known as jolt at times. Pulse, bounce, impulse, surge, super-acceleration and shock are less commonly used alternative expressions for jerk [57]. The formula for calculating jerk parameter is given by which on using Eqs. (13) and can be expressed in
Look-back time
In the past was the temporal gap between the production of light from the source and the reception of light on the Earth. As a result, the total time elapsed between the galaxy’s light beam emitting at time for a specific redshift and reaching us at a time for redshift is represented as [49] where, the value of can be calculated numerically. Astronomers might benefit from this technique. The farther away an object is, the further back in time we are
Conclusion
We studied the late-time behaviour of Universe in the background of the flat FLRW metric in gravity. The field equations are non-linear ordinary differential equations and are solved by parametrizing the scale factor in a hybrid form and then proceeding to analyse the obtained results, which describe the various stages of the evolution of the Universe. The model exhibits the point-type singularity. The volume of the model increases as . The positive values of the
CRediT authorship contribution statement
J.K. Singh: Conceptualization, Supervision, Methodology, Project administration, Software, Formal analysis, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Akanksha Singh: Methodology, Project administration, Software, Formal analysis, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. G.K. Goswami: Methodology, Project administration, Software, Formal analysis, Resources, Software,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
J.K.S. and A.S. express their thanks to Prof. Sushant G. Ghosh, CTP, Jamia Millia Islamia, New Delhi, India for fruitful discussions and suggestions. Authors also express their thanks to the referee for his valuable comments and suggestions.
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