A Unified Picture of Cosmic Evolution in a Gravity-Modified Framework

Abstract

In the present work a unified description of the early negative pressure dark energy during inflation and the late-time dark energy is posited within the framework of \(f\left( R \right) \) gravity. The innovative approach integrates seamlessly the two eras going through the interposing evolutionary phase that intermediates. The time rate of change of the scale factor describing cosmic expansion is specified by Hubble parameter H which gives variant phases. Considering the spatially flat and homogeneous geometry of FLRW universe, the Hubble rate is written as the function of the number of e-folds N which leads to construct a differential equation under certain constraints. The solution to the differential equation gives rise to an \(f\left( R \right) \) functional form which in turn is solved numerically. For various Hubble rates it is shown through visual simulations as how the transitions of various eras of cosmological dynamics occur. Secondly we explore a more plausible advanced f(R) framework that bridges the primordial inflationary phase with both pre-late and late-time contemporary dark energy dynamics emphasizing theoretical basis. The analysis involves mathematically sorting out the Friedmann equations utilizing redshift as a principal element therein and providing deep insights into key statefinder parameters that include the deceleration parameter and dark energy density. The study conclusively shows that the early dark energy exerts minimal influence on the late-time universe, thereby imposing stringent constraints on the models’s free parameters. We also study an effective gravitational constant in connection with \(f\left( R \right) \) gravity model, the analysis reveals its behaviour across different scales, exploring insights into cosmic structure formation and gravitational dynamics. Apart from it, the analysis shows the minimal gravity coupling of dark energy eras to the cosmic evolution as structure formation within it. However, we see that while the dark energy era is realized, the early dark energy does not contribute sufficiently to influence the late-time phase of dark energy which leads to accelerate the expansion rate of the universe. The free parameters in the model which significantly implicate the mechanism are subject to stringent constraints. Our results indicate to a great deal of important new proposals in apprehending the cosmic evolution specifically concerning the character of a modified theory and the way dark energy plays its role in shaping the universe.