Comparative Analysis of Perturbed $f(R)$ Gravity and Perturbed Rastall Gravity Models in Describing Cosmic Evolution from Early to Late Universe Relative to the $Λ$CDM Model

Abstract: This study conducts a meticulous examination of the cosmological implications inherent in Rastall gravity and $f(R)$ gravity models, assessing their efficacy across distinct cosmic epochs, from early universe structure formation to late-time acceleration. In the initial stages, both models exhibit commendable compatibility with observed features of structure formation, aligning with the established $\Lambda$CDM model. The derived Jeans' wavenumbers for each model support their viability. However, as the cosmic timeline progresses into the late universe, a discernible disparity surfaces. Utilizing the Markov Chain Monte Carlo method, we reconstruct the deceleration parameter $(q)$ and identify Deceleration - Acceleration redshift transition values. For $f(R)$ gravity, our results align closely with previous studies, emphasizing its superior ability to elucidate the recent cosmic acceleration. In contrast, Rastall gravity exhibits distinct redshift transition values. Our rigorous analysis underscores the prowess of $f(R)$ gravity in capturing the observed cosmic acceleration, positioning it as a compelling alternative to the conventional $\Lambda$CDM model. The discernible shifts observed in the peaks of the CMB power spectrum and evolution of deceleration parameter (q) for both $f(R)$ gravity and Rastall gravity models in the Early and Late universe, in relation to the $\Lambda $CDM model, provide compelling evidence supporting the proposition that these alternative gravity models can account for the anisotropy of the universe without invoking the need for dark energy.

• The paper compares perturbed f(R) and Rastall gravity models against the standard ΛCDM cosmology to describe cosmic evolution from early to late universe epochs using observational data.
• Both alternative gravity models demonstrate reasonable compatibility with early universe structure formation by producing devoted Jeans wavenumbers similar to ΛCDM.
• f(R) gravity shows better alignment with observed late-time cosmic acceleration and cosmological data than Rastall gravity, presenting a potentially viable alternative without invoking dark energy.

Comparative Analysis of Perturbed $f(R)$ Gravity and Perturbed Rastall Gravity Models

The paper presents a thorough investigation of the cosmological applications of Rastall gravity and $f(R)$ gravity models, assessing their respective capabilities across different cosmic epochs, from the early stages of universe structure formation to the more recent epoch characterized by late-time acceleration. This comparative study considers their alignment with observational data via the standard $\Lambda$CDM model as a benchmark. The analysis utilizes sophisticated statistical methodologies, such as the Markov Chain Monte Carlo (MCMC) algorithm, to derive parameters like the deceleration parameter $q$ and deceleration to acceleration redshift transition values. These techniques allow the authors to probe these alternatives to the $\Lambda$CDM model with significant detail and precision.

The analysis emphasizes that both Rastall gravity and $f(R)$ gravity demonstrate a reasonable degree of compatibility with the observed features of early universe structure formation. Both models produce devoted Jeans wavenumbers, suggesting that they can reliably describe the gravitational clumping processes necessary for structures like galaxies and clusters to form. An essential aspect of the study includes the derivation of various critical points in cosmic evolution provided by these models, with an in-depth numerical analysis featuring parameter explorations such as those for the sound speed $c_s$, where detailed exploration broadens the understanding of the stability and potential oscillation of density perturbations within these frameworks.

A key finding arises in the consideration of the performance of these models in explaining late-time acceleration, a well-established observational fact linked with dark energy. The $f(R)$ gravity model presents a closer alignment with observed cosmic acceleration metrics, highlighting its efficacy as a potentially viable alternative to the $\Lambda$CDM paradigm. This alignment emphasizes $f(R)$ gravity's flexibility in accommodating scalar degree modifications to the scalar curvature, allowing it to provide a coherent fit to current cosmological data without the inevitable introduction of DE intrinsic to the $\Lambda$CDM model.

Contrarily, Rastall gravity is showcased to leverage nonstandard conservation principles of energy momentum, yet it arrives at redshift transitions distinct from $f(R)$ and $\Lambda$CDM, signifying potentially novel dynamics. This includes the modification of traditional conservation laws and gravitational interaction mechanics, translating to divergences in deduced transition redshifts. This study’s results underscore its limitations in matching late-time cosmological acceleration as compared to $f(R)$ gravity.

The paper's methodological framework extends into a numerical examination of these models against various data sets, including Pantheon supernovae, cosmic microwave background (CMB), and baryonic acoustic oscillations (BAO) data. The results from these comprehensive analyses are consistent across datasets, reinforcing the robustness of the fit for $f(R)$ gravity across multiple observational lines of evidence. Simulation outcomes cover the reconstructed cosmic metrics and establish the pertinence of non-standard theories to cosmic phenomena traditionally attributed to dark energy in $\Lambda$CDM.

The results invite speculation on broader theoretical implications, suggesting that $f(R)$ gravity's ability to describe acceleration without dark energy might align with paradigmatics potentially derivable from quantum gravity theories or other fundamental alterations of gravitational understanding. Future observations, particularly large-scale structure surveys and precision cosmology, may offer further opportunities to constrain these models or gauge new departures from traditional gravitational paradigms, addressing current tensions in cosmological datasets.

In conclusion, by systematically juxtaposing Rastall and $f(R)$ gravity, the paper elucidates key differences in their cosmological implications, significantly contributing to the ongoing discourse regarding the most apt theoretical constructs for describing universal dynamics beyond General Relativity. These contributions provide avenues for challenging the currently observed Hubble tension and assessing the dynamic nature of the universe, suggesting that alternative gravitational models, without invoking the concept of dark energy, may sufficiently elucidate cosmos' acceleration phenomena within strong theoretical and observational constraints.