[Paper Review] Cosmological constraints on $R^2$-corrected Appleby-Battye model

Bruno Ribeiro, Armando Bernui+1 more|arXiv (Cornell University)|May 10, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates the $R^2$-corrected Appleby-Battye (R²-AB) model, a viable $f(R)$ gravity theory with one free parameter $b$, to test its consistency with cosmological data. Using SNe Ia, H(z), and RSD datasets, MCMC analyses yield $b = 2.18^{+5.41}_{-0.55}$ in the joint analysis, showing the model fits current observations well despite weak constraint on $b$, and it remains consistent with Solar System tests and de Sitter evolution.

ABSTRACT

Nowadays, efforts are being devoted to the study of alternative cosmological scenarios in which modifications of General Relativity have been proposed to explain the late cosmic acceleration without assuming the existence of dark energy. In this scenario, we investigate the $R^2$-AB model, which consists of an $f(R)$ model with only one extra free parameter, $b$, in addition to the 6 of the flat-$Λ$CDM. Regarding this model, it was already shown that a positive value for $b$ is required for the model to be consistent with Solar System tests, moreover, the condition for the existence of a de~Sitter state requires $b \ge 1.6$. To impose observational constraints on the $R^2$-AB model we consider three datasets: 31 $H(z)$ measurements from Cosmic Chronometers (CC), 20 $[{fσ}_{8}](z)$ measurements from Redshift-Space Distortion (RSD), and the most recent type Ia Supernovae (SNe Ia) sample from Pantheon+. Next, we perform two different analyses: we have considered only SNe Ia data and the combined likelihood SNe+CC+RSD. The first one has provided $b=2.28^{+6.52}_{-0.55}$, while the second one $b=2.18^{+5.41}_{-0.55}$. In the first case it was necessary to set the absolute magnitude $M_B = -19.253$ from SH0ES collaboration, while in the second we did a marginalization over the Hubble constant $H_0$ in the normalized growth function. We have also observed that the $H_0-M_B$ degeneracy was broken by adding CC data to the SNe data. Additionally, we perform illustrative analyses that compare this $f(R)$ model with the flat-$Λ$CDM model, considering several values of the parameter $b$, for diverse cosmological functions like the Hubble function $H(z)$, the equation of state $w_{ m eff}(z)$, the parametrized growth rate of cosmic structures $[fσ_8](z)$, and $σ_8(z)$. We conclude that the model fits well the data, but the parameter $b$ was not unambiguously constrained.

Motivation & Objective

• To assess the viability of the $R^2$-corrected Appleby-Battye (R²-AB) model as an alternative to $\Lambda$CDM in explaining late-time cosmic acceleration.
• To constrain the single free parameter $b$ of the R²-AB model using observational cosmological datasets.
• To compare the model's predictions for background and perturbation observables with those of the flat-$\Lambda$CDM model.
• To investigate whether the $H_0$--$M_B$ degeneracy in SNe Ia data is broken by inclusion of $H(z)$ and RSD data.
• To evaluate the model's consistency with phenomenological criteria such as Solar System tests and de Sitter attractor behavior.

Proposed method

• Constructs the $R^2$-AB model as a modified gravity theory with $f(R) = R + bR^2$, where $b$ is the sole free parameter beyond standard $\Lambda$CDM parameters.
• Performs Markov Chain Monte Carlo (MCMC) statistical analyses using three datasets: 31 H(z) measurements from Cosmic Chronometers, 20 $f\sigma_8(z)$ from Redshift-Space Distortions, and the Pantheon+ SNe Ia sample.
• Conducts two independent analyses: one using only SNe Ia with fixed $M_B = -19.253$ from SH0ES, and another combining SNe Ia, H(z), and RSD with marginalization over $H_0$.
• Compares model predictions for $H(z)$, $w_{\text{eff}}(z)$, $\sigma_8(z)$, and $f\sigma_8(z)$ across different $b$ values against observational data.
• Applies the Fisher matrix method to assess parameter degeneracies and constraints, particularly focusing on $H_0$--$M_B$ degeneracy.
• Validates model consistency with theoretical constraints: $b \geq 1.6$ for de Sitter attractor and $b > 0$ for Solar System compatibility.

Experimental results

Research questions

• RQ1 What is the observational constraint on the $R^2$-AB model parameter $b$ using current cosmological data?
• RQ2 How does the inclusion of $H(z)$ and RSD data affect the $H_0$--$M_B$ degeneracy in SNe Ia analyses?
• RQ3 How well does the $R^2$-AB model reproduce the background and perturbation evolution of the $\Lambda$CDM model?
• RQ4 What is the level of tension between the $R^2$-AB model and the $\Lambda$CDM model in terms of $H(z)$, $w_{\text{eff}}(z)$, and $f\sigma_8(z)$ evolution?
• RQ5 Does the $R^2$-AB model satisfy key phenomenological constraints such as Solar System tests and de Sitter attractor behavior?

Key findings

• The joint analysis of SNe Ia, H(z), and RSD data yields a best-fit value of $b = 2.18^{+5.41}_{-0.55}$, indicating the model is consistent with current data.
• The analysis using only SNe Ia data gives $b = 2.28^{+6.52}_{-0.55}$, with the $H_0$--$M_B$ degeneracy broken by the addition of $H(z)$ and RSD data.
• The $R^2$-AB model's predictions for $H(z)$, $w_{\text{eff}}(z)$, $\sigma_8(z)$, and $f\sigma_8(z)$ closely match those of the flat-$\Lambda$CDM model for the best-fit $b$ values.
• The model remains consistent with the de Sitter attractor condition $b \geq 1.6$ and Solar System constraints requiring $b > 0$.
• Despite good fit, the parameter $b$ is not tightly constrained, with large credible intervals indicating the need for improved data or new observables.
• The model's one-parameter structure makes it more efficient than other $f(R)$ models with multiple free parameters, yet still faces challenges in parameter degeneracy resolution.