Observational constraints using Bayesian statistics and Deep Learning in $f(Q)$ gravity

TL;DR

The paper investigates FRW cosmologies in $f(Q)$ gravity using a cubic-like $f(Q)$ form and a novel $H(z)$ parameterization to describe the transition from deceleration to late-time acceleration. It benchmarks Bayesian (MCMC) and deep-learning inference within the CoLFI framework, including a hybrid Mixture Neural Network, trained on hyperellipsoid-sampled data. Observational constraints from $H(z)$ measurements, Pantheon SN Ia data, and BAO support a consistent cosmology with a transition redshift around $z_t \nexists$; the neural methods yield posteriors competitive with traditional MCMC while offering likelihood-free efficiency. The work demonstrates the viability and advantages of integrating ML-based, likelihood-free inference for constraining modified gravity models like $f(Q)$ gravity and highlights potential benefits for addressing $H_0$ tensions in cosmology.

Abstract

This study investigates the evolution of Friedmann-Robertson-Walker (FRW) cosmological models within the $f(Q)$ gravity framework, utilizing a specific $f(Q)$ formulation and a novel Hubble parameter $H(z)$ parameterization to probe the universe's accelerating expansion. A central aspect is the application of advanced machine learning techniques for cosmological parameter estimation, alongside comparisons with traditional Bayesian (MCMC) methods. We employ a hybrid Mixed Neural Network (MNN), which synergistically combines Artificial Neural Networks (ANNs) and Mixture Density Networks (MDNs), to enhance the accuracy and robustness of parameter constraints. This MNN architecture is integrated into the CoLFI (Cosmological Likelihood-Free Inference) framework. CoLFI facilitates likelihood-free inference, a significant methodological advancement that provides an efficient and robust alternative, particularly for complex models with computationally expensive or intractable likelihood functions. Training efficiency for the neural networks is optimized by generating data via hyperellipsoid sampling. The $f(Q)$ model, constrained using these diverse approaches, successfully describes a universe transitioning from an early decelerating phase to the current accelerated expansion, with a computed transition redshift of $z_t = 0.60$. The physical and kinematic properties of the model are discussed, underscoring the efficacy of the MNN-CoLFI methodology and its consistency with MCMC results, while highlighting its advantages for obtaining observational constraints in $f(Q)$ gravity.

Figures (19)

• Figure 1: The two-dimensional contours depicting the $1\sigma$ and $2\sigma$ confidence regions.
• Figure 2: The integrated representation of two-dimensional contours at the $1\sigma$ and $2\sigma$ confidence levels.
• Figure 3: The 2D contour and marginalized distributions of $H_0$ and $n$, displaying the contours of 1$\sigma$ and 2$\sigma$.
• Figure 4: The left panel of above figure shows the variation of H(z) of our model with redshift z and its comparison with $\Lambda$CDM model for OHD data while the right panel of above figure exhibits the variation of distance modulus $\mu(z)$ of our model with redshift z and its comparison with $\Lambda$CDM model for Pantheon sample of SN Ia data.
• Figure 5: For Bayesian and Deep Learning Statistics, the development of energy density $\rho$ and pressure $p$ with redshift $z$ for $\gamma = 1$.