A comprehensive analysis of observational cosmology in f(Q) gravity with deep learning and MCMC method

Our goal in this study is to build FRW cosmological models inside the $f\left(Q\right)$ theory of gravity framework by using Bayesian statistics and deep learning method. We investigate the universe’s accelerating behaviour for a specific version of the $f\left(Q\right)$ gravity model using a novel, straightforward parameterization of the Hubble parameter in the form $H=H_0{(1+z)}^{1+q_0-q_1}exp\left(q_{1}z\right)$. The corresponding free parameters in $H\left(z\right)$ are limited between 1$\sigma$ and 2$\sigma$ confidence bounds using the ${\chi }^2$-minimization procedure. The results show that all the numbers we got are in the ballpark of what cosmological observations would predict. In our model, we examined the physical behaviour of the cosmos using characteristics such as energy density, pressure, and equation of state. We analysed kinematic factors including Hubble parameter, acceleration parameter, and universe age in our model. In our concept, the deceleration parameter $q\left(z\right)$ represents the universe’s transition from deceleration to acceleration. We employ a novel approach for parameter estimation by utilizing a mixed neural network (MNN) that combines artificial neural networks (ANN) and mixture density networks (MDN). This new methodology leverages the strengths of ANN, MDN, and MNN to enhance the accuracy of parameter estimation.