Investigating early and late-time epochs in $ f(Q) $ gravity

In the following work, a new hybrid model of the form $ f(Q)=Q(1+a)+b\frac{Q_0^2}{Q} $ has been proposed and confronted using both early as well as late-time constraints. We first use conditions from the era of Big Bang Nucleosynthesis (BBN) in order to constrain the models which are further used to study the evolution of the Universe through the deceleration parameter. This methodology is employed for the hybrid model as well as a simple model of the form $ \alpha_1 Q+\alpha_2 Q_0 $ which is found to reduce to $\Lambda$CDM. The error bar plot for the Cosmic Chronometer (CC) and Pantheon+SH0ES datasets which includes the comparison with $\Lambda$CDM, has been studied for the constrained hybrid model. Additionally, we perform a Monte Carlo Markov Chain (MCMC) sampling of the model against three datasets -- CC, Pantheon+SH0ES, and Baryon Acoustic Oscillations (BAO) to find the best-fit ranges of the free parameters. It is found that the constraint range of the model parameter ($a$) from the BBN study has a region of overlap with the ranges obtained from the MCMC analysis. Finally, we perform a statistical comparison between our model and the $\Lambda$CDM model using AIC and BIC method.