The ability of Big Bang Nucleosynthesis theory to accurately predict the primordial abundances of helium and deuterium, as well as the baryon content of the Universe, is considered one of the most significant achievements in modern physics. In the present study, we consider two highly motivated hybrid $f(T)$ models and constrain them using the observations from the Big Bang Nucleosynthesis era. In addition, using late-time observations of Cosmic Chronometers and Gamma-Ray Bursts, the ranges of the model parameters are confined which are in good agreement with early time bounds. Subsequently, the common ranges obtained from the analysis for early and late time are summarized. Further, we verify the intermediating epochs by investigating the profiles of cosmographic parameters using the model parameter values from the common range. From this study, we find the considered teleparallel models are viable candidates to explain the primordial-intermediating-present epochs.

In the following work, a new hybrid model of the form $$ f(Q)=Q(1+a)+b\frac{Q_0^2}{Q} $$f ( Q ) = Q ( 1 + a ) + b 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 $$ \alpha _1 Q+\alpha _2 Q_0 $$α 1 Q + α 2 Q 0 which is found to reduce to $$\Lambda $$Λ CDM. The error bar plot for the Cosmic Chronometer (CC) and Pantheon+SH0ES datasets which includes the comparison with $$\Lambda $$Λ 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 $$\Lambda $$Λ CDM model using AIC and BIC method.

As a classical approach, the dynamics of the Universe, influenced by its dark components, are unveiled through prior modifications of Einstein’s equations. Cosmography, on the other hand, is a highly efficient tool for reconstructing any modified theory in a model-independent manner. By employing kinematic variables, it offers a profound explanation for cosmic expansion. Although the cosmographical approach has been highly successful in several geometric theories in recent years, it has not been extensively explored in coupled gravities. With this in mind, we intend to constrain an extended teleparallel gravity model, f ( T , T ) , through cosmographic parameters. We utilize Taylor series expansion, assuming a minimally coupled form, to constrain the unknowns involved in the series. To achieve this, we conduct a Markov Chain Monte Carlo (MCMC) analysis using three different data sets (cosmic chronometers, baryon acoustic oscillations, and Pantheon+SH0ES). The constrained results obtained from MCMC are then compared and verified using various cosmological parameters. Finally, we compare the resulting models with three well-known f ( T , T ) models.

The symmetric teleparallel framework brings about the possibility of alleviating cosmological tensions. The current burning issue in cosmological studies is the increase in discrepancies in measurements from several surveys. Here, we have focused on H 0 and S 8 tensions, which are important factors in describing the evolution of the Universe from primordial perturbation to late-time acceleration. Additionally, the consistency of the sound horizon is verified against the Planck results. The f (Q) gravity model is constrained using recently obtained data. Implementing gravitational wave data to study late-time acceleration is one of the key features of our study. Since standard sirens show promising results, the implementation of gravitational waves to probe dark energy is an interesting study. Through our work, we introduce this possibility by performing statistical MCMC analysis for late-time cosmological evolution. Also, the H 0 and S 8 tensions are explored utilizing gravitational wave data alongside other prominent datasets, such as the latest DESI BAO, redshift space distortion, cosmic chronometers, Pantheon+SH0ES, and cosmic microwave background data. With the results obtained, we analyzed the profile of cosmological parameters. Finally, the study presents the tension of the model with observations, which is found to have a much lower magnitude compared to the current trend. Thus, the considered f(Q) model alleviates tension, making it the best candidate for further investigation.

The H ₀ and S ₈ tensions highlight critical discrepancies in modern cosmology, challenging the standard Lambda cold dark matter model. The H ₀ tension arises from conflicting measurements between local probes and early-Universe predictions. Similarly, the S ₈ tension, related to the amplitude of matter clustering, exposes inconsistencies between cosmic microwave background data and weak gravitational lensing surveys. The primary goal of this work is to address these tensions by modifying the underlying geometric framework. Specifically, we test two f(T) gravity models using cosmic chronometers, baryonic acoustic oscillations, gamma-ray bursts, and Pantheon+SH0ES data sets and compare the results with gravitational-wave data for validation. Both teleparallel models demonstrate promising performance in simultaneously alleviating the H ₀ and S ₈ tensions.