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In the past few years, f(Q) theories have drawn a lot of research attention in replacing Einstein’s theory of gravity successfully. The current study examines the novel cosmological possibilities emerging from two specific classes of f(Q) models using the parametrization form of the equation of state (EoS) parameter as $$\omega \left( z\right) =-\frac{1}{1+3\beta \left( 1+z\right) ^{3}}$$ ω z = - 1 1 + 3 β 1 + z 3 , which displays quintessence behavior with the evolution of the Universe. We do statistical analyses using the Markov chain Monte Carlo (MCMC) method and background datasets like Type Ia Supernovae (SNe Ia) luminosities and direct Hubble datasets (from cosmic clocks), and Baryon Acoustic Oscillations (BAO) datasets. This lets us compare these new ideas about the Universe to the $$\Lambda $$ Λ CDM model in a number of different possible ways. We have come to the conclusion that, at the current level of accuracy, the values of their specific parameters are the best fits for our f(Q) models. To conclude the accelerating behavior of the Universe, we further study the evolution of energy density, pressure, and deceleration parameter for these f(Q) models.

f (Q, T) theory of gravity is very recently proposed to incorporate within the action Lagrangian, the trace T of the energy-momentum tensor along with the non-metricity scalar Q. The cosmological application of this theory in a spatially flat isotropic and homogeneous Universe is well-studied. However, our Universe is not isotropic since the Planck era and therefore to study a complete evolution of the Universe we must investigate the f (Q, T) theory in a model with a small anisotropy. This motivated us to presume a locally rotationally symmetric (LRS) Bianchi-I spacetime and derive the motion equations. We analyse the model candidate f (Q, T) = αQ n+1 + βT, and to constrain the parameter n, we employ the statistical Markov chain Monte Carlo (MCMC) method with the Bayesian approach using two independent observational datasets, namely, the Hubble datasets, and Type Ia supernovae (SNe Ia) datasets.

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