In this paper, we present a spatially homogeneous and anisotropic Bianchi type-I cosmological model with a viscous bulk fluid in [Formula: see text] gravity where [Formula: see text] and [Formula: see text] are the Ricci scalar and trace of the energy-momentum tensor, respectively. The field equations are solved explicitly using the hybrid law of the scale factor, which is related to the average Hubble parameter and gives a time-varying deceleration parameter (DP). We found the deceleration parameter describing two phases in the universe, the early deceleration phase [Formula: see text] and the current acceleration phase [Formula: see text]. We have calculated some physical and geometric properties and their graphs, whether in terms of time or redshift. Note that for our model, the bulk viscous pressure [Formula: see text] is negative and the energy density [Formula: see text] is positive. The energy conditions and the [Formula: see text] analysis for our spatially homogeneous and anisotropic Bianchi type-I model are also discussed.

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In this paper, we examine the accelerated expansion of the Universe at late‐time in the framework of ffalse(Qfalse)$f(Q)$ gravity theory in which the non‐metricity scalar Q describes the gravitational interaction. To this, we propose a new parametrization of the Hubble parameter using a model‐independent way and apply it to the Friedmann equations in the FLRW Universe. Then we estimate the best fit values of the model parameters by using the combined datasets of updated Hfalse(zfalse)$H(z)$ consisting of 57 points, the Pantheon consisting of 1048 points, and BAO datasets consisting of six points with the Markov Chain Monte Carlo (MCMC) method. The evolution of deceleration parameter indicates a transition from the deceleration to the acceleration phase of the Universe. In addition, we investigate the behavior of statefinder analysis and Om diagnostic parameter, Further, to discuss other cosmological parameters, we consider a ffalse(Qfalse)$f(Q)$ model, specifically, f(Q)=Q+mQn$f(Q) =Q+mQ^{n}$, where m and n are free parameters. Finally, we find that the model supports the present accelerating Universe, and the EoS parameter behaves like the quintessence model.

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In this work, we study the interaction between the dark matter (DM) component and the dark energy (DE) component using the Tsallis holographic dark energy (THDE) density expression for a Bianchi type-II space–time within the framework of general relativity (GR). To obtain the exact solutions of Einstein’s field equations, we use two constraints: (i) the expansion scalar [Formula: see text] of the Universe is proportional to the component [Formula: see text] of the shear tensor [Formula: see text], i.e. [Formula: see text] and (ii) we assume that the scale factor follows the Hybrid Expansion Law (HEL). We have discussed some geometrical and physical parameters of our model such as the deceleration parameter (DP) and the equation of state (EoS) parameter. In these parameters, we plot their behavior in terms of redshift [Formula: see text]. We observe that the DP evolves from the early decelerating phase to the current accelerating phase with a current value consistent with the observation data. The EoS parameter evolves from a state of a stiff-matter fluid-dominated era [Formula: see text] for high redshift [Formula: see text] to a [Formula: see text]CDM era [Formula: see text] in the later times. Also, we have established a correspondence between the THDE model and the tachyon scalar field dark energy model. We have reconstructed the potential and the tachyon scalar field, which describes the current accelerated expansion of the Universe. Finally, using 51 values of observed Hubble measurement and the technique [Formula: see text]-test, we found the best fit value for the model parameters [Formula: see text], [Formula: see text] and [Formula: see text] (current value of the Hubble parameter). These results are consistent with the new measurements of [Formula: see text].[Formula: see text]