In this paper, the proposed dark energy, Tsallis holographic dark energy (THDE), infrared cut-off with the Hubble horizon has been investigated in the Bianchi-I (axially symmetric) anisotropic model with a hybrid expansion law. It has been observed that the THDE is in tune with the accelerating Universe with equation of state (EoS) parameter ($\omega_{T}$ < $−1/3$) in the $k$-essence region. We have used the statefinder diagnostic in our model. In addition, we try to accommodate the perspective of dark energy by the avenue of reconstructing the evolution of scalar field potential. We have considered the $k$-essence for the analysis of this reconstruction, showing the accelerated expansion at present.

In this work, we study a cosmological model based on the cosmological principle which exhibits a transition from deceleration to acceleration. We consider baryonic matter dark energy (DE), and ‘curvature’ energy. Both baryonic matter and DE have variable equations of state. It is assumed that DE interacts with and transforms energy to baryonic matter. A Friedmann–Robertson–Walker (FRW) Universe filled with two fluids has been discussed. The model is shown to satisfy current observational constraints. This Universe is at present in a phantom phase after passing through a quintessence phase in the past. Various cosmological parameters regarding the accelerating Universe have been presented. The evolution of DE, Hubble, deceleration parameters, etc. have been described with the aid of figures. Our theoretical results have been compared with the SNe Ia related Union 2.1 compilation 581 data and we have observed that our derived model is in good agreement with the current observational constraints. We have also explored the physical properties of the model.

We investigate a new class of LRS Bianchi type-II cosmological models by revisiting the paper of Mishra et al (Int. J. Theor. Phys. 52, 2546 (2013)) by considering a new deceleration parameter (DP) depending on the time in string cosmology for the modified gravity theory suggested by Sáez–Ballester (Phys. Lett. 113, 467 (1986)). We have considered the energy–momentum tensor proposed by Letelier (Phys. Rev. 28, 2414 (1983)) for bulk viscous and perfect fluid under some assumptions. To make our models consistent with recent astronomical observations, we have used the scale factor (Sharma et al, Astron Astrophys. 19, 55 (2018), Garg et al, Int. J. Geo. Meth. Mod. Phys. 16, 1950007 (2019)) $a(t) = \rm{exp} [ \frac{1}{\beta}\sqrt{2\beta t + k}]$, where $β$ and $k$ are positive constants and it provides a time-varying DP. By using the recent constraints ($H_{0} = 73.8$ and $q_{0} = −0.54$) from SN Ia data in combination with BAO and CMB observations (Giostri et al, JCAP 3, 27 (2012), arXiv:1203.3213v2[astroph. CO]), we affirm $\beta = 0.0062$ and $k = 0.000016$. For these constraints, we have substantiated a new class of cosmological transit models for which the expansion takes place from the early decelerated phase to the current accelerated phase. Also, we have studied some physical, kinematic and geometric behaviour of the models, and have found them consistent with observations and well-established theoretical results. We have also compared our present results with those of Mishra et al (Int. J. Theor. Phys. 52, 2546 (2013)) and observed that the results in this paper are much better, stable under perturbation and in good agreement with cosmological reflections.