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In this work, the Renyi holographic dark energy (RHDE)and its behaviour has been explored with the anisotropic and spatially homogeneous Bianchi type-I Universe in the framework of $f(G)$ gravity. We use IR cutoff as the Hubble and Granda-Oliveros (GO) horizons. To find a consistent solutions of the field equations of the models, it is assumed that the deceleration parameter is defined in terms of function of Hubble parameter $H$. With reference to current cosmological data, the behaviors of the cosmological parameters relating to the dark energy model are evaluated and their physical significance is examined. It is observed that for both the models, the equation of state parameter approaches to $-1$ at late times. However, the RHDE model with the Hubble horizon exhibits stability from the squared sound speed, but the RHDE model with the GO horizon exhibits instability. In both the models, deceleration parameter and statefinder diagnostic confirm the accelerated expansion of the Universe and also correspond to the $\Lambda$CDM model at late times.

Despite having a reasonably successful account of accelerated cosmology, understanding the early evolution of Universe has always been difficult for mankind. Our promising strategy is based on a novel class of symmetric teleparallel theories of gravity called $f(Q)$, in which the gravitational interaction is caused by the non-metricity scalar $Q$, which may help to solve some problems. We consider the locally rotationally symmetric (LRS) Bianchi type-I spacetime cosmological models and derive the motion of equations to study the early evolution of the cosmos. By assuming the Hybrid Expansion Law (HEL) for the average scale factor, we are able to determine the solutions to the field equations of Bianchi type-I spacetime. We discuss the energy density profile, the equation of state, and the skewness parameter and conclude that our models preserve anisotropic spatial geometry during the early stages of the Universe with the possibility of an anisotropic fluid present. However, as time goes on, even in the presence of an anisotropic fluid, the Universe may move towards isotropy due to inflation while the anisotropy of the fluid dims away at the same time. It is seen from the squared speed of sound that Universe shows phantom nature at the beginning then approaches to dark energy at present epoch. We analyze both geometrical and physical behaviors of the derived model., Comment: arXiv admin note: text overlap with arXiv:2208.08877

In this work a new law of varying deceleration parameter of third degree have been proposed. The solutions of the modified field equations have been derived under the newly proposed law of the deceleration parameter. Model exhibits the Big-bang singularity at cosmic time ($t=0$) and shows Big Rip at ($t=n$) then it re-enter the phase of initial singularity at $t=2n$ and ends its cyclic behavior at $t=3n$. The evolution of the physical and dynamical parameters of the Universe have been studied and the graphical representation has also been shown. Further $Om(z)$ diagnostic parameter and the energy conditions have also been studied together with their graphical representations.

Interaction of dark energy in the anisotropic Locally Rotationally Symmetric (LRS) Bianchi type-I metric is investigated in the context of modified f(R,T) theory of gravity, where R is the Ricci scalar and T is the trace of stress energy momentum tensor. We choose the particular form of the functional f(R,T)=f_1 (R,T)+f_2 (R,T) then we find the exact solutions of the field equations by applying inhomogeneous equation of state, p= {\omega}\r{ho}-{\Lambda}(t) and a generalized form of hybrid expansion law. Transition of deceleration to acceleration is observed in this model. It is also observed that the Universe shows accelerated expansion at late epoch. The derived model overlaps with {\Lambda}CDM at late time which is in agreement with present observation. Energy conditions of the derived model are also investigated. From the plot, we observe the age of Universe t_0=13.821 Gyr for the observed H_0=70.07Kms^(-1) Mpc^(-1). The physical and geometrical behaviours of these models are also discussed., Comment: 12 pages, 10 figures. Manuscript has been accepted for publication in Advances in High Energy Physics