Anisotropic Cosmic Expansion Inspired by Some Novel Holographic Dark Energy Models in f ( Q ) $f(Q)$ Theory

The present work discusses the topic of cosmic evolution in an intriguing framework of $f\left(Q\right)$ theory of gravity (with $Q$ as a non-metricity (NM) scalar which controls the gravitational interaction) by using some recently proposed holographic dark energy (HDE) models. To achieve this goal, the dynamical equations for locally rotationally symmetric (LRS) Bianchi type-I (BI) geometry are formulated with matter contents as a mixture of dust and anisotropic fluids. By assuming that the time-redshift relation follows a Lambert function, the cosmological model is constructed by using Rényi HDE (RHDE), Sharma–Mittal HDE (SMHDE) and Generalized HDE (GHDE) as separate cases where Hubble horizon is taken as an infrared (IR) cutoff. Cosmological characteristics of these models are then examined through graphs of energy densities, skewness parameter $\left(\gamma \right)$, deceleration, and EoS parameters. The evolution of the EoS parameter is also studied, i.e., ${\omega }^{{}^{\prime }}$ to discuss the dynamical characteristics of constructed DE models and assess the stability of models via the squared speed of sound parameter. It is found that the $\omega -{\omega }^{{}^{\prime }}$ plane shows the freezing region for RHDE and GHDE models while the thawing region for the SMHDE case. Also, it is concluded that all constructed models exhibit cosmologically viable and stable behavior.