Cosmological dynamics and structure formation in a generalized mass-to-horizon entropy-inspired modified gravity

In this article, our goal is to investigate the cosmological dynamics and structure formation in a modified cosmological framework inspired by a generalized mass-to-horizon entropy relation and consistent with the Clausius relation. Invoking the gravity-thermodynamics conjecture leads to alterations in the Friedmann equations as well as Hubble parameter evolution. The effects of the generalized entropy on various cosmographic parameters and on the growth of the linear matter perturbations by constructing perturbed field equations via employing spherical collapse formalism in a flat Friedmann-Lema\^{i}tre-Robertson-Walker background have been explored. We discuss a novel and well-known diagnostic approach to differentiate various cosmological models vis-\`a-vis flat and non-flat $\Lambda$CDM frameworks, and find that the generalized mass-to-horizon entropy-inspired modified cosmology ($n\ne 1$) successfully passes all the litmus tests by falsifying both the flat and non-flat $\Lambda$CDM paradigms. It is shown that this model also satisfies the requirements for the Universe to achieve thermodynamic equilibrium in the distant future. We also study the halo mass function and cluster number counts in this modified gravity scenario. All the results are compared with the fiducial $\Lambda$CDM profile, showing that the additional entropic correction influences the expansion history, the growth rate of structures, and the abundance of collapsed halos. We observe that the more massive collapsed structures are less abundant and form at later epochs, which is expected from the hierarchical model of large-scale structure formation.}