Can structure formation distinguish between holographic dark energy models?

Abstract

Structure formation has been used to test the viability of the IR cutoff in the THDE model, and the future event horizon was found to be a viable IR cutoff (Phys. Rev. D {\bf 113}, 123531 (2026)). Motivated by this, in this paper, using the combined Pantheon+ SN Ia, OHD, DESI DR2, and CMB distance prior datasets, we constrain the model parameters for the HDE, THDE, and BHDE models, and use parameter sets from the MCMC samples to analyze the evolution of $fσ_{8}(z)$ and $S_{8}(z)$ for these models. We find that the evolutionary curves of both $fσ_{8}(z)$ and $S_{8}(z)$ for HDE and BHDE are nearly indistinguishable, while the curves for the THDE model lie lowest. The $χ^{2}$ statistic shows that the THDE model yields the lowest $χ^{2}$ value for both $fσ_{8}$ and $S_{8}$ measurements. Finally, we construct the ($fσ_{8}$,$S_{8}$) plane as a diagnostic to analyze the trajectories of these models, and find that the trajectory of the THDE model can achieve the current ($fσ_{8}$,$S_{8}$) data region. These results demonstrate that structure formation can distinguish THDE from HDE and BHDE, and suggest that structure formation can serve as a useful probe for distinguishing HDE models and can be extended to other HDE scenarios.