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Recent observations from weak gravitational lensing (WL) surveys indicate asmoother Universe compared to the predictions of the Cosmic MicrowaveBackground (CMB). This inconsistency is commonly referred to as the $\sigma_8$tension or $S_8$ tension, where $\sigma_8$ represents the presentroot-mean-square matter fluctuation averaged over a sphere of radius $8 h^{-1}\mathrm{Mpc}$, and $S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$. In this article,we investigate a kind of general Dirac-Born-Infeld (DBI) Lagrangian referred toas \textit{surface-type DBI} (sDBI) model. We find that up to the linear order,the constraints on the sDBI model with high-redshift probe (CMB) andlow-redshift probes (WL and Galaxy Clustering, GC) yield $S_8=0.7448_{-0.21}^{+0.031}$ and $0.7426_{-0.085}^{+0.054}$, respectively.Remarkably, these values not only demonstrate self-consistency but also alignwith the values obtained from the majority of low-redshift probes. Furthermore,we present a discussion on exploring the non-linear effects of this model,which holds the potential to address additional challenges associated with ColdDark Matter (CDM) on small scales.

Relieving the $S_8$ Tension: Exploring the Surface-type DBI Model as a Dark Matter Paradigm