Linear matter density perturbations in the ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM model: Examining growth dynamics and addressing the $S_8$ tension

We investigate linear matter density perturbations in the ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM, in which the $\mathrm{\Lambda }$ is replaced by late-time ($z\sim 2$) mirror AdS-dS transition, resulting in distinct growth dynamics. We use two complementary approaches: (i) determining the initial density contrast and its evolution rate for a given collapse scale factor, (ii) computing the collapse scale factor for a specified initial density contrast and evolution rate. We derive analytical solutions for the growth rate $f={\mathrm{\Omega }}_{\mathrm{m}}^{\gamma }$ and growth index $\gamma$ in both models. Prior to the transition, the AdS-like $\mathrm{\Lambda }$ reduces cosmic friction, causing linear matter density perturbations to grow more rapidly than in $\mathrm{\Lambda }$CDM; this effect is most pronounced just before the transition, with a growth rate around $15\mathrm{\%}$ higher than $\mathrm{\Lambda }$CDM around $z\sim 2$. After the transition, ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM behaves similarly to $\mathrm{\Lambda }$CDM but features a larger cosmological constant, leading to higher $H(z)$ and greater cosmic friction that more effectively suppresses growth. Before the transition, $\gamma$ remains below both the $\mathrm{\Lambda }$CDM and EdS values ($\gamma \approx 6\mathrm{/}11$); during the transition, it increases rapidly and then grows gradually, paralleling $\mathrm{\Lambda }$CDM while remaining slightly higher in the post-transition era-though overall, it stays near $\gamma \sim 0.55$. Using the Planck best-fit values, ${\mathrm{\Omega }}_{\mathrm{m}0}=0.28$ for ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM and ${\mathrm{\Omega }}_{\mathrm{m}0}=0.32$ for $\mathrm{\Lambda }$CDM, we find $f=0.49$ and $f=0.53$, respectively. ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM predicts a value of $f=0.48$, recently obtained from LSS data when $\gamma$ is treated as a free parameter in $\mathrm{\Lambda }$CDM. This suggests that ${\mathrm{\Lambda }}_{\mathrm{s}}$CDM may resolve the structure growth anomaly, without deviating from $\gamma \sim 0.55$.