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

We investigate linear matter density perturbations in the $\Lambda_{\rm s}{\rm CDM}$ scenario, in which the $\Lambda$ is replaced by one that undergoes a late-time ($z\sim2$) mirror AdS-dS transition, resulting in distinct growth dynamics that shape cosmic structure evolution. We begin our analysis by developing a systematic method to track perturbation growth using two complementary approaches: (i) determining the initial density contrast and its evolution rate for a given collapse scale factor, and (ii) computing the collapse scale factor for a specified initial density contrast and evolution rate. We derive analytical solutions for the growth rate $f=\Omega_{\rm m}^\gamma$ and growth index $\gamma$ in both models, reinforcing the theoretical foundation of our approach. Our analysis indicates that prior to the transition, during the AdS-like phase---the AdS-like $\Lambda$ in $\Lambda_{\rm s}{\rm CDM}$ reduces cosmic friction, causing linear matter density perturbations to grow more rapidly than in $\Lambda{\rm CDM}$; this effect is most pronounced just before the transition, with a growth rate approximately $15%$ higher than that of $\Lambda{\rm CDM}$ around $z\sim2$. After the transition, $\Lambda_{\rm s}{\rm CDM}$ behaves similarly to $\Lambda{\rm CDM}$ but features a larger cosmological constant, leading to higher $H(z)$ and greater cosmic friction that more effectively suppresses growth. Before the transition, the growth index $\gamma$ remains below both the $\Lambda{\rm CDM}$ and Einstein-de Sitter values ($\gamma\approx6/11$); during the transition, it increases rapidly and then grows gradually, paralleling $\Lambda{\rm CDM}$ while remaining slightly higher in the post-transition era-though overall, it stays near $\gamma\sim0.55$, as in the $\Lambda{\rm CDM}$ model. Using the Planck best-fit values, namely $\Omega_{\rm m0}=0.28$ for $\Lambda_{\rm s}{\rm CDM}$ and $\Omega_{\rm m0}=0.32$ for $\Lambda{\rm CDM}$, we find that the corresponding growth rates at $z=0$ are $f=0.49$ and $f=0.53$, respectively. Notably, $\Lambda_{\rm s}{\rm CDM}$ predicts a value closer to $f=0.48$, recently obtained from LSS data when $\gamma$ is treated as a free parameter in $\Lambda{\rm CDM}$. This suggests that $\Lambda_{\rm s}{\rm CDM}$ may naturally resolve the structure growth anomaly, without deviating from $\gamma \sim 0.55$. The analysis of linear matter perturbations underscores $\Lambda_{\rm s}{\rm CDM}$'s potential to resolve multiple cosmological tensions within a unified framework, motivating further exploration of its implications for nonlinear structure formation and observational tests.

Structure

- data: Contains the data that is used in the mathematica\fs8_contour.nb file
- log: Contains the output files from the programs
- mathematica: Contains the mathematica files
- models: Contains the functions related to the calculation of the model parameters
- src: Contains the main functions which performs the calculations

Requirements

In Windows OS you can install the requirements via

py -m pip install matplotlib numpy scipy SciencePlots

meanwhile, in GNU/Linux

python3 -m pip install matplotlib numpy scipy SciencePlots

Attribution

If you find this code useful, please cite our research

BibTeX Format:

@article{Akarsu:2025ijk,
    author = {Akarsu, \"Ozg\"ur and \c{C}am, Arman and Paraskevas, Evangelos A. and Perivolaropoulos, Leandros},
    title = "{Linear matter density perturbations in the $\Lambda_{\rm s}$CDM model: Examining growth dynamics and addressing the $S_8$ tension}",
    eprint = "2502.20384",
    archivePrefix = "arXiv",
    primaryClass = "astro-ph.CO",
    month = "2",
    year = "2025"
}

Contributors

Arman Çam & Evangelos A. Paraskevas