The case for a low dark matter density in dynamical dark energy model from local probes

In this work we investigate, through a Bayesian study, the ability of a local low matter density $\Omega_{\rm M}$, in discrepancy with the value usually inferred from the CMB angular power spectrum, to accommodate observations from local probes without being in tension with the local values of the Hubble constant $H_0$ or the matter fluctuation $\sigma_8$ parameters. For that, we combine multiple local probes, with the criteria that they either can constrain the matter density parameter independently from the CMB constraints, or can help in doing so after making their relevant observations more model independent by relaxing their relevant calibration parameters. We assume however, either a dynamical dark energy model, or the standard $\Lambda$CDM model, when computing the corresponding theoretical observables. We also add, in almost all of our Monte Carlo runs, the latest Baryonic acoustic oscillations (BAO) measurements from the DESI year one release to our core group. We found that, within $\Lambda$CDM model, for different combinations of our probes, we can accommodate a low matter density along with the $H_0$ and $\sigma_8$ values usually obtained from local probes, providing we promote the sound drag $r_s$ component in BAO calculations to a free parameter, and that even if we combine with the Pantheon+ Supernova sample. Assuming $w_0w_a$CDM, we also found that relaxing $r_s$ allow us to accommodate $\Omega_{\rm M}$, $H_0$ and $\sigma_8$ within their local values, with still however a preference for $w_0w_a$ values far from $\Lambda$CDM. However, when including Pantheon+ Supernova sample, we found that the latter preference for high matter density pushes $\sigma_8$ to much smaller values, mitigating by then a low matter density solution to the two common tensions. We conclude that a low matter density value, helps in preserving the concordance within $\Lambda$CDM model. (abridged)