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

In this paper, we investigate, through a Bayesian study, the ability of a local low matter density [Formula: see text], 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 [Formula: see text] or the matter fluctuation [Formula: see text] 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 [Formula: see text]CDM model, when computing the corresponding theoretical observables. We also add the latest Baryonic Acoustic Oscillations (BAO) measurements from the DESI year one release to some of our MCMC runs. We found that, within [Formula: see text]CDM model, for different combinations of our probes, we can accommodate a low matter density along with the [Formula: see text] and [Formula: see text] values usually obtained from local probes, providing we promote the sound drag [Formula: see text] component in BAO calculations to a free parameter, and that even if we combine with the Pantheon [Formula: see text] Supernova sample, an addition that was found in previous work to mitigate our concordance. Assuming [Formula: see text]CDM, we also found that relaxing [Formula: see text] allow us to accommodate [Formula: see text], [Formula: see text] and [Formula: see text] within their local values, with still however a preference for [Formula: see text] values far from [Formula: see text]CDM. However, when including Pantheon+ Supernova sample, we found that the latter preference for high matter density pushes [Formula: see text] to values much smaller than the ones usually obtained from local probes, mitigating by then a low matter density solution to the two common tensions. We conclude therefore that a low matter density value helps in preserving the concordance within [Formula: see text]CDM model, even with recent BAO DESI measurements or extended to high redshift supernovae sample, while the dynamical dark energy model ultimately fails to solve all tensions at once within our low density matter hypothesis.