Non-comoving baryons and cold dark matter in cosmic voids

We examine the fully relativistic evolution of cosmic voids constituted by baryons and cold dark matter (CDM), represented by two non-comoving dust sources in a $\Lambda$CDM background. For this purpose, we consider numerical solutions of Einstein's field equations in a fluid-flow representation adapted to spherical symmetry and multiple components. We present a simple example that explores the frame-dependence of the local expansion and the Hubble flow for this mixture of two dusts, revealing that the relative velocity between the sources yields a significantly different evolution in comparison with that of the two sources in a common 4-velocity (which reduces to a Lemaitre-Tolman-Bondi model). In particular, significant modifications arise for the density contrast depth and void size, as well as in the amplitude of the surrounding over-densities. We show that an adequate model of a frame-dependent evolution that incorporates initial conditions from peculiar velocities and large-scale density contrast observations may contribute to understand the discrepancy between the local value of $H_0$ and that inferred from the CMB.