An Attractor-Repeller model of the Local Universe : The $Λ$-Szekeres spacetime and Perturbation Theory

Abstract

The standard linearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) framework provides an accurate description of the Universe on large scales but cannot fully capture the non-linear inhomogeneities that characterize the local Universe. In this work, we investigate whether an exact inhomogeneous solution of the Einstein field equations can provide a more realistic description of the nearby cosmic environment. To this end, we employ the quasi-spherical Szekeres class I solution with $Λ$ and focus on its axisymmetric subclass, motivated by recent observations indicating an axisymmetric expansion rate of the local Universe. We determine light propagation by numerically integrating the Sachs optical equations, thereby obtaining the luminosity distance and its evolution with redshift. We further compute the multipoles of the covariant cosmographic parameters and apply the formalism to an attractor-repeller configuration representative of the local matter distribution. We find that the resulting covariant cosmographic parameters are consistent with current observational constraints. Finally, we derive the gravitational potential and peculiar velocity field generated by the Szekeres density distribution, establishing the connection between the exact relativistic solution and its Newtonian counterparts.