Cosmography with $Λ$-Szekeres Models

TL;DR

The paper addresses the challenge of model-independent cosmography in anisotropic and inhomogeneous space-times by deriving the luminosity-distance expansion to $4^{ ext{th}}$ order in redshift and testing it within axially symmetric $Λ$-Szekeres models. It employs covariant cosmography formalism with a generalisation of the Hubble law to define effective parameters ($oldsymbol{rak{H}}, oldsymbol{rak{Q}}, oldsymbol{rak{R}}, oldsymbol{rak{J}}, oldsymbol{rak{S}}$) and uses ray tracing in Szekeres geometries to compare exact distances with the cosmographic series. The study quantifies directional variations of the cosmographic parameters, assesses the convergence of the cosmographic expansion, and evaluates Padé approximants, finding the $[2/1]$ Padé form often provides the best high-$z$ behavior within these structures. The results indicate reliable cosmography up to $z oughly0.01$ inside local inhomogeneities, with convergence depending on the density profile, informing the interpretation of local Universe tensions and the applicability of covariant cosmography to real data.

Abstract

The cosmological tensions present in the $Λ$ cold dark matter model that have emerged and strengthened over recent years motivate model independent approaches to analysing data. Cosmography is useful for interpreting data in cosmology without imposing assumptions about the field equations of gravity or the matter content in the Universe. Some cosmography methods, denoted covariant cosmography, go even further and stay agnostic to the underlying space-time metric. Due to their high level of generality, covariant cosmography methods can incorporate the anisotropies and inhomogeneities in the observer's vicinity, and may in turn inform about the associated curvature of the relevant structures in our cosmic neighbourhood. Thus, covariant cosmography is a powerful model-independent tool for analysing cosmological data while also enabling the mapping of our local cosmic neighbourhood. In order to be able to explore the covariant cosmography framework to its fullest, it must be tested in tractable models and simulations. In this paper we derive the cosmography of luminosity distance to fourth order in redshift and investigate it in the special case of axially symmetric Szekeres models. We compare the numerical results for the distance-redshift relations of synthetic observers placed within the Szekeres structures with the predictions from the cosmography, and comment on the found level of approximation of the cosmography in relation to other results in the litterature.

Figures (10)

• Figure 1: Parameter values for model 1 and model 2. These parameters are used within \ref{['eq:expdelta']} to control the behaviour of the Szekeres mass profile \ref{['eq:massfunc']}.
• Figure 2: Selected observer positions in Szekeres spherical coordinates.
• Figure 3: The initial present time 2D density profiles of model 1 and model 2, normalised to the background FLRW density $\Delta\rho = (\rho-\bar{\rho})/\bar{\rho}\,$. These profiles are plotted from the coordinate centre of the Szekeres model, with $r$ being the radial distance from the coordinate centre. In terms of the angular Szekeres coordinate $\theta$, solid $\theta=\pi$ lines corresponding to the angular position of the maximum over-density magnitude, while dashed $\theta=2\pi$ lines correspond to the angular position of the minimum over-density magnitude. Dash-dotted lines give the over-density magnitude located $90^\circ$ from these extrema. The parameter values of each profile are given in table \ref{['tab:model_params']}.
• Figure 4: Variation of the covariant cosmographic parameters (\ref{['eq:effh']}) and (\ref{['eqs:effparams']}) for observer 1 in model 1 and model 2.
• Figure 5: Variation of the ray traced $d_L$ for observer 1 in models 1 and 2. This is presented as full sky maps for hypothetical sources at five equally spaced redshifts between $z=0.002$ and $z=0.01$, with the observer located at $z=0$.