Covariant cosmography in the presence of local structures: comparing exact solutions and perturbation theory

TL;DR

This work assesses covariant cosmography (CC) in a fully relativistic, nonperturbative setting by placing an off-center observer in a Lemaître–Tolman–Bondi (LTB) spacetime and computing the exact luminosity distance via the Sachs equation. It compares CC reconstructions (including Hubble, deceleration, curvature and jerk) to the exact LTB distances, identifying regimes where CC remains reliable and where nonperturbative effects become essential near local inhomogeneities. It then builds a precise dictionary between linearized LTB (LLTB) and Linear Perturbation Theory (LPT) in conformal Newtonian gauge, linking CC multipoles to those from standard perturbation theory and clarifying gauge-related differences in the Hubble monopole. The results show CC outperforms linear perturbation theory for moderate central overdensities near the observer, while LPT remains accurate at larger separations; both converge to the exact solution at sufficiently large distances. These findings provide a principled framework for interpreting local expansion-rate anisotropies without assuming global homogeneity, and they chart a path toward data-driven, non-FLRW cosmography with a fully non-perturbative metric description.

Abstract

Recent observational evidence of axially symmetric anisotropies in the local cosmic expansion rate motivates an investigation of whether they can be accounted for within the Lemaître-Tolman-Bondi (LTB) framework with an off-center observer. Within this setting, we compute the exact relativistic luminosity distance via the Sachs equation and compare it with the approximate expression obtained from the covariant cosmographic approach (including Hubble, deceleration, jerk and curvature parameters). This comparison allows us to identify the regimes in which the covariant cosmographic method remains reliable. In addition, we compare the LTB relativistic distance for small inhomogeneities with the corresponding result derived from linear perturbation theory (LPT) in the standard cosmological model. This analysis establishes a precise correspondence between the LTB and LPT approaches, offering a consistent dictionary for the interpretation of the observed anisotropies of the large-scale gravitational field. This analysis will be instrumental in interpreting expansion-rate anisotropies, facilitating investigations of the local Universe beyond the FLRW framework with a fully non-perturbative metric approach.

Figures (10)

• Figure 1: Off-center LTB metric configuration. The observer is located at a distance of $\chi_o$ from the center along the z-axis. The angle $\psi$ denotes the separation between the direction of the center and the observer’s line of sight $\boldsymbol{n}$.
• Figure 2: Radial scaling profiles of $\delta$, $\tilde{\delta}$, $\tilde{\Omega}_{m0}$, $\Omega_{k0}$ and $H_0$ are shown for $\delta_c = 2.5$ and $R_s=37.4$ Mpc.
• Figure 3: Luminosity distance for the off-center observer in the LTB$_{M1}$ model, calculated using eq. (\ref{['dida']}). The scaling with redshift along two different line-of-sight directions is shown: towards the center of the density peak ($\psi=0$, solid red line) and in the antipodal direction ($\psi=\pi$, solid blue line).
• Figure 4: The comparison between the exact luminosity distances for the LTB$_{M1}$ model and the distance reconstructed using the covariant cosmographic (CC) expansion in the two panels. The relative error in the cosmographic approximation is displayed for various expansion orders, along two directions: toward the center of the density peak ( left panel), and in the antipodal direction ( right panel). The dashed vertical line indicates the center of the mass overdensity.
• Figure 5: Relative error (in %) in estimation of luminosity distance for an off-center observer using covariant cosmography up to $\mathcal{O}(z^3)$. The white "X" marks the center of the overdensity which is at a distance of 200 Mpc away from the observer.