Expanding covariant cosmography of the local Universe: incorporating the snap and axial symmetry

Studies show that the model-independent, fully non-perturbative covariant cosmographic approach is suitable for analyzing the local Universe $(z\lesssim 0.1)$. However, accurately characterizing large and inhomogeneous mass distributions requires the fourth-order term in the redshift expansion of the covariant luminosity distance $d_L(z,\mathit{n})$. We calculate the covariant snap parameter $\mathbb{S}$ and its spherical harmonic multipole moments using the matter expansion tensor and the evolution equations for lightray bundles. The fourth-order term adds 36 degrees of freedom, since the highest independent multipole of the snap is the 32-pole (dotriacontapole) $(\mathrm{\ell }=5)$. Including this term helps to de-bias estimations of the covariant deceleration parameter. Given that observations suggest axially symmetric anisotropies in the Hubble diagram for $z\lesssim 0.1$ and theory shows that only a subset of multipoles contributes to the signal, we demonstrate that only 12 degrees of freedom are needed for a model-independent description of the local universe. We use an analytical axisymmetric model of the local Universe, with data that matches the Zwicky Transient Facility survey, in order to provide a numerical example of the amplitude of the snap multipoles and to forecast precision.