Cosmography of the Local Universe by Multipole Analysis of the Expansion Rate Fluctuation Field

We establish a relationship between the multipoles of the expansion rate fluctuation field $\eta,$ which capture in an accurate way deviations from isotropy in the redshift-distance relation, and the multipoles of the covariant cosmographic parameters-Hubble $\mathbb{H}_o$, deceleration $\mathbb{Q}_o$, jerk $\mathbb{J}_o$, and curvature $\mathbb{R}_o$. These parameters, derived from the third-order expansion of the luminosity distance with respect to redshift in a generic spacetime, provide model-independent insight into the geometry and symmetries of the cosmic line element without requiring the Cosmological Principle. Moreover, we demonstrate that although this approach is fully nonperturbative and does not rely on concepts such as peculiar velocities, it has the potential to constrain the motion of the matter frame at the observer's position relative to the cosmic microwave background. We use two analytical axisymmetric models, motivated by observational evidence, to test the formalism and the effectiveness of the third-order expansion of the luminosity distance. These models also help to predict the precision with which future surveys, like the Zwicky Transient Facility, will constrain the covariant cosmographic parameters in the local ($z<0.1$) Universe.