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

We explore the possibility of characterizing the expansion rate on local cosmic scales (z ≲ 0.1), where the cosmological principle is violated, in a model-independent manner, i.e. in a more meaningful and comprehensive way than is possible using the H_0 parameter of the Standard Model alone. We do this by means of the expansion rate fluctuation field η, an unbiased Gaussian observable that measures deviations from isotropy in the redshift-distance relation. We show that an expansion of η in terms of covariant cosmographic parameters, both kinematic (expansion rate ℍ_o, deceleration ℚ_o and jerk 𝕁_o) and geometric (curvature ℝ_o), allows for a consistent description of metric fluctuations even in a very local and strongly anisotropic universe. The covariant cosmographic parameters critically depend on the observer's state of motion. We thus show how the lower order multipoles of η_ℓ (ℓ≤ 4), measured by a generic observer in an arbitrary state of motion can be used to disentangle expansion effects that are induced by observer's motion from those sourced by pure metric fluctuations. We test the formalism using analytical, axis-symmetric toy models which simulate large-scale linear fluctuations in the redshift-distance relation in the local Universe and which are physically motivated by available observational evidences. We show how to exploit specific features of η to detect the limit of validity of a covariant cosmographic expansion in the local Universe, and to define the region where data can be meaningfully analyzed in a model-independent way, for cosmological inference. We also forecast the precision with which future data sets, such as ZTF, will constrain the structure of the expansion rate anisotropies in the local spacetime