Cosmography for a General Spacetime Centred at Arbitrary Redshift

With upcoming surveys providing large volumes of highly precise observational data across a wide range of redshifts, it is increasingly important to have tools that can translate observational data into geometric and dynamical information without imposing a predetermined cosmological model. General cosmographic expansions centred at arbitrary redshift provide exactly such a tool. We here present the formalism for general cosmographic expansions centred at an arbitrary redshift, valid for 4-dimensional Lorentzian spacetimes. We then apply the expansion formalism to test its ability to reproduce the redshift-distance relation in two examples of large-scale cosmic structures (an underdensity and an overdensity) modelled by the Lema^itre-Tolman-Bondi metric, where we examine the effect of choosing different redshift intervals for the cosmographic series expansions. This quantifies the extent to which cosmographic coefficients inferred from redshift-distance observations retain their interpretation as local geometric and dynamical quantities, as is expected in standard FLRW cosmology. Similarly to earlier results, we here find that in more general spacetimes the coefficients instead become effective parameters reflecting the finite observational range probed. Lastly, we discuss possible strategies for using the expansions to constrain dynamical and geometric quantities.