The gauge-invariant formulation of the local expansion rate driven by the local average density in an inhomogeneous universe

The Hubble tension casts a blight on the standard cosmology. As a possible solution to the problem, the local variation of the expansion rate has been proposed where the spatial averaging over a finite domain was introduced in order to restore the local Friedmannian behavior in an inhomogeneous cosmology. So far, however, the approaches are limited to the particular choices of the gauges, and it has been unclear whether the results are gauge invariant. In this paper, we present the gauge-invariant formulation of the local expansion rate, which is driven by the spatial average of the gauge-invariant inhomogeneous density. We show that the local cosmological parameters in the finite domain may change from the global parameters, and the relations between them are expressed by the gauge-invariant averaged density.