Limits on the Inferred Hubble Constant Bias from a Local McVittie Gravitational Field

Abstract

We examine the exact propagation of light in a McVittie spacetime to quantify the observational bias introduced on the inferred Hubble constant $H_0$ by a localized central mass embedded in an expanding asymptotically de Sitter background. By integrating the Sachs optical equations for radial null geodesics, we derive linearized analytical expressions for the areal and luminosity distances to first order in the dimensionless Hawking-Hayward coupling parameter $\mu =H_{\text{dS}}{m}_H$. We evaluate the magnitude of this local gravitational field effect across realistic astrophysical scales, ranging from supermassive black holes (${\text{Sgr A}}^{\ast }$) to galaxy cluster halos. We show that while a McVittie central mass induces a coordinate-invariant path-focusing effect on incoming photons, the quantitative correction for local galactic parameters ($\mu \sim 10^{-16}$) is negligible ($\sim 10^{-15}\%$), proving that a localized galactic mass alone cannot resolve the $\sim 8\%$ Hubble tension. We establish strict upper bounds on the extent to which local spacetime inhomogeneity can bias low-redshift distance ladder measurements.

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