Effect of inhomogeneity of the Universe on a gravitationally bound local system: A no-go result for explaining the secular increase in the astronomical unit

TL;DR

This paper investigates whether cosmic inhomogeneity, modeled by the LTB solution, can explain the observed secular increase in the astronomical unit, concluding it cannot account for this phenomenon.

Contribution

The study derives the leading order effects of LTB inhomogeneity on planetary motion and shows these effects are insufficient to explain the astronomical unit increase.

Findings

Inhomogeneity effects are negligible for AU increase.

Both homogeneous and inhomogeneous models produce similar perturbations.

Inhomogeneity cannot explain the observed secular increase in AU.

Abstract

We will investigate the influence of the inhomogeneity of the universe, especially that of the Lema{\^i}tre-Tolman-Bondi (LTB) model, on a gravitationally bound local system such as the solar system. We concentrate on the dynamical perturbation to the planetary motion and derive the leading order effect generated from the LTB model. It will be shown that there appear not only a well-known cosmological effect arisen from the homogeneous and isotropic model, such as the Robertson-Walker (RW) model, but also the additional terms due to the radial inhomogeneity of the LTB model. We will also apply the obtained results to the problem of secular increase in the astronomical unit, reported by Krasinsky and Brumberg (2004), and imply that the inhomogeneity of the universe cannot have a significant effect for explaining the observed $d{\rm AU}/dt = 15 \pm 4 ~{\rm [m/century]}$.