Interpretation of geodesy experiments in non-Newtonian theories of gravity

TL;DR

This paper explores how deviations from Newtonian gravity, modeled by a Yukawa potential, affect geodesy experiments and discusses the implications for testing modified gravity theories using Earth-based and satellite data.

Contribution

It generalizes the Earth's gravitational potential multipolar expansion to include Yukawa interactions, revealing limitations of Newtonian assumptions in geodesy and proposing new constraints on fifth forces.

Findings

Yukawa interactions cause multipolar coefficients to depend on Earth's radius.

Geodesy data limitations restrict testing gravity modifications in space.

Current constraints suggest Yukawa fifth forces are negligible at altitudes above a few hundred kilometers.

Abstract

The tests of the deviations from Newton's or Einstein's gravity in the Earth neighbourhood are tied to our knowledge of the shape and mass distribution of our planet. On the one hand estimators of these "modified" theories of gravity may be explicitly Earth-model-dependent whilst on the other hand the Earth gravitational field would act as a systematic error. We revisit deviations from Newtonian gravity described by a Yukawa interaction that can arise from the existence of a finite range fifth force. We show that the standard multipolar expansion of the Earth gravitational potential can be generalised. In particular, the multipolar coefficients depend on the distance to the centre of the Earth and are therefore not universal to the Earth system anymore. This offers new ways of constraining such Yukawa interactions and demonstrates explicitly the limits of the Newton-based interpretation of geodesy experiments. In turn, limitations from geodesy data restrict the possibility of testing gravity in space. The gravitational acceleration is described in terms of spin-weighted spherical harmonics allowing us to obtain the perturbing force entering the Lagrange-Gauss secular equations. This is then used to discuss the correlation between geodesy and modified gravity experiments and the possibility to break their degeneracy. Finally we show that, given the existing constraints, a Yukawa fifth force is expected to be sub-dominant in satellite dynamics and space geodesy experiments, as long as they are performed at altitudes greater than a few hundred kilometres. Gravity surveys will have to gain at least two orders of magnitude in instrumental precision before satellite geodesy could be used to improve the current constraints on modified gravity.