Celestial Ephemerides in an Expanding Universe

TL;DR

This paper extends celestial dynamics theory to an expanding universe, deriving cosmological corrections to light propagation and proposing a method to measure the Hubble constant locally within the solar system.

Contribution

It formulates the Newtonian limit of Einstein's equations in an expanding universe and identifies missing cosmological terms in current light propagation models.

Findings

Equations of motion reduce to classical Newtonian form with conformal transformations.

Light propagation equations miss terms proportional to the Hubble constant.

Cosmological corrections enable local measurement of the Hubble constant.

Abstract

Post-Newtonian theory was instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the solar system with an unparalleled precision. The cornerstone of the theory is the postulate that the solar system is gravitationally isolated from the rest of the universe and the background spacetime is asymptotically flat. The present article extends this theoretical concept and formulates the principles of celestial dynamics of particles and light moving in gravitational field of a localized astronomical system embedded to the expanding Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. We formulate the precise mathematical concept of the Newtonian limit of Einstein's field equations in the conformally-flat FLRW spacetime and analyze the geodesic motion of massive particles and light in this limit. We prove that by doing conformal spacetime transformations, one can reduce the equations of motion of particles and light to the classical form of the Newtonian theory. However, the time arguments in the equations of motion of particles and light differ from each other in terms being proportional to the Hubble constant, H. This leads to the important conclusion that the equations of light propagation used currently by Space Navigation Centers for fitting range and Doppler-tracking observations of celestial bodies are missing some terms of the cosmological origin that are proportional to the Hubble constant, H. We also prove that the Hubble expansion does not affect the atomic time scale used in creation of astronomical ephemerides. We derive the cosmological correction to the light travel time equation and argue that their measurement opens an exciting opportunity to determine the local value of the Hubble constant, H, in the solar system independently of cosmological observations.