Application of Time Transfer Function to McVittie Spacetime: Gravitational Time Delay and Secular Increase in Astronomical Unit

TL;DR

This paper calculates gravitational time delay in McVittie spacetime using the time transfer function method, examining cosmological effects on light propagation and concluding they cannot explain the observed increase in the astronomical unit.

Contribution

It applies the time transfer function approach to a time-dependent spacetime, providing a new analysis of cosmological effects on solar system light propagation.

Findings

Cosmological expansion effects are 9 orders of magnitude smaller than observed AU increase.

The method circumvents null geodesic integration, simplifying calculations.

Cosmological effects cannot account for the secular increase in the astronomical unit.

Abstract

We attempt to calculate the gravitational time delay in a time-dependent gravitational field, especially in McVittie spacetime, which can be considered as the spacetime around a gravitating body such as the Sun, embedded in the FLRW (Friedmann-Lema\^itre-Robertson-Walker) cosmological background metric. To this end, we adopt the time transfer function method proposed by Le Poncin-Lafitte {\it et al.} (Class. Quant. Grav. 21:4463, 2004) and Teyssandier and Le Poncin-Lafitte (Class. Quant. Grav. 25:145020, 2008), which is originally related to Synge's world function $\Omega(x_A, x_B)$ and enables to circumvent the integration of the null geodesic equation. We re-examine the global cosmological effect on light propagation in the solar system. The round-trip time of a light ray/signal is given by the functions of not only the spacial coordinates but also the emission time or reception time of light ray/signal, which characterize the time-dependency of solutions. We also apply the obtained results to the secular increase in the astronomical unit, reported by Krasinsky and Brumberg (Celest. Mech. Dyn. Astron. 90:267, 2004), and we show that the leading order terms of the time-dependent component due to cosmological expansion is 9 orders of magnitude smaller than the observed value of $d{\rm AU}/dt$, i.e., $15 \pm 4$ ~[m/century]. Therefore, it is not possible to explain the secular increase in the astronomical unit in terms of cosmological expansion.