Application of time transfer function to McVittie spacetime: gravitational time delay and secular increase in astronomical unit

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-LemaItre-Robertson-Walker) cosmological background metric. To this end, we adopt the time transfer function method proposed by Le Poncin-Lafitte et al. (Class Quantum Gravity 21:4463, 2004) and Teyssandier and Le Poncin-Lafitte (Class Quantum Gravity 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 dAU/dt, i.e., 15 +/- 4 (m/century). Therefore, it is not possible to explain the secular increase in the astronomical unit in terms of cosmological expansion.