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

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