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Time Delay in Robertson-McVittie Spacetime and its Application to Increase of Astronomical Unit

We investigated the light propagation by means of the Robertson-McVittie solution which is considered to be the spacetime around the gravitating body embedded in the FLRW (Friedmann-Lema{î}tre-Robertson-Walker) background metric. We concentrated on the time delay and derived the correction terms with respect to the Shapiro's formula. To relate with the actual observation and its reduction process, we also took account of the time transformations; coordinate time to proper one, and conversely, proper time to coordinate one. We applied these results to the problem of increase of astronomical unit reported by Krasinsky and Brumberg (2004). However, we found the influence of the cosmological expansion on the light propagation does not give an explanation of observed value, $d{\rm AU}/dt = 15 \pm 4$ [m/century] in the framework of Robertson-McVittie metric.