Time-Delay Effect on the cOsmic Background Radiation by Static Gravitational Potential of Clusters

We present a quantitative analysis of the time-delay effect on the cosmic background radiation (CBR) by static gravitational potential of galaxy clusters. This is primarily motivated by growing observational evidence that clusters have essentially experienced no-evolution since redshift $z\approx1$, indicating that the contribution of a time-dependent potential to CBR anisotropy discussed in literature could be rather small for the dynamically-relaxed clusters. Using the softened isothermal sphere model and the universal density profile for the mass distribution of rich clusters, we calculate the CBR anisotropy by the time-delay effect and compare it with those generated by the thermal and kinematic S-Z effects as well as by the transverse motion of clusters. While it is unlikely that the time-delay effect is detectable in the current S-Z measurement because of its small amplitude of $10^{-6}$-$10^{-7}$ and its achromaticity, it nevertheless leads to an uncertainty of $\sim10$ in the measurement of the kinematic S-Z effect of clusters. Future cosmological application of the peculiar velocity of clusters to be measured through the S-Z effect should therefore take this uncertainty into account.