Time delay effect on the cosmic background radiation from static gravitational potential of clusters

We present a quantitative analysis of the time delay effect of the static gravitational potential of galaxy clusters on the cosmic background radiation (CBR). This is primarily motivated by growing observational evidence that clusters have essentially experienced no evolution since redshift z approximate to 1, indicating that the contribution of a time-dependent potential to the CBR anisotropy discussed in the literature could be rather small for 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 Sunyaev-Zeldovich (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 similar to 10% 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.