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 g ravitational potential of galaxy clusters on the cosmic background radiatio n (CBR). This is primarily motivated by growing observational evidence that clusters have essentially experienced no evolution since redshift z approx imate 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 a nd the universal density profile for the mass distribution of rich clusters , we calculate the CBR anisotropy by the time delay effect and compare it w ith those generated by the thermal and kinematic Sunyaev-Zeldovich (S-Z) ef fects, as well as by the transverse motion of clusters. While it is unlikel y 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 n evertheless leads to an uncertainty of similar to 10% in the measurement of the kinematic S-Z effect of clusters. Future cosmological application of t he peculiar velocity of clusters to be measured through the S-Z effect shou ld therefore take this uncertainty into account.