The anisotropy of the cosmic background radiation due to mass clustering in a closed universe

We consider the effects on the Cosmic Background Radiation (CBR) of a single, spherically symmetric structure using the Tolman-Bondi solutions in a closed universe. The exact frequency shift of a photon passing through the center of a lump which is located between the last scattering surface and an observer is calculated. An approximate expression of the frequency shift Delta v(0)/v(0) to the first order of delta(x) (the density contrast) is given. We find that there are several redshifts of the perturbing lump where the net frequency shift is zero. With increasing Omega, the first positions of zero frequency shift move toward lower redshift; the larger the value of Omega, the further they move. At the vicinity of the last scattering surface, where the universe is close to flat, the frequency shift will always be the same as that for a flat universe.