Anisotropies of Cosmic Background Radiation from a Local Collapse

We present an exact solution of the anisotropies of cosmic background radiation (CBR) from a local collapse described by a spherical over-dense region embedded in a flat universe, with the emphasis on the relationship between the dipole $(\Delta {\sf T}/{\sf T})_d$ and the quadrupole $(\Delta {\sf T}/{\sf T})_q$ anisotropy. This result has been used to examine the kinematic quadrupole correction $(\Delta {\sf T}/{\sf T})_q=(\Delta {\sf T}/{\sf T})_d^2/2$, which is usually applied to remove the contamination of the quadrupole produced by local density inhomogeneities when finding the cosmic amplitude of the quadrupole at the surface of last scattering. We have found that the quadrupole of local collapse origin cannot always be approximately described by the kinematic quadrupole. Our numerical result shows that the difference between the kinematic and local quadrupoles depends on the size and matter density in the peculiar field, and the position of the observer. For a given dipole, the local quadrupole can be different from the kinematic quadrupole by a factor as large as 3. Therefore, the kinematic quadrupole correction remains an uncertain factor in the determination of the amplitude of a cosmic quadrupole. Nevertheless, a preliminary analysis shows that this uncertainty might not dramatically change the cosmological origin of the COBE-DMR's quadrupole, unless a huge peculiar gravitational field is assumed.