How Cosmic Background Correlations at Large Angles Relate to Mass Autocorrelations in Space

Abstract

The Sachs-Wolfe effect is known to produce large angular scale fluctuations in the Cosmic Microwave Background Radiation (CMBR) due to gravitational potential fluctuations. We show how the angular correlation function of the CMBR can be expressed explicitly in terms of the mass autocorrelation function $ξ(r)$ in the Universe. We derive analytic expressions for the angular correlation function and its multipole moments in terms of integrals over $ξ(r)$ or its second moment, $J_3 (r)$, which does not need to satisfy the sort of integral constraint that $ξ(r)$ must. We derive similar expressions for bulk flow velocity in terms of $ξ$ and $J_3$. One interesting result that emerges directly from this analysis is that, for angles $θ$, there is a substantial contribution to the correlation function from a wide range of distance $r$ and that the radial shape of this contribution does not vary greatly with angle.