From Local Velocities to Microwave Background

The mass density field as extracted from peculiar velocities in our cosmological neighborhood is mapped back in time to the CMB in two ways. First, the density power spectrum ($P_k$) is translated into a temperature angular power spectrum of sub-degree resolution ($C_l$) and compared to observations. Second, the local density field is translated into a temperature map in a patch on the last-scattering surface of a distant observer. A likelihood analysis of the Mark III peculiar velocity data have constrained the range of parameters for $P_k$ within the family of COBE-normalized CDM models (Zaroubi et al 1996), favoring a slight tilt in the initial spectrum, $n<1$. The corresponding range of $C_l$'s is plotted against current observations, indicating that the CMB data can tighten the constraints further: only models with ``small'' tilt ($n\sim 0.9$) and ``high'' baryonic content ($Ω_b \sim 0.1$) could survive the two data sets simultaneously. The local mass density field that has been recovered from the velocities via a Wiener method is convolved with a Boltzmann calculation to recover $10'$ resolution temperature maps as viewed from different directions. The extent of the CMB patch and the amplitude of fluctuations depend on the choice of cosmological parameters, e.g., the local $100\hmpc$ sphere corresponds to $90'$ to $30'$ at the CMB for $Ω$ between 1 and 0 respectively. The phases of the temperature map are correlated with those of the density field, contrary to the contribution of the Sachs-Wolfe effect alone. This correlation suggests the possibility of an inverse reconstruction of the underlying density field from CMB data with interesting theoretical implications.