Cross‐correlating Cosmic Microwave Background Radiation Fluctuations with Redshift Surveys: Detecting the Signature of Gravitational Lensing

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Abstract
Density inhomogeneities along the line of sight distort fluctuations in the cosmic microwave background. Usually, this effect is thought of as a small second-order effect that mildly alters the statistics of the microwave background fluctuations. We show that there is a first-order effect that is potentially observable if we combine microwave background maps with large redshift surveys. We introduce a new quantity that measures this lensing effect, T(δθ ∇ T), where T is the microwave background temperature and δθ is the lensing due to matter in the region probed by the redshift survey. We show that the expected signal is first order in the gravitational lensing bending angle, (δθ)21/2, and find that it should be easily detectable, signal-to-noise ratio ~ 15-35, if we combine the Microwave Anisotropy Probe satellite and Sloan Digital Sky Survey data. Measurements of this cross-correlation will directly probe the "bias" factor, i.e., the relationship between fluctuations in mass and fluctuations in galaxy counts.