Magnification-temperature correlation: The dark side of integrated Sachs-Wolfe measurements

Abstract

Integrated Sachs-Wolfe (ISW) measurements, which involve cross-correlating the microwave background anisotropies with the foreground large-scale structure (e.g. traced by galaxies/quasars), have proven to be an interesting probe of dark energy. We show that magnification bias, which is the inevitable modulation of the foreground number counts by gravitational lensing, alters both the scale dependence and amplitude of the observed ISW signal. This is true especially at high redshifts because (1) the intrinsic galaxy-temperature signal diminishes greatly back in the matter-dominated era, (2) the lensing efficiency increases with redshift and (3) the number count slope generally steepens with redshift in a magnitude limited sample. At $z\ensuremath{\gtrsim}2$, the magnification-temperature correlation dominates over the intrinsic galaxy-temperature correlation and causes the observed ISW signal to increase with redshift, despite dark energy subdominance---a result of the fact that magnification probes structures all the way from the observer to the sources. Ignoring magnification bias therefore can lead to (significantly) erroneous conclusions about dark energy. While the lensing modulation opens up an interesting high $z$ window for ISW measurements, high redshift measurements are not expected to add much new information to low redshift ones if dark energy is indeed the cosmological constant. This is because lensing introduces significant covariance across redshifts. The most compelling reasons for pursuing high redshift ISW measurements are to look for potential surprises such as early dark energy domination or signatures of modified gravity. We conclude with a discussion of existing measurements, the highest redshift of which is at the margin of being sensitive to the magnification effect. We also develop a formalism which might be of more general interest: to predict biases in estimating parameters when certain physical effects are ignored in interpreting observations.