Lensing corrections to features in the angular two-point correlation function and power spectrum

Abstract

It is well known that magnification bias, the modulation of galaxy or quasar source counts by gravitational lensing, can change the observed angular correlation function. We investigate magnification-induced changes to the shape of the observed correlation function $w(\ensuremath{\theta})$, and the angular power spectrum ${C}_{\ensuremath{\ell}}$, paying special attention to the matter-radiation equality peak and the baryon wiggles. Lensing effectively mixes the correlation function of the source galaxies with that of the matter correlation at the lower redshifts of the lenses distorting the observed correlation function. We quantify how the lensing corrections depend on the width of the selection function, the galaxy bias $b$, and the number count slope $s$. The lensing correction increases with redshift and larger corrections are present for sources with steep number count slopes and/or broad redshift distributions. The most drastic changes to ${C}_{\ensuremath{\ell}}$ occur for measurements at high redshifts ($z\ensuremath{\gtrsim}1.5$) and low multipole moment ($\ensuremath{\ell}\ensuremath{\lesssim}100$). For the source distributions we consider, magnification bias can shift the location of the matter-radiation equality scale by 1%--6% at $z\ensuremath{\sim}1.5$ and by $z\ensuremath{\sim}3.5$ the shift can be as large as 30%. The baryon bump in ${\ensuremath{\theta}}^{2}w(\ensuremath{\theta})$ is shifted by $\ensuremath{\lesssim}1%$ and the width is typically increased by $\ensuremath{\sim}10%$. Shifts of $\ensuremath{\gtrsim}0.5%$ and broadening $\ensuremath{\gtrsim}20%$ occur only for very broad selection functions and/or galaxies with $(5s\ensuremath{-}2)/b\text{ }\ensuremath{\gtrsim}\text{ }2$. However, near the baryon bump the magnification correction is not constant but is a gently varying function which depends on the source population. Depending on how the $w(\ensuremath{\theta})$ data is fitted, this correction may need to be accounted for when using the baryon acoustic scale for precision cosmology.