The Flat Sky Approximation to Galaxy Number Counts

We derive and test an approximation for the angular power spectrum of galaxy number counts in the flat sky limit. The standard density and redshift space distortion (RSD) terms in the resulting approximation are distinct to the Limber approximation, providing an accurate result for multipoles as low as $\ell\simeq10$, where the corresponding Limber approximation is completely inaccurate. At equal redshift the accuracy of the density and RSD (standard) terms is around 0.2% for $z<3$ and 0.5% at $z=5$, even to $\ell<50$. At unequal redshifts, if we consider the total power spectrum, the precision is better than 5% only for very small redshift differences, $\delta <\delta_0 (\simeq 3.6\times10^{-4}(1+z)^{2.14})$ where the standard terms are well-approximated, or for large enough redshift differences $\delta >\delta_1 (\simeq 0.33(r(z)H(z))/(z+1))$ where the lensing terms dominate. The flat sky expressions for the pure lensing and the lensing-density cross-correlation terms are equivalent to the Limber approximation. For arbitrary redshift differences, the Limber approximation achieves an accuracy of 0.5% (above $\ell\simeq 40$ for pure lensing and $\ell\simeq 80$ for density-lensing). Besides being very accurate, the flat sky approximation is computationally much simpler and can therefore be very useful for data analysis and forecasts with MCMC methods. This will be particularly crucial for upcoming galaxy surveys that will measure the power spectrum of galaxy number counts.