In this paper we derive and test the flat sky approximation for galaxy number counts. We show that, while for the lensing term it reduces to the Limber approximation, for the standard density and redshift space distortion it is different and very accurate already at low ell while the corresponding Limber approximation completely fails. At equal redshift the accuracy of the standard terms is around 0.2% at low redshifts and 0.5% for redshift z=5, even to low ell. At unequal redshifts the precision is less impressive and can only be trusted for very small redshift differences,
\Delta z<\Delta z_0
\simeq 3.6\times10^{-4}(1+z)^{2.14}
|
, but the lensing terms dominate for
\Delta z>\Delta z_1 \simeq 0.33(r(z)H(z))/(z+1)
|
. The Limber approximation achieves an accuracy of 0.5% above
for the pure lensing term and above
for the lensing-density cross-correlation. Besides being very accurate, the flat sky approximation is also very fast and can therefore be useful for data analysis and forecasts with MCMC methods.
