The flat sky approximation to galaxy number counts

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