On the equivalence between galaxy angular correlation function and power spectrum in constraining primordial non-Gaussianity

We investigate the angular power spectrum ($C_\ell)$ and angular correlation function ($w(\theta)$) of galaxy number density field in the presence of the local-type primordial non-Gaussianity (PNG), explicitly accounting for the integral constraint in an all-sky survey. We show that the PNG signature in $C_{\ell}$ is confined to low multipoles in the linear regime, whereas its signature in $w(\theta)$ extends across a wide range of angular scales, including those below the nonlinear scale. Therefore, the equivalence between $C_\ell$ and $w(\theta)$ can be violated when scale cuts of multipoles or angular scales -- for example, to mitigate systematic effects -- are applied in the analysis. Assuming samples of photometric galaxies divided into multiple redshift bins in the range $0<z<7$, we forecast the precision of constraining the PNG parameter ($f_{\rm NL}$) from the hypothetical measurements of $C_\ell$ or $w(\theta)$ assuming different scale cuts in the multipoles or angular scales, respectively. Our results imply that the PNG information can be extracted from $w(\theta)$ on relatively small angular scales such as $\lesssim 10$ degree for a high-redshift galaxy sample or from $w(\theta)$ measured in a survey with partial area coverage.