Unbiased analysis of primordial non-Gaussianity: the multipoles of the full relativistic power spectrum

A major goal of ongoing and future cosmological surveys of the large-scale structure is to measure local type primordial non-Gaussianity in the galaxy power spectrum through the scale-dependent bias. General relativistic effects have been shown to be degenerate with this measurement, therefore requiring a non-Newtonian approach. In this work, we develop a consistent framework to compute integrated effects, including lensing convergence, time delay, and integrated Sachs–Wolfe, along with the local relativistic projection and wide-separation corrections in the multipoles of the power spectrum. We show that, for a $\mathit{\text{Euclid}}$-like H$\alpha$-line galaxy survey and a MegaMapper-like Lyman-break galaxy survey, ignoring these effects leads to a bias on the best fit measurement of the amplitude of primordial non-Gaussianity, $f_{\mathrm{N}\mathrm{L}}$, of around $3\sigma$ and $20\sigma$ respectively. When we include these corrections, the uncertainty in our knowledge of the luminosity function leads to further uncertainty in our measurement of $f_{\mathrm{N}\mathrm{L}}$. In this work, we show that this degeneracy can be partly mitigated by using a bright-faint multi-tracer analysis, where the observed galaxy sample is subdivided into two separate populations based on luminosity, which provides a $15$–$20\mathrm{\%}$ improvement on the forecasted constraints of local type $f_{\mathrm{N}\mathrm{L}}$. In addition, we present a novel calculation of the full multi-tracer covariance with the inclusion of wide-separation corrections~-- all of these results are implemented in the $\mathit{\text{Python}}$ code $\text{textsc}CosmoWAP$.