Impact and measurability of linear relativistic effects in galaxy surveys

The three-dimensional galaxy power spectrum is a powerful probe of primordial non-Gaussianity (PNG) and additional general relativistic (GR) effects on large scales, which can be constrained by current and upcoming large-scale structure surveys. In this work, we forecast the measurability of local PNG and linear-order relativistic effects in the spherical Fourier-Bessel (SFB) power spectrum for DESI, Euclid, and SPHEREx surveys. A Chebyshev-decomposition scheme is employed to accelerate the multi-tracer SFB power-spectrum calculations. Fisher forecasts establish baseline constraints and test the sensitivity of $f_{\rm NL}$ constraints to SFB mode cuts and to the marginalization over primordial cosmological parameters, while simulated Bayesian inference is used to quantify the impact and measurability of relativistic effects. We find that neglecting GR effects can bias $f_{\rm NL}$ constraints at the $1$-$3\sigma$ level for Euclid and SPHEREx. The degeneracy between GR and PNG terms is strongly tracer dependent, with the degradation of $\sigma(f_{\rm NL})$ ranging from a few percent to nearly a factor of two when the GR amplitudes are varied. Lensing can be detected at high significance for several tracers, while multi-tracer analyses substantially improve the measurability of the Doppler term. Assuming GR, we show that relativistic clustering partially breaks the exact $b_\phi f_{\rm NL}$ product degeneracy present in Newtonian linear power-spectrum analyses, although the resulting $b_\phi$ constraints remain weak for the survey configurations considered. The joint PNG-GR inference consistently propagates uncertainty in $b_\phi$ into the marginalized $f_{\rm NL}$ constraint. This firmly establishes the path toward extracting cosmological information from ultra-large-scale galaxy clustering.