Speeding up the detectability of the harmonic-space galaxy bispectrum

We present a method that allows us for the first time to estimate the signal-to-noise ratio (SNR) of the harmonic-space galaxy bispectrum induced by gravity, a complementary probe to already well established Fourier-space clustering analyses. We show how to do it considering only $\sim1000$ triangle configurations in multipole space, corresponding to a computational speedup of a factor $\mathcal{O}(10^2)-\mathcal{O}(10^3)$, depending on the redshift bin, when including mildly non-linear scales. Assuming observational specifications consistent with forthcoming spectroscopic and photometric galaxy surveys like the Euclid satellite and the Square Kilometre Array (phase 1), we show: that given a single redshift bin, spectroscopic surveys outperform photometric surveys; and that -- due to shot-noise and redshift bin width balance -- bins at redshifts $z\sim1$ bring higher cumulative SNR than bins at lower redshifts $z \sim 0.5$. Our results for the largest cumulative SNR $\sim 15$ suggest that the harmonic-space bispectrum is detectable within narrow ($\Delta z \sim 0.01$) spectroscopic redshift bins even when including only mildly non-linear scales. Tomographic reconstructions and inclusion of highly non-linear scales will further boost detectability with upcoming galaxy surveys. In addition, we discuss how, using the Karhunen-Lo\`eve transform, a detection analysis only requires a $1 \times 1$ covariance matrix for a single redshift bin.