Measuring $f_{\mathrm{NL}}$ with the SPHEREx Multi-tracer Redshift Space Bispectrum

The bispectrum is an important statistics helpful for measuring the primordial non-Gaussianity parameter $f_{\mathrm{NL}}$ to less than order unity in error, which would allow us to distinguish between single and multi-field inflation models. The Spectro-Photometer for the History of the Universe, Epoch of Reionization and Ices Explorer (SPHEREx) mission is particularly well-suited for making this measurement with its $\sim$100-band all-sky observations in the near-infrared. Consequently, the SPHEREx data will contain galaxies with spectroscopic-like redshift measurements as well as those with much larger errors. In this paper, we evaluate the impact of photometric redshift errors on $f_{\mathrm{NL}}$ constraints in the context of an updated multi-tracer forecast for SPHEREx, finding that the azimuthal averages of the first three even bispectrum multipoles are no longer sufficient for capturing most of the information (as opposed to the case of spectroscopic surveys shown in the literature). The final SPHEREx result with all five galaxy samples and six redshift bins is however not severely impacted because the total result is dominated by the samples with the best redshift errors, while the worse samples serve to reduce cosmic variance. Our fiducial result of $\sigma_{f_{\mathrm{NL}}} = 0.7$ from bispectrum alone is increased by $18\%$ and $3\%$ when using $l_{\mathrm{max}}=0$ and 2 respectively. We also explore the impact on parameter constraints when varying the fiducial redshift errors, as well as using subsets of multi-tracer combinations or triangles with different squeezing factors. Note that the fiducial result here is not the final SPHEREx capability, which is still on target for being $\sigma_{f_{\mathrm{NL}}} = 0.5$ once the power spectrum will be included.