Can we actually constrain $f_{\rm NL}$ using the scale-dependent bias effect? An illustration of the impact of galaxy bias uncertainties using the BOSS DR12 galaxy power spectrum

The scale-dependent bias effect on the galaxy power spectrum is a very promising probe of the local primordial non-Gaussianity (PNG) parameter $f_{\rm NL}$, but the amplitude of the effect is proportional to $f_{\rm NL}b_{\phi}$, where $b_{\phi}$ is the linear PNG galaxy bias parameter. Our knowledge of $b_{\phi}$ is currently very limited, yet nearly all existing $f_{\rm NL}$ constraints and forecasts assume precise knowledge for it. Here, we use the BOSS DR12 galaxy power spectrum to illustrate how our uncertain knowledge of $b_{\phi}$ currently prevents us from constraining $f_{\rm NL}$ with a given statistical precision $\sigma_{f_{\rm NL}}$. Assuming different fixed choices for the relation between $b_{\phi}$ and the linear density bias $b_1$, we find that $\sigma_{f_{\rm NL}}$ can vary by as much as an order of magnitude. Our strongest bound is $f_{\rm NL} = 16 \pm 16\ (1\sigma)$, while the loosest is $f_{\rm NL} = 230 \pm 226\ (1\sigma)$ for the same BOSS data. The impact of $b_{\phi}$ can be especially pronounced because it can be close to zero. We also show how marginalizing over $b_{\phi}$ with wide priors is not conservative, and leads in fact to biased constraints through parameter space projection effects. Independently of galaxy bias assumptions, the scale-dependent bias effect can only be used to detect $f_{\rm NL} \neq 0$ by constraining the product $f_{\rm NL}b_{\phi}$, but the error bar $\sigma_{f_{\rm NL}}$ remains undetermined and the results cannot be compared with the CMB; we find $f_{\rm NL}b_{\phi} \neq 0$ with $1.6\sigma$ significance. We also comment on why these issues are important for analyses with the galaxy bispectrum. Our results strongly motivate simulation-based research programs aimed at robust theoretical priors for the $b_{\phi}$ parameter, without which we may never be able to competitively constrain $f_{\rm NL}$ using galaxy data.