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.
