Primordial non-Gaussianity without tails -- how to measure fNL with the
  bulk of the density PDF
  We investigate the possibility to detect primordial non-Gaussianity by
analysing the bulk of the probability distribution function (PDF) of late-time
cosmic density fluctuations. For this purpose we devise a new method to predict
the impact of general non-Gaussian initial conditions on the late-time density
PDF. At redshift $z=1$ and for a smoothing scale of 30Mpc/$h$ our predictions
agree with the high-resolution Quijote N-body simulations to $\sim 0.2\%$
precision. This is within cosmic variance of a $\sim 100(\mathrm{Gpc}/h)^3$
survey volume. When restricting to this 30Mpc/$h$ smoothing scale and to mildly
non-linear densities ($\delta[30\mathrm{Mpc}/h] \in [-0.3, 0.4]$) and also
marginalizing over potential ignorance of the amplitude of the non-linear power
spectrum an analysis of the PDF for such a survey volume can still measure the
amplitude of different primordial bispectrum shapes to an accuracy of
\smash{$\Delta f_{\mathrm{NL}}^{\mathrm{loc}}=\pm 7.4\ ,\ \Delta
f_{\mathrm{NL}}^{\mathrm{equi}}=\pm 22.0\ ,\ \Delta
f_{\mathrm{NL}}^{\mathrm{ortho}}=\pm 46.0$} . When pushing to smaller scales
and assuming a joint analysis of the PDF with smoothing radii of 30Mpc/$h$ and
15Mpc/$h$ ($\delta[15\mathrm{Mpc}/h] \in [-0.4, 0.5]$) this improves to
\smash{$\Delta f_{\mathrm{NL}}^{\mathrm{loc}}=\pm 3.3\ ,\ \Delta
f_{\mathrm{NL}}^{\mathrm{equi}}=\pm 11.0\ ,\ \Delta
f_{\mathrm{NL}}^{\mathrm{ortho}}=\pm 17.0\ $} - even when marginalizing over
the non-linear variances at both scales as two free parameters. Especially,
such an analysis could simultaneously measure $f_{\mathrm{NL}}$ and the
amplitude and slope of the non-linear power spectrum. However, at 15Mpc/$h$ our
predictions are only accurate to $\lesssim 0.8\%$ for the considered density
range. We discuss how this has to be improved in order to push to these small
scales and make full use of upcoming surveys with a PDF-based analysis.
