  The fluctuating Gunn-Peterson approximation (FGPA) is a commonly-used method
to generate mock Lyman-$\alpha$ (Ly$\alpha$) forest absorption skewers at
Cosmic Noon ($z\gtrsim 2$) from the matter-density field of $N$-body
simulations without running expensive hydrodynamical simulations. Motivated by
recent developments in 3D IGM tomography observations as well as matter density
field reconstruction techniques applied to galaxy redshift samples at $z\sim
2$, we examine the possibility of observationally testing FGPA by directly
examining the relationship between the Ly$\alpha$ transmission and the
underlying matter density field. Specifically, we analyze the EAGLE, Illustris,
IllustrisTNG and Nyx cosmological hydrodynamic simulations, that were run with
different codes and sub-grid models. While the FGPA is an excellent description
of the IGM in lower-density regions, the slope of the transmission-density
distribution at higher densities is significantly affected by feedback
processes causing the FGPA to break down in that regime. Even without added
feedback, we find significant deviations caused by hydrodynamical effects
arising from non-linear structure growth. We then proceed to make comparisons
using realistic mock data assuming the sightline sampling and spectral
properties of the recent CLAMATO survey, and find that it would be challenging
to discern between the FGPA and hydrodynamical models with current data sets.
However, the improved sightline sampling from future extremely large telescopes
or large volumes from multiplexed spectroscopic surveys such as Subaru PFS
should allow for stringent tests of the FGPA, and make it possible to detect
the effect of galaxy feedback on the IGM.
