Lagrangian or Eulerian; Real or Fourier? Not All Approaches to
  Large-Scale Structure Are Created Equal
  We present a pedagogical systematic investigation of the accuracy of Eulerian
and Lagrangian perturbation theories of large-scale structure. We show that
significant differences exist between them especially when trying to model the
Baryon Acoustic Oscillations (BAO). We find that the best available model of
the BAO in real space is the Zel'dovich Approximation (ZA), giving an accuracy
of <~3% at redshift of z=0 in modelling the matter 2-pt function around the
acoustic peak. All corrections to the ZA around the BAO scale are perfectly
perturbative in real space. Any attempt to achieve better precision requires
calibrating the theory to simulations because of the need to renormalize those
corrections. In contrast, theories which do not fully preserve the ZA as their
solution, receive O(1) corrections around the acoustic peak in real space at
z=0, and are thus of suspicious convergence at low redshift around the BAO. As
an example, we find that a similar accuracy of 3% for the acoustic peak is
achieved by Eulerian Standard Perturbation Theory (SPT) at linear order only at
z~4. Thus even when SPT is perturbative, one needs to include loop corrections
for z<~4 in real space. In Fourier space, all models perform similarly, and are
controlled by the overdensity amplitude, thus recovering standard results.
However, that comes at a price. Real space cleanly separates the BAO signal
from non-linear dynamics. In contrast, Fourier space mixes signal from short
mildly non-linear scales with the linear signal from the BAO to the level that
non-linear contributions from short scales dominate. Therefore, one has little
hope in constructing a systematic theory for the BAO in Fourier space.
