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We present a pedagogical systematic investigation of the accuracy of Eulerianand Lagrangian perturbation theories of large-scale structure. We show thatsignificant differences exist between them especially when trying to model theBaryon Acoustic Oscillations (BAO). We find that the best available model ofthe BAO in real space is the Zel'dovich Approximation (ZA), giving an accuracyof <~3% at redshift of z=0 in modelling the matter 2-pt function around theacoustic peak. All corrections to the ZA around the BAO scale are perfectlyperturbative in real space. Any attempt to achieve better precision requirescalibrating the theory to simulations because of the need to renormalize thosecorrections. In contrast, theories which do not fully preserve the ZA as theirsolution, receive O(1) corrections around the acoustic peak in real space atz=0, and are thus of suspicious convergence at low redshift around the BAO. Asan example, we find that a similar accuracy of 3% for the acoustic peak isachieved by Eulerian Standard Perturbation Theory (SPT) at linear order only atz~4. Thus even when SPT is perturbative, one needs to include loop correctionsfor z<~4 in real space. In Fourier space, all models perform similarly, and arecontrolled by the overdensity amplitude, thus recovering standard results.However, that comes at a price. Real space cleanly separates the BAO signalfrom non-linear dynamics. In contrast, Fourier space mixes signal from shortmildly non-linear scales with the linear signal from the BAO to the level thatnon-linear contributions from short scales dominate. Therefore, one has littlehope in constructing a systematic theory for the BAO in Fourier space.

Lagrangian or Eulerian; Real or Fourier? Not All Approaches to Large-Scale Structure Are Created Equal