Recovering the Primordial Density Fluctuations: A Comparison of Methods

We present a comparative study of different methods for reversing the gravitational evolution of a cosmological density field to recover the primordial fluctuations. We test six different approximate schemes in all: linear theory, the Gaussianization technique of Weinberg, two different quasi-linear dynamical schemes, a hybrid dynamical-Gaussianization method, and the path interchange Zeldovich approximation (PIZA) of Croft & Gaztañaga. The final evolved density field from an N-body simulation constitutes our test case. We use a variety of statistical measures to compare the initial density field recovered from it to the true initial density field, using each of the six different schemes. These include point-by-point comparisons of the density fields in real space, and the individual modes in Fourier space, as well as global statistical properties such as the genus, the probability distribution function (PDF) of the density, and the distribution of peak heights and their shapes. We find linear theory to be substantially less accurate than the other schemes, all of which reverse at least some of the nonlinear effects of gravitational evolution even on scales as small as 3 h−1 Mpc. The Gaussianization scheme, while being robust and easy to apply, is the least accurate after linear theory. The two quasi-linear dynamical schemes, which are based on Eulerian formulations of the Zeldovich approximation, give similar results to each other and are more accurate than Gaussianization, although they break down quite drastically when used outside their range of validity, the quasi-linear regime. The complementary beneficial aspects of the dynamical and the Gaussianization schemes are combined in the hybrid method, which uses a dynamical scheme to account for the bulk displacements of mass elements and corrects for any systematic errors using Gaussianization. We find this reconstruction scheme to be more accurate and robust than either the Gaussianization or dynamical method alone. The final scheme, the PIZA, performs substantially better than the others in all point-by-point comparisons. The PIZA does produce an oversmoothed initial density field, with a smaller number of peaks than expected, but it recovers the PDF of the initial density with impressive accuracy on scales as small as 3 h−1 Mpc.