  We present a comparative study of six different methods for reversing the
gravitational evolution of a cosmological density field to recover the
primordial fluctuations: linear theory, the Gaussianization mapping scheme, two
different quasi-linear dynamical schemes based on the Zel'dovich approximation,
a Hybrid dynamical-Gaussianization method and the Path Interchange Zel'dovich
Approximation (PIZA). 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, the individual modes in
Fourier space, as well as global statistical properties such as the genus, the
PDF of the density, and the distribution of peak heights and their shapes. We
find linear theory to be the most inaccurate of all the schemes. The
Gaussianization scheme is the least accurate after linear theory. The two
quasi-linear dynamical schemes 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. We find this
Hybrid scheme to be more accurate and robust than either Gaussianization or the
dynamical method alone. The PIZA scheme performs substantially better than the
others in all point-by-point comparisons. However, it produces an oversmoothed
initial density field, with a smaller number of peaks than expected, but
recovers the PDF of the initial density with impressive accuracy on scales as
small as 3Mpc/h.
