  One way of recovering information about the initial conditions of the
Universe is by measuring features of the cosmological density field which are
preserved during gravitational evolution and galaxy formation. In this paper we
study the total number density of peaks in a (galaxy) point distribution
smoothed with a filter, evaluating its usefulness as a means of inferring the
shape of the initial (matter) power spectrum. We find that in numerical
simulations which start from Gaussian initial conditions, the peak density
follows well that predicted by the theory of Gaussian density fields, even on
scales where the clustering is mildly non-linear. For smaller filter scales, $r
\simlt 4-6 \hmpc$, we see evidence of merging as the peak density decreases
with time. On larger scales, the peak density is independent of time. One might
also expect it to be fairly robust with respect to variations in biasing, i.e.
the way galaxies trace mass fluctuations. We find that this is the case when we
apply various biasing prescriptions to the matter distribution in simulations.
If the initial conditions are Gaussian, it is possible to use the peak density
measured from the evolved field to reconstruct the shape of the initial power
spectrum. We describe a stable method for doing this and apply it to several
biased and unbiased non-linear simulations. We are able to recover the slope of
the linear matter power spectrum on scales $k \simlt 0.4 \hmpc^{-1}$. The
reconstruction has the advantage of being independent of the cosmological
parameters ($\Omega$, $\Lambda$, $H_0$) and of the clustering normalisation
($\sigma_8$). The peak density and reconstructed power spectrum slope therefore
promise to be powerful discriminators between popular cosmological scenarios.
