  In this paper I consider the nonlinear evolution of a rare density
fluctuation in a random density field with Gaussian fluctuations, and I
rigorously show that it follows the spherical collapse dynamics applied to its
mean initial profile. This result is valid for any cosmological model and is
independent of the shape of the power spectrum. In the early stages of the
dynamics the density contrast of the fluctuation is seen to follow with a good
accuracy the form $$\delta=(1-\delta_L/1.5)^{-1.5}-1,$$ where $\delta_L$ is the
linearly extrapolated overdensity. I then investigate the validity domain of
the rare event approximation in terms of the parameter $\nu=\delta_L/\sigma$
giving the initial overdensity scaled by the rms fluctuation at the same mass
scale, and find that it depends critically on the shape of the power spectrum.
When the power law index $n$ is lower than $-1$ the departure from the
spherical collapse is expected to be small, at least in the early stages of the
dynamics, and even for moderate values of $\nu$ ($\vert\nu\vert\ge 2$).
However, for $n\ge -1$ and whatever the value of $\nu$, the dynamics seems to
be dominated by the small--scale fluctuations and the subsequent evolution of
the peak may not be necessarily correlated to its initial overdensity. I
discuss the implications of these results for the nonlinear dynamics and the
formation of astrophysical objects.
