The Nonlinear Evolution of Rare Events

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.