Abstract
Abstract We study the non-linear gravitational clustering of collisionless particles in an expanding background using an integro-differential equation for the gravitational potential. In particular, we address the question of how the non-linear mode–mode coupling transfers power from one scale to another in the Fourier space if the initial power spectrum is sharply peaked at a given scale. We show that the dynamical equation allows self-similar evolution for the gravitational potential φk(t) in Fourier space of the form φk(t) = F(t)D(k) where the function F(t) satisfies a second-order non-linear differential equation. We analyse the relevant solutions of this equation, thereby determining the asymptotic time evolution of the gravitational potential and density contrast. The analysis suggests that both F(t) and D(k) have well-defined asymptotic forms indicating that the power transfer leads to a universal power spectrum at late times. The analytic results are compared with numerical simulations, showing good agreement over the range at which we could test them.
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