Gravitational instability in the strongly non-linear regime: a study of various approximations

We study the development of gravitational instability in the strongly non-linear regime. For this purpose we use a number of statistical indicators such as fillamentary statistics, the spectrum of overdense/underdense regions and the void probability function, each of which probes a particular aspect of gravitational clustering. We use these statistical indicators to discriminate between different approximations to gravitational instability which we test against N-body simulations. The approximations that we test are the truncated Zel'dovich approximation (TZ), the adhesion approximation (AA), and the frozen flow (FF) and linear potential (LP) approximations. Of these we find that FF and LP break down relatively early, soon after the non-linear length scale exceeds R_⋆ – the mean distance between peaks of the gravitational potential. The reason for this breakdown is easy to understand: particles in FF are constrained to follow the streamlines of the initial velocity field. Shell crossing is absent in this case and structure gradually freezes as particles begin to collect near minima of the gravitational potential. In LP, particles follow the lines of force of the primordial potential, oscillating about its minima at late times when the non-hnear length scale $$k^{-1}_{\text {NL}}\simeq R_\ast$$. Unlike FF and LP, the adhesion model (and to some extent TZ) continues to give accurate results even at late times when $$k^{-1}_{\text {NL}}\geqslant R_\ast$$. This is because both AA and TZ use the presence of long-range modes in the gravitational potential to move particles. Thus, as long as the initial potential has sufficient long-range power to initiate large-scale coherent motions, TZ and AA will remain approximately valid. In relation to AA, TZ suffers from a single major drawback - it underestimates the presence of small clumps. Similarly, it predicts the right mean density in large voids but misses subcondensations within them. The reason for this is clear: the artificial removal of power on scales smaller than $$k^{-1}_{\text {NL}}$$ in the initial potential in TZ, designed to prevent shell crossing, causes a substantial fraction of matter (which would have been clustered in N-body simulations) to lie within low-density regions at all epochs. On the other hand, TZ is very fast to implement and more accurately predicts the location of large objects at late times; AA more correctly represents the subcondensations but does not always accurately predict their positions.