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Weakly nonlinear Gaussian fluctuations and the Edgeworth expansion

Sep 1, 1993
39 pages
Published in:
  • Astrophys.J. 442 (1995) 39
e-Print:
Report number:
  • PRINT-93-0614 (IAS,PRINCETON),
  • IASSNS-AST-93-50
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Abstract: (ADS)
We use new analytical results and a series of N-body experiments to study the dynamcis of weakly nonlinear clustering in an Einstein-de Sitter universe. The initial density fluctuations are assumed to be Gaussian with power spectra of the form P(k)∝knP(k) \propto k^n with n=−1n = -1 and n=0n = 0. We focus on two spatially smoothed random fields: the mass density contrast, δ\delta, and the peculiar velocity divergence, θ\theta, with variances <δ>2=σ2<\delta>^2=\sigma^2 and <θ>2=σθ2<\theta>^2=\sigma_{\theta}^2. Using the perturbation theory, valid for σ≪1\sigma\ll1, we calculate the evolution of the 1-point probability distribution functions (PDFs) of δ\delta and θ\theta. We expand the PDF in Hermite polynomials and powers of \signa, using the so-called Edgeworth series. When σ<1\sigma<1, the simulations and perturbation theory agree precisely, demonstrating that the N-body method yields accurate results for the density field in the regime of weakly nonlinear clustering. The skewness and kurtosis of the density filed stay remarkably close to the values predicted by perturbation theory even when σ=2\sigma=2. For σ<1\sigma<1, the Edgeworth expansion properly describes the positive and negative tails of the PDF within  1/σ~1/\sigma standard deviations from zero. We show analytically that 'biased' galaxy formation preserves the relation <δ3>∝<δ2>2<\delta^3>\propto<\delta^2>^2 predicted by second-order perturbation theory, provided that the galaxy density is a local function of the underlying mass density. Our results should be useful in the analysis of large-scale density and velocity fields, allowing one to derive constraints on the nature of primordial fluctuations, the value of the cosmological density parameter, and the physical processes that govern galaxy formation.
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