3.2 Log-normal distribution

A probability distribution function (PDF) of the cosmological density fluctuations is the most fundamental statistic characterizing the large-scale structure of the Universe. As long as the density fluctuations are in the linear regime, their PDF remains Gaussian. Once they reach the nonlinear stage, however, their PDF significantly deviates from the initial Gaussian shape due to the strong non-linear mode-coupling and the non-locality of the gravitational dynamics. The functional form for the resulting PDFs in nonlinear regimes are not known exactly, and a variety of phenomenological models have been proposed  [34Jump To The Next Citation Point749Jump To The Next Citation Point25].

Kayo et al. [40Jump To The Next Citation Point] showed that the one-point log-normal PDF

1 [ {ln(1 + δ) + σ2∕2}2] 1 P(L1N)(δ) = ∘------exp − ----------2---1----- ----- (80 ) 2πσ21 2σ1 1 + δ
describes very accurately the cosmological density distribution even in the nonlinear regime (the r.m.s. variance σnl ≲ 4 and the over-density δ ≲ 100). The above function is characterized by a single parameter σ1 which is related to the variance of δ. Since we use δ to represent the density fluctuation field smoothed over R, its variance is computed from its power spectrum Pnl explicitly as
∫ 2 -1-- ∞ &tidle; 2 2 σnl(R ) ≡ 2π2 Pnl(k)W (kR )k dk. (81 ) 0
Here we use subscripts “lin” and “nl” to distinguish the variables corresponding to the primordial (linear) and the evolved (nonlinear) density fields, respectively. Then σ1 depends on the smoothing scale R alone and is given by
σ2(R ) = ln [1 + σ2 (R )]. (82 ) 1 nl
Given a set of cosmological parameters, one can compute σnl(R ) and thus σ1(R ) very accurately using a fitting formula for Pnl(k) (see, e.g., [67Jump To The Next Citation Point]). In this sense, the above log-normal PDF is completely specified without any free parameter.

Figure 3View Image plots the one-point PDFs computed from cosmological N-body simulations in SCDM, LCDM, and OCDM (for Standard, Lambda, and Open CDM) models, respectively [36Jump To The Next Citation Point40Jump To The Next Citation Point]. The simulations employ N = 2563 dark matter particles in a periodic comoving cube (100h −1 Mpc )3. The density fields are smoothed over Gaussian (left panels) and Top-hat (right panels) windows with different smoothing lengths: R = 2h −1 Mpc, 6h −1 Mpc, and 18h− 1 Mpc. Solid lines show the log-normal PDFs adopting the value of σnl directly evaluated from simulations (shown in each panel). The agreement between the log-normal model and the simulation results is quite impressive. A small deviation is noticeable only for δ ≲ − 0.5.

View Image

Figure 3: One-point PDFs in CDM models with Gaussian (left panels) and top-hat (right panels) smoothing windows: R = 2h −1 Mpc (cyan), 6h− 1 Mpc (red), and 18h −1 Mpc (green). The solid and long-dashed lines represent the log-normal PDF adopting σnl calculated directly from the simulations and estimated from the nonlinear fitting formula of [67Jump To The Next Citation Point], respectively. (Figure taken from [40Jump To The Next Citation Point].)

From an empirical point of view, Hubble [34] first noted that the galaxy distribution in angular cells on the celestial sphere may be approximated by a log-normal distribution, rather than a Gaussian. Theoretically the above log-normal function may be obtained from the one-to-one mapping between the linear random-Gaussian and the nonlinear density fields [9]. We define a linear density field g smoothed over R obeying the Gaussian PDF,

( ) (1) ----1--- -g2-- P G (g) = ∘2 -πσ2--exp − 2σ2 , (83 ) lin lin
where the variance is computed from its linear power spectrum:
1 ∫ ∞ σ2lin(R ) ≡ ---2 Plin(k) &tidle;W 2(kR )k2dk. (84 ) 2π 0
If one introduces a new field δ from g as
( ) 1 g ∘ --------2-- 1 + δ = ∘-------2 exp σ--- ln(1 + σnl) , (85 ) 1 + σnl lin
the PDF for δ is simply given by (dg ∕dδ)P (1)(g) G, which reduces to Equation (80View Equation).

At this point, the transformation (85View Equation) is nothing but a mathematical procedure to relate the Gaussian and the log-normal functions. Thus there is no physical reason to believe that the new field δ should be regarded as a nonlinear density field evolved from g even in an approximate sense. In fact it is physically unacceptable since the relation, if taken at face value, implies that the nonlinear density field is completely determined by its linear counterpart locally. We know, on the other hand, that the nonlinear gravitational evolution of cosmological density fluctuations proceeds in a quite nonlocal manner, and is sensitive to the surrounding mass distribution. Nevertheless the fact that the log-normal PDF provides a good fit to the simulation data, empirically implies that the transformation (85View Equation) somehow captures an important aspect of the nonlinear evolution in the real Universe.

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