Log-normal distribution

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 [34

, 74 , 9

, 25 ].

Kayo et al. [40 ] showed that the one-point log-normal PDF

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

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

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., [67

]). In this sense, the above log-normal PDF is completely specified without any free parameter.

Figure 3 plots the one-point PDFs computed from cosmological $N$ -body simulations in SCDM, LCDM, and OCDM (for Standard, Lambda, and Open CDM) models, respectively [36 , 40 ]. 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 $δ ≲ − 0.5$ .

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 [67

], respectively. (Figure taken from [40

].)

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,

where the variance is computed from its linear power spectrum:

If one introduces a new field $δ$ from $g$ as

the PDF for $δ$ is simply given by $(dg ∕dδ)P (1)(g) G$ , which reduces to Equation (80

At this point, the transformation (85 ) 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 (85 ) somehow captures an important aspect of the nonlinear evolution in the real Universe.