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
and the over-density
). The
above function is characterized by a single parameter
which is related to the variance of
. Since we use
to represent the density
fluctuation field smoothed over
, its variance is computed
from its power spectrum
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
depends on the smoothing scale
alone and is given by
Given a set of cosmological parameters, one can compute
and thus
very accurately using a
fitting formula for
(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
-body simulations in SCDM,
LCDM, and OCDM (for Standard, Lambda, and Open CDM) models,
respectively [36
, 40
]. The simulations
employ
dark matter particles in a periodic
comoving cube
. The density fields are
smoothed over Gaussian (left panels) and Top-hat (right panels)
windows with different smoothing lengths:
,
, and
. Solid lines show the log-normal PDFs adopting the
value of
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
.
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
smoothed over
obeying the Gaussian PDF,
where the variance is computed from its linear power spectrum:
If one introduces a new field
from
as
the PDF for
is simply given by
, 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
should be regarded as a nonlinear density field
evolved from
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.