3.4 Genus statistics
A complementary approach to characterize the clustering of the
Universe beyond the two-point correlation functions is the genus
statistics [26
]. This is a
mathematical measure of the topology of the isodensity surface. For
definiteness, consider the density contrast field
at the position
in the survey volume
. This may be evaluated, for instance, by taking the
ratio of the number of galaxies
in the
volume
centered at
to its average value
:
where
is its r.m.s. value. Consider the
isodensity surface parameterized by a value of
. Genus is one of the topological
numbers characterizing the surface defined as
where
is the Gaussian curvature of the isolated surface.
The Gauss-Bonnet theorem implies that the value of
is indeed an integer and equal to the number of holes minus 1. This
is qualitatively understood as follows: Expand an arbitrary
two-dimensional surface around a point as
Then the Gaussian curvature of the surface is defined by
. A surface topologically equivalent to a sphere (a
torus) has
(
), and thus Equation (99) yields
(
) which coincides with the number of
holes minus 1.
In reality, there are many disconnected
isodensity surfaces for a given
, and thus it is more
convenient to define the genus density in the survey volume
using the additivity of the genus:
where the
(
) denote the
disconnected isodensity surfaces with the same value of
. Interestingly the Gaussian density
field has an analytic expression for Equation (101):
where
is the moment of
weighted over the power spectrum of
fluctuations
and the smoothing function
(see, e.g., [4
]). It should be
noted that in the Gaussian density field the information of the
power spectrum shows up only in the proportional constant of
Equation (102), and its functional
form is deterimined uniquely by the threshold value
.
This
-dependence reflects the phase information which is
ignored in the two-point correlation function and power spectrum.
In this sense, genus statistics is a complementary measure of the
clustering pattern of Universe.
Even if the primordial density field obeys the
Gaussian statistics, the subsequent nonlinear gravitational
evolution generates the significant non-Gaussianity. To distinguish
the initial non-Gaussianity from that acquired by the nonlinear
gravity is of fundamental importance in inferring the initial
condition of the Universe in a standard gravitational instability
picture of structure formation. In a weakly nonlinear regime,
Matsubara derived an analytic expression for the non-Gaussianity
emerging from the primordial Gaussian field [49]:
where
are the Hermite polynomials:
,
,
,
,
, …. The three quantities
denote the third-order moments of
. This expression
plays a key role in understanding if the non-Gaussianity in galaxy
distribution is ascribed to the primordial departure from the
Gaussian statistics.