### 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.