All MFs can be expressed as integrals over the
excursion set. While the first MF is simply given by the volume
integration of a Heaviside step function normalized to the
total volume
,
For a 3-D Gaussian random field, the average MFs per unit volume can be expressed analytically as follows:
whereThe above MFs can be indeed interpreted as
well-known geometric quantities: the volume fraction , the total surface area
, the integral
mean curvature
, and the integral Gaussian curvature,
i.e., the Euler characteristic
. In our current
definitions (see Equations (101
, 108
), or
Equations (102
, 115
)), one can easily show
that
reduces simply to
. The MFs were
first introduced to cosmological studies by Mecke et al. [57
], and further
details may be found in [57, 32
]. Analytic
expressions of MFs in weakly non-Gaussian fields are derived
in [52].