Betti numbers of Gaussian fields
  We present the relation between the genus in cosmology and the Betti numbers
for excursion sets of three- and two-dimensional smooth Gaussian random fields,
and numerically investigate the Betti numbers as a function of threshold level.
Betti numbers are topological invariants of figures that can be used to
distinguish topological spaces. In the case of the excursion sets of a
three-dimensional field there are three possibly non-zero Betti numbers;
$\beta_0$ is the number of connected regions, $\beta_1$ is the number of
circular holes, and $\beta_2$ is the number of three-dimensional voids. Their
sum with alternating signs is the genus of the surface of excursion regions. It
is found that each Betti number has a dominant contribution to the genus in a
specific threshold range. $\beta_0$ dominates the high-threshold part of the
genus curve measuring the abundance of high density regions (clusters).
$\beta_1$ dominates the genus near the median thresholds which measures the
topology of negatively curved iso-density surfaces, and $\beta_2$ corresponds
to the low-threshold part measuring the void abundance. We average the Betti
number curves (the Betti numbers as a function of the threshold level) over
many realizations of Gaussian fields and find that both the amplitude and shape
of the Betti number curves depend on the slope of the power spectrum $n$ in
such a way that their shape becomes broader and their amplitude drops less
steeply than the genus as $n$ decreases. This behaviour contrasts with the fact
that the shape of the genus curve is fixed for all Gaussian fields regardless
of the power spectrum. Even though the Gaussian Betti number curves should be
calculated for each given power spectrum, we propose to use the Betti numbers
for better specification of the topology of large scale structures in the
universe.
