Probing Dark Energy with Alpha Shapes and Betti Numbers
  We introduce a new descriptor of the weblike pattern in the distribution of
galaxies and matter: the scale dependent Betti numbers which formalize the
topological information content of the cosmic mass distribution. While the
Betti numbers do not fully quantify topology, they extend the information
beyond conventional cosmological studies of topology in terms of genus and
Euler characteristic used in earlier analyses of cosmological models. The
richer information content of Betti numbers goes along with the availability of
fast algorithms to compute them. When measured as a function of scale they
provide a "Betti signature" for a point distribution that is a sensitive yet
robust discriminator of structure. The signature is highly effective in
revealing differences in structure arising in different cosmological models,
and is exploited towards distinguishing between different dark energy models
and may likewise be used to trace primordial non-Gaussianities.
  In this study we demonstrate the potential of Betti numbers by studying their
behaviour in simulations of cosmologies differing in the nature of their dark
energy.
