Persistent Homology and Non-Gaussianity
  In this paper, we introduce the topological persistence diagram as a
statistic for Cosmic Microwave Background (CMB) temperature anisotropy maps. A
central concept in `Topological Data Analysis' (TDA), the idea of persistence
is to represent a data set by a family of topological spaces. One then examines
how long topological features `persist' as the family of spaces is traversed.
We compute persistence diagrams for simulated CMB temperature anisotropy maps
featuring various levels of primordial non-Gaussianity of local type.
Postponing the analysis of observational effects, we show that persistence
diagrams are more sensitive to local non-Gaussianity than previous topological
statistics including the genus and Betti number curves, and can constrain
$\Delta f_{NL}^{\rm loc}= 35.8$ at the 68\% confidence level on the simulation
set, compared to $\Delta f_{NL}^{\rm loc}= 60.6$ for the Betti number curves.
Given the resolution of our simulations, we expect applying persistence
diagrams to observational data will give constraints competitive with those of
the Minkowski Functionals. This is the first in a series of papers where we
plan to apply TDA to different shapes of non-Gaussianity in the CMB and Large
Scale Structure.
