  We develop an analysis pipeline for characterizing the topology of large
scale structure and extracting cosmological constraints based on persistent
homology. Persistent homology is a technique from topological data analysis
that quantifies the multiscale topology of a data set, in our context unifying
the contributions of clusters, filament loops, and cosmic voids to cosmological
constraints. We describe how this method captures the imprint of primordial
local non-Gaussianity on the late-time distribution of dark matter halos, using
a set of N-body simulations as a proxy for real data analysis. For our best
single statistic, running the pipeline on several cubic volumes of size
$40~(\rm{Gpc/h})^{3}$, we detect $f_{\rm NL}^{\rm loc}=10$ at $97.5\%$
confidence on $\sim 85\%$ of the volumes. Additionally we test our ability to
resolve degeneracies between the topological signature of $f_{\rm NL}^{\rm
loc}$ and variation of $\sigma_8$ and argue that correctly identifying nonzero
$f_{\rm NL}^{\rm loc}$ in this case is possible via an optimal template method.
Our method relies on information living at $\mathcal{O}(10)$ Mpc/h, a
complementary scale with respect to commonly used methods such as the
scale-dependent bias in the halo/galaxy power spectrum. Therefore, while still
requiring a large volume, our method does not require sampling long-wavelength
modes to constrain primordial non-Gaussianity. Moreover, our statistics are
interpretable: we are able to reproduce previous results in certain limits and
we make new predictions for unexplored observables, such as filament loops
formed by dark matter halos in a simulation box.
