Precise scale range probed by the persistent homology summary statistic

Determine the precise range of physical length scales probed by the histogram-based persistent homology summary statistic constructed from birth and death counts of 0-, 1-, and 2-dimensional cycles in α-DTMℓ-filtrations of redshift-space halo catalogs (using nearest-neighbor parameter values k ∈ {1, 5, 15, 30, 60, 100}), so that the quantitative mapping between this statistic and the underlying spatial scales is established.

Background

In the paper, the authors build a summary statistic from persistent homology by constructing histograms of birth and death scales for 0-, 1-, and 2-cycles across multiple α-DTMℓ-filtrations with different nearest-neighbor parameters k. This statistic is then used in Fisher forecasts to compare information content against conventional two- and three-point statistics.

While they hypothesize that most of the information arises from scales larger than those used in the joint power spectrum and bispectrum analysis, they explicitly note that the precise scale range probed by the persistent-homology-based statistic is unclear, motivating a concrete determination of this mapping.