A like-for-like framework for persistent homology on galaxy surveys, and a topological anomaly in DESI BGS
ALESSANDRO MARCONI · Monthly Notices of the Royal Astronomical Society · 2026
ABSTRACT We apply three-dimensional persistent homology (PH) to the observed DESI Bright Galaxy Survey (BGS) Data Release 1 galaxy field (North Galactic Cap; 217 614 galaxies, $z \in [0.1,0.4]$) and show that naive PH comparisons between a bounded survey and periodic-box mocks are dominated by two systematics that can masquerade as physical anomalies. First, the mean $H_1$ persistence depends on the dimensionless smoothing parameter $\sigma _\mathrm{px}= R/\Delta x$, not on the physical scale R alone ($+8.42\sigma$ between $\sigma _\mathrm{px}= 0.216$ and 0.640), invalidating cross-survey comparisons at fixed R. Secondly, embedding a masked survey in a zero-padded cube with an unmasked filtration produces boundary-driven features – high mean persistence with a suppressed generator count – that are largely persistence-diagram truncation, not cosmology. We introduce a like-for-like framework (cut-sky mocks matched in footprint, selection, redshift-space distortions, density, weighting, and $\sigma _\mathrm{px}$, with a masked filtration) and apply it to DESI BGS against 2000 Quijote $w_0$CDM cosmologies. Most of the apparent signal dissolves. One feature survives: The DESI field contains fewer $H_1$ generators than all 2000 mocks (rank $1/2001$; look-elsewhere bound $p \lt 8/2001$), robust to a halo occupation distribution refit and more concentrated occupations, to weighting, to an angular fiber-incompleteness surrogate, and to a $\pm 20 {{\ \rm per\ cent}}$ variation of the satellite velocity dispersion. The in-range $w_0$ response is flat, so no sampled $w_0$ reproduces it. We report this deficit as an open anomaly, not a dark-energy detection, and release the pipeline as a template for PH analyses of observed galaxy fields.