Cosmic homogeneity: Bagla, Yadav & Seshadri's weak-clustering D_q formula miscounts pairs for q ≠ 2 #46

Question. cosmic-homogeneity/ (#26, #17) uses only D_2, where this changes nothing. But the note in #39 and the literature the study draws on speak of generalised dimensions D_q. Should the repository carry the corrected weak-clustering D_q, and say what it changes?

This came out of the exploration of #39's avenues. It was checked by an independent derivation, three controlled numerical tests, and a reading of the published simulation figures. All numbers below are from those checks.

The formula and the slip

Bagla, Yadav and Seshadri 2008 (BYS08; MNRAS 390, 829, App. B; arXiv:0712.2905) give, for weak clustering,

D_q ≈ 3 + (q/2) d xibar/d ln r.

Where it goes wrong. Their App. B expands the galaxy-centred count moments ⟨N^m⟩. It gives every one of the m(m+1)/2 pairs (the centre plus m neighbours) the one-point volume average xibar. Only the m centre–neighbour pairs carry xibar. The m(m−1)/2 neighbour–neighbour pairs, with both points integrated over the sphere, carry the shot-noise-free variance of counts in spheres, σ²(r) = V⁻² ∫∫ ξ.

The corrected first-order result (prime = d/d ln r; continuum limit):

D_q ≈ 3 + xibar′ + ((q − 2)/2) σ²′.

• The two agree only at q = 2, or at isolated radii where σ²′ = xibar′. Elsewhere they differ by ((q − 2)/2)(σ²′ − xibar′).
• BYS08's own shot-noise term already has the neighbour–neighbour coefficient m(m−1)/2, which is the Poisson part of σ² + 1/N̄. The clustering term should have the same structure.
• The correction is a one-line consequence of the standard count-moment expansion (Peebles 1980, §36). It is an erratum, not a new method.
• A second, minor inconsistency. BYS08 eq. 6 counts the centre and App. A keeps the resulting (q − 1)/N̄ term, but App. B drops it. So their weak-clustering formula does not reduce to their homogeneous one as ξ → 0.

Checks

Controlled tests. In each test, the corrected formula's residual shrinks as amplitude⁴, while BYS08's shrinks only as amplitude².

• An exactly solvable plane-wave Cox process.
• 256³ lognormal fields, with D_q measured directly:
  • corrected residual: 1e-4 or less for q = −2 to 6;
  • BYS08's residual: 1e-3 to 1e-2, about the size of D_q − 3 itself.
• A Poisson catalogue of about 4×10⁵ points with exact pair counts. The measured D_q matches exact theory, and its clustering part follows the corrected formula.

What changes for linear ΛCDM (Planck 2018, z = 0):

• σ²′/xibar′ is about 0.66–0.72, so the slope of D_q in q is about 0.7 of BYS08's.
• D_0 = 2.89, 2.96 and 2.98 at 20, 40 and 60 h⁻¹ Mpc. BYS08 gives exactly 3.
• D_q crosses 3 near q ≈ −0.8 to −1.0, not at q = 0.
• ΔD_4 is about 0.85 of BYS08's.

Published simulations agree with the corrected shape. Neither is decisive on its own.

• Yadav, Bagla and Khandai 2010 (arXiv:1001.0617), Fig. 1, read from its vector data. Outside the BAO region, the N-body ΔD_4 is 0.82–0.89 of the BYS08 formula; the authors describe the gap as "less than 10%". The corrected formula predicts 0.81–0.86.
• Bandyopadhyay, Choudhury and Seshadri 2017 (arXiv:1611.05554). Their z = 7 field gives D_q ≈ 2.995 − 0.0083 q, so q* ≈ −0.63, against −0.65 predicted.
  • All ten of their points lie below the BYS08 line.
  • Mocks of their estimator follow the corrected formula to rms 0.0009, against 0.0094 for BYS08.
  • Their quoted errors alone cannot exclude BYS08: χ² = 9.7 on 9 degrees of freedom.

What this does and does not change

Unchanged: anything that uses only D_2, which includes this repository's study.

Not specific to the correction: with a fixed threshold, a homogeneity scale R_H(q) depends on q under BYS08 too, because there ΔD_q ∝ q.

Specific to the correction:

• D_0 falls below 3.
• The crossing q* moves from 0 to about −0.8.
• The slope ratio is about 0.7.
• Yadav et al. 2010's argument that their homogeneity scale is "unique" for all q no longer holds. That argument rests on their eq. sigdq, Var{ΔD_q} ≃ (Dq/2)² Var{xibar − ξ}. The corrected r_H(q) trend is robust, but its values (about 120–450 or 140–670 h⁻¹ Mpc for q = −1 to 10) depend on the variance model, so they are not quoted as results.

Limits: when xibar is a few percent and q ≥ 3, second-order terms shrink the advantage over BYS08.

Novelty: no correction turned up among about 100 arXiv papers citing BYS08, Yadav et al. 2010 or BCS17. The published MNRAS texts and non-arXiv citers were not checked.

First steps

• Add D_q(r) from xibar and σ² to cosmic-homogeneity/homogeneity.js, with a test that D_2 equals the existing D2.
• Test it on a Poisson-sampled lognormal catalogue, including the shot-noise term.
• Tabulate q*(r), D_0(r) and the D_q slope against BYS08's. Say plainly where second order matters.
• Before any erratum-style note outside this repository, read the published MNRAS texts and the non-arXiv citers.