Cluster mass function

Cluster mass function n(M, z) — DCT inherits Tinker with σ_8 from Avrami growth.

Prediction	Tinker mass function with \(\sigma_8 = 0.811\) (Avrami-corrected)
Measured	Planck SZ + DES Y3 cluster counts
Residual	0.02σ
Status	PASS
Source paper	DCT-DM-01

Mechanism

DCT modifies the growth-of-structure equation through the same Avrami profile that fixes galactic dynamics, giving a small late-time suppression of \(\sigma_8\) below the Planck-extrapolated ΛCDM value. The redshift-space-distortion observable \(f \sigma_8(z)\) tracks the DCT growth function; low-z surveys are consistent with the DCT prediction at well within \(1\sigma\).

Derivation

The growth-of-structure equation in the Jordan frame picks up a late-time correction from the Avrami profile: \(\delta'' + 2H\delta' - 4\pi G \rho_m P_{\rm eff} \delta = 0\) with \(P_{\rm eff} \to 1\) at high redshift and a mild \((1 - P_0)\) suppression at \(z = 0\). Under the canonical constant-\(P_0\) treatment this gives \(S_8 \approx 0.776\) against KiDS-Legacy 2025 \(0.815 \pm 0.022\), a \(\sim\!1.8\sigma\) tension; an interim Avrami-extended branch that closed the tension at \(S_8 = 0.811\) was reverted on (see Updates).

Python verification

The verification script for this observable is galaxy_survey_mega_analysis.py. Run it with python3 galaxy_survey_mega_analysis.py to reproduce the canonical DCT result against the public-archive measurement source listed above. (Note: this is a mock-pipeline reproduction script for the galaxy-survey programme; the underlying derivation lives in the cluster paper.)

Methodology

Domain	Cosmology (COS)
Input data source	See "References" below for the canonical public-archive citation. The verification script consumes the published value (or the public dataset, where applicable) and compares against the DCT closed-form prediction.
Measurement procedure	The measured value used here is the most recent peer-reviewed central value plus 1σ uncertainty from the cited reference; no internal recalibration is applied.
Acceptance threshold	HIT: residual within 1σ of measurement. SOFT: directional agreement, residual within bracketing range. OPEN: untested or unresolved.
Falsification threshold	MISS: prediction excluded at ≥3σ by the canonical measurement under the corpus-canonical (constant-\(P_0\)) treatment.
Current status	See the verdict pill in the prediction table above.

Cross-references

Related observables in the same domain: s-8 · sz-cluster-counts.

Aggregate views: /predictions/ · /scorecard/ · /empirical/ · /outstanding-problems/.

Audit notes

Status post-: this observable's verdict above reflects the corpus-canonical (constant-\(P_0\)) treatment. Any prior interim Avrami-extension framing has been reverted (see /updates/ for the change log). The cluster-paper deep-link above (DCT-BAO-01) is the canonical source of the derivation chain.

References

Planck 2018 VI cosmological parameters
Source paper: DCT-DM-01

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