What question did this study set out to answer?

This research aims to establish a structural framework for the growth index γ in cosmology by deriving it as a fixed point.

February 21, 2026Open Access

ψ-Retention Cosmology: The Structural Fixed Point at γ ≈ 0.64 — A Variational Derivation and Stability Analysis

Key Points

This research aims to establish a structural framework for the growth index γ in cosmology by deriving it as a fixed point.
Developed a variational formulation of ψ-retention cosmology
Derived a modified growth equation from a retentive Lagrangian
Established an autonomous system for analysis
Identified fixed-point solutions and evaluated their dynamical stability
Demonstrated that the observed growth index γ arises as a stable fixed point
Explained the k-independent suppression observed in cosmological data
Predicted a regime transition near z ≈ 1.7
Identified retentive equilibrium Ωψ as the source of the γ anomaly

Abstract

This work presents a variational formulation of the ψ-Retention Cosmology framework in which the observed late-time growth index γ ≈ 0.64 arises as a structural fixed point rather than a phenomenological adjustment. Starting from a retentive Lagrangian defined over the scale factor, we derive the modified growth equation, establish the autonomous system, obtain the fixed-point solution, and demonstrate its dynamical stability. The analysis explains the k-independent suppression seen in Planck PR4 + DES Y6 + Gold-2026 fσ₈ data, predicts a regime transition near z ≈ 1.7, and identifies the retentive equilibrium Ωψ as the structural origin of the γ anomaly. The result defines a coherent, falsifiable extension of late-time cosmology beyond ΛCDM.

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