Testing Quantum Gravity via Cosmological Structure Formation: Spectral RG Flow in History-Dependent Gravity (Paper XLVII)

We investigate the observable consequences of the spectral renormalization group (RG) flow in History-Dependent Gravity (HDG), building on the fixed-point structure established in our previous work (Paper XLVI). The RG analysis reveals two relevant directions with critical exponents θ1≈11. 78θ1≈11. 78 and θ2≈1. 42θ2≈1. 42, of which only the latter governs infrared physics. We show that this leads to a running effective gravitational coupling characterized by a single RG-relevant mode. Upon identifying the RG scale with the cosmological expansion rate, k∼H (a) k∼H (a), this induces a time-dependent modification of Newton's constant. We compute the impact of this running on the growth of large-scale structure and on gravitational wave propagation. To rigorously assess the model, we perform a comprehensive Markov Chain Monte Carlo (MCMC) analysis using 14 current measurements of the growth rate observable fσ8 (z) fσ8 (z) from surveys spanning z∈0. 02, 2. 33z∈0. 02, 2. 33. Our analysis reveals a nuanced picture. While the primary driver for improved fits to late-time data is the freedom to lower the primordial normalization to σ8≈0. 60σ8≈0. 60, the HDG framework provides a genuine, additional dynamical mechanism for growth suppression, improving the fit by Δχ2≈38Δχ2≈38 compared to CDM with free ₈ Crucially, when constrained to the Planck normalization (σ8=0. 811σ8=0. 811), the data drive the HDG parameters toward the boundary of the physically admissible region (A≈−0. 99A≈−0. 99), resulting in a near-critical regime where Geff (z≫1) /G0≈0. 01Geff (z≫1) /G0≈0. 01. This serves as a profound diagnostic: observational data are actively probing the stability boundaries of the quantum gravity fixed point. The model also predicts a redshift-dependent deviation between gravitational-wave and electromagnetic luminosity distances (dLGW/dLEM≈0. 96dLGW/dLEM≈0. 96 at z=1z=1), testable by future detectors like LISA. This work establishes a direct, minimal, and predictive link between quantum gravitational fixed-point structure and late-time cosmological observables.