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∈[0.02,2.33]z∈[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 \Lambda$CDM with free $\sigma_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.