A Self-Gravitating Memory Kernel as a Scalar Alternative to Cold Dark Matter: CMB, Matter Power Spectrum, and the BAO-Scale Prediction

We implement a holographic memory kernel in the CLASS Boltzmann code and confront it with CMB and large-scale structure data. The framework, developed in Papers 1–7, replaces cold dark matter (CDM) perturbations with a scalar memory variable m(k,η) governed by two equations: a modified continuity equation that includes both standard gravitational infall and a baryon-tracking filter term, and the standard CDM Euler equation. At early times (z ≳ 10), gravitational self-interaction dominates the filter and the memory variable clusters identically to CDM, reproducing the Planck CMB acoustic spectrum (TT, TE, EE) to 0.01% precision. At late times and large scales, the filter term a(H0+H)(δb−m) becomes significant, partially coupling the memory to baryon acoustic oscillations. This produces a specific, falsifiable prediction: a 1–8% suppression of the matter power spectrum P(k) at the BAO scale (k ≈ 0.04 h/Mpc), growing with redshift as H(z)/H0. The suppression is 1% at z = 0, 2% at z = 1, and 5% at z = 5—detectable at 2σ in the most precise DESI DR1 bin (z = 0.93). The suppression may account for DESI's reported hints of evolving dark energy. A passive filter without gravitational self-interaction (Route A) fails catastrophically, shifting the first peak to l ≈ 427, but achieves a peak ratio P1/P2 = 2.43 (closer to Planck's 2.4 than ΛCDM's 2.21), demonstrating that the filtering mechanism produces correct peak height alternation. The self-gravitating version (Route D) preserves this mechanism while restoring correct peak positions through gravitational growth. The model is a scalar analogue of the Skordis-Zlošnik relativistic MOND theory: CDM-like at high redshift, MOND-like at low redshift, governed by a single set of equations.