RLMT: From Lindblad Relaxation to the Primordial Spectrum: A de Sitter Cosmogenesis Map, Symmetric Base Spectra, and a Measurable Relaxation Exponent

Abstract

RLMT-lite describes the primordial curvature spectrum as a mixture of three modes, P(k) = Σ_n w_n(k) P_n(k), with softmax weights and phenomenological base spectra. A previous paper (doi:10.5281/zenodo.23293066) found that current CMB data cannot fix its parameters and left the task of deriving them to theory. Here we derive the form of the spectrum from the dynamics. (1) For the representation-theoretic mode coefficients c ∝ (1,1,2) and transition rates symmetric under exchange of the two |m| = 1 modes, Lindblad relaxation from the maximally mixed state stays exactly on the Gibbs (softmax) family; for non-degenerate patterns it does not. (2) In a quasi-de Sitter cosmogenesis in which the Lindblad time is cosmic time and each mode freezes its weights at horizon exit, the weights become w_n = x/3 + (1 − x) G_n(γ) with x = min[1, (k/k_on)^(−ν)] and ν = Γ/H, the relaxation rate in units of the Hubble rate. (3) A de Sitter thermal bath gives γ = 2πκ/H; an Ohmic coupling α gives ν = αγ coth(γ/2). Gravitational coupling alone gives ν ~ 10⁻¹⁰, i.e. a pure power law, and the weak-coupling (Markov) condition requires ν ≲ 0.3. (4) The m → −m symmetry forces P_1 = P_2, and exact de Sitter removes the extra tilt of P_3, so the base spectra reduce to one amplitude ratio a_3 = P_3/P_1, for which zero-point fluctuations suggest 1/√2. With Planck low-ℓ, cut Planck high-ℓ and ACT DR6 (P-ACT-lite), the minimal model with a_3 = 1/√2 and free ν gives Δχ² = −4.0 relative to ΛCDM with two or three extra parameters (ΔAIC ≃ 0 to +2), but needs ν ≈ 0.5; the feature is a change of slope of about 2% across the ACT damping tail, beginning near ℓ ≈ 1900. The fully consistent model (ν ≤ 0.3) is indistinguishable from ΛCDM (Δχ² = −0.1), and the model does not explain the low-ℓ power deficit. A Fisher forecast shows that, with a_3 fixed by theory, the Simons Observatory can measure ν to σ(ν) ≈ 0.2–0.35, which tests the weak-coupling bound; with a_3 free it cannot. Code to reproduce all tables is included.