Ring-Down Oscillations in the Primordial Power Spectrum from Decoherence-Induced Bogoliubov Occupancy

Author: Raman Marozau · ORCID: 0009-0000-0241-1135 · Independent Researcher

Date: 2026-04-05

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Abstract

We report a data-conditioned inference of ring-down oscillations in the primordial scalar power spectrum $P_\zeta(k)$, arising from the Bogoliubov occupancy $\bar{n}_k$ computed via the Mukhanov–Sasaki solver within the ToE decoherence framework. Using BK18+Planck+BAO chains (2,842,467 samples) and the full CAMB pipeline, we find: (i) the occupancy-enhanced spectrum $P_\zeta(k) = (1+2\bar{n}_k) \cdot A_s \cdot (k/k_\ast)^{n_s-1}$ is compatible with Planck 2018 at the pivot ($n_s = 0.9669$, $0.5\sigma$; $A_s = 2.098 \times 10^{-9}$, $0.1\sigma$); (ii) ring-down modulation with effective amplitude $A_\text{eff} \sim \mathcal{O}(\varepsilon_H) \approx 1.25\%$ is predicted at $k \lesssim k_0$; (iii) the predicted naive per-mode signal-to-noise ratio for CMB-S4 at $0.1\%$ fractional precision is $\text{SNR} = 12.5$, making this a detection candidate pending full Fisher analysis. The ring-down amplitude is set by the first slow-roll parameter $\varepsilon_H$ and the Bogoliubov matching at $\eta_0 = -1/k_0$, with no additional free parameters. At the pivot scale, the occupancy enhancement is $1 + 2\bar{n}_k = 1.000000004$, recovering standard inflation as a null test.

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1. The Claim

The ToE decoherence mechanism produces a ring-down modulation of the primordial scalar power spectrum with:

(i) Effective amplitude $A_\text{eff}(k) = A_\text{ring}(k) \cdot e^{-\Gamma_k \Delta\eta}$, peaking at $A_\text{eff} \approx 1.25\%$ near $k \sim k_0$;

(ii) Oscillatory structure $\delta P_\zeta / P_\zeta = A_\text{eff} \cos(2 c_s k \eta_0 + \phi_k)$ with phase $\phi_k$ from Bogoliubov matching;

(iii) Predicted naive per-mode SNR = 12.5 for CMB-S4 ($\sigma(P_\zeta)/P_\zeta = 0.1\%$), SNR = 0.63 for Planck ($\sigma = 2\%$). A full Fisher forecast is needed to confirm;

(iv) Full compatibility with Planck 2018 at the pivot scale ($n_s$ within $0.5\sigma$, $A_s$ within $0.1\sigma$).

The ring-down amplitude is $\mathcal{O}(\varepsilon_H)$ — set by slow-roll, not by fitting. This is a falsifiable prediction testable with CMB-S4.

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2. What Is New

• Ring-down from first principles. The oscillation amplitude $A_\text{ring}(k)$ and phase $\phi_k$ emerge from a single call to the Mukhanov–Sasaki solver with Bogoliubov matching at $\eta_0 = -1/k_0$. No phenomenological template is assumed.

• Quantitative forecast. Naive per-mode SNR = 12.5 at CMB-S4 precision — a concrete, falsifiable number requiring confirmation via full Fisher analysis. If CMB-S4 achieves $0.1\%$ fractional precision on $P_\zeta(k)$ and no ring-down is detected, the ToE decoherence mechanism is ruled out.

• Occupancy enhancement profile. The full $\bar{n}_k$ profile from 60 MS solver modes shows enhancement from $\bar{n}_k \approx 0.16$ at $k = 5 \times 10^{-4}$ Mpc$^{-1}$ to $\bar{n}_k \sim 10^{-9}$ at the pivot — a smooth transition from ToE-dominated to standard inflation.

• CAMB pipeline verification. $H_0 = 67.66$ km/s/Mpc from the full CAMB calculation confirms background cosmology consistency.

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3. Physical Framework

The ring-down oscillations arise from the Bogoliubov particle production at the decoherence matching surface $\eta_0 = -1/k_0$.

When the first internal decoherence act occurs at conformal time $\eta_0$, the Mukhanov–Sasaki mode function $v_k(\eta)$ must be matched across the transition. Before $\eta_0$, the mode is in the Bunch-Davies vacuum; after $\eta_0$, it is in a mixed state described by Bogoliubov coefficients $(\alpha_k, \beta_k)$. The coefficient $\beta_k$ encodes particle production: $\bar{n}_k = |\beta_k|^2$ is the mean occupancy number.

The scalar power spectrum in the presence of occupancy is (manuscript sec03):

$P_\zeta(k) = \frac{(1 + 2\bar{n}_k) H_*^2}{8\pi^2 M_\text{Pl}^2 \varepsilon_{H*} c_s^*}$

The factor $(1 + 2\bar{n}_k)$ enhances the spectrum at scales affected by decoherence. The interference between the $\alpha_k$ and $\beta_k$ components produces an oscillatory modulation — the ring-down:

$\frac{\delta P_\zeta}{P_\zeta} = A_\text{eff}(k) \cos(2 c_s k \eta_0 + \phi_k)$

where $A_\text{ring}(k)$ is the amplitude from $|\beta_k|$ (set by horizon geometry at $\eta_0$), $\phi_k = \arg(\alpha_k \beta_k^*)$ is the phase from Bogoliubov matching, and the effective amplitude includes decoherence damping: $A_\text{eff}(k) = A_\text{ring}(k) \cdot e^{-(\Gamma/H) \Delta\eta}$, where $\Delta\eta = \eta_* - \eta_0$ is the conformal time between the decoherence act and horizon crossing.

The ring-down amplitude scales as $A_\text{eff} \sim \mathcal{O}(\varepsilon_H)$ because the Bogoliubov coefficient $|\beta_k|$ is determined by the slow-roll parameter through the mode equation. This is a structural prediction — the amplitude is set by the inflationary dynamics, not by fitting. For $\varepsilon_H = 0.01$, the peak effective amplitude is $A_\text{eff,max} \approx 1.25\%$.

This differs fundamentally from phenomenological feature models (e.g., step potentials, particle production templates) in that the oscillation frequency $2 c_s k \eta_0$, amplitude $A_\text{ring}(k)$, and phase $\phi_k$ are all computed from a single call to the Mukhanov–Sasaki solver with no additional free parameters beyond the ToE parameter set $(k_0, \varepsilon_H, \eta_H, c_s^*, \Gamma/H)$.

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4. Data

The observational data are drawn from the joint BK18+Planck+BAO analysis (NASA LAMBDA). The scalar spectral index $n_s$ and amplitude $A_s$ anchor the base power spectrum, while $r$ constrains the tensor sector. The ring-down modulation is computed on top of this observationally determined baseline.

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5. Method

5.1 ToE Parameters

The five ToE parameters define the decoherence mechanism. The IR feature scale $k_0$ determines where the ring-down peaks in $k$-space (and correspondingly at $\ell \sim 40$–$100$ in the CMB). The slow-roll parameters $\varepsilon_H$ and $\eta_H$ set the inflationary background, $c_s^*$ is the sound speed, and $\Gamma/H$ controls how rapidly the ring-down oscillations are damped.

5.2 Computation Pipeline

The computation combines three independent data streams: the CAMB Boltzmann solver for background cosmology, the Mukhanov–Sasaki solver for the occupancy profile $\bar{n}_k$ and ring-down quantities, and the BK18 chain posteriors for the observational baseline. The ring-down modulation and SNR forecasts are then derived from these inputs.

1. run_toe_calculation(DEFAULT_COBAYA_PARAMS) → CAMB pipeline: $H_0$, $\sigma_8$, $C_\ell$, $\bar{n}_k$ at pivot, $Q_\text{toe}$.
2. compute_ms_nbar(K_FINE, TOE_PARAMS) → MS solver on 60-mode $k$-grid: $\bar{n}_k$, $A_\text{ring}$, $\phi_k$ at each $k$.
3. load_bk18_chains() → BK18 posteriors: $n_s$, $r$, $A_s$.
4. Compute $P_\zeta(k) = (1+2\bar{n}_k) \cdot A_s \cdot (k/k_\ast)^{n_s-1}$.
5. Compute ring-down: $\delta P_\zeta / P_\zeta = A_\text{eff}(k) \cos(2 c_s k \eta_0 + \phi_k)$.
6. Compute SNR forecasts at Planck, CMB-S4, and ideal precision.

5.3 Ring-Down Physics

The ring-down arises from the Bogoliubov coefficient $\beta_k$ at the matching surface $\eta_0$:

• $A_\text{ring}(k)$: amplitude from $|\beta_k|$, set by horizon geometry at $\eta_0$
• $\phi_k = \arg(\alpha_k \beta_k^\ast)$: phase from Bogoliubov matching
• Damping: $e^{-(\Gamma/H) \cdot \Delta\eta}$ where $\Delta\eta = \eta_\ast - \eta_0$
• Effective amplitude: $A_\text{eff}(k) = A_\text{ring}(k) \cdot e^{-(\Gamma/H) \Delta\eta}$

No free parameters beyond the ToE parameter set.

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6. Results

6.1 Occupancy Profile

The Bogoliubov occupancy $\bar{n}_k$ is computed from the MS solver on a 60-mode $k$-grid. The occupancy enhancement factor $(1 + 2\bar{n}_k)$ directly multiplies the scalar power spectrum — values significantly above 1 indicate strong ToE modification of the primordial spectrum at that scale.

Note: $\bar{n}_k$ is non-monotonic. The peak at $k \approx 2.85 \times 10^{-3}$ Mpc$^{-1}$ (near $k_0$) arises from resonant Bogoliubov particle production at the matching surface $\eta_0$. The profile falls off both toward lower $k$ (fewer modes affected) and higher $k$ (modes deep sub-horizon at $\eta_0$).

6.2 Ring-Down Forecast

The ring-down detection forecast compares the peak effective amplitude $A_\text{eff,max} = 1.25\%$ against the fractional precision $\sigma(P_\zeta)/P_\zeta$ achievable by current and future CMB experiments. The naive per-mode SNR is the ratio of signal to noise at the peak.

Ring-down amplitude: $A_\text{eff,max} = 1.25\%$ at $k \sim k_0$.

Note: The SNR quoted here is a naive per-mode estimate: $\text{SNR} = A_\text{eff,max} / \sigma(P_\zeta/P_\zeta)$. This is an upper bound. A realistic detection forecast requires a full Fisher matrix analysis with $C_\ell$ covariance, foreground marginalization, and look-elsewhere correction, which may reduce the effective SNR.

6.3 Planck Compatibility

The ToE framework must be compatible with existing Planck 2018 constraints at the pivot scale. All four key observables ($n_s$, $A_s$, $r$, $H_0$) are within $0.5\sigma$ of the Planck values, confirming that the ring-down modulation at low $k$ does not spoil the fit at the pivot.

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7. Null Test: Pivot Scale

At $k_\ast = 0.05$ Mpc$^{-1}$:

• $\bar{n}_k = 1.91 \times 10^{-9}$
• Occupancy enhancement = $1.000000004$
• $A_\text{eff} = 0$ (ring-down fully damped)
• $P_\zeta(k_\ast) = 2.098 \times 10^{-9}$ (standard inflation recovered)

The ToE reduces to standard inflation at the pivot scale. This null test passes.

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8. Robustness

8.1 Ring-Down Amplitude Scaling

$A_\text{eff} \sim \mathcal{O}(\varepsilon_H)$. For $\varepsilon_H = 0.01$: $A_\text{eff,max} = 1.25\%$. This is a structural prediction — the amplitude is set by slow-roll, not by fitting.

8.2 Sensitivity to $\Gamma/H$

$\Gamma/H$ controls damping rate. Larger $\Gamma/H$ suppresses ring-down faster. At $\Gamma/H = 5$: ring-down visible at $k \lesssim 0.01$ Mpc$^{-1}$. At $\Gamma/H = 20$: ring-down confined to $k \lesssim 0.003$ Mpc$^{-1}$.

8.3 Sensitivity to $k_0$

$k_0$ sets the scale of the feature. Ring-down peaks near $k \sim k_0$. For $k_0 = 0.002$: feature at $\ell \sim 40$–$100$ in CMB.

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9. Falsification Criteria

9.1 Confirmation

CMB-S4 detects oscillatory modulation in $P_\zeta(k)$ at $k \lesssim 0.01$ Mpc$^{-1}$ with amplitude $\sim 1\%$ and frequency matching $2 c_s k \eta_0$.

9.2 Refutation

CMB-S4 achieves $0.1\%$ precision on $P_\zeta(k)$ and finds no ring-down modulation at $> 3\sigma$ — rules out $A_\text{eff} > 0.03\%$.

9.3 Inconclusive

CMB-S4 precision insufficient to reach $0.1\%$ at relevant $k$-scales.

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10. Limitations

Several limitations affect the scope of this inference. The most important is that the SNR estimate is a naive per-mode upper bound; a realistic forecast with foreground marginalization, $C_\ell$ covariance, and look-elsewhere correction may significantly reduce the effective detectability.

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11. Reproducibility

All results presented in this work are computed from a publicly available open-source pipeline implementing the Mukhanov–Sasaki solver with Bogoliubov matching for ring-down computation, the CAMB Boltzmann solver for background cosmology, and BK18+Planck+BAO public chains (NASA LAMBDA) for the observational baseline. The pipeline requires Python 3.8+, NumPy, SciPy, Cobaya, and CAMB. No manual parameter tuning is involved.

Code and data DOI: 10.5281/zenodo.19313505

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