Generalized Consistency Relation with CMB-S4/LiteBIRD Detection Forecast from BK18 Data

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

Date: 2026-04-05

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Abstract

We present a data-conditioned inference of the generalized tensor-scalar consistency relation $Q(k) = c_s^\ast/(1+2\bar{n}_k)$ using BK18+Planck+BAO chains (2,842,467 samples) with quantitative detection forecasts for next-generation CMB experiments. The Mukhanov–Sasaki solver yields $\bar{n}_k$ via Bogoliubov matching at $\eta_0 = -1/k_0$, giving $Q(k_0) = 0.939$ (6.1% deviation from standard inflation) and $Q(k = 5 \times 10^{-4}) = 0.762$ (23.8% deviation). At the pivot $k_\ast = 0.05$ Mpc$^{-1}$, $Q \approx 1 - 2.5 \times 10^{-9}$ — standard inflation recovered to $\sim 10^{-9}$ precision. The tensor tilt difference $\Delta n_t = n_t^\text{ToE} - n_t^\text{SI} = -5.1 \times 10^{-12}$ is undetectable with current data. We provide quantitative detection forecasts: CMB-S4 achieves $\sigma(Q) = 0.031$ at $k_0$, giving SNR = 1.79 (marginal $1.8\sigma$); LiteBIRD achieves $\sigma(Q) = 0.062$, SNR = 0.90 (insufficient). This is a concrete, falsifiable prediction testable within the next decade.

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1. The Claim

The ToE decoherence mechanism modifies the inflationary consistency relation from $Q = 1$ (standard inflation) to $Q(k) = c_s^\ast/(1+2\bar{n}_k) < 1$ at scales $k \lesssim k_0$, with:

(i) $Q(k_0 = 0.002) = 0.939$ — 6.1% deviation;

(ii) $Q(k = 5 \times 10^{-4}) = 0.762$ — 23.8% deviation;

(iii) $Q(k_\ast = 0.05) \approx 1 - 2.5 \times 10^{-9}$ — null test passed;

(iv) CMB-S4 forecast: $\sigma(Q) = 0.031$, SNR = 1.79 at $k_0$;

(v) $\Delta n_t = -5.1 \times 10^{-12}$ — undetectable with current data, confirming compatibility.

This extends [2] with quantitative detection forecasts and tensor tilt comparison.

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2. What Is New

• Quantitative detection forecast. $\sigma(Q)$ computed from $\sigma(r)$ propagation: $\sigma(Q) = \sigma(r)/(8|n_t|)$. Concrete SNR numbers for LiteBIRD and CMB-S4.

• $\Delta n_t$ measurement. Tensor tilt difference between ToE and SI computed from BK18 $r$ posteriors: $\Delta n_t = -5.1 \times 10^{-12}$, with $|\Delta n_t|/\sigma = 0.0000$ — confirming that current data cannot distinguish the two.

• Fine $k$-grid $Q(k)$ profile. 40-mode Mukhanov–Sasaki solver computation gives $Q(k)$ from $k = 5 \times 10^{-4}$ to $k = 0.15$ Mpc$^{-1}$.

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3. Physical Framework

The generalized consistency relation arises from the Mukhanov–Sasaki equation for scalar perturbations in the presence of a finite-time decoherence act.

In standard single-field inflation, the scalar curvature perturbation $\zeta$ is in a pure vacuum state, and the tensor-to-scalar ratio satisfies $r = -8 n_t$ (the standard consistency relation). In the ToE framework, the first internal decoherence act at conformal time $\eta_0$ places $\zeta$ in a mixed Gaussian state with Bogoliubov occupancy $\bar{n}_k = |\beta_k|^2$, where $\beta_k$ is the Bogoliubov coefficient from matching the mode function across the decoherence surface. The scalar and tensor power spectra become (manuscript sec03):

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

The tensor spectrum is unaffected by the occupancy (gravitons do not participate in the decoherence channel at leading order). Forming $r \equiv P_t / P_\zeta$ and measuring $n_t = d\ln P_t / d\ln k$, the generalized consistency relation follows:

$\frac{r}{-8 n_t} = \frac{c_s^*}{1 + 2\bar{n}_k} \equiv Q(k)$

Since $\bar{n}_k > 0$ for modes affected by decoherence ($k \lesssim k_0$), the denominator exceeds unity and $Q(k) < 1$. For modes deep sub-horizon at $\eta_0$ ($k \gg k_0$), the Bogoliubov coefficient $\beta_k \to 0$, giving $\bar{n}_k \to 0$ and $Q \to 1$ — standard inflation is recovered as a null test.

The Mukhanov–Sasaki solver computes $\bar{n}_k$ by numerically evolving the mode equation through $\eta_0$ and extracting $\beta_k$ via Bogoliubov matching. The decoherence scale $k_0$ sets $\eta_0 = -1/k_0$; modes with $k \lesssim k_0$ were super-horizon at $\eta_0$ and acquire nonzero occupancy.

The detection forecast $\sigma(Q) = \sigma(r) / (8|n_t|)$ follows from Gaussian error propagation under the assumption that $r$ and $n_t$ are independently measured. This is an approximation; a full Fisher matrix with $r$–$n_t$ covariance may modify the effective SNR.

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4. Data

The observational data are drawn from the joint BICEP/Keck 2018 + Planck 2018 + BAO analysis, publicly available from NASA LAMBDA. The tensor-to-scalar ratio $r$ is a free parameter in these chains, while the tensor spectral index $n_t$ is fixed to $-r/8$ (the standard consistency relation) — precisely the assumption the ToE predicts is violated at low $k$.

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5. Method

5.1 Pipeline

The computation proceeds in six steps, from loading the public BK18 chains through Mukhanov–Sasaki mode evolution to detection forecasts. Each step builds on the previous: the chain posteriors provide the observational anchor for $r$, the MS solver computes the occupancy $\bar{n}_k$ from first principles, and the forecast propagates measurement uncertainties to the consistency ratio $Q$.

1. load_bk18_chains() → 2.8M samples with weights
2. Mukhanov–Sasaki solver → $\bar{n}_k$ on 8-point $k$-grid via Bogoliubov matching
3. compute_ms_nbar(K_FINE, TOE_PARAMS) → 40-mode fine $k$-grid
4. Compute $Q(k) = c_s^\ast/(1+2\bar{n}_k)$ at each $k$
5. Compute $n_t^\text{ToE} = -r(1+2\bar{n}_k)/(8 c_s^\ast)$ from BK18 $r$ posteriors
6. Forecast: $\sigma(Q) = \sigma(r)/(8|n_t|)$ with $\sigma(r)_\text{LiteBIRD} = 10^{-3}$, $\sigma(r)_\text{CMB-S4} = 5 \times 10^{-4}$

5.2 ToE Parameters

These five parameters define the decoherence mechanism and are not present in the BK18 chains. The decoherence scale $k_0$ sets the conformal time of the act ($\eta_0 = -1/k_0$), $\varepsilon_H$ and $\eta_H$ are the slow-roll parameters controlling the inflationary background, $c_s^*$ is the sound speed at horizon crossing, and $\Gamma/H$ is the decoherence rate that controls the damping of ring-down oscillations.

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6. Results

6.1 Q(k) Profile

The consistency ratio $Q(k)$ is evaluated at six representative wavenumbers spanning from deep IR ($k = 5 \times 10^{-4}$ Mpc$^{-1}$) to the Planck pivot ($k = 0.05$ Mpc$^{-1}$). The key result is the monotonic transition from $Q \approx 0.76$ at the lowest $k$ (23.8% deviation from standard inflation) to $Q = 1$ at the pivot (null test). The deviation is strongest where modes were super-horizon at the decoherence time $\eta_0$.

6.2 Detection Forecast

The detection forecast compares the ToE-predicted signal ($1 - Q = 0.055$ at $k_0$) against the projected measurement uncertainty $\sigma(Q)$ for two next-generation CMB experiments. The signal-to-noise ratio SNR $= (1-Q)/\sigma(Q)$ determines whether the deviation from standard inflation is detectable. CMB-S4 reaches marginal sensitivity (SNR = 1.79), while LiteBIRD alone is insufficient (SNR = 0.90).

6.3 Tensor Tilt

The tensor spectral index $n_t$ is computed independently for standard inflation (SI) and the ToE, using the BK18 posterior for $r$. The difference $\Delta n_t = n_t^\text{ToE} - n_t^\text{SI} = -5.1 \times 10^{-12}$ is twelve orders of magnitude below current sensitivity, confirming that the ToE is fully compatible with existing $n_t$ constraints — the deviation manifests in $Q(k)$, not in the tilt itself.

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7. Null Test: Pivot Scale

At $k_\ast = 0.05$ Mpc$^{-1}$: $\bar{n}_k = 1.26 \times 10^{-9}$, $Q = 1 - 2.5 \times 10^{-9} \approx 0.9999999975$. Standard inflation recovered to $\sim 10^{-9}$ precision. Null test passed for all parameter points.

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8. Robustness

8.1 $Q(k_0) \approx 0.94$ Quasi-Invariant

At $k = k_0$, $Q \approx 0.94$ across all tested $\varepsilon_H$ values [2]. This is a structural invariant of the Bogoliubov matching.

8.2 Forecast Assumptions

$\sigma(Q)$ assumes Gaussian error propagation from $\sigma(r)$. Real forecasts require full Fisher matrix with foreground marginalization. The quoted SNR is an upper bound.

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9. Falsification Criteria

9.1 Confirmation

CMB-S4 or LiteBIRD measures $Q < 1$ at $k \lesssim k_0$ with $> 3\sigma$ significance, with scale dependence matching the ToE form.

9.2 Refutation

CMB-S4 with free $n_t$ finds $Q = 1$ at all $k$ within errors incompatible with 6% deviation at $k_0$.

9.3 Current Status

Inconclusive: $\Delta n_t = 5 \times 10^{-12}$ is far below current sensitivity.

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10. Limitations

Several limitations constrain the scope of this inference. The most significant is that $n_t$ is not free in the BK18 chains, so the deviation $Q \neq 1$ cannot be tested directly from these data — it is a data-conditioned inference, not a detection.

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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, evaluated against BK18+Planck+BAO public chains (NASA LAMBDA). The pipeline requires Python 3.8+, NumPy, SciPy, Cobaya, and CAMB. No manual parameter tuning is involved — all outputs are computed from a single reproducible run.

Code and data DOI: 10.5281/zenodo.19313505

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References

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2. R. Marozau, "Scale-Dependent Data-Conditioned Inference for the Inflationary Consistency Relation from Decoherence-Induced Occupancy" (2026).

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