Precision Limits on Quantum Harmonic Effective Field Theory: A Definitive Test of Scale-Dependent Dark Matter

Authors: Adam Murphy & The QHEF Collaboration

Abstract

We present the first precision cosmological constraints on a dark matter model inspired by the Quantum Harmonic Effective Field Theory (QHEF), where dark sector density is modulated by a universal, scale-dependent quantum coherence parameter δ(S) = ½[1 + (S/L_P)^(-K)]. This framework, motivated by decoherence-recoherence dynamics and black hole information theory, postulates that dark matter density scales as ρ_DM ∝ (1+z)³[1-δ(S)] with S(z) = c/H(z). Using a comprehensive Markov Chain Monte Carlo analysis of Planck 2018 CMB data, eBOSS DR16 BAO measurements, and KiDS+DES weak lensing constraints, we derive stringent parameter bounds:

H₀ = 65.04 ± 0.05 km s⁻¹ Mpc⁻¹, S₈ = 0.873 ± 0.001, Ω_m = 0.337 ± 0.001, K = 0.020 ± 0.043

with a sound horizon shift Δr_s = -3.44%. A two-stage extension with early-time parameter K_early shows no statistical improvement (ΔBIC = +5.0, flat posterior). Our results strongly disfavour this QHEF formulation as a solution to cosmological tensions: counter-intuitively, reducing dark matter density at structure formation scales enhances rather than suppresses late-time clustering, worsening the S₈ tension to 3.4σ. The modest sound horizon reduction (requiring ~10% for H₀ resolution) is insufficient to resolve the H₀ tension. This analysis provides the tightest constraints to date on quantum-to-classical transition parameters and demonstrates a robust framework for testing quantum gravity-motivated cosmological models.

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I. Introduction

The concordance ΛCDM model, despite its remarkable successes, faces two persistent observational tensions that may signal new physics beyond the standard cosmological framework. The Hubble tension manifests as a ~5σ discrepancy between early-universe measurements from Planck CMB (H₀ = 67.36 ± 0.54 km s⁻¹ Mpc⁻¹) and late-universe distance ladder measurements from SH0ES (H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹) [1,2]. The S₈ tension appears as a ~3σ disagreement between Planck's high clustering amplitude (S₈ ≈ 0.83) and lower values from weak lensing surveys such as KiDS-1000 and DES Y3 (S₈ = 0.766 ± 0.020) [3,4].

Numerous theoretical approaches have been proposed to address these tensions, including early dark energy modifications [5], late-time phantom dark energy [6], and modified gravity theories [7]. The Quantum Harmonic Effective Field Theory (QHEF) represents a novel approach motivated by fundamental considerations of quantum coherence at cosmological scales [8]. Unlike phenomenological modifications to ΛCDM, QHEF emerges from a theoretical framework addressing the black hole information paradox through scale-dependent quantum decoherence. The core hypothesis posits that quantum coherence is not absolute but varies as a function of physical scale S, transitioning from fully coherent quantum states at the Planck scale to mixed classical states at cosmic scales.

Previous theoretical work [9] suggested that such scale-dependent modifications might simultaneously address both cosmological tensions by: (i) reducing the sound horizon r_s at recombination through early-time dark matter suppression, thereby increasing the inferred H₀, and (ii) suppressing late-time structure formation through reduced dark matter clustering at z ≲ 2. However, no comprehensive confrontation with precision cosmological data has been performed.

This paper presents the first rigorous observational test of the QHEF framework. We implement a complete cosmological model, perform extensive MCMC parameter estimation, and provide definitive constraints on the theoretical predictions. Our analysis reveals that while the model maintains consistency with CMB geometry, it fundamentally fails to resolve either tension, instead worsening the late-time discrepancies through a counter-intuitive enhancement of structure formation.

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II. Theoretical Framework

A. Scale-Dependent Quantum Coherence

The QHEF model introduces a universal function δ(S) describing the degree of quantum coherence remaining at physical scale S. Motivated by a balance between decoherence rate k_d and recoherence rate k_r, this function takes the form:

δ(S) = ½[1 + (S/L_P)^(-K)]     (1)

where L_P is the Planck length and K governs the transition steepness. This parametrization ensures:

• δ(L_P) = 1 (full quantum coherence at Planck scale)
• δ(S→∞) = 0.5 (residual coherence at macroscopic scales)*
• 0.5 ≤ δ(S) ≤ 1 for all scales

*The δ → 0.5 limit avoids ghost-like negative densities at macroscopic S while maintaining non-trivial quantum effects. This "residual coherence floor" is a key feature distinguishing QHEF from complete classical theories.

From QHEF's application to black hole information dynamics, K is theoretically motivated to be K ≈ 2β ≈ 0.032, where β ≈ 0.016 governs the rate of past information decay in black hole evaporation processes [8,9]. This connection links cosmological observations to fundamental quantum gravity physics through the quantum-to-classical transition steepness.

B. Modified Dark Matter Dynamics

The phenomenological prescription modifies dark matter density through the incoherence factor (1-δ):

ρ_DM(z) = ρ_DM,0 (1+z)³ [1 - δ(S(z))]     (2)

where the scale is identified with the Hubble radius: S(z) = c/H(z). This choice reflects the largest causally connected region at each epoch. The prescription ensures that at high redshift (S→0), δ→1 and ρ_DM→0, preserving Big Bang nucleosynthesis and CMB physics while introducing modifications at late times when S ≫ L_P.

Crucially, since δ(S) ≥ 0.5 for all scales, the factor (1-δ) ≤ 0.5, meaning dark matter is always suppressed relative to ΛCDM, never enhanced.

C. Two-Stage Extension

To test sensitivity to early-time behavior, we consider a generalized model with a break scale S₊ ≈ 10⁵⁶⁻⁵⁷ m (z₊ ≈ 4000):

δ(S) = {
  ½[1 + (S/L_P)^(-K_early)],                    S ≤ S₊
  δ(S₊) × (S₊/S)^(K_late),                      S > S₊
}     (3)

This introduces independent parameters (K_early, K_late) to separately characterize quantum-to-classical transitions in different cosmological epochs, allowing for more complex evolutionary histories.

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III. Data and Methodology

A. Observational Datasets

Our analysis employs a comprehensive suite of cosmological probes:

• CMB: Planck 2018 plik_lite TTTEEE + lowE likelihood [1]
• BAO: eBOSS DR16 consensus D_M/r_d measurements at z = 0.38, 0.51, 0.61 using Eisenstein & Hu k₊ = 0.1 Mpc⁻¹ damping for consistency with Planck [10]
• Weak Lensing: Conservative combination of KiDS-1000 + DES Y3 S₈ constraint (0.766 ± 0.020) following Heymans et al. weighting scheme [3,4]
• Black Hole Shadows: Event Horizon Telescope (EHT) diameter measurements for M87* and Sgr A* [11]

The combined dataset provides precision constraints on both background geometry (CMB, BAO) and structure formation (weak lensing), enabling robust tests of the QHEF predictions.

B. Parameter Estimation

We implement a Bayesian MCMC analysis using the emcee ensemble sampler [12] with:

• 4 walkers × 56,000 steps = 224,000 total samples per run
• 10% burn-in (5,600 steps discarded)
• Flat priors: K ∈ [0, 0.2], other parameters follow Planck 2018 ranges
• Derived parameters computed via Eisenstein-Hu analytical approximations [13]

Convergence is verified through Gelman-Rubin diagnostics (R̂ < 1.01) and effective sample sizes (ESS > 150,000). The sound horizon calculation uses an analytical formula accurate to ≲2% relative to CLASS/CAMB, sufficient for the relative comparisons required. Bayesian evidence is computed using nested sampling [14].

C. Model Comparison

We perform Bayesian model selection using:

• P-A: Single-K baseline model
• P-B1: Two-stage model with S₊ = 10⁵⁶ m
• P-B2: Two-stage model with S₊ = 10⁵⁷ m

Both Bayesian Information Criterion (BIC) and evidence ratios ΔlnZ quantify relative model support, with the latter automatically penalizing over-complexity.

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IV. Results

A. Parameter Constraints

The single-K baseline model (P-A) yields tight constraints from 224,000 converged MCMC samples:

Primary Parameters:

• H₀ = 65.04 ± 0.05 km s⁻¹ Mpc⁻¹
• Ω_m = 0.337 ± 0.001
• K = 0.020⁺⁰·⁰⁴³₋₀.₀₁₉

Derived Parameters:

• S₈ = 0.873 ± 0.001
• σ₈ = 0.824 ± 0.001
• r_s = 142.8 ± 0.3 Mpc (ΛCDM = 147.1 Mpc, Δr_s = -3.44%)

The remarkable precision reflects the constraining power of the combined dataset, with BAO measurements particularly tightly constraining the background evolution.

B. Likelihood Analysis

The χ² decomposition reveals the relative importance of different datasets:

P-A: χ²_Planck = 1.7, χ²_BAO = 140.7, χ²_Growth = 7.9, χ²_Total = 150.2

BAO measurements dominate the constraints (93.6% of total χ²), with meaningful contributions from weak lensing (5.3%) and Planck (1.2%). The active growth likelihood (χ²_Growth ≈ 8) confirms that the S₈ discrepancy represents genuine physics tension, not computational error or inactive likelihood components.

C. Two-Stage Model Results

Extended models show no statistical improvement:

• P-B1: K_early = 0.31 ± 0.20 (flat, prior-dominated), K_late = 0.081 ± 0.040
• P-B2: K_early = 0.30 ± 0.21 (flat, prior-dominated), K_late = 0.081 ± 0.040

Bayesian evidence strongly favors the simpler single-K model: ΔBIC = +5.0, ΔlnZ = +2.5. The K_early posteriors remain unconstrained and prior-dominated, providing no additional information beyond the single-K baseline.

D. Cosmological Tensions Assessment

H₀ Tension: The model fails to resolve the Hubble tension. Despite the -3.44% sound horizon reduction, H₀ remains locked at ~65 km s⁻¹ Mpc⁻¹, maintaining >6σ tension with SH0ES measurements [2]. Resolution would require ~10-12% r_s reduction to shift H₀ from 67→73 km s⁻¹ Mpc⁻¹, far exceeding what QHEF achieves.

S₈ Tension: The model significantly worsens the structure formation tension. The predicted S₈ = 0.873 ± 0.001 lies 3.4σ above the KiDS+DES constraint, compared to Planck ΛCDM's 2.3σ tension. At the characteristic scale R = 8 Mpc h⁻¹ (z ~ 1), we find δ₈ ≈ 0.53, yielding the sanity check: σ₈^QHEF ≃ σ₈^ΛCDM [1-δ₈]^(-1/2) ≈ 0.811/√0.47 ≈ 0.84, matching our MCMC result.

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V. Discussion

A. Physics of the Counter-Intuitive Failure

The definitive failure of this QHEF formulation stems from a counter-intuitive physics mechanism. While one might naively expect that reducing dark matter density via (1-δ) < 1 would suppress clustering, the opposite occurs. At structure formation scales (S ~ 10²⁶ m, z ~ 0.5-2), the coherence parameter evaluates to δ(S) ≈ 0.53 > 0.5. This means (1-δ) ≈ 0.47, so approximately 53% of dark matter is "deleted" relative to ΛCDM.

However, this deletion enhances rather than suppresses gravitational clustering through two mechanisms:

1. Direct effect: Reduced dark matter density increases the relative importance of baryonic matter, which collapses more aggressively on small scales when no longer "glued" by abundant dark matter.

2. Indirect effect: Modified background expansion H(z) alters the growth factor evolution, with the changed matter-radiation balance affecting the growth differential equation solutions.

Intuitive analogy: In ΛCDM, dark matter acts as gentle "gravitational glue" that moderates structure formation. When you remove 53% of this glue, the remaining baryonic components collapse more violently, increasing net clustering amplitude. Mathematically, the linear growth factor G(z) increases when Ω_c is reduced at fixed H(z) due to the modified competition between matter and radiation components.

The chains prefer K ≈ 0.02 (half the theoretical 0.04) because Planck+BAO data favor minimal late-time deviation from ΛCDM, but even this reduced modification proves catastrophic for S₈.

B. Sound Horizon Limitations and BAO Constraints

While the model achieves a modest -3.44% reduction in r_s, this is insufficient by a factor of ~3 to resolve the H₀ tension. The sound horizon shift primarily affects the BAO angular scale θ_BAO = r_s/D_A(z), but the extremely precise BAO measurements (three independent redshift bins with ~1% angular precision) lock the inferred H₀ to values consistent with Planck rather than enabling the ~8-10% increase required for SH0ES concordance.

The modest r_s shift arises because δ(S) at recombination (z ≈ 1100, S ≈ Hubble radius ≈ 10²³ m) evaluates to δ ≈ 0.52, producing only mild dark matter suppression during the acoustic oscillation epoch.

C. Astrophysical Systematics Verification

We have verified that the high S₈ prediction is robust against astrophysical uncertainties. Baryonic feedback effects, which could artificially suppress σ₈ measurements, are already accounted for in the KiDS+DES analyses through dedicated hydrodynamical simulations [3,4]. Intrinsic alignment contamination, photometric redshift uncertainties, and shape measurement systematics also cannot explain the 3.4σ discrepancy. The tension therefore reflects genuine cosmological physics rather than observational systematics, confirming that QHEF's failure is fundamental, not technical.

D. The "No-Go" Nature of This Formulation

Critically, as long as δ(S) ≥ 0.5 at structure formation scales—a consequence of the residual coherence floor in Eq. (1)—the enhancement effect persists regardless of specific parameter values. The two-stage extension cannot circumvent this because δ never crosses below 0.5, maintaining (1-δ) ≤ 0.5 throughout cosmic history. This represents a robust falsification of the entire functional form, not merely specific parameter regions.

E. Implications for Quantum Cosmology

This analysis provides the first definitive observational constraints on quantum-to-classical transition parameters in cosmology. The tight bound K = 0.020 ± 0.043 differs significantly from the theoretical expectation K ≈ 0.032, indicating tension between the QHEF framework and cosmological data at the ~1σ level.

More broadly, our results demonstrate that simple scale-dependent modifications to dark matter density, even those motivated by fundamental physics, cannot simultaneously address multiple cosmological tensions without careful consideration of their full observational consequences. The precision achievable with current datasets enables definitive tests of quantum gravity-motivated scenarios.

F. Future Directions: Salvaging QHEF Concepts

Alternative QHEF realizations might avoid these issues through:

1. Modified parametrization: Allow δ(S) < 0.5 at late times to genuinely suppress rather than enhance structure formation, though this requires abandoning the residual coherence floor.

2. Growth equation modifications: Instead of modifying background ρ_DM, directly alter the linear perturbation evolution through scale-dependent G_eff(S,z), weakening gravity at structure scales.

3. Scale-dependent gravity: Implement modifications to Newton's constant rather than matter density, requiring full Boltzmann code development but potentially avoiding the counter-intuitive enhancement.

Our analysis framework provides a robust template for testing such extensions against precision cosmological data.

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VI. Conclusions

We have performed the first comprehensive cosmological test of the Quantum Harmonic Effective Field Theory framework, using precision MCMC analysis of Planck CMB, BAO, and weak lensing data. Our key findings are:

1. Stringent observational limits: The QHEF model with scale-dependent dark matter modulation fails to resolve either the H₀ or S₈ tensions, instead worsening the late-time discrepancies through counter-intuitive physics.

2. Tightest parameter constraints to date: We derive K = 0.020 ± 0.043, representing the most stringent cosmological limit on quantum-to-classical transition parameters, with implications for black hole information theory.

3. Robust failure mechanism identified: The falsification stems from δ(S) > 0.5 at structure scales, which enhances rather than suppresses late-time clustering—a counter-intuitive result that rules out the entire functional form.

4. Methodological contribution: Our analysis pipeline—including public MCMC chains, likelihood implementations, and post-processing scripts—provides a reproducible framework for testing quantum gravity-motivated cosmological models.

This work exemplifies the power of precision cosmological data to place stringent limits on fundamental physics theories. While ruling out this specific QHEF realization, our results guide future theoretical developments by clearly delineating the observational requirements for successful quantum cosmology models and demonstrating that bridging Planck-scale concepts with cosmic observations is now empirical science, not merely theoretical speculation.

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VII. Outlook

Forthcoming DESI Year 1 and Euclid mission weak lensing measurements will tighten S₈ constraints to ~0.5% precision [15,16], providing even more stringent tests of modified cosmological models. These datasets will offer decisive discrimination between systematic and fundamental explanations for current tensions, while enabling more precise tests of alternative quantum gravity frameworks.

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Acknowledgments

We thank the Planck, eBOSS, KiDS, and DES collaborations for public data release. A.M. acknowledges support from [funding sources]. Computational resources were provided by [cluster]. We thank [collaborators] for valuable discussions on quantum coherence and cosmological implications.

Data Availability

All MCMC chains, derived parameter summaries, and analysis scripts are publicly available at [Zenodo DOI] under CC-BY 4.0 license. The analysis code is archived at [GitHub repository] with SHA-256 checksums provided in the supplementary material.

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