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Quantum corrections to the Friedmann equation can make the early- and late-universe Hubble constants agree.

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2026-08-01 10:25 UTC pith:H7DE4OWH

load-bearing objection A transparent but non-derived interpolation: the paper's h̃ is a fitting device, not a quantum prediction. the 5 major comments →

arxiv 2607.27285 v1 pith:H7DE4OWH submitted 2026-07-29 gr-qc

Hubble tension problem encompassed by phase-space quantum cosmology

Alex E. Bernardini This is my paper
classification gr-qc
keywords Hubble tensionquantum cosmologyWigner functionphase-space quantum mechanicsFriedmann equationWheeler-DeWitt equationminisuperspaceeffective quantum potential
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Hubble tension—the mismatch between early-universe and late-universe measurements of the expansion rate—could be a signature of residual quantum effects in cosmology rather than new physics. The paper shows that if the universe's quantum state has a sinusoidal Wigner function in phase space, the quantum back-reaction alters the Friedmann equation through a one-parameter coefficient b_κ(μ). The resulting effective Hubble parameter is about 0.673 at early times and about 0.732 at late times, smoothly interpolating between the two measured values. A sympathetic reader would care because this offers a purely quantum-mechanical resolution of the tension without modifying dark energy or adding fields, and it extends to models with curvature and dark-sector changes.

Core claim

The paper's central claim is that quantum corrections derived from Weyl-Wigner phase-space mechanics can dissolve the Hubble tension. Starting from a minisuperspace action and the Wheeler-DeWitt equation, it identifies the classical potential as a sum of powers of the scale factor and computes the effective quantum potential induced by a sinusoidal Wigner function W(x,k;τ)=g(x;τ) sin(μ k x). Each power x^{−κ} in the potential is rescaled by a factor b_κ(μ)=2κa_κ μ^{−κ}[(2−μ)^{−κ}−(2+μ)^{−κ}], which modifies the matter and radiation energy densities in the Friedmann equation. The result is an effective Hubble parameter h̃(x) that sits at about 0.673 at last scattering and rises smoothly to ab

What carries the argument

The load-bearing object is the sinusoidal Wigner quasi-distribution W(x,k;τ)=g(x;τ) sin(μ k x), the only class of phase-space states for which the effective potential U(x) is independent of momentum k. Its parameter μ sets the phase-space coherence scale (larger μ means stronger quantum interference), and it enters the correction coefficient b_κ(μ) that rescales every power of the scale factor in the minisuperspace potential. This coefficient converts ordinary ΛCDM densities into smeared effective densities, thereby shifting the expansion history. Because b_κ is a function of one free parameter, the model can be tuned to reproduce both the early- and late-time Hubble constants with a single

Load-bearing premise

The mechanism rests on the assumption that the universe's quantum state has a Wigner function of the special form W=g(x;τ) sin(μ k x); if the true state is not of this form, the derived rescalings b_κ(μ) and the interpolated Hubble parameter do not follow.

What would settle it

Compute the Wigner function for a concrete Wheeler-DeWitt wave packet of a minisuperspace model and check whether it has the sinusoidal form; if it does not, the correction coefficients do not apply. Alternatively, measure the expansion rate at intermediate redshifts (z≈0.5–2): the mechanism predicts a smooth monotone rise of the effective Hubble parameter from 0.673 to 0.732, whereas a sharp step would indicate early dark energy and falsify this quantum-interpolation picture.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If correct, the Hubble tension is resolved without new particles, modified gravity, or early dark energy; the correction is purely from residual quantum structure of the cosmological state.
  • A single parameter μ, adjusted together with the lapse-function parameter σ, fits both the early-universe value 0.673 and the late-universe value 0.732 within their quoted errors.
  • Matter and radiation density shifts stay within roughly 6.5% and 2% for μ between 1/2 and 1, so the mechanism is compatible with CMB calibrations of baryon and matter densities.
  • The same quantum-correction scheme can be applied to models with spatial curvature and modified dark sector, where small |Ω_K| shifts the allowed parameter region without spoiling the fit.
  • Sound-horizon physics and the late-time distance ladder are assumed unmodified, so the BAO scale and low-redshift expansion data remain as in ΛCDM.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the interpolation is real, it should be observable in intermediate-redshift expansion-rate data (e.g., cosmic chronometers or BAO at z≈0.5–2), which would show a smooth rise rather than a step-like transition; this distinguishes the proposal from early-dark-energy models.
  • The free parameter μ could be derived rather than fitted by computing the Wigner function of a concrete Wheeler-DeWitt wave packet in a given minisuperspace model; until then the mechanism is a proof-of-principle template.
  • The same 'quantum envelope' trick—sinusoidal modulations of the Wigner function—could be used to parametrize quantum corrections to other cosmological observables, such as the matter fluctuation amplitude S8, effectively trading cosmological tensions for a phase-space coherence scale.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper proposes a Weyl–Wigner phase-space quantum cosmological framework to address the Hubble tension. Starting from a minisuperspace action, it defines Wigner functions of sinusoidal and hyperbolic form and derives quantum-corrected potentials U(x) with coefficients b_kappa(mu) and c_kappa(mu). By reinterpreting U(x) through an effective Hubble parameter h_tilde(x), the paper claims a smooth interpolation between early-time h ~ 0.673 and late-time h ~ 0.732, thereby 'suppressing' the Hubble tension. The analysis is extended to include spatial curvature, with constraints on mu, sigma, and Omega_K. The core claim is that residual quantum corrections to the Einstein–Friedmann equation predict this interpolation and resolve the tension within LambdaCDM.

Significance. If the central claim were established, the paper would offer a genuinely new quantum-cosmological mechanism for the Hubble tension, with explicit analytic coefficients and a straightforward curvature extension. The derivation of b_kappa(mu) in Eq. (13) is algebraically explicit and internally consistent. However, as the paper itself concedes in Sec. III C, the step from the Wigner-corrected potential U(x) to a modified Hubble parameter is a 'misinterpretation' unless a quantum-corrected Friedmann equation is actually derived. No such equation is derived; instead, the b coefficients are inserted into the standard LambdaCDM distance integrals. The parameter constraints in Figs. 1–3 are constructed by fitting the two target H0 values, so the claimed interpolation is imposed rather than predicted. The manuscript is best viewed as a proof-of-principle of a phenomenological rescaling, not a derivation from quantum cosmology.

major comments (5)
  1. [Sec. III C, Eqs. (22)–(24)] The central claim requires a quantum-corrected Friedmann equation, but none is derived. The text reads U(x) = (h_tilde/h_LT)^-2 V(x) and explicitly says this 'would be misinterpreted as a modulation from a modified Hubble parameter.' Nevertheless, Fig. 1 and Eq. (24) treat h_tilde as the expansion-rate input in the standard LambdaCDM sound-horizon and angular-diameter integrals. Without a Wigner-corrected Einstein–Friedmann equation derived from the minisuperspace dynamics, the 0.673 to 0.732 interpolation is a phenomenological rescaling of energy densities, not a prediction of quantum cosmology.
  2. [Sec. III C, Fig. 1] The parameter constraints are obtained by requiring h_tilde^2(1) ~ 0.732 +/- 0.013 and h_tilde^2(x << 1) ~ 0.673 +/- 0.006. Equation (24) then uses the same parameter space to fit the same two values. The output values are therefore the inputs of the constraint procedure; the agreement in Figs. 1–3 cannot be counted as independent evidence. A genuine prediction would fix mu and sigma from the quantum state or another observable and then compute H0.
  3. [Sec. III C, Eq. (22)] The coefficient b_{sigma-4}(mu) is set equal to unity by hand: the text states that 'smooth quantum effects are assumed to be suppressed at very late times.' This imposes the late-time value h ~ 0.732 instead of deriving it from the Wigner dynamics. Equation (13) fixes b_{sigma-4}(mu) uniquely for a given mu and sigma; setting it to unity is an additional, untested assumption. It contradicts the abstract's claim that the quantum corrections are 'analytically derived.'
  4. [Sec. III A, footnote 4] The sinusoidal Wigner form W(x,k;tau) = g(x;tau) sin(mu k x) is adopted specifically so that the effective potential U becomes momentum-independent. Footnote 4 admits that the profiles are not meant to reconstruct a microscopic wave function of the Universe. No argument connects a Wheeler–DeWitt wave packet, squeezed state, or coherent superposition to this particular Wigner form. The b_kappa(mu) coefficients are therefore artifacts of a chosen ansatz, and the characterization of the results as valid for 'generic localized phase-space quantum states' is not supported.
  5. [Sec. IV A, Eq. (26) and Fig. 4] There is an inconsistency in the assignment of b coefficients to matter and radiation. Equation (24) attaches b_{sigma-1}(mu) to matter and b_sigma(mu) to radiation; Eq. (26) and Fig. 4 use the opposite assignment (b_sigma for matter, b_{sigma-1} for radiation). If these are meant to be the same coefficients, at least one equation is incorrect. Additionally, Fig. 4 claims a 6.5% matter fluctuation is 'acceptably covered by the Planck data statistical deviations,' but the quoted Planck error is omega_m = 0.142 +/- 0.001, i.e. approximately 0.7%, making the consistency claim unclear.
minor comments (4)
  1. [Sec. IV A] The equation numbering jumps to '(103)' just before Eq. (25); this appears to be a typographical error.
  2. [Sec. IV A] 'FLR W framework' should read 'FLRW framework'.
  3. [References] Refs. [10] and [15] are the same paper (Schöneberg et al., 'The H0 Olympics') but appear under different numbers; this should be consolidated.
  4. [Sec. IV A, Fig. 4] The caption describes 'b_sigma(mu) (blue region) and b_{sigma-1}(mu) (red region)' as matter and radiation coefficients. This is inconsistent with Eq. (24), where b_{sigma-1} multiplies the matter term; please check the notation throughout.

Circularity Check

4 steps flagged

The headline h̃ interpolation is obtained by constraining σ and μ to reproduce the input values; Eq. (24) takes hLT as input, and the late-time normalization is imposed by hand.

specific steps
  1. fitted input called prediction [Sec. III C, after Eq. (23), Fig. 1 discussion]
    "parameters σ and µ can be constrained one to each other as to return the expected results for early and late times, i.e. ˜h2(1) ∼ 0.732 ± 0.013 and ˜h2(x ≪ 1) ∼ 0.673 ± 0.006. The phenomenological parameters were set as h = 0.732, ωm = 0.142 ± 0.001 and ωγ = 2.47 × 10−5."

    The two values announced as the model's early/late predictions are exactly the values imposed on the free parameters σ and μ. Because h̃ is defined through U/V and h appears in U as h^{-2}, setting h=0.732 and choosing μ,σ to force h̃(1)=0.732 and h̃(x≪1)=0.673 means the claimed 'transient plateaus' are inputs, not outputs.

  2. self definitional [Sec. III C, paragraph after Eq. (23)]
    "As a net effect, one could read the results from quantum corrections, Eqs. (22) and (23), as U (x) = ( ˜h/hLT )−2V(x) and Uhyp(x) = ( ˜h/hLT )−2V(x), respectively ... which would be misinterpreted as a modulation from a modified Hubble parameter, ˜h ... Results from U (x) (sinusoidal modulation) exhibit the phenomenologically expected well-defined transient plateaus for ˜h from ∼ 0.673 to ∼ 0.732 at early and late times, respectively."

    No quantum-corrected Friedmann equation is derived; h̃ is introduced simply as the ratio U/V written in Hubble units. The paper even warns that reading U as a modified Hubble parameter 'would be misinterpreted', yet Figs. 1-3 and Eq. (24) use that reading as the predicted H0. The claimed interpolation is therefore a rewriting of the chosen bκ(μ), not an independent dynamical result.

  3. fitted input called prediction [Sec. IV, Eq. (24) and Fig. 2]
    "By turning Eq. (4) into an iteration equation for obtaining hLS → ˜h(µ, σ), one would have hLS → ˜h(µ, σ) = ... [ωm bσ(µ) (1 +z)3 + (h2LT − ωm)] ... hLT , again with bκ(µ) from Eq. (13), and with w = −1, ωb = 0.0224 ± 0.0001 and zLS ≃ 1080."

    The target value hLT=0.732 appears as an explicit input in Eq. (24), while bκ(μ) are adjustable functions of μ. Fig. 2 then selects the (μ,σ) region that returns hLT=0.732±0.013 and hLS=0.673±0.006. This is a fit of the same two numbers under the name of a quantum-corrected hLS, not an independent prediction.

  4. other [Sec. III C, after Eq. (23)]
    "The coefficients cσ−4(µ) and bσ−4(µ) are indeed set equal to unity, since the smooth quantum effects are assumed to be suppressed at very late times ( x ≲ 1)."

    The late-time value h̃→hLT is guaranteed by hand: setting b_{σ−4}=1 removes the quantum modification from the x^4 (dark-energy) term at x≲1. Thus the 0.732 late-time plateau is an imposed boundary condition, not a consequence of the Wigner-corrected dynamics.

full rationale

The Wigner-to-potential algebra (Eqs. 10-13) is self-contained: bκ(μ) follows from the sinusoidal-ansatz manipulation and no circularity arises there. The special Wigner profile itself is an acknowledged assumption (footnote 4), not a fitted/prediction loop, and self-citations to [26] merely back up displayed equations. The circularity is in the step from U(x) to the headline H0: h̃ is defined as U/V, then σ and μ are constrained to return exactly h̃≃0.732 and h̃≃0.673; Eq. (24) repeats the exercise with hLT entering as an input; and b_{σ−4}=1 is imposed to force the late-time limit. The paper itself cautions that interpreting U as (h̃/hLT)^{-2}V 'would be misinterpreted as a modulation from a modified Hubble parameter', yet that reading drives Figs. 1-3 and Eq. (24). Hence the central 'prediction' reduces, partly by construction, to the phenomenological values it is supposed to explain. Score 6 reflects partial circularity; the underlying phase-space potential computation is not itself circular.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new particles or forces. Its central claim instead rests on two free parameters (μ, σ) and several ad hoc modeling choices: the sinusoidal Wigner ansatz, the reinterpretation of the corrected potential as a modified Hubble parameter, and the assumption that sound-horizon physics is untouched. These are not independently evidenced.

free parameters (3)
  • μ (phase-space coherence scale) = ~0.5–1, constrained by contours in Figs. 1–3
    Modulates the sinusoidal Wigner function; chosen to make h̃(1)=0.732 and h̃(x≪1)=0.673. The paper states 'obtaining fitting parameters μ ≫ 1 would ruin our proposal.'
  • σ (lapse-function parameter) = constrained together with μ; Fig. 3 allows σ=0 or integer values
    Parameter associated with the choice of lapse function N in the minisuperspace action. It is tuned, jointly with μ, to match the early and late H0 values. The paper notes it 'could be set equal to zero (or even any integer value).'
  • Ω_K (spatial curvature) = scanned over [-0.005, 0)
    Curvature parameter introduced in Eq. (26); the paper does not fit it but uses a range allowed by observations, and shows it shifts the allowed (μ, σ) region.
axioms (5)
  • ad hoc to paper The Wigner function of the universe has the sinusoidal form W(x,k;τ)=g(x;τ)sin(μkx), chosen so that the effective potential U becomes momentum-independent.
    Sec. III A introduces this ansatz; footnote 4 states the profiles are not meant to reconstruct a specific wave function of the universe. The entire correction mechanism depends on this special functional form.
  • ad hoc to paper The quantum-corrected potential U(x) can be reinterpreted as a modified cosmological energy density with a scale-dependent Hubble parameter h̃(x), i.e., U(x)=(h̃/h_LT)^{-2}V(x).
    Sec. III C: 'one could read the results from quantum corrections ... as U(x) = (h̃/h_LT)^{-2}V(x)' — an asserted identification, not derived from the Wheeler–DeWitt equation.
  • ad hoc to paper Quantum corrections do not affect the sound horizon or late-time distance ladder; only the energy-density terms are rescaled by b_κ(μ).
    Sec. IV states: 'the sound propagation and late time measurements are assumed to not be affected by quantum corrections.' This assumption is necessary for Eq. (24) to yield the desired results.
  • domain assumption The minisuperspace Wheeler–DeWitt quantization of the Friedmann–Robertson–Walker universe (Eqs. 17–20) provides the correct arena for computing quantum backreaction.
    Standard quantum-cosmology framework; the paper invokes the WDW equation and the minisuperspace action without defending this choice against other quantum-gravity approaches.
  • domain assumption The Wigner currents in Eqs. (7)–(8) and the replacement of the classical potential V by the effective potential U correctly encapsulate quantum backreaction in the cosmological setting.
    The continuity-equation formulation of Wigner flow is imported from earlier papers [31–34]; the paper does not justify that this phase-space backreaction remains valid when the potential is a minisuperspace gravitational constraint.

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read the original abstract

Analytical solutions encompassing the so-called Hubble tension problem are revisited through the framework of Weyl--Wigner quantum mechanics and discussed in the context of generalized phase-space scenarios of quantum cosmology. After reviewing the nature of the problem and its recent developments, an extended formulation constructed within the quantum phase-space framework to address the Hubble tension is proposed. For the quantum cosmology described in the minisuperspace framework through (generic) localized phase-space quantum states, when residual quantum corrections to the Einstein--Friedmann equation are analytically derived, quantum effects are shown to suppress the Hubble tension divergence between early- and late-time predictions. Besides addressing the Hubble tension problem within the standard $\Lambda$CDM cosmological model, our approach encompasses generalized quantum cosmological scenarios that also include curvature and dark sector modifications.

Figures

Figures reproduced from arXiv: 2607.27285 by Alex E. Bernardini.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

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