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arxiv: 2606.25184 · v1 · pith:46G556EWnew · submitted 2026-06-23 · 🌌 astro-ph.CO

A cross-epoch endpoint-consistency test of a single effective scaling from dark energy to inflation

Martin Drobczyk This is my paper

Pith reviewed 2026-06-25 22:39 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords dark energyinflationscaling relationendpoint consistencycosmological turnarounddensity-responsive gravityvacuum dilution
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The pith

A single power-law scaling from H0 to H* matches dark energy to inflation with γ ≃ 0.491 and β ≃ 0.68.

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

The paper formulates an endpoint-consistency test that extrapolates one effective scaling relation between the present-day dark-energy scale and the inflationary Hubble rate. Anchored by late-time closure relations, the extrapolation over a lever arm of 10^55 yields matching coefficients of order unity for the benchmark case. This produces a concrete prediction that the effective vacuum energy density dilutes faster than spatial curvature, opening the possibility of a cosmological turnaround in a closed universe at curvature densities around 10^{-4}. The test compresses uncertainties on the scaling exponent into a narrow band and supplies a full error budget including threshold and observational contributions.

Core claim

The central claim is that the power-law form M_U(μ) = β^{1/4} M_P (μ/M_P)^γ, normalized at μ = H0 via the DRG closure relations, when evaluated at the Starobinsky inflationary scale, produces a residual matching factor C_infl of order unity for γ ≃ 0.491 and β ≃ 0.68. Varying the exponent p in the potential shifts γ only mildly (0.45–0.52) and favors p ≃ 3–4. As a direct consequence the vacuum sector evolves as ρ_Φ ∝ a^{-6γ}, which exceeds the dilution rate of curvature for γ > 1/3 and therefore permits a turnaround at |Ω_K| ∼ O(10^{-4}).

What carries the argument

The effective scaling M_U(μ) = β^{1/4} M_P (μ/M_P)^γ that carries the extrapolation from the late-time anchor to the inflationary epoch and defines the residual C_infl.

If this is right

  • Matching at the inflationary scale selects γ ≃ 0.491 and β ≃ 0.68 for the benchmark V0 ∝ M_U^4 with c4 = O(1).
  • Changing the potential exponent p across [2,8] moves γ only mildly within 0.45–0.52 and favors natural values near p ≃ 3–4.
  • The vacuum energy density scales as ρ_Φ ∝ a^{-6γ} at late times, diluting faster than spatial curvature whenever γ > 1/3.
  • In a closed universe the faster dilution allows a cosmological turnaround at curvature densities |Ω_K| ∼ O(10^{-4}).

Where Pith is reading between the lines

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

  • Tighter late-time measurements of w0 and A_s would shrink the allowed window on γ without requiring new high-energy data.
  • The predicted dilution law ρ_Φ ∝ a^{-6γ} supplies a distinct late-time expansion history that can be contrasted with ΛCDM using distance-redshift surveys.
  • The same scaling supplies a concrete target for model-building attempts that seek a single effective description across the two epochs.

Load-bearing premise

The normalization is fixed at μ = H0 by the late-time dark energy closure relations of the density-responsive gravity framework.

What would settle it

A measurement showing that C_infl lies many orders of magnitude away from unity after the full uncertainty budget (including w0, A_s, H*, and threshold effects) is applied, or the absence of a turnaround signature at |Ω_K| ∼ 10^{-4} in Stage-IV data.

Figures

Figures reproduced from arXiv: 2606.25184 by Martin Drobczyk.

Figure 1
Figure 1. Figure 1: Left: normalization β(γ) from the late-time anchor. Right: matching factor Cinfl(γ). Shaded bands indicate the one-decade (Cinfl ∈ [0.1, 10]) and two-decade (Cinfl ∈ [0.01, 100]) viable windows. The black circle marks the best fit γ = 0.491. The red square marks the reference γ = 0.5. At the best fit, β ≃ 0.68 and Cinfl ≃ 1. 4.4. Sensitivity to operator scaling The baseline identification V RG 0 = c4 M4 U … view at source ↗
Figure 2
Figure 2. Figure 2: Viability region in the (γ, w0) plane for the baseline operator scaling V RG 0 = 3 M4 U (H∗). Solid line: Cinfl = 1. Dashed lines: one-decade naturalness window | ln Cinfl| < ln 10, i.e. Cinfl ∈ [0.1, 10]. Dotted lines: two-decade window | ln Cinfl| < ln 100, i.e. Cinfl ∈ [0.01, 100]. Circle: benchmark (γ = 0.491, w0 = −0.989). Dash-dotted: γ = 0.50 (naive walking). 15 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 3
Figure 3. Figure 3: Pilot Bayesian posterior on (γ, log10 β, H0, Ωm, ε∗, As) from a 64×2×104 -step emcee run with Planck 2018 anchors and the two likelihood constraints described in the text. The narrow γ posterior (top diagonal) sits at γ = 0.491 ± 0.002, matching the algebraic best-fit of Section 4.3. The (γ, log10 β) panel shows the expected anti-correlation along the late-time degenerate direction. Black solid line: algeb… view at source ↗
read the original abstract

A cross-epoch endpoint-consistency test is formulated for a single power-law effective scaling, $M_U(\mu)=\beta^{1/4}\,M_P(\mu/M_P)^\gamma$, connecting late-time cosmic acceleration to the inflationary energy scale. The normalization is anchored at $\mu=H_0$ by the late-time dark energy closure relations of the density-responsive gravity (DRG) framework and extrapolated to the inflationary Hubble rate $\mu=H_*$. Comparison with the CMB-normalized Starobinsky plateau defines the residual matching factor $C_{\rm infl}\equiv V_0^{A_s}/V_0^{\rm RG}$. The novelty is the formulation of the inflation - dark-energy connection as a quantitative endpoint test between two empirically anchored energy scales. The lever arm $H_*/H_0\sim 10^{55}$ compresses endpoint uncertainties into $\delta\gamma\propto [p\ln(H_*/H_0)]^{-1}\delta\ln X$, so requiring $C_{\rm infl}=\mathcal{O}(1)$ selects a narrow consistency band. For the benchmark $V_0\propto M_U^4$ with $c_4=\mathcal{O}(1)$, matching requires $\gamma\simeq 0.491$ and $\beta\simeq 0.68$, both $\mathcal{O}(1)$. Varying $p\in[2,8]$ shifts $\gamma$ only mildly (approximately $0.45$--$0.52$), with natural matching favoring $p\simeq 3$--$4$. We present the full uncertainty budget, including cumulative threshold effects ($\Xi$) and observational errors on the late-time anchor ($w_0$, $A_s$, $H_*$). A secondary consequence is that the effective vacuum sector dilutes as $\rho_\Phi\propto a^{-6\gamma}$ at late times, faster than spatial curvature for $\gamma>1/3$. In a closed universe, this permits a cosmological turnaround at $|\Omega_K|\sim\mathcal{O}(10^{-4})$, testable with Stage-IV surveys and qualitatively distinct from $\Lambda$CDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper formulates a cross-epoch endpoint-consistency test for the single power-law effective scaling M_U(μ)=β^{1/4} M_P (μ/M_P)^γ that connects the late-time dark-energy scale at μ=H_0 to the inflationary Hubble rate μ=H_*. The normalization is fixed at H_0 by the late-time closure relations of the density-responsive gravity (DRG) framework; extrapolation across the lever arm H_*/H_0∼10^{55} is then required to produce C_infl≡V_0^{A_s}/V_0^{RG}=O(1) when matched to the CMB-normalized Starobinsky plateau. For the benchmark V_0∝M_U^4 with c_4=O(1) this selects γ≃0.491 and β≃0.68; the values remain O(1) under variation of the exponent p∈[2,8]. A secondary consequence is that the effective vacuum energy dilutes as ρ_Φ∝a^{-6γ}, permitting a cosmological turnaround at |Ω_K|∼O(10^{-4}) in a closed universe.

Significance. If the DRG late-time closure relations hold and are independently validated, the large lever arm converts the endpoint-matching requirement into a narrow consistency band for the two free parameters γ and β, both of which emerge O(1). The same scaling supplies a concrete, falsifiable prediction (faster-than-curvature dilution allowing a turnaround at |Ω_K|∼10^{-4}) that is qualitatively distinct from ΛCDM and testable with Stage-IV surveys. The manuscript also supplies a full uncertainty budget that includes threshold effects Ξ and observational errors on w_0, A_s and H_*.

major comments (2)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: the normalization M_U(H_0) is taken directly from the late-time dark-energy closure relations of the DRG framework. No independent derivation or external validation of those relations is supplied in the present manuscript; without them the power-law has no empirical anchor and the subsequent extrapolation to H_* reduces to an internal consistency condition on parameters already fitted inside DRG.
  2. [Abstract] Abstract: the claim that the procedure constitutes an 'independent cross-epoch test' is undermined by the fact that both the anchor and the requirement C_infl=O(1) are internal to the DRG framework. The derived values γ≃0.491, β≃0.68 are therefore best described as the parameter region in which the DRG scaling remains consistent with Starobinsky inflation rather than a prediction tested against an external datum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for highlighting the need to clarify the scope of our cross-epoch test. We respond to the major comments below, emphasizing that the work assumes the DRG late-time relations as input from prior work and tests their consistency with inflation as an external scale.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] the normalization M_U(H_0) is taken directly from the late-time dark-energy closure relations of the DRG framework. No independent derivation or external validation of those relations is supplied in the present manuscript; without them the power-law has no empirical anchor and the subsequent extrapolation to H_* reduces to an internal consistency condition on parameters already fitted inside DRG.

    Authors: We agree that this manuscript does not re-derive or independently validate the DRG closure relations; these are adopted from the established DRG framework. The present analysis focuses on extrapolating the resulting power-law scaling to the inflationary epoch and requiring consistency with the Starobinsky model normalized by CMB data. The empirical content comes from matching to the independently observed inflationary scale, providing a test of whether the DRG-derived form remains viable across epochs. We are prepared to revise the abstract to explicitly state that the late-time relations are taken as given from DRG. revision: partial

  2. Referee: [Abstract] the claim that the procedure constitutes an 'independent cross-epoch test' is undermined by the fact that both the anchor and the requirement C_infl=O(1) are internal to the DRG framework. The derived values γ≃0.491, β≃0.68 are therefore best described as the parameter region in which the DRG scaling remains consistent with Starobinsky inflation rather than a prediction tested against an external datum.

    Authors: The inflationary scale and the condition C_infl = O(1) are not internal to DRG. Starobinsky inflation and its CMB normalization (A_s) are standard results from inflationary cosmology, independent of the DRG framework. The test is whether the DRG scaling, anchored at late times, produces a value of V_0 that matches this external inflationary datum within O(1). The narrow selection of γ ≈ 0.49 and β ≈ 0.68 is thus a consequence of this cross-epoch requirement. We note that the abstract in the manuscript describes it as an 'endpoint-consistency test' rather than explicitly 'independent', but we can adjust wording to avoid any ambiguity. revision: partial

Circularity Check

1 steps flagged

Normalization at H0 relies on DRG closure relations; γ,β values are consistency conditions within that framework

specific steps
  1. self citation load bearing [Abstract]
    "The normalization is anchored at μ=H0 by the late-time dark energy closure relations of the density-responsive gravity (DRG) framework and extrapolated to the inflationary Hubble rate μ=H*."

    The load-bearing normalization and scaling premise is taken directly from the DRG framework (prior work by the same author). The subsequent extrapolation and requirement that C_infl≡V0^{A_s}/V0^{RG}=O(1) then selects γ and β, rendering the 'consistency test' and O(1) values dependent on the internal DRG closure relations rather than independent external data.

full rationale

The derivation anchors the power-law scaling at μ=H0 using late-time dark energy closure relations from the DRG framework, then extrapolates across H*/H0∼10^55 and requires C_infl=O(1) to obtain γ≃0.491, β≃0.68. This makes the endpoint test a consistency condition on the DRG anchor rather than an independent cross-epoch prediction against external benchmarks. The central claim therefore reduces to the self-cited DRG relations for its empirical starting point, with the lever-arm compression applying only after that anchor is granted. No other circular steps are exhibited in the provided text.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the DRG closure relations at H0 (not re-derived here), the assumption that the same power-law form holds from H0 to H*, the choice of V0 scaling with M_U^4, and the requirement that C_infl remain O(1). These are not standard math axioms but domain assumptions imported from prior work.

free parameters (3)
  • γ
    Slope of the effective scaling; selected to make C_infl = O(1) after extrapolation.
  • β
    Prefactor of the effective scaling; selected to make C_infl = O(1).
  • p
    Auxiliary exponent in the potential scaling; varied in [2,8] and shown to affect γ only mildly.
axioms (2)
  • domain assumption Late-time dark energy closure relations of the DRG framework hold at μ = H0.
    Abstract paragraph 2; supplies the anchor point for the entire extrapolation.
  • domain assumption The same functional form M_U(μ) = β^{1/4} M_P (μ/M_P)^γ remains valid from H0 to H*.
    Core modeling choice stated in the first sentence of the abstract.

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discussion (0)

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Reference graph

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