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arxiv: 2605.20748 · v1 · pith:RKRRULWLnew · submitted 2026-05-20 · 🌌 astro-ph.CO · gr-qc

Probing late-time deviations from ΛCDM with a quadratic dark energy expansion

Sehjal Khandelwal , Abra\~ao J. S. Capistrano , Suresh Kumar , Rafael C. Nunes This is my paper

Pith reviewed 2026-05-21 02:40 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords quadratic dark energyHubble tensionlate-time deviationscosmological parametersBayesian model selectiondark energy phenomenologyCMB constraintssupernova data
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The pith

A quadratic dark energy expansion raises the inferred Hubble constant and is strongly favored over constant dark energy by combined datasets.

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

The paper tests a quadratic parametrization of dark energy that permits mild evolution only at late times while leaving early-universe physics unchanged. When this model is fit to Planck CMB data, ACT and SPT measurements, DESI galaxy clustering, and Pantheon Plus supernovae, the inferred value of the present-day expansion rate increases relative to the standard model. The higher rate narrows the gap with local distance-ladder measurements. Bayesian evidence across most dataset combinations favors the quadratic extension over a pure cosmological constant.

Core claim

The QDEE model adds two extra parameters that allow the dark energy density to deviate quadratically from a constant at late times. Constraints from Planck 2018, ACT DR6, SPT-3G, DESI DR2, and Pantheon Plus data shift the posterior for the Hubble constant upward while preserving consistency with early-universe observables. Bayesian model comparison yields strong evidence in favor of QDEE, and posterior predictive checks confirm that the model's predictions remain statistically compatible with the observations.

What carries the argument

The quadratic dark energy expansion (QDEE), a two-parameter phenomenological form for the late-time evolution of dark energy density that exactly recovers standard early-universe behavior.

If this is right

  • The inferred Hubble constant moves closer to local measurements without violating CMB constraints.
  • Bayesian evidence favors the quadratic extension over ΛCDM in multiple data combinations.
  • Posterior predictions from the model remain consistent with the observed data within uncertainties.
  • The framework keeps standard early-universe predictions intact while allowing controlled late-time changes.

Where Pith is reading between the lines

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

  • Confirmation would suggest that dark energy evolves slowly rather than remaining exactly constant after recombination.
  • The same parametrization could be tested against future surveys that tighten late-time expansion measurements.
  • It offers a minimal way to explore whether the Hubble tension arises from late-time physics rather than early-universe modifications.

Load-bearing premise

That the quadratic form supplies a sufficient description of late-time deviations and introduces no hidden inconsistencies when the listed datasets are combined.

What would settle it

Future data that force the quadratic parameters to be consistent with zero while returning a lower Hubble constant would remove the reported preference for the model.

Figures

Figures reproduced from arXiv: 2605.20748 by Abra\~ao J. S. Capistrano, Rafael C. Nunes, Sehjal Khandelwal, Suresh Kumar.

Figure 1
Figure 1. Figure 1: FIG. 1. Triangular plots showing the marginalized posterior distributions and parameter degeneracies for (a) the QDEE model [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Joint constraints in the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Reconstructed equation of state [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Log-Bayesian evidence of QDEE and CPL models [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Marginalized posterior distribution of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Posterior predictive distribution of the global [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Posterior predictive distribution of the maximum discrepancy statistic [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We investigate the observational viability of a quadratic dark energy expansion (QDEE) model as a phenomenological extension of the standard $\Lambda\mathrm{CDM}$ cosmological framework. This approach introduces the additional degrees of freedom that permit mild late-time deviations from a constant dark-energy component while preserving the standard early-Universe behavior. We constrain the model using a comprehensive combination of cosmological datasets, including Planck 2018 cosmic microwave background (CMB) measurements, Atacama Cosmology Telescope (ACT) Data Release 6 (DR6) and South Pole Telescope (SPT-3G) data, Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), and the Pantheon Plus type Ia supernova compilation. Our results show that the QDEE framework shifts the inferred Hubble constant toward higher values relative to $\Lambda\mathrm{CDM}$, partially alleviating the tension with local measurements while remaining consistent with early-Universe constraints. Bayesian model comparison indicates strong evidence in favor of this framework over standard $\Lambda\mathrm{CDM}$ across multiple dataset combinations. Posterior predictive checks further demonstrate that the model yields predictions consistent with the observed data within statistical uncertainties.

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 / 2 minor

Summary. The manuscript introduces a quadratic dark energy expansion (QDEE) as a phenomenological late-time extension to ΛCDM that adds degrees of freedom for mild deviations from constant dark energy density while exactly preserving standard early-Universe evolution. Using Planck 2018 CMB, ACT DR6, SPT-3G, DESI DR2, and Pantheon+ data, the authors report that the model shifts the inferred Hubble constant upward relative to ΛCDM, partially easing the H0 tension, yields strong Bayesian evidence favoring QDEE over ΛCDM for multiple dataset combinations, and passes posterior predictive checks.

Significance. If the central results hold, the work supplies a minimal, late-time-only parameterization that can be used to test whether the Hubble tension can be addressed without modifying early-Universe physics. The multi-probe dataset combination and explicit Bayesian model comparison are positive features that allow direct comparison with other extensions.

major comments (2)
  1. [§2] §2 (Model): the explicit functional form of the quadratic dark energy expansion (e.g., the expression for ρ_DE(a) or w_DE(a)) is not stated, so it is impossible to verify the claim that early-Universe behavior is exactly preserved or to assess possible degeneracies with the sound horizon or other parameters.
  2. [§4] §4 (Analysis and priors): prior ranges and functional forms for the two quadratic coefficients are not specified, nor are degeneracy checks between these coefficients and H0 reported; without these the quoted 'strong evidence' and the magnitude of the H0 shift cannot be reproduced or assessed for robustness.
minor comments (2)
  1. [Table 1] Table 1: column headers for the different dataset combinations are abbreviated without a legend, reducing readability.
  2. [Figure 3] Figure 3: the posterior contours for the quadratic coefficients are shown without the corresponding ΛCDM reference values overlaid, making direct visual comparison harder.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address each major comment in detail below and have revised the manuscript to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [§2] §2 (Model): the explicit functional form of the quadratic dark energy expansion (e.g., the expression for ρ_DE(a) or w_DE(a)) is not stated, so it is impossible to verify the claim that early-Universe behavior is exactly preserved or to assess possible degeneracies with the sound horizon or other parameters.

    Authors: We agree that the explicit functional form was not stated with sufficient mathematical detail in Section 2 of the original submission. In the revised manuscript we have added the precise expression for ρ_DE(a) (and the corresponding w_DE(a)) that defines the QDEE model. The parameterization is constructed so that the additional quadratic terms vanish or become negligible as a → 0, ensuring that the early-Universe evolution is identical to that of ΛCDM and that the sound horizon at recombination remains unchanged. We have also included a short discussion of the absence of degeneracies with early-Universe parameters arising from the strictly late-time nature of the extension. revision: yes

  2. Referee: [§4] §4 (Analysis and priors): prior ranges and functional forms for the two quadratic coefficients are not specified, nor are degeneracy checks between these coefficients and H0 reported; without these the quoted 'strong evidence' and the magnitude of the H0 shift cannot be reproduced or assessed for robustness.

    Authors: We acknowledge this omission. In the revised Section 4 we now explicitly state the prior ranges and functional forms adopted for the two quadratic coefficients. We have also added degeneracy diagnostics, including the correlation coefficients between the quadratic coefficients and H0 together with a supplementary figure of the relevant joint posteriors. These additions allow the quoted Bayesian evidence and the reported H0 shift to be reproduced and assessed for robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a quadratic dark energy parametrization as a phenomenological late-time extension, fits its parameters to standard datasets (Planck, ACT, SPT, DESI, Pantheon+), and performs Bayesian evidence comparison against ΛCDM. No derivation step reduces by construction to its inputs: the model is not self-defined in terms of its outputs, no fitted parameter is relabeled as an independent prediction, and no load-bearing claim rests on a self-citation chain or imported uniqueness theorem. The reported H0 shift and model preference emerge from posterior constraints on the extra parameter, which are externally falsifiable against the listed observations. This is a standard data-driven analysis with no internal reduction to tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a quadratic late-time expansion can be added without altering early-universe physics and that the listed datasets can be combined without additional unmodeled systematics. Two free parameters are introduced and fitted to data.

free parameters (1)
  • quadratic dark energy coefficients
    Two additional degrees of freedom that control the late-time deviation and are constrained by the cosmological data.
axioms (1)
  • domain assumption Early-universe cosmology remains exactly standard Lambda CDM
    The model is stated to preserve standard early-Universe behavior.

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