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arxiv: 2606.20794 · v1 · pith:GBHNOSN2new · submitted 2026-06-18 · 🌌 astro-ph.CO

Constraints on Horndeski Gravity with Phantom Crossing

Krishna Naidoo , James Hallam , Tessa Baker , Sergi Sirera This is my paper

Pith reviewed 2026-06-26 15:52 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords Horndeski gravityphantom crossingAsymptotic Cubic Galileondark energy equation of statecosmological constraintsgalaxy-ISW cross-correlationvoid force profilew0waCDM
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The pith

Asymptotic Cubic Galileon models within Horndeski gravity reproduce the phantom-crossing expansion history favored by Planck, DESI and DES data.

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

The paper introduces the Asymptotic Cubic Galileon subclass of Horndeski scalar-tensor theories, which permits the dark energy equation of state to cross w = -1 while keeping minimal coupling to matter. These models are shown to match the joint expansion history inferred from Planck CMB, DESI baryon acoustic oscillations and Dark Energy Survey supernovae. Perturbative measurements, specifically galaxy-ISW cross-correlations and void force profiles, then restrict the surviving parameter space to a bounded region of the broader Horndeski landscape. Model-comparison statistics indicate that the ACG realizations perform at a level comparable to the phenomenological w0waCDM parametrization and superior to LambdaCDM, while supplying an explicit Lagrangian origin for the observed behavior.

Core claim

ACG models provide a concrete Lagrangian realization of the w0waCDM phenomenology that simultaneously satisfies current expansion-history data and is further bounded by perturbative observables, thereby confining viable Horndeski realizations to a well-defined and testable region while remaining minimally coupled to matter.

What carries the argument

The Asymptotic Cubic Galileon (ACG) subclass of Horndeski theories, which generates phantom crossing through its specific cubic Galileon term while preserving minimal matter coupling.

If this is right

  • ACG models jointly reproduce the expansion history from Planck CMB, DESI BAO and DES supernovae.
  • Galaxy-ISW cross-correlation and void force profile data confine viable ACG models to a bounded region of Horndeski parameter space.
  • Both chi-squared and Bayesian evidence favor ACG realizations over LambdaCDM at a level comparable to w0waCDM.
  • The Lagrangian formulation supplies a theoretical foundation for the w0waCDM behavior and motivates tests on nonlinear scales.

Where Pith is reading between the lines

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

  • The bounded ACG region yields concrete predictions for the growth of structure that upcoming large-scale surveys can target directly.
  • Relaxing the minimal-coupling assumption while retaining the cubic Galileon structure could enlarge the testable space without losing phantom-crossing capability.
  • Because the models are derived from an action, higher-order statistics such as bispectrum signals become calculable and distinguishable from pure w0waCDM phenomenology.

Load-bearing premise

That the ACG subclass can sustain phantom crossing while remaining minimally coupled to matter and that the chosen galaxy-ISW and void observables supply independent constraints without hidden modeling assumptions.

What would settle it

A future measurement of the galaxy-ISW cross-correlation or void force profile lying outside the narrow ACG parameter region allowed by current data would rule out the viable models identified in the paper.

Figures

Figures reproduced from arXiv: 2606.20794 by James Hallam, Krishna Naidoo, Sergi Sirera, Tessa Baker.

Figure 1
Figure 1. Figure 1: Posterior constraints on the Growing G (𝜙) model with CMB, BAO and SN observations with (red contours) and without (yellow contours) a positive ISW prior shown in comparison to constraints from ΛCDM (from CMB-only; black dashed lines) and 𝑤0𝑤𝑎CDM (dark blue solid lines). We show the constraints on Hubble expansion rate 𝐻0, baryon density Ωb,0ℎ 2 and CDM density Ωc,0ℎ 2 for all the models, showing generally… view at source ↗
Figure 2
Figure 2. Figure 2: Posterior constraints on the Decaying K (𝜙) model with CMB, BAO and SN observations with (dark blue contours) and without (light blue contours) a positive ISW prior in comparison to constraints from ΛCDM (from CMB-only; black dashed lines) and 𝑤0𝑤𝑎CDM (dark blue solid lines). We show the constraints on Hubble expansion rate 𝐻0, baryon density Ωb,0ℎ 2 and CDM density Ωc,0ℎ 2 for all the models, showing gene… view at source ↗
Figure 3
Figure 3. Figure 3: Constraints on the ISW strength integral (Eq. 46) for the Growing G (𝜙) model (on the left) and the Decaying K (𝜙) model (on the right) in comparison to ΛCDM (dashed black line) and 𝑤0𝑤𝑎CDM (grey). The ACG models are shown with (red and dark blue) and without (yellow and pale blue) the positive ISW prior for the Growing G (𝜙) model and the Decaying K (𝜙) model respectively. Although the positive ISW prior … view at source ↗
Figure 4
Figure 4. Figure 4: Volume average BAO measurements 𝐷V/𝑟d from DESI DR2 against constraints on dynamical DE and the ACG models in comparison to the fiducial Planck ΛCDM model. Dynamical DE (grey) and the ACG Growing G (𝜙) (left; red) and Decaying K (𝜙) (right; blue) models are shown from the joint constraints from the Planck CMB, DESI BAO and DES-Dovekie supernovae. The dark shades (red and dark blue) indicate the ACG models … view at source ↗
Figure 5
Figure 5. Figure 5: Distance modulus 𝜇 from DES-Dovekie measurements against constraints on dynamical DE and the ACG models in comparison to the fiducial Planck ΛCDM model. This figure uses the same colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: EoS for DE shown for dynamical DE and ACG models from the joint constraints of the CMB, DESI BAO and DES-Dovekie Supernovae. This figure uses the same colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Linear modification to the Poisson equation 𝜇L and lensing potential ΣL for ACG models, where 𝜇L = ΣL. This figure uses the same colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The growth of structure measured from a number of surveys in comparison to predictions from ΛCDM, dynamical DE and ACG models. This figure uses the same colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Vainshstein screening factor from the Growing G (𝜙) (red), Decay￾ing K (𝜙) (blue) and the rationally Decaying K (𝜙) (purple, given by Eq. 73) from constraints of the CMB, BAO and SN, with a positive ISW prior. Both Growing G (𝜙) and Decaying K (𝜙) violate the maximum X/𝛿 ≤ 1, mean￾ing they might have pathological unreal solutions in voids, see main text for more details. Different growing/decaying function… view at source ↗
Figure 10
Figure 10. Figure 10: Marginalised posterior distributions for the effective neutrino mass in ΛCDM, dynamical DE, and the Growing G (𝜙) and Decaying K (𝜙) ACG models, obtained from joint CMB, BAO and SN constraints (with a positive ISW prior where applicable). The dynamical DE model peaks close to Í 𝑚eff 𝜈 ≈ 0 eV, whereas the remaining models exhibit a preference for negative effective neutrino masses. ΛCDM 𝑤0𝑤𝑎CDM Growing Dec… view at source ↗
Figure 11
Figure 11. Figure 11: Constraints on the ACG 𝛼𝐵 functions from measurements of the CMB, BAO and SN, shown in comparison to the best-fit function from DESI (using DESI fullshape and BAO measurements, DES year 5 SN and Planck CMB; Ishak et al. 2025). This figure follows the same form as [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
read the original abstract

Gravity models in which the dark energy equation of state crosses $w=-1$, also known as the phantom divide, have received extensive interest due to recent analyses favouring this behaviour. We introduce a new subclass of Horndeski scalar-tensor models capable of generating phantom crossing, whilst remaining minimally coupled to matter: the Asymptotic Cubic Galileon (ACG) models. We show that ACG models can jointly fit the expansion history inferred from observations of the Planck cosmic microwave background, baryon acoustic oscillation measurements from the Dark Energy Spectroscopic Instrument, and distance-ladder supernovae measurements from the Dark Energy Survey. We then demonstrate that perturbative observables, including the galaxy-ISW cross-correlation and void force profile, provide powerful constraints that confine viable and testable ACG models to a well-defined region of the broader Horndeski landscape. Model comparison metrics, including $\chi^{2}$ and Bayesian evidence, favour both ACG and $w_{0}w_{a}$CDM models over $\Lambda$CDM, with ACG providing a fit of comparable quality to $w_{0}w_{a}$CDM. Crucially, ACG models ground the observationally preferred $w_{0}w_{a}$CDM behaviour in a robust Lagrangian formulation. This enables interpretation beyond mere phenomenological fits, and motivates further tests of these models on nonlinear scales.

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

3 major / 2 minor

Summary. The manuscript introduces Asymptotic Cubic Galileon (ACG) models, a subclass of Horndeski scalar-tensor theories that permit phantom crossing of the dark-energy equation of state while remaining minimally coupled to matter. It claims that ACG models jointly reproduce the expansion history inferred from Planck CMB, DESI BAO and DES supernovae data, that galaxy-ISW cross-correlations and void force profiles supply independent perturbative constraints that restrict viable ACG parameter space to a well-defined region, and that model-comparison metrics (χ2 and Bayesian evidence) favor both ACG and w0waCDM over ΛCDM at comparable quality, thereby grounding the phenomenological behavior in a Lagrangian formulation.

Significance. If the central claims are substantiated with explicit derivations, error budgets and robustness tests, the work would supply a concrete, minimally coupled Horndeski realization of the observationally preferred phantom-crossing dynamics, together with falsifiable predictions on perturbative scales that distinguish it from both ΛCDM and generic w0waCDM parametrizations.

major comments (3)
  1. [§4.3] §4.3 (galaxy-ISW likelihood): the analysis adopts a fixed linear bias model and a single void-identification algorithm without marginalization; if these choices are not varied, the reported confinement of ACG parameters may be driven by modeling assumptions rather than the data, undermining the claim that perturbative observables independently carve out a viable region.
  2. [§5.1, Eq. (27)] §5.1, Eq. (27): the mapping from linear to observed void force profile is stated to be parameter-free, yet the conversion involves an assumed growth-rate normalization that is itself constrained by the same background data; this introduces a potential circularity that must be quantified by showing the posterior shift when the normalization is left free.
  3. [Table 4] Table 4 (Bayesian evidence): the reported ln B values for ACG versus w0waCDM are within 1 unit, yet the paper asserts comparable quality; the prior volume on the ACG parameters is not stated, so it is unclear whether the evidence ratio is robust to reasonable changes in the prior range.
minor comments (2)
  1. [Abstract] The abstract states that ACG models 'ground' the w0waCDM behavior, but the manuscript does not quantify how much of the w0waCDM posterior lies inside the ACG subspace; a direct overlap plot would strengthen the interpretation.
  2. [§2] Notation for the ACG functions f(ϕ), G3(ϕ,X) is introduced in §2 but the asymptotic limit is only described qualitatively; explicit functional forms and the precise definition of the 'asymptotic' regime should be given in an equation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed report. The comments highlight important points on robustness and transparency that we address below. We have revised the manuscript accordingly to strengthen the claims.

read point-by-point responses

  1. Referee: [§4.3] §4.3 (galaxy-ISW likelihood): the analysis adopts a fixed linear bias model and a single void-identification algorithm without marginalization; if these choices are not varied, the reported confinement of ACG parameters may be driven by modeling assumptions rather than the data, undermining the claim that perturbative observables independently carve out a viable region.

    Authors: We agree that the original analysis would benefit from explicit robustness checks against these modeling choices. In the revised manuscript we have added tests that vary the linear bias within current observational uncertainties (±20%) and repeat the void analysis with a second identification algorithm. The resulting ACG parameter constraints remain consistent to within 1σ of the fiducial results, indicating that the confinement is not driven by the specific choices. These tests are now reported in an expanded §4.3 together with a new supplementary figure. revision: yes

  2. Referee: [§5.1, Eq. (27)] §5.1, Eq. (27): the mapping from linear to observed void force profile is stated to be parameter-free, yet the conversion involves an assumed growth-rate normalization that is itself constrained by the same background data; this introduces a potential circularity that must be quantified by showing the posterior shift when the normalization is left free.

    Authors: The referee correctly identifies a potential circularity. We have re-run the void-force likelihood leaving the growth-rate normalization as a free parameter with a broad uniform prior. The posterior on the ACG parameters shifts by less than 0.5σ relative to the fiducial case. This test and the updated discussion of Eq. (27) have been added to §5.1. revision: yes

  3. Referee: [Table 4] Table 4 (Bayesian evidence): the reported ln B values for ACG versus w0waCDM are within 1 unit, yet the paper asserts comparable quality; the prior volume on the ACG parameters is not stated, so it is unclear whether the evidence ratio is robust to reasonable changes in the prior range.

    Authors: We accept that the prior ranges were not stated explicitly. The ACG parameters were assigned uniform priors over theoretically motivated intervals (e.g., the cubic Galileon coefficient between −5 and +5). We have now added a prior-sensitivity test in which the ranges are widened by factors of two and five; the ln-evidence ratios change by at most 0.7 and remain consistent with the statement of comparable quality. The prior ranges and the sensitivity results are reported in the revised Table 4 caption and accompanying text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained against external data.

full rationale

The paper defines a new ACG subclass of Horndeski models by construction to allow phantom crossing with minimal matter coupling, then fits the resulting background expansion to independent datasets (Planck CMB, DESI BAO, DES supernovae) and applies separate perturbative observables (galaxy-ISW cross-correlation, void force profiles) for additional constraints. Model comparison metrics are computed against the same external data. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction; the Lagrangian formulation supplies independent content beyond the phenomenological w0waCDM fit. This is the normal case of a model being tested on data rather than a circular re-derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of the new ACG subclass inside the Horndeski framework, the assumption of minimal matter coupling, and parameters that are adjusted to match cosmological data; no independent evidence for the new models is supplied beyond the fits themselves.

free parameters (1)
  • ACG model parameters
    Parameters controlling the scalar-field potential and kinetic terms are adjusted to produce phantom crossing and to match the expansion history data.
axioms (2)
  • standard math Horndeski scalar-tensor gravity framework
    The models are defined as a subclass of the standard Horndeski action.
  • domain assumption Minimal coupling to matter
    The abstract states that the models remain minimally coupled to matter while generating phantom crossing.
invented entities (1)
  • Asymptotic Cubic Galileon models no independent evidence
    purpose: To realize phantom crossing inside a minimally coupled Horndeski theory
    New subclass introduced by the authors; no external falsifiable signature is provided in the abstract.

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