Exploring cosmological imprints of phantom crossing with dynamical dark energy in Horndeski gravity

Rajeev Kumar Jain; Ujjwal Upadhyay; Yashi Tiwari

REVIEW 4 major objections 5 minor 2 cited by

Exploring cosmological imprints of phantom crossing with dynamical dark energy in Horndeski gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A Horndeski scalar-tensor dark energy model can cross the phantom divide and go negative at high redshift without instabilities, and combined cosmological data fit it as well as LambdaCDM.

desk verdict A solid existence proof for stable phantom crossing in a Horndeski model, but the data-driven preference for nonzero couplings is contingent on an unvaried field normalization and should be treated cautiously. read the letter →

arxiv 2412.00931 v2 pith:6TMANZ3N submitted 2024-12-01 astro-ph.CO gr-qchep-th

Yashi Tiwari , Ujjwal Upadhyay , Rajeev Kumar Jain This is my paper · ORCID

classification astro-ph.COgr-qchep-th

keywords dynamicaldarkenergyHorndeskigravityphantomcrossingnegativedensitycosmologicaltensionsscalar-tensortheorygrowthofstructureMarkovchainMonteCarlo

topics Dark Energy

open problems Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a specific Horndeski scalar-tensor model, in which dark energy is a scalar field nonminimally coupled to gravity and carrying derivative self-interactions, can reproduce two features that are impossible in general relativity without pathologies: the dark energy equation of state crossing the phantom divide $w=-1$, and a negative dark energy density at high redshifts. The authors show that both features arise from the dynamics of the field rather than from a phenomenological parameterization, while the kinetic self-interaction keeps the perturbations free of ghost and gradient instabilities. Confronting the model with Planck 2018 CMB data, BAO and $f\sigma_8$ measurements, and PantheonPlus supernovae, they find a fit statistically comparable to $\Lambda$CDM ($\Delta\chi^2 = -0.6$) and a $1\sigma$ preference for positive values of all three coupling parameters. A sympathetic reader would care because stable phantom-like behaviour is exactly the kind of late-time modification often invoked to address the Hubble and $S_8$ tensions, and because recent large-scale-structure data have made dynamical dark energy a live question.

What carries the argument

The load-bearing object is the Horndeski Lagrangian with the three chosen functions $G_2 = X - V_0\phi$, $G_3 = c_1\phi + c_2X$, and $G_4 = \frac{1}{2} + c_3\phi$, with $G_5=0$ so that gravitational waves propagate at the speed of light. The nonminimal coupling $G_4 = \frac{1}{2} + c_3\phi$ produces the high-redshift negative energy density through the $-6c_3\phi H^2$ term in $\rho_\phi$, while the derivative self-interaction $G_3$ controls late-time phantom behaviour; in particular $c_2$ maintains $Q_s>0$, so the phantom regime is reached without a ghost. The argument is carried by the Horndeski second-order action for perturbations, whose stability conditions $Q_s>0$ and $c_s^2>0$ the model is required to satisfy throughout the evolution.

What would settle it

Recompute the background and perturbation evolution with the same $G_i$ functions but with initial conditions varied over, say, $\phi_i\in[1,20]$ and $\phi'_i\in[10^{-12},10^{-5}]$, retuning $V_0$ so that today's dark energy density is fixed: if the phantom crossing and negative high-redshift density disappear in any stable region of parameter space, the model's headline features are an artifact of the chosen initial data rather than a property of the Lagrangian.

Watch

Extended reading notes

Core claim

The central claim is that the Lagrangian $\mathcal{L}_\phi = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi) - (c_1\phi + \frac{1}{2}c_2\,\partial_\mu\phi\,\partial^\mu\phi)\,\Box\phi + R(\frac{1}{2}+c_3\phi)$, with $V(\phi)=V_0\phi$, defines a Horndeski subclass (equivalently $G_2=X-V$, $G_3=c_1\phi+c_2X$, $G_4=\frac{1}{2}+c_3\phi$, $G_5=0$) in which phantom crossing and negative dark energy at high redshift occur without instabilities. For positive $c_3$, the term $-6c_3\phi H^2$ in the effective energy density dominates at early times, making $\rho_\phi$ negative while the total density stays positive; at low redshift the field's density turns positive and, for positive $c_1$, the equation of state enters the regime $w<-1$. The paper reports that the combined likelihood analysis of Planck 2018 CMB, BAO/$f\sigma_8$, and PantheonPlus data gives $\Delta\chi^2 = -0.6$ relative to $\Lambda$CDM, with posterior means $c_1 = 6.27^{+0.42}_{-2.3}$, $10^{-8}c_2 = 4.17^{+1.7}_{-3.0}$, and $c_3 = 0.00042^{+0.00012}_{-0.00040}$, all positive within $1\sigma$. The model also predicts a suppressed growth rate at low redshift, but in the joint analysis it does not actually resolve the $H_0$ and $S_8$ tensions.

Load-bearing premise

The phenomenology depends on fixed initial conditions for the scalar field at $z\sim1000$, namely $\phi_i=10$ and $\phi'_i=10^{-10}$, and on the linear potential $V(\phi)=V_0\phi$ with $V_0$ tuned by a shooting method; these choices are not varied in the MCMC, so if different initial data or a different potential shape remove the phantom crossing or the negative-density epoch, the reported parameter preferences could change.

Editorial extensions

If this is right

If the model's stability and fit claims hold, a stable fundamental-field realization of phantom crossing exists, so late-time modifications of the expansion history need not be dismissed as ghost-ridden.
The statistically comparable fit to $\Lambda$CDM means current data do not exclude this class of modified-gravity dark energy, and the weak preference for positive $c_3$ keeps the negative high-redshift density scenario observationally alive.
Because the joint analysis sharply limits the model's ability to raise $H_0$ or lower $S_8$, testing a modified-gravity dark energy model on perturbations is essential; background-only fits can overstate its tension-solving power.
Upcoming full large-scale-structure data releases and gravitational-wave speed measurements would discriminate the model's predictions for $w(z)$ and structure growth from $\Lambda$CDM.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's claim that 'this class of models' exhibits phantom crossing and negative densities is conditional on the fixed initial conditions $\phi_i=10$, $\phi'_i=10^{-10}$ at $z\sim1000$ and the linear potential $V_0\phi$; the paper does not test whether other initial data or potential shapes preserve these features, so the generality of the mechanism is unverified.
If negative high-redshift dark energy is real, it would suppress the early expansion rate and could feed structure formation; the model provides a concrete field-theoretic template for studying that effect, including its possible link to early massive galaxies.
A natural extension would be to rerun the same likelihood analysis with newer BAO data or with varied initial conditions; the $1\sigma$ preference for nonzero couplings found here could sharpen or vanish, giving a sharp test of whether the negative-density feature is data-driven.
The contrast between the individual-dataset hints of tension relief and the joint-analysis null result suggests that other modified-gravity models claiming to resolve both tensions should be checked with full CMB perturbation likelihoods before conclusions are drawn.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does what it says on the tin: it gives a concrete Horndeski action whose background dynamics cross the phantom divide and yield negative dark energy density at high z, and it checks that scalar perturbations are free of ghosts and gradient instabilities. That part is clean. The new work relative to the authors' earlier paper is the full perturbation analysis and the combined MCMC constraints from Planck18, PantheonPlus, and BAO/f_sigma8. The fits are standard and honestly reported: Delta chi^2 = -0.6 against LCDM with three extra parameters, so the model does not resolve the H0 or S8 tensions in the joint analysis. The paper says so plainly. Credit is due for that honesty and for providing the stability conditions and the hi_class implementation details.

Where I part ways with the abstract's emphasis is the 'preference for non-zero values of the model parameters.' That preference is real only within the chosen convention phi_i = 10 at z~1000. The nonminimal coupling enters as c3*phi, so the physical strength of c3 is set relative to the field value. If you chose phi_i = 1, the best-fit c3 would shift by an order of magnitude, and the 1-sigma positive preference could easily disappear. The stress-test note is right about that. The paper does not sample or vary the initial conditions, nor does it test a different potential shape, so the claimed data-driven preference is conditional. It is not a fatal flaw—the existence proof is independent of this—but it does undercut the 'preference' language.

Two smaller issues. The Gelman-Rubin threshold R-1 < 0.05 is loose by modern standards; R-1 < 0.01 would be more convincing for a nine-parameter posteriors. And no chains or code are released, so the quantitative claims are not directly verifiable. These are fixable in revision, not reasons to reject.

Who should read this: anyone working on modified-gravity dark energy or phantom crossing will find a useful worked example with explicit stability conditions. It is a legitimate data point in the 'can Horndeski do this without instabilities?' discussion. It deserves serious referee time; the authors should be asked to release chains and to show how the c3 posterior shifts under different phi_i choices. If that sensitivity is mild, the paper becomes considerably stronger. As it stands, accept for review with that request.

Referee Report

4 major / 5 minor

Summary. The paper studies a Horndeski dark energy model defined by G2 = X − V0φ, G3 = c1φ + c2X, G4 = 1/2 + c3φ, and G5 = 0. It demonstrates that this model can produce phantom crossing and negative scalar-field energy density at high redshifts while maintaining Qs > 0 and cs^2 > 0, and it studies the impact on the growth rate, matter power spectrum, and CMB temperature spectrum. The authors perform an MCMC analysis with Planck18, BAO/fσ8, and PantheonPlus data, finding a fit comparable to ΛCDM (Δχ² = −0.6) and reporting a 1σ preference for positive nonzero c1, c2, and c3.

Significance. If the results are sound, the paper provides a concrete, non-parametric dark energy construction in which phantom crossing and negative energy density arise from a fundamental scalar-tensor action rather than from a phenomenological parameterization. The use of the public hi_class and MontePython codes and of standard likelihoods is a strength, and the stability conditions are stated explicitly. The quantitative claim of an observational preference for nonzero couplings is, however, weak and conditional on several unvaried choices, so the paper's main value is as a proof-of-concept and a starting point for more robust tests. The comparison with ΛCDM is honest, and the analysis correctly emphasizes that perturbation data, especially CMB, strongly constrain the model.

major comments (4)

[Sec. II and Appendix A] The sign convention for X is inconsistent. Section II defines X = ∂µϕ∂µϕ/2, and Appendix A uses the FLRW metric ds² = −dt² + a²dx², for which ∂µϕ∂µϕ = −φ̇², so X = −φ̇²/2. With this convention, the kinetic term in G2 = X − V is negative, contradicting the claim that the c1 = c2 = c3 = 0 limit is canonical quintessence. The sign inconsistency propagates to Eq. (5) and to the stability conditions (A20)–(A21), where the signs of the c1 and c2 terms depend on this convention. Since the central claims about c2 ensuring Qs > 0 and about the sign of the c3 contribution to ρϕ rest on these signs, the authors must adopt a single, explicit convention (e.g., X = −(1/2)∂µϕ∂µϕ) and re-derive Eqs. (5), (A20), and (A21) accordingly.
[Sec. III and Table I] The reported 1σ preference for nonzero c1, c2, and c3 is conditional on the fixed initial conditions φi = 10 and φ̇i = 10⁻¹⁰ set at z ≈ 1000 and on the linear potential V = V0φ. The physical strength of the nonminimal coupling is set by the combination c3φ, so scaling φi by an order of magnitude rescales the effective coupling; the posterior on c3 would shift correspondingly. The paper does not vary φi, φ̇i, or the potential shape, and no robustness tests are presented. Without such tests, the data-driven preference for these couplings cannot be regarded as a robust model prediction, even though the theoretical capability of the model to exhibit phantom crossing is not in question.
[Sec. V.A and Table I] The MCMC convergence criterion R − 1 ≲ 0.05 is considerably weaker than the standard threshold of 0.01, and no effective sample sizes or number of chains are reported. Given the wide, asymmetric posterior for c1 (best fit 9.85, mean 6.27 with 1σ lower bound 3.97) and the marginal detection of c3, the quoted 1σ intervals may not be converged. The claim of a preference for nonzero parameters should be verified with a stricter convergence criterion and with a report of effective sample sizes.
[Sec. V.A] The prior ranges for c1, c2, and c3 are not specified. The text states that sampling starts around zero and that the parameters can take both positive and negative values 'without imposing strict bounds,' but the actual ranges used in the Monte Python runs are not given. This is essential for reproducibility and for interpreting the marginal posteriors, especially since wide or unbounded priors can slow convergence and influence the reported means and credible intervals.

minor comments (5)

[Sec. IV.A and Fig. 2] The statement that the equation-of-state singularity at ρϕ = 0 is 'physically acceptable' is asserted rather than demonstrated. Since the paper's central claim is the absence of instabilities, it would be useful to show explicitly that the perturbation variables, not just Qs and cs², remain finite through the crossing.
[Table I and Fig. 8] The '1σ preference' for c3 is marginal: Table I gives c3 = 0.00042 +0.00012/−0.00040, so the 68% interval excludes zero only at its lower edge. The wording in Sec. V.B ('preference for a positive, non-zero value of all the model parameters within 1σ') overstates the strength of the evidence; the authors should report the credible interval explicitly and temper the claim.
[Fig. 8] The axis label '10+8c2' appears to be a typographical error for '10⁸ c2'; this should be fixed for clarity.
[Sec. V.B] The comparison with ΛCDM uses only Δχ² = −0.6. Since the model has three extra parameters, a model-selection criterion such as AIC or BIC would clarify whether the modest χ² improvement is penalized; this would also strengthen the statement that the model does not outperform ΛCDM.
[Sec. III] The units of c2 are given as Mpc² and the paper states that c1 and c3 are dimensionless; this should be stated consistently in Table I, where the quoted quantity is 10⁻⁸c2, and in the prior description.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phantom-crossing and negative-density results are derived from the stated Horndeski action and stability conditions, and the MCMC preferences are fitted parameters rather than predictions.

full rationale

The paper's central claim, that the Horndeski subclass with G3 = c1*phi + c2*X and G4 = 1/2 + c3*phi can exhibit phantom crossing and negative scalar-field energy density at high redshifts without ghost or gradient instabilities, is derived from the explicit Lagrangian (Eq. 3), the general Horndeski background equations (Eqs. A2-A10), and the perturbation stability conditions (Eqs. A12-A19). These features are consequences of the model equations, not of a fitted quantity or of a self-citation. The MCMC analysis in Sec. V fits c1, c2, and c3 to Planck18, BAO/f_sigma8, and PantheonPlus data and reports posterior preferences; these are fitted parameters presented as data constraints, not independent predictions. The authors cite their previous work [52] for model provenance and an earlier H0-tension result, but the present constraints are recomputed with hi_class and MontePython, and the central derivation does not reduce to that citation. The strongest caveat is conditional rather than circular: the reported 1-sigma preference for nonzero c3 is set relative to the unvaried initial condition phi_i = 10 and the linear potential, so varying those choices could shift the posterior, but this is a robustness and prior-sensitivity concern, not a reduction of the derivation to its own inputs.

Assumptions & free parameters 11 free parameters · 4 assumptions · 0 invented entities

The model introduces no new particles or forces; it is a scalar field with nonminimal coupling and self-interactions. The free parameters c1, c2, c3 are fitted to data, V0 is tuned to fix the present-day dark energy density, and the initial field values are fixed ad hoc. The main axioms are standard FLRW and Horndeski perturbation theory, the specific Lagrangian choice, and the trust in hi_class and Monte Python implementations.

free parameters (11)

c1 = 6.27 (mean; 1 sigma interval [3.97, 6.69])
Coupling in G3 = c1*phi; fitted to Planck18+BAO/f_sigma8+PantheonPlus data; controls the late-time equation of state.
c2 = 4.17 x 10^8 Mpc^2 (mean of 10^-8*c2 = 4.17)
Coupling in G3 = c2*X; fitted to data; required for Qs > 0 to avoid ghost instability.
c3 = 0.00042 (mean; 1 sigma interval [0.00002, 0.00054])
Nonminimal coupling G4 = 1/2 + c3*phi; fitted to data; positive value gives negative dark energy density at high redshifts.
V0 = Not sampled; calibrated by shooting
Coefficient of linear potential V(phi) = V0*phi; tuned to match present-day dark energy density, so it is fixed by the background rather than predicted.
Initial field values phi_i and phi'_i = phi_i = 10, phi'_i = 10^-10
Set at z ~ 1000 so that c1=c2=c3=0 reduces to canonical quintessence consistent with LambdaCDM. These fixed choices are not varied in the analysis.
Omega_b h^2 = 0.02233
Baryon density; fitted with Planck18 data.
Omega_c h^2 = 0.1208
Cold dark matter density; fitted with Planck18 data.
n_s = 0.9647
Scalar spectral index; fitted with Planck18 data.
10^9 A_s = 2.1093
Primordial amplitude; fitted with Planck18 data.
h = 0.6784
Dimensionless Hubble constant; fitted with Planck18 data.
tau = 0.0547
Reionization optical depth; fitted with Planck18 data.

assumptions (4)

standard math The universe is described by a flat FLRW metric with matter, radiation, and a scalar field whose action is the Horndeski Lagrangian, relying on standard Friedmann equations and linear perturbation theory.
Used throughout Sec. II and Appendix A, following Horndeski [47] and Kobayashi [48].
ad hoc to paper The specific Lagrangian (Eq. 3) with G2 = X - V0*phi, G3 = c1*phi + c2*X, G4 = 1/2 + c3*phi, G5 = 0 is taken as the dark energy model.
This form is chosen to realize phantom crossing and negative energy density; it is not derived from a more fundamental principle. Introduced in Sec. III.
domain assumption The hi_class code correctly implements the Horndeski background and perturbation equations, including the stability conditions Qs > 0 and c_s^2 > 0, and the MCMC sampling uses the stated likelihoods.
The paper relies on hi_class [55, 56] and Monte Python [65, 66] without providing the model implementation or verifying the code output independently.
domain assumption The scalar field initial conditions at z about 1000 are phi_i = 10 and phi'_i = 10^-10 in Planck units.
Stated in Sec. III; chosen so the c1=c2=c3=0 limit mimics LambdaCDM. The dependence of results on these values is not tested.

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Pith. "Pith review of Exploring cosmological imprints of phantom crossing with dynamical dark energy in Horndeski gravity." pith.science (2026). https://pith.science/paper/6TMANZ3N

@misc{pith2026241200931,
  author       = {Pith},
  title        = {Pith review of: Exploring cosmological imprints of phantom crossing with dynamical dark energy in Horndeski gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TMANZ3N}},
  note         = {Machine review of arXiv:2412.00931}
}

abstract

In the current era of precision cosmology, the persistence of cosmological tensions, most notably the Hubble tension and the $S_8$ tension, challenges the standard $\Lambda$CDM model. To reconcile these tensions via late-time modifications to expansion history, various features such as phantom crossing in the dark energy equation of state, a negative energy density at high redshifts, etc., are favoured. However, these scenarios cannot be realized within the framework of GR without introducing ghost or gradient instabilities. In this work, we investigate a dynamical dark energy scenario within the framework of Horndeski gravity, incorporating nonminimal coupling to gravity and self-interactions. We highlight that the model can exhibit novel features like phantom crossing and negative dark energy densities at high redshifts without introducing any instabilities. For this specific Horndeski model, we perform a comprehensive analysis of the background evolution along with the effects on perturbations, examining observables like growth rate, matter and CMB power spectrum. To check the consistency of the model with the observational data, we employ MCMC analysis using BAO/$f\sigma_8$, Supernovae, and CMB data. While the model does not outperform the standard $\Lambda$CDM framework in a combined likelihood analysis, there remains a preference for non-zero values of the model parameters within the data. This suggests that dynamical dark energy scenarios, particularly those with non-minimal couplings, merit further exploration as promising alternatives to GR, offering rich phenomenology that can be tested against a broader range of current and upcoming observational datasets.

Figures

Figures reproduced from arXiv: 2412.00931 by the authors.

**Figure 3.** CMB temperature power spectrum (upper panel) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Matter power spectrum (upper panel) and relative [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: The evolution of fσ8(z) for different values of the model parameters (colored). The black dashed line represents the ΛCDM model for reference. power spectrum is particularly relevant to the growth tension. An alternative way to study the effect on the evolution of mat…

**Figure 6.** Figure 6: 1D and 2D posterior distributions for H0 and Ωm obtained for the model (solid) and the ΛCDM (dashed), using PantheonPlus (green) and BAO/fσ8 (navy) data. 66 70 74 H0 0.80 0.85 0.90 S8 0.80 0.82 0.84 0.86 0.88 8 0.24 0.26 0.28 0.30 0.32 m 0.24 0.27 0.30 0.33 m 0.81 0.85…

**Figure 7.** Figure 7: 1D and 2D posterior distribution for the cosmolog [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: 1D and 2D posterior distribution for a subset of cosmological parameters and the model parameters [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Sincew denotes a derived quantity used for convenience to describe the behavior of dark energy, a singularity does not cause any unphysical behavior in the underlying dynamics

This is physically acceptable, as both energy den- sity and pressure contribute similarly to gravitational effects, and the absence of one does not inherently lead to any issues. Sincew denotes a derived quantity used for convenience to describe the behavior of dark energy, a singularity does not cause any unphysical behavior in the underlying dynamics. T...

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