One Field, Two Cosmic Eras: Curvaton Model Offers New Take on Phantom Dark Energy

Dark energy may not be the frozen, unchanging force that cosmologists have assumed for more than two decades. New measurements from the Dark Energy Spectroscopic Instrument, or DESI, have reignited a debate about whether the mysterious component driving the accelerating expansion of the universe is truly constant, or whether it evolves over cosmic time. Now, a theoretical study published in The European Physical Journal C by Bichu Li and Lei-Hua Liu of Jishou University proposes a strikingly economical answer: a single scalar field, born in the first instants of the universe as a curvaton, could still be shaping cosmic expansion today, and its gravitational coupling to spacetime curvature could allow dark energy to behave in ways that ordinary models cannot.

The stakes come from a subtle but consequential feature of the DESI data. When the latest baryon-acoustic-oscillation measurements are combined with cosmic microwave background constraints and Type Ia supernova distances, the data can be summarized by a phenomenological form in which the dark energy equation of state, written as w(a) = w0 + wa(1 − a), prefers w0 slightly greater than −1 and wa negative. That combination implies that the effective equation of state crosses the so-called phantom divide at w = −1 at some point in the recent expansion history. For theorists, this crossing is notoriously difficult. A canonical scalar field with a standard kinetic term can never push w below −1, while a field with a wrong-sign kinetic term can do so but at the cost of ghost instabilities, quantum states of negative norm that signal a breakdown of the theory. Stable phantom-crossing models therefore demand extra structure: additional degrees of freedom, effective-field-theory operators, modified gravity, or scalar-tensor dynamics.

Li and Liu’s proposal builds on the running-curvaton framework, in which a light scalar field called the curvaton contributes to the primordial curvature perturbation that seeded all structure in the universe. In the original, minimally coupled version of the model, the same field was expected to survive to the present day and act as dark energy, but its late-time behavior resembled thawing quintessence, a form of dynamical dark energy that cannot naturally reproduce the DESI-motivated phantom-crossing region. The authors’ key move is to add a Jordan-frame non-minimal coupling, a term of the form ξχ²R that ties the curvaton field directly to the Ricci scalar, a measure of spacetime curvature. This coupling is not exotic by the standards of modern field theory; scalar fields coupled to curvature have been studied for decades in inflation and scalar-tensor gravity, and such theories belong to the well-studied Horndeski class. The novelty lies in embedding the coupling within the running-curvaton scenario in a way that preserves the model’s early-universe successes.

That preservation is the first technical hurdle the paper clears. During slow-roll inflation, the Ricci scalar is approximately 12H², where H is the Hubble expansion rate, so the non-minimal coupling shifts the effective mass of the curvaton fluctuation from its original value to m_eff² = 6H²(g0 + 2ξ), where g0 parametrizes the inflaton-curvaton interaction. Crucially, the observable spectral tilt of the curvaton perturbations depends only on the combination g0 + 2ξ. The authors show that this shift can be absorbed, at leading order, by retuning the mass parameter: one simply defines an observed coupling g0_obs = g0 + 2ξ and adjusts g0 accordingly. The predicted spectral index n_χ − 1 ≈ 4g0_obs then matches the minimally coupled model exactly. Similarly, the local non-Gaussianity parameter f_NL, which measures departures from Gaussian statistics in the primordial perturbations, is controlled mainly by the curvaton’s energy fraction at decay, with f_NL ≈ 5/(4r_dec) in the sudden-decay approximation. Because spacetime is nearly flat during radiation domination, R ≈ 0, the curvature-induced mass correction is suppressed precisely when it matters most for non-Gaussianity, and the Planck satellite’s observational bounds remain satisfied. Tensor perturbations fare equally well: since the curvaton is a subdominant spectator during inflation, the correction to the primordial gravitational-wave amplitude is of order ξχ²/M_P², negligible for sub-Planckian field values.

The second hurdle is the late-time dynamics, and this is where the non-minimal coupling earns its keep. After the inflaton decays, the surviving part of the curvaton potential, a phenomenological form V(χ) = V1(1 − V0 e^(−λχ/M_P)), drives the field’s dark-energy-like evolution. When the modified Friedmann equations are recast in the standard Einstein form, the effective dark-energy pressure acquires geometric terms involving the coupling, such as −ξχ²(2Ḣ + 3H²) and −2ξ(χχ̈ + χ̇²). The sum of the effective dark-energy density and pressure, which determines whether w is above or below −1, becomes ρ+p = (1 − 2ξ)χ̇² + 2ξ(Hχχ̇ − χχ̈ − χ²Ḣ). The second group of terms can become sufficiently negative during late-time evolution to push the effective equation of state across the phantom divide, all without introducing a phantom kinetic term. The theory retains a canonical kinetic structure, and because the Horndeski function G4 depends on the field but not on its kinetic variable, the speed of gravitational waves remains exactly luminal, c_T² = 1, consistent with the landmark constraint from the neutron-star merger GW170817.

To test whether these theoretical trajectories are actually compatible with observations, the authors performed a Markov-chain Monte Carlo likelihood analysis using numerical solutions of the model’s homogeneous background equations. The observational input combined three pillars: a reduced CMB prior on the acoustic angular scale and the physical baryon and total matter densities, the official DESI DR2 baryon-acoustic-oscillation data vector with its full 13-by-13 covariance matrix, and the Pantheon+ supernova compilation with 1580 objects, deliberately excluding the SH0ES Cepheid calibration to avoid entangling the analysis with the Hubble-tension debate. The sampled parameters were the dimensionless Hubble parameter h, the baryon and cold dark matter densities, and four microscopic quantities of the curvaton sector: the coupling ξ, the potential slope λ, the potential normalization V0, and the initial field value χ_i. The late-time normalization V1 was fixed for each sample by a shooting condition requiring the normalized expansion rate to equal unity today.

The results are encouraging but carefully circumscribed. The combined data yield H0 = 67.88 km/s/Mpc with an uncertainty of roughly half a percent, a matter density Ωm = 0.3072, and dark energy parameters w0 = −0.922 and wa = −0.205, placing the posterior squarely in the DESI-motivated phantom-crossing region. The best-fit point improves the chi-square relative to a flat ΛCDM reference fit by Δχ² = −6.56, but because the curvaton model carries four additional microscopic parameters, the Akaike information criterion actually favors ΛCDM by ΔAIC = +1.44. The authors are explicit about this: the model is compatible with the distance data and can modestly improve the best fit, but the analysis does not constitute a decisive statistical preference over the standard cosmological constant. The posterior bands show the median equation of state starting above −1 today and evolving toward a mildly phantom regime at intermediate redshift, while the expansion-rate history H(z)/H0 remains tightly constrained, demonstrating that the phantom-crossing mechanism does not spoil the distance information.

Equally instructive is what the data cannot yet pin down. The derived background quantities, including H0, Ωm, the sound horizon r_d, w0 and wa, are compressed sharply by the likelihood, but the microscopic parameters ξ, λ, V0 and χ_i remain broad, skewed and long-tailed in the marginalized posterior. A quantitative compression diagnostic makes the point vivid: the 68 percent posterior width of ξ spans about 61 percent of its allowed prior range, whereas the corresponding ratios for h and the cold dark matter density are only about 2 to 3 percent. A supplementary run widening the prior on ξ from 5 to 8 broadens the high-ξ tail further without the chain running away, confirming that the current bound is prior-conditional. This degeneracy is not a failure of the mechanism but a genuine limitation of background-level data, which constrain the expansion history far more tightly than the underlying microphysics that produces it.

The authors are equally candid about the analysis’s boundaries. The CMB information was compressed to three early-background numbers, so the calculation does not include the non-minimal coupling’s effects on the full temperature and polarization spectra, CMB lensing, the late integrated Sachs-Wolfe effect, or scalar perturbation evolution. A complete test will require implementing the model in a Boltzmann code, the standard machinery used to compute CMB anisotropies from first principles. Local gravity constraints also demand a separate post-Newtonian or environmental analysis: in high-density regions the scalar mass receives a contribution proportional to the local matter density, suggesting possible Yukawa suppression of fifth-force effects analogous to chameleon or symmetron screening, but no definitive Solar-System viability claim is made. Within those limits, the study delivers a clean proof of concept: the same field that may have shaped the seeds of galaxies in the infant universe could, through its intimate coupling to spacetime curvature, be steering the accelerated expansion we observe today, and it can do so while crossing the phantom divide that has long frustrated simpler models of dark energy.

Subject of Research: A non-minimally coupled running curvaton model linking primordial perturbations to phantom-crossing dynamical dark energy constrained by DESI, CMB and supernova data

Article Title: Non-minimally coupled running curvaton for DESI-motivated dynamical dark energy

Article References: Li, B., & Liu, L.-H. (2026). Non-minimally coupled running curvaton for DESI-motivated dynamical dark energy. The European Physical Journal C, 86(10), Article 1156. https://doi.org/10.1140/epjc/s10052-026-16396-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16396-3

Keywords: dark energy, curvaton, non-minimal coupling, DESI, phantom divide, cosmology, scalar-tensor gravity, baryon acoustic oscillations, Pantheon+, cosmic microwave background, Horndeski theory, MCMC