The research, carried out by Jiayuan Huang, Tonghua Liu, and Chenggang Shao of Yangtze University, is notable for its methodological discipline. Rather than fixing the sound horizon—the baryon drag ruler that underpins baryon acoustic oscillation measurements—to a value inferred from early-universe cosmic microwave background data, the team treated it as a free parameter. This choice means the analysis relies purely on late-time information: cosmic chronometer measurements of the Hubble parameter, three independent Type Ia supernova compilations (Pantheon+, DES-Dovekie, and DESY5), the DESI DR2 baryon acoustic oscillation dataset, and an extended compilation of redshift-space distortion measurements that probe the growth of cosmic structure. Five different data combinations were analyzed using Bayesian Markov chain Monte Carlo sampling, and the competing models were scored against Lambda-CDM with the Akaike and Bayesian information criteria.

The three models share a common intellectual ancestry in the holographic principle, a deep idea from quantum gravity suggesting that the information content of a region of space is bounded by its surface area, not its volume. In holographic dark energy, the vacuum energy density is capped by the mass of a black hole of the same size, producing a density proportional to the inverse square of an infrared cutoff length. The original version of the model, proposed by Miao Li in 2004, identifies that cutoff with the future event horizon of the universe, introducing a dimensionless parameter c that controls the dark energy’s equation of state. When c exceeds one, dark energy behaves like quintessence, a slowly evolving field; when c equals one, the model approaches a de Sitter state, indistinguishable at the background level from a cosmological constant; and when c falls below one, the dark energy becomes phantom-like, with an equation of state more negative than minus one.

The results for this flagship model are striking in their precision. Across all data combinations that include DESI DR2 baryon acoustic oscillations, the holographic parameter converges to values extremely close to unity, with the full dataset including redshift-space distortions yielding c = 1.011 with an uncertainty of about 0.07. The Hubble constant comes out at roughly 67.3 to 68.0 kilometers per second per megaparsec, and the matter density at about 0.27. Physically, this means the data are steering the model directly toward the de Sitter boundary—the razor’s edge between quintessence-like and phantom-like behavior—rather than into a robust phantom regime. In other words, the best-fitting holographic universe looks, at the level of its expansion history, remarkably like the standard cosmological constant it was designed to replace.

The agegraphic model takes a different route to the same holographic inspiration. Instead of a spatial horizon, it ties the dark energy density to the conformal age of the universe, drawing on spacetime quantum fluctuations and the Karolyhazy uncertainty relation. This construction makes the model unusually predictive: the matter density is not a free parameter at all but is fixed by the flatness condition and the normalization of the expansion rate once the Hubble constant and the model parameter n are specified. The analysis finds n stably pinned at about 2.78 to 2.81 across all baryon acoustic oscillation combinations, indicating that the data consistently select the same age scale for the onset of late-time acceleration. But this economy comes at a price. The model prefers a Hubble constant of only 66.5 to 66.8 kilometers per second per megaparsec, and the information criteria generally disfavor it relative to Lambda-CDM once DESI DR2 data are included, with the full dataset giving a difference of 3.51 in both criteria.

Ricci dark energy is the most radical of the three. It identifies the infrared cutoff with the Ricci scalar curvature of spacetime itself, linking the dark energy density directly to the Hubble parameter and its time derivative. This gives the model an analytical expansion history and considerable flexibility at the background level, and it shows: the Akaike criterion actually favors Ricci dark energy over Lambda-CDM in all baryon acoustic oscillation combinations, with differences reaching nearly ten units for the DESY5 supernova sample. The Ricci parameter gamma is tightly constrained at about 0.53 to 0.55. Yet this statistical success conceals a serious physical problem. The model persistently prefers a matter density of only 0.215 to 0.219—roughly nine sigma below the Planck 2018 value of 0.315. Because the curvature-linked dark energy can absorb part of the distance behavior normally attributed to matter, the model achieves its good fit by effectively hiding matter inside its dark energy component, a trade-off that clashes with early-universe and structure-formation expectations.

The growth of cosmic structure provides an independent cross-check, and here the picture becomes more nuanced. Redshift-space distortion measurements of the quantity f-sigma-8, drawn from surveys including 6dFGS, SDSS, WiggleZ, VIPERS, FastSound, and eBOSS, probe how fast matter fluctuations grow over cosmic time. When these data are added, the fitted present-day fluctuation amplitude sigma-8 comes out at 0.875 for holographic dark energy, 0.893 for agegraphic dark energy, and a notably high 0.961 for Ricci dark energy. The combination of low matter density and high fluctuation amplitude in the Ricci case is physically suggestive: with less matter in the background, a larger fluctuation amplitude is needed to match the observed growth data. The authors are careful to stress, however, that this is a phenomenological consistency check, not a full perturbative stability analysis. Whether Ricci dark energy is genuinely viable at the level of linear perturbations depends on the complete dynamical trajectory and remains an open question for future work.

One of the most consequential findings concerns the Hubble tension itself. Because baryon acoustic oscillation measurements constrain ratios of distances to the sound horizon, freeing that horizon induces a strong negative correlation with the inferred Hubble constant, with Pearson coefficients between roughly minus 0.93 and minus 0.97 across all models and data combinations. Yet the fitted sound horizon values remain close to the standard scale of about 147 megaparsecs, meaning none of the three holographic models resolves the tension by shrinking the baryon acoustic ruler. All three prefer Hubble constants in the 66.5 to 68 range—comfortably near the Planck value of 67.4 and far below the SH0ES local measurement of 73.04. The reconstructed equation-of-state evolution reinforces this conclusion: holographic dark energy stays above minus one with a present-day value near minus 0.9, agegraphic dark energy is quintessence-like by construction, and Ricci dark energy hovers near minus 0.8 today with rapid evolution at higher redshift. No model shows robust evidence of crossing the phantom divide.

The bottom line is a sobering but honest verdict on one of cosmology’s most elegant theoretical programs. Holographic dark energy emerges as the most balanced performer of the three, statistically competitive with Lambda-CDM by the Akaike criterion—though not decisively so once the Bayesian criterion penalizes its extra parameter—while agegraphic dark energy is disfavored and Ricci dark energy’s apparent success is undermined by its anomalously low matter density. The study also demonstrates the power of the DESI DR2 dataset to drive model parameters toward sharp, physically meaningful values rather than vague allowed ranges. As the authors note, the near-de Sitter solution preferred for holographic dark energy is increasingly degenerate with a cosmological constant at the background level, so the decisive test will come from perturbation-level observables: weak lensing, full-shape galaxy clustering, and the integrated Sachs-Wolfe effect. For now, the holographic idea survives, but the universe, as measured by our best instruments, still looks stubbornly like one governed by a simple constant.

Subject of Research: Observational constraints on holographic, agegraphic, and Ricci dark energy models using late-time cosmological data

Article Title: Late-time cosmological constraints on three holographic dark energy models with DESI DR2 BAO and Type Ia supernovae

Article References: Huang, J., Liu, T., & Shao, C. (2026). Late-time cosmological constraints on three holographic dark energy models with DESI DR2 BAO and Type Ia supernovae. The European Physical Journal C, 86(9), Article 1042. https://doi.org/10.1140/epjc/s10052-026-16300-z

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16300-z

Keywords: dark energy, holographic principle, DESI DR2, baryon acoustic oscillations, Type Ia supernovae, Hubble tension, cosmology, Lambda-CDM, cosmic chronometers, redshift-space distortions, phantom divide, Markov chain Monte Carlo

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

Torsion is not some exotic bolt-on to general relativity. It is the part of the affine connection that Einstein’s theory sets to zero by hand. In Einstein–Cartan gravity, torsion is sourced by the intrinsic, quantum-mechanical spin of matter, and its effects normally become competitive only at the most extreme fermion densities imaginable. The theory has a celebrated pedigree: in the 1970s, physicists showed that torsion could avert the initial singularity, replacing the Big Bang with a cosmic bounce. But a recent proposal by researchers Yun and Lee went much further, claiming that in Einstein–Cartan cosmology the scalar torsion mode could rescue a famously broken model of dark energy, produce late-time acceleration, and even allow the equation of state to cross the so-called phantom divide, with possible relevance to the DESI anomaly. The new paper shows, with brutal clarity, that the claim fails, and that it fails for reasons reaching well beyond this particular model.

The technical heart of the argument is a single scaling relation. If the matter sector of the universe is separately conserved, the homogeneous torsion scalar Phi must obey a dilution equation that forces it to fall as the inverse cube of the scale factor. When this mode enters the Friedmann constraint, it appears as a negative energy density scaling as the inverse sixth power of the scale factor, a stiff component with an effective equation of state equal to plus one. This combination is an old acquaintance in disguise: it is precisely how the spin–spin contact interaction of a Weyssenhoff spin fluid enters Einstein–Cartan cosmology, where it powers the nonsingular bounce at high density. The physics under scrutiny is therefore the classical Einstein–Cartan bounce, examined at the opposite end of cosmic history, and the question is how much room the data leave for it there.

The first casualty is the holographic dark energy component itself. Holographic dark energy ties the dark energy density to an infrared cutoff length scale, and the most natural choice, the Hubble radius, famously fails in ordinary general relativity: because the density is proportional to the square of the Hubble rate, it merely tracks the dominant component and can never drive acceleration. That failure is why the field migrated to the future event horizon, with all its causality troubles. Yun and Lee claimed torsion changes this verdict. The new analysis shows it does not. Because the holographic density is proportional to H squared, it drops out of the deceleration parameter identically: the deceleration parameter is the same function of matter and torsion with or without it. Whatever acceleration the model produces belongs to torsion alone, and the cancellation is structural, not an accident of the quadratic choice. Any cutoff built from the Hubble rate alone inherits its dynamics from the other components instead of supplying its own, torsion or no torsion.

So where does the acceleration actually come from? From the bounce, and only from the bounce. The constraint that the Hubble rate squared must remain nonnegative enforces a turning point at which the expansion rate vanishes, the classical Einstein–Cartan bounce. The entire accelerating regime is confined to a transient window around that bounce, spanning less than a factor of about 1.6 in scale factor. The universe exits the bounce super-accelerating, crosses the phantom divide partway through the window, stops accelerating at its edge, and then settles into behaving like cold matter with a rapidly fading memory of the bounce. The window through which torsion could accelerate the universe closes before any observer can look through it. Dragging that window to redshifts where acceleration is actually observed would force the Hubble rate to vanish in our recent past, and the measured expansion history says emphatically otherwise.

What turns this qualitative observation into hard numbers is a requirement almost embarrassing in its modesty: the universe must contain its own past. Because the torsion term grows as the sixth power of one plus redshift going backward in time, faster than matter, radiation, or anything else, demanding that the expansion rate remain real up to some redshift the universe demonstrably reached yields a ladder of nested upper bounds on the torsion density parameter. A fit to the official DESI DR2 baryon acoustic oscillation likelihood gives a bound of 8.7 times ten to the minus four at 95 percent confidence. The mere existence of the spectroscopically confirmed galaxy JADES-GS-z14-0 at redshift 14.32 tightens this to 8.4 times ten to the minus five. The cosmic microwave background pushes it to 3.1 times ten to the minus ten. And Big Bang nucleosynthesis crushes it to 5 times ten to the minus twenty-four. Each rung rests on strictly weaker assumptions than the one before, and the last one settles the question beyond any reasonable doubt.

The DESI fit itself is a model of care. The authors used the released data vector and covariance exactly as distributed with the collaboration likelihood, thirteen entries spanning the brightest galaxy sample and six anisotropic tracers, and validated their pipeline by reproducing the published DESI Lambda-CDM fit to full precision before opening the torsion parameter. The result: the data express no preference whatsoever for torsion, with a negligible improvement in chi-squared. Even saturating the weakest bound would place the torsion-induced turning point at a redshift of roughly six, leaving no universe for the quasars and galaxies we observe beyond that epoch. The baryon acoustic data run out of constraining power exactly where the existence bounds take over, a striking convergence of independent lines of evidence.

The imprint on today’s dark energy equation of state makes the irrelevance concrete. The torsion contribution shifts the effective equation of state by an amount bounded by twice the torsion density parameter divided by the dark energy density. Even the weakest rung of the ladder caps this shift at a few parts in a thousand, on the phantom side of minus one, where DESI does not want it. The DESI preference for evolving dark energy asks for deviations of order a quarter on the quintessence side. That is a gap of two orders of magnitude at best, and twenty-two at worst, with the wrong sign throughout. In the Granda–Oliveros cutoff, an alternative infrared choice that includes the Hubble derivative, the verdict inverts in a darkly comic way: torsion does not prevent the big-rip singularity, it deepens it, adding to the super-acceleration instead of moderating it as it becomes ever more irrelevant on approach to the rip.

Every no-go theorem is a list of assumptions read backwards, and this one has a short list. The load-bearing assumption is adiabaticity: the requirement that matter be separately conserved, which forces the inverse-cube scaling. The authors show in an appendix that this scaling is nothing deeper than dilution, the kinematics of a comoving spin fluid whose particle number falls as the volume grows. Escaping the no-go therefore costs real physics. One must let matter and torsion exchange energy, turning the conservation identity into the definition of an interacting dark sector, and demand a spin density that refuses to dilute, through progressive spin alignment, condensation, or some other mechanism operating at late times. In Einstein–Cartan gravity torsion is algebraic and cannot outlive its source, so the mechanism must live in the matter sector explicitly, making a fermionic dark matter component with evolving spin alignment the natural candidate. Either sign of the spin contact term lands on the same verdict: no late-time acceleration from the adiabatic mode, with the sign merely deciding whether the early universe gets a bounce in compensation.

The authors are careful to delimit their target. Torsion cosmology at large survives this paper. Dynamical torsion in Poincaré gauge theory, where propagating modes can oscillate and drive late-time acceleration, remains untouched, as do interacting matter–torsion scenarios and the steady-state vectorial mode, which has already been confronted with DESI, supernova and microwave background data with results consistent with zero at sub-percent precision. But the message for the holographic dark energy program is stark. The Hubble cutoff’s Einstein–Cartan incarnation inherits, intact, the defect identified two decades ago: proportionality to the Hubble rate squared makes it a spectator. Torsion adds a bounce, and the bounce brings a brief phantom display, but every part of it stays locked behind a wall at high redshift. The window through which torsion could have accelerated the universe sits, and always sat, on the far side of that wall. If torsion has something to say about dark energy, it will have to say it where adiabaticity breaks, and that is precisely where the authors say their work is heading next.

Subject of Research: A no-go result for torsion-based holographic dark energy in Einstein–Cartan cosmology

Article Title: No late-time role for adiabatic torsion: a no-go result for Hubble-cutoff holographic dark energy in Einstein–Cartan cosmology

Article References: Izaurieta, F., Lepe, S., & Quinzacara, C. (2026). No late-time role for adiabatic torsion: a no-go result for Hubble-cutoff holographic dark energy in Einstein–Cartan cosmology. The European Physical Journal C, 86(9), Article 1088. https://doi.org/10.1140/epjc/s10052-026-16337-0

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16337-0

Keywords: dark energy, torsion, Einstein–Cartan gravity, holographic dark energy, DESI, cosmology, no-go theorem, phantom divide, Big Bang nucleosynthesis, cosmic microwave background, baryon acoustic oscillations, Hubble radius