Modified Gravity Realizations of Quintom Dark Energy after DESI DR2

This paper asks whether the “quintom” form of dynamical dark energy—specifically an equation-of-state (EoS) parameter that crosses the phantom divide at $w=-1$ from below—can be realized naturally within modified gravity, and whether such a realization is consistent with the latest observational indications from DESI DR2 baryon acoustic oscillations (BAO) when combined with supernovae (SNe) and CMB constraints. The question matters because the standard $\Lambda$CDM model corresponds to $w=-1$ with no crossing, while multiple recent analyses of BAO/SNe/CMB data have reported statistically significant preferences for evolving dark energy, including quintom-like behavior. If the data truly prefer $w$ crossing, then either new degrees of freedom must be introduced or gravity itself must be modified; moreover, simple single-scalar-field models are generally constrained by “No-Go” results that prevent phantom-divide crossing.

The authors’ significance contribution is twofold. First, they provide a theoretical “realization” pathway: starting from a general effective field theory (EFT) formulation of dark energy in metric-affine geometry (MAG), they show how two specific modified gravity theories—$f(T)$ (teleparallel/torsion-based) and $f(Q)$ (non-metricity-based)—can be mapped into the EFT language in unitary gauge. Second, they confront a concrete $f(T)$ model (and, in the coincident gauge, the corresponding $f(Q)$ model) with observational data using a non-parametric Gaussian-process (GP) reconstruction of the BAO-derived distance function, from which they reconstruct the dark-energy EoS $w(z)$. Their reconstruction is reported to exhibit quintom-type evolution with phantom-divide crossing from below, consistent with DESI DR2 hints.

Methodologically, the paper proceeds in three layers. (1) Geometry/EFT layer: The authors review metric-affine gravity where the metric and affine connection are independent (Palatini formalism). They define non-metricity $Q_{\alpha \mu \nu }={\nabla }_{\alpha }{g}_{\mu \nu }$ and torsion $T^{\alpha }{}_{\mu \nu }={\Gamma }^{\alpha }{}_{\nu \mu }-{\Gamma }^{\alpha }{}_{\mu \nu }$, and express the Ricci scalar in MAG in terms of Levi-Civita curvature plus torsion/non-metricity scalars and boundary terms. They then construct the most general EFT action in unitary gauge, writing the background-relevant part as a time-dependent combination of operators built from $\mathring{R}$, torsion scalar $T$, non-metricity scalar $Q$, and additional “00”-type operators (e.g., $Q^0$, ${\tilde{Q}}^0$, $T^0$, $Q^{000}$), plus a matter sector minimally coupled to the metric (Jordan frame). (2) Model mapping layer: They show that for $f(T)$ gravity, choosing a torsion-aligned slicing (so that $\delta T=0$ in unitary gauge) collapses the expansion to linear perturbative structure, allowing a direct identification of EFT functions such as $\Psi (t)$ and $d(t)$. They derive effective dark-energy pressure and density in $f(T)$ cosmology and obtain an explicit EoS formula $w_{f(T)}$ in terms of $f(T)$, $f_T$, and $f_{TT}$. For $f(Q)$, they work in the coincident gauge where $Q=-6H^2$ and show the analogous EFT identifications (with the background evolution matching $f(T)$ under $T\to Q$). (3) Observational layer: They specify a particular $f(T)$ functional form designed to allow quintom behavior, then reconstruct $w(z)$ and the normalized dark-energy density $f_{de}(z)={\rho }_{de}(z)/{\rho }_{de,0}$ using GP reconstruction of the latest DESI DR2 BAO data combined with SNe and a CMB “effective BAO point” at $z_{\ast }\simeq 1089$. The GP uses a squared-exponential kernel $k(x,x^{\prime })={\sigma }_f^2\exp [-(x-x^{\prime }{)}^2/(2l^2)]$ with hyperparameters inferred from data. From the reconstructed comoving angular diameter distance $D_M(z)$, they obtain $H(z)=c/D_M^{\prime }(z)$ and then infer $w(z)$ and $f_{de}(z)$ via background relations.

On the results side, the paper reports qualitative and some quantitative model-selection outcomes. The key qualitative observational result is that the reconstructed $w(z)$ “always crosses $-1$ from below,” being in the phantom regime at higher redshift and moving into the quintessence regime at late times, corresponding to “quintom-B” evolution. For the reconstructed normalized dark-energy density, they report that $f_{de}(z)$ decreases with increasing redshift and remains positive. They also reconstruct the implied $f(T)$ function from the observationally favored quintom behavior.

For quantitative comparison, they perform parameter estimation (via MCMC) for three model classes: their quintom $f(T)$ model, a quadratic $f(T)$ model, and $\Lambda$CDM. They use information criteria AIC and BIC, defined as $\mathrm{AIC}=-2\ln {\mathcal{L}}_{\max }+2p_{\mathrm{tot}}$ and $\mathrm{BIC}=-2\ln {\mathcal{L}}_{\max }+p_{\mathrm{tot}}\ln N_{\mathrm{tot}}$. For the DESI DR2 + CMB + Pantheon+ dataset combination, Table 1 reports: - Quintom model: AIC $=97.875$, BIC $=119.65$ - Quadratic model: AIC $=97.151$, BIC $=113.841$ - $\Lambda$CDM model: AIC $=100.983$, BIC $=111.87$ Thus, AIC slightly favors the quadratic model over the quintom model, while BIC favors $\Lambda$CDM (and also favors quadratic over quintom). The authors interpret this as a consequence of BIC’s stronger penalty for model complexity.

The paper also includes a theoretical stability check at the perturbation level for the chosen $f(T)$ model. They study the evolution of the Newtonian potential ${\Phi }_k$ in Fourier space in a dark-energy-dominated, pure-gravitational sector, assuming no anisotropic stress so $\Psi =\Phi$. The mode equation is written as ${\ddot{\Phi }}_k+\Gamma {\dot{\Phi }}_k+{\Omega }^2{\Phi }_k=0$. They argue that instability corresponds to negative ${\Omega }^2$, and they rewrite ${\Omega }^2$ in terms of $f$, $f_T$, and $f_{TT}$: ${\Omega }^2=\frac{-\frac{f}{4}-T^2{f}_{TT}}{f_T+2T{f}_{TT}}\ge 0$. They introduce a dimensionless diagnostic ${\omega }_d^2=-{\Omega }^2/T$ to track the sign and report that suitable parameter choices can realize quintom behavior without inducing gravitational instability (as illustrated in their figure).

Limitations are present both explicitly and implicitly. Observationally, the GP reconstruction is sensitive to kernel choice and hyperparameter priors, and the paper uses an “effective BAO point” compression of CMB information rather than full CMB likelihoods. Theoretically, the stability analysis is restricted to the pure gravitational sector and neglects matter perturbations; while the authors argue that adding matter would not remove an existing instability, this still leaves open questions about full coupled perturbation stability. Additionally, for $f(Q)$ they restrict to the coincident gauge; they note that beyond-coincident branches can introduce non-monotonic $Q$ evolution and complicate the time-slicing choice, so their mapping may not cover all $f(Q)$ realizations of quintom behavior.

Practically, the results suggest that modified gravity frameworks like $f(T)$ and (in the coincident gauge) $f(Q)$ can accommodate quintom-like dark energy evolution consistent with DESI DR2-inspired reconstructions. This should matter to cosmologists and gravitational theorists working on interpreting DESI BAO/SNe/CMB evidence for dynamical dark energy, and to those building EFT-based or geometry-based models that can reproduce $w=-1$ crossing while remaining stable. Even though BIC does not strongly prefer the quintom model over $\Lambda$CDM, the paper’s central message is that the modified-gravity structure provides a plausible mechanism for the data-favored dynamical behavior, motivating further model building and more comprehensive perturbation and likelihood analyses with future DESI releases and other probes (e.g., growth of structure, lensing).

Overall, the paper’s core contribution is the combination of (i) a unified EFT-in-MAG theoretical mapping showing how $f(T)$ and $f(Q)$ can realize quintom behavior, and (ii) a GP-based observational reconstruction using DESI DR2 BAO + SNe + CMB constraints that yields $w(z)$ crossing the phantom divide from below, consistent with “quintom-B” evolution.