Non-parametric exploration of minimally coupled gravity with phantom crossing

TL;DR

The paper investigates whether minimally coupled Horndeski theories of the KGB class can realize phantom crossing without conformal coupling, in light of DESI-era hints for $w$ crossing $-1$. Using mochi_class with a manifestly stable EFT basis, the authors non-parametrically sample stable EFT functions and background histories, generating ~250k KGB models and selecting six (M1–M6) that satisfy background and late-time growth criteria, including a positive galaxy–ISW cross-correlation. They predict late-time observables (CMB TT and lensing, RSD through $f\sigma_8$, cosmic shear, galaxy–ISW) and find that the selected KGB models can closely mimic $ ext{ΛCDM}$ expansion while enhancing the growth and lensing signals in a manner broadly consistent with current data, particularly for lensing and ISW, without requiring non-minimal conformal coupling. The results demonstrate viable phantom-crossing solutions within minimally coupled KGB, suggesting that derivative interactions offer a competitive explanation for the data and motivating future full Bayesian analyses with non-parametric EFT priors to quantify preference over conformally coupled scenarios.

Abstract

Recent measurements of the baryon acoustic oscillations by the Dark Energy Spectroscopic Instrument (DESI), especially when combined with cosmic microwave background (CMB) and supernova data, favor a late-time dark energy equation of state that crosses $w=-1$, which has been argued to point toward non-minimal conformal coupling in Horndeski gravity. We test this interpretation by performing a non-parametric exploration of the minimally coupled, luminal Horndeski subclass known as kinetic gravity braiding (KGB). Using mochi_class and its manifestly stable effective field theory (EFT) basis implementation, we efficiently scan a broad class of models in which the EFT functions are allowed to vary freely in time, while enforcing the absence of ghost and gradient instabilities from the outset. We identify a set of KGB models that realize phantom crossing and remain broadly consistent with current probes of the background and linear large-scale structure, including CMB temperature and lensing power spectra, redshift-space distortions, cosmic shear, and the cross-correlation between galaxies and the Integrated Sachs-Wolfe effect. Our results demonstrate that viable phantom-crossing solutions exist without conformal coupling, motivating future full Bayesian analyses of this model class with non-parametric EFT priors.

Figures (6)

• Figure 1: Left: Evolution of the stable basis functions $\{D_{\rm kin}, c_{s}^2\}$ for the six selected KGB models. In all cases, the sound speed starts out subluminal ($c_{s}^2 < 1$) and becomes superluminal ($c_{s}^2 > 1$) at late times. For convenience, we also show their product, $c_{s{\rm N}}^2$, which is more directly related to the modification of the Newtonian potential (see Eq. \ref{['eq:mu_infinity']}). Right: Derived parameters braiding, $\alpha_{B}$, and kineticity, $\alpha_{K}$, for the same models shown in the left panel.
• Figure 2: Left: Evolution of background quantities for the six selected KGB models. From top to bottom we show the present-day normalized dark energy density $f_{\rm DE}$, the transverse comoving distance $D_{M}$ relative to the DESI DR2 best-fit $w_{0}w_{a}$CDM prediction, and the corresponding Hubble distance $D_{H}$ relative to the same fiducial model. Models M1–M4 share this best-fit expansion history (blue), while M5 and M6 follow the two mirage dark energy backgrounds (orange and red). Right: Linear-theory predictions for the quasi-static modification of the Newtonian potential, $\mu_{\infty}$ (top), and for the growth function, $D(z)$, relative to the DESI DR2 best-fit $w_{0}w_{a}$CDM model (bottom). Coloured curves correspond to the six KGB models M1–M6, while black lines show the three reference $w_{0}w_{a}$CDM cosmologies used throughout this work.
• Figure 3: Top: Predictions for the RSD parameter combination $f\sigma_{8}$ for the six KGB models M1–M6 (solid colored lines), along with their matched phenomenological models parameterized by $\mu_{0}$ (see Eq. \ref{['eq:mu_sigma_pheno']}; dashed lines) and the three $w_{0}w_{a}$CDM cosmologies (black lines). Black points with error bars show the DESI DR1 measurements reported in Fig. 14 of Ref. Adame2025. Bottom: Ratio of the predicted $f\sigma_8$ for each model to the DESI DR2 best-fit $w_0w_a$CDM cosmology. Line styles and colors are the same as in the top panel.
• Figure 4: Left: CMB temperature power spectrum for the six KGB models M1–M6 (colored lines) and for the three reference $w_{0}w_{a}$CDM cosmologies (black lines). Gray points with error bars are the CamSpec bandpower estimates derived from the reanalysis of the Planck PR4 temperature maps Efstathiou2021. The lower panel shows residuals with respect to the DESI DR2 best-fit $w_{0}w_{a}$CDM prediction. Here we use the conventional rescaling $\mathcal{D}_\ell^{TT} \equiv \ell(\ell+1)C_\ell^{TT}/(2\pi)$. Right: CMB lensing convergence power spectrum for the six KGB models M1–M6 (colored lines) and for the three reference $w_{0}w_{a}$CDM cosmologies (black lines). Gray points with uncertainties are the Planck 2018 lensing bandpower measurements obtained using the minimum-variance (MV) estimator over the conservative multipole range $8 \leq \ell \leq 400$PlanckCollaboration2019. The lower panel shows ratios relative to the DESI DR2 best-fit $w_{0}w_{a}$CDM prediction.
• Figure 5: Ratios of the cosmic shear auto-power spectra for the four DES Y3 tomographic bins, shown relative to the DESI DR2 best-fit $w_0w_a$CDM prediction. Solid colored curves show the six KGB models M1–M6; black lines correspond to the three reference $w_0w_a$CDM cosmologies; and dashed colored curves in the bottom right panel display the matched phenomenological models parameterized by $\Sigma_0$ (with $\mu_0$ fixed to the values used in Fig. \ref{['fig:fsigma8']}). Gray bands indicate the 68% uncertainties on each tomographic auto-spectrum for a DES Y3-like survey, expressed as $\pm\sigma[C_\ell^{\gamma\gamma}]/C_\ell^{\gamma\gamma}$, including only the Gaussian contributions.