Observational constraints on 3-forms dark energy

TL;DR

The paper investigates whether a 3-form field with a Gaussian potential can act as phantom-like dark energy and help alleviate the Hubble tension. It develops a dynamical-system description, identifies an LSBR attractor, and constrains the model using a comprehensive data set (Planck CMB, DESI BAO, Pantheon+, Cepheid calibrations, and DES Y1). The analysis shows that the 3-form model raises the inferred H0 relative to
CDM, reducing tension with late-time measurements while keeping perturbations well-behaved; however, achieving the phantom regime relies on initial conditions. Overall, the Gaussian 3-form emerges as a viable, theoretically motivated dark-energy candidate that can address observational tensions without destabilizing cosmological perturbations, albeit with some fine-tuning in its early evolution.

Abstract

3-forms are natural candidates for describing the late-time accelerated expansion of the Universe, as they can inherently reproduce a positive cosmological constant when lacking an evolving potential. When such a potential is present, a 3-form field may exhibit either quintessence-like or phantom-like behaviour. In this paper, we consider a 3-form model with a Gaussian potential, which features stable, ghost-free phantom-like behaviour within its convergence region and leads to an LSBR late-time attractor. We constrain this model observationally by performing a Markov Chain Monte Carlo (MCMC) analysis employing a comprehensive cosmological dataset, including Planck PR4 cosmic microwave background (CMB) data, DESI DR1 baryon acoustic oscillation (BAO) measurements, Pantheon+ Type Ia supernovae data, low-$z$ Cepheid calibrators, and DES Y1 large-scale structure observations. We demonstrate that the 3-form model successfully increases the predicted Hubble parameter of CMB and BAO data from $67.89\pm0.36{\rm km/s/Mpc}$ of $Λ$CDM model to $68.29^{+0.56}_{-0.61}{\rm km/s/Mpc}$ without fine-tuning of the model parameters, thus reducing the tension with the late-time observation. Furthermore, we verify the sub-dominance of the 3-form field perturbation via both analytical and numerical analyses. Thus, the 3-form field does serve as a promising candidate of phantom-like dark energy from both theoretical and observational points of view.

Figures (7)

• Figure 1: Representative trajectories of the Gaussian 3-form system, chosen to illustrate how the model can mimic $\Lambda$CDM at early-time while deviating at late-time. We set $\bar{V}=1$ and $\xi\simeq 4.48$, and choose initial conditions such that the redshift of matter–radiation equality matches that of $\Lambda$CDM. Top row: Evolution of the compact variables $(v,y,\mathfrak{h},r,s)$ and of the two equations of state, $w_{\rm tot}$ and $w_{\chi}$, for a small-field trajectory that passes close to the matter saddle $A_2$ (branch $D_1^{+}\!\to A_2 \to B^{+}$). Bottom row: Same quantities for a large-field trajectory passing close to the matter saddle $D_2^{+}$ (branch $D_1^{+}\!\to D_2^{+} \to B^{+}$). In both cases, the 3-form dynamics eventually drive the system away from matter domination and towards the LSBR attractor $B^{+}$, producing a late-time phantom crossing ($w_{\rm tot}< -1$) and an asymptotic approach to the LSBR regime with $w_{\rm tot}\to -1^{-}$.
• Figure 2: 68% and 95% C.L. posterior distribution of the early-time parameters from CMB (black), CMB + BAO (blue), CMB + BAO + SNe (red), CMB + BAO + SNe + low-$z$ (green), and CMB + BAO + SNe + low-$z$ + DES Y1 (yellow) data, with $\Lambda$CDM model ($\Lambda$) the dashed contours and the 3-form dark energy model ($\chi$) the solid contours. The result demonstrates the superior stability of the 3-form dark energy for early-time parameters under the influence of the late-time observation.
• Figure 3: 68% and 95% C.L. posterior distribution of the late-time quantities, with $\Lambda$CDM model ($\Lambda$) the dashed contours and the 3-form dark energy model ($\chi$) the solid contours. Left: Inferred from CMB (black), CMB + BAO (blue), CMB + BAO + SNe (red), CMB + BAO + SNe + low-$z$ (green), and CMB + BAO + SNe + low-$z$ + DES Y1 (yellow) data. Right: Inferred from CMB (Black), BAO (blue), SNe + low-$z$ (red), and the combined (green, identical to the one on left) data. The strong tension between astrophysical data of SNe + low-$z$ and cosmological data of CMB + BAO presented in $\Lambda$CDM model posterior of $H_0$ is moderately reduced in the 3-form dark energy model.
• Figure 4: Scatter plot of the extended parameters for the 3-form dark energy model, colour-coded by $\Omega_{\rm m0}$, with CMB on the top and SNe + low-$z$ on the bottom.
• Figure 5: $f\sigma_8$ (top), dark energy EoS $w_{\rm DE}$ (centre) and dynamical variables (bottom) as functions of $z$ for 600 samples drawn from individual fits, against CMB + BAO (left) and CMB + BAO + SNe + low-$z$ (right) dataset. In bottom figures, colour represents variables of section \ref{['sec: dyn_var']}: blue: $v$, purple: $s$, green: $\mathfrak h$, red: $r$, yellow: $y$, magemta: $\kappa\sqrt{V/3}/H$. In top and centre figures, the colour indicates the initial field strength $\log_{10}(a_i^3 \sqrt\xi \kappa \chi_i)$: (red: $>-3/4$, blue: $-3/4 \sim -5/4$, yellow: $-5/4 \sim -7/4$, green: $<-7/4$). In top figures, the $f\sigma_8$ data are taken from table 2 of Avila:2022xad.