A new unified dark sector model and its implications on the $σ_8$ and $S_8$ tensions

In this paper, we introduced the Unified Three-Form Dark Sector (UTFDS) model, a unified dark sector model that combines dark energy and dark matter through a three-form field. In this framework, the potential of the three-form field acts as dark matter, while the kinetic term represents dark energy. The interaction between dark matter and dark energy is driven by the energy exchange between these two terms. Given the dynamical equations of UTFDS, we provide an autonomous system of evolution equations for UTFDS and perform a stability analysis of its fixed points. The result aligns with our expectations for a unified dark sector. Furthermore, we discover that the dual Lagrangian of the UTFDS Lagrangian is equivalent to a Dirac-Born-Infeld (DBI) Lagrangian. By fixing the parameter $\kappa X_0$ to 250, 500, 750, we refer to the resulting models as the $\overline{\rm UTFDS}$ model with $\kappa X_0$=250, 500, 750, respectively. We then place constraints on these three $\overline{\rm UTFDS}$ models and the $\Lambda$CDM model in light of the Planck 2018 Cosmic Microwave Background (CMB) anisotropies, Redshift Space Distortions (RSD) observations, Baryon Acoustic Oscillation (BAO) measurements, and the $S_8$ prior chosen according to the KiDS1000 Weak gravitational Lensing (WL) measuement. We find that the $\overline{\rm UTFDS}$ model with $\kappa X_0$=500 is the only one among the four models where both $\sigma_8$ and $S_8$ tensions, between CMB and RSD+BAO+WL datasets, are below 2.0$\sigma$. Furthermore, the tensions are relieved without exacerbating the $H_0$ tension. Although both the CMB and RSD+BAO+WL datasets provide definite/positive evidence favoring $\Lambda$CDM over the $\overline{\rm UTFDS}$ model with $\kappa X_0$=500, the evidence is not strong enough to rule out further study of this model.