Scaling solutions in three-form cosmology

Abstract

A hybrid three-form model of dark energy is developed in order to identify scaling solutions, a long-sought feature in three-form cosmology. Exploiting Hodge dualities, the theory is formulated in terms of two scalar functions that are associated with the conjugate momentum, and the three-form dual vector in an isotropic background. The resulting Lagrangian yields a stable scaling attractor where the three-form energy density tracks the dominant background fluid. A dynamical mechanism is also identified that naturally drives the system out of this regime toward a late-time accelerated phase distinguishable from a cosmological constant. This constitutes the first realization of scaling behavior within a three-form dark energy framework.

Figures (5)

• Figure 1: Phase portrait of the dynamical system showing the scaling fixed point (a) in red, for $M^2<M_a^2$. Its coordinates are given by Eqs. \ref{['eq:coord_x']} and \ref{['eq:coord_y']}. The physical region is the interior of the green arc-circle $x^{2} + y^{2} < 1$. Left panel: $n = 6$ and $\gamma = 4/3$. Right panel: $n = 4$ and $\gamma = 1$.
• Figure 2: Phase portrait for $\gamma=1$ and $n=4$. For $M^2>M_a^2$, the scaling Point $(a)$ becomes nonphysical, and the dynamical system is attracted toward the three-form dominated configuration in Point $(b)$. The other three-form domination Point $(c)$ is a repeller.
• Figure 3: Region of stability of the scaling regime and three-form-dominated universe, in the parameter space $(n,S^2)$, for $\gamma=1$. The values $n=(0,2)$ are excluded. The boundary $M_a^2$ between the two attractors is given by Eq. \ref{['eq:S_a1']}.
• Figure 4: Left panel: Evolution of the energy densities of radiation $(\rho_r)$, matter ($\rho_m$), and the three-form $(\rho_A)$, for $\lambda^2=0.1$, $n=1.9$, and $m=0.5$. The black dotted curve corresponds to the analytic solution of $\rho_A$ in Eq. \ref{['eq:solution_rho_A']} during the matter-dominated era. Right panel: Evolution of the three-form equation of state $(w_A)$ for the same parameters.
• Figure 5: Evolution of the fractional energy densities of radiation $(\Omega_r)$, matter $(\Omega_m)$ and the three-form $(\Omega_A)$, for $\lambda^2=0.1$, $n=1.9$, and $m=0.5$.