From Dark Radiation to Dark Energy: Unified Cosmological Evolution in K-essence Models

Abstract

We study a class of Unified Dark Matter (UDM) models based on generalized K-essence, where a single scalar field with non-canonical kinetic terms accounts for dark radiation, dark matter, and dark energy. Starting from the purely kinetic Lagrangian proposed by Scherrer (2004), we extend the analysis to quadratic and exponential scalar potentials and explore their phenomenology. All models are implemented in a modified version of \texttt{Hi\_CLASS} and confronted with data from \textit{Planck} 2018, DESI DR1, and Big Bang Nucleosynthesis. The scenarios reproduce the full sequence of cosmic epochs: an early radiation-like phase, a matter-dominated era, and late-time accelerated expansion. The new models predict slightly higher values of the Hubble constant compared to $Λ$CDM, thereby partially alleviating the respective tensions from $\sim 4.4 σ$ to $\sim 3.4 σ$. The quadratic potential requires an ultralight mass that makes it effectively indistinguishable from the Scherrer solution. Overall, generalized K-essence provides a minimal and observationally viable realization of UDM, offering a unified description of the dark sector with distinctive signatures in both early- and late-time cosmology.

Figures (11)

• Figure 1: Solutions to $X(a)$ to the purely kinetic Lagrangian. As the evolution equation \ref{['third']} is cubic, it has three different solutions. Two of which are only valid for $a>\sqrt[6]{27\theta^2/4}$ (red dashed and blue dotted curves), and one valid for all $a>0$ (black continuous curve). Assuming that the field starts evolving with $X\ll X_0$ at $a\ll 1$ the physical solution is the black continuous line given by the expression \ref{['sol_cual']}.
• Figure 2: Late-time phase portraits in the $(\varphi,v)$ plane for (top left) the purely kinetic sector $V=0$, (top right) the quadratic potential, and (bottom) the exponential potential. The light-blue shaded band indicates the ghost domain $v^2\le 1$ (i.e. $X\le X_0$), excluded by the no-ghost condition $F_X>0$ (equivalently $X>X_0$). The gray shaded region corresponds to $h^2(\varphi,v)\le 0$, where the Friedmann equation would give an imaginary expansion rate and the phase-space flow is not physically defined. The dashed horizontal lines mark the kinetic singularity $3v^2=1$, i.e. $v=\pm 1/\sqrt{3}$, where the evolution equations become singular and the flow splits into disconnected branches.
• Figure 3: Compactified late-time phase portrait for the exponential potential in the $(u,v)$ plane, with $u=e^{-\gamma\varphi}$. The light-blue shaded band indicates the ghost domain $v^2\le 1$, while the gray shaded region corresponds to $h^2(u,v)\le 0$ (no real $H$). The dashed horizontal lines mark the kinetic singularity $v=\pm 1/\sqrt{3}$.
• Figure 4: Redshift evolution of the scalar field equation of state $w_\phi$ for various values of $\theta$. Vertical dashed lines indicate the epochs of BBN and recombination. We see that the field behaves as radiation at high redshifts, transitioning to a matter-like behavior before recombination. Therefore the bounds over extra radiation components on BBN are very important con constrain the model.
• Figure 5: Redshift evolution of the effective number of relativistic species $N_{\mathrm eff}$. The gray shaded region denotes the upper bound from BBN constraints Cooke:2017cwoAver:2015izaPeimbert:2016bdgPitrou:2018cgg. As the scalar field becomes non-relativistic, its contribution to the radiation density vanishes, restoring the standard value of $N_{\mathrm eff}$.