Essentials of k-essence

TL;DR

k-essence introduces a scalar field with non-linear kinetic terms to drive late-time cosmic acceleration via dynamical attractor solutions, addressing the cosmic coincidence problem without fine-tuning. The framework yields trackers that mimic the background during radiation and de Sitter or k-attractors near matter domination, enabling acceleration without fixed initial conditions; it also predicts distinct late-time behaviors, including possible finite-duration acceleration via a late dust tracker. The paper provides detailed classifications, evolution scenarios through radiation and matter epochs, and concrete toy models illustrating both eternal and finite acceleration, along with simpler alternative Lagrangians. It foregrounds observational signatures such as a time-varying equation of state $w_k(z)$ and a non-unit sound speed $c_S^2$, linking non-linear kinetic terms typical of string/supergravity effective actions to potentially measurable cosmological effects.

Abstract

We recently introduced the concept of "k-essence" as a dynamical solution for explaining naturally why the universe has entered an epoch of accelerated expansion at a late stage of its evolution. The solution avoids fine-tuning of parameters and anthropic arguments. Instead, k-essence is based on the idea of a dynamical attractor solution which causes it to act as a cosmological constant only at the onset of matter-domination. Consequently, k-essence overtakes the matter density and induces cosmic acceleration at about the present epoch. In this paper, we present the basic theory of k-essence and dynamical attractors based on evolving scalar fields with non-linear kinetic energy terms in the action. We present guidelines for constructing concrete examples and show that there are two classes of solutions, one in which cosmic acceleration continues forever and one in which the acceleration has finite duration.

Figures (11)

• Figure 1: A sample function $g(y)$. Boldface letters denote the corresponding attractors; their positions are given on the $y$-axis. The tangent to the curve at a radiation tracker, such as ${\bf R}$, goes through $4y_R/3$, whereas the tangent to the curve at the de Sitter point ${\bf S}$ goes through the origin.
• Figure 2: Phase diagram for case A$_r$ during the radiation-dominated epoch. Phase lines flow in the direction shown by the arrows, dashed horizontal lines determine the $y$ coordinate of attractor solutions and boldface labels the corresponding attractor points. The dotted line shows the points where $\varepsilon_k/\varepsilon_{tot}=r^2(y)$.
• Figure 3: Phase diagram of a model of the type B$_r$ during the radiation-dominated phase. In the relevant region of the diagram all trajectories can be traced back to a common origin. Some of the phase trajectories converge to the radiation tracker ${\bf R}$, while others, after approaching the de Sitter point ${\bf S}$ finally reach the ${\bf K}$-attractor. The saddle point ${\bf x}$ "separates" both types of trajectories.
• Figure 4: Phase diagram of a model of the type C$_r$ during radiation domination, with same notation as in Fig. 3.
• Figure 5: Phase diagram of a model of type A$_d$ during the matter-dominated epoch. All trajectories have a common origin and all of them finally reach the ${\bf K}$-tracker. Trajectories which "skim" the line $\varepsilon_k/\varepsilon_{tot}\approx 0$ reach this attractor after going through a nearly de Sitter stage (the S-attractor).