Attractor scenarios and superluminal signals in $k$-essence cosmology

Cosmological scenarios with $k$-essence are invoked in order to explain the observed late-time acceleration of the Universe. These scenarios avoid the need for fine-tuned initial conditions (the “coincidence problem”) because of the attractorlike dynamics of the $k$-essence field $\varphi$. It was recently shown that all $k$-essence scenarios with Lagrangians $p=L(X){\varphi }^{-2}$, where $X\equiv \frac{1}{2}{\varphi }_{,\mu }{\varphi }^{,\mu }$, necessarily involve an epoch where perturbations of $\varphi$ propagate faster than light (the “no-go theorem”). We carry out a comprehensive study of attractorlike cosmological solutions (“trackers”) involving a $k$-essence scalar field $\varphi$ and another matter component. The result of this study is a complete classification of $k$-essence Lagrangians that admit asymptotically stable tracking solutions, among all Lagrangians of the form $p=K(\varphi )L(X)$. Using this classification, we select the class of models that describe the late-time acceleration and avoid the coincidence problem through the tracking mechanism. An analogous “no-go theorem” still holds for this class of models, indicating the existence of a superluminal epoch. In the context of $k$-essence cosmology, the superluminal epoch does not lead to causality violations. We discuss the implications of superluminal signal propagation for possible causality violations in Lorentz-invariant field theories.