The Effective Theory of Quintessence: the w<-1 Side Unveiled

TL;DR

The paper develops a general effective field theory for quintessence perturbations, incorporating higher-derivative operators to capture the full range from standard $k$-essence to Ghost Condensate. It derives the quadratic action in terms of background quantities and explores stability, showing that the $w_Q<-1$ regime can be rendered stable with appropriate higher-derivative terms and that phantom-divide crossing can occur without pathologies. The authors map theoretical constraints onto the quintessential plane $(1+w_Q)\Omega_Q$ vs. $c_s^2$, and discuss observational implications, including how a vanishing sound speed would enhance dark-energy clustering and modify gravity on cosmological scales. The work provides a framework for comparing data with models that cross the phantom divide or reside in the $w_Q<-1$ sector, highlighting the importance of $c_s^2\approx 0$ in interpreting observations.

Abstract

We study generic single-field dark energy models, by a parametrization of the most general theory of their perturbations around a given background, including higher derivative terms. In appropriate limits this approach reproduces standard quintessence, k-essence and ghost condensation. We find no general pathology associated to an equation of state w_Q < -1 or in crossing the phantom divide w_Q = -1. Stability requires that the w_Q < -1 side of dark energy behaves, on cosmological scales, as a k-essence fluid with a virtually zero speed of sound. This implies that one should set the speed of sound to zero when comparing with data models with w_Q < -1 or crossing the phantom divide. We summarize the theoretical and stability constraints on the quintessential plane (1+w_Q) vs. speed of sound squared.

Figures (4)

• Figure 1: The quintessential plane $1+w_Q$ vs. $c_s^2$ in the case of $k$-essence. If we require the absence of ghosts, the sign of the spatial kinetic term is fixed to be the same as $1+w_Q$, so that one has to worry about gradient instabilities for $1+w_Q < 0$. For $1+w_Q>0$ one has superluminal propagation for $M^4 <0$.
• Figure 2: On the quintessential plane, we show the theoretical constraints on the equation of state and speed of sound of quintessence, in the presence of the operator $\bar{M}$. Instability regions are dashed. Where $1+w_Q$ and $c_s^2$ have opposite sign we have a ghost-like instability corresponding to negative kinetic energy. For $w_Q<-1$, the dashed regions in the left-lower panel is unstable by gradient $(c_s^2 \lesssim - H \bar{M} /M^2)$ and Jeans $((1+w_Q)\Omega_Q \lesssim -1)$ instabilities, while the strip close to the vertical axis corresponds to the stability window (\ref{['stabilitywindow']}). Furthermore, the strip around the horizontal axis given in eq. (\ref{['strip']}) corresponds to the Ghost Condensate. Above this region, in the right-upper panel, we find standard $k$-essence.
• Figure 3: Example of phantom divide crossing, as given by eqs. (\ref{['Hdt']}) and (\ref{['Ht']}), where we have defined $H_c\equiv H_* - \mu^4/(2mM_{\textrm{Pl}}^2)$. Left figure: behavior of $H$ and $\dot H$; the crossing of $w_Q=-1$ takes place at $t=m^{-1}$ when $\dot H=0$ and $H=H_c$. Right figure: trajectory on the quintessential plane.
• Figure 4: The two scaling regimes as a function of the momentum $k$ together with the scaling dimensions of some of the operators. Above: the Ghost Condensate regime where $\omega \sim k^2/M$; below: the regime where $\omega \sim c_s k$.