Quintessence

Definition

Work in units with $c = \hbar = 1$ and with metric signature $(-,+,+,+)$. Let $g$ be a spacetime metric with scalar curvature $R$, let $G$ be Newton's constant, and let $V \colon \R \to \R$ be a smooth function, called the potential. A quintessence field is a real scalar field $\phi$ with the following action, which couples it minimally to gravity: $$S = \int \left[ \frac{R}{16 \pi G} - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi\, \partial_\nu \phi - V(\phi) \right] \sqrt{-\det g}\; d^4x + S_{\mathrm{m}}.$$ Here $S_{\mathrm{m}}$ is the action of the other matter, and it does not depend on $\phi$. Take a spatially flat Friedmann–Lemaître–Robertson–Walker background with scale factor $a(t)$ and Hubble rate $H = \dot a / a$. A homogeneous field $\phi(t)$ on it has energy density and pressure $$\rho_\phi = \tfrac{1}{2} \dot\phi^2 + V(\phi), \qquad p_\phi = \tfrac{1}{2} \dot\phi^2 - V(\phi),$$ and it obeys $$\ddot \phi + 3 H \dot \phi + V'(\phi) = 0.$$ Its equation of state parameter is $w_\phi = p_\phi / \rho_\phi$. Quintessence is the use of such a field as the dark energy of the universe.