 Mass scale of quadratic quintessence

A homogeneous scalar with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. If it is to provide potential-dominated dark energy today rather than rapidly oscillating matter, $m\lesssim H_0\sim 10^{-33}\mathrm{eV}$ is required. Its present density gives ${\varphi }_0\simeq \sqrt{6{\Omega }_X}{M}_{\mathrm{Pl}}{H}_0/m$. Taking $m\sim H_0$ gives a Planck-scale field; a much smaller mass requires a larger field value, so density alone does not fix a unique mass.

// Target: cosmology.bigb