Possibility of `Hubble waves'

 $$\omega=\dot{\theta}=H(t):=\frac{d\log\epsilon(t)}{dt},$$

$$\epsilon(t)=\epsilon_0 e^\theta.$$

Applying the transformation

$$\eta(t):= \epsilon^i (t)=\tilde{\epsilon_0}e^{i\theta},$$

so, in particular

$$\boxed{\eta(t)=\tilde{\epsilon_0}e^{iHt}}$$