The pesky 2iπ of MOND-FLRW

 My biggest obstacle so far to derive MOND from FLRW in a completely satisfactory manner, is that from (flat) FLRW metric

$$\frac{dv}{dt}_{\text{null}}=-\frac{cH}{\epsilon(t)},$$

whereas for MOND to work, I need

$$\frac{dv}{dt}_{\text{null}}=-\frac{1}{2i\pi}\frac{cH}{\epsilon(t)}.$$

Currently one of my few ideas is:

$$\omega= H(t).$$

Recall that \(\omega=\dot{\theta}\), and that

$$H(t)=\frac{d}{dt}\log \epsilon(t),$$

so

$$\frac{d\theta}{dt}=\frac{d}{dt}\log \epsilon(t),$$

yielding

$$\epsilon=\epsilon_0 e^\theta .$$

Applying the condition that 

$$\epsilon(t_0)=1,$$

and allowing \(\theta\in \mathbb{C}\), yields

$$\theta_0 = 2i n\pi-\log\epsilon_0,\ \ n\in\mathbb{Z}.$$