form (ignoring ${\cal O}(\pi^5)$ terms, see [@EFT2008]) $$\frac{M_3^4}{6} \left(g^{00} + 1 \right)^3 = - \frac{2}{3} M_3^4 \left[ 2 \dot{\pi}^3 + 3 \dot{\pi}^4 
- \frac{3}{a^2} \dot{\pi}^2 (\partial_i \pi)^2 \right] \; .$$

It is important to recognize that, thanks to the EFT formulation, it is very easy to identify a regime in which the Stueckelberg field has non-negligible self-interactions without violating the slow-roll nature of the background solution. Since the symmetry arguments can not fix the value (or sign) of the coefficient $M_3$ (which should be determined from observations, see [@Planck-2013-NG] for the latest limits), one can write $M_3^4 (t) = - c_3 (t) M^4$ where $M$ is a mass scale characterizing the interaction. An extra shift symmetry can be imposed requiring that the time dependence of $c_3$ is negligibly weak.

In this regime, the action of the $\pi$ field becomes (see [@SZ2010]) $$\begin{aligned}
 \label{S_SZ2010}
 S=\int d^4x \; a^3 \left[ -\dot{H} M_{\rm Pl}^2 \left(\dot\pi^2-\frac{1}{a^2}(\partial_i\pi)^2\right) + \right. \\ \nonumber
\left. \frac{2}{3}c_3 M^4\left(2\dot\pi^3+3\dot\pi^4-3\frac{1}{a^2}\dot\pi^2(\partial_i\pi)^2\right)\right]\ ,\end{aligned}$$

This is perhaps the simplest possible interacting cosmological perturbation theory. It is also observationally interesting (see [@Planck-2013-NG]). This is the most general kind of interactions which are not slow roll suppressed and which are leading order when we impose the 