power law $$\Delta_{\zeta, {\rm Free}}^2 (k) = A_s \left( \frac{k}{k_0} \right)^{n_s - 1} \; ,$$ where $A_s$ is the spectral index while $n_s$ is the spectral amplitude and $k_0$ is a pivot scale, see [@2009arXiv0907.5424B]. This implies that $$\label{eq:G:zeta:free}
 \tilde{\cal G}_{\rm Free}(k) = \frac{A_s}{4 \pi k_0^3} \left( \frac{k}{k_0} \right)^{n_s - 4} \; .$$

The leading interactions of the metric fluctuations are typically due to cubic operators, so, it is expected that the loop correction to $\tilde{\cal G}$ shall be of the form (notice that the mass dimension of $\tilde{\cal G}$ is $-3$) $$\tilde{\cal G}_{\rm 1-loop} = k^3 (\tilde{\cal G}_{\rm tree})^2 \times ({\rm factors}) \; .$$ The factors on the RHS can involve various non-trivial logarithmic runnings e.g. $\log a \sim Ht$ or $\log kL$ [@2011JCAP...01..023G; @2010JCAP...08..006B] or $\log (H/\mu)$ [@SZ2010], while a running of the form $\log k/\mu$ is not possible since this shall not leave $\tilde{\cal G}$ invariant under the transformation $x \rightarrow \Lambda x$, $k \rightarrow k/\Lambda$ and $a \rightarrow a/\Lambda$ [@SZ2010].

The $\dot{\pi}^3$ theory
------------------------

While evaluating correlations for an interacting quantum field on an accelerating universe, one can encounter, apart from the familiar UV divergences, relatively unfamiliar divergences too. To avoid having to deal with these 