(i) the equations of motion to order n of \delta q, \delta p, \delta N^\mu, \delta \Pi_\mu are generated by

\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle H_{n}^{\rm can}:=\sum\limits_{k=2}^{n+1}H_{\rm can}^{(k)}. \nonumber \end{align*}

(ii) the perturbation up to order n of {{\mathcal O}}_f, given by

\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle {{\mathcal O}}_{f,n}:=\sum\limits_{k=0}^n {{\mathcal O}}^{(k)}_f, \nonumber \end{align*} is a Dirac observable with respect to H_{\rm can}, and hence C_\mu,\Pi_\mu, up to terms of order at least n + 1

Cosmological perturbations from quantum fluctuations to large scale structure Cosmology and Particle Physics. Proc., CCAST (World Laboratory) Symp./Workshop Nanjing, P.R / China, 30 June-12 July) pp 1-64

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