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The construction of gauge-invariant variables for any order perturbations isdiscussed. Explicit constructions of the gauge-invariant variables forperturbations to 4th order are shown. From these explicit construction, therecursive structure in the definitions of gauge-invariant variables for anyorder perturbations is found. Through this recursive structure, thecorrespondence with the fully non-linear exact perturbations is brieflydiscussed.

Recursive structure in the definitions of gauge-invariant variables for any order perturbations