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

Understand

The construction of gauge-invariant variables for any order perturbations is discussed.

• Explicit constructions of the gauge-invariant variables for perturbations to 4th order are shown.
• From these explicit construction, the recursive structure in the definitions of gauge-invariant variables for any order perturbations is found.
• Through this recursive structure, the correspondence with the fully non-linear exact perturbations is briefly discussed.