'Constraint consistency' at all orders in Cosmological perturbation
  theory
  We study the equivalence of two - order-by-order Einstein's equation and
Reduced action - approaches to cosmological perturbation theory at all orders
for different models of inflation. We point out a crucial consistency check
which we refer to as 'Constraint consistency' that needs to be satisfied. We
propose a quick and efficient method to check the consistency for any model
including modified gravity models. Our analysis points out an important feature
which is crucial for inflationary model building i.e., all `constraint'
inconsistent models have higher order Ostrogradsky's instabilities but the
reverse is not true. In other words, one can have models with constraint lapse
function and shift vector, though it may have Ostrogradsky's instabilities. We
also obtain the single variable equation for non-canonical scalar field in the
limit of power-law inflation for the second-order perturbed variables.
