# Tensor perturbation inflation

During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2$.

I know that the eqn.motion governing GW's from inflation (tensor perturbations) is

$$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$

derived from varying $$S^{(2)}~=~\int \frac{a^{2}(t)}{2}~\partial^{\mu}h ~\partial_{\mu}h ~d^{4}x.$$

See paper 1 and paper 2. Surely this isn't valid during inflation because it isn't de-Sitter. I have read lots of papers/lectures where quantum fluctuations are found by deriving equations of motion from a metric with an $a(t)$ and then Fourier decomposed. Surely during inflation the metric can't contain an $a(t)$. So why do all these lectures/papers use it?

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Related question by OP: physics.stackexchange.com/q/56444/2451 – Qmechanic Mar 12 '13 at 23:18

ok sorry, just to be clear could the metric $dt^2 - a^2 dx^2$ describe inflation? I am just getting confused because the above derivation seems to use that mtric in a perturbed form but surely that is only valid in flat space-time after inflation? – user21119 Mar 12 '13 at 22:20
There is a simple coordinate transformation from the metric you are using to an FRW metric with a particular $a$ function. They are physically the same. – Michael Brown Mar 13 '13 at 0:56