# 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

De Sitter space is a special case of the Robertson-Walker spacetime. If you want, you can even work out what the coordinate transformation is.

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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
I just don;t understand how the above eqn can be valid for perturbations created during inflation because de-sitter space is similar to minkowski en.wikipedia.org/wiki/De_Sitter_space and yet that derivation seemed to involved a non de-sitter metric (at least to my understanding). –  user21119 Mar 12 '13 at 22:24
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
@user21119: yes. There is a coordinate system that covers a patch of De Sitter space in which the metric has that form. –  Jerry Schirmer Mar 13 '13 at 15:20