Thursday 29 October 2015

homework and exercises - Varying an action (cosmological perturbation theory)


I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action


$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$


And this is the solution,



$$\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h~=~0. $$


This is what I get


$$\partial_{0}(a^{2}\partial_{0}h)-\partial_{0}(a^{2}\nabla h)-\nabla(a^{2}\partial_{0}h)+\nabla^{2}(ha^{2})~=~0.$$


I don't really see my mistake, perhaps I am missing something. (dot represents $\partial_{0}$)


It is this problem (see Lectures on the Theory of Cosmological Perturbations, by Brandenburger):


enter image description here




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