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TL;DR: The factor of 2 comes from the fact that there are 2 Fabry-Pérot cavities; where you used $K_-^2 P_{\mathrm{arm}}/h\nu$, you should have used $G_- P_{\mathrm{arm}}/h\nu$ because that's the number of photons hitting the mirrors; and the factor of $K_-$ is a transfer function between coupled oscillators (the fluctuating pressure noise and the ...

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