Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 4 / iii / Solution

{title2=$\delta\Phi_{(p)}=M_{\rm Pl}\psi_{(p)}$}

If the tensor metric perturbation is $h_{ij}=2E_{ij}$ and the polarization tensors have contraction $M^{(p)}_{ij}M^{(q)*ij}=\delta_{pq}$, the quadratic action for each Fourier amplitude is $(M_{\rm Pl}^2/2)\int a^2(|\psi_{(p)}'|^2-k^2|\psi_{(p)}|^2)$. Thus each helicity is normalized like a real massless scalar by the displayed rescaling. Its power per polarization is $(H_k/(2\pi M_{\rm Pl}))^2$. Factors of two in other tensor spectra depend on the definitions of the metric amplitude, the polarization norm and whether both helicities are summed.


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 Incoming links: Tensor normalization with unit-norm polarization tensors