= Vacuum-normalized Newtonian spectrum in stiff contraction
{title2=$|\Phi_k|^2\sim3\hbar/(8\pi k^4\tau^4)$}
The momentum constraint relates the incoming scalar mode $a^{-1}\sqrt{\hbar/(2k)}e^{-ik\tau}$ to the <Newtonian-potential Hankel modes in stiff contraction>. Large-argument Hankel asymptotics give $|N_k|^2=3\pi\hbar/(32k^2)$. The small-argument behavior $H_1^{(1)}(z)\sim-2i/(\pi z)$ then gives the displayed spectrum. Its dimensional power is $k^{-4}$ and its power per logarithmic interval is proportional to $k^{-1}$. These are Newtonian-potential spectra; a curvature-perturbation spectrum need not have the same exponent.
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