Newtonian-potential Hankel modes in stiff contraction
= Newtonian-potential Hankel modes in stiff contraction
{title2=$\Phi_k=N_k(-\tau)^{-1}H_1^{(1)}(-k\tau)$}
In <conformal Newtonian gauge>, a free scalar has no linear anisotropic stress. Eliminating its perturbation with the momentum constraint gives $\Phi_k''+3\Phi_k'/\tau+k^2\Phi_k=0$. Substitution of $\Phi_k=(-\tau)^{-1}y(-k\tau)$ gives the order-one <Bessel equation>. The displayed <Hankel function> has positive incoming frequency $e^{-ik\tau}$ at early times.