Cosmological contraction with a free homogeneous scalar ($a(\tau )=(-\tau {)}^{1/2},{\varphi }_0=\sqrt{3/2}\log (-\tau )$)

In units $8\pi G=1$, a free homogeneous scalar has $\rho =P={\dot{\varphi }}_0^2/2$ and therefore a stiff equation of state. Its equation gives $a^2{\varphi }_0^{\prime }=\text{constant}$, while the Friedmann equation gives $3{\mathcal{H}}^2={\varphi }_0^{\prime 2}/2$. The displayed contracting branch satisfies both equations for negative conformal time. Constant normalizations and an additive scalar shift can be restored.

In conformal Newtonian gauge, a free scalar has no linear anisotropic stress. Eliminating its perturbation with the momentum constraint gives ${\Phi }_k^{\prime \prime }+3{\Phi }_k^{\prime }/\tau +k^2{\Phi }_k=0$. 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.

The momentum constraint relates the incoming scalar mode $a^{-1}\sqrt{\hslash /(2k)}{e}^{-ik\tau }$ to the Newtonian-potential Hankel modes in stiff contraction. Large-argument Hankel asymptotics give $|N_k{|}^2=3\pi \hslash /(32k^2)$. 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.