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 Incoming links: Newtonian-potential Hankel modes in stiff contraction

For the exact mode equation write

z = - k τ > 0

and

Φ_{k} = (- τ)^{- 1} y_{k} (z)

. Substitution gives

z^{2} y_{k}^{''} + z y_{k}^{'} + (z^{2} - 1) y_{k} = 0.

Φ_{k} = N_{k} (- τ)^{- 1} H_{1}^{(1)} (- k τ) .

For large

z

H_{1}^{(1)} (z) \sim 2/ (π z) e^{i (z - 3 π /4)}

. Part (d) and the normalized scalar incoming mode give

A_{k} = \frac{C}{2 ik} \frac{ℏ}{2 k}, ∣ A_{k} ∣^{2} = \frac{3ℏ}{16 k ^{3}}, ∣ N_{k} ∣^{2} = \frac{πk}{2} ∣ A_{k} ∣^{2} = \frac{3 π ℏ}{32 k ^{2}} .

At late times,

H_{1}^{(1)} (z) \sim - 2 i / (π z)

- 2 i N_{k} / (πk τ^{2})

⟨ ∣ Φ_{k} ∣^{2} ⟩ \sim \frac{3ℏ}{8 π k ^{4} τ ^{4}}, power of k = - 4.

Fourier-volume conventions can modify the overall normalization but not this exponent. The dimensionless power per logarithmic interval is

k^{3} ⟨ ∣ Φ_{k} ∣^{2} ⟩ \propto k^{- 1}

, corresponding to

n_{s} = 0

if written as

k^{n_{s} - 1}

. The question asks the dimensional mode variance's exponent, which is

- 4

; it is also important not to substitute a curvature-perturbation spectrum for the specified cosmological Newtonian potential.

The momentum constraint relates the incoming scalar mode

a^{- 1} ℏ/ (2 k) e^{- ik τ}

∣ N_{k} ∣^{2} = 3 π ℏ/ (32 k^{2})

. The small-argument behavior

H_{1}^{(1)} (z) \sim - 2 i / (π z)

then gives the displayed spectrum. Its dimensional power is

k^{- 4}

and its power per logarithmic interval is proportional to

k^{- 1}