Solution

In the far past,

∣ c k τ ∣ ≫ 1

, the term

2/ τ^{2}

is negligible and the rescaled mode is a quantum harmonic oscillator of angular frequency

c k

. The stated choice of its ground state is the Bunch-Davies vacuum.

At late times, after the mode is far outside the Hubble scale,

∣ c k τ ∣ ≪ 1

. Its variance becomes

P_{k} = \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}} (1 + c^{2} k^{2} τ^{2}) ⟶ \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}} .

Since

P_{k}

has dimensions of length cubed, the dimensionless power is proportional to

k^{3} P_{k}

. More precisely, with the standard isotropic Fourier convention,

Δ^{2} (k) = \frac{k ^{3} P _{k}}{2 π ^{2}} ⟶ \frac{ℏ H _{inf}^{2}}{4 π ^{2} c ^{3}},

which is independent of the wavenumber

k

. Thus equal logarithmic intervals in

k

carry equal late-time power. This is the sense in which inflation naturally generates a scale-invariant inflationary power spectrum.

Solved by gpt-5.6-sol high.

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