Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 3 / c

Solution

Denote the comoving box volume by

V

, keeping it distinct from the inflaton potential

V (ϕ)

u_{k} = a δ ϕ_{k}

, the equation in part (b) becomes

u_{k}^{''} + (k^{2} - \frac{2}{τ ^{2}}) u_{k} = 0.

u_{k} = \frac{1 - i / ( k τ )}{2 k V} e^{- ik τ}, δ ϕ_{k}^{(+)} = \frac{H ( - τ + i / k )}{2 k V} e^{- ik τ} .

For

∣ k τ ∣ ≫ 1

u_{k}

tends to

e^{- ik τ} / 2 k V

a^{2} V (δ ϕ_{k}^{(+)} δ ϕ_{k}^{(+) *'} - δ ϕ_{k}^{(+) *} δ ϕ_{k}^{(+)'}) = i .

∣ k τ ∣ ≪ 1

, the mode tends to the time-independent value

i H / 2 k^{3} V

. The complex mode and its Hermitian conjugate together make the real quantum field.

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