Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 4 / a

Table of contents
- Solution a

Solution

The creation and annihilation operators obey the canonical commutation relations

[a_{k}, a_{k^{'}}^{†}] = (2 π)^{3} δ^{(3)} (k - k^{'}), [a_{k}, a_{k^{'}}] = [a_{k}^{†}, a_{k^{'}}^{†}] = 0.

Since

δ ϕ = f / a

, the vacuum two-point function identifies

Δ_{δ ϕ}^{2} (k, τ) = \frac{k ^{3}}{2 π ^{2}} \frac{∣ f _{k} ( τ ) ∣ ^{2}}{a ^{2}} = \frac{H ^{2}}{4 π ^{2}} (1 + k^{2} τ^{2}) .

On superhorizon scales,

∣ k τ ∣ = k / (a H) ≪ 1

, so

Δ_{δ ϕ}^{2} = (\frac{H}{2 π})^{2} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version