Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 3 / b / i

Table of contents
- Solution i

Solution

Along the unperturbed photon path, combine the temperature and gravitational-redshift terms and use

T^{'} = - Γ

:

\frac{d}{d τ} (Θ + Ψ) + Γ (Θ + Ψ) = Φ^{'} + Ψ^{'} + Γ (Θ_{0} + Ψ - n \cdot v_{e}) .

The integrating factor is

e^{- T}

, and

g = Γ e^{- T}

. Neglecting the exponentially hidden initial boundary term gives the Cosmic microwave background line-of-sight solution

(Θ + Ψ)_{0} = \int^{τ_{0}} d τ e^{- T} (Φ^{'} + Ψ^{'}) + \int^{τ_{0}} d τ g (Θ_{0} + Ψ - n \cdot v_{e}) .

For instantaneous recombination,

g (τ) = δ (τ - τ_{dec})

, and the observer potential contributes only an unobservable monopole. Therefore

Θ (τ_{0}, 0, n) = Θ_{0, dec} + Ψ_{dec} - n \cdot v_{e, dec} + \int_{τ_{dec}}^{τ_{0}} d τ (Φ^{'} + Ψ^{'}) .

The first two terms form the ordinary Sachs-Wolfe effect, the velocity term is the Doppler CMB anisotropy at last scattering, and the integral is the Integrated Sachs-Wolfe effect produced by evolving potentials.

 Ancestors (11)

 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