The Cosmic Background Radiation — Question 8

Question 8

An acoustic angular scale combines a ruler with a distance

In a supplied flat-cosmology model, the comoving sound horizon is 144 Mpc and the transverse comoving distance to last scattering is 13800 Mpc. Define $\theta_s=r_s/D_M$ and $\ell_A=\pi/\theta_s$.

1. Calculate theta in radians and degrees, and the acoustic multipole scale.

2. If the sound horizon stays fixed but DM rises by 5 percent, find the fractional changes of theta and ellA.

Question 8 - Solution

An acoustic angular scale combines a ruler with a distance

(a) Method and calculation

$$\boxed{\theta_s=144/13800=.0104348\ {\mathrm{rad}}=.597869^\circ}.$$$$\boxed{\ell_A=\pi(13800)/144=301.069}.$$

(b) Interpretation and further calculation

$$\boxed{\theta_s^\prime/\theta_s=1/1.05=.952381}.$$$$\boxed{\ell_A^\prime/\ell_A=1.05}.$$

Physical check

The acoustic scale is not exactly the first temperature-peak multipole because phase shifts and detailed plasma dynamics affect peak locations.