Let's break down the problem step by step. --- ## Step 1: Understand the Angular Scale Formula In a flat universe, the angular scale (θ) for the first acoustic peak in the CMB is approximately: \[ \theta \approx \frac{r_s}{D_A} \] Where: - \( r_s \) = size of the sound horizon at recombination. - \( D_A \) = angular diameter distance to the surface of last scattering. For the first peak, we usually approximate: \[ \theta \approx \frac{c_s t_*}{D_A} \] where \( c_s \) is the sound speed, and \( t_* \) is the time at recombination. The ratio \( \frac{r_s}{D_A} \) is typically about 1 degree for our universe when \( c_s = c/\sqrt{3} \). **Explanation:** The angular scale is proportional to the sound speed. If the sound speed decreases, the sound horizon shrinks, and the angular scale of the peak becomes smaller. --- ## Step 2: Substitute the New Sound Speed Given: - In our universe, \( c_s = \frac{c}{\sqrt{3}} \), angular scale ≈ 1°. - In this fictitious universe, \( c_s = \frac{c}{x} \) and \( x = 9 \). The angular scale will be: \[ \theta' = \theta \times \frac{c_s'}{c_s} = 1^\circ \times \frac{c/x}{c/\sqrt{3}} = 1^\circ \times \frac{\sqrt{3}}{x} \] Plug in \( x = 9 \): \[ \theta' = 1^\circ \times \frac{\sqrt{3}}{9} \approx 1^\circ \times .19245 \approx .19^\circ \] **Explanation:** We scaled the 1° result by the ratio of the new sound speed to the old one. --- ## **Final Answer** \[ \boxed{.19} \] So, the approximate angular scale is **.19 degrees**.