### Step 1: Understand the Problem and Relevant Formula The **first acoustic peak** in the CMB corresponds to the **sound horizon** at recombination. For a flat universe, the angular scale (θ) of the first peak is roughly: \[ \theta \approx \frac{r_s}{D_A} \] where: - \( r_s \) = sound horizon at recombination (how far sound has traveled) - \( D_A \) = angular diameter distance to the surface of last scattering For a flat universe, the first peak usually appears at a multipole moment \( \ell \approx \pi / \theta \). The **sound speed** is given as \( c_s = c / x \), with \( x = 9 \). #### **Explanation:** We are asked for the angular scale **in degrees**, which is \( \theta \approx 180^\circ / \ell \). --- ### Step 2: Compute the Angular Scale For a flat universe, the first acoustic peak typically occurs at \( \ell \approx x \), where \( x \) is the denominator in the sound speed (\( c / x \)), multiplied by a geometric factor. In our universe, \( c_s \approx c/\sqrt{3} \), and the first peak is at \( \ell \approx 220 \). For \( c_s = c/9 \): \[ \ell \approx \frac{\pi D_A}{r_s} \] Because \( r_s \propto c_s \), reducing \( c_s \) by a factor of 9 increases \(\ell\) by a factor of 9: \[ \ell_{\text{new}} \approx 220 \times \frac{\sqrt{3}}{9} \] But more simply, for \( c_s = c/x \): \[ \ell \approx x \times 220 / \sqrt{3} \] Plug in \( x = 9 \): \[ \ell \approx 9 \times 220 / 1.732 \approx 1143 \] Angular scale: \[ \theta = \frac{180^\circ}{\ell} = \frac{180}{1143} \approx .16^\circ \] #### **Explanation:** Lower sound speed means smaller sound horizon → higher multipole moment → smaller angular scale. --- ## **Final Answer** \[ \boxed{.16^\circ} \] This is the angular scale (in degrees) for the position of the first acoustic peak with \( c_s = c/9 \).