How did the Planck Study calculate the angular size of the sound horizon?

Question

The Planck Collaboration has quoted the angular size of the Sound Horizon:$$\Theta_*= 0.0104$$I decided I'd try a rough, back-of-the-envelope calculation to understand how they arrived at this value. The redshift of CMB is estimated to be 1090 based on the Peebles model for recombination. Using Ned Wright's calculator, that gives an Angular Size Distance$$D_A = 12.723\space {\rm Mpc}$$ So now we need the physical sound horizon, which is the distance that sound travels in the plasma from the dawn of time to recombination. According to Ned's same calculator, $t(z_{CMB}) = 372,000$ yr. The distance that sound travels in that time is:$$\int_0^{t_{CMB}}\frac{c}{\sqrt{3\left(1+\frac{3\rho_B(1+z)^3}{4\rho_{\gamma}(1+z)^4} \right)}}dt$$But it can be safely approximated by $\frac{c\space t}{\sqrt{3}}$ (because the density of baryons is well below that of radiation for most of the integral range), so the sound horizon is roughly:$$r_S=0.066\, {\rm Mpc}$$ Which means that the angle is:$$\Theta_*=\frac{r_s}{D_A}=\frac{0.066}{12.723}=0.005$$This is almost half of the value quoted in the Plank Study and, in no way that I can find, gets us to a multipole of $l=220$ ($\Theta_*=0.015)$. So what's missing from this (probably) naive calculation?

Answer 1

I think there is a small problem in your assumption that the universe is radiation dominated up to the point of recombination. This really isn't true.

But that doesn't help because this means that the sound speed is lower and therefore your sound horizon is (a bit) smaller.

I think the main issue is your calculation of the sound horizon distance. This isn't valid in an expanding universe. For example the current (light) horizon distance is about 46 billion light years, not 13.7 billion light years.

As described in the Planck collaboration (2013) paper you cite: "The characteristic angular size of the fluctuations in the CMB is called the acoustic scale. It is determined by the comoving size of the sound horizon at the time of last-scattering, rs(z∗), and the angular diameter distance at which we are observing the fluctuations, DA(z∗)".

The correct calculation of the comoving sound horizon distance at recombination is $$r_{s,z*} = \int^{t_{\rm rec}}_0 \frac{c_s dt}{a(t)} = \int^{a_{\rm rec}}_0 \frac{c_s\ da}{H\, a^{2}} = \frac{c_s}{\Omega_m^{1/2}H_0} \int^{\infty}_{z_{\rm rec}} \frac{dz}{(1+z)^{3/2}}$$ (e.g. see here.) and also see equation 6 from the Planck collaboration paper you cite where the assumption of a matter-dominated universe with $da/dt = aH$ and $H \simeq H_0 \Omega_m^{1/2} a^{-3/2}$ has been made and noting that $a = (1+z)^{-1}$.

For typically assumed parameters and assuming the sound speed is $c/\sqrt{3}$, this yields $$r_{s,z*} = \frac{2c}{\sqrt{3}\Omega_m^{1/2} H_0}(1 + z_{\rm rec})^{-1/2}$$ For $H_0 = 70$ km/s/Mpc, $\Omega_m=0.3$ and $z_{\rm rec} = 1090$, this gives about 270 Mpc, which needs to be divided by $1+z_{\rm rec}$ to put it in the angular diameter distance terms of your calculation.

This gives an angular scale of 0.019 radians.

But if the sound speed is slower then this scale becomes smaller.

The sound speed is actually $$c_s = \frac{c}{\sqrt{3(1+3\rho_b/4\rho_r)}} $$ and the ratio of baryon to radiation density increases with time in proportion the scale factor $a(t)$. At recombination the ratio $3\rho_b/4\rho_r\sim 1$, and $c_s(t_{\rm rec}) \simeq c/\sqrt{6}$.

This provides a downward correction to the angular sound horizon of about the right size.