## Wednesday, November 13, 2013

### slow acoustic waves

the scale of baryon acoustic oscillation is set by the distance a sound wave can travel from the end of inflation until recombination. it is very important in this context that the sound wave is driven by photon pressure which makes the wave so fast. could you estimate the size of the BAO-scale if the acoustic wave traveled as a normal sound wave, i.e. driven by adiabatic compression of an ideal gas?

1. This is tricky, because the sound speed is drastically dropping during the expansion, and simultaneously the already travelled distance is "stretched" by the continuing expansion.
Uli Bastian

2. in contrast to radiation pressure driven sound waves in the early universe, whose speed is

c_s^2 = \frac{c^2}{3(1+Q)}

with $Q = 3\rho_b/4/\rho_\gamma$ with the baryon-density $\rho_b$ and the photon sensity $\rho_\gamma$, which approximates $c/\sqrt{3}$ for high radiation density, adiabatic sound waves driven by their thermal equation of state travel much slower,

c_s^2 = \gamma\frac{k_BT}{m}

putting in typical numbers yields a factor of almost $10^4$ between these two propagation velocities, implying that the BAO-scale in the case of adiabatically driven sound waves of $\sim10$ kpc with correspondingly tiny patches in the CMB-sky at $\ell\simeq 10^6$.