I am confused about length scales and baryon acoustic oscillations. I am clearly missing something very simple.

The length scale found in the $SDSS$ and $2dF$ data is quoted as being $150Mpc$. Scaling back to the surface of last scattering, is a redshift of about $1100$ and hence an S of ~1/1100 putting that scale at $0.14Mpc$ at recombination.

However, I have read that the scale is the acoustic horizon and the speed of BAOs in the plasma about $\frac{c}{2}$ or roughly $1.7e8ms^{-1}$. If recombination happens $379 000$ years into history, that puts the acoustic horizon at $215 000$ light years or $0.065Mpc$ – just about half the figure of $0.14Mpc$.

What am I missing?


The scale-factor of the universe changes significantly over that period of time, so you can't calculate distance as simply $d = v \cdot t$. You have to actually integrate over the expanding spacetime metric, i.e.

$$s = \int_{z_1}^{z_2} \frac{c_s}{H(z)} dz$$

Where $c_s$ is the speed of sound (and I think that should be $c/\sqrt{3}$ instead of $c/2$, right?); and $H(z)$ is the evolving Hubble parameter.


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