Getting the wrong multipole for 1st acoustic peak

[DoobleD]

I'm trying to do a simple calculation, but there must be something wrong.

The wavelength $\lambda_1$ corresponding to first acoustic peak of the CMB is related to the sound horizon at last scattering, $d_{hs}$, by :

$\lambda_1 = 2d_{hs}$ (see for instance slide 14 on Wayne Hu PDF slides).

Now, the multipole $l$ of the first acoustic peak can be related to its wavelength and to the distance to last scattering surface, $D$, by :

$l_1 = \frac{2 \pi}{\lambda_1} D$ (see slide 15)

From that I deduce the following equation :

$l_1 = \frac{\pi}{d_{hs}}D$

I find in the litterature that $D \approx 14000 Mpc$, and $d_{hs} \approx 150 Mpc$. I plug those values into the previous equation, and I find $l_1 \approx 293$, which is quite far from the $l_1 \approx 200$ I should get for the first peak. What's wrong ?

 

[DoobleD]

I get the values for distance to last scattering surface and sound horizon here. I wonder however if 150 Mpc for the sound horizon is not in comoving coordinates, while I should use the physical distance instead (which I don't know) ?

EDIT : I just realized that at the very end of that WMAP values document, they basically give the exact same formula, $l = \frac{\pi}{d_{hs}}D$. And with the values they gives, I get $l = 299$. Why am I not getting 200 ?

 

[DoobleD]

I just found the exact same question asked by someone else on physics.stackexchange : https://physics.stackexchange.com/questions/222993/how-is-the-first-acoustic-peak-calculated-in-cmb

The guy also finds $l \approx 300$ instead of 200.

 

[bapowell]

You should use comoving quantities. See https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/, perhaps start after Eq. 17. I don't compute the numerical values for the sound horizon and distance to last scattering in Mpc, but instead leave them as a ratio. This might help, though.

 

[DoobleD]

You should use comoving quantities. See https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/, perhaps start after Eq. 17. I don't compute the numerical values for the sound horizon and distance to last scattering in Mpc, but instead leave them as a ratio. This might help, though.

Thank you for the help. I'm still puzzled by my problem, but great article.

 

[DoobleD]

I'm back on this issue.

Same problem with again another source. We know that $\theta_s = 0.0104$ (slide 4), and $l = \pi / \theta_s$ (slide 18), so we get $l = 302$ instead of around 200. Exactly the same issue as in the https://redirect.viglink.com/?format=go&jsonp=vglnk_152494367988315&key=6afc78eea2339e9c047ab6748b0d37e7&libId=jgjovf9v010009we000DLcrw2gf0c&loc=https%3A%2F%2Fwww.physicsforums.com%2Fthreads%2Fgetting-the-wrong-multipole-for-1st-acoustic-peak.923207%2F&v=1&out=https%3A%2F%2Fphysics.stackexchange.com%2Fquestions%2F222993%2Fhow-is-the-first-acoustic-peak-calculated-in-cmb&ref=https%3A%2F%2Fwww.physicsforums.com%2Fsearch%2F81044411%2F&title=Getting%20the%20wrong%20multipole%20for%201st%20acoustic%20peak%20%7C%20Physics%20Forums&txt=https%3A%2F%2Fphysics.stackexchange.com%2Fquestions%2F222993%2Fhow-is-the-first-acoustic-peak-calculated-in-cmb post I linked earlier actually.

 

[DoobleD]

Nevermind, this very question has been already answered here. Thanks to @George Jones.