# I Getting the wrong multipole for 1st acoustic peak

1. Aug 19, 2017

### 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 ?

2. Aug 19, 2017

### 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 ?

Last edited: Aug 19, 2017
3. Aug 19, 2017

### DoobleD

4. Aug 21, 2017

5. Sep 5, 2017

### DoobleD

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

6. Apr 28, 2018

### 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 stackexchange post I linked earlier actually.

7. Apr 28, 2018

### DoobleD

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