# How is the first peak of the CMB calculated?

1. Dec 8, 2015

### Dr. Strange

This is a follow up to an old thread: https://www.physicsforums.com/threads/the-first-acoustic-peak-in-cmb.475412/. In the final draft of the Plank study - http://arxiv.org/pdf/1303.5076v3.pdf - they quote the sound horizon at 1.04147 x 10-2. This equates to a multipole moment of 302. However, if you look at the diagram of the power curve, the first peak is at roughly 220.

In the earlier post, someone suggested that there was a phase shift, but if you actually calculate the phase shift from the referenced document - using the r* from the Plank study - you get a factor of 0.5. So 302 x 0.5 = 150 which still doesn't equal a multipole moment of 220. So what is the relation between the quoted sound horizon of 0.0104147 and the first acoustic peak at a moment of 220?

2. Dec 8, 2015

3. Dec 8, 2015

### Dr. Strange

Thanks for the post, but I've been through that NED document a couple of times. It doesn't help with the issue. I'd like to know how the sound horizon quoted in the Plank document, 0.014147 radians, relates to the first peak of the power spectrum at 220 in Figure 1 of the same document.

4. Dec 9, 2015

### George Jones

Staff Emeritus

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5. Dec 9, 2015

### Dr. Strange

I'm sorry, but you seem to have misunderstood my question. I have a formula (see the reference in the original post). The one you reference appears to be the same form. I already took the numbers from Plank, plugged them into the formula and got a multipole movement of 152.2, so apparently I'm unable to pick the right numbers out of the Plank report. I'm looking for someone who can tell me what numbers from the final Plank report I can use to go from a quoted sound horizon of 0.014147 to a moment of roughly 220.

6. Dec 9, 2015

### Chalnoth

Btw, my bet is that this whole issue is a matter of different conventions being used for the parameters in the different sources. I'll see if I can't look into it in more detail when I get home later.

7. Dec 10, 2015

### George Jones

Staff Emeritus
Since the first peak of 220 appears in the data, shouldn't this be turned around? Shouldn't the question be "How are the Planck data (all of the data, not just the peak at 220) used to to calculate the sound horizon at last scattering?"

A second, related question might be "Why is the calculated sound horizon different than the first peak?"

Computer code (see 2.1.5 in the Planck paper) that integrates Boltzmann equations for interactions is used to generate test CMB data for model universes that are parametrized by six or so parameters. These model data sets are statistically fitted to the the actual data to find a best-fit set of parameters, i.e., to find a best-fit universe. This best-fit universe is then used to calculate the sound horizon via equation (6) of the Planck paper.

The sound horizon is different from the first peak in the data because of reason given in the pdf that I attached above. See the paragraph that begins "the shift $\Delta l$"