This is a follow-up to a really basic question I had: How is the first acoustic peak calculated in CMB? Plank quotes an angular size of the sound horizon at 0.0104, but that corresponds to an multipole of $l=\frac{\pi}{0.0104}=302$, but the first peak on the chart is clearly around $l=220$.

Edit: Apparently there's a "Shift Parameter" that changes the angular distance, $D_A(z*)$. This parameter appears to be linked to the amount of Dark Energy, however, it appears that this parameter has already been factored in by the Plank Collaboration in their calculation of the angular scale.

  • $\begingroup$ I looked around for the reference I remember this from, but can't find it. So take what I'm about to say lightly. I seem to recall that the first peak is shifted slightly away from what we'd expect for a flat universe. Interestingly, there is no shift if you assume minute amounts of closed curvature (though, that does present other problems elsewhere). I think I read that the shift parameter accounts for this by slightly mimicking the effects of having a closed universe on the BAO peaks. $\endgroup$
    – Jim
    May 23, 2017 at 12:47
  • $\begingroup$ I think this is treated as a measure of dark energy because the more you have to shift the peaks towards a closed universe, the more dark energy must be there to shift everything towards looking like an open universe (yeah, I didn't follow that bit myself). I'm not posting this as an answer because I can't find the reference any more and I'm really skeptical about my memory on this topic $\endgroup$
    – Jim
    May 23, 2017 at 12:48
  • $\begingroup$ I found this paper: aanda.org/articles/aa/pdf/2007/31/aa7292-07.pdf which describes how the 'Shift Parameter' is calculated. The results, $l=303$ is nearly identical with the value quoted in the Plank Study, $l=302$. It still leaves us with the question, how do you go from $l=303$ to $l=220$ in the charts of the power spectrum. $\endgroup$
    – user32023
    Dec 24, 2018 at 21:15

2 Answers 2


This question is partially answered by Relation between multipole moment and angular scale of CMB .

The relationship that $\theta \sim \pi/l$ is just an approximate relation and there is no direct one-to-one mapping between angular scale and multipole moment. Angular variations with a set value of $\theta$ contribute power over a range of spherical harmonics defined in terms of $l$ and $m$.

The approximate relationship is derived by just noting that there are $2l$ nodes around the equator. The angular separation of these nodes is $\pi/l$. But if you go to a "latitude" $\delta$ on your sphere and draw a line of constant $\delta$, then you also intercept $2l$ nodes, but the angular separation of each node is now $\pi \sin \delta/l$.

The solid angle of sky visible at any $\delta$ is $2\pi \cos \delta\ d\delta$, so an "average" value of $\delta$ is $2/\pi$ and $\sin \bar{\delta} = 0.594$, so an alternative "average" value for angular scale (along circumferences of constant $\delta$) is $$\Delta \theta \simeq 0.594 \frac{\pi}{l}$$

However at the same time you still have $2l$ nodes along a full $2\pi$ angle along a line of longitude, each node separated by an angle $\pi/l$.

The two approximations bracket the actual value of the peak $l$ and so perhaps a better (best?) hand-waving correspondence between $\theta$ and $l$ would be to take the geometric mean: $$ \bar{\Delta \theta} \simeq \sqrt{0.594}\, \frac{\pi}{l} = 0.77\, \frac{\pi}{l}$$

This factor of 0.77 is indeed quite close to $220/302 = 0.73$.

Your thoughts about the "shift parameter" are a red herring. The shift parameter is just something that can be measured from the CMB. It is closely related to where the first acoustic peak occurs and is theoretically defined in terms of a combination of cosmological parameters - it is strongly dependent on the curvature of the universe but only weakly dependent on the cosmological constant or equation of state for dark energy.


This is an interesting question, I hadn't come across the shift parameter, its conspicuous absence from the Planck 2015 paper (https://arxiv.org/abs/1502.01589) suggests that this effect is taken into account in a different way by the Planck collaboration (I would be interested to hear suggestions about this).

In the (somewhat limited) sources I managed to find, these slides, http://bccp.berkeley.edu/o/beach_program-oops/presentations11/COTB11Melchiorri2.pdf, suggest that the shift parameter has a simple geometric interpretation,
$R=\frac{H_0}{\sqrt{\Omega_m}}\int_0^{z_{dec}}\, \frac{dz}{H(z)}$
(from slide 21), but there is still no physical intuition for what is causing this shift.

  • $\begingroup$ That expression suggests it's simply an effect of the evolution of the universe since the CMB was emitted! $\endgroup$
    – Tom Clarke
    May 24, 2017 at 9:32
  • $\begingroup$ What surprises me is the number of people who are absolutely convinced that the CMB peaks are incontrovertible evidence that the L-CDM is correct, yet I've not found one person who can explain in physical terms why the first peak is off by a significant amount without this fudge (excuse me, shift) factor. $\endgroup$
    – user32023
    May 29, 2017 at 18:39

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