I'm trying to wrap my head around dipole anisotropies of the Cosmic Microwave Background (CMB). I think my confusion is rooted in a misunderstanding of how multipole expansion is used in describing the anisotropy.

I know the CMB exhibits a basic anisotropy because 'we' are moving with respect to its rest frame. This manifests itself in a region of 'hot' redshifted CMB - which we are moving towards - and a 'cold' CMB region, which we are moving away from. This allows us to deduce the direction and speed of our local galactic cluster w.r.t to the CMB rest frame. There is also a smaller anisotropy that represents the unevenness of the early universe which shows no overall pattern.

This paper suggests that while the kinematic movement accounts for 99% of the dipole anisotropy, there is a secondary effect from these primordial fluctuations in the dipole:

First Paper

Yet in this paper there is no mention of this. Instead, this source says these inflationary anisotropies only manifest themselves at higher 'multipoles'. This, in my mind, makes sense as describing all the unevenness requires a higher expansion. enter image description here

I think my main question is ultimately this: is the dipole anisotropy a result only of our relative movement and the doppler effect ?

Further to this, what does it mean to say that 'anisotropies only become relevant at '$ l>1 $ ', such as in a graph like this: enter image description here

I understand this is in reference to spherical harmonics. For $ l = 100 $, does this mean that the anisotropy can only be explained with multiple expansions of the spherical harmonics?


1 Answer 1


So I think there are two questions here: 1) What are the dominant contributions to the observed CMB Dipole and 2) What can these measurements tell us about the universe.

For (1), as you stated, the dominant contribution is the doppler shift relative to the CMB rest frame. Our movement results in a hot spot in the direction we are moving towards, and a cold spot in the opposite direction. There is also a contribution from the primordial fluctuations, as these fluctuations occur across all scales (with the sourcing amplitude determined by the $A_s$ and power-law slope by $n_s$).

For (2), the meaning of anisotropies is tied up in their relative error. At low $\ell$ there is significant error due to galactic dust as well as "cosmic variance." Cosmic variance is a fundamental limit in our measurements of these very large scale fluctuations since we only have one universe to study; if we only can ask one person in a country we would have a very hard time figuring out the average age of people, but if we can ask 10,000 we could have a very good estimate of the average and whole distribution of possible ages in that country. In the case of the primordial CMB dipole we are only measuring one single mode in one universe, so it is susceptible to huge error and so it isn't particularly "relevant" for CMB analysis.

The plot you show is very old, where the dominant source of error comes from noise in the map itself; measuring very cold things (i.e. 2.72 K) is very hard when your detector generates heat and warms up what you are trying to observe! More modern measurements are dominated by noise from foregrounds (i.e. dust and galaxies), as well as cosmic variance over most of the range of interest. Here is a more modern version from the Planck Sattilite: enter image description here

The green shaded region represents the cosmic variance region; i.e. for the same cosmological parameters we would expect the CMB anisotropies to vary within this region. While not shown explicitly, you can extrapolate to see that $\ell = 1$ has huge error purely from this variation.

  • $\begingroup$ Thank-you but I'm still not quite sure what it means by multi-pole moment - would you mind explaining this further ? Thanks ever so much though ! $\endgroup$
    – jolene
    Feb 3, 2021 at 17:24
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    $\begingroup$ Essentially the multi-pole moments are ways to decompose some spherical signal into different scales. I really like this animation for showing what different parts of the power-spectra "mean" background.uchicago.edu/~whu/metaanim.html $\endgroup$ Feb 3, 2021 at 19:52
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    $\begingroup$ This is a classic figure to describe what each mode "looks like" burro.case.edu/Academics/Astr328/Notes/CMB/multipoles.png $\endgroup$ Feb 3, 2021 at 19:54
  • $\begingroup$ This is exactly what I didn't get - thank you ! $\endgroup$
    – jolene
    Feb 4, 2021 at 22:27

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