I am currently reading a book about Astrophysics and also tried to find some information about it on the web, but was not able to figure it out. That is why I ask you guys now. Any help is greatly appreciated :)

I understand that the CMBR of approximately $3\,\textrm{K}$ has fluctuations of $\frac{\Delta T}{T}\approx 10^{-5}$ depending in which direction of the universe you look. I also understand that his has to be some structural information (about possible pertubations) from the last scattering surface when matter an light decoupled. However, I just don't have any idea how you are able to determine the Hubble constant and the curvature constant ($k$) from this. In most sources I read it was just stated that if we assume the universe is flat ($k=0$) than we can find the Hubble constant from WMAP data and that the WMAP data agrees well with a flat universe. I don't understand that.

I am looking for a concrete explanation how the information about $H_0$ and $k$ can be extracted from the WMAP data.

  • 1
    $\begingroup$ You might find the first two links in background.uchicago.edu/index.html helpful. Also, this introductory paper is quite good (it deals with SNIA measurements rather than CMB data though): arxiv.org/abs/hep-ph/9906447v1 Finally, for a thorough and rigorous derivation have a look into the book "Physical Foundations of Cosmology" by V. Mukhanov, in particular the last chapter. $\endgroup$
    – Photon
    Commented Mar 26, 2019 at 16:27
  • $\begingroup$ You might also want to check this other question physics.stackexchange.com/questions/431780/…, the only thing you need to know is how $k$ is related to $\Omega$ which you can find in any of the links from @Photon, but the answer lies in the "shape" of the spectrum as is explained briefly in the stackExchange question. $\endgroup$
    – ohneVal
    Commented Mar 29, 2019 at 13:54

1 Answer 1


I’ll talk about the Planck data, just because it’s more recent, but all of this applies to WMAP as well. If you look at the Planck results (Table 1 here), you’ll see that both curvature and Hubble parameter are in the second group of quantities. This is because you do not actually measure these parameters from the data (contrary to what you may have heard). All you do is assume that the universe is flat, based off of local measurements, and from this deduce that the sum of energy densities must equal one. Hence any deviation from one we ascribe to dark energy, but it does not actually have any effect. This makes sense - how could dark energy do anything at only 300,000 years after the Big Bang. Once you have these quantities, the Hubble parameter is just given by Friedman’s equation.

  • $\begingroup$ I believe sadly this answer is incomplete. It is true that for the estimation of some of the parameters a flat universe is assumed. However WMAP and Planck observations allow for constraints in the composition of the universe, that is the amount of ordinary matter, dark matter and dark energy. $\endgroup$
    – ohneVal
    Commented Mar 29, 2019 at 13:37
  • $\begingroup$ Yes they do. You can measure the density of dark and baryonic matter. They do not allow for direct measurements of the dark energy content. That is inferred by subtracting the previous two measurements from what the total must be in a flat, FRW universe. $\endgroup$
    – gmarocco
    Commented Mar 30, 2019 at 11:19

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