# CMB variance as percentage - does that make any sense?

Reading a popular science book that is pretty accurate and in depth, but stumbled upon this line (translated):

The variation, that the CMB pattern reveals amounts to less than one hundredth of a percent.

I can’t figure out what that’s supposed to mean. What’s the reference magnitude here? Can’t be absolute temperature, that would be nonsensical. Can’t be referring to rms variance or anisotropy either, that’s completely different values.

Any idea? Or is it just badly worded?

• Can’t be absolute temperature, that would be nonsensical. Why? Apr 10, 2020 at 17:43

It means that the maximum amplitude of relative variations in the absolute temperature seen in different patches of the sky is about 1 part in ten thousand. This is frequently plotted in terms (roughly) of the square of the temperature fluctuations versus the angular scale of those fluctuations (see below).

As to how exactly this ubiquitous picture of the CMB power spectrum translates into a rough temperature variation amplitude, I refer you to the as-yet unsatisfactorily (to me) answered question: How to translate from the CMB Power spectrum to a spectrum of temperature variations

Edit: In fact forget the power spectrum, just look at the temperature map (minus the mean temperature) on the sky and note the variation compared to the mean value of 2.7K; about 0.01%. (Note though, this map has also taken out the dipole component due to our motion with respect to the CMB, which causes a larger, smooth variation).

• Got it, thanks. That makes a little more sense. Thanks for the link to your question, I’ll have to wrap my head around that later.
– lthz
Apr 10, 2020 at 18:23
• I think the answer you had in that question is about as good as anyone can do. I did once look at the maths when I was an undergrad, but all I now remember is thinking it was brutal and I don't want to look at it again! Apr 10, 2020 at 18:44
• The CMB power spectrum is the departure from isotropy. @CharlesFrancis Apr 10, 2020 at 19:58
• @RobJeffries, You are right. Apr 10, 2020 at 21:10

It is variation in temperature. Why would that not make sense?

• Well my intuition was that absolute temperature and variation shouldn’t have anything to do with each other. Absolute temp. of the CMB obviously changes over time, whereas structure doesn’t. But I basically just wanted to know where the author pulled the number from to dig into it.
– lthz
Apr 10, 2020 at 18:39
• Homogeneity is an assumption. Perfect homogeneity cannot be expected. There should be random variations. This is to show that the random fluctuations are small on the scale at which we can measure them. Apr 10, 2020 at 18:42
• @lthz The variation in temperature we’re talking sbout is over space, not over time. It is anisotropy — slightly different temperatures in different directions. Apr 10, 2020 at 18:43
• They take the dipole anisotropy out (due to Earth's movement relative to CMB frame) before they calculate random fluctuations, which I think we are talking of here. Apr 10, 2020 at 18:47
• The timescale for the expansion of the universe is so large that there is no chance of measuring a change from one millenia to the next, let alone tomorrow! The space variation tells us something about the assumption of homogeneity, which is fundamental to cosmology (although not to physics). Apr 10, 2020 at 19:35