# How, exactly, does the precise measurement of the CMB's polarization modes and temperature fluctuations tell us the value of Hubble's 'constant'?

Amidst all the talk a year and change ago about the value of the Hubble parameter reached by the Planck satellite team, and how it's value differed from the value reached by the 'distance-ladder' team(s), I've yet to read an explanation of how precise measurement of the cosmic microwave background's 'temperature' fluctuations and E-mode polarization(s) ultimately give you a value for Hubble's 'constant'...

I presume the answer is a bit complicated, or I probably would have found it somewhere...

Also, the Planck satellite shut down or whatever in 2013, and this news came out only last year...

Is it okay to 'edit' this question to put it back at the top?

I still don't have a real answer, and the recent news articles and videos about Wendy Freedman and her attempts to resolve the tensions surrounding the discrepancies in the measurements of Hubble's 'constant' always imply that astrophysicists and cosmologists consider the Planck CMB measurement-based cosmological model to be 'better', more reliable, based on sounder, better-understood science than the distance-ladder method...

• The short, but unsatisfactory, answer is that the power spectrum of angular fluctuations in the CMB is sensitive to the value of $H_0$. But you'll want to know why... Nov 12 '20 at 12:27
• Yes, I DO want to know why.... Thank you for whatever help you can provide though, Rob.... Dec 7 '20 at 1:20
• A recent issue of 'New Scientist' magazine devoted its Cover story to the discrepancy between the Planck/CMB value of Hubble's 'constant' and the cosmic-distance-ladder/type-1a-supernova value... November 28-December 4, 2020 issue titled Beyond Space-Time by Stuart Clark... Dec 7 '20 at 1:25

## 1 Answer

The peaks in the temperature and polarization spectra determine the angular size of the sound horizon at the time of recombination fairly accurately:

$$\theta = \frac{r}{D(z)}$$

The sound horizon, which is represented by r, is the comoving distance a sound wave could travel from the beginning of the universe to recombination and is a standard ruler is any given cosmological model.

D(z) is the comoving distance from a present - day observer to the epoch of recombination and is dependent upon the redshift-dependent expansion rate H(z).

In $$\lambda$$CDM cosmology, fitting $$H_0$$ this way is also dependent on the fractional matter energy density today as seen below. The range of cosmologies that can work to fit the CMB are fairly narrow.

• Hi! I edited your answer using LaTeX formatting and added a link for the cosmic sound horizon. Jul 25 at 17:51
• I saw that, thank you! Jul 25 at 18:20