# Future redshifting and effect on the 'pitch' of CMB radiation

After discovering this question exploring the sound of a blackbody, I started wondering about the sound of the Cosmic Microwave Background radiation from the Big Bang, specifically what the current pitch might be at its peak frequency and how that might evolve over time due to redshifting. Here's what I've been able to deduce so far:

$$pitch_{CMB} = 160.4\,GHz \approx D_{33} = 157.7\,GHz \\ pitch_{CMB}\,lowered\,by\,30\,octaves = {160.4\, GHz \over 2^{30}} = 149.38\, Hz \approx D_3 = 146.83\,Hz$$

Building on this result:

• How much time will need to elapse before the CMB redshift will cause this pitch to drop by an octave (or even by a semitone)?
• How much variability would this pitch have depending on the dipole anisotropy of the CMB from our perspective as well as due to more generalized anisotropies in the CMB?
• What pitch would the peak value of this blackbody have had at earlier points in the history of the Universe?

I've tried to apply some of the information in this question and elsewhere, but would greatly appreciate hearing from someone with more expertise in the subject area if this unorthodox comparison has piqued your interest like it has mine.

• When you say "pitch", do you simply mean frequency? Doesn't that word imply how humans perceive the sound of some spectrum?
– pela
Jul 8, 2016 at 5:10
• @pela Yes, I'm interested in the frequency but would also like to map it onto the musical scale. The answer to my question could be as simple as naming a few frequencies, but I'm also looking for the rate of change of the 'key' of the Universe and for the 'key' of the early Universe and wasn't sure of the best way to translate those rates or determine an accurate starting point.
– Alec
Jul 8, 2016 at 12:25

The wavelength of the CMB increases linearly with the scale factor $a$, which is defined as 1 today, so the "sound" of the CMB has dropped an octave when the Universe has doubled in size (in all three directions), i.e. when $a = 2$, which will happen when it is roughly 25 billion years old (see e.g. Fig 1 of Davis & Lineweaver 2004).

The magnitude of the dipole of the CMB is $3.4\,\mathrm{mK}$ (Kogut et al. 1993), i.e. $\sim10^{-3}$ of the peak value. The primordial fluctuations are even lower, roughly a factor $10^{-5}$ of the peak value. The dipole is partly due to our motion round the Milky Way, and so will fluctuate somewhat with the period of a Galactic year, which is 225 million years.

The figure shows how the peak frequency evolved in the past. Its current value is 160.4 Ghz. To calculate how a human would perceive the sound at a given time, I suppose you'll have to multiply the spectrum with the frequency-dependent transmission function of the human ear, but that's physiology which I know nothing about.

The figure was produced by calculating the frequency at a given time as $$\nu(t) = \frac{160.4\,\mathrm{GHz}}{a},$$ where $a$ runs from $1/1100$ to $4$, and calculating the age by integrating the Friedmann equation, assuming a Planck 2015 cosmology.

• Wow, very helpful and intriguing. I've corrected the formula above.
– Alec
Jul 8, 2016 at 14:46
• @Alec: Okay, I removed the part about the wrong value, and edited to plot to go a bit into the future.
– pela
Jul 8, 2016 at 14:53
• Thank you! The points are very helpful :) Do you know the frequency value at the first point where CMB is emitted?
– Alec
Jul 8, 2016 at 14:56
• @Alec: You're welcome :). The CMB was emitted at redshift z = 1100, i.e. when the scale factor was a = 1/(1+z) = 0.0009, so according to the equation given, the frequency was 160.4 GHz / 0.0009 ~ 180,000 GHz. Note that if the spectrum is expressed as a function of wavelength, the peak wavelength doesn't correspond to the peak frequency, since spectra are expressed per unit wavelength or frequency bin.
– pela
Jul 8, 2016 at 15:04
• If you plug in more precise numbers (z = 1089, and peak$_{\sf today}$ = 160.23 GHz), it's actuallly closer to D$\sharp_{43}$. :)
– pela
Jul 9, 2016 at 15:39