<p>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 <a href="http://arxiv.org/abs/astro-ph/0310808" rel="nofollow noreferrer">Davis &amp; Lineweaver 2004</a>).</p>

<p>The magnitude of the dipole of the CMB is $3.4\,\mathrm{mK}$ (<a href="http://arxiv.org/abs/astro-ph/9312056" rel="nofollow noreferrer">Kogut et al. 1993</a>), 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.</p>

<p>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.</p>

<p><a href="https://i.sstatic.net/EX0pT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EX0pT.png" alt="CMBfreq"></a></p>

<p>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 <a href="https://en.wikipedia.org/wiki/Friedmann_equations" rel="nofollow noreferrer">Friedmann equation</a>, assuming a <a href="http://arxiv.org/abs/1502.01589" rel="nofollow noreferrer">Planck 2015</a> cosmology.</p>

CC BY-SA 3.0
2016-07-08T14:46:50.170Z
Wow, very helpful and intriguing. I've corrected the formula above.
Alec
CC BY-SA 3.0
2016-07-08T14:53:33.113Z
@Alec: Okay, I removed the part about the wrong value, and edited to plot to go a bit into the future.
pela
CC BY-SA 3.0
2016-07-08T14:56:28.830Z
Thank you! The points are very helpful :) Do you know the frequency value at the first point where CMB is emitted?
Alec
CC BY-SA 3.0
2016-07-08T15:04:07.683Z
@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
CC BY-SA 3.0
2016-07-08T15:16:44.780Z
That comes out to approximately E_43, if I'm not mistaken. Thanks!
Alec
CC BY-SA 3.0
2016-07-09T15:39:41.407Z
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
CC BY-SA 4.0
2019-01-26T06:05:16.890Z
160.23 GHz / (2^30) = 149.225816 Hz Closer to D3 (146.83) than D#3 (155.56)
Right? Anyone know the secondary frequencies?
Is it like a music chord or does the peak frequency really stand out?
John Klemesrud
CC BY-SA 4.0
2019-01-29T07:34:40.190Z
@JohnKlemesrud Yes, _today_'s frequency is close to a D (D$_{33}$, to be specific), as the OP writes in his question, but the frequency when it was _emitted_ was closer to a D♯ (D♯$_{43}$, to be specific}. The ratio between these two notes is (close to) the CMB redshift. There isn't really any "secondary" frequencies, as the CMB is nothing like a chord. It's more like white noise (not exactly white since there _is_ a peak). You can see the full spectrum [here](https://en.wikipedia.org/wiki/Cosmic_microwave_background#/media/File:Cmbr.svg).
pela
CC BY-SA 3.0
2016-07-08T14:31:54.110Z
2016-07-08T14:52:28.700Z
2016-07-08T14:52:28.700Z
pela
<p>After discovering <a href="https://physics.stackexchange.com/questions/216276/what-does-a-blackbody-sound-like">this question</a> 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 <em>pitch</em> 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:</p>

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

<p>Building on this result:</p>

<ul><li>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)?</li>
<li>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?</li>
<li>What pitch would the peak value of this blackbody have had at earlier points in the history of the Universe?</li>
</ul>

<p>I've tried to apply some of the information in <a href="https://astronomy.stackexchange.com/questions/8763/does-the-cosmic-microwave-background-change-over-time">this question</a> 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.</p>

CC BY-SA 3.0
2016-07-08T05:10:05.463Z
When you say "pitch", do you simply mean frequency? Doesn't that word imply how humans perceive the sound of some spectrum?
pela
CC BY-SA 3.0
2016-07-08T12:25:03.840Z
@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
<cosmology><big-bang-theory><redshift><cosmic-microwave-background><black-body>
Future redshifting and effect on the 'pitch' of CMB radiation
Alec
# Future redshifting and effect on the 'pitch' of CMB radiation

(Asked by: username_0 on 2016-07-07)

After discovering [this question](https://physics.stackexchange.com/questions/216276/what-does-a-blackbody-sound-like) 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](https://astronomy.stackexchange.com/questions/8763/does-the-cosmic-microwave-background-change-over-time) 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.

**Tags:** cosmology, big-bang-theory, redshift, cosmic-microwave-background, black-body
> **Comments:**
> **username_1**: When you say "pitch", do you simply mean frequency? Doesn't that word imply how humans perceive the sound of some spectrum?  
> **username_0**: @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.  

---

## 1 answers

### Answer by username_1 on 2016-07-08. Score: 3 | Accepted answer

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](http://arxiv.org/abs/astro-ph/0310808)).

The magnitude of the dipole of the CMB is $3.4\,\mathrm{mK}$ ([Kogut et al. 1993](http://arxiv.org/abs/astro-ph/9312056)), 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.

[![CMBfreq](https://i.sstatic.net/EX0pT.png)](https://i.sstatic.net/EX0pT.png)

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](https://en.wikipedia.org/wiki/Friedmann_equations), assuming a [Planck 2015](http://arxiv.org/abs/1502.01589) cosmology.

> **Comments:**
> **username_0**: Wow, very helpful and intriguing. I've corrected the formula above.  
> **username_1**: @Alec: Okay, I removed the part about the wrong value, and edited to plot to go a bit into the future.  
> **username_0**: Thank you! The points are very helpful :) Do you know the frequency value at the first point where CMB is emitted?  
> **username_1**: @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.  
> **username_0**: That comes out to approximately E_43, if I'm not mistaken. Thanks!  
> **username_1**: If you plug in more precise numbers (z = 1089, and peak$_{\sf today}$ = 160.23 GHz), it's actuallly closer to D$\sharp_{43}$. :)  
> **username_2**: 160.23 GHz / (2^30) = 149.225816 Hz Closer to D3 (146.83) than D#3 (155.56) Right? Anyone know the secondary frequencies? Is it like a music chord or does the peak frequency really stand out?  
> **username_1**: @JohnKlemesrud Yes, _today_'s frequency is close to a D (D$_{33}$, to be specific), as the OP writes in his question, but the frequency when it was _emitted_ was closer to a D♯ (D♯$_{43}$, to be specific}. The ratio between these two notes is (close to) the CMB redshift. There isn't really any "secondary" frequencies, as the CMB is nothing like a chord. It's more like white noise (not exactly white since there _is_ a peak). You can see the full spectrum [here](https://en.wikipedia.org/wiki/Cosmic_microwave_background#/media/File:Cmbr.svg).  

---

