<p>The square of CMB temperature fluctuations represented in spherical coordinates is proportional to <span class="math-container">$C_l$</span>, where <span class="math-container">$C_l$</span> is the angular power spectrum (<a href="https://academic.oup.com/ptep/article/2014/6/06B101/1561631" rel="nofollow noreferrer">reference</a>).</p>
<p>And according to <a href="https://arxiv.org/pdf/0803.0834.pdf" rel="nofollow noreferrer">this review paper</a>, <span class="math-container">$C_l$</span> usually absorbs the <span class="math-container">$T_\mathrm{CMB}$</span> and the extra quantites are due to &quot;historical reasons&quot;, which I imagine are just factors from <a href="https://en.wikipedia.org/wiki/Spherical_harmonics" rel="nofollow noreferrer">spherical harmonics</a>.</p>

CC BY-SA 4.0
2022-07-26T00:49:59.397Z
2022-07-26T00:49:59.397Z
terribleGuitarPlayer
<p>A way of thinking about it is that a point in the plotted &quot;angular power spectrum&quot; represents an averaged sum over the whole sky of the temperature fluctuation in one part multiplied by the temperature fluctuation in another part separated by some angle from the first - represented on the x-axis of the graph by the multiple moment, where <span class="math-container">$\theta \simeq 180^\circ/l$</span> (and is shown along the top x-axis).</p>
<p>This means that if the fluctuations have a characteristic angular scale, this reveals itself as a peak in the power spectrum at the corresponding multiple moment.</p>
<p>Since the units of the temperature fluctuations from the mean are Kelvins, then the power spectrum, which is a sum over the product of fluctuation pairs is in Kelvin<span class="math-container">$^2$</span>.</p>
<p>The units of <span class="math-container">$\mu$</span>K<span class="math-container">$^2$</span> are used because the fluctuations are of order <span class="math-container">$10^{-5}$</span> K and so their product is of order thousands of <span class="math-container">$\mu$</span>K<span class="math-container">$^2$</span>.</p>
<p>The <span class="math-container">$l(l+1)/2\pi$</span> are just unitless normalising factors to make the power spectrum roughly horizontal.</p>

CC BY-SA 4.0
2022-07-26T06:48:33.760Z
2022-07-26T07:25:23.290Z
2022-07-26T07:25:23.290Z
ProfRob
<p>From what I have read and seen, the minute temperature fluctuations in the CMB are measured in microKelvin, or μK.</p>
<p>However, many charts and graphs show μK<sup>2</sup>, or 'microKelvin-squared'.</p>
<p>Do they simply mean 'square degrees', as in the 41,252.96° in a sphere?</p>
<p>An example is seen in this <a href="https://en.wikipedia.org/wiki/Cosmic_microwave_background#/media/File:PowerSpectrumExt.svg" rel="nofollow noreferrer">WMAP</a> power spectrum:</p>
<p><a href="https://i.sstatic.net/ih9BS.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ih9BS.jpg" alt="WMAP" /></a></p>

CC BY-SA 4.0
2022-07-11T05:17:43.047Z
I have a hunch that it's [variance](https://en.wikipedia.org/wiki/Variance) like $\sigma^2$.
uhoh
CC BY-SA 4.0
2022-07-11T05:44:51.120Z
Also this question would profit from exemplary citation and link to a source where this is seen or used
planetmaker
CC BY-SA 4.0
2022-07-11T12:41:45.333Z
Quoting out of context makes it difficult to provide an exact answer.  The "general" answer is that this unit represents, as @uhoh said, $\sigma$ since it's described as a 'fluctuation".  Otherwise it might  belong to a specific equation within which temperature probably appears more than once.
Carl Witthoft
CC BY-SA 4.0
2022-07-24T09:37:54.083Z
I voted to re-open, since I think it's rather clear that the question is about the reason for the unit of the power spectrum of the CMB.
pela
<astrophysics><cosmology><dark-matter><cosmic-microwave-background>
What is a 'square' Kelvin degree? μ$K^2$? In terms of the cosmic microwave background's (CMB's) temperature fluctuations?
Kurt Hikes
# What is a 'square' Kelvin degree? μ$K^2$? In terms of the cosmic microwave background's (CMB's) temperature fluctuations?

(Asked by: username_0 on 2022-07-11)

From what I have read and seen, the minute temperature fluctuations in the CMB are measured in microKelvin, or μK.

However, many charts and graphs show μK2, or 'microKelvin-squared'.

Do they simply mean 'square degrees', as in the 41,252.96° in a sphere?

An example is seen in this [WMAP](https://en.wikipedia.org/wiki/Cosmic_microwave_background#/media/File:PowerSpectrumExt.svg) power spectrum:

[![WMAP](https://i.sstatic.net/ih9BS.jpg)](https://i.sstatic.net/ih9BS.jpg)

**Tags:** astrophysics, cosmology, dark-matter, cosmic-microwave-background
> **Comments:**
> **username_1**: I have a hunch that it's [variance](https://en.wikipedia.org/wiki/Variance) like $\sigma^2$.  
> **username_2**: Also this question would profit from exemplary citation and link to a source where this is seen or used  
> **username_3**: Quoting out of context makes it difficult to provide an exact answer.  The "general" answer is that this unit represents, as @uhoh said, $\sigma$ since it's described as a 'fluctuation".  Otherwise it might  belong to a specific equation within which temperature probably appears more than once.  
> **username_4**: I voted to re-open, since I think it's rather clear that the question is about the reason for the unit of the power spectrum of the CMB.  

---

## 2 answers

### Answer by username_6 on 2022-07-26. Score: 4 | Accepted answer

A way of thinking about it is that a point in the plotted "angular power spectrum" represents an averaged sum over the whole sky of the temperature fluctuation in one part multiplied by the temperature fluctuation in another part separated by some angle from the first - represented on the x-axis of the graph by the multiple moment, where $\theta \simeq 180^\circ/l$ (and is shown along the top x-axis).

This means that if the fluctuations have a characteristic angular scale, this reveals itself as a peak in the power spectrum at the corresponding multiple moment.

Since the units of the temperature fluctuations from the mean are Kelvins, then the power spectrum, which is a sum over the product of fluctuation pairs is in Kelvin$^2$.

The units of $\mu$K$^2$ are used because the fluctuations are of order $10^{-5}$ K and so their product is of order thousands of $\mu$K$^2$.

The $l(l+1)/2\pi$ are just unitless normalising factors to make the power spectrum roughly horizontal.


---

### Answer by username_5 on 2022-07-26. Score: 2

The square of CMB temperature fluctuations represented in spherical coordinates is proportional to $C\_l$, where $C\_l$ is the angular power spectrum ([reference](https://academic.oup.com/ptep/article/2014/6/06B101/1561631)).

And according to [this review paper](https://arxiv.org/pdf/0803.0834.pdf), $C\_l$ usually absorbs the $T\_\mathrm{CMB}$ and the extra quantites are due to "historical reasons", which I imagine are just factors from [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics).


---

