<p>Having now looked at the paper by <a href="https://phy-act1.princeton.edu/public/saiola/act_dr4_A20.pdf" rel="noreferrer">Aiola et al. (2020)</a>, it emerges that for that map, they filtered the data to exclude low frequency multipoles with <span class="math-container">$|l|&lt;150$</span>, corresponding to about 1 degree. This filtering was done to all the maps in the paper and will be responsible for the dramatic &quot;hole&quot; in your Fourier transform.</p>
<p>As for the high frequency rings, well they might correspond to high <span class="math-container">$l$</span> peaks in the angular power spectrum of the CMB, but other things to consider:</p>
<p>I don't know if the image has already gone through some deconvolution process. If not, then your CMB signature will also be multiplied by the FT of the angular response function of the telescopes (probably a Bessel-like function with some high frequency features).</p>
<p>The finite size of the map will introduce apodisation effects, whereby your Fourier transform is the convolution of the CMB transform with the FT of the apodisation function (a narrow 2D <span class="math-container">${\rm sinc}$</span> function.</p>
<p>You also have the pixel size to worry about. Pixelation will be convolution of the true signal with a square top hat kernel. This will also introduce a multiplicative function in to the net Fourier transform of the FT of the pixel kernel (i.e. multiplying by a broad 2D <span class="math-container">${\rm sinc}$</span> function). I think I see this as a cross-shaped pattern on the image.</p>

CC BY-SA 4.0
2020-07-16T19:21:21.903Z
That's an *E-mode* map, not a T-mode map. The peaks in the EE power spectrum are more prominent than in the TT power spectrum you're plotting.
amaurea
CC BY-SA 4.0
2020-07-16T19:33:56.060Z
The CMB TT spectrum does not actually have a hole at low l, the way it looks like in the plot you show. That plot is actually showing `l²*C(l)`, not just `C(l)`. That makes it look like there is more power around l=200 than l=10, but there isn't.
amaurea
CC BY-SA 4.0
2020-07-16T22:48:12.610Z
Agreed (about the $l^2$ factor).
ProfRob
CC BY-SA 4.0
2020-07-16T07:56:46.473Z
2020-07-16T22:43:32.637Z
2020-07-16T22:43:32.637Z
ProfRob
<p>For that specific E-mode map we have applied a Wiener filter to highlight the high SN modes (those &quot;rings&quot;).</p>
<p>I also further apply the following filter: <span class="math-container">$((1 + (kx/5)^{-4})^{-1}) * ((1 + (k/150)^{-4})^{-1})$</span>. This second filter gives the &quot;hole&quot; and a &quot;thin&quot; vertical line in your 2D PS. The image above is just for PR purposes.</p>
<p>In Aiola et al. (2020) the maps shown are made by applying the kx and k filter, but we do not apply any Wiener filter (again the filtering is done for visualization purposes).</p>
<p>The original raw maps used for the analysis are available here: <a href="https://lambda.gsfc.nasa.gov/product/act/actpol_prod_table.cfm" rel="noreferrer">https://lambda.gsfc.nasa.gov/product/act/actpol_prod_table.cfm</a></p>
<p>As amaurea points out, that is a E-mode map and not a temperature map.</p>

CC BY-SA 4.0
2020-07-17T01:11:42.533Z
I see! [1](https://arxiv.org/abs/2007.07288), [2](https://phy-act1.princeton.edu/public/saiola/act_dr4_A20.pdf) :-) Thanks you very much and *Welcome to Stack Exchange!*
uhoh
CC BY-SA 4.0
2020-07-16T19:24:00.610Z
2020-07-17T06:52:15.060Z
2020-07-17T06:52:15.060Z
Simone Aiola
<p>The BBC's <a href="https://www.bbc.com/news/science-environment-53420433" rel="nofollow noreferrer">Desert telescope takes aim at ageing our Universe</a> contains the image below of the <a href="https://en.wikipedia.org/wiki/Cosmic_microwave_background" rel="nofollow noreferrer">Cosmic Microwave Background</a> from the <a href="https://en.wikipedia.org/wiki/Atacama_Cosmology_Telescope" rel="nofollow noreferrer">Atacama Cosmology Telescope</a> or ACT. It looks like this is plotted with a <a href="https://www.kennethmoreland.com/color-maps/" rel="nofollow noreferrer">diverging colormap</a> which makes sense since they've likely subtracted off the average temperature and probably at least its dipole component if not more (<a href="https://en.wikipedia.org/wiki/Cosmic_microwave_background#CMBR_dipole_anisotropy_(%E2%84%93_=_1)" rel="nofollow noreferrer">CMBR dipole anisotropy (ℓ = 1)</a></p>
<p>But looking at it closely, I thought I noticed a strange regularity to the spatial scale, so I downloaded it and imported into Python and took the Fourier transform to see what I'd find.</p>
<p>Despite this being an ugly and very unscientific analysis, I'm still seeing both a sharp cutoff below a spatial frequency of 1.0 inverse degrees, and &quot;ringing&quot; or <span class="math-container">$J_1$</span> Bessel-like ripples at higher frequencies.</p>
<p>Are either or both of these profound and important to the analysis, or an instrumental artifact, or something else?</p>
<hr />
<p>Image in the BBC news item is a JPEG and you can see the subfield discontinuities (part of the way the JPEG wavelets are implemented) once the the thee RGB color channels are separated, but this is a square grid and not related to the clearly circular patterns in the Fourier transform.</p>
<ul>
<li>Python script: <a href="https://pastebin.com/t52VbqeD" rel="nofollow noreferrer">https://pastebin.com/t52VbqeD</a></li>
<li>Exported image converted to PNG: <a href="https://i.sstatic.net/BC87Y.png" rel="nofollow noreferrer">https://i.sstatic.net/BC87Y.png</a></li>
</ul>
<p><a href="https://i.sstatic.net/5jZPF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5jZPF.png" alt="unscientific hack analysis" /></a></p>

CC BY-SA 4.0
2020-07-16T07:43:56.913Z
Try and see what you get with an image of white noise?
planetmaker
<observational-astronomy><radio-astronomy><python><cosmic-microwave-background><mathematics>
Why does the Fourier transform of this CMB image have a hole in it?
uhoh
# Why does the Fourier transform of this CMB image have a hole in it?

(Asked by: username_0 on 2020-07-16)

The BBC's [Desert telescope takes aim at ageing our Universe](https://www.bbc.com/news/science-environment-53420433) contains the image below of the [Cosmic Microwave Background](https://en.wikipedia.org/wiki/Cosmic_microwave_background) from the [Atacama Cosmology Telescope](https://en.wikipedia.org/wiki/Atacama_Cosmology_Telescope) or ACT. It looks like this is plotted with a [diverging colormap](https://www.kennethmoreland.com/color-maps/) which makes sense since they've likely subtracted off the average temperature and probably at least its dipole component if not more ([CMBR dipole anisotropy (ℓ = 1)](https://en.wikipedia.org/wiki/Cosmic_microwave_background#CMBR_dipole_anisotropy_(%E2%84%93_=_1))

But looking at it closely, I thought I noticed a strange regularity to the spatial scale, so I downloaded it and imported into Python and took the Fourier transform to see what I'd find.

Despite this being an ugly and very unscientific analysis, I'm still seeing both a sharp cutoff below a spatial frequency of 1.0 inverse degrees, and "ringing" or $J\_1$ Bessel-like ripples at higher frequencies.

Are either or both of these profound and important to the analysis, or an instrumental artifact, or something else?

---

Image in the BBC news item is a JPEG and you can see the subfield discontinuities (part of the way the JPEG wavelets are implemented) once the the thee RGB color channels are separated, but this is a square grid and not related to the clearly circular patterns in the Fourier transform.

* Python script: <https://pastebin.com/t52VbqeD>
* Exported image converted to PNG: <https://i.sstatic.net/BC87Y.png>

[![unscientific hack analysis](https://i.sstatic.net/5jZPF.png)](https://i.sstatic.net/5jZPF.png)

**Tags:** observational-astronomy, radio-astronomy, python, cosmic-microwave-background, mathematics

> **Comments:**
> **username_1**: Try and see what you get with an image of white noise?  

---

## 2 answers

### Answer by username_4 on 2020-07-16. Score: 19 | Accepted answer

For that specific E-mode map we have applied a Wiener filter to highlight the high SN modes (those "rings").

I also further apply the following filter: $((1 + (kx/5)^{-4})^{-1}) \* ((1 + (k/150)^{-4})^{-1})$. This second filter gives the "hole" and a "thin" vertical line in your 2D PS. The image above is just for PR purposes.

In Aiola et al. (2020) the maps shown are made by applying the kx and k filter, but we do not apply any Wiener filter (again the filtering is done for visualization purposes).

The original raw maps used for the analysis are available here: <https://lambda.gsfc.nasa.gov/product/act/actpol_prod_table.cfm>

As username_3 points out, that is a E-mode map and not a temperature map.

> **Comments:**
> **username_0**: I see! [1](https://arxiv.org/abs/2007.07288), [2](https://phy-act1.princeton.edu/public/saiola/act_dr4_A20.pdf) :-) Thanks you very much and *Welcome to Stack Exchange!*  

---

### Answer by username_2 on 2020-07-16. Score: 20

Having now looked at the paper by [Aiola et al. (2020)](https://phy-act1.princeton.edu/public/saiola/act_dr4_A20.pdf), it emerges that for that map, they filtered the data to exclude low frequency multipoles with $|l|<150$, corresponding to about 1 degree. This filtering was done to all the maps in the paper and will be responsible for the dramatic "hole" in your Fourier transform.

As for the high frequency rings, well they might correspond to high $l$ peaks in the angular power spectrum of the CMB, but other things to consider:

I don't know if the image has already gone through some deconvolution process. If not, then your CMB signature will also be multiplied by the FT of the angular response function of the telescopes (probably a Bessel-like function with some high frequency features).

The finite size of the map will introduce apodisation effects, whereby your Fourier transform is the convolution of the CMB transform with the FT of the apodisation function (a narrow 2D ${\rm sinc}$ function.

You also have the pixel size to worry about. Pixelation will be convolution of the true signal with a square top hat kernel. This will also introduce a multiplicative function in to the net Fourier transform of the FT of the pixel kernel (i.e. multiplying by a broad 2D ${\rm sinc}$ function). I think I see this as a cross-shaped pattern on the image.

> **Comments:**
> **username_3**: That's an *E-mode* map, not a T-mode map. The peaks in the EE power spectrum are more prominent than in the TT power spectrum you're plotting.  
> **username_3**: The CMB TT spectrum does not actually have a hole at low l, the way it looks like in the plot you show. That plot is actually showing `l²*C(l)`, not just `C(l)`. That makes it look like there is more power around l=200 than l=10, but there isn't.  
> **username_2**: Agreed (about the $l^2$ factor).  

---

