LSS Mocks - User contributions [en-ca]

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T21:50:11Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection angles calculated using the <math> \kappa </math> maps.
Since 'lenspix' produces <math> \psi_lm </math> maps (and I checked that their values are legitimate), I used equations from the appendix of Lewis (2005), https://arxiv.org/pdf/astro-ph/0502469.pdf as a reference to calculate the deflection angles.

There, it is mentioned that 'the lensed temperature at a position <math> (\theta, \phi) </math> is given by the unlensed temperature at position <math> (\theta', \phi + \Delta \phi) </math>, where:

<math> \cos \theta' = \cos d \cos \theta - \sin d \sin \theta \cos \alpha </math> & <math> \sin \Delta \phi = \frac{\sin\alpha \sin d}{\sin \theta'} </math>, and the deflection vector is <math> \nabla \psi = d_{\theta} \mathbf{e_{\theta}} + d_{\phi} \mathbf{e_{\phi}} = d \cos \alpha \mathbf{e_{\theta}} + d \sin \alpha \mathbf{e_{\phi}} </math>. (''' ''Lewis'' ''') - It seems I should be able to be derive this from the cosine law on the sphere, but I wasn't able to yet.

However, using the diagram below and spherical trignometry, I got:
(Pythagorean theorem) <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>, <math> \theta = \theta' - [\cos^{-1} \frac{\cos\theta}{\cos \Delta \phi}] </math>
(Sine Law) <math> \frac{\sin \Delta \phi}{\sin \alpha} = \frac{\sin d}{\sin 90} </math>, <math> \Delta \phi = [\sin^{-1} (\sin\alpha \sin d)] </math> (''' ''Right'' ''')

[[File:Deflection_diagram_0629.png|450px|]]

In the last case, I simply used <math> \theta' - \theta = d_{\theta} </math> and <math> \Delta \phi = d_{\phi} </math>. (''' ''Euclidean'' ''')

I made maps for the three cases above - I will refer to them as 'Lewis', 'Right', and 'Euclidean (Approximation).'
<Note> Since I could not preserve the signs (+, -) of the deflection directions for 'Lewis' and 'Right', I took the absolute values of the calculated angles and moved them in the same direction as for 'Euclidean.'

'''3 versions of Deflected maps'''

The characteristics of the maps (unlensed, the 3 deflected maps, and 'lenspix' lensed) are as below. (I'll add the values for the 'Euclidean' deflected map as soon as possible)

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Deflected (Euclidean)
|Deflected (Right)
|Deflected (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

For the deflected maps, we see that their variances are slightly smaller than the unlensed map, while their skewness' and kurtosis' are slightly larger. For the 'lenspix' lensed map, we see again a substantial decrease in the variance and increases in the mean, skewness and kurtosis (by a big amount).

'''Histograms (Unlensed vs. Lenspix vs. Euclidean'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

This is a comparison of the histograms - there is a slight difference (a tiny shift to the right) between the unlensed and (Euclidean) deflected histograms. As before, the lenspix-lensed map has negative pixels.

'''Maps''' (lensed by total <math> \kappa </math>, units are (MJy/sr))

Below are visualizations of the deflection in comparison to the unlensed and lenspix-lensed map. Order: Unlensed, Deflected, Lensed. Longitude range: <math> (0, 2\pi) </math>, Latitude range: <math> (-\frac{\pi}{2}, \frac{\pi}{2} ) </math>.

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

For the low-latitude visualizations, we see that all 3 deflected maps simulate the galaxies/pixels moving quite well - at higher latitudes, especially -80 deg, we see that Lewis' equation definitely simulates the deflection better.

'''N-Point Statistics''' (3-point, 4-point, then 2-point)

'''Euclidean (Not scaled)'''
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_sigma_total_ratio_545GHz.png|450px|]]

'''Euclidean (Scaled so the <math> C_{\ell} </math> is unchanged)''' 3-point: divided by (<math> <\sigma^2> </math> - ratio)^1.5, 4-point: divided by (<math> <\sigma^2> </math> - ratio)^2
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_Euclid_scaled.png|450px|]] [[File:Kurt_ylim.png|450px|]]

'''Right (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_simple_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_simple_scaled.png|450px|]]

'''Lewis (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_lewis_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_lewis_scaled.png|450px|]]

'''Lenspix'''
[[File:Unlensed_vs_lensed_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_sigma_total_ratio_545GHz.png|450px|]]

It is hard to see the difference between the three deflected versions - however, the scaled plots display some similarities with the 'lenspix' lensed map's at <math> \ell > 4000 </math> (especially for the 4-point).

'''Summary'''
Through this exercise, we can see that the statistics given by 'lenspix' seem to make sense, even if this does not directly resolve the 'Negative Pixel' problem.

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T21:49:51Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection angles calculated using the <math> \kappa </math> maps.
Since 'lenspix' produces <math> \psi_lm </math> maps (and I checked that their values are legitimate), I used equations from the appendix of Lewis (2005), https://arxiv.org/pdf/astro-ph/0502469.pdf as a reference to calculate the deflection angles.

There, it is mentioned that 'the lensed temperature at a position <math> (\theta, \phi) </math> is given by the unlensed temperature at position <math> (\theta', \phi + \Delta \phi) </math>, where:

<math> \cos \theta' = \cos d \cos \theta - \sin d \sin \theta \cos \alpha </math> & <math> \sin \Delta \phi = \frac{\sin\alpha \sin d}{\sin \theta'} </math>, and the deflection vector is <math> \nabla \psi = d_{\theta} \mathbf{e_{\theta}} + d_{\phi} \mathbf{e_{\phi}} = d \cos \alpha \mathbf{e_{\theta}} + d \sin \alpha \mathbf{e_{\phi}} </math>. (''' ''Lewis'' ''') - It seems I should be able to be derive this from the cosine law on the sphere, but I wasn't able to yet.

However, using the diagram below and spherical trignometry, I got:
(Pythagorean theorem) <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>, <math> \theta = \theta' - [\cos^{-1} \frac{\cos\theta}{\cos \Delta \phi}] </math>
(Sine Law) <math> \frac{\sin \Delta \phi}{\sin \alpha} = \frac{\sin d}{\sin 90} </math>, <math> \Delta \phi = [\sin^{-1} (\sin\alpha \sin d)] </math> (''' ''Right'' ''')

[[File:Deflection_diagram_0629.png|450px|]]

In the last case, I simply used <math> \theta' - \theta = d_{\theta} </math> and <math> \Delta \phi = d_{\phi} </math>. (''' ''Euclidean'' ''')

I made maps for the three cases above - I will refer to them as 'Lewis', 'Right', and 'Euclidean (Approximation).'
<Note> Since I could not preserve the signs (+, -) of the deflection directions for 'Lewis' and 'Right', I took the absolute values of the calculated angles and moved them in the same direction as for 'Euclidean.'

'''3 versions of Deflected maps'''

The characteristics of the maps (unlensed, the 3 deflected maps, and 'lenspix' lensed) are as below. (I'll add the values for the 'Euclidean' deflected map as soon as possible)

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Deflected (Euclidean)
|Deflected (Right)
|Deflected (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

For the deflected maps, we see that their variances are slightly smaller than the unlensed map, while their skewness' and kurtosis' are slightly larger. For the 'lenspix' lensed map, we see again a substantial decrease in the variance and increases in the mean, skewness and kurtosis (by a big amount).

'''Histograms (Unlensed vs. Lenspix vs. Euclidean'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

This is a comparison of the histograms - there is a slight difference (a tiny shift to the right) between the unlensed and (Euclidean) deflected histograms. As before, the lenspix-lensed map has negative pixels.

'''Maps''' (lensed by total <math> \kappa </math>, units are (MJy/sr))

Below are visualizations of the deflection in comparison to the unlensed and lenspix-lensed map. Order: Unlensed, Deflected, Lensed. Longitude range: <math> (0, 2\pi) </math>, Latitude range: <math> (-\frac{\pi}{2}, \frac{\pi}{2} ) </math>.

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

For the low-latitude visualizations, we see that all 3 deflected maps simulate the galaxies/pixels moving quite well - at higher latitudes, especially -80 deg, we see that Lewis' equation definitely simulates the deflection better.

'''N-Point Statistics''' (3-point, 4-point, then 2-point)

'''Euclidean (Not scaled)'''
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_sigma_total_ratio_545GHz.png|450px|]]

'''Euclidean (Scaled so the <math> C_{\ell} </math> is unchanged)''' 3-point: divided by (<math> <\sigma^2> </math> - ratio)^1.5, 4-point: divided by (<math> <\sigma^2> </math> - ratio)^2
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_Euclid_scaled.png|450px|]] [[File:Kurt_ylim.png|450px|]]

'''Right (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_simple_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_simple_scaled.png|450px|]]

'''Lewis (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_lewis_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_lewis_scaled.png|450px|]]

'''Lenspix'''

[[File:Unlensed_vs_lensed_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_sigma_total_ratio_545GHz.png|450px|]]

It is hard to see the difference between the three deflected versions - however, the scaled plots display some similarities with the 'lenspix' lensed map's at <math> \ell > 4000 </math> (especially for the 4-point).

'''Summary'''
Through this exercise, we can see that the statistics given by 'lenspix' seem to make sense, even if this does not directly resolve the 'Negative Pixel' problem.

June 29th, 2020 - CIB Lensing: Lenspix - Smoothing and Negative Pixels

2020-06-29T21:48:33Z

Jlee: Created page with " Since we saw that 'lenspix' reduces the variance of the Websky CIB map, we agreed I should try smoothing the unlensed map with beam comparable to the size of a pixel. Howeve..."

Since we saw that 'lenspix' reduces the variance of the Websky CIB map, we agreed I should try smoothing the unlensed map with beam comparable to the size of a pixel. However, using an FWHM a bit smaller than the width of an average pixel would not preserve the power spectrum at high <math> \ell </math>'s, and the variance would become much smaller than the 'lenspix'-lensed map.

So, I subsequently made the beam smaller and smaller, until the smoothing had virtually no effect on the 2, 3, 4-point statistics and the variance of the smoothed map was comparable to that of the 'lenspix'-lensed map. (FWHM = <math> 10^{-7} </math> rad)

{| class="wikitable"
!colspan="4"|Map Characteristics
|-
|
|Unlensed
|Smoothed (FHWM = 1e-7)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|5.959061
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| -1.321831
| -0.322438
|-
|# of pixels smaller than 0
|0
|1167158
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|0.0258813
|0.0264728
|-
|Skewness
|1.720313
|1.150339
|1.776150
|-
|Kurtosis
|23.366768
|11.763371
|35.614250
|}

There are discrepancies in the minimum, maximum, skewness and kurtosis values of the smoothed map with the 'lenspix'-lensed map - I believe part of this is because 'lenspix' must smooth the map with an even smaller beam, and also because I smoothed the unlensed map which already had the 400mJy fluxcut imposed (rather than smoothing, and then removing the bright pixels).

'''Histograms'''
[[File:Smoothing_hist.png|450px|]] [[File:Sm_vs_lpix.png|450px|]]

However, the unlensed-smoothed histogram quite resembles the 'lenspix'-lensed histogram (except for a slight shift to the right for the lensed histogram).

'''N-point statistics''' (2, 3, 4-point)

Below is a sanity check to show that the smoothing did not change the n-point statistics.

[[File:Unlensed_vs_lensed_sigma_total_ratio_545GHz_smoothed.png|450px|]] [[File:Unlensed_vs_lensed_bl_total_ratio_545GHz_smoothed.png|450px|]] [[File:Unlensed_vs_lensed_kurt_total_ratio_545GHz_smoothed.png|450px|]]

'''Summary'''
It is now clear that the negative pixels in the 'lenspix'-lensed map won't affect our statistical analysis - whether 'lenspix' lenses point sources accurately may be a different matter (hopefully we don't need to worry about this).

File:Unlensed vs lensed bl total ratio 545GHz smoothed.png

2020-06-29T21:44:02Z

Jlee:

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T21:28:28Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection angles calculated using the <math> \kappa </math> maps.
Since 'lenspix' produces <math> \psi_lm </math> maps (and I checked that their values are legitimate), I used equations from the appendix of Lewis (2005), https://arxiv.org/pdf/astro-ph/0502469.pdf as a reference to calculate the deflection angles.

There, it is mentioned that 'the lensed temperature at a position <math> (\theta, \phi) </math> is given by the unlensed temperature at position <math> (\theta', \phi + \Delta \phi) </math>, where:

<math> \cos \theta' = \cos d \cos \theta - \sin d \sin \theta \cos \alpha </math> & <math> \sin \Delta \phi = \frac{\sin\alpha \sin d}{\sin \theta'} </math>, and the deflection vector is <math> \nabla \psi = d_{\theta} \mathbf{e_{\theta}} + d_{\phi} \mathbf{e_{\phi}} = d \cos \alpha \mathbf{e_{\theta}} + d \sin \alpha \mathbf{e_{\phi}} </math>. (''' ''Lewis'' ''') - It seems I should be able to be derive this from the cosine law on the sphere, but I wasn't able to yet.

However, using the diagram below and spherical trignometry, I got:
(Pythagorean theorem) <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>, <math> \theta = \theta' - [\cos^{-1} \frac{\cos\theta}{\cos \Delta \phi}] </math>
(Sine Law) <math> \frac{\sin \Delta \phi}{\sin \alpha} = \frac{\sin d}{\sin 90} </math>, <math> \Delta \phi = [\sin^{-1} (\sin\alpha \sin d)] </math> (''' ''Right'' ''')

[[File:Deflection_diagram_0629.png|450px|]]

In the last case, I simply used <math> \theta' - \theta = d_{\theta} </math> and <math> \Delta \phi = d_{\phi} </math>. (''' ''Euclidean'' ''')

I made maps for the three cases above - I will refer to them as 'Lewis', 'Right', and 'Euclidean (Approximation).'
<Note> Since I could not preserve the signs (+, -) of the deflection directions for 'Lewis' and 'Right', I took the absolute values of the calculated angles and moved them in the same direction as for 'Euclidean.'

'''3 versions of Deflected maps'''

The characteristics of the maps (unlensed, the 3 deflected maps, and 'lenspix' lensed) are as below. (I'll add the values for the 'Euclidean' deflected map as soon as possible)

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Deflected (Euclidean)
|Deflected (Right)
|Deflected (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

For the deflected maps, we see that their variances are slightly smaller than the unlensed map, while their skewness' and kurtosis' are slightly larger. For the 'lenspix' lensed map, we see again a substantial decrease in the variance and increases in the mean, skewness and kurtosis (by a big amount).

'''Histograms (Unlensed vs. Lenspix vs. Euclidean'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

This is a comparison of the histograms - there is a slight difference (a tiny shift to the right) between the unlensed and (Euclidean) deflected histograms. As before, the lenspix-lensed map has negative pixels.

'''Maps''' (lensed by total <math> \kappa </math>, units are (MJy/sr))

Below are visualizations of the deflection in comparison to the unlensed and lenspix-lensed map. Order: Unlensed, Deflected, Lensed. Longitude range: <math> (0, 2\pi) </math>, Latitude range: <math> (-\frac{\pi}{2}, \frac{\pi}{2} ) </math>.

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

For the low-latitude visualizations, we see that all 3 deflected maps simulate the galaxies/pixels moving quite well - at higher latitudes, especially -80 deg, we see that Lewis' equation definitely simulates the deflection better.

'''N-Point Statistics''' (3-point, 4-point, then 2-point)

'''Euclidean (Not scaled)'''
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_sigma_total_ratio_545GHz.png|450px|]]

'''Euclidean (Scaled so the <math> C_{\ell} </math> is unchanged)''' 3-point: divided by (<math> <\sigma^2> </math> - ratio)^1.5, 4-point: divided by (<math> <\sigma^2> </math> - ratio)^2
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_Euclid_scaled.png|450px|]] [[File:Kurt_ylim.png|450px|]]

'''Right (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_simple_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_simple_scaled.png|450px|]]

'''Lewis (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_lewis_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_lewis_scaled.png|450px|]]

'''Lenspix'''

[[File:Unlensed_vs_lensed_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_sigma_total_ratio_545GHz.png|450px|]]

It is hard to see the difference between the three deflected versions - however, the scaled plots display some similarities with the 'lenspix' lensed map's at <math> \ell > 4000 </math> (especially for the 4-point).

'''Summary'''

Through this exercise, we can see that the statistics given by 'lenspix' seem to make sense, even if this does not directly resolve the 'Negative Pixel' problem.

File:Sm vs lpix.png

2020-06-29T21:27:07Z

Jlee:

File:Smoothing hist.png

2020-06-29T21:26:56Z

Jlee:

File:Unlensed vs lensed kurt total ratio 545GHz smoothed.png

2020-06-29T21:26:27Z

Jlee: Jlee uploaded a new version of File:Unlensed vs lensed kurt total ratio 545GHz smoothed.png

File:Unlensed vs lensed kurt total ratio 545GHz smoothed.png

2020-06-29T21:26:11Z

Jlee:

Jason Logbook

2020-06-29T21:24:02Z

Jlee:

[[June 29th, 2020 - CIB Lensing: Lenspix - Smoothing and Negative Pixels]]

[[June 29th, 2020 - CIB Lensing: Lenspix - Deflection]]

[[June 1st, 2020 - CIB Lensing: Differences Caused by Cosmological Parameters - Plots for Meeting]]

[[June 1st, 2020 - CIB Lensing: Shell-Added Statistics Sanity Check - Plots for Meeting]]

[[May 4th, 2020 - CIB Lensing: Shell by Shell Comparisons (Field Contribution)]]

[[May 4th, 2020 - CIB Lensing: Scrambled vs. Original (Field Contribution)]]

[[May 4th, 2020 - CIB Lensing: Shell by Shell Comparisons (ell less than 8000)]]

[[May 4th, 2020 - CIB Lensing: Field Contribution to Kappa (Shells)]]

[[Apr 19th, 2020 - CIB Lensing: Statistics of Kappa Shells (New Post Above)]]

[[Apr 19th, 2020 - CIB Lensing: Shell by Shell Comparisons (Up to ell_max)]]

[[Apr 19th, 2020 - CIB Lensing: Power Spectrum Conservation Law and N-Side Effect]]

[[Apr 19th, 2020 - CIB Lensing: Histograms & Negative Pixels]]

[[Apr 19th, 2020 - CIB Lensing: Maps & Statistics]]

[[Apr 3rd, 2020 - CIB Lensing: A First Glance]]

[[Apr 3rd, 2020 - Shell-Edge Effects? - Different Initial Parameters Probably]]

[[Feb 27th, 2020 - Exploring the Stochasticity of Websky CIB Maps - II: Enhancement of the 3-Point Compared to the 2-point]]

[[Feb 24th, 2020 - Exploring the Stochasticity of Websky CIB Maps - II: Halo & Galaxy Levels]]

[[Feb 7th, 2020 - Exploring the Stochasticity of Websky CIB Maps - I: Pixel-Level]]

[[Dec 29th, 2019 - Websky CIB vs. Halo Scrambled CIB (Sanity Check on the New Map)]]

[[Nov 24th, 2019 - Comparison Between Lacasa, Planck & Websky at 545GHz & What We Can Learn From Lacasa vs. SPT]]

[[Nov 13th, 2019 - Scaling the Bispectra Using Different Flux Cuts & Power Spectra Sanity Check]]

[[Nov 3rd, 2019 - Sanity Check on Shell-Split-Then-Added Map]]

[[Oct 27th, 2019 - Filtered Variance, 'Sims' and Planck power spectra]]

[[Oct 17th, 2019 - Discrepancies between Planck, SPT, and 'Sims' Bispectra?]]

[[Oct 11th, 2019 - Comparison with Planck at All (3) Frequencies (Reduced Bispectrum, New Maps)]]

[[Oct 10th, 2019 - Comparison with Planck at All (3) Frequencies (Reduced Bispectrum, Old Maps)]]

[[Sept 30th, 2019 - Masking & Removing 1st 100Mpcs]]

[[Sept 6th, 2019 - Effect of Negative Pixels (Lenspix Projection) on the Reduced Bispectrum]]

[[Sept 1st, 2019 - Lenspix Inducing Negative Pixels?]]

[[Sept 1st, 2019 - CIB non-Gaussianity: Whitening II]]

[[Aug 26th, 2019 - CIB non-Gaussianity: Whitening of the Maps]]

[[Aug 18th, 2019 - CIB Bispectrum: Comparison With Crawford Plot (217GHz, Poisson Part)]]

[[Aug 13th, 2019 - CIB Lensing: Effect on the Reduced Bispectrum After Masking Brightest Pixels]]

[[Aug 12th, 2019 - CIB Bispectrum: A Few Pixels in the 0th Shell Going Wild?]]

[[Aug 9th, 2019 - CIB Lensing: Shell By Shell Analysis (All Shells, Prescription Vs. Halo Only)]]

[[July 30th, 2019 - CIB Lensing: Finding the Bias for Theory vs. Field + Bias * Halo (All Shells) (Log Fit)]]

[[July 29th, 2019 - CIB Lensing: Finding the Bias for Theory vs. Field + Bias * Halo (All Shells)]]

[[July 29th, 2019 - CIB Lensing: Finding the Bias Factor (for Power Spectra): Theory/Halo (All Shells)]]

[[July 29th, 2019 - CIB Lensing: Kappa Reduced Bispectra Ratio (by Mass Cut) (Selected Redshifts)]]

[[July 29th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Reduced Bispectra (Selected Shells & Multipoles)]]

[[July 26th, 2019 - CIB Lensing: Kappa Power Spectra Ratio (by Mass Cut) (Selected Redshifts)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles, y-axis set to zero, goes up to higher mass cut)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles, y-axis set to zero) (WRONG)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles) WRONG]]

[[July 26th, 2019 - CIB Lensing: Halo vs. Field Kappa Power Spectra (All Shells)]]

[[July 19th, 2019 - Reduced Bispectrum, 0th Shell Additional Plots]]

[[July 18th, 2019 - CIB Lensing: Shell By Shell Analysis (All Shells)]]

[[July 18th, 2019 - CIB Lensing: Shell By Shell Analysis (Selected Shells)]]

[[July 18th, 2019 - CIB Lensing: Total Maps Analysis]]

[[July 18th, 2019 - CIB Total Maps Analysis]]

[[July 14th, 2019 - CIB Lensing: Is the Lensing Being Done Properly? (for NSIDE 4096) (Not always true!)]]

[[July 14th, 2019 - CIB Smoothing: Iterations]]

[[July 14th, 2019 - CIB Lensing: Smoothing and Interpolation]]

[[July 7th, 2019 - CIB Lensing: Basic Analysis II (Plots)]]

[[July 3rd, 2019 - CIB Halo Scramble: Solved (Edited)]]

[[June 27th, 2019 - CIB Lensing: Basic Analysis (545GHz)]]

[[June 16th, 2019 - Detectability of the Reduced Bispectrum (545GHz)]]

[[June 9th, 2019 - Accuracy of the Wigner 3j Approximation]]

[[June 4th, 2019 - Reduced Bispectrum by Frequency (Fitted Plots)]]

[[June 4th, 2019 - New Plots: Other Frequencies (217GHz, 353GHz)/Comparisons with Planck Data and Lacasa Model (Revised)]]

[[June 2nd, 2019 - New Plots: Other Frequencies (217GHz, 353GHz)/Comparisons with Planck Data and Lacasa Model]]

[[May 28th, 2019 - Investigating the Discrepancy Between Poisson and Sims 3rd Moments]]

[[May 23th, 2019 - Plots for Meeting (Updated Plots)]]

[[May 21th, 2019 - Analysis without the Brightest Galaxies]]

[[May 21th, 2019 - New Plots with NSIDE 4096]]

[[May 15th, 2019 - Flux Re-normalization & Poisson variance/skewness]]

[[May 8th, 2019 - Some analysis on the sims map and the expected Poisson bispectrum]]

[[May 6th, 2019 - Calculating the Poisson variance and skewness (first attempt)]]

[[April 21, 2019 - The origin of the excess kurtosis (revised) ]]

[[March 1, 2019 - Halo maps analysis ]]

[[Feb 20, 2019 - Comparison of Skewness & Kurtosis between sims and Gaussian maps (edited)]]

[[Feb 14, 2019 - Filtering of Unlensed, Poisson, and Gaussian maps]]

[[Feb 13, 2019 - Top hats and Poisson maps]]

[[Jan 10, 2019 - Band filtering - (Cls and 32bands)]]

[[Jan 9, 2019 - Basic Gaussian filtering (revised)]]

[[Jan 7, 2019 - An attempt at Gaussian filtering (incorrect)]]

[[Jan 6, 2019 - Basic statistics of the CIB - non-Gaussianity?]]

[[Nov 7, 2018 - Analysis of the New Maps]]

[[Oct 28, 2018 - Revisions Made]]

[[Oct 17, 2018 - Lensing the CIB]]

[[Sep 26, 2018 - Excercise1]]

File:Unlensed vs lensed sigma total ratio 545GHz smoothed.png

2020-06-29T21:23:40Z

Jlee:

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T21:23:02Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection angles calculated using the <math> \kappa </math> maps.
Since 'lenspix' produces <math> \psi_lm </math> maps (and I checked that their values are legitimate), I used equations from the appendix of Lewis (2005), https://arxiv.org/pdf/astro-ph/0502469.pdf as a reference to calculate the deflection angles.

There, it is mentioned that 'the lensed temperature at a position <math> (\theta, \phi) </math> is given by the unlensed temperature at position <math> (\theta', \phi + \Delta \phi) </math>, where:

<math> \cos \theta' = \cos d \cos \theta - \sin d \sin \theta \cos \alpha </math> & <math> \sin \Delta \phi = \frac{\sin\alpha \sin d}{\sin \theta'} </math>, and the deflection vector is <math> \nabla \psi = d_{\theta} \mathbf{e_{\theta}} + d_{\phi} \mathbf{e_{\phi}} = d \cos \alpha \mathbf{e_{\theta}} + d \sin \alpha \mathbf{e_{\phi}} </math>. (''' ''Lewis'' ''')

However, using the diagram below and spherical trignometry, I got:
(Pythagorean theorem) <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>, <math> \theta = \theta' - [\cos^{-1} \frac{\cos\theta}{\cos \Delta \phi}] </math>
(Sine Law) <math> \frac{\sin \Delta \phi}{\sin \alpha} = \frac{\sin d}{\sin 90} </math>, <math> \Delta \phi = [\sin^{-1} (\sin\alpha \sin d)] </math> (''' ''Right'' ''')

[[File:Deflection_diagram_0629.png|450px|]]

In the last case, I simply used <math> \theta' - \theta = d_{\theta} </math> and <math> \Delta \phi = d_{\phi} </math>. (''' ''Euclidean'' ''')

I made maps for the three cases above - I will refer to them as 'Lewis', 'Right', and 'Euclidean (Approximation).'
<Note> Since I could not preserve the signs (+, -) of the deflection directions for 'Lewis' and 'Right', I took the absolute values of the calculated angles and moved them in the same direction as for 'Euclidean.'

'''3 versions of Deflected maps'''

The characteristics of the maps (unlensed, the 3 deflected maps, and 'lenspix' lensed) are as below. (I'll add the values for the 'Euclidean' deflected map as soon as possible)

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Deflected (Euclidean)
|Deflected (Right)
|Deflected (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

For the deflected maps, we see that their variances are slightly smaller than the unlensed map, while their skewness' and kurtosis' are slightly larger. For the 'lenspix' lensed map, we see again a substantial decrease in the variance and increases in the mean, skewness and kurtosis (by a big amount).

'''Histograms (Unlensed vs. Lenspix vs. Euclidean'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

This is a comparison of the histograms - there is a slight difference (a tiny shift to the right) between the unlensed and (Euclidean) deflected histograms. As before, the lenspix-lensed map has negative pixels.

'''Maps''' (lensed by total <math> \kappa </math>, units are (MJy/sr))

Below are visualizations of the deflection in comparison to the unlensed and lenspix-lensed map. Order: Unlensed, Deflected, Lensed. Longitude range: <math> (0, 2\pi) </math>, Latitude range: <math> (-\frac{\pi}{2}, \frac{\pi}{2} ) </math>.

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

For the low-latitude visualizations, we see that all 3 deflected maps simulate the galaxies/pixels moving quite well - at higher latitudes, especially -80 deg, we see that Lewis' equation definitely simulates the deflection better.

'''N-Point Statistics''' (3-point, 4-point, then 2-point)

'''Euclidean (Not scaled)'''
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_deflected_sigma_total_ratio_545GHz.png|450px|]]

'''Euclidean (Scaled so the <math> C_{\ell} </math> is unchanged)''' 3-point: divided by (<math> <\sigma^2> </math> - ratio)^1.5, 4-point: divided by (<math> <\sigma^2> </math> - ratio)^2
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_Euclid_scaled.png|450px|]] [[File:Kurt_ylim.png|450px|]]

'''Right (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_simple_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_simple_scaled.png|450px|]]

'''Lewis (Scaled so the <math> C_{\ell} </math> is unchanged)
[[File:Unlensed_vs_deflected_bl_total_ratio_545GHz_lewis_scaled.png|450px|]] [[File:Unlensed_vs_deflected_kurt_total_ratio_545GHz_lewis_scaled.png|450px|]]

'''Lenspix'''

[[File:Unlensed_vs_lensed_bl_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_kurt_total_ratio_545GHz.png|450px|]] [[File:Unlensed_vs_lensed_sigma_total_ratio_545GHz.png|450px|]]

It is hard to see the difference between the three deflected versions - however, the scaled plots display some similarities with the 'lenspix' lensed map's at <math> \ell > 4000 </math> (especially for the 4-point).

'''Summary'''

Through this exercise, we can see that the statistics given by 'lenspix' seem to make sense, even if this does not directly resolve the 'Negative Pixel' problem.

File:Kurt ylim.png

2020-06-29T21:14:34Z

Jlee:

File:Unlensed vs lensed kurt total ratio 545GHz.png

2020-06-29T21:12:14Z

Jlee:

File:Unlensed vs lensed bl total ratio 545GHz.png

2020-06-29T21:12:00Z

Jlee:

File:Unlensed vs lensed sigma total ratio 545GHz.png

2020-06-29T21:11:48Z

Jlee:

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T20:48:27Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection angles calculated using the <math> \kappa </math> maps.
Since 'lenspix' produces <math> \psi_lm </math> maps (and I checked that their values are legitimate), I used equations from the appendix of Lewis (2005), https://arxiv.org/pdf/astro-ph/0502469.pdf as a reference to calculate the deflection angles.

There, it is mentioned that 'the lensed temperature at a position <math> (\theta, \phi) </math> is given by the unlensed temperature at position <math> (\theta', \phi + \Delta \phi) </math>, where:

<math> \cos \theta' = \cos d \cos \theta - \sin d \sin \theta \cos \alpha </math> & <math> \sin \Delta \phi = \frac{\sin\alpha \sin d}{\sin \theta'} </math>, and the deflection vector is <math> \nabla \psi = d_{\theta} \mathbf{e_{\theta}} + d_{\phi} \mathbf{e_{\phi}} = d \cos \alpha \mathbf{e_{\theta}} + d \sin \alpha \mathbf{e_{\phi}} </math>. (Lewis)

However, using the diagram below and spherical trignometry, I got:
(Pythagorean theorem) <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>, <math> \theta = \theta' - [\cos^{-1} \frac{\cos\theta}{\cos \Delta \phi}] </math>
(Sine Law) <math> \frac{\sin \Delta \phi}{\sin \alpha} = \frac{\sin d}{\sin 90} </math>, <math> \Delta \phi = [\sin^{-1} (\sin\alpha \sin d)] </math> (Right)

[[File:Deflection_diagram_0629.png|450px|]]

In the last case, I simply used <math> \theta' - \theta = d_{\theta} </math> and <math> \Delta \phi = d_{\phi} </math>. (Euclidean)

I made maps for the three cases above - I will refer to them as 'Lewis', 'Right', and 'Euclidean (Approximation)'

'''3 versions of Deflected maps'''

The characteristics of the maps (unlensed, the 3 deflected maps, and 'lenspix' lensed) are as below.

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Deflected (Euclidean)
|Deflected (Right)
|Deflected (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

'''Histograms'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

'''My attempt at deriving Lewis' equation'''

Considering a unit sphere and the right spherical triangle, the lengths of the triangle are <math> \theta' - \theta </math> (d_theta), <math> \Delta \phi </math> (d_phi), and <math> d </math>. The cosine law gives - <math> \cos \Delta \phi = \cos (\theta' - \theta) \cos d + \sin(\theta' - \theta) \sin d \cos \alpha </math> and the Pythagorean theorem gives <math> \cos d = \cos \Delta \phi \cos (\theta' - \theta) </math>. Assuming <math> \cos \Delta \phi, \cos d, \cos (\theta' - \theta) \ne 0 </math> (deflection angles are small), <math> \cos \Delta \phi = \frac{\cos d}{\cos (\theta' - \theta)} = \cos (\theta' - \theta) \cos d + \sin(\theta' - \theta) \sin d \cos \alpha </math>, <math> 1 = \cos^2(\theta' - \theta) + \sin(\theta' - \theta) \cos(\theta' - \theta) \frac{\sin d}{\cos d} \cos \alpha </math>. Then, <math> 1 - \cos^2(\theta' - \theta) = \sin^2(\theta' - \theta) = \sin(\theta' - \theta) \cos(\theta' - \theta) \frac{\sin d}{\cos d} \cos \alpha </math>. As long as <math> \theta' \ne \theta </math>, <math> \sin(\theta' - \theta) = \cos(\theta' - \theta) \frac{\sin d}{\cos d} \cos \alpha </math>
[[File:Deflection_diagram_0629.png|450px|]]

June 29th, 2020 - CIB Lensing: Lenspix - Deflection

2020-06-29T20:04:26Z

Jlee: Created page with " Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the galaxies by their corresponding deflection..."

File:Deflection diagram 0629.png

2020-06-29T19:58:28Z

Jlee:

Jason Logbook

2020-06-29T19:54:01Z

Jlee:

[[June 29th, 2020 - CIB Lensing: Lenspix - Deflection]]

[[June 1st, 2020 - CIB Lensing: Differences Caused by Cosmological Parameters - Plots for Meeting]]

[[June 1st, 2020 - CIB Lensing: Shell-Added Statistics Sanity Check - Plots for Meeting]]

[[May 4th, 2020 - CIB Lensing: Shell by Shell Comparisons (Field Contribution)]]

[[May 4th, 2020 - CIB Lensing: Scrambled vs. Original (Field Contribution)]]

[[May 4th, 2020 - CIB Lensing: Shell by Shell Comparisons (ell less than 8000)]]

[[May 4th, 2020 - CIB Lensing: Field Contribution to Kappa (Shells)]]

[[Apr 19th, 2020 - CIB Lensing: Statistics of Kappa Shells (New Post Above)]]

[[Apr 19th, 2020 - CIB Lensing: Shell by Shell Comparisons (Up to ell_max)]]

[[Apr 19th, 2020 - CIB Lensing: Power Spectrum Conservation Law and N-Side Effect]]

[[Apr 19th, 2020 - CIB Lensing: Histograms & Negative Pixels]]

[[Apr 19th, 2020 - CIB Lensing: Maps & Statistics]]

[[Apr 3rd, 2020 - CIB Lensing: A First Glance]]

[[Apr 3rd, 2020 - Shell-Edge Effects? - Different Initial Parameters Probably]]

[[Feb 27th, 2020 - Exploring the Stochasticity of Websky CIB Maps - II: Enhancement of the 3-Point Compared to the 2-point]]

[[Feb 24th, 2020 - Exploring the Stochasticity of Websky CIB Maps - II: Halo & Galaxy Levels]]

[[Feb 7th, 2020 - Exploring the Stochasticity of Websky CIB Maps - I: Pixel-Level]]

[[Dec 29th, 2019 - Websky CIB vs. Halo Scrambled CIB (Sanity Check on the New Map)]]

[[Nov 24th, 2019 - Comparison Between Lacasa, Planck & Websky at 545GHz & What We Can Learn From Lacasa vs. SPT]]

[[Nov 13th, 2019 - Scaling the Bispectra Using Different Flux Cuts & Power Spectra Sanity Check]]

[[Nov 3rd, 2019 - Sanity Check on Shell-Split-Then-Added Map]]

[[Oct 27th, 2019 - Filtered Variance, 'Sims' and Planck power spectra]]

[[Oct 17th, 2019 - Discrepancies between Planck, SPT, and 'Sims' Bispectra?]]

[[Oct 11th, 2019 - Comparison with Planck at All (3) Frequencies (Reduced Bispectrum, New Maps)]]

[[Oct 10th, 2019 - Comparison with Planck at All (3) Frequencies (Reduced Bispectrum, Old Maps)]]

[[Sept 30th, 2019 - Masking & Removing 1st 100Mpcs]]

[[Sept 6th, 2019 - Effect of Negative Pixels (Lenspix Projection) on the Reduced Bispectrum]]

[[Sept 1st, 2019 - Lenspix Inducing Negative Pixels?]]

[[Sept 1st, 2019 - CIB non-Gaussianity: Whitening II]]

[[Aug 26th, 2019 - CIB non-Gaussianity: Whitening of the Maps]]

[[Aug 18th, 2019 - CIB Bispectrum: Comparison With Crawford Plot (217GHz, Poisson Part)]]

[[Aug 13th, 2019 - CIB Lensing: Effect on the Reduced Bispectrum After Masking Brightest Pixels]]

[[Aug 12th, 2019 - CIB Bispectrum: A Few Pixels in the 0th Shell Going Wild?]]

[[Aug 9th, 2019 - CIB Lensing: Shell By Shell Analysis (All Shells, Prescription Vs. Halo Only)]]

[[July 30th, 2019 - CIB Lensing: Finding the Bias for Theory vs. Field + Bias * Halo (All Shells) (Log Fit)]]

[[July 29th, 2019 - CIB Lensing: Finding the Bias for Theory vs. Field + Bias * Halo (All Shells)]]

[[July 29th, 2019 - CIB Lensing: Finding the Bias Factor (for Power Spectra): Theory/Halo (All Shells)]]

[[July 29th, 2019 - CIB Lensing: Kappa Reduced Bispectra Ratio (by Mass Cut) (Selected Redshifts)]]

[[July 29th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Reduced Bispectra (Selected Shells & Multipoles)]]

[[July 26th, 2019 - CIB Lensing: Kappa Power Spectra Ratio (by Mass Cut) (Selected Redshifts)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles, y-axis set to zero, goes up to higher mass cut)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles, y-axis set to zero) (WRONG)]]

[[July 26th, 2019 - CIB Lensing: Minimum Mass vs. (Halo) Kappa Power Spectra (Selected Shells & Multipoles) WRONG]]

[[July 26th, 2019 - CIB Lensing: Halo vs. Field Kappa Power Spectra (All Shells)]]

[[July 19th, 2019 - Reduced Bispectrum, 0th Shell Additional Plots]]

[[July 18th, 2019 - CIB Lensing: Shell By Shell Analysis (All Shells)]]

[[July 18th, 2019 - CIB Lensing: Shell By Shell Analysis (Selected Shells)]]

[[July 18th, 2019 - CIB Lensing: Total Maps Analysis]]

[[July 18th, 2019 - CIB Total Maps Analysis]]

[[July 14th, 2019 - CIB Lensing: Is the Lensing Being Done Properly? (for NSIDE 4096) (Not always true!)]]

[[July 14th, 2019 - CIB Smoothing: Iterations]]

[[July 14th, 2019 - CIB Lensing: Smoothing and Interpolation]]

[[July 7th, 2019 - CIB Lensing: Basic Analysis II (Plots)]]

[[July 3rd, 2019 - CIB Halo Scramble: Solved (Edited)]]

[[June 27th, 2019 - CIB Lensing: Basic Analysis (545GHz)]]

[[June 16th, 2019 - Detectability of the Reduced Bispectrum (545GHz)]]

[[June 9th, 2019 - Accuracy of the Wigner 3j Approximation]]

[[June 4th, 2019 - Reduced Bispectrum by Frequency (Fitted Plots)]]

[[June 4th, 2019 - New Plots: Other Frequencies (217GHz, 353GHz)/Comparisons with Planck Data and Lacasa Model (Revised)]]

[[June 2nd, 2019 - New Plots: Other Frequencies (217GHz, 353GHz)/Comparisons with Planck Data and Lacasa Model]]

[[May 28th, 2019 - Investigating the Discrepancy Between Poisson and Sims 3rd Moments]]

[[May 23th, 2019 - Plots for Meeting (Updated Plots)]]

[[May 21th, 2019 - Analysis without the Brightest Galaxies]]

[[May 21th, 2019 - New Plots with NSIDE 4096]]

[[May 15th, 2019 - Flux Re-normalization & Poisson variance/skewness]]

[[May 8th, 2019 - Some analysis on the sims map and the expected Poisson bispectrum]]

[[May 6th, 2019 - Calculating the Poisson variance and skewness (first attempt)]]

[[April 21, 2019 - The origin of the excess kurtosis (revised) ]]

[[March 1, 2019 - Halo maps analysis ]]

[[Feb 20, 2019 - Comparison of Skewness & Kurtosis between sims and Gaussian maps (edited)]]

[[Feb 14, 2019 - Filtering of Unlensed, Poisson, and Gaussian maps]]

[[Feb 13, 2019 - Top hats and Poisson maps]]

[[Jan 10, 2019 - Band filtering - (Cls and 32bands)]]

[[Jan 9, 2019 - Basic Gaussian filtering (revised)]]

[[Jan 7, 2019 - An attempt at Gaussian filtering (incorrect)]]

[[Jan 6, 2019 - Basic statistics of the CIB - non-Gaussianity?]]

[[Nov 7, 2018 - Analysis of the New Maps]]

[[Oct 28, 2018 - Revisions Made]]

[[Oct 17, 2018 - Lensing the CIB]]

[[Sep 26, 2018 - Excercise1]]

June 29th, 2020 - CIB Lensing: Lenspix - Deflection and Smoothing

2020-06-29T19:52:42Z

Jlee:

Because we were seeing negative pixels in lensed CIB maps produced by 'lenspix,' I tried simulating the lensing by deflecting the

'''3 versions of Deflected maps'''

{| class="wikitable"
!colspan="6"|Map Characteristics
|-
|
|Unlensed
|Lensed (Euclidean)
|Lensed (Right)
|Lensed (Lewis)
|Lensed (Lenspix)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| 0.000000
| 0.000000
| 0.000000
| -0.322438
|-
|# of pixels smaller than 0
|0
|0
|0
|0
|1477969
|-
|Mean
|0.3179518
|0.3179518
|0.3179518
|0.3179518
|0.3180850
|-
|Variance
|0.0331789
|
|0.0321213
|0.0320472
|0.0264728
|-
|Skewness
|1.720313
|
|1.740189
|1.742209
|1.776150
|-
|Kurtosis
|23.366768
|
|24.801155
|24.912688
|35.614250
|}

'''Histograms'''
[[File:Unlensed_vs_lenspix.png|450px|]] [[File:Unlensed_vs_deflected.png|450px|]]

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif||]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon264lat-20.gif||]] [[File:Right_lon264lat-20.gif||]] [[File:Lewis_lon264lat-20.gif||]]

'''Longitude [220, 221] Latitude [60, 61] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon220lat60.gif||]] [[File:Right_lon220lat60.gif||]] [[File:Lewis_lon220lat60.gif||]]

'''Longitude [44, 45] Latitude [-80, -79] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon44lat-80.gif||]] [[File:Right_lon44lat-80.gif||]] [[File:Lewis_lon44lat-80.gif||]]

File:Unlensed vs deflected bl total ratio 545GHz lewis scaled.png

2020-06-29T19:32:08Z

Jlee:

File:Unlensed vs deflected kurt total ratio 545GHz lewis scaled.png

2020-06-29T19:31:17Z

Jlee:

File:Unlensed vs deflected bl total ratio 545GHz scaled.png

2020-06-29T19:30:59Z

Jlee:

File:Unlensed vs deflected kurt total ratio 545GHz simple scaled.png

2020-06-29T19:30:14Z

Jlee:

File:Unlensed vs deflected bl total ratio 545GHz simple scaled.png

2020-06-29T19:29:55Z

Jlee:

File:Unlensed vs deflected kurt total ratio 545GHz Euclid scaled.png

2020-06-29T19:29:06Z

Jlee:

File:Unlensed vs deflected bl total ratio 545GHz Euclid scaled.png

2020-06-29T19:28:45Z

Jlee:

File:Unlensed vs deflected kurt total ratio 545GHz.png

2020-06-29T19:27:20Z

Jlee:

File:Unlensed vs deflected bl total ratio 545GHz.png

2020-06-29T19:27:18Z

Jlee:

File:Unlensed vs deflected sigma total ratio 545GHz.png

2020-06-29T19:27:05Z

Jlee:

File:Unlensed vs deflected.png

2020-06-29T19:26:25Z

Jlee:

File:Unlensed vs lenspix.png

2020-06-29T19:26:18Z

Jlee:

June 29th, 2020 - CIB Lensing: Lenspix - Deflection and Smoothing

2020-06-29T19:14:07Z

Jlee:

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif|Euclid_lon0lat0.gif]] [[File:Right_lon0lat0.gif||]] [[File:Lewis_lon0lat0.gif||]]

'''Longitude [264, 265] Latitude [-20, -19] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:5deg_Lensing_0419.gif|500×440px|]] [[File:1deg_Lensing_0419.gif|500×440px|]]

'''Statistics'''
{| class="wikitable"
!colspan="5"|Map Characteristics
|-
|
|Unlensed
|Lensed (total)
|Lensed (field-only)
|Lensed (halo-only)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| -0.394870
| -0.391796
| -0.421715
|-
|# of pixels smaller than 0
|0
|3834246
|3841808
|2942385
|-
|Mean
|0.317096
|0.317329
|0.317523
|0.317467
|-
|Variance
|0.050519
|0.040714
|0.040736
|0.040718
|-
|Skewness
|2.378954
|2.428554
|2.425379
|2.531022
|-
|Kurtosis
|24.764697
|35.934237
|35.901710
|36.194444
|}

File:Euclid lon0lat0.gif

2020-06-29T19:12:52Z

Jlee: Jlee uploaded a new version of File:Euclid lon0lat0.gif

File:Euclid lon44lat-80.gif

2020-06-29T19:12:50Z

Jlee:

File:Euclid lon220lat60.gif

2020-06-29T19:12:49Z

Jlee:

File:Euclid lon264lat-20.gif

2020-06-29T19:12:47Z

Jlee:

File:Lewis lon0lat0.gif

2020-06-29T19:12:06Z

Jlee: Jlee uploaded a new version of File:Lewis lon0lat0.gif

File:Lewis lon44lat-80.gif

2020-06-29T19:12:04Z

Jlee:

File:Lewis lon220lat60.gif

2020-06-29T19:12:02Z

Jlee:

File:Lewis lon264lat-20.gif

2020-06-29T19:12:01Z

Jlee:

File:Right lon0lat0.gif

2020-06-29T19:11:31Z

Jlee: Jlee uploaded a new version of File:Right lon0lat0.gif

File:Right lon44lat-80.gif

2020-06-29T19:11:28Z

Jlee:

File:Right lon220lat60.gif

2020-06-29T19:11:27Z

Jlee:

File:Right lon264lat-20.gif

2020-06-29T19:11:25Z

Jlee:

June 29th, 2020 - CIB Lensing: Lenspix - Deflection and Smoothing

2020-06-29T16:44:38Z

Jlee:

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif|Euclid_lon0lat0.gif]] [[File:Right_lon0lat0.gif|500x440px|]] [[File:Lewis_lon0lat0.gif|500x440px|]]

[[File:5deg_Lensing_0419.gif|500×440px|]] [[File:1deg_Lensing_0419.gif|500×440px|]]

'''Statistics'''
{| class="wikitable"
!colspan="5"|Map Characteristics
|-
|
|Unlensed
|Lensed (total)
|Lensed (field-only)
|Lensed (halo-only)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| -0.394870
| -0.391796
| -0.421715
|-
|# of pixels smaller than 0
|0
|3834246
|3841808
|2942385
|-
|Mean
|0.317096
|0.317329
|0.317523
|0.317467
|-
|Variance
|0.050519
|0.040714
|0.040736
|0.040718
|-
|Skewness
|2.378954
|2.428554
|2.425379
|2.531022
|-
|Kurtosis
|24.764697
|35.934237
|35.901710
|36.194444
|}

June 29th, 2020 - CIB Lensing: Lenspix - Deflection and Smoothing

2020-06-29T16:44:10Z

Jlee:

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif|500x440px|]] [[File:Right_lon0lat0.gif|500x440px|]] [[File:Lewis_lon0lat0.gif|500x440px|]]

[[File:5deg_Lensing_0419.gif|500×440px|]] [[File:1deg_Lensing_0419.gif|500×440px|]]

'''Statistics'''
{| class="wikitable"
!colspan="5"|Map Characteristics
|-
|
|Unlensed
|Lensed (total)
|Lensed (field-only)
|Lensed (halo-only)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| -0.394870
| -0.391796
| -0.421715
|-
|# of pixels smaller than 0
|0
|3834246
|3841808
|2942385
|-
|Mean
|0.317096
|0.317329
|0.317523
|0.317467
|-
|Variance
|0.050519
|0.040714
|0.040736
|0.040718
|-
|Skewness
|2.378954
|2.428554
|2.425379
|2.531022
|-
|Kurtosis
|24.764697
|35.934237
|35.901710
|36.194444
|}

June 29th, 2020 - CIB Lensing: Lenspix - Deflection and Smoothing

2020-06-29T16:43:47Z

Jlee:

'''Maps''' (lensed by total <math> \kappa </math>)

'''Longitude [0, 1] Latitude [0, 1] ''' (Euclidean, Right Triangle, Lewis Equation)

[[File:Euclid_lon0lat0.gif|500x400px|]] [[File:Right_lon0lat0.gif|500x400px|]] [[File:Lewis_lon0lat0.gif|500x400px|]]

[[File:5deg_Lensing_0419.gif|500×440px|]] [[File:1deg_Lensing_0419.gif|500×440px|]]

'''Statistics'''
{| class="wikitable"
!colspan="5"|Map Characteristics
|-
|
|Unlensed
|Lensed (total)
|Lensed (field-only)
|Lensed (halo-only)
|-
|Max values (MJy/sr)
|6.408424 (= 400 mJy)
|6.408424
|6.408424
|6.408424
|-
|Min values (MJy/sr)
| 0.000000
| -0.394870
| -0.391796
| -0.421715
|-
|# of pixels smaller than 0
|0
|3834246
|3841808
|2942385
|-
|Mean
|0.317096
|0.317329
|0.317523
|0.317467
|-
|Variance
|0.050519
|0.040714
|0.040736
|0.040718
|-
|Skewness
|2.378954
|2.428554
|2.425379
|2.531022
|-
|Kurtosis
|24.764697
|35.934237
|35.901710
|36.194444
|}

File:Right lon44lat20.gif

2020-06-29T16:42:05Z

Jlee: