Why the angular diameter distance starts to decrease after a certain redshift (around z = 1.5)?

The angular diameter of something is the angular diameter it had when the light was emitted and started on its way to us.

Because pure expansion does not change angles. If you think diagrammatically, the lightrays are not spread apart or squinched together by expansion. They are just stretched out longer.

But the angular-size distance is just based on the angular diameter of some standard ruler (like a 100 thousand lightyear galaxy), and since the angle spread of the incoming light does not change the angular size distance equals the instantaneous proper distance to the object measured on the day when the light was emitted.

And that is smaller than the presentday distance by a factor of z+1.

So think about this example: the matter that emitted the CMB which we are now detecting is now about 45 billion LY from us. But the redshift of that ancient light is z = 1090. So the distances have increased by a factor of z+1 = 1091. Not to be too fussy, distances have increased by a factor of 1100. So when the light was emitted, that matter was only about 41 million LY from our matter! When the ancient light was emitted, the matter was much much closer

Reference:
https://www.physicsforums.com/threads/angular-diameter-distance.501108/
https://forum.cosmoquest.org/showthread.php?103185-angular-diameter-distance

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赤方偏移、宇宙年齢、距離の表(Plankパラメータ)

赤方偏移 宇宙年齢 ルックバック 角度距離 光度距離
     _(Gy)  _タイム(Gy)  (Gly) _ _(Gly) _
0.000   13.798    0.000    0.000     0.000
0.010   13.654    0.143    0.143     0.145
0.050   13.102    0.696    0.680     0.749
0.100   12.454    1.344    1.281     1.550
0.200   11.288    2.510    2.291     3.299
0.500   8.604     5.194    4.231     9.519
0.700   7.310     6.488    4.948    14.298
1.000   5.862     7.935    5.538    22.152
1.500   4.278     9.520    5.844    36.522
2.000   3.284    10.514    5.775    51.972
3.000   2.148    11.649    5.3080    84.928
4.000   1.541    12.257    4.7875   119.69
5.000   1.173    12.624    4.3243   155.67
6.000   0.931    12.866    3.9301   192.57
7.000   0.762    13.036    3.5967   230.19
8.000   0.639    13.159    3.3134   268.38
9.000   0.545    13.253    3.0706   307.06
10.00   0.472    13.325    2.8608   346.16
15.00    0.269    13.529    2.1336   546.19
20.00    0.178    13.619    1.7039   751.44
25.00    0.129    13.668    1.4202   960.09
50.00  46.63×10-3  13.751    0.7808   2031.0
100   16.46×10-3  13.781    0.4149   4232.4
500   1.341×10-3  13.796    0.0891   22374
1000   0.430×10-3  13.797    0.0452   45316
1090*   0.373×10-3  13.797    0.0415   49455
* 宇宙晴れあがり期

ref: https://blogs.yahoo.co.jp/karaokegurui/70712013.html

出典:谷口義明、岡村定矩、祖父江義明 編 『銀河I-銀河と宇宙の階層構造』 日本評論社 シリーズ現代の天文学[第2版]第4巻 2018年8月発行
P.362~364 付表
Plank宇宙論パラメータ H0=67.7kms-1Mpc-1,Ωm0=0.309,ΩΛ0=0.691に基づく,赤方偏移と宇宙年齢,ルックバックタイム,角度距離,光度距離の関係(Gy=10億年,Gly=10億光年=3.06×108pc)

Latex grammers

LaTeX – コマンド一覧

http://www1.kiy.jp/~yoka/LaTeX/latex.html
http://www.latex-cmd.com/

LaTeX Math Symbols

http://web.ift.uib.no/Teori/KURS/WRK/TeX/symALL.html

big O notation

http://texblog.org/2014/06/24/big-o-and-related-notations-in-latex/

ドイツ語、スペイン語

http://www.biwako.shiga-u.ac.jp/sensei/kumazawa/tex/accent.html
http://mikilab.doshisha.ac.jp/dia/seminar/latex/doc/uml.html

LaTeXのsplitで数式モード内の長い数式を折り返す

http://www.irohabook.com/latex-split

\begin{equation*}
\begin{split}
\sum_{k=1}^{6}k&=1+2+3\\
&\quad+4+5+6\\
&=21\\
&=3\times{7}
\end{split}
\end{equation*}

 

 

Python programming

Scipy Lecture Notes

http://www.turbare.net/transl/scipy-lecture-notes/index.html

クラス定義の基本

https://python.keicode.com/lang/oop-basics.php

pythonで外部のシェルスクリプトやコマンドを実行する方法

https://qiita.com/komeiy/items/d6b5f25bf1778fa10e21

  1. os.system(‘ls -la’)
    コマンドが成功すれば0が返ってきます。
  2. commands.getoutput(“ls -la”)
    実際に実行した結果が返ってきます。

PythonでGaussian Fitting (for scatter plot)

https://omedstu.jimdo.com/2018/07/16/python%E3%81%A7gaussian-fitting/

Python: fit with error on both axis

https://micropore.wordpress.com/2017/02/07/python-fit-with-error-on-both-axis/
https://www.physics.utoronto.ca/~phy326/python/odr_fit_to_data.py

FFT

https://seesaawiki.jp/met-python/d/fft

How to calculate the likelihood of curve-fitting in scipy?

https://stackoverflow.com/questions/23004374/how-to-calculate-the-likelihood-of-curve-fitting-in-scipy

P-value from Chi sq test statistic in Python

https://stackoverflow.com/questions/11725115/p-value-from-chi-sq-test-statistic-in-python

Evaluating a model fit with chi-square

http://www.astroml.org/book_figures/chapter4/fig_chi2_eval.html

How to split strings into text and number?

https://stackoverflow.com/questions/430079/how-to-split-strings-into-text-and-number

 

(Issue)

np.savetxt: Problem with strings being converted to unicode

https://github.com/nipy/nipype/issues/2038

np.savetxt(out_file, np.ones((vols,), dtype=np.int), fmt=str("%i"), newline=str(" "))