{"id":7145,"date":"2023-12-19T17:19:50","date_gmt":"2023-12-19T08:19:50","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=7145"},"modified":"2024-12-24T10:04:40","modified_gmt":"2024-12-24T01:04:40","slug":"python-%e3%81%a7-planck-2018-results-%e3%81%8b%e3%82%89%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2%e3%81%a8%e7%89%a9%e8%b3%aa%e5%af%86%e5%ba%a6%e3%82%92%e8%a8%88%e7%ae%97%e3%81%99%e3%82%8b","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/7145\/","title":{"rendered":"Python \u3067 Planck 2018 results \u304b\u3089\u5b87\u5b99\u5e74\u9f62\u3068\u7269\u8cea\u5bc6\u5ea6\u3092\u8a08\u7b97\u3059\u308b"},"content":{"rendered":"

Python \u3092\u96fb\u5353\u3068\u3057\u3066\u4f7f\u3063\u3066\u5b87\u5b99\u5e74\u9f62\u3084\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6\u3092\u8a08\u7b97\u3059\u308b\u4f8b\u3002Python \u3067\u306a\u304f\u3066\u3082\uff0c$\\sqrt{x}$ \u3084 $\\tanh^{-1} x$ \u304c\u3067\u304d\u308b\u96fb\u5353\u304c\u3042\u308c\u3070\u3044\u3044\u3067\u3059\u3002<\/p>\n

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Planck 2018 \u306e\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf<\/h3>\n

Planck 2018 results<\/a> \u306e\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n

\\begin{eqnarray}
\nH_0 &=& (67.4 \\pm 0.5) \\ \\mbox{km\/s\/Mpc} \\\\
\n\\Omega_{\\rm m} &=& 0.315 \\pm 0.007 \\\\
\n\\Omega_{\\Lambda} &=& 1 – \\Omega_{\\rm m}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u5b87\u5b99\u5e74\u9f62<\/h3>\n

$\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1$ \u3059\u306a\u308f\u3061 $k = 0$ \u306e\u5834\u5408\u306e\u5b87\u5b99\u5e74\u9f62\u306f\u3053\u3053<\/a> \u306b\u66f8\u3044\u3066\u3044\u308b\u3088\u3046\u306b<\/p>\n

$$
\nH_0 t_0 = \\frac{2}{3(\\sqrt{1-\\Omega_{\\rm m} })}\\tanh^{-1} \\sqrt{1-\\Omega_{\\rm m} } \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} < 1$$<\/p>\n

\u3088\u308a<\/p>\n

$$t_0 = \\frac{1}{H_0} \\times \\frac{2}{3(\\sqrt{1-\\Omega_{\\rm m} })}\\tanh^{-1} \\sqrt{1-\\Omega_{\\rm m} }$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom sympy<\/span> import *<\/span> <\/span>\r\n\r\nimport<\/span> numpy<\/span> as<\/span> np<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d1\u30fc\u30bb\u30af\u306e\u5b9a\u7fa9\u3068\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570<\/h3>\n
$$1 \\,\\mbox{pc} \\equiv \\frac{\\mbox{au}}{\\mbox{1 arcsec}} = \\frac{648000}{\\pi} \\,\\mbox{au}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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au<\/span> =<\/span> 149597870700<\/span> *<\/span> m<\/span>\r\n# Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002<\/span>\r\nMpc<\/span> =<\/span> 648000<\/span>\/<\/span>float<\/span>(<\/span>pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>\r\nkm<\/span> =<\/span> 1E3<\/span> *<\/span> m<\/span>\r\n\r\nH0<\/span> =<\/span> 67.4<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>\r\nH0<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
$\\displaystyle \\frac{2.1842852410855 \\cdot 10^{-18}}{s}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u306e\u9006\u6570\u3092\u5104\u5e74\u3067\u8868\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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okunen<\/span> =<\/span> 60<\/span>*<\/span>60<\/span>*<\/span>24<\/span>*<\/span>365.25<\/span> *<\/span> 1E8<\/span> *<\/span> s<\/span>\r\nprint<\/span>(<\/span>'\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f <\/span>%.1f<\/span> \u5104\u5e74'<\/span> %<\/span> (<\/span>1<\/span>\/<\/span>H0<\/span>\/<\/span>okunen<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30cf\u30c3\u30d6\u30eb\u5e74\u9f62\u306f 145.1 \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u3067\u8a08\u7b97\u3059\u308b\u5b87\u5b99\u5e74\u9f62<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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def<\/span> t0<\/span>(<\/span>H<\/span>,<\/span> Omega<\/span>):<\/span>\r\n    \"\"\"<\/span>\r\n    H0 (km\/(s Mpc)) \u3068 Omegam \u304b\u3089\u5b87\u5b99\u5e74\u9f62\u3092\u5104\u5e74\u3067\u8fd4\u3059\u3002<\/span>\r\n    \"\"\"<\/span>\r\n    au<\/span> =<\/span> 149597870700<\/span> *<\/span> m<\/span>\r\n    # Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002<\/span>\r\n    Mpc<\/span> =<\/span> 648000<\/span>\/<\/span>float<\/span>(<\/span>pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>\r\n    km<\/span> =<\/span> 1E3<\/span> *<\/span> m<\/span>\r\n    H0<\/span> =<\/span> H<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>\r\n    \r\n    okunen<\/span> =<\/span> 60<\/span>*<\/span>60<\/span>*<\/span>24<\/span>*<\/span>365.25<\/span> *<\/span> 1E8<\/span> *<\/span> s<\/span>\r\n\r\n    # SymPy \u306e\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3002<\/span>\r\n    t<\/span> =<\/span> 2<\/span>\/<\/span>(<\/span>3<\/span>*<\/span>H0<\/span>*<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>))<\/span>*<\/span>atanh<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>))<\/span>\/<\/span>okunen<\/span>\r\n    print<\/span>(<\/span>'\u5b87\u5b99\u5e74\u9f62\u306f <\/span>%.1f<\/span> \u5104\u5e74'<\/span> %<\/span> t<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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H_0<\/span> =<\/span> 67.4<\/span>\r\nOmega_m<\/span> =<\/span> 0.315<\/span>\r\nt0<\/span>(<\/span>H_0<\/span>,<\/span> Omega_m<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b87\u5b99\u5e74\u9f62\u306f 138.0 \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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NumPy \u3067\u8a08\u7b97\u3059\u308b\u5b87\u5b99\u5e74\u9f62<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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def<\/span> T0<\/span>(<\/span>H<\/span>,<\/span> Omega<\/span>):<\/span>\r\n    \"\"\"<\/span>\r\n    H0 (km\/(s Mpc)) \u3068 Omegam \u304b\u3089\u5b87\u5b99\u5e74\u9f62\u3092\u5104\u5e74\u3067\u8fd4\u3059\u3002<\/span>\r\n    \"\"\"<\/span>\r\n    au<\/span> =<\/span> 149597870700<\/span> *<\/span> m<\/span>\r\n    # Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002<\/span>\r\n    Mpc<\/span> =<\/span> 648000<\/span>\/<\/span>float<\/span>(<\/span>np<\/span>.<\/span>pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>\r\n    km<\/span> =<\/span> 1E3<\/span> *<\/span> m<\/span>\r\n    H0<\/span> =<\/span> H<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>\r\n    \r\n    okunen<\/span> =<\/span> 60<\/span>*<\/span>60<\/span>*<\/span>24<\/span>*<\/span>365.25<\/span> *<\/span> 1E8<\/span> *<\/span> s<\/span>\r\n    \r\n    # NumPy \u306e\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3002<\/span>\r\n    t<\/span> =<\/span> 2<\/span>\/<\/span>(<\/span>3<\/span>*<\/span>H0<\/span>*<\/span>np<\/span>.<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>))<\/span>*<\/span>np<\/span>.<\/span>arctanh<\/span>(<\/span>np<\/span>.<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>))<\/span>\/<\/span>okunen<\/span>\r\n    print<\/span>(<\/span>'\u5b87\u5b99\u5e74\u9f62\u306f <\/span>%.1f<\/span> \u5104\u5e74'<\/span> %<\/span> t<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u554f\u984c 1<\/h3>\n
\u8aa4\u5dee\u3092\u8003\u3048\u3066\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u6700\u5927\u5024\uff0c\u6700\u5c0f\u5024\u3092\u305d\u308c\u305e\u308c $H_{\\rm max}, \\ H_{\\rm min}$\uff0c\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u6700\u5927\u5024\uff0c\u6700\u5c0f\u5024\u3092\u305d\u308c\u305e\u308c $\\Omega_{\\rm max}, \\ \\Omega_{\\rm min}$ \u3068\u3059\u308b\u3002<\/p>\n
\\begin{eqnarray}
\nH_{\\rm max} &=& 67.4 + 0.5\\\\
\nH_{\\rm min} &=& 67.4 – 0.5\\\\
\n\\Omega_{\\rm max} &=& 0.315 + 0.007\\\\
\n\\Omega_{\\rm min} &=& 0.315 – 0.007\\\\
\n\\end{eqnarray}<\/p>\n
\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u5927\u304d\u304f\u3059\u308b\u3068\uff0c\u5b87\u5b99\u5e74\u9f62 $t_0$ \u306e\u5024\u306f\u5927\u304d\u304f\u306a\u308b\u306e\u304b\u5c0f\u3055\u304f\u306a\u308b\u306e\u304b\uff0c\u307e\u305f\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u5927\u304d\u304f\u3059\u308b\u3068\uff0c\u5b87\u5b99\u5e74\u9f62 $t_0$ \u306e\u5024\u306f\u5927\u304d\u304f\u306a\u308b\u306e\u304b\u5c0f\u3055\u304f\u306a\u308b\u306e\u304b\uff0c\u3053\u306e\u3078\u3093\u3092\u7406\u89e3\u3057\u305f\u4e0a\u3067\uff0c$t_0$ \u306e\u6700\u5927\u5024\uff0c\u6700\u5c0f\u5024\u3092\u6c42\u3081\uff0c$138 \\pm ??$ \u5104\u5e74\u306e\u5f62\u306b\u8868\u305b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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# \u89e3\u7b54\u4f8b\uff1a<\/span>\r\n# \u5b87\u5b99\u5e74\u9f62\u306f H0 \u306b\u53cd\u6bd4\u4f8b\u3057\uff0cOmega_m\u306e\u5358\u8abf\u6e1b\u5c11\u95a2\u6570\u3060\u304b\u3089...<\/span>\r\n# \u5b87\u5b99\u5e74\u9f62\u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f<\/span>\r\nH_0<\/span> =<\/span> 67.4<\/span> -<\/span> 0.5<\/span>\r\nOmega_m<\/span> =<\/span> 0.315<\/span> -<\/span> 0.007<\/span>\r\nprint<\/span>(<\/span>'\u6700\u5927\u3068\u306a\u308b'<\/span>,<\/span> end<\/span>=<\/span>''<\/span>)<\/span>\r\nT0<\/span>(<\/span>H_0<\/span>,<\/span> Omega_m<\/span>)<\/span>\r\n\r\n# \u5b87\u5b99\u5e74\u9f62\u304c\u6700\u5c0f\u3068\u306a\u308b\u306e\u306f<\/span>\r\nH_0<\/span> =<\/span> 67.4<\/span> +<\/span> 0.5<\/span>\r\nOmega_m<\/span> =<\/span> 0.315<\/span> +<\/span> 0.007<\/span>\r\nprint<\/span>(<\/span>'\u6700\u5c0f\u3068\u306a\u308b'<\/span>,<\/span> end<\/span>=<\/span>''<\/span>)<\/span>\r\nT0<\/span>(<\/span>H_0<\/span>,<\/span> Omega_m<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6700\u5927\u3068\u306a\u308b\u5b87\u5b99\u5e74\u9f62\u306f 139.9 \u5104\u5e74\r\n\u6700\u5c0f\u3068\u306a\u308b\u5b87\u5b99\u5e74\u9f62\u306f 136.1 \u5104\u5e74\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u81e8\u754c\u5bc6\u5ea6<\/h3>\n
\u3053\u3053<\/a> \u306b\u66f8\u3044\u3066\u3044\u308b\u3088\u3046\u306b<\/p>\n
$$
\n\\rho_{\\rm cr}\\equiv \\frac{3 H_0^2}{8\\pi G}
\n$$<\/p>\n
\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6<\/h3>\n
\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6 $\\rho$ \u306f
\n$$\\rho = \\rho_{\\rm cr} \\Omega_{\\rm m}$$\u304b\u3089\u6c42\u3081\u308b\u3002<\/p>\n
\u4e07\u6709\u5f15\u529b\u5b9a\u6570<\/a> $G$ \u306f<\/p>\n
$$ G = 6.674 \\times 10^{-11} \\, \\frac{\\mbox{m}^3}{\\mbox{kg}\\cdot\\mbox{s}^2}$$<\/p>\n
\u307e\u305f\uff0c$H_0$ \u306f\u5225\u30da\u30fc\u30b8<\/a>\u306b\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b $s^{-1}$ \u306e\u6b21\u5143\u3092\u6301\u3064\u306e\u3067\uff0c\u9069\u5207\u306b\u5358\u4f4d\u3092\u5408\u308f\u305b\u3066\u8a08\u7b97\u3059\u308c\u3070\u78ba\u304b\u306b $\\rho_{\\rm cr}$ \u306f
\n\u8cea\u91cf\u5bc6\u5ea6 $\\mbox{kg}\/\\mbox{m}^3$ \u306e\u6b21\u5143\u3092\u3082\u3064\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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var<\/span>(<\/span>'kg'<\/span>)<\/span>\r\nG<\/span> =<\/span> 6.674E-11<\/span> *<\/span> m<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>kg<\/span> *<\/span> s<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\nau<\/span> =<\/span> 149597870700<\/span> *<\/span> m<\/span>\r\n# Mpc \u3060\u304b\u3089 1E6 pc \u3067\uff0ckm \u3060\u304b\u3089 1E3 m\u3002<\/span>\r\nMpc<\/span> =<\/span> 648000<\/span>\/<\/span>float<\/span>(<\/span>pi<\/span>)<\/span> *<\/span> au<\/span> *<\/span> 1E6<\/span>\r\nkm<\/span> =<\/span> 1E3<\/span> *<\/span> m<\/span>\r\nH0<\/span> =<\/span> 67.4<\/span> *<\/span> km<\/span>\/<\/span>(<\/span>s<\/span> *<\/span> Mpc<\/span>)<\/span>\r\nOmega_m<\/span> =<\/span> 0.315<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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rho<\/span> =<\/span> 3<\/span>*<\/span>H0<\/span>**<\/span>2<\/span>\/<\/span>(<\/span>8<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>*<\/span>G<\/span>)<\/span> *<\/span> Omega_m<\/span>\r\nrho<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\frac{2.68797019689748 \\cdot 10^{-27} kg}{m^{3}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u554f\u984c 2<\/h3>\n
\u73fe\u5728\u306e\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6\u306f\u6b21\u306e\u3046\u3061\u306e\u3069\u308c\u306b\u6700\u3082\u8fd1\u3044\u304b\u3002<\/p>\n
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  1. $1\\,\\mbox{cm}^3$ \u306b\u96fb\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n
  2. $1\\,\\mbox{cm}^3$ \u306b\u967d\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n
  3. $1\\,\\mbox{m}^3$ \u306b\u96fb\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n
  4. $1\\,\\mbox{m}^3$ \u306b\u967d\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n
  5. $1\\,\\mbox{km}^3$ \u306b\u96fb\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n
  6. $1\\,\\mbox{km}^3$ \u306b\u967d\u5b50\u304c\u6570\u500b\u7a0b\u5ea6<\/li>\n<\/ol>\n
    \u53c2\u8003<\/h4>\n
    \u967d\u5b50\uff0c\u96fb\u5b50\u306e\u8cea\u91cf\u3092\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u7b49\u304b\u3089\u78ba\u8a8d\u3057\u3066\u304a\u304f\u3053\u3068\u3002<\/p>\n