{"id":4418,"date":"2022-12-08T12:04:09","date_gmt":"2022-12-08T03:04:09","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4418"},"modified":"2023-03-14T16:49:13","modified_gmt":"2023-03-14T07:49:13","slug":"planck-2018-results-%e3%81%8b%e3%82%89-gnuplot-%e3%81%a7%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2%e3%81%a8%e5%ae%87%e5%ae%99%e7%a9%ba%e9%96%93%e3%81%ae%e7%89%a9%e8%b3%aa%e5%af%86%e5%ba%a6%e3%82%92%e8%a8%88","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4418\/","title":{"rendered":"Planck 2018 results \u304b\u3089 gnuplot \u3067\u5b87\u5b99\u5e74\u9f62\u3068\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6\u3092\u8a08\u7b97\u3059\u308b"},"content":{"rendered":"

gnuplot \u3092\u95a2\u6570\u96fb\u5353\u3068\u3057\u3066\u4f7f\u3046\u4f8b\u3002\u4f55\u3082 gnuplot (\u3084 Maxima) \u3067\u306a\u304f\u3066\u3082 $\\sqrt{x}$\u00a0 \u3068 $\\tanh^{-1} x$ \u304c\u3067\u304d\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

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

\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570\u306f\u6b74\u53f2\u7684\u7d4c\u7def\u3082\u3042\u308a\uff0c\u5358\u4f4d\u304c $\\mbox{km\/s\/Mpc}$ \u3068\u306a\u3063\u3066\u3044\u308b\u3002
\n$\\mbox{Mpc} = 10^6 \\mbox{pc}$ \u3092 $\\mbox{m}$ \u3067\u8868\u3057\u3066\uff0c\u5358\u4f4d\u3092\u305d\u308d\u3048\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n

\u3053\u3061\u3089<\/a>\u306b\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b<\/p>\n

$$1 \\mbox{pc} \\equiv \\frac{\\mbox{au}}{\\theta} = \\frac{648000}{\\pi} \\mbox{au}$$<\/p>\n

\u3060\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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au<\/span> =<\/span> 149597870700<\/span>\r\npc<\/span> =<\/span> 648000<\/span>\/<\/span>pi<\/span> *<\/span> au<\/span>\r\nMpc<\/span> =<\/span> pc<\/span> *<\/span> 1E6<\/span>\r\nkm<\/span> =<\/span> 1E3<\/span>\r\n\r\nH0<\/span> =<\/span> 67.4<\/span> *<\/span> km<\/span>\/<\/span>Mpc<\/span>\r\n\r\nf<\/span>(<\/span>Omega<\/span>)<\/span> =<\/span> atanh<\/span>(<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>))<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>Omega<\/span>)<\/span>\r\n\r\nokunen<\/span> =<\/span> 60<\/span>*<\/span>60<\/span>*<\/span>24<\/span>*<\/span>365.25<\/span>*<\/span>1E8<\/span>\r\n\r\nt0<\/span> =<\/span> 2.<\/span>\/<\/span>(<\/span>3<\/span> *<\/span> H0<\/span>)<\/span> *<\/span> f<\/span>(<\/span>0.315<\/span>)<\/span>\/<\/span>okunen<\/span>\r\n\r\nprint<\/span> sprintf<\/span>(<\/span>\"\u5b87\u5b99\u5e74\u9f62\u306f %.0f \u5104\u5e74\"<\/span>,<\/span> t0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
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\u5b87\u5b99\u5e74\u9f62\u306f 138 \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
$$\\rho_{\\rm cr}\\equiv \\frac{3 H_0^2}{8\\pi G}$$<\/p>\n
\u7269\u8cea\u5bc6\u5ea6<\/h3>\n
$$\\rho = \\rho_{\\rm cr} \\Omega_{\\rm m}$$<\/p>\n
\u4e07\u6709\u5f15\u529b\u5b9a\u6570<\/h3>\n
$$G = 6.674 \\times 10^{-11} \\, \\frac{\\mbox{m}^3}{\\mbox{kg}\\cdot\\mbox{s}^2}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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Omega<\/span> =<\/span> 0.315<\/span>\r\nG<\/span> =<\/span> 6.674E-11<\/span>\r\n\r\nrhoc<\/span> =<\/span> 3<\/span>*<\/span>H0<\/span>**<\/span>2<\/span>\/<\/span>(<\/span>8<\/span>*<\/span>pi<\/span>*<\/span>G<\/span>)<\/span>\r\n\r\nprint<\/span> sprintf<\/span>(<\/span>\"\u81e8\u754c\u5bc6\u5ea6\u306f  %8.3g (kg\/m^3)\"<\/span>,<\/span> rhoc<\/span>)<\/span>\r\nprint<\/span> sprintf<\/span>(<\/span>\"\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6\u306f  %8.3g (kg\/m^3)\"<\/span>,<\/span> rhoc<\/span> *<\/span> Omega<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
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\u81e8\u754c\u5bc6\u5ea6\u306f  ???e-?? (kg\/m^3)\r\n\u5b87\u5b99\u7a7a\u9593\u306e\u7269\u8cea\u5bc6\u5ea6\u306f  ???e-?? (kg\/m^3)\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6f14\u7fd2<\/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