{"id":4425,"date":"2022-12-08T15:24:18","date_gmt":"2022-12-08T06:24:18","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4425"},"modified":"2022-12-13T22:24:19","modified_gmt":"2022-12-13T13:24:19","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e8%a7%92%e5%be%84%e8%b7%9d%e9%9b%a2%e3%81%8c%e6%a5%b5%e5%a4%a7%e3%81%a8%e3%81%aa%e3%82%8b%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e8%a7%92%e5%be%84%e8%b7%9d%e9%9b%a2\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e8%a7%92%e5%be%84%e8%b7%9d%e9%9b%a2%e3%81%8c%e6%a5%b5%e5%a4%a7%e3%81%a8%e3%81%aa%e3%82%8b%e8%b5%a4%e6%96%b9%e5%81%8f%e7%a7%bb%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/","title":{"rendered":"\u88dc\u8db3\uff1aMaxima \u3067\u89d2\u5f84\u8ddd\u96e2\u304c\u6975\u5927\u3068\u306a\u308b\u8d64\u65b9\u504f\u79fb\u3092\u6c42\u3081\u308b"},"content":{"rendered":"

\u30b0\u30e9\u30d5\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u89d2\u5f84\u8ddd\u96e2 $d_A(z)$ \u306f\u6975\u5927\u5024\u3092\u6301\u3063\u3066\u3044\u308b\u3088\u3046\u3060\u3002$d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b $z$ \u3092\u6c42\u3081\u3066\u307f\u308b\u3002
\n<\/p>\n

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\u30b0\u30e9\u30d5\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u89d2\u5f84\u8ddd\u96e2 $d_A(z)$ \u306f\u6975\u5927\u5024\u3092\u6301\u3063\u3066\u3044\u308b\u3088\u3046\u3060\u3002$d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b $z$ \u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n

\"dA\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306e\u89d2\u5f84\u8ddd\u96e2<\/h3>\n

$H_0$ \u3092\u7701\u7565\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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dA<\/span>(<\/span>z<\/span>, Omega<\/span>)<\/span>:=<\/span> 2\/<\/span>(<\/span>Omega<\/span>**<\/span>2*<\/span>(<\/span>1+<\/span>z<\/span>)<\/span>**<\/span>2<\/span>)<\/span> *<\/span> \r\n  (<\/span>2-<\/span>Omega<\/span>+<\/span> Omega<\/span>*<\/span>z<\/span> -<\/span>(<\/span>2-<\/span>Omega<\/span>)<\/span>*<\/span>sqrt<\/span>(<\/span>1+<\/span>Omega<\/span>*<\/span>z<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}{\\it dA}\\left(z , \\Omega\\right):=\\frac{2}{\\Omega^2\\,\\left(1+z\\right)^2}\\,\\left(2-\\Omega+\\Omega\\,z+\\left(-\\left(2-\\Omega\\right)\\right)\\,\\sqrt{1+\\Omega\\,z}\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\Omega_{\\rm m} = 1$ \u306e\u5834\u5408\u306e $d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b\u8d64\u65b9\u504f\u79fb<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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dA<\/span>(<\/span>z<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\frac{2\\,\\left(-\\sqrt{z+1}+z+1\\right)}{\\left(z+1\\right)^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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ddA<\/span>:<\/span> diff<\/span>(<\/span>dA<\/span>(<\/span>z<\/span>, 1<\/span>)<\/span>, z<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}-\\frac{2\\,\\sqrt{z+1}-3}{\\sqrt{z+1}\\,\\left(z^2+2\\,z+1\\right)}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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solve<\/span>(<\/span>ddA<\/span> =<\/span> 0<\/span>, z<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}\\left[ z=\\frac{5}{4} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\Omega_{\\rm m} = 0.1$ \u306e\u5834\u5408\u306b\u306f\u6570\u5024\u7684\u306b\u6c42\u3081\u308b<\/h4>\n
$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u3067\u3042\u3063\u3066\u3082\uff0c$d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b $z$ \u306f3\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u305d\u3053\u3067\uff0c\u89e3\u306e\u516c\u5f0f\u304b\u3089\u53b3\u5bc6\u89e3\u3092\u6c42\u3081\u308b\u3088\u308a\u3082\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6570\u5024\u89e3\u3092\u6c42\u3081\u3066\u307f\u308b\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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ddA<\/span>:<\/span> diff<\/span>(<\/span>dA<\/span>(<\/span>z<\/span>, 1\/<\/span>10<\/span>)<\/span>, z<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}-\\frac{\\sqrt{z+10}\\,\\left(2\\,10^{\\frac{3}{2}}\\,z+74\\,10^{\\frac{3}{2}}\\right)-570\\,z-7410}{\\sqrt{z+10}\\,\\left(\\sqrt{10}\\,z^3+3\\,\\sqrt{10}\\,z^2+3\\,\\sqrt{10}\\,z+\\sqrt{10}\\right)}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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\/* \u89d2\u5f84\u8ddd\u96e2\u306e\u30b0\u30e9\u30d5\u304b\u3089\u6975\u5927\u306f 2 \u304b\u3089 5 \u307e\u3067\u306e\u9593\u306b\u3042\u308a\u305d\u3046 *\/<\/span>\r\nfind_root<\/span>(<\/span>ddA<\/span> =<\/span> 0<\/span>, z<\/span>, 2<\/span>, 5<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}2.835430200415257\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u4f8b<\/h4>\n
$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\uff0c$d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b $z$ \u30923\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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ddA<\/span>:<\/span> diff<\/span>(<\/span>dA<\/span>(<\/span>z<\/span>, Omega<\/span>)<\/span>, z<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}-\\frac{\\sqrt{\\Omega\\,z+1}\\,\\left(2\\,\\Omega\\,z-6\\,\\Omega+8\\right)+\\left(3\\,\\Omega^2-6\\,\\Omega\\right)\\,z-\\Omega^2+6\\,\\Omega-8}{\\sqrt{\\Omega\\,z+1}\\,\\left(\\Omega^2\\,z^3+3\\,\\Omega^2\\,z^2+3\\,\\Omega^2\\,z+\\Omega^2\\right)}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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\/* \u5206\u5b50\u304c\u30bc\u30ed\u306b\u306a\u308c\u3070\u3044\u3044\u306e\u3060\u304b\u3089... *\/<\/span>\r\ntmp<\/span>:<\/span> num<\/span>(<\/span>ddA<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}-\\sqrt{\\Omega\\,z+1}\\,\\left(2\\,\\Omega\\,z-6\\,\\Omega+8\\right)-\\left(3\\,\\Omega^2-6\\,\\Omega\\right)\\,z+\\Omega^2-6\\,\\Omega+8\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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\/* \u30eb\u30fc\u30c8\u306e\u9805\u3092\u4e8c\u4e57\u3057\u3066 3\u6b21\u65b9\u7a0b\u5f0f\u306b *\/<\/span>\r\neq<\/span>:<\/span> (<\/span>tmp<\/span> -<\/span> (<\/span>-<\/span>(<\/span>3*<\/span>Omega<\/span>**<\/span>2-<\/span>6*<\/span>Omega<\/span>)<\/span>*<\/span>z<\/span>+<\/span> Omega<\/span>**<\/span>2<\/span> -<\/span> 6*<\/span>Omega<\/span> +<\/span> 8<\/span>))<\/span>**<\/span>2<\/span> \r\n         -<\/span> (<\/span>-<\/span>(<\/span>3*<\/span>Omega<\/span>**<\/span>2-<\/span>6*<\/span>Omega<\/span>)<\/span>*<\/span>z<\/span>+<\/span> Omega<\/span>**<\/span>2<\/span> -<\/span> 6*<\/span>Omega<\/span> +<\/span> 8<\/span>)<\/span>**<\/span>2<\/span> =<\/span> 0<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}4\\,\\Omega^3\\,z^3+\\left(12\\,\\Omega^3-9\\,\\Omega^4\\right)\\,z^2+\\left(6\\,\\Omega^4-12\\,\\Omega^3\\right)\\,z-\\Omega^4+12\\,\\Omega^3-16\\,\\Omega^2=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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sols<\/span>:<\/span> solve<\/span>(<\/span>eq<\/span>, z<\/span>)<\/span>$\r\nsols<\/span>[<\/span>3]<\/span>, ratsimp<\/span>;\r\nev<\/span>(<\/span>sols<\/span>[<\/span>3]<\/span>, Omega<\/span>=<\/span>1<\/span>)<\/span>;\r\nev<\/span>(<\/span>sols<\/span>[<\/span>3]<\/span>, Omega<\/span>=<\/span>0.1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}z=\\frac{\\left(27\\,\\Omega^4-144\\,\\Omega^3+\\sqrt{1-\\Omega}\\,\\left(64\\,\\Omega^2-192\\,\\Omega+128\\right)+272\\,\\Omega^2-256\\,\\Omega+128\\right)^{\\frac{2}{3}}+\\Omega^{\\frac{1}{3}}\\,\\left(3\\,\\Omega-4\\right)\\,\\left(27\\,\\Omega^4-144\\,\\Omega^3+\\sqrt{1-\\Omega}\\,\\left(64\\,\\Omega^2-192\\,\\Omega+128\\right)+272\\,\\Omega^2-256\\,\\Omega+128\\right)^{\\frac{1}{3}}+\\Omega^{\\frac{2}{3}}\\,\\left(9\\,\\Omega^2-32\\,\\Omega+32\\right)}{4\\,\\Omega^{\\frac{1}{3}}\\,\\left(27\\,\\Omega^4-144\\,\\Omega^3+\\sqrt{1-\\Omega}\\,\\left(64\\,\\Omega^2-192\\,\\Omega+128\\right)+272\\,\\Omega^2-256\\,\\Omega+128\\right)^{\\frac{1}{3}}}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{12}$}z=\\frac{5}{4}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{13}$}z=2.835430200415417\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1$ \u306e\u5834\u5408\u306e\u89d2\u5f84\u8ddd\u96e2<\/h3>\n
$H_0$ \u3092\u7701\u7565\u3057\u3066…<\/p>\n
\\begin{eqnarray}
\nd_A &=& \\frac{1}{1+z} \\int_0^z \\frac{dz}{\\sqrt{(1-\\Omega_{\\rm m}) + \\Omega_{\\rm m} (1+z)^3} }
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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f<\/span>(<\/span>z<\/span>, Omega<\/span>)<\/span>:=<\/span> 1\/<\/span>(<\/span>sqrt<\/span>((<\/span>1-<\/span>Omega<\/span>)<\/span>+<\/span>Omega<\/span>*<\/span>(<\/span>1+<\/span>z<\/span>)<\/span>**<\/span>3<\/span>))<\/span>;\r\nrombergtol<\/span>:<\/span> 1E<\/span>-<\/span>12$\r\n\r\n\/* dA \u306e\u5fae\u5206 *\/<\/span>\r\nddA<\/span>(<\/span>z<\/span>, Omega<\/span>)<\/span>:=<\/span> -<\/span> 1\/<\/span>(<\/span>1+<\/span>z<\/span>)<\/span>**<\/span>2<\/span> *<\/span> romberg<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>)<\/span>, x<\/span>, 0<\/span>, z<\/span>)<\/span> +<\/span> \r\n                  1\/<\/span>(<\/span>1+<\/span>z<\/span>)<\/span> *<\/span> f<\/span>(<\/span>z<\/span>, Omega<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}f\\left(z , \\Omega\\right):=\\frac{1}{\\sqrt{1-\\Omega+\\Omega\\,\\left(1+z\\right)^3}}\\]<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}{\\it ddA}\\left(z , \\Omega\\right):=\\frac{-1}{\\left(1+z\\right)^2}\\,{\\it romberg}\\left(f\\left(x , \\Omega\\right) , x , 0 , z\\right)+\\frac{1}{1+z}\\,f\\left(z , \\Omega\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\Omega_{\\rm m} = 0.1$ \u306e\u5834\u5408\u306b\u6570\u5024\u7684\u306b\u6c42\u3081\u308b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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\/* \u89d2\u5f84\u8ddd\u96e2\u306e\u30b0\u30e9\u30d5\u304b\u3089\u6975\u5927\u306f 1 \u304b\u3089 3 \u307e\u3067\u306e\u9593\u306b\u3042\u308a\u305d\u3046 *\/<\/span>\r\nfind_root<\/span>(<\/span>ddA<\/span>(<\/span>z<\/span>, 1\/<\/span>10<\/span>)<\/span> =<\/span> 0<\/span>, z<\/span>, 1<\/span>, 3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}2.0793683111316\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Planck 2018 results $\\Omega_{\\rm m} = 0.315$ \u306e\u5834\u5408<\/h4>\n
Wikipedia \u82f1\u8a9e\u7248\u306e Angular diameter distance<\/a> \u306b\u3088\u308b\u3068\uff0c$\\Lambda \\mbox{CDM}$ \u30e2\u30c7\u30eb\u3067\u306f $z$ \u304c\u7d04 $1.5$ \u3050\u3089\u3044\u306e\u3068\u3053\u308d\u3067 $d_A(z)$ \u306e\u6975\u5927\u304c\u3042\u308b\u3088\u3046\u306a\u8a18\u8ff0\u304c\u3042\u308b\u3002\u8abf\u3079\u3066\u307f\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b $z = 1.59$ \u3042\u305f\u308a\u304c\u6975\u5927\u5024\u3092\u4e0e\u3048\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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find_root<\/span>(<\/span>ddA<\/span>(<\/span>z<\/span>, 315\/<\/span>1000<\/span>)<\/span> =<\/span> 0<\/span>, z<\/span>, 1<\/span>, 5<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}1.587997010828216\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u30b0\u30e9\u30d5\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u89d2\u5f84\u8ddd\u96e2 $d_A(z)$ \u306f\u6975\u5927\u5024\u3092\u6301\u3063\u3066\u3044\u308b\u3088\u3046\u3060\u3002$d_A(z)$ \u304c\u6975\u5927\u3068\u306a\u308b $z$ \u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>
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