{"id":7461,"date":"2024-02-02T13:40:34","date_gmt":"2024-02-02T04:40:34","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7461"},"modified":"2024-02-02T13:44:50","modified_gmt":"2024-02-02T04:44:50","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9asympy-%e3%81%a8-spb-%e3%81%a7%e3%82%b9%e3%82%b1%e3%83%bc%e3%83%ab%e5%9b%a0%e5%ad%90%e3%81%ae%e3%82%b0%e3%83%a9%e3%83%95%e3%82%92%e6%8f%8f%e3%81%8f","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\/%e5%ae%87%e5%ae%99%e8%ab%96%e3%83%91%e3%83%a9%e3%83%a1%e3%83%bc%e3%82%bf%e3%81%a8%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2\/%e8%a3%9c%e8%b6%b3%ef%bc%9asympy-%e3%81%a8-spb-%e3%81%a7%e3%82%b9%e3%82%b1%e3%83%bc%e3%83%ab%e5%9b%a0%e5%ad%90%e3%81%ae%e3%82%b0%e3%83%a9%e3%83%95%e3%82%92%e6%8f%8f%e3%81%8f\/","title":{"rendered":"\u88dc\u8db3\uff1aSymPy \u3068 SPB \u3067\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f"},"content":{"rendered":"<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u5c0e\u51fa\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\uff1a<\/p>\n<ul>\n<li><a href=\"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\/%e5%ae%87%e5%ae%99%e8%ab%96%e3%83%91%e3%83%a9%e3%83%a1%e3%83%bc%e3%82%bf%e3%81%a8%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%b9%e3%82%b1%e3%83%bc%e3%83%ab%e5%9b%a0%e5%ad%90%e3%81%ae%e8%a7%a3\/\">\u88dc\u8db3\uff1a\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u89e3<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p><!--more--><\/p>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e-import\">\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[1]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"kn\">from<\/span> <span class=\"nn\">sympy<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<span class=\"c1\"># 1\u6587\u5b57\u5909\u6570\u306e Symbol \u306e\u5ba3\u8a00\u304c\u7701\u7565\u3067\u304d\u308b<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy.abc<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<span class=\"c1\"># \u5186\u5468\u7387<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy<\/span> <span class=\"kn\">import<\/span> <span class=\"n\">pi<\/span>\r\n<span class=\"c1\"># SymPy Plotting Backends (SPB)<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">spb<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n\r\n<span class=\"c1\"># \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n<span class=\"o\">%<\/span><span class=\"k\">config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[2]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u3067\u306f\u306a\u304f\u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\u306e\u305f\u3081<\/span>\r\n<span class=\"kn\">import<\/span> <span class=\"nn\">matplotlib.pyplot<\/span> <span class=\"k\">as<\/span> <span class=\"nn\">plt<\/span>\r\n\r\n<span class=\"n\">config<\/span> <span class=\"o\">=<\/span> <span class=\"p\">{<\/span>\r\n    <span class=\"s1\">'axes.labelsize'<\/span><span class=\"p\">:<\/span> <span class=\"s1\">'x-large'<\/span><span class=\"p\">,<\/span>\r\n    <span class=\"s1\">'mathtext.fontset'<\/span><span class=\"p\">:<\/span> <span class=\"s1\">'cm'<\/span>\r\n<span class=\"p\">}<\/span>\r\n<span class=\"n\">plt<\/span><span class=\"o\">.<\/span><span class=\"n\">rcParams<\/span><span class=\"o\">.<\/span><span class=\"n\">update<\/span><span class=\"p\">(<\/span><span class=\"n\">config<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"$t-=-0$-\u304b\u3089\u306e-$a(t)$-\u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b\">$t = 0$ \u304b\u3089\u306e $a(t)$ \u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h3>\n<p>\u7570\u306a\u308b $\\Omega_{\\rm m}$ \u306e\u5834\u5408\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u5909\u5316\u306e\u30b0\u30e9\u30d5\u3092\uff0c$t=0$ \u3067\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(t)$ \u306e\u50be\u304d\u3092\u63c3\u3048\u3066\u63cf\u304f\u5834\u5408\u3002\u540c\u3058\u3088\u3046\u306b\u30d3\u30c3\u30b0\u30d0\u30f3\u3067\u59cb\u307e\u3063\u305f\u5b87\u5b99\u306e\u81a8\u5f35\u304c\uff0c$\\Omega_{\\rm m}$ \u306e\u5024\u306b\u3088\u3063\u3066\u305d\u306e\u5f8c\u306e\u81a8\u5f35\u306e\u4ed5\u65b9\u306b\u9055\u3044\u304c\u3042\u3089\u308f\u308c\uff0c\u3042\u308b\u5834\u5408\u306b\u306f\u9014\u4e2d\u3067\u81a8\u5f35\u304c\u6b62\u307e\u3063\u3066\u53ce\u7e2e\u306b\u8ee2\u3058\u305f\u308a\uff0c\u3042\u308b\u5834\u5408\u306b\u306f\u6c38\u4e45\u306b\u81a8\u5f35\u304c\u7d9a\u3044\u305f\u308a\u3059\u308b\u3093\u3060\u306a\u3041&#8230; \u3068\u3044\u3046\u3053\u3068\u3092\u7406\u89e3\u3059\u308b\u306e\u306b\u9069\u5207\u306a\u30b0\u30e9\u30d5\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-&gt;-1$-\u3059\u306a\u308f\u3061-$k-&gt;-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} &gt; 1$ \u3059\u306a\u308f\u3061 $k &gt; 0$ \u306e\u5834\u5408<\/h4>\n<p>$\\Omega_{\\rm m} \\rightarrow \\Omega$ \u3068\u3057\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\na_1(u, \\Omega) = \\frac{a}{a_0}<br \/>\n&amp;=&amp; \\frac{\\Omega}{2 (\\Omega -1)}<br \/>\n\\left(1 -\\cos\\left(\\sqrt{\\Omega-1} u\\right)\\right)\\\\<br \/>\nt_1(u, \\Omega) = H_0 t &amp;=&amp; \\frac{\\Omega}{2 (\\Omega -1) }<br \/>\n\\left(u -\\frac{\\sin\\left(\\sqrt{\\Omega-1} u\\right)}{\\sqrt{\\Omega-1}}\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308a\u305d\u3046\u3060\u304c\uff0c$|u| \\ll 1$ \u3067\u306e\u632f\u308b\u307e\u3044\u304c\u5f8c\u8ff0\u306e $\\Omega_{\\rm m} = 1$ \u306e\u5834\u5408\u306e\u3088\u3046\u306b\uff08$\\Omega$ \u306e\u5024\u306b\u3088\u3089\u305a\u306b\uff09<\/p>\n<p>\\begin{eqnarray}<br \/>\na_1 &amp;\\simeq&amp; \\frac{u^2}{4} \\\\<br \/>\nt_1 &amp;\\simeq&amp; \\frac{u^3}{12}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308b\u305f\u3081\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7e26\u6a2a\u7b49\u500d\u3067\u7e2e\u5c3a\u3092\u5909\u3048\u3066\u3084\u308c\u3070\u3088\u3044\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\na_1(u, \\Omega) \\equiv \\frac{a}{a_0} \\times \\Omega^{-1}<br \/>\n&amp;=&amp; \\frac{1}{2 (\\Omega -1)}<br \/>\n\\left(1 -\\cos\\left(\\sqrt{\\Omega-1} u\\right)\\right)\\\\<br \/>\nt_1(u, \\Omega) \\equiv H_0 t \\times \\Omega^{-1}&amp;=&amp; \\frac{1}{2 (\\Omega -1) }<br \/>\n\\left(u -\\frac{\\sin\\left(\\sqrt{\\Omega-1} u\\right)}{\\sqrt{\\Omega-1}}\\right)<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'Omega'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">a1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">))<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">u<\/span><span class=\"p\">))<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">))<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"o\">-<\/span><span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">u<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-&lt;-1$-\u3059\u306a\u308f\u3061-$k-&lt;-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} &lt; 1$ \u3059\u306a\u308f\u3061 $k &lt; 0$ \u306e\u5834\u5408<\/h4>\n<p>\u540c\u69d8\u306b<\/p>\n<p>\\begin{eqnarray}<br \/>\na_2(u, \\Omega) \\equiv \\frac{a}{a_0}\\times \\Omega^{-1}<br \/>\n&amp;=&amp; \\frac{1}{2 (1-\\Omega)}<br \/>\n\\left(\\cosh\\left(\\sqrt{1-\\Omega} u\\right) -1\\right)<br \/>\n\\\\<br \/>\nt_2(u, \\Omega) \\equiv H_0 t \\times \\Omega^{-1}&amp;=&amp; \\frac{1}{2 (1 -\\Omega) }<br \/>\n\\left(\\frac{\\sinh\\left(\\sqrt{1-\\Omega} u\\right)}{\\sqrt{1-\\Omega}}- u\\right)<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">a2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">))<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">cosh<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">u<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">t2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">))<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">sinh<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">u<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">u<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-=-1$-\u3059\u306a\u308f\u3061-$k-=-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} = 1$ \u3059\u306a\u308f\u3061 $k = 0$ \u306e\u5834\u5408<\/h4>\n<p>\\begin{eqnarray}<br \/>\na_3(u) \\equiv \\lim_{\\Omega\\rightarrow 1} a_1(u, \\Omega) &amp;=&amp; \\frac{u^2}{4} \\\\<br \/>\nt_3(u) \\equiv \\lim_{\\Omega\\rightarrow 1} t_1(u, \\Omega) &amp;=&amp; \\frac{u^3}{12}<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Limit<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a1(u, Omega)'<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">),<\/span> \r\n   <span class=\"n\">limit<\/span><span class=\"p\">(<\/span><span class=\"n\">a1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">),<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[5]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\lim_{\\Omega \\to 1^+} a_{1}{\\left(u,\\Omega \\right)} = \\frac{u^{2}}{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Limit<\/span><span class=\"p\">(<\/span><span class=\"s1\">'t1(u, Omega)'<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">),<\/span> \r\n   <span class=\"n\">limit<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">),<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[6]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\lim_{\\Omega \\to 1^+} t_{1}{\\left(u,\\Omega \\right)} = \\frac{u^{3}}{12}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">a3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">u<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"o\">\/<\/span><span class=\"mi\">4<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">t3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">u<\/span><span class=\"o\">**<\/span><span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"mi\">12<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[8]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Omega1<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">1.1<\/span>\r\n<span class=\"n\">Omega2<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">0.9<\/span>\r\n<span class=\"c1\"># 0 &lt;= u &lt;= u1 \u307e\u3067 <\/span>\r\n<span class=\"n\">u1<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">\/<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega1<\/span> <span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">trange<\/span> <span class=\"o\">=<\/span> <span class=\"n\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"mf\">1.02<\/span>\r\n\r\n<span class=\"n\">p<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot_parametric<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"p\">(<\/span><span class=\"n\">t2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"n\">a2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">t3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span>        <span class=\"p\">),<\/span> <span class=\"n\">a3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span>        <span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"n\">a1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"n\">xlim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">trange<\/span><span class=\"p\">),<\/span> <span class=\"n\">ylim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">42<\/span><span class=\"p\">),<\/span> \r\n    <span class=\"n\">use_cm<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">show<\/span><span class=\"o\">=<\/span><span class=\"kc\">False<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p<\/span><span class=\"o\">.<\/span><span class=\"n\">ax<\/span>\r\n\r\n<span class=\"c1\"># \u5ea7\u6a19\u8ef8\u306e\u76ee\u76db\u3092\u975e\u8868\u793a\u306b<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axes<\/span><span class=\"o\">.<\/span><span class=\"n\">xaxis<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ticks<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">]);<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axes<\/span><span class=\"o\">.<\/span><span class=\"n\">yaxis<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ticks<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">]);<\/span>\r\n\r\n<span class=\"c1\"># \u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_title<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\Lambda}=0$ \u306e\u5834\u5408\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u767a\u5c55\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_xlabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$t$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ylabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$a(t)$\"<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u66f2\u7dda\u306e\u8fd1\u304f\u306b\u30e9\u30d9\u30eb\u3068\u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">60<\/span><span class=\"p\">,<\/span> <span class=\"mi\">38<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm<\/span><span class=\"si\">{m}<\/span><span class=\"s2\">} &lt; 1\\ \\ (k &lt; 0)$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'x-large'<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">60<\/span><span class=\"p\">,<\/span> <span class=\"mi\">18<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm<\/span><span class=\"si\">{m}<\/span><span class=\"s2\">} = 1\\ \\ (k = 0)$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'x-large'<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">60<\/span><span class=\"p\">,<\/span> <span class=\"mf\">10.5<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm<\/span><span class=\"si\">{m}<\/span><span class=\"s2\">} &gt; 1\\ \\ (k &gt; 0)$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'x-large'<\/span><span class=\"p\">});<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-7520\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Spb-A-fig1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"$t-=-t_0$-\u3067-$a(t_0)$-\u3068-$H_0-=-\\frac{\\dot{a}}{a}|_{t_0}$-\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b\">$t = t_0$ \u3067 $a(t_0)$ \u3068 $H_0 = \\frac{\\dot{a}}{a}|_{t_0}$ \u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h3>\n<p>\u7570\u306a\u308b $\\Omega_{\\rm m}$ \u306e\u5834\u5408\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u5909\u5316\u306e\u30b0\u30e9\u30d5\u3092\uff0c\u73fe\u5728\u6642\u523b $t=t_0$ \u3067\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(t)$ \u306e\u50be\u304d\u3092\u63c3\u3048\u3066\u63cf\u304f\u5834\u5408\u3002\u73fe\u5728\u6642\u523b\u3067\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u50be\u304d\u3092\u8868\u3059\u30cf\u30c3\u30d6\u30eb\u5b9a\u6570 $H_0$ \u306e\u5024\u304c\u540c\u3058\u3067\u3082\uff0c\u6642\u9593\u3092\u9061\u308b\u3068\u3084\u304c\u3066 $a(t)$ \u304c\u30bc\u30ed\u306b\u306a\u308b\u6642\u523b\u3059\u306a\u308f\u3061\u5b87\u5b99\u5e74\u9f62\u304c\u7570\u306a\u308b\u306e\u3060\u306a\u3041&#8230; \u3068\u3044\u3046\u3053\u3068\u3092\u7406\u89e3\u3059\u308b\u306e\u306b\u4fbf\u5229\u306a\u30b0\u30e9\u30d5\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-&gt;-1$-\u3059\u306a\u308f\u3061-$k-&gt;-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} &gt; 1$ \u3059\u306a\u308f\u3061 $k &gt; 0$ \u306e\u5834\u5408<\/h4>\n<p>\\begin{eqnarray}<br \/>\n\\frac{a}{a_0} \\equiv x<br \/>\n&amp;=&amp; \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}<br \/>\n\\left(1 -\\cos\\left(\\sqrt{k} \\eta\\right)\\right)<br \/>\n\\\\<br \/>\nH_0 t &amp;=&amp; \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)^{\\frac{3}{2}} }<br \/>\n\\left(\\sqrt{k} \\eta -\\sin\\left(\\sqrt{k} \\eta\\right)\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u304b\u3089<br \/>\n\\begin{eqnarray}<br \/>\n\\cos\\sqrt{k} \\eta &amp;=&amp; 1 &#8211; \\frac{2 (\\Omega_{\\rm m} -1)}{\\Omega_{\\rm m}}x\\\\<br \/>\n\\sqrt{k} \\eta &amp;=&amp; \\cos^{-1} \\left(1 &#8211; \\frac{2 (\\Omega_{\\rm m} -1)}{\\Omega_{\\rm m}}x \\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>$\\Omega_{\\rm m} \\rightarrow \\Omega$ \u3068\u3057\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\therefore\\ \\ T_1(x, \\Omega) &amp;\\equiv&amp; H_0 t = \\frac{\\Omega}{2 (\\Omega -1)^{\\frac{3}{2}} }<br \/>\n\\left(\\sqrt{k} \\eta -\\sin \\sqrt{k} \\eta \\right) \\\\<br \/>\n&amp;=&amp; \\frac{\\Omega}{2 (\\Omega -1)^{\\frac{3}{2}} }<br \/>\n\\left(\\cos^{-1} \\left(1 &#8211; \\frac{2 (\\Omega -1)}{\\Omega} x \\right)<br \/>\n&#8211; \\sqrt{1 &#8211; \\left(1 &#8211; \\frac{2 (\\Omega -1)}{\\Omega} x \\right)^2} \\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u3057\u3066\uff0c$H_0 t$ \u3092 $x$ \u3067\u8868\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[9]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">T1<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">))<\/span> <span class=\"o\">*<\/span> \r\n            <span class=\"p\">(<\/span><span class=\"n\">acos<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span> <span class=\"o\">-<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Omega<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> \r\n             <span class=\"o\">-<\/span> <span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Omega<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-&lt;-1$-\u3059\u306a\u308f\u3061-$k-&lt;-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} &lt; 1$ \u3059\u306a\u308f\u3061 $k &lt; 0$ \u306e\u5834\u5408<\/h4>\n<p>\\begin{eqnarray}<br \/>\n\\frac{a}{a_0} \\equiv x<br \/>\n&amp;=&amp;\\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m} )}<br \/>\n\\left(\\cosh \\sqrt{k} \\eta -1\\right)a<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u304b\u3089<br \/>\n\\begin{eqnarray}<br \/>\n\\cosh\\sqrt{k} \\eta &amp;=&amp; 1 + \\frac{2 (1-\\Omega_{\\rm m})}{\\Omega_{\\rm m}}x \\\\<br \/>\n\\sqrt{k} \\eta &amp;=&amp; \\cosh^{-1} \\left(1 + \\frac{2 (1-\\Omega_{\\rm m})}{\\Omega_{\\rm m}}x\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>$\\Omega_{\\rm m} \\rightarrow \\Omega$ \u3068\u3057\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\therefore\\ \\<br \/>\nT_2(x, \\Omega) &amp;\\equiv&amp; H_0 t = \\frac{\\Omega}{2 (1-\\Omega)^{\\frac{3}{2}} }<br \/>\n\\left(\\sinh \\sqrt{k} \\eta &#8211; \\sqrt{k} \\eta \\right) \\\\<br \/>\n&amp;=&amp; \\frac{\\Omega}{2 (1-\\Omega)^{\\frac{3}{2}} }<br \/>\n\\left(<br \/>\n\\sqrt{\\left(1 + \\frac{2 (1-\\Omega)}{\\Omega}x \\right)^2 &#8211; 1} &#8211;<br \/>\n\\cosh^{-1} \\left(1 + \\frac{2 (1-\\Omega)}{\\Omega}x\\right) \\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u3057\u3066\uff0c$H_0 t$ \u3092 $x$ \u3067\u8868\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[10]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">T2<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"p\">(<\/span><span class=\"n\">Omega<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">))<\/span> <span class=\"o\">*<\/span> \r\n            <span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">((<\/span><span class=\"mi\">1<\/span><span class=\"o\">+<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Omega<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span> \r\n             <span class=\"o\">-<\/span><span class=\"n\">acosh<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Omega<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$\\Omega_{\\Lambda}-=-0,-\\Omega_{\\rm-m}-=-1$-\u3059\u306a\u308f\u3061-$k-=-0$-\u306e\u5834\u5408\">$\\Omega_{\\Lambda} = 0, \\Omega_{\\rm m} = 1$ \u3059\u306a\u308f\u3061 $k = 0$ \u306e\u5834\u5408<\/h4>\n<p>\\begin{eqnarray}<br \/>\n\\frac{a}{a_0} = x &amp;=&amp; \\left(\\frac{3}{2}H_0 t \\right)^{\\frac{2}{3}}<br \/>\n\\end{eqnarray}$$\\therefore\\ \\ T_3(x) \\equiv H_0 t = \\frac{2}{3} x^{\\frac{3}{2}}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[11]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">T3<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">2<\/span><span class=\"o\">\/<\/span><span class=\"mi\">3<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[12]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Omega1<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">2<\/span>\r\n<span class=\"n\">Omega2<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">0.1<\/span>\r\n\r\n<span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"n\">plot_parametric<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"p\">(<\/span><span class=\"n\">T2<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T2<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} &lt; 1\\ \\ (k &lt; 0)$\"<\/span><span class=\"p\">,<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">T3<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span>        <span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T3<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span>        <span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} = 1\\ \\ (k = 0)$\"<\/span><span class=\"p\">,<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">T1<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T1<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} &gt; 1\\ \\ (k &gt; 0)$\"<\/span><span class=\"p\">,<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"n\">xlim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mf\">1.6<\/span><span class=\"p\">),<\/span> <span class=\"n\">ylim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n    <span class=\"n\">use_cm<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> <span class=\"n\">show<\/span><span class=\"o\">=<\/span><span class=\"kc\">False<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p<\/span><span class=\"o\">.<\/span><span class=\"n\">ax<\/span>\r\n<span class=\"c1\"># \u30b0\u30ea\u30c3\u30c9\u3092 dotted \u3067<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">grid<\/span><span class=\"p\">(<\/span><span class=\"n\">which<\/span><span class=\"o\">=<\/span><span class=\"s2\">\"major\"<\/span><span class=\"p\">,<\/span> <span class=\"n\">linestyle<\/span> <span class=\"o\">=<\/span> <span class=\"s1\">'dotted'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_title<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\Lambda}=0$ \u306e\u5834\u5408\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u767a\u5c55\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_xlabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$H_0 (t -t_0)$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ylabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$a(t)\/a_0$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">legend<\/span><span class=\"p\">(<\/span><span class=\"n\">prop<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"s2\">\"size\"<\/span><span class=\"p\">:<\/span><span class=\"s2\">\"large\"<\/span><span class=\"p\">})<\/span>\r\n\r\n<span class=\"c1\"># \u73fe\u5728 t=t0 <\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axhline<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">color<\/span><span class=\"o\">=<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"n\">dashes<\/span><span class=\"o\">=<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> <span class=\"n\">linewidth<\/span><span class=\"o\">=<\/span><span class=\"mf\">0.6<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axvline<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">color<\/span><span class=\"o\">=<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"n\">dashes<\/span><span class=\"o\">=<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> <span class=\"n\">linewidth<\/span><span class=\"o\">=<\/span><span class=\"mf\">0.6<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u526f\u76ee\u76db\u3082\u8868\u793a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">minorticks_on<\/span><span class=\"p\">()<\/span>\r\n<span class=\"c1\"># \u526f\u76ee\u76db\u306b\u306f grid \u3092\u3064\u3051\u306a\u3044<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">grid<\/span><span class=\"p\">(<\/span><span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">which<\/span><span class=\"o\">=<\/span><span class=\"s2\">\"minor\"<\/span><span class=\"p\">);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-7521\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Spb-A-fig2.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"$k-=-0,-\\-\\Omega_{\\Lambda}-&gt;-0$-\u306e\u5834\u5408\u306e\u8ffd\u52a0\">$k = 0, \\ \\Omega_{\\Lambda} &gt; 0$ \u306e\u5834\u5408\u306e\u8ffd\u52a0<\/h3>\n<p>\u5b87\u5b99\u5b9a\u6570\u304c\u3042\u308b\u5834\u5408\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u5909\u5316\u306b\u3064\u3044\u3066\u3082\u8ffd\u52a0\u3057\u3066\u304a\u304f\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{a}{a_0} = x &amp;=&amp;<br \/>\n\\left\\{\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}}}<br \/>\n\\sinh\\left(\\frac{3\\sqrt{1-\\Omega_{\\rm m}}}{2} H_0 t\\right)\\right\\}^{\\frac{2}{3}} \\end{eqnarray}<\/p>\n<p>\u3088\u308a $\\Omega_{\\rm m} \\rightarrow \\Omega$ \u3068\u3057\u3066<\/p>\n<p>$$T_4(x, \\Omega) \\equiv H_0 t = \\frac{2}{3\\sqrt{1-\\Omega}}<br \/>\n\\sinh^{-1} \\left( \\sqrt{\\frac{1-\\Omega}{\\Omega}} x^{\\frac{3}{2}}\\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[13]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">T4<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">2<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">))<\/span> <span class=\"o\">*<\/span> <span class=\"n\">asinh<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">((<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span> <span class=\"o\">*<\/span> <span class=\"n\">x<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u307e\u305f\uff0c$t = 0$ \u304b\u3089\u306e $a(t)$ \u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306b\u3059\u308b\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308c\u3070\u3088\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[14]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">a4<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">))<\/span><span class=\"o\">*<\/span><span class=\"n\">sinh<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">u<\/span><span class=\"o\">**<\/span><span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"mi\">12<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span><span class=\"p\">))<\/span><span class=\"o\">**<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">\/<\/span><span class=\"mi\">3<\/span><span class=\"p\">)<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">t4<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">u<\/span><span class=\"o\">**<\/span><span class=\"mi\">3<\/span><span class=\"o\">\/<\/span><span class=\"mi\">12<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$t-=-0$-\u304b\u3089\u306e-$a(t)$-\u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b\">$t = 0$ \u304b\u3089\u306e $a(t)$ \u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[15]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Omega1<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">1.1<\/span>\r\n<span class=\"n\">Omega2<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">0.7<\/span>\r\n<span class=\"c1\"># 0 &lt;= u &lt;= u1 \u307e\u3067 <\/span>\r\n<span class=\"n\">u1<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"o\">\/<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">Omega1<\/span> <span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">trange<\/span> <span class=\"o\">=<\/span> <span class=\"n\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"mf\">1.1<\/span>\r\n\r\n<span class=\"n\">p<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot_parametric<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"p\">(<\/span><span class=\"n\">t4<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mf\">0.999<\/span><span class=\"p\">),<\/span> <span class=\"n\">a4<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mf\">0.999<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'purple'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">t2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"n\">a2<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">t3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span>        <span class=\"p\">),<\/span> <span class=\"n\">a3<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span>        <span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"n\">a1<\/span><span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">u<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">u1<\/span><span class=\"p\">),<\/span> <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"n\">xlim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">trange<\/span><span class=\"p\">),<\/span>  <span class=\"n\">ylim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">120<\/span><span class=\"p\">),<\/span> \r\n    <span class=\"n\">use_cm<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">show<\/span><span class=\"o\">=<\/span><span class=\"kc\">False<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p<\/span><span class=\"o\">.<\/span><span class=\"n\">ax<\/span>\r\n\r\n<span class=\"c1\"># \u5ea7\u6a19\u8ef8\u306e\u76ee\u76db\u3092\u975e\u8868\u793a\u306b<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axes<\/span><span class=\"o\">.<\/span><span class=\"n\">xaxis<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ticks<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">]);<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axes<\/span><span class=\"o\">.<\/span><span class=\"n\">yaxis<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ticks<\/span><span class=\"p\">([<\/span><span class=\"mi\">0<\/span><span class=\"p\">]);<\/span>\r\n\r\n<span class=\"c1\"># \u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_title<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u767a\u5c55\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_xlabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$t$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ylabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$a(t)$\"<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u66f2\u7dda\u306e\u8fd1\u304f\u306b\u30e9\u30d9\u30eb\u3068\u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">80<\/span><span class=\"p\">,<\/span> <span class=\"mi\">75<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$k=0, \\Omega_{\\Lambda} &gt;0$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'purple'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'large'<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">80<\/span><span class=\"p\">,<\/span> <span class=\"mi\">62<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$k&lt;0, \\Omega_{\\Lambda} =0$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'large'<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">80<\/span><span class=\"p\">,<\/span> <span class=\"mi\">30<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$k=0, \\Omega_{\\Lambda} =0$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'large'<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">text<\/span><span class=\"p\">(<\/span><span class=\"mi\">80<\/span><span class=\"p\">,<\/span> <span class=\"mi\">9<\/span><span class=\"p\">,<\/span> <span class=\"sa\">r<\/span><span class=\"s2\">\"$k&gt;0, \\Omega_{\\Lambda} =0$\"<\/span><span class=\"p\">,<\/span> \r\n        <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">,<\/span> <span class=\"s1\">'size'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'large'<\/span><span class=\"p\">});<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-7522\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Spb-A-fig3.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$t-=-t_0$-\u3067-$a(t_0)$-\u3068-$H_0-=-\\frac{\\dot{a}}{a}|_{t_0}$-\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b\">$t = t_0$ \u3067 $a(t_0)$ \u3068 $H_0 = \\frac{\\dot{a}}{a}|_{t_0}$ \u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[16]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Omega1<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">2<\/span>\r\n<span class=\"n\">Omega2<\/span> <span class=\"o\">=<\/span> <span class=\"mf\">0.3<\/span>\r\n\r\n<span class=\"n\">p<\/span><span class=\"o\">=<\/span><span class=\"n\">plot_parametric<\/span><span class=\"p\">(<\/span>\r\n    <span class=\"p\">(<\/span><span class=\"n\">T4<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T4<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">4<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">, \\ \\Omega_{\\Lambda} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">$\"<\/span> <span class=\"o\">%<\/span> <span class=\"p\">(<\/span><span class=\"n\">Omega2<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'purple'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">T2<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T2<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega2<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">, \\ \\Omega_{\\Lambda} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">$\"<\/span> <span class=\"o\">%<\/span> <span class=\"p\">(<\/span><span class=\"n\">Omega2<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'blue'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">T3<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span>        <span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T3<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span>        <span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">, \\ \\Omega_{\\Lambda} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">$\"<\/span> <span class=\"o\">%<\/span> <span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"p\">(<\/span><span class=\"n\">T1<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">)<\/span><span class=\"o\">-<\/span><span class=\"n\">T1<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">Omega1<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"sa\">r<\/span><span class=\"s2\">\"$\\Omega_{\\rm m} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">, \\ \\Omega_{\\Lambda} =<\/span><span class=\"si\">%.1f<\/span><span class=\"s2\">$\"<\/span> <span class=\"o\">%<\/span> <span class=\"p\">(<\/span><span class=\"n\">Omega1<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">),<\/span> \r\n     <span class=\"p\">{<\/span><span class=\"s1\">'color'<\/span><span class=\"p\">:<\/span><span class=\"s1\">'red'<\/span><span class=\"p\">}),<\/span> \r\n    <span class=\"n\">xlim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"mf\">1.6<\/span><span class=\"p\">),<\/span> <span class=\"n\">ylim<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">4<\/span><span class=\"p\">),<\/span> \r\n    <span class=\"n\">use_cm<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> <span class=\"n\">show<\/span><span class=\"o\">=<\/span><span class=\"kc\">False<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p<\/span><span class=\"o\">.<\/span><span class=\"n\">ax<\/span>\r\n<span class=\"c1\"># \u30b0\u30ea\u30c3\u30c9\u3092 dotted \u3067<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">grid<\/span><span class=\"p\">(<\/span><span class=\"n\">which<\/span><span class=\"o\">=<\/span><span class=\"s2\">\"major\"<\/span><span class=\"p\">,<\/span> <span class=\"n\">linestyle<\/span> <span class=\"o\">=<\/span> <span class=\"s1\">'dotted'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u73fe\u5728 t=t0 <\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axhline<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">,<\/span> <span class=\"n\">color<\/span><span class=\"o\">=<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"n\">dashes<\/span><span class=\"o\">=<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> <span class=\"n\">linewidth<\/span><span class=\"o\">=<\/span><span class=\"mf\">0.6<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">axvline<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"n\">color<\/span><span class=\"o\">=<\/span><span class=\"s1\">'black'<\/span><span class=\"p\">,<\/span> <span class=\"n\">dashes<\/span><span class=\"o\">=<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">,<\/span> <span class=\"mi\">3<\/span><span class=\"p\">),<\/span> <span class=\"n\">linewidth<\/span><span class=\"o\">=<\/span><span class=\"mf\">0.6<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u6570\u5f0f\u30d5\u30a9\u30f3\u30c8\u8a2d\u5b9a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_title<\/span><span class=\"p\">(<\/span><span class=\"s2\">\"\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u6642\u9593\u767a\u5c55\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_xlabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$H_0 (t - t_0)$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">set_ylabel<\/span><span class=\"p\">(<\/span><span class=\"sa\">r<\/span><span class=\"s2\">\"$a(t)\/a_0$\"<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">legend<\/span><span class=\"p\">(<\/span><span class=\"n\">prop<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"s2\">\"size\"<\/span><span class=\"p\">:<\/span> <span class=\"s2\">\"large\"<\/span><span class=\"p\">})<\/span>\r\n\r\n<span class=\"c1\"># \u526f\u76ee\u76db\u3082\u8868\u793a<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">minorticks_on<\/span><span class=\"p\">()<\/span>\r\n<span class=\"c1\"># \u526f\u76ee\u76db\u306b\u306f grid \u3092\u3064\u3051\u306a\u3044<\/span>\r\n<span class=\"n\">ax<\/span><span class=\"o\">.<\/span><span class=\"n\">grid<\/span><span class=\"p\">(<\/span><span class=\"kc\">False<\/span><span class=\"p\">,<\/span> <span class=\"n\">which<\/span><span class=\"o\">=<\/span><span class=\"s2\">\"minor\"<\/span><span class=\"p\">);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-7523\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/Spb-A-fig4.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5c0e\u51fa\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\uff1a<\/p><p><a class=\"more-link btn\" href=\"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\/%e5%ae%87%e5%ae%99%e8%ab%96%e3%83%91%e3%83%a9%e3%83%a1%e3%83%bc%e3%82%bf%e3%81%a8%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2\/%e8%a3%9c%e8%b6%b3%ef%bc%9asympy-%e3%81%a8-spb-%e3%81%a7%e3%82%b9%e3%82%b1%e3%83%bc%e3%83%ab%e5%9b%a0%e5%ad%90%e3%81%ae%e3%82%b0%e3%83%a9%e3%83%95%e3%82%92%e6%8f%8f%e3%81%8f\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n<ul>\n<li>\u88dc\u8db3\uff1a\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u89e3<\/li>\n<\/ul>\n","protected":false},"author":33,"featured_media":0,"parent":1483,"menu_order":9,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7461","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7461","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=7461"}],"version-history":[{"count":7,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7461\/revisions"}],"predecessor-version":[{"id":7524,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7461\/revisions\/7524"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1483"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=7461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}