{"id":1483,"date":"2022-01-26T12:42:26","date_gmt":"2022-01-26T03:42:26","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1483"},"modified":"2024-07-22T11:20:45","modified_gmt":"2024-07-22T02:20:45","slug":"%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","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\/","title":{"rendered":"\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u5b87\u5b99\u5e74\u9f62"},"content":{"rendered":"

\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span> $H_0$ \u3084\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span> $\\Omega_{\\rm m}, \\ \\Omega_{\\Lambda}$ \u306e\u5c0e\u5165\u3068\uff0c\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u304b\u3089\u6c42\u3081\u308b\u5b87\u5b99\u5e74\u9f62 <\/strong><\/span>\\(t_0\\)\uff08\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u304c \\(a=0\\) \u304b\u3089\u00a0 \\(a_0 = a(t_0)\\) \u306b\u306a\u308b\u307e\u3067\u306e\u6642\u9593\uff09\u3002<\/p>\n

\u7279\u306b\uff0c$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306e\u5b87\u5b99\u5e74\u9f62 \\(t_0\\) \u306f\uff0c<\/p>\n

\\begin{eqnarray}
\nH_0 t_0 &=& -\\frac{1}{\\Omega_{\\rm m} -1}+\\frac{\\Omega_{\\rm m}}{(\\Omega_{\\rm m}-1)^{\\frac{3}{2}} }
\n\\tan^{-1}\\sqrt{\\Omega_{\\rm m}-1} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} > 1\\\\
\nH_0 t_0 &=& \\frac{1}{1-\\Omega_{\\rm m}}-\\frac{\\Omega_{\\rm m}}{(1-\\Omega_{\\rm m})^{\\frac{3}{2}} }
\n\\tanh^{-1}\\sqrt{1-\\Omega_{\\rm m}} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} < 1
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c\\(\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1\\) \u3059\u306a\u308f\u3061 \\(k = 0\\) \u306e\u5834\u5408\u306f\uff0c<\/p>\n

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

\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

<\/p>\n

\n

\u81a8\u5f35\u5b87\u5b99<\/strong><\/span>\u3092\u3042\u3089\u308f\u3059 FLRW \u8a08\u91cf<\/strong><\/span>\u306f\uff0c\u6642\u9593\u306e\u307f\u306e\u95a2\u6570\u3067\u3042\u308b\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50<\/strong><\/span> \\(a(t)\\) \u3068\uff0c\u7a7a\u9593\u5ea7\u6a19\u306e\u307f\u306e\u95a2\u6570\u3067\u3042\u308b3\u6b21\u5143\u5b9a\u66f2\u7387\u7a7a\u9593\u306e\u8a08\u91cf<\/strong><\/span> \\(\\gamma_{ij}\\) \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f\u3002\uff08\\(c\u00a0 = 1\\) \u3068\u3059\u308b\u3002\uff09
\n$$ds^2 =-dt^2 + g_{ij}dx^i dx^j =\u00a0 -dt^2 + a^2(t) \\gamma_{ij} dx^i dx^j$$<\/p>\n

\u5b87\u5b99\u7a7a\u9593\u3092\u6e80\u305f\u3059\u7269\u8cea\u304c\u7269\u8cea\u5bc6\u5ea6 \\(\\rho\\) \uff08\u5727\u529b $P$ \u304c\u30bc\u30ed\uff09\u306e\u30c0\u30b9\u30c8<\/strong><\/span>\u306e\u5834\u5408\uff0c\u5b87\u5b99\u5b9a\u6570<\/strong><\/span> \\(\\Lambda\\) \u3082\u5165\u308c\u305f\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u65b9\u7a0b\u5f0f\u306f\uff0c
\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho + \\frac{\\Lambda}{3}$$<\/p>\n

$$\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\rho + \\frac{\\Lambda}{3} $$\u307e\u305f\uff0c
\n$$\\dot{\\rho} + 3\\frac{\\dot{a}}{a} \\rho = 0, \\ \\ \\therefore\\ \\ \\rho \\propto \\frac{1}{a^3}$$<\/p>\n

\u3053\u3053\u3067 $k$ \u306f\uff08\u30cb\u30e5\u30fc\u30c8\u30f3\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u306f\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u7a4d\u5206\u306e\u969b\u306b\u3042\u3089\u308f\u308c\u308b\u7a4d\u5206\u5b9a\u6570\u3068\u3044\u3046\u610f\u5473\u3057\u304b\u306a\u304b\u3063\u305f\u304b\u3082\u77e5\u308c\u306a\u3044\u304c\uff09\u4e00\u822c\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u306f\uff0c\u4e00\u69d8\u7b49\u65b9\u306a3\u6b21\u5143\u7a7a\u9593\u306e\u66f2\u7387\u5b9a\u6570<\/strong><\/span>\u3068\u3044\u3046\u610f\u5473\u3092\u6301\u3064\u3002$k > 0$ \u306e\u5834\u5408\u306f\u9589\u3058\u305f\u7a7a\u9593\uff0c$k<0$ \u306e\u5834\u5408\u306f\u958b\u3044\u305f\u7a7a\u9593\uff0c\u3061\u3087\u3046\u3069 $k = 0$ \u306e\u5834\u5408\u306f\u5e73\u5766\u306a\u7a7a\u9593\u3068\u3044\u3046\u3002<\/p>\n

\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf<\/h3>\n

\u8fd1\u63a5\u3057\u305f2\u70b9\u9593\u306e\u7a7a\u9593\u7684\u8ddd\u96e2\u3092
\n$$\\ell \\equiv \\sqrt{g_{ij} dx^i dx^j} = a(t) \\sqrt{\\gamma_{ij} dx^i dx^j}$$\u3068\u3059\u308b\u3068\uff0c\u305d\u306e\u6642\u9593\u5909\u5316\u306f $\\dot{\\gamma}_{ij} = 0$ \u3067\u3042\u308b\u304b\u3089
\n$$\\dot{\\ell} = \\frac{\\dot{a}}{a} \\ell$$\u3067\u3042\u308b\u3002<\/p>\n

\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247<\/strong><\/span> \\(v = H_0 r\\) \u3068\u306e\u6bd4\u8f03\u304b\u3089
\n$$H_0 = \\frac{\\dot{a}}{a}\\Bigg|_{t=t_0}$$<\/p>\n

\u5b87\u5b99\u306e\u81a8\u5f35\u7387\u3092\u3042\u3089\u308f\u3059 \\(\\displaystyle \\frac{\\dot{a}}{a}\\) \u306e\u73fe\u5728 \\(t = t_0\\) \u3067\u306e\u5024\u304c\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span> \\(H_0\\) \u3067\u3042\u308b\u3002<\/p>\n

\uff08\u30cf\u30c3\u30d6\u30eb\u300c\u5b9a\u6570\u300d\u3068\u3082\u547c\u3070\u308c\u308b\u3060\u308d\u3046\u304c\uff0c\u6642\u9593\u306b\u4f9d\u5b58\u3057\u306a\u3044\u300c\u5b9a\u6570\u300d\u3068\u3044\u3046\u3088\u308a\u306f\uff0c\u6642\u9593\u306b\u4f9d\u5b58\u3059\u308b\u95a2\u6570\u306e\u73fe\u5728\u6642\u523b\u3067\u306e\u5024\u3068\u3044\u3046\u3053\u3068\u3060\u304b\u3089\u30cf\u30c3\u30d6\u30eb\u300c\u30d1\u30e9\u30e1\u30fc\u30bf\u300d\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3002\uff09<\/p>\n

\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf<\/h3>\n

\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho + \\frac{\\Lambda}{3}$$<\/p>\n

\u3092\u73fe\u5728\u306e\u6642\u523b \\(t =t_0\\) \u3067\u8a55\u4fa1\u3059\u308b\u3068<\/p>\n

$$H_0^2 + \\frac{k}{a_0^2} = \\frac{8\\pi G}{3} \\rho_0 + \\frac{\\Lambda}{3}$$
\n\u3067\u3042\u308b\u3002\u3053\u3053\u3067\uff0c$a_0 \\equiv a(t_0), \\ \\rho_0 \\equiv \\rho(t_0)$<\/p>\n

\u73fe\u5728\u306e\u5b87\u5b99\u81a8\u5f35\u304c\u30c0\u30b9\u30c8\u306e\u7269\u8cea\u5bc6\u5ea6\u306e\u307f\u306b\u3088\u3063\u3066\u30c9\u30e9\u30a4\u30d6\u3055\u308c\u3066\u3044\u308b\u306a\u3089\u3070\uff08\\(k = 0, \\ \\Lambda = 0\\) \u3068\u3044\u3046\u3053\u3068\uff09\uff0c\u305d\u306e\u3068\u304d\u306e\u7269\u8cea\u5bc6\u5ea6\u3092\u81e8\u754c\u5bc6\u5ea6<\/strong><\/span> \\(\\rho_{\\rm cr}\\) \u3068\u3044\u3046\u304c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002
\n$$H_0^2 = \\frac{8\\pi G}{3} \\rho_{\\rm cr}, \\quad\\therefore\\ \\ \\rho_{\\rm cr}\\equiv \\frac{3 H_0^2}{8\\pi G}$$<\/p>\n

\u73fe\u5728\u306e\u7269\u8cea\u5bc6\u5ea6 \\(\\rho_0\\) \u3068\u81e8\u754c\u5bc6\u5ea6\u306e\u6bd4\u3092\u3068\u308a\uff0c
\n$$\\Omega_{\\rm m} \\equiv \\frac{\\rho_0}{\\rho_{\\rm cr}} = \\frac{8\\pi G \\rho_0}{3 H_0^2}$$
\n\u3053\u306e \\(\\Omega_{\\rm m}\\) \u3092\uff08\u30c0\u30b9\u30c8\uff09\u7269\u8cea\u306e\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3068\u3044\u3046\u3002<\/p>\n

\u307e\u305f\uff0c\u5b87\u5b99\u5b9a\u6570\u3092\u5b8c\u5168\u6d41\u4f53\u3068\u307f\u306a\u3057\u305f\u3068\u304d\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u00a0 \\(\\rho_{\\Lambda}\\) \u306b\u3064\u3044\u3066\u3082 $\\displaystyle \\frac{\\Lambda}{3} = \\frac{8 \\pi G}{3} \\rho_{\\Lambda}$ \u3067\u3042\u308b\u304b\u3089\uff0c\u540c\u69d8\u306b
\n$$\\Omega_{\\Lambda} \\equiv \\frac{\\rho_{\\Lambda}}{\\rho_{\\rm cr}} = \\frac{\\frac{\\Lambda}{8 \\pi G}}{\\frac{3 H_0^2}{8\\pi G}}= \\frac{\\Lambda}{3 H_0^2}$$<\/p>\n

\u3053\u308c\u3089\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u3042\u3089\u305f\u3081\u3066 \\(t = t_0\\) \u3067\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068
\n$$H_0^2 + \\frac{k}{a_0^2} = H_0^2 \\left(\\Omega_{\\rm m} + \\Omega_{\\Lambda}\\right)$$
\n$$\\therefore\\ \\frac{k}{a_0^2} =H_0^2 \\left(\\Omega_{\\rm m} + \\Omega_{\\Lambda} -1 \\right)$$<\/p>\n

\\(H_0, \\Omega_{\\rm m}, \\Omega_{\\Lambda}\\) \u3092\u307e\u3068\u3081\u3066\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3002<\/p>\n

\u6e1b\u901f\u30d1\u30e9\u30e1\u30fc\u30bf<\/h4>\n

\u3061\u306a\u307f\u306b\uff0c\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf\u304c
\n$$H_0 = \\frac{\\dot{a}}{a}\\Bigg|_{t=t_0}$$\u306e\u3088\u3046\u306b\uff0c1\u968e\u5fae\u5206\uff08\u81a8\u5f35\u901f\u5ea6\uff09\u3092\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u3067\u5272\u3063\u305f\u91cf\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u306b\u5bfe\u5fdc\u3057\u3066\uff0c2\u968e\u5fae\u5206\uff08\u81a8\u5f35\u306e\u52a0\u901f\u5ea6\u30fb\u6e1b\u901f\u5ea6\uff09\u3092\u8868\u3059\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u3057\u3066
\n$$q_0 \\equiv -\\frac{1}{H_0^2} \\frac{\\ddot{a}}{a}\\Bigg|_{t=t_0}$$\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u305f $q_0$ \u3092\u6e1b\u901f\u30d1\u30e9\u30e1\u30fc\u30bf<\/strong><\/span>\u3068\u547c\u3073\uff0c20\u4e16\u8a18\u306e\u6559\u79d1\u66f8\u3067\u306f\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u305f\u304c\uff0c\u6700\u8fd1\u306e\u30c6\u30ad\u30b9\u30c8\u3067\u306f\u3042\u307e\u308a\u898b\u306a\u304f\u306a\u3063\u305f\u304b\u3082\u77e5\u308c\u306a\u3044\u3002<\/p>\n

$$\\frac{\\ddot{a}}{a} = -\\frac{4\\pi G}{3} \\rho + \\frac{\\Lambda}{3} $$<\/p>\n

\u306e\u5f0f\u3092 $t = t_0$ \u3067\u8a55\u4fa1\u3057\u3066\u3084\u308b\u3068\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n

$$q_0 = -\\frac{1}{2} \\Omega_{\\rm m} + \\Omega_{\\Lambda}$$<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u672c\u7a3f\u3067\u306f $\\Omega_{\\rm m} $ \u3068 $ \\Omega_{\\Lambda}$ \u3092\u4f7f\u3046\u3053\u3068\u306b\u3057\u3066\uff0c$q_0$ \u306f\u4f7f\u308f\u306a\u3044\u65b9\u5411\u3002<\/p>\n

\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u66f8\u3044\u305f\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/h3>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u4efb\u610f\u6642\u523b \\(t\\) \u3067\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306f
\n\\begin{eqnarray}
\n\\left(\\frac{\\dot{a}}{a} \\right)^2 &=& \\frac{8\\pi G}{3} \\rho + \\frac{\\Lambda}{3} -\\frac{k}{a^2} \\\\
\n&=& H_0^2 \\left\\{\\frac{8\\pi G \\rho_0}{3 H_0^2}\\left(\\frac{a_0}{a}\\right)^3
\n+ \\frac{\\Lambda}{3 H_0^2} \\right\\} -\\frac{k}{a_0^2}\\left(\\frac{a_0}{a}\\right)^2\\\\
\n&=& H_0^2 \\left\\{\\Omega_{\\rm m} \\left(\\frac{a_0}{a}\\right)^3 + \\Omega_{\\Lambda}
\n+ \\left(1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}\\right)\\left(\\frac{a_0}{a}\\right)^2\\right\\}
\n\\end{eqnarray}<\/p>\n

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

\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247\u304b\u3089 \\(\\dot{a} > 0\\) \u3067\u3042\u308b\u3002\u307e\u305f\u7269\u8cea\u5bc6\u5ea6\u304c\u30bc\u30ed\u3067\u306a\u3044\u306a\u3089 \\(\\rho > 0\\) \u306a\u306e\u3067\uff0c\\(\\ddot{a} < 0\\) \u3068\u306a\u308a\uff0c\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 \\(a(t)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0a\u306b\u51f8\u3068\u306a\u308b\u3002\u3053\u306e\u3053\u3068\u306f\uff0c\u6709\u9650\u306e\u904e\u53bb\uff0c\\(a=0\\) \u3068\u306a\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u3064\u307e\u308a\u5b87\u5b99\u306b\u306f\u59cb\u307e\u308a\u304c\u3042\u3063\u305f\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u304b\u3089\u5b87\u5b99\u5e74\u9f62<\/strong><\/span> \\(t_0\\)\uff08\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u304c \\(a = 0\\) \u304b\u3089 \\(a = a_0 = a(t_0)\\) \u306b\u306a\u308b\u307e\u3067\u306e\u6642\u9593\uff09 \u3092\u6c42\u3081\u308b\u3002\u5e38\u8b58\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u306b \\(\\displaystyle \\left(\\frac{a}{a_0}\\right)^2\\) \u3092\u304b\u3051\u3066\u5e73\u65b9\u6839\u3092\u3068\u308b\u3068
\n$$\\frac{d}{dt}\\left(\\frac{a}{a_0}\\right) =
\nH_0 \\sqrt{\\Omega_{\\rm m} \\left(\\frac{a_0}{a}\\right)
\n+\\left(1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}\\right)
\n+ \\Omega_{\\Lambda} \\left(\\frac{a}{a_0}\\right) ^2}$$<\/p>\n

$\\displaystyle x \\equiv \\frac{a}{a_0}$ \u3068\u3044\u3046\u5909\u6570\u3092\u4f7f\u3048\u3070\uff0c\u4e0a\u306e\u5f0f\u306f<\/p>\n

\\begin{eqnarray}\\frac{dx}{dt}&=&
\nH_0 \\sqrt{\\Omega_{\\rm m} \\left(\\frac{1}{x}\\right)
\n+\\left(1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}\\right)
\n+ \\Omega_{\\Lambda} x^2}\\\\
\n&=& H_0 \\frac{\\sqrt{\\Omega_{\\rm m} + (1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}) x+ \\Omega_{\\Lambda} x^3}}{\\sqrt{x}}
\n\\end{eqnarray}<\/p>\n

\u5909\u6570\u5206\u96e2\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u306a\u3063\u3066\uff0c<\/p>\n

\\begin{eqnarray}H_0 dt
\n&=&\u00a0 \\frac{\\sqrt{x}}{\\sqrt{\\Omega_{\\rm m} + (1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}) x+ \\Omega_{\\Lambda} x^3}} dx
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067<\/p>\n

\\begin{eqnarray}
\nf(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda}) &\\equiv& \\frac{\\sqrt{x}}{\\sqrt{\\Omega_{\\rm m} + (1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}) x+ \\Omega_{\\Lambda} x^3}}
\n\\end{eqnarray}<\/p>\n

\u3068\u304a\u3051\u3070\uff0c<\/p>\n

\\begin{eqnarray}
\nH_0 t_0 &=&\u00a0 \\int_0^1 f(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda}) dx
\n\\end{eqnarray}<\/p>\n

$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408<\/h4>\n

\\(\\Omega_{\\rm m} > 1\\) \u3092\u60f3\u5b9a\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nH_0 t_0 &=&\u00a0 \\int_0^1 f(x, \\Omega_{\\rm m}, 0) dx\\\\
\n&=& \\int_0^1\\frac{\\sqrt{x}}{\\sqrt{\\Omega_{\\rm m} -(\\Omega_{\\rm m}-1 ) x}} dx \\\\
\n&=& \\frac{1}{\\sqrt{\\Omega_{\\rm m}-1}}
\n\\int_0^1\\frac{\\sqrt{x}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} -x}} dx
\n\\end{eqnarray}<\/p>\n

\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u6570\u5909\u63db\u3092\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\frac{\\sqrt{x}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} -x}} &\\equiv& t \\\\
\n\\therefore \\ \\ x &=& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} \\frac{t^2}{1+t^2} \\\\
\ndx &=& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} \\frac{2t}{(1+t^2)^2} dt
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ H_0 t_0 &=& \\frac{1}{\\sqrt{\\Omega_{\\rm m}-1}}
\n\\int_0^1\\frac{\\sqrt{x}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} -x}} dx \\\\
\n&=& \\frac{\\Omega_{\\rm m}}{(\\Omega_{\\rm m}-1)^{\\frac{3}{2}}}
\n\\int_0^{\\sqrt{\\Omega_{\\rm m}-1}} \\frac{2t^2}{(1+t^2)^2} dt \\\\
\n&=& \\frac{\\Omega_{\\rm m}}{(\\Omega_{\\rm m}-1)^{\\frac{3}{2}}}
\n\\Bigl[\\tan^{-1} t -\\frac{t}{1 + t^2} \\Bigr]_0^{\\sqrt{\\Omega_{\\rm m}-1}}
\n\\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306e\u7a4d\u5206\u306f\uff0c\u3053\u3053<\/a><\/strong><\/span>\u3067\u793a\u3057\u3066\u304a\u304d\u307e\u3057\u305f\u3002<\/p>\n

\u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u3092\u5b9f\u884c\u3057\u305f\u7d50\u679c\u3092\u66f8\u304f\u3068<\/p>\n

$$H_0 t_0 = -\\frac{1}{\\Omega_{\\rm m} -1}+\\frac{\\Omega_{\\rm m}}{(\\Omega_{\\rm m}-1)^{\\frac{3}{2}} }
\n\\tan^{-1}\\sqrt{\\Omega_{\\rm m}-1} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} > 1$$<\/p>\n

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

\u3053\u3053\u3067\uff0c$\\Omega_{\\rm m} < 1$ \u306e\u5834\u5408\u306b\u306f $\\tan^{-1} (i x) = i \\tanh^{-1} x$ \u306e\u95a2\u4fc2\u3092\u4f7f\u3063\u3066 $\\Omega_{\\rm m} > 1$ \u306e\u7d50\u679c\u304b\u3089\u5c0e\u3044\u305f\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u3053\u308c\u3089\u306e\u95a2\u4fc2\u306b\u3064\u3044\u3066\u306f\uff0c\u5225\u9014\u6388\u696d\u3067\u7fd2\u3063\u305f\u3088\u306d\u3047\u3002\uff08\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/a>\uff09<\/p>\n

\u7279\u306b\uff0c$\\Omega_{\\rm m} = 1$\u00a0 \u3059\u306a\u308f\u3061 $k = 0$ \u306e\u3068\u304d\u306b\u306f\u4e0a\u5f0f\u3067 \\(\\Omega_{\\rm m} \\rightarrow 1\\) \u306e\u6975\u9650\u3092\u3068\u308c\u3070\u3088\u304f\u3066\uff0c\u7d50\u679c\u306f
\n$$H_0 t_0\u00a0 = \\frac{2}{3} \\quad\\mbox{for}\\ \\ \\Omega_{\\rm m} = 1$$<\/p>\n

 <\/p>\n

\u4ee5\u4e0b\u306e\u8a08\u7b97\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c<\/p>\n

$$\\displaystyle \\frac{\\partial}{\\partial \\Omega_{\\rm m}}f(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda}) < 0 \\quad\\quad\u00a0 \\mbox{for} \\ 0 < x < 1$$<\/p>\n

\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda})\\)\uff0c\u3057\u305f\u304c\u3063\u3066\u7a4d\u5206\u3068\u3057\u3066\u306e\u5b87\u5b99\u5e74\u9f62 \\(t_0\\) \u306f \\(\\Omega_{\\rm m}\\) \u306e\u5358\u8abf\u6e1b\u5c11\u95a2\u6570 \u3067\u3042\u308b\u304b\u3089<\/p>\n

$$H_0 t_0\u00a0 \\left \\{ \\begin{array}{cl}
\n< \\frac{2}{3}\u00a0 & \\mbox{for}\\\u00a0 \\Omega_{\\rm m} > 1 \\\\
\n= \\frac{2}{3}\u00a0 & \\mbox{for}\\\u00a0 \\Omega_{\\rm m} = 1 \\\\
\n> \\frac{2}{3}\u00a0 & \\mbox{for}\\\u00a0 \\Omega_{\\rm m} < 1\\end{array} \\right.$$\u3067\u3042\u308b\u3002\u6700\u5927\u5024\u3068\u306a\u308b\u306e\u306f $\\Omega_{\\rm m} \\rightarrow 0$ \u306e\u3068\u304d\u3067<\/p>\n

$$H_0 t_0 = \\int_0^1 f(x, 0, 0) dx = \\int_0^1\\frac{\\sqrt{x}}{\\sqrt{ x}} dx = 1$$<\/p>\n

\\(\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1\\) \u3059\u306a\u308f\u3061 \\(k = 0\\) \u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\nH_0 t_0 &=&\u00a0 \\int_0^1 f(x, \\Omega_{\\rm m}, 1-\\Omega_{\\rm m}) dx\\\\
\n&=& \\int_0^1\\frac{\\sqrt{x}}{\\sqrt{\\Omega_{\\rm m} -(\\Omega_{\\rm m} -1) x^3}} dx \\\\
\n&=& \\frac{1}{\\sqrt{\\Omega_{\\rm m} -1}}
\n\\int_0^1\\frac{\\sqrt{x^3}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{ \\Omega_{\\rm m} -1} -x^3}} \\frac{dx}{x}
\n\\end{eqnarray}<\/p>\n

\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u6570\u5909\u63db\u3092\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\frac{\\sqrt{x^3}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{ \\Omega_{\\rm m} -1} -x^3}} &\\equiv& t \\\\
\n\\therefore\\ \\ x^3 &=& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} \\frac{t^2}{1+t^2} \\\\
\n3 x^3 \\frac{dx}{x} &=& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} \\frac{2 t^2}{(1+t^2)^2}\\frac{dt}{t} \\\\
\n&=& 2 x^3 \\frac{dt}{t (1 + t^2)} \\\\
\n\\therefore\\ \\ \\frac{dx}{x} &=& \\frac{2}{3} \\frac{dt}{t (1 + t^2)}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ H_0 t_0
\n&=& \\frac{1}{\\sqrt{\\Omega_{\\rm m} -1}}
\n\\int_0^1\\frac{\\sqrt{x^3}}{\\sqrt{\\frac{\\Omega_{\\rm m}}{ \\Omega_{\\rm m} -1} -x^3}} \\frac{dx}{x} \\\\
\n&=& \\frac{1}{\\sqrt{\\Omega_{\\rm m} -1}}\\int_0^{\\sqrt{ \\Omega_{\\rm m} -1}} t \\cdot \\frac{2}{3} \\frac{dt}{t (1 + t^2)}\\\\
\n&=& \\frac{2}{3\\sqrt{\\Omega_{\\rm m} -1}} \\Bigl[\\tan^{-1} t \\Bigr]_0^{\\sqrt{ \\Omega_{\\rm m} -1}}
\n\\end{eqnarray}<\/p>\n

\u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u3092\u5b9f\u884c\u3057\u305f\u7d50\u679c\u3092\u66f8\u304f\u3068<\/p>\n

$$H_0 t_0 = \\frac{2}{3(\\sqrt{\\Omega_{\\rm m} -1})}\\tan^{-1} \\sqrt{\\Omega_{\\rm m} -1} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} > 1$$<\/p>\n

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

\u3053\u3053\u3067\uff0c$\\Omega_{\\rm m} < 1$ \u306e\u5834\u5408\u306b\u306f $\\tan^{-1} (i x) = i \\tanh^{-1} x$ \u306e\u95a2\u4fc2\u3092\u4f7f\u3063\u3066 $\\Omega_{\\rm m} > 1$ \u306e\u7d50\u679c\u304b\u3089\u5c0e\u3044\u305f\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u3053\u308c\u3089\u306e\u95a2\u4fc2\u306b\u3064\u3044\u3066\u306f\uff0c\u5225\u9014\u6388\u696d\u3067\u7fd2\u3063\u305f\u3088\u306d\u3047\u3002\uff08\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/a>\uff09<\/p>\n

\u7279\u306b\uff0c$\\Omega_{\\rm m} = 1$\u00a0 \u3059\u306a\u308f\u3061 $\\Omega_{\\Lambda} = 0$ \u306e\u3068\u304d\u306b\u306f
\n$$H_0 t_0\u00a0 = \\frac{2}{3} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} = 1$$<\/p>\n

 <\/p>\n

\u4ee5\u4e0b\u306e\u8a08\u7b97\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c<\/p>\n

$$\\displaystyle \\frac{\\partial}{\\partial \\Omega_{\\Lambda}}f(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda}) > 0 \\quad\\quad\u00a0 \\mbox{for} \\ 0 < x < 1$$<\/p>\n

\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x, \\Omega_{\\rm m}, \\Omega_{\\Lambda})\\)\uff0c\u3057\u305f\u304c\u3063\u3066\u7a4d\u5206\u3068\u3057\u3066\u306e\u5b87\u5b99\u5e74\u9f62 \\(t_0\\) \u306f \\(\\Omega_{\\Lambda}\\) \u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3067\u3042\u308b\u3002<\/p>\n

\u306a\u304a\uff0cMaxima \u3067\u306f\u306a\u304b\u306a\u304b\u304d\u308c\u3044\u306b\u3057\u3066\u304f\u308c\u306a\u304b\u3063\u305f\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u4eba\u529b\u3067\u6574\u3048\u305f\u3002
\n$$\\tan^{-1} (x) + \\tan^{-1}\\left(\\frac{1}{x}\\right) = \\frac{\\pi}{2}$$<\/p>\n

\u3044\u3063\u305f\u3093\uff0c\\(\\Omega_{\\rm m}>1\\) \u306e\u5834\u5408\u304c\u89e3\u3051\u308c\u3070\uff0c\\(\\Omega_{\\rm m}<1\\) \u306e\u5834\u5408\u3082
\n$$\\tan^{-1} (i x) = i \\tanh^{-1} x$$\u306a\u3069\u306e\u516c\u5f0f\u3092\u4f7f\u3048\u3070\u3059\u3050\u306b\u6c42\u307e\u308b\u3060\u308d\u3046\u3002<\/p>\n

\u7d50\u5c40\uff0cMaxima \u3060\u3051\u3058\u3083\u3042\u30c0\u30e1\u3067\u4eba\u529b\u3067\u6574\u3048\u308b\u5fc5\u8981\u304c\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3057\u305f\u3002<\/p>\n

\u5b87\u5b99\u5e74\u9f62\u306e\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf\u4f9d\u5b58\u6027\u306e\u30b0\u30e9\u30d5<\/h3>\n

\"\"<\/a><\/p>\n

 <\/p>\n

Maxima-Jupyter \u3067\u306e\u8a08\u7b97\u4f8b<\/h3>\n
\n
\n
<\/div>\n
\n
\n

\u5b87\u5b99\u5e74\u9f62 $t_0$<\/h4>\n

$$ H_0 t_0 = \\int_0^1 \\frac{\\sqrt{x}}{\\sqrt{\\Omega_{\\rm m} + (1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}) x + \\Omega_{\\Lambda} x^3}} dx$$<\/p>\n

Maxima \u306e\u8868\u8a18\u306e\u90fd\u5408\u4e0a\uff0c
\n$\\Omega_{\\rm m} \\rightarrow \\Omega, \\quad \\Omega_{\\Lambda} \\rightarrow \\Omega_1$ \u3068\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
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In\u00a0[1]:<\/div>\n
\n
\n
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f<\/span>(<\/span>x<\/span>, Omega<\/span>, Omega1<\/span>)<\/span>:=<\/span> sqrt<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>sqrt<\/span>(<\/span>Omega<\/span> +<\/span> (<\/span>1<\/span> -<\/span> Omega<\/span>-<\/span>Omega1<\/span>)<\/span>*<\/span>x<\/span>+<\/span>Omega1<\/span>*<\/span>x<\/span>**<\/span>3<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}f\\left(x , \\Omega , \\Omega_{1}\\right):=\\frac{\\sqrt{x}}{\\sqrt{\\Omega+\\left(1-\\Omega-\\Omega_{1}\\right)\\,x+\\Omega_{1}\\,x^3}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
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$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408<\/h4>\n
$\\Omega_{\\Lambda} = 0, \\ \\Omega_{\\rm m} > 1$ \u306e\u5834\u5408\u306f Maxima \u306f\u7a4d\u5206\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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\/* \u5ff5\u306e\u305f\u3081\uff0cassume() \u7b49\u3067\u306e\u8a2d\u5b9a\u3092\u5168\u3066\u5fd8\u308c\u3055\u305b\u308b\u3002*\/<\/span>\r\nforget<\/span>(<\/span>facts<\/span>())<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> ><\/span> 1<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span> =<\/span>\r\n ans<\/span>:<\/span> integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/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{${\\it \\%o}_{4}$}\\int_{0}^{1}{\\frac{\\sqrt{x}}{\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}}\\;dx}=\\frac{\\pi\\,\\sqrt{\\Omega-1}\\,\\Omega}{2\\,\\Omega^2-4\\,\\Omega+2}-\\frac{\\arctan \\left(\\frac{1}{\\sqrt{\\Omega-1}}\\right)\\,\\sqrt{\\Omega-1}\\,\\Omega+\\Omega-1}{\\Omega^2-2\\,\\Omega+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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factor<\/span>(<\/span>ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}-\\frac{2\\,\\arctan \\left(\\frac{1}{\\sqrt{\\Omega-1}}\\right)\\,\\sqrt{\\Omega-1}\\,\\Omega-\\pi\\,\\sqrt{\\Omega-1}\\,\\Omega+2\\,\\Omega-2}{2\\,\\left(\\Omega-1\\right)^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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$\\Omega_{\\Lambda} = 0, \\ 0 < \\Omega_{\\rm m} < 1$ \u306e\u5834\u5408\u306f\uff0c\u610f\u5473\u4e0d\u660e…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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\/* \u5ff5\u306e\u305f\u3081\uff0cassume() \u7b49\u3067\u306e\u8a2d\u5b9a\u3092\u5168\u3066\u5fd8\u308c\u3055\u305b\u308b\u3002*\/<\/span>\r\nforget<\/span>(<\/span>facts<\/span>())<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> ><\/span> 0<\/span>)<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> <<\/span> 1<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}\\int_{0}^{1}{\\frac{\\sqrt{x}}{\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}}\\;dx}=-\\lim_{x\\downarrow 0}{\\frac{\\sqrt{1-\\Omega}\\,\\Omega\\,\\log \\left(\\frac{\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}\\,\\left| 2\\,\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}-2\\,\\sqrt{1-\\Omega}\\,x\\right| -\\sqrt{1-\\Omega}\\,x\\,\\left| 2\\,\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}-2\\,\\sqrt{1-\\Omega}\\,x\\right| }{2\\,\\Omega\\,x}\\right)}{2\\,\\Omega^2-4\\,\\Omega+2}-\\frac{\\left(2\\,\\Omega-2\\right)\\,\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x+\\Omega}}{2\\,\\Omega^2-4\\,\\Omega+2}}+\\frac{\\log \\left(-\\frac{\\left| 2\\,\\sqrt{1-\\Omega}-2\\right| \\,\\sqrt{1-\\Omega}-\\left| 2\\,\\sqrt{1-\\Omega}-2\\right| }{2\\,\\Omega}\\right)\\,\\sqrt{1-\\Omega}\\,\\Omega}{2\\,\\Omega^2-4\\,\\Omega+2}-\\frac{\\Omega}{\\Omega^2-2\\,\\Omega+1}+\\frac{1}{\\Omega^2-2\\,\\Omega+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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$\\Omega_{\\Lambda} = 0, \\ \\Omega_{\\rm m} = 1$ \u306e\u5834\u5408\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, 1<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\frac{2}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u4e00\u898b\uff0c\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3063\u3066\u6016\u305d\u3046\u3060\u3051\u3069\uff0c\u3061\u3083\u3093\u3068$\\Omega_{\\rm m} > 1$ \u306e\u89e3\u306e\u6975\u9650\u3092\u3068\u3063\u3066\u3082\u6b63\u3057\u3044\u7b54\u3048\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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limit<\/span>(<\/span>ans<\/span>, Omega<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\frac{2}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1$ \u3059\u306a\u308f\u3061 $k = 0$ \u306e\u5834\u5408<\/h4>\n
$\\Omega_{\\rm m} > 1$ \u306e\u5834\u5408\u306f Maxima \u306f\u7a4d\u5206\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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\/* \u5ff5\u306e\u305f\u3081\uff0cassume() \u7b49\u3067\u306e\u8a2d\u5b9a\u3092\u5168\u3066\u5fd8\u308c\u3055\u305b\u308b\u3002*\/<\/span>\r\nforget<\/span>(<\/span>facts<\/span>())<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> ><\/span> 1<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 1-<\/span>Omega<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span> =<\/span>\r\n ans<\/span>:<\/span> integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 1-<\/span>Omega<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}\\int_{0}^{1}{\\frac{\\sqrt{x}}{\\sqrt{\\left(1-\\Omega\\right)\\,x^3+\\Omega}}\\;dx}=\\frac{\\pi\\,\\sqrt{\\Omega-1}}{3\\,\\Omega-3}-\\frac{2\\,\\arctan \\left(\\frac{1}{\\sqrt{\\Omega-1}}\\right)\\,\\sqrt{\\Omega-1}}{3\\,\\Omega-3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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factor<\/span>(<\/span>ans<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}-\\frac{2\\,\\arctan \\left(\\frac{1}{\\sqrt{\\Omega-1}}\\right)-\\pi}{3\\,\\sqrt{\\Omega-1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$0 < \\Omega_{\\rm m} < 1$ \u306e\u5834\u5408\u306f\uff0c\u610f\u5473\u4e0d\u660e…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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\/* \u5ff5\u306e\u305f\u3081\uff0cassume() \u7b49\u3067\u306e\u8a2d\u5b9a\u3092\u5168\u3066\u5fd8\u308c\u3055\u305b\u308b\u3002*\/<\/span>\r\nforget<\/span>(<\/span>facts<\/span>())<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> ><\/span> 0<\/span>)<\/span>$\r\nassume<\/span>(<\/span>Omega<\/span> <<\/span> 1<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 1-<\/span>Omega<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, 1-<\/span>Omega<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}\\int_{0}^{1}{\\frac{\\sqrt{x}}{\\sqrt{\\left(1-\\Omega\\right)\\,x^3+\\Omega}}\\;dx}=\\frac{\\log \\left(-\\frac{\\left| 2\\,\\sqrt{1-\\Omega}-2\\right| \\,\\sqrt{1-\\Omega}-\\left| 2\\,\\sqrt{1-\\Omega}-2\\right| }{2\\,\\Omega}\\right)\\,\\sqrt{1-\\Omega}}{3\\,\\Omega-3}-\\frac{\\sqrt{1-\\Omega}\\,\\left(\\lim_{x\\downarrow 0}{\\log \\left(\\frac{\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x^3+\\Omega}\\,\\left| 2\\,\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x^3+\\Omega}-2\\,\\sqrt{1-\\Omega}\\,x^2\\right| -\\sqrt{1-\\Omega}\\,x^2\\,\\left| 2\\,\\sqrt{x}\\,\\sqrt{\\left(1-\\Omega\\right)\\,x^3+\\Omega}-2\\,\\sqrt{1-\\Omega}\\,x^2\\right| }{2\\,\\Omega\\,x}\\right)}\\right)}{3\\,\\Omega-3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5b87\u5b99\u5e74\u9f62\u306e\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf\u4f9d\u5b58\u6027<\/h4>\n
\u88ab\u7a4d\u5206\u95a2\u6570 $f(x, \\Omega, \\Omega_1)$ \u306f $\\Omega$ \u306e\u5358\u8abf\u6e1b\u5c11\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, Omega1<\/span>)<\/span>, Omega<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{20}$}-\\frac{\\left(1-x\\right)\\,\\sqrt{x}}{2\\,\\left(\\Omega_{1}\\,x^3+\\left(-\\Omega_{1}-\\Omega+1\\right)\\,x+\\Omega\\right)^{\\frac{3}{2}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$0 < x < 1$ \u3067\u306f $\\displaystyle \\frac{\\partial f}{\\partial \\Omega} < 0$<\/p>\n
\u5f93\u3063\u3066\uff0c$H_0$ \u3084 $\\Omega_{\\Lambda}$ \u304c\u540c\u3058\u306a\u3089 $\\Omega_{\\rm m}$ \u304c\u5c0f\u3055\u3044\u65b9\u304c\u5b87\u5b99\u5e74\u9f62\u304c\u4f38\u3073\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u3061\u306a\u307f\u306b\uff0c$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\uff0c$\\Omega_{\\rm m}$ \u304c\u5c0f\u3055\u3044\u307b\u3069\u5b87\u5b99\u5e74\u9f62\u306f\u4f38\u3073\u308b\u306e\u3067\uff0c\u305d\u306e\u6700\u5927\u5024\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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tlong<\/span>:<\/span> integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, 0<\/span>, 0<\/span>)<\/span>, x<\/span>, 0<\/span>, 1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{21}$}1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3064\u307e\u308a\uff0c$$t_0 \\rightarrow \\frac{1}{H_0}$$\u304c\u6700\u5927\u5024\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u88ab\u7a4d\u5206\u95a2\u6570 $f(x, \\Omega, \\Omega_1)$ \u306f $\\Omega_1$ \u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u304b\u3089\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, Omega<\/span>, Omega1<\/span>)<\/span>, Omega1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{22}$}-\\frac{\\sqrt{x}\\,\\left(x^3-x\\right)}{2\\,\\left(\\Omega_{1}\\,x^3+\\left(-\\Omega_{1}-\\Omega+1\\right)\\,x+\\Omega\\right)^{\\frac{3}{2}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$0 < x < 1$ \u3067\u306f $\\displaystyle \\frac{\\partial f}{\\partial \\Omega_1} > 0$<\/p>\n
\u5f93\u3063\u3066\uff0c$H_0$ \u3084 $\\Omega_{\\rm m}$ \u304c\u540c\u3058\u306a\u3089 $\\Omega_{\\Lambda}$ \u304c\u5927\u304d\u3044\u65b9\u304c\u5b87\u5b99\u5e74\u9f62\u304c\u4f38\u3073\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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 <\/p>\n","protected":false},"excerpt":{"rendered":"
\u30cf\u30c3\u30d6\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf $H_0$ \u3084\u5bc6\u5ea6\u30d1\u30e9\u30e1\u30fc\u30bf $\\Omega_{\\rm m}, \\ \\Omega_{\\Lambda}$ \u306e\u5c0e\u5165\u3068\uff0c\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u304b\u3089\u6c42\u3081\u308b\u5b87\u5b99\u5e74\u9f62 \\(t_0\\)\uff08\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u304c \\(a=0\\) \u304b\u3089\u00a0 \\(a_0 = a(t_0)\\) \u306b\u306a\u308b\u307e\u3067\u306e\u6642\u9593\uff09\u3002<\/p>\n
\u7279\u306b\uff0c$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306e\u5b87\u5b99\u5e74\u9f62 \\(t_0\\) \u306f\uff0c<\/p>\n
\\begin{eqnarray} H_0 t_0 &=& -\\frac{1}{\\Omega_{\\rm m} -1}+\\frac{\\Omega_{\\rm m}}{(\\Omega_{\\rm m}-1)^{\\frac{3}{2}} } \\tan^{-1}\\sqrt{\\Omega_{\\rm m}-1} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} > 1\\\\ H_0 t_0 &=& \\frac{1}{1-\\Omega_{\\rm m}}-\\frac{\\Omega_{\\rm m}}{(1-\\Omega_{\\rm m})^{\\frac{3}{2}} } \\tanh^{-1}\\sqrt{1-\\Omega_{\\rm m}} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} < 1 \\end{eqnarray}<\/p>\n
\u307e\u305f\uff0c\\(\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1\\) \u3059\u306a\u308f\u3061 \\(k = 0\\) \u306e\u5834\u5408\u306f\uff0c<\/p>\n
\\begin{eqnarray} H_0 t_0 &=& \\frac{2}{3(\\sqrt{\\Omega_{\\rm m} -1})}\\tan^{-1} \\sqrt{\\Omega_{\\rm m} -1} \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} > 1\\\\ H_0 t_0 &=& \\frac{2}{3(\\sqrt{1-\\Omega_{\\rm m} })}\\tanh^{-1} \\sqrt{1-\\Omega_{\\rm m} } \\quad \\mbox{for}\\ \\ \\Omega_{\\rm m} < 1\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n","protected":false},"author":33,"featured_media":0,"parent":1430,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1483","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\/1483","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=1483"}],"version-history":[{"count":55,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1483\/revisions"}],"predecessor-version":[{"id":9265,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1483\/revisions\/9265"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1430"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=1483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}