{"id":1634,"date":"2022-02-01T14:34:03","date_gmt":"2022-02-01T05:34:03","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1634"},"modified":"2024-07-22T10:14:39","modified_gmt":"2024-07-22T01:14:39","slug":"%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","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%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\/","title":{"rendered":"\u88dc\u8db3\uff1a\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u89e3"},"content":{"rendered":"

<\/p>\n

\u307e\u305a\uff0c\u5b87\u5b99\u5b9a\u6570\u3082\u5165\u308c\u305f\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306f
\n$$\\left(\\frac{\\dot{a}}{a}\\right)^2 + \\frac{k}{a^2} = \\frac{8\\pi G}{3} \\rho_0\\left(\\frac{a_0}{a}\\right)^3 + \\frac{\\Lambda}{3}$$<\/p>\n

\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf<\/p>\n

$$H_0 = \\frac{\\dot{a}}{a}\\Bigg|_{t=t_0}, \\ \\ \\Omega_{\\rm m} = \\frac{8\\pi G \\rho_0}{3 H_0^2},
\n\\ \\ \\Omega_{\\Lambda}\u00a0 = \\frac{\\Lambda}{3 H_0^2}$$<\/p>\n

\u4ee5\u4e0b\u306e\u5f0f\u3082\u4f7f\u3044\u307e\u3059\u3002<\/p>\n

$$\\frac{k}{H_0^2 a_0^2} = \\Omega_{\\rm m} + \\Omega_{\\Lambda} -1$$<\/p>\n

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

\\begin{eqnarray}
\n\\left(\\frac{\\dot{a}}{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

\u3053\u308c\u3092\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(t)$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u307f\u308b\u3002<\/p>\n

 <\/p>\n

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

$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306f\uff0c\u5225\u4ef6<\/a>\u3067\u3082 $dt = a d\\eta$\u00a0 \u3068\u3057\u3066\u5c0e\u5165\u3057\u3066\u3044\u308b\u5171\u5f62\u6642\u9593 $\\eta$ \u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

$$\\left\\{ \\frac{d}{d\\eta}\\left(\\frac{a}{a_0}\\right)\\right\\}^2
\n=H_0^2 a_0^2 \\left\\{\\Omega_{\\rm m} \\left( \\frac{a}{a_0}\\right) -(\\Omega_{\\rm m} -1)\\left( \\frac{a}{a_0}\\right)^2\\right\\}
\n$$<\/p>\n

$\\displaystyle x \\equiv \\frac{a}{a_0}$ \u3092\u4f7f\u3046\u3068\uff08$\\Omega_{\\rm m} > 1$ \u3092\u4eee\u5b9a\u3057\u3066\uff09
\n\\begin{eqnarray}
\n\\left(\\frac{dx}{d\\eta} \\right)^2 &=&\u00a0 \\left\\{\\Omega_{\\rm m} x -(\\Omega_{\\rm m} -1) x^2\\right\\} \\\\
\n&=& H_0^2 a_0^2 (\\Omega_{\\rm m} -1)
\n\\left\\{\\left(\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)} \\right)^2 –
\n\\left(x -\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}\\right)^2\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\uff0c\u5909\u6570\u3092<\/p>\n

\\begin{eqnarray}
\n\\bar{\\eta} &\\equiv&\u00a0 H_0 a_0 \\sqrt{\\Omega_{\\rm m} -1} \\eta = \\sqrt{k} \\eta\\\\
\n\\bar{x} &\\equiv& x -\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)} \\\\
\n\\frac{1}{b} &\\equiv& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}
\n\\end{eqnarray}<\/p>\n

\u3068\u304a\u304d\u304b\u308c\u3070<\/p>\n

$$\\left(\\frac{d\\bar{x}}{d\\bar{\\eta}} \\right)^2 = \\frac{1}{b^2} -\\bar{x}^2$$
\n\u3068\u3044\u3046\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u5225\u30da\u30fc\u30b8<\/strong><\/span><\/a>\u3067\u3082\u8aac\u660e\u3057\u3066\u3044\u308b\u3088\u3046\u306b\uff0c\u305f\u3060\u3061\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\bar{x} = \\frac{a}{a_0} -\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)} = \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}\\sin(\\bar{\\eta} + C)
\n\\end{eqnarray}<\/p>\n

$\\eta\u00a0 = 0$ \u3059\u306a\u308f\u3061 $\\bar{\\eta} = 0$ \u3067 $a = 0$\u00a0\u00a0 \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3092\u3064\u3051\u308b\u3068\uff0c\u7a4d\u5206\u5b9a\u6570 $C$ \u306f
\n$$\\sin C = -1, \\quad\\therefore\\ \\ C = -\\frac{\\pi}{2}$$
\n$$\\sin(\\bar{\\eta} -\\frac{\\pi}{2}) = -\\cos\\bar{\\eta}$$\u3092\u4f7f\u3046\u3068\uff0c\u6700\u7d42\u7684\u306b<\/p>\n

\\begin{eqnarray}
\n\\frac{a}{a_0} &=& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}
\n\\left(1 -\\cos\\bar{\\eta}\\right)\\\\
\n&=& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}
\n\\left(1 -\\cos\\left(\\sqrt{k} \\eta\\right)\\right)
\n\\end{eqnarray}<\/p>\n

\u6642\u9593\u5ea7\u6a19 $t$ \u306f $dt = a(\\eta) d\\eta$ \u3088\u308a<\/p>\n

\\begin{eqnarray}
\nH_0 t &=& H_0 a_0 \\int_0^{\\eta} \\frac{a}{a_0} d\\eta\\\\
\n&=&\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)^{\\frac{3}{2}} }
\n\\left(\\sqrt{k} \\eta -\\sin\\left(\\sqrt{k} \\eta\\right)\\right)
\n\\end{eqnarray}<\/p>\n

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

\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(t)$ \u306f\u5171\u5f62\u6642\u9593 $\\eta$ \u3092\u5a92\u4ecb\u5909\u6570\u3068\u3057\u3066\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3002\u4eca\u5f8c\u306e\u305f\u3081\u306b<\/p>\n

$$\\sqrt{k} \\eta = H_0 a_0 \\sqrt{\\Omega_{\\rm m} -1} \\eta \\equiv \\sqrt{\\Omega_{\\rm m} -1} u$$<\/p>\n

\u3068\u3057\u3066\uff0c\u5a92\u4ecb\u5909\u6570 $u$ \u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{a}{a_0}
\n&=& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}
\n\\left(1 -\\cos\\left(\\sqrt{k} \\eta\\right)\\right)\\\\
\n&=&
\n\\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)}
\n\\left(1 -\\cos\\left(\\sqrt{\\Omega_{\\rm m} -1} u\\right)\\right) \\\\
\nH_0 t &=& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)^{\\frac{3}{2}} }
\n\\left(\\sqrt{k} \\eta -\\sin\\left(\\sqrt{k} \\eta\\right)\\right) \\\\
\n&=& \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1) }
\n\\left(u -\\frac{\\sin\\left(\\sqrt{\\Omega_{\\rm m} -1} u\\right)}{\\sqrt{\\Omega_{\\rm m} -1}}\\right)
\n\\end{eqnarray}<\/p>\n

\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a$ \u306f $\\sqrt{k} \\eta = \\pi$\uff0c\u3059\u306a\u308f\u3061 $H_0 t = \\frac{\\Omega_{\\rm m}}{2 (\\Omega_{\\rm m} -1)^{\\frac{3}{2}} } \\pi$ \u3067\u6700\u5927\u5024 $\\frac{a}{a_0} = \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}$ \u306b\u306a\u308a\uff0c\u305d\u306e\u5f8c\uff0c\u53ce\u7e2e\u306b\u8ee2\u3058\u308b\u3002<\/p>\n

$t = 0$ \u3067 $a = 0$ \u304b\u3089\u59cb\u307e\u3063\u305f\u5b87\u5b99\u304c\uff0c\u3075\u305f\u305f\u3073 $a=0$ \u307e\u3067\u306e\u6642\u9593\u306f $H_0 t = \\frac{\\Omega_{\\rm m}}{ (\\Omega_{\\rm m} -1)^{\\frac{3}{2}} } \\pi$ \u3067\u3042\u308b\u3002<\/p>\n

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

\u4e0a\u306e\u5f0f\u304c\u305d\u306e\u307e\u307e\u4f7f\u3048\u3066<\/p>\n

\\begin{eqnarray}
\n\\frac{a}{a_0}
\n&=& -\\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m} )}
\n\\left(1 -\\cos\\left(i \\sqrt{|k|} \\eta\\right)\\right)\\\\
\n&=&\\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m} )}
\n\\left(\\cosh\\left(\\sqrt{|k|} \\eta\\right) -1\\right) \\\\
\n&=&\\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m} )}
\n\\left(\\cosh\\left(\\sqrt{1-\\Omega_{\\rm m}}\u00a0 u\\right) -1\\right) \\\\
\n\\\\
\nH_0 t &=& -\\frac{1}{i} \\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m})^{\\frac{3}{2}} }
\n\\left(i \\sqrt{|k|} \\eta -\\sin\\left(i\\sqrt{|k|} \\eta\\right)\\right)\\\\
\n&=& \\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m})^{\\frac{3}{2}} }
\n\\left(\\sinh\\left(\\sqrt{|k|} \\eta\\right) -\\sqrt{|k|} \\eta\\right) \\\\
\n&=& \\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m}) }
\n\\left(\\frac{\\sinh\\left(\\sqrt{1-\\Omega_{\\rm m}} u\\right)}{\\sqrt{1-\\Omega_{\\rm m}} } -u\\right)
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u5834\u5408\uff0c\u5b87\u5b99\u306f\u7121\u9650\u306b\u81a8\u5f35\u3092\u7d9a\u3051\u308b\u304c\uff0c\u5341\u5206\u5927\u304d\u306a $\\eta$ \u306b\u5bfe\u3057\u3066\u306f<\/p>\n

\\begin{eqnarray}
\n\\frac{a}{a_0} &\\rightarrow& \\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m} )} \\frac{1}{2} \\exp\\left(\\sqrt{1-\\Omega_{\\rm m}}\u00a0 u\\right) \\\\
\nH_0 t &\\rightarrow& \\frac{\\Omega_{\\rm m}}{2 (1-\\Omega_{\\rm m})^{\\frac{3}{2}} }\\frac{1}{2} \\exp\\left(\\sqrt{1-\\Omega_{\\rm m}}\u00a0 u\\right) \\\\
\n\\therefore\\ \\ \\frac{a}{a_0} &\\simeq& \\sqrt{1-\\Omega_{\\rm m}} H_0 t \\propto t
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(t)$ \u306f\u6642\u9593 $t$ \u306b\u6bd4\u4f8b\u3059\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n

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

\u4e0a\u306e\u5f0f\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u3066 $k \\rightarrow 1$\u00a0 \u306e\u6975\u9650\u3092\u3068\u308c\u3070\u3088\u3044\u3002<\/p>\n

$$\\lim_{\\Omega_{\\rm m} \\rightarrow 1} \\frac{a}{a_0} = \\frac{u^2}{4}$$<\/p>\n

$$\\lim_{\\Omega_{\\rm m} \\rightarrow 1} H_0 t = \\frac{u^3 }{12} = \\frac{2}{3} \\left( \\frac{u^2}{4}\\right)^{\\frac{3}{2}}$$<\/p>\n

\u3053\u308c\u304b\u3089 $u$ \u3092\u6d88\u53bb\u3057\u3066<\/p>\n

$$\\therefore\\ \\ \\frac{a}{a_0}\u00a0 = \\left(\\frac{3}{2} H_0 t\\right)^{\\frac{2}{3}}$$<\/p>\n

\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u30b0\u30e9\u30d5\u306e\u4f8b<\/h4>\n

$t = 0$ \u304b\u3089\u306e $a(t)$ \u306e\u7acb\u3061\u4e0a\u304c\u308a\u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h5>\n

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

$t = t_0$ \u3067 $a(t_0)$ \u3068 $H_0 = \\frac{\\dot{a}}{a}|_{t_0}$ \u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h5>\n

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

 <\/p>\n

 <\/p>\n

$k=0$ \u306e\u5834\u5408<\/h3>\n

$k = 0, \\ 0<\\Omega_{\\rm m} < 1$ \u3064\u307e\u308a $\\Omega_{\\Lambda} > 0$ \u306e\u5834\u5408<\/h4>\n

$1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{\\dot{a}}{a} \\right)^2
\n&=& H_0^2 \\left\\{\\Omega_{\\rm m} \\left(\\frac{a_0}{a}\\right)^3 + (1 -\\Omega_{\\rm m})
\n\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3092 $\\displaystyle x \\equiv \\frac{a}{a_0}$ \u3092\u4f7f\u3063\u3066\u8868\u3059\u3068\uff0c<\/p>\n

$$\\frac{dx}{dt} = H_0 \\sqrt{1-\\Omega_{\\rm m}}\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}} \\frac{1}{x} + x^2}$$
\n$$\\therefore\\ \\\u00a0 \\sqrt{1-\\Omega_{\\rm m}} H_0 t = \\int_0^x \\frac{\\sqrt{x} dx}{\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}} + x^3}}$$<\/p>\n

\u3053\u306e\u7a4d\u5206\u3092\u884c\u3046\u305f\u3081\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u6570\u5909\u63db\u3092\u4f7f\u3046\u3002
\n$$x^3 \\equiv \\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}} y^2, \\quad \\therefore\\ \\
\nx^{\\frac{3}{2}} = \\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}}} y$$<\/p>\n

\u4e21\u8fba\u306e\u5fae\u5206\u3092\u3068\u308b\u3068
\n$$\\sqrt{x} dx = \\frac{2}{3} \\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}}} dy$$<\/p>\n

\u3088\u3063\u3066\u61f8\u6848\u306e\u7a4d\u5206\u306f<\/p>\n

\\begin{eqnarray}
\n\\int_0^x \\frac{\\sqrt{x} dx}{\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}} + x^3}}&=&
\n\\frac{2}{3} \\int_0^y \\frac{dy}{\\sqrt{1 + y^2}} \\\\
\n&=& \\frac{2}{3} \\sinh^{-1} y
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066
\n$$\\sqrt{1-\\Omega_{\\rm m}} H_0 t =\\frac{2}{3} \\sinh^{-1} y$$\u6700\u7d42\u7684\u306b $\\displaystyle x = \\frac{a}{a_0}$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{a}{a_0} &=& \\left\\{\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}}}
\n\\sinh\\left(\\frac{3\\sqrt{1-\\Omega_{\\rm m}}}{2} H_0 t\\right)\\right\\}^{\\frac{2}{3}} \\\\
\n&=&\\left\\{\\sqrt{\\frac{\\Omega_{\\rm m}}{1-\\Omega_{\\rm m}}}
\n\\sinh\\left(\\frac{3}{2} \\sqrt{\\frac{\\Lambda}{3}} t\\right)\\right\\}^{\\frac{2}{3}}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u5834\u5408\uff0c\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306f\u6642\u9593\u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3068\u306a\u308b\u304c\uff0c\u7279\u306b $t$ \u304c\u975e\u5e38\u306b\u5927\u304d\u3044\u3068\u304d\u306b\u306f
\n$$a \\propto \\left\\{\\sinh\\left(\\frac{3}{2} \\sqrt{\\frac{\\Lambda}{3}} t\\right) \\right\\}^{\\frac{2}{3}} \\simeq
\n\\left\\{\\frac{1}{2}\\exp\\left(\\frac{3}{2} \\sqrt{\\frac{\\Lambda}{3}} t\\right) \\right\\}^{\\frac{2}{3}} $$\u3068\uff0c\u307e\u3055\u306b\u6307\u6570\u95a2\u6570\u7684\u306b\u81a8\u5f35\u3059\u308b\u3002<\/p>\n

\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u30b0\u30e9\u30d5\u306e\u4f8b<\/h4>\n

$t = t_0$ \u3067 $a(t_0)$ \u3068 $H_0 = \\frac{\\dot{a}}{a}|_{t_0}$ \u3092\u63c3\u3048\u305f\u30b0\u30e9\u30d5\u306e\u4f8b<\/h5>\n

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

$k = 0, \\ 1<\\Omega_{\\rm m} $ \u3064\u307e\u308a $\\Omega_{\\Lambda} < 0$\u306e\u5834\u5408<\/h4>\n

\u4e0a\u306e\u5f0f\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u3066
\n\\begin{eqnarray}
\n\\frac{a}{a_0} &=& \\left\\{\\frac{1}{i} \\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}}
\n\\sinh\\left(i \\frac{3\\sqrt{\\Omega_{\\rm m}-1}}{2} H_0 t\\right)\\right\\}^{\\frac{2}{3}}\\\\
\n&=& \\left\\{\\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}}
\n\\sin\\left(\\frac{3\\sqrt{\\Omega_{\\rm m}-1}}{2} H_0 t\\right)\\right\\}^{\\frac{2}{3}}
\n\\end{eqnarray}<\/p>\n

$k = 0, \\ \\Omega_{\\rm m} =1$ \u3064\u307e\u308a $\\Omega_{\\Lambda} = 0$\u306e\u5834\u5408<\/h4>\n

\u4e0a\u306e\u5f0f\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u3066 $\\Omega_{\\rm m} \\rightarrow 1$ \u306e\u6975\u9650\u3092\u3068\u308c\u3070
\n\\begin{eqnarray}
\n\\lim_{\\Omega_{\\rm m} \\rightarrow 1}\\frac{a}{a_0}
\n&=& \\left(\\frac{3}{2} H_0 t\\right)^{\\frac{2}{3}}
\n\\end{eqnarray}<\/p>\n

$\\Omega_{\\rm m} = 0$ \u306e\u5834\u5408<\/h3>\n

\u7269\u8cea\u304c\u4f55\u3082\u306a\u3044\u5b87\u5b99\u3068\u3044\u3046\u306e\u306f\u610f\u7fa9\u304c\u3042\u308b\u306e\u304b\u3069\u3046\u304b\u306f\u5225\u306b\u3057\u3066\uff0c$\\Omega_{\\rm m} = 0$ \u306e\u5834\u5408\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f<\/p>\n

\\begin{eqnarray}
\n\\left(\\frac{\\dot{a}}{a} \\right)^2
\n&=& H_0^2 \\left\\{ \\Omega_{\\Lambda} + (1-\\Omega_{\\Lambda})\\left(\\frac{a_0}{a}\\right)^2
\n\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3092 $\\displaystyle x \\equiv \\frac{a}{a_0}$ \u3092\u4f7f\u3063\u3066\u8868\u3059\u3068\uff0c<\/p>\n

$$\\frac{dx}{dt} = H_0 \\sqrt{\\Omega_{\\Lambda}} \\sqrt{\\frac{1-\\Omega_{\\Lambda}}{\\Omega_{\\Lambda}}\u00a0 + x^2}$$<\/p>\n

$0 < \\Omega_{\\Lambda} < 1$ \u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\n\\sqrt{\\Omega_{\\Lambda}}H_0 t &=& \\int_0^x \\frac{dx}{\\sqrt{\\frac{1-\\Omega_{\\Lambda}}{\\Omega_{\\Lambda}} + x^2 }} = \\sinh^{-1} \\left( \\sqrt{\\frac{\\Omega_{\\Lambda}}{1-\\Omega_{\\Lambda}} }x\\right)
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066<\/p>\n

$$ \\frac{a}{a_0} = \\sqrt{\\frac{1-\\Omega_{\\Lambda}}{\\Omega_{\\Lambda}} }\\sinh \\left(\\sqrt{\\Omega_{\\Lambda}}H_0 t \\right)$$<\/p>\n

$\\Omega_{\\Lambda} = 1$ \u306e\u5834\u5408<\/h4>\n

$$\\frac{dx}{dt} = H_0\u00a0 x$$\u304b\u3089
\n$$\\frac{a}{a_0} = \\exp\\left(H_0 t\\right)$$<\/p>\n

\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 \\(a(t)\\) \u304c\u30bc\u30ed\u3068\u306a\u308b\u6642\u523b\u3092\u300c\u5b87\u5b99\u306e\u306f\u3058\u307e\u308a<\/strong><\/span>\u300d\u3068\u3059\u308b\u3068\uff0c\u3053\u306e\u5834\u5408 $a(t) = 0$ \u3068\u306a\u308b\u306e\u306f $t \\rightarrow \\infty$ \u3064\u307e\u308a\u7121\u9650\u306e\u904e\u53bb\u3068\u306a\u308b\u3002<\/p>\n

$\\Omega_{\\Lambda} > 1$ \u306e\u5834\u5408<\/h4>\n

$\\displaystyle x \\equiv \\frac{a}{a_0}$ \u3092\u4f7f\u3063\u3066\u8868\u3057\u305f\u5f0f<\/p>\n

$$\\frac{dx}{dt} = H_0 \\sqrt{\\Omega_{\\Lambda}} \\sqrt{x^2 -\\frac{\\Omega_{\\Lambda}-1}{\\Omega_{\\Lambda}}\u00a0 }$$<\/p>\n

\u3088\u308a\uff0c$$x = \\frac{a}{a_0} \\geq \\sqrt{\\frac{\\Omega_{\\Lambda}-1}{\\Omega_{\\Lambda}} }$$\u3068\u306a\u308a\uff0c\u5b87\u5b99\u306f $a=0$ \u304b\u3089\u59cb\u307e\u3089\u306a\u3044\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

$\\displaystyle X \\equiv \\sqrt{\\frac{\\Omega_{\\Lambda}}{\\Omega_{\\Lambda}-1}} x, \\ \\ d\\tau \\equiv \\sqrt{\\Omega_{\\Lambda}} H_0 dt$ \u3068\u304a\u3051\u3070\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{dX}{d\\tau} &=& \\sqrt{X^2 -1} \\\\
\n\\frac{dX}{\\sqrt{X^2 -1} } &=& d\\tau \\\\
\n\\cosh^{-1} X &=& \\tau + C \\\\
\nX &=& \\cosh (\\tau + C) \\\\
\n\\sqrt{\\frac{\\Omega_{\\Lambda}}{\\Omega_{\\Lambda}-1}} \\frac{a}{a_0} &=& \\cosh (\\sqrt{\\Omega_{\\Lambda}} H_0\u00a0 t + C) \\\\ \\ \\\\
\n\\therefore \\ \\ \\frac{a}{a_0} &=& \\sqrt{\\frac{\\Omega_{\\Lambda}-1}{\\Omega_{\\Lambda}}}\\cosh (\\sqrt{\\Omega_{\\Lambda}} H_0\u00a0 t + C)
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u3088\u3046\u306a\u3075\u308b\u307e\u3044\u3092\u3059\u308b\u5b87\u5b99\u30e2\u30c7\u30eb\u306f\uff0c\u30ab\u30c6\u30ca\u30ea\u30fc\u5b87\u5b99<\/strong><\/span>\u3068\u3088\u3070\u308c\u308b\u3002\u300c\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3082\u53c2\u7167\u3002<\/p>\n

\u756a\u5916\u7de8\uff1a$\\Omega_{\\rm m} = 0, \\Omega_{\\Lambda} =0$ \u306e\u5834\u5408<\/h3>\n

\u30c0\u30b9\u30c8\u7269\u8cea\u3082\u306a\u3044\uff08$\\Omega_{\\rm m} = 0$\uff09\u3057\uff0c\u5b87\u5b99\u5b9a\u6570\u3082\u306a\u3044\uff08$\\Omega_{\\Lambda} = 0$\uff09\uff0c\u3064\u307e\u308a\u771f\u7a7a\u3068\u3044\u3046\u3053\u3068\u3002\u305d\u306e\u5834\u5408\u306e\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\u65b9\u7a0b\u5f0f\u306f
\n\\begin{eqnarray}
\n\\left(\\frac{\\dot{a}}{a} \\right)^2
\n&=& H_0^2 \\left(\\frac{a_0}{a}\\right)^2
\n\\end{eqnarray}<\/p>\n

$\\dot{a}>0$ \u3068\u3057\u3066\u89e3\u304f\u3068\uff0c
\n$$ \\frac{a}{a_0} = H_0 t$$<\/p>\n

$$\\frac{k}{H_0^2 a_0^2} = \\Omega_{\\rm m} + \\Omega_{\\Lambda} -1 \\Rightarrow -1 $$<\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\u8ca0\u306e\u66f2\u7387\u9805\u304c\u5b87\u5b99\u81a8\u5f35\u3092\u5f15\u304d\u8d77\u3053\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u3053\u306e\u5b87\u5b99\u30e2\u30c7\u30eb\u3092\u30df\u30eb\u30f3\u5b87\u5b99<\/strong><\/span>\u3068\u547c\u3076\u3002\u30df\u30eb\u30f3\u5b87\u5b99\u306e\u8a08\u91cf\u306f FLRW \u8a08\u91cf\u3067\u3042\u308b\u304b\u3089\uff0c3\u6b21\u5143\u7a7a\u9593\u306e\u8a08\u91cf\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u3092\u3044\u304f\u3064\u304b\u66f8\u304f\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\nds^2 &=& -dt^2 + \\left( H_0 a_0 t\\right)^2 \\left(\\frac{dr^2}{1 + |k| r^2} + r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2) \\right) \\\\
\n&=&-dt^2 + \\left( H_0 a_0 t\\right)^2 \\left(d\\chi^2\u00a0 +\\left(\\frac{\\sinh(\\sqrt{|k|}\\chi)}{\\sqrt{|k|}}\\right)^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2) \\right)
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u89e3\u306f\uff0c\u4e00\u69d8\u7b49\u65b9\u3060\u304b\u3089\u5f53\u7136\u7403\u5bfe\u79f0\u3067\u771f\u7a7a\u3060\u3051\u3069\uff0c\u9759\u7684\u3067\u306a\u3044<\/strong><\/span>\u4f8b\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n

\u300c\u7403\u5bfe\u79f0\u771f\u7a7a\u306a\u3089\u9759\u7684<\/strong><\/span>\u300d\u3068\u3044\u3046\u6709\u540d\u306a\u5b9a\u7406\u304c\u3042\u308b\u304c\uff0c\u30df\u30eb\u30f3\u5b87\u5b99\u306f\u305d\u306e\u4f8b\u5916\uff1f\u3068\u3044\u3046\u304b\u53cd\u4f8b\uff1f\u306b\u306a\u3063\u3066\u3044\u308b\u3088\u3046\u306b\u601d\u3048\uff0c\u3059\u308f\uff01\u30d0\u30fc\u30b3\u30d5\u306e\u5b9a\u7406\uff0c\u7834\u308c\u305f\u308a!? \u3068\u65e9\u5408\u70b9\u3057\u306a\u3044\u3088\u3046\u306b\u3002MTW \u306e 27.11 \u306b\u3061\u3083\u3093\u3068\u66f8\u3044\u3066\u3042\u308a\u307e\u3057\u305f\u3002<\/p>\n

\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c \\( H_0 a_0 = 1\\) \u3068\u3059\u308b\u3068\uff0c\\( k = -1\\) \u3068\u306a\u308b\u306e\u3067\uff0c\u30df\u30eb\u30f3\u5b87\u5b99\u306e\u8a08\u91cf\u306f<\/p>\n

\\begin{eqnarray}
\nds^2
\n&=&-dt^2 + t^2 \\left(d\\chi^2\u00a0 +\\sinh^2 \\chi (d\\theta^2 + \\sin^2\\theta d\\phi^2) \\right)
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u304c\uff0c\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u7c21\u5358\u306a\u5ea7\u6a19\u5909\u63db\u3067\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf\u306b\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\nr &\\equiv& t \\sinh \\chi \\\\
\n\\tau &\\equiv& t \\cosh \\chi
\n\\end{eqnarray}<\/p>\n

\u5fae\u5206\u3092\u3068\u308b\u3068<\/p>\n

\\begin{eqnarray}
\ndr &=& dt \\sinh \\chi\u00a0 + t \\cosh \\chi\\, d\\chi \\tag{1}\\\\
\nd\\tau &=& dt \\cosh \\chi + t \\sinh \\chi\\, d\\chi\\tag{2}
\n\\end{eqnarray}<\/p>\n

\u3053\u306e\u307e\u307e \\(-d\\tau^2 + dr^2\\) \u3092\u8a08\u7b97\u3057\u3066\u3082\u3044\u3044\u3067\u3059\u3057\uff0c\u3042\u3048\u3066 \\( (2) \\times \\cosh \\chi -(1) \\times \\sinh \\chi\\) \u304b\u3089<\/p>\n

$$d\\tau \\cosh\\chi -dr \\sinh \\chi = dt$$<\/p>\n

\u3068 \\( (1) \\times \\cosh \\chi -(2) \\times \\sinh \\chi\\) \u304b\u3089<\/p>\n

$$dr \\cosh \\chi -d\\tau \\sinh \\chi = t d\\chi$$<\/p>\n

\u3092\u8a08\u7b97\u3057\u3066\uff0c<\/p>\n

\\begin{eqnarray}
\nds^2
\n&=&-dt^2 + t^2 d\\chi^2\u00a0 +t^2 \\sinh^2 \\chi (d\\theta^2 + \\sin^2\\theta d\\phi^2) \\\\
\n&=& -\\left(d\\tau \\cosh\\chi -dr \\sinh \\chi\\right)^2 + \\left( dr \\cosh \\chi -d\\tau \\sinh \\chi\\right)^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2) \\\\
\n&=& -d\\tau^2 + dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\uff0c\u3042\u3042\uff01\u30df\u30f3\u30b3\u30d5\u30b9\u30ad\u30fc\u8a08\u91cf\u306b\u306a\u308b\u306e\u3060\u306a\u3041\uff0c\u3068\u601d\u3063\u3066\u3082\u3088\u3044\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":1483,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-1634","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\/1634","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=1634"}],"version-history":[{"count":65,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1634\/revisions"}],"predecessor-version":[{"id":9259,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/1634\/revisions\/9259"}],"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=1634"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}