{"id":1551,"date":"2022-01-28T18:25:53","date_gmt":"2022-01-28T09:25:53","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=1551"},"modified":"2024-07-29T16:55:14","modified_gmt":"2024-07-29T07:55:14","slug":"%e8%a7%92%e5%be%84%e8%b7%9d%e9%9b%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\/%e8%a7%92%e5%be%84%e8%b7%9d%e9%9b%a2\/","title":{"rendered":"\u89d2\u5f84\u8ddd\u96e2"},"content":{"rendered":"

\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u30cc\u30eb\u6e2c\u5730\u7dda\u3067\u3042\u308b\u5149\u7dda\u306e\u675f\uff0c\u3064\u307e\u308a\u5149\u7dda\u675f<\/strong><\/span>\uff08ray bundle\uff09\u306e\u4f1d\u64ad\u304b\u3089\u5b9a\u7fa9\u3055\u308c\u308b\u89d2\u5f84\u8ddd\u96e2<\/strong><\/span> angular diameter distance \u306b\u3064\u3044\u3066\u3002<\/p>\n

FRLW \u6642\u7a7a\u306b\u304a\u3051\u308b\u5149\u306e\u4f1d\u64ad\u306e\u304a\u3055\u3089\u3044<\/h3>\n

\u3044\u304f\u3064\u304b\u306e\u30da\u30fc\u30b8\u306b\u6563\u4e71\u3057\u3066\u3044\u308b\u7d50\u679c\u3092\uff0c\u3053\u3053\u306b\u304a\u3055\u3089\u3044\u3057\u3066\u304a\u304f\u3002<\/p>\n

\\begin{eqnarray}
\nds^2
\n&=& a^2(\\eta) \\left\\{-d\\eta^2 + d\\chi^2 + \\sigma^2(\\chi)\\left(d\\theta^2 + \\sin^2\\theta d\\phi^2 \\right) \\right\\}\\\\
\n\\sigma(\\chi) &\\equiv& \\frac{\\sin(\\sqrt{k} \\chi)}{\\sqrt{k}}\\\\
\nk&=& H_0^2 a_0^2 \\left(\\Omega_{\\rm m} + \\Omega_{\\Lambda} -1 \\right)\\\\
\n1 + z &=& \\frac{a_0}{a(\\eta)}
\n\\end{eqnarray}<\/p>\n

\u52d5\u5f84\u65b9\u5411\u306b\u4f1d\u64ad\u3059\u308b\u5149\u306e\u7d4c\u8def\u3092\u8d64\u65b9\u504f\u79fb $z$ \u3067\u8868\u3059\u5f0f\u306f\uff0c<\/p>\n

$$ \\chi = \\eta_0 -\\eta = \\frac{1}{H_0 a_0} \\int_0^z \\frac{dz}{\\sqrt{\\Omega_{\\Lambda} +\\left(1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}\\right)(1+z)^2 + \\Omega_{\\rm m} (1+z)^3}}$$<\/p>\n

\u89d2\u5f84\u8ddd\u96e2\u306e\u5b9a\u7fa9<\/h3>\n

\u52d5\u5f84\u65b9\u5411\u306b\u5782\u76f4\u306a2\u6b21\u5143\u9762\u306e\u6642\u523b $\\eta\\ (<\\eta_0)$ \u306b\u304a\u3051\u308b\u9762\u7a4d $dS$ \u3092\uff0c\u73fe\u5728 $\\eta = \\eta_0$ \u306b\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u898b\u8fbc\u3080\u7acb\u4f53\u89d2\u3092 $d\\Omega$ \u3068\u3059\u308b\u3068<\/p>\n

$$dS \\equiv d^2_A \\,d\\Omega$$<\/p>\n

\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u8ddd\u96e2 $d_A$ \u304c\u89d2\u5f84\u8ddd\u96e2<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u8ddd\u96e2\u3067\u3042\u308b\u3002<\/p>\n

\"\"<\/p>\n

\uff08\u89d2\u5f84\u8ddd\u96e2<\/strong><\/span>\u306e\u5b9a\u7fa9\u3068\u3057\u3066\uff0c1\u6b21\u5143\u7684\u9577\u3055 $\\ell$ \u3092\u898b\u8fbc\u3080\u89d2\u5ea6\u3092 $\\delta\\theta$ \u3068\u3057\u305f\u3068\u304d\u306b\uff0c$\\ell \\equiv d_A \\delta\\theta$ \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u8ddd\u96e2 $d_A$ \u3068\u3044\u3046\u306e\u3092\u63a1\u7528\u3057\u3066\u3044\u308b\u30c6\u30ad\u30b9\u30c8\u3082\u6563\u898b\u3055\u308c\u308b\u3002\u4e00\u69d8\u7b49\u65b9\u5b87\u5b99\u30e2\u30c7\u30eb\u3067\u306f\uff0c\u3053\u306e2\u3064\u306e\u5b9a\u7fa9\u306f\u540c\u7b49\u3067\u3042\u308b\u3002FLRW \u6642\u7a7a\u3057\u304b\u8003\u3048\u306a\u3044\u306e\u3067\u3042\u308c\u3070\u7c21\u5358\u306a\u65b9\u3092\u4f7f\u3048\u3070\u3088\u3044\u304c\uff0c\u3053\u3053\u3067\u306f\u3088\u308a\u4e00\u822c\u7684\u3068\u79c1\u304c\u8003\u3048\u308b\u307b\u3046\u306e\uff0c\u9762\u7a4d\u3092\u898b\u8fbc\u3080\u7acb\u4f53\u89d2\u304b\u3089\u306e\u5b9a\u7fa9\u3092\u63a1\u7528\u3059\u308b\u3053\u3068\u306b\u3059\u308b\u3002\u5f8c\u3067\u5b9a\u7fa9\u3059\u308b\u5149\u5ea6\u8ddd\u96e2 $d_L$ \u306b\u5bfe\u3057\u3066\u4efb\u610f\u306e\u6642\u7a7a\u3067 reciprocity theorem<\/strong><\/span> $d_L = (1+z)^2 d_A$ \u304c\u6210\u308a\u7acb\u3064\u306e\u3082\uff0c\u9762\u7a4d\u3092\u898b\u8fbc\u3080\u7acb\u4f53\u89d2\u304b\u3089\u5b9a\u7fa9\u3057\u305f $d_A$ \u3067\u3042\u308b\u3002\u306a\u306e\u3067\uff0c\u89d2\u5f84\u8ddd\u96e2 angular diameter distance \u3068\u533a\u5225\u3057\u3066\u9762\u7a4d\u8ddd\u96e2 area distance \u3068\u3067\u3082\u547c\u3093\u3060\u307b\u3046\u304c\u3044\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u3002\uff09<\/p>\n

\u89d2\u5f84\u8ddd\u96e2\u306e\u89b3\u6e2c<\/h4>\n

\u5b9f\u969b\u306b\u89b3\u6e2c\u3059\u308b\u91cf\u306f\uff0c\u5929\u4f53\u306e\u898b\u304b\u3051\u306e\u5927\u304d\u3055\u3092\u8868\u3059\u7acb\u4f53\u89d2\uff08\u3042\u308b\u3044\u306f1\u6b21\u5143\u7684\u9577\u3055\u3092\u898b\u8fbc\u3080\u89d2\uff09\u3067\u3042\u308b\u304b\u3089\uff0c\u9060\u65b9\u306e\u5929\u4f53\u306e\u5b9f\u969b\u306e\u5927\u304d\u3055\uff08standard scale, \u6a19\u6e96\u3082\u306e\u3055\u3057\uff09\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u3053\u3068\uff0c\u307e\u305f\u89b3\u6e2c\u88c5\u7f6e\u306e\u5206\u89e3\u80fd\u3092\u3082\u3063\u3066\u3057\u3066\uff0c\u305d\u306e\u5927\u304d\u3055\u30fb\u5f62\u304c\u89b3\u6e2c\u3067\u308f\u304b\u308b\u7a0b\u5ea6\u306e\u5927\u304d\u3055\u3067\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308b\uff08\u5b9f\u969b\u306e\u5927\u304d\u3055\u304c\u304c\u5c0f\u3055\u3044\u3068\uff0c\u70b9\u3068\u3057\u304b\u89b3\u6e2c\u3055\u308c\u306a\u3044\uff0c\u3053\u308c\u3067\u306f\u30c6\u30f3\u3067\u30c0\u30e1\uff09\u3002<\/p>\n

\u89d2\u5f84\u8ddd\u96e2\u3092\u8d64\u65b9\u504f\u79fb\u3068\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u3059<\/h3>\n

\u4e0a\u3067\u518d\u63b2\u3057\u305f FLRW \u8a08\u91cf\u3067\u306f\uff0c\u52d5\u5f84\u65b9\u5411\u3064\u307e\u308a $\\chi$ \u65b9\u5411\u306b\u5782\u76f4\u306a2\u6b21\u5143\u9762\u306e\u9762\u7a4d\u8981\u7d20 $dS$ \u306f<\/p>\n

\\begin{eqnarray}
\ndS &\\equiv& \\sqrt{{}^{(2)}\\! g}\\, d\\theta d\\phi\\\\
\n&=& \\sqrt{\\det \\begin{pmatrix}
\ng_{\\theta\\theta} & 0\\\\
\n0 & g_{\\phi\\phi}
\n\\end{pmatrix}}\\, d\\theta d\\phi\\\\
\n&=& a^2(\\eta) \\sigma^2(\\chi) \\sin\\theta\\, d\\theta d\\phi\\\\
\n&=& a^2(\\eta) \\sigma^2(\\chi)\\,d\\Omega \\\\ \\ \\\\
\n\\therefore\\ \\ d_A &\\equiv& a(\\eta) \\sigma(\\chi)
\n\\end{eqnarray}<\/p>\n

\u4ee5\u4e0a\u304c FLRW \u306e\u5834\u5408\u306e $d_A$ \u3092\u3082\u3063\u3068\u3082\u624b\u3063\u53d6\u308a\u65e9\u304f\u6c42\u3081\u308b\u65b9\u6cd5\u3002\u7b54\u3048\u304c\u308f\u304b\u308c\u3070\u3088\u3044\u306e\u3067\u3042\u308c\u3070\uff0c\u3053\u308c\u3067\u3088\u3057\u3002\u4ee5\u4e0b\u306f\uff0c\u3088\u308a\u6301\u3063\u3066\u56de\u3063\u305f\u65b9\u6cd5\uff08\u3088\u308a\u4e00\u822c\u7684\u306a\u6642\u7a7a\u3067\u3082\u6210\u308a\u7acb\u3064\u65b9\u6cd5\uff09\u3067\u6c42\u3081\u308b\u3084\u308a\u304b\u305f\u3002<\/p>\n

\u4e0a\u3067\u6c42\u3081\u305f\u7d50\u679c\u3067\u3042\u308b $d_A = a(\\eta) \\sigma(\\chi)$ \u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a $d_A$ \u306b\u5bfe\u3059\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u304b\u3089\u3082\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\uff08FLRW \u4ee5\u5916\u3067\u306f\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068 expansion $\\theta$ \u306e\u30c8\u30e9\u30f3\u30b9\u30dd\u30fc\u30c8\u65b9\u7a0b\u5f0f\u304b\u3089\u6c42\u3081\u308b\u3053\u3068\u306b\u306a\u308b\u3002\uff09<\/p>\n

\u307e\u305a\uff0c\u5225\u30da\u30fc\u30b8<\/a>\u3067\u5149\u7dda\u675f\u306e\u5fae\u5c0f\u9762\u7a4d $dS$ \u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\u3002<\/p>\n

$$\\frac{d}{dv} dS= k^{\\mu}_{\\ \\ ;\\mu} S = 2 \\theta \\, dS$$<\/p>\n

\u3053\u308c\u3068\uff0c$d_A$ \u306e\u5b9a\u7fa9\u5f0f $dS \\equiv d^2_A \\,d\\Omega$ \u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{dv} dS &=& \\frac{d}{dv} \\left(d^2_A \\right) \\,d\\Omega \\\\
\n&=& 2 d_A \\frac{d}{dv}\\left( d_A \\right) \\,d\\Omega \\\\
\n&=& 2 \\theta \\, d^2_A \\,d\\Omega \\\\ \\ \\\\
\n\\therefore\\ \\ \\frac{d}{dv}\\left( d_A \\right) &=& \\theta\\, d_A
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u304c $d_A$ \u306b\u5bfe\u3059\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u3002FLRW \u306e\u5834\u5408\u306b\u306f expansion $\\theta$ \u306f\u5225\u30da\u30fc\u30b8<\/a>\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6c42\u3081\u308c\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n

$$\\theta = \\frac{1}{2} k^{\\mu}_{\\ \\ ;\\mu} = \\frac{1}{a\\sigma} \\frac{d}{dv} \\left( a \\sigma\\right)$$<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0cFLRW \u3067\u306f $d_A$ \u306f<\/p>\n

$$\\frac{1}{d_A} \\frac{d}{dv}\\left( d_A \\right) = \\frac{1}{a\\sigma} \\frac{d}{dv} \\left( a \\sigma\\right)$$<\/p>\n

\u3088\u308a\uff0c\u7a4d\u5206\u5b9a\u6570 $K$ \u3092\u4f7f\u3063\u3066<\/p>\n

$$d_A(v) = K \\, a(\\eta(v))\\,\\sigma(\\chi(v))$$<\/p>\n

\u3068\u306a\u308b\u3002\u7a4d\u5206\u5b9a\u6570 $K$ \u306f\uff0c\u5f8c\u306e\u7d50\u679c\u3092\u4f7f\u3063\u3066<\/p>\n

$$ d_A \\simeq \\frac{1}{H_0} z\\quad\\mbox{for}\\ \\ |z| \\ll 1$$<\/p>\n

\u3068\u306a\u308b\u3088\u3046\u306b $K=1$ \u3068\u9078\u3093\u3067\u304a\u3053\u3046\u3002\u30a2\u30d5\u30a3\u30f3\u30d1\u30e9\u30e1\u30fc\u30bf $v$\u00a0 \u3068\u5ea7\u6a19 $\\eta, \\, \\chi$ \u306e\u95a2\u4fc2\u306f\uff0c
\n$v = 0$ \u304c\u73fe\u5728\uff08$z = 0$\uff09\u3067 $\\eta(0) = \\eta_0, \\\u00a0 \\chi(0) = 0$\uff0c
\n$v = v$ \u304c\u904e\u53bb\uff08\u8d64\u65b9\u504f\u79fb $z$\uff09\u3067 $\\eta(v) = \\eta, \\ \\chi(v) = \\chi$
\n\u3068\u306a\u308b\u3088\u3046\u306b\u3068\u3063\u3066\u3044\u308b\u306e\u3067\uff0c<\/p>\n

$$d_A = a(\\eta) \\sigma(\\chi)$$<\/p>\n

\u3068\u306a\u308b\u3002<\/p>\n

\u6b21\u306b\uff0c$k > 0$ \u306e\u5834\u5408\u306b $d_A$ \u3092\u8d64\u65b9\u504f\u79fb $z$\u00a0 \u3068\u5b87\u5b99\u8ad6\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u8868\u3059\u3002\u307e\u305a\uff0c<\/p>\n

\\begin{eqnarray}
\nd_A &=& a(\\eta) \\sigma(\\chi) \\\\
\n&=& \\frac{ a(\\eta)}{a_0} \\frac{a_0}{\\sqrt{k}} \\sin\\left(\\sqrt{k} \\chi\\right)\\\\
\n&=& \\frac{1}{H_0 (1 + z) \\sqrt{\\Omega_{\\rm m} + \\Omega_{\\Lambda} -1}}\\sin\\left(\\sqrt{k} \\chi\\right)
\n\\end{eqnarray}<\/p>\n

$\\sin\\left(\\sqrt{k} \\chi\\right)$ \u306e\u4e2d\u8eab\u306e $\\sqrt{k} \\chi$ \u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\sqrt{k} \\chi &=& \\sqrt{\\Omega_{\\rm m} + \\Omega_{\\Lambda} -1} \\int_0^z \\frac{dz}{\\sqrt{\\Omega_{\\Lambda} +\\left(1 -\\Omega_{\\rm m} -\\Omega_{\\Lambda}\\right)(1+z)^2 + \\Omega_{\\rm m} (1+z)^3}}
\n\\end{eqnarray}<\/p>\n

$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306e\u89d2\u5f84\u8ddd\u96e2<\/h4>\n

$\\Omega_{\\Lambda} = 0$ \u306e\u5834\u5408\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\sqrt{k} \\chi &=& \\sqrt{\\Omega_{\\rm m}\u00a0 -1}
\n\\int_0^z \\frac{dz}{\\sqrt{ \\Omega_{\\rm m} (1+z)^3-\\left(\\Omega_{\\rm m} -1\\right)(1+z)^2 }}\\\\
\n&=& \\int_0^z \\frac{dz}{(1+z) \\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}(1+z)-1}}
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u6570 \\(y\\) \u3092\u5b9a\u7fa9\u3059\u308b\u3068\u7f6e\u63db\u7a4d\u5206\u3067\u304d\u308b\u3002<\/p>\n

\\begin{eqnarray}
\ny &\\equiv& \\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}(1+z)-1}
\n= \\sqrt{\\frac{1+\\Omega_{\\rm m}z}{\\Omega_{\\rm m}-1}}\\\\
\n1+y^2 &=& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1} (1+z) \\\\
\n\\therefore\\ \\ (1+z) &=& \\frac{\\Omega_{\\rm m}-1}{\\Omega_{\\rm m}} (1+y^2) \\\\
\ndz &=& \\frac{\\Omega_{\\rm m}-1}{\\Omega_{\\rm m}} \\cdot 2 y dy
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\sqrt{k} \\chi
\n&=& \\int_0^z \\frac{dz}{(1+z) \\sqrt{\\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm m}-1}(1+z)-1}} \\\\
\n&=& \\int_{y_0}^{y_z} \\frac{2 y dy}{(1+y^2) y} \\\\
\n&=& \\int_{y_0}^{y_z} \\frac{2 dy}{(1+y^2)} \\\\
\n&=& \\Bigl[ 2 \\tan^{-1} y\\Bigr]_{y_0}^{y_z} \\\\
\n&=& 2 \\tan^{-1} (y_z)
\n-2 \\tan^{-1} (y_0) \\\\
\n&=& 2 \\tan^{-1} \\left(\\frac{\\sqrt{1 + \\Omega_{\\rm m} z}}{\\sqrt{\\Omega_{\\rm m} -1}} \\right)
\n-2 \\tan^{-1} \\left(\\frac{1}{\\sqrt{\\Omega_{\\rm m} -1}} \\right)
\n\\end{eqnarray}<\/p>\n

\u4e2d\u8eab\u306e $\\sqrt{k} \\chi$ \u304c\u308f\u304b\u308c\u3070 $\\sin\\left(\\sqrt{k} \\chi\\right)$ \u306f<\/p>\n

\\begin{eqnarray}
\n\\sin\\left(\\sqrt{k} \\chi\\right)&=& \\sin\\bigl(2 \\tan^{-1} (y_z)
\n-2 \\tan^{-1} (y_0) \\bigr) \\\\
\n&=& \\sin\\left( 2 \\tan^{-1} (y_z)\\right) \\cos\\left(2 \\tan^{-1} (y_0) \\right) \\\\
\n&&-\\cos\\left( 2 \\tan^{-1} (y_z)\\right) \\sin\\left(2 \\tan^{-1} (y_0) \\right)
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u516c\u5f0f\uff081\u5e74\u751f\u306e\u3068\u304d\u306b\u6559\u3048\u305f\u3053\u3068\u306b\u306a\u3063\u3066\u3044\u308b<\/a>\uff09<\/p>\n

\\begin{eqnarray}
\n2 \\tan^{-1} y &=& \\cos^{-1} \\left(\\frac{1 -y^2}{1 + y^2} \\right)\\\\
\n&=& \\sin^{-1} \\left(\\frac{ 2y}{1 + y^2} \\right)
\n\\end{eqnarray}<\/p>\n

\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\sin\\left(\\sqrt{k} \\chi\\right)&=& \\frac{2\\sqrt{\\Omega_{\\rm m}-1}\\sqrt{1 + \\Omega_{\\rm m} z}}{\\Omega_{\\rm m} (1+z)}\u00a0 \\frac{\\Omega_{\\rm m}-2}{\\Omega_{\\rm m}} \\\\
\n&&-\\frac{\\Omega_{\\rm m}-2 -\\Omega_{\\rm m} z}{\\Omega_{\\rm m} (1+z)}\u00a0\u00a0 \\frac{2\\sqrt{\\Omega_{\\rm m} -1}}{\\Omega_{\\rm m}}\\\\
\n&=& \\frac{2\\sqrt{\\Omega_{\\rm m} -1}}{\\Omega_{\\rm m}^2 (1+z)}
\n\\left\\{2 -\\Omega_{\\rm m} + \\Omega_{\\rm m} z -(2-\\Omega_{\\rm m}) \\sqrt{1 + \\Omega_{\\rm m} z}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u6700\u7d42\u7684\u306b\u89d2\u5f84\u8ddd\u96e2 $d_A$ \u306f<\/p>\n

\\begin{eqnarray}
\nd_A &=& \\frac{1}{H_0 (1 + z) \\sqrt{\\Omega_{\\rm m}\u00a0 -1}}\\sin\\left(\\sqrt{k} \\chi\\right) \\\\
\n&=& \\frac{2}{H_0 \\Omega_{\\rm m}^2 (1+z)^2} \\left\\{2 -\\Omega_{\\rm m} + \\Omega_{\\rm m} z -(2-\\Omega_{\\rm m}) \\sqrt{1 + \\Omega_{\\rm m} z}\\right\\}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u3002$\\Omega_{\\Lambda}=0, \\Omega_{\\rm m} > 1$ \u3059\u306a\u308f\u3061 $k > 0$ \u306e\u5834\u5408\u306b\u5c0e\u3044\u305f\u3053\u306e\u5f0f\u306f\uff0c$\\Omega_{\\Lambda}=0$ \u3067\u3042\u308c\u3070\uff0c$0 < \\Omega_{\\rm m} \\leq 1$ \u3059\u306a\u308f\u3061 $k \\leq 0$ \u306e\u5834\u5408\u3067\u3082\u305d\u306e\u307e\u307e\u6210\u308a\u7acb\u3064\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u3053\u306e\u3053\u3068\u306f \\(d_A\\) \u306e\u5b9a\u7fa9\u306b\u3042\u308b<\/p>\n

$$\\sigma(\\chi) = \\frac{\\sin\\left(\\sqrt{k} \\chi\\right)}{\\sqrt{k}}$$<\/p>\n

\u304c $k \\leq 0$ \u306e\u3068\u304d\u306b\u3082\u305d\u306e\u307e\u307e\u4f7f\u3048\u308b<\/a>\u3053\u3068\u304b\u3089\u304d\u3066\u3044\u308b\u3002<\/p>\n

$z \\ll 1$ \u306e\u5834\u5408\u306e\u8fd1\u4f3c\u5f0f<\/h5>\n

$z \\ll 1$ \u306e\u5834\u5408\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3067\u304d\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nd_A
\n&=& \\frac{2}{H_0 \\Omega_{\\rm m}^2 (1+z)^2} \\left\\{2 -\\Omega_{\\rm m} + \\Omega_{\\rm m} z -(2-\\Omega_{\\rm m}) \\sqrt{1 + \\Omega_{\\rm m} z}\\right\\} \\\\
\n&\\simeq& \\frac{2}{H_0 \\Omega_{\\rm m}^2} \\left\\{2 -\\Omega_{\\rm m} + \\Omega_{\\rm m} z -(2-\\Omega_{\\rm m}) \\left(1 + \\frac{1}{2}\\Omega_{\\rm m} z\\right)\\right\\} \\\\
\n&=& \\frac{2}{H_0 \\Omega_{\\rm m}^2} \\left\\{ \\frac{1}{2}\\Omega_{\\rm m}^2 z\\right\\}\\\\
\n&=& \\frac{z}{H_0}
\n\\end{eqnarray}<\/p>\n

\u305d\u3046\u3044\u3048\u3070\uff0c\u3082\u3068\u3082\u3068\u3053\u3046\u306a\u308b\u3088\u3046\u306b\u7a4d\u5206\u5b9a\u6570 $K$ \u3092\u6c7a\u3081\u3066\u3044\u305f\u306e\u3067\u3042\u3063\u305f\u3002\u5149\u901f $c$ \u3092\u3042\u304b\u3089\u3055\u307e\u306b\u66f8\u304f\u3068<\/p>\n

$$d_A = \\frac{c z}{H_0}, \\quad\\therefore\\ \\ cz = H_0\\, d_A$$<\/p>\n

\u3068\u306a\u308a\uff0c\u30cf\u30c3\u30d6\u30eb\u30fb\u30eb\u30e1\u30fc\u30c8\u30eb\u306e\u6cd5\u5247\u3068\u898b\u6bd4\u3079\u308c\u3070\uff0c\u5f8c\u9000\u901f\u5ea6 $v$ \u3068\u8d64\u65b9\u504f\u79fb $z$ \u306e\u95a2\u4fc2\u306f<\/p>\n

$$ v = cz$$<\/p>\n

$\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1$ \u306e\u5834\u5408\u306e\u89d2\u5f84\u8ddd\u96e2<\/h4>\n

$\\Omega_{\\rm m} + \\Omega_{\\Lambda} = 1$ \u3064\u307e\u308a $k = 0$ \u306e\u5834\u5408\u306e\u89d2\u5f84\u8ddd\u96e2\u306f $\\sigma(\\chi) = \\chi$ \u3067\u3042\u308b\u304b\u3089
\n\\begin{eqnarray}
\nd_A &=&a(\\eta) \\sigma(\\chi) \\\\\u00a0 &=& a(\\eta) \\chi \\\\
\n&=& \\frac{1}{1+z} a_0 \\chi \\\\
\n&=& \\frac{1}{H_0 (1+z)} \\int_0^z \\frac{dz}{\\sqrt{(1-\\Omega_{\\rm m}) + \\Omega_{\\rm m} (1+z)^3} }
\n\\end{eqnarray}<\/p>\n

$z \\ll 1$ \u306e\u5834\u5408\u306e\u8fd1\u4f3c\u5f0f<\/h5>\n

$z \\ll 1$ \u306e\u5834\u5408\u306b\u306f\u3084\u306f\u308a<\/p>\n

$$d_A \\simeq \\frac{c z}{H_0}$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u3092\u7c21\u5358\u306b\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u30d2\u30f3\u30c8\uff1a<\/p>\n

$$\\int_0^z f(x) \\,dx \\simeq f(0) \\,z \\qquad \\mbox{for}\\ \\,\u00a0 z\\ll 1$$<\/p>\n

\u89d2\u5f84\u8ddd\u96e2\u306b\u95a2\u3059\u308b\u53c2\u8003\u6587\u732e<\/h3>\n

20\u4e16\u7d00\u306e\u6614\u306e\u82e5\u304b\u308a\u3057\u9803\u306b\u3084\u3063\u305f Fukugita, Futamase, Kasai, and Turner (1992) \u306e\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3042\u305f\u308a\u306b\u3082\u3046\u5c11\u3057\u4e00\u822c\u5316\u3057\u305f\u89d2\u5f84\u8ddd\u96e2\u306e\u5f0f\u304c\u8f09\u3063\u3066\u3044\u308b\u3002\u61d0\u304b\u3057\u304f\u306a\u3063\u305f\u306e\u3067\u3002<\/p>\n