FLRW \u6642\u7a7a\u306b\u304a\u3051\u308b\u5149\u306e\u4f1d\u64ad\u3092\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f\u304b\u3089\u89e3\u304f\u3002<\/p>\n
FLRW \u6642\u7a7a<\/strong><\/span>\u3067\u3042\u3063\u3066\u3082\uff0c\u5149\u304c\u4f1d\u64ad\u3059\u308b\u7d4c\u8def\u306f\u30cc\u30eb\u6e2c\u5730\u7dda<\/strong><\/span>\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u5225\u30da\u30fc\u30b8<\/a>\u3067\u8ff0\u3079\u305f\u3088\u3046\u306b\uff0c<\/p>\n <\/p>\n \u4e00\u69d8\u7b49\u65b9\u306a\u81a8\u5f35\u5b87\u5b99\u30e2\u30c7\u30eb\u3092\u8868\u3059 FLRW \u6642\u7a7a\u306e\u8a08\u91cf\u306f\u4ee5\u4e0b\u306e\u8868\u793a\u3092\u63a1\u7528\u3059\u308b\u3002<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067\uff0c$a(\\eta) d\\eta \\equiv dt$ \u3067\u5b9a\u7fa9\u3055\u308c\u308b $\\eta$ \u306f\u5171\u5f62\u6642\u9593<\/strong><\/span> (conformal time<\/strong><\/span>) \u3068\u547c\u3070\u308c\u308b\u6642\u9593\u5ea7\u6a19\u3067\u3042\u308b\u3002$\\displaystyle \\frac{d a}{dt} = \\dot{a}$\u00a0 \u3068\u8868\u8a18\u3057\u3066\u3044\u308b\u306e\u3067\uff0c$t$ \u3067\u306e\u5fae\u5206\u3068\u533a\u5225\u3057\u3066\uff0c$\\eta$ \u3067\u306e\u5fae\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u8a18\u3059\u308b\u3002<\/p>\n $$\\frac{d a}{d\\eta} = a’$$<\/p>\n \u307e\u305f\uff0c$\\sigma(\\chi)$ \u306f\u300c\u5b9a\u66f2\u7387\u7a7a\u9593\u8a08\u91cf\u306e\u3044\u304f\u3064\u304b\u306e\u8868\u793a\u4f8b<\/a>\u300d\u306b\u66f8\u3044\u305f\u3088\u3046\u306b\uff0c<\/p>\n $$\\sigma(\\chi) = \\frac{\\sin \\left( \\sqrt{k} \\chi\\right)}{\\sqrt{k}}$$<\/p>\n \u4e0a\u8a18\u306e\u3088\u3046\u306b\u7dda\u7d20\u3092\u66f8\u304f\u3053\u3068\u306b\u3088\u3063\u3066\uff0c\u5ea7\u6a19\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3068\u3063\u3066\u3044\u308b\u3053\u3068 \\begin{eqnarray} \u3067\u3042\u308a\uff0c\u305d\u308c\u4ee5\u5916\u306e\u6210\u5206\u306f\u5168\u3066\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u4e00\u6319\u306b\u8868\u3057\u3066\u3044\u308b\u3002<\/p>\n $$\\frac{dk_{\\nu}}{dv} = \\frac{1}{2} g_{\\lambda\\mu, \\nu} k^{\\lambda} k^{\\mu}$$<\/p>\n \u3053\u306e\u5f0f\u304b\u3089\uff0c\u4e00\u822c\u306b\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u6210\u5206 $g_{\\lambda\\mu}$ \u304c\u5ea7\u6a19 $x^{\\nu}$ \u4f9d\u5b58\u6027\u3092\u3082\u305f\u306a\u3044\u5834\u5408\u306f\uff0c$$ g_{\\lambda\\mu, \\nu} = 0 \\quad\\Rightarrow\\quad \\frac{d k_{\\nu}}{dv} = 0 \\quad\\Rightarrow\\quad\u00a0 k_{\\nu} = \\mbox{const.} $$\u3068\u306a\u308a \\(k_{\\nu} \\) \u6210\u5206\u304c\u4fdd\u5b58\u91cf\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n FLRW \u8a08\u91cf\u306e\u6210\u5206\u306f\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 $a(\\eta)$ \u3092\u901a\u3057\u3066 $x^0 = \\eta$ \u306b\u4f9d\u5b58\u3057\u3066\u3044\u308b\u304c\uff0c\u30cc\u30eb\u6761\u4ef6\u3088\u308a<\/p>\n \\begin{eqnarray} $$\\therefore\\ \\ k^0 = \\frac{k_0}{g_{00}} = \\frac{\\omega_c}{a^2}$$<\/p>\n \u307e\u305f\uff0cFLRW \u8a08\u91cf\u306e\u6210\u5206\u306f $x^3 = \\phi$ \u3092\u542b\u307e\u306a\u3044\u306e\u3067<\/p>\n \\begin{eqnarray} \u7a7a\u9593\u304c\u4e00\u69d8\u7b49\u65b9\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u4eca\u5f8c\u306f\u52d5\u5f84\u65b9\u5411\u306e\u5149\u306e\u4f1d\u64ad\u3092\u8003\u3048\u308c\u3070\u5341\u5206<\/strong><\/span>\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066 $\\displaystyle k^2 = \\frac{d\\theta}{dv}$ \u306b\u3064\u3044\u3066\u306f<\/p>\n $$g_{22} \\frac{d k^2}{dv} + \\frac{d g_{22}}{dv} k^2 = a^2 \\sigma^2 \\sin\\theta\\, \\cos\\theta \\left(k^3\\right)^2$$<\/p>\n \u3067\u3042\u308b\u304b\u3089\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 $v = 0$ \u3067 $\\displaystyle \\theta = \\frac{\\pi}{2}, k^2 = 0$ \u3068\u3059\u308b\u3068 $\\displaystyle \\frac{dk^2}{dv} = 0$ \u3067\u3042\u308a\uff0c\u305d\u306e\u5f8c\u3082\u5e38\u306b $k^2 = 0$ \u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\uff0c\u3053\u306e\u521d\u671f\u6761\u4ef6\u3092\u63a1\u7528\u3057\uff0c \u3053\u306e\u3053\u3068\u306f<\/p>\n \u7b49\u65b9\u6027\u306b\u3088\u308a\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f<\/strong><\/span> \u3053\u3068\u3092\u521d\u671f\u5024\u554f\u984c\u3068\u3057\u3066\u793a\u3057\u305f\u3082\u306e\u3067\u3042\u308b\u3002\u3067\u3082\uff0c\u7a7a\u9593\u304c\u4e00\u69d8\u7b49\u65b9\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u4eca\u5f8c\u306f\u52d5\u5f84\u65b9\u5411\u306e\u5149\u306e\u4f1d\u64ad\u3092\u8003\u3048\u308c\u3070\u5341\u5206<\/strong><\/span>\u3067\u3042\u308b\u304b\u3089\uff0c$k^2=0$ \u3068\u3059\u308b\uff0c\u3068\u3057\u305f\u307b\u3046\u304c\u7c21\u5358\u3002<\/p>\n \u3055\u3066\uff0c\u3053\u308c\u307e\u3067\u306e\u3068\u3053\u308d\uff0c\u308f\u304b\u3063\u305f\u306e\u306f<\/p>\n $$k^0 = \\frac{\\omega_c}{a^2}, \\quad k^2 \\Rightarrow 0, \\quad k^3 \\Rightarrow 0$$<\/p>\n \u6b8b\u308a\u306e $\\displaystyle k^1 = \\frac{d\\chi}{dv}$ \u306b\u3064\u3044\u3066\u306f\uff0c\u30cc\u30eb\u6761\u4ef6\u3088\u308a<\/p>\n \\begin{eqnarray} \u3053\u308c\u304c\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u52d5\u5f84\u65b9\u5411\u306e\u5149\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u3067\u3042\u308b\u3002<\/p>\n FLRW \u6642\u7a7a<\/strong><\/span>\u3092\u6e80\u305f\u3059\u5b8c\u5168\u6d41\u4f53\u3068\u5171\u306b\u904b\u52d5\u3059\u308b\u89b3\u6e2c\u8005\uff0c\u3064\u307e\u308a\u5171\u52d5\u89b3\u6e2c\u8005<\/strong><\/span>\u306e4\u5143\u901f\u5ea6<\/strong><\/span> \\(u^{\\mu}\\) \u306f $\\displaystyle u^0 = \\frac{1}{a}$ \u3068\u306a\u308b\u306e\u306f\u898f\u683c\u5316\u6761\u4ef6 $g_{\\mu\\nu} u^{\\mu} u^{\\nu} = g_{00} (u^0)^2 = -1$ \u3088\u308a\u3002<\/p>\n 4\u5143\u901f\u5ea6<\/strong><\/span> \\(\\boldsymbol{u}\\) \u306e\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570<\/strong><\/span> \\(\\omega\\) \u306f\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u3082\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n $$\\omega \\equiv\u00a0 – k_{\\mu} u^{\\mu}$$ FLRW \u6642\u7a7a<\/strong><\/span>\u4e2d\u306e\u5171\u52d5\u89b3\u6e2c\u8005<\/strong><\/span>\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570\u306f\uff0c<\/p>\n $$\\omega = -k_0 u^0 = \\frac{\\omega_c}{a(\\eta)} \\propto \\frac{1}{a(\\eta)}$$ \u7279\u306b\u6642\u523b \\(\\eta = \\eta_e\\) \u306b\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u632f\u52d5\u6570\u3092 \\(\\omega_e\\) \u3068\u6e2c\u5b9a\u3057\u305f\u5149\u3092\uff0c\u73fe\u5728\u6642\u523b \\(\\eta = \\eta_0 > \\eta_e\\)\u00a0 \u306b\u5225\u306e\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u6e2c\u5b9a\u3059\u308b\u3068\u304d\u306e\u632f\u52d5\u6570\u3092 \\(\\omega_0\\) \u3068\u3059\u308b\u3068\uff0c\u81a8\u5f35\u5b87\u5b99\u3067\u306f\uff0c\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 \\(a(\\eta)\\) \u306f\u6642\u9593\u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\uff08$a(\\eta_e) < a(\\eta_0) \\equiv a_0$\uff09\u3067\u3042\u308b\u305f\u3081\uff0c \u5149\u306e\u6ce2\u9577 \\(\\lambda\\) \u306f\u632f\u52d5\u6570\u306b\u53cd\u6bd4\u4f8b\u3059\u308b\u304b\u3089\uff0c\u5149\u304c\uff08\u904e\u53bb\u306b\uff09\u653e\u305f\u308c\u305f\u3068\u304d\u306e\u6ce2\u9577\u3092 $\\lambda_e$\uff0c\u305d\u308c\u3092\u73fe\u5728\uff08$\\eta = \\eta_0$\uff09\u89b3\u6e2c\u3057\u305f\u3068\u304d\u306e\u6ce2\u9577\u3092 $\\lambda_0$ \u3068\u3059\u308b\u3068\uff0c\u4efb\u610f\u306e\u6642\u7a7a\u306b\u304a\u3044\u3066<\/strong><\/span>\uff0c<\/p>\n \\begin{eqnarray} FLRW \u6642\u7a7a<\/strong><\/span>\u306b\u304a\u3044\u3066\u306f<\/p>\n \\begin{eqnarray} \u30b9\u30b1\u30fc\u30eb\u56e0\u5b50 \\(a(\\eta)\\) \u304c\u5c0f\u3055\u304b\u3063\u305f\u904e\u53bb\u306b\u653e\u51fa\u3055\u308c\u305f\u5149\u3092\u73fe\u5728\u89b3\u6e2c\u3059\u308b\u3068\uff0c\u6ce2\u9577\u304c\u4f38\u3073\u3066\uff08$z>0$\uff09 \u89b3\u6e2c\u3055\u308c\u308b\u3002\u53ef\u8996\u5149\u3067\u306f\u6ce2\u9577\u306e\u9577\u3044\u5149\u306f\u8d64\u304f\u898b\u3048\u308b\u306e\u3067\uff0c\u305f\u3068\u3048\u53ef\u8996\u5149\u3067\u306a\u304f\u3066\u3082\uff0c\u5b87\u5b99\u81a8\u5f35\u306b\u3088\u3063\u3066\u4e00\u822c\u306b\u5149\uff08\u96fb\u78c1\u6ce2\uff09\u306e\u6ce2\u9577\u304c\u4f38\u3073\u3066\u89b3\u6e2c\u3055\u308c\u308b\u3053\u3068\u3092\u5149\u306e\u8d64\u65b9\u504f\u79fb<\/strong><\/span>\u3068\u547c\u3093\u3067\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n <\/p>\n \u6642\u523b \\(\\eta\\) \u306b\u52d5\u5f84\u5ea7\u6a19 \\(\\chi\\) \u306e\u4f4d\u7f6e\u304b\u3089\u653e\u51fa\u3055\u308c\u305f\u5149\u3092\u73fe\u5728\u6642\u523b \\(\\eta_0\\) \u306b\u52d5\u5f84\u5ea7\u6a19 \\(\\chi = 0\\) \u306e\u4f4d\u7f6e\u3067\u89b3\u6e2c\u3057\u305f\u3068\u304d\u306e\u8d64\u65b9\u504f\u79fb \\(z\\) \u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u8868\u3055\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n $$1 + z = \\frac{a_0}{a(\\eta)}$$<\/p>\n \u307e\u305f\uff0c$\\eta \\rightarrow \\eta_0$ \u3059\u306a\u308f\u3061 $d\\eta > 0$ \u306e\u3068\u304d $\\chi \\rightarrow 0$ \u3059\u306a\u308f\u3061 $d\\chi < 0$ \u3067\u3042\u308b\u304b\u3089\uff0c\u81a8\u5f35\u5b87\u5b99\u306b\u304a\u3051\u308b\u52d5\u5f84\u65b9\u5411\u306e\u5149\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u306f\uff0c<\/p>\n $$\\frac{d\\chi}{d\\eta} = \\pm 1 \\Rightarrow -1$$<\/p>\n \u3067\u3042\u308b\u3002\u3053\u308c\u3089\u3092\u4f7f\u3046\u3068\uff0c$\\eta$ \u3084 $\\chi$ \u3068\u3044\u3046\u5ea7\u6a19\u5024\u3092\u8d64\u65b9\u504f\u79fb $z$ \u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n\n
\n$$k^{\\mu} = \\frac{dx^{\\mu}}{dv}$$<\/li>\n
\n$$\\frac{dk_{\\nu}}{dv} = \\frac{1}{2} g_{\\lambda\\mu, \\nu} k^{\\lambda} k^{\\mu}$$<\/li>\n<\/ul>\nFLRW \u6642\u7a7a<\/h3>\n
\nds^2 &=& -dt^2 + a^2(t) \\gamma_{ij} dx^i dx^j \\\\
\n&=& -a^2(\\eta) d\\eta^2 + a^2(\\eta) \\Bigl(d\\chi^2 + \\sigma^2(\\chi)\\left(d\\theta^2 + \\sin^2\\theta d\\phi^2 \\right) \\Bigr)
\n\\end{eqnarray}<\/p>\n
\n$$x^{\\nu} = (x^0, x^1, x^2, x^3) = (\\eta, \\chi, \\theta, \\phi)$$
\n\u304a\u3088\u3073\u8a08\u91cf\u30c6\u30f3\u30bd\u30eb\u306e\u30bc\u30ed\u3067\u306a\u3044\u6210\u5206\u306f<\/p>\n
\ng_{00} &=& -a^2(\\eta) \\\\
\ng_{11} &=& a^2(\\eta) \\\\
\ng_{22} &=& a^2(\\eta) \\sigma^2(\\chi)\\\\
\ng_{33} &=& a^2(\\eta) \\sigma^2(\\chi) \\sin^2\\theta
\n\\end{eqnarray}<\/p>\n\u6e2c\u5730\u7dda\u65b9\u7a0b\u5f0f<\/h3>\n
$k^0$ \u306e\u89e3<\/h4>\n
\n\\frac{d k_0}{dv} &=& \\frac{1}{2} g_{\\lambda\\mu, 0} k^{\\lambda} k^{\\mu}\\\\
\n&=& \\frac{a’}{a} g_{\\lambda\\mu} k^{\\lambda} k^{\\mu} = 0 \\\\
\n\\therefore\\ \\ k_0 &=& \\mbox{const.} \\equiv – \\omega_c
\n\\end{eqnarray}<\/p>\n$k^3$ \u306e\u89e3<\/h4>\n
\nk_3 &=& \\mbox{const.} \\equiv \\ell\\\\
\n\\therefore\\ \\ k^3 &=& \\frac{\\ell}{g_{33}} = \\frac{\\ell}{a^2 \\sigma^2 \\sin^2\\theta}
\n\\end{eqnarray}<\/p>\n
\n$$\\ell = 0\\quad\\Rightarrow\\quad k^3 = 0$$\u3068\u3059\u308b\u3002<\/p>\n$k^2$ \u306f\u521d\u671f\u6761\u4ef6\u304b\u3089<\/h4>\n
\n$$\\theta = \\frac{\\pi}{2}, \\quad k^2 = \\frac{d\\theta}{dv} = 0$$\u3068\u3059\u308b\u3002<\/p>\n
\n\u8d64\u9053\u9762\u4e0a\u306b\u8ecc\u9053\u3092\u5236\u9650\u3067\u304d\u308b<\/strong><\/span><\/p>\n$k^1$ \u306f\u30cc\u30eb\u6761\u4ef6\u304b\u3089<\/h4>\n
\ng_{\\mu\\nu} k^{\\mu} k^{\\nu} &=& -a^2 \\left(k^0 \\right)^2 + a^2 \\left(k^1 \\right)^2 \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ k^0 &=& \\pm k^1 \\\\
\n\\frac{d\\eta}{dv} &=& \\pm \\frac{d\\chi}{dv} \\\\
\n\\therefore\\ \\ \\frac{d\\chi}{d\\eta} &=& \\pm 1
\n\\end{eqnarray}<\/p>\nFLRW \u6642\u7a7a\u4e2d\u306e\u5171\u52d5\u89b3\u6e2c\u8005<\/h3>\n
\n$$ u^{\\mu} = \\frac{dx^{\\mu}}{d\\tau} = (u^0, 0, 0, 0) = \\left(\\frac{1}{a}, 0, 0, 0 \\right)$$<\/p>\n\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u5149\u306e\u632f\u52d5\u6570<\/h3>\n
\n4\u5143\u30d9\u30af\u30c8\u30eb\u540c\u58eb\u306e\u5185\u7a4d\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u632f\u52d5\u6570\u306f\uff0c\u5f53\u7136\u306a\u304c\u3089\u5ea7\u6a19\u306e\u53d6\u308a\u65b9\u306b\u3088\u3089\u306a\u3044\u4e0d\u5909\u30b9\u30ab\u30e9\u30fc\u91cf\u3067\u3042\u308b\u3002<\/p>\n
\n\u3068\u306a\u308a\uff0c\u540c\u3058\u5149\u6e90\u304b\u3089\u653e\u305f\u308c\u305f\u540c\u3058\u5149\u3092\u89b3\u6e2c\u3057\u3066\u3044\u3066\u3082\uff0c\u73fe\u5728\u6642\u523b\u3067\u5171\u52d5\u89b3\u6e2c\u8005\u304c\u89b3\u6e2c\u3059\u308b\u632f\u52d5\u6570 \\(\\omega\\) \u306f\u5149\u304c\u653e\u51fa\u3055\u308c\u305f\u6642\u523b\u306e\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306b\u53cd\u6bd4\u4f8b\u3059\u308b<\/strong><\/span>\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n\u5b87\u5b99\u81a8\u5f35\u306b\u3088\u308b\u8d64\u65b9\u504f\u79fb<\/h3>\n
\n$$\\frac{\\omega_0}{\\omega_e} = \\frac{a(\\eta_e)}{a_0} < 1$$
\n\u3068\u306a\u308b\u3002<\/p>\n
\nz &\\equiv&\u00a0 \\frac{\\lambda_0-\\lambda_e}{\\lambda_e} \\\\
\n&=& \\frac{\\omega_e}{\\omega_0} – 1 \\\\
\n\\therefore\\ \\ 1+ z &=& \\frac{\\omega_e}{\\omega_0}
\n\\end{eqnarray}
\n\u3068\u66f8\u3044\u3066\uff0c\u6ce2\u9577\u306e\u5909\u5316\u306e\u5272\u5408\u3092\u3042\u3089\u308f\u3059 $z$ \u3092\u8d64\u65b9\u504f\u79fb<\/strong><\/span>\u3068\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n
\n1+ z &=& \\frac{\\omega_e}{\\omega_0} = \\frac{a_0}{a(\\eta_e)}
\n\\end{eqnarray}<\/p>\n\u8d64\u65b9\u504f\u79fb\u3068\u5ea7\u6a19\u5024\u3068\u306e\u95a2\u4fc2<\/h3>\n