Fény terjedése inhomogén univerzumban
A Sachs-féle optikai egyenletek:
$
\begin{eqnarray}
\frac{d\Theta}{d\lambda}+\Theta^2+\frac{1}{2}w^2=-\frac{1}{2}R_{jk}u^ju^k
\end{eqnarray}
$
$
\begin{eqnarray}
\frac{dw_{\alpha\beta}}{d\lambda}+2\Theta w_{\alpha\beta} =C_{ ijkl}L^i_\alpha
u^ju^kL^l_\beta
\end{eqnarray}
$
$
\begin{eqnarray}
w^2=w_{\alpha\beta}w_{\beta\alpha}
\end{eqnarray}
$
$
\begin{eqnarray}
\frac{dA}{d\lambda}=2\Theta A
\end{eqnarray}
$
$
\begin{eqnarray}
I=L\frac{\Omega}{4\pi}\frac{\left(u^{t}(t_0)\right)^2}{\left(u^{t}(t_1)\right)^2A(t_0)}
\end{eqnarray}
$
$
\begin{eqnarray}
z=\frac{u^t(t_1)}{u^t(t_0)}-1
\end{eqnarray}
$
$
\begin{eqnarray}
g_{ij}=g_{ij}^{(0)}+g_{ij}^{(1)}+g_{ij}^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
\Gamma_{jk}^i=\Gamma_{jk}^{i(0)}+\Gamma_{jk}^{i(1)}+\Gamma_{jk}^{i(2)}
\end{eqnarray}$
$
\begin{eqnarray}
R_{ij}=R_{ij}^{(0)}+R_{ij}^{(1)}+R_{ij}^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
C_{ijkl}=C_{ijkl}^{(1)}+C_{ijkl}^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
u^{i}=u^{i(0)}+u^{i(1)}+u^{i(2)}
\end{eqnarray}$
$
\begin{eqnarray}
w^{i}=w^{i(0)}+w^{i(1)}+w^{i(2)}
\end{eqnarray}$
$
\begin{eqnarray}
L^{i}_\alpha=L^{i(0)}_\alpha+L^{i(1)}_\alpha+L^{i(2)}_\alpha
\end{eqnarray}$
$
\begin{eqnarray}
\Theta=\Theta^{(0)}+\Theta^{(1)}+\Theta^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
w_{\alpha \beta}=w_{\alpha \beta}^{(1)}+w_{\alpha \beta}^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
A=A^{(0)}+A^{(1)}+A^{(2)}
\end{eqnarray}$
$
\begin{eqnarray}
t(\lambda)=t^{(0)}(\lambda)+t^{(1)}(\lambda)+t^{(2)}(\lambda)
\end{eqnarray}$
$
\begin{eqnarray}
r(\lambda)=r^{(0)}(\lambda)+r^{(1)}(\lambda)+r^{(2)}(\lambda)
\end{eqnarray}$
$
\begin{eqnarray}
\vartheta(\lambda)=\vartheta^{(0)}+\vartheta^{(1)}(\lambda)+\vartheta^{(2)}(\lambda)
\end{eqnarray}$
$
\begin{eqnarray}
\phi(\lambda)=\phi^{(0)}+\phi^{(1)}(\lambda)+\phi^{(2)}(\lambda)
\end{eqnarray}
$
$
\begin{eqnarray}
z=z^{(0)}+z^{(1)}+z^{(2)}
\end{eqnarray} $
$
\begin{eqnarray}
z^{(0)}=\left(\frac{t_0}{t_1}\right)^{\frac{2}{3}}-1\;,\\
z^{(1)}=a(t_0)\left[u^{t(1)}(t_1)-(1+z^{(0)})u^{t(1)}(t_0)\right]\;,\\
z^{(2)}=a(t_0)\left[u^{t(2)}(t_1)-(1+z^{(0)})u^{t(2)}(t_0)\right]\\
-a^4(t_0)u^{t(1)}(t_0)\left[u^{t(1)}(t_1)-(1+z^{(0)})u^{t(1)}(t_0)\right]
\end{eqnarray}$
$
\begin{eqnarray}
\ln{I}-\ln{I^{(0)}(z)}=
\left[\frac{z^{(1)}}{(1+z^{(0)})\left(\sqrt{1+z^{(0)}}-1\right)}
-\frac{(1+z^{(0)})A^{(1)}}{9t_0^2\left(\sqrt{1+z^{(0)}}-1\right)^2}
\right]
\\
+\left[\frac{z^{(2)}}{(1+z^{(0)})\left(\sqrt{1+z^{(0)}}-1\right)}
-\frac{(1+z^{(0)})A^{(2)}}{9t_0^2\left(\sqrt{1+z^{(0)}}-1\right)^2}\right.\\\left.
+\frac{\left(2-3\sqrt{1+z^{(0)}}\right)z^{(1)2}}{4(1+z^{(0)})^2\left(\sqrt{1+z^{(0)}}-1\right)^2}
+\frac{(1+z^{(0)})^2A^{(1)2}}{162t_0^4\left(\sqrt{1+z^{(0)}}-1\right)^4}
\right]
\end{eqnarray}
$
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