New look at 'Einstein rings' around distant galaxies just got us closer to solving the darkmatter debate ConversationEDU NatureAstronomy
\begin{eqnarray} {\tilde{\delta }}_{\gamma }^{\prime }&=&-{\tilde{\delta }}_{\gamma }-\frac{4}{3}{\tilde{t}}_{\gamma \rm b}x^{2}-\frac{2\Theta }{3}, \end{eqnarray}
\begin{eqnarray} {\tilde{\delta }}_\mathrm{ b}^{\prime }&=&-{\tilde{\delta }}_\mathrm{ b}-{\tilde{t}}_{\gamma \rm b}x^{2}-\frac{\Theta }{2}, \end{eqnarray} \begin{eqnarray} {\tilde{t}}_{c}^{\prime }&=&-2{\tilde{t}}_{c}-\frac{\left}{\left}{\tilde{t}}_{c}, \end{eqnarray} \begin{eqnarray} \Theta ^{\prime }&=&-\frac{\left}{\left}\Theta -\frac{6\left}{\left^{2}}\nonumber \\ &&-\,\frac{12x\left}{\kappa \left}-\frac{32\pi G a^{2}\rho _{{\rm a}} x^{2} \tilde{\delta }_{{\rm a}}}{C^{2}\kappa ^{2}}\nonumber \\ &&-\,\frac{72\pi G a^{2}\rho _{{\rm a}}x^{2}}{C^{2}\kappa ^{2}}{\tilde{u}}_{{\rm a}}\left\left, \end{eqnarray}
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