If light is emmited from galaxy A at $t_e$ and received at galaxy B at $ t_o.$ The flux at B is $$ (1) -- \phi(t_o)=a^2(t_e)L(t_e)/4πd_L^2$$ $d_L$ being the current distance between A and B In the comoving frame flux at B at $t_o$ is $$(2) -- \eta=L(t_e)/4πx^2, $$ x being the comoving distance Now in terms of discrete energy level (e.g if photons are emmited from A). Flux at A at $ t_e $ is $$\phi(t_e)=n_ecE(t_e)$$,$n_e $being the no. density of photons at $ t_e.$ Flux recieved at B at $ t_o $ is $$\phi(t_o)=n_oca(t_e)E(t_e)$$ $n_o$ being the no.density at $t_o$. $$n_o=n_ea^3(t_e)/a^3(t_o).$$ $$\phi(t_o)=n_ea^3(t_e)^4cE(t_e)$$ ,for $a(t_o)=1$ present time. In the comoving frame $$\eta=\phi(t_e)$$. hence $$\phi(t_o)=a(t_e)^4\eta$$. From (1) and (2) $d_L$ comes out $$x/a(t_e)$$ e.g $$d_L=x(1+z)$$ e.g luminosity distance. . Is it so??


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