# Luminosity distance

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??