Difference between revisions of "Light and distances"

The equation of the photon's worldline is $ds^{2}=0$. Let us consider the a trajectory with observer placed at the origin. For spatially flat metric in terms of conformal-comoving variables the equation is \[ds^{2}=a^{2}(t)(d\eta^{2}-d\chi^{2})=0.\] Using the relation between differentials \begin{equation}\label{Deta(Dz)} d\eta=\frac{d\eta}{dt}\frac{dt}{da}\frac{da}{dz}dz=-\frac{\dot{a}}{a}dz=-\frac{dz}{H(z)}, \end{equation} we get \[\chi(z)=\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]

If $\chi(t)$ is the comoving distance to the source, then the proper distance is \[R(t)=a(t)\chi(t),\] or in terms of redshift \[R(z)=\frac{1}{1+z}\chi(z),\] where the scale factor is normalized by the value at the moment of observation. Using the result of the previous problem, then we obtain \[R(z)=\frac{1}{1+z}\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]

In this case \[H(z)=H_{0}(1+z)^{3/2}\] and evaluation of the integral gives \[\chi(z)=\frac{2}{H_0}\big[1-(1+z)^{-1/2}\big].\]

\[V\equiv \frac{dR}{dt}=\frac{da}{dt}\chi=H(z)R(z)=\frac{H(z)}{1+z}\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]

Energy emitted by the source per time $dt_{e}$ is $Edt_{e}$. Due to redshift, which reduces the energy of each photon emitted, during the time the scale factor changes from $a_{emission}=a_{e}$ to $a_{observation}=a_{o}$, this energy becomes \[Edt_{e}\cdot \frac{1}{1+z}=Edt_{e}\frac{a_e}{a_o}.\] The observed luminosity is the energy detected per given interval of local time $dt_{o}$ by observers that cover the sphere of the proper radius $R$, to which the photons travel by the time of observation. This local time of observation is subject to cosmological dilation: the worldline of one photon is given by $d\chi^{2}-d\eta^{2}=0$, and thus its equation is $\chi=\eta-\eta_{0}$. As the comoving distance between the source and detector is constant, the conformal time the photons travel is also the same, so for two photons emitted and detected one after the other \[d\eta=\frac{dt_{o}}{a_{o}}=\frac{dt_e}{a_e}.\] Then the observed luminosity is \[L=\frac{Edt_{e}\frac{a_e}{a_o}}{dt_{o}}=E\cdot \Big(\frac{a_e}{a_o}\Big)^{2},\] and the detected energy flux is \[F=\frac{L}{4\pi R^2}.\] On the other hand, the luminosity distance is defined though \[F=\frac{E}{4\pi d_{L}^{2}},\] so we obtain \begin{equation}\label{dL} d_{L}=\frac{a_o}{a_e}R=(1+z)R \end{equation}

\[d_{L}(z)=(1+z)(1-\Omega_0)^{-1/2}\cdot H_{0}^{-1}\Sigma\Big[(1-\Omega_0)^{1/2}H_{0} \int\limits_{0}^{z}\frac{dz'}{H(z')}\Big],\] where \[\Sigma(z)=\left\{\begin{array}{ll}\sin x,&\quad \Omega_{0}>1,\\ x,&\quad \Omega_{0}=1\\ \sinh x,&\quad \Omega_{0}<1.\end{array}\right.\]

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