Light and distances
Contents
- 1 Problem 1: comoving distance in a flat Universe
- 2 Problem 2: the proper distance
- 3 Problem 3: comoving distance in Einstein-de Sitter
- 4 Problem 4: recession velocity
- 5 Problem 5: luminosity distance in a flat Universe
- 6 Problem 6: generalization to arbitrary curvature
- 7 Problem 7: multi-component flat Universe
- 8 Problem 8: luminocity distance in terms of deceleration parameter
- 9 Problem 9: Einstein-de Sitter
- 10 Problem 10: small redshifts limit
- 11 Problem 11: the closed and open dusty Universes
- 12 Problem 12: angular diameter distance in terms of redshift
- 13 Problem 13: in terms of $q(z)$
- 14 Problem 14: a two-parametric expansion
- 15 Problem 15: another expansion
- 16 Problem 16: maximum of angular diameter distance in Einstein-de Sitter
- 17 Problem 17: the maximum in Friedman models
Problem 1: comoving distance in a flat Universe
Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe
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 \[\chi(z)=\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]
Problem 2: the proper distance
Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$
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 \[R(z)=\frac{1}{1+z}\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]
Problem 3: comoving distance in Einstein-de Sitter
Solve the previous problem for a flat Universe with domination of non-relativistic matter (the Einstein-de Sitter model)
In this case \[H(z)=H_{0}(1+z)^{3/2}\] and evaluation of the integral gives \[\chi(z)=\frac{2}{H_0}\big[1-(1+z)^{-1/2}\big].\]
Problem 4: recession velocity
Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe
\[V\equiv \frac{dR}{dt}=\frac{da}{dt}\chi=H(z)R(z)=\frac{H(z)}{1+z}\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]
In cosmology the are other types of distances used, besides the proper and comoving one. One of the most frequently used is the photometric distance. Let $E$, $[E]=J/s$, be the internal absolute luminosity of some source. The observer on Earth detects energy flux $F$, $[F]=J/s\cdot m^2$. The luminosity distance to the source $d_{L}$ is then defined through
\[F=\frac{E}{4\pi d_{L}^{2}}.\]
Thus this would be the distance to the observed object, given its absolute and observed luminosities, in a flat and stationary Universe. Non-stationarity and curvature imply that $d_{L}$ in general does not coincide with the proper distance.
Problem 5: luminosity distance in a flat Universe
Express the luminosity distance in terms of observed redshift for a spatially flat Universe
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}
Problem 6: generalization to arbitrary curvature
Generalize the result of the previous problem to the case of arbitrary curvature
\[d_{L}(z)=(1+z)(1-\Omega_0)^{-1/2}\cdot H_{0}^{-1}\Sigma\Big[(1-\Omega_0)^{1/2}H_{0} \int\limits_{0}^{z}\frac{dz'}{H(z')}\Big],\] where \[\Sigma(z)=\left\{\begin{array}{ll}\sin x,&\quad \Omega_{0}>1,\\ x,&\quad \Omega_{0}=1\\ \sinh x,&\quad \Omega_{0}<1.\end{array}\right.\]
Problem 7: multi-component flat Universe
Find the expression for the luminosity distance for the multi-component flat Universe
solution
Problem 8: luminocity distance in terms of deceleration parameter
Express the luminosity distance in a flat Universe in terms of the redshift dependence of deceleration parameter $q(z)$
solution
Problem 9: Einstein-de Sitter
Express the luminosity distance in terms of redshift for the Einstein-de Sitter model
solution
Problem 10: small redshifts limit
Show that in the first order by $z\ll 1$ luminosity distance is $d_{L}\approx z /H_{0}$ and find the second order correction
solution
Problem 11: the closed and open dusty Universes
Derive the luminosity distance as function of redshift for the closed and open models of the Universe, dominated by non-relativistic matter (dust)
solution
Another distance used is the angular diameter distance. It is defined through the angular dimension of the object $\delta \theta$ and its proper transverse size $\delta l$ as
\[d_{A}=\frac{\delta l}{\delta \theta}.\]
Again, in a stationary and flat Universe this is reduced to the ordinary distance, while in general they differ.
Problem 12: angular diameter distance in terms of redshift
Express the angular diameter distance in terms of the observed redshift
solution
Problem 13: in terms of $q(z)$
Express the angular diameter distance in terms of $q(z)$
solution
Problem 14: a two-parametric expansion
Find $d_{A}(z)$ in a flat Friedman Universe for the linear two-parametric expansion$^*$ \[q(z)=q_{0}+q_{1}(z).\] $^*$ J. Lima et al. arXiv:0905.2628
solution
Problem 15: another expansion
Find $d_{A}(z)$ in a flat Friedman Universe for the linear two-parametric expansion \[q(z)=q_{0}+q_{1}\frac{z}{1+z}.\]
solution
Problem 16: maximum of angular diameter distance in Einstein-de Sitter
Find the redshift for which the angular diameter distance of an object in the Einstein-de Sitter Universe reaches its maximum
solution
Problem 17: the maximum in Friedman models
Find the redshift for which the angular diameter distance reaches the maximum in the closed and open Friedman models$^*$ $^*$ Juri Shtanov, Lecture notes on theoretical cosmology, 2010
solution
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