Light and distances
Contents
- 1 Problem 1: proper distance
- 2 Problem 2: comoving distance in a flat Universe
- 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
- 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: 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 $z$
solution
Problem 2: comoving distance in a flat Universe
Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe
solution
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)
solution
Problem 4: recession velocity
Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe
solution
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
solution
Problem 6: generalization to arbitrary curvature
Generalize the result of the previous problem to the case of arbitrary curvature
solution
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
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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