# Light and distances

### Problem 1: comoving distance in a flat Universe

Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe

### 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 $z$

### 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)

### Problem 4: recession velocity

Determine the recession velocity caused by the cosmological expansion for an object with redshift $z$ in a flat Universe

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

### Problem 6: generalization to arbitrary curvature

Generalize the result of the previous problem to the case of arbitrary curvature

### Problem 7: multi-component flat Universe

Find the expression for the luminosity distance for the multi-component flat Universe

### 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 $q(z)$

### Problem 9: Einstein-de Sitter

Express the luminosity distance in terms of redshift for the Einstein-de Sitter model

### 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

### 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)

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

### Problem 13: in terms of $q(z)$

Express the angular diameter distance in terms of $q(z)$

### 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

### Problem 15: another expansion

Find $d_{A}(z)$ in a flat Friedman Universe for the linear two-parametric expansion $q(z)=q_{0}+q_{1}\frac{z}{1+z}.$

### 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

### 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