Difference between revisions of "Light and distances"

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(Problem 17: the maximum in Friedman models)
(Problem 17: the maximum in Friedman models)
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Tor the closed model  
+
For the closed model  
\[\Sigma(\chi)=\sin(\chi)\]
+
\[\Sigma(\chi)=\sin\chi\]
 
and
 
and
 
\[a(\eta)=\frac{\alpha}{2}(1-\cos\eta),\]
 
\[a(\eta)=\frac{\alpha}{2}(1-\cos\eta),\]
 
where $\alpha$ is some constant. Then
 
where $\alpha$ is some constant. Then
\[d_{A}(\eta)=a_{e}\Sigma(\chi)\sim (1-\cos\eta_{e})\sin(\eta_{o}-\eta_{e}).\]
+
\[d_{A}(\eta)=a_{e}\Sigma(\chi)\sim (1-\cos\eta_{e})\sin(\eta_{o}-\eta_{e}),\]
and condition
+
where subscripts "e" and "o" denote emission and observation respectively, and condition
 
\[\frac{dd_{A}}{d\eta_{e}}=0\]
 
\[\frac{dd_{A}}{d\eta_{e}}=0\]
is transformed to
+
is transformed, after renaming $\eta_{e}\equiv \eta$, to
\[0=\sin\eta \sin(\eta_o -\eta)-(1-\cos\eta)\cos(\eta_o -\eta)\sim \cos(2\eta-\eta_o)-\cos(\eta-\eta_o) \sim \sin \frac{3\eta-2\eta_o}{2}\sin\frac{\eta}{2}.\]
+
\[0=\sin\eta\; \sin(\eta_o -\eta)-(1-\cos\eta)\cos(\eta_o -\eta)\sim \cos(2\eta-\eta_o)-\cos(\eta-\eta_o)
 +
\sim \sin \frac{3\eta-2\eta_o}{2}\sin\frac{\eta}{2}.\]
 
So the maximum $d_A$ is realized for $\eta_{e}=\frac{2}{3}\eta_{o}$. The corresponding redshift is found from
 
So the maximum $d_A$ is realized for $\eta_{e}=\frac{2}{3}\eta_{o}$. The corresponding redshift is found from
\[1+z_{\max}=\frac{a_o}{a_e}=\frac{1-\cos \eta_o}{1-\cos\eta_e}.\]
+
\[1+z_{\max}=\frac{a_o}{a_e}=\frac{1-\cos \eta_o}{1-\cos\eta_e}\frac{1-\cos \eta_o}{1-\cos\frac{2}{3}\eta_o}.\]
  
 
In the open model, following the same procedure, one obtains the same result with trigonometric functions replaced by hyperbolic ones.</p>
 
In the open model, following the same procedure, one obtains the same result with trigonometric functions replaced by hyperbolic ones.</p>
 
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Revision as of 12:57, 7 October 2012


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