Difference between revisions of "Light and distances"

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(Problem 10: small redshifts)
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=== Problem 1: proper distance ===
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=== Problem 1: comoving distance in a flat Universe===
Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$
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Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe
 
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     <p style="text-align: left;"> solution</p>
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     <p style="text-align: left;"> 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
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\[ds^{2}=a^{2}(t)(d\eta^{2}-d\chi^{2})=0.\]
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Using the relation between differentials
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\begin{equation}\label{Deta(Dz)}
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d\eta=\frac{d\eta}{dt}\frac{dt}{da}\frac{da}{dz}dz=-\frac{\dot{a}}{a}dz=-\frac{dz}{H(z)},
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\end{equation}
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we get
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\[\chi(z)=\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]
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=== Problem 2: comoving distance in a flat Universe===
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=== Problem 2: the proper distance ===
Find the comoving distance to a galaxy as function of redshift in a spatially flat Universe
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Determine the "physical" distance -- the proper distance measured along the hypersurface of constant cosmological time -- to an object that is observed with redshift $z$
 
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     <p style="text-align: left;"> solution</p>
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     <p style="text-align: left;"> If $\chi(t)$ is the comoving distance to the source, then the proper distance is
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\[R(t)=a(t)\chi(t),\]
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or in terms of redshift
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\[R(z)=\frac{1}{1+z}\chi(z),\]
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where the scale factor is normalized by the value at the moment of observation. Using the result of the previous problem, then we obtain
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\[R(z)=\frac{1}{1+z}\int\limits_{0}^{z}\frac{dz'}{H(z')}.\]
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Revision as of 13:59, 5 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

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