Dynamics of the Universe in terms of redshift and conformal time

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Revision as of 12:46, 14 October 2012 by Igor (Talk | contribs) (Problem 11.)

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Problem 1.

Express the first Friedman equation in terms of redshift and analyze the contribution of different terms in different epochs.


Problem 2.

Find the conformal time as function of the scale factor for a Universe with domination of a) radiation and b) non-relativistic matter.


Problem 3.

Find the relation between time and redshift in the Universe with dominating matter.


Problem 4.

Derive $a(\eta)$ for a spatially flat Universe with dominating radiation.


Problem 5.

Express the cosmic time through the conformal time in a Universe with dominating radiation.


Problem 6.

Derive $a(\eta)$ for a spatially flat Universe with dominating matter.


Problem 7.

Find $a(\eta)$ for a spatially flat Universe filled with a mixture of radiation and matter.


Problem 8.

Suppose a component's state parameter $w_i=p_i/\rho_i$ is a function of time. Find its density as function of redshift.


Problem 9.

Derive the Hubble parameter as a function of redshift in a Universe filled with non-relativistic matter.


Problem 10.

The redshift of any object slowly changes with time due to acceleration (or deceleration) of the Universe's expansion. Find the rate of change of redshift $\dot{z}$ for a Universe with dominating non-relativistic matter.



Problem 11.

The Universe is known to have become transparent for electromagnetic waves at $z\approx 1100$ (in the process of formation of neutral hydrogen, recombination), i.e when it was $1100$ times smaller than at present. Thus in practice the possibility of optical observation of the Universe is limited by the so-called optical horizon: the maximal distance that light travels since the moment of recombination. Find the ratio of the optical horizon to the particle one for a Universe dominated by matter.



Problem 12.

Derive the particle horizon as function of redshift for a Universe filled with matter and radiation with relative densities $\Omega_{m0}$ and $\Omega_{r0}$.



Problem 13.

Show that any signal emitted from the cosmological horizon will arrive to the observer with infinite redshift.