Dynamics of the Universe in terms of redshift and conformal time

Problem 1: the first Friedman equation

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

Problem 2: $\eta(a)$ in one-component Universe

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: $t(z)$ for matter domination

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

Problem 4: $a(\eta)$ for radiation domination

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

Problem 5: $t(\eta)$ for dominating radiation

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

Problem 6: $a(\eta)$ for dominating dust

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

Problem 7: $a(\eta)$ for radiation and dust

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

Problem 8: variable EoS parameter

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: $H(z)$ for dominating dust

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

Problem 10: $\dot z$ for dominating dust

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: optical horizon

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: particle horizon

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: time dilation at cosmological horizon

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