Entropy of Expanding Universe

Problem 1

Transform the energy conditions for the flat Universe to conditions for the entropy density (see [1])

Problem 2

Find the entropy density for the photon gas.

Problem 3

Use the result of the previous problem to derive an alternative proof of the fact that $aT=const$ in the adiabatically expanding Universe.

Problem 4

Find the adiabatic index for the CMB.

Problem 5

Show that the ratio of CMB entropy density to the baryon number density $s_\gamma/n_b$ remains constant during the expansion of the Universe.

Problem 6

Estimate the current entropy density of the Universe.

Problem 7

Estimate the entropy of the observable part of the Universe.

Problem 7

Why is the expansion of the Universe described by the Friedman equations adiabatic?

Problem 8

Show that entropy is conserved during the expansion of the Universe described by the Friedman equations.

Problem 9

Show that the entropy density behaves as $s\propto a^{-3}$.

Problem 10

Using only thermodynamical considerations, show that if the energy density of some component is $\rho=const$ than the state equation for that component reads $p=-\rho$.

Problem 11

Show that the product $aT$ is an approximate invariant in the Universe dominated by relativistic particles.

Problem 12

Show that for a system of particles in thermal equilibrium, $\frac{dp}{dT} =\frac{1}{T} \left(\rho +p\right)$

Problem 12

Show that for the substance with the equation $p=w\rho$ $T\propto \rho ^{\frac{w}{w+1} }$

Problem 13

Show that for the substance with the equation $p=w\rho$ $T\propto a^{-3w}$

Problem 14

Obtain a generalization of Stefan-Boltzmann law for cosmological fluid, described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n}$.

Problem 15

Show that the velocity of sound for cosmological fluid described by generalized polytropic equation of state $p=w\rho +k\rho ^{1+1/n}$ vanishes at the point where the temperature is extremum.