Entropy of Expanding Universe

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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.