Problem 1: Newtonian interpretation
Show that the first Friedman equation can be treated as the energy conservation law in Newtonian mechanics. Use this equation to classify the solutions describing different dynamics of the Universe.
Problem 2: thermodynamic derivation
Obtain the conservation equation for the expanding Universe using only thermodynamical considerations.
Let $E$ be the total energy inside the volume $V$, and $\rho$ is the energy density in the Universe, then
\[dE = d(\rho V)= Vd\rho+\rho dV.\]
In the first principle of thermodynamics $dQ = dE+pdV$, as the Universe is a closed system by definition, it is evident that $dQ=0$. Taking into account that $dV = 4\pi a^2da$, it is easy to see that
Problem 3: the lost energy of the Universe
A photon's wavelength is redshifted due to the Universe' expansion. Estimate the rate of change of the energy of the Universe due to this process.