# Peculiarities of Thermodynamics in Early Universe

### Problem 1

Find the temperature dependence for the Hubble parameter in the early flat Universe.

### Problem 2

Find the time dependence of the temperature of the early Universe by direct integration of the first Friedman equation.}

### Problem 3

Prove that results of the problems #time1 and #time1 are equivalent.

### Problem 4

Determine the energy density of the Universe at the Planck time.

### Problem 5

Show that at Planck time the energy density of the Universe corresponded to $10^{77}$ proton masses in one proton volume.

### Problem 6

What was the temperature of radiation-dominated Universe at the Planck time?

### Problem 7

Determine the age of the Universe when its temperature was equal to $1\ MeV$.

### Problem 8

In the first cyclic accelerator - the cyclotron (1931)- particles were accelerated up to energies of order $1MeV$. In the next generation accelerators - the bevatrons - energy was risen to $1GeV$. In the last generation accelerator - the LHC -protons are accelerated to energy of $1\ TeV$. What times in the Universe history do those energies allow to investigate?

### Problem 9

Show that in the epoch when the energy density of the Universe was determined by ultra-relativistic matter and effective number of internal degrees of freedom did not change, held $\dot{T}/T\propto -T^2$.

### Problem 10

Estimate the baryon-antibaryon asymmetry $A\equiv(n_b-n_{\bar{b}})/n_{\bar{b}}$ in the early Universe.

### Problem 11

Determine the monopoles' number density and their contribution to the energy density of the Universe at the great Unification temperature. Compare the latter with the photons' energy density at the same temperature.

### Problem 12

At what temperature and time does the contribution of monopoles into the Universe energy density become comparable to the contribution of photons?