The Phase diagram of the Universe

• Exercise sheet 3

The Matter density vs Dark Energy density discussion can be found at

• Zee: VIII.2

but similar material is, once again, present in all GR books in the references.

Analyzing Friedman's equation for various values of the relative densities Omega_i, and eliminating the radiation contribution, which today is negligible (and will remain so in the future), one sees that we can restrict the discussion to the dependence on two variables, Ω_M and Ω_Λ. We discussed various boundaries on the plot Ω_M vs Ω_Λ:

• the flat Universe straight line, the boundary between spherical, and hyperbolic geometry
• the 2Ω_Λ-Ω_M=0 boundary between accelerating and decelerating Universes
• the boundary between Universes with and without a Big Bang
• the boundary between Universes that eventually will re-collapse into a Big Crunch and Universes that will expand forever

We also gave rough estimates of the age of the Universe,as well as of the particle horizon, based on the assumption that most of its lifetime it has been Matter dominated, and saw that the order of magnitude for both quantites is set by the (inverse of) H_0, the value of the Hubble parameter today.

Thermodynamics of the Universe

The basic thermodynamics derivations are from

• Rubakov and Gobrunov: sec. 5.1

Starting from the Einstein-Bose and Fermi-Dirac distributions of particles in equilibrium, we deduced the relations between energy density, pressure, number density and temperature.

We also saw an example of a rough, order of magnitude estimate of the temperature below which a particular interaction (electromagnetic) is expected to actively regulate the thermal equilibrium. We have, thereby, described the condition for particles that interact via electromagnetism to be in thermal equilibrium: the interaction rate has to be much smaller than the expansion rate of the Universe, H, expressed in terms of the temperature.