Astroparticle Physics

202 9 The Early Universe9.3.1 Digression on Thermal EquilibriumHaving derived the relations for quantities such as numberand energy density as a function of temperature, it is worthasking when one expects them to apply. In order for a systemto be characterized by a temperature, there must exist interactionsbetween the particles that allow their numbers andmomentum distribution to adjust to those of thermal equilibrium.Furthermore, one has to wait long enough for equi-librium to be attained, namely, much longer than the timescale of the individual microscopic interactions.Now in any change in the temperature, the microscopicinteractions must take place quickly enough for the thermaldistribution to adjust. One can express this condition by requiringthat the rate Γ of the reaction needed to maintainequilibrium must be much greater than the fractional changein the temperature per unit time, i.e.,thermal equilibriumconditionsΓ ≫|Ṫ/T| . (9.41)But for a system of relativistic particles one has from (9.25)that T ∼ 1/R, soṪ/T =−Ṙ/R =−H . Therefore, (9.41)is equivalent to the requirementΓ ≫ H, (9.42)interaction ratescross sectionwhere the absolute value has been dropped since the expansionrate is always assumed to be positive. For a given temperatureit is straightforward to use (9.38) to determine H .It is basically proportional to T 2 , ignoring the small temperaturedependence of g ∗ .The reaction rate Γ is the number of interactions of aspecified type per unit time per particle. It is the reciprocal ofthe mean time that it will take for the particle to undergo theinteraction in question. It can be calculated as a function ofthe particle’s speed v, the number density of target particlesn, and the interaction cross section σ byΓ = n〈σv〉 , (9.43)where the brackets denote an average of σv over a thermaldistribution of velocities.If thermal equilibrium has been attained, one can findthe number density n using the appropriate formulae fromSect. 9.2. To find the cross section, one needs to bring in theknowledge of particle physics.