Astroparticle Physics

400 B Results from Statistical Physics: Thermodynamics of the Early Universegeneral expressionfor the pressure∂E∂V = p ( ) −p 1E L 3L 2 = −p23EV .(B.43)Putting this into (B.40) provides the general expression forthe pressure,P = 1 ∫ p23V E f(p) d3 p.(B.44)In the relativistic limit the particle’s rest mass can beneglected, so, E = √ p 2 + m 2 ≈ p. Equation (B.44) thenbecomesP = 1 ∫Ef(p) d 3 p.(B.45)3VBut the total energy U is (B.39)∫U = Ef(p) d 3 p(B.46)pressurein the relativistic limitand ϱ = U/V, so the final result for the pressure for a gasof relativistic particles is simplyP = ϱ 3 .(B.47)non-relativistic limit:ideal gas lawvacuum energy densityfrom a cosmological constantnegative pressureThis is the well-known result from blackbody radiation, butone realizes here that it applies for any particle type in therelativistic limit T ≫ m.In the non-relativistic limit, the pressure is given by theideal gas law,P = nT .(B.48)In this case, however, the energy density is simply ϱ = mn,so for T ≪ m one has P ≪ ϱ and in the acceleration andfluid equations one can approximate P ≈ 0.Finally, the case of vacuum energy density from a cosmologicalconstant can be treated,ϱ v =Λ8πG .(B.49)If one takes U/V = ϱ v as constant, then the pressure is( ) ∂UP =− =− U ∂V S V =−ϱ v . (B.50)Thus, a vacuum energy density leads to a negative pressure.