Astroparticle Physics

182 8 Cosmology8.5 The Fluid Equation“The universe is like a safe to whichthere is a combination – but the combinationis locked in the safe.”Peter de VriesThe Friedmann equation cannot be solved yet because onedoes not know how the energy density ϱ varies with time. Instead,in the following a relation between ϱ, its time derivative˙ϱ, and the pressure P will be derived. This relation,called fluid equation, follows from the first law of thermody-namics for a system with energy U, temperature T ,entropyS, andvolumeV ,fluid equationadiabatic expansionFor the second term in (8.24) one getsdV= d dt dt R3 = 3R 2 Ṙ. (8.27)Putting (8.26) and (8.27) into (8.24) and rearranging termsgives˙ϱ + 3Ṙ (ϱ + P) = 0 , (8.28)Rwhich is the fluid equation. Unfortunately, this is still notenough to solve the problem, since an equation of state re-lating ϱ and P is needed. This can be obtained from the lawsof statistical mechanics as will be shown in Sect. 9.2. Withthese ingredients the Friedmann equation can then be usedto find R(t).equation of statedU = T dS − P dV . (8.23)The first law of thermodynamics will now be applied to avolume R 3 in our expanding universe. Since by symmetrythere is no net heat flow across the boundary of the volume,one has dQ = T dS = 0, i.e., the expansion is adiabatic.Dividing (8.23) by the time interval dt then givesdUdt+ P dVdtThe total energy U is= 0 . (8.24)U = R 3 ϱ. (8.25)The derivative dU/dt is thereforedUdt= ∂U∂RṘ + ∂U∂ϱ ˙ϱ = 3R2 ϱṘ + R 3 ˙ϱ . (8.26)