de Sitter space or Λ.D.: aΛ.d.≈exp
t q H02ΩΛ , (1.44)
having used that a(t = 0) = 0 in the integration. The fate of the universe can be deduced from these equations.
1.3.3 Thermodynamics
In the previous sections we have made the simplifying assumption that the only relevant thermodynamical properties of matter in calculating the dynamics of spacetime are density andpressure. This is indeed almost certainly a good assumption in many cases. For example, the large scale distribution of matter can for the most part be described in terms of the perfect fluid approximation, because we can neglect particle free-streaming and diffusion (Liddle & Lyth, 1993). Let us now find the relationship between density and pressure and justify the perfect fluid assumption.
The requirement for local thermodynamic equilibrium (LTE) in an expanding universe is that the interaction rate between the particles (often denoted by Γ(t)) must be much greater than the rate of the expansion of the universe, H(t). For a relativistic (mx TD) or non-
relativistic (mx TD) species that decouples from theheat bath14 while still in equilibrium,
the distributions function remains self-similar with expansion. This is not the case with semi- relativistic species for whichmx∼TD. Fortunately not many cosmologically relevant species
fall into the last category.
A perfect fluid
A perfect fluid fully characterised byρ(x, t),S(x, t) andV(x, t), its energy density distribu- tion, its entropy per unit mass and its vector field of 3-velocities (see also ch. 6 ofMukhanov, 2005). Assuming that it is in thermal equilibrium gives the equation of state for the perfect fluid, which describes the relationship between these three quantities: P = P(ρ, S). Taylor expanding the perturbed pressure to linear order,
P0+δP =P(ρ+δρ, S+δS) , (1.45)
yields the familiar
δP =c2sδρ+σδS , (1.46) where clearly c2s ≡ ∂P ∂ρ S and σ ≡ ∂P ∂S ρ . (1.47) 14
The additional assumption that the matter in the universe can be described as anadiabatic perfect fluid, means that δS = 0, i.e. no perturbations in the total entropy. This results in that for a perfect fluid, to linear order, w ≈ c2s. The assumption of adiabaticity in an expanding volume is correct for a species in true thermal equilibrium, which can be shown by the consideration of the first and second laws of thermodynamics (see for example sec. 3.4 in Kolb& Turner,1994).
Then, for the linear but nonrelativistic case, P ρc2 and the speed of sound, cs c
and effectivelywm≈0.
In everyday, Newtonian physics, the pressure, P, depends on the 3-velocity via the Euler equation, which describes movement of fluid in a gravitational potential (Mukhanov, 2005, p.267, eqn. 6.8) and the gravitational potential, φ, depends on ρ via the Poisson equation, which describes how mass sources the gravitational potential (Mukhanov, 2005, eqn. 6.10). Together with the continuity equation, which describes how matter flows and conservation of entropy, which describes the conservation of entropy if we neglect dissipation (Mukhanov, 2005, eqns. 6.3 and 6.9), these five equations form a complete set of equations describing a perfect fluid.
The Euler equation, the continuity equation and conservation of entropy all emerge from the conservation of the energy-momentum tensor, which is a consequence of intrinsic property called theBianchi identitythat emerges from the symmetries of the Riemann curvature tensor that we encountered in the previous sections. The Poisson equation is a combination of the 0−0 and the integrated 0−i component of the perturbed Einstein field equations. Such perturbed equations will be described in the next chapter. The equation of state is the only equation here that does not directly emerge from Einstein’s theory, but requires the assumption of the ideal gas (perfect fluid).
Phase space distribution
In order to see what happens in the relativistic case, we must extend our analysis to the phase space15 distribution of particles of our perfect fluid. Generally, for a spatially homogeneous weakly-interacting fluid of particles with g internal degrees of freedom, we write down (Kolb & Turner,1994, sec. 3.3):
ρ = g (2π)3 Z E(pi)f0(pi)dpi P = g (2π)3 Z | p|2 3Ef0(pi)dpi , (1.48) where pi =dxi/ds is the physical momentum as defined in the previous section (eq 1.23), f0
is the homogeneous one-particle phase space distribution function and E=P0 is the energy
of the particles of the fluid. As before, we can calculate the magnitude of the 4-momentum, |P|2 =PµPµ=E2−a2|p|2 =m2 , (1.49)
|p|2 = δ
ijpipj being the magnitude of the physical momentum and m being the mass of the
particles making up the fluid and as always, the speed of light c = 1. Neglecting for the
15
The phase space for a system of N particles is the 6N-dimensional space of positions and momenta of all the particles.
moment the expansion of the universe16,a= 1, this yields the familiar expression:
E2=|p|2+m2 . (1.50) In thermal equilibrium, the phase space function is given by either the Fermi-Dirac or Bose-Einstein distribution, depending on whether the particles are fermions or bosons re- spectively:
f0(pi) =
1
exp{(E−µ)/T} ±1 , (1.51) whereT is the temperature of the gas of fermions (+1) or bosons (−1) and µis its chemical potential.
For a relativistic gas of either fermions or bosons not interacting chemically (T µ), inserting eq. (1.51) into eq. (1.48) yields the equation of state wr=1/3, since relativistic
means that T m, yielding P = ρ/3. Not spending too much time on the details let us also mention that from the above equations one finds that for a non-degenerate17 relativistic species of matter, the energy density also found from multiplying the number density of particles by the average energy per particle,hEi is
ρ∝nhEi ∝gT4 . (1.52)
In fact, in a universe dominated by relativistic species, we can characterise the temperature by referring to the energy densty (or vice versa):
ρr=
π2 30g∗T
4 , (1.53)
where we have replaced the number of the degrees of freedom, g for different species by the summation over all the relativistic species,nto get the effectivenumber of relativistic degrees of freedom: g∗(T) = bosons X x gx Tx T 4 + 7 8 fermions X x gx Tx T 4 . (1.54)
The factor of 7/8 arises from simplifying eq. (1.48) for a relativistic fermion species. Note that this equation is different in characterising the relation between pressure and temperature if the contribution of non- and semi-relativistic species is significant.
We have already shown that for a nonrelativistic gas the equation of state is vanishing, but let us note here that the expressions in eq. (1.48) simplify the same way regardless of whether the gas is made up of bosons or fermions.
We will use these results in thesec. 1.3.1 and revisit the phase space distribution without assuming homogeneity insec. 2.2.4.