Astroparticle Physics

180 8 Cosmologywhere k is the curvature parameter and R the scale factor.The metric tensor is obtained as a solution to the Einsteinfield equations,R µν − 1 2 Rg µν = 8πGT µν + Λg µν , (8.19)where R µν is the Ricci tensor, R is the Ricci scalar (not tobe confused with the scale factor R), both of which are specificfunctions of g µν . On the right-hand side, Λ is the cos-mological constant and T µν is the energy–momentum tensor.An isotropic and homogeneous universe implies that T µνis of the formcosmological constantT µν = diag(ϱ, −P,−P,−P) , (8.20)energy–momentum tensorvacuum energy densitywhere ϱ is the energy density, P is the pressure, and ‘diag’means a square matrix with diagonal elements given by(8.20) and zeros everywhere else.Combining the Robertson–Walker metric and the T µνfrom (8.20) together with the field equations results in a differentialequation for the scale factor R,Ṙ 2R 2 + k R 2 = 8π 3 Gϱ + Λ 3 . (8.21)This is essentially the Friedmann equation that was foundfrom Newtonian gravity, but with three differences. First, ϱhere represents the energy density, not just the mass density.It includes all forms of energy, including, for example, photons.Second, the curvature parameter k really represents thecurvature of space, which is why it has this name. In theNewtonian case k was simply a measure of the total energy.Third, there is an additional term in the equation fromthe cosmological constant Λ. It can be absorbed into ϱ bydefining the vacuum energy density ϱ v asϱ v =Λ8πG(8.22)and then regarding the complete energy density to includeϱ v . It is called the vacuum energy because such a term ispredicted by quantum mechanics to arise from virtual particlesthat ‘fluctuate’ in and out of existence from the vacuum.