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Solutions chapter 4 - Cosmology and particle astrophysics

Solutions chapter 4 - Cosmology and particle astrophysics

Solutions chapter 4 - Cosmology and particle

Solutions chapter 4 - Cosmology and particle astrophysics 4.4 Einstein’s equations are Rµν − 1 2 gµνR − Λgµν = 8πGTµν, (1) where the Ricci tensor Rµν and the Ricci scalar R = g µν Rµν are functions of derivatives of the metric and it’s inverse. The constant Λ is the so-called cosmological constant and it has usints of the inverse of length squared. The stress energy tensor for a perfect fluid is given by eq. (4.13) in Bergström and Goobar: Tµν = (p + ρ) uµuν − pgµν where uµ is the four-velocity of the fluid element (and as such, it will depend on the observer!). Evaluated in the rest frame of the fluid, where u µ = (1, 0, 0, 0) and gµν = ηµν, this becomes ⎛ ρ ⎜ Tµν = ⎜ 0 ⎝ 0 0 p 0 0 0 p 0 0 0 ⎞ ⎟ ⎠ . (3) 0 0 0 p If we move the cosmological constant to the right hand side of eq. (1), we see that we can define an effective stress energy tensor for the cosmological constant: T Λ µν = Λ 8πG gµν. If we evaluate this in a frame where gµν = ηµν and compare with eq. (3), we see that we can identify ρΛ = Λ 8πG , pΛ = − Λ 8πG = −ρΛ. (4) The Robertson-Walker metric is ds 2 = dt 2 − a (t) 2 dr 2 1 − kr 2 + r2 dΩ 2 (2) . (5) Using this metric, together with the perfect fluid form of the stress energy tensor, in Einstein’s equations, gives the following two equations: 2ä a + 2 ˙a a + k 8πG = a2 3 ρtot, (6) 2 ˙a + a k a2 = −8πGptot. (7) Here we have included the cosmological constant in the energy density and pressure: ρtot = ρm + ρrad + ρΛ, (8) ptot = pm + prad + pΛ. (9) 1

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