Views
5 years ago

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

The Pittsburgh Particle physics, Astrophysics and Cosmology Center ...
Modern Cosmology - Astronomy & Astrophysics Group - University ...
Neutrino Masses in Astrophysics and Cosmology International ...
Dark Matter Particle Physics in Cosmological Simulations
Teresa Montaruli - Fermilab Center for Particle Astrophysics
Lars Bergström, Cosmology and Particle Astrophysics, 2011 2011 ...
Quantum Phenomena in the Realm of Cosmology and Astrophysics
18. BIG-BANG COSMOLOGY - Particle Data Group
Particle Methods in Astrophysical Fluid Dynamics
Astrophysical Smooth Particle Hydrodynamics.pdf
High-Energy Astrophysics Lecture 4: Particle acceleration and ...
A1Y Cosmology - Astronomy & Astrophysics Group
2. astrophysical constants and parameters - Particle Data Group
2. astrophysical constants and parameters - Particle Data Group
Chapter 12 Cosmology: The Homogeneous Universe
4 Relativistic Dynamics of Particles - Smoot Group Cosmology
Lessons in Particle Physics - Center for Gravitation and Cosmology
15. THE POCKET COSMOLOGY - Particle Data Group
Lecture 21 - University of Toronto, Particle Physics and Astrophysics
Slides - Fermilab Center for Particle Astrophysics
Supersymmetry and cosmology - Institute of Nuclear and Particle ...
Andrew MacFadyen - Center for Cosmology and Particle Physics ...
Toolkit sheet - Particle Physics and Particle Astrophysics
Introduction-Cosmology-Ryden_tmp
Applications of Cosmological Perturbation Theory.pdf
PH5870, General relativity and cosmology
Astronomy 220 Marr's Cosmology Notes Spring 2012 - Union College
Introduction to Cosmological Perturbation Theory