Lecture 2

Question 1

Acceleration equation

Answer 1

differentiating the Friedmann equation with respect to time

d/dt [ (ȧ/a)^2] = d/dt [8πGρ/3 - kc^2/a^2]

giving

ä/a = - 4πG/3 (ρ + 3P/c^2)

Question 2

the acceleration equation is independent of

Answer 2

curvature

Question 3

The Robertson-Walker metric

Answer 3

using an Einstein summation convention

cartesians: ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2

spherical polars: ds^2 = -c^2dt^2 + dr^2 + r^2dθ^2 + r^2sin^2θdφ^2

Question 4

The shortest distance between two points is given by

Answer 4

the line θ = 0, φ = 0: the radial geodesic

Question 5

The proper distance between r = 0 and r = R at fixed time dt = 0 is

Answer 5

D = a(t) (R ∫ 0) dr/√(1-kr^2) = a(t0)f(r)

Question 6

The cosmological constant

Answer 6

Λ is a term that if sufficiently large acts against gravitational attraction giving an accelerating universe

Question 7

Cosmological constant

Answer 7

ΩΛ = Λ/3H^2

Question 8

Matter

Answer 8

Ωm = 8πGρ/3H^2

Question 9

Curvature

Answer 9

Ωk = -kc^2/a^2H^2

Question 10

density, geometry and the cosmological constant are interconnected

Answer 10

Ωm + ΩΛ + Ωk = 1