8.5

Question 1

Universe Models

Answer 1

• Based on Friedmann equation
• Old but provide a simple picture of basic cosmological models

Question 2

Assumptions for Friedmann Equation

Answer 2

• Spherical shell with radius a
• Inside volume with constant density

Question 3

Potential Energy

Answer 3

E_pot = - 4/3 · π · G · ρ · a²

Question 4

Kinetic Energy

Answer 4

E_kin = a_dot² / 2

Question 5

Total Mechanical Energy

Answer 5

E = E_kin + E_pot

Question 6

Metric Energy

Answer 6

E_metric = - k · c² / 2

Question 7

Friedmann Equation

Answer 7

a_dot² / a² + k · c² / a² = 8 · π · G · ρ / 3

Question 8

Mechanics Analogy to Friedmann Equation

Answer 8

Projectiles with positive/negative total energy escape/fall back due to gravity.

Question 9

Energy and Universe Expansion

Answer 9

• k = -1: Universe expands forever, E > 0
• k = +1: Universe collapses, E < 0
• k = 0: Universe on borderline, expansion stops for a = ∞, E = 0

Question 10

Critical Density (fromula not number)

Answer 10

ρ_c = 3 · H² / (8 · π · G)

Question 11

Present Day Critical Density

Answer 11

ρ_c,0 ≈ 1 * 10^-26 kg/m³
≈ 6 protons/m³

this is also already on a different card (not sure where)

Question 12

Density Parameter

Answer 12

Ω = ρ₀ / ρ_c,0

Question 13

Temporal Evolution (list of parameters)

Answer 13

Parameters evolve with time:

• a(t)
• a_dot(t)
• H(t)
• ρ(t)
• Ω(t)

Question 14

Energy Density

Answer 14

energydensity = matter density for a matter dominated universe
ρ · c² = ρ₀ · c² · (a₀ / a)³

Question 15

Critical Solution

in the context of the Friedman Equation

Answer 15

• k = 0
• Universe expands ∝ t^2/3 (expands forever but gets slower)

Question 16

Closed Solution

Answer 16

• k = +1
• Universe:
  1. expands
  2. reaches max point
  3. collapses

Question 17

Open Solution

Answer 17

• k = -1
• Universe expands faster than for k = 0

Question 18

Einstein Equation

Answer 18

a_dot² / a² + k · c² / a² = 8 · π · G · ρ / 3 + Λ / 3