T2: 4. Cosmology

Question 1

Define isotropy

Answer 1

There is no preferred direction in space; the universe looks the same from all angles. (Idea of spherical symmetry)

Question 2

Define Homogeneity (in space)

Answer 2

The universe is the same at all spacial points; i.e. given two spacial points, transformations between them leaves the metric invariant

Question 3

State the Copernican principle

Answer 3

Space is isotropic and homogeneous

Question 4

What does homogeneous and isotropic impart on the spatial part of the metric?

Answer 4

H. The time component can have have no spacial dependence: time must behave the same at all points in space.

I. There are no time, space cross terms; there cannot be a preferred direction.

Question 5

State the Hubble constant H_0

Answer 5

H_0 = a^dot (t_1)/a(t_1)

Question 6

State the three values of κ and what kind of universe they correspond to

Answer 6

κ =
0 - flat space
1 - closed universe (sphere s3)
-1 - open universe (hyperbolic)

Question 7

What form does the equation of state take?

Answer 7

p=ωρ

Question 8

What values do p and ω take for matter?

Answer 8

p = 0
ω = 0

Question 9

What values do p and ω take for radiation?

Answer 9

p=ρ/3
ω = 1/3

Question 10

What values do p and ω take for vacuum?

Answer 10

p=-ρ
ω =-1

Question 11

State the Friedmann equation

Answer 11

(a^dot/ a)^2 + κ/a^2 = (8πG/3) ρ

Question 12

State the Raychaudri equation

Answer 12

(a^ddot/ a) = -(4πG/3) (ρ+3p)

Question 13

State the conservation equation

Answer 13

ρ^dot + 3(a^dot/ a)(ρ+p) = 0

Question 14

What condition does homogeneity impart on the metric?

Answer 14

Constant Ricci scalar

Question 15

What constant value does the Ricci scalar take on in FRW metric?

Answer 15

6κ

Question 16

How is the metric/Ricci tensor/Ricci scalar affected by a positive scalar?

Answer 16

scaled by λ; unchanged; scaled by λ^-1

Question 17

What is the general spacial metric for homogeneous, isotropic manifolds?

Answer 17

dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2

Question 18

What is the general spacial metric for homogeneous, isotropic manifolds?

Answer 18

dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2

Question 19

What spacial geometry does a closed universe have? Describe it?

Answer 19

Three-sphere, parameterised by (χ,θ,ϕ). For each χ ∈ [0,π] there is a two-sphere. The two spheres get bigger til π/2 and then smaller.

Positive curvature

Question 20

What coordinate transformations do we use for a closed universe?

Answer 20

χ = sin^-1(r)

dχ = dr/sqrt(1-r^2)

Question 21

What spacial geometry does an open universe have? Describe it?

Answer 21

Negatively curved manifold; a hyperbolic space parameterised by (χ,θ,ϕ). For each χ∈[0,∞) there is a two-sphere, however they keep getting bigger to ∞.

Question 22

What coordinate transformations do we use for an open universe?

Answer 22

χ = sinh(r)

dχ = dr/sqrt(1+r^2)

Question 23

A maximally symmetric space in d dimensions has how many killing vectors?

Answer 23

N_d = d(d+1)/2

Question 24

Define comoving observers

Answer 24

Observers that are at rest with respect to spacial coordinates; still moving through time.

Question 25

What is the ratio of time interval to scale factor for cosmological redshift?

Answer 25

a(t_2)/a(t_1) = Δt_2/Δt_1

Question 26

Give the formula for cosmological redshift parameter z

Answer 26

z = Δt_2/Δt_1 -1

Question 27

How does the parameter z relate to how much a object has been redshifted?

Answer 27

More z implies more redshift since the receiving interval has increased.

Question 28

For a small t_2 - t_1, what is the instantaneous distance between observers

Answer 28

a(t_1)r_1

Question 29

Define the Hubble constant H_0

Answer 29

The ratio of a^dot (t_1)/a(t_1) at present

Question 30

Given a small distance between observers, give the redshift formula

Answer 30

z ≈ H_0 d

where d = t_2 - t_1

Question 31

How do we find the Friedmann and Raychaudhuri equations?

Answer 31

Friedmann: R_tt part of Ricci tensor

Raychaudri: R_ij part of Ricci tensor (all proportional, find R_θθ)

Question 32

How do we find the conservation equation from Freidmann and Raychaudhuri?

Answer 32

Multiply Friedmann by a^2 and take time derivative. Eliminate a^ddot with raychauduri.

Question 33

Why is the conservation equation so-called?

Answer 33

It is analogous to writing ∇_μ T^μν = 0: the conservation of the stress tensor.

Question 34

What does finding a reverse chain rule on continuity tell us?

Answer 34

ρ ~ ρ_0/(a^3(1+w))

Question 35

How does density dilute for matter and radiation universes?

Answer 35

Matter: ρ ~ ρ_0/a^3

Radiation: ρ ~ ρ_0/a^4

Question 36

How does density dilute for a vacuum universe? What does this tell us about the Friedmann eq?

Answer 36

It remains constant. We add an extra term to the RHS of sol Friedmann Λ/3 and take density as a func of a.

Question 37

How do scale factors for matter/radiation, and vacuum domination differ?

Answer 37

Matter/radiation increases (and for k=1 decreases) polynomialy; vacuum increases exponentially

Question 38

How do we define dimensionless density parameters?

Answer 38

Take the Friedmann eq with a little matter, radiation and cosmological constant. Divide by H^2 and notice each term must equal 1.

Question 39

What does the sign of the density parameter Ω indicate about our universe

Answer 39

Ω > 1 → κ=1 : closed

Ω = 1 → κ=0 : flat space

Ω < 1 → κ=-1 : open

Question 40

What is dark matter? How did we find it?

Answer 40

An undetected particle that does not give off ordinary light. It makes us for the discrepancy between predicted and observed matter density.

Question 41

Give a speedy timeline of l’universe

Answer 41

Big bang → radiation domination → matter domination (w/recombination) → vacuum domination