T2: 4. Cosmology Flashcards

1
Q

Define isotropy

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Define Homogeneity (in space)

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

State the Copernican principle

A

Space is isotropic and homogeneous

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

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

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

State the Hubble constant H_0

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

What form does the equation of state take?

A

p=ωρ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What values do p and ω take for matter?

A

p = 0
ω = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What values do p and ω take for radiation?

A

p=ρ/3
ω = 1/3

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

What values do p and ω take for vacuum?

A

p=-ρ
ω =-1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

State the Friedmann equation

A

(a^dot/ a)^2 + κ/a^2 = (8πG/3) ρ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

State the Raychaudri equation

A

(a^ddot/ a) = -(4πG/3) (ρ+3p)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

State the conservation equation

A

ρ^dot + 3(a^dot/ a)(ρ+p) = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What condition does homogeneity impart on the metric?

A

Constant Ricci scalar

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

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

A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

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

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

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

A

dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2

18
Q

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

A

dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2

19
Q

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

A

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

20
Q

What coordinate transformations do we use for a closed universe?

A

χ = sin^-1(r)

dχ = dr/sqrt(1-r^2)

21
Q

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

A

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 ∞.

22
Q

What coordinate transformations do we use for an open universe?

A

χ = sinh(r)

dχ = dr/sqrt(1+r^2)

23
Q

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

A

N_d = d(d+1)/2

24
Q

Define comoving observers

A

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

25
What is the ratio of time interval to scale factor for cosmological redshift?
a(t_2)/a(t_1) = Δt_2/Δt_1
26
Give the formula for cosmological redshift parameter z
z = Δt_2/Δt_1 -1
27
How does the parameter z relate to how much a object has been redshifted?
More z implies more redshift since the receiving interval has increased.
28
For a small t_2 - t_1, what is the instantaneous distance between observers
a(t_1)r_1
29
Define the Hubble constant H_0
The ratio of a^dot (t_1)/a(t_1) at present
30
Given a small distance between observers, give the redshift formula
z ≈ H_0 d where d = t_2 - t_1
31
How do we find the Friedmann and Raychaudhuri equations?
Friedmann: R_tt part of Ricci tensor Raychaudri: R_ij part of Ricci tensor (all proportional, find R_θθ)
32
How do we find the conservation equation from Freidmann and Raychaudhuri?
Multiply Friedmann by a^2 and take time derivative. Eliminate a^ddot with raychauduri.
33
Why is the conservation equation so-called?
It is analogous to writing ∇_μ T^μν = 0: the conservation of the stress tensor.
34
What does finding a reverse chain rule on continuity tell us?
ρ ~ ρ_0/(a^3(1+w))
35
How does density dilute for matter and radiation universes?
Matter: ρ ~ ρ_0/a^3 Radiation: ρ ~ ρ_0/a^4
36
How does density dilute for a vacuum universe? What does this tell us about the Friedmann eq?
It remains constant. We add an extra term to the RHS of sol Friedmann Λ/3 and take density as a func of a.
37
How do scale factors for matter/radiation, and vacuum domination differ?
Matter/radiation increases (and for k=1 decreases) polynomialy; vacuum increases exponentially
38
How do we define dimensionless density parameters?
Take the Friedmann eq with a little matter, radiation and cosmological constant. Divide by H^2 and notice each term must equal 1.
39
What does the sign of the density parameter Ω indicate about our universe
Ω > 1 → κ=1 : closed Ω = 1 → κ=0 : flat space Ω < 1 → κ=-1 : open
40
What is dark matter? How did we find it?
An undetected particle that does not give off ordinary light. It makes us for the discrepancy between predicted and observed matter density.
41
Give a speedy timeline of l'universe
Big bang → radiation domination → matter domination (w/recombination) → vacuum domination