T2: 4. Cosmology Flashcards
Define isotropy
There is no preferred direction in space; the universe looks the same from all angles. (Idea of spherical symmetry)
Define Homogeneity (in space)
The universe is the same at all spacial points; i.e. given two spacial points, transformations between them leaves the metric invariant
State the Copernican principle
Space is isotropic and homogeneous
What does homogeneous and isotropic impart on the spatial part of the metric?
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.
State the Hubble constant H_0
H_0 = a^dot (t_1)/a(t_1)
State the three values of κ and what kind of universe they correspond to
κ =
0 - flat space
1 - closed universe (sphere s3)
-1 - open universe (hyperbolic)
What form does the equation of state take?
p=ωρ
What values do p and ω take for matter?
p = 0
ω = 0
What values do p and ω take for radiation?
p=ρ/3
ω = 1/3
What values do p and ω take for vacuum?
p=-ρ
ω =-1
State the Friedmann equation
(a^dot/ a)^2 + κ/a^2 = (8πG/3) ρ
State the Raychaudri equation
(a^ddot/ a) = -(4πG/3) (ρ+3p)
State the conservation equation
ρ^dot + 3(a^dot/ a)(ρ+p) = 0
What condition does homogeneity impart on the metric?
Constant Ricci scalar
What constant value does the Ricci scalar take on in FRW metric?
6κ
How is the metric/Ricci tensor/Ricci scalar affected by a positive scalar?
scaled by λ; unchanged; scaled by λ^-1
What is the general spacial metric for homogeneous, isotropic manifolds?
dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2
What is the general spacial metric for homogeneous, isotropic manifolds?
dσ^2 = (dr^2)/(1-κr^2 ) + r^2 dΩ^2
What spacial geometry does a closed universe have? Describe it?
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
What coordinate transformations do we use for a closed universe?
χ = sin^-1(r)
dχ = dr/sqrt(1-r^2)
What spacial geometry does an open universe have? Describe it?
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 ∞.
What coordinate transformations do we use for an open universe?
χ = sinh(r)
dχ = dr/sqrt(1+r^2)
A maximally symmetric space in d dimensions has how many killing vectors?
N_d = d(d+1)/2
Define comoving observers
Observers that are at rest with respect to spacial coordinates; still moving through time.