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.
What is the ratio of time interval to scale factor for cosmological redshift?
a(t_2)/a(t_1) = Δt_2/Δt_1
Give the formula for cosmological redshift parameter z
z = Δt_2/Δt_1 -1
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.
For a small t_2 - t_1, what is the instantaneous distance between observers
a(t_1)r_1
Define the Hubble constant H_0
The ratio of a^dot (t_1)/a(t_1) at present
Given a small distance between observers, give the redshift formula
z ≈ H_0 d
where d = t_2 - t_1
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)
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.
Why is the conservation equation so-called?
It is analogous to writing ∇_μ T^μν = 0: the conservation of the stress tensor.
What does finding a reverse chain rule on continuity tell us?
ρ ~ ρ_0/(a^3(1+w))
How does density dilute for matter and radiation universes?
Matter: ρ ~ ρ_0/a^3
Radiation: ρ ~ ρ_0/a^4
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.
How do scale factors for matter/radiation, and vacuum domination differ?
Matter/radiation increases (and for k=1 decreases) polynomialy; vacuum increases exponentially
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.
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
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.
Give a speedy timeline of l’universe
Big bang → radiation domination → matter domination (w/recombination) → vacuum domination
