Cosmology

Question 1

Friedmann-Robertson-Walker Metric

Definition

Answer 1

• isotropic - the same in all directions
• homogeneous - the same in all places
• expanding
• for timelike paths dτ = dt, so t is real clock time everywhere, this MUST be true for homogeneity

Question 2

Friedmann-Robertson-Walker Metric

Lightlike Radial Paths

Answer 2

-radial => dθ = dφ = 0
-lightlike => ds² = 0
=>
c²dt² = a² dr²/[1-kr²]
=>
cdt/a = dr/√[1-kr²]
-where all functions of time are now on the LHS
-this is now a relation between comoving distances

Question 3

Redshift

Description and Distance

Answer 3

-consider a light source at a fixed coordinate r1
-it emits a photon packet between times t1 -> t1 + Δt1
-sometime later we detected these at to -> to + Δto
-if we choose that the photons were released at r=0 then the distance travelled by the photons released at t1 is identical to that travelled by those released at t1 + Δt1
=>
[t1,t0]∫ cdt/a = [t1+Δt1,to+Δt0] ∫ cdt/a
= [r1, 0] ∫ dr/√[1-kr²]

Question 4

Redshift

Small Δt

Answer 4

-for small Δt the integral:
[t1,t0]∫ cdt/a = [t1+Δt1,to+Δt0] ∫ cdt/a
= [t1,t0]∫ cdt/a + cΔto/a(to) - cΔt1/a(t1)
-cancel terms:
cΔto/a(to) = cΔt1/a(t1)
=>
Δto/a(to) = Δt1/a(t1)

Question 5

Redshift

Frequency

Answer 5

-frequency is proportional to 1/Δt
-as the universe expands, a(t) gets larger so a photon experiences a frequency shift:
νo = ν1 a1/a0

Question 6

Redshift

Wavelength

Answer 6

• wavelength is proportional to a
• as the universe expands a(t) gets larger so a photon experiences a wavelength shift
• light redshifts as the universe expands

Question 7

Redshift

Formal Definition, z

Answer 7

z = λo - λe / λe
-where λo is the observed wavelength, λe is the emitted wavelength
-since λo/λe = ao/ae:
z = ao/ae - 1
-this is as observed by us at z=0

Question 8

Cosmological Constant

Definition

Answer 8

• Einstein introduced the cosmological constant because he wanted a static stable universe
• any small perturbation in a static universe leads to collapse
• adding a term in the Field Equation that acts like a negative pressure to counteract this

Question 9

Friedmann-Robertson-Walker Metric

Equation of State

Answer 9

• treat as a perfect fluid but with U^μ = (c,0,0,0)

- i.e. the fluid is perfectly at rest since we are using comoving coordinates

Question 10

How are the Friedmann equations derived?

Answer 10

-using the Friedmann-Robertson-Walker metric, sub in the Ricci and Stress tensors into the Field Equations

Question 11

The Fluid Equation

When can it be solved?

Answer 11

-can solve the fluid equation if we know the equation of state

Question 12

The Fluid Equation

Equation of State

Answer 12

p = wρc²
-where w is the equation of state parameter:
w = 0 => dust (non-interacting matter)
w = 1/3 => radiation
w = -1 => comological constant

Question 13

The Fluid Equation

Solution

Answer 13

ρ ∝ a^[-3(1+w)]
-and
ρ = ρo a^[-3(1+w)]
-where ρo is the density at the present day and the scale factor ao is taken to be 1

Question 14

Age of the Universe

Answer 14

-sub the solution to the fluid equation into the Friedmann equation

Question 15

Hubble Parameter

Answer 15

H = a’/a

-and at the present day: Ho = ao’/ao

Question 16

Density Parameter

Answer 16

Ω = ρ/ρcrit
-where ρcrit = H²/[8πG/3], i.e. the value of ρ when k=0
-and at the present day:
Ωo = ρo/ρcrit,o

Question 17

Ωo and K

Answer 17

Ωo > 1 => k is positive

• high density -> closed spherical spatial sections
• eventually reach the point where the universe contains enough stuff for its own self-gravity to balance expansion

Ωo < 1 => k is negative

• low density -> unbound hyperbolic sections, potentially infinite in size
• if w > -1/3 you get free expansion since the effect of the mass/energy density has diluted -> 0 more rapidly than the curvature term

Question 18

Is the universe stable?

Answer 18

• from the Acceleration Equation if we have real stuff in the universe ( 1+3w > 0) the universe is always decelerating
• expansion turns into contraction

Question 19

w < -1/3

Answer 19

w < - 1/3
=>
a’’ / a > 0 (since 1+3w < 0)
-so the expansion accelerates
w = -1 referes to ρ = const. (cosmological)
w < -1 violates the strong energy condition (energy density constant or increasing as universe expands)

Question 20

Ages of w ≥ -1/3 Universes

Answer 20

-the age depends only on ρ, Ωo and Ho
-can derive an exact expression if you ignore curvature
-then age is proportional to 1/√[Ωo]
=>
-low density universe is older at the same value of scale factor

Question 21

Ruler Distance

Answer 21

Druler = ∫ a dr/√[1-kr²]

Question 22

Co-moving Physical Distance

Answer 22

Dphys = Druler / a = ∫ dr/√[1-kr²]

Question 23

Ruler Distance and Hubble’s Law

Answer 23

V = Druler’
= a’ Druler/a
= a’/a Druler
V = H Druler
-Hubble’s Law - galaxies nearby expand linearly with distance
-this velocity depends on the coordinate r and t i.e. id frame dependent

Question 24

Photons Moving Radially

Answer 24

cdt/a = dr/√[1-kr²] = dDphys
-want to integrate this to give the comoving distance:
∫ cdt/a = c ∫ dt/da da/a = c ∫ da/a’a
-where a’ can be found from the Friedmann equation

Question 25

Dphys and r

Answer 25

• r is a real radial coordinate analogous to what we see in flat space
• at small r, Dphys -> r but in general Dphys is not a ‘simple’ radial coordinate
• if we identify v=cz and z > 1 => v > c
• but this is because we are trying to apply a LOCAL property of the radial coordinates to a situation where that simply does not apply
• this is a classic example of a ‘non’-problem in GR that results from making flawed (Newtonian inspired) assumptions

Question 26

Dynamics of the Cosmological Constant

FRW with Λ only

Answer 26

(a’/a)² = Λc²/3 = H²
=>
a = ai exp[√(Λc²/3) (t-ti)]
-boundary conditions: from ti to present time, t
-this is exponential expansion starting at time ti

Question 27

Dynamics of the Cosmological Constant

Distances

Answer 27

-for t > > ti :
Dphys = √(3/Λc²) c/a1
-the horizon size is [0,t] ∫cdt/a
-so if ai is the point at which Λ becomes dominant, this is just the ~ horizon at that time
-what we can see stays the same size, but the objects in it expand away beyond that

Question 28

Conservation of Energy

Answer 28

-photons redshift when the universe expands so they lose energy, is energy conserved?
-on a very local basis it is since:
T^vμ_;μ = 0
-in GR, ‘global’ energy is conserved if the geometry itself is constant, but even the term global makes little sense in an infinite expanding universe
-GR does not (in most cases) have a good concept of energy to be conserved, nor does it conserve it
-this is a result of moving to curved spacetimes that change dynamically rather than a sign that GR is ‘wrong’