Friedmann-Robertson-Walker Metric
Definition
- 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
Friedmann-Robertson-Walker Metric
Lightlike Radial Paths
-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
Redshift
Description and Distance
-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]
Redshift
Small Δt
-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)
Redshift
Frequency
-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
Redshift
Wavelength
- 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
Redshift
Formal Definition, z
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
Cosmological Constant
Definition
- 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
Friedmann-Robertson-Walker Metric
Equation of State
- 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
How are the Friedmann equations derived?
-using the Friedmann-Robertson-Walker metric, sub in the Ricci and Stress tensors into the Field Equations
The Fluid Equation
When can it be solved?
-can solve the fluid equation if we know the equation of state
The Fluid Equation
Equation of State
p = wρc² -where w is the equation of state parameter: w = 0 => dust (non-interacting matter) w = 1/3 => radiation w = -1 => comological constant
The Fluid Equation
Solution
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
Age of the Universe
-sub the solution to the fluid equation into the Friedmann equation
Hubble Parameter
H = a’/a
-and at the present day: Ho = ao’/ao
Density Parameter
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
Ωo and K
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
Is the universe stable?
- from the Acceleration Equation if we have real stuff in the universe ( 1+3w > 0) the universe is always decelerating
- expansion turns into contraction
w < -1/3
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)
Ages of w ≥ -1/3 Universes
-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
Ruler Distance
Druler = ∫ a dr/√[1-kr²]
Co-moving Physical Distance
Dphys = Druler / a = ∫ dr/√[1-kr²]
Ruler Distance and Hubble’s Law
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
Photons Moving Radially
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
