5. Homogeneous and Isotropic Spacetimes

Question 1

Perfect Fluid

Answer 1

-the only energy-momentum tensor compatible with homogeneity and isotropy

Question 2

Equation of State

Answer 2

-relates density and pressure:

P_ = wc²ρ, wϵℝ

Question 3

Possible Matter Content of the Universe

Answer 3

• dust/cold matter/collisionless matter/non-relativistic -> P_=0, w=0
• photons/radiation/hot matter/relativistic -> w=1/3
• dark energy -> w=-1

Question 4

The Fluid Equation for Perfect Fluids

Answer 4

-sub in P_ = wc²ρ
=>
ρ ∝ a^[-3(1+w)]

Question 5

Density and Scalre Factor Relationship for Dust, Radiation and Dark Energy

Answer 5

• dust => w=0 => ρ ∝ a^(-3)
• radiation => w=1/3 => ρ ∝ a^(-4)
• dark energy => w=-1 => ρ ∝ a^0, const.

Question 6

The Acceleration Equation

Answer 6

-need ρ>0 and P_≥0
=>
a''<0 
-so the universe CANNOT be static
-have either a'<0, contraction or a'>0 expansion

Question 7

The Freidmann Equation

Answer 7

• rearrange for (a’/a)²

- since ρ>0, if ϰ=0,-1 then a’ can never be zero so if expanding the universe will continue to expand forever

Question 8

Empty Universe

Answer 8

• no radiation, no dust, no dark energy => ρ=0, P=0
• sub into Friedmann equation
• assuming a²≠0, a’²=-ϰc²

Question 9

Empty Universe

ϰ=0

Answer 9

a’=0

-no curvature and no matter should recover Minkowski spacetime

Question 10

Empty Universe

ϰ=-1

Answer 10

-the Milne cosmological model
-assuming expansion:
a’ = c
a(t) = ct + K

Question 11

Single Component Cosmological Models

Dust

Answer 11

ρ>0 and P_=0
ρ∝a³ so ρa³ = constant
-let
M = 4πρa³/3
-solve Friedmann equation for a'
a'² = c²(A²/a - ϰ)
-where A² = 2GM/c²

Question 12

Single Component Cosmological Models
Dust
ϰ=0

Answer 12

a(t) = ao [t/to]^(2/3)

-where ao=a(to) and to is the present time

Question 13

Single Component Cosmological Models
Dust
ϰ=-1

Answer 13

ct = A²( 1/2 sinhx - x)

-where a = A²sinh²x

Question 14

Single Component Cosmological Models
Dust
ϰ=1

Answer 14

ct = A²(x - 1/2 sin(2x))

Question 15

Single Component Cosmological Models

Radiation

Answer 15

P_ = 1/3 ρc²
-fluid equation => ρa^4 = constant
-let ρa^4 = ρo ao^4
-Friedmann equatio
=>
a'² = 8πGρoao^4/3a² - ϰc²
-in very early universe when a<<1, the first term dominates the equation and curvature does not matter i.e. same behaviour in the early universe for all ϰ

Question 16

Single Component Cosmological Models
Radiation
ϰ=0

Answer 16

a(t) = ao [t/to]^(1/2)
-expansion forever
H(t) = a'(t)/a(t) = 1/2t
=>
a(t) = ao [2Hot]^(1/2)

Question 17

Redshift

Equation

Answer 17

Z = [λo - λe]/λe
-where, λo is the wavelength of the photon as observer and λe is the wavelength at source
-special relativity defines redshift as:
Z = √[ (1 + v/c)/(1 - v/c) ] - 1
-where v is the speed of the source relative to the observer

Question 18

Redshift for Galaxies

Answer 18

-can show that for v<
Ho = 100h km/s (Mpc)^(-1)
-where h is uncertainty in Ho

Question 19

Modern Interpretation of Hubble’s Expansion

Answer 19

-position your coordinate system so the observer is at the origin and the galaxy is moving towards them along the z axis, i.e. θ=0, φ=0
-sub this into the metric, only remaining terms are in a(t) and the comoving coordinate χ
-consider movement of photon 1 emitted at (te, χe) and observed at (to, χo), by choice of coordinates, χo=0
-then consider photon 2 emitted a bit later at (te+dte, χe) and observed at (to+dto, χo)
=>
dto/a(to) = dte/a(te)
-so for light waves:
To/a(to) = Te/a(te)
-where T is the period of the wave, T=λ/c

Question 20

Generalised Hubble Law

Answer 20

-always true assuming isotropy and homogeneity
HZ = a(to)/a(te)
-expansion => a(to) > a(te)

Question 21

Recovering the Hubble Result From the Generalised Hubble Law

Answer 21

-if redshift is small, z«1,
-sub into metric and integrating line element from observer to source
-use MacLaurin expansion on a(te) and only keep linear terms
=>
Z = Ho/c l

Question 22

Hubble Constant and the Age of the Universe

Answer 22

-for ϰ=0, a(t) ∝ t^α
=> a’(t) = α/t a(t)
-so, a’/a = α/t = H(t)
-at present, Ho = α/to and to is the current age of the universe:
to = α/Ho
-using α=2/3 for dust or α=1/2 for radiation and the observed value of Ho:
to ~ billions of years

Question 23

How can we determine ϰ if we know the matter content of the universe?

Answer 23

-take the Friedmann equation, sub in H(t)=a’/a and rearrage:
ρ - 3/8πG H² = ϰ 3c²/8πGa²
-since 3c²/8πGa² is positive, the sign of the LHS tells us the sign of ϰ

Question 24

Critical Density

Answer 24

-define
ρc = ρc(t) = 3/8πG H²
-so that
ρ - ρc = ϰ 3c²/8πGa²
-then ϰ is +1 if ρ>ρc, 0 if ρ=ρc and 1 if ρ

Question 25

Critical Parameter

Answer 25

-define
Ω = ρ/ρc
=>
1 - Ω = ϰc²/a²H²