COSMOLOGY – PHYS 30392
SINGLE-COMPONENT UNIVERSES Part II
http://www.jb.man.ac.uk/~gp/
giampaolo.pisano@manchester.ac.uk
Giampaolo Pisano - Jodrell Bank Centre for Astrophysics
The University of Manchester - March 2013
SINGLE-COMPONENT UNIVERSES
Energy Density Evolution
→ Curvature Only Universes
Spatially Flat Universes
Matter Only Universes
Radiation Only Universes
Lambda Only Universes
References: Ryden, Introduction to Cosmology - Par. 5.2
Proper distance: Another derivation 1/2
- Earlier we have derived the proper distance of a galaxy assuming to know
its co-moving coordinates (r, θ, φ ), at a fixed time t0 :
x
r
d p (t0 ) = a (t0 ) ∫ dr = a (t0 )r - Current Proper Distance
0
( r,θ,φ )
x
(0,0,0)
- However, we measure red-shifts and we infer when the light was emitted
We need to find an expression involving emission and observation times
- During the travel from the galaxy to us, the light will follow a null geodesic:
→ ds 2 = −c 2 dt 2 + a(t ) 2 [dr 2 + Sκ (r ) 2 dΩ 2 ] = 0
ds 2 = 0
(θ , ϕ ) = const → dΩ = 0
2
2
2
c dt = a(t ) dr
2
→ cdt = a (t )dr
→ c
dt
= dr
a(t )
Proper distance: Another derivation 2/2
- Suppose that the light is emitted by the galaxy at te and observed at t0,
the null geodesic will satisfy:
r
d p (t0 )
dt
→ c∫
= ∫ dr = r =
te a (t )
0
a(t0 )
x
t0
te
x
t0
dt
d p (t0 ) = ∫ dr = c ∫
- Current Proper Distance
0
te a (t )
r
t0
Valid in any universe whose geometry is described by a R-W metric
The proper distance depends on how the scale factor a(t) varies with time
- We will apply this formula to different types of universes
Curvature Only Universes: Friedmann equation solutions
- Let’s consider a simple empty Universe:
No matter No radiation No cosmological constant
Λ=0
- The Friedmann equation becomes:
2
κ c2 1
Λ
 a&  8π G
ε(t)
−
+
  =
3c 2
R02 a(t)2 3
a
ε=0
κ c2
a& = − 2
R0
2
- Let’s the different possible solutions:
κ=0
a& = 0
Empty, static, spatially flat Universe
(Geometry described by Minkowski metric)
κ=1
a& = Im # Positively curved empty Universe forbidden
κ = -1
a& = ±
c
R0
Negatively curved empty Universe must
expand or contract
Curvature Only Universes: Negatively curved solution
Λ=0
- Let’s consider an expanding empty Universe:
a& =
c
R0
da c
→
=
dt R0
t
a(t ) =
t0
→ da =
κ =-1
ε=0
c
dt
R0
R0
with: t0 =
c
There is no gravitational force and a(t) increases linearly with time
- The Hubble parameter is:
a&
1 t0
1
H (t ) =
=
=
a
t0 t
t
t0 =
1
H0
The age of the Universe is exactly equal to the Hubble time
Empty Negatively Curved Universe: Properties
- Plotting the scale factor with the time:
κ = -1
a(t)
ε =0
∝t
Λ=0
Scale Factor vs Time
in a Negatively curved,
Empty universe
1/H0
t0
t
Note
- An empty expanding universe can be used when ε << εc :
The linearity in a(t) is a good approximation of the real one
Empty Negatively Curved Universe: Proper distance
κ =-1
ε=0
- Let’s calculate the proper distance in this type of universe:
Λ=0
t
a(t ) =
→
t0
t 0 dt
dt
= ct0 ∫
d p (t0 ) = c ∫
te a (t )
te t
t0
t0
d p (t0 ) = ct0 ln
te
- Reminding the redshift - scale factor relation:
a (t0 )
t0
1
=
1+ z =
=
a (te )
a(te )
te
- We can express the proper distance in terms of redshift:
c
d p (t0 ) = ct0 ln(1 + z ) =
ln(1 + z )
H0
In an empty universe we can see objects currently at arbitrary large distances
SINGLE-COMPONENT UNIVERSES
Energy Density Evolution
Curvature Only Universes
→ Spatially Flat Universes
Matter Only Universes
Radiation Only Universes
Lambda Only Universes
References: Ryden, Introduction to Cosmology - Par. 5.3
Spatially Flat Universes: Friedmann equation solutions 1/2
- Let’s consider a flat universe with a single component w:
ε ≠0
κ=0
and
Λ=0
- We have derived the energy density relation with a(t): →
εw =
- The Friedmann equation becomes:
2
κ c2 1
Λ
 a&  8π G
=
ε(t)
−
+
 
3c 2
R02 a(t)2 3
a
→
a& 2 8π G
− 3− 3 w
=
ε
a
0
a2
3c 2
8πGε0 −(1+3w)
a& =
a
2
3c
2
ε w, 0
a 3(1+ w)
κ=0
ε≠0
Spatially Flat Universes: Friedmann equation solutions 2/2
- Let’s make a guess that the solution of the equation has the form:
Λ=0
a ∝ tq
- Inserting into the equation, we can find q:
a& 2 =
8πGε0 −(1+3w)
a
2
3c
→ 2q − 2 = −(1 + 3w)q
a& ∝ qt q-1 → a& 2 ∝ t 2q-2
a − (1+3w) ∝ t − (1+3w) q
→ 3q(1 + w) = 2
2
q=
3 + 3w
(with w ≠ -1)
- The scale factor will be:
t 
a(t ) =  
 t0 
2 /( 3+ 3 w )
- General Scale Factor for w Component
Spatially Flat Universes: Properties
- From the scale factor we can derive:
κ=0
ε≠0
Λ=0
2
1
 a& 
H0 =   =
- Hubble constant
 a t =t0 3(1 + w) t0
2
1
t0 =
- Age of the Universe
3(1 + w) H 0
- Inverting the above equation:
- The proper distance to the most distant object we can see today is:
te = 0
t0 : today
d hor (t0 ) = c ∫
t0
0
dt
c
2
=
- Horizon distance
a(t ) H 0 1 + 3w
w > -1/3 Finite Horizon: we can see a finite portion of the Universe
(Visible Universe ≡ all points causally connected with observer)
w ≤ -1/3 Infinite Horizon: we can see every point in space
(Very distant objects will be redshifted)
SINGLE-COMPONENT UNIVERSES
Energy Density Evolution
Curvature Only Universes
Spatially Flat Universes
→ Matter Only Universes
Radiation Only Universes
Lambda Only Universes
References: Ryden, Introduction to Cosmology - Par. 5.4
Liddle, Introduction to Modern Cosmology - Par. 5.3.1
Matter Only Universes: Properties 1/2
- Let’s consider a flat universe containing only non-relativistic matter:
ε ≠0
κ=0
w≅0
and
Λ=0
- We have:
t 
a(t ) =  
 t0 
t0 =
d hor (t0 ) =
2 /( 3+ 3 w )
2
1
3(1 + w) H 0
c
2
H 0 1 + 3w
t 
a(t ) =  
 t0 
t0 =
Einstein - de Sitter
universe
2/3
2
3H 0
2c
d hor (t0 ) = 3ct0 =
H0
- Scale Factor
- Age of the Universe
- Horizon distance
Matter Only Universes: Properties 2/2
- In addition:
  te  2c 
dt
1 


d p (t0 ) = c ∫
= 3ct0 1 −   =
1−
- Proper distance


t e (t / t ) 2 / 3
1+ z 
0
  t0  H 0 
t0
κ=0
a(t)
ε≠0
∝ t 2/3
w≅0
Λ=0
Scale Factor vs Time
in a Flat,
Matter dominated universe
2/(3H0)
t0
t
SINGLE-COMPONENT UNIVERSES
Energy Density Evolution
Curvature Only Universes
Spatially Flat Universes
Matter Only Universes
→ Radiation Only Universes
Lambda Only Universes
References: Ryden, Introduction to Cosmology - Par. 5.5
Liddle, Introduction to Modern Cosmology - Par. 5.3.2
Radiation Only Universes: Properties 1/2
- Let’s consider a flat universe containing relativistic matter:
ε ≠0
κ=0
w = 1/3
and
Λ=0
- We have:
t 
a(t ) =  
 t0 
t0 =
2 /( 3+ 3 w )
2
1
3(1 + w) H 0
c
2
d hor (t0 ) =
H 0 1 + 3w
1/ 2
t 
a(t ) =  
 t0 
t0 =
1
2H 0
d hor (t0 ) = 2ct0 =
c
H0
- Scale Factor
- Age of the Universe
- Horizon distance
Note: It is coincident with the
Hubble distance
Radiation Only Universes: Properties 2/2
- In addition:
dt
t e (t / t )1 / 2
0
d p (t0 ) = c ∫
t0
  t 1/ 2  c 
z 
= 2ct0 1 −  e   =
1
−
- Proper distance


  t0   H 0  1 + z 
κ=0
a(t)
ε≠0
∝ t 1/2
w=1/3
Λ=0
Scale Factor vs Time
in a Flat,
Radiation dominated universe
1/(2H0)
t0
t
SINGLE-COMPONENT UNIVERSES
Energy Density Evolution
Curvature Only Universes
Spatially Flat Universes
Matter Only Universes
Radiation Only Universes
→ Lambda Only Universes
References: Ryden, Introduction to Cosmology - Par. 5.6
Lambda Only Universes: Friedmann equation solution
- Let’s consider a flat, lambda dominant universe:
ε =0
κ=0
de Sitter universe
and
Λ≠0
w = -1
- The Friedmann equation becomes:
2
κ c2 1
Λ
 a&  8π G
+
  = 2 ε(t) − 2
3c
R0 a(t)2 3
a
→ d (ln a ) = H 0 dt
a(t ) = e
a& 2 8π G
8π GεΛ
&
→
a
=
a → a& = H 0 a
→ 2 =
ε
Λ
2
2
3c
a
3c
H 0 ( t −t0 )
a&
8πGεΛ
=
- Scale Factor
a
3c 2
Note: H(t) independent from t
with: H 0 =
A flat universe with only a cosmological constant is exponentially expanding
Lambda Only Universes: Properties
- In addition:
t0 = ∞ - Age of the Universe
d hor (t0 ) = ∞ - Horizon distance
d p (t0 ) = c ∫
dt
t0
te
e
H 0 ( t −t 0 )
=
c H 0 ( t0 −te )
c
e
−1 =
z - Proper distance
H0
H0
[
]
κ=0
a(t)
ε=0
Λ≠0
∝e
w = -1
H 0 ( t −t 0 )
Scale Factor vs Time
in a Flat,
Lambda dominated universe
¶
t0
t
Single-Component Universes: Summary
κ=0
ε=0
Λ≠0
κ =-1
ε=0
w = -1
Λ=0
κ=0
ε≠0
w≅0
Λ=0
κ=0
ε≠0
w=1/3
Λ=0
All these universes continue to expand forever if they expand at t = t0
Next Topic: MULTIPLE-COMPONENT UNIVERSES