COSMOLOGY – PHYS 30392
MULTIPLE-COMPONENT UNIVERSES
Λ
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
Multiple-Component Universes: Friedmann equation 1/2
- Earlier we have derived the following form for the:
2
8
G
c
1
π
κ
- Friedmann Equation
ε
H (t ) 2 =
(
t
)
−
2
2
2
3c
R0 a(t )
( Here the energy density ε(t) includes all the components, including Λ )
- We derived also the relation between κ and Ω:
1 − Ω(t ) = −
κ c2
2
0
2
R a(t ) H (t )
2
→
at present
- We defined also the critical density today as:
εc,0
3c 2
2
=
H0
8π G
−
κ c2
2
0
R H
2
0
= 1 − Ω0
Multiple-Component Universes: Friedmann equation 2/2
- Substituing into the Friedmann equation:
→
1 − Ω0
H (t ) 2
8π G
= 2 2 ε (t ) +
2
H0
3c H 0
a(t ) 2
→
H (t ) 2 ε (t ) 1 − Ω 0
=
+
2
ε c ,0 a(t ) 2
H0
- Considering only matter, radiation and Λ:
→
 1 − Ω0
H2
1  ε r ,0 ε m,0
 4 + 3 + ε Λ ,0  +
=
2
H 0 ε c,0  a
a
a2

H 2 Ω r ,0 Ω m,0
1 − Ω0
= 4 + 3 + Ω Λ ,0 +
2
H0
a
a
a2
Ω r ,0
ε r ,0
=
ε c,0
Ω m,0
ε m,0
=
ε c,0
Ω Λ ,0
- Friedmann Equation
ε Λ,0
- Density parameters
=
ε c,0
MULTIPLE-COMPONENT UNIVERSES
→ Matter and Curvature
Matter and Lambda
Matter, Curvature and Lambda
Radiation and Matter
Benchmark Model
References: Ryden, Introduction to Cosmology - Par. 6.1
Matter and Curvature: Solutions 1/4
- We have already studied a flat, matter dominated universe:
ε
κ=0
≠0
w≅0
and
Λ=0
This universe expands outward forever as t2/3
Big Chill
- Let’s consider now curved universes containing matter:
ε
κ≠0
≠0
w≅0
and
Λ=0
The fate of these universes depends on the density parameter Ω0
Matter and Curvature: Solutions 2/4
- In a curved, matter-dominated universe:
Ω m,0 = Ω 0
H 2 Ω0 1 − Ω0
= 3 +
2
H0 a
a2
- Assume an initially expanding universe →
- It will it stop expanding when →
H0 > 0
H (t ) = 0
Ω0
Ω0 1 − Ω0
+ 1 − Ω0 = 0
→
+
=0 →
3
2
a
a
a
amax =
Ω 0 - Scale factor
Ω 0 − 1 at maximum
The solution depends on the value of Ω0
Matter and Curvature: Solutions 3/4
Ω0 > 1
amax =
Ω0
Ω0 − 1
κ = +1
- The equation for amax has solutions:
The universe will collapse after
‘ a finite time
( Finite in space and time)
Big Crunch
- After maximum time reversal of expansion phase
- During contraction the distant galaxies will look ‘blue-shifted’
Ω0 < 1
κ = -1
- Initially matter-dominated curvature-dominated:
‘
The universe will expand forever
( Infinite in space and time)
Big Chill
Matter and Curvature: Solutions 4/4
κ≠ 0
ε≠0
- In summary:
w≅0
Λ=0
κ = -1
κ=0
κ = +1
Ryden
Fig.6.1
Table 6.1
MULTIPLE-COMPONENT UNIVERSES
Matter and Curvature
→ Matter and Lambda
Matter, Curvature and Lambda
Radiation and Matter
Benchmark Model
References: Ryden, Introduction to Cosmology - Par. 6.2
Matter and Lambda: Solutions 1/3
- Let’s consider a flat universe containing both matter and Λ:
ε
κ=0
≠0
w≅0
and
Λ≠0
- Flat universe implies:
Ω0 = 1 → Ω Λ ,0 = 1 − Ω m,0
H 2 Ω m,0
= 3 + (1 − Ω m, 0 )
2
H0
a
The solution depends on the value of Ωm,0
H 2 Ω m,0
= 3 + (1 − Ω m, 0 )
H 02
a
Matter and Lambda: Solutions 2/3
Ωm,0 < 1
ΩΛ,0 > 0
- The RHS of the equation is always positive:
‘
The universe will expand forever
Ωm,0 > 1
ΩΛ,0 < 0 Big Chill
Λ<0
- Negative cosmological constant Attractive force
‘
The universe will cease to expand and collapse
after a relatively short finite time
Big Crunch
Matter and Lambda: Solutions 3/3
κ=0
ε≠0
- In summary:
w≅0
Λ≠0
Ωm,0=0.9
ΩΛ,0=0.1
Ωm,0 < 1
Ωm,0=1.0
ΩΛ,0=0
Ωm,0 > 1
Ωm,0=1.1, ΩΛ,0=−0.1
Ryden Fig.6.2
Observations exclude the possibility to live in a universe with Λ < 0
MULTIPLE-COMPONENT UNIVERSES
Matter and Curvature
Matter and Lambda
→ Matter, Curvature and Lambda
Radiation and Matter
Benchmark Model
References: Ryden, Introduction to Cosmology - Par. 6.3
Matter, Curvature and Lambda: Solutions 1/3
- Let’s consider a curved universe containing both matter and Λ:
ε
κ≠0
≠0
w≅0
and
Λ≠0
- Choosing different values for: → (Ω Λ , 0 , Ω m, 0 )
H 2 Ω m,0 1 − Ω m,0 − Ω Λ ,0
= 3 +
+ Ω Λ ,0
2
2
H0
a
a
- Friedmann Equation
A wide range of bahaviours is possible for a(t)
There are cases where an initially contracting universe
reaches a minimum and then expands again:
Big Bounce
Matter, Curvature and Lambda: Solutions 2/3
Big
Chill
- Solution examples:
κ≠0
ε≠0
w≅0
Λ≠0
κ=0
κ = -1
κ = +1
Big
Bounce
Loitering
κ = +1
Big
Crunch
Ryden Fig.6.4
There are other cases where the universe enters a phase
where a(t)~const for a long period of time :
Loitering
Matter, Curvature and Lambda: Solutions 3/3
Big Crunch
a(0) = 0 → amax → a(tCrunch ) = 0
Big Chill
a(0) = 0 → a(∞) = ∞
Loitering
Universes below dividing line between
Big Bounce - Big Chill
κ = +1
κ = -1
Ryden Fig.6.3
Big Bounce
a(0) ≠ 0
→ a(t bounce ) = amin → a(∞) = ∞

a& (0) < 0
MULTIPLE-COMPONENT UNIVERSES
Matter and Curvature
Matter and Lambda
Matter, Curvature and Lambda
→ Radiation and Matter
Benchmark Model
References: Ryden, Introduction to Cosmology - Par. 6.4
Radiation and Matter: Solutions
- Let’s consider a flat universe containing both matter and radiation:
ε
κ=0
≠0 and
w≅0
w = 1/3
Λ=0
- Around the time of radiation-matter equality:
H 2 Ω r ,0 Ω m,0
= 4 + 3
2
H0
a
a
- Friedmann Equation
- Solving the equation we obtain:
(
a ≈ (1.5
a ≈ 2 Ω r , 0 H 0t
)
1/ 2
Ω m , 0 H 0t
)
[a << arm ] - Radiation dominated
2/3
[a >> arm ] - Matter dominated
MULTIPLE-COMPONENT UNIVERSES
Matter and Curvature
Matter and Lambda
Matter, Curvature and Lambda
Radiation and Matter
→ Benchmark Model
References: Ryden, Introduction to Cosmology - Par. 6.5
Benchmark Model: Ingredients
- It assumes a flat universe that contains radiation, matter & Λ:
κ=0
ε≠0
w ≅ 0, 1/3
Λ≠0
List of Ingredients
------------------------
0.046
0.240
0.286
0.714
(WMAP)
Important Epochs
Ω 0 = Ω r ,0 + Ω m ,0 + Ω Λ ,0 ≅ 1.004 ± 0.004 - Total Density parameter
Benchmark Model: Solution
κ=0
ε≠0
- The scale factor as a function of time results:
Matter
phase
w ≅ 0, 1/3
Λ≠0
Cosmological
Constant
phase
Radiation
phase
Ryden Fig.6.5
As anticipated, our Universe was initially radiation dominated, then
matter dominated and recently entered a phase Λ dominated
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