Cosmological Phase Transitions · 2005-02-03 · Cosmological Phase Transitions Origin and...
Transcript of Cosmological Phase Transitions · 2005-02-03 · Cosmological Phase Transitions Origin and...
Cosmological Phase Transitions
Origin and Implications
Guy Raz
A lecture in Particle Cosmology
Summer 2002
References:
E. W. Kolb and M. S. Turner,“The Early Universe,”
T. Vachaspati, hep-ph/0101270
R. H. Brandenberger, Rev. Mod. Phys. 57, 1 (1985).
M. Quiros,hep-ph/9901312.
Cosmological Phase Transitions 1
Invitation
Turning up the heat
Simple bare scalar potential:
V (φ) = −1
2m2φ2 +
λ
4!φ4 .
Cosmological Phase Transitions 2
Invitation
Turning up the heat
Simple bare scalar potential:
V (φ) = −1
2m2φ2 +
λ
4!φ4 .
Effective potential:
V Teff(φ) = −1
2m2φ2 +
λ
4!φ4
+
(
−m2 + λφ2
2
)2
64π2
[
log
(
−m2 + λφ2
2
µ2
)
− 1
2
]
Cosmological Phase Transitions 2
Invitation
Turning up the heat
Simple bare scalar potential:
V (φ) = −1
2m2φ2 +
λ
4!φ4 .
Effective potential at Finite temperature:
V Teff(φ) = −1
2m2φ2 +
λ
4!φ4
+
(
−m2 + λφ2
2
)2
64π2
[
log
(
−m2 + λφ2
2
µ2
)
− 1
2
]
+
(
−m2 + λφ2
2
)
T 2
24+−π290
T 4 + . . .
Cosmological Phase Transitions 2
Content
Outline
• How come?
• So what?
Cosmological Phase Transitions 3
Content
Outline
• How come?
• So what?
• Finite temperature field theory.
• The effective scalar potential.
• Phase transitions.
• Topological defects.
• Summary
Cosmological Phase Transitions 3
How come? - Finite temperature field theory
The finite temperature background
Replace the vacuum by a thermal bath:
〈0| O |0〉 =⇒∑
n
e−βE(n) 〈n| O |n〉 ,
where β ≡ 1/T .
Cosmological Phase Transitions 4
How come? - Finite temperature field theory
The finite temperature background
Replace the vacuum by a thermal bath:
〈0| O |0〉 =⇒∑
n
e−βE(n) 〈n| O |n〉 ,
where β ≡ 1/T .
The finite-temperature n-point Green’s functions:
Gβn(x1, . . . , xn) =
Tr[
e−βHT [φ(x1) . . . φ(x2)]]
Tr[e−βH]
Cosmological Phase Transitions 4
How come? - Finite temperature field theory
What makes it easy
e−βH is the time evolution operator.
The (analytic continued) thermal Green’s function has periodic
boundary conditions:
Tr[
e−βHφ(y0, ~y)φ(x0, ~x)]
= Tr[
φ(x0, ~x)e−βHφ(y0, ~y)
]
= Tr[
e−βHeβHφ(x0, ~x)e−βHφ(y0, ~y)
]
= Tr[
e−βHφ(x0 − iβ, ~x)φ(y0, ~y)]
.
Cosmological Phase Transitions 5
How come? - Finite temperature field theory
What makes it easy
e−βH is the time evolution operator.
The (analytic continued) thermal Green’s function has periodic
boundary conditions:
Tr[
e−βHφ(y0, ~y)φ(x0, ~x)]
= Tr[
φ(x0, ~x)e−βHφ(y0, ~y)
]
= Tr[
e−βHeβHφ(x0, ~x)e−βHφ(y0, ~y)
]
= Tr[
e−βHφ(x0 − iβ, ~x)φ(y0, ~y)]
.
So, for example:
Gβ2 (x0, ~x) = Gβ
2 (x0 − iβ, ~x) .
This is the “imaginary time formalism”.
Cosmological Phase Transitions 5
How come? - Finite temperature field theory
Thermal Feynman rules
Periodicity in normal space implies discretization of momentum
space.
The Feynman rules:
Propagator:i
p2 −m2with pµ ≡ (
2πin
β, ~p)
Loops:i
β
∞∑
n=−∞
∫
d3k
(2π)3
Vertices:β
i(2π)3δ(Σni)δ
3(Σki)
Cosmological Phase Transitions 6
How come? - The effective scalar potential
The effective scalar potential∗
∗You won’t learn it here, though.
starting with the (connected) generating functional
W [j] = −i log[∫
Dφ exp
(
i
∫
d4x(L+ Jφ)
)]
.
Cosmological Phase Transitions 7
How come? - The effective scalar potential
The effective scalar potential∗
∗You won’t learn it here, though.
starting with the (connected) generating functional
W [j] = −i log[∫
Dφ exp
(
i
∫
d4x(L+ Jφ)
)]
.
We define its Legandre transform
Γ[φ̄] =
(
W [J ]−∫
d4xJφ̄
)∣
∣
∣
∣
J: δW
δJ=φ̄
.
Cosmological Phase Transitions 7
How come? - The effective scalar potential
The effective scalar potential∗
∗You won’t learn it here, though.
starting with the (connected) generating functional
W [j] = −i log[∫
Dφ exp
(
i
∫
d4x(L+ Jφ)
)]
.
We define its Legandre transform
Γ[φ̄] =
(
W [J ]−∫
d4xJφ̄
)∣
∣
∣
∣
J: δW
δJ=φ̄
.
The effective scalar potential is obtained by assuming a constant
configuration and removing the space-time volume:
Γ[φ̄] =
∫
d4x Veff(φ̄) .
Cosmological Phase Transitions 7
How come? - The effective scalar potential
Why bother?∗
∗You still won’t learn it here.
Γ[φ̄] has some very nice properties:
Cosmological Phase Transitions 8
How come? - The effective scalar potential
Why bother?∗
∗You still won’t learn it here.
Γ[φ̄] has some very nice properties:
• Veff(φ̄) is the energy density when 〈a|φ |a〉 = φ̄.
⇓The minimum of Veff is the vacuum energy and φ̄
at the minimum is the expectation value.
Cosmological Phase Transitions 8
How come? - The effective scalar potential
Why bother?∗
∗You still won’t learn it here.
Γ[φ̄] has some very nice properties:
• Veff(φ̄) is the energy density when 〈a|φ |a〉 = φ̄.
⇓The minimum of Veff is the vacuum energy and φ̄
at the minimum is the expectation value.
• It is easy to calculate ! (perturbatively!!!).
Veff(φ̄) = −∑
n
1
n!Γ(n)(p1 = 0, . . . , pn = 0)φ̄n
To one loop:
Cosmological Phase Transitions 8
How come? - The effective scalar potential
Calculating the effective scalar potential
A toy model:
V0(φ) = −1
2m2φ2 +
λ
4!φ4 .
Cosmological Phase Transitions 9
How come? - The effective scalar potential
Calculating the effective scalar potential
A toy model:
V0(φ) = −1
2m2φ2 +
λ
4!φ4 .
To one loop:
Veff(φ̄) = V0(φ̄) + i
∞∑
n=1
∫
d4k
(2π)41
2n
[
λφ̄2/2
k2 +m2
]n
= V0(φ̄) +1
2
∫
d4k
(2π)4log
[
k2 −m2 +λφ̄2
2
]
Cosmological Phase Transitions 9
How come? - The effective scalar potential
Calculating the effective scalar potential
A toy model:
V0(φ) = −1
2m2φ2 +
λ
4!φ4 .
To one loop:
Veff(φ̄) = V0(φ̄) + i
∞∑
n=1
∫
d4k
(2π)41
2n
[
λφ̄2/2
k2 +m2
]n
= V0(φ̄) +1
2
∫
d4k
(2π)4log
[
k2 −m2 +λφ̄2
2
]
Switching the temperature on
Veff(φ̄) = V0(φ̄) +1
2β
∞∑
n=−∞
∫
d3k
(2π)3log
[
(
2πn
β
)2
+ ~k2 −m2 +λφ̄2
2
]
.
Cosmological Phase Transitions 9
How come?
The potential (some lowest terms)
The result:
V Teff(φ̄) = −1
2m2φ̄2 +
λ
4!φ̄4
+
(
−m2 + λφ̄2
2
)2
64π2
[
log
(
−m2 + λφ̄2
2
µ2
)
− 1
2
]
+
(
−m2 + λφ̄2
2
)
T 2
24+−π290
T 4 + . . .
Cosmological Phase Transitions 10
How come?
The potential (some lowest terms)
The result:
V Teff(φ̄) = −1
2m2φ̄2 +
λ
4!φ̄4
+
(
−m2 + λφ̄2
2
)2
64π2
[
log
(
−m2 + λφ̄2
2
µ2
)
− 1
2
]
+
(
−m2 + λφ̄2
2
)
T 2
24+−π290
T 4 + . . .
The quadratic coefficient:
1
2
[
−m2
(
1− λ
64π2
)
+λ
24T 2
]
φ̄2 .
We get symmetry restoration at large enough T .
Cosmological Phase Transitions 10
So what?
Phase transitions
The effective potential is
temperature dependent.
Surely, as the universe cools
down, first (discontinues) or
second (continues) order phase
transition result.
Cosmological Phase Transitions 11
So what?
Phase transition - good and bad
Phase transitions are needed
• To produce thermal non-equilibrium.
(On the walls of expanding bubbles which nucleate after 1st
order phase transition occur).
• For inflation.
(To allow for the inflaton’s dynamics).
Cosmological Phase Transitions 12
So what?
Phase transition - good and bad
Phase transitions are needed
• To produce thermal non-equilibrium.
(On the walls of expanding bubbles which nucleate after 1st
order phase transition occur).
• For inflation.
(To allow for the inflaton’s dynamics).
Unfortunately, phase transitions also tend
• To produce unwanted relics in the form of topological defect:
• Domain walls.
• Cosmic strings.
• Magnetic monopoles.
Cosmological Phase Transitions 12
So what? - Topological defects
Domain walls
Spontaneous breaking of a discrete symmetry. For real φ:
L =1
2(∂µφ)
2 − 1
4λ(φ2 − σ2)2 .
The vacuum is at 〈φ〉 ≈ ±σ
Cosmological Phase Transitions 13
So what? - Topological defects
Domain walls
Spontaneous breaking of a discrete symmetry. For real φ:
L =1
2(∂µφ)
2 − 1
4λ(φ2 − σ2)2 .
The vacuum is at 〈φ〉 ≈ ±σSuppose that φ(z = −∞) = −σ and φ(z =∞) = +σ.
There is a stable “kink” solution:
Cosmological Phase Transitions 13
So what? - Topological defects
Domain walls
Spontaneous breaking of a discrete symmetry. For real φ:
L =1
2(∂µφ)
2 − 1
4λ(φ2 − σ2)2 .
The vacuum is at 〈φ〉 ≈ ±σSuppose that φ(z = −∞) = −σ and φ(z =∞) = +σ.
There is a stable “kink” solution:
The “false-vacuum”solution is
“frozen”.
A surface energy density results:
η ≡ 2√2
3λ1/2σ3 .
Cosmological Phase Transitions 13
So what? - Topological defects
Domain walls
Typical causal length . the horizon at phase transition.
Cosmological Phase Transitions 14
So what? - Topological defects
Domain walls
Typical causal length . the horizon at phase transition.
The unavoidable result: A typical one per horizon abundance
(ρW ∝ a−1).
Domain wall network:
Cosmological Phase Transitions 14
So what? - Topological defects
Domain walls
Typical causal length . the horizon at phase transition.
The unavoidable result: A typical one per horizon abundance
(ρW ∝ a−1).
Domain wall network:
The contribution to energy
density is much larger than
the critical density.
Domain walls are excluded!
(A solution is needed).
Cosmological Phase Transitions 14
So what? - Topological defects
Cosmic strings
Spontaneous breaking of a U(1) symmetry (GUT):
L = DµφDµφ† − 1
4FµνF
µν − λ(φ†φ− σ2/2)2 .
The vacuum is at 〈φ〉 ≈ (σ/√2)eiθ.
Cosmological Phase Transitions 15
So what? - Topological defects
Cosmic strings
Spontaneous breaking of a U(1) symmetry (GUT):
L = DµφDµφ† − 1
4FµνF
µν − λ(φ†φ− σ2/2)2 .
The vacuum is at 〈φ〉 ≈ (σ/√2)eiθ.
Continuity require along any closed contour ∆θ = 2πn.
For n 6= 0 there must be a point inside with 〈φ〉 = 0.
Cosmological Phase Transitions 15
So what? - Topological defects
Cosmic strings
Spontaneous breaking of a U(1) symmetry (GUT):
L = DµφDµφ† − 1
4FµνF
µν − λ(φ†φ− σ2/2)2 .
The vacuum is at 〈φ〉 ≈ (σ/√2)eiθ.
Continuity require along any closed contour ∆θ = 2πn.
For n 6= 0 there must be a point inside with 〈φ〉 = 0.
A energy density per unit length result:
µ ∼ πσ2 .
Cosmological Phase Transitions 15
So what? - Topological defects
Cosmic strings
We expect again: A one per horizon abundance (ρS ∝ a−2).
Cosmic string network:
Cosmological Phase Transitions 16
So what? - Topological defects
Cosmic strings
We expect again: A one per horizon abundance (ρS ∝ a−2).
Cosmic string network:
However:
• String may be chopped by
mutual interactions.
• Small string loops can
radiate gravitationally.
Cosmological Phase Transitions 16
So what? - Topological defects
Cosmic strings
We expect again: A one per horizon abundance (ρS ∝ a−2).
Cosmic string network:
However:
• String may be chopped by
mutual interactions.
• Small string loops can
radiate gravitationally.
It might be OK!
In fact, cosmic string can be involved in structure formation.
They may be observed by gravitational lensing.
Cosmological Phase Transitions 16
So what? - Topological defects
Magnetic monopoles
The unbroken symmetry is incontractable.
For example, a non-abelian symmetry (GUT):
L = DµφDµφ† − 1
4FµνF
µν − λ(φ†φ− σ2/2)2 .
A “hedgehog” configuration, 〈φi〉 −−−→r→∞
σ r̂i
Cosmological Phase Transitions 17
So what? - Topological defects
Magnetic monopoles
The unbroken symmetry is incontractable.
For example, a non-abelian symmetry (GUT):
L = DµφDµφ† − 1
4FµνF
µν − λ(φ†φ− σ2/2)2 .
A “hedgehog” configuration, 〈φi〉 −−−→r→∞
σ r̂i
• Must have a 〈φ〉 = 0 inside.
• Looks like a magnetic monopole: Bi =12εijkFjk = r̂i
er2 .
• Has energy associated with it of: mM = 4πσ
e.
Cosmological Phase Transitions 17
So what? - Topological defects
Magnetic monopoles
Again: A one per horizon abundance (ρS ∝ a−3).
Cosmological Phase Transitions 18
So what? - Topological defects
Magnetic monopoles
Again: A one per horizon abundance (ρS ∝ a−3).
• The contribution to energy density is much larger than the
critical density.
• (GUT) Magnetic monopole will dominate the universe before
nucleosynthsis.
• Monopoles will dissipate the magnetic fields of galaxies.
Cosmological Phase Transitions 18
So what? - Topological defects
Magnetic monopoles
Again: A one per horizon abundance (ρS ∝ a−3).
• The contribution to energy density is much larger than the
critical density.
• (GUT) Magnetic monopole will dominate the universe before
nucleosynthsis.
• Monopoles will dissipate the magnetic fields of galaxies.
Monopoles are excluded!
(A solution is needed).
Cosmological Phase Transitions 18
So what? - Topological defects
Topological defects - summary
Dimension Would appear in Prospects
Domain wall. 2 ? Excluded.
Cosmic strings. 1 GUT phase Not excluded.
transition. may be important.
Magnetic 0 GUT phase Excluded.
monopoles. transition.
Cosmological Phase Transitions 19
So what? - Topological defects
Topological defects - summary
Dimension Would appear in Prospects
Domain wall. 2 ? Excluded.
Cosmic strings. 1 GUT phase Not excluded.
transition. may be important.
Magnetic 0 GUT phase Excluded.
monopoles. transition.
Possible ways out:
• Inflation.
• No GUT phase transition.
• Multiple (complex) phase transitions.
Cosmological Phase Transitions 19
Summary
• Spontaneous broken symmetry is restored at high enough
temperature.
• If topology allows, 2, 1 and 0 dimensional defects will form
with abundance of about one per horizon.
• 2 and 0 dimensional defects are bad news. Their energy density
is too large and they have various other unwanted effects.
Inflation may solve these problems.
• Due to their interactions, 1 dimensional defects are OK. They
may even help explain structure formation.
Cosmological Phase Transitions 20