Cosmological Phase Transitions · 2005-02-03 · Cosmological Phase Transitions Origin and Implications Guy Raz A lecture in Particle Cosmology Summer 2002 References: E.W.KolbandM.S.Turner,\TheEarlyUniverse,"

$Page 1: Cosmological Phase Transitions · 2005-02-03 · Cosmological Phase Transitions Origin and Implications Guy Raz A lecture in Particle Cosmology Summer 2002 References: E.W.KolbandM.S.Turner,\TheEarlyUniverse,"$
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 .


Invitation

Turning up the heat


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

]


Invitation

Turning up the heat


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 + . . .


Content

Outline

• How come?

• So what?


Content

Outline

• How come?

• So what?

• Finite temperature field theory.

• The effective scalar potential.

• Phase transitions.

• Topological defects.

• Summary


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 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]



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)]

.



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”.



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)


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)

)]

.






W [j] = −i log[∫

Dφ exp

(

i

∫

d4x(L+ Jφ)

)]

.

We define its Legandre transform

Γ[φ̄] =

(

W [J ]−∫

d4xJφ̄

)∣

∣

∣

∣

J: δW

δJ=φ̄

.






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) .



Why bother?∗

∗You still won’t learn it here.

Γ[φ̄] has some very nice properties:



Why bother?∗



• 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.



Why bother?∗



• 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:



Calculating the effective scalar potential

A toy model:

V0(φ) = −1

2m2φ2 +

λ

4!φ4 .




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

]




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

]

.


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 + . . .


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 .


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.


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).


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.


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



Domain walls


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:



Domain walls


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 .



Domain walls

Typical causal length . the horizon at phase transition.



Domain walls


The unavoidable result: A typical one per horizon abundance

(ρW ∝ a−1).

Domain wall network:



Domain walls


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).



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.



Cosmic strings



4FµνF

µν − λ(φ†φ− σ2/2)2 .


Continuity require along any closed contour ∆θ = 2πn.

For n 6= 0 there must be a point inside with 〈φ〉 = 0.



Cosmic strings



4FµνF

µν − λ(φ†φ− σ2/2)2 .


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 .



Cosmic strings

We expect again: A one per horizon abundance (ρS ∝ a−2).

Cosmic string network:



Cosmic strings



However:

• String may be chopped by

mutual interactions.

• Small string loops can

radiate gravitationally.



Cosmic strings



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.



Magnetic monopoles

The unbroken symmetry is incontractable.

For example, a non-abelian symmetry (GUT):


4FµνF

µν − λ(φ†φ− σ2/2)2 .

A “hedgehog” configuration, 〈φi〉 −−−→r→∞

σ r̂i



Magnetic monopoles

The unbroken symmetry is incontractable.

For example, a non-abelian symmetry (GUT):


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.



Magnetic monopoles

Again: A one per horizon abundance (ρS ∝ a−3).



Magnetic monopoles


• 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.



Magnetic monopoles


• 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).



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.



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.


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.


Documents

Cosmological Phase Transitions · 2005-02-03 · Cosmological Phase Transitions Origin and Implications Guy Raz A lecture in Particle Cosmology Summer 2002 References: E.W.KolbandM.S.Turner,\TheEarlyUniverse,"