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Lecture-07Phase Transitions and Inflation

Ping HeITP.CAS.CN

2006.05.31

http://power.itp.ac.cn/~hep/cosmology.htm

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1.2 10 GeV

1.4 10

p

p p

p

p p

pp B

B

Gt s

c

Gl ct cm

c

cm g

G

cE m c

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E cT k K

k G

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7.1 Phase Transition Spontaneous symmetry breaking (SSB):

SSB can be used both in quantum field theory (particle physics) and in phase transition of statistical physics.

In QFT, SSB provides a mechanism for the unification of interactions,as T decreases, new type of interactions will emerge.

1510 GeV

23 10 GeV

Strong W E

(3) (2) (1)SU SU U

(2) (1)SU U

(1)U

GUT( (5))SU SSB

SSB

4

GUT transition 1510 GeV

supersymmetry transition (possible) 310 GeV

electroweak transition 210 GeV

quark-hadron transition110 GeV

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While from the side of thermodynamics, the universe will experience a series of phase transitions, just like water from vapor to liquid to ice;

In both cases, SSB is implemented by Higgs mechanism, in which

there exists a scalar filed : (1) In particle physics, it is called a Higgs Field (Boson, spin=0); (2) In statistical physics, it is called order parameter.

Higgs field: a scalar field with self-interaction, like this:

2 4(Eq-7.1)( ) , ( 0, 0)V

symmetry for

The above expression can also be used to describe 1st order phaseTransition.

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1st order phase transition, discontinuous transition: (1) latent heat, (2) nucleation of bubbles.

2th order phase transition, continuous transition: (1) less dramatic, no-latent heat, (2) quantified by correlation-length .

F()

cT T

cT T

cT T

F )

cT T

cT T

: Higgs potential

ground state, or vacuum state: the state of the minimum energy.

When , the vacuum expectation valuecT T 0

2nd order1st order

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7.2 Topological Defects

The type of defect produced in a symmetry-breaking phase transitiondepends on the symmetry and how it is broken in a complicated fashion.

Topological defects are relics of phase transitions

Typically, there will be the following types of topological defects: (1) magnetic monopoles; (2) cosmic strings; (3) domain walls; (4) textures.Vacuum that experiences SSB is not perfect, that is, topologicaldefect.

SSB of SU(5) monopoles;SSB of U(1) cosmic strings;SSB of Z2 domain walls, etc.

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Example-1, the formation of domain wall (畴壁 ), by SSB of Z2.

At the boundary between different

magnetism area, <>=0, this is just a topological defect, which is a wall-like structure, with a small thickness,so that it has energy and mass, andcontributes to the cosmic matter density.

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Example-2, the formation of cosmic strings, by SSB of a U(1).

A closed path encompasses a string-like structure, in which

<>=0, with a small transverse dimension, so that it has energy and mass, andcontributes to the cosmic matter density. Such a string must be eitherclosed or infinite.

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We see that all kinds of topological defects have mass and energy, whichcontribute to the total cosmic mass and energy density.

Any GUT in which electromagnetism (described by U(1) gauge group)is contained, with a gauge theory involving SSB of a higher symmetry, e.g., SU(5), can provide a natural explanation for the quantization ofelectrical charge and this implies the existence of magnetic monopoles.

Monopoles are point-like defects in the Higgs

filed which appears in GUTs. See the figure,

arrows indicate the 3-D orientation of in theinternal symmetry space of the theory, while the location of the arrows represents a positionin ordinary space.

Example-3, the formation of monopoles, by SSB of GUT gauge symmetry.

Not exactly a point, but also has a small finitesize, with mass and energy.

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We discuss the mass and density of monopoles.

In electrostatic units, monopoles has a magnetic charge

(Eq-7.2), with is Dirac charge, 68.52n D D D

cg ng g g e

e

and a mass

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(Eq-7.3)4 10M X X

cm m m

e

where X is the Higgs boson that mediates the GUT interaction, with mass

1/ 2 1(Eq-7.4)( ) 10X GUT GUTm e c m m

GUTm is the energy corresponding to the GUT SSB, for typical GUTs,

e.g., SU(5), we have 14 1510 10 GeVGUTm , so that

1610 GeVMm

Furthermore, the size of the monopoles is

28(Eq-7.5)10 cmM

X

rm c

very heavy

1GeVp nm m

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Cosmological monopole problem

At T=TGUT, GUTSU(3)XSU(2)XU(1), if is the characteristic dimensionof the domain during symmetry breaking, the maximum number density ofmonopoles has the following relation.

(Eq-7x .63

,ma )1, with ~ ,10Mn p p

Since any single domain should be causally connected, we have

1/ 22

(Eq-7.7)( ) 2 0.6 ( ) pH

B

T cr t ct g T

k T

where Tp is the Planck temperature. It turns out that, at TGUT (~10^15GeV),

3* 1/ 2

10(Eq-7.8)

( )( ) 10

0.6GUT GUT

M GUTp

g T Tn p n T n

T

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Cosmic strings, however, assuming their existence, may be a solutionrather than a problem because they may be responsible for generating primordial fluctuations which give rise to galaxies and clusters, but justa minority of cosmologists believe this.

0 0 (Eq-7.9)M bn n

Any subsequent physical processes are very inefficient to reducing the ratio of /Mn n, so the present-day number density of monopoles isequal to, or greater than the baryon density.

So the density parameter of monopoles is

16(Eq-7.10)10M

M bp

m

m

Similar case for domain walls. So we can see that monopoles anddomain walls represent a problem to cosmology, which was theessential stimulus for inflationary cosmology.

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7.3 Problems with the Standard Cosmological Model

(1) The problem of horizon (homogeneity);(2) The problem of flatness;(3) The problem of the origin of structure;(4) The problem of monopole.

The problem of monopole has been addressed previously.

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7.3.1 The cosmological horizon problem (Same as homogeneity problem)

Big-bang singularity , with limited light speed existence of particle horizon

0(Eq-7.11)( ) ( )

( )

t

H

cdtR t a t

a t

comoving particle horizon: (Eq-7.12)( ) ( ) / ( )H Hr t R t a t

Hubble radius (length):32

(Eq-7.13)2 rad-dominated

( )mat-dominated( )H

ctcD t

ctH t

comoving Hubble radius (length): ( ) ( ) /H Hd t D t a

(Eq-7.14)2 ( ) rad-dominated

( )3 2 ( ) mat-dominated

HH

H

ct D tR t

ct D t

In the above, we use 1/ 2 2 /3( ) (radiation), or ( ) (matter)a t t a t t

So, ( ) ( )H HR t D t

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and the size of particle horizon at last-scattering surface:

3/ 2 3/ 20 0 (Eq-7.17)( ) 3 3 (1 ) 3 , 1000H ls ls ls ls lsR z ct ct z ct z z

In the above calculation, we have used2/3

2/3 0 0 (Eq-7.18)( ) , and 1ls lsls ls

a ta t t z z

a t

The angular-diameter distance 0 (Eq-7.15)( )1

zA

a rd z

z

so at the last scattering surface 1

0 (Eq-7.16)( ) 3A ls lsd z ct z

The angle subtended by particle horizon at the last-scattering surface:

1/ 2 1/ 20 (Eq-7.19)1.8 degH

lsA

Rz

d P

QSince no causal connection betweenP and Q, how TP=TQ?

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7.3.2 The flatness problem

From Friedmann equation, we have (see Lecture-02):

1K so, we have

22(Eq-7.20)

matter1 11

radiationK

a

aa

Since, 10 17 60

0 10 10 10 plt yr s t , so at the Planck epoch,

6011 10

, that is 60

(Eq-7.21)( ) 1 10plt

So, the present-day non-flatness evolves from the discrepancy of at the initial (Planck) time.

6010

A fine-tuning problem!

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7.3.3 The problem of structure formationAt the Planck time, any physical scaleshould be , so( )phy pl plL t l

00

1/ 2 80

2/3 2 180

(Eq-7.22)( ) ( )

( / ) 10 (radiation)

( / ) 10 10 Mpc (matter)

phy phy plpl

pl pl

pl pl

aL t L t

a

t t l km

t t l km

That is, the scale of the largest structure at the present-day should notbe larger than 100km!

tt0

Hd tscale

( )a t

tpl

lpl

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Inflation: an extremely fast expansion at the birth of the Universe,was driven by the vacuum energy of some material fields.

Modern cosmology predicts that a short period of inflation occurred at the extremely early epoch, which can immediately overcome orexplain the above problems in the standard cosmology:

7.4 Inflationary Cosmology

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7.4.1 inflation driven by vacuum energy

Friedmann equation, with flatgeometry:

2

(Eq-7.23)8

3

a G

a

For the very early universe:

(Eq-7.24)vac

Before the phase transition, that is , the vacuum energy can beneglected, so that the dynamics is totally controlled by radiation.

cT T

1/ 2(Eq-7.25)a t

The temperature decreases as the expansion of the universe, when

cT T , phase transition will not promptly begin, and the universe

is in metastable state, which is overcooled.

F()

cT T

cT T

cT T

: Higgs potential

1st order

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The expansion of the universe rapidly reduced the density of radiation

(T^4), while leaving the vacuum energy unchanged, so that the energydensity is vacuum-dominated. The dynamic equation is

2 1/ 24 2 4

(Eq-7.26)8 8 8

, where3 3 3vac c c

a G G GT H H T

a

Here, Tc is a constant, so that H is also a constant, and hence

(Eq-7.27)Hta e

For GUT, Tc=10^15GeV, so that 1 3510H s This is inflation.

If inflation lasts for merely , then a will increase times3310 s 100e

Definition of e-foldings: (Eq-7.28)ln f

i

aN

a

For the above example, the e-foldings N=100

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7.4.2 evolution of scale and temperature

Lo

g (

ph

ysic

al

scal

e)

Lo

g (

com

ovi

ng

sc

ale)

Inflation

physicalHubblelength

comovingHubblelength

physicalscale

comovingscale

Log (time)

Log (time)

(1) Horizon is nearly unchangedduring inflation and otherwiseincreasing.(2) A physical scale is initiallyinside the horizon, but crossesoutside some time before theend of inflation, reentering longafter inflation is over.

|scale |

|

amp

|

Comoving wave number

(Eq-7.29)2 2

/c pc p

k akL L a

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T(t)a(t)

T(t)a(t)

Standard cosmology

Inflationary cosmology

inflation reheating

(1) Because of the inflation, the temperature dramatically decreased. Its evolution is adiabatic, so aT~ const, that is, ;

(2) When inflation stopped, the latent heat of the vacuum energy was released;

(3) The universe was re-heated again to the temperature of about Tc;

(4)Reheating is non-adiabatic, hence, large amount of entropy was generated during reheating.

(5) After the end of reheating, the universe restored to the adiabatic expansion.

~ HtT e

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7.4.3 inflationary solution to homogeneity problem (horizon problem)

Lo

g (

ph

ysic

al

scal

e)

Inflation

physicalHubblelength

physicalscale

Log (time)

lss

For example, the super-horizon scale at the last-scatteringsurface (lss) was causally connected before inflation, thus inthis way, the homogeneity problem is overcome.

For this, the e-foldingsshould be larger thanN>60. (details omitted)

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7.4.4 inflationary solution to flatness problem

From Eq-7.20

22(Eq-7.20)

matter1 11

radiationK

a

aa

At the inflationary stage, density is dominated by vacuum energy,which is constant. That is,

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(Eq-7.30)1 1

1 K aa

Assuming, the e-foldings N=100, that is, a increases 43 orders of magnitude. From Eq-7.30, we can see that decreases for86 orders of magnitude. After the cancellation of 60 orders of magnitudeof pre-inflationary discrepancy, there are still net 26 orders of magnitudeof decrease left.

1 1/

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It not just a solution to the flatness problem, it is also a prediction of flatness, from Eq-7.30, we see that

2

(Eq-7.31)1 1/

1 1/f f

ii

a

a

where i, and f indicate the start and end of inflation, since

100 43(Eq-7.32)~ 10f

i

ae

a

even remarkably deviates from 1, then after inflation, i

(Eq-7.33)1f

The prediction is model-dependent, there is also open-inflationary model.

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7.4.5 inflationary solution to structure formation problem

00

1/ 2 80

2/3 2 180

(Eq-7.22)( ) ( )

( / ) 10 (radiation)

( / ) 10 10 Mpc (matter)

phy phy plpl

pl pl

pl pl

aL t L t

a

t t l km

t t l km

Since a increases ~43 orders during inflation, so

From Eq-7.22, we know that

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0 25(Eq-7.34)

10 Mpc (radiation)( )

10 Mpc (matter)phyL t

The largest structure at the present-day universe is galaxy cluster, whichhas typically the size of about ~ Mpc. Thus, structure formation problemis overcome! The current idea about the formation of structure is:Quantum fluctuations on microscopic scales during the inflationary epochcan, by virtue of the enormous expansion, lead to fluctuations on very largescales today.

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7.4.6 inflationary solution to monopole problem

Monopoles are generated during the phase transition by SSB of GUT.If inflation took place after (or during) the phase transition, then sincescale factor increases by 43 orders, so the density of monopoles isdiluted by ~130 orders of magnitude. That is, from Eq-7.10

3

16 130 114(Eq-7.35)( ) ( ) 10 10 10f

M Mi

af i

a

The same is true for cosmic strings and all other topological defects.

However, if there were other phase transitions after the epoch ofinflation, defects could have been formed again.

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7.5 Dynamics of Scalar Fields

The previous example is based on Guth’s “old inflation” model (1981), which was settled upon 1st order phase transition, and is now abandoned,because that, being a 1st order transition, it occurs by a process of bubblenucleation. So that:(1) They are too small to be identified with our observable universe, and(2) They are carried apart by the expanding phase too quickly for them to coalesce and produce a large bubble which is identified with our universe. So that(3) The end state of this model would therefore be a highly chaotic universe, quite the opposite of what is intended.

7.5.1 Guth’s “Old Inflation” model

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7.5.2 “New Inflation” model (Linde 1982; Albrecht & Steinhardt 1982)

To obtain inflation, we need material with the unusual property of anegative pressure. Such a material is a scalar, describing spin=0particles. The scalar field responsible for inflation is often called theinflaton.

The Lagrangian density of the scalar field is

(Eq-7.36)1

( )2

L V

Gives a contribution to the energy-momentum tensor of the formT

(Eq-7.37)T Lg

For a homogeneous state, the spatial gradient terms vanishes, andbecomes the type of the perfect fluid:

T

2

2

(Eq-7.38)( )2

( )2

V

p V

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The equation of motions of is:

2 2

(Eq-7.39)

8 1( )

3 2

3 ( ) 0

GH V

H V

F )

cT T

cT T

: Higgs potential

2nd order

To produce a long enough period of inflation and a rapid reheating after

inflation, the potential V() should be like the following of the right one.

Such a potential may be implemented by super-symmetry theories.

( )V

0

32

2

(Eq-7.40)

1( ) 1,

16

1( ) 1

8

V

G V

V

G V

This is a 2nd order phase transition. As the temperature of the Universelowers, the state should slowly roll down to the minima of the potential. That is, the potential should satisfy the slow-roll conditions:

With slow-roll approximation, Eq-7.39 becomes

2

(Eq-7.41)

8( )

3

3 ( )

GH V

H V

After the slow-rolling phase the field falls rapidly into the minimum at 0and undergoes oscillations, and a rapid liberation of energy which wastrapped in the false vacuum, that is re-heating. The oscillations are damped

by the creation of particles coupled to field.

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Problems with this “New Inflation” model, is that it suffers severefine-tuning problems.

(1)The potential must be very flat near the origin to produce enough inflation and to avoid excessive fluctuations due to the quantum field.

(2) The field is assumed to be in thermal equilibrium with the other

matter fields before the onset of inflation, which requires that be coupled fairly strongly to the other fields. But the coupling constant would induce corrections to the potential which would violate the previous constraint.

The ending of the inflation is

(Eq-7.42)1, 1

It is unlikely to achieve thermal equilibrium in a self-consistent way thatinflation can start under this slow-roll conditions.

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7.5.3 “Chaotic Inflation” model (Linde 1983)

It is an improvement of the “New Inflation” model, which is also basedupon a scalar field, but the potential takes some simplest form, e.g.,

2 2(Eq-7.43)

1( )

2V m

Here m is the mass of the filed. Slow-roll conditions can be satisfied if

2(Eq-7.44)1/ 4 G

Chaotic inflation model assumes that at some initial time, perhaps just

after the Planck time, thefield varied from place to place in an arbitrarymanner. If any region satisfies the above conditions it will inflate andeventually encompass our observable universe. The results are locallyflat and homogeneous, but on scales larger than the horizon the universe ishighly curved and inhomogeneous. In this model, no need for GUT or

super-symmetry, and no requirement for any phase transition. The field at the Planck time is completely decoupled from all other physics.

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7.5.4 Other inflationary models

(1) “Stochastic Inflation” model (Linde 1994), also called eternal inflation. The basic idea is the same as the chaotic one in that the universe is globally extremely inhomogeneous. The stochastic inflation model

takes into account quantum fluctuations during the evolution of . In this case, the universe at any time will contain regions which are just entering into an inflationary phase. The picture of this universe is a continuous branching process in which new mini-universes expand to produce locally smooth Hubble patches within a highly chaotic background universe. (2) “Open inflation” model (Coles and Ellis 1994). Before WMAP results, The universe may have the possibility that it is open, so that open inflation model is constructed, in which after this kind of inflation, the universe is homogeneous but is curved by invoking a kind of quantum tunneling from a meta-stable false vacuum state immediately followed by a 2nd order of phase transition of inflation. The tunneling creates a bubble inside which the space-time resembles an open universe. After WMAP results, less interesting now.(3) Many …

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rse, Addison-Wesley Publishing Company, 1993

• P. Coles & F. Lucchin, Cosmology, 2nd edtion, John Wiley & Sons, 2002

• A.R. Liddle & D.H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, 2000

• L. Bergstrom & A. Goobar, Cosmology and Particle Astrophysics, Springer, 2004

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• 俞允强，热大爆炸宇宙学，北京大学出版社， 2001

• 范祖辉， Course Notes on Physical Cosmology, See this site.