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Lecture-07 Phase Transitions and Inflation

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Lecture-07Phase Transitions and Inflation

Ping He

ITP.CAS.CN

2006.05.31

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

7.0 Preliminary: Planck Era

SSB

SSB

Strong

E

W

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.

GUT transition

supersymmetry transition (possible)

electroweak transition

quark-hadron transition

symmetry for

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 f:

(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 f with self-interaction, like this:

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

Transition.

F(f)

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

F (f )

When , the vacuum expectation value

f

f : Higgs potential

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 .

1st order

2nd order

7.2 Topological Defects

Topological defects are relics of phase transitions

The type of defect produced in a symmetry-breaking phase transition

depends on the symmetry and how it is broken in a complicated fashion.

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, topological

defect.

SSB of SU(5) monopoles;

SSB of U(1) cosmic strings;

SSB of Z2 domain walls, etc.

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

At the boundary between different magnetism area, <f>=0, this is just a topological defect, which is a wall-like structure, with a small thickness,

so that it has energy and mass, and

contributes to the cosmic matter density.

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

A closed path encompasses a string-like structure, in which <f>=0, with a small transverse dimension, so that it has energy and mass, and

contributes to the cosmic matter density. Such a string must be either

closed or infinite.

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 of

electrical charge and this implies the existence of magnetic monopoles.

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

Monopoles are point-like defects in the Higgs

filed f which appears in GUTs. See the figure,

arrows indicate the 3-D orientation of f in the

internal symmetry space of the theory, while

the location of the arrows represents a position

in ordinary space.

Not exactly a point, but also has a small finite

size, with mass and energy.

We see that all kinds of topological defects have mass and energy, which

contribute to the total cosmic mass and energy density.

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

e.g., SU(5), we have

, so that

We discuss the mass and density of monopoles.

In electrostatic units, monopoles has a magnetic charge

and a mass

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

Furthermore, the size of the monopoles is

very heavy

Cosmological monopole problem

At T=TGUT, GUTSU(3)XSU(2)XU(1), if x is the characteristic dimension

of the domain during symmetry breaking, the maximum number density of

monopoles has the following relation.

Since any single domain should be causally connected, we have

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

Any subsequent physical processes are very inefficient to reducing the

ratio of

, so the present-day number density of monopoles is

equal to, or greater than the baryon density.

So the density parameter of monopoles is

Similar case for domain walls. So we can see that monopoles and

domain walls represent a problem to cosmology, which was the

essential stimulus for inflationary cosmology.

Cosmic strings, however, assuming their existence, may be a solution

rather than a problem because they may be responsible for generating

primordial fluctuations which give rise to galaxies and clusters, but just

a minority of cosmologists believe this.

7.3 Problems with the Standard Cosmological Model

- The problem of horizon (homogeneity);
- The problem of flatness;
- The problem of the origin of structure;
- The problem of monopole.

The problem of monopole has been addressed previously.

7.3.1 The cosmological horizon problem (Same as homogeneity problem)

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

comoving particle horizon:

Hubble radius (length):

In the above, we use

comoving Hubble radius (length):

So,

P

Q

The angular-diameter distance

so at the last scattering surface

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

In the above calculation, we have used

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

Since no causal connection between

P and Q, how TP=TQ?

, so at the Planck epoch,

Since,

So, the present-day non-flatness evolves from the discrepancy of

at the initial (Planck) time.

7.3.2 The flatness problem

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

so, we have

, that is

A fine-tuning problem!

scale

t

t0

lpl

tpl

7.3.3 The problem of structure formation

At the Planck time, any physical scale

should be , so

That is, the scale of the largest structure at the present-day should not

be larger than 100km!

7.4 Inflationary Cosmology

Modern cosmology predicts that a short period of inflation occurred

at the extremely early epoch, which can immediately overcome or

explain the above problems in the standard cosmology:

Inflation:

an extremely fast expansion at the birth of the Universe,

was driven by the vacuum energy of some material fields.

F(f)

f

The temperature decreases as the expansion of the universe, when

, phase transition will not promptly begin, and the universe

is in metastable state, which is overcooled.

1st order

7.4.1 inflation driven by vacuum energy

Friedmann equation, with flat

geometry:

For the very early universe:

f : Higgs potential

Before the phase transition, that is , the vacuum energy can be

neglected, so that the dynamics is totally controlled by radiation.

The expansion of the universe rapidly reduced the density of radiation

(r ~ T^4), while leaving the vacuum energy unchanged, so that the energy

density is vacuum-dominated. The dynamic equation is

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

This is inflation.

For GUT, Tc=10^15GeV, so that

If inflation lasts for merely , then a will increase times

Definition of e-foldings:

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

Log (physical scale)

physical

scale

physical

Hubble

length

Log (time)

Log (comoving scale)

Inflation

comoving

Hubble

length

| amp |

comoving

scale

|scale |

Log (time)

7.4.2 evolution of scale and temperature

(1) Horizon is nearly unchanged

during inflation and otherwise

increasing.

(2) A physical scale is initially

inside the horizon, but crosses

outside some time before the

end of inflation, reentering long

after inflation is over.

Comoving wave number

T(t)

Standard cosmology

a(t)

T(t)

Inflationary cosmology

a(t)

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;
- 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.

Log (physical scale)

physical

Hubble

length

physical

scale

Log (time)

Inflation

lss

7.4.3 inflationary solution to homogeneity problem (horizon problem)

For example, the super-horizon scale at the last-scattering

surface (lss) was causally connected before inflation, thus in

this way, the homogeneity problem is overcome.

For this, the e-foldings

should be larger than

N>60. (details omitted)

7.4.4 inflationary solution to flatness problem

From Eq-7.20

At the inflationary stage, density is dominated by vacuum energy,

which is constant. That is,

Assuming, the e-foldings N=100, that is, a increases 43 orders of

magnitude. From Eq-7.30, we can see that decreases for

86 orders of magnitude. After the cancellation of 60 orders of magnitude

of pre-inflationary discrepancy, there are still net 26 orders of magnitude

of decrease left.

It not just a solution to the flatness problem, it is also a prediction of

flatness, from Eq-7.30, we see that

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

even remarkably deviates from 1, then after inflation,

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

7.4.5 inflationary solution to structure formation problem

From Eq-7.22, we know that

Since a increases ~43 orders during inflation, so

The largest structure at the present-day universe is galaxy cluster, which

has typically the size of about ~ Mpc. Thus, structure formation problem

is overcome! The current idea about the formation of structure is:

Quantum fluctuations on microscopic scales during the inflationary epoch

can, by virtue of the enormous expansion, lead to fluctuations on very large

scales today.

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 since

scale factor increases by 43 orders, so the density of monopoles is

diluted by ~130 orders of magnitude. That is, from Eq-7.10

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

However, if there were other phase transitions after the epoch of

inflation, defects could have been formed again.

7.5 Dynamics of Scalar Fields

7.5.1 Guth’s “Old Inflation” model

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 bubble

nucleation. 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.

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

For a homogeneous state, the spatial gradient terms vanishes, and

becomes the type of the perfect fluid:

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

To obtain inflation, we need material with the unusual property of a

negative pressure. Such a material is a scalar, describing spin=0

particles. The scalar field responsible for inflation is often called the

inflaton.

The Lagrangian density of the scalar field f is

F (f )

f : Higgs potential

The equation of motions of f is:

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

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

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

2nd order

This is a 2nd order phase transition. As the temperature of the Universe

lowers, 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

After the slow-rolling phase the field f falls rapidly into the minimum at f0

and undergoes oscillations, and a rapid liberation of energy which was

trapped in the false vacuum, that is re-heating. The oscillations are damped

by the creation of particles coupled to f field.

The ending of the inflation is

- Problems with this “New Inflation” model, is that it suffers severe
- fine-tuning problems.
- 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 f is assumed to be in thermal equilibrium with the other
- matter fields before the onset of inflation, which requires that f be
- coupled fairly strongly to the other fields. But the coupling constant
- would induce corrections to the potential which would violate the
- previous constraint.

It is unlikely to achieve thermal equilibrium in a self-consistent way that

inflation can start under this slow-roll conditions.

7.5.3 “Chaotic Inflation” model (Linde 1983)

It is an improvement of the “New Inflation” model, which is also based

upon a scalar field, but the potential takes some simplest form, e.g.,

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

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

after the Planck time, the f field varied from place to place in an arbitrary

manner. If any region satisfies the above conditions it will inflate and

eventually encompass our observable universe. The results are locally

flat and homogeneous, but on scales larger than the horizon the universe is

highly curved and inhomogeneous. In this model, no need for GUT or

super-symmetry, and no requirement for any phase transition. The field f

at the Planck time is completely decoupled from all other physics.

7.5.4 Other inflationary models

- “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 f.
- 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 …

- E.W. Kolb & M.S. Turner, The Early Universe, 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

- 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.