Lecture-07 Phase 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):.
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Lecture-07Phase Transitions and Inflation
7.0 Preliminary: Planck Era
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.
supersymmetry transition (possible)
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
ground state, or vacuum state: the state of the minimum energy.
F (f )
When , the vacuum expectation value
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 .
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;
Vacuum that experiences SSB is not perfect, that is, topological
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
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
, 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 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):
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,
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!
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:
an extremely fast expansion at the birth of the Universe,
was driven by the vacuum energy of some material fields.
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.
7.4.1 inflation driven by vacuum energy
Friedmann equation, with flat
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)
Log (comoving scale)
| amp |
7.4.2 evolution of scale and temperature
(1) Horizon is nearly unchanged
during inflation and otherwise
(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
Log (physical scale)
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
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
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
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.
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
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