# Inflationary Universe A possible

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```					                  Inflationary
Universe:
A possible solution to the horizon and flatness
problems.
Alan H. Guth
1981

Sudeep Das

“Greatest Hits” Seminar             October 11,
2004
Plan of the Talk

• Rush through Big Bang Cosmology…
• And stumble upon
• Flatness , Horizon & Monopole Problems.
• Enter Guth…
• Inflation can solve these problems,
• But it raises others…
• New Inflation
• Chaotic Inflation
The Standard Model of the Very Early Universe
Assume that the Universe is homogenous and isotropic, and therefore
described by the Robertson-Walker Metric:

where k =+1, -1 or 0 for a closed, open and flat universe, respectively,
and an energy-momentum tensor:

I am assuming the speed of light c=1
The Standard Model of the Very Early Universe
With these, the Einstein’s Equations governing the evolution of
the universe become:

Freidman Eqns.
where,

is the Hubble parameter.
At any time, one can define a critical density
The Standard Model of the Very Early Universe
With the equation of state taken as:
where w = 1/3 for relativistic particles.
w=          for NR matter.
and                 for a Cosmological Constant ,
We get, for a flat universe the following behavior:

The early universe was radiation dominated.
The Standard Model of the Very Early Universe
Thermodynamics of the ultra-relativistic plasma:
Number density:

The entropy density is defined as:

Where,
The Standard Model of the Very Early Universe
Thermodynamics of the ultra-relativistic plasma:
Number density:

The entropy density is defined as:

and the following entropy conservation is assumed:
implying
For T>>m, so that g(T) is a constant, we get
The Standard Model of the Very Early Universe
Since               , one can trade T for a in the Freidmann Equations.
In terms of T , the first of these, i.e.

becomes

where,

with
which is conserved.
The Puzzles: Flatness Problem
Consider the Friedmann Eqn:

WMAP results give            =1.02+/-0.02 today.
Since, in the radiation or matter dominated epochs,
positive

Hence,                         and
can only increase with time and hence
So,
has to be stupendously close to 1 in the
beginning. HOW CLOSE?
The Puzzles: Flatness Problem

Guth takes,                 today.
He gets                                                      today.
Since, S is assumed conserved, its value at early times can be
estimated by its value today.
Since,                  the bound on a leads to,
The Puzzles: Flatness Problem
So,           1.
2.
Giving,
and

Taking
One gets,
So, by demanding    to be within an order of magnitude of today,
we find                   that has to be exceedingly fine tuned in the
The Puzzles: Horizon Problem
Particle Horizon or Physical Horizon: Physical distance that light
can travel starting from Big Bang ( t = 0 ).
Consider a ray of light traveling radially in a flat universe.
Since, light travels along null geodesics:

So, the coordinate distance traversed in time t (comoving horizon):

The physical horizon distance is, therefore,
Since               , this turns out to be
The Puzzles: Horizon Problem
One can see the Horizon Problem vividly by considering the
isotropy of the cosmic microwave background.
At decoupling the comoving distance
over which casual interactions occur
is 180 h-1 Mpc. (subtends about 1
degree on the sky today).
This is much less than the comoving
decoupling 5820 h-1 Mpc.

How could these patches know that they had to be at the same temperature?
The Puzzles: Horizon Problem
So, size of the region in causal contact at time t is
Compare this with the size L(t) of a region that grows into our
observable universe. Using conservation of entropy,

We can take

The ratio of the volumes is,

Again with
Meaning 10 83 causally disconnected
regions. Yet homogenous? Horizon Problem
The Puzzles: Monopole Problem
If Grand Unification occurs with a simple guage group, any
spontaneous breaking of the symmetry down to the Standard
Model will lead to magnetic monopoles.
T he expected relic abundance of monopoles works out be,

This is far too big. Greater by 14 orders of magnitude than
observational constraints
There are other unwanted model-dependent relics.
A clue to the Solutions
A crucial assumption that led us to the flatness and horizon problems
was the conservation of entropy S. What if that was grossly wrong?

What if                                ?
A clue to the Solutions: Flatness Problem
In establishing the Flatness Problem, we used :

Which gave:

Eventually giving,

To obviate the flatness problem one needs
A clue to the Solutions: Horizon Problem
For the Horizon Problem we estimated the length that grows into the
scale of the observed universe as:

The length scale in causal contact (horizon) was

So, the horizon problem can be obviated if
Good ! But how does it work?
How do we get such a huge entropy multiplication?
•A first order phase transition in the Early Universe.
•Nucleation rate of new phase<< expansion rate.
•Universe supercools below the critical temperature Tc to a much
lower temperature Ts , whence the transition takes place.
•Latent heat is released, and temperature goes up again to Tr ~ Tc
•Entropy density is multiplied by a factor                   with the
scale factor remaining constant. So,
Guth’s Scenario

As the universe
Universe lingers                                       cools, the true
in the so-called                                       minimum gets
metastable                                             deeper.
“False Vacuum”                                          At T~Ts, the
with energy                                           universe tunnels
density                                               to the “True
Vacuum”.

Figure from A. Albrecht and P. Steinhardt, Phys. Rev. D, 48, 1220 (1981).
What is inflating here?
Of course, entropy inflates…but wait…
Consider again the old Freidmann Equation, but with the new
energy density.

As the universe, supercools the 2nd and 3rd terms fall off, leaving:

which has soln.,
with
This implies an exponentially inflating scale factor
What is inflating here?

We should be able to see this behavior directly from Freidmann Eqns.
Stress-energy tensor for the “false vacuum” is:

This makes the Friedmann Eqn.:
Changing the point of view
We saw that Guth attacked the problem from the entropy point of
view.

What if, we forget all about phase transitions and entropy and ask:

If we somehow create an accelerated expansion, will the problems
of Big Bang theory be solved?

The answer is, as it should be, yes.
Changing the point of view: Flatness Problem
Define Inflation to be an epoch with
Recall that flatness problem emerged from the increasing nature of
the RHS of:

However, during inflation,

so that even if                 started from a value away from 0,
it can be brought sufficiently close to zero by end of inflation so that
Flatness problem does not arise.
Changing the point of view: Flatness Problem
Curvature is flattened out by the huge expansion.
Changing the point of view: Horizon Problem
The physical horizon was shown to be,

The comoving horizon is 1/(a H) and is a decreasing function of time
during inflation.

Therefore, the comoving horizon shrinks during inflation.
Solving Horizon Problem in Comoving Coordinates

start

now
end

Smooth patch

Liddle and Lyth, Cosmological Inflation and Large Scale Structure
Inflation in the Abstract
Inflation is an add-on to the standard Big Bang scenario.
Following equivalent descriptions hold:
An era of accelerated expansion,

An era of shrinking horizon,

An era of a non-standard equation of state,
What went wrong with Guth’s Theory?
Guth’s phase transition scenario surely conforms to these requirements.
But there was a serious problem.
•The bubbles of true vacuum will form at
various times in various places in the false
vacuum.
•They will have great difficulty merging
because the space separating them will still
be expanding exponentially.
•The phase transition will never be
completed even if the bubbles grow at the
speed of light.
Guth & Weinberg (1983)
What came after Guth’s Theory?
•New Inflation (Linde 1982, Albrecht & Steinhardt 1982):
Roll down a flat potential.

•Chaotic Inflation (Linde 1983)
Slide through a viscous medium.
New Inflation
Linde (1982), Albrecht and Steinhardt (1982).
Still a phase transition.
Most of the interesting part of
inflation takes place away from the
false vacuum.
The field rolls down slowly.
Space inside bubbles of new phase
expand exponentially.

One bubble can grow to a size ~ 103000cm >> scale of obs. Universe~ 1028cm
The whole universe can be accommodated in one such bubble.
Chaotic Inflation (Linde 1983)
Chaotic Inflation can occur even with simplest form of potentials like

The equation of motion for the field has a “drag” term due to expansion of
the universe,

This term slows down the velocity of the field.

If                       ,the equation of state is that required for Inflation.
Chaotic Inflation
Simple!!
No supercooling/tunneling from
false vacuum. No plateau.
No thermal equilibrium!
In 10-35 s, a Plank size region
blows up to~
cms !
Initial Universe may be thought of as having chaotically distributed values

of field. Inflation took place only where was large.
At the end of inflation, the field oscillates and decays into particle-pairs, which
interact and thermalize at some temperature. Standard Big Bang takes over from
here.
Eternal Inflation
Quantum Fluctuations can “kick the rolling ball back up the hill” at
some points.
These regions expand faster than the parent regions.
In many points within this region, the expansion decays into
standard big bang evolution (universes as our own), quantum
fluctuations again causes some other point to inflate even faster.
This process goes on and on and the Universe reproduces itself.
Essentially all inflationary models turn out to be “eternal”.
Still not fully understood !
• The inflationary paradigm is essential in theory of cosmic evolution because it is a
single class of theories which consistently answers,
• Why is the Universe so flat?
•Why is the Universe so homogenous and isotropic?
•Why aren’t there any magnetic monopoles?
•How did the Hubble Expansion start?
•How do the density perturbations and anisotropies in the CMB appear?.
Scale invariance.               (Katie’s Talk!!)
•Guth’s 1981 paper is important because inflation was invented in it.
•So far, observations have been consistent with predictions of the simplest
inflationary models. WMAP:            =1.02+/-0.02. (Flatness)
Spectral index n=0.93+/-0.03 ( 1 for scale invariance)

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