The following
content is provided
under a Creative
Commons license.
Your support will help MIT
OpenCourseWare continue
to offer high quality
educational resources for free.
To make a donation or to
view additional materials
from hundreds of MIT courses,
visit MIT OpenCourseWare
at ocw.mit.edu.
PROFESSOR: OK, in that
case, let's get going.
Today's lecture will be
mainly on the blackboard.
I have a few slides
I want to show.
And what we want to talk about
is the inflationary universe
model.
So I'll start by describing
the mechanism of inflation,
how it happens.
Inflation is based
on the physics
of scalar fields and gravity.
As I think we've said, in
present day particle theory,
and by present day I mean
not yet string theory,
all particles are
described as fields,
quantum excitations
of the field.
The analogy that most people
are at least qualitatively
familiar with, is
the photon, which
is a quantum excitation of
the electromagnetic field.
But in fact, to describe
a relativistic theory
of interacting
particles, the only way
we really know for
any kind of particle
is to introduce a field
and describe the particle
as a quantized
excitation of the field.
So when we talk
about a scalar field,
that's the quantum
representation
of some kind of a
scale or particle.
And scalar in this
case, means spinless,
same in all directions.
A scalar field then,
is just a number
defined at each point in space.
The only scalar field that we've
actually seen in nature so far,
is the Higgs field.
And indeed, inflation
is very much
modeled on the Higgs field.
Although the field
that drives inflation,
which is by definition,
called the inflation,
is probably not the Higgs
field of the standard model.
Although recently, actually,
in the past few years,
people have written a
number of papers proposing
that maybe the Higgs field
of the standard model
could, in fact, be the
field that drives inflation.
So we don't know.
It's an open question.
But in any case, the
field drives inflation
is some kind of
cousin, at least,
of the Higgs field
of the standard model
and that has many of
the same properties.
In particular, the
properties of a scalar field
are pretty much summarized by
its potential energy function.
Energy density is a function
of the value of the field.
And there are two kinds of
potential energy functions
that I like to talk about.
One is the kind of that is used
in New inflationary models.
And potential energy
versus field value.
And it has a plateau with a peak
at someplace, which is usually
assumed to be phi equals zero
and a potential energy function
which may or may not be
symmetric about phi equals zero
but I'll assume it is,
just for simplicity.
And the second type, which
I'd like to talk about,
mainly for comparison--
This is really
the one that will
be interesting.
But one could also imagine
a potential energy function
which really has a local
minimum someplace, which
is not the global minimum.
And I'll draw it with
the local minimum
at the origin and the
global minima, two of them
degenerate, elsewhere.
And this, again, is a
graph of mi versus phi.
And the reason I'm
drawing this is partly
for historical interest.
This was what was used in the
original inflationary model.
It is my original paper.
It does not work.
But we'll want to talk
about why it does not work.
In both cases we're
interested in a state, which
can be called a false vacuum,
which is a state where
the scalar field is just
sitting at phi equals zero.
In the case of the second
of these potentials,
phi equals zero being
a local minimum,
is classically
completely stable.
If one had a scalar field
in some region of space
just sitting in that
minimum, there'd
be no place where energy could
come from that would drive it
out of that minimum
over the barrier.
In the second case, the
field is classically stable.
But it's still possible for it
to quantum mechanically tunnel
through the barrier.
And that process has been
calculated and understood.
Originally by Sidney
Coleman and collaborators.
And an important feature
of that tunneling
is that does not
happen globally.
You might think that there
would be some probability that
suddenly everywhere
in the universe
the scalar field would
tunnel over the barrier
and go down to the other side.
The probability of
that is zero, as you
might realize if you
thought about a bit more.
There's just no
way that the scalar
field that far
over there is going
to know the tunnel at the
same time as the scalar field
far over there.
So the timely happens
locally and it really
happens in a very
small region, which
then tunnels over the
barrier and the scalar field
start rolling down on the other
side in this small region.
And then that region grows.
The scalar field, as it
rolls over the barrier,
pulls the scalar field nearby.
And the region
grows with a speed
that rapidly approaches
the speed of light.
In the New
inflationary potential,
where we have actually
a local maximum here,
the situation is
classically meta stable,
in the sense that the smallest
possible fluctuation can start
the field rolling down the hill.
And in particular,
quantum fluctuations
will, if nothing
else, start the field
rolling down the hill in
some finite amount of time.
We're interested
in the case where
the amount of time it takes
for quantum fluctuations
to push the field off the hill
is relatively long compared
to time scales involved
in the early universe.
And the key time scale
involved in the early universe
is the Hubble time.
And the Hubble time is just
driven by the energy density.
So one can calculate
the Hubble time.
And one is interested
in the case
for building
inflationary models,
where the top of the hill is
smooth enough, gentle enough,
has a small enough
second derivative
so that the amount of time it
will take for the scalar field
to roll off the hill
is long compared
to the Hubble time
of the universe.
So in both cases,
for short times,
the scalar field is just
stuck at the origin.
And that's what's
important, as far as
what we want to talk about next.
So the characteristic
of this state
called the false vacuum is
that the scalar field is
pinned at a high energy state.
Scalar field is pinned
at a high energy density.
And by pinned, I mean in advance
you just can't change quickly.
In general, particle
physicists use the word vacuum
to mean the state of lowest
possible energy density.
When we call this
a false vacuum,
we're really using the word
false in the sense of the word
temporary.
These states are
temporary vacuums,
in that for some period of time,
which is long by early universe
standards, the energy
density can't get any lower.
So it's acting like a vacuum.
Now what are the
consequences of that?
The important
consequence of that
is that the pressure has
to be large in negative
and in fact, equal to the
negative of the energy density.
And there are two ways we can
convince ourselves of that.
The first is that if we remember
the cosmological equation
that we derived somewhere in the
middle of course for row dot.
We learned that row dot is
equal to minus 3 a dot over a
where a is the scale
factor times row
plus the pressure
over c squared.
Now what we're saying here, is
that as the universe expands,
the scalar field is just stuck
at this false vacuum value.
The energy density is stuck at
the energy density associated
with that value of the
field, the potential energy
density of the field itself.
And therefore, row dot will be
zero as the universe expands.
And if row dot is
zero, we can just
read off from this equation
what the pressure has to be.
Row dot equals zero implies
that the pressure is just
equal to minus the energy
density times c squared,
which is another way of
saying it's minus-- Excuse me,
the mass density
times c squared,
which is another way of saying
it's minus the energy density.
I'm using u for energy density
and row for mass density.
And they just differ by
a factor of c squared.
So this is the straightforward
equation method
of seeing the answer here.
But if you want to explain this
to your roommates or somebody
who is not taking
this class, there's
also a simple argument based
on a thought experiment, which
I think is worth
keeping in mind.
And we've used this
argument before, actually,
in similar contexts.
If we imagine a piston chamber
in our thought experiment.
And in our piston
chamber, we're going
to put false vacuum
on the inside.
And the false vacuum will
have an energy density,
I'll call it u sub f.
And on the outside.
we're going to have
a zero energy vacuum.
Now we've learned
that since 1998, we've
known that our vacuum is
not a zero energy vacuum.
We seem to be seeing a non-zero
vacuum energy in our universe.
However, even if that's
true, the vacuum energy
of our universe is
incredibly small
compared to the false
vacuum energy density
that we're talking about in
terms of the early universe.
So you could still very
well approximate it as zero
and not worry about it.
So that's what
we'll be doing here.
So we'll think of
the outside as being
either a fictitious vacuum,
which by definition, has zero
energy density and
we can talk about it
even if it doesn't exist.
Or we could think of it
as being the real vacuum
in our universe, which has
an energy density which
is approximately
zero on this scale.
Now what we want to
do is just imagine
pulling out that piston.
So we have now created
an extra region
on the interior of
the piston chamber.
And we're going to be
assuming that we've somehow
rigged the walls of the chamber
so that the false vacuum will
be stretched as we
pull out the piston.
The piston is attached to
the false vacuum in some way.
So this entire area inside
region is now false vacuum.
And therefore, the volume
of the false vacuum region
has enlarged.
And if we call the
extra region here
delta v, the volume
of that region,
we now have a situation where
the energy has increased
by the energy density of the
false vacuum times delta v
by changing the
volume of the chamber.
Now, energy has to be conserved.
So this energy has to be
equal to the work that
was done by whatever force
pulled out on this piston.
We won't need to specify who
was pulling on the piston,
but the work done when
one pulls out on a piston
is just equal to
minus p times delta v,
the work done by the person
pulling on the piston.
So the normal case, the
pressure would be positive,
the piston would be
pushing out on the person
holding the piston and the
interior would be doing work
on the person pulling out.
And it would be positive if
the pressure were positive.
The work done on
the person, but this
is supposed to be the
work done on the gas.
And that's minus
p times delta v.
So if energy is conserved,
the work done on the gas
has to be equal to the
change in energy of the gas.
And the change in energy
of the gas is that.
So conservation
of energy implies
that delta e equals delta
w or use of f times delta v
equals minus p times delta
v, which of course implies
that p is equal to minus
use of f, as we said before.
So the point is that if the
energy inside the piston
is going to increase, the
person pulling out on the piston
had better be doing work, had
better be doing positive work.
And if the pressure
inside were positive,
the person pulling
out on the piston
would be doing negative work.
The piston would
be pushing on him.
So for it to make sense
here, with the energy
in the piston increasing,
the person pulling out
has to really be pulling
against a suction.
He has to do work to pull out.
And a suction means
a negative pressure,
if we have zero
pressure outside.
Pressure inside
has to be negative.
So we could reach this
conclusion either of two ways
and we get the same conclusion.
The pressure is just equal
to minus the energy density.
Yes?
AUDIENCE: I'm a little
confused about why
the energy's increasing inside.
Because why couldn't
you just say
the energy density decreases
with the increased volume?
PROFESSOR: OK the question
is why couldn't you
just say that the
energy density would
decrease with the
increased volume.
That certainly is
what will happen
if you have normal gas inside.
What makes this particular
false vacuum odd
is the origin of this
energy density, which
is the potential energy
density of the field.
So if we were talking
about the situation,
for example, which
is the clearest cut,
the only way the energy
density here could go down
is if the scalar field
goes up over the barrier
and then comes down over here.
And there's no way to drive
it there, except to wait
for a quantum fluctuation,
which is a very slow process.
And similarly, here
there's no barrier.
So it can't just roll down.
But that takes a
certain amount of time.
And we're assuming that all the
things we're talking about here
are happening on
a time that's fast
compared to the amount of time
it takes for the scalar field
to roll.
So what makes this peculiar
false vacuum special is that it
cannot lower its
energy density quickly.
And that's what the word
false vacuum implies.
And there are states like that.
And then those
states necessarily
have a negative
pressure, or pressure
that's equal to good accuracy
to minus the energy density.
Or I say to good accuracy only
because the energy density
could change a
little bit slowly,
at least for the top case.
But it's limited how
much it can change.
OK, now what are
the consequences
of this cosmologically?
Well we've also learned that
we could write the second order
Friedman equation, which is the
equation really tells us what
the force of gravity is doing.
A double dot is equal minus
4 pi over 3 g times row
plus 3 p over c squared.
Now for the false vacuum,
p is equal to minus row
c squared minus
the energy density.
And that means that this term is
negative and three times as big
as that term.
So for the false
vacuum, this quantity,
which we normally think
of as being positive,
becomes negative.
I should write factor of a here.
And that means that
instead of gravity slowing
down the expansion the
universe for a false vacuum,
the expansion is accelerated.
And that's also what
we're seeing today
with the vacuum
energy, which behaves
the same way as
this false vacuum
and produces gravitational
repulsion in exactly
the same way.
So false vacuum implies
gravitational repulsion.
OK, this basically is the
mechanism of inflation.
So we're sort of finished
with this chapter.
Are any questions
before I go on about how
this gravitational
repulsion arises?
Michael.
AUDIENCE: So, for the top vacuum
that you've drawn up there,
where there's no barrier,
you just roll slowly,
are we assuming that it
takes a long time for it
to begin to roll or
after it's started
rolling that it also takes
a very long time to reach
the bottom.
PROFESSOR: I guess I'd
say both, Begin to roll
is not that well defined.
Because it may have it
an infinitesimal velocity
from the time you
start discussing it.
And then that
infinitesimal velocity
gets bigger and bigger.
But I think what we're saying is
that the whole process, however
you divide it up, it's going
to take a long time compared
to the time it takes for
the exponential expansion
to set in.
OK.
So now, I'd like to
take this physics
and just put a
scenario around it.
And we'll call the new
inflationary scenario
because that's what it is.
Maybe now I should
mention a little bit more
about the history here.
When I wrote my
original paper, I
was assuming a
potential of something
like this, because it seemed
generic and created inflation
and I was able to understand
that inflation would solve
a number of cosmological
problems, which
are the problems that
we've talked about.
And we'll come back
to talk about how
inflation solves them.
But
But, one still has
to end inflation.
In this model,
inflation would end only
by the tunneling
of the scalar field
through the barrier,
which as I said,
happens in small
regions which then grow.
Those regions are spherical
so they're called bubbles.
And the whole process really is
very much the way water boils.
When you boil water, it forms
very small bundles initially
and the bundles grow and then
start colliding with each other
and making a big frothy mess.
And it turns out that that's
exactly what would happen
in the early universe
if you had this model.
When I first started
thinking about it,
I hoped that these bundles
could collide with each other
while they're still
small and merge
into a uniform, hot region of
the new phase, a phase where
the scalar field is
not there, but there.
But that turned
out to be the case.
It turned out that the bubble
formation process produced
horrible inhomogeneities
that there did not
seem to be anyway to cure.
And that then was
the downfall of
the original inflationary model.
But a few years later, Andrei
Linde in the Soviet Union,
and independently, Albrecht
and Steinhardt in the US,
proposed what came to be called
the new inflationary model,
which started with a
different assumption
about what the underlying
potential energy
function for the
scalar field was.
Instead of assuming
something like this,
which might be called
generic in some sense,
they instead assumed
something that's
a little bit more special,
a potential energy
function with a
very flat plateau
somewhere, which, well, we
normally put in the middle.
And this has the advantage,
that the inflation ends, not
by bubbled nucleation by
tunneling, but instead,
by just small fluctuations
building up and pushing
the scalar field down the hill.
And what makes it
work, basically
is that those small fluctuations
have some spatial correlations
built into them.
So over some small region, which
I will calla coherence region,
the fluctuations are
essentially uniform.
And the other
important feature is
that once the scalar
field starts to roll,
it still has some nearly
flat hill to roll on.
So a significant
amount of inflation
happens after this homogeneous
coherence region forms.
So the initial coherence
region can be microscopic,
but it is then stretched
by the inflation that
continues as the
scalar field rolls down
the hill towards the bottom.
So that process of stretching
the coherence region
after it has already formed is
what makes this model workable,
while this model was not.
So that's the basic story of
how new inflation succeeded
in allowing inflation
to end gracefully,
is the phrase that was used.
The problems associated
with this model
came to be called the
graceful exit problem.
And this is the first solution
to the graceful exit problem.
They're now other solutions.
But they're very
similar actually.
So I'll just write
here that's it's
a modification of the
original inflationary model
to solve this graceful
exit problem problem.
Now I should say a little bit
about how inflation starts.
But I can only say a
little bit about it
because the bottom line
really is we don't know.
We still don't have any real
theory of initial conditions
for cosmology, whether
it's inflationary cosmology
or any kind of cosmology.
The nice feature of
inflation is that it
allows a significantly broader
set of initial conditions
than is required, for example,
in the standard cosmological
model, where, as we discussed,
the needed initial conditions
are very precisely specified.
I might say a few things though,
about ideas people have had.
One idea, which I think
sounds very reasonable,
is due to Andrei Linde.
And it's a vague
idea, so it really
needs to be more precise
before it could really
be considered a theory.
But this is just the idea
that the universe started out
with some kind of chaotic
random initial conditions.
And then the hope is simply that
inflation will start somewhere.
That somewhere in the initial
chaotic distribution there'll
be a place where the
scalar field will
have the right properties,
the right configuration
to initiate inflation.
There are also
models by Vilenkin,
Alex Vilenkin of Tufts, and
independently, Andrei Linde,
who by the way, is at Stanford.
They both worked on
models where the universe
could begin by a quantum
tunneling process,
starting from
absolutely nothing.
I wrote here absolutely
nothing and that's
more nothing than nothing.
You might think of nothing
as just empty space.
But from the point of view
of general relativity,
as you already know
enough to understand,
empty space is not
really nothing.
Empty space is really
a dynamical system.
Empty space can bend and
twist and stretch and do
all kinds of complicated things.
It's really no different,
in some basic sense,
from a big piece of rubber.
So nothingness
really is intended
to mean a state where there's
not only no matter present,
but also no space and
no time, really nothing.
One way to think
of it, perhaps, is
as a closed universe, the
limit as the size of the closed
universe goes to zero so
that there's nothing left.
None of these
theories are precise.
We don't really know how to
precisely formulate them.
In this tunneling
from nothing, one
is talking about tunneling
in the context where
the structure of space itself
changes during the tunneling
process.
So it's tunneling in the
context of general relativity.
And we don't really have
a successful quantum
theory of general relativity.
So these ideas are very
speculative and quite vague.
But they do indicate
some possibilities
for how the universe
might have started.
An idea closely related to
this tunneling from nothing
is the Hartle and Hawking--
This is Jim Hartle, of UC Santa
Barbara, who's also the
author of a general relativity
textbook now.
And Stephen Hawking,
who you must
know from Cambridge University.
They proposed something
called the wave
function of the universe.
From their point of view,
it's self contradictory
to talk about the
universe having an origin
because before the origin of
the universe, space and time
we're not even defined.
And therefore, you could not
think of there being a time
before the universe was created.
And therefore universe
didn't actually get created.
It just is and has some
earliest possible time.
And that's what this wave
function of the universe
formalism reflects.
But otherwise, it's
pretty similar, really,
to the idea of
tunneling from nothing.
The idea is that the universe
had some kind of a quantum
origin, which determined the
initial state of the universe.
In any case, for the purpose
of inflation, what we really
need to assume, and this could
be an assumption which follows
from any of these
theories, we need
to assume that the
early universe contained
at least a patch, and
we don't know exactly
know how big the patch
has to be, but greater
than or about equal to the
inverse Hubble constant times
the speed of light,
the Hubble length.
And this initial patch
also has to be expanding,
or else it would just collapse.
It really has to
be expanding faster
than a certain threshold.
But I won't try to put that
all into the one sentence.
Oh, I didn't say
patch of what yet.
Whoops.
That's where the average
value of phi is about zero.
And by average, I mean averaging
over rapid fluctuations,
if there are any.
And if one has this, no
matter where one got it from,
inflation will begin.
And once inflation
begins, it doesn't
matter much how it begins.
To see what happens next, it's
easiest to at least pretend,
that to a good
approximation, you
can treat a small
region of this patch
as if it were
homogeneous and behaving
like a Robertson-Walker universe
of the type we know about.
Then we can write the first
order Friedman equation,
which is a little
bit more informative
than the second order one.
I'm going to leave out
the curvature term.
We'll argue later that the
curvature term becomes small
quickly.
But for our first
pass, we'll just
assume that the universe
is described by something
as simple as the
Friedman Robertson-Walker
equation for a flat universe.
For row, we're just
going to put row sub f.
We'll assume that
our space is just
dominated by this false
vacuum energy density.
And this can easily be solved.
It just says that the
first derivative divided
by the function
itself is a constant.
Just take the square
root of this equation.
And that is an equation
which just immediately get
solved and gives
you an exponential.
So you find that for late
times, you just get a of t
behaving as a constant times
an exponential of time where
the exponential time constant,
which I'm calling chi.
Chi is just the square
root of this coefficient.
The square root of 8 pi
g over 3 times row sub f.
So clearly this is the
solution to that equation.
And for late times, it is the
solution that will dominate.
Actually it is
the only solution,
the way I've already
simplified this.
But if we started with the
full system of equations,
there'd be other solutions with
different initial conditions.
But this is always
what you would
be led to, for late times.
The exponential
expansion would dominate.
So that's what
innovation basically is.
It's a period of
exponential expansion.
There are a few
features of inflation,
which helps to understand
why it is so robust.
That is, why no
matter how it starts,
it leads to the same result.
So one feature of inflation
I'd like to mention
is the cosmological no-hair.
Some people call it a
theorum and some people
call it a conjecture.
I think the more precise
statement about this theorem
is that you can prove it as
a theorem perturbatively.
That is, if all initial
deviations are small,
you can really
prove it, but people
think it's true, even
beyond perturbation theory.
And in that case
it's a conjecture.
But it's basically the statement
that if one has a system with p
equals minus row c squared
and row is greater than zero,
if that describes the
matter, then essentially
any metric that you
start with will evolve
into this exponentially
expanding flat metric.
Any system will evolve
to locally resemble
a flat exponentially
expanding space time.
And the word locally there
is needed to make it true.
If, for example, you start
with a closed universe, as just
a simple example, which has
this kind of matter filling it.
It will start to
grow exponentially.
It will always stay
a closed universe.
It will never become
literally flat.
But as it gets bigger and
bigger, any piece of it
will look flatter and flatter.
And it will keep getting bigger
and bigger exponentially fast
forever.
So it will rapidly
approach a space
which looks like an
exponentially expanding
flat space.
Now this exponentially
expanding flat space time
has a name, which is
de Sitter space, named
after a Dutch astronomer.
It was discovered
early on in the history
of general relativity.
1917, I think, was the date
that de Sitter wrote his paper
about de Sitter space.
It has some very
interesting properties,
which De Sitter do not
notice all of them.
In spite of the fact
that I'm describing it
as a flat exponentially
expanding space time,
that's not the only
possible description.
It turns out that
the same space time,
by changing what you
call equal time services,
can be described as
either an open or closed
Robertson-Walker universe,
completely homogeneous
in both cases.
So it's very weird.
But the easy way to think
of it for most purposes,
is as this flat exponentially
expanding picture.
OK Next thing I want to point
out about de Sitter spaces is
that they have what are
called event horizons.
Now early in the course
we talked about horizons
and didn't really try
to quantify the name.
The horizons that we
used to talk about
are technically called
particle horizons.
Those are horizons
that have to do
with the past history
of the universe
and are related to the fact
that since the universe has
only a finite past history,
or as a cosmological model
at least, there's a finite
distance that light could
have traveled up
until this time.
And we cannot have any way of
seeing anything that's further
away than that maximum distance
that light could have traveled.
That's the particle horizon.
These event horizons
are different.
They're related really, to
the future of the universe,
rather than the past.
It's a statement that,
because of the fact
that these universes are
exponentially expanding,
if two events that happen
at a particular time
are separated from each other
by more than a certain distance,
then the light
from one will never
reach the future
evolution of the other.
And one can see that by
looking at the total coordinate
distance that light could
travel between any two times.
So I going to let
delta r of t1 t2
be equal to the
coordinate distance
that light travels
from t1 to t2.
And I'm going to
assume that a of t
is given by exactly
this formula.
And I'll write out
const because I
don't want to write
it too many times.
I could give it a one variable
symbol if I wanted to.
This delta r of t1 t2
is just the integration
of the coordinate
velocity of light
from t1 to t2 of c
divided by a of t dt.
The coordinate velocity of light
is just c divided by a of t.
We've seen that formula before.
And this can easily be done
by putting in what a of t is.
And we get c over
the constant that
appeared in that
formula, whatever
it is, times chi, the
exponential expansion
rate, times e to the minus chi t
1 minus e to the minus chi t 2.
And now, the question we
want to ask ourselves is,
suppose we let this
light ray travel
for an arbitrarily long
amount of time, which
means taking t2 to infinity.
And the important feature
of this expression
is that as t2 goes to
infinity, the expression
approaches a finite value.
The second term just disappears.
And you're left
with the first term.
So no matter how long
you wait, anything
that started out with a
coordinate separation larger
than that value, that
asymptotic value,
will just never be
reached by the light pulse
that you've sent.
And that's what this
event horizon is.
And it's easy to
see what it actually
amounts to numerically.
If you want to know how far
away the object has to be now,
in physical terms, so that its
coordinate distance is larger
than the maximum we get
here, we know how to do that.
The maximum value can be
written as just the limit as t2
goes to infinity
of delta of r1 r2.
And we have the expression
for it right here.
It's just the first
piece of this answer.
And this is the
coordinate distance.
If we want to know the present
physical distance of something
which is at that
coordinate distance,
we would just multiply it
by the present scale factor.
And present here means, t1
and t2 are the arguments here,
and we just want to multiply
by a of t1 to get the physical
distance of an object
which is at this boundary,
the boundary of what we'll be
able to receive a light ray
from and what we won't.
So this is the event horizon
distance, physical distance,
and it's just equal to
c times chi inverse.
When you multiply
by a of t1, you
cancel the constant
of the denominator
and you cancel the
e minus chi t1.
And you're just left
with c times chi inverse.
Which is the Hubble length.
It's the inverse
Hubble constant times
the speed of light, which
is the Hubble length.
So anything that is further away
one Hubble length, from us now,
if that object emits a light
ray, we will never receive it.
And that's called
the event horizon.
Now the reason this is
important is nothing travels
faster than light.
And that means that
in a de Sitter space,
everything is limited in
how far it can ever get.
And an important
implication of that
is that if, in our full space,
which may not be entirely
de Sitter space, if we
have a de Sitter region,
but junk outside that,
which we don't understand,
don't know how to predict,
could be anything,
we would still know, even
without knowing what's outside,
that whatever's
outside can never
penetrate into the
de Sitter region
by more than one event horizon,
by more than one Hubble length.
So the interior of
the de Sitter region
is protected from
anything on the outside.
And that is a rigorous
theorem of general relativity,
this protection.
And that means that once you
have a sizable region of de
Sitter space, no matter
what's going on outside,
it's never going to disappear.
It will always be protected
by this event horizon.
I should give you
now a few sample
numbers associated
with this scenario.
And here I have to say
that we don't really
know very accurately what are
the right numbers to give here.
So I think the word sample
numbers was well chosen there.
What we don't know is what
energy scale inflation actually
happened at in the
history of universe.
Turns out that the
consequences are pretty much
identical for most questions,
or all the questions
that they have been able so far
to investigate observationally,
regardless of what energy
scale inflation happened.
Inflation was
originally invented
in the context of
Grand Unified Theories.
And I think that's still
a very plausible context
in which inflation
might have happened.
And the sample
numbers I'll give you
will be numbers associated
with Grand Unified Theories.
And what starts the whole
story is the energy scale
of Grand Unified Theories,
which is about 10
to the 16 GeV billion
electron volts.
And this number is arrived at by
measuring, at accessible energy
with accelerators,
the interaction
strengths of the three
fundamental interactions
of the standard model
of particle physics.
The standard model
of particle physics
is based on three different
gauge groups, su 3, su 2
and u1.
Each one of those
gauge groups has
associated with it an
interaction strength.
And they can be measured.
And that's where we
start this discussion.
Then once you measure them
at accessible energies, which
is like 100 GeV, or
something like that,
then you can
theoretically extrapolate
to much higher energies.
And what is found is
that to good accuracy,
the three actually
meet at one point.
And that is the
underlying basis, really,
of Grand Unified Theories.
That's what allows the
possibility that all three
interactions are really just a
manifestation of one underlying
interaction, where the one
underlying interaction is made
to look like three
interactions at lower energies
through this process
called spontaneous symmetry
breaking, which was talked about
a little bit in a lecture I
gave the time before
last, I think, in,
probably in Scott's lecture.
Now this meeting
of the three lines
is decent in the context
of what is literally
the standard model
of particle physics.
But if one modifies the standard
model of particle physics
by incorporating supersymmetry,
a symmetry between fermions
and bosons, and that involves
adding a lot of extra particles
because none of the
particles that we know of
make up a fermion boson pair.
So in a supersymmetric model
for every known particle,
you introduce a new
unknown particle.
Which would be it's
supersymmetric partner.
In that minimal
supersymmetric extension
of the standard model,
the meeting of the lines
works much better.
So it's a piece of evidence
in favor of supersymmetry.
In any case, where
the lines meet
to good approximation in either
one of these two discussions,
whether it's supersymmetric or
not, is it about 10 to 16 GeV.
So that becomes the
fundamental math scale
of the end unified theories.
Hold on a second.
That' what I'm looking for.
Now once one has
this mass scale,
one can figure out an
appropriate mass density.
And that's what we're
really interested in,
what would be an
appropriate mass
density for a false vacuum
in a grand unified theory.
And one can develop
that, and we really
don't know how to do any better.
Because as I've told
you, we don't know really
know how to calculate
vacuum energies anyway.
But as a dimensional
analysis answer,
we can get the answer because
it is really uniquely determined
by dimensional
analysis up to factors.
If one wants to make an
energy density out of E gut
plus constants of
physics, the only way
to do that is to take E
gut to the fourth power
and divide it by h bar
cubed c to the fifth.
And you can convince
yourself at home
that that gives you
an energy density.
And you could even evaluate it
numerically, by mass density,
excuse me.
And this is about
equal to 2.3 times 10
to the 81 grams per
centimeter cubed.
So it's a fantastically
high mass density,
10 to the 81 grams
per centimeter cubed.
And if one puts this
into the formula for chi,
the exponential
expansion rate, chi
turns out to be about 2.8 times
10 to the minus 38 seconds.
And c times chi inverse,
the Hubble length,
it turns out to be about 8 times
10 to the minus 28 centimeters.
So all these numbers, off
scale by human standards.
And that's just a
feature of the fact
the Grand Unified Theories are
off scale by human standards.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Do I
have this backwards?
No, this is incredibly small.
This is 10 to the minus 28.
AUDIENCE: So then
it's chi [INAUDIBLE]
PROFESSOR: I'm sorry.
Hold on.
Yeah, no this--
AUDIENCE: Chi should
be [INAUDIBLE]
PROFESSOR: Chi
inverse is a time.
C times the time is a distance.
So I think that's right.
AUDIENCE: So is chi inverse
10 to the [INAUDIBLE]
PROFESSOR: Yeah, if we're in
cgs units, Chi inverse by itself
would differ by a
factor of 10 to the 10.
So it would be 10
to the minus 38, Hm.
Hold on.
This must be chi inverse.
AUDIENCE: Oh, OK.
PROFESSOR: There is
an inconsistency here.
You are right.
Yes, that's chi inverse.
This is time.
And then this just
multiplies by c.
OK so the way this
scenario would work is,
we would start with the early
universe with some patch
or order of magnitude this size.
Which I might point out
is 14 orders of magnitude
smaller than the size of
a single proton, which
would be about 10 to the
minus 13 centimeters.
So 15 orders of
magnitude, maybe.
And then we would
need enough inflation,
so that at the end of
inflation, the patch should
be on the order of maybe one
to 10 centimaters or more
It has to be at least about this
big, but could be much bigger.
There's no problem with
the being much bigger.
Much bigger would just
mean there's more inflation
than you minimally needed.
There's no problem with
having too much inflation.
And then it's a
matter of checking
and a calculation, which
I'll tell you the answer of.
If we want to go from some
size of the end of inflation
to the present universe--
And that's really what
we're interested in, ultimately,
getting the present universe.
--there'd be a further
coasting expansion
from the end of
inflation until now,
which can just be calculated
by using the idea that a times
temperature, scale factor times
temperature, is a constant.
So the increase in
the scale factor
is proportional to decrease
in the temperature.
And the reheat temperature
of this model--
Maybe I didn't describe
reheating exactly,
I'll describe it
quickly in words.
At the end of inflation,
the scalar field
is destabilized by
these fluctuations
and rolls down the hill, then
oscillates about the bottom.
And when it oscillates
about the bottom,
we need to take into account the
fact that this field interacts
with other fields.
And it then gives its energy
to the other fields, basically
the standard model
fields ultimately,
heating them up, producing
the hot soup of particles
that we think of as the starting
point for the conventional Big
Bang Theory.
So this reheating
process at the end
of inflation as
the inflaton field
oscillates about its minimum,
reproduces the starting point
of the conventional
Big Bang Theory.
And it produces a
temperature which
is comparable to the
temperature that you started at,
which is the temperature
scale of the theory.
So if it's Grand
Unified Theory scales,
we would reheat to a temperature
of order 10 to the 16 GeV.
And then, to ask what will
be the expansion factor
between then and now, it would
be 10 to the 16 GeV times
the Boltzmann constant
times 2.7 Kelvin.
This is the ratio of
the temperature then
to the temperature now,
both expressed as energies.
And then we might
want to multiply this
by 10 centimeters, if we say
that at the end of inflation,
the universe was 10
centimeters across.
Size at end of inflation.
This, I worked out at
home, is about 450 times 10
to the 9th light years.
And we would want
something like 40 times 10
to the 9th light years to
explain the present universe.
So this is about
10 times too big.
And that's OK.
It means that we could
get by with one centimeter
and 10 centimeters is
being a bit generous.
So inflations would start
with this tiny patch.
At the end of
inflation the patch
would have grown
to one or maybe 10
or perhaps more,
centimeters in length.
And then by coasting
up to today it
becomes something that's
larger than the region
that we now observe.
And that's basically
how the scenario works.
Any questions about those
numbers or the general pattern
of what we're
talking about here?
OK.
What I want to talk about
next, and this will pretty much
be where we'll stop,
although a few other things
we might mention
if we have time,
I want to talk about
how it solves the three
cosmological problems
that we have discussed
of the conventional
Big Bang model.
And the explanations are
actually quite simple.
So we can go through
them pretty quickly.
First we had the horizons
slash homogeneity problem.
Remember that was caused, or
could be stated as, the problem
that the early universe
expanded so fast
that the different
pieces of it did not
have time to talk to each other.
And, in particular, when the
cosmic microwave background was
released, points at opposite
sides of the universe
were separated from each other
by about maybe 50 horizon
distances, we calculated.
And that means there's no way
they could have communicated
with each other, and
therefore no way we
could explain how
they turned out
to have the same temperature
at the same time.
In this case, what
we've done is,
we've inserted into the
history of the universe
an extra phase of evolution,
the inflationary phase.
And if we go back
to the beginning
of the inflationary
phase, we see
that that problem
is just not there.
And if it's not there,
it doesn't develop later.
At the beginning of
the inflationary phase,
by assumption, the region that
we're starting to talk about
was about horizon
length in size.
And if we had enough inflation
to produce 10 centimeters out
of that, that was 10
times more than we needed,
it would mean that the entire
observed universe would
be coming from a region
that would be only
about a tenth of the size
of this Hubble length.
So that would therefore
be well inside the horizon
at that time.
And that means that if you
allow a little bit of leeway
with these numbers by having a
little bit of extra inflation,
there can be plenty of time
for the entire region that's
going to become our
personally observed region,
to come to a uniform temperature
by the ordinary processes
of thermal equilibrium.
Because they're much less than
the horizon distance apart.
And then once the uniformity is
established, before inflation,
when the region that
we're talking about
is incredibly tiny, inflation
takes over and stretches
that tiny region
so that today, it's
large enough to encompass
everything that we see.
And therefore
everything that we see
had a causally
connected past and had
time at the early stages to come
to uniform temperature, which
is then preserved as
the whole thing expands.
So that gives a very
simple explanation
for the homogeneity problem.
Basically before inflation
the region was tiny.
Second on our list was
the flatness problem.
And the basis of that
problem was the calculation
that we did about how omega
minus 1 evolves in time.
And we discovered
that omega minus 1
always grows in magnitude
during conventional evolution
of the universe.
And that therefore, for omega
minus 1 to be small today,
it would have to be amazingly
small in the early universe,
as small as 10 to the minus
18 at one second after the Big
Bang to be consistent with
present measurements of omega
minus 1.
The key element there was
this unstable equilibrium
and the fact omega L minus
1 always grew with time.
And that depended on
the Friedman equations.
During inflation, the
Friedman equations
in some schematic sense
were the same equations,
but the rows that go
into it are different.
So the equations
basically are different.
And if we look at the key
equation, the first order
Friedman equation,
H squared equals
8 pi over 3 G row minus
Kc squared over a squared.
This was the
equation that we used
to derive this flatness problem.
We could see
immediately, if we now
think about it, during
the inflationary process,
things are completely reversed.
Omega is driven towards
1, and exponentially fast.
And the way I see that is to
just ask what is this equation
do during inflation.
And during inflation,
we just replace row
by this constant
value row sub f,
the energy density of the
false vacuum is fixed.
And that means that during
inflation, this term is fixed.
This term is falling off
like 1 over a squared.
And a is growing
exponentially with time.
So that means that
this term is decreasing
relative to that term
by a huge factor,
by the square of the
expansion factor.
So in our sample
numbers over there,
we were talking about an
overall expansion from 10
to the minus 27
centimeters to 10.
That's expansion by a
factor of 10 to the 28.
In that case, during
inflation, this term
decreases by a factor
of 10 to the 56
while this term
remains constant.
And that means that
by the end inflation,
this is completely
negligible and this equation
without this extra term means
you have a flat universe.
So during inflation,
the universe
is driven towards flatness,
like one over a squared, which
is 1 over the square of this
exponential expansion factor,
so very, very rapidly.
And finally, the third of the
problems that we talked about
was the monopole problem.
We argued, originally
Kibble argued,
that you'd expect approximately
one of these monopoles
to form per horizon volume,
just because the monopoles are
basically twists in
the scalar fields.
And there's no way the scalar
fields can organize themselves
on distances larger than
the horizon distance.
So you'd expect
something on order
of-- It's a crude argument.
--but something on
the order of 1 not
in the scalar field
per horizon volume.
And that led to far too
many magnetic monopoles,
fantastically too many.
And the formation of one
monopole per horizon volume
is hard to avoid.
I don't know of any
way of avoiding it.
But what gets us out
of the problem here,
is that we can easily arrange
in our inflationary model
for the bulk of the
inflation to happen
after the monopoles form.
And that means
the monopoles will
be diluted by the
exponential expansion that
will occur after
the monopoles form.
The rest of the
matter is not diluted,
because when
inflation takes place
it's at a constant
energy density
so the amount of other
stuff that will be produced
is not diminished by
this extra expansion.
But the monopolies,
which will produce first,
will be thinned out
by the expansion.
So the basic idea here
is that the volume goes
by a factor of the order,
using our sample numbers,
it's linearly growth by
a factor of 10 to the 28.
Volumes go like cubes
of linear distances.
So 10 to the 28 cubed is
10 to the 84, I think.
Probably right.
And that means that we
can dilute these monopoles
by a fantastic factor and make
everything work, if we just
arrange for the
monopoles to be produced
before the exponential
expansion sets in.
OK.
Finally, and I think
this is probably
the last thing we
will talk about,
another problem that we
could have talked about
and we'll talk about
the solution of it
now, even though we never really
talked about as a problem,
is the small scale
nonuniformities
of the universe.
And if we look out
around the universe
we don't see a uniform
mass distribution,
we see stars and stars collected
in galaxies and galaxies
collected in clusters
and clusters collected
in super clusters, a very
complicated array of structure
in matter.
Those are all nonuniformities.
And we think we understand how
they evolve from early times
because we also see in the
cosmic microwave background
radiation, small fluctuations,
which we can now actually
measure to very high
degree of accuracy.
Those small fluctuations
provide the seeds
for the structure in the
universe that happens later
because of the fact that the
universe is gravitationally
unstable.
So at very early times
what we're seeing directly
in the CMB, these
nonuniformities
were only at the level
of one part in 100,000.
But nonetheless,
in regions where
there was slightly
more mass density,
that pulls in
slightly more matter,
producing still stronger
gravitational field
pulling in more matter and that
amplifies the fluctuations.
And that affect, we believe,
is enough to account
for all the structure that
we see in the universe
as originating from these tiny
ripples on the cosmic microwave
background.
But that still
leaves the question
of where do these tiny
ripples come from.
And in conventional
cosmology, one really
had no idea where
they come from.
One knew they had to be there
even before they were seen
because we had to account for
the structure in the universe
and how it evolved.
When they finally were seen,
it was seen just right.
Everything fit together.
In inflationary cosmology,
the exponential expansion
tends to smooth everything out.
And for a while, those
of us working on it
were very worried that inflation
would produce a perfectly
smooth universe and we'd
have no way of accounting
for the small fluctuations
that were needed to explain
the existence of
stars and galaxies.
But then it was realized
that quantum theory
can come to our rescue.
Classically, inflation
would smooth everything out
and produce a uniform
mass density everywhere.
But quantum
mechanically, because
quantum mechanical theories are
fundamentally probabilistic,
the classical prediction
of a uniform density
turns into a quantum mechanical
prediction of an almost uniform
density, but with some
places being slightly higher
than that uniform density, other
places being slightly lower.
And qualitatively,
that's exactly what
we see in the cosmic microwave
background radiation.
And furthermore, we can
do it quantitatively.
One can actually
calculate the effects
of these quantum fluctuations.
And that's what I
want to show you now.
The actual data on that,
which is just gorgeous.
Shown here is the
Planck seven year data.
Shown here is the
Planck seven year data,
where what's being
plotted is the amplitude
of the fluctuations versus
the angular wave length.
One is seeing these as
a pattern on the sky.
So the wavelength you see as
an angle, not as a distance.
And long wave lengths
are at the left.
Short waive lengths
are at the right.
It's really done as
a multiple expansion,
if you know what that means.
And those numbers are
showing on the top.
And the data points are
shown as these black bars
with their appropriate errors.
And the red line is the
theoretical prediction
due to inflation, putting
in the amount of dark matter
that we need to fit
the data that we also
measure from the supernovae.
And it's absolutely gorgeous.
So I have a little
Eureka guy to show you
how happy I was when
I saw this graph.
And with the help
of Max Tegmark,
we've also put on
this graph what
other theories of
cosmology would give.
So if we had an
open universe, where
omega was just 0.2 or 0.3,
as many people believed
before 1998, we would have
gotten this yellow line.
If we had inflation
without dark energy,
making omega equal what out of
matter, out of ordinary matter,
we would get this
greenish line, which
also doesn't fit
the data at all.
And there's also something
called cosmic strings
that we haven't talked about.
It was for a time, thought
to be a possible source
of the fluctuations
in the universe.
But once this data came in, that
became completely ruled out.
Now this is not quite
the latest data.
The latest data come from
the Planck satellite.
And it was released last March.
And I don't have that
plotted on the same scale,
but this is the latest
data which, as you see,
fits even more gorgeously
than the data from WMAP.
The more accurately
it gets measured,
the better it fits the
theoretical expectations.
Now I should mention for
truth in advertising,
that this data is to some
extent fit to the model.
It's actually a
six parameter fit.
But of those six parameters,
I don't have time
to talk about them in
detail, but four of them
are pretty much determined
by other features.
Two of them are just
fit to the data.
And one of them
is something that
changes the shape a little bit.
It's the opacity of
the space between us
and the surface of
last scattering.
An important parameter that's
fit that you should know about,
is the height of the curve.
The height of the curve
can, in principle,
be predicted by
inflation if you knew
the full details of
this potential energy
function that I've erased
for the scalar field.
But we don't.
We just have some
qualitative idea
about what it might look like.
So the height of the
curve is fit to the data.
But nonetheless, the location of
all these peaks and everything
really just come out of the
theory and it's just gorgeous.
And it works wonderfully.
So the bottom line
is I think inflation
does look like a very good
explanation for the very
early universe.
It's kind of bizarre since
it talks about times like 10
to the minus 35 seconds
after the Big Bang which
seemed like a totally incredible
extrapolation from physics
that we know.
But nonetheless,
marvelously, it produces
data that agrees fantastically
with what astronomers are now
measuring.
So we'll stop there.
I want to thank you all
for being in the class.
It's really been a
fun class to teach.
I have very much enjoyed
all of your questions
and enjoyed getting
to know you and hope
to continue to see you around.
Thank you.
