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PROFESSOR: OK, in that case,
let's begin in our usual way
by going through a review
of last time's lecture.
Last time, we talked
really about two
calculational problems.
One was the calculation of
the age of the universe,
taking into account a
universe model which
has matter, radiation,
vacuum energy, and curvature.
And we got the general formula.
And then for the same type
of cosmological model,
we also calculated how
one finds the brightness
of a distant source--
the energy flux in terms
of the redshift of that source.
So first, the age of the
universe calculation--
that really just depends on the
first-order Friedman equation,
which I've rewritten here.
We put three terms
on the right hand
side for the mass density-- a
matter term, a radiation term,
and a vacuum energy term.
And we know-- and this is
the important ingredient--
we know how each depend
on the scale factor.
Non-relativistic
matter falls off like 1
over the cube of
the scale factor.
Radiation falls off like
1 over the fourth power
of the scale factor.
And vacuum energy
is just constant.
Next step that we did was just
to rewrite this equation, where
we put in the explicit
time dependence
in the form of this x
which is the ratio of a
of t to the present value of
the scale factor-- a of t 0.
And furthermore, we
expressed the matter density
in terms of the present
contribution to omega.
And rewriting equation in that
language, it takes that form.
And then, I pulled a fast one.
I said we could also
write this last term
to look pretty much
like the others.
It just is a constant that
falls off like 1 over a squared.
So if you define
omega sub k 0, which
is exactly what you need to
make this look like that,
and in terms of omega
sub k 0, all four terms
have the same characteristic.
They're just a constant
times a power of x.
So this is, then, the rewriting
of the Friedman equation one
more time, just using this
new definition of how we're
going to treat the
curvature of the universe.
And simply by looking
at this formula
and applying it
to x equals 1, you
can see that that
becomes then 1 is
equal to the sum
of these omegas.
And that can be thought
of as a clearer, perhaps,
definition of what
omega sub k 0 is.
It's just 1 minus all of the
other contributions to omega.
So it's how much the actual
mass density of the universe
differs from the
critical density.
Then once we have
this equation, which
is the equation
which tells us what
x dot is as a function of x,
we could just rewrite that
by bringing dt to one
side of the equation
and dx to the other and
integrating both sides.
And that leads to
our final result.
The age of the universe is
simply given by that integral.
And this is a very
neat expression
for the age of the
universe in terms
of the present value of
the Hubble expansion rate
and each contribution to omega
in terms of its present value.
And you just plug those
into this formula.
In general, you have to do
the integral numerically,
because the integral's
a little too complicated
to have an analytic expression.
And that will give you
the age of universe
for any model that
meets this description.
So any questions about that
calculation before we go on?
OK, very good.
The next calculation
we did last time
was the calculation of
radiation flux versus redshift.
And this is exactly
what the astronomers
were measuring in 1998
when they concluded
that the universe
was accelerating.
They were looking at distant
supernova type 1a explosions.
They made the assumption
that all supernova type
1a explosions have the same
intrinsic power output.
That's based roughly on
observation and guesswork.
There's not really a
good theory for it,
so it's mostly a matter of being
consistent with observations.
But then they could
calculate for any given model
in terms of these different
omegas what you expect in terms
of received radiation as
a function of redshift.
And they compared their
data with the models--
and I'll show you
that data shortly--
and found that the
models only fit
if one had a significant
component of vacuum
energy causing
universe to accelerate.
So to do the calculation, we
need a metric for the universe.
And I considered only
the closed universe case.
There's also the flat
case and the open case,
which are similar.
And you'll actually be asked to
do those on the homework set.
So the metric for
a closed universe
can be written this
way, where sine psi is
the square root of k
times r, to relate it
to the other way-- the more
standard way-- of writing
the Robertson Walker metric.
But for our purposes
for this calculation,
it's easiest to do it
this way, because we're
going to be interested in
radio trajectories of photons.
And this metric simplifies
the radial direction
as much as it can be.
It's just d psi squared.
Oh, it's the
computer that froze.
You never know with Windows.
I think we're in business now.
Back to where we were.
We have the metric.
Now what we want to do
is imagine a light source
being received by a detector.
And we put the light
source in the center
of our coordinate system.
We put the detector at some
distance corresponding to psi
equals psi sub D, where psi is
our radial coordinate and psi
sub D is the radial
coordinate of the detector.
We imagine a whole sphere
with the same radius
as the detector,
because we expect
the source to be
spherically symmetric.
And therefore, the light
emitted by the source
will be uniformly
spread over that sphere.
And that will allow us to
calculate how much of it
will hit the detector.
The fraction
hitting the detector
will just be the area
of the detector divided
by the area of the sphere.
The area of the detector
is whatever it is.
We call it capital A.
The area of the sphere
is 4 pi times the
radius of the sphere.
And the radius of the sphere
in physical coordinates
is the scale factor squared
times the sine squared
of psi sub D, coming
from the metric.
It's the radius that appears
in the angular part that
counts, because it's the angles
that we're integrating over
to get the area of the sphere.
So the radius is
just a tilde squared
times sine squared is
the radius squared.
Then we also need to
remember something
we've said a number of times
previously in this class, which
is that when the photons
travel from the source
to the detector, their
intensity is suppressed
by two powers of 1 plus z,
two powers of the redshift.
And one of those
factors in 1 plus z
comes from redshifting
each photon.
The frequency of each
photon is redshifted,
and that means that the energy
of each photon is redshifted--
goes down by a
factor of 1 plus z.
But in addition, the rate
of arrival of the photons
is essentially a clock
which is also time dilated.
So the rate of arrival of the
photons as seen by the observer
is suppressed by another
factor of 1 plus z.
So putting all that together,
the received energy flux,
which is the power received
divided by the area,
is just the power emitted by
the source divided by 4 pi.
We get this factor
of 1 plus z squared,
due to what we just discussed.
And then the a squared of
t sine squared psi sub D.
So it's just the total
power times that fraction
that we receive times the two
factors of 1 over 1 plus z.
And this then is essentially
the final answer,
except we want to know how to
evaluate a tilde squared of t 0
and sine squared of psi
sub D in terms of things
that we more directly measure.
So to do that, a
tilde of t 0 turns out
to be easy, because
it really is just
related by the definition of
omega sub k 0 to omega sub k 0.
So this formula is
just a rewriting
of the definition
of omega sub k 0.
To figure out what
psi is, we want
to integrate along
the line of sight
to be able to figure
out the time of emission
in terms of psi.
And that time of
emission could then
be related to the redshift,
because the redshift is just
the ratio of the scale factors
between reception and emission.
So we look first at the metric.
And say we're
going to be looking
at null geodesics in
the radial direction.
And null means ds
squared equals 0,
and that's minus c
squared dt squared
plus a tilde squared of
t times d psi squared.
And that implies
immediately that the psi dt
is just equal to the speed
of light divided by a tilde.
And then we can get
the total increment
in psi between the source
and us by integrating
between the time of
emission-- the time
of the source-- to the
present time-- t sub 0.
And then it's just a matter
of changing variables
to express the variable
of integration.
Instead of st, we could express
it as z-- the redshift itself.
And that brings in a factor of
h, because h is a dot over a.
And I showed the
manipulations last time,
but it brings in a factor of h.
But we know what h is
as a function of z.
It comes from the
Friedman equation.
And that then gives us an
expression for psi of z sub
s as an integral over z.
And writing in what h of
z is and what a tilde of z
is from that expression,
the expression for psi of z
becomes the equation
that's boxed.
Just a matter of algebraic
substitutions involving
the Friedman equation, which
determines what h of z is.
And then putting
everything together,
J is just given by this
expression, where all I've done
is to substitute the
expression for a tilde of c 0.
And sine squared
psi is still here,
but it gets evaluated
according to that formula.
And putting these together,
we have a complete calculation
of the received radiation
flux as a function
of cosmological parameters--
the omegas and the h 0--
and the redshift of the source.
And that's the end
of the calculation.
And that's where we
finished last time.
So any questions about
that calculation?
OK, fine.
In that case, moving on, the
next thing I wanted to show you
was some real data.
So here are some real data
from one of those two teams
that made the original
announcements in 1998.
This is from the High-Z
Supernova Search Team.
And I should write some
definitions on the blackboard.
The vertical axis there
is essentially brightness.
But you wouldn't expect the
astronomers to just call
it brightness, because they
like to use fancier words.
So they write it as
little m minus capital
M-- measured in magnitudes,
they put in parentheses.
And little m minus
capital M has the name,
it's called the
distance modulus,
meaning it's a way of
measuring distance.
They think of brightness as
a way of measuring distance,
which indeed is what
it's being used for.
And it's defined as
5 times the logarithm
base 10 of d sub L
over 1 megaparsec,
which means the luminosity
distance-- I'll define this
in more detail in
a second-- d sub L
is the luminosity distance--
distance as inferred
from the luminosity.
And they're measuring
it in megaparsecs
and taking the
logarithm base 10.
And then, by convention,
there's an offset here of 25.
Why not?
So this is the definition
of the distance modulus.
And d sub L is defined by
the relationship of what
J would be in a flat
Euclidean universe
if you were receiving
that luminosity.
So J is equal to
the actual power
output of the source divided
by 4 pi d sub L squared.
This defines d sub L. So
d sub L is the distance
that that source would have
to be at in a static Euclidean
universe for you to see
it with the brightness
that you actually see.
This, I guess, completes
the definitions,
but we can put these together.
And m minus M is then
equal to minus 5/2
times the logarithm
base 10 of 4 pi J times
1 megaparsec squared,
divided by the actual power
output of the source, and
then, of course, plus 25.
So this relates this
distance modulus
to the energy flux and the power
output of the original source.
There's also on this
slide the acronym MLCS.
MLCS stand for multi-color
light curve shape.
And what that refers to is the
High-Z Supernova Search Team
invented a method of
compensating, to some extent,
for small variations
in the actual power
output of the
supernovae type 1a.
Instead of assuming
that they all
have exactly the
same brightness,
they discovered by looking at
nearby supernovae of this type
that there's a correlation
between the absolute brightness
of the supernovae and the
shape of the light curve--
that is, light versus time.
So they were careful to
measure the light versus time
for the supernovae that
they used in this study.
And they used that as a way
of applying a small correction
to what they interpreted
as the intrinsic brightness
of each supernova.
And the results
are these points.
[INAUDIBLE] the top
are the raw points,
and three different curves
for three different models.
And they characterize the
models in the same way
we would-- in terms of different
contributions to omega.
So the top model is the
cosmological constant dominated
model, where omega
sub lambda, which
is what we've been calling
omega sub vac is 0.76.
And it's a flat model,
so 0.24 for omega matter.
And radiation is ignorable.
They compared that with the
middle model of these three,
which was a model that
had no vacuum energy,
and omega matter of 0.2.
That was essentially
the dominant model
at the time, the belief that
the universe was open then
had about a critical
density of 1/5 or 1/4.
And then they also
compared it with a model
where omega was 1-- entirely of
matter with no vacuum energy.
And that was this
dashed curve, which
is the lower of
these three curves.
And when the data
is just plotted,
it's a little hard to see
how much difference there
is between the three curves.
So they re-plotted the data,
plotting the middle curve
as a straight line
by construction.
And then they plotted
deviations from that line.
And they did that for both
the theoretical curves
and the data.
And in this magnified
picture, you
can see a little bit better
that this top curve fits
things the best.
And that's what they call
the lambda CDM model.
It corresponds to
omega m equals 0.24.
Omega lambda equals 0.76.
So it's the model with
a cosmological constant,
with a vacuum energy.
And lambda CDM stands for
lambda and cold, dark matter.
And cold, dark
matter is just what
we've been calling
non-relativistic matter.
So the claim is that these data
points, even though there's
a fair amount of scatter,
fit the top curve much,
much better than they
fit the middle curve
or the bottom curve.
And statistically, that's true.
It really is a much better
fit, even though by eye, it's
not that clear what's going on.
I think by eye it looks clear
that the top one fits it better
than others, but
it's not that clear
how important the difference is.
But nonetheless, the astronomers
were thoroughly convinced
that this was a real effect.
There was considerable
discussion
about possible
systematic errors.
And I guess next, I'll say
a few words about that.
First of all, I
should maybe just
clarify a little bit
better what's being seen.
What's being seen is that for
a given redshift, this curve,
which basically shows brightness
in a funny, funny way, where
dimmer is upward, larger
values of little m minus M--
there's a minus sign
in this formula--
means a dimmer galaxy, one
that looks further away.
And basically, when astronomers
see this, they think distance.
So larger values
means further away.
So what's being seen is that
these distant supernovae
are a little bit
dimmer than what
you would expect in either
of the other two models,
either of the models that
do not have vacuum energy.
And the amount by
which they're dimmer
is a few tenths of a magnitude.
And each tenth of a magnitude
corresponds to about 10%
in brightness.
So what they're saying is that
these distant if supernovae,
if we assume they
really fit this curve,
are 20% to 30%
dimmer than you would
have expected in other models.
It might be worth saying a
little bit about why dimmer
is the right sign to correspond
to acceleration, which
is by no means totally
obvious, I don't think.
So we're plotting--
AUDIENCE: What year is this?
PROFESSOR: What year?
This was old data that
was published in 1998.
It has gotten better.
Now it's much more
unambiguous that this works.
So this is distance as
inferred by brightness.
So this is basically
what's being plotted.
If one thinks about a
fixed z in which way
that you go-- up or down--
I find that totally cryptic.
I don't really parse that
very well in my own head.
But it's much clearer if you
think about the other way.
You could think about a
galaxy-- or a supernova
in this case-- at a
fixed distance, and ask,
suppose I compare different
models-- ones that accelerate
and models that
don't accelerate.
So if we fix the distance
and say, what would we
expect for the
redshift of a given
galaxy, in an accelerating
model versus a non-accelerating
model-- remember, the
redshift is basically
a measure of the
velocity, or at least it's
strongly influenced by the
velocity of the object.
So if the universe
is accelerating,
it means that the universe was
expanding slower in the past
than you would have
thought otherwise.
It's speeded up to reach
its present expansion rate.
So an accelerating
universe is a universe
that was expanding
slower in the past.
And slower in the past means
that a galaxy at a given
distance would have
been moving slower,
and hence would have
had a lower value of z.
So the effect of acceleration
for a given distance-- we'll
fix the distance-- should be
to move the line that way,
towards lower z.
And by moving the dot that way,
it puts it above the curve.
So it's the same as
shifting things up,
which is the more
natural way of describing
what's seen in the graph.
The points are higher
than the curve.
So the bottom line, though,
is that what they're saying
is distant supernovae are
20%, 30% dimmer than you
might have thought.
And from that, they want to
infer that the universe is
accelerating, which is a
rather dramatic conclusion.
So naturally, you
want to ask, are there
other things that can cause
supernovae to look dimmer?
And of course, there
are other things
that can cause
supernovae to look dimmer
than you might have thought.
And there are two
main ideas that
were discussed at the time.
One of them is just plain dust.
If you're looking at something
through a dusty atmosphere,
it looks dimmer than
it would otherwise.
And that is a
genuine possibility
that was strongly considered.
The arguments against
dust were mainly twofold.
The first is that
dust very rarely
absorbs uniformly
across the spectrum.
Dust usually-- depending on
the size of the dust grains--
absorbs more blue
light than red light,
leaving more red
light coming through.
So the effect of seeing
something through dust
is normally to cause
it to look more red.
And this reddening was not seen.
The spectrum of the light
from the existing supernovae
was analyzed very carefully.
And the spectrum
of the distant ones
looked just like the spectrum of
the nearby ones-- appropriately
redshifted, of course, but
otherwise not distorted
in any way.
There was no sign
of this reddening.
Now, it's possible to have
what the astronomers refer
to as gray dust, which is by
definition dust that absorbs
uniformly across the spectrum
of what you're looking at.
But the grains have
to be unusually large.
And nobody was ever
able to figure out
a source for dust
grains of that sort.
So based partly on
theoretical grounds
and partly on what
nobody has ever found,
there's no evidence
for dust grains
that would possibly
cause dimming
that would look this way,
that would be dimming
that was uniform
across the spectrum.
Yes?
AUDIENCE: How do you
tell the difference
between reddened light from
dust and redshifted light?
PROFESSOR: OK, how do
you tell the difference
between reddened light from
dust and plain old redshift?
The difference is that the
plain old redshift uniformly
shifts everything
by the same factor.
So the whole spectrum is
just moved down uniformly
towards the red.
This reddening
effect really means
that the blue part
of the spectrum
is depressed relative
to the red part.
So the shape of the
spectrum is changed.
So one argument is that
we don't see reddening,
and we don't know any way to
make dust that would be gray.
The second argument is that
if dust was a major factor,
presumably most of the
dust that would be relevant
would be dust in the same galaxy
as the supernova explosion
itself, because there's
not that much dust
in intergalactic space.
And if dust in the galaxy of the
supernova itself were relevant,
then-- let me draw a
little picture here.
So if dust in what's
called the host
galaxy-- the galaxy which
has the supernova in it--
then you would have
a picture where
there would be a ball of
dust filling the galaxy.
And the supernova that you're
looking at might be there,
or it might be there.
And let's say we're
looking from over here.
So depending on where the
supernova was in the galaxy,
we would see very different
amounts of intervening dust.
And if dust were
causing this dimming,
it would mean we would be
seeing a significant scatter
in the amount of dimming
depending on where
the supernova happened
to be in its host galaxy.
And that spread was not seen.
The spread that one
sees in that curve
could be measured and calibrated
against known uncertainties
in the brightness of
supernovae and then
the detection apparatus.
And the spread that
was seen was just what
you expect without any
additional spread associated
with a dusty galaxy
acting as the host.
So no evidence for the
spread of brightnesses
that would be expected
from a dusty host.
Another item that
was considered--
these are the main
arguments against dust--
another argument
that was considered,
another possible source of
dimming, is galactic evolution.
And there, the main effect
that people worried about
was the production of
heavy chemical elements
during the life of a galaxy.
As you've certainly learned
about from your reading--
I don't know if we've talked
about it in class or not--
the early universe was essential
all hydrogen and helium.
Heavier elements were
made later in stars
that produce
supernovae explosions.
And these supernovae
explosions gradually
cause galaxies to become
more and more enriched
with heavy elements.
And by heavy, I mean
anything heavier than helium.
And that could
affect, in principle,
the behavior of
supernovae explosions.
So the evidence
against that was simply
that every other
characteristic that astronomers
could measure of
these supernovae
in the distant
galaxies looked exactly
like what was seen
for nearby galaxies.
So no evidence for any
kind of evolution was seen.
And there are many
properties you
could measure that are
independent of distance,
like the shape of the
spectrum and things like that,
and the pattern of the
light curve versus time.
So all those characteristics
that astronomers can measure
seem to be exactly the same
for the very distant supernovae
which happened
billions of years ago,
and the more nearby ones
that happened recently.
And furthermore,
among the nearby ones,
there's a big spread of
abundances of heavy chemical
elements, just because
different galaxies
have had different histories.
So among the nearby
ones, you could
look for is there
an effect caused
by the relative abundance
of heavy elements,
and astronomers didn't find any.
So there was no sign
that galactic evolution
could be playing a role
here, even though one
does need to worry about it.
So the point is that
distant supernovae 1a
look like nearby ones.
I'll call that a in my outline.
And b is that among the nearby
1a's, heavy element abundance
had no perceptible effect.
So the dominant opinion
gradually shifted,
and now I think it's almost 100%
that this acceleration is real.
The acceleration, by
the way, is further
confirmed by measurements
of fluctuations
in the cosmic background
radiation measurements that
have been done by some
ground-based experiments,
and also the satellite
experiments of WMAP
and now Planck, which
measure the anisotropies--
the ripples-- in the cosmic
background radiation.
It's hard to see what
those ripples would
have to do with the
amount of vacuum energy.
But it does turn out-- and
we'll talk more about this
a little bit later--
that we really
do have a detailed theory
of what makes these ripples.
We can calculate what the
spectrum of those ripples
should look like.
And the calculations
depend on parameters
which include the
amount of vacuum energy.
And in order to
make things work,
one does have to put
in essentially exactly
the same amount of vacuum
energy as has been detected
in these supernova
1a observations.
So everything fits
together very tightly.
And I think now, just about
everybody is convinced
that the universe
really is accelerating.
The acceleration
could, in principle,
have at least two different
causes that we can talk about.
One is vacuum energy, which is
the one that I'm focusing on,
which is the
simplest explanation.
The other possibility that is
discussed in the literature
is something called
quintessence,
which is a made-up word.
And what it refers
to is the possibility
that the acceleration
of the universe today
could be caused by a mechanism
which is really in principle
exactly the same as what
we talk about for inflation
in the early universe and
will be talking about later.
Specifically, there could be
a slowly evolving scalar field
which is essentially uniform
throughout the universe,
and changing slowly with time so
it looks like it's a constant.
And it could be the energy
density of that scalar field
that is looking to us as
if it were vacuum energy.
But that's the
minority point of view.
And that introduces
extra parameters
that don't seem to be necessary.
But it's up for grabs.
Nobody really knows.
OK, any questions about
what we just talked about?
In that case, let me go
on to my next topic, which
is I want to talk
a little bit more
about the physics
of vacuum energy.
What is it that we
understand about it,
and why is it that
most physicists say
it's the least understood
issue in physics?
We really don't
understand vacuum energy,
even though we do understand
why it might be nonzero.
Where we're totally at a loss
is trying to make any sense out
of the value of the energy
density that is actually
observed.
So where does vacuum energy come
from in a quantum field theory?
There are basically, I would
say, three contributions.
Maybe I should say in
quantum field theory.
The other context in which
this might be discussed
would be string theory.
I may or may not say
something about string theory,
but I won't say much.
But in quantum field
theory, there are basically,
I think, three contributions.
The first is the
easiest to understand,
which is quantum fluctuations
in bosonic fields,
where the best
example is the photon,
or the electromagnetic field.
Now, in a classical
vacuum, e and b-- the
the electric and
magnetic fields--
would just be 0, because that's
the lowest possible energy
density.
But just as you are
probably aware that
there's an uncertainty principle
in quantum mechanics which
tells you that the momentum and
position of a particle cannot
be well-defined
at the same time,
it is also true that e and
b cannot be well-defined
at the same time.
So the uncertainty principles
applied to the field theory
imply that e and b cannot
just be 0 and stay 0.
E and b are constantly
fluctuating.
And that means
that there's energy
associated with
those fluctuations.
And the mathematics of it is
actually incredibly simple.
If one imagines the
fields inside a box,
to be able to at least
avoid the infinity of space,
the fields inside a
box could be described
in terms of standing waves,
where each standing wave is
either a half wavelength or
a full wavelength across.
And by the way, you'll be doing
a homework problem on this.
And each standing
wave has the physics
of a harmonic oscillator.
It oscillates sinusoidally
with time, the wave.
And when one works out
the mathematics, and even
the quantum mechanics,
it's exactly the same
as a harmonic oscillator.
So each standing wave
has a zero-point energy.
You may know that the zero-point
energy of a harmonic oscillator
is not 0, but it's 1/2 h
bar omega, or 1/2 h nu,
depending on whether you're
using nu or omega to describe
the frequency of the oscillator.
So each standing wave
contributes 1/2 h bar omega.
And then the problem is how
many standing waves are there?
And the answer is, there's
an infinite number of them,
because there's no limit to how
short the wavelength can be.
So there's no limit to
how many ups and downs you
can have in your
standing wave from one
end of the box to the next.
So the answer you
get is infinite.
It diverges.
Now, the fact that it
diverges at short distances
can be used as an excuse for
getting the problem wrong.
Obviously, it's wrong.
The answer's not infinite.
But we have an excuse,
because we certainly
know there are wavelengths
that are short enough
that we don't understand the
physics at those length scales
anymore.
We're basing everything on
extrapolating from wavelengths
that we can actually
measure in the laboratory.
So one could
imagine that there's
some wavelength beyond which
everything we're saying here
is nonsense, and we don't have
to keep adding up 1/2 h bar
omega anymore,
because the arguments
that justify the 1/2 h
bar omega no longer apply.
So we can use that as a
cutoff for the calculation.
And a typical
cutoff-- by typical, I
mean typical in arguments
that physicists talk about, so
typical in physics speak.
So a cutoff that's
often invoked here
is the Planck scale,
which is the square root
of h bar times G
divided by c cubed.
And that has units
of length, and it's
equal to about 1.6 times 10
to the minus 33 centimeters.
And what makes the
scale significant
is it's the scale at which we
expect the effects of quantum
gravity to start
to be important.
And we know that
this quantum field
theory that we're
talking about does not
include the effects of gravity.
And we don't really
even know how
to modify it so that it would
include the effects of gravity.
So the quantum
effects of gravity
are still something
of a mystery.
So it makes sense to
cut the theory off,
if not earlier, at least
at the Planck scale.
Yes?
AUDIENCE: So I would imagine
what we're doing, in order
to say that we have a standing
wave, we have to have a box.
And then in order to realize
the fact that the universe may
be large, you just take the
limit as the box gets large.
But is it really OK to do that?
I mean, to treat
an infinite system
as the limit of a finite system?
PROFESSOR: OK, the
question is-- what
we're going to be
doing here is I
talked about putting the
standing waves in a box.
And then at the end, we're
going to take the limit
as the box gets
bigger and bigger.
And the question is, is
that really a valid way
of treating the infinite space?
And the answer is,
in this case, it is.
I'm not sure how solid
an argument I can make.
Certainly what one does
find is what you'd expect,
that as you make the
box bigger and bigger,
the energy that you get is
proportional to the size
of the box.
So you're calculating
an energy density.
And probably the most precise
thing I can say at the moment
is that if it were not true,
if the answer you got really
depended on the way in which
the space was infinite,
then you'd be learning something
about the infinite universe
by doing an experiment
in the lab, which
is a little far-fetched.
That is, if you do an
experiment in a lab,
it really doesn't
tell you anything
about whether the universe
is infinite or turns back
on itself and is closed.
And calculations
certainly do show
that you get the
same-- you could do,
for example, a closed
universe without a box.
And you get the
same energy density,
as long as the universe was
big, as we're getting this way.
So I think there's a pretty
solid calculational evidence
that what you get does
not depend on the box.
Yes?
AUDIENCE: Going
off that question,
do we use the maximum
size of our box
as the size of our
observable universe, then?
PROFESSOR: OK, the
question is, what do we
use as the maximum
size of the box?
Is it the size of the
observable universe?
The answer really is
that what you find
is that you get an energy
density that's independent
of the size of the box,
as long as the box is big.
And it's that energy density
that we're looking for.
We don't claim to know anything
about the total energy.
And we don't really
need to know anything
about the total energy.
Everything that we formulated
here in terms of energy
densities.
Now, the catch is, that
if one puts in this cutoff
and takes into account only
the energies of 1/2 h bar
omega going up to this cutoff
and stopping there-- or down
to the cutoff if one's thinking
of length as the measure--
you could then ask, do
we get an energy density
that's in any way close to
what the astronomers tell us
the vacuum energy actually is?
And the answer is
emphatically no.
We don't get anything close.
We're in fact off by about
120 orders of magnitude,
which even in cosmology is
a significant embarrassment,
which is why physicists consider
this question of the vacuum
energy density to be such
an incredible mystery.
We really have no
idea how to get
a number as small
as what we observe.
Let us go on to talk
about other contributions,
because they are certainly
important in the way we
think about things.
So far I have number one, right?
So next comes two.
And that is the quantum
fluctuations of Fermi fields,
where the best-known example
here is the electron.
Now, in quantum field
theory, I should point out
that all particles are described
by fields, not just the photon.
The electron is described
by a field also.
It's called the electron field.
And because the electron is
a fermion and not a boson,
the electron field has
somewhat different properties
than bosonic fields,
reflecting the fact
that the fermions themselves
obey the exclusion principle.
It turns out that
for fermions, there
are also quantum fluctuations.
They're also of order
1/2 h bar omega.
But actually, it's a
little bit different.
They're in some sense h
bar omega and not 1/2.
But what's peculiar is that
for electrons, the contribution
is negative.
And the origin of this
negativity I think
has a fairly simple
explanations,
although the explanation is
not ever given, actually.
The exploration that's used
in quantum field theory books
involves looking at
equation 47 and seeing
that there's an
anticommutator there.
And because the
fields anticommute,
there's a minus sign.
And that means the
energy is negative.
And that is basically the way
it's described in textbooks.
That certainly says
where the minus sign
appears in which equation.
But I don't think it's
really an explanation of what
the minus sign is talking about.
But I think there is an
explanation of what the minus
sign is talking about, which
goes back to the old picture
that Dirac himself
introduced when he first
invented the Dirac equation.
When Dirac first invented
the Dirac equation,
he was trying to
interpret it more or less
in the same language as
the Schrodinger equation.
We don't quite do that anymore.
But in doing that,
Dirac discovered
that his Dirac
equation, which was
the natural relativistic
generalization
of the Schrodinger equation
to a particle which
has spin 1/2, which I'm not sure
how Dirac knew it had spin 1/2,
but in any case,
it's the equation
for a particle of spin 1/2.
And what he found was that if
you just look at the energy
the spectrum that the
equation itself gives you,
it's symmetric about 0.
So if we plot energy going this
way, if there's a state here,
there's also a state
there at negative energy.
And if there's a
state there, there's
another state there,
exactly opposite it.
It's completely
symmetric up and down.
Now, the interpretation
that Dirac gave to that
was not that there are a lot of
ways of making negative energy.
He realized that the
vacuum is by definition
the state of lowest
possible energy.
And if you can lower the
energy by adding a particle
to these negative
energy states, that
would mean that there'd be a
way of lowering the energy,
and the state would
not be the vacuum.
So the vacuum, Dirac
proposed, is the state
in which all of these negative
energy levels are filled.
And the action of putting
all these x's on the picture
is often called
filling the Dirac sea.
S-E-A-- sea, where sea refers
to this ocean of negative energy
states, which is infinite.
It just keeps going down.
You can imagine filling all of
them to describe the vacuum.
Then if you ask what is
the physics after you've
done that-- what are
the possible excitations
of the vacuum, what
states does this theory
contain other than the vacuum?
And the answer is that
there could be occupations
of these positive energy states,
and those are called electrons.
It's also possible
to remove-- if you
put in the right
amount of energy--
one of the negative energy
states, which is filled,
but we could take away
the particle that's there.
And the absence of a
particle there-- a hole
in the negative energy
sea-- is a positron.
So electrons are there.
The e plus is a hole
in the Dirac sea.
Now, the difficulty
with this picture,
and the reason why it's
not often use these days,
is that it makes it
look like there's
an intrinsic difference between
electrons and positrons.
Nonetheless, Dirac
was perfectly aware
that when you went
through the math,
they were completely symmetric.
The fact that you
described it this way
is just really a feature
of your description,
but it doesn't make any
measurable difference.
So a positron really is just a
perfect image of an electron,
but with the opposite
charge, with otherwise
all the same
physical properties.
And there's ways
of describing it
where you don't make this
distinction between particles
and holes.
But the particle hole way
is I think the easiest
way of understanding where the
negative energy is coming from.
The negative energy came by
saying that the energy was 0
before we filled
any of these levels.
And as you fill the
negative energy sea,
you're lowering
energy all the time.
And it's that contribution
which makes up
the infinite negative
contribution coming
from the Fermi fields.
And the algebra is
certainly exactly right.
The energy that
people write down
for the negative energy
of the Fermi fields--
what they get by anticommuting
two operators in equation 37--
is exactly the
expression you get
for what it takes to
fill the Dirac sea.
Yes?
AUDIENCE: Are we
pretty confident
that the smallness
of the vacuum energy
can't come from cancellations
between the bosonic
and the Fermi fields?
PROFESSOR: OK, the question
is, are we confident that
the cancellation cannot come
from the cancellations between
the Fermi fields and
the bosonic fields.
No, we're all confident that
it cannot come from that.
it very likely does
come from that.
But we are confident that we
have no idea why that happens.
And therefore,
it's a big mystery.
Certainly our ignorance
allows for any answer.
Because we have a positive
infinite contribution,
we're just going to cut
off and make it large.
And we're going to have a
negative contribution, which
we're going to cut off and
make it large in magnitude
but negative.
And then we're
going to add them,
and we have no idea
what we're going to get.
But the fact that
we get something
that gets incredibly close to
0, and not something that's
at all the same magnitude as the
pieces you're adding together--
the positive piece or
the negative piece--
means there's something going
on that we don't understand.
There's a cancellation that's
happening that we cannot
explain.
Now, I should maybe
add that there's
one context where we would
expect a cancellation.
And that is, there
are theories that
are what are called
supersymmetric,
which have a perfect symmetry
between bosons and fermions,
which would relate the
positive energy from the photon
to the infinite negative energy
you would get from particles
called photinos, which would
be the supersymmetric partner
of the photons-- a
spin 1/2 particle
that's a mirror
image of a photon
but has a fermionic character.
So in an exactly
supersymmetric theory,
you would get an
exact cancellation
between the positive and
the negative contributions.
And the answer has to be 0 in an
exactly supersymmetric theory.
However, the world is clearly
not exactly supersymmetric.
This photino has
never been seen.
And there'd be a particle
called the selectron, which
would be the scalar partner
of the electron, which
also has not been seen.
And every known
particle would have
a partner, which
has not been seen.
There are no supersymmetric
pairs which are known.
So supersymmetry is
still a possibility
as a broken symmetry of nature.
And a lot of people think-- for
pretty good reasons, I thin--
that it's very likely
that the world does
have an underlying
supersymmetry.
But as long as the
supersymmetry is broken,
it no longer guarantees
this cancellation.
And you could estimate
what the mismatch is.
And it does make things
a little bit better here.
If we just take this
Planck scale cutoff,
we miss an energy density by a
factor of about 10 to the 120.
If we apply
supersymmetry and make
an estimate of what the
supersymmetry breaking scale is
and what effect that has on the
mismatch of these calculations,
then it gets reduced.
Instead of being 120 order
of magnitude problems,
it's got a 50 order of
magnitude problem, which
is a lot better,
but not good enough.
Now, I do want to mention
a third contribution here
for completeness.
The third one is
likely be finite,
so it's not as problematic
as the other two.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Same thing.
Planck scale.
AUDIENCE: Oh, OK.
PROFESSOR: The
third contribution
is that some fields
are believed to have
nonzero values in the vacuum.
And the famous example of
that is the Higgs field,
for which the particle
associated with the Higgs
was discovered a
year ago at CERN,
after over 50 years
of looking for it.
And the Higgs field is
maybe the only field
that's part of the
standard model that
has a nonzero expectation value,
a nonzero value in the vacuum.
But in more
sophisticated theories
like grand unified
theories, there
are many more fields that have
nonzero values in the vacuum.
So that's a likely extension
of our standard model
of particle physics.
So the bottom line is that it's
easy for particle physicists
to understand why the vacuum
energy should be nonzero,
but damned hard to have any
idea of why it has the value
that it has.
We'll talk maybe at
the end of the course
about the possibility that
the value of the vacuum energy
density is, quote,
"anthropically selected."
That is one possible
explanation,
which maybe shows how
desperate physicists
are to look for an
explanation here.
One possible explanation
begins with ideas
from string theory, where
string theory tells us
that there isn't just
one kind of vacuum,
but in fact, a huge number
of different types of vacuum,
perhaps 10 to the 500.
And that would
mean that if there
were sort of random values for
these infinite numbers that
get cut off, that get cut
off with different values--
and there are other ways
of looking at the vacuum
energy in string
theories-- you'd
expect coming out
of string theory
that the typical vacuum energy
would be about the same as what
you get when you
cut off the quantum
fluctuations of the
electromagnetic field
at the Planck scale.
That is, the typical
vacuum energy coming out
of a string theory would be
at the Planck scale, which
is this huge number
compared to what we observe.
But string theory
would be predict
that there would be a spread
of numbers going essentially
from plus the Planck scale
to minus the Planck scale,
with everything in between.
There'd be a tiny fraction
of those vacua that
would have a very small vacuum
energy like what we observe.
That's what you'd expect
from string theory--
a large number, but a tiny
fraction of vacua that
would be in that integral.
And then the only
problem would be
to explain why we might likely
be living in such an unusually
small fraction of the set
of all possible vacuums.
And the answer to
that that's discussed
is that it may be
anthropically selected.
That is, life may only
form when the vacuum
energy is incredibly small.
And that is not built
entirely from whole cloth.
There is some
physics behind that.
We know that this vacuum energy
affects the Friedman equation,
which means it affects the
expansion rate of the universe.
So if we had a Planck
scale vacuum energy,
that would cause the
universe to essentially blow
apart at the time scale of the
Planck scale, which is about 10
to the minus 40
something seconds,
due to the huge repulsion
that would be created
by that positive vacuum energy.
And conversely, if there
was a huge negative vacuum
energy on the order
of the Planck scale,
the universe would just
implode on a time scale
of order of the
Planck scale-- 10
to the minus 40
something seconds.
So assuming that life takes
billions of years to evolve
and assuming nothing
else about life,
one can conclude that
life can only exist
in the very narrow band
of possible vacuum energy
densities which are
incredibly small, like the one
that we're living in.
So it could be that we're
here only because there
isn't any life anyplace else.
So all living things see a very,
very small value of this vacuum
energy density, even though
if you plunk yourself down
at a random place
in this multiverse,
you'd be likely to see
a vacuum energy that's
near the Planck scale.
OK, I'm done talking
about this for now.
Any further questions about
it before we leave the topic?
I had suggested that we go
on to talk about problems
with the conventional
big bang model,
but, there is actually
something else I wanted to do.
I don't know how long
it will take exactly,
but I have a little historical
interlude to talk about here.
We've been talking about
the Friedman equations
and how they're modified by the
cosmological constant, which
of course is an item that was
very dear to Einstein's heart.
So I'd like to tell
you a little history
story about Albert Einstein
and Alexander Friedman,
which I think is
very interesting.
The punchline of the story is
that Einstein made pretty much
of a fool out of
himself on this.
And the reason why I like
the story is maybe twofold.
One is, I find it
very comforting
to know that even perhaps the
greatest physicist of all time
can make dumb mistakes
just like the rest of us
make dumb mistakes.
I think that's a very comforting
thing to keep in mind.
And the other moral
of the story is,
I think, the
importance of trying
to be open-minded about
issues in physics.
Einstein was very much convinced
that the universe was static,
and so convinced that,
in fact, he really made
stupid mistakes trying to
defend his static universe.
So this will be a story
of such a mistake.
So those are the two people.
Friedman was a Russian natural--
he was really a meteorologist.
They didn't really have that
many theoretical physicists
back in those days.
But as a meteorologist,
he was an expert
in solving partial
differential equations,
and got himself interested
in general relativity, which
was a new theory at this point.
And in 1922, he published
an actual physics paper,
I think the first physics
paper he ever published,
and one of two.
He wrote basically two papers
about the Friedman equations--
one for closed universes,
and one for open universes.
So the first of those
papers was published
in June 29, 1922 in the premier
physics journal of the day--
the Zeitschrift fur
Physik, a German journal.
And almost immediately--
or a few months later,
when Einstein noticed
this article--
Einstein submitted a comment
about the article claiming
that the article was
entirely wrong, just
mathematically wrong.
And the article was titled
"Remark on the Work of A.
Friedmann 'On the Curvature
of Space' " by A. Einstein,
Berlin, received
September 18, 1922.
Looking at these dates--
the original article
was received in June
1922, and Einstein
was responding by September
18, a few months later.
And this is a translation,
which comes from a book called
Cosmological Constants,
which is basically
a book of famous
articles in cosmology,
like Friedman's, and all
these original articles.
It's a great book if you
can still get a copy of it.
It's no doubt out of print.
It was written by Jeremy
Bernstein and Gary Feinberg.
And I'm taking the
translation from there,
because this was
written in German.
I don't know German.
"The works cited contains
a result concerning
a non-stationary world
which seems suspect to me.
Indeed, those
solutions do not appear
compatible with the
field equations."
And I guess A is the label
of the field equations
as they appeared in
Friedman's paper.
"From the field equation,
it follows necessarily
that the divergence of the
matter tensor Tik vanishes."
That is, energy
momentum is conserved
as a four-vector quantity.
"This along with ansatzes C and
D"-- equations from the paper--
leads, according to Einstein,
to an equation which we can all
recognize the meaning of-- the
partial of rho with respect
to x sub 4-- time-- is 0.
Einstein convinced himself
that the equations of general
relativity led to the conclusion
that rho cannot change with
time.
And he then goes on to
say "which together with 8
implies that the world radius
R"-- that's the scale factor.
That's what we call a of
t-- "is constant in time.
The significance of
the work, therefore,
is to demonstrate
this constancy."
All Friedman does
once you correct
his equations,
according to Einstein,
was prove that the only
cosmological solution is rho
equals a constant, which was
Einstein's static solution.
This was entirely wrong-- no
basis whatever in mathematics.
But it took a while
before Einstein
got himself straightened out.
And he did actually
publish this.
The sequence of events was, June
29, Friedman submits his paper.
September 18, Einstein submits
his rebuttal to the paper.
Friedman didn't learn about this
until the following December.
Friedman had a friend who played
a key role in this story-- Yrui
Krutkov, who was visiting
in Berlin during this time.
And Friedman actually
learned from Krutkov
that Einstein had
submitted a rebuttal.
So Friedman apparently was able
to track it down and read it.
And he wrote a detailed
letter to Einstein
explaining to Einstein
what he got wrong,
which is a gutsy thing
to do, but Friedman
was right in this case.
But Einstein was traveling and
actually never read the letter,
at least not until much later.
Then the following May,
Krutkov and Einstein
are both at a conference in
Leiden, a conference that they
were both attending, which was
a farewell lecture by Lorentz,
who was retiring at that time.
So they met and started
talking, and continued talking.
And we know most about it
from a series of letters
that Krutkov wrote to his
sister back in Saint Petersburg.
And according to those
letters-- and I'm now
quoting from a rather lovely
book called Alexander A.
Friedmann-- The Man who
Made the Universe Expand,
by Tropp, Frenkel, and Chernin.
Krutkov wrote to his sister
that on Monday, May 7,
1923, "I was reading,
together with Einstein,
Friedman's article in the
Zeitschrift fur Physik.
And then on May 18, he
wrote, "I defeated Einstein
in the argument about Friedmann.
Petrograd's honor is saved!"
Petrograd is what we now
call Saint Petersburg,
and where they were all from--
that is, Friedman and Krutkov.
And then shortly
after that, on May 31,
Einstein submitted a
retraction of his refutation
of Friedman's paper.
And the retraction
is-- again, I'm
quoting from Cosmological
Constant, which
translates all these
nice papers into English.
Einstein wrote, very briefly,
"I have in an earlier note
criticized the cited
work-- Friedmann 1922.
My objection rested
however, as Mr. Krutkov
off in person and a
letter from Mr. Friedmann
convinced me, on a
calculational error.
I am convinced that
Mr. Friedmann's results
are both correct and clarifying.
They show that in addition
to the static solution
to the field
equations, there are
time varying solutions with a
spatially symmetric structure."
Anyway, the expanding universe
that we now talk about.
Einstein did have to
admit, ultimately,
that algebra is algebra, and you
can't really futz with algebra.
And the Einstein equations
do not imply that rho cannot
change with time, and
that Friedman was right.
There's an interesting twist
on this retraction letter.
This is just a photo of
Einstein at this time period,
and Krutkov.
There's an interesting twist
on the retraction letter, which
is that the original
draft still exists.
I forget what museum it's in.
But it's quoted in
another marvelous book
about this history
called The Invented
Universe, by Pierre Kerzberg.
And I Xeroxed this
from the book.
And this is the original draft.
And notice there
are some cross-outs.
And the last cross-out, which
followed this explanation
that there is this expanding
solution-- in Einstein's
original draft, he wrote
but then crossed out
"a physical significance can
hardly be ascribed to them."
So his initial instinct,
even after having
been convinced that these
were a valid solution
to the equations, was to say
that they couldn't possibly
be physical, because
they're not physical.
The universe is static.
But somehow, before
he submitted it,
he did realize that
there wasn't actually
any solid logic
behind that reasoning.
So logic did prevail, and
he decided that he really
had no right to say
that the solution has
no physical significance, which
is a good thing, because now,
of course, it is the
solution that we consider
physically significant-- the
expanding solution of Friedman.
So [INAUDIBLE] is
a mystery, I think.
OK, we have just a couple
minutes left in the class.
So I think that is
nearly enough time for me
to at least introduce what
I want to talk about next.
What we'll be talking about next
time-- and I'll just introduce
it now-- are a set of
two problems associated
with the conventional
big bang theory.
And by the conventional
bang big bang theory,
I mean basically the theory
we've been talking about,
but in particular,
the big bang theory
without inflation, which we
will be talking about later.
But so far, we've been talking
about the big bang theory
without inflation.
And the two problems
that we'll talk about
are called the horizon
or horizon homogeneity
problem, and the
flatness problem.
Both of these are
problems connected
with the initial conditions
necessary to make the model
work.
So this horizon
homogeneity problem
is a problem about
trying to understand
the uniformity of the observed
universe, which we've just
put in as part of our
initial conditions.
The model that we've
constructed was just completely
homogeneous and isotropic
from start to present.
The evidence for the
uniformity of the universe
shows up most strongly,
as I think we said before,
in the cosmic background
radiation, which
can be measured to
fantastic precision.
And this radiation is known to
be uniform in all directions
to an accuracy of
one part in 100,000,
which is really a phenomenal
level of accuracy.
Now, what makes this
hard to understand
in the conventional
big bang theory is
that if instead
of just putting it
in as an assumption about
the initial conditions,
you try to get it out
of any kind of dynamics,
that turns out to be impossible.
And in particular, a calculation
that we'll do next time
is we'll imagine
tracing back photons
from the cosmic background
radiation arriving at the Earth
today from two opposite
directions in the sky.
Now, the phenomenology is
that those photons come
with exactly the same
temperature to an accuracy
of one part in
100,000, and that's
what we're trying to explain.
Now, we all do know
that systems do
come to a uniform temperature.
If you heated the air
in this room in a corner
and then let the
room stand, the heat
would scatter
throughout the room,
and the room would come
to a uniform temperature.
If you take a hot slice
of pizza out of the oven,
it gets cool, as
everybody knows.
So there is this
so-called zeroth law
of thermodynamics which
says that everything tends
to come to a
uniform temperature.
And it's a fair question to
ask, can we perhaps explain
the uniformity of the
universe by invoking
this zeroth law
of thermodynamcs?
Maybe the universe
just had time to come
to a uniform temperature.
But one can see immediately
when one looks at details
that that's not the case.
Within the context of
our conventional model
of cosmology, the
universe definitely
did not have time to come
to a uniform temperature.
And the easiest way
to drive that home
will be a calculation that we
will do first thing next time,
which is that we will
trace back photons coming
from opposite
directions in the sky
and ask, what would
it take for them
to have been set equal
to the same temperature
when they were first emitted?
And what we'll find is
that when we trace them
back to their emission sources,
that those emissions took place
at two points which were
separated from each other
by about 50 horizon distances.
So assuming that physical
influences are limited
by the speed of
light-- and according
to everything that we know about
the laws of physics, that's
true-- there is no way that the
emission of that photon coming
from that direction could
have had any causal connection
with the emission
of the photon coming
from the other direction.
So if the uniformity
had to be set up
by physical processes
that happened
after the initial singularity,
there's just no way
that that emission could
have known anything
about what was
going on over there,
and no way they could have
arranged to be emitting photons
at the same energy
at the same time.
Now, everything
does work if you're
willing to just assume that
everything started at uniform.
But if you're not
willing to assume that,
and want to try to derive the
uniformity of the universe
as a dynamical
consequence of processes
in the early
universe, there's just
no way to do it in the
conventional big bang
theory because of this
causality argument.
And later, we'll see that
inflation gets around that.
Yes?
AUDIENCE: How do we know
that the homogeneity wasn't
just created when the
universe was smaller,
in such a way that the
speed of light limit
wouldn't be violated, and
that it would just maintain
[INAUDIBLE]?
PROFESSOR: OK, the
question is, how do we
know that the uniformity
wasn't established when
the universe was very small,
and then the speed of light
might not have to be violated?
Well, the point is
that if the dynamics is
the conventional big bang
model, what we'll show
is that there's
not really enough.
No matter how early you
imagine it happening,
it still is 50 horizon
distances apart.
And there's no way that those
points could've communicated,
no matter how close
you come to t equals 0.
Now, you are of course
free to assuming anything
you want about the
singularity at t equals 0.
So if you want to just assume
that somehow the singularity
homogenized
everything, that's OK.
But there's no theory behind it.
That's just speculation.
But it is satisfactory
speculation.
There's nothing it contradicts.
But the beauty of inflation
is that it does, in fact,
provide a dynamical explanation
for how this uniformity could
have been created, which,
at least to many people,
is better than just
speculating that somehow it
happened in the singularity.
OK, I think that's it for now.
I will tell you about
the other problem
we'll talk about
next time next time.
And I will see you
all next Tuesday.
