>> Let's take a minute to review the results
from the previous two lectures on cosmology.
Homogeneity and isotropy tell us that the metric
can be written in the Robertson-Walker form,
that's ds squared equals minus dt squared
plus a of t squared times dr squared divided
by 1 minus kr squared plus
r squared d omega squared.
So that's the Robertson-Walker metric --
-- where this parameter k is equal to either
plus 1, that's for the closed cosmology
where space is at 3 sphere, minus 1,
that the opens cosmology where space is
at 3 hyperboloid, or zero where space is flat.
And this function a of t
is called the scale factor.
Now when we apply the Einstein equations to the
Robertson-Walker metric, we obtained an equation
for the scale factor, which is a
dot squared equals minus k plus 8 pi
over 3 times rho times a squared.
This is the Friedmann Equation --
-- where rho is the total energy density.
And in the standard cosmological model,
we assume the energy density is comprised
of three components, an ordinary
matter component,
radiation component, and
a dark energy component.
So this is m stands for matter.
So this is baryonic matter,
atoms, and molecules,
and so forth and includes dark matter.
R stands for radiation and D is the dark energy.
Now currently in our universe, so our universe
now, the matter component comprises about 32%
of the total energy density and
32% it's 28% of dark matter.
So only about 4% is baryons,
atoms, and so forth.
Very small percentage is
radiation, maybe 1/100 of a percent.
And the remainder, 68%, roughly is dark energy.
So these are the percentages now.
But if we go back in time, the
percentages were different.
And we can see that by recalling that the
matter energy density is proportional to 1
over a cubed, the radiation energy
is a constant over a to the 4th,
and the dark energy component
is just a constant.
Of course, we also know the universe is
expanding, so a is getting larger with time.
Or set another way, if we go
back in time, a becomes smaller.
So, as we go back in time, these components of
the total energy become more and more important.
In fact, if the universe began in a
Big Bang where a is equal to zero,
then in the early moments of the universe, the
most important component of the energy according
to this model would be the radiation component.
Now in the field of cosmology,
it's common practice
to introduce a set of rescaled variables.
So we define t tilde to be t
divided by the Hubble time t sub H.
And we define a rescaled scale factor a
tilde of t tilde equals a of t divided
by the present value of the scale
factor, which we call a sub zero.
As an aside, let me remind you that the
Hubble parameter is defined as a dot over a.
So this is a function of time t. And the
present value when we denote the present time
by t sub zero, the present value of the
Hubble parameter we call H sub zero.
That's just H evaluated at t sub zero.
That's the present value of a dot divided by
the present value of a. And the Hubble time,
t sub H is just 1 over the present
value of the Hubble parameter.
For our universe, observations
show that the present value
of the Hubble parameter is approximately
72 kilometers per second per mega parsec.
And the inverse of H naught, t sub
H, is approximately 14 billion years.
Now in terms of these rescaled variables, the
Friedmann equation can be rewritten as follows.
A tilde dot squared equals omega sub k plus
omega sub m over a tilde plus omega sub R
over a tilde squared plus
omega D times a tilde squared.
Now, these constants omega m, omega
R, and omega D are just proportional
to the present day values of the energy
density of matter, radiation, and dark energy.
So, omega m in particular is defined
as 8 pi times the present value
of the matter energy density.
So we'll call that rho sub m0 divided
by 3 times the present value of
the Hubble parameter squared.
Omega R is 8 pi rho r0 divided by 3H0 squared.
And omega D is 8 pi rho D0
divided by 3H0 squared.
Now this constant omega k is defined
as minus k divided by a0 squared,
so a0 is the present value of the
scale factor times H0 squared.
In these four constants, the four omegas
are not independent from one another.
They satisfy 1 equals the sum.
So 1 equals omega k plus omega m plus omega R
plus omega D. We can now solve these equations
for various choices of the omega parameters
to obtain various model universes.
And these model universes are usually referred
to as Friedman-Robertson-Walker
cosmological models.
Or FRW cosmologies.
Let me describe in a little more detail how
we can build one of these FRW cosmologies.
The first step, we need to specify numerical
values for omega m, omega R, omega D,
and the present value of the
Hubble parameter H sub zero.
And of course, from these values,
we can determine omega k. The next step
is to solve the Friedmann Equation.
So to solve this differential
equation, we take the square root
of both sides and use separation of variables.
And integrating, we have t tilde equals the
integral of da tilde divided by the square root
of omega k plus omega m over
a tilde plus omega R
over a tilde squared plus
omega D times a tilde squared.
Now, this integral determines t
tilde as a function of a tilde.
So the next step is to invert that relation
to obtain a tilde as a function of t tilde.
And of course, it might not be possible
to do this inversion analytically.
You might have to do it numerically.
Now when you carry out this integral
there's of course a constant of integration
and you're free to choose that however you like.
But remember, the present
day value of a tilde is 1.
So we can define a step four the present value
of t tilde, we'll call that t tilde sub zero
as the value of t tilde when a tilde equals 1.
So we just set a tilde equal 1 in this
expression to find t tilde sub zero.
So this is the value of t tilde.
Now, meaning at the present time, and likewise,
most of these cosmologies will have a Big Bang.
So we can define the t tilde value of the
Big Bang by just setting a tilde equals zero.
So this is the value of t tilde at the Big Bang.
The next step is to determine whether this
model cosmology is open, flat, or closed.
So open, flat, or closed.
And we do this by examining omega
k. So remember omega k is defined
to be minus k divided by a0
squared times H0 squared.
So if omega k is positive, that tells us
that the parameter little k is minus 1,
which means we have an open cosmology.
So space is a hyperboloid.
And furthermore, we can use the numerical value
of omega k to determine the present day value
of the scale factor a. So we find a0 equals 1
divided by the square root of omega k times H0.
And remember, H0 was one of
the quantities we specified
at the beginning, so we know
its numerical value.
Now, if omega k is less than zero, that
tells us this k parameter is plus 1
and our cosmology is closed.
And again, we can solve for the present day
value of the scale factor, a0 equals 1 divided
by square root of absolute
value of omega k times H0.
And, of course, if omega k is equal to zero,
that tells us the k parameter is
zero and the universe is flat.
Now in this case, we can't solve
this equation for the present value
of the scale factor for a sub zero.
And that's for good reason.
Because in the flat case, the present value of
a doesn't have any intrinsic geometric meaning.
So remember, the metric for space
time, the Robertson-Walker metric looks
like minus dt squared plus a of
t squared times, in this case,
the flat metric dr squared
plus r squared d omega squared.
And now just by a simple change of
coordinates, we can rescale the scale factor.
In other words, we can replace this scale factor
with some constant times the old scale factor,
and absorb that constant into
r into a new coordinate r,
which is the old coordinate divided by
c. And this leaves the metric unchanged.
So the scale factor itself is only
defined within an overall constant.
And in particular, the present day value
of the scale factor can be anything we
like by just adjusting this constant c. So,
in particular, in this case, the flat case,
it's convenient to simply choose the present
day value of the scale factor to be 1.
And now the final step in constructing
our cosmological model is to write
down the scale factor, which from the definition
of a tilde is equal to a0 times a tilde taken
as a function of t divided
by tH, the Hubble time.
So the Hubble time is just 1 over the
present value of the Hubble parameter.
And we can also find the present day value
of the time t in ordinary units like years.
It's related to the present value of t tilde
by just multiplying by the Hubble time,
and also the time of the Big Bang,
would be the t tilde times t sub H. Now,
let's spend a little bit of time discussing some
of the implications of these
cosmological models.
These FRW models.
These models assume a matter content
for the universe that's a mixture
of ordinary baryonic matter plus dark
matter, radiation, and dark energy.
The density of matter scales like 1 over a
cubed, the density of radiation like 1 over a
to the 4th, and the dark energy
density is a constant independent
of the scale factor a. Now we know the
scale factor is currently increasing,
the universe is expanding.
And almost all of these cosmological
models tell us that if we go back in time,
the scale factor shrinks to zero.
So the universe starts with the Big Bang.
In particular, the cosmological model
that best fits our present observations
is one that begins with the Big Bang.
So let's go ahead and restrict ourselves
to models that begin with the Big Bang.
So we'll just assume there's a Big Bang.
So sometime in the past, the
scale factor shrinks to zero.
Now at the present time, the universe
is about 1/3 matter, 2/3 dark energy,
and a very small percentage radiation.
But if we go back in time, as a goes to
zero, these components become more important.
In fact, [inaudible] limit as a goes to zero,
the most important contribution
will be the radiation
that will become dominant as we go back in time.
So as a goes to zero, so
that's an early universe just
after the Big Bang, the radiation dominated.
One thing we know about radiation
is that the density
of radiation is proportional to
the 4th power of temperature.
And this can be computed from
analyzing blackbody radiation.
So combining these two results
for the radiation energy density,
we see that in the early universe, temperatures
scaled like 1 over the scale factor a.
So in particular, as a goes to zero, going
back in time, the temperature goes to infinity.
Let me draw a spacetime diagram
that shows some of these features.
So, this is the Big Bang and this is now.
So, time is running up in this diagram,
and also the scale factor is
increasing upwards in this diagram.
The temperature of the universe
decreases as we go forward in time,
so it increases going backwards in time.
So, this is temperature.
And also the energy density and radiation
and energy density and baryonic matter prior
to about 400,000 years, the universe was so
hot and dense that atoms weren't able to form.
So this is the time in which matter begin
to condense into hydrogen and helium.
This is referred to as the time of decoupling.
It's also referred to as the recombination time.
The decoupling time is important
because this is the time
which the universe became largely
transparent to electromagnetic radiation.
Prior to this time, the universe was opaque.
So electromagnetic radiation was absorbed
by the ionized matter almost
as soon as it was released.
So when we look out into the universe today
and we see the cosmic microwave background
radiation, what we're seeing is the photons
that were released at the decoupling time.
So here's our world line.
And the photons we see today came from
these events at the time of decoupling.
So this radiation constitutes
the cosmic microwave background.
So CMB is shorthand for cosmic
microwave background.
And the current temperature of this
radiation is in the microwave range.
It's about 2.7 Kelvin.
But when it was released, the temperature
of the radiation was about 3,000 degrees.
So, the radiation has been redshifted.
And this corresponds to redshift distance
of about 3,000 divided by
2.7 is approximately 1,100.
So this is the basic picture of
the evolution of our universe.
This is sometimes referred
to as the standard model,
where the major events are the
Big Bang and recombination.
Let me make a few comments about the Big Bang.
Big Bang is the time in which the scale factor
goes to zero, the density goes to infinity,
and the temperature goes to infinity.
But the Big Bang didn't occur
at some location in space.
The Big Bang occurred everywhere in space.
Everywhere here is the Big Bang.
Sometimes you hear the Big Bang described or
pictured as an infinitesimally small dense ball
of matter that explodes outward and expands.
Of course, then you have to
ask, what is it expand again to?
Empty space?
Well, this is of course wrong.
The Big Bang occurred it all of space.
In fact, space is homogeneous, so
it must have occurred everywhere.
Now the question how big was
the universe at the Big Bang,
or just after the Big Bang,
is a different question.
So, here's our world line.
Here's another cluster of galaxies,
another cluster of galaxies.
And as we go back in time, the
scale factor is shrinking to zero.
So the distance between these clusters
of galaxies is shrinking to zero.
But space itself, if it's an open or a
flat cosmology, then space is infinite.
So even in the limit as we approach, the
Big Bang, space is still infinitely large.
Now if space is a closed cosmology,
if space is three-sphere, then yes,
the volume of that three-sphere does go
to zero as you approach a equals zero.
Now, there are several problems with a standard
model, things that we don't understand.
And I'd like to discuss two
of them in particular.
The first is referred to
as the flatness problem.
Recall that the energy density of ordinary
matter, radiation, and dark energy,
the present day values are
proportional to these omega parameters.
And omega m was about 0.32, omega D is
about 0.68, and omega R is very small
on the order of 10 to the minus 5 or so.
And these parameters must satisfy 1 equals omega
k plus omega m plus omega R plus omega D. Now,
the fact that these three add up so close
to 1 means omega k is very close to zero,
which tells us that the universe
is flat or very close to flat.
So, these values which come from the present
day observations of the energy density
of the universe imply that the universe is
nearly flat, meaning space has a geometry
that to a high degree of
approximation is a flat geometry.
Now, to understand why this is a problem,
let's take a look at the Friedmann Equation
that's a dot squared equals minus k plus 8 pi
over 3 times the total energy
density times a squared.
Now let's define the value of the
energy density that makes k equals zero.
So the value of rho that makes k equals
zero which corresponds to flat space.
We'll call that the critical value, rho
sub c. So solving this equation for rho
with k equals zero, we find rho sub c equals 3
over 8 pi times a dot squared over a squared.
And of course, that's just the Hubble parameter.
So this is 3 over 8 pi times H squared.
Now with this definition, we can rewrite the
Friedmann Equation as rho sub c equals minus 3k
over 8 pi times a squared plus rho.
So all I've done is I've multiplied
this equation by 3 over 8 pi
and also divided through by a squared.
Now this equation can be rearranged to give
rho c minus rho times a squared equals minus 3k
over 8 pi.
Now, let me divide this factor by
rho and multiply this factor by rho.
So what we have is rho c over rho minus 1
times rho times a squared equals this constant,
this is a constant in time.
Now here's the problem.
The product of this factor and this
factor remain constant in time.
But as we go back in time, this factor, so
back in time, this factor goes to infinity.
Remember, rho is going to be
dominant by radiation in the past,
and this is proportional to 1 over a to the 4th.
So a squared divided by a to the 4th is
going to blow up as a shrinks to zero.
So this factor goes to infinity,
is multiplied by this factor,
which must then go to zero
as we go back in time.
So as we go back in time, the energy density
in the universe had to become closer and closer
to the critical density, the value of
the density that makes the universe flat.
So the fact that the present day density
is close to the critical density means
that in the past, it must have been
infinitesimally close to the critical density.
So the initial conditions for the universe,
the conditions that came out of the Big Bang,
must have been very finely
tuned or finely balanced,
to give us the nearly flat
universe that we see today.
So another way of expressing the
flatness problem is to ask the question,
how could the initial conditions --
-- be so finely balanced --
-- to give us a present a
universe that's nearly flat?
Now the next problem I want to discuss with the
standard model is called the horizon problem.
The horizon problem is a generic problem
for all of these Big Bang cosmologies.
But I think it will be interesting
to describe it
in concrete terms using the present day
observational data for our universe.
So as an example, let's take our universe,
which you'll recall has omega m equal about 1/3,
omega radiation is close to zero,
omega dark energy is about 2/3,
and omega k is approximately zero.
By the way, this particular model is
often referred to as the lambda CDM model.
The lambda in the name comes
from the dark energy.
Remember, dark energy contributes
to the Einstein equations just
like a cosmological constant.
And the symbol for a cosmological
constant is lambda.
So this refers to the dark energy and
the CDM refers to cold dark matter.
So remember, most of the
matter content is dark matter
with a small fraction of that baryonic matter.
And remember, this component of
the universe is cold in the sense
that we modeled it as a pressureless dust.
So the name lambda CDM comes from the fact
that we're modeling the energy density
of the universe is primarily dark
energy, and cold dark matter.
And sometimes people will include a small
amount of radiation in these models.
But for simplicity, let's just take these
as the parameters of our cosmological model.
And to complete the specification of
the model, let's take the Hubble time,
which is 1 over the present
value of the Hubble parameter
to be 14 billion years at 14
times 10 to the 9th years.
Now for this model, we can compute
the scale factor analytically.
Just following the steps outlined earlier in
this lecture, we find a of t is equal to 1
over square root of 2 times hyperbolic
sinh of square root of 3/2 times t
over th, all of this to the power 3/2.
We also find for this solution that
the time with the Big Bang is zero,
that's the time that a is equal to zero.
The current time, the now time, is square root
of 2/3 times arcsinh of square
root of 2 times th.
And that's approximately 0.936 times th.
And the current value of the scale factor is 1.
Now recall the time of decoupling
is about 400,000 years.
This is the time atoms formed and the
photons were released that we currently see
as the cosmic microwave background radiation.
So this is about 2.9 times 10
to the minus 5 Hubble times.
And the scale factor at the time of decoupling
was about 8.5 times 10 to the minus 4.
So it was quite small compared
to the current value of 1.
Let me draw another spacetime diagram.
So here's the time of the Big Bang,
here's decoupling, and here's now.
So this is Big Bang time,
which is of course zero.
This is the time of decoupling and this is now.
This is our world line.
And here are the photons that we see
released at the time of decoupling.
Now in this two-dimensional diagram,
these horizontal lines are radial lines.
So let's place ourselves at the origin so
that coordinate value little r equals zero.
And this event will take to be coordinate value
capital R, little r equals capital R. What I'd
like to do now is compute the proper
distance between this event and this event.
So as we look out into the sky, we
see the cosmic microwave background
in one direction coming from this event.
And if we look in the opposite direction,
we see the photons coming from this event.
So let's call that distance
D. So this is the distance --
the proper distance between cosmic microwave
background radiation photons that we see
when we look in opposite directions in the sky.
Now I'm going to put a little subscript p on
here, because it's the distance as determined
by the past light cone or past light
cone at the time of decoupling.
Now the distance between us at
the time of decoupling, this event
and this event is just half
of the distance that we want.
So this is D sub p divided by 2.
So how do we compute this distance?
Well, let's recall that the Robertson-Walker
metric along t equal constant slice
of our constant radial direction is
just a of t squared times dr squared.
More generally, it's dr squared
divided by 1 minus kr squared.
But remember, we're assuming the
universe is flat, so k is equal to zero.
So that means we can ignore this factor.
So now taking the square root
of both sides and integrating,
what we find is the proper distance
Dp over 2 is equal to a at the time
of decoupling times the integral of dr which
is just capital R. And now we find capital R
by integrating the Robertson-Walker
metric along this null ray.
So along a null direction, ds squared is zero.
So that's zero equals minus dt squared
plus a of t squared times dr squared.
In the end, we drop the factor 1 minus
kr squared since k is equal to zero,
and also d theta and d phi are zero since
this null ray is along a radial direction.
Now at separate variables,
what we find is dt divided by a
of t equals dr. And of course, we can integrate.
So we integrate from zero to
capital R. And the t integral goes
from the decoupling time to the present time t0.
The right hand-side here is, of
course, capital R. So what we get
by combining these two results is D
sub p equals 2 times a at the time
of decoupling times the integral from
t decoupling to t0 of dt divided by a
of t. Now let's follow this result away for
a moment and consider a similar calculation
where we start with the Big Bang.
And this is the decoupling time.
And let's consider the future null
cone of an event at the Big Bang.
So we can think of this as the path of light
rays that are released at the Big Bang,
and reach the decoupling surface.
Of course, light can't really
travel freely during this period
in the history of the universe.
But the future null cone still represents the
region of causal influence for this event.
So let's compute this distance
from this event to this event.
Let's call it D sub f, the subscript
f refers to the future null cone here.
This is the distance of causal
influence of an event at the Big Bang.
So let's let this have value r equals zero,
this is coordinate value capital R. Now
by integrating ds equals a of t dr, we find
that df over 2, that's the proper distance
from this event to this event, equals a
at the decoupling time times capital R.
And now integrating along this null direction,
the null directions has phi minus dt squared
plus a of t squared dr squared equals zero.
We find the integral of dt over a of t
equals the integral dr, and that's from zero
to capital R, which of course gives
capital R. And the t integration ranges
from the Big Bang time, which
is zero to the decoupling time.
So that's from zero to t decoupling.
And now combining these results, you find
the Df equals 2 times the scale factor
at the decoupling time times the integral
from zero to t decoupling of dt over a
of t. Now let's compute this
result along with our result
for the distance D sub p using this
expression for the scale factor, right?
Well, when we do that, we find that the distance
D sub p, this is for the lambda CDM model,
D sub p is about 70 million light years.
D sub f is about two million light years.
So the point being D sub p is much, much greater
than D sub f. Let's take a look
at what this is telling us.
So here's the Big Bang.
Here's the time of decoupling.
And here's now, so this is t0,
t decoupling, and t Big Bang.
Here's our world line.
Here's the light that we see from the decoupling
time that's released at the time of decoupling.
In this distance, this event to this
event, is about 70 million light years.
On the other hand, if we look at the future
light cones of events coming from the Big Bang,
they span a distance of only
about two light years.
So, for example, this would be one light cone,
here's another light cone, another light cone,
and so forth, and similarly over here.
So you can see that since these light cones
only span about two million light years,
at the time of decoupling, the set
of events in the very early universe
that could have causally influenced
these events are widely separated.
So for example, for this event,
the set of events that could causally
influence this event are here,
the set of events that could causally
influence this event are these.
The point is that these two events,
let's call them E1 and E2 were never
in causal contact with one another.
So this is the horizon problem.
The problem is to explain why the cosmic
microwave background looks the same
in all directions.
So it's not only the same temperature
but also the small fluctuations
in temperature have the same spectrum.
So how can that be when these regions of
the universe were never in causal contact,
so there was never an opportunity for
those parts of the universe to come
to any kind of thermal equilibrium?
And as I said before, this horizon
problem is a generic problem
of all of these cosmological models.
Not just of the lambda CDM model.
So how do we solve these problems
with the standard cosmology,
in particular the flatness
problem and the horizon problem?
Currently, the most popular explanation
is that the universe underwent a period
of rapid inflation just after the Big Bang.
Let's see how this works.
Here's a graph of the scale factor as a function
of time for our simple lambda CDM model.
So this is scale factor on the vertical axis.
And the curve for the lambda CDM
model looks something like this.
This is lambda CDM.
We're currently about right here on the curve.
So this is t0 and the scale factor
right now is 1 in this model.
And we're just beginning to enter this period
of exponential growth driven by the dark energy.
Now, the origin of the graph where a equals
zero is, of course, the time of the Big Bang.
And by the way, you can easily check
that for this model, for small t,
the scale factor varies like t to the 2/3 power.
Now on a linear timescale, the time
of decoupling would be really
close to the Big Bang.
But I'm going to stretch this out in this
picture and put the decoupling time here,
just so that I can show what's going on
in the very early universe more clearly.
So this is not a linear timescale.
So the idea behind inflation is that
the universe underwent a brief period
of very rapid expansion shortly
after the Big Bang.
So, let's say the brief period is here.
And this curve is modified
to look something like this.
So, now it begins this period of inflation.
And it grows very rapidly and then continues
with the current model we see today.
So, this would be the lambda
CDM model with inflation.
So, this is the brief period of inflation --
-- which was thought to take place around 10
to the minus 35 seconds after the Big Bang.
And during this time, the scale factor
increases by 30 orders of magnitude or more.
So what could cause the universe
to expand like this?
Well, one possibility is that the
energy density goes through a period
where it's dominated by a constant component.
So inflation could be caused by
period with constant energy density,
constant meaning independent
of the scale factor.
An example of the constant
energy density is dark energy.
So this is like dark energy, otherwise
known as the cosmological constant.
But in this case, in the inflationary
period, it might be distinct.
So it might be distinct from the
dark energy that we have today.
So if we have a period in the early
universe where rho is equal to a constant,
then the Friedman Equation tells us, remember,
a dot squared equals minus k plus 8 pi
over 3 times rho times a squared.
And we'll take k to be zero, then a dot
squared is proportional to a squared
since rho is a constant, which
tells us that, of course,
a dot is proportional to
a, so a is an exponential.
So during this period of inflation
with a constant energy density,
the scale factor grows exponentially.
Now how does this solve the flatness problem?
Recall that we can write the Friedman
Equation as the critical density
over the density minus 1 times the
density times a squared equals a constant,
where this critical density is
the value that the density needs
to be for the universe to be flat.
Now without inflation and going back in
time, we have rho a squared goes to infinity.
That's because rho is dominated
by the radiation component,
which goes like 1 over a to the 4th power.
So, since this factor goes to
infinity, this factor must go to zero.
So in other words, rho becomes
closer and closer to rho c,
the critical value as you go back in time.
But we know that rho is approximately
rho c now, so it must have been much,
much closer to rho c the
critical value in the past.
And the problem is to explain this
fine tuning of the energy density.
But with inflation, we have a period where
going back in time rho a square goes to zero.
And this allows rho to diverge from the
critical density going back in time.
Said the other way around, the initial value of
rho near the Big Bang does not need to be close
to the critical density rho c. So it does not
need to be close to rho c. As the universe goes
through the inflation period, rho a squared
increases by many orders of magnitude,
which means this factor decreases
by many orders of magnitude.
So inflation drives rho to
the critical value rho c.
And universe we see today is very nearly flat.
How does inflation solve the horizon problem?
Well, if you'll recall, the size of
our past event horizon at the time
of decoupling was given by this proper distance
called D sub p, that's 2 times a at the time
of decoupling times the integral from the
decoupling time to the present time of dt over a
of t. And the size of the future light cone
for events near the Big Bang is 2 times a
at the time of decoupling times the
integral from the time of the Big Bang
to the decoupling time dt over a of t.
And without inflation, we have the problem
that Dp is much, much greater than Df.
What about with inflation?
Well, you'll notice these quantities Dp and
Df depend on the scale factor at the time
of decoupling and integrals over the inverse
of the scale factor over different ranges
from decoupling time to the present time for
Dp and from Big Bang to decoupling time for Df.
But the curve with and without inflation,
you'll see it's only modified four times
between the Big Bang and the end of inflation.
So in particular, the scale factor at the
time of decoupling isn't changed by inflation.
And, in fact, this entire integral
over the inverse of the scale factor
from the decoupling time to
the present time is the same.
So said another way are the size of our past
event horizon isn't changed by inflation.
So that means Dp is the same within
without inflation, but Df is larger.
And this is because during the time
between the Big Bang and inflation,
the scale factor is 30 orders
of magnitude or more smaller.
So that makes this integral much larger.
Even if the time between the
Big Bang and inflation is small,
if the scale factor is small enough, we
can make the integral as large as we like.
So Df can be as large as we like by
adjusting these parameters in a suitable way.
And we can avoid this horizon problem,
but Dp being much larger than Df.
Let me sketch a spacetime diagram showing these
future and past light cones with inflation.
So here's the time of the Big Bang.
Here's the decoupling time.
Here's our current time.
So this is t naught, t decoupling, t Big Bang.
Here's our world line.
Here's the cosmic microwave background radiation
that we see released from
the time of decoupling.
So these are the events I call E1 and E2.
And the distance between E1 and E2 is
what I call D sub p. Now, the light --
the future light cones from events at the Big
Bang, with inflation, they're very much larger
than they were without inflation.
So here would be a future light cone.
Here's another future light cone.
And these have width D sub f, and so forth.
You can draw a few more in here.
So you can see in this case, there's plenty
of overlap here at the time of the Big Bang,
sets of events that can influence
both E1 and E2.
Well, these are events in the
causal past of both E1 and E2.
So there's plenty of time for information to
be exchanged back and forth between the regions
of the universe that evolve to E1 and E2.
So with inflation, the cosmic microwave
background radiation that we see today comes
from regions of the universe that were able
to come into thermal equilibrium in the past.
There's one question we still need to address.
How do we arrange for a period
of inflation for a period
which rho is approximately
constant in the early universe?
Most schemes for inflation rely on the use
of a scalar field called the inflaton field.
So with a scalar field phi, the action for
the inflaton field is integral d4x square root
of minus g, that's the volume
element in curved spacetime,
minus 1/2 g mu nu del mu phi
del nu phi minus v sub t of phi.
This function v of t is the potential.
And it depends on temperature.
T. So, now the equations of motion that come
from this action are del mu del mu phi
equals partial v sub t with respect to phi.
And the stress energy momentum tensor
for this inflaton field is del mu phi del nu
phi minus 1/2g mu nu del sigma phi del sigma phi
minus g mu nu times v sub t of phi.
In one of the early proposals for inflation,
sometimes called the new inflation,
the potential looks like the following.
So, here's the phi axis, here's
a potential v sub t of phi.
This is the phi axis.
For high temperatures, the potential
curve looks something like a parabola.
So, this is high temperature.
And for low temperature, the
curve looks something like this.
So this is low temperature.
And for our immediate temperatures,
it looks something in between.
So before inflation began, the temperature was
high and the potential curve looked like this.
And the solution for phi, to the equations
of motion, is just phi sitting at zero,
sitting at the bottom at this potential.
So phi equals zero.
Phi equals a constant means the left-hand side
here is zero, and phi sitting at the bottom
of this potential well means
dv d phi equals zero.
So this is a solution and you can see
that the stress energy momentum tensor
from for phi equals zero, this
term is zero, this term is zero,
we simply have t mu nu equals minus g
mu nu times v sub t at phi equal zero.
So this is just like the stress energy momentum
tensor for dark energy or cosmological constant.
With energy density equal to minus p sub
t at phi equals zero, which is a constant.
So this is the solution before inflation,
or perhaps we should say at
the beginning of inflation.
But after inflation, the temperature drops.
So the potential energy curve looks like this.
And the solution then shifts to here.
Let's call this value of phi
-- or let's call this phi hat.
So, this is the solution after inflation.
We have low temperature, phi is equal
to the constant value of phi hat,
and the stress energy momentum tensor is minus
g mu nu times v sub t evaluated at phi hat.
So in this case the energy density is minus
v sub t at phi hat, which could be much --
very much smaller than the
energy density before inflation.
It could be that this contribution to the
energy density today is insignificant,
or it could be that this is the
dark energy that we see today.
This particular scheme for inflation that I just
described sometimes called the new inflation has
some problems of its own.
So, although inflation solves some of
the problems with standard cosmology,
we don't yet have a real understanding
of how inflation would come about.
There are lots of other schemes for
inflation that have been proposed,
but none of them are truly satisfactory.
Nevertheless, the idea of
inflation is pretty compelling.
And it appears to solve the flatness problem,
the horizon problem and as
well as other problems.
So as a result, inflation has been
around for a long time for about 30 years
and it probably isn't going away anytime soon.
