21. Problems of the Conventional (Non-inflationary) Hot Big Bang Model

21. Problems of the Conventional (Non-inflationary) Hot Big Bang Model


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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, in that
case, let’s take off. There’s a fair
amount I’d like to do before the end of the term. First, let me quickly review
what we talked about last time. We talked about the
actual supernovae data, which gives us brightness as a
function of redshift for very distant objects and
produced the first discovery that the model is not fit very
well by the standard called dark matter model, which
would be this lower line. But it fits much better
by this lambda CDM, a model which involves a
significant fraction– 0.76 was used here– and significant
fraction of vacuum energy along with cold, dark
matter and baryons. And that was a model
that fit the data well. And this, along
with similar data from another group
of astronomers, was a big bombshell
of 1998, showing that the universe appears
to not be slowing down under the influence of gravity
but rather to be accelerating due to some kind of
repulsive, presumably, gravitational force. We talked about the
overall evidence for this idea of an
accelerating universe. It certainly began with
the supernovae data that we just talked about. The basic fact
that characterizes that data is that the most
distant supernovae are dimmer than you’d
expect by 20% to 30%. And people did cook up
other possible explanations for what might have made
supernovae at certain distances look dimmer. None of those really
held up very well. But, in addition, several other
important pieces of evidence came in to support this idea
of an accelerating universe. Most important, more
precise measurements of the cosmic background
radiation anisotropies came in. And this pattern
of antisotropies can be fit to a
theoretical model, which includes all of the parameters
of cosmology, basically, and turns out to give a very
precise setting for essentially all the parameters of cosmology. They now really all have their
most precise values coming out of these CMB measurements. And the CMB measurements
gave a value of omega [? vac ?], which
is very close to what we’ll get from the supernovae,
which makes it all look very convincing. And furthermore, the
cosmic background radiation shows that omega total is
equal to 1 to about 1/2% accuracy, which is very hard
to account for if one doesn’t assume that there’s a
very significant amount of dark energy. Because there just
does not appear to be nearly enough of anything
else to make omega equal to 1. And finally, we pointed out
that this vacuum energy also improves the age calculations. Without vacuum energy,
we tend to define that the age of the universe
as calculated from the Big Bang theory always ended up
being a little younger than the ages of the oldest
stars, which didn’t make sense. But with the vacuum
energy, that changes the cosmological calculation of
the age producing older ages. So with vacuum energy of the
sort that we think exists, we get ages like 13.7
or 13.8 billion years. And that’s completely
consistent with what we think about the ages
of the oldest stars. So everything fits together. So by now, I would say that
with these three arguments together, essentially,
everybody is convinced that this acceleration is real. I do know a few people
who aren’t convinced, but they’re oddballs. Most of us are convinced. And the simplest explanation
for this dark energy is simply vacuum energy. And every measurement
that’s been made so far is consistent with the
idea of vacuum energy. There is still an
alternative possibility which is called
quintessence, which would be a very slowly
evolving scalar field. And it would show up, because
you would see some evolution. And so far nobody has
seen any evolution of the amount of dark
energy in the universe. So that’s basically
where things stand as far as the observations of
acceleration of dark energy. Any questions about that? OK, next, we went on to talk
about the physics of vacuum energy or a
cosmological constant. A cosmological constant
and vacuum energy are really synonymous. And they’re related
to each other by the energy
entering the vacuum being equal to this expression,
where lambda is what Einstein originally called the
cosmological constant and what we still call
the cosmological constant. We discussed the fact
that there are basically three contributions
in a quantum field theory to the
energy of a vacuum. We do not expect it to
be zero, because there are these complicated
contributions. There are, first of all,
the quantum fluctuations of the photon and
other Bosonic fields, Bosonic fields
meaning particles that do not obey the Pauli
exclusion principle. And that gives us a
positive contribution to the energy, which
is, in fact, divergent. It diverges because every
standing wave contributes. And there’s no lower bound to
the wavelength of a standing wave. So by considering shorter
and shorter wavelengths, one gets larger and
larger contributions to this vacuum energy. And in the quantum field
theory, it’s just unbounded. Similarly, there are
quantum fluctuations to other fields like
the electron field which is a Fermionic
field, a field that describes a particle that obeys
the Pauli exclusion principle. And those fields behave
somewhat differently. Like the photon, the
electron is viewed as the quantum
excitation of this field. And that turns out to be by
far, basically, the only way we know to describe
relativistic particles in a totally consistent way. In this case, again,
the contribution to the vacuum
energy is divergent. But in this case, it’s
negative and divergent, allowing possibilities of
some kind of cancellation, but no reason that
we know of why they should cancel each other. They seem to just be
totally different objects. And, finally, there
are some fields which have nonzero
values in the vacuum. And, in particular, the Higgs
field of the standard model is believed to have a nonzero
value even in the vacuum. So this is the basic story. We commented that if we
cut off these infinities by saying that we don’t
understand things at very, very short wavelengths, at
least one plausible cut off would be the Planck scale,
which is the scale associated with where we think quantum
gravity becomes important. And if we cut off at
this Planck scale, the energies become
finite but still too large compared to what we observe
by more than 120 orders of magnitude. And on the homework set
that’s due next Monday, you’ll be calculating
this number for yourself. It’s a little bit more than
120 orders of magnitude. So it’s a colossal failure
indicating that we really don’t understand what controls
the value of this vacuum energy. And I think I
mentioned last time, and I’ll mention it a
little more explicitly by writing it on
the transparency this time, that the situation
is so desperate in that we’ve had so much trouble trying to
find any way of explaining why the vacuum energy should be so
small that it has become quite popular to accept the
possibility, at least, that the vacuum energy is
determined by what is called the anthropic
selection principal or anthropic selection effect. And Steve Weinberg
was actually one of the first people who
advocated this point of view. I’m sort of a recent convert
to taking this point of view seriously. But the idea is
that there might be more than one possible
type of vacuum. And, in fact, string
theory comes in here in an important way. String theory seems
to really predict that there’s a colossal number
of different types of vacuum, perhaps 10 to the 500 different
types of vacuum or more. And each one would have
its own vacuum energy. So with that many, some of them
would have a, by coincidence, near perfect
cancellation between the positive and negative
contributions producing a net vacuum energy that
could be very, very small. But it would be a tiny fraction
of all of the possible vauua, a fraction like 10
to the minus 120, since we have 120
orders of magnitude mismatch of these ranges. So you would still have to
press yourself to figure out what would be the explanation
why we should be living in such an atypical vacuum. And the proposed answer is that
it’s anthropically selected, where anthropic means
having to do with life. Whereas, the claim is
made that life only evolves in vacuua which
have incredibly small vacuum energies. Because if the vacuum energy is
much larger, if it’s positive, it blows the universe apart
before structures can form. And it it’s negative,
it implodes the universe before there’s time
for structures to form. So a long-lived universe
requires a very small vacuum energy density. And the claim is that
those are the only kinds of universes that support life. So we’re here because it’s
the only kind of universe in which life can
exist is the claim. Yes? AUDIENCE: So, different
types of energies, obviously, affect the
acceleration rate and stuff of the universe. But do they also affect, in any
way, the fundamental forces, or would those be the
same in all of the cases? PROFESSOR: OK, the question
is, would the different kinds of vacuum affect the kinds
of fundamental forces that exist besides the force
of the cosmological constant on the acceleration of
the universe itself? The answer is, yeah, it would
affect really everything. These different vacuua would be
very different from each other. They would each have
their own version of what we call the standard
model of particle physics. And that’s because the standard
model of particle physics would be viewed as
what happens when you have small perturbations
about our particular type of vacuum. And with different
types of vacuum you get different time
types of small perturbations about those vaccua. So the physics really could
be completely different in all the different vacuua that
string theory suggests exist. So the story here,
basically, is a big mystery. Not everybody accepts
these anthropic ideas. They are talked about. At almost any
cosmology conference, there will be some session where
people talk about these things. They are widely
discussed but by no means completely agreed upon. And it’s very much
an open question, what it is that explains
the very small vacuum energy density that we observe. OK, moving on, in
the last lecture I also gave a quick historical
overview of the interactions between Einstein
and Friedmann, which I found rather interesting. And just a quick summary here,
in 1922 June 29, to be precise, Alexander Friedmann’s first
paper about the Friedmann equations and the dynamical
model of the universe were received at [INAUDIBLE]. Einstein learned about it
and immediately decided that it had to be wrong and
fired off a refutation claiming that Friedmann had gotten
his equations wrong. And if he had gotten
them right, he would have discovered that
rho dot, the rate of change of the energy density,
had to be zero and that there was nothing but
the static solution allowed. Einstein then met a friend
of Friedmann’s Yuri Krutkov at a retirement lecture
by Lawrence in Leiden the following spring. And Krutkov convinced
Einstein that he was wrong about
this calculation. Einstein had also
received a letter from Friedmann,
which he probably didn’t read until this
time, but the letter was apparently also convincing. So Einstein did finally retract. And at the end of May
1923, his refraction was received at
Zeitschrift fur Physik. And another interesting
fact about that is that the original handwritten
draft of that retraction still exists. And it had the curious
comment, which was crossed out, where Einstein suggested that
the Friedmann solutions could be modified by the phrase,
“a physical significance can hardly be ascribed to them.” But at the last minute,
apparently, Einstein decided he didn’t really
have a very good foundation for that statement
and crossed it out. So I like the
story, first of all, because it illustrates
that we’re not the only people
who make mistakes. Even great physicists
like Einstein make really silly mistakes. It really was just a
dumb calculational error. And it also, I think,
points out how important it is not to allow
yourself to be caught in the grip of some firm idea
that you cease to question, which apparently
is what happened to Einstein with his belief
that the universe was static. He was so sure that
the universe was static that he very quickly
looked at Friedmann’s paper and reached the incorrect
conclusion that Friedmann had gotten his calculus wrong. In fact, it was Einstein
who got it wrong. So that summarizes
the last lecture. Any further questions? OK, in that case, I think
I am done with that. Yeah, that comes later. OK, what I want to
do next is to talk about two problems associated
with the conventional cosmology that we’ve been learning
about and, in particular, I mean cosmology
without inflation, which we have not
learned about yet. So I am talking
about the cosmology that we’ve learned about so far. So there are a total
of three that I want to discuss
problems associated with conventional cosmology
which serve as motivations for the inflationary
modification that, I think, you’ll be learning about
next time from Scott Hughes. But today I want to
talk about the problems. So the first of 3 is sometimes
called the homogeneity, or the horizon problem. And this is the problem
that the universe is observed to be
incredibly uniform. And this uniformity
shows up most clearly in the cosmic microwave
background radiation, where astronomers have now made
very sensitive measurements of the temperature as a
function of angle in the sky. And it’s found that that
radiation in uniform to one part in 100,000,
part in 10 to the 5. Now, the CMB is
essentially a snapshot of what the universe looked
like at about 370,000 years after the Big Bang at the
time that we call t sub d, the time of decoupling. Yes? AUDIENCE: This measurement,
the 10 to the 5, it’s not a limit
that we’ve reached measurement technique-wise? That’s what it actually
is, [INAUDIBLE]? PROFESSOR: Yes, I was
going to mention that. We actually do see fluctuations
at the level of one part in 10 to the five. So it’s not just a limit. It is an actual observation. And what we interpret is
that the photons that we’re seeing in the CMB have been
flying along on straight lines since the time of decoupling. And therefore, what
they show us really is an image of what
the universe look like at the time of decoupling. And that image is an image
of the universe which is almost a perfectly smooth
mass density and a perfectly smooth temperature–
it really is just radiation– but tiny
ripples superimposed on top of that uniformity
where the ripples have an amplitude of order
of 10 to the minus 5. And those ripples are
important, because we think that those are the
origin of all structure in the universe. The universe is
gravitationally unstable where there’s a positive
ripple making the mass density slightly higher than average. That creates a slightly
stronger than average gravitational field
pulling in extra matter, creating a still stronger
gravitational field. And the process cascades
until you ultimately have galaxies and
clusters of galaxies and all the complexity
in the universe. But it starts from these
seeds, these minor fluctuations at the level of one
part in 10 to the five. But for now we want
to discuss is simply the question of how
did we get so uniform. We’ll talk about how the
non-uniformities arise later in the context of inflation. The basic picture is
that we are someplace. I’ll put us here in
a little diagram. We are receiving photons,
say, from opposite directions in the sky. Those little arrows represent
the incoming patterns of two different
CMB photons coming from opposite directions. And what I’m interested
in doing to understand the situation with regard
to this uniformity is I’m interested in tracing
these photons back to where they originated
at time t sub d. And I want to do
that on both sides. But, of course, it’s symmetric. So I only need to
do one calculation. And what I want
to know is how far apart were these two points. Because I want to explore
the question of whether or not this uniformity
in temperature could just be mundane. If you let any object
sit for a long time, it will approach a
uniform temperature. That’s why pizza gets cold when
you take it out of the oven. So could that be responsible
for this uniformity? And what we’ll see
is that it cannot. Because these two points
are just too far apart for them to come in
to thermal equilibrium by ordinary thermal
equilibrium processes in the context of the
conventional big bang theory. So we want to calculate how
far apart these points were at the time of emission. So what do we know? We know that the temperature
at the time of decoupling was about 3,000
Kelvin, which is really where we started with our
discussion of decoupling. We did not do the
statistical mechanics associated with this statement. But for a given density,
you can calculate at what temperature
hydrogen ionizes. And for the density that we
expect for the early universe, that’s the temperature at
which the ionization occurs. So that’s where
decoupling occurs. That’s where it becomes neutral
as the universe expands. We also know that
during this period, aT, the scale factor
times the temperature is about equal to
a constant, which follows as a consequence
of conservation of entropy, the idea that the universe
is near thermal equilibrium. So the entropy does not change. Then we can calculate
the z for decoupling, because it would just be the
ratio of the temperatures. It’s defined by the ratio
of the scale factors. This defines what you mean
by 1 plus z decoupling. But if aT is about
equal to a constant, we can relate this to the
temperatures inversely. So the temperature of decoupling
goes in the numerator. And the temperature today
goes into the denominator. And, numerically,
that’s about 1,100. So the z of the cosmic
background radiation is about 1,100, vastly larger
than the red shifts associated with observations of
galaxies or supernovae. From the z, we can calculate
the physical distance today of these two locations, because
this calculation we already did. So I’m going to call l sub p
the physical distance between us and the source of
this radiation. And its value today– I’m
starting with this formula simply because we
already derived it on a homework set– it’s 2c h
naught inverse times 1 minus 1 over the square
root of 1 plus z. And this calculation was done
for a flat matter dominated universe, flat matter dominated. Of course, that’s
only an approximation, because we know our
real universe was matter dominated at the
start of this period. But it did not remain
matter dominated through to the present at
about 5,000 or 6,000 years ago. We switched to a situation where
the dark energy is actually larger than the
non-relativistic matter. So we’re ignoring
that effect, which means we’re only going to
get an approximation here. But it will still be easily
good enough to make the point. For a z this large, this
factor is a small correction. I think this ends up
being 0.97, or something like that, very close to 1,
which means what we’re getting is very close to 2c h
naught inverse, which is the actual horizon. The horizon corresponds
to z equals infinity. If you think about
it, that’s what you expect the horizon to be. It corresponds to
infinite red shift. And you don’t see
anything beyond that. So if we take the best value
we have for h naught, which I’m taking from the
Planck satellite, 57.3 kilometers per second
per megaparsec, and put that and the value
for z into this formula, we get l sub p of t naught
of 28.2 billion light years times 10 to the 9 light year. So it’s of course larger
than ct as we expect. It’s basically 3ct for a
matter dominated universe. And 3ct is the same
as 2ch 0 inverse. Now, what we want
to know, though, is how far away was this
when the emission occurred, not the present distance. We looked at the
present distance simply because we had a formula
for it from our homework set. But we know how to extrapolate
that backwards, l sub t at time t sub d. Distances that are fixed
in co-moving space, which these are,
are just stretched with the scale factor. So this will just
be the scale factor at the time of decoupling
divided by the scale factor today times the
present distance. And this is, again, given by
this ratio of temperatures. So it’s 1 over
1,100, the inverse of what we had over there. So the separation
at this early time is just 1,100 times smaller
than the separation today. And that can be
evaluated numerically. And it gives us 2.56 times
10 to the seven light years, so 25 million light years. Now, the point is that
that’s significantly larger than the horizon
distance at that time. And remember, the
horizon distance is the maximum possible
distance that anything can travel limited
by the speed of light from the time of the
big bang up to any given point in cosmic history. So the horizon at
time t sub d is just given by the simple formula
that the physical value of the horizon
distance, l sub h phys, l sub horizon physical, at
time t sub d is just equal to, for a matter dominated
universe, 3c times t sub d. And that we can evaluate,
given what we have. And it’s about 1.1 times 10 to
the sixth light years, which is significantly less than
2.56 times 10 to the seven light years. And, in fact, the ratio of
the two, given these numbers, is that l sub p of t sub d
over l sub h is also of t sub d is about equal to 23,
just doing the arithmetic. And that means if we go
back to this picture, these two points of emission
were separated from each other by about 46 horizon distances. And that’s enough to
imply that there’s no way that this point could have known
anything whatever about what was going on at this point. Yet somehow they knew to
emit these two photons at the same time at
the same temperature. And that’s the mystery. One can get around this
mystery if one simply assumes that the singularity
that created all of this produced a perfectly homogeneous
universe from the very start. Since we don’t understand
that singularity, we’re allowed to attribute
anything we want to it. So in particular, you can
attribute perfect homogeneity to the singularity. But that’s not really
an explanation. That’s an assumption. So if one wants to be able
to explain this uniformity, then one simply cannot do it
in the context of conventional cosmology. There’s just no
way that causality, the limit of the speed of
light, allows this point to know anything about what’s
going on at that point. Yes? AUDIENCE: How could a
singularity not be uniform? Because If it had
non-uniform [INAUDIBLE], then not be singular? PROFESSOR: OK, the
question is how can a singularity
not be uniform? The answer is,
yes, singularities can not be uniform. And I think the way one can
show that is a little hard. But you have to imagine a
non-uniform thing collapsing. And then it would just be
the time reverse, everything popping out of the singularity. So you can ask, does
a non-uniform thing collapse to a singularity? And the answer to that
question is not obvious and really was debated
for a long time. But there were theorems
proven by Hawking and Penrose that indeed not only do the
homogeneous solutions that we look at collapse but in
homogeneous solutions also collapse to singularities. So a singularity does
not have to be uniform. OK, so this is the story
of the horizon problem. And as I said, you
can get around it if you’re willing to just
assume that the singularity was homogeneous. But if you want to have a
dynamical explanation for how the uniformity of the
universe was established, then you need some model
other than this conventional cosmological model that
we’ve been discussing. And inflation will
be such a twist which will allow a
solution to this problem. OK, so if there
are no questions, no further questions,
we’ll go on to the next problem
I want to discuss, which is of a similar nature
in that you can get around it by making strong assumptions
about the initial singularity. But if one wants,
again, something you can put your hands on,
rather than just an assumption about a singularity, then
inflation will do the job. But you cannot solve the
problem in the context of a conventional
big bang theory, because the mechanics of the
conventional big bang theory are simply well-defined. So what I want to
talk here is what is called the flatness
problem, where flatness is in the sense of
Omega very near 1. And this is
basically the problem of why is Omega today
somewhere near 1? So Omega naught is the
present value of Omega, why is it about equal to 1? Now, what do we
know first of all about it being about equal to 1? The best value from
the Planck group, this famous Planck
satellite that I’ve been quoting a lot
of numbers from– and I think in all cases,
I’ve been quoting numbers that they’ve established
combining their own data with some other pieces of data. So it’s not quite
the satellite alone. Although, they do give numbers
for the satellite alone which are just a little
bit less precise. But the best number they give
for Omega naught is minus 0.0010 plus or minus 0.0065. Oops, I didn’t put
enough zeroes there. So it’s 0.0065 is the error. So the error is just a
little bit more than a half of a percent. And as you see, consistent
with– I’m sorry, I meant this to be 1. Hold on. This is Omega naught minus 1
that I’m writing a formula for. So Omega naught is very
near 1 up to that accuracy. What I want to emphasize in
terms of this flatness problem is that you don’t need to know
that Omega naught is very, very close to 1 today,
which we now do know. But even back when inflation
was first invented around 1980, in circa 1980 we
certainly didn’t know that Omega was so
incredibly close to 1. But we did know that
Omega was somewhere in the range of about
0.1 and 0.2, which is not nearly as close to 1 as what we
know now, but still close to 1. I’ll argue that the
flatness problem exists for these numbers
almost as strongly as it exists for those numbers. Differ, but this is still a
very, very strong argument that even a number like
this is amazingly close to 1 considering what
you should expect. Now, what underlies this is the
expectations, how close should we expect Omega to be to 1? And the important
underlying piece of dynamics that controls this
is the fact that Omega equals 1 is an unstable
equilibrium point. That means it’s like a
pencil balancing on its tip. If Omega is exactly
equal to 1, that means you have a flat universe. And an exactly
flat universe will remain an exactly
flat universe forever. So if Omega is
exactly equal to 1, it will remain exactly
equal to 1 forever. But if Omega in
the early universe were just a tiny
bit bigger than 1– and we’re about
to calculate this, but I’ll first qualitatively
describe the result– it would rise and would
rapidly reach infinity, which is what it reaches if
you have a closed universe when a closed universe
reaches its maximum size. So Omega becomes infinity and
then the universe recollapses. So if Omega were
bigger than 1, it would rapidly approach infinity. If Omega in the early universe
were just a little bit less than 1, it would rapidly
trail off towards 0 and not stay 1 for
any length of time. So the only way to
get Omega near 1 today is like having a pencil
that’s almost straight up after standing there
for 1 billion years. It’d have to have
started out incredibly close to being straight up. It has to have
started out incredibly close to Omega equals 1. And we’re going to
calculate how close. So that’s the set-up. So the question we want to
ask is how close did Omega have to be to 1 in
the early universe to be in either one of
these allowed ranges today. And for the early
universe, I’m going to take t equals one second
as my time at which I’ll do these calculations. And, historically, that’s
where this problem was first discussed by Dicke and
Peebles back in 1979. And the reason why one
second was chosen by them, and why it’s a sensible time
for us to talk about as well, is that one second
is the beginning of the processes of
nucleosynthesis, which you’ve read about in
Weinberg and in Ryden, and provides a real test of
our understanding of cosmology at those times. So we could say that we
have real empirical evidence in the statement that the
predictions of the chemical elements work. We could say that we have
real empirical evidence that our cosmological model
works back to one second after the Big Bang. So we’re going to
choose one second for the time at which we’re
going to calculate what Omega must’ve been then for it to
be an allowed range today. How close must Omega have been
to 1 at t equals 1 second? Question mark. OK, now, to do this
calculation, you don’t need to know anything
that you don’t already know. It really follows as a
consequence of the Friedmann equation and how matter and
temperature and so on behave with time during radiation
in matter dominated eras. So we’re going to start with
just the plain old first order Fiedmann equation, h squared
is equal to 8 pi over 3 g Rho minus kc squared
over a squared, which you have seen many, many
times already in this course. We can combine that
with other equations that you’ve also
seen many times. The critical density is just
the value of the density when k equals 0. So you just solve
this equation for Rho. And you get 3h
squared over h pi g. This defines the
critical density. It’s that density which makes
the universe flat, k equals 0. And then our standard
definition is that Omega is just defined to be
the actual mass density divided by the critical mass density. And Omega will be the quantity
we’re trying to trace. And we’re also going
to make use of the fact that during the era that
we’re talking about, at is essentially
equal to a constant. It does change a little bit
when electron and positron pairs freeze out. It changes by a
factor of something like 4/11 to the 1/3 power
or something like that. But that power will be of
order one for our purposes. But I guess this is
a good reason why I should put a
squiggle here instead of an equal sign as an
approximate equality, but easily good enough
for our purposes, meaning the corrections
of order one. We’re going to
see the problem is much, much bigger
than order one. So a correction of order
one doesn’t matter. Now, I’m going to start by using
the Planck satellite limits. And at the end, I’ll just make
a comment about the circa 1980 situation. But if we look at the
Planck limits– I’m sorry. Since I’m going to write an
equation for a peculiar looking quantity, I should
motivate the peculiar looking quantity first. It turns out to be useful
for these purposes. And this purpose
means we’re trying to track how Omega
changes with time. It turns out to be useful
to reshuffle the Friedmann. And it is just an
algebraic reshuffling of the Friedmann equation
and the definitions that we have here. We can rewrite the
Friedmann equation as Omega minus 1 over Omega
is equal to a quantity called a times the temperature
squared over Rho. Now, the temperature didn’t even
occur in the original equation. So things might look a
little bit suspicious. I haven’t told
you what a is yet. a is 3k c squared over 8
pi g a squared t squared. So when you put the a into this
equation, the t squares cancel. So the equation doesn’t
really have any temperature dependence. But I factored things
this way, because we know that at is
approximately a constant. And that means that this
capital a, which is just other things which
are definitely constant times, a square t
square in the denominator, this a is approximately
a constant. And you’ll have to
check me at home that this is exactly equivalent
to the original Friedmann equation, no approximations
whatever, just substitutions of Omega and the
definition of Rho sub c. So the nice thing about
this is that we can read off the time dependence
of the right-hand side as long as we know the time
dependence in the temperature and the time dependence
of the energy density. And we do for matter dominated
and radiation dominated eras. So this, essentially,
solves the problem for us. And now it’s really
just a question of looking at the
numerics that follow as a consequence
of that equation. And this quantity,
we’re really interested in just Omega minus 1. The Friedmann equation gave
us the extra complication of an Omega in the denominator. But in the end, we’re going
to be interested in cases where Omega is very,
very close to 1. So the Omega in the denominator
we could just set equal to one. And it’s the Omega minus
1 in the numerator that controls the value of
the left-hand side. So if we look at these Planck
limits, we could ask how big can that be? And it’s biggest if the error
occurs on the negative side. So it contributes to
this small mean value which is slightly negative. And it gives you 0.0075
for Omega minus 1. And then if you put that in the
numerator and the same thing in the denominator, you
get something like 0.0076. But I’m just going
to use the bound that Omega naught minus 1
over Omega is less than 0.01. But the more accurate
thing would be 0.076. But, again, we’re not really
interested in small factors here. And this is a one signa error. So the actual error could
be larger than this, but not too much
larger than this. So I’m going to divide the time
interval between one second and now up into two integrals. From one second to
about 50,000 years, the universe was
radiation dominated. We figured out earlier that the
matter to radiation equality happens at about 50,000 years. I think we may have gotten
47,000 years or something like that when we calculated it. So for t equals 1
second to– I’m sorry, I’m going to do it
the other order. I’m going to start with the
present and work backwards. So for t equals 50,000
years to the present, the universe is
matter dominated. And the next thing
is that we know how mattered dominated
universe’s behave. We don’t need to recalculate it. We know that the scale
factor for a matter dominated flat universe goes
like t to the 2/3 power, I should have a
portionality sign here. a of t is proportional
to t to 2/3. And it’s fair to assume
flat, because we’re always going to be talking about
universes that are nearly flat and becoming more and more
flat as we go backwards, as we’ll see. And again, this isn’t an
approximate calculation. One could do it more
accurately if one wanted to. But there’s really no need
to, because the result will be so extreme. The temperature behaves like
one over the scale factor. And that will be true for
both the matter dominated and a radiation
dominated universe. And the energy density
will be proportional to one over the scale factor cubed. And then if we put those
together and use the formula on the other blackboard and ask
how Omega minus 1 over Omega behaves, it’s proportional
to the temperature squared divided by the energy density. The temperature
goes like 1 over a. So temperature squared
goes like 1 over a squared. But Rho in the denominator
goes like 1 over a cubed. So you have 1 over a squared
divided by 1 over a cubed. And that means it just goes
like a, the scale factor itself. So Omega minus 1 over
Omega is proportional to a. And that means it’s
proportional to t to the 2/3. So that allows us to
write down an equation, since we want to
relate everything to the value of Omega
minus 1 over Omega today, we can write Omega
minus 1 over Omega at 50,000 years is about equal
to the ratio of the 2 times the 50,000 years and today,
which is 13.8 billion years, to the 2/3 power
since Omega minus 1 grows like t to the 2/3. I should maybe have
pointed out here, this telling us that Omega
minus 1 grows with time. That’s the important feature. It grows t to the 2/3. So the value at 50,000
years is this ratio to the 2/3 power times
Omega minus 1 over Omega today, which I can
indicate just by putting subscript zeros on my Omegas. And that makes it today. And I’ve written this as
a fraction less than one. This says that Omega
minus 1 over Omega was smaller than it
is now by this ratio to the 2/3 power, which
follows from the fact that Omega minus 1 over Omega
grows like t to the 2/3. OK, we’re now halfway there. And the other half is similar,
so it will go quickly. We now want to go
from 50,000 years to one second using the
fact that during that era the universe was
radiation dominated. So for t equals 1
second to 50,000 years, the universe is
radiation dominated. And that implies
that the scale factor is proportional to t to the 1/2. The temperature is,
again, proportional to 1 over the scale factor. That’s just
conservation of entropy. And the energy density goes
one over the scale factor to the fourth. So, again, we go
back to this formula and do the corresponding
arithmetic. Temperature goes like 1 over a. Temperature squared goes
like 1 over a squared. That’s our numerator. This time, in the
denominator, we have Rho, which goes like
one over a to the fourth. So we have 1 over a
squared divided by 1 over a to the fourth. And that means it
goes like a squared. So we get Omega
minus 1 over Omega is proportional to a squared. And since goes like
the square root of t, a squared goes like t. So during the
radiation dominated era this diverges even
a little faster. PROFESSOR: It goes like
t, rather than like t to the 2/3, which is a
slightly slower growth. And using this fact, we
can put it all together now and say that Omega minus
1 over Omega at 1 second is about equal to 1
second over 50,000 years to the first
power– this is going like the first
power of t– times the value of Omega minus 1
over Omega at 50,000 years. And Omega at 50,000 years,
we can put in that equality and relate everything
to the present value. And when you do that,
putting it all together, you ultimately find that Omega
minus 1 in magnitude at t equals 1 second is less than
about 10 to the minus 18. This is just putting
together these inequalities and using the Planck value for
the present value, the Planck inequality. So then 10 to the minus 18
is a powerfully small number. What we’re saying is that to
be in the allowed range today, at one second
after the Big Bang, Omega have to have
been equal to 1 in the context of this
conventional cosmology to an accuracy of
18 decimal places. And the reason
that’s a problem is that we don’t know
any physics whatever that forces Omega
to be equal to 1. Yet, somehow Omega
apparently has chosen to be 1 to an accuracy
of 18 decimal places. And I mention that the argument
wasn’t that different in 1980. In 1980, we only knew
this instead of that. And that meant that instead
of having 10 to the minus 2 on the right-hand side
here, we would have had 10 differing by three
orders of magnitude. So instead of getting
10 to the minus 18 here, we would have gotten
10 to minus 15. And 10 to minus 15 is, I
guess, a lot bigger than 10 to minus 18 by a
factor of 1,000. But still, it’s an
incredibly small number. And the argument really
sounds exactly the same. The question is, how
did Omega minus 1 get to be so incredibly small? What mechanism was there? Now, like the
horizon problem, you can get around it by
attributing your ignorance to the singularity. You can say the
universe started out with Omega exactly equal
to 1 or Omega equal to 1 to some extraordinary accuracy. But that’s not really
an explanation. It really is just a
hope for an explanation. And the point is
that inflation, which you’ll be learning about
in the next few lectures, provides an actual explanation. It provides a
mechanism that drives the early universe towards Omega
equals 1, thereby explaining why the early universe
had a value of Omega so incredibly close to 1. So that’s what we’re going
to be learning shortly. But at the present time,
the takeaway message is simply that for Omega to
be in the allowed range today it had to start out unbelievably
close to 1 at, for example, t equals 1 second. And within
conventional cosmology, there’s no explanation for
why Omega so close to 1 was in any way preferred. Any questions about that? Yes? AUDIENCE: Is there
any heuristic argument that omega [INAUDIBLE]
universe has total energy zero? So isn’t that, at
least, appealing? PROFESSOR: OK the question
is, isn’t it maybe appealing that Omega should equal
1 because Omega equals 1 is a flat universe,
which has 0 total energy? I guess, the point is that
any closed universe also has zero total energy. So I don’t think Omega
equals 1 is so special. And furthermore, if you look at
the spectrum of possible values of Omega, it can be positive–
I’m sorry, not with Omega. Let me look at the
curvature itself, little k. Little k can be
positive, in which case, you have a closed universe. It can be negative,
in which case, you have an open universe. And only for the one
special case of k equals 0, which really is one
number in the whole real line of possible numbers, do
you get exactly flat. So I think from that
point of view flat looks highly special and not at
all plausible as what you’d get if you just grabbed
something out of a grab bag. But, ultimately, I think there’s
no way of knowing for sure. Whether or not Omega equals 1
coming out of the singularity is plausible really depends
on knowing something about the singularity, which we don’t. So you’re free to speculate. But the nice thing
about inflation is that you don’t
need to speculate. Inflation really does
provide a physical mechanism that we can understand
that drives Omega to be 1 exactly
like what we see. Any other questions? OK, in that case,
what I’d like to do is to move on to
problem number three, which is the magnetic monopole
problem, which unfortunately requires some background
to understand. And we don’t have too much time. So I’m going to go through
things rather quickly. This magnetic
monopole problem is different from the other two
in the first two problems I discussed are just problems
of basic classical cosmology. The magnetic
monopole problem only arises if we believe that
physics at very high energies is described what are called
grand unified theories, which then imply that these magnetic
monopoles exist and allow us a root for estimating
how many of them would have been produced. And the point is
that if we assume that grand unified theories
are the right description of physics at very
high energies, then we conclude that far too
many magnetic monopoles would be produced if we had just the
standard cosmology that we’ve been talking about
without inflation. So that’s going to be the
thrust of the argument. And it will all go
away if you decide you don’t believe in grand
unified theories, which you’re allowed to. But there is some evidence
for grand unified theories. And I’ll talk about
that a little bit. Now, I’m not going to have
time to completely describe grand unified theories. But I will try to tell
you enough odd facts about grand unified theories. So there will be kind of
a consistent picture that will hang together, even though
there’s no claim that I can completely teach you grand
unified theories in the next 10 minutes and then talk
about the production of magnetic monopoles and
those theories in the next five minutes. But that will be
sort of the goal. So to start with, I
mentioned that there’s something called the
standard model of particle physics, which is
enormously successful. It’s been developed
really since the 1970s and has not changed too
much since maybe 1975 or so. We have, since 1975, learned
that neutrinos have masses. And those can be incorporated
into the standard model. And that’s a recent addition. And, I guess, in
1975 I’m not sure if we knew all three
generations that we now know. But the matter, the
fundamental particles fall into three
generations, these particles of a different type. And we’ll talk about them later. But these are the quarks. These are the
spin-1/2 particles, these three columns on the left. On the top, we have the quarks,
up, down, charm, strange, top, and bottom. There are six different
flavors of quarks. Each quark, by the way, comes
in three different colors. The different colors
are absolutely identical to each other. There’s a perfect
symmetry among colors. There’s no perfect
symmetry here. Each of these quarks is a little
bit different from the others. Although, there are
approximate symmetries. And related to each
family of quarks is a family of
leptons, particles that do not undergo
strong interactions in the electron-like
particles and neutrinos. This row is the neutrinos. There’s an electron neutrino,
a muon neutrino, and a tau neutrino, like
we’ve already said. And there’s an electron,
a muon, and a tao, which I guess we’ve
also already said. So the particles on the left are
all of the spin-1/2 particles that exist in the standard
model of particle physics. And then on the right, we
have the Boson particles, the particles of
integer span, starting with the photon on the top. Under that in this
list– there’s no particular order
in here really– is the gluon which is
the particle that’s like the photon but the
particle which describes the strong interactions, which
are somewhat more complicated and electromagnetism, but still
described by spin-1 particles just like the photon. And then two other spin-1
particles, the z0 and the w plus and minus, which
are a neutral particle and a charged
particle, which are the carriers of the
so-called weak interactions. The weak interactions
being the only non-gravitational interactions
that neutrinos undergo. And the weak interactions
are responsible for certain particle decays. For example, a neutron can
decay into a proton giving off also an electron–
producing a proton, yeah– charge has to be
conserved, proton is positive. So it’s produced with
an electron and then an anti-electron neutrino to
balance the so-called electron lepton number. And that’s a weak direction. Essentially, anything
that involves neutrinos is going to be weak interaction. So these are the characters. And there’s a set
of interactions that go with this
set of characters. So we have here a
complete model of how elementary particles interact. And the model has been
totally successful. It actually gives
predictions that are consistent with every
reliable experiment that has been done since the
mid-1970s up to the present. So it’s made particle
physics a bit dull since we have a theory that
seems to predict everything. But it’s also a
magnificent achievement that we have such a theory. Now, in spite of the fact that
this theory is so unbelievably successful I don’t think I know
anybody who really regards this as a candidate even or the
ultimate theory of nature. And the reason for that
is maybe twofold, first is that it does not
incorporate gravity, it only incorporates
particle interactions. And we know that gravity exists
and has to be put in somehow. And there doesn’t seem to be any
simple way of putting gravity into this theory. And, secondly– maybe there’s
three reasons– second, it does not include any good
candidates for the dark matter that we know exists
in cosmology. And third, excuse me– and this
is given a lot of importance, even though it’s an aesthetic
argument– this model has something like 28 free
parameters, quantities that you just have to go out and measure
before you can use the model to make predictions. And a large number
of free parameters is associated, by theoretical
physicists, with ugliness. So this is considered
a very ugly model. And we have no real
way of knowing, but almost all
theoretical physicists believe that the
correct theory of nature is going to be simpler and
involve many fewer, maybe none at all, free knobs
that can be turned to produce different
kinds of physics. OK, what I want to
talk about next, leading up to grand
unified theories, is the notion of gauge theories. And, yes? AUDIENCE: I’m sorry, question
real quick from the chart. I basically heard
the explanation that the reason for the
short range of the weak force was the massive mediator that is
the cause of exponential field decay. But if the [INAUDIBLE]
is massless, how do we explain
that to [INAUDIBLE]? PROFESSOR: Right,
OK, the question is for those of you
who couldn’t hear it is that the short range
of the weak interactions, although I didn’t talk about
it, is usually explained and is explained by the
fact that the z and w naught Bosons are very heavy. And heavy particles
have a short range. But the strong interactions
seem to also have a short range. And yet, the gluon is
effectively massless. That’s related to a
complicated issue which goes by the name of confinement. Although the gluon is
massless, it’s confined. And confined means that
it cannot exist as a free particle. In some sense, the
strong interactions do have a long range in that
if you took a meson, which is made out of a quark
and an anti-quark, in principle, if
you pulled it apart, there’d be a string
of gluon force between the quark
and the anti-quark. And that would produce
a constant force no matter how far
apart you pulled them. And the only thing
that intervenes, and it is important
that it does intervene, is that if you pulled
them too far apart it would become
energetically favorable for a quark anti-quark pair
to materialize in the middle. And then instead of having a
quark here and an anti-quark here and a string
between them, you would have a quark here
and an anti-quark there and a string between them,
and an anti-quark here and– I’m sorry, I
guess it’s a quark here and an anti-quark there
and a string between those. And then they would
just fly apart. So the string can
break by producing quark anti-quark pairs. But the string can never just
end in the middle of nowhere. And that’s the essence
of confinement. And it’s due to the
peculiar interactions that these gluons
are believed to obey. So the gluons behave
in a way which is somewhat uncharacteristic
of particles. Except at very short distances,
they behave very much like ordinary particles. But at larger distances,
these effects of confinement play a very significant role. Any other questions? OK, onward, I want talk
about gauge theories, because gauge
theories have a lot to do with how one gets
into grand unified theories from the standard model. And, basically, a gauge
theory is a theory in which the fundamental fields
are not themselves reality. But rather there’s a
set of transformations that the fields can
undergo which take you from one description
to an equivalent description of the same reality. So there’s not a
one to one mapping between the fields and reality. There are many different
field configurations that correspond to
the same reality. And that’s basically
what characterizes what we call a gauge theory. And you do know one
example of a gauge theory. And that’s e and m. If e and m is expressed in terms
of the potentials Phi and A, you can write e in terms
of the potential that way and b as the curl of A, you
could put Phi and A together and make a four-vector
if you want to do things in a
Lorentz covariant way. And the important point,
though, whether you put them together or not, is
that you can always define a gauge transformation
depending on some arbitrary function Lambda, which is a
function of position and time. I didn’t write in the
arguments, but Lambda is just an arbitrary function
of position and time. And for any Lambda you
can replace 5 by 5 prime, given by this line, and a by
a prime, given by that line. And if you go back
and compute e and b, you’ll find that
they’ll be unchanged. And therefore, the
physics is not changed, because the physics really
is all contained in e and b. So this gauge transformation
is a transformation on the fields of the theory–
it can be written covariantly this way– which leaves
the physics invariant. And it turns out that
all the important field theories that we know
of are gauge theories. And that’s why it’s
worth mentioning here. Now, for e and m, the gauge
parameter is just this function Lambda, which is a function
of position and time. And an important
issue is what happens when you combine
gauge transformations, because the succession
of two transformations had better also be a
symmetry transformation. So it’s worth understanding
that group structure. And for the case of e
and m, these Lambdas just add if we make
successive transformations. And that means the
group is Abelian. It’s commutative. But that’s not always the case. Let’s see, where am I going? OK, next slide actually
comes somewhat later. Let me go back to
the blackboard. It turns out that the important
generalization of gauge theories is the generalization
from Abelian gauge theories to non-Abelian ones,
which was done originally by Yang and Mills
in 1954, I think. And when it was first proposed,
nobody knew what to do with it. But, ultimately, these
non-Abelian gauge theories became the standard model
of particle physics. And in non-Abelian
gauge theories the parameter that describes the
gauge transformation is a group element, not just
something that gets added. And group elements
multiply, according to the procedures of some group. And in particular,
the standard model is built out of three groups. And the fact that there
are three groups not one is just an example
of this ugliness that I mentioned and is
responsible for the fact that there’s some significant
number of parameters even if there were no
other complications. So the standard model is based
on three gauge groups, SU3, SU2, and U1. And it won’t really be
too important for us what exactly these groups are. Let me just mention
quickly, SU3 is a group of 3 by 3 matrices which are
unitary in the sense that u adjoint times u is equal
to 1 and special in the sense that they have determinant 1. And the set of all
3 by 3 matrices have those properties
form a group. And that group is SU3. SU2 is the same
thing but replace the 3 in all those
sentences by 2s. U1 is just the group of phases. That is the group
of real numbers that could be written
as either the i phi. So it’s just a complex number of
magnitude 1 with just a phase. And you can multiply those
and they form a group. And the standard model
contains these three groups. And the three groups
all act independently, which means that if you
know about group products, one can say that the full
group is the product group. And that just means that a full
description of a group element is really just a set
of an element of SU3, and an element of SU2,
and an element of U1. And if you put together three
group elements in each group and put them
together with commas, that becomes a group element
of the group SU3 cross, SU2 cross U1. And that is the gauge
group of the standard model of particle physics. OK, now grand unified theories,
a grand unified theory is based on the idea that
this set of three groups can all be embedded in
a single simple group. Now, simple actually has a
mathematical group theory meaning. But it also, for
our purposes, just means simple, which is
good enough for our brush through of these arguments. And, for example–
and the example is shown a little bit in
the lecture notes that will be posted
shortly– an example of a grand unified
theory, and indeed the first grand unified
theory that was invented, is based on the full gauge group
SU5, which is just a group of 5 by 5 matrices which are
unitary and have determinant 1. And there’s an easy way to embed
SU3 and SU2 and U1 into SU5. And that’s the way that
was used to construct this grand unified theory. One can take a 5 by 5 matrix–
so this is a 5 by 5 matrix– and one can simply take
the upper 3 by 3 block and put an SU3 matrix there. And one can take the
lower 2 by 2 block and put an SU2 matrix there. And then the U1 piece–
there’s supposed to be a U1 left
over– the U1 piece can be obtained by giving
an overall phase to this and an overall phase
to that in such a way that the product of
the five phases is 0. So the determinant
has not changed. So one can put an e to the i2
Phi there and a factor of e to the minus i3 Phi
there for any Phi. And then this Phi
becomes the description of the U1 piece of
this construction. So we can take an arbitrary
SU3 matrix, and arbitrary SU2 matrix, and an arbitrary
U1 value expressed by Phi and put them together
to make an SU5 matrix. And if you think about
it, the SU3 piece will commute with the SU2
piece and with the U1 piece. These three pieces will all
commute with each other, if you think about
how multiplication works with this construction. So it does exactly what we want. It decomposes SU5. So it has a subgroup of
SU3 cross SU2 across U1. And that’s how the simplest
grand unified theory works. OK, now, there are important
things that need to be said, but we’re out of time. So I guess what we need
to do is to withhold from the next problem set,
the magnetic monopole problem. Maybe I was a bit over-ambitious
to put it on the problem set. So I’ll send an email
announcing that. But the one problem
on the problem set for next week about
grand unified theories will be withheld. And Scott Hughes will pick up
this discussion next Tuesday. So I will see all of
you– gee willikers, if you come to my office
hour, I’ll see you then. But otherwise, I may not
see you until the quiz. So have a good Thanksgiving
and good luck on the quiz.

3 thoughts on “21. Problems of the Conventional (Non-inflationary) Hot Big Bang Model

  1. Around 22:00 prof. Guth says that around 5 or 6 thousand years ago dark energy started to dominate over non relativistic matter. Other sources say this happened around 4 or 5 billion years ago. Is the latter the correct statement ?

    Awesome course BTW, thanks once more to MIT OCW, and prof. Alan Guth for making this material available for all.

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