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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.

Very good and clear

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.

That grand unified theory at the end ๐ฎ๐ฅ