Where are we?Running Hubble backwardsRelativistic cosmologyBack to the big bangThe hot bang and the CMBR
Where are we?
Having done a lightning introduction to particle physics before break and an introduction to theHubble expansion in the last lecture, we are now going to work our way back towards the beginningof the universe. This is sort of fun, for its own sake, but it will also, I claim, get us closer to ananswer to the question we asked at the very beginning of the course — why is Newton’s third lawa formula for acceleration. We won’t get there today, but we should make it by Thursday.
Running Hubble backwards
Let us start by going back to the laws we derived to describe the evolution of the Hubble expansion.
If we put ourselves at the origin and take ✁ to be the position of some distant galaxy, the Hubblelaw can be written as
where ✆✟✞ ✠ ✡ is the Hubble parameter. Then, assuming that we can use Newton’s theory of gravityand ignore relativity, we found that gravity slows the Hubble expansion
Then by considering the conservation of the mass inside a sphere with radius ✁
relativity), we found that we could integrate (2) to obtain
depends on the initial conditions. Again, this derivation was valid for a
universe dominated by slowly moving matter, so long as the cosmological principle is satisfied.
It is conventional and convenient to get rid of the vectors in (1)-(3). We can do this by defining
between points in the universe. For example, we could take
the distance between us and the distant galaxy. But the notion is more general, as it has to bebecause we want to use it to describe the universe at earlier times, before we or even galaxies werearound.
is just an arbitrary scale that measures the relative size of the universe. The value of
doubles, that means that the size of the universe has doubled!
and that fact that the direction of ✁ doesn’t change with time, we can write
. As I have said before, there are theoretical
reasons (that we will discuss next time) to think that this is a good approximation. We can thenfollow the Hubble evolution backwards towards the big bang, at least for a while.
, we can take the square root of (7) and write
Note that have taken the positive sign, corresponding to expansion.
To determine the time evolution of ✪ , we need a relation between ✪ and . This relation depends
on what kind of stuff our universe is made of at the time of interest. For slowly moving matter,we can simply use the fact
is a constant, because the density is inversely proportional to the
This is a differential equation that we can solve for
(or course, the result is only meaningful as
is a constant set by initial conditions. As expected, the density decreases as the universe
expands and increases as we run the tape backwards. Since
The simple behavior implied by (14) does not continue all the way back to the beginning of things.
There is good evidence that we have left out something very important — the big bang was hot aswell as dense. Heat complicates matters in a couple of ways. Heat is the random motion of theconstituents of matter. The higher the temperature, the faster the motion. But once stuff is movingaround, we have not only density, but also pressure — we left pressure out of our simple analysisof the Hubble dynamics. Furthermore, as we go back farther in time, the temperature gets higherand higher, and eventually, the random motion of the particles approaches the speed of light. Thenwe cannot ignore relativity! In this regime, as we know from our study of relativity, all sorts ofbizarre things happen. Particles are created and destroyed. Energy and momentum are conserved,but not mass. We must be careful.
A complete understanding of what happens when things get relativistic requires that we gener-
alize Newton’s theory of gravity to incorporate special relativity. The resulting theory is Einstein’sgeneral relativity. We are not going to discuss this here, except for one result, which will be all weneed. It turns out that (7) remains correct in general relativity if
density, but as the relativistic energy density divided by
. This makes sense in that if we go back to low temperatures and things come
to rest, the energy density just becomes the mass density. But in the relativistic regime, it just turnsout that gravity affects all forms of energy, not just mass (that is rest energy), so we have to use theenergy density instead in (7). It is worth writing (7) again, now that we know how general it is:
This is called the Friedmann equation. The parameter
Let’s see what becomes of (9) and (6) in the presence of pressure and relativity. First consider
(9). Here we can consider the total energy in a cube with side ❳ , and how it changes with time.
Remember that ❳ is some distance that changes along with the stuff that the universe is made outof. On the average, no stuff comes into or goes out of the cube, or at least as much stuff comes inas goes out. We could think of it as having impermeable sides or pistons on all sides or somethinglike that. Because we are now intepreting
the pressure, the energy changes with the volume — the pressure
which reduces the energy. Quantitatively, the work done by the pressure in an infinitesimal change
of the boundary is a surface integral over the boundary of the force
. The force of pressure has magnitude equal to the area
pressure , and its direction is normal to the boundary. Thus the work done is the pressure times
the area times the perpendicular motion of boundary, which is equal to
Notice that this reduces to (9) when the pressure vanishes.
Using (20), we can find the appropriate generalization of (6) as follows. Multiply (16) by ✈ ④ ,
and differentiate with respect to time. ❶
This is the relativistic generalization of (6), to which it reduces when ⑦✎⑤✻➃ and ✉ is the rest energy.
Back to the big bang
Armed with the results of the previous section, we can go back a bit further into the history ofthe universe. But now that we have included the effect of pressure, we also have to understandthe relation between the energy density ✉ and the pressure ⑦ for relativistic stuff. An importantexample of relativistic stuff is a gas of photons — particles of light — radiation. We can computethe pressure in any shape container, so consider the pressure of a gas of photons with energy density
in a cubical container of side ✈ . The pressure of such a gas arises because the photons bounce
off the sides of the container. Suppose the sides are lined up with the coordinate axes. A photonwith energy
and momentum ⑦ , if all the components of ⑦ are nonzero, bounces around off all the
sides. Each time it hits a side, the component of momentum perpendicular to the side changes sign,but the others remain unchanged. Consider the force on the sides perpendicular to ➇ . When the
photon bounces off this side, it imparts an impulse ❻✥➈
. The time it takes to get back to the same
. The contribution of this photon to the impulse per unit time, which
The contribution to the pressure is the force (24) divided by the area ✈ ④ . Then we get the totalpressure by summing over all the photons
But the pressure is the same on each side, so because the velocity of each photon is 1, the resultmust be
The pressure of a relativistic gas is 1/3 the energy density (in relativistic units, with
Now let us apply (26) to understand how the energy density of a gas of photons changes with
is constant. Thus the energy density of a relativistic gas falls like ➞❄➠❄➟ as
the universe expands, faster than the energy density of nonrelativistic matter, which falls like ➞ ➠▲➡ .
It may help to understand this result to realize that this means that the energies of the individualrelativistic particles must be falling like
, because their number density clearly falls like ➞ ➠▲➡ .
What is happening is the that energies are red-shifted down as the universe expands. The largerthe universe is, the farther away the relativistic particles reaching some particular point are comingfrom. But the farther away they came from, the more red-shifted they are (because of the Hubbleexpansion). If the particles have mass, the process eventually stops when the particles becomenonrelativistic. But for photons, and other massless particles, it continues forever.
Furthermore, going in the other direction, this means that if we follow the history of a relativis-
tic gas back in time, it not only gets more dense as we go back, but because the energies of theindividual particles are increasing, the temperature increases as well. It turns out that the averageenergy is proportional to the temperature, so the temperature of the relativistic gas goes like
The hot bang and the CMBR
There is one more component to the now standard picture of the hot big bang. One assumes thatat some time in the early history of the universe, all of the particles were in thermal equilibriumat an enormously high temperature. What thermal equilibrium means is that the particles collidefrequently enough that their motions are thoroughly randomized. You might think that this ran-domness would make it hard to understand how such a hot universe works. But in fact, exactly theopposite is true. In thermal equilibrium, all the important properties are determined on the averageby a single parameter — the temperature.1 You can them follow the subsequent evolution of theuniverse, at least on the average, using the tools we have developed above.
Let’s assume that the hot big bang picture is correct, and think about what the universe looked
like when it was very hot. First of all, normal matter, made of neutral atoms certainly didn’t exist,because the high energy collisions would have completely ionized all the atoms. The universewould be a plasma. Furthermore, since particle number is not conserved in relativistic collisions,particles and their antiparticles can be produced and destroyed. So for example, while it seems thatin the universe today, there are a lot more electrons than there were positrons, long ago when theuniverse was less than a millionth its current size, there almost as many positrons as electrons. Thesmall excess of electrons that eventually became the electrons in our atoms was quite unimportantat early times. For the same reason, at even earlier times, there was a lot of other stuff around inthe early moments of the universe that we don’t see much of today — heavy unstable particles
1This seemingly paradoxical situation is beautifully explained in one of the best popular science books I know of,
Steven Weinberg’s The First Three Minutes
, which I recommend for those of you who want to learn more about
the subject. You will also learn (much) more about the connection between temperature and randomness if you take
which today we can make only at large accelerator laboratories and which quickly decay backinto ordinary stuff were as common in the very early universe as electrons. These heavy particledisappear when the universe cools to a temperature such that the typical particle energy is belowtheir mass. It is all quite strange — but simple, in a funny way, because everything is more or lessfixed just by the temperature.
Now why would anyone believe this? We cannot, after all, go back and do experiments on the
early universe. Why is this discussion science? The answer is that we can almost see it! At least wecan look back toward the beginning of the universe by looking far away in the universe, because thelight from far away regions of the universe has taken a long time to get to us. But we can’t look backall the way. Once the universe gets so hot that atoms dissociate into ions, photons cannot go veryfar without colliding with electrons — the universe becomes opaque. Thinking about this in theother direction is even more interesting. As the universe cools to below the temperature at whichatoms dissociate (a few thousand degrees C), it becomes transparent to photons, which means thatphotons fall out of thermal equilibrium. From then on, most of the photons just move freely, nevercolliding with anything again. This “gas” of photons continues to behave like relativistic stuff,while the atoms are nonrelativistic. Thus as the universe continues to expand, the energy densityin the photons gets less and less important to the overall Hubble evolution, but the photons are stillthere, getting more and more red shifted as time goes on. This gas of photons from the formation ofatoms, roughly 100,000 years after the big bang, is the Cosmic Microwave Background Radiation(CMBR). A tiny fraction of these photons hit the earth and can be detected. Much of what weactually know about the early universe comes from studies of the CMBR. The first obvious thingto do is to measure the temperature, which turns out to be about 2.7 C, which is about 1000 times
smaller than the temperature at which atoms come apart into ions. This means, since photontemperature and energy is inversely proportional to , means that the universe today is about 1000
times bigger today than it was when atoms first formed. There is actually much more to thisstatement than meets the eye. It is an important prediction of the hot big bang model that CMBRlooks like it has a temperature at all. The reason it does, even though the photons are no longercolliding very much, and are not in thermal equilibrium, is that the random distribution of photonenergies that was present when the universe first became transparent is still there — just all theenergies have been scaled down. This prediction has been confirmed by looking at the CMBR inmany different regions of photon energy, and checking that the distribution of energies is what onewould expect in a thermal distribution.
We cannot directly see the universe at scales smaller than 1/1000 the current scale. However,
we can use the tools we have discussed to follow the universe back to small sizes and highertemperatures and energies. We can go back about a factor of a trillion (
such high energies that have not directly seen the physics in laboratory experiments, so we arereasonably confident that we have the picture right back that far. And there are some observableconsequences. For example, most of the nuclei of light elements (deuterium, helium, etc) wereformed back when the universe was a million times smaller and protons and neutrons first started tostick together. The hot big bang picture predicts a specific pattern of abundances of these elements,which can be checked by looking for them out in the universe. This is much less direct and moreproblematic than direct observation, because one worries about what has happened to these nucleiin the 10 billion years since they were formed. Nevertheless, there is interesting information to be
Eventually, however, we get back to such high temperatures that we really don’t know how the
physics works. We’ll talk more about this on Thursday!
Here are some web links to more information about the Cosmic Microwave Background:This is a very nice site maintained by a faculty member at Chicago.
A simpler site, full of news and history.
2Again, see Steven Weinberg’s The First Three Minutes
for more details.
The cosmological constant
There is one more possibly important thing to say about the energy density of the vacuum. In thediscussion above, we have implicitly assumed that it is only the false vacuum that has a nonzeroenergy density. But why should this be true. Why should the true vacuum, the true lowest energystate of the universe, the state with absolutely nothing in it — why should this vacuum have zeroenergy density. This question is only meaningful in the presence of gravity. In Newtonian mechan-ics, as you know, only differences in energy make a difference. We can always redefine the energyby subtracting a constant from everything. But as we have seen with the false vacuum, in thepresence of gravity, this is not true. The energy density and the negative pressure associated withit have a gravitational effect. And no one has ever been able to come up with a convincing reasonwhy such an energy density should not be there. In fact, it was suggested by Einstein himself, earlyin the history of general relativity. It is called the ‘cosmological constant.”
Einstein originally suggested the cosmological constant for what turned out to be the wrong
reason. He felt, for philosophical reasons, that the universe should not change with time. He didn’tknow about the Hubble expansion, which was discovered later. But with ordinary gravity, thereis no way to make a universe that stands still. If it starts at rest, gravity will cause it to contract.
But as we have seen, and energy density of the vacuum produces a repulsive gravitational effectbecause of its negative pressure. This can, if properly chosen, exactly cancel the effect of gravity,allowing for a stationary universe.
Of course this original motivation for the cosmological constant has long since gone away. It is
sometimes referred to as “Einstein’s mistake” because it seems very ugly to theorists like me. I’llcome back below to why it seems so ugly. Nevertheless, in the last couple of years, astronomershave published evidence that it might not be a mistake after all. By studying type IIa supernovas2in distant galaxies, astronomers have apparently found evidence that the Hubble expansion is notslowing down as fast as one would expect if the energy density of the universe is dominated by anykind of matter. In fact, if anything, it looks as if the expansion is speeding up!
What they actually see is that the brightness of distant supernovas with red shifts between ❱❳❲ ❨
is lower that what would be expected by computing their distance using their red shift. This
interpreted to mean billions of years ago when this light was emitted, the relation between redshift and distance was different. For a given red shift, the objects were further away then than wewould expect on the basis of the Hubble constant determined by looking at nearer (and thus morerecently viewed) objects. Since we have to to farther out to get the same red shift, that means thatthe universe was expanding more slowly then. But that means that the expansion of the universe
2These are supernovas that occur when a white dwarf star gobbles up matter and increases in mass beyond the
point where the pressure from atomic matter that keeps the star from collapsing can compete the gravity that is tryingto squeeze it. The result is a huge explosion that at least theoretically should always produce about the same amountof energy and should therefore provide a “standard candle” that allows astronomers to determine its distance bymeasuring its brightness.
has been speeding up since the light from these supernovas was emitted, billions of years ago. Thefigure at the end of this lecture is taken from one of the original papers, which you can find atIt shows a plot of magnitude (a measure of brightness - upis dimmer) versus redshift.
If all this is right, it implies that about 70% of the energy density of the universe, which it seems
is roughly equal to the critical density required for flatness, comes from the energy of the vacuum.
This is a bizarre, crazy result that I have a lot of trouble believing, for reasons that I will comeback to. I would not be surprised if the result changed as more data is accumulated. However, ithas been around for several years now as the number of supernovas on which the estimate is basedhas grown from a few to hundreds. There is also some indirect support for this view from theconsistency of the measured anisotropy of the CMBR with a model in which 30% of the energydensity of the universe is in the form of cold dark matter and the rest is energy density. I still don’treally believe it, but who knows. Keep an eye on the New York Times, where stories of this sortare usually first reported.
Here are some links to more information.
An elementary introduction: A recent discovery, strengthening the case for acceleration:
The laws of physics and the Taylor expansion
Now finally, I want to return to the question that we discussed at the very beginning of the course.
? Why is Newton’s law a formula for acceleration? Of course, we know that
Newton’s law is not right. It ignores special relativity and quantum mechanics, for example. Butin fact, our current theories of relativistic quantum mechanics are based on Lagrangians that arereally rather straightforward generalizations of those that we use to derive Newton’s law. TheLagrangians that we use to describe the world still depend just on coordinates and their first timederivatives. The equations of motion are thus still equations for “acceleration.” So in a certainsense, Newton has survived the revolutions of special relativity and quantum mechanics, and thequestion still remains an interesting one. I certainly don’t know the answer to this question. But Ithink that it is related to a much deeper question, which may be the central mystery of the way theuniverse works.
This mystery takes a bit of explaining. We have already talked about particle physics units, in
and are set equal to 1 because they are built into the way the universe works. If we adopt
these sensible units, then all dimensional quantities can be related. For example, we can expresseverything in terms of mass. The properties of our world are primarily determined by the masses
of the electron and the proton, and a few numbers, like the fine structure constant,
Almost all of the physics of the everyday world involves combinations of these basic parameters.
But one thing that is different is gravity. Gravity, you remember, is described by the gravitationalconstant
, which in particle physics units is proportional to
(0.5,0.5) (0, 0)
( 1, 0 ) (1, 0)
(Hamuy et al,
FIG. 1.— Hubble diagram for 42 high-redshift Type Ia supernovae from the Supernova Cosmology Project, and 18 low-redshift Type Ia supernovae from the
Calán/Tololo Supernova Survey, after correcting both sets for the SN Ia lightcurve width-luminosity relation. The inner error bars show the uncertainty due tomeasurement errors, while the outer error bars show the total uncertainty when the intrinsic luminosity dispersion, 0.17 mag, of lightcurve-width-corrected Type Iasupernovae is added in quadrature. The unfilled circles indicate supernovae not included in Fit C. The horizontal error bars represent the assigned peculiar velocityuncertainty of 300 km s−1. The solid curves are the theoretical m
) for a range of cosmological models with zero cosmological constant: (Ω
on top, (1, 0) in middle and (2,0) on bottom. The dashed curves are for a range of flat cosmological models: (ΩM, ΩΛ) = (0, 1) on top, (0.5, 0.5) second from top,(1, 0) third from top, and (1.5,-0.5) on bottom.
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