Invitation to a Contemporary Physics (2004)

loops which may flow into other dimensions, known as the bulk, must be SM singlets. In a superstring theory, they contain the graviton, the dilaton, the antisymmetric tensor Bµν , and their super-partners. Whether the right-handed neutrino is among them is a matter of conjecture.

There is another kind of duality known as S-duality. It interchanges gS with 1/gS, so weak coupling becomes strong coupling. This is a more di cult subject so we will not venture into it. Su ce to say that with the studies of S-duality and T -duality transformations, one is able to change any of the five superstring theories into any other, so it seems that if these two transformations are taken to be a basic symmetry in our fundamental theory, then the five superstring theories are simply di erent manifestations of one and the same theory. There is very little known about what this fundamental theory is, except that it is believed to be an 11-dimensional theory, rather than the ten for strings. It is known as the M theory.

9.8 Further Reading

http://pdg.lbl.gov/. This is the Particle Data Group website. It contains a comprehensive listing of all known particles and their properties. It also contains a lot of other useful information on particle physics.

9.9 Problems

9.1Use http://pdg.lbl.gov/2002/contents tables.html — the Particle Data Group webpage — to produce a list of particles whose rest energy mc2 is less than 1 GeV. Construct a table listing the name of the particle, its rest energy, and its spin (called J there), and electric charge. For fermions, list only the particles with fermionic number +1.

9.2Find the range R = c/mc2 of the forces obtained from the exchange of ρ, ω, η, and η .

9.3Let us assume the energy required to pull a quark-antiquark a distance r apart from their equilibrium position is V = σr, with σ = (400 MeV)2/ c. At what r is this energy large enough to create a π+π− pair?

9.4Let q be the quark number, Q the electric charge, and I3 the z-component of the strong isospin of a u, d quark, or their antiquarks. Show that the relation Q = I3 + q/6 is valid in each case.

9.5The measured magnitudes of the CKM matrix elements in Eq. (9.1) can be found on the webpage http://pdg.lbl.gov/2002/kmmixrpp.ps. Take in each case the central value in the range of values given. Find the parameters λ, A, ρ, η in the Wolfenstein parameterization of the VKM matrix (footnote 3) to fit approximately the measured values.

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light towards the blue color and is therefore called a blue-shift. Again, the amount of shift indicates the velocity of the object; red-shift for recession, and blue-shift for approach. It is by measuring such shifts in the light reaching us that the velocity of stars and galaxies is determined.

Astronomers use not just visible light, but electromagnetic radiation at other wavelengths as well. Even though this radiation is beyond the visible range, the terms red-shift and blue-shift are still used.

If λ is the wavelength at the source, receding at a velocity v, and if λ is the wavelength observed on earth, then the quantity z = (λ /λ)−1 measures the amount of shift, and is called the red-shift factor. It is approximately equal to v/c when v is small, where c 3 × 108 m/s is the speed of light. For the galaxies measured by Hubble, v/c is indeed small.

To make red-shift a useful tool, we must know the original color of the light in order to figure out the amount of shift. Since we are unable to go to the star to find out, we must resort to other means. For this we make use of a remarkable property of the atoms: the existence of spectral lines.

Just like finger prints and dental records, which vary from individual to individual, and can be used to identify a person, atoms and molecules can also be distinguished by their ‘finger prints’ — in the form of their characteristic spectral lines. An object, when it is heated, emits two kinds of electromagnetic radiation. One contains a broad spectrum of all wavelengths. It reflects the ambient temperature of the surroundings but not the identity of the atom (see Sec. 10.5.2 and Appendix C). The other kind appears at discrete and fixed wavelengths characteristic of the atom. When analyzed through a spectrograph, this kind of light appears as a series of lines on a photographic plate, hence they are called spectral lines. The spectral lines are caused by the electrons of the atom jumping from one discrete orbit to another. The wavelengths characterize the energy di erences of the orbits, which characterize the atom. It is for this reason that spectral lines can be used as a tool to identify atoms.

By looking at the light coming from a star or a galaxy, two interesting points were discovered.

First, except for possible Doppler shifts, the spectral lines of light from distant stars are identical to the spectral lines of the same atoms on earth. This must mean that the physics on a distant star is identical to the physics on earth! This is an extremely important discovery, for it allows us to apply whatever knowledge of physics we learn here to other corners of the universe.

Secondly, the spectral lines from many galaxies are red-shifted (but seldom blueshifted). This means that these galaxies are flying away from us. The velocity of this flight can be determined by measuring the amount of red-shift.

Between 1912 and 1925, V. M. Slipher measured the red-shifts for a number of galaxies. Using Slipher’s compilation of red-shifts of galaxies, the receding velocities can be computed. With his own measurement of distances, Hubble was then able

to establish his famous law that the receding velocity increases linearly with the distance of the galaxy.

This law revolutionizes our thinking of the universe, as will be discussed in the rest of this chapter.

10.1.2Astronomical Distances

The second requirement to establish Hubble’s law is a way to measure astronomical distances.

Distances between cities are measured in kilometers (km), but this is too small a scale in astronomy. Astronomical distances are huge. It takes light, traveling with a speed c = 3 ×105 km/s, a speed which allows it to go around the earth more than seven times in a second, 500 seconds to reach the earth from the sun, and some four years to reach the nearest star. The appropriate unit to measure such vast distances is either a light-year, or a parsec. A light-year is the distance covered by light in one year. The nearest star is four light-years from us, the center of our Milky Way galaxy is some twenty-six thousand light-years away. In ordinary units, a light-year is equal to 9.46 × 1012 km, or in words, almost ten million million kilometers.

One parsec is 3.26 light-years. The name comes from the way astronomical distances are measured. Distances to nearby stars are measured by triangulation. As the earth goes around its orbit, the position in sky of a nearby star seems to shift against the background of distant stars, for much the same reason that nearby scenery appears to move against distant mountains when you look out of the window of your car. If the star is at a distance such that the shift from the average position is 1 second of arc each way, then the star is said to have a parallax angle of one second, and the distance of the star is defined to be 1 parsec. This distance turns out to be 3.26 light-years. A thousand parsecs is called a kiloparsec (kpc), and a million parsecs is called a megaparsec (Mpc). It is this last unit that has been used in the value of the Hubble constant above.

The more distant the star, the smaller the parallax angle, so after some 20 pc or so, the parallax angle will become too small for this method of measuring distances to be useful. Other methods are required.

One such method was discovered by Henrietta Leavitt and Harlow Shapley at the beginning of the twentieth century. It makes use of the special properties of a class of stars, called the Cepheid variables.

During his exploration in 1520, Magellan found two luminous ‘clouds’ in the sky of the southern hemisphere. These two ‘clouds’ turn out to be two nearby irregular galaxies, nowadays called the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC). In 1912, Leavitt found in the SMC twenty five variable stars of the type known as the Cepheid variables. A variable star is a star whose brightness changes with time. A Cepheid variable is a kind of variable star which has a fixed period of brightness variation, and this period usually falls within

390 Cosmology

a range of a few days to a month or more. They are called the Cepheid variables because a typical variable star of this type is found in the constellation Cepheus.

By studying these Cepheid variables in the SMC, Leavitt found that the period of variation of these twenty five Cepheid variables follows a simple law. The period of a star is related to its average brightness observed on earth: the longer the period, the brighter the star.

The brightness of astronomical objects is measured in magnitudes. It is a logarithmic scale. If star A has a magnitude x and star B has a magnitude x − y, then star B is (100)y/5 (2.512)y times brighter than star A. The larger the magnitude, the dimmer is the object. The brightest star in the sky, Sirius, has a magnitude −1.47. Our sun has a magnitude −26.7.

These are the apparent magnitudes, which measures the brightness of these objects seen on earth. This apparent brightness depends on the intrinsic brightness of the object, as well as its distance R from us. Light emitted from a star is spread over a spherical shell whose area grows with R2, thus the apparent brightness falls as 1/R2. The intrinsic brightness of an object is measured in absolute magnitudes. It is defined to be the apparent magnitude if the object is put 10 pc away. The absolute magnitude of Sirius turns out to be +1.4. The sun, whose apparent brightness is so much greater than Sirius, turns out to have a much larger absolute magnitude of +4.8.

If we know the distance of the star from earth and its apparent brightness as seen here, then we can use this relation to compute its intrinsic brightness. This is important because it must be the intrinsic brightness that is controlled by physics of the star, and not the apparent brightness.

At the time of Leavitt’s discovery, the distance to the SMC was not known. In 1917, Harlow Shapley was able to use a statistical method of stellar motions to estimate this distance, as well as the distances to some globular clusters containing Cepheids. The distance to each individual star in a galaxy or a globular cluster is di erent for di erent stars, but since a galaxy or a globular cluster is very far away from us, the di erence in distances between individual stars within a galaxy or a globular cluster is negligible in comparison, so we may treat them all to be at some average distance to the galaxy or globular cluster.

This distance measurement allowed Harlow Shapley to convert the apparent brightness of these stars to their intrinsic brightness. Like Leavitt, he plotted the period of these Cepheids against their brightness, but this time he used the intrinsic brightness. In this way he found that the period of variation of a Cepheid variable bears a definite relationship with its intrinsic brightness, whether the star is in one galaxy or another globular cluster. In other words, he found this to be a universal physical law independent of the location of the star. This celebrated law is called the period–luminosity relation for the Cepheid variables.

Once established, this period–luminosity relation can be used as a yardstick to measure the distance to a new Cepheid variable. For that reason Cepheid variables can be regarded as a standard candle. What one has to do is to measure its period

as well as its apparent brightness on earth. From the period one gets its intrinsic brightness through this relation. Knowing now both the intrinsic and the apparent brightness, one can calculate its distance R.

In 1923, Hubble found twelve Cepheid variables in a nearby nebula in the Andromeda constellation. This nebula is actually a galaxy and is now called the Andromeda galaxy, or by its Messier catalog number M31. He also found twentytwo other Cepheid variables in another nebula, M33. In this way, he determined both of their distances to be several hundreds of kiloparsecs away, placing them well outside of the confines of our Milky Way galaxy, and settling the debate once and for all in favor of these objects being distant galaxies in their own right, rather than some luminous clouds within our own galaxy.

It also allowed him to establish Hubble’s law.

The distance to nearby galaxies can be determined by the brightness of their Cepheid variables. The farthest Cepheid variable observed to date was through the Hubble Space Telescope, in 1994, in a galaxy called M100 in the Virgo cluster of galaxies. Its distance is determined to be 56 ± 6 million light-years.

In recent years, a much brighter standard candle has been found to measure greater distances. This is the type Ia supernova. As discussed in Chapter 8, a supernova is an exploding star. It comes in two types. Type II results from the collapse of a single massive star at the end of its life, while Type Ia is always associated with a binary star system. One of the two stars in such a binary system is a white dwarf (see Chapter 8), and the other is a live star which keeps on dumping hydrogen from its atmosphere onto the nearby white dwarf. When the accumulated weight becomes too large for the white dwarf to support, a collapse and a subsequent explosion takes place. Since the two kinds of explosions have di erent origins and the stars involved have di erent histories, the spectral lines seen in their explosions also have di erent chemical compositions. For Type II supernova, the outermost atmosphere consists of hydrogen, so hydrogen lines are usually seen. For Type Ia, hydrogen dumped on the surface of the white dwarf collapses and fuses, so no hydrogen lines are present. The light curves of the two, as the supernova brightens and then dims, also have di erent characteristics. It is through such di erences that one can distinguish a Type Ia supernova from a Type II supernova.

After the explosion, it takes less than a month for a Type Ia supernova to reach its peak brightness. At its peak, it can be several billion times brighter than our sun, though being so much further away than our sun, of course we do not see it on earth as bright as the sun. The time it takes the brightness to decay away is found empirically to depend on the peak brightness: the brighter it is the longer it takes to decay. This is then very much like the Cepheid variables, but instead of using the period of brightness variation to determine the intrinsic brightness we can now use the decay time of a Type Ia supernova to determine its intrinsic brightness, once the intrinsic brightness of some of them is determined by other means. So like the Cepheid variable, it can be used as a standard candle to measure the distance.

Since the supernova is so much brighter than a star, we can use it to reach a much greater distance.

With this method one can reach galaxies much farther than those seen by Hubble. The farthest Type Ia supernova to date has a red-shift factor of z = 1.7. The result of such deep sky measurements will be discussed in a later section.

10.1.3Summary

The velocity of a galaxy can be measured by the amount of red-shift in its spectral lines. Its distance can be measured using either Cepheid variables or Type Ia supernova as standard candles, or some other means not discussed here.

By measuring the velocity and the distance of relatively nearby galaxies, Edwin Hubble established the most important law in modern cosmology, that galaxies are receding from us with velocities proportional to their distances. The latest value of Hubble’s constant is H0 = 72 km/s/Mpc.

10.2The Big Bang

Since galaxies are flying away from us, they must have been closer a million years ago, and closer still a billion years ago. Running this backwards, at some time past all galaxies must come together to a single point. That point can be taken to be the beginning of the universe. The ‘explosion’ that sets every galaxy flying outward has come to be known as the Big Bang.

We can estimate how long ago that happened, if we assume the velocity v of every galaxy to be constant in time. If t0 is the present time, then the present distance of the galaxy from us is s = vt0. Comparing this with Hubble’s law, s = vH0−1, we arrive at the conclusion that the present age of the universe is t0 = H0−1. If we take H0 = 72 km/sec/Mpc, then H0−1 turns out to be about 14 billion years.

However, v cannot be a constant because there are gravitational pulls from the rest of the universe, and there is also pressure exerted by the other galaxies. Both cause a deceleration, making v today to be smaller than that in the past. Taking this into account, we shall see that the age of the universe is reduced to (2/3)H0−1. With the above value of H0, the age comes out to be about 9 billion years. This causes a serious problem, because we know of stars in globular clusters which are 11 or 12 billion years old, and it does not make sense to have a star to be older than the universe it lives in! This serious dilemma has been resolved by a recent observation which we will come back to in the next section.

For now, let us discuss some of the implications of Hubble’s law, and the Big Bang. Galaxies fly away from us in every direction. Does it mean that we are at the center of the universe? Well, yes, if it helps our ego to think so. However, we had better realize that everyone else in other galaxies can claim the same honor as well. This is because we know from observation that the large-scale structure of

our universe is homogeneous and isotropic. There can therefore be no distinction between a galaxy here and a galaxy in another part of the universe. To help us understand this point, we may imagine the universe to be a very large loaf of raisin bread being baked in the oven, with the galaxies being the raisins, and the expansion of the universe simulated by the rise of the bread. Whichever raisin you live on, all the other raisins are moving away from you isotropically as the bread rises, so every raisin can claim to be at the center of the universe.

The idea that the universe started from a Big Bang also brings about many other interesting questions. What caused the explosion, and what happened before the explosion? Are space and time already there when the universe exploded, or are they created together with the universe? Why is the universe created in a threedimensional space, and not in ten dimensions, for example? If it is ten dimensions, why are three of them big and the other seven so small to be unobservable? How does the universe evolve after creation, and how is it going to end up? There are many more questions of this kind that can be asked. We do not know most of the answers, but we know a few. Those will be discussed in the rest of this chapter.

10.2.1Summary

The universe as we know it began in an explosion, known as the Big Bang, some ten to twenty billion years ago. This picture of the universe brings up many interesting philosophical and physical problems, some of which will be discussed in the rest of the chapter.

10.3After the Big Bang

In this section we discuss how the size of the universe evolves under the influence of gravity and pressure. Derivations and quantitative details are relegated to the last subsection.

10.3.1Gravity and Pressure

The distribution of matter in the universe, averaged over a scale of hundreds of millions of light-years, looks the same everywhere. For that reason we may model the universe as a homogeneous fluid, with the galaxies1 playing the role of molecules of the fluid.

A homogeneous fluid at any time t is specified by an energy density2 ρ(t), and a pressure P (t). If r is the present distance of a galaxy (from us), then the uniform expansion of the universe is described by a common scale factor a(t), so that the

1Note that real galaxies are not formed until much later, but even in the early universe, it is convenient to refer to the chunks of matter that make up the fluid to be ‘galaxies.’

2According to the theory of relativity, the equivalent mass density is ρ(t)/c2.

3

2

1

Figure 10.1: The position of a galaxy (black dot) in an expanding universe at three successive times. Darker shading indicates a higher density.

distance of the galaxy at time t is s(t) = a(t)r. In particular, this fixes the normalization of a(t) to be a(t0) = 1 at the present time t0. The present distance r is known as the comoving distance.

Such a model of the big-bang universe is known as a Friedmann–Robertson– Walker (FRW) universe.

The black dots in Fig. 10.1 represent a galaxy at three di erent times, t1 < t2 < t3. As the universe expands, the scale factor a(t) gets bigger and the density ρ(t) becomes smaller. This is indicated in the diagram by the di erent size of the spheres, and the di erent gray-scale shadings. The black-dot galaxy on the surface of the sphere is surrounded on all sides by other galaxies. Only galaxies inside the sphere exert a net gravitational pull on the black-dot galaxy. This pull can be represented by a mass M placed at the center of the sphere, with M equal to the total mass of all the galaxies inside the sphere. The gravitational pull from galaxies outside the sphere all cancel one another. The isotropic expansion of the universe may now be thought to be the expansion of the spherical surfaces.

The expansion of the sphere is a ected by two factors: gravitational pull from the galaxies inside the sphere, and pressure exerted on the sphere by the galaxies outside to resist its expansion. Both tend to slow down the expansion of the sphere and the outward flight of the black-dot galaxy. The total e ect of these two is a force proportional to ρ(t) + 3P (t), as we shall demonstrate later. The first term is due to gravity, while the second term comes from pressure.

Right after the Big Bang, the universe is hot and the energy density is high. This causes the galaxies to have a highly relativistic random motion, thereby exerting a large pressure on the sphere and the surrounding galaxies. This pressure is related to the energy density by the equation P = ρ/3. Such an equation is known as

an equation of state. This period is known as the radiation-dominated era because

√

the galactic motions are relativistic, like light. In this epoch, a(t) grows like t and ρ(t)a3(t) falls like 1/a(t), as we shall show. The total energy inside a sphere of comoving radius r is proportional to ρa3. This energy falls because part of it is used up to do work to counter the pressure exerted by the galaxies as the sphere expands.

As the universe expands, its energy is being shared by a larger volume, so its temperature drops. Eventually, the universe will cool down su ciently so that