. 20
( 28)


and (e). “4
We can calculate what this redshifted black-
body radiation will look like. To do this, we note
log Bv(3,000)
that the photons will be preserved; their frequen-
cies will just change in a known way. For a black-
log Bv(1,500)
body at a temperature T, the energy density in
photons with frequencies between and d is
log Bv(750)
given by the Planck function,
U1 , T 2 d a3 ba bd
8h 1
eh kT
c 1 “9
To find the number of photons per unit vol-
ume in frequencies between and d , we take 10 11 12 13 14 15 16
the energy density and divide by the energy per
Log ν
photon, h . This gives
Fig 21.2. Spectrum of the radiation in the universe as it
n1 , T 2 d a ba bd
8 1
(21.2) expands.We plot log intensity vs. log frequency, for three
c3 eh kT
temperatures. 3000 K was the temperature when radiation
We assume that the radiation is emitted at and matter decoupled. 1500 K is when the scale factor
increased by a factor of 2 after decoupling. 750 K is
some epoch t, with scale factor R, and we detect it
another factor of 2 increase in the scale factor.
at the reference epoch t0. The observed wave-
length 0 is related to the emitted wavelength
by equation (20.29), as
Multiplying by h 0 gives us back the energy
0 density spectrum for radiation detected between
0 and 0 d 0:
c/ , the observed frequency
Since is related
to the emitted frequency by 3
U1 0, T 2d a ba bd
8h 1
0 0
3 h kRT
1R e 1

This looks just like a blackbody spectrum at tem-
This means that
perature T0 RT. Therefore, the radiation will
R (21.3)
still have a blackbody spectrum, but will appear
In addition, photons emitted in the frequency cooler by a factor of R. In Fig. 21.2, we see a few
range d will be observed in the frequency range examples of how the spectrum evolves at differ-
d 0, given by ent temperatures.
The redshift of the background radiation
d (21.4)
has an interesting consequence on the evolu-
The photons emitted between and d are tion of the universe. This is illustrated in Fig.
now observed between 0 and 0 d 0. Also, all 21.3. The energy density of the matter in the
universe is matc2, where mat is the density of
volumes increase by a factor of 1/R . Combining all
of these, we now have the number of observed pho- matter. We have already seen in Chapter 20 that
the density is proportional to 1/R3. The number
tons per unit volume with frequencies between 0
and 0 d 0 as density of photons in the universe is also pro-
portional to 1/R3. However, the redshift means
n1 0, T 2d R3 a ba b a h ba b
8 1
that the energy per photon is proportional to
c3 kRT
e 1

1/R, so the energy density of radiation in the
a ba bd
8 universe is proportional to 1/R4. This means that
3 h RkT
c e 1 the energy density of radiation (integrated over

21.1.2 Observations of the cosmic
background radiation
Alpher and Herman (in 1948) gave an equation to
Matter calculate the temperature of the background radi-
ation, and found a value of about 5 K. The actual
Energy Density

value that you predict depends on knowing cer-
tain parameters, such as the Hubble parameter.
These parameters were, obviously, not as well
determined in 1948 as they are now. If one puts in
the current values of the various parameters, the
predicted value of the temperature of the back-
ground radiation drops slightly, to 2.7 K. Even with
the slightly higher value, it seemed that this radi-
ation would be very hard to detect. Since most of
the radiation would be expected in the radio part
of the spectrum, Alpher and Herman talked to
radio astronomers, but the radio astronomers saw
Fig 21.3. Density of radiation and matter as the universe many difficulties. First, the signals would be very
ages. On the horizontal axis we plot the age of the universe. weak. Also, radio observations are easiest when
On the vertical axis we plot the energy density of matter in you can compare the radiation in one direction
blue, and radiation in red.
with that in another direction. Since the back-
ground radiation would be the same everywhere,
this basic technique could not be used. Also, at the
the whole spectrum) drops more quickly than
time, there was a general feeling that the steady-
the energy density of matter. In the early uni-
state theory is correct, so there was not much
verse, the energy density of radiation was greater
motivation to carry out this difficult experiment.
than that of matter. We say that the universe
Two physicists at the Bell Telephone Labora-
was radiation-dominated at that time. Now the
tories, Arno Penzias and Robert Wilson (Fig. 21.4), acci-
opposite is true; we live in a matter-dominated
dentally detected this radiation and reported their
universe. The time when the radiation drops
observations in 1965. Penzias and Wilson were
below the matter energy density is called the
unaware of the work of Alpher and Herman. They
crossover time.
were using a very accurate radio telescope for both
To understand the distribution of this radia-
communications and radio astronomy. To have
tion on the sky, we go back to our expanding
accurately calibrated results, they had to under-
surface analogy (Fig. 21.1). At the instant before
stand all sources of noise (interference) in their sys-
the electrons and protons recombine, making
tem. They found an unaccounted“for source of
the universe transparent, matter at all points is
noise at a very low level. The noise seemed to be
emitting photons in all directions. Since the
coming either from their system, or from every-
matter becomes transparent, the photons con-
where in the sky. After carefully analyzing their
tinue running around in all directions on the
system (including disassembling and reassembling
surface of our expanding sphere. We see a
certain parts, and even cleaning out bird drop-
steady stream of photons, not a brief flash as
pings), they were confident that the noise was
might be expected if we were looking at a local-
coming from everywhere in the sky. They only had
ized explosion. The fact that the radiation is
a measurement at one wavelength, so they could
moving in all directions means that we see it
not confirm the shape of the spectrum. However,
coming from all directions. The radiation should
they found that the intensity corresponds to a
appear isotropic. Also, cosmic background pho-
blackbody at a temperature of about 3 K.
tons reaching us today are coming from one
At the same time, a group at Princeton, unaware
light day farther away than those that reached
of the work of Alpher and Herman, rederived their
us yesterday.



log(Iν [MJy/sr])

FIRAS data with 50σ errorbars
’1 Ground-Based Data
2.725 K Blackbody
’2 ’1 0 1
log([1 cm]/»)

Fig 21.5. Spectrum of the cosmic background radiation as
taken from ground-based observations. Most are at centime-
ter wavelengths where the Earth™s atmosphere is very trans-
parent. Measurements are the black data points with error
bars.The smooth blue curve through those points is the
best ¬t to the data.The red curve represents the COBE
satellite (see below).The horizontal axis is essentially
frequency plotted as log(1/ ), where is in centimeters.
[© Edward L.Wright, used with permission]

However, until it was shown unambiguously
that the spectrum is that of a blackbody, there
were still some ways for steady-staters to produce
something like the background radiation. As Fig.
Fig 21.4. Arno A. Penzias and Robert W.Wilson, who dis-
21.5 shows, the discovery and subsequent confir-
covered the cosmic background radiation in 1967, in front of
mations of the background radiation were on the
the telescope they used, at Bell Laboratories in New Jersey.
[Reprinted with permission from Lucent Technologies™ long wavelength side of the peak of a 3 K black-
Bell Labs] body spectrum (which is at about 1 mm). The most
convincing observations would be to show that the
prediction, and set out to make the measurement. spectrum does, indeed, turn over at wavelengths
Someone familiar with both the Bell Laboratories shorter than 1 mm.
and Princeton work put those two groups However, even at wavelengths shorter than
together. It was clear that Penzias and Wilson had 1 cm, the Earth™s atmosphere becomes sufficiently
found the cosmic background radiation. That opaque that ground-based radio observations of
confirmation came when the Princeton group sufficient precision are virtually impossible. More
carried out their observations at a different wave- recently, observations from space have helped clar-
length, and found that the radiation is present ify the situation. Before we discuss those, however,
and that the spectrum is consistent with that of an interesting experiment performed in the late
a blackbody. For their painstaking work in detect- 1960s (and still being improved upon) is worth
ing this important signal, Penzias and Wilson some mention. This experiment involves optical
were awarded the Nobel Prize for Physics in 1977. observations of interstellar cyanogen (CN), which
The cosmic background radiation is of such was first discovered in 1939.
significance in cosmology that its discovery led to The basic idea behind the CN experiment is
extensive efforts to measure its properties as accu- shown in Fig. 21.6. The optical absorption lines are
rately as possible. An immediate consequence of observed when the CN makes an electronic transi-
its discovery was the death, for all practical pur- tion from its ground state to the first excited state.
poses, of steady-state theories of the universe. In Chapter 14 we saw that, for any electronic state,

Low Temp High Temp

Starlight Passing
Through Cloud 2 1 2
1 1





Interstellar Cloud with
CN Molecules
Wavelength Wavelength
(a) Low Temp High Temp
Fig 21.6. Using interstellar CN to study the cosmic background radiation. (a) We see what observations are done.The cloud
with the CN is in between the Earth and a bright star.We take a spectrum of the star and see absorption lines from the CN.
(b) We see how the strength of the absorption lines changes as we increase the temperature of the background radiation. In the
upper left we see the energy levels. Electrons jumping from one level to another give us the big change in energy. Much smaller
changes occur when the rate of rotation of the molecule changes. (These rotational energy shifts are exaggerated in the dia-
gram.) Transitions 1 and 2 both correspond to the same jump by the electron, but they occur with the molecule in slightly dif-
ferent rotational states. In the upper right, we see the populations of the levels at low and high temperatures.The thicker the
black line, the more populated the level.The red arrows indicate how strong the absorption is.The thicker the line, the stronger
the absorption. In the lower panels, the resulting low and high temperature spectra are shown.

where g1 and g2 are the statistical weights of the
a molecule can be in many different rotational
two rotational levels, and E21 is the energy dif-
states. Normally, interstellar molecules are stud-
ied by radio observations of rotational transitions ference between the two rotational levels. The
within the same electronic state. The effect of the energy difference between the two levels in CN
rotational energy levels is to split the optical elec- corresponds to the wavelength of 2.64 mm. If
tronic transition into several lines. For example, the CN is in a cloud of low density, there will be
there are lines corresponding to a transition from very few collisions with hydrogen to get mole-
a given rotational state of the ground electronic cules from the ground state to the next rota-
state to the same rotational state in the excited tional state. (Collisions with electrons may also
electronic state. The wavelengths of the various be important.) If there are no collisions, the CN
transitions are slightly different. In practice, two can only get to the higher rotational state by
transitions are observed, one from the ground absorbing radiation. If the CN is far from any
rotational state and the other from the first excited star, the only radiation available is from the cos-
rotational state. mic background. Under these circumstances,
The relative strength of the two optical lines is the populations will adjust themselves so the
equal to the ratio of the populations in the two temperature in equation (21.6) is the tempera-
rotational states (in the ground electronic state), ture of the cosmic background radiation at
n2/n1. This is given by the Boltzmann equation as 2.64 mm. (Of course, if the cosmic background
radiation has a blackbody spectrum, its temper-
n2 ature at 2.64 mm should be the same as at any
E21 kT
e (21.6)
other wavelength.)
n1 g1

So, the CN sits in space, sampling the cosmic
background radiation at a wavelength of 2.64 mm.
This is a wavelength at which we could not
directly measure the background temperature
from the ground. The CN then modifies the light
passing through the cloud. The relative intensity
of these optical signals contains the information
that the CN has collected on the cosmic back-
ground radiation. The optical signals easily pene-
trate the Earth™s atmosphere. All we have to do is
detect them and decode them. To decode them,
we take the relative intensities as giving us n2/n1.
We then solve equation (21.6) for T. As a farther Fig 21.7. Diagram showing COBE satellite. [NASA
Goddard Space Flight Center and the COBE Science
refinement, we can check to see if there is any
Working Group]
excitation of the CN due to collisions by electrons
or atoms or molecules in the cloud. If there were,
then the T that we measure would be greater this. This shows us that, though many important
than the background temperature. But, if this discoveries in astronomy are made unexpectedly,
were happening, there would be a weak emission simple luck is not enough. Confronted with the
line at 2.64 mm. By looking for such a line and unexpected, the observer still must be able to rec-
not finding it, we can rule out significant colli- ognize that something important is happening,
sional excitation of the CN. and then be able to follow up the results.
The most careful application of these tech- The cosmic background radiation was consid-
niques gives a temperature of the cosmic back- ered so important that NASA decided to devote a
ground radiation at 2.64 mm of 2.80 K, with an whole satellite to its study. Thus, the Cosmic
uncertainty of about 1%. This agrees well with Background Explorer Satellite, or COBE, was launched
the best direct longer wavelength measurements, in 1989. A diagram of the satellite is shown in Fig.
indicating that, as we approach the peak, the 21.7. It carried instruments for making direct
radiation keeps its blackbody nature. It is possi- measurements of the radiation at wavelengths
ble to study the next rotational transition up at that were blocked by the atmosphere. The leader
1.32 mm, giving the background temperature of the COBE team was John Mather of NASA™s
even closer to the peak. The lower level of the sec- Goddard Spaceflight Center.
ond rotational transition does not have much The COBE result for the spectrum is shown in
population, so this is a very difficult experiment. Fig. 21.8. Measurements at many wavelengths are
However, the results are consistent with the shown. Notice that the error bars are quite small.
blackbody nature continuing to the peak. There For comparison, we see the differences between
is another use to the CN technique. If we can observations and a 2.725 K blackbody. The agree-
observe CN in distant objects, we will be seeing it ment is spectacular. First, it is clear that the spec-
as it was in the past, and so the cosmic back- trum is truly that of a blackbody. The temperature
ground temperature will be higher, and the CN of the background radiation has now been deter-
excitation temperature will be higher. This gives mined to better than 1%. There can be little doubt
us a check on our theories of how the tempera- that this is the radiation predicted in 1948 by
ture of the universe varies with z. Alpher and Herman.
There is an interesting aside to this story. When
21.1.3 Isotropy of the cosmic
interstellar CN was discovered in the late 1930s, the
background radiation
people who observed it also observed both optical
spectral lines. They noted that the relative amounts Earlier in this chapter, we said that the cosmic
of absorption corresponded to a temperature of background radiation should appear the same no
about 3 K, but did not attach any significance to matter which way you look. That is, the radiation

Cosmic Microwave Background Spectrum
Intensity (10’18 ergs s’1 cm’2 Hz’1 sr’1)

3000 Spectrum Measured


Residual Relative to 2.725 K Blackbody

10 15 20
Frequency (cm’1)
Fig 21.8. Diagram showing COBE spectrum of the back-
ground radiation with best ¬t to 2.725 K.The errors are so
Fig 21.9. Origin of the dipole anisotropy of the cosmic
small that they do not show up on the data.The upper part
background radiation.We see an observer in the center,
of the diagram has the best ¬t spectrum, and the lower part
moving to the right.This produces a Doppler shift to the
has the error bars on an expanded scale. [NASA Goddard
blue on the right, to the red on the left.
Space Flight Center and the COBE Science Working Group]

anisotropy is characterized by a hot pole and a
cold pole, with smooth variations in between, we
should appear isotropic. The first observations
call it the dipole anisotropy. We should point out
found the radiation to be isotropic, but did not
that such an anisotropy doesn™t violate special
test this to a high degree of accuracy. More sensi-
relativity by providing a preferred reference
tive observations have looked for small deviations
frame for the universe. We just happen to be
from perfect isotropy, called anisotropies. In describ-
measuring our velocity with respect to the matter
ing any anisotropies in the background radiation,
that last scattered the background radiation. In
there are two relevant quantities: (1) how strong
fact, if we accurately determined Hubble™s con-
is the deviation, and (2) over what angular scale
stant separately in different directions, we should
on the sky is the deviation seen?
measure the same anisotropy. (In fact, we do, in
One local source of anisotropy is the com-
that this shows up in redshift surveys, primarily
bined motion of the Earth, Sun, galaxy, local
as our motion towards the great attractor, dis-
group and local supercluster with respect to the
cussed in Chapter 18.)
general Hubble flow, as indicated schematically
As Fig. 21.10 shows, this dipole anisotropy has
in Fig. 21.9. This motion produces a Doppler shift,
been observed by COBE. When we correct for the
which varies with direction. If our net motion is
motions of the Earth and Sun within the galaxy,
with a speed v, in a particular direction, then the
we find a Doppler shift corresponding to a
radial velocity away from the direction we are
motion of 600 km/s towards the great attractor.
heading is
There have also been searches for anisotropies
vr v cos (21.7)
on smaller angular scales. The goal of these stud-
ies is to learn more about the structure of the uni-
In the direction in which we are heading (
verse when the temperature was about 3000 K.
0 ), we get a maximum blueshift, and the radia-
After all, the background radiation retains an
tion appears slightly hotter. In the opposite direc-
almost perfect record of that era. In looking for
tion ( 180 ), we get a maximum redshift, and
these anisotropies, we recognize that the universe
the radiation appears slightly cooler. Since this

Searches for these fluctuations have been car-
ried out almost since the background radiation
was discovered. However, the fact that they are so
weak has made them quite elusive. The situation
was resolved by COBE. In addition to measuring
the spectrum of the radiation, COBE was also able
to measure the distribution of radiation on the
sky. The result of that map is shown in Fig. 21.11.
In it, red regions are cooler than average, and blue
regions are hotter than average. You can see that
Fig 21.10. COBE measurements of the dipole anisotropy
there is a mottled appearance, with structure all
in the cosmic background radiation.This plot shows the
over the map. However, the signals are so weak,
whole sky on a single plot, with the galactic plane going hori-
and the analysis is so difficult, that each bright
zontally across the middle. Bright regions are slightly hotter
spot you see does not exactly correspond to one
than the average temperature; dark regions are slightly
cooler than the average temperature. [NASA Goddard Space bright area in the early universe. Each spot still
Flight Center and the COBE Science Working Group] has some important experimental uncertainties.
However, the experimenters can use a statistical
analysis of all of the bright spots, and tell us about
is not perfectly homogeneous. We are here, and
their average properties. When this result was
our galaxy, cluster and supercluster are here. We
announced, cosmologists and the general public
can also see many other galaxies and clusters. For
were quite excited. These fluctuations are so
these objects to exist, there must have been very
small that they are even a challenge to a satellite
small seeds out of which they grew. That is, there
designed for the purpose. So, while the spectrum
must have been small concentrations of material
of the background radiation (like that in Fig. 21.8)
around which more material could be attracted
was measured with great accuracy very early in
gravitationally. These small concentrations would
the mission, the world had to wait another year to
also have been slightly hotter than their surround-
hear about the fluctuations. Of course, COBE can
ings. This means that the radiation from these
only sample on relatively large angular scales.
regions should appear slightly stronger than that
In order to study the anisotropies on a smaller
from the surroundings. Theoreticians have sug-
scale, a balloon mission was launched over
gested that these fluctuations in brightness will
be very small, only about 10 5 of the strength of Antarctica in 1998. The project was known as
the radiation itself. Balloon Observations of Millimetric Extragalactic

Fig 21.11. COBE measurements
of the small-scale anisotropies in
the cosmic background radiation.
The plot shows the whole sky
viewed as two hemispheres. In this
false color image, red areas are
slightly cooler than the average and
blue areas are slightly hotter.
Notice that the largest ¬‚uctuations
are 100 K, out of a total 2.7 K
signal. [NASA Goddard Space Flight
Center and the COBE Science
Working Group]
North Galactic Hemisphere South Galactic Hemisphere

’100 µK +100 µK

Radiation and Geophysics, BOOMERANG. The relative amplitude of fluctuations on different
results are shown in Fig. 21.12. In Fig. 21.12(a), we angular scales. Extrapolating these fluctuations
show the map of roughly 3% of the sky that was back to the time of decoupling, this tells us about
observed. In Fig. 21.12(b), we look at how theoret- the seeds of various scales of structure that we
ical simulations show how the fluctuations see, as described in Chapter 18. In Fig. 21.12(d), we
should look for different geometries of the uni- see how the BOOMERANG data fit into various
verse (closed, flat, open). A more detailed analysis other constraints on various cosmological param-
of this type provides very strong evidence that eters. Remember, flat universes are ones for which
the universe is flat. In Fig. 21.12(c), we look at the 1.

’300 µK 300 µK 6
’300 ’200 ’100 0 100 200 300

Relative amplitude


7 1 1/2 1/3

Angular scale (degrees)

Fig 21.12. Balloon (BOOMERANG) measurements of smaller-scale anisotropies. (a) This image is almost 1800 square
degrees, with the false color the same as in Fig. 21.11. (b) Models of ¬‚uctuations for different geometries of the universe.
(c) Relative intensities of the ¬‚uctuations on various scales. (d) We see how the BOOMERANG data ¬t into various other
constraints on various cosmological parameters. [The BOOMERANG Collaboration]

There is one remaining interesting problem Fig 21.14. Relationship
with the isotropy of the background radiation. between linear and angular
It is actually too isotropic. For the background to scales that could have communi-
be isotropic, the early universe, apart from the cated in the lifetime of the
fluctuations that would become galaxies, must
have been quite uniform. We even incorporate
this uniformity into our cosmological models.
However, as illustrated in Fig. 21.13, conditions
can only be identical at different locations if


Light Signal
(a) (b)
they have some way of communicating with
each other. Something must have regularized
the structure in the early universe. However,
such communication can travel no faster than
the speed of light. Two objects separated by a
distance greater than that which light can travel
in the age of the universe cannot affect, or cause
events to happen at, each other. This is called
the causality problem.
Region Reached by
We can use Fig. 21.14 to show the scales on
Signal Before Decoupling
which this is a problem. We must relate the cur-
rent angular separation between two points with
Fig 21.13. Causality problems in the early universe. In this their linear separation at the time of decoupling.
sequence, we look at the section of the universe that can We now let the two points be separated by an
be affected by light signals starting at some point just after
angle 0 (where 0 V 1 rad). If the points are a dis-
the expansion started, traveling at the speed of light until
tance d from us, their current separation is
decoupling. (a) The light signals start at the center of the grid.
(b) As the universe expands (indicated by the larger grid), x0 0d
the light signals spread out in all directions.The green
If we are just seeing the light from these
shaded area shows the region through which these signals
points, it must have been traveling for t0, the cur-
have passed. (c) The expansion continues and the light signals
rent age of the universe. (We ignore the small dif-
spread farther. Eventually decoupling is reached.The green
shaded area shows that region containing material that could ference between the current age of the universe
have absorbed the radiation before decoupling.This is the and the time since decoupling, since decoupling
only part of the universe that could have been affected
occurred very early in the history of the universe.)
before decoupling by the conditions at the center of the grid
This means that
at the beginning. (d) After decoupling, no more material can
be affected by the spreading light signals.The region that has d ct0
been affected (shaded in purple) now simply expands with
the universe.This is indicated by the fact that the shaded
area in (d) covers the same number of grid sections as in (c).
x0 0ct0

If we take our reference epoch to be now, so There is another source of small-scale anisotropy
that R(t0) 1, and we let R be the scale factor at in the background radiation that is not actually
the time of decoupling, then the separation cosmological. It has to do with the interaction of
between the two points at the time of decoupling the radiation with clusters of galaxies as it passes
is R times their current separation, or
Before After
x 0ct0R

The time for light to travel the distance x is x/c,

γ e e

For these two points to communicate, t(x)
must be less than or equal to t, the age at decou-
pling. The farthest the two points can be is when
t(x) t, so

t 0t0R

We now solve for 0, the maximum current
angular separation between the two points that
could have been causally connected before
t Rt0 (21.8)

Example 21.1 Causality in the early universe
If the age of the universe at decoupling was 105 yr,
and the current age is 1.5 1010 yr, find the maxi-
mum current angular separation between two
CBR Photons

points that can be causally connected.

To use equation (21.8), we have to find R, the scale
factor at the time of decoupling. We know that the
temperature of the background radiation at any
time t, T(t), is proportional to 1/R(t). The back-
ground temperature is 3 K now and was 3000 K at
decoupling, so R at decoupling must have been
10 3. This means that
1105 yr2 Cluster of Galaxies
110 2 11.5 10 yr 2
0 3 10
Fig 21.15. (a) Compton and inverse Compton scattering.
7 10 rad
In Compton scattering, a gamma-ray strikes a low energy
electron and becomes a lower frequency photon, with the
This is approximately 0.4 .
excess energy going to the electron. In inverse Compton
The above example tells us that points that scattering a low energy photon scatters off a high energy
appear more than half a degree apart on the sky electron, with the photon gaining energy and the electron
cannot be causally connected. The background losing it. (b) The Sunyaev“Zeldovich effect. As low energy
radiation should not appear smooth on scales cosmic background photons strike the hot gas within a clus-
ter of galaxies, inverse Compton scattering takes some of
larger than half a degree. However, it clearly does
the photons from the low energy side of the 3 K blackbody
appear smooth on large scales. We will come back
peak and transfers them to the high energy side.
to this question in Section 21.3.

through. The galaxies in a cluster don™t present a T[K] 109
very large target, so most of the interaction is H
with the hot, low density intracluster gas. This gas 4He

log(mass fraction)
is hot enough to be almost completely ionized.
The background radiation interacts with the elec- 5 3He
trons in the gas. 3H

In Fig. 21.15(a) we look at what happens when n
photons scatter off electrons, a process called
10 7Li
Compton scattering. (A. H. Compton worked out the
theory of the process and studied it in the labo-
100 101 102 103 104
ratory as a demonstration of the photon nature of
t [sec]
light.) Generally, in Compton scattering, the pho-
ton initially has more energy than the electron, Fig 21.16. The formation of light elements in the early
so some energy is transferred from the photon to universe. On the horizontal axis, we show time. On the
the electron. Since the photon loses energy, the vertical axis we show the abundance of that isotope as
wavelength increases. However, it is also possible time goes on. Each isotope is represented by a line of a
different color, as marked. [© Edward L. Wright, used with
for the electron to start out with more energy
than the photon. In this case, called inverse
Compton scattering, energy is transferred from the
electron to the photon. The wavelength of the
formed. The 4He is very stable, and, in those three
photon decreases.
The electrons in the intracluster gas have high minutes, 25% of the mass of the universe was tied
up in 4He. The 3He makes up only 10 4 to 10 5 of
energies, and the photons for the background
radiation have very low energies. When they inter- the total mass. There is no stable nucleus with
A 5. This served as an effective barrier to the
act, we have inverse Compton scattering. The pho-
tons get a large boost in energy, going in as radio copious formation of still heavier elements.
wavelength photons and coming out at shorter We now look in a little more detail at what
determined the abundance of 4He. To do this we
wavelengths. As shown in Fig. 21.15(b), this means
that if we look in the directions of clusters of galax- must look at the conditions that existed when
the temperature was 109 K. This was probably at
ies at radio wavelengths, some of the background
radiation will have been removed. The background 200 to 300 s. We need to know the density
radiation should appear slightly weaker in the then, and the relative number of neutrons and
directions of clusters of galaxies, since some of the protons. Of course this depends on what hap-
radio photons have been shifted to higher energy. pened before this time, when temperatures were
This is called the Sunyaev“Zeldovich effect, after the even higher. We will see later in this chapter how
two Russian astrophysicists who proposed it. This we describe the physics in the first fraction of a
effect is very hard to detect, but it has now been second, but for He production, we only note that
1010 K (before 1 s) there was a thermal
for T
equilibrium between radiation and matter. At
these temperatures, the energy density of radia-
21.2 Big-bang nucleosynthesis tion is much greater than that of matter (as dis-
cussed in the previous section). So the evolution
of the scale factor, R(t), is described by general rel-
In the first three minutes of its existence, the uni-
verse was hot enough for nuclear reactions to ativity, where the energy density of radiation is
the dominant energy. The evolution of R also
take place. Fig. 21.16 shows how the abundances
of various light elements changed during this depends on the number of different types of light
particles, NL, such as neutrinos. These light parti-
time. Protons and neutrons combined to form
deuterium (2H), and two isotopes of helium (3He cles are important, because they move at or near
and 4He). The deuterium is very reactive, and the speed of light, so their motions dominate the
most of it was used up as quickly as it could be equation of state for this relativistic gas. In a

modification of the original Alpher and Herman ber of photons is not affected by the redshift,
only their energy.) The ratio is about 10 10. We
paper, a few years later Alpher, Follin and
Herman found that the best fit for this number will talk more below about the significance of
is three. (Alpher and Herman treated the neu- this ratio.
tron abundance as a free parameter in their In terms of these various factors, it is conven-
model, whereas Alpher, Follin and Herman cal- ient to write these dependencies as
culated it by following the nuclear physics
1010 nB
10.2302 10.0011 2ln a b
through the first seconds.) We will see below Y
that these correspond to three types of neutri-
10.013 2 1NL 32
nos. At 1010 K, that equilibrium breaks down as
10.014 2 1t1 2
the neutrons are frozen in at about 0.1 the num-
10.6 min2
ber of protons. So we then need to follow nuclear
reactions through 109 K. The final mixture of elements produced in the
big bang depends on the density of material at t
The reactions that affected neutron abundance
1 s, when the reactions started. We can relate the
density at one second, (1 s), to the density now,
ndp e
0, by knowing R(1 s). Fig. 21.17 shows the results
of theoretical calculations of nucleosynthesis for
e ndp

ndp e

The first reaction, with the right arrow is just the
decay of a free neutron, which has a half-life of
10.6 min. The rates of all of these reactions can be 10
expressed in terms of that half-life.
Number Abundance Relative to Hydrogen

Because deuterium is so reactive, most of the 10
neutrons that go into making deuterium will even- ’3
tually end up in 4He. The important reactions are:
2 2
H d 3H
H p
2 2
H d 3He ’5
H n
S 10
3 2
H d 4He
H n
S ’6 D
3 2
H d 4He
He p
’7 Li-7
If that is the case, then the mass fraction in
He is

2nn ’9
np n n
2a b a1 b
nn nn
0.001 0.01 0.1 1
np np 2
The final abundance should also depend on Fig 21.17. How the abundances of certain light elements
nB, the number density of baryons (particles depend on the density of the universe. On the horizontal
such as neutrons and protons that participate in axis, we plot the density of baryons (particles which can
nuclear reactions) and n , the number density of undergo nuclear reactions), expressed as the baryon density
photons. relative to the closure density. On the vertical axis, we plot
the abundances of the various isotopes.The shaded area
Even though each of these densities decreases
represents the best estimate of density from the abundance
as the universe expands, they each go as 1/R3, so
data. [© Edward L.Wright, used with permission]
their ratio doesn™t change. (Remember the num-

models with different densities. These densities deuterium. This means that the current D/H is
are expressed as the current density that would less than the primordial value. On Earth, the D/H
abundance ratio is about 10 4. This means that
correspond to that early density. The stability of
He makes its abundance relatively insensitive to for every deuterium nucleus there are 10 000
the density. However, the deuterium abundance hydrogen nuclei. From Fig. 21.17, we see that this
corresponds to a current density of 3 10 31 g/cm3,
drops off sharply with increasing density. This is
because the deuterium is so reactive that a higher which is much less than crit. Of course, the abun-
density provides it with more opportunities to dances in the Solar System may not be typical of
react and be destroyed. There is a smaller density the rest of the galaxy.
dependence in the 3He. The next step is to study interstellar D, in
The heavier elements (especially 7Li) become whatever forms it is found. Direct observations of
more important at higher densities. This is atomic D can be done in the Lyman line (and
because the high densities provide more reac- other Lyman series lines) in the ultraviolet. These
tions capable of building up the heavier ele- lines are observed in absorption against stars.
ments. For densities greater than crit, C, N and O Since interstellar extinction is so high in the
become somewhat abundant, but are still many ultraviolet, we cannot study clouds that are very
orders of magnitude below their current far away. The results in our general area of the
observed abundances. This means that their cur- galaxy are similar to the Solar System value.
rent abundances of these heavier elements must Optical lines in the lines of sight to certain
be explained by production in stars. quasars give a value of about one D nucleus for
The density dependence of the relative abun- every 40 000 H nuclei. This gives us an idea of val-
dances of various nuclei provides us with a way of ues outside our galaxy.
determining the density of the universe. We can Another possibility is the observation of the
measure the relative abundances of certain equivalent of the 21 cm line from atomic D. For
nuclei, such as D/H. We then use a diagram, like deuterium, this line is shifted from 21 cm to
Fig. 21.17, to find the current density to which 90 cm. Unfortunately, at this wavelength syn-
that ratio corresponds. In measuring the current chrotron radiation from the galaxy, whose inten-
sity goes up approximately as 2, provides strong
abundances of any nucleus, it is important to
account for any modification that has taken place interference.
since the big bang. Most important, the effect of A final possibility is the radio observation of
nuclear processing of D in stars must be taken molecules containing deuterium, such as DCN.
into account. The abundances that we see now are When we measure the abundances of these mol-
the result of big-bang and stellar nucleosynthesis. ecules, we find that the D abundance seems
When we want to talk about the results of big- quite high. However, we now know that this is a
bang nucleosynthesis alone, we speak of primor- result of chemical reactions (especially ion“
dial abundances. molecule reactions) involving D and H proceed-
It is important to note that this type of study ing at different rates for certain molecules. Thus,
only gives us the density of nuclear matter (pro- the abundances of the molecules don™t reflect
tons and neutrons) in the universe. That is the true D/H abundance ratio. The observation of
because it is only the nuclear matter that could molecules with D substituted for H tells us more
have participated in nuclear reactions. For exam- about interstellar chemistry than it does about
ple, if the dark matter is in the form of neutrinos cosmology.
(which only interact via the weak interaction), The general conclusion of all of these D/H
then it may be that the nuclear matter is insuffi- experiments is that the density of nuclear matter
cient to close the universe, but the dark matter is is probably about 5% of the critical density. If this
still sufficient. result is correct, it doesn™t mean that the uni-
Most of the studies relating abundances to the verse is open. It just means that nuclear matter is
density of the universe have involved deuterium. not sufficient to close it. There is an additional
note concerning 3He. It also has an abundance
The net effect of stellar processing is to destroy

that depends on density, though not as strongly explained by saying that the elements are not fun-
as the D abundance does. If an atom of 3He has damental, but are made up of even smaller struc-
one electron removed, a single electron remains. tures “ nuclei and electrons. We have seen that
This ion behaves somewhat like a hydrogen atom. the nuclei are made up of protons and neutrons.
It has a transition analogous to the 21 cm line, at
21.3.1 Fundamental particles
a wavelength of 9 cm. This line has been detected
in galactic HII regions, but more detailed analy- Since the 1950s, physicists studying elementary
ses are needed before these observations can particles have been able to use accelerators, like
yield cosmologically significant information. that shown in Fig. 21.18, to bring particles
In this section, we have discussed only what together at high energy. (Before the advent of
happened between one second and three min- accelerators, naturally produced cosmic rays
utes. Earlier than one second the universe was so were used.) When the energy of the collisions is
hot that the internal structure of the neutrons just right, particles can be created out of the
and protons is important. Before we can under- excess energy. As accelerators with more energy
stand what happened when the universe was less became available, increasing numbers of funda-
than one second old, we must look at some impor- mental particles were found. Physicists began to
tant features of elementary particle physics. suspect that these particles were no more funda-
mental than the 92 elements are.
To follow these results, we divide the various
21.3 Fundamental particles particles into groups, according to their ability to
and forces interact via the strong nuclear force. This is
shown in Table 21.1. Particles that do not interact
by the strong force, but still obey the Pauli exclu-
A major goal of physics over the centuries has
sion principle, are called leptons. The most famil-
been the search for the most fundamental build-
iar lepton is the electron. The strongly interacting
ing blocks of matter. We have progressed from
particles are called hadrons. We further divide the
earth, air, fire and water, through the atoms of
hadrons into the more massive baryons, and the
Democritus, to Mendeleev™s realization that the
less massive mesons. The most familiar baryons
elements showed a regularity that was eventually

Fig 21.18. Aerial view of
Fermilab, the large particle
accelerator outside Chicago.The
ring has a diameter of over 6 km.
[Fermilab photo]

Classification of elementary
Table 21.1. Quark properties.
Table 21.2.
Name Symbol Charge Mass (GeV)
Hadrons Leptons
Up u (2/3)e 0.35
Mesons Baryons
Down d (1/3)e 0.35
Strange s (1/3)e 0.55
(pion) n (neutron) e, e (electron,
Charm c (2/3)e 1.8
Top t (2/3)e 4.5
( others) p (proton) , (muon,
Bottom b (1/3)e 40.0
( others) , (tau, neutrino)
and a neutron is udd. A meson is combination of
a quark and an antiquark (not necessarily as the
are the proton and neutron, and the most famil-
same type as the quark.) For example, a positive
iar meson is the pion.
pion is ud, where the bar represents an antiparti-
The leptons appear to be the simplest group.
cle. The negative pion is ud, and the neutral pion
In addition to the electron, the (mu) and (tau)
is uu. All known hadrons (and there are over 100)
particles are heavier versions of the electron.
can be constructed by these simple rules. Note
Their masses are 207 and 3660 times that of the
that, though there are six quarks, the only ones
electron, respectively. Neutrinos are also leptons.
we need for everyday life (that is to make the pro-
We think that there are three types of neutrinos,
ton and neutron) are u and d. Similarly, the only
one to go with each of the other leptons (e, , ).
leptons that we need for everyday life are the elec-
That is, in reactions for which we see an electron,
tron and the electron neutrino.
we will see an electron neutrino, and so on. All of
The quark theory, originally proposed by
the evidence to date indicates that the leptons
Murray Gell-Mann (who received the 1969 Nobel
are truly fundamental. They seem to have no
Physics Prize for this work) was immediately
internal structure. We still don™t know if the six
quite successful, but there were a few important
leptons are all that there are. There is some evi-
problems left. One was why only the particular
dence to suggest that there are no others, but we
combinations mentioned are allowed. The other
have been surprised before.
is that, despite considerable effort, no one has
The hadrons do appear to have internal struc-
been able to detect a free quark.
ture. This can be seen in experiments that have
sufficiently high energy to probe the charge dis-
21.3.2 Fundamental forces
tribution within a proton. We now think that the
hadrons are composed of particles called quarks. Along with the quest for fundamental particles,
physicists are also trying to understand the forces
In the original theory of quarks, there were only
with which the particles interact. The concepts of
three; now six have been found. That makes six
the forces and particles are intimately tied
quarks and six leptons, a balance which theorists
together. Without forces, particles would have no
seem to like.
meaning, since we would have no way of detecting
The properties of the quarks are given in Table
the particles. Our current thinking is that there
21.2. Notice that they have fractional charges,
are four fundamental forces, summarized in Table
coming in units of (plus and minus) one-third
and two-thirds of the fundamental charge, e. 21.3. They are arranged in order of strength, and
their strength is given relative to the strong
However, the quarks can only combine in ways
nuclear force. Short range forces are felt prima-
that produce integral net charges. Each quark has
rily on the scales of nuclei. Long range forces have
its own antiquark. All of the properties of a given
a 1/r2 falloff in the force, which allows them to be
antiquark (except mass) are the negative of those
felt over a large distance.
for the corresponding quark.
It also appears that particles are necessary as
In the quark theory, any baryon is a combina-
carriers of the forces, as summarized in Table 21.3.
tion of three quarks. For example, a proton is uud

Forces and particles.
Table 21.3.
Force Relative strength Range Carrier

Strong Nuclear 1 short pion
Electromagnetic 10 long photon
Weak Nuclear 10 short W ,W , Z
Gravity 10 long graviton (?)

For example, quantum electrodynamics (QED) is the The weak nuclear force, also a short range
theory that describes the electromagnetic force as force, is carried by three particles. Two are the
being carried by photons. These photons may be positive and negative W (for weak), and the third
real, or may only exist briefly on energy that can is a neutral particle, called the Z. These particles
be borrowed because of the uncertainty principle are much more massive than the pion, and the
(discussed in Chapter 8). These photons that live weak force is also a short range force. The W and
on borrowed energy are called virtual photons. The Z particles have recently been detected with
fact that the photon is massless leads to the elec- masses 80 and 90 times that of the proton.
tromagnetic force being long range. QED has been
21.3.3 The role of symmetries
tested in many ways to very high accuracy, and is
a very successful theory. The theory was devel- In any study of particles and forces, symmetries
oped in the 1940s by a number of physicists. One play a very important role. Symmetries are
of the leaders in the field was Richard Feynman important in many areas of physics. When we say
who came up with a way of visualizing the theory that a system has a particular symmetry, we
that also allowed him to carry out what had pre- mean that the system looks the same after a cer-
viously been very difficult calculations, which tain transformation. For example, spherical sym-
could be tested experimentally. Feynman shared metry means that the system looks the same,
the 1965 Nobel Prize in Physics for this work. even if we rotate it through any angle, about any
It is speculated that the gravitational interac- axis through one particular point (the center of
tion is carried by a massless particle, called the the sphere). Recognizing symmetries can greatly
graviton. It is presumed to be massless, because simplify the solving of a problem. If a problem
gravity has the same long range behavior as elec- has a certain symmetry, then the solution must
tricity. However, no gravitons have ever been have the same symmetry. For example, in Fig.
detected, and this theoretical framework is still 21.19, we show a spherically symmetric charge
being developed. We will see that the absence of distribution. If we want to find the electrical
quantum mechanical theory of gravity provides a forces around it, those forces must have the same
limitation on how far back we can go in probing symmetric appearance.
the big bang. Symmetries have an even deeper importance
The strong nuclear force is carried by the pions. in physics. Whenever there is a symmetry, there
Of course, we now know that the pions themselves is some quantity that is constant throughout the
are not fundamental. They are each made of a problem. This means that there is a conservation
quark and an antiquark. Since the pion has a law. For example, the fact that the laws of physics
mass, the strong nuclear force has a short range. cannot be changed by rotating our coordinate
In fact, the mass of the pion can be inferred from system leads to the conservation of angular
the range of the force (see Problem 21.9). The momentum. The fact that the laws of physics are
more massive the carrier, the greater the energy not changed by the translation of the origin of a
that must be temporarily ˜borrowed™ to produce coordinate system leads to the conservation of
the virtual particle. This means that the virtual linear momentum. The fact that the laws of
particle lives a shorter length of time, and can physics don™t change with time leads to the con-
travel a shorter distance in that lifetime. servation of energy.

strange quark(s) was included to incorporate this
+ property.
One interesting property of the weak interac-
tion is that it doesn™t obey all the conservation
laws that the other forces do. Before this was real-
ized, it was thought that conservation laws were
absolute. The first symmetry found to be broken
+ Wrong was that concerning parity, which has to do with
the behavior of a system under a mirror reflec-
tion. It was realized by T. D. Lee and C. N. Yang in
1957 that it is possible to set up a beta decay
experiment (beta decay taking place via the weak
interaction) and the mirror image experiment,
and achieve different results. This experiment
was carried out by C. S. Wu a year later, and Yang
+ Correct and Lee shared the 1959 Nobel Prize in Physics for
their prediction. Two other symmetries violated
by the weak interaction are charge conjugation (the
interchange of particles and antiparticles) and
time reversal.
(c) Sometimes, we find situations which are
inherently symmetric, but somehow lead to an
Fig 21.19. Symmetry in an electricity problem. Suppose
asymmetric result. They are called spontaneous
we wish to calculate the electric ¬eld due to the spherical
symmetry breaking. For example, suppose we toss a
charge distribution in (a).The resulting ¬eld must have the
same symmetry as the charge distribution.The result in (b) coin in the air. As the coin is spinning, it has an
is clearly wrong because, if we rotate the page, the charge equal probability of being heads or tails. As long
distribution still looks the same, but the ¬eld direction
as the coin stays in the air, the situation is sym-
changes.The type of distribution in (c) has the proper sym-
metric between heads and tails. However, once
metry. A consideration of symmetry allows us to eliminate
the coin falls, the symmetry is broken. It is either
unreasonable answers.
heads or tails. Another example is a ferromagnet,
shown in Fig. 21.20. If the magnet is heated above
We can understand the various forces by its critical temperature, it has spherical symme-
understanding what symmetries they have, or, try. There is no preferred direction. When we cool
equivalently, what conservation laws they obey. In the material, it becomes a permanent magnet.
general, a process will take place as long as it does- Until it cools, all directions are equally probable.
n™t violate some conservation law. If a reaction Once it cools, one direction is selected and the
that you think should take place does not, it whole magnet cools, pointing in that direction.
means that there is some conservation law that This is an example of a situation that is symmet-
you might not be aware of, and that the reaction ric as long as it is hot enough, but cooling the sys-
would violate that conservation law. For example, tem breaks the symmetry. Some circumstances in
before the quark theory had been proposed, there nature are symmetric as long as there is enough
was a group of particles that should have decayed energy.
by the strong nuclear force, but did not. Because
21.3.4 Color
of this strange behavior, these particles were
We have already mentioned two problems with
called ˜strange™ particles. It was proposed that
the quark theory. One is that there was no expla-
there must be some property of these strange par-
nation for why the only allowed combinations
ticles which had to be conserved. The particular
are three quarks or one quark and one antiquark.
decays would then have violated this conservation
The other is that we have not been able to detect
law. When the quark theory was proposed, the

Each flavor of quark comes in each of the three
colors. The six antiquarks come in corresponding
The rule for combining quarks is that we can
have only colorless combinations, like those shown
in Fig. 21.21. One way of having a colorless com-
bination is to have a quark of one color and an

High Temperature (a)


Low Temperature (b)

Fig 21.20. Spontaneous symmetry breaking in a ferromag-
net. (a) High temperature, above the critical temperature.
Each arrow shows one iron atom™s magnetic dipole orienta-
tion. Each is randomly oriented, so there is no preferred
direction. (b) Low temperature. Now all the dipoles line up, u u
so there is one preferred direction.
d d
u d

any free quarks. There was another problem with
Proton Neutron
the original theory. Quarks have the same spin as
electrons. Therefore, they should also obey the
u d
Pauli exclusion principle. However, some parti-
cles are observed which are clearly combinations u
of identical quarks, uuu for example, all in the
ground state. This appears to be a violation of the Pion Antipion
exclusion principle.
The solution was to postulate an additional
quark property that could be different for each of Fig 21.21. (a) Allowed quark combinations.These are the
allowed color (and anticolor) combinations. Remember that
the three quarks in a baryon. This property is
each quark comes in six possible ¬‚avors (and each antiquark
called color. This is just a convenient name, and
in six anti¬‚avors) that give the particles their basic proper-
has nothing to do with real color that we see. The
ties. In the top row, we see the combinations of three colors
color property is more like electric charge, except
(or three anticolors) that give the baryons (and antibaryons).
that it has three possible values (plus three anti- In the bottom row we see combinations of a color and anti-
values) instead of one. (Electric charge is thought color that give the mesons. (b) Quark combinations for
of as having one value and one antivalue.) Quark familiar particles.The proton is uud and the neutron is udd.
colors can be red (R), green (G) and blue (B). These In each case there is one quark of each color. (It doesn™t
colors relate to the ways in which quarks com- matter whether the u or d is any particular color.) The pion
is ud (with the u being some color and the d being the cor-
bine, and not to the properties of the particles
responding anticolor).The antipion is ud (with the d being
they make up. To distinguish them from colors,
some color and the u being the corresponding anticolor).
we call the six quark types (u, d, s, c, t, b) flavors.

antiquark with the corresponding anticolor. For QCD. While the photons that carry the electro-
example, a pion is ud, where the possible color magnetic force have no electric charge them-
combinations are RR, GG, or BB. This explains the selves, the gluons that carry the color force have
mesons. The other possibility is to have one quark a color charge. This means that the gluons can
of each color. So for a proton (uud), the possible interact with each other and with quarks. They
colors for three quarks are RBG, RGB, BGR, BRG, can also change the colors of the quarks they
GRB, or GBR. We can see from this that a free interact with. This makes the theory very diffi-
quark would not be colorless, and is therefore not cult mathematically. The detailed calculations
allowed by the theory. So the introduction of that have characterized the success of QED have
color has solved the three major problems men- not yet been possible for QCD.
tioned above. Another interesting feature comes out of QCD.
The properties involving color are more than a The force between two quarks does not fall off
set of ad hoc rules. They have actually been derived with distance between the quarks. This means that
from a mathematical theory, derived only on the the complete separation of two quarks requires an
assumptions of the types of symmetry that the infinite amount of energy. Even if we had the
theory should have. This theory is called quantum energy available to drive two quarks far apart, we
chromodynamics, or QCD. It is an analog to QED. In could not isolate a quark. This is illustrated in Fig.
this theory, the strong force between hadrons is 21.22. As you pulled the two quarks apart, you
no longer a fundamental force. The real force is would put enough energy into the system that you
called the color force, acting between the quarks. would simply create quark“antiquark pairs out of
Just as electric charge is a measure of the ability the energy. The new quark and antiquark would
of particles to exert and feel the electromagnetic bind with the two quarks you were trying to pull
force, color measures the ability of particles to
exert or feel the color force.
There are other analogies between the electric
force and the color force. The requirement that
particles be colorless is analogous to the state-
ment that matter should be electrically neutral.
For the most part, matter is neutral unless we
provide a lot of energy to ionize atoms. As we will
see below, the color force is so strong that the
analogous process is not possible for quarks.
There is another similarity. In the 19th century
physicists thought that the force between neutral
molecules, called the van der Waals force, was a
fundamental force. After the development of the
atomic theory in the early 20th century, it became
clear that this was nothing more than the resid-
ual force between electrons and protons within
the molecules. In the same way particle physicists (c)
now realize that the strong force between two par-
Fig 21.22. Quark con¬nement.The ¬gure shows a quark
ticles is the residual of the color force among the and antiquark in a meson, as we try to separate them. In
quarks making up the particles. each part, the yellow arrows show the force between the


. 20
( 28)