effects. This still leaves us a factor of five short of closing the universe.

We have said that the best way to measure the mass of any object is to measure its gravitational effect on something. If we want to determine the mass of the Earth, we measure the acceleration of gravity near the surface. Therefore, instead of trying to find all of the matter needed to close the universe, we can look for its gravitational effects. We can try to measure the actual slowing down of the expansion of the universe to see if — R is large enough to stop the expansion. When we do this, we are determining the current value of the deceleration parameter from its original definition (equation 20.23). Using the fact that R(t0) = H0 (equation 20.12), it becomes q0 =

We don't actually try to measure R(t0). What we try to measure is the current rate of change of the Hubble constant H(t0). We would therefore like to express R(t0) in terms of H0 and H(t0). We start with equation (20.11):

Differentiating both sides with respect to t gives

Setting t = t0, and remembering that R (t0) = H0, this becomes

Substituting this into equation (20.42), we have q0 =

Equation (20.44) tells us that if we can measure the rate of change of the Hubble parameter, we can determine q0. Unfortunately, measuring H(t0) is not easy. This should not be a surprise since measuring H0 is not easy. In principle, we can measure H(t0) by taking advantage of the fact that we see more distant objects as they were a long time ago. If we can determine H for objects that are five billion light years away, then we are really determining the value of H five billion years ago. If we include near and distant objects in a plot of Hubble's law, we should be able to see deviations from a straight line as we look farther back in time.

The difficulty comes in the methods used for measuring distances to distant objects. In our discussion of the extragalactic distance scale, we saw that for the most distant galaxies we can see, we cannot look at individual stars, such as Cepheids, within a galaxy. Instead, we must look at the total luminosity of a galaxy. We already know that the luminosities of galaxies change as they evolve. Galactic cannibalism provides us with the most spectacular example of this, but even normal galaxies change in luminosity with time. Therefore, if we calibrate the distance scale using the luminosities of nearby galaxies, we cannot apply this to more distant galaxies, precisely because we are seeing them as they were in the past. Before we can interpret observations of distant objects, we must apply theoretical evolutionary corrections. These corrections can be so large that they can make an apparently closed universe appear open or an open universe appear closed.

When we discussed the extragalactic distance scale, we mentioned that supernovae might provide very useful standard candles. In this case, the more reliable standards are those that arise in close binary systems (type I). Nearby examples tell us how to relate the light curve to the peak luminosity. So if we can compare the observed brightness with the known luminosity, we can calculate the distance. HST has been particularly useful for studying these objects, and can detect faint ones at cosmologically significant distances. These results suggest that the expansion of the universe may not be slowing very much. In fact, a few points are consistent with an acceleration in the expansion. This would be consistent with a non-zero cosmological constant. However, before these preliminary results are accepted, many more such objects need to be observed, and we need to examine whether these particular standard candles are not standard as thought.

An alternative approach is to measure the curvature of space-time by surveying the universe on a large scale. One way of carrying this out is with radio source counts. We divide the universe into shells, such as for our discussion of Olbers's paradox earlier in this chapter. We then count the number of radio sources in each shell. We use radio sources because we can see them far away. (With large, sensitive optical telescopes, optical counts are now being used also.) If the geometry of space-time is flat (Euclidean) the number of sources per shell will go up as r2. If the geometry of space-time is curved, that curvature will become more apparent as we survey larger regions. Therefore, as we look farther away, we would expect to see deviations from the r2 dependence. What is actually varying is the relationship between r and surface area. Of course, as we look far enough to see such deviations, we are also looking far back in time, and we are seeing radio sources as they were. Again, evolutionary corrections are necessary. The results so far are consistent with a flat universe.

There are also more indirect methods that have proved fruitful. These involve an understanding of the formation of elements in the big bang, and will be discussed in the next chapter. The results of these so far support a universe which is open. In addition, they only give information on the density of material that can participate in nuclear reactions, and may not include the dark matter.

One of the interesting aspects of this whole problem is that we should be so close to the boundary. Of all the possible values for the density of the universe, ranging over many orders of magnitude, we seem to be tantalizingly close to the critical density. Cosmologists have wondered whether this is accidental, or whether it is telling us something significant about the universe. They have noted that if Q is not exactly unity, then it evolves away from unity as the universe becomes older. This means that for the actual density to be pretty close to the critical density

Gyr from now

H Scale factor vs. time for the cosmological model which best fits current data.The model has a Hubble constant of 65 km/s/Mpc and QM = 0.3, QA = 0.7. [© Edward L.Wright, used with permission]

H Scale factor vs. time for the cosmological model which best fits current data.The model has a Hubble constant of 65 km/s/Mpc and QM = 0.3, QA = 0.7. [© Edward L.Wright, used with permission]

now, it had to be very close to the critical density in the past.

If there is a non-zero cosmological constant, then it is possible to have a flat universe. Remember, we defined the density parameter for matter in equation (20.27) as

We saw that if there is a non-zero A, then we can define an effective density due to that A, as A/8-n-G, so we can define a density parameter associated with A as

Then the total density parameter for the universe would be

Fig. 20.10 shows R vs. t for what is currently the best estimate of the model universe with Qtot = 1 (Qm = 0.3, Qa = 0.7). We will discuss the future of the universe in the next chapter when we look at the big bang.

Cosmology is the study of the universe at the largest scale. It asks about the large-scale structure of the universe and how it has evolved. When we are talking about cosmological scales, the smallest building blocks are the galaxies.

One of the fascinating things about cosmology is that we can do normal astronomical observations to answer cosmological questions. The consideration of the simple question, "Why is the sky dark at night?" leads us to the profound conclusion that the universe must have a finite size, a finite age, or both.

In explaining the large-scale structure of the universe, we start with the simplest assumptions, that the universe is homogeneous and isotropic. This means that (on cosmological scales), the average properties are the same from place to place, and the appearance is the same in all directions. A class of theories that also include the assumption that the universe doesn't change with time, so-called steady-state theories, are no longer supported by observational evidence. Cosmological models in which the universe is becoming less dense as it expands have an era in the early universe when it was hot and dense. This era is called the big bang, and models that include it are called big-bang cosmological models.

In keeping track of the expansion of the universe, it is useful to deal with the scale factor, which tells us how much distances between galaxies have changed from one time to another.

As the universe expands, the wavelengths of all the photons in the universe increase by the same amount as the scale factor increases. This is called the cosmological redshift. It is the redshift that produces Hubble's law. In treating the structure of the universe, we must be sure to account for the effects of general relativity. The geometry of the universe may behave differently from that of a normal flat surface.

Each particle in the universe feels the gravitational attraction of all the other particles in the universe. Whether the universe will expand forever, or the expansion will eventually reverse, depends on the total density of material in the universe. This has turned out to be very hard to measure, though we have been able to determine that there is not nearly enough luminous matter to close the universe. So, if the universe is closed, then the dark matter must be responsible. The evolution of the universe may also be affected by a non-zero cosmological constant.

*20.1. How large a scale do we have to look at before the cosmological principle can be applied? How does this scale compare with the distance over which light could have reached us in the age of the universe?

20.2. Restate the argument in our discussion of Olbers's paradox using galaxies instead of stars as the sources of light.

20.3. In our discussion of Olbers's paradox, does it matter whether we talk about the appearance of the daytime or night time sky?

*20.4. Suppose that we were trying to invoke interstellar dust as a way out of Olbers's paradox by saying that it is the scattering by the dust that blocks out the distant light, not absorption. The dust will therefore not heat. Why doesn't this argument help?

20.5. How does the universe having a finite size or age save us from Olbers's paradox?

20.6. Does our motion towards the great attractor violate the part of the cosmological principle that the universe should appear isotropic?

20.7. What is the observational evidence that the universe is expanding?

20.8. Does Hubble's law rule out the steady-state models?

*20.9. What observations can we do to verify that the universe is isotropic?

20.10. If we lived in a contracting universe, would we still observe a cosmological redshift?

*20.11. How can we measure the curvature of the universe without getting outside it?

*20.12. What is the universe expanding into?

20.13. What are the advantages of using the scale factor R(t) to keep track of the expansion of the universe?

20.14. Where is the center of the universe?

20.15. What are the various interpretations of the quantity k, discussed in this chapter?

20.16. If the universe is closed, can we distinguish a "one-time" universe from an oscillating universe?

20.17. Of the methods described for deciding whether the universe is open or closed, which ones rely on measuring the gravitational effects of all of the matter in the universe?

20.18. If the universe is expanding, how is it possible for that expansion to reverse?

20.19. Why do we say that if the universe is closed, then the dark matter must do it?

Problems

For all problems, unless otherwise stated, use H0 = 70 km/s/Mpc.

20.1. Suppose we detect radiation that was emitted by some galaxy far away. In the time the radiation traveled to reach us, its wavelength doubled. What happened to the scale factor of the universe in that time?

20.2. How much brighter would the sky be if it were uniformly filled with Suns, rather than the one we have? (Hint: think of the solid angle covered by the Sun relative to the whole sky.)

20.3. Show that, if the universe were infinite in age and extent the cosmological redshift is not sufficient to get us out of Olbers's paradox.

20.4. Estimate the rate of continuous creation required to keep the density constant (at

' g/cm3). Express your answer in protons/yr/km3.

20.5. Show that the density of the universe is proportional to 1/R3(t).

20.6. For the case k < 0, find an expression for R(t) valid for large R. What are the limits on R for your expression to be valid?

20.7. Show that equation (20.24) follows from equations (20.11), (20.16) and (20.17).

20.8. Show that the density parameter O is twice the deceleration parameter q.

20.9. Rewrite equations (20.17) and (20.19) in terms of the density parameter, substituting for the critical density from equation (20.26).

20.10. If the current density of the universe is 1 X 10-29 g/cm3, what value would be needed for the cosmological constant A in order for the universe to be static?

20.11. Show that equation (20.30) can be obtained by the appropriate use of a Taylor series.

20.12. What values of H(t0) would be needed to make q0 equal to (a) 0, (b) 1/2, (c) 1?

20.13. Derive an expression for the critical density (equation 20.26) without introducing the deceleration parameter q0.

20.14. Show that if the distances are given by Hubble's law, then the distance modulus is given by equation (20.33) without the last term on the right.

20.15. Compare the distances obtained using Hubble's law and equation (20.33) with q0 = 1/2 for objects with z = (a) 0.1, (b) 1.0, (c) 3.5, (4) 5.0.

20.16. Estimate the distance modulus, m - M, for objects with z = (a) 0.1, (b) 1.0, (c) 3.5, (4) 5.0.

20.17. Estimate the age of the universe at the time when radiation was emitted from objects with z = (a) 0.1, (b) 1.0, (c) 3.5, (4) 5.0.

20.18. Suppose we observe an object that is 10 Mpc away. At what wavelength is the Ha line observed if (a) the object has no other motion, (b) the object has an additional motion away from us at 1000 km/s, (c) the object has an additional motion towards us at 1000 km/s.

20.1. For the case k = 0, find an expression for R vs. t, 20.4. For k = 0, plot a graph of the distance modulus and plot a graph of your result. (m - M) vs. z.

20.2. For k = 0, what is the difference between the cur- 20.5. Using equation (20.41), plot a graph of t in rent age of the universe and the current value of gigayears vs. z, for H0 = 50, 75 and 100 km/s/Mpc. the Hubble parameter?

20.3. For k = 0, how different is the Hubble parameter for objects with z = 0.1, 1, 3, 5 and 103?

Chapter 21

In the preceding chapter, we noted that Lemaitre first pointed out that if the universe is expanding, then there must have been an era in the past when it was much denser than it is now. This hot, dense early era was named the big bang by Fred Hoyle, a steady-state cosmologist, in an attempt to ridicule the theory. The theory survived the ridicule, the name remained, and we now refer to all cosmological models with an evolving universe as 'big-bang cosmologies'. In this chapter, we will see what we can learn about conditions in the big bang, and what the relationship is between those conditions and the current state of the universe.

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