The quantity in the square bracket of equation [37] is independent of R(t) and will have the same form for the entire history of the Universe— according to this closed-Universe model.

We can simplify metric [37] even further when we model the space part in the rounded brackets by a unit sphere (Figure 6).

The angle 0 (the direction in the spatial plane we are considering) is measured around a latitude of this unit sphere from some arbitrary initial direction, and the variable u is the perpendicular distance from the axis to the point being described. The variable u can be replaced by the angle 0 measured from the upward vertical axis (Figure 7). For a unit sphere, the distance from the north pole along a longitude is also measured by 0 in radians. From Figure 7, we have a relation between « and the new angle 0

Figure € Location of an event A on the unit sphere using the coordinates u and <t>.

Arc length = 0 fora unit sphere I

and we can use a familiar trigonometric identity to write de = -^- =-— = —in— [40]

QUERY 17 Metric for the closed Universe model. Substitute recent expressions into [37] to show that the metric can be simplified to the form dx2 = R2(t) [dr\2-(dQ2+sm2Q rf<|>2)] [41]

The metric [41] carries a powerful and simple description of the closed model Universe with our usual restriction to two dimensions on a spatial plane. Here <|> is the usual direction on that plane with respect to some arbitrary direction of zero angle, and r is coded in the variable 8 through equations [14] and [38]. The size of the Universe is R(t), given by equation [27], and the corresponding "angle" i\ is related to t by equation [29]. Every location in two spatial dimensions in this model Universe at a fixed time is described by a position on a unit sphere multiplied by that radius. The metric [41] relates adjacent events in the model closed Universe. The wristwatch time dx between this pair of events is a product of the current size R(t) of the Universe and the spacetime separation between the two events given by the expression in square brackets in equation [41].

According to this model (refer to Figures 4 and 5), when tj reaches the value 7i then the Universe reaches its point of maximum expansion, and when T| reaches 2ti the Universe has again contracted to the Big Crunch.

Now we can analyze how light moves in our closed model Universe. For light dx = 0 and the metric [41] collapses to the expression

The right side of this equation has a simple interpretation: It is the distance between two nearby points on the unit sphere. Equation [42] says that in an increment of time as measured by t|, light moves an equal increment of distance on the unit sphere as measured by the angles 0 and <|>.

Our everyday language trips us up when we apply it to the Universe as a whole. Space and time were created at the Big Bang. Equation [41] tells us that at the beginning, when the radius R(t) is zero, the proper time dx (as well as the proper distance do) between any two events is zero. We are present at the Big Bang wherever we are in the spatial plane represented by the unit sphere (Figure 6). The Big Bang occurs everywhere on the unit sphere at t = r| = 0.

Equation [42] helps us to describe how we continue to see the big bang at later times. Take our position to be at the north pole of the unit sphere and ask what we see as we look outward. It takes time for light from the Big Bang to reach us from other portions of the unit sphere. The light we see will move along longitudes of the unit sphere, namely with d§ = 0 in equation [42]. As the time parameter increases, the available angle over which the light travels also increases, so that

This is an easy equation to integrate! Figure 8 shows the ring on the unit sphere representing the source of light from the Big Bang that we see at the time parameter t|.

Let dti be the period of a light wave emitted at time t\ when the size of the Universe is described by R(ti). Let this light wave propagate a great distance, such as that from S to O in Figure 8. What will be the observed period dt2 at reception, when the Universe has expanded to R(t2)? Equation [42], page G-17, says that at all stages of expansion light moves equal distances along the unit sphere in equal units of the time parameter ti, so that dr\ remains constant as the wave propagates. Equation [21] on page G-9 says that dt = R(t)dr\. Hence for a larger R(t), we have a larger dt, a longer period, a redder light signal. This cosmic red shift is a generalization in expanding curved spacetime of the Doppler shift of light in flat spacetime.

Figure 8 Unit sphere representing the location of all events that occur on a plane In space The Big Bang occurs everywhere on this unit sphere att = 0. The heavy ring shows the source of light from the Big Bang that we at location O see at a time given by the time parameter ti The light path from S to O represents all light paths along fixed longitudes from the source ring to our location.

Light path

Figure 8 Unit sphere representing the location of all events that occur on a plane In space The Big Bang occurs everywhere on this unit sphere att = 0. The heavy ring shows the source of light from the Big Bang that we at location O see at a time given by the time parameter ti The light path from S to O represents all light paths along fixed longitudes from the source ring to our location.

QUERY 18 Seeing the whole Universe. Continue predictions implied by the construction in Figure 8. Pay attention to light which travels directly to us at position O along lines of longitude, for which dty = 0.

A. How long after the Big Bang will light have reached us from all parts of the Universe? Express your answer in the "time angle" t].

B. At the time found in part A, what is the state of expansion of the Universe?

C. Suppose that it takes some time after the Big Bang for galaxies to condense out of the primordial gases. At the time found in part A, will we have seen every galaxy in the Universe that lies on the spatial plane represented by the unit sphere? Why or why not?

QUERY 19 When will we see it all? How many billion years after the Big Bang will we have been able to receive light from the entire Universe, according to this closed-Universe Friedmann model? From part B of Query 18, this occurs when the Universe has reached it maximum expansion. Assume that the mass density is twice the critical mass density of five hydrogen atoms (effectively five protons) per cubic meter at the time of maximum expansion. In brief, the mass density at maximum expansion corresponds to 10 protons/meters3. Assume this rough approximation is numerically exact for purposes of the following calculations.

A. Express the characteristic radius R of the Universe in terms of its mass parameter M at the moment of maximum expansion, at which time we will have received light from all of the universe.

B. Use the result of part A plus equation [18], page G-8 to find an expression for the mass parameter M as a function of the mass density p at the moment of maximum expansion.

C. Use equation [29], page G-10 and the value of t] at maximum expansion to derive an expression for the time t at this state of expansion. Verify that the geometric units are the same on both sides of this equation.

D. Compute the value of the density pconv in conventional units kilogram/ meter3. Convert to geometric units using the conversion factor G/c2 (page 2-14 and inside the back cover). The result to one significant digit is 1 x 10~53 meter-2. Find the result to three significant digits.

E. Find the value of M from your result of part B and substitute your value from part D into the expression for t in part C. The result to one digit is 2 x 1026 meters of time. Find the result to three significant digits.

F. Convert your result in part E to seconds and then to years (conversion factors inside the back cover). According to this model of the universe (and our assumed value of the mass density at maximum expansion), how many billion years after the Big Bang will we have received light from the entire Universe?

Accelerating Expansion of the Universe?

The results of Riess, Filippenko, and their coworkers.

Is the rate at which the Universe expands actually increasing with time? That ¡s the conclusion of two separate groups, one group including Adam Riess, Alexei V. Filippenko, and eighteen other people, the other group including S. Perlmutter and thirty-one other people.

These observers use light from an exploding supernova of a particular kind, the so-called Type la supernova. They believe that the Type la supernova results when a white dwarf gradually accretes mass from a large binary companion, finally reaching a mass at which the white dwarf becomes unstable and explodes into a supernova. The "slow fuse" on the gradual accretion process may lead to almost the same size explosion on each such occasion, giving us a "standard candle" of the same intrinsic brightness, provided that nuclear burning has left every white dwarf with the same nuclear composition. If so, the brightness of the explosion as seen from Earth provides a measure of the distance to the supernova. The cosmic red shift of the light tells us how fast the supernova is receding. Because supernovas are so bright, they can be seen at a very great distance, which brings us information about the Universe much of the way back to the Big Bang.

The figures display observed brightness on the vertical scale versus red shift on the horizontal scale. Calibration of the vertical scale is in stellar magnitude, a logarithmic measure of observed brightness. The larger the magnitude number, the dimmer is the observed star. (The vertical scales of the two graphs differ by a standard magnitude, which moves the curve up and down without changing its shape.) The horizontal scale, also logarithmic, is calibrated in what is called the red-shift factor z, defined implicitly in the following equation

Here ^observed's ^e wavelength of light observed from Earth, while Xemitted is the wavelength of the light emitted from the source. The emitted wavelength is known if one knows the emitting atom, identified from the pattern of different wavelengths.

If the data points in the figure lie along a straight line, then the expansion velocity is proportional to distance, as one would expect if nothing slowed down matter blasted out of the Big Bang. But the data points in the figure appear to lie slightly above the straight line for the most distant supernovas (upper right on the diagrams). This could mean that the most distant supernovas (highest on the vertical axis) are moving away slower than expected (farther to the left on the horizontal axis than expected). We are watching motions of these most distant supernovas as they were long ago, because it takes light such a long time to reach us. Could this mean that the expansion rate long ago was slower than it is now? And how significant is the apparent deviation from the straight line? You be the judge.

Whatever you think about the claimed change in expansion rate, the use of supernovas as standard candles for small and intermediate distances leads to a new and improved value for the average expansion rate of the Universe From this average expansion rate, Riess, Filippenko, and their colleagues derive a time of (14.2 ± 1.7) x 109 years for the age of the Universe. The result derived by Perlmutter and company is (14.5 ± 1.0) x 109 years.

The results on previous pages illustrate the predictions for our closed-model Universe. Unfortunately they cannot correspond to reality. The mass-energy density in the early Universe was dominated by radiation and extreme pressure, whereas our simple Friedmann models consist of pressure-free dust. Moreover, in the plasma soup (swarm of uncombined electrons and nuclei) in the early Universe, light was continually absorbed and scattered and could not propagate in a straight line. Because of these effects, light could not move directly to us from the Big Bang. At present the so-called cosmic background radiation provides our earliest view of the Universe. The cosmic background radiation is microwave radiation that permeates the Universe, emitted just before the moment at which electrons in the cooling soup combined with protons to form atoms. With the formation of atoms, the Universe suddenly became transparent to electromagnetic radiation. This occurred about 300 000 years after the Big Bang. The cosmic red shift reduces the characteristic temperature of the background radiation that we observe from thousands of degrees Kelvin to 2.73 degrees Kelvin.

So at present we could not see the Big Bang directly, through the blanket of early plasma, whatever the predictions of our model. Neutrinos and gravity waves are much less affected by plasma than is light. So future observations made using neutrinos or gravity waves may penetrate the wall of early plasma, letting us peek farther back toward our ultimate origin. Selah.

How can physics live up to its true greatness except by a new revolution in outlook which dwarfs all its past revolutions? And when it comes, will we not say to each other, "Oh, how beautiful and simple it all is! How could we ever have missed it so long!"

—John Archibald Wheeler

When shall I cease from wondering?

—Galileo Galilei

9 References and Acknowledgments

Edmund Bertschinger made many important suggestions for this project.

Initial quote, Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, W. H. Freeman and Company, San Francisco (now New York), 1971, page 707. Slightly edited by JAW.

P. J. E. Peebles, Principles of Physical Cosmology, Princeton University Press, 1993.

For a brief biography of A. Friedmann and an account of his work, see Alan H. Guth, The Inflationary Universe, Helix Books, Addison-Wesley, Reading MA, 1997, ISBN 0-201-32840-2, pages 41-46.

For translations of original papers by Hubble, Friedmann, and others (and Einstein's changing reactions to the Friedmann model), see Cosmological Constants: Papers in Modern Cosmology, edited by Jeremy Bernstein and Gerald Feinberg, Columbia University Press, New York, 1986, ISBN 0-231-06376-8.

"Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant," Adam G. Riess, Alexei V. Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M. Garnavich, Ron L. Gilliland, Craig H. Hogan, Saurabh Jha, Robert P. Kirshner, B. Leibudgut, M. M. Phillips, David Riess, Brian P. Schmidt, Robert A. Schommer, Nicholas B. Suntzeff, and John Tonry, Astronomical Journal, Volume 116, pages 1009-1038 (September 1998).

"Measurements of Í2 and A from 42 High-Redshift Supernovae," S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabro, A. Goodbar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-LaPuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, Astrophysical Journal, Volume 517, Number 2, pages 565-586 (1999).

Selah Used many places in the Biblical Psalms. Exact meaning uncertain. Our favorite definition is from Bucer's Psalms in the year 1530: "This word Selah signifyeth ye sentence before to be pond'red with a depe affecte, long to be rested upon and the voyce there to be exalted." Oxford English Dictionary, 2nd edition, Clarendon Press, Oxford, 1989.

Final quote by Wheeler from John Archibald Wheeler, "On Recognizing 'Law Without Law'," American Journal of Physics, Volume 51, pages 398-404 (1983).

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