The evidence for the expansion of the universe comes from the change in wavelength of the light emitted by distant galaxies. We analyzed a similar effect as the relativistic Doppler shift, which we can write in terms of wavelength as
..... 1
where v represents the relative velocity between the source of the light and the observer. Here λ' is the wavelength we measure on Earth and λ* is the wave length emitted by the moving star or galaxy in its own rest frame.
The light emitted by a star such as the Sun has a continuous spectrum. As light passes through the star's atmosphere, some of it is absorbed by the gases in the atmosphere, so the continuous emission spectrum has a few dark absorp tion lines superimposed. Comparison between the known wavelengths of these lines (measured on Earth for sources at rest relative to the observer) and the Doppler-shifted wavelengths allows the speed of the star to be deduced from Equation 1.
Of the stars in our galaxy, some are found to be moving toward us, with their light shifted toward the shorter wavelengths (blue), and others are mov ing away from us, with their light shifted toward the longer wavelengths (red).
The average speed of these stars relative to us is about 30 km/s (10-4 c). The change in wavelength for these stars is very small. Light from nearby galaxies, those of our "local" group, again shows either small blue shifts or small red shifts.
However, when we look at the light from distant galaxies, we find it to be systematically red shifted, and by a large amount. Some examples of these measurements are shown in Figure . We do not see a comparable num ber of red and blue shifts, as we would expect if the galaxies were in random motion. All of the galaxies beyond our local group seem to be moving away from us.
The cosmological principle asserts that the universe must look the same from any vantage point, and so we must conclude that any other observer in the universe would draw the same conclusion: The galaxies would be observed to recede from every point in the universe.
Hubble's Law
In the 1920s, astronomer Edwin Hubble was using the 100-inch telescope on Mount Wilson in California to study the wispy nebulae. By resolving indi vidual stars in the nebulae, Hubble was able to show that they are galaxies like the Milky Way, composed of hundreds of billions of stars. Hubble also was able to observe variable stars in the distant galaxies whose brightness oscillated with periods of days. Using a scale of period vs. brightness for the variable stars in nearby galaxies developed in 1908 by astronomer Henrietta Swan Leavitt, Hubble deduced the distances to the remote galaxies. Finally, plotting his deduced distances against the recessional velocities obtained from the red shifts, Hubble established the empirical relationship between distance and speed known as Hubble's law:
v = Ho d
The proportionality constant H0 is known as the Hubble parameter.
Figure a shows a plot of Hubble's data. Although the points scatter quite a bit (due primarily to uncertainties in the distance measurements), there is a definite indication of a linear relationship. (Hubble's distance calibra tion was incorrect, so the labels on the horizontal axis do not correspond to the actual distances to the galaxies.)
More modern data based on observing supernovas in distant galaxies are shown in Figure b. There is again clear evidence for a linear relationship, and the slope of the line gives a value of the Hubble parameter of about 72 km/s/Mpc*, within a range of about ±10%. The Hubble parameter can also be determined from a variety of other cosmo logical experiments. These agree with the supernova data, and the best current value is
The uncertainty in this value is on the order of +4%.
*A parsec, pc, is a measure of distance on the cosmic scale; it is the distance that corresponds to one angular second of parallax. Because parallax is due to the Earth's motion around the Sun, the parallax angle 2a is the diameter 2R of the Earth's orbit divided by the distance d to the star or galaxy. Thus, a = R/d radians, which gives 1 pc = 3.26 light-years = 3.084 x 1013 km. One megaparsec, Mpc, is 106 pc.
The Hubble parameter has the dimension of inverse time. As we show later, H0-1 is a rough measure of the age of the universe. The best value of Ho gives an age of 14 x 109 y. If the speed of recession has been changing, the true age can be less than H¹.
How does the Hubble law show that the universe is expanding?
Consider the unusual universe represented by the three-dimensional coordinate system shown in Figure a, where each point represents a galaxy. With the Earth at the origin, we can determine the distance d to each galaxy. If this universe were to expand, with all the points becoming farther apart, as in Figure b, the distance to each galaxy would be increased to d'. Suppose the expansion were such that every dimension increased by a constant ratio k in a time t; that is, x' = kx, and so forth. Then d' = kd, and a given galaxy moves away from us by a distance d'-d in a time t, so its apparent recessional speed is
If we compare two galaxies 1 and 2,
a relationship identical with Hubble's law, Eq. Thus, in an expanding universe, it is perfectly natural that the farther away from us a galaxy might be, the faster we observe it to be receding.
Notice also from Figure a that this is true no matter which point we hap pen to choose as our origin. From any point in the "universe" of Figure 15.3, the other points would be observed to satisfy Eq. 15.4 and thus also Hubble's law. We can further demonstrate this with two analogies.
If we glue some spots to a balloon and then inflate it, every spot observes all other spots to be moving away from it, and the farther away a spot is from any point, the faster its separation grows.
For a three-dimensional analogy, consider the loaf of raisin bread shown in Figure rising in an oven. As the bread rises, every raisin observes all the others to be moving away from it, and the speed of recession increases with the separation.
The correct interpretation of the cosmological redshifts requires the tech niques of general relativity, which we discuss later in this chapter. According to general relativity, the shift in wavelength is caused by a stretching of the entire fabric of spacetime. Imagine small photos of galaxies glued to a rubber sheet. As the sheet is stretched, the distance between the galaxies increases, but they are not "in motion" according to the terms we usually use in physics to describe motion. However, the stretching of the space between the galaxies causes the wavelength of a light signal from one galaxy to increase by the total amount of the stretching before it is received at another galaxy. This is very different from the usual interpretation of the Doppler formula (Eq). (In fact, for some galaxies the wavelength shift is so large that the special relativ ity formula would imply a recessional speed greater than the speed of light!) At low speeds, the Doppler interpretation of the redshift (that is, calculat ing a speed from the Doppler formula and using that speed in Hubble's law) gives results that correspond with those based on an expansion of spacetime.
However, for very large cosmological redshifts, a more correct analysis must be based on the stretching model:
where Ro represents a "size" or distance scale factor of the universe at the present time and R represents a similar factor at the time the light was emitted.
The expansion of the universe has been widely accepted since Hubble's discoveries in the 1920s. There are, however, two interpretations of this expan sion. (1) If the galaxies are separating, long ago they must have been closer together. The universe was much denser in its past history, and if we look back far enough we find a single point of infinite density. This is the "Big Bang" hypothesis, developed in 1948 by George Gamow and his colleagues. (2) The universe has always had about the same density it does now. As the galaxies separate, additional matter is continuously created in the empty space between the galaxies, to keep the density more or less constant. This is the "Steady State" hypothesis, proposed also in 1948 by astronomer Fred Hoyle and others. New galaxies created from this new matter would make the uni verse look the same not only from all vantage points but also at all times in the present and future. (To keep the density constant, the rate of creation need be only about one hydrogen atom per cubic meter every billion years.)
Both hypotheses had their supporters, and during the 1940s and 1950s, the experimental evidence did not seem to favor either one over the other. In the 1960s, the new field of radio astronomy revealed the presence of a universal background radiation in the microwave region, which is believed to be the remnant radiation from the Big Bang. This single observation has propelled the Big Bang theory to the forefront of cosmological models.
