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Physical Cosmology
By Phillip James Edwin Peebles PRINCETON UNIVERSITY PRESS
Copyright © 1971 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08108-3
CHAPTER 1
GOLDEN MOMENTS IN COSMOLOGY 1912-1950
a) The Expansion of the Universe
The first observational basis of modern cosmology was the discovery of the nature of the extragalactic nebulae, or galaxies, and of their distribution and motion. Hubble gives an admirable account of this discovery in Realm of the Nebulae.
The apparent radial motion of the nebulae is indicated by the observed frequency shift in their spectra. In 1912 Slipher obtained the first positive measurement of this effect for an extragalactic nebula, the Andromeda Nebula, M 31. He found a velocity of approach of 300 km sec-1, consistent with the best modern value. Slipher continued painstaking accumulation of data on apparent radial velocities of the nebulae, and by 1924 Eddington could list in his Mathematical Theory of Relativity apparent velocities obtained by Slipher for 41 objects. Of these 36 are positive, that is, the observed spectrum is shifted toward the red, as if the nebulae were moving away from us.
These data found their way into a book on relativity theory because of a suspicion that the tendency toward apparent recession of the nebulae may have something to do with the cosmological model invented by deSitter in 1917. DeSitter followed the two assumptions Einstein had adopted in his cosmological model: a) The Universe is homogeneous and isotropic, so that the Universe appears the same in any direction or from any spot; b) The Universe is unchanging, the mean mass density is constant and the curvature of space is constant. Einstein found that his original general relativity theory would not admit a solution consistent with these two conditions. Therefore he added to his gravitational field equations a new long-range effect, the Cosmological Constant Λ term, which with suitable adjustment of the parameters would serve to balance the gravitational attraction among matter and make a static model. DeSitter observed that he could obtain another solution with constant non-zero curvature simply by removing all the matter from the model. The solution is stationary just because there is no matter to move about. If now some test particles were introduced in this DeSitter model the Λ term would cause the particles to "scatter," or accelerate away from each other. It is this effect that people thought might have some connection with the observed recession of the nebulae. The theoretical Sturm und Drang on the subject of the Λ term has been chronicled by North.
All of this discussion was highly uncertain because people could not be sure whether the spiral nebulae were separate "island universes," star systems like our own Milky Way Galaxy, or perhaps only lesser objects swarming around the Galaxy. It was known from star counts as functions of apparent magnitude that the Galaxy is a finite flattened star system, thought to be about centered on the Solar System – this picture was called the Kapetyn universe. This was well accepted by 1917, so Einstein's assumption that the system of stars about us extends out without boundary was clearly and literally false, but it is today thought to be accurately valid when applied not to stars but to the large-scale mean distribution of galaxies. This brilliant coup by Einstein was not entirely a good thing, however, for it helped inspire later cosmologists similarly to attempt to deduce the nature of the Universe from pure thought, with at best questionable results.
The first step in the discovery of the nature of the spiral nebulae was Shapley's determination (in about 1918) of the distance scale in the Galaxy. Shapley made use of Cepheid and RR Lyrae variable stars, for which the luminosity is a periodic function of time. It was known from the variable stars at very nearly common distance from us in the Magellanic clouds that the period of variability is correlated with absolute magnitude (total luminosity, ergs sec-1). This enables one to determine the relative distances of different Cepheid variables from their periods and apparent magnitudes (brightness in the sky, ergs cm-2 sec-1). To fix the absolute distances (determine the zero point of the period-luminosity relation) Shapley used the method of statistical parallaxes, which had been applied earlier by Hertzsprung and others. There are some nearby Cepheid variables for which one has both proper motions (rate of angular motion across the sky) and radial velocities. If the radial and transverse velocities are the results of random motions of the stars then evidently the average radial motion (km sec-1 ) and proper motion (radians sec-1 ) ought to be in proportion to the absolute distance to the star (km). Shapley determined the distances to globular clusters (dense nearly spherical star clusters) by isolating in these systems Cepheid-like variables. He argued that the globular clusters form a sort of halo around the Galaxy, and, because most of the globulars are concentrated in one part of the sky, that we are well away from the center of the system. Shapley concluded that the center of the Galaxy is some 20 kpc away. This number has now been reduced to around 10 kpc, the main correction being for interstellar absorption.
The distances to the nearer spiral nebulae were convincingly established with the discovery of Cepheid variable stars in these systems. This was accomplished with the help of the 100" telescope on Mount Wilson, completed in 1918. Apparently the first Cepheid variable identified in an extragalactic nebula was in M 31, the Andromeda Nebula, the nearest giant galaxy and the other dominant member of the Local Group of galaxies. In the Annual Report of the Mount Wilson Observatory for 1924 it was announced that Hubble had found several variable stars in M 31, at least the brightest of which had a light curve typical of the Cepheid variables. Interestingly enough the report did not state the distance to M 31 this would imply (~ 300 kpc according to Shapley's calibration of the period-luminosity relation) perhaps because in the same Report there is tentative confirmation of van Maanen's suggestion that the spiral nebulae may exhibit appreciable proper motions of rotation or streaming along the spiral arms. The suggested proper motion was ~ 3 × 10-3 arc sec/year. This with the observed internal radial velocities within the nebulae indicated that they were typically 1 kpc away, well within the Galaxy. Subsequent developments proved that this was in error.
The next year Hubble presented data on variable stars in another object, the irregular nebula NGC 6822. Eleven stars had light curves typical of Cepheids. The observed periods and apparent magnitudes were related in the standard way, so Hubble had every reason to suppose that these stars were ordinary Cepheid variables, but at unusually great distance. From Shapley's normalization of the period-luminosity relation based on galactic Cepheids Hubble concluded that the distance to NGC 6822 is 214 kpc (this later was increased by a factor of 2, as described in Chapter III). Hubble pointed out that this put the nebula well outside the Galaxy, and made it about comparable in linear extent and luminosity to the Magellanic Clouds, which are satellites of the Galaxy. It is interesting that two years earlier Shapley had arrived at the same conclusion from morphological arguments. He argued that NGC 6822 looks rather similar to the Magellanic Clouds but the angular size is smaller, and that star clusters in NGC 6822 look like those in the Magellanic Clouds but smaller and dimmer, and finally that one can apparently distinguish some bright individual stars in NGC 6822 but they are fainter than the brightest stars in the Magellanic Clouds. Each of these points suggested to Shapley that NGC 6822 is about 300 kpc away. There seemed to be good reason, therefore, to believe that NGC 6822 is an independent star system, roughly on a par with the Milky Way Galaxy, although smaller in radius by a factor of perhaps 40.
Having the distances to some nearby galaxies from the Cepheid variables, Hubble could find the distances to other galaxies with the help of another indicator, the brightest star in the galaxy. The brightest stars seem to have approximately constant absolute magnitude, and they are a more powerful distance indicator just because they can be picked out at greater distance. Hubble calibrated the absolute magnitude of the brightest star in the few galaxies to which he had distances from Cepheid variables. For the larger sample of galaxies in which he could pick out what seemed to be individual bright stars he formed an estimate of the typical absolute magnitude of a galaxy, which he could then use to carry his distance estimates to still more remote galaxies.
In 1929 Hubble published his discovery that the apparent recession velocity of a galaxy, as determined from Slipher's observations of the shift of the spectral lines, and corrected for the peculiar velocity of the Solar System (mainly, this is the rotation velocity about the center of the Galaxy), is directly proportional to his estimate of the distance of the galaxy,
v = Hl,
H [congruent to] 500 km sec-1 Mpc-1. (1)
The constant of proportionality H (Hubble used K) is now called Hubble's constant, and as described in Chapter III the best modern value is a factor 5-10 lower than (1).
The linear connection between redshift and distance was anticipated by some theoretical work prior to 1929. In 1923 Weyl had remarked that in a deSitter model the motion of a set of test particles with appropriately chosen initial conditions will at any later time satisfy a linear relation of the form (1) when l is small enough. Robertson rediscovered this in 1928, and he even remarked that Slipher's redshift data taken together with the data Hubble was accumulating on distances to the nebulae roughly verify the linear expression. In Measure of the Universe North suggests that Hubble was influenced by Robertson in arriving at his law of the general recession. Certainly Robertson was at Caltech in 1927-1929, and Hubble was aware of the "deSitter effect" because he mentions it in an uncharacteristically theoretical last paragraph of his paper. However, in view of Hubble's healthy distrust of theorists (as expressed for example in Realm of the Nebulae), I would be surprised if he had in mind anything more than the simplest reasonable expression of his results.
The deSitter model with appropriate initial values for the test particles is a special limiting case of the evolving homogeneous and isotropic cosmological models. These models all imply the relation (1), where in general H could be positive or negative (galaxies receding or approaching). The evolving models were derived independently by Friedman and Lemaître. In Lemaître's discussion the line element in comoving coordinates (that is, each galaxy has fixed spatial coordinates r, θ, φ) may be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where R is a constant and the function a(t) now is called the expansion parameter. On substituting the indicated components of the metric tensor into Einstein's gravitational field equations, Lemaître obtained a differential equation for the expansion parameter,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)
where the mass density ρ includes the mass equivalent of any energy present (like electromagnetic radiation), and Λ is Einstein's Cosmological Constant. Lemaître's second differential equation needed to specify the cosmological model is
dρ/dt + 3(ρ + P/c2) [??]/a = 0. (4)
He correctly identified this as the expression of local energy conservation, the term 3P [??]/a representing the work done per unit volume against the pressure P of the fluid.
Lemaître was clearly aware of the observational significance of his evolving cosmological model. He recognized that the galaxies might map out the large-scale structure of the Universe. He derived the linear relation (1) when l is small, and he stated without reference a value for the constant of proportionality, H = 600 km sec-1 Mpc-1. This is so close to Hubble's value that there must have been communication of some sort between the two. Although the theory of the evolving universe clearly was anticipated by Friedman, it was Lemaître who had the great good fortune to have derived the expanding cosmological model at the right time, when the basic phenomenon, the law of the general recession of the galaxies, was just becoming clarified, and Lemaître recognized the significance of this phenomenon. According to the usual criterion for establishing credit for scientific discoveries Lemaître deserves to be called the "Father of the Big Bang Cosmology."
The dramatic agreement of theory and observation in the expression (1) was considered strong evidence in favor of Lemaître's picture. Thus Einstein promptly advised that his original cosmological model be abandoned, and with it the Cosmological Constant he had introduced for the purpose of constructing the model. There was very soon some discussion of possible interconnections of the supposed expansion of the Universe with other phenomena, as is the proper course in a physical science. Before describing this let us consider a few subsequent points of clarification of the theory of the Lemaître cosmological model.
b) Nature of the Lemaître Model
Robertson showed that, in a general relativity world picture assumed to be homogeneous and isotropic, the most general line element is the expression (2) with the constant R-2 allowed to be positive or negative or zero. In this expression the coordinates are comoving, that is the spatial coordinates are tied to the matter. Later (1935) Robertson and Walker independently demonstrated that in any homogeneous and isotropic world model based on a Riemannian geometry the expression (2), with the constant R-2 and the function a(t) freely chosen, is the most general possible line element (cf. Sec. VI-a-i). For this reason (2) is known as the Robertson-Walker line element.
There is a direct connection between Hubble's law and the assumed homogeneity and isotropy of the Universe. According to Robertson and Walker homogeneity and isotropy imply the line element (2). The line element fixes the proper time or distance interval (as measured with real rods and clocks) between two points (events) in space-time in terms of the coordinate interval between the points. For example, if two points are separated by the interval dt = dθ = dφ = 0, dr > 0, the proper distance between the points is dl = adr (1 – r2/R2)-1/2. We can fix the origin of coordinates on any chosen galaxy, like our own. Then the proper distance to another galaxy at coordinate distance r << [absolute value of R] is
l(t) = a(t)r. (5)
Because the radial coordinate r of a chosen galaxy is fixed, the coordinates being comoving, the velocity of recession of the galaxy relative to our own is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
which is Hubble's law.
Apparently Milne first pointed out that the form of Hubble's law is an immediate consequence of the assumed homogeneity and isotropy of the Universe. He showed this by an even simpler argument, which in modified form may be stated as follows. Let us consider three galaxies separated by distances so large that local irregularities like the Local Group may be ignored, but so small that we can ignore relativistic effects in the expansion. If the Universe is expanding in a homogeneous and isotropic way the triangle defined by the three galaxies must at all times remain similar to the original triangle. This means that the length of each side has to scale by the same factor, say a(t), as the Universe expands. By extending the net to a fourth galaxy, a fifth, and so on, we see that a(t) has to be a universal scale factor (when local irregularities are ignored). Thus the distance between two galaxies satisfies
l(t) = l0 a(t), (7)
where l0 is independent of time. Hubble's law follows as before (eq. 6).
(Continues...)
Excerpted from Physical Cosmology by Phillip James Edwin Peebles. Copyright © 1971 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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