Astronomers have long been acquainted with certain rather peculiar-looking
Celestial objects which had come to be called “spiral nebulae”. In contrast to other known nebulosities, which usually have irregular shapes and look more or less like clouds in the sky, spiral nebulae always have well-developed structures, consisting of a lenticular central body with a pair of spiral arms winding around it (Plates I and II). Until about a quarter of a century ago, the spiral nebulae were more or less generally believed to be located among the stars of our Milky Way system and were thought to be possible examples of stars giving birth to their own planetary systems according to the classical picture of Kant and Laplace.
However, in 1925, all these views were completely overthrown by a great discovery made by Edwin P. Hubble, an astronomer at Mount Wilson Observatory. Studying the Great Spiral Nebula of Andromeda, which is visually the largest of them all and therefore the most accessible to observation, he noticed that its spiral arms contain a number of extremely faint stars whose brightness changes periodically, following a simple sine law. Such stars, called “Cepheid variables” (after Delta Cephei, the first star in which such variability was noticed), are well known in our Milky Way system, and their periodic changes in luminosity are explained as the result of periodic pulsations of their giant bodies. A simple correlation exists between the period of these pusations and the absolute luminosity of the star in question: the brighter the star the longer the period of pulsation. This so-called “period-luminosity relation” established by the Hardvard astronomer Harlow Shapley is a powerful tool for measuring the distances of pulsating stars which are too far away to show a parallax displacement. By measuring directly the pulsation period of a given star, we can arrive at a definite conclusion about its absolute brightness. This, in combination with the visual brightness, tells us the actual distance of the star.
The observed pulsation periods of the Cepheids found by Hubble in the spiral arms of the Andromeda nebula indicated that they must possess very high absolute luminosities. On the other hand, they are so faint visually as to be at the limit of visibility. The inevitable conclusion was that they – and also the nebula itself – must be extremely far away. The distance to the Andromeda nebula estimated by that method worked out to almost 1 million light-years, that is, about a hundred times the diameter of the entire Milky Way system! Other spiral nebulae, which are visually smaller and fainter than the one in Andromeda, must be much farther away. If the spiral nebulae are really that far off, they must also much larger than originally suspected; in fact they must be about as large as the Milky Way system itself!
Thus Hubble’s discovery removed the spiral nebulae from their former humble position as common members of our galaxy and enthroned them as independent galaxies in their own right, scattered through the vast expanse of the universe. It became clear that these objects have nothing to do with ordinary nebulae (like the one in Orion), which are really only large clouds of dust floating in interstellar space. The spiral nebulae are formed by many billions of individual stars which blur together into one faintly luminous mass only because of their exceedingly great distance from the observer. More recently this conclusion was proved directly by another Mount Wilson astronomer, Walter Baade, who was able to resolve photographically the central body of the Andromeda nebula and those of its two companions into myriads of tiny luminous dots representing the individual stars from which these distant systems are built. It seems advisable to change the old terminology, and instead of talking about spiral nebulae to talk about spiral galaxies.
Hubbles’s discovery proved to be the germ of still more remarkable progress in our knowledge about the nature of the universe. It had been known for some time that spectral lines in the light emitted by spiral nebulae show a shift toward the red end of the spectrum. Interpreted in terms of ordinary Doppler effect, this meant that these objects were moving away from the observer. As long as these objects were believed to be members of our stellar system, one had to conclude that they had some peculiar motion among the stars, being driven from the central regions of the Milky Way toward its periphery. With the new broadening of horizons a completely new picture emerged: the entire space of the universe, populated by billions of galaxies, is in a state of rapid expansion, with all its members flying away from one another at high speed.
If the expansion of the space of the universe is uniform in all directions, an observer located in any one of the galaxies will see all other galaxies running away from him at velocities proportional to their distances from the observer. This can be easily demonstrated by gluing a number of pieces of paper (cut in the shape of galaxies, if desired) to the surface of a rubber ballon and blowing up the ballon to larger and larger size. An observer located in any one of these model galaxies will see that all the others run away from him and he may be inclined to believe (incorrectly) that he is at the center of expansion. A three-dimensional model of an expanding space is given by the “jungle gym” often found on children’s playgrounds. We need only imagine that the jungle gym extends in all directions without any limit, and is built of telescoping tubes so that the distances between the children sitting at various intersections are gradually increasing.
In addition to regular recession velocities resulting from the uniform expansion of the entire system, galaxies also possess individual random motion similar to the thermal motion of gas molecules. Since the velocities of that random motion are comparable in magnitude to the recession velocities of neighboring galaxies, the two kinds of motion may sometimes counteract each other and thereby produce confusing results. The component of the random velocity of a neighboring galaxy directed toward us may happen to be larger than the corresponding recession velocity of that galaxy. In that case the galaxy is moving toward us, and will show a violet shift in the lines of its spectrum. This is actually observed in the case of the Great Spiral Nebula of Andromeda. However, at greater distances the everincreasing recession velocities soon become too large to be reversed by the random “thermal” motion of individual galaxies, and the expansion of the whole system becomes quite evident.
In addition to the mutual recession and the individual random motion, galaxies also show various degrees of rotation around their axes. A very few galaxies apparently do not rotate at all and possess regular spherical form. Other galaxies rotate with various speeds, shown by various degrees of elongation of their elliptical bodies. Most of the galaxies, however, rotate so fast that some of the material flows out from their equatorial bulges, forming the characteristic pattern of spiral arms. It is interesting to note that the average kinetic energy of their random translatory motion. This fact agrees with the general law of statistical mechanics, which also holds for the translatory and rotational motions of molecules in an ordinary gas.
The theory of the expanding universe
It was first noticed by the Belgian scientist George Lemaitre that the observed expansion of the system of galaxies agrees with the cosmological conclusions of the general theory of relativity. It is true that Einstein’s original model of the so-called “spherical universe” is mechanically static and does not provide either for expansion or contraction. However, the Russian mathematician A. Friedmann pointed out that the static nature of Einstein’s universe was the result of an algebraic mistake (essentially a division by zero) made in the process of its derivation. Friedmann then went on to show that the correct treatment of Einstein’s basic equations leads to a class of expanding and contracting universes. In particular, the “spherical universe” originally conceived by Einstein was shown to be dynamically unstable, and apt to start contracting or expanding at the slightest provocation. Within the scope of this book we cannot delve either into a detailed mathematical treatment of the fundamental tensor equations of the Einstein theory, or into their application to the problems of cosmology. Fortunately, however, many properties of the relativistic cosmological models can be described and understood on the basis of classical Newtonian theory with the use of very little mathematics.
Imagine the limitless space of the universe with a multitude of galaxies scattered through it more or less uniformly. Can these galaxies stay at rest with respect to one another? Apparently not, since the force of Newtonian gravity acting between individual galaxies will pull them together so that the entire system will collapse. If we assume that the galaxies were not originally in a state of rest but were moving away from one another at high initial velocities, then the distances between the neighboring galaxies will continue to increase and the entire system will expand. We have here a situation comparable to that of a sports reporter inspecting a photograph showing a football high in the air above the heads of the players. Even though the picture does not show any motion, he is sure that the ball is either still rising or already falling. It could not have been hanging motionless above the field-unless, of course, it was resting on top of a pole or suspended on a string. In the same way the assumption of a static system of galaxies under mutual gravitational attraction necessitates the introduction of additional forces which prevent the galaxies from drawing together. We must imagine a system of struts which counteract the forces of gravity and keep the galaxies apart. The mathematical apparatus of the general theory of relativity (which is nothing but a glorified generalization of the old Newtonian theory of gravity) leads to exactly that conclusion. In order to obtain a model of a static universe (this was before Hubble’s discovery), Einstein was forced to introduce into his general equations an additional term containing the so-called “cosmological constant” which was physically equivalent to the assumption of a universal repulsion between material bodies. In contrast to other physical forces which always decrease with distance, cosmological repulsion was assumed to be very weak over short distances but important at intergalactic distances. Einstein’s introduction of this force was not quite arbitary, since, as it was shown later, this term represents a logical mathematical generalization from the original equations of general relativity. When the fact that our universe is not static but is rapidly expanding was recognized, the introduction of cosmological constant became superfluous. However, as we see later, this constant may still be of some help in cosmology, even though the primary reason for its introduction has vanished.
The discovery that our universe is expanding provided a master key to the treasure chest of cosmological riddles. If the universe is now expanding, it must have been once upon a time in a state of high compression. The matter which is now scattered through the vast empty space of the universe in tiny portions which are individual stars must at that time have been squeezed into a uniform mass of very high density. It must have been subjected to extremely high temperatures, since all material bodies are heated when compressed and cooled when expanded. At present the possible maximum density of this compressed primordial state of matter is not accurately known. The nearest guess is that the over-all density of the universe at that time was comparable to that of nuclear fluid, tiny droplets of which form the nuclei of various atoms. This would make the original pre-expansion density of the universe a hundred thousand billon times greater than the density of water; each cubic centimeter of space contained at that time a hundred million tons of matter! In such a highly compressed state all the matter which is now within the reach of the 200-inch telescope must have occupied a sphere only thirty times as large as the sun. But since the universe is, and always was, infinite, the space outside of that sphere was also occupied by matter, the matter, the matter which now lies beyond the reach of the 200-inch telescope.
The fact that material occupying an infinite space can be squeezed or expanded and still occupy the same infinite space is one of the so-called “paradoxes of infinity”. It is best illustrated by an example given by a famous German mathematician, David Hilbert, in one of his lectures.
“Imagine”, said Hilbert, “a hotel with a finite number of rooms, all rooms being occupied. When a new client arrives, the room clerk must turn him down with regrets. But let us imagine a hotel with an infinite number of rooms. Even if all these rooms are occupied, the room clerk will be glad to accommodate a new customer. All he has to do is to move the occupant of the first room into the second, the occupant of the second into the third, the occupant of the third into the fourth, und so weiter… . Thus the new customer can get into the first room. Imagine now a hotel with an infinite number of rooms, all occupied”, continued Hilbert, “and an infinite number of new customers. The room clerk will be glad to oblige. He will move the occupant of the first room into the second, the occupant of the second into the fourth, the occupant of the third into the sixth, und so weiter… . Thus every second room (all odd numbers) will now be free to accommodate the infinity of new customers.”
In exactly the same way that an infinite hotel can accommodate an infinite number of customers without being overcrowded, an infinite space can hold any amount of matter and, whether this matter is packed far tighter than herrings in a barrel or spread as thin as butter on wartime sandwich, there will always be enough space for it.
What started the expansion?
We can now ask ourselves two important questions: why was our universe in such a highly compressed state, and why did it start expanding? The simplest, and mathematically most consistent, way of answering these questions would be to say that the Big Squeeze which took place in the early history of our universe was the result of a collapse which took place at a still earlier era, and that the present expansion is simply an “elastic” rebound which started as soon as the maximum permissible squeezing density was reached. As was indicated in the previous section, we do not know exactly what was the density reached at the maximum of compression, but according to all indications this density could have been very high indeed. Most likely the masses of the universe were squeezed together to such an extent that any structural features which may have been existing during the “pre-collapse era” were completely obliterated, and even the atoms and their nuclei were broken up into the elementary particles (protons, neutrons, and electrons) from which they are built. Thus nothing can be said about the pre-squeeze era of the universe, the era which may properly be called “St. Augustine’s era”, since it was St. Augustine of Hippo who first raised the question as to “ what God was doing before He made heaven and earth”. As soon as the density of the masses of the universe reached its maximum value, the direction of motion was reversed and the expansion started, so that very high densities could have existed only for a very short time. During the earlier or later stages of this expansion, various differentiation processes must have taken place in the cosmic masses, processes which resulted in the present highly complex structure of our universe.
The date of the Big Squeeze
We have seen that Hubble’s observations indicate that the galaxies scattered through the vast space of the universe seem to be running away from us, and the farther they are the faster they run. The relationship between recession velocity and distance is given by Hubble’s law:
If the distances are expressed in light-years, and the velocities in kilometers per second, the numerical value of the constant in Hubble’s equations is 1.8*10^(-4). Thus, for example, a galaxy located 1 million light-years away recedes with the velocity of 180 kilometers per second. If, as is customary in physics, we measure distances in centimeters and velocities in centimeters per second, the numerical value of the constant in Hubble’s law becomes 1.9*10^(-7). In order to find out at what date in the past all galaxies were packed tightly together, we need only to divide their present mutual distances by the velocities of their recession. Since the recession velocities are directly proportional to the distances, the result of the division is always the same, no matter whether we take two neighboring or two distant galaxies. It is simply the inverse value of the constant in Hubble’s law. Thus we get for the date the value
1/1.9*10^(-17)=5.3*10^16 sec=1.7*10^9 yr
If we imagine the expansion process in reverse, so that the galaxies eventually run into one another, the difference between the moment at which the galaxies coalesce and the moment at which the stars and atoms forming these galaxies are destroyed will not exceed 1 per cent of the value given. This results from the fact that the distances between the galaxies right now are hundreds of times greater than their diameters. We may accept the value of 1.7*10^9 years as the best astronomical estimate of the date of the Big Squeeze.
Comparing this figure with the various other estimates of the age of the universe (see Chapter I), we find that it falls short of the average. In particular it is only about one-half of the figure obtained by Holmes from the study of the relative abundances of radiogenic lead isotopes.
This discrepancy between astronomical and geological estimates could be remedied by a modification of the theory of the expansion process. As Lemaitre first proposed, one could introduce the cosmological constant originally used by Einstein for a model of a static universe. This cosmological constant corresponds physically to the introduction of repulsive forces acting between galaxies over long distances. The presence of such forces would make our universe expand with ever-increasing velocity and change the position of zero point in time. If the expansion process is accelerated, the recession velocities of the neighboring galaxies would have been smaller in the past than today, so that the date of the beginning would be shifted back in time. Assuming such a small numerical value as 10^(-33) sec^(-1) for the cosmological constant LAMDA, one could bring Hubble’s original value into agreement with the geological estimate.
Another and much more radical modification of the expansion theory was recently proposed by two English mathematicians, H. Bondi and T. Gold. This modification (enthusiastically endorsed by the British astronomer Fred Hoyle, who used it as the cornerstone of his “new cosmogony”) is based on the assumption that the thinning of matter in the universe caused by continuous expansion is compensated for by continuous creation of new matter, taking place uniformly throughout intergalactic space. To compensate fully for expansion not more is required than the production of one new hydrogen atom per gallon of expanding space once every 250 million years, so the creative genie would not overstrain himself doing the job. According to these views, the older galaxies are gradually receding farther and farther, but all the time new galaxies are being formed by condensation of newly created matter in the widening spaces between them. Thus the show goes on without a beginning and without an end. If we were to make a motion picture representing the views of Bondi, Gold, and Hoyle, and run it backward, it would seem at first that all the galaxies on the screen were going to pile up as soon as we reached the date of 1.7 billion years ago. But as the film continued to run backward, we would notice that the nearby galaxies, which were approaching our Milky Way system from all sides, threatening to squeeze it into a pulp, would fade out into thin space long before they became a real danger. And before the second-nearest neighbors could converge on us (at about 3.4 billion years back in time), our own galaxy would fade out too. While this point of view provides for the origin and evolution of individual galaxies, it considers the universe itself as being eternal, though with a constantly changing galactic population.
This hypothesis, which may look very attractive to those who find it difficult philosophically to imagine a beginning in time, experiences serious difficulties on theoretical as well as on observational grounds. Because it fails to provide for the original epoch of the Big Squeeze, it is unable to account for the origin of atomic species (see Chapter III). Unless, of course, one assumes (and why not!) that the atoms of the various chemical elements and their isotopes were formed in proper proportions by the postulated process of continuous creation of matter. Hoyle himself is inclined to believe that, while hydrogen atoms are continuously created from nothing, atoms of heavier elements are cooked later in the hot interiors of stars, and scattered through space by violent stellar explosions (supernovae). This point of view has so far failed to provide us with a satisfactory quantitative explanasion of the observed abundances of chemical elements and is, in the opinion of the author, artificial and unreal.
Though the theoretical objections to the hypothesis of the steady-state universe are to a large extent a matter of opinion, observational evidence which seems to contradict these view is a different story. Several years ago two American astronomers, J. Stebbins and A. E. Whitford, decided to study color distribution in far-distant galaxies. They picked out four groups, or clusters, of galaxies located in the constellations of Virgo, Coma Berenices, Coron a Borealis, and Bootes. (The estimated distances are 6, 40, 140, and 240 million light-years, respectively.) To their own surprise, and that of everybody else, Stebbins and Whitford found that the color of these galaxies differs according to their distances from us – the father away they are, the redder they look. This general change of color, which is different from the Doppler red shift of spectral lines, could of course be explained by assuming the presence of dust in intergalactic space. The phenomenon would then be similar to the reddening of the setting sun near the horizon. But, in order to explain the observed reddening by the presence of dust, one would have to introduce so much dust that one would run into difficulties with other astronomical facts. Besides, the most recent observations seem to indicate that the reddening is confined to elliptical (armless) galaxies and that no such effect is present in spirals. If the reddening were due to intergalactic dust, the effect would not depend on the type of galaxy. It is now almost unanimously agreed among astronomers that the observed reddening of distant galaxies must be the result of the relative numbers of red and blue stars which form them. But when we look at a galaxy which is, say, 300 million light-years away, we see it as it was 300 million years ago. Since the stellar populations of galaxies may be expected to evolve with time (see Chapter V) it is logical to conclude that galaxies could have contained more Red Giant stars in their youth than they do at a more mature age. If such an interpretation of the observed reddening is accepted (and there does not seem to be any alternative), we have no choice but to agree that the general properties of galaxies in the past differed from those of today. This strongly contradicts the hypothesis of a steady-state universe as proposed by Bondi, Gold, and Hoyle.
Fortunately it seems that neither the introduction of a cosmological constant (universal repulsion) nor the hypothesis of a steady-state universe (continuous creation of matter) will be actually needed in order to remove the contradiction between Holmes’ and Hubbles’s figures for the age of the universe. In fact, recent studies by the German astronomer A. Behr indicate that the commonly accepted values for intergalactic distances are not correct, and must be critically revised in the light of new observational evidence. Taking into account such technical points as the “dispersion of galactic luminosities”, the “deviation from the black body distribution”, and the “space reddening”, Behr came to the conclusion that the actual distances between the galaxies must be about twice as great as had been assumed so far. This will reduce by one-half the numerical coefficients in Hubble’s formula and thereby raise his estimate of the time of the Big Squeeze from the original 1.7 billion to about 3.4 billion years.
Will the expansion ever stop?
The question of whether the present expansion of our universe will continue forever or will at some time in the future stop, and then turn into a collapse, resembles the question of whether a rocket fired from the surface of the earth will continue to travel in space or stop and fall back on our heads. In the case of the rocket everything depends on the velocity which it acquired. If its velocity is more than 11.2 kilometers per second (the so-called “escape velocity”, unattainable at present) the rocket will defy gravity and never come back; for any lesser velocity the rocket will inevitably come to a stop at a certain altitude and then fall back. In the first case we say that the kinetic energy of the rocket was greater than the potential of the terrestrial gravitational field; in the second case the situation is reversed.
The case of galaxies receding from one another with given velocities against the forces of mutual gravity which try to pull them together is similar to the example of the rocket. The question is simply whether the inertial force of the galactic recession or the pull of their gravitational fields is more powerful. Simple calculations, presented in the Appendix, indicate that at the present epoch gravitational pull between galaxies is negligibly small as compared with their inertial velocities of recession. Here we have a case similar to that of a rocket moving away from the earth with a velocity far higher than escape velocity. The distances between the neighboring galaxies are bound to increase beyond any limit, and there is no chance that the present expansion will ever stop or turn into a collapse.
Is our universe finite or infinite?
This seems to be the proper moment to raise the question of the total size of our universe. Is it finite, having a volume of so many cubic feet, or so many cubic light-years, as Einstein once suggested, or does it extend without limit in all directions as visualized by food old Euclidian geometry? Within the distance of 1 billion light-years covered by the 200-inch telescope of Palomar Observatory the galaxies (about 1 billion of them) seem to be scattered through space in a more or less uniform fashion. But suppose astronomers go on building 400-inch, 800-inch, 1600-inch, etc., telescopes. What will they find? Einstein’s general theory of relativity and gravitation leads to two possible mathematical alternatives.
The first is that the space of the universe may “curve in” in the manner of the surface of the earth (positive curvature) and finally close upon itself in an “antipodal point”. This is Einstein’s closed universe (which can be either static or expanding), in which, barring space obscuration, one would be able to inspect the haircut on the back of one’s neck, if one could manage to wait for several billion years while the light travels around the universe. Or else the space of the universe may “curve out”, like the surface of a western saddle.
Which of these two possibilities must be ascribed to our universe, only observational evidence can decide. And the existing evidence seems to be strongly in favor of the second possibility, namely, the limitless infinite universe. From the purely mathematical point of view, whether the geometry of the universe is “open” or “closed” will directly determine its behavior in the course of time. It can be shown that a closed Einsteinian universe can expand only to a certain limit, beyond which the expansion will go over into contraction. But an open universe is bound to expand forever without any limit. One can say that if the universe is closed and periodic in space (as the surface of a sphere), it is also periodic in time and is subject to alternating expansions and contractions (pulsating universe). On the other hand, if the universe is open and aperiodic in space (as the surface of a paraboloid), it will not repeat itself in time either. Thus the same arguments which, in the previous section, led us to accept the limitless expansion of the universe also lead us to accept its limitless extension.
Of course these conclusions, which are based entirely on observational evidence within a sphere with a radius of 1 billion light-years, should not be necessarily extrapolated into a trans-Palomarian universe which is beyond our empirical knowledge. We cannot exclude the possibility that at some distance farther out the properties of space may change so radically that our present knowledge no longer applies. One may imagine, for example, that, although within the observed distances space does not show the slightest tendency to close, at greater distances it may suddenly “change its mind” and begin to close. The discussion of such a possibility is naturally outside the scope of empirical science. In this connection an interesting point of view expressed many years ago by the Swedish astronomer C. V. L. Charlier may be mentioned. It might be called the hypothesis of unlimited complexity. Charlier suggested that, just as the multitude of stars surrounding our sun belongs to a single cloud known as our galaxy, galaxies themselves form a much larger cloud, only a small part of which falls within the range of our telescopes. This implies that if we could go farther and farther into space we would finally encounter a space beyond galaxies. However, this supergiant galaxy of galaxies is not the only one in the universe, and much, much farther in space other similar systems can be found. In their turn these galaxies of galaxies cluster in still larger units ad infinitum! Intriguing as it is, this picture of an ever-increasing aggregation of matter is unfortunately outside the possibility of observational study.
Nebular counts, and the confusion between distances and ages
There is another method for finding out whether the space of the universe curves “in” or “out”, a method based on pure geometry and quite independent of the theory of expansion in time. It is connected with the question of how much space is available within a given distance and can be best understood by referring to a two-dimensional example. Suppose we cut out a circular piece of leather from an ordinary football and try to flatten it on the surface of a table. It is obvious that this cannot be done without stretching the leather at the outer parts of our piece. Since the original surface curved in, there is not enough, material at its peripheral portions. If we now repeat the same experiment with a piece of leather cut from a Western saddle, the situation is entirely different. There is too much leather at the rim and we would have to shrink it to make it flat. If the two surfaces are covered with an originally uniform pattern of dots, we find that after flattening there will be a rarification of dots near the rim of the spherical surface and a crowding of dots near the rim of the saddle surface. Speaking mathematically, the area of a circle drawn on a spherical surface increases more slowly than the square of the radius; on a saddle surface it increases faster than the square of the radius.
A similar situation exists in the case of three-dimensional curved space, even though, being inside of that space, we cannot visualize it as easily as we can two-dimensional surfaces which we can look at from the outside. Instead of dots on the leather we now have galaxies in space, presumably in uniform distribution. If space is finite and curves in, the number of galaxies must increase more slowly than the cube of that distance. If space is infinite and curves out, the number of galaxies should increase faster than the cube of that distance.
Such “galactic counts” have been made by Hubble, who found, in contradiction to the conclusions we reached in the previous section, that the space of the universe is curving in, and very fast indeed. However, Hubble’s result depends entirely on the correctness of the estimates of galactic distances. If, as was suggested by Behr, these distances are actually twice as large as previously thought, galactic counts lead to the opposite result, and the universe is found to be open and infinite. It must also be remembered that all the estimates of galactic distances are based on the assumption that the galaxies possess a constant luminosity; in fact, the distances are measured simply by using the inverse-square law for the visual brightness of a distant source. Since we see distant galaxies as they were hundreds of millions of years ago, the results of galactic counts will be substantially different if galaxies change their luminosities with time. As we have seen, such changes are to be expected, since, according to the observations of Stebbins and Whitford, the stellar content of individual galaxies seems to show definite evolutionary changes.
The attempt to combine Hubble’s method of galactic counts and the studies of Stebbins and Whiford on evolutionary changes of galactic luminosities places us in an unfortunate dilemma. We cannot use the method of galactic counts for the study of the curvature of space unless we first know how much and how fast the galaxies change their luminosity with age. On the other hand, a fruitful study of evolutionary changes of galactic luminosities is possible only if we have a reliable method for estimating their distances, which in turn requires a knowledge of the geometry of the universe. The only way to proceed seems to be to estimate the geometry of the universe from the theory of expansion (as was done in the previous section) and to use these results to decipher the observations pertaining to evolutionary changes of galactic luminosities.
Early stages of expansion
Mathematical studies of the expansion process, presented in some detail in the Appendix, indicate that the constant in Hubble’s law changes slowly with the progressing evolution of the universe. For comparatively early stages of the expansion, the value of Hubble’s constant is connected with the mean density of the universe by the relation:
[Hubble’s constant]=5.8*10^(-7)[mean density]
It must be pointed out that the mean density of the universe includes not only the density of ordinary matter but also the mass-density of radiation (light visible and invisible) filling space. We know that, according to Einstein’s famous principle of “ equivalence of mass and energy”, radiation possesses a certain weight which can be expressed numerically by dividing its energy by the square of the velocity of light. In everyday life the weight of radiation is so small that it might as well be neglected. Specifically, the weight of the visible light in a brightly illuminated room is negligible when compared with the weight of the air filling the same soon. In the cosmos, however, the situation is different, not so much because the weight of radiation is higher, as because the mean density of matter is so low. According to a well-known formula of classical physics (the so-called Stephan-Boltzmann formula), the amount of radiant energy per unit volume of space at temperature T (absolute temperature counted from the zero point at -273 degrees centigrade) is equal to 7.6*10^(-15)*T^4. Dividing this by the square of the velocity of light (c^2=9*10^20) we find for the weight of that radiant energy the value of 8.5*10^(-36)*T^4 grams per cubic centimeter. At normal room temperature (about 300 degrees absolute), the weight of radiation (in this case heat rays) is only 10^(-25) grams per cubic centimeter. In interstellar space, which is heated by stars and has a constant temperature of about 100 degrees absolute (near the temperature of liquid air), the density of radiation (very cold heat raysl) is 10^(-27) grams per cubic centimeter. Small as this is, it forms about 0.1 per cent of the density of interstellar gas, which is 10^(-24) grams per cubic centimeter.
From the laws of classical physics, we can derive the fact that the density of radiation in an expanding volume will drop faster than the density of matter in the same volume. We then have to assume that during the earlier stages of expansion the weight of the radiation in each volume of space exceeded that of the matter in the same volume. During these epochs ordinary matter did not count, and the main role was played by intensely hot radiation.
One may almost quote the Biblical statement: “In the beginning there was light”, and plenty of it! But, of course, this “light” was composed mostly of high-energy X rays and gamma rays. Atoms of ordinary matter were definitely in the minority and were thrown to and fro at will by powerful streams of light quanta.
The relation previously stated between the value of Hubble’s constant and the mean density of the universe permits us to derive a simple expression giving us the temperature during the early stages of expansion as the function of the time counted from the moment of maximum compression. Expressing that time in seconds and the temperature in degrees, we have:
Thus when the universe was 1 second old, 1 year old, and 1 million years old, its temperatures were 15 billion, 3 million, and 3 thousand degrees absolute, respectively. Inserting the present age of the universe (t=10^17 sec) into that formula, we find
T present=50 degrees absolute
which is in reasonable agreement with the actual temperature of interstellar space. Yes, our universe took some time to cool from the blistering heat of its early days to the freezing cold of today!
While the theory provides an exact expression for the temperature in the expanding universe, it leads only to an expression with an unknown factor for the density of matter. In fact, one can prove that
[density of matter]=constant/[time]^3/2
We see in Chapter III that the value of that constant may be obtained from the theory of the origin of atomic species.