The Universe Of Friedmann

The theory of relativity allows us to complete our description of space with a variable radius by introducing here some dynamic considerations.

As before, we shall represent it as being in the interior of a sphere, the center of which is a point which we can choose arbitrarily. This sphere is not the boundary of the system, it is the edge of the map or of the diagram which we have made of it. it is the place at which the two opposite, half straight lines are soldered into a closed straight line.

Cosmic repulsion is manifested as a force proportional to the distance to the center of the diagram. As for the gravitational attraction, it is known that, in the case of distribution involving spherical symmetry around a point, and that is certainly the case here, the regions farther away from the center than the point being considered have no influence upon its motion; as for the interior points, they act as though they were concentrated at the center.

By virtue of the homogeneity of the distribution of matter, the density is constant, the force of attraction which results is thus proportional to a distance, just as is cosmic repulsion.

Therefore, a certain density exists, which we shall call the density of equilibrium or the cosmic density, for which the two forces will be in equilibrium.

These elementary considerations permit recognition, in a result which calculation gives and which is contained in Friedmann’s equation:

( dR)2 2M R2 dt =-i+ if+;P'

The last term represents cosmic repulsion (it is double the function of the forces of this repulsion). T is a constant depending on the value ofthe cosmological constant and able to replace this. The next-to-iast term is double the potential of attraction due to the interior mass. The radius of space R is the distance from the origin of a point of angular distance (] =i. if one multiplied the equation by ~, one would have the<. corresponding equation for a point at any distance.

That which is remarkable in Friedmann’s equation is the first term -i. The elementary considerations which we have just advanced would allow us to assign it a value which is more or less constant; it is the constant of . energy in the motion which takes place under the action of two forces. The complete theory determines this constant and thus links the geometric properties to the dynamic properties.

Einstein’s Equilibrium

SINCE BY VIRTUE OF EQUATIONS, the radius R is set constant, the state of the universe in equilibrium, or Einstein’s universe, is obtained. The conditions of the universe in equilibrium are easily deduced from Friedmann’s equation:

T 31 T RE = fj; PE = 4'1T T2; M = fj'

In these formulas, the distances are calculated in light-time, which amounts to taking the velocity of light c as equal to unity, but, in addition, the unit of mass is chosen in such a way that the constant-of gravitation may also be equal to unity.

It is easy to pass on to the num~rical values of c.g.s. by re-establishing in the formulas the constants c and G in such a manner as to satisfy the equations of dimension. in particular, if one takes T as being equal to 2 Xi09 years, as we shall suppose in a moment, one finds that the density PE is equal toi0-27 gram per cubic centimeter.

These considerations can be extended to a region in which the distribution is no longer homogeneous and where even the spherical symmetry is no longer verified, provided that the region under consideration be of small dimension.

In a small region, Newtonian mechanics is always a good approximation.

Naturally, it is necessary, in applying Newtonian mechanics, to take account of cosmic repulsion but, aside from this easy modification, it is perfectly legitimate to utilize the intuition acquired by the practice of classic mechanics and its application to systems which are more or less complicated.

Among other things, it can be noted that the equilibrium of which we have just spoken is unstable and that the equilibrium can even be disturbed in one sense, in one place, and in the opposite sense in another region.

Perhaps it is necessary to mention here that Friedmann’s equation is only rigorously exact if the mass M remains constant. While one takes account of the radiation which circulates in space and also of the characteristic velocities of the particles which cross one another in the manner of molecules in a gas and, as in a gas, give rise to pressure, it is necessary to consider the work of this pressure during the expansion of space, in the evaluation ofthe mass or the energy. But it is apparent that such an effect is generally negligible, as detailed researchers elsewhere have shown.

The Significance Of Clusters Of Nebulae

We are now in a position to take up again the description which we had begun of the expansion of space, following the disintegration of the primeval atom. We had shown how, in a first period of rapid expansion, gaseous clouds must have been formed, animated by great, proper velocities. We are now going to suppose that the mass M is slightly larger than fj'

The second member of Friedmann’s equation will thus be able to become smaller, but it will not be able to vanish. Thus, we may distinguish three phases in the expansion of space.

The first rapid expansion will be followed by a period of deceleration, during the course of which attraction and repulsion will virtually bring themselves into equilibrium. Finally, repulsion will definitely prevail over attraction, and the universe will enter into the third phase, that of the resumption of expansion under the dominant action of cosmic repulsion.

Let us consider the phase of slow expansion in more detail.

The gaseous clouds are undoubtedly not distributed in a perfectly uniform manner. Let us consider in a region sufficiently small, and that only from the point of view of classical mechanics, the conflict between the forces of repulsion and attraction which almost produces equilibrium.

We easily see that as a result oflocal fluctuations of density, there will be regions where attraction will finally prevail over repulsion, in spite of the fact that we have supposed that, for the universe in its entirety, it is the contrary which takes place.

These regions in whi.ch attraction has prevailed will thus fall back upon themselves, while the universe will be entering upon a period of renewed expansion. We shall obtain a universe formed of regions of condensations which are separated from one another. Will not these regions of condensations be elliptical or spiral nebulae? We shall come back to this question in a moment

Let us note that, although it is of rare occurrence, it will be possible for large regions where the density or the speed of expansion differ slightly from the average to hesitate between expansion and contraction, and remain in equilibrium, while the universe has resumed expansion.

Could these regions not be identified with the clus~ers of nebulae, which are made up of several hundred nebulae located at relative distances from one another, which are a dozen times smaller than those of isolated nebulae? According to this interpretation, these clusters are made up of nebulae which are retarded in the phase of equilibrium; they represent a sample of the distribution of matter, as it existed everywhere, when the radius of space was a dozen times smaller than it is at present, when the universe was passing through equilibrium.

The Findings Of De Sitter

This interpretation gives the explanation for a remarkable coincidence upon which de Sitter insisted strongly, in the past. Calculating the radius of the universe in the theory which bears his name, that is, ignoring the presence of matter and introducing into the formulas the value T subH given by the observation of the expansion, he obtained a result which scarcely differs from that which is obtained, in Einstein’s totally different theory of the universe, by introducing into the formulas the observed value of the density of matter.

The explanation of this coincidence is, according to our interpretation of the clusters of nebulae, that, for a value of the radius which is a dozen times the radius of equilibrium, the last term in Friedmann’s formula greatly prevails over the others.

The constant T which figures in it is therefore practically equal to the observed value T subH: but since, in addition, the clusters are a fragment of Einstein’s universe, it is legitimate to use the relationship existing between the density and the constant T for them.

For T = TH one finds, as we have seen, that the density in the clusters must be 10(27) gram per cubic centimeter, which is the value given by observation. This observation is based on counts of nebulae and on the estimate of their mass indicated by their spectroscopic velocity of rotation.

In addition to this argument of a quantitative variety, the proposed interpretation also takes account of important facts of a qualitative order. it explains why the clusters do not show any marked central condensations and have vague forms, with irregular extensions, all things which it would be difficult to explain if they formed dynamic structures controlled by dominant forces, as is manifestly the case for the starclusters or the elliptical and spiral nebulae.

it also takes into account a manifest fact which is the existence oflarge fluctuations of density in the distribution ofthe nebulae, even outside the clusters.

This must be so, in fact, ifthe universe has just passed through a state of unstable equilibrium, a whole gamut of transition between the properly-termed clusters which are still in equilibrium, while passing through regions where the expansion, without being arrested, has nevertheless been retarded, in such a manner that these regions have a density which is greater than the average.

This interpretation permits the value of the radius at the moment of equilibrium to be determined at a billion light-years, and thus 10(10) lightyears for the present value of the radius. Since American telescopes prospect the universe as far as half a billion light-years, one sees that this observed region already constitutes a sample of a size which is not at all negligible compared to all space; hence, it is legitimate to hope that the values of the coefficient of expansion TH and of the density, obtained for this restricted domain, are representative of the whole.

The only indeterminacy which exists is that which relates to the degree of approximation with which the situation of equilibrium has been approached. it is on this value which the estimate of the duration of expansion depends. Perhaps it will be possible to estimate this value by means of statistical considerations regarding the relative frequency of the clusters, compared to the isolated nebulae.

The Proper Motion Of Nebulae

Now we must come back to the question of the formation of nebulae from the regions of condensation. We have seen that the characteristic velocities, or the relative velocities of gaseous clouds, which cross one another in the same place, must have been very large. Since certain of them, because of a density which is a little too large, form a nucleus of condensation, they will be able to retain the clouds which have about the same velocity as this nucleus.

The proper velocity of the cloud so formed will hence be determined by the velocity of the nucleus of condensation. The nebulae formed by such a mechanism must have large relative velocities. in fact, that is what is observed in the clusters of nebulae. in the one which has been best studied, that of Virgo, the dispersion of the velocities about the mean velocity is 650 kilometers per second.

The proper velocity must have been the proper velocity of all the nebulae at the moment of passage through equilibrium. For isolated nebulae, this velocity has been reduced to about one-twelfth, as a result of expansion, by the same mechanism which we have explained with reference to the formation of gaseous clouds.

The Formation Of Stars

The density of the clouds is, on the average, the density of equilibrium 10(27). For this density of distribution, a mass such as the Sun would occupy a sphere of one hundred light-years in radius. These clouds have no tendency to contract. in order that a contraction due to gravitation can be initiated, their density must be notably increased. This is what can occur if two clouds happen to collide with great velocities. Then the collision will be an inelastic collision, giving rise to ionization and emission of radiation. The two clouds will flatten one another out, while remaining in contact, the density will be easily doubled and condensation will be definitely initiated. It is clear that a solar system or a simple or multiple star may arise from such a condensation, through known mechanisms. That which characterizes the mechanism to which we are led is the greatness of the dimensions of the gaseous clouds, the condensation of which will form a star. This circumstance takes account of the magnitude of the angular momentum, which is conserved during the condensation and whose value could only be nil or negligible if the initial circumstances were adjusted in a wholly improbable manner. The least initial rotation must give rise to an energetic rotation in a concentrated system, a rotation incompatible with the presence of a single body but assuming either multiple stars turning around one another or, simply, one star with one or several large planets turning in the same direction.

The Distribution Of Densities Of Nebulae

Here is the manner in which we can picture for ourselves the evolution of the regions of condensation. The clouds begin by falling toward the center, and by describing a motion of oscillation following a diameter from one part and another of the center. in the course of these oscillations, they will encounter one another with velocities of several hundreds of kilometers per second and will give rise to stars.

At the same time, the loss of energy due to these inelastic collisions will modify the distribution of the clouds and stars already formed in such a manner that the system will be further condensed. it seems likely that this phenomenon could be submitted to mathematical analysis.

Certain hypotheses will naturally have to be introduced, in such a way as to simplify the model, so as to render the calculation possible and also so as artificially to eliminate secondary phenomena. There is scarcely any doubt that there is a way of thus obtaining the law of final distribution of the stars formed by the mechanism described above. Since the distribution of brilliance is known for the elliptical nebulae and from that one can deduce the densities in these nebulae, one sees that such a calculation is susceptible of leading to a decisive verification of the theory.

One of the complications to which i alluded, a moment ago, is the eventual presence of a considerable angular momentum. in excluding it, we have restricted the theory to condensations respecting spherical symmetry, that is, nebulae which are spherical or slightly elliptical. it is easy to see what modification will bring about the presence of considerable angular momentum. it is evident that one will obtain, in addition to a central region analogous to the elliptical nebulae, a flat system analogous to the ring of Saturn or the planetary systems, in other words, something resembling the spiral nebulae. in this theory, the spiral or elliptical character of the nebula is a matter of chance; it depends on the fortuitous value of the angul~ momentum in the region of condensation. it can no longer be a question of the evolution of one type into another. Moreover, the same thing obtains for stars where the type of the star is determined by the accidental value of its mass, that is, of the sum of the masses of the clouds whose encounter produced the star.

Distribution Of Supergiant Stars

if the spirals have this origin, it must follow that the stars are formed by an encounter of clouds in two very distinct processes. in the first place, and especially in the central region, the clouds encounter one another in their radial movement, and this is the phenomenon which we have invoked for the elliptical nebulae. Kapteyn’s preferential motion may be an indication of it.

But besides this relatively rapid process, there must be a slower process of star formation, beginning with the clouds which escaped from the central region as a result of their angular momentum. These will encounter one another in a to-and-fro motion, from one side to another of the plane of the spiral.

The existence of th.ese two processes, with different ages, is perhaps the explanation of the fact that supergiant stars are not found in the elliptical nebulae or in the nucleus of spirals, but that one observes them only in the exterior region of the spirals. in fact, it is known that the stars radiate energy which comes from the transformation of their hydrogen into helium.

The supergiant stars radiate so much energy that they could only maintain this output during a hundred million years. it should be understood, thus, that, for the oldest stars, the supergiants may be extinct for lack of fuel, whereas they still shine where they have been recently formed.