Towards a Unified Cosmology

13.1: The Problem Defined
In the last chapter it was shown that, if matter originates continuously, as is asserted in (A3), a cloud must occasionally form in extragalactic space. But this in itself does not go far towards proving that a cosmological model based on (A3) resembles actuality. To prove this one must do much more: One must show that in such a model the undifferentiated accumulation of hydrogen that I have called a 'cloud' will evolve into the kind of structure to which the name galaxy has been given. The simple accumulation of hydrogen must acquire a mass of the same order of magnitude as is possessed by the observed galaxies; its distance from its nearest neighbours must be of the same order of magnitude; it must behave in a similar manner; it must have a similar volume, density, shape, detailed structure. Why, one is led to ask, should all this happen?

13.2: The Variables that Influence Growth
There is a substantial risk of drawing wrong inferences when constructing a cosmological model on the basis of any selected hypotheses, and one reason for this is that a large variety of circumstances must influence the evolution of an extragalactic cloud, and that most of them interact with each other in complicated ways. Hence great care must be taken to con- sider all these circumstances and to assess their effects. It will help towards the understanding of what is to follow if they are listed here. Let it be pointed out, therefore, that what happens to a cloud after it has begun to form on an astronomical summit must depend on:
(a) The rate of origin of new matter in the region, which is at present unknown.
(b) The rate of extinction of matter in the region, whichis also unknown.
(c) The known expansion of space, which causes the gravitational pull of distant nebulae on the substance of the cloud to decrease with time.
(d) Changes, if any, in the masses of neighbouring distant galaxies, which must occasion changes in the gravitational pull that they exert on the substances of the cloud.
(e) Changes in the density of the cloud.
(f) The gravitational field of the cloud itself, which must fundamentally influence the distribution of forces in the region where the cloud is, revers- ing the local potential gradient and causing particles to fall towards the cloud that previously fell towards a distant galaxy.
(g) Irregularities in the potential gradient around an astronomical summit.
(h) Shrinking of the cloud under the influence of its own gravitational field.
(i) Rotation of the cloud, which introduces centrifugal forces.

The length of the above list serves as a warning against premature treatment of the subject in mathematical terms. It is at the stage of qualitative consideration that one is best able to ensure that none of the interacting circumstances is overlooked, and so I shall here, as in Chapter 12, restrict mathematical treatment to the unavoidable minimum. The cosmological model can be checked for its detailed quantitative resemblance to actuality more easily after, than before, its broad qualitative features have been inferred. The small amount of mathematics introduced here will therefore be no more than is needed in order to test whether certain inferred quantities are of the right order of magnitude.

13.3: The Cloud's Income and Loss Account
If the inferred cosmological model is to resemble actuality, one thing that must certainly happen to it is that the incipient cloud must grow in extent and density during its transition from cloud to galaxy, and for this to happen it must receive more particles than it loses. Let us consider both the cloud's sources of income and its sources of loss. They can be thought of as analogous to profit and loss in accountancy; but here income and loss are more precise terms. The sources and amounts of both differ greatly according to whether the Hypothesis of Asymmetrical or of Symmetrical Impermanence is used as a basis.

Sources of Income: According to both hypotheses there are two sources of income. The first is origins within the region occupied by the cloud, and the second is existing matter that the cloud attracts from its surroundings by virtue of its own gravitational field. According to both hypotheses the first source of income is constant per unit volume. (This is probably obvious, but the reason will be fully given in Chapter 15.) But the amount of income from origins is not the same in the two hypotheses. According to Asymmetrical Impermanence it is always the 500 atoms of hydrogen per cubic kilometre per year estimated by McCrea. But according to Symmetrical Impermanence this number is the difference between the rate of origin and the average rate of extinctions per unit volume for the whole universe. The gross rate of origins is greater and the net income depends on the local rate of extinctions, as will be explained in Chapter 14.

Remembering the two distinct sources of income, one can speak of growth by the origin of new matter and growth by the capture of existing matter. While the cloud is still small and tenuous it has little mass and cannot attract matter towards itself in competition with the more massive neighbouring galaxies; so growth by capture cannot occur. On the contrary, the cloud does not even retain all the particles that originate within it. Some of these are attracted away towards the neighbouring galaxies; in the metaphor of an astronomical landscape they fall away down the slope. But in so far as origins cause the cloud to become more massive, it competes more and more successfully with the distant galaxies and growth by capture occurs eventually and then at an increasing rate. Growth of an extragalactic cloud thus occurs in two successive stages. During the first it depends entirely on the excess of origins over extinctions in the region occupied by the cloud. At the second it depends to an increasing extent on capture of particles from beyond its fringe.

Sources of Loss: Two sources of loss are postulated by Symmetrical Impermanence: loss by falling down the slope and loss by extinction. But only one source of loss is postulated by the Hypothesis of Asymmetrical Impermanence, i.e. the gravitational fields of the neighbouring galaxies, which are the cause of the potential gradient down which some of the particles in the cloud fall during the first stage of growth. When the second stage has begun, this source of loss ceases and there are the two sources of income: the 500 atoms of hydrogen per cubic kilometre per year and the particles captured from outside the cloud.

13.4: Income and Loss During the First Stage of Growth
Now 500 atoms of hydrogen is not much. It would not go far towards fattening a microbe. Even if the 500 were all retained in the cloud from the moment when it began to form, it would take about four million million years before the cloud gained one molecule of hydrogen per cubic centimetre, and this number would still leave the cloud very tenuous.

How minute a quantity 500 atoms of hydrogen is can be appreciated from another consideration. Some advocates of hypothesis (A2) have been claiming recently that the universe was created about seven thousand million years ago. A cloud that was acquiring 500 atoms of hydrogen per cubic kilometre per year during this span of time would now have a density such that a volume equal to that of the earth would weigh about one gramme.

But in our model the rate at which the cloud gains mass is much less than even this very low value. For only a small fraction of the newly originating atoms are retained in it. When the cloud is just beginning to form, nearly the whole lot of them are lost, for the potential gradient is such that the number of particles falling down the slope is just, but only just, less than the number that originate in the region. In this model the periods of time taken by a volume equal to that of the earth to increase its mass by one gramme is much more than seven thousand million years. It is not until the gradient has become nearly flat that most of the atoms are retained and that this maximum rate of growth is achieved.

In this model the rate of flattening is, moreover, slow, for the potential gradient is subjected to two opposing tendencies. The first is the expansion of space. As the neighbouring nebulae, which are the cause of the potential gradient, move further away, the force per unit mass with which they attract particles out of the region diminishes, which is another way of saying that the metaphorical slope flattens.

But for the model based on Asymmetrical Impermanence there is an opposing tendency. It will shortly be shown that in this model the retreating galaxies are themselves rapidly becoming more massive. Hence the flattening effect of increasing distance is at least partly offset by the steepening effect of the growing masses of the galaxies. It does not seem certain that the potential gradient decreases at all in this model; but if it does it must happen very slowly. However, it will be shown in Appendix B that this opposing tendency need not operate for the model based on Symmetrical Impermanence.

There is yet another circumstance that retards the growth of the cloud during the first stage. This is the fact that the loss rate is a direct function of the density, as can be inferred from equation (12a) in Section 12.4. For every gradient there is a limiting density above which the rate of loss down the slope exceeds the rate of income. The cloud cannot become more dense until the gradient has become flatter.

The cumulative effect of these various retarding factors is that, in our model the first stage of growth is exceedingly slow. The cloud will, however, become significantly more massive during the second stage of growth, when it captures matter from outside. This will cause its density to increase: and the density will also increase as the cloud shrinks under its own gravitational field. One must expect the cloud to be much more extensive at first than the galaxy into which it evolves.

For the model based on Symmetrical Impermanence the rate of income during the first stage of growth is greater by an amount at present unknown. The 500 atoms of hydrogen per cubic kilometre per year (or whatever the correct figure may be) are a net and not a gross rate. They are the average excess of origins over extinctions for the whole of the observable universe and are not relevant to any particular locality.

The region where the cloud has just begun to form has been depleted of matter until that moment, and so there cannot be many particles to become extinct in the region and the local net rate of origins must be very nearly equal to the gross rate. Hence the model based on Symmetrical Impermanence permits a more rapid growth during the first stage. One could only assess the significance of this distinction if one knew the gross rates of origins and extinctions.

13.5: End of First Stage
The moment when the first stage of growth is ended and the second stage begins can be defined precisely in terms of the fringe of the cloud. This is the surface around the incipient cloud at which the income from new origins balances the loss from extinctions and from falling away down the slope. (See page 102.) It is the boundary within which the density of the cloud increases and it is associated everywhere with a specific potential gradient, which I have called the critical one. The end of the first stage is the moment when particles just beyond the fringe of the cloud reverse their direction and fall towards, instead of away from, the cloud. This happens at the moment when the potential gradient at the fringe reverses, when the gravitational pull exerted by the cloud at its fringe just suffices to cancel the pull exerted by a neighbouring galaxy. It is the moment when the astronomical reversal zone of the cloud coincides with its fringe. After the cloud has become massive enough for this to happen, no more particles fall out of it.

In the metaphor of an astronomical landscape, the zone at which the potential gradient reverses is, it will be remembered, a ridge from which the ground slopes to either side. This ridge begins as the lip of a crater at the astronomical summit and moves gradually outwards as the mass of the cloud increases. The diagrams in Fig. 3 will help to make the course of events clear.

Fig. 3. the two stages of growth of an incipient galaxy. Above: First stage
Below: Second stage.

At first, when the cloud has only just begun, the crater is small. There is cloud on each side of its lip, as is shown in the diagram that represents the first stage. Particles do not fall out of this part of the cloud that is within the crater; but they do fall away from the outer part. But as the lip moves outwards, becoming more like a mountain ridge, less and less of the cloud is outside it and the loss rate from falling down the slope decreases. The moment when the ridge, or reversal zone, reaches the fringe of the cloud is the moment when loss from this source comes to an end and the second stage of growth begins. As the cloud's mass increases still further, this zone moves out beyond the fringe. The situation is then as shown in the diagram for the second stage. There is a growing region surrounding the cloud from which particles fall towards it, and its mass increases by capture and not only by origins within it.

Should the cloud eventually acquire a mass equal to that of a neighbouring galaxy, the boundary will be half way between the cloud and this galaxy; it will, as mentioned already, be analogous to a mountain pass over which one must travel in order to go from the cloud to the galaxy. A model that resembles reality must be one in which the reversal zone will eventually arrive at approximately such a half-way position, and this depends in turn on the rate at which the slope around the summit flattens. It will be shown later, when the subject is treated quantitatively, that by astronomical standards this rate is very rapid.

In this model the incipient cloud remains tenuous and spreads quickly until the product of its density and its volume suffices to attract particles that are just outside it and that would, before that moment, have been attracted towards the neighbouring nebula. Thereafter its volume may decrease by shrinking, but its mass per unit volume must increase at a greater rate.

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