Inhomogeneous cosmological models

2:30 My work investigating inhomogeneous cosmologies was given by Christian and Sachs, a rigorous procedure, but their work is, they use approximations and one doesn't have any intuitive picture of what's going on. So what I did when I read the Vogel's paper was to produce a simple cosmological model which would accommodate this data. Well, the necessary exact solutions were found a long time ago by Tolman in 1934, and they were also studied by Bondi in 1947. They are spherically symmetric dust flows using Einstein's equations. And the idea of the work is to take a Tolman model, the dust flow, an inhomogeneous dust flow of spherical symmetry, Use it then to arrange that one has this on some space-like hypersurface, T for small. One uses this as an initial condition on a space-like hypersurface. Now, strictly speaking, of course, this, when you look out into space, you're looking back on the null code, so one ought to set this problem of this density This is a much more difficult problem, and I didn't tackle that. I took the easier problem of setting this initial datum on this space-like hypersurface, t equals naught. Now, one has to be rather careful to distinguish between two sorts of density. Which depends, which is obtained by averaging over sphere of given volume, B, and there's the density rho, which is the point density over a small volume, a volume tending to zero, and that's the volume, that's the density which occurs in Einstein's equations, and it's not a very precise notion, cosmologically speaking, but one might think of this as being over perhaps...

5:00 And this volume velocity as referring to volumes with greater radius than that. Well, the solution of Einstein's equations that I need, the Tollman-Dust flow, quite a simple one, is an arbitrary function of R bar. And there's only one field equation. I put a cosmological constant here. There's only one field equation. So that is another arbitrary function. The capital F is another arbitrary function of r bar. Dash means d by d r bar, and the dot means d by dt. And capital R, which is the scale function, is a function of r bar and t. Density is given by that.

7:30 Well, this is quite an easy equation to integrate. And it has three cases, calling as F squared is greater than, equal to, or less than one. And in order to get some feeling for the problem in a simple way as possible, I took the case where F squared is equal to one. So F squared is identically one throughout the whole of the space-time. And this means that every section t equals constant is spatially flat. Every section t equals constant. So, we now get a simple model, and one can integrate this equation in this case very simply, and one gets the following. I should say at this point that the general solution of the Toplin problem contains three arbitrary functions of R, which I've shown too, and another one arises in the integration process of this equation. The problem consists as two arbitrary functions of a radial coordinate. And in this procedure, I put one of the arbitrary functions equal to one, and I've left essentially

10:00 with one function, as we shall see in a moment. And now I can eliminate H by introducing a new coordinate. One has to assume, make some assumption about H being a monotonic function of R bar. Otherwise one shouldn't do this, but I won't go into that now. Then one has now that R, having done that, is equal to the new R, f of r, this is a new function, to the power two-thirds, and that is an arbitrary function still, and the problem now is to choose f to simulate de Vauquelin's density observations on the space-like hypersurface t equals naught. We have to choose this f to fit in with de Vauquelin's density. Now we are at the moment making observations at time t equals naught. And there, one can easily check that the space has a Euclidean metric, and so this volume density that I spoke of, is simply what it would be in ordinary Euclidean space at the present time, volume density, and this is what evokes those, presumably.

12:30 All of these terms are very important to us because they are very, very important to us in the world of mathematics, in the world of science, in the world of physics, in the world of mathematics, in the world of physics, in the world of mathematics, in the world of physics, in the world of mathematics, in the world of physics, in the world of mathematics, in the world of physics, in the world of mathematics. Using the fact that f of 0 has to be 0, otherwise one has a singularity in space-time. Are you saying t equals 0 at the present time? Yes, taking t equals 0 at the present time. It has to choose capital evidence at one's disposal, and one can choose it so that it simulates the vocoder's density function. In fact, it's convenient to take f equal to 0, where a and k are constants. Then this gives 4 thirds a squared upon 1 plus kr cubed to the half and you see that for large r, when this term kr cubed is greater than 1, then this is going like r to the minus 3 halves and that is a good enough approximation to the Bokehler's law which... Remember, have the density going like r to the minus 1.7. So this is a suitable function, a suitable choice of f, as it will give us something like the vocalist's law.

15:00 So let's make that choice and see what sort of model we get. Well the metric is given here. x is this function 1 plus k r cubed. There are in fact singular higher surfaces, as I mentioned, its density is singular when this term vanishes, that is to say, when t is equal to that, then the density is infinite and so is the metric. The metric is also singular. There's another infinity when this term vanishes here. So there are two singular hyper-services in the model. Shouldn't there be a minus in front of the L-shaped rooms? Well, no, that's... Oh, it's all in the brackets. I see, I'm sorry. Yeah, that's in the big brackets. Now we can plot the... we can draw the model in the following way. This is here and now. This is the section we've called now, when T was law. These are the world lines of particles in the model in this coordinate system. This is a co-moving system, so each particle retains its value of r, which is why these lines are all parallel. Now these are the two singular hypersurfaces I spoke of. This one here, when t equals minus 8 to minus 1x to the quarter, that gives you what the singularity known as, what the human would describe here by 1. It turns out the other singularity I mentioned, when the other factor in the denominator of the disc vanishes, this gives another different curve, but it lies below the first one, so it's of no physical influence.

17:30 So in this model, we have the Big Bang is on this stone. It's not a simultaneous throughout the universe, of course. It is, in fact, an implosion. The singularity takes place first at distant regions and it implodes and occurs lastly at the origin, occurs lastly at the origin. Now, apart from this, this big bang at the surface, one can check that the model is This represents the backward null cone from the here and now and one can, without actually integrating the null geodesics, one can Find out what they do, and one can convince oneself that the fast now cone in fact insects the Big Bang hypersurface, so in this model, the observer could see the Big Bang hypersurface. This wouldn't necessarily be so with all models like this. You might have a model where the Big Bang is like this, and so the backward now cone would not insect it, the observer would see it. This is a very long time ago actually, this problem is not to scale, and the big bang occurs as seen from O, something like 10 to the 16th years ago. And then of course the density somewhere here, the density which the observer is seeing on his backward null code must start to increase, because here it's integral. So the Baudelaire's rule can't go on indefinitely if you consider it as you should do on the backward null code.

20:00 Well, if you consider the model with various limits, you get some more insight into it. If you consider, if you put k equal to zero, you get the Einstein-de Sitter. I hope everybody's familiar with the Einstein-de Sitter model. It's the simplest model. It has flat spatial sections, and the scale factor is t to the 4 thirds. But r is actually t to the 2 thirds. R squared is t to the fourth of x. Similar to this Robertson-Walker model. And if we put k, this promises k equals 0, and make a trivial transformation of the time, we get the Einstein-Sitter model. As we consider the limit as t times infinity, we get this for the metric, and that for the density. The density becomes homogeneous, and this limit is t times infinity. In this limit, the universe is homogeneous, and this is, in fact, the Einstein-de Sitter model again, which one can obtain by a coordinate transformation. So, as t tends to infinity, the model tends to the Einstein-de Sitter model once again. A useful parameter for studying in homogeneous models is what's called density contrast, the logarithmic derivative of density. And if you study it for this model, you find that it starts off at infinity, at t equals naught at infinity, and then monotonically decreases to zero, of course, at Moscow, and showing that the degree of inhomogeneity gradually diminishes.

22:30 Well, one can get some numerical estimates for these constants. We will look again at the diagram. Big bang at the origin took place when t was minus 8 and minus 1, and so one needs something like that. The parameter k measures the degree of inhomogeneity of the universe, and so it becomes important when kr cubed is approximately 1. If you look at these data, see where they show the inhomogeneity is commencing, this gives you some idea of the value one ought to take for k. Anyway, by arguments of this sort, and in order to put some numbers in, I eventually took 12. And then, having those parameters, one can study the redshift distance. Redshift luminosity distance formula to see how that fits in. Now there one can expressly use the word Christian and Sachs because they worked out the redshift in a precise way as if you were observing along the back of an alcove for an arbitrary cosmological model. And I won't go into the details.

25:00 But you find that if d is the luminosity distance, then up to order d squared, you get the following. Z is the ratio. Now this is in fact the same as the Einstein-de Sitter model, and the inhomogeneity has not yet entered. And the inhomogeneity enters in the higher terms in the expansion procedure of Christian and Sachs. One gets an interesting conclusion because when K does eventually enter the expression for the redshift, you find that it enters it in a very powerful way and that the result of this investigation was that we would get some nonlinear terms which would be extremely powerful and would get the facts. Terms of order d to the fourth, the luminosity distance to the fourth, and these would, for a fairly small d, be comparable with the linear term. For a fairly small d, a fairly small redshift, even for a redshift of, say, about 0.1, these nonlinear terms would be important. Now this is certainly not absurd, because the linear formula for the redshift against distance It is accurate out of two of quite high redshifts to the order of 0.4, and so this particular model wouldn't satisfy the redshift luminosity distance observations. That, by no means, disposes of de Boeckler's theory, of course, because this was a very particular model which had this simplifying assumption built in. And what one could do if one were going to take the vocalist very seriously, one could choose a different value of f squared so that one trying to fit the redshift observations, use the value of this function here, capital F, to fit the densities with the density of observations and this function here to fit the redshift and

27:30 In this way, one might be able to do better. You will begin to get these inhomogeneities unconsciously close. This seems a bit contradictory, because you'd think that if the inhomogeneities were close, then it would chart more on the redshift relation, wouldn't it? Well, the redshift... It's rather puzzling why one gets this behaviour of the redshift. The fact is that... If you look at the expansion of the model, the invariant, where u is the core velocity, you find that this model has a very pronounced effect on this invariant. And it's due to this that it has a pronounced effect on the redshift, because the redshift is really measuring this. Ordering k, you could effect it. And when it's seen, I'm saying it could go up by 1, I think that when it's seen, even there, quite a long way. Well, k is small by a factor of 10 or something. It's possible, it's possible. What were the reasons for taking that, then? Well, the reason was that this was when, looking at the Vogler's data, this is when they showed that the inhomogeneities were beginning to appear. It's a bit difficult when you're quite close to say that whether the universe is inhomogeneous or not, because it obviously is on a small scale.

30:00 Density law is to begin... So this term, the model begins to depart from Einstein, you said that, when this term is beginning to get dark, say anywhere below one. So surely then if k is very small, that shows that it's near the Einstein principle, right? Yes, that's right. Yes, yes. But isn't that a step in the right direction? If I said k was 10 to the minus 79, the vocaloid might say, well, I can see calculus on a smaller scale than that, but then you would be erring on the direction of conventional mathematics. Well, if you make a smaller, that means that one needs a greater value of r before these terms become important. And that means that you're getting to the edge of the range of the local earth observations. These observations only go out to a certain number. But then it's always a cube root, you see. So if you make it, if you bring it down by the factor of ten, you're only changing the distance scale by the cube root of that. So it's probably not going to make much difference. Whereas it would say it would change. I didn't know that it would, you see, you've got 10 to the 4th. Anyway, I think perhaps I should look at that again.

32:30 I think that as far as the redshift is concerned, the model does differ very considerably from the Einstein-De Sitter model in its expansion properties, and that's what's causing the difference in the redshift. But the lifetime is 10 to the 16th year? Well, I didn't go any further into this because You heard that I didn't write any more about this density law, and the other reserves I spoke to were in favor of it, so I didn't do any more on that, but recently I considered in-home genius psychological models from a different point of view, namely the evolution problem. Can I ask one question? The next question is about R. This one, yes? Is that power of two-thirds meant to be just on an H? No, that's on the whole right. How do you get to the second center? It seems to be distributed over this? No, the one over there. I think that's right.

35:00 Now, we come back to the two-thirds, and essentially you want to make that into the real R, so you get an R here, and you're left with a 1 there, 1 plus T times C, which is R, and you take the H out of the whole bracket, yes. This universe is isotropic, isn't it? Well, only to the people at the center. It's the greatest metric model. So if we, in order to, if this represented our universe, we'd have to suppose that we pretty nearly understand that, and we would see it as isotropic. The other observers wouldn't. No, no, not yet. Towards the end of its lifetime, it becomes nearly isotropic, and it gets more and more isotropic. Well now I want to adopt a more orthodox view of the universe, and suppose that it is homogeneous and isotropic now, and ask, was it always so? Can the isotopy and homogeneity be a product of evolution from some different condition, or do we have to suppose that the universe has always been isotropic and homogeneous? In that case, the initial conditions of the universe must be very special ones. Well, this question was raised by Mizner in 1968, who had a nice idea. He said, well, perhaps... No matter what the universe, how the universe started, there was some homogenizing process which reduced it to what it is now, and so a business idea was, no matter what the initial conditions were, then the universe, they would all evolve to the same situation, maybe the one we see now. And there were a lot of investigations of the mixed master universe, which I think was originated to study this idea, and not much progress was made.

37:30 But recently Collins and Hawking have put forward the opposite point. They said that they think that the initial conditions of the universe for it to evolve to the one we now see must have been very special. And we know what they said. First of all, they show that the set of spatially homogeneous cosmological models which approach isotropy at infinite times is of measure zero, and they conclude that there's only a small set of initial conditions that would give rise to the isotropic universe in our sea. Now they are studying there... These are spatially homogeneous models, anisotropic models, and they're seeing whether these anisotropic models become isotropic, and they don't. Then they go on and they say, although their work does not prove that there is no open set of inhomogeneous initial data, which gives rise to models that approach homogeneity and isotropy, it makes it appear very improbable since one would expect inhomogeneities So they make the conjecture that even if they had considered the inhomogeneous models as well, it wouldn't have made any difference in their conclusions. Well this struck me as rather implausible, so I used these Tolman models to investigate this conjecture. So the problem now is starting with an inhomogeneous model, can one get a... Well, we've already seen in this model here that one does, in some special cases anyway, get such a model, because this inhomogeneous one does evolve to the Robertson-Walker model. Well, I start with this potentially the same, it is the same metric, and the same equation here, putting the cosmological constant zero again, same equation for the density.

40:00 And one can make the general assumption that the density is positive, the human density is positive, and then one finds that one has this function f dash of r has to be positive, and you also make assumptions about the regularity of the space-time at the origin, and you find that You get, this would be r bar, you get that term. So the important thing is that f of r bar has to be greater than or equal to naught and it's monotonically increasing. Now one can consider the three models, three sorts of volume given by those values of x squared, and let's consider the simplest sort first. So it'll be like the work I did before, except that I shan't be making the specialization for the function f. So we have this metric here, and I use a slightly different notation this time. When I integrate this, I write this as follows, where f and beta are arbitrary functions of r bar,

42:30 And so the model contains one arbitrary function, eta. And now the model looks like that. Y is defined here, dash means d by the r, as usual, density is given by that. And this function, beta, could be used, as before, to give you an arbitrary density distribution on some space like hypersurface, for example. To study the density contrast, I mentioned before, sigma, which is d by dr of log rho, minus 2 by eta dash upon t plus beta, minus y dash. The metric is singular for two big bands. These services can, they don't have to be like the one I showed before. They could intersect one another. So what I do is to say the model is defined only for T greater than whichever of these hyper-services is uppermost.

45:00 As the greater of, well supposedly I call that 1 and 2, greater than the maximum of 1 and 2, the value of t that is, then this model is valid only for t greater than sigma. Now let's decompress this into this, as a big bang, as expected, because one or other of these denominators will vanish there, but as t tends to infinity, The density contrast goes from infinity, starts off to infinity, and goes to zero. In fact, if one studies the density, given by the appropriate specialization in that relation, we find that 8 pi rho is 4 over 3 t squared plus or minus 3. And so, this is tending to the Einstein-de Sitter-dense. This is the Einstein-de Sitter-dense. And indeed, if you consider the metric, the metric that I have there, and let t tend to infinity, you'll find the metric itself tends to the Einstein-de Sitter-metric. Tends to the Einstein-de Sitter-metric if one ignores terms of order.

47:30 T to the minus 1. There are some terms of all the T to the minus 1 in there, which I recall. So the conclusion of this part of the work is the following. Framing Coulomb's model of S squared equal to 1, and everywhere a positive density, necessarily evolves to the Einstein decider. Whatever its initial condition, a rather surprising conclusion. Well, then one can consider the other models, S squared greater than one, for instance, this is what I call the hyperbolic models. Could I ask, have you put any numerical values into the sort of times involved in the integration of that? No, I haven't. I'm sure one could do that. Of course, one hasn't got any particular constraints. The only thing one's got, really, is to get an appropriate time-sensitive bang, really, isn't it? I think there'll be no difficulty in fitting some reasonable numbers into that case. I haven't done it. Now, the hyperbolic models, one has F squared greater than one, and this is the case where the dust is escaping from its own gravitational field. This, the case I consider F squared equal to one, is a parabolic case, where the...

50:00 It just has the escape velocity. This classification of these models is not perhaps quite obvious, but it was given very clearly by Bondi in his 1947 paper, conceiving these models from the Newtonian point of view of escape velocity. The metric I'm talking about is the one right at the beginning. Now, that's this one here. r is the scale function. The arbitrary functions are f, little f, and theta. Three arbitrary functions, of which one would be removed by a scale change of r. You see a parameter u in the beams. Specializations which give the corresponding Robertson-Walker model are if one puts F squared equal to 1 plus R squared, capital F equal to 4Kr cubed, and beta equal to 0. That gives the Robertson-Walker model, which many people expect, contains just a constant cap, hyperbolic Robertson-Walker model. Now, in order to study this, to get some feel for this, let's consider it at large times. At large times, you take shine u and cross u approximately equal to a half-heat of u,

52:30 and much greater than u, and then one gets the following expression for r. r is then equal to that. T plus the function beta is equal to that. And so one gets approximately that r is simply equal to a function of r times the time. And so in this coordinate system, the co-moving particles are moving uniformly. And one could say that at this stage, the gravitation is not having any effect. They have escaped now from the gravitational field and are moving uniformly. That's a loose way of putting it. If you examine the density contrast in this model, you get a term which is constant, plus terms of order, theta minus one, and so on. And so this is not usually zero, so I can say that not all these models evolve to the corresponding Robertson-Walker model. The corresponding Robertson-Walker model, of course, the density contrast is zero because it's homogeneous. So I can see that not all these models evolve to the Robertson-Walker model. This is a particular value. This is B. It's another constant. And then in that case, what happens is that you get this for the in-home use. And the scale function is given by that. And the time is given by that. And you can make a choice by a scale transformation. You can arrange for the little f squared to be 1 plus r squared. And then one has precisely the Robertson-Walker type of model except for one function, namely beta.

55:00 So this is the Robertson-Walker model, but for the function beta. So it's a Robertson-Walker model with one function of the distance coming into it. And that is the most general of the model of this type, which will evolve to the homogeneous isotropic model. So in this case, you see, if they all evolved, that would mean you would have two arbitrary functions to play with, but in fact you would have just one. It turns out that the density for large time doesn't contain beta, the density contrast obviously doesn't, and so beta is really a function which conditions the Big Bang in a way. It determines the Big Bang hypersurface, and one could say that at large times this model has forgotten its Big Bang altogether, and beta is no longer important. Well, the other case, F squared less than one, is entirely different because those models, they have a finite history, they collapse after a finite time, The question of letting the time tend to infinity can't arise, and so the problem doesn't really apply to those. Well, everything I've described is referred to spherical symmetry, and recently with Mrs. Tommy Murrah, I've been studying models which don't have spherical symmetry, but you can consider plane, models with We found that they do. But when you go to models with less symmetry than that, then the equations become very difficult to solve and we haven't made a great deal of progress yet.

57:30 One other idea that we've had is that one might use some perturbation process. To study this problem, usually the problem one has is the following. With perturbations in cosmology, one starts with a Robinson-Walker model, and one asks, could galaxies form? So let's give the model some perturbations and see whether the perturbations grow at a sufficient rate to form galaxies. And an enormous amount of work has been done on that, and the problem still hasn't got a successful conclusion. One would apply the perturbations backward. Of course, one would say, well, the universe is homogeneous and isotropic now. Let us apply these perturbations backwards and see how quickly they grow if you go backwards in time. This would give you some idea whether the present state could have evolved from a more inhomogeneous state. It's a very paper-bearing standard of sex analysis, and it enables you to investigate inhomogeneities from the observational point of view in a very comprehensive way, but by an approximation procedure, so therefore it's a little difficult to see what's going on.

1:00:00 I think the question of stability is very important, because for a full reputation of the Hawking-Williams point of view, one has to show that there is an open set of conditions which will produce energy data. There's only got one claim, which is that it's a certain measure zero. But didn't the movie say what they meant by a certain measure zero, did they? Would they, do you think, if one showed that one had an axially symmetric universe with one arbitrary function, wouldn't there be a representation? They weren't very precise with that, and I find it important to refute that.