Constraints on the Cosmological Constants

0:00 Thank you. Thank you. I'm not sure if I could give this thought in a highly unlike the magic one, and a very I will be using units in which speed or speed of light I'll use this sign for a very crude quality because that's the most I shall bring.

2:30 concerned with. Now, in, well, in ordinary physics, the main, well, if you work in terms of this sort of unit, then at least you can be fairly sure that any number that turns up really in physical discussion rarely is, are real. and most of these are physical parameters when you get down to it those which are fundamental are the present stuff and the fundamental and the well perhaps the most important is the electric which is charged on the electron, which is all about as well, so that it's squared in the same way. Other important coupling constants... Well, there are a lot of other coupling constants which I won't say very much about today because they're concerned with short-range nuclear reactions and I'll be almost exclusively talking about things which are banked on long-range forces which are dominant most effects in astronomy. In long-range forces, well, there are only basically two long-range forces namely electric forces and gravitational forces and so the only coupling constants you can really have four long-range effects are the charge coupling constant, this thing, and the gravitational coupling constant, which is just the masses of the particles in these units. I mean, the main thing is the mass of the electrons, the mass of the electrons, and the mass of the electrons, like this. Now, nearly all the coupling constants, both these ones and the coupling constants do have come into short range phenomena, such as the strong coupling constants, and not too far in order to make a difference from unity.

5:00 But this particular, in this basic, the basic habitational pumping plants, I don't like the lecture, maybe you don't know, is now ridiculous, the practical unit. It won't be practical that is coming around 10 to 90, 10 to 90, 10 to 90, 10 to 90, 10 to 90, or so. And this, well, this is the only number in ordinary physics, which is astronomical, in the sense that the astronauts are the ones who normally uses the word. Now, the basic coincidence is listed in Bondi's book based on the observation that some of the astronomical numbers that are actually observed It turned out to a degree, in surprisingly high accuracy, with multiples, with exact multiples of this coupling constant. Do you mean the power? Yes, it's a power. Well, there are three such things which I'll take one by one. Now, the first one, which I'll talk most about, well, I'll give it the longest time to discuss it, although it's not the main thing I can say the most about it, not the most spectacular, really, is the observation that the number of particles, well, number of variants, say, in a typical star is always about at the order of mp to the minus 3, something like typically 10 to the 58 or so particles. And, for example, the Sun in particular has that number of protons, and all the other stars that one sees have that number within 1 or 2,000 or 10 more or less.

7:30 And this was noticed particularly according to Gandhi by Yodhan in the, I guess, not excited about this, in the late 1930s. And Jordan apparently said, well, this is too surprising to me, but there must be a reason. There must be some mechanism by which matter is created in bursts or ups, coming in with about this number of cards. Now, this, as Bondi said, although he didn't actually derive it, this is not a very reasonable, well, this is not a very reasonable idea, and in fact, this number is the number of particles in a typical star. Now, what body didn't actually do is to show that when you go through the theory of star formation, the number you actually get, it's not just that it happens to be 10 to the 60, it's that the reason why it's 10 to the 60 is precisely because it is the inverse cube of this mass. So it's not a coincidence at all. Jordan was at least right in guessing that there wasn't a real connection between the proton mass and these numbers. What I'll start off by doing for the next few minutes is to just run through the basic arguments which are all in these standard astronomy books. It's just that in these standard books they didn't have such detail that a lot of astronomers don't even know why it actually comes out to do this. So what I'm going to run through is a very, very simple argument. It's quite the same, why wives don't actually are formed, or always, with numbers, about the number. In order to understand this, you have to just work through the basic theory of the equilibrium of massive bodies.

10:00 Now, the simplest thing to work out as a basic landmark is the ground state of a certain number of particles. If I take the number to the chosen number of n variables, I can ask myself what is the ground state of a certain number of the variables. And if I take one variant, the ground state is a proton. Well, it depends actually. I think it says something about the electron number. It doesn't mean how it should happen through the electron. But if I take a number such as the number of protons in the Earth, then the ground state is some state not very different from the Earth. It might be the absolute ground state, I guess, but the Earth is too far from its ground state, and the ground state is an intense spherical body. Now, to work out where the ground states lie, to begin with, when you don't have too many particles, You can effectively ignore gravity, but the gravitational constant is so very weak and the equilibrium is essentially determined by electromagnetic forces. The main forces in ordinary matter are the electromagnetic forces between the atomic nuclei on one hand and the electrons on the other hand, which tend to pull the things together. indefinitely is the further exclusion principle which says that if you try putting these two words together they have to get up into high momentum therefore high energy states so there's always a minimum of energy um with a certain mean separation where the um where the effects the exclusion effect, the principal effect of the energy out, and the electrostatic attraction cancels. It's always attractive, and the forces are attractive always between nearest names, of course, because they always are instant, so it's the opposite charges in their lives.

12:30 And the forces cancel out altogether for particles which are not the nearest names, So you can see very easily that they must establish an equilibrium. And the equilibrium distance can only depend on these numbers that are the masses concerned. In fact, the distance between the electron, because it's the lightest particles, which, when you push them together, first start to get into high energy. so the normal density is the electron mass which determines so you get a number depending on the electron mass and the electric charge which is the main separation between particles and that is just what it comes to with the raw radius which is And from that, you see that what I'll plot here is a plot of the number density of the variance per unit volume, I'll plot against the total number n of variance. And you find that the number of negatives therefore comes out to be just the inverse cube of this, so you have a certain number here, which is in fact about e to the sixth and the cube, which which is the basic, the number density in ordinary code mantra. Numerically, it comes out to be minus seventy-five or something. I'm plotting on a logarithmic scale, so that 10 to the minus 60 Eta.

15:00 And this gives you the equilibrium density for any small object, whether it be a hydrogen atom or an organic molecule or something like a cannonball or a piece of chalk or something like the earth. And all these things, how about this density, which isn't very far off in many systems of units, it's unit density, but CC, and that's really why the units are chesed into it, this unit density is maybe far off in a certain moment, or less than one ground, because it's C, I can't get exactly it, of course, different constituents, and it's not pure I, it's not exactly in this state, so there's a certain fluctuation, materials about that, but it shows up much on a logarithmic clock at this scale. Well, this is where all objects lie. Until you get up to such a mass that the gravitational forces begin to be dominant. And the reason why they do so is, of course, the electric force is, although vastly stronger between individual particles, only operating between neurons by navel because of the cancellation. whereas the gravitational force acts between all the particles of the body, obviously there could be a new limitation to nearby neighbors. And sooner or later, the two things cancel out. I mean, the two things become equal. And above that, above an assertion of a critical number, you have to balance the gravitation force directly against the generancy effect, the exclusion principle, rather than balancing the electric force. Well, this, I don't know about that case, you get basically what's a white circle degenerate star, a white dwarf star.

17:30 So this happens at a critical mass, which is about the critical mass which I suppose was probably basically discovered by Fowler, who I think really first started thinking about these deuterium gases and their role in astronomy. I mean, not Willie Fowler, and earlier Fowler, and above that case, you get an equilibrium with a, on a line, which is, it's a parallel, along this logarithmic top, it's a straight line, which is in the slope of the 90s too. And this goes on until... Well, basically until the particles become relativistic. You see, in the basic Fermi gas state, you can think of the electrons as being a non-relativistic degenerate gas, and therefore with a pressure given by... I'll add it over here. The pressure is... is 1 over M-E-N to the 5. So, well, M is... I'm looking very prudely in order of magnitudes. The number density of electrons is the same as the number of density of variance. So you have this five-thirds power lower of pressure against density, which works so long as the electrons are non-relativistic. But once the mean energy of the electrons becomes comparable with their rest-mass energy,

20:00 this switches over to a relativistic law, B and N equal to N to A. Now, when you've got a relativistic body, you've got a state of marginal equilibrium. In order to calculate where these things are very crudely, what I use is the virion theorem which tells you that in order for a body to be in equilibrium, the pressure, or the typical pressure, or the central pressure, must be given by roughly the four-thirds power of the mean density times the two-thirds power of the mass. This is a formula which you can derive very crudely by equating the typical pressure gradient, P over R, supporting the body against gravity with the product of the density of times the typical gravitation force squared and giving rough estimates. Now, so long as you've got a gamma law where the gas index gamma, so long as you've got gas index government is greater than four-thirds, one can always get stability of the body because if the body is not dense enough for equilibrium, it will contract a bit, but as it contracts, the pressure will go up faster, the pressure of the gas will go up faster than the pressure required for equilibrium. Whereas if you've got an index head that's less than four-thirds, then you have an unstable situation in which if it's too dense for equilibrium, they'll be runaway to collapse. And if it's not dense enough for equilibrium, they'll be runaway to think we'll just expand out and dissipate itself. And it just so happens, for reasons which in a fundamental sense, but anyway, it always comes out that the condition of the characteristic of a relativistic gas is precisely the gas law of this four-thirds power,

22:30 which corresponds to nuclear, to neutral equilibrium. So when you have a relativistic gas of this kind, which comes out when you get to the mass NP, or the variant number, NP minus 3, you have neutral equilibrium, which from then on, Well, for a body of that mass, you can have an equilibrium in principle at any density you like. For a body above this mass, you just can't have an equilibrium at all. So on this very naive theorem of so long as you're considering electron pressure holding up the body, these are the equilibrium states. Now, this is not quite exactly what happens here. This is the incident of the Chandrasekharan. When Chandrasekharan first ended this conclusion, I got a headache and refused to believe it. But nobody has been able to think of anything very wrong with it since then. Except in detail. In fact, what happened was more dramatic. is that when you get up to very high densities, which this corresponds to... When you get up to these densities, you finally get to the point at which the Fermi energy of the electrons is sufficient to crush them into the protons and form neutrons. So you start forming a neutron star, and the equilibrium line, or the actual equilibrium line, that turns backwards, like this, which is in fact, of course, unstable in Unnapp's capabilities, this way up. So there are invincible equilibrium states here, which are unstable in any way. And you don't re-establish equilibrium till a much greater density when the things are all compressed into, or nearly all the electrons being pushed into neutrons, and you get a parallel curve in which what you equate to this is something, a pressure problem of the same form, but with one of the neutron banks instead, which is the same as the neutron banks near another.

25:00 instead of the electron mass. In other words, you come to the stage where you've got a degenerate neutron gas balancing its pressure against gravity instead of a degenerate electron gas. And this gives you a parallel locus on this logarithmic diagram, and the things will come around the biscuit And again, exactly the same number. The number at which things become relatively dependent on the mass of the particle producing the weight, not on the mass producing the degenerates as we're producing the pressure. So it makes no difference to the mass whether it's electrons holding the thing up or neutrons holding the thing up. In both cases, it's a variance of the protons or neutrons, which of course isn't the way. So I think we come to relativistic, and we'll get back to this phenomenon. Well, this is, roughly speaking, of course, if one wants to be accurate, there will be smoothed up curves here, which is something very important, on a rather written experiment, and then we complicate the picture. Now, all this that I've said so far, and using this formula, has been based on implicitly on using Newtonian theory of gravity. In working this out, one should always check to make quite sure that the conditions are Newtonian. And the condition for Newtonian theory to be plausibly correct is that the gravitational potential should be of a particle, well, the field potential should be basically less than c squared. and the arm, well C is one of the firsts, should be that greater than one. Now, um... Well, in other words, on the point of the general relativity, the great non-euternal theory, the condition of the radius should be large compared of the object and center should be large compared to the Schwarzschild radius. When you can plot the logos on which the mass is comparable with the radius on this diagram,

27:30 a varying line of mass depends just on the density and the radius, and it turns out to be a curve of, well, a straight line of the moderately at the same slope as these other curves, but with the opposite side. And this locus, which is where things become general alchivistic, is where one expects the black holes to lie. So you can, if you like, think of black holes as also brown states or micro-lubrium states of a body. And it says that you have, as it's on this picture, a complete set of ground states of bodies with various physical variant numbers. Now, the actual scars that you see shining, of course, not in their ground states. They're hot. And therefore, from a physical point, you can say, one of the excited states. up here above this curve. And for a given mean temperature of a hot body, you have a corresponding logos of the corresponding excited states of different ovarian elements, so with high Actually, it's surprising that the low temperatures become higher up and the high temperatures become lower down. Low temperatures actually correspond to a higher degree of excitation than high temperatures, paradoxically, because the low temperature is only sufficient to hold the body up if in fact it's in a very diffused state. So the gravitational forces are not too small. so you have and of course you have therefore the paradox you should call it a paradox but anyway you have a well known fact that a star or something cools down

30:00 and it loses its energy and contracts it heats up in a normal sense the basic formula For the thermal equilibrium, I thought by replacing these gas formulas, these degenerate formulas, by thermal temperature formulas, of which there's a non-relativistic formula, according to which P is roughly equal to N times P. If you want a set of non-relativistic particles, it's just the number of particles times the temperature. And if things get relativistic, then it doesn't... Well, if the mean energy of the particle is larger than the rest of the particles, in other words, and then you find that P is just T to the core. It doesn't matter, in this case, how many particles there are, because if there aren't enough, you will just create more. For example, if you're in particular electrons, and your thermal energy is way above the rest-mass energy of electrons, then you will just create electron and positrons, which will likely contribute to the pressure. So you need to remember two electrons, depending on that many different between electrons and positrons, and the total pressure will be given by this formula, in many cases. Now, when you equate these formulae up with the formulae in density, you have two equilibrium curves, depending on which of these two formulae you use, the relativistic or the non-relativistic. The relativistic one, well, the non-relativistic formula, is equating that to that density, which is m to the minus a half t cubed, and the relativistic one...

32:30 So, this is the non-relativistic algorithm. The non-relativistic one is in those cubes of a mass of the particles contributing in the weight, n to the minus 2 tq. And the relative is the one in the n to the minus 1 tq. In both cases, the density goes as the equilibrium density goes as the cube of its temperature, mass-dependence. And the boundary, the dividing line between the two, comes when they are of course equal, which as you can see by looking at this formula, is when the mass is equal to the inverse square of the pressure on mass, or in other words, when the varying number is the inverse cube. In other words, it comes precisely at the same critical number here. So below, for that is less than this, the equilibrium will be achieved by effectively a non-relativistic pressure and above this, you will have essentially relativistic pressure. At low densities, the main pressure will be contributed merely by ordinary photons. So you'll have a volume of an ordinary radiation supported star. At high densities, you'll also have pressure due to electron-positron chaos and other things as you go up. And the velocities have the, anything, you can't just have a slope known as 2 on the logarithmic diagram, well, never anything. At least in the non-relativistic region they have the same slope. And one can plot the isotherms for particular temperatures, and they'll plot the three critical temperatures which are the greatest important. The three temperatures which are of biggest importance are the temperature for ionization of the body in the first place, which is known when the energy ends given up by ordinary Rydberg energy.

35:00 There'll be a high temperature when you actually start doing air creation, when the energy becomes comfortable with the electron mass, and in between there'll be an important temperature, which is the temperature at which thermonuclear reactions take place, and that's roughly the temperature at which the energy is sufficient for penetration of the Foulon-Repulsion barrier which is given by a formula similar to the ionization temperature, for example, the proton mass in its plane. So this is a thermonuclear. And this is pairs. Now, the ionization temperature isoflone actually starts off and presides to this point. You see, one way of looking at what's happening here is to say that this is the critical mass of which in a cold-body pressure ionization of the matter sets in. Well, from a thermal point of view, it's also where thermal ionization sets in, so that the equilibrium list just branches off from this junction point like this. And then when it gets to the relativistic changeover point, it changes in slope and goes off like this, and this is ionization. Now, again, this is, from the point of view of a couple body, this point is where the electrons start getting rather listed. And for the same reason, it's where the thermal focus of the pair-graduation temperature comes in. Now, this exists only in the relativistic region, of course, and it has the same slope as this, and goes off like that. And in between, you have this temperature, at which the temperature is just right for the thermonuclear reaction. Now, in a simple theory of what happens when you have an object like a star or something, well, if you have a body which you form in roughly an equilibrium at a certain temperature

37:30 and then leave to see what happens, basically what will happen is it will lose some of this energy by radiation and will drift down in diagrams like this to higher and higher densities until it stops. If it's a small object, it will just drift down and down and down. very hot and we'll never get this out of that temperature at all. And we'll end up at the cold lakets, equilibrium lakets down here somehow, like right here, or maybe even smaller. If it's larger, it will come to a point at which it becomes ionized, and which will get to stay down its cooling, because at this point it will become a fake. but it will still go uncool and it will form a protostar and it will cool fairly rapidly until it gets to such a temperature that thermonuclear reactions take place and then it will stop contracting and automatically thermonuclear reactions have very high temperature sensitivity so they will automatically adjust themselves until the energy is being radiated, well, it's being generated just sufficiently to replace the energy lost by radiation from the object of the cells. And it will stay like that for a long, long time, and that's the state of the Sun right now. And then when a large amount of the protons that actually invert the helium, that contraction takes place. So, that's what happens in a very simple picture at these higher levels. However, when it's finally got there, it might come down and talk to a white dwarf star, or it might, if it's on this side, collapse in the form of a black black hole, or you might use it as a neutron star. It's, for example, it could be easy to get to a neutron star. But in the early 60s, theoreticians believed more strongly in the existence of black holes, which are rather hard to miss out there, than in the existence of neutron stars, which are, well, it depends on, if you wanted to see how a neutron star is in the form, So we have to start looking at fine details of this diagram, which I'll be ignoring.

40:00 However, they've actually, since we've observed neutron stars and pulsed stars, now there's a lot of people potentially... Well, this doesn't yet explain why they... Now, the observed star that we see nearly all about there, this is the main sequence. In other words, very close to this critical mass of yodans, but not much higher. And the basic reason why it's an example is that higher masses, or indeed lower masses, is concerned with the instability. Now, the simple, the conventional idea of Cedar formation is that you start with a very diffuse private galaxy formed by condensation out of the expanding Big Bang. And there's no understanding of how and why that works. But once you've got an ex-, let me go on, not too diffuse clouded gas, the subsequent stages are understood, not in detail, but at least in a crude way, you expect gravitational instabilities of clouds above the gene plane to grow, a leading contraction. And basically, you expect the larger, the largest sizes of stairs to start having gravitation and stability first. So to begin with, you would expect to start forming very large glass, very highly massive glass glass, rather than blowing things up here. So you'd think you'd be forming the stars way up at this end. But, things like that are always effective in all that unstable, but also because of this basic inequality between the relativistic pressure and the pressure acquired for equilibrium, you can see that when you're equating a pressure given by a relativistic 4 thirds power law and in fact it's the same as thermal pressure it's still effective at some of the same phenomena when you've got a relativistic gas

42:30 whether it's thermal or degenerative pressure you're always in a state of marginal instability and the slightest corrections making the stuff realistic are almost done to introduce real instabilities. For some of them, of course, what happens if you have a detailed style model is that some of the corrections, some part of the model may have even more stabilized certain parts, and the corrections in other parts are unstable as other parts, but any part of it is unstable is effectively unstable. unstable. So that the things that the pregnant stars that start forming up here always turn out to be unstable one way or the other. They can either start having some stars forming out as blobs within the bigger things, or what very often might happen is that the central condensation will rapidly heat up and produce such a strong radiation pressure that the are blown off. But whichever way it happens, the thing is likely to either reduce its mass or split up, so that instead of having things up here, you end up by forming things smaller and smaller, and this will go in a sort of cascade, and the thing will divide up into smaller and smaller units, until you're now pretty close to the non-car characteristic region, where you get a significant increase of gamma above 4 thirds. And in this region, of course, the whole thing is very strongly stimled, which they have to run through a star with a 5 thirds of gamma long. So when the breakup of the Frodo-South-Hound has got to the point where you produce the masses as small as this, further breakup doesn't happen. And so you end up with things of this size. You don't put many things of this smaller subsidiary side effects. You get things like the Earth and asteroids thrown off as a small detail as a byproduct for the formation of the sun. So you do get a few of these things, but the bulk of the matter, it's fairly easy to see why the bulk of the matter ends up in units of about this size, neither bigger nor smaller. So that is why, and this number So it's a gravitational phenomenon that's dominating all these things. It's a gravitational phenomenon that comes from the program which fixes the number, and it gives this basic formula of rich, unimpressed, yonah.

45:00 So all that is very easy to see and understand. Now, the next committee that's been finally listed was the one that got excited. And this was the observation if you work out the Hubble expansion rate of the universe, it comes out to be about the same number in fundamental units. Now, this is roughly the, in the normal 30s, there is the age of the universe. And Dirac, at about the same time as you are, in the late 30s, thought that this is an extraordinary coincidence, this number, which could have been anything. which depends purely on Hubble's observation of the expansion rate of the galaxies, should come out to be so close to this microscopic number. And Dirac said, this is so striking that there must be some fundamental reasoning for it. And then he suggested what the fundamental reasoning was. But what he did say was, if it's true, it must be true of all kinds. And as this is roughly the age of the universe, and the age is always getting older with time, therefore, when the right-put big G must be getting smaller, but if I put G equal to 1 at the beginning, What it comes to translated into this location is that it means that the frame on that is getting smaller as the universe gets older. Well, people have done, actually, there's a certain amount of experimental evidence against this, not that, I mean, there are still some viable carriers according to each the gravitation may be getting mica, but it's very hard to see how it could be getting mica quite as fast as you are suggesting, but I reckon that very drastic effect on the sun and the solar system in its past history was detected.

47:30 Dickey's theory of gravity, the France Dickey's theory also effectively means that there's a weakening gravitation constant, but not by any means of such a rapid weakening as the algorithm had. And Dickey's theory, I suppose, isn't compatible with the facts. However, I don't know if you think of the facts about Geo. Well, I think that the new motion is that they are going to be able to do this with something that means that this is rather only going to be larger than some critical number, which mainly means to achieve if it's decreased even slower than that, and therefore even more compatibly with the facts. Well, however, as far as I knew, there wasn't any satisfactory explanation of this. In fact, well, the explanation, the character, this was in the late 60s, I thought it was original, but afterwards discovered that Dickey had the same idea several years earlier. was that let's just ask ourselves in advance what this number could possibly have been. Now, the age of the universe at the time we come along to start worrying about it can't It can't be anything. It isn't something fairly alternative random. As a minimum, it's thought to be old enough for ourselves as observers or people or some such thing. something of this sort has got to come along before it can start asking itself what the issue of the universe is. And without having too much of a detailed knowledge of exactly

50:00 what is necessary and sufficient for an intelligent life, it does seem that one can guess some very reasonable constraints, and one of them is that you must have a certain number of heavy elements to make life, you can't make people out of pure hydrogen chemistry, which is a really interesting possibility, well, this seems impossible enough, and according of the Big Bang. The only elements that you get out of the original history are hydrogen and helium, insignificance, I don't know if you have a very large component or something, but none of the important elements like carbon and oxygen, which are biochemical chemistry. And the standard idea is that these things are all formed in stars during the course of the evolution, and then they get thrown out back into the universe when the stars And across this stage, you get down here, the instability has become exclusive, particularly going down this corner here, a very exclusive stage, and a fair amount of the mass that's been processed in here gets thrown out into the ambient environment and forms new stars, new planets, and the ordinary heavy elements we've got here, pretty well, presumably, formed its stars which burned out before it was formed. Now, if we calculate the time in which it takes this star to burn, and it doesn't depend on the rate of thermonuclear reactions. in the star, which depends basically on these low mass levels, on the scattering cross-section of photons by electrons, which depends on the electron, that's the radius, which is

52:30 this number. And so you get a simple formula depending on just these numbers here of all the rate of velocity of a star and balancing that against the mass, you can estimate roughly the age of a typical star. And, well, I'm ignoring a few small correction powers that use of things like unstructured plants, which are very narrowed in the fifth level of accuracy. The formula you get to the age of the sky is the chip. And so, therefore, like you said, well, we hardly expected the age of the universe to be much, much less than this when we came along. Now, the burning time is moderately sensitive to the massive stars. A very massive stars, in fact, really are very much faster than others, and have lifetimes several thousand times, well, a thousand or so times less than medium-sized stars, which themselves have lifetimes a thousand times a so shorter than massive stars. It doesn't make a thousand or less, it doesn't make much difference in the numbers we're talking about. But it means that, for example, it's quite possible to have started and burned out in a small fraction of the sun's age already. Now, there's an upper limit also. You couldn't expect people to come at your distance very long after and you got beyond the lifetime of the burning of the smaller stars, which is this factor that increased by a thousand dollars and diminished by the answer there, because the present picture is that the matter is gradually being processed through the intergalaxi, It's being processed in these stellar, well, it's being processed in stellar, and it ending up, gradually, in the form of the inner momentum, cold white balls here, and neutron stars here, or black balls, down there. And the time scale for doing this process is of the order of the burning lifetime of the smaller, longer-lived stars.

55:00 What one feels is that after the galaxy, what it would appear is that after the galaxy is significantly older than this time scale, that we are plus 1,000 characters in, then one would expect it would have been practically burnt out and there would be very little activity going on anymore. All the matter would have been converted into things like this, so that the total active lifetime of the galaxy can't be more than 1,000 countries or longer than this kind of space. So therefore, if we have the law of the code, I have to call it, some of the data, some of the data, we'll probably mess it up with you. We've come to think of the thought, but it is very simple, and I think it's a serious consideration. Speaking with it, at the age of the universe, it had to be about a number, if I take it back to a thousand or so, and that even before, in principle, even before I was going to have look at the galaxies and measure this, could have guessed what the answer had to be within this accuracy. Isn't it where you just know it's going to get it? Oh, well. I don't know. The point is that you can think of it more than probability distribution, I think. But, yeah, and if I do think very large, then it seems very normal. It has to have been possible for life, the position that there are so few stuff around, and you see that there might be, on those these very low times, and at the end we find it very hard for people who might have lucked by, you might have lucked by, and you might have lucked by everything going very favorable to evolution, and you might have low life, but basically we expect it to be more likely to count in some intermediate times here, with a sort of normal probability distribution, falling off. Now the median time that you get is definitely I should think we're only getting a thousand or so. I don't think there's so much in trouble with that. Even the fact that a thousand or so of this number.

57:30 Whereas now if you want to go up there's a non, perhaps a non-zero probability, stretching way out there's a million or so, more or less. I have this coming, I'm trying to work out the details. That is the sort of picture that you would be making here in the form of a distribution and on. Well, anyway, I mean, certainly what you would have guessed would have been this. If you had been up here and you found that the whole time was somewhere up here, you could have said either, well, this is very surprising. Well, you would have said this is surprising. You could have said, no, it could have been, that's just the way you were, but you came in rather late. That's why we are all the same. Or, you could have said, no, this is, if you were really far, if you would have said this is just too odd to be explained that way, it may mean that one of our fundamental, more likely perhaps one of our fundamental assumptions is not. And the fundamental assumptions I believe in this would just be ordinary everyday fundamental assumptions. What I'm saying is that that could be used as evidence that some new assumptions, such as varying gravitation, or some revolution, or some radical departure from conventionalist statistics. But the fact that we are vain in the region where you have thought less likely to do it, is that it needs to happen either to be able to have a confirmation of conventional ideas, not at all the justification we don't have, like Dirac and other people have done, and say you must have some slightly, radically new explanation. Of course, you could mean out there, but it's still okay if you are close to the map of the university. Yeah, well, the point is, you've viewed the amount of data that we've done. Well, there's been a lot of conclusions. I mean, you see there are a lot of conclusions of this feature.