Cosmology | Lecture 8

this program is brought to you by Stanford University it's a what Omega is do you see that oh I'll make a big ole maker yes yeah um right okay so people you ask me what Omega is and Omega is an important quantity so let's let's discuss it you go back to the freedom in robertson-walker equations and those have the form a dot over a squared we know about that and on the right hand side we have various things we could have a vacuum energy let's talk about today or at least for an era well after the radiation dominated era ended at the time which includes the present today when the equation of State outs wait a minute oh wait a minute all right so somebody asked me what capital Omega in other words there are several capital omegas incidentally one of them simply represents the sphere the two sphere or the three sphere that's another that's a different Omega in cosmology there was another Omega which has to do with the percentage of matter or the percentage of energy in different forms that come in different forms okay so let's the percentage of the energy that goes on the right hand side of the expansion equation the Friedman Robertson Walker equation the left hand side of course is the Hubble constant of the Hubble parameter squared the right hand side can contain first of all a vacuum energy let's see it's not cold we shouldn't call lambda Rho North 8 pi G eight PI G over 3 times the vacuum energy then if we're talking about the era R which includes today when radiation was not very important and which mostly the energy density of the universe apart from the vacuum energy is dominated by ordinary nonrelativistic slowly moving objects such as galaxies then there's another term in here which is our let's say Rho matter it also multiplies the 8 pi G over 3 so let's put a bracket around it the vacuum energy the part which doesn't change with expansion and that of course just to remind you this part here is some constant let's call it m divided by 8 cubed so it dilutes with expansion there might also be a radiation part but let's ignore it it's very very small after the period when the Sun when the universe was very very hot when it cooled below the temperature of the Sun let's say radiation was not so important and the matter part of the energy density was the important part and let's see is there anything we'll missing yeah we're missing plus 1 over R plus K a minus K of a squared that's the curvature part that's the part which is K equals +1 for a positively curved universe k equals minus 1 for a negatively curved universe and k equals 0 for a for a flat universe let's uh mostly concentrate on for the moment on the flat universe then all Omega is is it's the fraction of energy well that's for this way yeah it's just a fraction of energy carried by the different components okay it must add up to one for the simple reason that sorry I'm saying it's not quite right okay not quite right yeah what I want to do is get rid of this and put it over on the other side 3 over 8 pi G let's get rid of it over here and let's forget this for the moment now um these two have to add up to this this is the square of the Hubble constant they've got to add up to this but not if K is anything is either plus 1 or minus 1 if K is plus 1 or minus 1 they don't quite have to add up to the left-hand side here the thing to do you have to understand this the thing to do is to divide this equation by what's on the left hand side let's just simplify it in kora Hubble squared this is the 8 pi/3 over G but let's ignore that for the sake of argument for here it's a double squared times a number and that's equal to the sum of three terms here's the way to think about it's a sum of three terms Rho naught plus Rho matter minus K over a squared and this is either plus 1 over a squared or minus 1 over a squared or whichever and then divide it by Hubble squared divide it by Hubble squared 1 divided by Hubble squared divided by Hubble squared divided by Hubble squared these things have to add up to 1 that's just the trivial I just divided by the left hand side of the equation to get 1 here all right but notice that's not the only thing we could say we can say if K is equal to 0 these two things have to add up to 1 the energy density in the vacuum together with the energy density and matter must add up to 1 otherwise space can't be flat now first of all since Rho naught and Rho matter very differently with time there's no way that this can be constant so the only way you can have it add up to 1 is if no matter over h squared if there's no not if H if H were constant but H is not constant H is not constant in general and since H is not constant in general there's really no reason why this cannot add up to one in general if H is varying with time so because H varies with time row not over row matter over H squared could add up to one but it can only add up to one if K is equal to zero that's what the equations tell you that if Rho naught plus Rho matter over H squared add up to one and what do you do you measure the different components of the energy you add them up you divide by the observed Hubble constant squared and here's a prediction if space is flat if this is not important then these have to add up to one the energy density or the vacuum energy density plus the matter energy density divided by the square of the observed Hubble constant must add up to one now these things this is called Omega this is called Omega and the first term here is called Omega zero or sometimes Omega vacuum or sometimes Omega lambda to stand for cosmological constant and the second term here is just called Omega matter Rho divided by H squared is called Omega okay first conclusion is if space is flat Omega naught plus Omega M must add up to one so in that sense if K is equal to 0 Omega naught and Omega matter are simply the fraction of energy carried by either vacuum energy or matter energy that's what they are they can change with time but they have to change with time in such a way as to add up to 1 but if on the other hand space is curved if space is curved then we can put on the right hand side 1 plus K of a over a squared H squared and then the sums of the Omegas do not have to exactly add up to one adding them up and then remind you what this means you first measure to the best of your ability astre astronomically in various ways you measure the vacuum energy you measure the matter energy and you divide it by hubble squared the sum the sum may or may not add up to one if it adds up to one that's an indication that space is flat so when people say space is flat it implies that the sum of these two have to add up to 1 and in that case they really are just a fraction of energy carried by matter and by vacuum ok if these two don't quite add up to 1 for example if they add up to something bigger than 1 that says that K is positive it implies that the universe must be positively curved so if you measure Hubble and you measure Rho and the ratio is bigger than 1 then space must be positively curved if it's less than 1 then K must be negative in space must be negatively curved so this is what Omega is omega is a measure young yes it includes dark matter yeah well yellow we could we could add in here Rho radiation but it's such a small fraction that it doesn't that it doesn't matter so in practice it's just matter and vacuum yeah is there some dark energy the Hubble constant itself is not a measure of dark energy it's the way that the Hubble constant changes with time the fact that it doesn't seem to be changing with time is which is the indication of dark matter dark energy dark energy yeah the fact that the change the time rate of change of the Hubble constant is decreased which is the same thing as saying that these that we're on this branch of exponential expansion in other words if a dot over a is becoming constant that's equivalent to saying a is exponentially increasing so the observed apparent tendency toward exponential increase is the measurement of is the direct evidence for Rho naught okay I mean how is that separate from H ah well okay it's not it's not you're asking observational e how do you tell the difference observational e what you actually measure is K what you actually measure is K and K tells you what row north is observation Li you measure the fact geometry geometrically we haven't had a chance to get into this but geometrically you measure the fact that on large scale space is flat that tells you that K is equal to 0 once you know K is equal to 0 and you'll have also measured the matter energy then you know what Rho naught has to be given now that you know Rho naught is you can then predict the way the universe expands with time you find out that it's not 0 and it says that it's accelerated so there is a predictive there is a prediction in there from the cat fact that space looks flat it tells you that Rho naught R or that ya this you measure that's yeah you measure this by measuring the ordinary material in the universe that you can see plus the part that you can't see which is the dark matter which you measure gravitationally by the orbits of stars around galaxies yeah that's right we have three different several different ways to measure this several ways to measure this they're both measured and they add up to something close to one sorry yes they add up to something doc they add up to something we also measure the Hubble constant and it all adds up but on top of that the measurement of the curvature itself is equal to zero yeah so we have some cross-checks there are various cross checks and the cross checks involve the acceleration of the universe which tells us that the direct measurement of matter and dark matter which tells us that the direct measure of the Hubble constant which just looks out the different distant galaxies and tells you how fast they're moving indeed they add up to about 1 and that tells us that space ought to be flat and then you measure the curvature of space by triangulating it and you find that it's flat ok to a good approximation so so those are the pieces of evidence that are telling us that however we got that way we don't know what space is very very flat to a couple of percent why is it so flat well the theory is that it inflated being flat just means it's very big it's very big and very smooth very big and very smooth and very homogeneous on scales how big on scales of billions of light years on scales of billions of light years in fact really tens of billions of light years it's very flat very homogeneous how did it get wet that way despite the speculation or the conjecture and by now it's more than a conjecture there's good empirical evidence for it that the universe underwent very early period before what we normally call a Big Bang the universe will underwent a period of exponential expansion and that exponential expansion just stretched it out stretched it out so much that it appeared very very flat now you can overdo a good thing the universe is very flat and that time of the Big Bang it was very smooth but who will perfectly smooth it had been perfectly smooth then there would be no way for structure to form a perfectly smooth universe one which is perfectly symmetric will stay perfectly symmetric the equations of gravity and so forth will not create lumpiness out of perfect smoothness and so we know that the universe wasn't perfectly smooth and if we trace backward from the current knowledge of galaxies and the current knowledge of the of the degree to which the universe is not homogeneous if we trace that backward we find out that at the time when the universe first became transparent at the time that radiation decoupled from matter and so forth the time that we normally call a Big Bang there was something like about 1 of 1 part in 10 to the 5th anisotropy and in homogeneity that means if we were to have plotted at that time the energy density of mass density as a chin up position we would have found that it's very very constant almost perfectly flat but and of course I can't draw this on the blackboard with about one part in 10 to the fifth there was a bit of a wiggle enos in it willingness in it at different scales that wig leanness now that's much much smaller than the current now by now stuff is concentrated into galaxies and it's highly anisotropic Nessa in homogeneous on small scales so what happened and this is something that we can follow starting with this primordial little bit of fluctuation where it came from we care where we have some ideas but where it came from we don't ask at the moment we say supposing it was there what happened next well the little over densities is the regions of over density they were surrounded by regions of under density let's go through those of potted and then other regions of over density and so forth other regions of over density the regions of over density gravitated and pulled things toward them in fact they pull the matter immediately in between these together of these two regions into the two regions increased the over density and increased the magnitude of the under density so um it's a sort of self reinforcing feedback that the more flock the more and in homogeneity you have the more gravity will lead it to become yet more in homogeneous this is different than the air in this room if we created a perturbation in the air in the room a little wave like disturbance that wave like disturbance will eventually disappear it'll disappear it'll come to thermal equilibrium and the room will again become homogeneous and it's content of air okay for example so gravity is different than other forces it tends to lead to this instability we're a little over density will increase itself in fact if you just took a completely homogeneous universe and put in just a little bit of over density it would start sucking the material from nearby and create an under density the under density would start to grow and the over density would start to grow and eventually the thing would collapse and form perhaps a galaxy yeah ah that's a good question okay that's a good question that that is a good question why did it happen at the time that radiation became unimportant compared to the other part the reason has to do with pressure remember we talked about we talked about the relation between pressure and energy density wrote on the blackboard pressure is equal to something called W W is just a just a parameter times the energy density I think I call it Epsilon does anybody remember what I called it the density of energy well maybe a row yeah we call it row sure sure we did okay so let's see how much you can remember okay for radiation what is w ordinary radiation electromagnetic radiation filling a room this room is not filled with electromagnetic radiation it's mainly filled with atoms but blackbody radiation for blackbody radiation the pressure this is not momentum its pressure anybody remember the formula one third the energy density was pretty high and therefore at that early time and therefore there was a good deal of pressure what positive pressure does is it tends to do exactly what happens in this room positive pressure tends to homogenize things if there's positive pressure positive pressure is good for homogenizing things it pushes everything out and tends to create a homogeneous background that's in fact why the air in the room is homogeneous the positive press now what about matter-dominated or looks before we do that what about vacuum dominated anybody remember vacuum dominated yeah - one but I just asked you that - see if you remember all right but what about what about matter-dominated zero no pressure at all that means that if there was no pressure whatever in this room and there was no gravity let's start with no pressure and no gravity if there were no pressure in other words if the air molecules exerted no pressure how could that be what would it take for the air for the molecules to tala to exert no pressure aloft of zero velocity right zero velocity they'd be standing perfectly still obviously if they stand perfectly still and you create a bit of in homogeneity it's just frozen in it doesn't do anything why because the molecules are standing perfectly still P equals zero means everything's standing still ah but now let's add a little bit of gravity on top of that if we add a little bit of gravity on top of that then the standing still is not enough to overcome the tendency of gravity to pull things toward the center so it's a kind of competition gravity versus pressure pressure keeps things from imploding gravity can overcome pressure if it's strong enough and cause something to implode now very simply during the radiation-dominated era the pressure was so high that it simply prevented the gravity from doing its work and the causing causing implosion radiation pressure was very high and it just prevented it just prevented lumps from forming in the same way that it does in this room here eventually the radiation pressure became negligible the pressure due to the motion of galaxies motion of other things was also negligible and gravity became the most important part gravity became more important than the pressure and at that point stuff started to the contract don't you need a dissipation mechanism yes well actually you need a dissipation mechanism to cause the ordinary baryonic matter to to collide not to not to have the dark matter create halos so the first thing that happened is the dark matter the fluctuations in the dark matter and the fact that the pressure was so small started to create lumps of dark matter now the dark matter created lumps and the ordinary matter sort of follow those lumps but then the dissipative mechanisms caused the the ordinary matter to start to collect near the center of those lumps and the dark matter didn't dissipate so just stay there out there as big halos so you don't need dissipation you don't need I know what you're thinking I know what you're thinking there's a form of dissipation in cosmology which I was going to talk about in a few minutes there's a form of dissipation in cosmology which doesn't have to do with the normal things it has to do with expansion expansion creates its own kind of dissipation and the expansion replaced dissipation since the dark matter is effectively an ideal gas yeah but the idea of gas serum and pepper travel we still see it this original configuration no here was its original configuration and gravity started it moving a little bit gravity started it condensing into lumps well expansion eats up to pedasi I was going to talk a little bit about the relationship between dissipation and expansion and we'll see it because we'll need it for something else but just in an expanding time-dependent universe are you don't need dissipation to slow things down so we'll come to it was the dark matter that the Dark Matter exists when there was nothing but energy is dark matter though what do you mean nothing but energy all there ever is is energy are various forms you mean radiation energy no it was there yeah in fact the there are two time scales which are close to each other but not exactly the same there's the time scale or the time at which the universe became transparent and there's a separate time scale which is the time scale when the matter energy density in other words the nonrelativistic dark matter that energy density became larger than the than the radiation energy those are two crossover points one of them is a place where it cold enough for atoms to recombine and the other is a place where the matter density in the dark matter exceeded the radiation pressure and allowed allowed contraction to start happening the contractions started happening a little bit before maybe a factor of ten in time that the sounds like that is a little bit on the scale of these things happened a little bit before the universe became transparent so unfortunately we can't look back to it I've been a little bit before universals maybe another factor of 10 smaller or that the gravity started to do its work on the dark matter if the dark matter hadn't been there and it was only ordinary matter then the start of the contracting phase not the contraction of the universe but the contraction of these lumps would have happened about the same time just more or less accidentally about the same time that the universe became transparent and we might have been able to look back and actually see it happening but we can't so it's the dark matter which provides yeah one or two two more questions and then I want to yeah a little collegey to give us the elements greater than helium the elements greater than helium developed inside stars for the most part the reason I haven't talked about it very much is because it's not really considered in a certain extent part of ordinary cosmology it happened too late it happened it's really part of astrophysics it's really part of the dynamics of stars almost all of the heavy elements except for a tiny tiny bit which was produced during the Big Bang you never say none I mean never say never but during the Big Bang virtually nothing but hydrogen and helium were produced oh yeah a bit of that that's a form of hydrogen yeah right a bit of deuterium hydrogen and helium everything else was almost completely negligible although a little bit little trace bits of it the actual current abundance of it was mainly made during the stellar evolution in thermonuclear processes so in some sense it's not considered part of cosmology it's part of part of astrophysics that's probably taking a narrow view of cosmology position rock solidity a little emotion something else oh they also had some angular momentum yeah but that's was just inherited from from these fluctuations that they started with those are density fluctuations but they also had some motion there's also velocity fluctuations density and velocity and when we actually see in the sky is temperature fluctuations across the sky which are remnants of these little bits of over density and under density and also a little bit of their motion the motion of the gas at that time either toward us away from us also tended to produce some variation in the apparent temperature on the sky so let's just remember I mean we look back we look way far out to the surface of last scattering and some of that surface of last scattering is a little bit over dense some of it is a little under dense the over dense regions tend to look a little bit hotter to us know the over dense regions tend to look a little bit colder to us excuse me the reason is because the over dense regions have a little more mass they have a little more gravity and because they have gravity the photons that come from them are somewhat red shifted more than they would be from the other regions so from the over dense region where gravity is a little bit stronger it's pulling the photons back so we see looking from here the over dense regions the photons look a little bit less energetic from the under dense regions they look a little bit extra energetic but there's also another thing going on this gas is in motion some of the motion may be toward us or away from us so there's also a bit of Doppler shift and the Doppler shift also things which are moving toward us look hotter things which are moving away from us look colder all of this gets folded in and it's complicated difficult numerical calculations to start with a given initial condition and to evaluate what you know what what flows out of that experts do that and the conclusion basic conclusion was at the time that the universe became transparent or even before that there was about one part in 10 to the fifth variation are in the energy density of the universe so on the one hand it was very very smooth it was also the universe is also known to be very flat so it appears as if it expanded a great deal but on the other hand if it had expanded infinitely you might have thought that it would earn out these variations here so could it be that these variations are due to the fact that it didn't grow by an enormous amount well I wish that were the case but it's not the case on these fluctuations are there because of quantum mechanics they're quantum the remnants of quantum mechanics and they would be there even if the universe had expanded by an infinite amount or an infinite of course the silly even it expanded by 10 to the 10 to the 10 to the 10 to the 10th that would not have erased these fluctuations those fluctuations would have been created anew by quantum mechanics so I thought we would talk about that a little bit and get into a little bit why why it is that people believe this crazy theory that the universe exponentially expanded and remember what a clock what's that ah I don't know there are from time to time people who will put forward other ideas and usually the excuses well it's just another idea booth to to provide a straw man in other words a a foil R yeah right I don't know anybody who doesn't believe it that may be a bad sign that there's too many people believing too many believers you know I my friend BOM Paige who is a evangelical Christian he's also a very famous physicist and a great one and we argue all the time about whether such things is whether Jesus can be proved to be the Son of God and so forth and one of his arguments is how could so many people believe it if it weren't true I mean so maybe inflation is like that oh I don't think so I think there's some I think there's some spectacular empirical success over the idea but in order to tell you what that empirical success is I have to tell you a little bit more about it let's remember the idea starts with the existence of a scalar field and it's a scalar field that has a potential energy which is just a function of the field strength all right so we have a Phi which is some scalar field what it's doing there and how it comes into the physics of particle physics and so forth is unknown it's some scalar field qualify and it's there for no other purpose other than to provide some vacuum energy the vacuum energy or the energy of this field it has one component which is just due to the usual energy of an oscillating field like the electromagnetic field the electromagnetic field has energies for two sources one comes from its time dependence that's the electric energy and the other comes from its space dependence it's gradients and that's it's magnetic energy but this field has another form of energy which is just potential energy V can this be seen I'm writing with blue on blue no okay how about pink on blue V of v so the energy in this field if I wanted to I could write it down or many of you have done some of these classes would recognize it there's a term in the energy density which contains time derivative squared there's another term which contains spatial derivative squared this is the energy density it's conventional to put a factor of 2 in there so vibration of the field that's this one gives energy variation of the field and space that's this one gives energy density and then one more term a potential energy V of Phi and everybody will agree that scalar fields typically do have such potentials an example would be the Higgs field which some of you may or may not know about but that's one which we empirically have good reason to believe in and that has a V Fi so VF I is not an unusual thing or peculiar thing but a shape is a little bit unusual and contrived contrived meaning to say that you'd have to maybe tune some parameters via Phi is a function of Phi well there are zillions of functions around most of them are not very simple one is a simple function Phi is a simple function Phi squared is it not too complicated a function Phi cubed is pretty simple if you start adding things like 1 plus Phi plus Phi squared you can make some pretty complicated function and so it is with vfi it's a complicated looking function meaning to say it's not the thing that you might have written down on your first you know on your first calculus examination but nevertheless it has to have a shape which looks something like this it has to be fairly flat over some range over some range it has to be fairly flat not absolutely flat don't want it to be absolutely flat you want to have a little bit of a slope a little bit of slope downward and then at some critical value of Phi the bottom drops out and it falls away like so now it doesn't quite fall to zero it falls to something very small though and in plunk units in natural units units or a plunk units plunk units our units in which C is equal to one that's not hard to think about Planck's constant is equal to one a Newton's constant or equal to one those are called Planck units the units in which the natural unit of length is the Planck length which is 10 to the minus 33 centimeters the natural unit of time is 10 to the minus 42 centres seconds and the natural unit of mass anybody know the clonk mass is how much ten to the minus five grams all right so it's some unit so some natural units in which the most basic constants of nature are equal to one this falls down to something very small about 10 to the minus 123 that's the vacuum energy today up here it was larger there was still small and plunk units we know that it could not have been large on the Planck scale it was perhaps much much bigger than this but much much smaller than plunk the universe started what does it mean to say the universe over here what does it mean to say with this is not a place incidentally this is a value of the field what does it mean to say the universe started at this point when as I started I don't mean to say that it literally started at that point I mean to say we can trace it back to a certain starting point which was more or less like this and the details of this incidentally are not well known this is more of a cartoon of a situation whose details are really somewhat unknown but more or less like this what does it mean to say the universe started at a particular point on this plot well it means to say that the universe started with the field value being something sort of having some value filling space now it's not important whether it varied from place to place at this point because what's going to happen is the universe is going to expand so much that wrinkles and Phi will get enormous ly stretched out and the region where we sit is deep in the interior of one of these very stretched out regions and so for practical purposes we might as well assume that after a little while up here the universe expanded by so much that our part of it our patch of it and perhaps much more than that was very very ironed out why did the universe expand so fast well that's the robertson-walker equation of Freedman robertson-walker equation and VF i VF I incidentally is something which does not what's the word I'm looking for does not dilute so it's like vacuum energy in fact the V of Phi was perfectly flat it would would be exactly the same as a vacuum energy if for some reason the universe just sat at a point in other words if I did not change it to sat there then the value of the potential energy at that point would be vacuum energy so as long as this is pretty flat the universe will be exponentially expanding with an exponential factor with a Hubble constant which will be proportional to the square root of this vacuum energy the square root of V Fi I'm leaving out eight pies and that sort of thing let's just put proportional to the Hubble constant will expand proportional to the square root of V Fi and the universe will exponentially expand like e to the HT that will continue YUM what is fine fine what's the space is a filled front some version of space yep but what you feel but what is that space do you know anything about that space-time you don't have oh yes we do absolutely this is that is Einstein's field equation you yeah yeah oh yes yes yes this is Einsteins field equation and if we wanted to simplify it or we were just right for Rho naught plus Rho matter we would just write V via Phi for the energy density that's good enough as long as Phi is slowly varying what happens to these pieces of the energy well as long as Phi is moving slowly with time this is not important and as long as it's been stretched out as long as things have been stretched out a great deal then the gradient of Phi is small so these are unimportant and via Phi is the big thing in the energy while you're up on the top here then of course you eventually slide down the shallow hill let me talk about the dynamics of that in a moment you slide down this shallow hill until you get to the edge and then plop down you go in this picture cosmology all of cosmology is a roll from up on the top of a potential up here somewheres down to the bottom almost everything that we know about almost everything we know about and certainly everything we can observe happened after the universe plopped to the bottom down here with the exception of this exponential expansion almost all of which in fact all of which would took place up on the top here going over the edge is where the vacuum energy was converted into the energy of particles radiation and other more familiar forms of energy but not all of it some little bit of it was left over as vacuum energy why we don't know we don't know this is just the picture that emerges from from lots and lots of detailed observation so when the big you went over the edge of the hill here off the edge of the cliff that's the place where vacuum energy this vacuum energy not this vacuum energy here but this big vacuum energy up here was converted into ordinary energy ordinary things yeah including dark matter and other things I can't think of any known process of the converts vacuum energy matter-energy this yeah this field like any other field has quanta associated with it when the field rolls down to here and oscillates back and forth those quanta have energy this is a kind of harmonic oscillator it oscillates back and forth here the energy of those harmonic oscillators is another way of speaking about the quanta the energy of the quanta of that field so when this field which is called the inflict on it has a name once you name it you no longer ask why it exists you just say it's the in flick on the in photon field goes over the edge and it starts ringing down here starts oscillating those oscillations are another way of talking about the particles the field quanta which are the field quanta associated with this particular field all right now those field quanta may or may not be stable if they were perfectly stable that's all that would exist there would be no way of transferring that energy to other kinds of energy but typically the situation with scalar particles like the Higgs particle or other other scalar fields is they're unstable and they decay so one of these particles forming the sloshing stuff here let's call it the in photon can decay what can it decay into who knows photons electrons B plus C minus and that's the way the energy gets converted from the sloshing of this in photon field into ordinary ordinary particles now do we know enough about this field to say that this must happen no but we know enough about nature to say that it must happen most of the energy in the universe is composed of either dark matter or one thing this could be the sloshing around in here which is a form of energy could be the dark matter but we still have to get some of it out into photons and the protons and neutrons and so forth so some decays of this type must have happened that's the only way to get it to transfer but you know there's all kind there's all kinds of processes that can convert this sloshing stuff into into ordinary particles dark matter could be decaying if there if in fact this is connected with dark matter we don't know what dark matter is with the precision yeah that could be happening today today in this picture we're really stuck down at the bottom here the sloshing ended but now I'm gonna show if I get to it I'll show you why the sloshing ended so what we know is as the vacuum energy is really the residual effect of this fact yeah yeah if you think you're some kind there may be some the IR connection Marinol let's get to it let's get to an important point yet you might ask if well first of all if you started up on the top here and you fell over the edge wouldn't you start by energy conservation wouldn't you swing back and forth here and perhaps come back up onto the top or at minimum when you expect to endlessly swing back and forth here is there any reason why this energy would get degraded as if there was dissipation as if they were friction which would cause this field to slide down to the bottom now think of a mechanical analogy the mechanical analogy is a particle rolling on a potential like this why does it come to rest down here does it come to rest down here the answer is it better come to rest down here because if it didn't our universe would be filled with this enormous quantity of energy and it's not it's pretty much empty space how did it get to be empty space where did all that energy go and I'm going to tell you it dissipated and so what do you mean it dissipate it dissipated into what well there's a form of dissipation which takes place in cosmology that doesn't have to do with genuine friction it resembles friction very much but it is actually due to the expansion of space so I thought I would take a couple of minutes and show you where that comes from to do that you'll have to recall something which not everybody in the room may know but it's Lagrangian equations of motion this is the expression for an energy let's suppose let's suppose that the field did not vary with position in space just to make the story simple let's ignore the gradients of the field with respect to position in space why because it's gotten itself stretched out so much that this is unimportant let's assume that's unimportant all right this is the potential energy this looks like a kinetic energy of something it's not the motion of a particle but mathematically it's proportional to the square of a time derivative it looks like a kinetic energy of a degree of freedom which in this case is a field and this looks like a potential energy what will be the Lagrangian this would be the energy what would be the Lagrangian of such a thing energy is T plus V Lagrangian is T minus V T stands for kinetic energy so the lagrangian so the Lagrangian is T minus V now actually this is not the Lagrangian this is the Lagrange density you have to multiply it by the volume of space this was not the potential energy this was the potential energy density to get the potential energy you have to multiply it by the volume of space right same here this was energy density so what's the volume of space well a cubed a cubed of time so now if we know how a cube behaves with time we have the Lagrangian of this field Phi ok let's work out its equation of motion do you remember how to work out the equation of motion for Lagrangian you take partial of L with respect to Phi dot that's a of T cubed times Phi dot and then take the time derivative of that that's the left hand side what about the right hand side anybody remember PVDF I like a force doc dvdx like dvdx except is the a cubed here a cubed DV or DV we find DVD phi is the slope and it's the force term which creates what would be an ordinary acceleration if a was constant with time but a is not constant with time so we have a little bit of work to do to take this derivative to take the derivative what is the derivative well the first term the D by DT acts on the Phi dot and that gives us a cubed times Phi double dot which is like the acceleration not the acceleration of the expansion of the universe but it's just the acceleration of this field Phi that second time derivative of Phi that's one term what's the other term the other term comes from differentiating the a cubed and that's 3a squared of T times what times a dot times v dot and that's equal to a cubed DV d phi but i want to divide both sides of the equation by a cube just to get rid of as many a cubes as I can all right so on the right hand side I'll just have DV d Phi and on the left hand side I'll get rid of this a cubed I'll get rid of a squared but it will leave me with a 1 over a in the denominator so let's put 1 over a a dot over a what are you what's a dot over a is another name for it not the square root of h @h right just H so this is the Hubble constant now have you ever seen or this is - there should be a minus here - DVD 5 yeah - DVD 5 force is proportional to minus the derivative the potential energy have you ever seen an equation of motion with a single fight down in it like that viscosity viscosity a ball or a rock falling through water this is essentially the equation of motion of a rock falling through water here's the gravitational force on it DVD Phi and it's proportional to the slope here here is the acceleration and here is the viscous drag force here is the viscous drag force so believe it or not this is called hubble friction hubble friction it's not real friction it's due to the time dependence of the size of the universe that's where it came from it came from differentiating a but it resembles very very closely are the equation of motion for an object moving through a viscous fluid and if V has a more or less uniform slope it's like a object falling through a viscous fluid in under the presence of a uniform force what happens under those circumstances to the a falling object it comes to some terminal velocity it comes to some terminal velocity very quickly trying to drop a rock in the water and it very quickly comes to terminal velocity so the universe starts sliding down this hill if this is shallow enough it goes very slowly down the hill yeah and as it goes down the hill viscous friction or Hubbell viscosity or whatever it should be called huh boom damping uh brings it very quickly to a terminal velocity and it just slides down this hill slowly till it gets to the end during the time that it's sliding down the hill or sliding down a hill which is a very very shallow nature there's some vacuum energy and that vacuum energy causes almost exact exponential expansion when I say almost exact it's not exactly because there's some slope here so the vacuum energy changes from one time to another but if this is reasonably flat the universe will grow exponentially approximately with a constant H here and then it will come to the end and fall down and how many times does the size of the universe double well that depends on how shallow this is how broad it is and how high it is the height of it goes into the coefficient here goes into the time constant here the width of it and the slope of it determine just how long it stays up on the top of the hill where so by making it broad enough and flat enough you can make the universe expand as many times as you like during this period of inflation okay so then you would say if it exponentiate by many many many orders of magnitude why doesn't that why doesn't that make why doesn't that iron out all of the variation in the field why does it have any variation at all by the time it gets over the edge here what is that variation that variation is that small little bit that's necessary for galaxies to form so in the one hand you've explained the flatness of the universe and the homogeneity but it looks as though you might have overdone it you might have overdone it and lost the possibility of explaining the existence of galaxies now marvelously that's not exactly right so to understand why it's not exactly right we have to have we have to take some things from quantum mechanics let's suppose that the universe was not expanding and just the simplest the simplest possibility let's forget the expansion of the universe for a moment but let's add quantum mechanics to this picture of a field a field filling space with a field because of the expansion but we're going to forget the expansion but because of the expansion the field has gotten extremely ironed out so it's almost perfectly homogeneous but quantum mechanics makes it fluctuate quantum mechanics makes it fluctuate one way of seeing the fact that quantum mechanics makes it fluctuate is just by looking at the similarity of the field of its energy plus vfi by looking at the similarity of such a field with the motion of an ordinary particle this is like kinetic energy and some potential energy over here plus another term over here why is it it's kind of like a harmonic oscillator this VF I could conceivably be a harmonic oscillator potential but there's also a harmonic oscillator potential in here hidden in here why is it that harmonic oscillators have zero-point energy they have zero-point energy because of the uncertainty principle you cannot have Phi being fixed and locked and at the same time have the velocity of Phi we fixed the lock it's the uncertainty principle the uncertainty principle in the same way that the uncertainty principle tells you that there's zero-point energy of a harmonic oscillator ground state energy whatever it's called are the same mathematics tells you that Phi is necessarily even in its ground state even when all the energy is sucked out of it it's still oscillating with zero-point energy so that means if you were to look microscopically at this field you would see it constantly in a state of agitation even just in the state of absolutely lowest energy okay why don't why don't we notice those fluctuations and for the most part we don't notice them because they were too rapid they take place too rapidly for most for most observations and we average over them okay we average over them most such fluctuations are fairly short wavelength and the short wavelengths oscillate very fast and because they oscillate fast we don't notice them but in cosmology the situation is a little bit different and to understand how the situation is different you have to take into account the expansion of the universe that's the main new thing the expansion of the universe okay let's draw the universe horizontal in space as always vertical is time and X goes horizontally X is the coordinate X it doesn't measure distance it measures coordinate along here here are the various galaxies for example I'm putting them equally spaced which of course they're not and of course the world is three-dimensional not one dimensional but these are the galaxies i've approximated them by saying they're equally spaced and they're just vertical lines but the distance between galaxies is not the difference in X but what is it a X a being scale factor so distance is a of t times Delta X where Delta X might be the distance in coordinate between two points so if a is expert so let's imagine now that a is exponentially expanding e to the HT times Delta X that means the distance between galaxies on the average if they're far enough apart is expanding exponentially Delta X is of course not expanding exponentially Delta X is just the fixed coordinate positions of the galaxies you can think of it as their names but it's the scale factor which is increasing exponentially now what happens to a wave in here what happens to a typical wave as the universe expands it gets stretched out but in these coordinates it's not stretched and these coordinates it stays more or less the same with time it has a fixed wavelength and X but that means an exponentially growing wavelength in proper distance the proper distance of the wave more or less has the same profile it may be oscillating but it's not expanding on this diagram because this diagram already has the expansion built in as the expansion of the distance between these points okay but it oscillate s' the field oscillates in general it oscillates unless it's wavelength is too long if it's wavelength is too long it doesn't oscillate so let me explain why it doesn't oscillate um supposing we have why does a wave why does a wave oscillate first of all why does an ordinary electromagnetic wave oscillate think of a rope stretched between two points and perhaps the Rope has some curvature over here maybe it does maybe it doesn't let's just um may have some curvature not why does the rope over here get pulled down it gets pulled down by its neighbors by the neighboring segments of rope which would typically be down here but what if the neighboring segment of rope was up here which we're going to get pulled up alright so the only way to know which way it's going to get pulled is to know what the neighbors are doing in particular well yes the north neighborhood what the neighbors are doing but supposing the size of the wave here was about as big as the horizon as about as big as the horizon remember what the horizon is the horizons distance beyond which things are moving away from you faster than the speed of light you can never get any information from beyond the horizon that's a principle nothing can communicate from behind the horizon so here's a piece of rope that if the scale is big enough for the variation of the position of the rope one could imagine that that this piece of rope is out beyond the horizon supposing this piece of rope was out beyond the horizon can it have any influence on this piece no no so it better be that the tendency to pull on this piece of rope or this piece of field this little piece of field in here that had better get be tremendously diminished when the wavelength of the wave becomes about equal to the size of the horizon when the wavelength is longer than the size of the horizon then the pieces which are pulling a piece over here are out beyond the horizon and the answer is if you follow the mathematics that when waves get longer wavelength in the size of the horizon they freeze they're not pulled back by the neighbors and they just free use the motion becomes frozen why should I worry about waves are more than I'm worried about quantum fields my goodness what are we talking about waves longer than the size of the horizon well the point is that the universe is expanding so take a wave of a certain wavelength and push it ahead in time on this graph it has the same wavelength but the distance between these points exponentially grows so no matter how small the wavelength is to start with eventually as the universe grows and stretches each hidden wave each zero-point oscillation will eventually have a wavelength longer than the size of the horizon when that happens the wave stops oscillating it just stops oscillating it is no longer pull back and it freezes it just freezes so if we wait long enough let's suppose this wavelength here is um I don't know 10 to the minus 8 centimeters some small wavelength and it's oscillating like mad because it's being pulled back into place by its neighbors but as time goes on this 10 to the minus 8 centimeters grows to the point where it may be as big as the horizon will be as big as the horizon when that happens all of a sudden the wave freezes so the zero point also and an expanding universe and exponentially expanding universe the zero point oscillations become frozen first the longer wavelength ones become frozen all right so eventually this way of over here will become frozen but then what about shorter wavelengths shorter wavelengths will take a longer time but they will also eventually become frozen even the very shortest wavelengths that you can imagine will eventually become frozen yeah but it kind of asymptotically building a very cool thank you yeah that's a good point all right so let's draw what the horizon would look like yeah the horizon what is the horizon the horizon let's let's work out what the size of the horizon the horizon is a distance that corresponds to the speed of light using the Hubble formula all right so it's a velocity sorry distance I always have I always have to work this out distance is sorry velocity is equal to Hubble and climbs distance arm the place the distance at which this is the speed of light that's called the horizon and that distance therefore is the speed of light provided by the Hubble constant that's the distance that corresponds to the distance to the horizon so that's the distance that what did you ask me you asked me a question oh yeah right right yeah yes yes it does the distance is a fixed number the distance to the horizon is fixed it's just C over H and in an exponentially expanding universe H is fixed all right but the distance is on this graph are exponentially increasing so a fixed distance on this graph looks smaller and smaller a fixed distance of Hubble size R this is called Hubble size this distance here would look smaller and smaller because it's being compared with exponentially growing distances so that means that the horizon size would look like this exponentially getting smaller and smaller with time and that's why each wave will eventually it's frozen in becomes frozen when its wavelength gets longer than the horizon sighs okay or when the horizon in coordinate distance is shrunken to be smaller than the right okay so each way it gets frozen and they pile up on top of each other they pile up on top of each other each one frozen and what was originally quantum fluctuations very rapidly oscillating become large-scale fluctuations of the field which grow and which pile up on top of each other and which just freeze at the end of this by the time you get to here all these waves are piled up and made some sort of variation in the field some sort of frozen variation in the field and so the next question is when this field which now varies from notice notice it does not matter how long it's been inflating it could have been inflating so long that the universe is 10 to the 10 to the 10 to the 10 to the 10 we go on all night that's how bear that's how much bigger it is that's how much it would be ironed out but nevertheless as time goes on new fluctuations get frozen new waves become frozen more and more of them become frozen and so the end result is is always a variation that's kind of a fossil remnant of early vacuum fluctuations that got frozen into the system and they get frozen in pile up on top of each other and become a large-scale massive macroscopic variation of the field yeah what happens what yeah it's scaling down it's diluting at the same time energy density is diluting as it expands so in fact I mean it doesn't get it doesn't get out of control from the point of view of energy density but it just keeps making the you know more oscillations top piled up on tomorrow more oscillations and the effect is a kind of stationary steady state the variation that's it's yeah where as time goes on you erase earlier well you stretch them out yeah and new ones keep coming in right right you stretch them out so that they're bigger than the horizon you look at one time to look at it two hours later right that's the same right that's called a scale invariant stressor spectrum scale invariant spectrum and one of the things that has been known about the initial you see right but it sounds like they want to fight first rate out they were awesome fibrin stretch but the noxious fluctuation okay remember there was the horizon to do lets there were two different horizon scales there was the horizon up here and is the horizon down here the horizon up here just means the distance scale associated with the vacuum energy in here okay the Hubble scale down here means the distance scale associated with this energy density they're very different untended write a small horizon up here alright these ways become ossified when they move during this era the waves get ossified when they get bigger than the horizon associated with this energy scale this could be some microscopically small thing from our point of view right then they get frozen they're frozen that's right that's right our horizon grows what happens at first is that these waves get bigger than the temporary it's really a false horizon but a temporary horizon over here but then the horizon grows when it rolls down here the horizon suddenly turns that way and these waves come back into the horizon and they start to oscillate again they start to fluctuate again but by that time they're stretched out on enormous ly large scales and they're piled up so you have a huge pileup of waves macroscopic structures and those macroscopic structures start to slosh around again in fact that's really what happens they start to slosh around again the words we use is they go out of the horizon and come back into the horizon so they expand out beyond this temporary horizon and then get swallowed up by the larger horizon at which point they start to slosh around again right but as I said by the time they start sloshing around again they're big macroscopic right I'm take a long time to swing feel the spot here all those footbe foldings hurting will get very small amount of time where we don't mind well well ELISA say is that the last week like you say maybe see oh that would be very very small amount of time I mean by macroscopic standards yes very verse part of it I don't know if you answers in the tiny name how did you calculate power spectrum you get honey well just from this picture just just from this picture that the waves keep different wavelengths keep passing out of the horizon but at each stage the same thing is happening exactly the same thing is happening as the previous stage and it's on a logarithmic scale one can calculate it and see that at each stage it looks the same except everything's scaled up and it reaches the sort of stationary phase that's so that's a little too complicated to go into but it does it does achieve this scale invariant behavior where the same more or less the same amplitude of oscillation for every wavelength that's not exactly true but but close to it and the result is a particular structure of frozen in a soul a frozen in density variations okay now let's let's go a little bit further and talk about what happens when you go over the edge here when you go over the edge those fluctuations get magnified not magnified in spatial size but magnified in amplitude or in energy density the energy density the energy variation from place to place gets magnified and it gets magnified proportional or inversely proportional to how slowly the field no inversely proportional to how fast the field was rolling down here the flatter it is the smaller the terminal velocity as it's rolling down here the smaller the terminal velocity you might think this is a little bit counterintuitive but the smaller the terminal velocity the bigger the variation in the energy density will be after you fall off the edge here so let me just show you why that is I talked about a little bit last time I was a little too tired last time so let me just say a little bit more about it okay so the field is varying from place to place because of these quantum fluctuations let's say to the right is red and to the left is blue then there will be regions which are redder regions which are use green is green back here and red over here so there will be greener regions and red regions and green regions and red regions now I don't really mean back here and up here I just mean small deviations from here to here it's all small fluctuations this like this is an effect that has to be fairly small otherwise what Jennifer created too much random mess in the universe so it's a small effect but there are greener regions slightly to the left and red are regions to slightly to the right naturally the redder regions fall off the edge before the greener regions okay so Paul hang a little red region over here falls off the edge of the cliff it falls off the edge of the cliff and all of this vacuum energy let's suppose all of it for simplicity all of it got converted to radiation energy that's a simple model that's a simplified story but let's suppose it just got all turned into photons and so forth so let's see the redder region after because it was a little bit ahead it was a little bit ahead moving toward the edge of the cliff it their first and fell off the edge so this rado region would have all of its energy turned into a kind of energy which dilutes with expansion the greener energy which is still up on the top here is still vacuum energy it does not dilute okay so what happens to the energy density in here the energy density in there decreases it decreases just by the standard rules by the time you down off this cliff you're in a world now which is pretty much like ordinary cosmology ordinary material radiation or whatever if it were radiation we know the energy density in there we go as one over a to the fourth the scale factor in that region or the expansion factor in that region would be growing and the effect would be that the energy density in there decreased what about the energy density in the greener region after it fell off the edge of the cliff it would be higher because it fell off later so by the time a bit of green energy goes over the edge the red energy has already been red shifted it has already been red shifted or diluted by expansion whereas the greener regions are not yet diluted but if you wait a little more the redder regions will get even more diluted the green region will start to get diluted and the even greener region will come and fall over the edge and make a yet higher energy density region R this is the way that oh and what is this effect proportional to well the slower the average motion the longer the time scale between different pieces falling over the edge if everything is moving really slowly even if one point is slightly ahead of the other here it is moving extremely slowly there will be a large gap in time between the time the first piece rolls off and time the second piece rolls off that means that the red energy will get highly diluted by the time the green energy falls off so the flatter this is and the slower moves the larger will be the variations in the energy density from place to place the final upshot is a spectrum of air fluctuations of energy density which is the remnant of all of this which depends it depends on the height of this it depends on the width doesn't depend much on the width but it depends on the slope a little bit why the slope because the terminal velocity depends on the slope and the slower the terminal velocity the bigger will be the fluctuations doesn't depend much on the width so for that reason we can't say how many e foldings took place but it depends on the height and the slope in such a way that the shallower the slope the bigger the fluctuations also the higher it is the bigger the fluctuations the higher it is the more energy density the larger will be the fluctuations and the flatter it is a formula that you can work out that will tell you how big the fluctuations are but the point is that the detailed result of such an analysis fits extremely well with with empirical data no unfortunately you cannot tell with a height and slope is you can only tell a combination of the height in the slope there's one combination of the height and the slope that you can predict okay and well okay that's not quite true okay not quite true arm if the height is high enough if the height is high enough then there are in addition to the production of these density fluctuations there are production during the gravitational era here are doing this inflationary era there are also gravitational waves produced and so admixed into this density fluctuation and the temperature fluctuation on the sky there may be also gravitational waves that are visible if there are gravitational waves that will tell us how high it is but if it's too low then there won't be visible gravitational waves and that's possibly the case very likely the case so it's going to wait until the gravitational waves can be detected from this period and that will give us some idea of how high it is most likely it will just tell us it's lower than a certain amount if the energy density we're any bigger I'll tell you the if implant units in plunk units if the energy density were bigger than about 10 to the minus 14th which may sound small but it's a huge energy density bigger than about 10 to the minus 14th then we would have already been swamped by gravitational waves from it so we know that the energy density is less than ten to the minus 14th because we were not swamped by gravitational waves on the other hand if it's less than about ten to the minus sixteen then we have no chance of seeing the gravitational waves and so there's a narrow little window which we have no reason to believe is something like that I think narrow little window where you could see gravitational waves 10 to the minus 14th might actually be 10 to minus 2 I don't remember the precise numbers but it's in that ballpark there's only a narrow region of parameter space where you could see these mast on the fourth mass to the fourth yeah why is it mass to the fourth so let me explain why it's mass to the fourth first of all um if you said C equal to 1 and H bar equal to 1 those are those are natural things to do then mass is the same as inverse length all right massive the same as inverse light so that's just the mass an inverse length or the same thing what do you ask me I forgot or what are the units of the energy all right its energy density first of all energy density so its energy divided by length cubed and if 1 over length is the same as energy than its energy to the fourth okay so it's M plunk to the fourth that the unit's right and and empirically we know from the lack of gravitational waves in the spectrum of in the cosmic background spectrum that it's less than maybe it's ten to the minus twelfth I don't remember exactly but uh and the problem is that if it was another decade or two smaller than that you would have no chance of seeing it so you're caught between being swamped by it well you're not swamped by it and then it kind a couple of decades down and you couldn't see it oh yeah yeah we would have seen we would have seen a lot more structure in the microwave background yeah yeah excessive freezing out of these long ways and undergoing a random walk yeah it doesn't undergo a random walk at any point at any point that undergoes a random walk it's exactly what it does but of course there undergoes different random walks at different places and that's what makes the variation it yeah it's a statistical process that's exactly right so if you focused on one particular point in space you would see one mode freeze out and the next mode freeze out the next mode freeze out with random phase between them and it would make a random walk that's exactly right southern question is preoccupied people for a long time this is what happened at equals zero is irrelevant there is no people well irrelevant or none no okay so the right the well let's put it this way the success of this theory and has been highly successful and in correlating a lot of data on different wavelength scales when I say it correlates lots of data it correlates data about fluctuations on very different scales scales comparable to the whole size of the sky variations across the sky which are which are large scaled down to very small scales of a thousand times smaller than the size of the whole sky and it correlates all those fluctuations density fluctuations basically into one set of parameters governing this potential here so in that sense it's highly successful the upside of it is it washes out whatever the initial conditions were because this stretching washes out what came before and only the quantum fluctuations which are produced later irrelevant so the success of it is that it washes out what came before but what's one eras success becomes another eras disaster it means now that we accept all of this and we all think it's true next thing we want to know is what happens earlier and all of the things which made this a very successful theory work against us in finding out what happened earlier so inflation when it occurs tends to wipe out the past and as I said that's both good and bad particularly bad for those who want to understand what might have led up to this this is a the points this is a ridiculous starting point this energy density while it's easy to accept that the universe might have started with a very high energy density that's easy to accept big deal the problem is the energy density was very very high in ordinary units joules per cubic meter or whatever it was enormous ly high energy even on particle scale units much higher energy density than for example the density of energy inside a proton so it was enormous ly high energy on the other hand it was very small in any natural units it was 10 to the minus 14th or something or 10 to the minus 12 or smaller so it's neither here nor there so to speak it's neither very big and they're very small it doesn't sound like it's the sort of place that natural starting point would be natural starting point might be with the energy density for example being plunk in being one unit in plunk units 10 to the minus 13 or 10 to the minus 14th doesn't sound like it's the starting point now of course that could be wrong I don't know but there's no natural explanation of this whole structure and why we might have started over here and whether or not we might have down from some much much higher energy density up here that's all lost that's all experimentally lost because inflation just stretch things out so much that we lost it could the exponential have gone on you mean into the remote past no no no now this is an extremely odd story you say what would have happened if the energy density was absolutely constant out here for example well this function has to do something right it could increase in this manner here in which case it would say in the remote past something different was happening than down here okay exactly what was it well we then go up to here and we say could there have been a plateau up here that went on and on indefinitely well we might as well say could there have been a plateau over here that went on and on indefinitely all right so what are the consequences then of a universe with an absolutely constant energy density like this let's put a way out here with an absolutely constant energy density well when I the I told you that the sitter space was an exponentially expanding universe that is correct at late times in fact real de sitter space is a universe which looks like this which contracts and then expands that's the real honest solution of Einstein's equations with a cosmological constant it contracts and then it expands it contracts to some minimum size the minimum size happens to be the Hubble scale and then it starts expanding and it expands exponentially I wrote that they - I wrote the that the that the scale factor was like the exponential of HT that's not exactly right can you think of a function which contracts and then expands and and the lake x looks like exponential hyperbolic hyperbolic cosh yeah cosh HT that's the real solution of Einstein's equations so the real solution of Einstein's equations is not an eternally exponentially expanding universe but it's one which contracts and expands now should we believe the question comes up should we believe in this contracting phase all right should we believe in the contracting phase that came before the expansion I will just leave that as a question mark I will leave that as a question mark or should we believe that somehow we started in some other way unknown I think it mister I think that is mystery down here we don't know very much about that we really don't have a theory of the beginnings if this plateau really gave way to a much higher mountain here it would mean that this might have started much smaller in fact if it gave way to a mountain which is as high as the plunk scale we might even want to say it started as the smallest possible thing 1 plunk length on the side the fact is we don't know we don't know very much about any of this no no why do you say that is all it raised it well this means you have to be clever yeah it's of course all a race if there was too many fallings right if we are very very lucky and there were 63 a foldings then we have a chance of seeing it whatever chance of seeing some of this I'm trying to think out what the double damping does to the field equations particular does that change the quantum of the field equations as time goes on what it is is it's the red shifting of the field quanta it's equivalent to the red shifting of the field quanta in other words it appears that energy is being lost but one way of thinking about it is the energy is just being stretched so it's completely equivalent to the red shifting mention my pet that is costing us less lenses begin first the unity units the units of length and mass are inverse to each other the formula for example for the Compton what's the formula for the comp and wavelength quantum mechanical wavelength of a particle of mass M Compton wavelength lambda compound anybody know kind of h-bar in it it's quantum mechanical well over momentum but we're talking about now a particle of a given mass so it has to have units of mass times velocity M times C h-bar over MC is the Compton wavelength so we know that H bar over MC has units of a length now if we work in units in which C is one which we've all gotten very used to by now and if we were also quantum mechanically inclined we might also set H bar equal to one we don't have to we can just say that lambda is inverse to a mass but if we do set H bar equal to one which is a common thing to do then wavelength has units of one over mass so length has units of one over mass mass has units of one over length stretch out frozen a yes oh they continually be as this process goes on waves become frozen they become stretched their energy is degraded and they're replaced by new ones but the net effect is pointed out was a kind of random walk where the field just becomes statistical noisy mess and on every scale and every length scale so what's depleted by the expansion gets replaced by by new waves coming in and and freezing in on top of those previous ones is that like the slowing down of coordinate time when you purchase a vetericyn which approaches that rise of a black hole yeah you get a more more red shipping and being then it's connected with a connected that's yeah it is but there yeah anyway it makes sense on out of a question of how probable inflation was in a sense of look okay various parameters for the yeah yeah there's a space of parameters the first-order not first-order in any technical sense but the first-order and thinking about it is the width of this is the height of it and there's a slope of it the slope remember the slope comes in in a prominent way because slower the smaller the slope the slower the field moves and slower it moves the more the bigger the fluctuations these are three parameters the height let's call it the not the height of it the slope of it and the width of it I don't know Delta Phi now um generally speaking most physicists are of the opinion Delta Phi incidentally has units and has units of mass generally speaking physicists get very uncomfortable when a thing with the unit of mass gets bigger a fundamental thing becomes bigger in mass than the Planck mass it very uncomfortable about that and most of us would probably maintain that the scale of variation of a field like Phi should never be bigger than about one unit and plunk mass that may or may not be true but if it's true it tells us one piece of information another piece of information is that the energy density should never be bigger than one in Planck units that's something we really believe I won't get into the reasons for it but that's there's a lot of reason for that and so having some kind of bounds on these we're really talking about bound on the slope here we're talking about what's the probability that the slope is shallow enough that 62e foldings happen or sixtieth foldings happen what's the probability and the answer is about two percent that you have to adjust this slope to about two percent in order to get it to roll slowly enough here to get the required number of e foldings all right it's two percent is that how how unlikely or how likely is that well it's two percent now there's another interesting element to it there is another but I won't have time for it so I'm not going to try to get into it there is another element to it the other element ah the question is is inflation if we would need to get into issues of anthropic questions we have two puzzles cosmological puzzles both of them are fine-tuning puzzles one of them is a mild fine-tuning it's only 2% in some way of thinking about it it's it's only 2% the other is the fine-tuning of this 10 to the minus 123 down here took big fine-tuning puzzles both of them can be resolved in one simple explanation whether you like it or not that the universe had to be such that structure formed in it structure meaning galaxies stars planets and so forth and here's a way to think about it a positive cosmological let's let's take let's take a particular case positive cosmological constant and open negatively curved universe let's take that as our model we could think of any other combination we like to get the same answer but let's take positive cosmological constant in open universe okay so what is the positive cosmological constant from a Newtonian point of view from an antonio point of view it's a repulsion it's a repulsion a repulsive force a universal repulsive force which is proportional to distance that's what a cosmological constant is if we went back to the Newtonian description of things that we started with the very early part of the quarter and added in to the usual Newtonian gravitational force between objects another term which was proportional to a thing called lambda times the distance between objects that would mimic the cosmological constant and a positive cosmological constant which is what we see we correspond to a universal repulsion but because it increases as a function of distance I it had better be small that better be small enough that it doesn't affect the solar system in fact it better be small enough that it doesn't affect the galaxies in fact it better be small enough that it doesn't affect seriously big clusters of galaxies and in fact experimentally it's known to only be important that the scale of that at 10 billion light-years so it's very small repulsion but what would happen if it was a hundred times bigger 10 to the minus 123 is how big it is right it's because it's so small that it only has an effect on incidentally what if it was 10 to the minus 10 that repulsion yeah it would be easily enough to blow apart an electron what if it was 10 to the minus 100 is 'le big enough to blow the earth out of just easily easily by many orders of magnitude all right so what if it was 10 to the minus 120 1 instead of 10 to the minus 123 well not galaxies today you would not be potent enough to blow up the galaxies today but it would be potent enough to prevent this one part in 10 to the 5 fluctuations from condensing into galaxies so this very mild variation that was there left over from inflation ah it would only take a little bit of universal repulsive cosmological constant term that prevent it from condensing into galaxies this was Weinberg Steven Weinberg's discovery in 1987 that a cosmological constant roughly two orders of magnitude bigger than the 10 to the minus 123 which would have prevented galaxies from forming at the time it was thought that the cosmological constant was zero and he said no there's no reason for that whether it just has to be small enough to allow galaxies to form and he was right that experimentally it wasn't zero it was just small enough to allow galaxies to form okay there's a similar logic that goes with the curvature of the universe let's take the case of negative curvature they remember what K means from the Newtonian point of view or a particular K being whether you're positive negatively or zero curved it had to do with the energy of recession escape velocity K equals zero meant everything is moving with exactly the escape velocity K greater than zero would mean sorry K greater than zero means less than the escape velocity K less than zero means greater than it so if we take the case of K less than zero which for one reason or another has become the favored case but it wouldn't really matter if K were less than zero it would mean that everything was shot out with greater than the escape velocity okay well that sounds bad it sounds like too much velocity can also prevent galaxies from four two things can prevent a rock which are going out from returning one of them is if you're above the escape velocity the other would be an additional outward force that you didn't know about which might have been proportional to distance but very small okay both the cosmos or the positive cosmological constant and negative curvature would both work in the direction of preventing galaxies from forming now the cosmological constant is very small what about the negative possibility of negative curvature if the universe is slightly negatively curved today it was much more negatively curved in the past right because curvature radius of curvature gets smaller so was much more negatively curved in the past we go back to the past when galaxies form and we say the universe must have not been too negatively curved what could have caused it not to be negatively curved if it started out negatively curved what can balute curvature expansion curvature yeah inflation inflation alright so you can say how many 'if Olding zuv inflation had to take place in order that the universe flattened itself out enough before this era when structure was formed how many foldings had to have taken place in order that the outward thrust of things didn't prevent galaxies from forming and the answer is about 59.5 ii foldings we know that there was at least 62 we know that for us to be here it had to be more than 60 not 60 not 59 point five all right so one explanation then is that we can just throw away the possibility that there was less than 50 9.5 e foldings and then we can ask the conditional probability what's the probability given that we're here to ask the question well it's given that there were more than 59.5 you foldings then we can ask what's the probability that there was more than 62 in other words we can take this parameter space and we can say let's cut out all the parts of it where there weren't enough he foldings enough expansion to get rid of the large outward thrust which would prevent galaxies from forming throw it away and ask what the remaining part of the parameter space given that there were more than 62 he fold more than 59 foldings what's the probability that was more than 62 and then the answer becomes maybe 0.9 or something so it depends on how you ask the question what the probability of inflation was if you ask the question conditional that we're here to ask a question then the answer is that's very probable if you ask the question from a more a priori point of view what's the volume in the parameter space then it's less than 2% so it depends on how you ask the question the same is true for the cosmological constant if you ask what's the a priori probability that a number is less than 10 to the minus 123 well it's 10 to the minus 123 but what's the probability that that if we're here then the number becomes of order 1 so these are the great puzzles these are the great confusions the great puzzles the great debates which are going on and that brings us to the end of the great debate you you

Channel: Stanford

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Keywords: quantum, physics, mechanics, cosmology, math, science, sphere, capital, omega, potential, energy, form, theory, dark, matter, vacuum, galaxy, movement, curve, flat, universe, density, space, constant, time, exponential, expansion, light, year, big, bang, stretch, inh

Id: 9R57sdG61tA

Length: 118min 41sec (7121 seconds)

Published: Fri Jun 12 2009