Cosmology Lecture 5

stanford university okay let's uh i want to review a little bit and then discuss the equation of state two equations of state well three equations of state and where we get our information about them where our knowledge not cosmological knowledge but our knowledge based on basic theory basic physics where that knowledge about the equation of state comes from but before i do i just want to very very quickly remind you where we are where we're going the basic question that we want to answer in cosmology is what is the history of the universe and to the extent that the universe can be thought of as homogeneous and isotropic it really boils down to what is the time history of the scale factor if we know the time history of the scale factor we know an awful lot about the history of the universe we can test it and we can observe it in various ways and so that's the question that's uh one or it's not the only question but it is one overriding question that if you want to do cosmology you better have under your control what is a of t as a function of time how does it evolve i'll just remind you quickly we we studied some models there was the matter dominated model and in the matter dominated model a of t expanded like t to the two thirds in the radiation dominated universe a of t expands like t to the one half both of these are models neither one of them is exactly correct today at late times this is almost exactly correct at very early times we believe that this was more correct and there was a transition between them we talked about it at length uh we're going to when we when we get to observational cosmology we're going to talk a great deal about how we know anything about this how we know anything about this and what the various meanings of them are but not yet tonight okay we talked about also the importance of the equation of state that the radiation and the matter dominated universe are two examples of universes which evolve under different conditions which can be characterized by an equation of state the equation of state incidentally is what tells us how the energy density which is on the right hand side of the friedman equations of the cosmological equations it tells us how the energy density changes with changes in the scale factor for example it tells us in the matter dominated case matter dominated it tells us that rho is equal to some constant let's call it rho naught divided by a cubed and the a cubed is just the volume of a piece of space as it expands the density is the amount of energy in it divided by the volume and that's just something over a cubed in the radiation dominated case which we're going to talk about extensively today where where the equation of state comes from rho goes like rho naught divided by a to the fourth this is radiation dominated now the difference between these two originates and the difference between the relationship between pressure i'm going to i'm going to review these things quickly now uh in the relationship between pressure and energy there are many many the relation between pressure and energy density is called the equation of state in cosmology it's also part of the equation of state of a statistical mechanical system which usually involves other variables like temperature but in cosmology we make a simplifying assumption that uh that the energy density and the pressure are simply related and for some for simplicity and because it covers a lot of interesting cases we take the equation of state to be pressure is equal to some number called w and w is a number it characterizes the fluid whatever it happens to be times the energy density rho let me just go back again and show you how the equation of state tells you these two different types of things right the equation of state for matter dominated let's start with that one first the equation of state for matter dominated what matter dominator just means particles or galaxies non-relativistic matter and stuff which is in its own local frame moving slowly okay what is the energy density the energy density basically comes from just let's call it the e equals m c squared energy of a particle at rest the rest energy of the particle since we tend to set c equal to one in this class it's just energy is equal to the mass of a particle that's the energy of a particle at rest but if we're thinking about particles which are all moving slowly oh let's put back the c squared let me put back the c squared for a minute the energy density is the number of particles per unit volume times the energy of a particle and it has this big fat c squared in front of it it's relative the c-squared is the speed of light and it gives a huge magnitude to the energy of even a very light particle a dust grain because of its mass a tiny dust grain has enough energy in it to cause a big explosion okay if it were annihilated so the energy density due to the particles is large on the other hand the particles are moving slowly where does pressure come from now first of all the particles are moving very slowly by comparison with the speed of light their motion is non-relativistic what does pressure come from well pressure comes from particles hitting the walls of a system if we were just to think about a simple ordinary gas and a volume of space pressure comes just by particles hitting the wall it's proportional or related to the velocity of the particles that hit the wall it contains the mass mass is important if a bowling ball hits the wall it creates a bigger force on the wall than if a ping-pong ball does but whatever is hitting the wall it's hitting the wall slowly because all of these particles are moving slowly which means the pressure does not contain the speed of light in the formula for it what does it contain it contains the velocities of particles instead and because it doesn't contain the speed of light typically the pressure is much much smaller for ordinary non-relativistic particles much much smaller than the energy density that's the approximation that the pressure is approximately zero compared to the energy density and that corresponds to w equals zero so for non-relativistic matter density w is equal to zero for radiation w is equal to a third and we'll prove that in a little while but let me just again go back and quickly remind you how we use that one of the things that you need to know in order to work out the equations of cosmology is how the energy density depends on ae this is the equation that tells us so let me just go back very briefly and remind you how that worked we began with a box of gas now the equation of state can be analyzed by laboratory methods if we ha of course if the if the uh if the gas that makes up the universe is made up out of galaxies it's not so easy to put a bunch of galaxies in a box but galaxies are just particles they're just particles from our point of view you can put particles in a box and in the box you can investigate the relationship between the energy and the pressure and that's what we are interested in so let's take a box with a certain pressure the pressure on the walls of the box let's take the pressure on this wall of the box here it's equal to the force on that wall divided by the area force per unit area is the thing that's called pressure or force is equal to pressure times area force is equal to pressure times area now let's suppose we expand the box a little bit we expand the box a little bit increase its volume a little bit let's say we increase its volume by increasing this side by amount dx keeping everything else the same what happens to the uh to the energy inside the box well the if the box is exerting pressure on the walls and we move the walls then the gas inside the box does some work on the walls work is equal to force times distance and so there's a little bit of work that's done and what's the work that's done the work is equal to the force on the wall this is the work the force times the distance that it's displaced and that's equal to the pressure times the area times little dx the area times dx this is the area of let's say this side of the box over here this is the area this side of the box over here box expands the area times dx that's the change in volume of the box all right so the work done is the pressure times the change in the volume of the box a times dx is the change in the volume of the box that's of course a famous equation i just want to write it down because i like it all right what happens to the energy in the box the end the gas has done some work if the gas does work then the energy in the box must decrease if something does work then its energy must decrease so that means that the change in energy in the box the e must be minus the pressure times dv that's our equation so let's go through it very quickly to remind ourselves how how this tells you anything about how energy density scales with scale factor we start with the e equals minus p p dv and then we remember that the energy inside the box e is equal to the energy density times the volume energy density times the volume is the energy inside the box and so let's consider the left hand side the change in the energy of the box is equal to two terms just ordinary calculus the energy density times the change in volume plus the volume times the change in the energy density both things change in general both things change when you expand the box a little bit the energy density changes certainly and the volume changes the net change in the energy is the sum of two of them and that has to be equal to minus the pressure minus the pressure times dv okay but now let's plug in the equation the hypothetical equation of state something we haven't really justified yet but let's plug in our guess for an equation of state that pressure is equal to w times energy density and that just changes p to minus the number w times the energy density now take all the terms with dv and put them on one side of the equation and all the terms would be row and put them on the other side there's only one term with the row it's v d rho that's this term over here and that's equal to on the right hand side minus we have both cases we have rho dv from here we get a 1 and from here we get a w so this is a famous equation well not yet it's the preliminary to a famous equation let's get all the stuff with rho on one side and all the stuff one plus w that's just a number remember that one plus w is just a number whatever it is it's a number let's get all the stuff with rho on one side and all the stuff with v on the other side so that means divide by row divide by row to remove the row from here and to put it in the denominator over here and do the same with v divide by v to get rid of the v over here and put it over here the equation is getting more famous but it's not quite famous yet okay d rho over rho that's the differential of the logarithm of rho dv over v that's the differential of the logarithm of v so this equation says that the logarithm of the density of the energy density is equal to minus 1 plus w times the logarithm of the volume or that the energy density is one divided by the volume to the one plus w power we're allowed to put a constant here now it's a famous equation the energy density you may not recognize it but but it's uh it is famous the energy density is proportional to with a constant of proportionality to the volume of the box to the power one plus w all right but the volume of the box is proportional to the cube if we imagine i did this problem by expanding the box along one axis but you could expand the box uniformly along all the axes and you get exactly the same thing it was not important that the volume increased by only increasing one dimension here we could have increased it isotropically same equation here and if we increase the box isotropically we can think of it that the volume of the box is proportional to this cube of the scale factor the volume of a box of space is proportional to the cube of the scale factor and so that is equal to some constant which we can call rho naught but i'll just call it constant divided by the scale factor cubed to the one plus w y cubed because the volume is the scale factor cube if you're one of these crazy people who likes to do cosmology in different numbers of dimensions then this cubed could become the fourth power it could become the second power and so forth but otherwise it would be the same but if you're a sensible three-dimensional person this is the formula and this formula now is famous okay let's just remind ourselves again for matter dominated where the pressure is almost zero because things are moving slowly where the pressure is almost zero that corresponds to w equals zero pressure is equal to zero times the energy density in that case we just get rho goes like one over a cubed and that's this formula over here for radiation which i simply told you the answer for but we'll work it out tonight for radiation w is equal to a third if w is equal to a third then one plus w is four thirds and this becomes a constant over a to the fourth one plus w is four-thirds times three is just four four-thirds times three is four and we get one over eight to the fourth so okay that's uh that was review now let's come to the question of why w is equal to a third for radiation radiation is massless particles radiation means photons we could think of it also as electromagnetic waves uh we get the same answer incidentally but let's think of it as photons the characteristic feature of photons that makes it different that makes it different than uh than um the non-relativistic matter is that the photons are moving fast and in fact they're moving with the speed of light okay so let's work out the equation of state let's work it out and work it out in detail the equation of state for a box filled with photons here's our box it's three-dimensional but i'm not good at drawing three-dimensional boxes so we'll just draw a two-dimensional box and it's filled uniformly with lots of photons and of course this is an instantaneous picture of it but the photons are whizzing around with the speed of light what's more they're bouncing off the walls they're bouncing off the walls we'll assume that when they bounce off the walls they bounce off and lose no energy and exert pressure on the walls we need to know a couple of things all of which i think we've talked about in the past uh first of all photons have energy let's call what i'm going to do is pretend but then i'll tell you why it didn't matter i'm going to pretend all of the photons have the same energy now for a box of photons in thermal equilibrium it's approximately true as a matter of fact that they all have roughly the same energy but nothing i'm doing really depends on that and we'll see why but for the moment let's just pretend they all have the same energy let's call the energy per photon or per particle let's call it epsilon i'm not calling it epsilon with any deep motive often epsilon is used for a small number well the energy of a photon is a small number but that's not why i used it i used it because it looks like e but i want to save e for the total energy so epsilon is the energy per particle or energy per photon uh energy photon on the average okay what about the momentum of a proton we're going to need the momentum why do we need the momentum because forces what forces are is their response to the change of momentum if i throw a tennis ball at the wall the tennis ball has some momentum when it reflects back it has the opposite momentum there's been a change of momentum of the tennis ball there's also been a transfer of momentum to the wall and that transfer of momentum per unit time transfer of momentum per unit time is the force on the wall so we need to know something about the momenta of uh of photons right and the momentum of a photon i normally would call p the problem with p is i'm already using p for pressure so we're running into this problem that the number of letters of the alphabet is bounded by 26. therefore i use greek letters the momentum of a photon i'm going to call pi it is not 3.14159 it's just the momentum of a particle and it's a little vector it's a vector it has three components that's the momentum of a characteristic particle in there uh of course the momentum could be in any direction and one of the assumptions is if i look in any little volume of the box on the average the momentum could be in any direction that if i look at the velocity or momentum distribution anywheres in the box it's isotropic as many particles going in every direction as in every other direction and that's a good assumption that's a fair assumption uh that can be justified using statistical mechanics all right so pi the vector pi is the momentum the magnitude of the momentum we can just call pi or put some bars around it the magnitude of it or the absolute value of it is just called pi it's the magnitude of the vector pi and the relationship between the energy of a particle and its momentum if i keep around the speed of light then the energy of a massless particle a photon is equal to the speed of light times the magnitude of the momentum whoops not p not p pi instead of writing the bars i'm not going to write the bars i'm just going to write pi but when i mean the vector pi i'll put a little vector symbol on top so pi is the magnitude of the momentum equals momentum of a particle times c energy is pi times c that's the relationship between the energy of a massless particle and its momentum and since we set c equal to one the energy is just the magnitude of the momentum next what about the number of particles the number of photons in the box and better yet the number density let's let nu nu for number let nu be the number of particles number of photons per unit volume the density of photons it's not the density of energy what is the density of energy in this language the energy per particle times the number of particles so epsilon times nu would be rho we'll come back to that epsilon times nu will be rho let's calculate the pressure now to calculate the pressure we have to have a proper theory of what pressure is so here's the walls of a hypothetical box that's a wall the boundary of a box the gas is on the left side of the box the gas of photons and let's take a little volume here i'll tell you what this is let's consider a little time interval delta t take a little time interval delta t and what i'm going to be interested in is how many particles hit the boundary of the box and transfer momentum to it in the time interval delta t now the answer is a particle will hit the boundary in time delta t if it's close enough oh incidentally what's the velocity of these particles one one or c yeah c but we'll take it to be one all right if the particle is moving horizontally to the left where does it have to be in here in order that it will hit the boundary within time delta t and the answer is quite clear if delta x is less than or if delta x is equal to delta t and i make this little interval here delta x then any particle moving to the right with horizontal velocity will hit the wall in time delta t but what if it's not quite moving to the right what if it's moving at an angle theta so let's take a particle moving in angle theta then it will hit the wall of the box let me get the equation straight if delta x is equal to delta t times cosine of the angle theta if the cosine of the angle of theta is one that means it's horizontal then the particle will hit the corner of the wall of the box if it's in if it's within delta x equals delta t on the other hand supposing cosine theta supposing theta is perpendicular is vertical supposing theta is 90 degrees what's the cosine of 90 degrees zero and that's of course correct if the particles are moving almost vertically they will only hit the box if delta x is very small they'll have to be very close to hit the box in time delta t so this is the condition all particles within a distance delta x will hit the wall of the box if delta x is equal to delta t times cosine theta now let's take particles moving at angle delta theta okay supposing one particle hits the wall of the box how much momentum does it transfer how much horizontal momentum does it transfer to the wall of the box well the magnitude of the momentum is epsilon let's call this um yeah let's say delta pi the change in the x component what we're thinking about now is a particle which hits the wall and bounces off and it transfers some x momentum to the wall how much well the magnitude of the momentum that it started with was epsilon that's the magnitude of the momentum its component along the x-axis is epsilon cosine theta so that's the component of the momentum and how much momentum is transferred twice that much why is it twice that much because it starts moving with a certain momentum it bounces back and the amount of change of momentum is twice its momentum so the change in the momentum of that particle along the x axis is twice epsilon cosine of theta now let's divide that by delta t i'll tell you in a moment why we're dividing it by delta t oh yeah that's right twice epsilon cosine theta let's divide it by delta t why am i dividing it by delta t because the force on the wall is the change of momentum per unit time is the transfer of momentum per unit time that's newton's equations the change the force on an object is the time rate of change of its momentum and so this is the force exerted for each particle that hits the wall twice epsilon cos theta over over delta t ah good now how do we find the full force we have to calculate how many particles hit the wall this is what we get per particle hitting the wall how many particles hit the wall how many particles moving at angle cosine theta hit the wall in a time delta t well a particle will hit the wall if it's in within delta x how many of them are there within delta x the answer is the number of particles let's put the number of particles that will hit the wall in that time is going to be delta x times the area of the wall times the area that's the volume of this little region times the number of particles per unit volume n delta x n equals delta x area times the number of particles and um let's see if i left anything out of this nothing okay so we should multiply this by the number of particles that hit the wall in time delta t and that's pressure will equal twice epsilon cosine theta divided by delta t times the number which is delta x area number of particles per unit volume but delta x is delta t cosine theta so delta x over delta t is cosine theta so we get a formula the pressure due to particles moving at angle theta is twice epsilon cosine theta delta x over delta t is another factor of cosine theta cosine squared theta times nu there's one mistake in this formula the factor of two why is there a mistake in the factor of two and the answer is simple a particle if it's in here and moving toward the right will hit the wall but one moving toward the left won't so half the particles per unit volume are unavailable to hit the wall really we should only count those particles whose x component of velocity is toward the wall the other particles moving in the opposite direction are not going to hit the wall so we've really over estimated by a factor of 2 and that's correct that is correct we've overestimated by a factor of 2 because in putting in here the full number of particles per unit volume i've put too many in so we just wipe out the two here and now we have a correct formula epsilon times nu what is that the energy per unit volume or row we're getting there the pressure is equal to the energy per unit volume that's rho times this cosine squared theta now wait a minute what the hell do we do we're getting we're getting an answer that depends on the angle but of course this is the pressure due to particles moving at a particular angle what we need to do is integrate up the effect of all the different angles that the particle could be moving at or better yet we can ask it's equivalent what is the average of the square of the cosine of theta for an for for for the particles if we average over all particles what is the average value of the value of cosine squared of theta there are particles moving at theta near zero there are particles moving near theta equals pi there are other ones here there are other ones here what is the average value of cosine squared if i asked you what the average value of cosine would you would say zero but it's cosine squared so we have to ask what is the average value and here's now here's here's the problem the x-axis here is the one perpendicular to the wall there are particles flying about at every angle in the room all possible directions we want to know on the average what is the square of the cosine of the angle okay it's an easy problem it's an easy mathematical problem it has a very simple solution here let's leave that there it's less than one isn't it why is it less than one because the cosine never gets bigger than one okay so surely less than one but we can we can calculate it rigorously we suppose every particle the direction of every particle is characterized by a little unit vector in three dimensions the unit vector let's call it n and it has three components and x and y and n z and it represents the little unit vector along the direction of motion of the particle three components here's the x-axis and i maintain that n x is just cosine theta i think that's obvious that the x component of the unit vector is just cosine theta now here's something which is true and x squared plus ny squared plus nz squared equals one that's just the fact that this is a unit vector now let's average let's average this equation over all possible directions what will that give us that will give us the average of nx squared plus the average of ny squared plus the average of nz squared but nx squared and x and y squared and then z squared they're all equivalent they're just uh related by rotation if the gas is isotropic locally isotropic so that the velocity distribution is the same in every direction then the average of nx squared let's average it bracket just means average and then x squared plus and y squared plus nz squared is just one and if they're all equal that tells me that the average of nx squared is just one-third huh if there were four directions of space it would be one-fourth if there were two directions of space it would be one-half if there was only one direction of space it would be one so what have we found we found that the average of the cosine squared of theta is equal to one-third pressure equals one third row that's the derivation of the equation of state for radiation oh did i really make any mistake when i uh when i said all the particles have the same energy no this could be thought of as the contribution from particles of a given energy but for every energy each contribution is such that the pressure from that contribution is equal to the energy density from that contribution if you add them all up it doesn't matter you get the same answer so yeah perfect um balls yeah if it has a black wall that absorbs and then re-emits the photon if it is a black uh yes the answer would be the same for a black wall which re-emits the photons but you might ask what wall are we talking about what wall are we talking about there's no wall out there in space so there's no these photons are not reflecting off a wall nor are they being absorbed by the wall what are they really doing they're going right through they go right through the wall but on the other hand for everyone that goes through on the average is one coming from the opposite direction so on the average the wall of the box really does behave as though the particles that go out you lose their momentum the particles that come in you get their momentum and it really does behave the same way so um it really doesn't matter what your model for the relation for the for the origin of pressure is it's always the same radiation pressure is the third the energy density this is just a little example in a box with reflecting walls in a box with absorbing walls it re-radiates if you're in thermal equilibrium then it's the same the radiation in the university is mostly almost all the microwave background and it is in thermal equilibrium so uh okay that uh that then ties up a bunch of little ends it ties all of this together we now understand this we understand that we understand that we understand that uh this yeah and we're ready to move on to new kinds of equations of state yes um did the results any of the results change if we use quantum statistics no no no no no none at all right the cross section for photons are hitting each other is really really small it uh it does it does make a tiny tiny change in the equation of state it does but it's really really tiny it actually depends it actually depends on the um on the temperature and density if the temperature and density are high enough so that when the particles when the photons collide they can produce electron positron pairs then the equation of state will change and uh that's the main effect that's the main effect that's but that's exceedingly high temperatures exceedingly way way way beyond and the temperatures that we're talking about are very low uh well they by that by comparison with that they're very very low so the cross-section for two photons to interact and to scatter off each other is negligible but it in principle would affect things yeah oh uh what happened to the area is when i calculated the pressure it's the force per unit area did i leave it out did i throw it away somewhere oh yes um oh over here pressure pressure pressure this should have been force good this should have been force total force was the force due to one particle times the number of particles that hit it and then force divided by area is pressure so let's divide out the area does that answer your question did that answer the question yeah okay my uh my omission good any questions do you find that hard or easy not too bad okay now can energy density ever be negative yeah under certain circumstances it can but not not under any circumstances we will ever be interested in or at least not for the moment uh energy density well i take that back yes energy density can be negative no i take it back it it can be negative and we'll even talk about negative energy densities but more familiar pressure can be negative okay so let's discuss under what circumstances pressure can be negative are there any pressure any situations where pressure is negative yes is yes negative pressure has another term another name it's called tension or in particular in one dimension think about a one-dimensional world here's the one-dimensional world we make a box in the one-dimensional world a box is just a little line interval and we can imagine particles flying around in it particles flying around in it will exert flying around and bouncing off it will exert forces on the wall and when they bounce off they will push obviously push the wall out and correspond to positive pressure doesn't make any sense to think about negative pressure sure it does here's an example of negative pressure instead of particles flying around imagine that the two ends here were connected by a string or a spring just imagine there was a spring in space that connected the two ends of the box when you pull them apart the spring pulls back it doesn't push the walls of the box out it pulls the walls of the box in that's tension the tension of the spring is effectively a negative pressure when you for positive pressure when you increase the size of the box you do some work on the wall of the box and the energy decreases on the inside for tension when you pull against it you increase the energy on the inside so if for some reason you had negative uh negative pressure but let's say positive energy then w would be negative that's a possibility we should think about it in fact if it wasn't absolutely central to the cosmology i wouldn't even be telling you about it so it is central that pressure can be negative even if energy density is positive pressure tends to be pressure will be positive if you have a bunch of particles moving around which don't interact with each other very much and mainly just bounce off the wall if the particles are attracting each other they're pulling themselves together and if they also attract the wall they will pull the wall in so there are certainly circumstances where pressure can be negative and as i said it corresponds to tension we're going to talk about an example called vacuum energy where pressure can be negative where pressure typically is negative okay so let's talk about a special kind of energy density that's called vacuum energy it is a consequence of quantum field theory but we need we don't need to know where it comes from to describe it vacuum energy is just a energy that we assign to empty space we don't need to know where it comes from just to say i can in my bookkeeping none of my bookkeeping i can conjecture that just empty space with nothing in it has energy now we know where that energy comes from in quantum mechanics it comes from zero point energy of fluctuation it comes from zero point energy of harmonic oscillators which represent the the quanta of the field we know where it comes from but whatever it is it's energy that's simply there in empty space it's as if this blackboard had a uniform energy density on it and nothing i would do well i could put some extra particles in but nothing i could do short of putting in more material and so forth will change the energy of that blackboard it's just a fixed thing that uh that's there now i take a box it could be a box with fictitious walls or it could be a box with real walls how much energy vacuum energy in this case now is there in the box the answer is whatever the vacuum energy density is times the volume of the box and vacuum energy has the special property that the vacuum energy density is universal constant it does not change when you change the size of the box the density it's just a characteristic of empty space and as long as the box only has empty space in it the vacuum energy density is fixed yes so when you make the box you limit the number of modes that can be inside outside you do so it would seem like a density would be less inside than outside the limit that's why no that's why the energy is less than the energy in all of space if i didn't limit the number of modes i would just be talking about the energy in all of space and surely the energy in the box is less than the energy in all of space uh no the energy density doesn't change it's the total energy which changes because of what you said and it's there is there is a casimir force but that that that is only important when the walls of the box are really really close together other than the casimir force that uh and that's that's not important unless the distance between the walls of the box is comparable or smaller than the wavelength of the uh of the radiation or or really really close really really close it's not important for our purposes for our purposes now we're just talking about an energy density which is there it's always there no matter what we do and the box uh the box doesn't affect the energy density okay let's give it a name let's see um let's call it row naught naught stands for the vacuum the vacuum energy density and i'm going to also call it all right lambda rho naught let me get it straight rho naught is equal to another constant it's just another name for it lambda but i'm going to put a factor in and the factor is 3 over 8 pi g energy density is energy density we know what we mean by it 3 and 8 pi g are numerical constants this defines lambda it is the definition of lambda there's a name for lambda it's called the cosmological constant the relation between the two left side and the right side is trivial it's just the definition you'll see why you'll see why uh why it's useful to define lambda you know what it's useful why is it useful i'll remind you why it might be useful let me just remind you about the friedman equation the friedman equation says that a dot over a squared is equal to eight pi g over three times rho plus of plus maybe one over a squared or something but it always comes in eight pi g over three well eight eight pi g over three times rho 8 pi g over 3 times rho naught is equal to lambda that's why the 8 pi 3 over g is there lambda appears nicely in the equation rho appears less nicely but nevertheless let's just uh think of rho as energy density okay let's uh incidentally for vacuum energy we know immediately what the relation between the vacuum energy density is and the scale factor there is no relation no matter how big or small you make the box the vacuum energy is always the same it's a universal energy density in the vacuum and it doesn't change when you change the size of the universe the density of it so we already know the answer to how it varies but let's ask let's just ask for fun about its equation of state what kind of equation of state does it correspond to and it does correspond to an equation of state let's go let's work this backward work this backward now and let's work it backward for the special case of a vacuum energy density so the energy is equal to rho naught times v and rho naught does not change so d rho is zero and that's equal to minus w rho naught dv can you read off what w is not very hard row naught d v is minus w rho naught dv with a little bit of calculation maybe a half an hour is thought w is equal to minus one w is equal to minus 1 for vacuum energy when you read in various places that astronomers are measuring w and they're discovering that w is close to minus one this is what they're talking about they're talking about vacuum energy the closer the experimental evidence is to w equals minus one what they're really talking about is that saying that the energy density of the universe is like vacuum energy it doesn't dilute it doesn't dilute when you expand space doesn't dilute because it's a property of empty space to begin with all right that's uh that's vacuum energy it can be positive or negative in either case the pressure and the energy density have opposite sign that's the meaning of w equals minus one so if the energy density of the vacuum is positive the pressure is negative if the energy density is negative the pressure is positive that's a characteristic of vacuum energy it's not into uh you know um after a while when you think about it it becomes uh familiar and it's something that's not all that crazy but when you think when you're trying to think about it in terms of particles and in terms of um you know the usual things that you're used to thinking about causing pressure first of all negative pressure may seem a little odd but especially odd is this fact that the energy density and the pressure have opposite sign but what it comes down to is that uh there's just this little derivation here negative pressure positive energy density or the opposite that's the equation of state for an empty universe if there is a energy density now what the value of rho naught yeah this is the equation of state for an empty universe assuming yeah that it's governed by vacuum energy now what the value of rho naught is that's something we don't know how to compute um it uh there are too many contributions to it they come from all sorts of quantum fields that we may not have discovered yet they come from high energies they come from low energies uh one would have to have a pretty exact theory of all of the quantum fields in nature to be able to compute what rho naught is and we haven't got the vagus idea of why it is what it is the numerical value of it um we'll talk about the numerical value of it what i will tell you it is extremely small but uh what are the implications of it what are the implications of it it's a form of energy density in the vacuum and it competes with the other energy densities but let's study the special case where the only energy density in the universe is vacuum energy just like we studied the pure matter dominated case and then we studied the pure radiation dominated case and then we mixed the two of them and we said radiation dominates early matter dominates late let's isolate out just what pure vacuum energy density would do all right so let's go back to the equations governing the expansion of the universe and see how vacuum energy would influence things there are two cases well there are actually six cases the six cases are lambda which is proportional to the energy density is equal to positive or negative plus or minus of course there's an infinite number of cases when i say plus or minus it could be any number any value but let's distinguish positive and negative energy density and positive and and the three possible values of k remember what k is k is the curvature of space it's either positive it's either plus one minus one or zero plus one for spherical space zero for flat space minus one for hyperbola for hyperbolic space so we have k equals minus 1 plus 1 or 0. that makes six cases all together and what are the equations the equations are the good old friedman equations let's write them down a dot over a squared which is also i'll remind you the square of the hubble constant instantaneously a dot over a squared is equal first of all 8 pi over 3 g times the energy density but now the energy density is just the constant energy density of the empty vacuum and then minus k plus or minus 1 or 0 divided by a squared that's our equation and that's the equation we'd like to solve before i do so let's just take advantage now of our definition that eight pi g over three is called lambda that's why that's why lambda was introduced it was introduced to get rid of this nasty 8 pi 3 over g 8 pi g over 3 and just call it lambda it's called the cosmological constant it was introduced by einstein who later rejected it and uh then famously oh it's also called dark energy it's also the thing which the newspaper is called dark energy dark because it doesn't glow yeah so as i said we can have lambda equals plus minus or and we can and also lambda can also equal zero incidentally plus minus or zero but uh we can take the various cases if it's zero we've already done the various things now let's start with lambda equals positive let's take the case lambda equals positive and also k equals positive there are fewer cases that may that are relevant i'll show you some cases which don't make any sense first of all supposing lambda is negative and supposing k is positive then both sides of this equation are negative but this side is positive and so it can't make sense right so the case the case uh lambda negative and k positive that doesn't make any sense there's a number of other cases that don't make any sense or at least that don't have any solutions but let's take one which does have a solution the simplest case this is by far the simplest case let's take k equals zero we'll come back to the other cases let's take k equals zero that's the flat universe space is flat and the scale factor satisfies this equation here and let's solve it to solve it we take the square root a dot is equal to the square root of lambda times a all right so what's the solution a dot means dadt let's write it out the adt and this is the equation who's such that the time rate of change of something is proportional to that something what's the solution of such an equation exponential growth okay now notice that if lambda were negative we would be having a problem here immediately would make no sense so lambda being negative and k equals zero no good doesn't make sense no solution but lambda equals positive and k equals zero there is a solution and what is the solution the solution is that a grows exponentially with time a is equal to some constant doesn't matter what constant you choose uh it actually doesn't matter they all give the same answer uh for the for the geometry times e to the t but what's the coefficient in front of t square root of lambda right that's an interesting case the universe exponentially expands all right so that's a consequence of vacuum energy positive vacuum energy and no cur let's we're doing the case with no curvature k equals zero in that case the universe exponentially expands let's calculate the hubble constant remember what the hubble constant is the hubble console oh i don't have to calculate double constant a dot over a square a dot over a is the hubble constant and must be the square root of lambda so the hubble constant in this case not generally but in this case the hubble constant is just the square root of lambda and we can also write that the scale factor exponentially increases some constant doesn't matter what constant c e to the hubble constant times time this is a space time which exponentially expands and is called the sitter space the sitter space sitter was a dutch physicist astronomer and he discovered this solution of einstein's equations with a cosmological constant and it's named after him we still call it he discovered at some time i think it was about 1917 i'm not sure very very shortly after einstein and uh this is one version of the sitter space exponential expansion um yeah so in this case um the home constant is actually a constant even with respect to time right that's that's right this is the unique geometry with a hubble constant which is constant that's a little bit that's this is uh this is a little bit ambiguous and um i will try to explain to you why it's ambiguous let's let's hold off on that this geom all right if you want a technical set of words this geometry is not geodesically complete there are trajectories back into the past which go to the infinite past and a finite proper time that means some of the geometry is missing but we'll take that up separately we'll take it up when we get to it and uh it's it's a problematic question of whether there's a big bang in this kind of space or not however this kind of space doesn't exist by itself there's no reason why we shouldn't put back other kinds of matter into it and when we do put other kinds of matter into it things change in particular they change at early time let's see um let's imagine let's see what goes on here let's let's put in rhonaut which is just lambda but let's also put in some other kinds of matter other kinds of matter might be some radiation that would be some constant over a to the fourth so we would be adding to the cosmological constant some matter now the very early universe is the time when the universe was small the late universe is the time when it's big when a is very large when it's big enough this will become smaller than lambda and eventually when the universe gets big enough lambda will dominate and the universe will exponentially expand on the other hand very early times is when a is small when a is small it will be more important than lambda so very early times the vacuum energy is not important very late times it dominates everything that's why when we um make our observations we're in the we're in the process meaning to say the universe is in the process now of making the transition from being matter dominated let's put a cubed here it's in the transition region where these two are more or less competing with each other right so we're not yet seeing genuine exponential expansion it's too early there's still competition from this term over here even though this one is bigger well it's no they're more yeah this one is a little bigger this one's bigger but they're competing but we're beginning to see a transition from this behavior here to this behavior over here that's what these curves that i've drawn repeatedly look like they they show something which looks very much at early times like matter dominated but over the last one or two or three billion years we begin to see a deviation from it and the deviation is pointing in this direction why is it called accelerating it's called accelerating for the simple reason that if a increases exponentially and we calculate the acceleration that just means the second time derivative it's also increasing exponentially the derivative of an exponential is just another exponential the second derivative is just another exponential and so the x so the universe is not only expanding but it's expanding in an exponential way but in an accelerated way it can it could be accelerated without being exponential incidentally but okay so what's the truth the truth is that observation at the present time confirms acceleration more precision the more precision we get the more it looks like it's uh beginning to exponentially accelerate okay any questions yes so um you seem to say that the uh the positive vacuum energy could be associated with the ground state of quantum fields so so then how do you explain negative vacuum energy oh no um if you calculate the vacuum energy of a quantum field it will be positive for bosons and negative for fermions so it's just a fact of fact in mathematics that the vacuum energy for for bosons is a half h-bar omega for each fluctuating mode and it's minus a half h-bar omega for each fermionic uh mode but we're not going to try to answer the question where the vacuum energy comes from and what it's due to um what is much weirder than having vacuum energy is having no vacuum energy there is no known theory uh no known theory that's in any way consistent with the world as we know it which would predict zero vacuum energy so when people talk about the mysterious dark energy what they should be saying is it's very mysterious that there's so little of it um we can discuss in what sense it's numerically very small but it must be numerically very small in some sense if it took so long to discover it really no no i mean that i mean that if it were big enough to cause an exponential expansion that we could see in this room of course it would blow everything apart it would be a disaster for us but if it had any appreciable size then we would have discovered it the fact we did discover it but it was very hard to discover and it took enormous um enormously big telescopes seeing to the end of the universe in effect and uh that's an indication that in some sense it's a very small number and it is yeah say that it's problematic to extrapolate that back in time to the big bang what about extrapolating forward to a big rift or something like that well this a big rip i i never followed very much about the big rip it seemed to me one of these ideas which uh the press liked more than any physicist i knew but um um the big rip as i understand it is what would happen if w was even more negative than minus one and there's no known sensible theory where w was more negative than minus one uh nevertheless we could talk about it but i'm not sure what you had in mind when you asked me the question just if it was consistent with the big rip with w equal to negative one as it is i think the rip is not uh w less than minus one or is minus two or something like that at least if i'm getting my terminology right i never pay too much attention to it maybe it's worth paying attention to incidentally experiment focuses as focusing in more and more on w equals minus 1. it's within you know it's somewhere between minus 1.1 and point minus 0.9 and with diminishing error bars but i think uh it will never be a uh a high precision number but i think they can do they can narrow it down more it could it could it could that's correct and reason because every fermion comes along with a boson so they cancel each other exactly when i said when i chose my words i was thinking about exactly that i said no theory that agrees with everything we know about nature and what we know about nature is that fermions and bosons are not exactly matched so yeah what's the story with this factor of 10 to the 21 discrepancy between the calculated zero point energy and the measured expansion rate 10 to 121. sorry a lot well there's only one most of the constants of nature that we usually call constants of nature are not terribly uh fundamental the mass of the electrons thought to be just a sort of consequence of more complicated crap there are the really fundamental constants of nature are c speed of light planck's constant and the gravitational constant why do i say they're fundamental because there's a sense in which they're about universal things um yeah i mean let me slow down and discuss this a little bit what is universal about c nothing nothing in the world can move faster than speed c no signal can go faster than the speed of light so it really does have a universality to it it's not conditional on saying well we're going to be using um oatmeal to send messages it uh it's it's it's fundamental you can't get past it that's right what about planck's constant planck's constant is also universal it has to do with the uncertainty principle no matter what object you're talking about it doesn't matter if it's a bowling ball or an electron uncertainty in position times uncertainty and momentum is always greater than or equal to planck's constant period so it has a certain universal aspect to it the newton constant is also very universal again think of the law of gravity all objects all exert forces between them gravitationally which are equal to the product of their masses the distance between them squared and newton's constant so it's the use of the word all in all of those three cases which says um that there's something deep and fundamental there there are other constants that we sometimes talk about let's say the ratio of the electron well the electron mass to the proton mass um it's probably it is probably true that all protons and electrons have the same ratio at the mass but the zillions of particles lots of different particles the ratios of masses are not in any special way universal so we tend to think of g h bar and c is very fundamental now out of g h bar and c out of those constants you can make an energy density you can make an energy first of all you can make a unit of energy it's called the plonk energy the plunk energy corresponds to the energy of a mass of about 10 to the minus 5 grams in other words a macroscopic mass well a microscopic macroscopic mass not like an elementary particle but like a little bit of dust a little bit of dust if it were to annihilate the energy that would be released is the plunk energy or the plunk mass and it's about a tank of gasoline or something like that it's a lot it's a lot of energy there's a plunk length the plunk length is a very small length and there's a plunk time there they are the units they are the units of length time and uh and mass that you can make out of g h bar and c or another way of saying it is that the units of length mass and time that correspond to setting g h bar and c equal to 1. now if you can make a unit of length block length and it's 10 to the minus 33 centimeters you can also make a unit of volume the unit of volume is 10 to the minus 99 centimeters if you have a unit of mass you have a unit of energy it's one planck energy and then you have a unit of energy density one plunk mass per cubic plunk length that's the natural unit the universal unit or the unit uh that uh that and incidentally how big is that let's see um the plunk volume is tiny the plunk mass is pretty big that's a huge energy density vastly vastly bigger than anything that we ever experience in on the ordinary world but on the other hand when i say it's huge i mean it's huge by comparison you know with us ordinary creatures it's the unit of energy density it's the only unit of energy density that that occurs in very very basic physics how big is the vacuum energy density the vacuum energy density is nowhere near as big as that and it's about 123 orders of magnitude smaller okay so somewheres from somewhere unknown to us there is a tiny tiny energy density of the vacuum which is 123 orders of magnitude smaller than what we might guess nobody knows how to calculate it but what we might guess offhand is that it would be approximately one in natural units well we would never have guessed that because we never would have been here to guess it if it were true but but if you were to take your random guess about what a set of laws of nature would produce it would be 123 orders of magnitude bigger than what we see so we know for this course we will not well maybe we will ask the question but at least at the present time we won't ask the question where this vacuum energy comes from but we should take note of that of what is of what is mysterious about it what is mysterious about it is not that it's there it's that it's almost not there that uh that it's the lack of it which is the mysterious fact okay uh yeah i want to do another example but uh another another case you were saying it's you know what's surprising about it is that how little there is of it or whatever but but then but when they discovered it they were sort of surprised it was even there at all that's that was that was more psychological than anything else um you know at some time in the history of astronomy it was possible to um to try to detect it let's say at the level of 10 to the minus 100. uh 10 to the minus 100 is rather big incidentally i think it's probably much too big but sometime in the history of astronomy 10 to the minus 100 could be discovered but not 10 to the minus 101 101 it was too small all right astronomers did not discover it at the level of 10 to the minus 100 it seemed to be zero at that level so they pushed ahead and they looked for it at the level of 10 to the minus 101 still zero 10 to the minus 102 still zero 10 to the minus 103 10 to the minus 120 still zero 10 to the minus 121 still zero you got the feeling that maybe this thing really is zero for reasons that we don't know this was a attitude that had affected almost everybody in both the physics astronomy astrophysics community einstein himself didn't uh well he didn't think he thought of it in different ways but it was just the fact that it was so small and each time another decimal was added to the knowledge of it it was still zero it just got people convinced that it might be that it was zero there must be some reason you know the logic was a crazy logic the cosmological constant seems to be exactly zero it must be a consequence of the right theory of nature if we had the right theory of nature and of course everybody knows the right theory of nature or string theory and therefore it must be a constant consequence of string theory ha we just explained it okay that that mental thing was really there it really was there that since we now have a theory of gravity and quantum mechanics and we know that the cosmological constant is zero then it must predict it and if it predicts it we win we're successful um the best laid plans of mice and men you know and uh and it didn't turn out that way right could change over time that's measuring w now you can't obviously you can't um discover whether it will make some sudden or semi-sudden change after a trillion years no way to do that but you can try to discover whether over the last uh billion or two billion years it might have changed by a small amount and that's equivalent to measuring w more precisely so w is measured to about 10 percent now and it's minus 1 to within 10 percent that's evidence that at least over the relatively short term a few billion years it hasn't changed much hasn't changed by more than a few percent um i think we'll never be able to nail it completely does the dimension that the observer is in make a difference in the value of w and since we're inherently in the third dimension at the present does that explain why we have that disparity like if we were in the fourth or say 11th dimension would we see these things differently or no not really they're pretty similar it's it doesn't depend much on dimension so the shift that we see in the expansion universe wouldn't be any different if we were in the 11th dimension right now pretty similar details would be different general pattern would be the same all right let's do another case just for fun let's do lambda positive i'm not sure where my equation is i lost empty board okay let's do the case lambda positive but instead of k equals zero let's do k equal to plus one so this is the spherical universe the positively curved universe but with a positive cosmological constant let's try that one that's an interesting case all right here's how we work it out here's how we can uh intuit the rough properties of it and then i'll give you the exact solution a dot over a squared is equal to the energy density rho and then if k is equal to plus one we get minus k over a squared that's our equation with k equal to plus 1 oops this is not right i don't mean row here do i i mean 8 pi g over 8 pi g over 3 times rho but that's the same as lambda so here's our equation and let's see if we can make some sense out of it one way to make sense out of it is to try is to try to see if it has the same structure as an equation where we may already have a great deal of familiarity with yes this does and i'm going to show you what the what the first multiply it by a squared multiply everything by a squared and we get a dot squared is equal to and let's let's write minus minus lambda a squared i've transposed this to the left side equals minus one you got that did i do it right yeah yeah okay now think of a as the coordinate of a particle all right it hap it has to be positive a is always positive but let's ignore that for a moment and just think of it as the coordinate of a particle on a line we usually call it x but let's just call it a this would then be proportional to the kinetic energy of the particle a dot squared on the other hand this minus lambda a squared could be like a potential energy potential energy what would the potential energy be it would be minus the constant lambda times a squared that would be a potential energy of a slightly unusual kind it would be negative because of the minus sign and it would increase with a squared so it would be an upside down parabola a dot squared minus m lambda a squared equals minus one what does that say that says the total energy kinetic of this fictitious particle the total energy kinetic plus potential is just equal to minus one so the vertical axis here is energy why is the vertical axis energy because it's potential energy imagine that you had a particle now moving in this kind of potential which had a total energy equal to minus one that's over here how would that particle move well there's pretty much only one kind of motion it can have you can throw it in from far away it'll hit the wall over here and bounce back think of that think of this as a high hill you can throw something up the hill it will get so far and then it will go back down the hill or you can start it another way to think about it is start it with total energy equal to minus one that's over here and let it go it'll just fall back down but then you can say let's take let's take the falling down reverse it and think of a kind of bounce where you go up come to rest and go back down a gets bigger and bigger or your a starts big shrinks comes to rest and then goes back down that's what this equation is describing if we solve this equation basically it is the equation for a motion of a particle that comes up a hill and back down a hill what does a look like as a function of time if i plot time this way and a this way then it starts out big in the past that negative time it shrinks and goes back up in other words it's some kind of new kind of cosmology where the universe shrinks reaches some size bounces and comes back out all right can we solve it exactly yes it's not hard to solve exactly i'm going to tell you what the solution is you can work it out yourself and check that this is the solution one over square root of lambda times the hyperbolic cosine of square root of lambda t that's the exact solution but let me just remind you what hyperbolic cosine this is a this figure here is the graph for hyperbolic cosine hyperbolic cosine is the same as this is the same as 1 over 2 square root of lambda times e to the square root of lambda t plus e to the minus square root of lambda t that's what the hyperbolic cosine is it's a symmetric function of time the same for negative time and for positive time symmetric it at late times this piece of it is not important e to the minus squared of lambda t that goes to zero but this piece exponentially increases so it exponentially increases on this side and on the opposite side it exponentially increases into the past if you wanted to draw this universe if you wanted to make a picture of this universe most people would be inclined to draw the following way let's put time upward now here's time going upward at time equals zero the scale factor is as small as it will ever get so at time equals zero the universe is a small sphere remember k being positive says a spherical geometry let's draw that as a small circle over here as time goes forward the scale factor increases and it increases exponentially so the universe increases increases like that but as time goes into the past this solution is simply reflected and looks like this so this is a strange kind of universe which exponentially increases in the future remember that the flat case also exponentially increased in the future the flat case had only e to the square root of lambda t it did not have this term here so at very very late times basically at late times they both just look exponentially expanding and they look very similar to each other but the whole geometry from negative infinite past to positive infinite past is usually described as a bounce a bounce now we don't believe that the lower half of this means anything but nevertheless this is the mathematical structure that if we only had vacuum energy positive vacuum energy and we had k equals plus one the universe would be a bounce wouldn't just wouldn't expand wouldn't you contract or it would contract bounce and expand this is called also the sitter space now strangely the flat case and this case are really secretly the same geometry i'll try to explain to you that when we get to it they're really not different but we that will take some time to explore they look very different but they're not um is there any case other case that's yeah let's do one other case let me think um negative cosmological constant let's try a case with negative cosmological constant and see what we can learn all right lambda is now negative let's describe that by putting a minus sign in front of it over here in fact for simplicity let's be really simple and just set lambda or the absolute value of lambda equal to one just to be really simple are just too many symbols here let's call it minus one cosmological constant that happens to be minus one in some units or another minus 1 over a squared but it's really minus k over a squared minus k over a squared now if k is 1 in this case in other words if space is positively curved this equation is nonsense on the left hand side a positive thing on the right hand side a negative thing if k is positive so the answer is there is no solution uh with a cosmological constant for the positive for the negative cosmological constant for the posit for the positively curved case but for the negatively curved case there is let's see if we can figure out what it looks like let's take k to be minus 1 in which case this becomes plus 1 over a squared and let's do the same job on it that we did here's our equation let's convert it to a simple mechanical system let's see what mechanical system this corresponds to with positive cosmological constant it corresponded to a potential energy which just you know went off into the basement let's see what happens now let's multiply by a squared so that we get a dot squared no a squared in the denominator equals minus a squared plus one got that multiplied by a squared let's transpose the a squared to the other side it becomes a dot squared plus a squared if i had kept lambda around it would multiply this but let's not let's just keep it plus a squared equals one now what kind of system are we talking about if this is kinetic energy and this is potential energy the potential energy is plus a squared what kind of system has a potential energy which increases quadratically with displacement harmonic oscillator right so this equation here is actually the energy conservation equation for energy equal to plus one for a harmonic oscillator with in this case a unit spring constant the energy is the potential energy the total energy is equal to plus one what kind of motion do you get in particular what kind of motion do you get if you start at a equals zero a has to always be positive so half of this really doesn't mean anything a is by definition positive but you start at a equals zero at the big bang and you shoot the marble up the hill and what does it do it comes back down the universe expands and crashes this is with k equals minus one this is with the open universe okay this is the opposite of what we might have expected the open universe but with k with negative cosmological constant negative cosmological constant causes it instead of expanding exponentially it just expands and comes back and collapses a big crunch even though it's open and infinite it still comes back and crunches so be thankful that we don't live in a universe with too large a negative cosmological constant in fact be thankful that we don't live in a universe with too large a positive cosmological consonant either case would be deadly in one case the flow would be so large outward that it would just grab everything and uh you know you hold on to your wife's hand or your husband's hand you wouldn't be able to overcome the outward flow if lambda was too big and positive if it was negative and too big you would just have a crunch okay that uh that's that's the theory of vacuum energy in a nutshell and uh um there are more cases if you're free to look through them some of them make sense some of them don't make sense uh but they all have their own characteristic behavior and they're interesting and you can always analyze them by translating them into some sort of mechanical system and thinking about the conservation of energy for that system good okay these are einstein's equations there are einstein's appraisers these are einstein's equations applied to i'm not sure what that would mean a sphere with complex radius i mean not that i know of i don't i don't know of any application of these ideas to complex all right no um for more or less accidentally all of the interesting cases are like that um but keep in mind if you add two different fluids together it won't be true anymore and we do add two different fluids and we just say it one range behaves this way other range behaves this way but um no a general fluid will not have that property uh i'll tell you what it means uh the it means that the speed of sound in the material is constant and doesn't and doesn't depend on density but why that should be true it's certainly not true for all possible things okay for more please visit us at stanford.edu

Channel: Stanford

Views: 82,473

Keywords: quantum physics, science, mathematics, universe, cosmology, Leonard Susskind

Id: GgKGPyFL-TM

Length: 105min 41sec (6341 seconds)

Published: Wed Feb 20 2013