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Cosmology | Lecture 5

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Automatisierte Medienanalyse

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Steiger university nor are
you a little bit and then discuss the equation of state do we equations of state well 3 equations estate and where we get our information about them where our knowledge not a cosmological knowledge but our knowledge no based on basic theory basic a physics where that knowledge about the equations that comes from but before I do it if I'm very very quickly remind you of where we are aware of our and the basic question that we want answers cosmetology as 1 of the history of the universe and to the extent that the universe commit homogeneous and isotropic it really boils what is kind history of the scale factor if another time history of the scale factor we know an awful lot about the history of the universe we can test it and go before we can observe it in various ways so that's a question that at 1 always the only question but has 1 overriding question that if you want to do cosmology you better have a control what is a of tea as a function of time has evolved I'll remind you quickly Wheatley studied some marbles there was matter of dominated marble and the matter dominated model he expanded like keep the two-thirds In the radiation dominated universe a expands like Tito the 1 half both of these models 1 remembers exactly correct art please times this is almost exactly correct and very early times we believe that this was more correct and there was a transition between them we talked about it the brain Ali ritual we get observational cosmology we're going to talk a great deal about how we know anything about this probably only thing about this with the various meanings of them are our but but tonight OK we talk about also will be importance Of the equation of state that the radiation as a matter dominated universe are 2 examples of universes which evolved under different conditions which Comey characterized by an equation of state the equation estate incidentally is what tells us how the energy density which is on the right hand side of the Friedman equations the cosmological equations it tells us how told me energy density changes with changes in the scale factor for example tells us in the matter dominated case never dominated trousers that role it is equal to some constantly core remarks provided by a Q and a
Q is just the volume of a piece of space as it expands the density is the amount of energy in it provided by volume and that's just something over a Q in the radiation dominated case which we talk about extensively where reviewed equation of state comes from all our role goes like are divided by before this is radiation dominated now the damage between these 2 0 originates in the difference between the relationship between pressure from on got to review these things quickly narrow are in the relationship between pressure and energy density there are many many relation between pressure in energy densities called the equation 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 because knowledge we make a simplifying assumptions that after that the energy density and be pressure are simply related and was what the simplicity and because it covers a lot of interesting cases we take the equation mistake to be pressured is equal to some number called W W was a number it characterizes the fluid whatever happens to be times the energy density role we just go back again and show you how the equation of state tells you these 2 different types of things the equation estate matter dominated let's start with our 1st equation of state from that dominated what mattered nominated his means particles or galaxies nonrelativistic matters and stuff which is in its own local frame moving slowly of was the energy density the energy density basically comes from just what's called E equals M C squared but energy of a particle at rest the rest energy of the particle since we think cassette equal to 1 in this class just energy is equal to the mass of a particle that's the energy of the particle arrests but if we're thinking about particles which are all moving slowly but was put back to square labor practices from the energy density is the number of particles per unit volume times the energy of a particle and it has is big fat facts squared from its role there the C squared the speed of light and a huge magnitude to the energy of Eve Avery light particles of dust grains because of its massive trading dust grain has enough energy in the cause of the explosion case the were annihilated so the energy density due to the particles as large really in the particles a moving slowly with pressure come from 1st of all the particles of moving very slowly by comparison with the speed of light emotions nonrelativistic what this pressure come from more pressure comes from particles hitting the walls of a system of just a think about a simple ordinary gas the volume of space pressure comes just by particles hitting the wall this proposal related to the velocity of the particles of hit the wall contains the mass mess is important bowling ball hits the wall it creates a bigger force on wall think I'm more hours but whatever it is hitting the wall is hitting the wall slowly because of these particles are moving slowly which means the pressure does not contain the speed of light the formula for it was a container contains the velocities of particles instead and because it doesn't contain the speed of light typically that pressure is much much smaller ordinary nonrelativistic particles much much smaller than the energy density that's the approximation the pressure is approximately 0 compared to the energy density and that corresponds to W equals 0 so the nonrelativistic a matter density W was equal to 0 for radiation W was equal to a 3rd will prove that while but just again go back and quickly remind you how we use that 1 of the things that you need to know in order to work out the equations of cosmology is how the energy density depends on any of this is the equation that tells us so we just go back very briefly remind you have at work Her we began with a box of gas and the equation the state can be analyzed by laboratory it we have our of course if these gas that makes up the universe is made up of galaxies it's not so easy to put a bunch of galaxies in a box but galaxies it is particles that particles from our point of view you but particles in a box in in the box you can investigate the relationship between the energy and the pressure that's what we are interested in Seoul let's take a box with a certain pressure the pressure on the walls of the box with sector pressure on this wall box here at a quarter of on the wall divided by the area William areas
of things score pressure off forces a quota pressure plant area forces eagled pressure turns area now let's suppose we expand the box a little better we expand the box a little bit increases volume a little bit but say we increase volume by increasing this side by amount DX giving everything else is what happens to the video the energy inside the box well if the boxers 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 force times distant little were stunned and watch the work is done the work is equal but before us some wall of the work force times the distance that displays and that's people the pressure currency area times ruled BX the area times DX is the area of course this side of the box area the size of a box over here box expand the area times D X that's change in volume of the box sell the work done the pressure times change in the volume of the box 8 times DX a change in the volume of a box of course a famous equation I just want to write it down because I like it what happens to the energy in the box because the the the guests has done some work at the gas does work mainly in the Duma box was decreased something does work that its energy must be critics so that means the change in energy in the box he must be minus the pressure turns DVD that's our equations so let's go through it very quickly to remind ourselves how how this tells you anything about how all energy density scales with scale factor for start with was minus and only remembered that energy inside the box he is equal to the energy density times the volume energy density kinds of volume is energy inside the box and so let's consider the left-hand side changing the energy of the box is eagled the took turns just ordinary calculus the energy density times a change in volume plus the volume car the change in the energy density both things change in general both things change when you express the box a little bit the energy density changes going on volume changes the net change in energy is the sum of 2 of them and that has to be equal from minus the pressure but minus the pressured OK but don't plug in the equation hypothetical equation of state something we haven't really justified yet but let's plug-in are guests for an equation of state the that pressure is equal to W. Hinds energy density and that just changes P minus the number W. Hinds energy density now take older turns with DVD and put them on 1 side of the equation and all the terms of the euro and put them on the the other side is only 1 term would be Rome Vijay Rao was over here and that's equal to or on the right-hand side minus we have both cases we have rolled DVD From Healy get a wonder how from year we get MW serve a famous equation while not yet with the it's the preliminary to a famous equation a let's get all the stuff with roll on 1 side and all was that we don't want us is the number remember the 1 plus W. Just the number where it is number but all the stuff withdrawn 1 side and all the stuff with me and the other side so that is divided by a role right by road to remove the rubble from here and the public in the denominator over here the same they provide might be to get rid of the V over here and product over here equation is getting more famous but it's not quite famous yet bureau over Rome that's the differential of the larger remember world over that's the differential of the library the movie so this equation says that the logarithm of the density of the energy density is equal to minus 1 plus W. plans logarithm of our volume for but the energy density is 1 divided by the volume of a 1 plus W Powell for a lot constant here finish equation v energy density human recognizes that a 3rd its the energy density is
proportional to the constant of proportionality the volume of the box to the power 1 plus a review but the volume of the box is proportional to the Cuomo every imagine I did this problem by expanding the box along 1 axis but you could expand the box uniformly along all the ax season you get exactly the way thing it was not important the volume increased by only increasing 1 dimension year we could have increased tip isotropically same equation here if we increase the box isotropically we could think of it that the volume of the box is proportional to the queue wound on the scale factor volume of boxes based was proportional killed the scale factor and that is equal some constant which we call Rome boaters car constant provided by the scale factor Q will go 1 plus W Y killed because of volume was a scare scale factor Q however these crazy people who likes to do well because cosmetology in different numbers of dimensions Mendez Hewitt could become the 4th power could become the 2nd power and but otherwise it would be the same hour but a furor a sensible three-dimensional per person it is the formula this formula now is that OK let's through remind ourselves again for matter dominated with pressure is almost 0 because things are moving slowly with the pressure is on 0 0 that corresponds to W equals 0 pressure is equal to 0 times the energy density in that case we just get rolled goes like 1 over a cube and that's this formula over here for radiation which I simply told you the answer but will work out tonight for radiation W was equal to a 3rd if doubly was equal to 3rd then 1 plus W is for thirds and this becomes a constant only 8 of the former 1 plus W was for 3rd times 3 is just 4 4 3 times 3 for only get 1 of 8 to the floor some at that worst review now I come to the question of why W was equal to a 3rd radiation raises massless particles radiation means we could figure also was electromagnetic waves Ali give the same answer incidentally but let's think of it as 4 times the characteristic feature of photons that makes a different makes it different then and and be nonrelativistic matter is the photons of moving fast and that a moving with the speed of light so that's worked out the equation of state would get out about there in detail the equation of state from box filled with 4 times are box is three-dimensional but I'm not good during three-dimensional boxes soldiers wrote two-dimensional box and failed uniformly with lots of our times and of course this is an instantaneous picture Ave but the photons whizzing around with the speed of light what were the bouncing off the walls the bouncing off the walls so when the bounce off the walls they bounced off and lose no energy can exert pressure on the walls we cannot or couple things for which I think we've talked about the Paris bomb but 1st of all photons energy let's call it could do was pertain but then not tell you why it didn't matter I'm going to pretend all of the 4 have the same energy now for box of photons and thermal equilibrium at approximately true was a matter of fact they all have roughly the same energy but nothing doing really depends on that will see white but for the moment but just pretend they all have the same energy let's call the energy per photon apart particle call epsilon and Epsilon will be motive often epsilon is used for a small number of but photon is a small number but that's not why I use if I use because it looks like e but I want to see E for the total energy so epsilon is energy per particle our energy per photon energy part of a all time average what about the momentum of our product
Oregon mean the momentum writer we need the momentum because forces voices are is response to the change of momentum by thought a tennis ball the wall the tennis ball has some momentum when a reflect back it has the opposite momentum is a change of momentum of a tennis ball there's also been a transfer of momentum for the and that transfer of momentum car unit time to transfer momentum per unit time is the 4th along so we need to know something about the moment Volvo of photons and the momentum of photon I normally would call P E the problem with P is already using keeper pressure so sewer running into this problem that the number of letters of the alphabet is bounded by 26 therefore I use Greek letters the momentum of thought time and their call hi is not 3 . 1 4 1 5 Marine it's just the momentum of a particle and a little vector vector has 3 components that's the momentum of the characteristic particle in their 5 the momentum could be in any direction 1 of the assumptions is looking any little volume of a box on the the average the momentum could be in any direction that if I look at the velocity of momentum distributions anyway as in the box isotropic as many particles going in every direction the other direction and that's a good assumption that offer assumptions were justified in using statistical mechanics are supplying vector plight is a momentum the magnitude of the moment written just call parlay corporate boras around it the magnitude of the of the absolute value of his just called the magnitude of vector plight and the relationship between the energy of a particle and its momentum if I keep around the speed of light and the energy of a massless particle of photon is equal to the speed of light times the magnitude of the momentum Brooks not P. wrapped beat fight ahead instead of writing bars are not going to write bars and just going to write parlayed but when I meet the vector pipe up through vectors among top suppliers is the magnitude of the moment equals momentum of a bit of a particle foci energy is Pike state that's the relationship between the energy of a massless particle and momentum and we set Siegel the warned me energy is just the magnitude our momentum next where about the number of particles and No. 4 terms in the box and better yet the number density Wexler no more no phone number you'll be the number of particles number of photons per unit volume the density of vote on the matter density of energy was density of energy in this language the energy per particle times the number of particles so epsilon times new would be wrong will combative at epsilon terms new would wrote but calculator pressure now calculated pressure we have to have a proper Feria work pressure is so the walls of a hypothetical box vessel wall boundary box gas on the left side of the box guest of photons and let's take a little volume here for Europe is consider a little kind of Delta T he had a little time interval Delta T going to be the 1st in is how many particles hit the boundary of the box and transfer momentum in the time interval delta T now we at article will head the boundary in times of the of it's close enough always Bentley what's the velocity of these particles 1 1 IOC young take to be 1 if the particles movie horizontally Kabul left where doesn't have to be in here In order that it will hit the boundary with kind tree and the answer is quite clear that Delta X is less than or or got the ax is equal to adopt I make this will end the beleaguered Delta X any particle moving to the right with horizontal velocity
will the war in time Dr. but was it's not quite moving from a right others moving an angle theta source said a particle moving him angle theta bid it will hit the wall of the box the equation straight if Delta X is equal to dealt the tee times cosine of the angle if the cosine of the angle of fate is 1 that means it's horizontal then the particle will hit the core course of the wall of the box if it's in which would be in doubt the Texaco's built on the other hand supposing cosigned opposing is perpendicular is vertical supposing fate is 90 degrees west coast 90 degrees 0 and that's of course correct if the particles a moment almost vertically they will only hit the box if Delta X is very small very close the box in time Delta tape so this is a condition all particles With into a distance Delta X will hit the wall of the box if dealt the X is equal to that of the terms cos theta now let's take particles moving angle dealt the data Bohon 1 particle hits the wall of the box much momentum does a transfer from much horizontal momentum as a transfer to wall of the box wealth will be the magnitude of the momentum is epsilon Rothko over so young dealt the party change in X component thinking about hours particle which hits the wall and bounces off transfers some extra momentum to the wall are
much well the magnitude of the momentum that started with was epsilon at the magnitude of the moment of its component along the x-axis is epsilon cosigned data us component of a momentum How much momentum is transferred twice that much wiser twice that much because it starts moving with a certain momentum it bounces back and the amount the siege momentum is twice moment so the change the the moment about particle along the X actors is twice epsilon cursorily figure now let's divide that I don't like to tell your moment why dividing if I don't see all the at 4 per cent data provided by Delta why dividing it by Delta T because the force on the wall is a change of momentum per unit time is a transfer of momentum per unit time as news equation the change before Son object is the kind rate of change of momentum so this is not exerted fro each particle that the twice epsilon coast data over over Dr. take our threat but now however find the full force we have to calculate how many particles hit the wall we get per particle hitting the wall primary particles hit the wall particles moving a hat angle cosigned later hit the wall in the kind of see well a particle will hit the wall units in within facts harmony of the more they will dealt the X year the number of particles of the number of particles will hit the wall in that time is going to be X times area of the wall and the area fatso volume of this region times the number of particles per unit volume and dump X air any equals don't X area times the number of particles and armed with 2 thinking I would not so we should multiplied by the number of particles that hit the wall in time to take and the pressure will equal twice epsilon cars bidder the by by Delta T times the number which is dealt X area number of particles per unit volume but don't X is delta T course I played so don't X Over delta T cosigned later so formula the pressure don't the particles moving angle theta there's twice epsilon cross-eyed later the X over key is another factor of course made a across squared later no as 1 mistaken this formula a factor of 2 2 Why mistaken the tool the answer is simple a particle its it here and moving toward the right will hit the wall but when moving toward the left wall saw half the particles per unit volume are unavailable to hit the wall really we should only count those particles whose X component of velocity it poured the war the other particles moving in the opposite direction are not going to hit the wall so we really are overestimated by a factor of 2 and that's correct that is correct we overestimated by a factor of 2 figures and putting in he the full number of particles per unit volume put too many and so it is wiped out the 2 here and now we have a correct formula Absolon times known was that the energy per unit volume or Raul we're getting there the pressure is equal to the energy per unit volume that's Rome times this cosigned squared they now in a way that we deal with getting away getting an answer that depends on a angle of class this Is the pressure due to particles moving at a particular angle will Fidel is 8 the great up the effect of all the different angles particle could be moving act or better yet we can adjust its equivalent what is the average of the square of the cosigner favor foreign foreign troops for the particle if we average over all particles what Is the average value of the value of cosine squared of fate their particles moving it faded near 0 their particles moving favor articles pie there are the ones here there are the ones here where is it average value of cosine square if I ask you would have advised cosine would you say 0 but it's cosigned squares or have to ask what is he average value when he is now here's a user problem the X-axis here the 1 perpendicular to the wall particles flying about every angle in the room all possible directions we want to know on the average What is the squalor of the cosine of the angle take it easy problems too easy mathematical problem and has a very simple solution here with sleep it's less than 1 is it why is a lesson 1 because of cost I never gets bigger than 1 markets would surely lasted war but now we can we can calculate it rigorously we supposed every particle the direction of every particle is characterized by a little unit vector in 3 dimensions unit vector which caught a hanger I'm has 3 components and X and why and then and it represents the little unit vector along the direction of motion of a particle 3 components yeah the X-axis I maintain that is is just co-starring I think it's obvious that we X component of the unit vector is just coastline played now he is something which is true an X squared plus
NY squared plus ends E squared equals 1 that's just the fact that this is a unit vector knowledge average but Amberidge's equation all possible direction whether that give us that will give us average of an X squared plus the average of NY 1 squared plus the average frenzy square but an ex clinic and white Quinn Lindsay square all equivalent adjust related by rotation of the gasses isotropic locally isotropic so that the velocity distributions saying that every direction than the average of Annex squared with average age bracket just means average of Annex square plus NY square placentae squared 1 the role equal that tells me that the average of index squared is just 1 one-third our if your for their actions a space it would be 1 4 the bill to directions space would be 1 half of those only 1 direction the space it would be 1 what we found the average sign square the figure is equal to one-third pressure equals 1 3rd wrote that the derivation of the equation the state for radiation all that I really make any mistake when idea when said Ola particles have the same energy know this could be thought of as the contribution from particles of a given energy but the every energy each contribution is such that the pressure from that contribution is equal to energy density from that contribution mall up it doesn't matter you get the same answer so young box that now what were it is a black tie yes and would be the same for black which we're real photons but you might ask well what war we talking about what Wall we talking about is no wall our parents based on stove is no these photons I'm not reflecting off wall nor would be absorbed by what they really going to go electoral we go right to the wall by the other hand for everyone it goes through on the averages 1 coming from the opposite direction so I only an average the wall of the box really does behave as tho the particles a go out you lose their momentum the particles a committee you get their momentum and really does behave the same way sorrow for it really doesn't matter what you'll marvel the relations for the year the origin of pressures it's always the same radiation pressures energy density this is just a little example in a box with reflecting walls a box of absorbing walls every radiates if you thermal equilibrium that sustain our these radiation in the universe is mostly almost all we microwave background indeed is the remote Colibri OK that end ties up a bunch of little ends it ties all of this together we now understand this we understand that the thing that we understand that Our missed young but and were ready to to move on to new kinds of recruiters of state did did it results in a results changes the use statistic the level that will be 2 of the crust at all times and to those really really small head back it does it does make a crying need tiny changes the equation of state it does but it's really really kind it actually depends depends on the yield on the temperature and density if the temperature and density of high enough so that uh where the particles with photons collide could produce electron-positron pairs in the equation of state will change and that's the main fact that the main effect but that's exceedingly high temperatures exceeding the way way way beyond and the temperatures that were talking about every Lola well received by that by comparison with about their very low so the cross section the two-photon still interact and scatter each other's negligible but only to 8 in principle would affect things Yum Yum don't what happened the areas on a calculated the pressures of force per unit area about the way someone yes I'm only yet pressure pressure pressure Mr. that it should enforce total force was the forest to 1 particle times the number of particles that hit and then us divided by area is pressure so it's provide happy area designed to question that of Clifton yup Kirkuk my year but omission Greg any questions if you define FOR easy but the now can energy density ever be negated now under certain circumstances it can but not not under any circumstances we will ever be interested in or at least not for the moment are energy density well I think that back yes energy density can be negative not I take it back it it can be negative even talk about negative energy densities but more familiar pressure communicative so let's discuss under what circumstances pressure could be negative any pressure many situations where pressures negative yes negative pressure as another Terranova Maine Hey the called attention are in particular In 1 dimension think about a one-dimensional World is a one-dimensional world we make a box in the one-dimensional world of boxes just lying in the
and we can imagine particles flying around particles flying around that it will exert flying around bouncing off it will exert forces on the wall and when they bounce off they push obviously push wall Our and correspond the positive pressure His make any sense to think about negative pressure show does his an example of negative pressure instead of particles flying around imagined that the 2 ends here or connected by a string or spring the margin there was a spring in space that connected the 2 ends the box when you pull them apart the sprinkles back it doesn't push their walls of box out old walls of a boxy in that tension retention of the spring is effectively under negative pressure when you for positive pressure when you increase the size of a box you do some work on more a box and the energy decreases on the inside but attention when you pull against you increase energy on inside so if the some reason you had negative negative pressure but what's it positive energy then w would be negative that's a possibility we should think about it thank effect if it wasn't absolutely central to cosmology it would need to tell 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 in March and mainly leaders bounce off the wall of the particles are attracting each other the pulling themselves together and if they also attract the wall they will pull in so there certainly circumstances where pressure can be negative resisted it corresponds to pensioners will want to talk about an example called vacuum energy with pressure can be negative pressure typically is negative OK so let's talk about a special kind of energy density it's called vacuum energy it is a consequence of quantum field theory but we really don't need to know where it comes from describe a vacuum energy is just energy that we assigned to empty space we don't need to know where it comes from just to say I can it bookkeeping bookkeeping I can conjecture that just empty space with nothing in it has energy now we know where that energy comes from quantum mechanics comes from 0 . energy of fluctuation comes from 0 . energy of harmonic oscillators which represent the video of the quantum field win or comes from for whatever it is it's energy that simply there an empty space it's as if the blackboard had a uniform energy density on and nothing I would do well I could put some extra particles but nothing I could do a short of putting in more material and so forth will change the energy of that black just off fixed things that loads but now I think a box economy about a 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 answers were ever the vacuum energy density is times the volume of a box and vacuum energy as a special property that the vacuum energy density is universal constant it does not change when you change the size of a box with density adjuster characteristic of empty space and as long as the box only has empty space in the vacuum energy density is 6 yes said make a lot of it set aside you to so the same day that I knew it and that's what I know that's why these energy is less than dual space if I didn't limit the number of it would just be talking about the energy in all of space surely the energy in the box is less than the energy all space the top Our nor energy density doesn't change its total revenue which changes because of what you said that is there a chasm force but that that is only important when the walls of the boxer really really close together other than the chasm of forests about the next steps on important on the last the distance between the walls of a box is comparable smaller than the wavelength of the year of the radiation it really really close recalls it's not important for our purposes for our purposes now which is talking about energy density which is there have always been a matter what we do and the boxer the box doesn't affect the energy density focus given name street on but call a runoff not stands for Slovakia vacuum energy density and I'm going to move rip also Court right when that road lottery get straight round on his equal tho another constant as just another name for it my number Barbara pro factor in and the fact their ears re- already 8 pie Jean energy density is energy density we know what we mean by 3 and they pie G on numerical constants this defines land is the definition of His a name for land that it's called the cosmological constant relation between toast left side and the right side is trivial it's just a definition you'll see you'll see why why it's useful to the final and they're not its useful life is useful remind you what might be useful he does remind you about me Friedman equation the Freeman equations says a doctor over 80 square it is equal pay 8 time GE almost 3 times plus the fast may be 1 of a square but always comes in a
tight G over 3 well 8 8 by over 3 times Rome 8 by G over 3 times role not cocoa and that's why a pipe 3 of which lender appears nicely in the equation role appears less nicely but nevertheless it's just think of role as energy density OK let's a incidentally the vacuum energy we know immediately what these relations between the vacuum energy density is scale factor there is no relation no matter how big or small you make a box vacuum energy is always the same as 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 however varies but let's escalates is best for fun about equation of state what kind of equations state is a correspondent room but it does correspond to an equation of state let's go let's work this backward workers backward and work at backward for the special case of vacuum energy density so the energy is able to run kinds V and run does not change so dear old 0 and that's equal to minus W rome . DVD Muriel for W is covered are run on TV is minus W not DVD a little bit of calculation maybe a half an hour's thought at all W is equaled the miners 1 W is equal to minus 1 vacuum energy when you read In a various places that astronomers are measuring W and they're discovering that W. is close to minus 1 this is what they're talking about talking about vacuum closer the experimental evidences W. equals minus 1 what they really talking about is it saying that energy density of the universe it's like a vacuum energy it doesn't do it doesn't die lewd when you expand space doesn't I look because the property of empty space to begin alright that's so that's vacuum energy it can be positive or negative in either case pressure and the energy density have opposite signs that the meaning of W minus 1 of the energy density of vacuum was positive the pressures negative if the energy density is negative the pressures positive that's a characteristic of actor budget to add he after a while when you think about it becomes familiar and that's something not all like crazy but would you think Italy China and think about it in terms all particles and terms a of usual things that you use that thinking about causing pressure 1st of all negative pressure may seem a little odd but especially on his factor the energy density and the pressure have opposite signed but what it comes down to is that just as derivation here negative pressure positive energy density or the opposite they were there that's the equation state Fernand B. University energy density now what the value of rural North yet this is the equation of state of the universe assuming that governed by recommend G not what the value of row North Korea's that's something we don't know how the computer it go there too many contributions to it they come from all sorts of quantum field that we may not discovered yet come from high energies Eicon from low energies Our 1 where half an hour of pretty exact theory of all the quantum fields in nature to be able to compute what Rome markets and we haven't got the vaguest idea of why it is what it the numerical value of it arm will talk about the numerical value of it or I will tell you it is extremely small but so what are the implications of it where are the implications of the performer of energy density in the vacuum and compete for the other energy densities but let's study the special case With only energy density in the universe is vacuum energy just like we studied the pure matter dominated Jason than we studied the pure radiation nominated case in the mix the 2 of them and we said radiation dominates early matter dominates late let's isolate out just what pure vacuum energy density would do so let's go back the equations governing the expansion of the universe and see how vacuum energy would inform things took a are actually 6 cases of 6 cases are lander which is proportional to the energy density is equal positive or negative plus or minus because as an infinite number of cases rise it for a minus a community number and value but what distinguished positive and negative energy density any positive and and the 3 possible values of Kate with carriers Cayzer curvature of space the positive plus 1 minus 1 0 0 plus 1 for spherical space 0 4 flats space minus 1 hyperbole hyperbolic space so we have K Eagles minus 1 plus 1 or 0 Remics seek cases altogether want the equations the equations of the good old Friedman equations right there on a diet already squared which is also remind you the square of the Hubble constant instantaneously 8 . 0 8 squared is equal restore a part of 3 for JET part the energy density but now the energy density is just the constant energy density of EMD vacuum and then my plus or minus 1 0 0 divided by a square that equation Tennessee equation we like to solve before I do so it's just take advantage now of our definition that 8 pie IG over 3 years called that's why that's why when the was introduced was introduced to get rid of this nasty by 3 over J a party 3 and just
call it land call the cosmological constant it was introduced by Einstein will later of rejected it yeah yeah yeah famously Oh which also called dark energy plus other things which the newspapers call dark energy power start because a dozen go now so I as I said we could have when the eagles plus minus or anemic and also led the could also Zero incidentally minus or 0 but that we can take various cases had 0 we've already done the various things now let's start with land equals positive let's take the case land was positive and also Katie equals positive move there are fewer cases that may armed robbery lot of cases which don't make any sense 1st of all suppose that is negative and supposing Keyes positive them both sides of this equation of negative but this side is positive and so I can accept so the case the case when the negative Inc a positive that doesn't make any sense there's a number of other cases that don't make any sense or don't a solutions but let's take 1 which does have a solution on the simplest case by far the simplest case let's take a equals 0 will come back to be of the cases of thick cable 0 the flat universe spaces flat end of the scale factor satisfies this equation here must solve it that we take a square root equal to the square root of land mines any but so what's the solution
a . means DEAD tried out BAD and this is the equation was such that the time rate of change of something is proportional that something was the social such an equation exponential growth notice that a slender 1 negative we would be having a problem here immediately would make no sense sold land the being negative and equals 0 no God was accepts no solution of land they cause positive and they're cable 0 there is a solution for the Bush and solution that a grows exponentially with time aII is equal to some constant doesn't matter what constitute chose a actually doesn't matter they'll give the same answer Our 1st geometry times the steeple what's the coefficient in front of 280 square of land president dressing case the universe exponentially explained so as a consequence of vacuum energy positive acting managing no progress it would do would do in a case with no curvature Cagle 0 in that case the universe exponentially expands let's calculate the Hubble constant member the Hubble constant is the Hubble comes over have to calculate the Hubble constant 8 . or a square a doubt over a is the Hubble constant but hold the 3rd it must be square so the Hubble constant in this case not generally but in this case the Hubble constant is just the squared of Lambda Omega also right that the scale factor exponentially increases some constant Bosnia or force them each of the Hubble constant terms time this is a space time which exponentially expands and is called de Sitter space research at Ford the sooner or very Dutch physicist astronomer and he discovered this solution of Einstein's equations with the cosmological constant named after him which took or we discover it sometime I was about 19 17 much better shortly after Einstein and missed it 1 version of the airspace exponential expansion on young based on that last year but right that's that's right this is you need geometry live a Hubble constant which is constant what at a little bit the resources of this is a little bit ambiguous and I will try to explain to you why it's ambiguous lech with hold off on there this GM Arab if you want to technical were this geometry is not due basically complete there are trajectories back into the past which go to the infinite past and find a proper turning means some of the geometry is missing but will pick that up separately will pick it up we get to it and it's it's a problematic question whether it was a big bang in this kind of space not however this kind of space doesn't exist by itself on school is no reason why we shouldn't put back other kinds of matters into it and do put and other kinds of matter and tour things change and particularly change early time let's see Armed with magic with see grows on here it's best wrote which is just Lander but let's also put in some other kinds of matter other kinds of matter might be summed radiation that would be some constant all 8 before off so it would be a link the cosmological constant summer matter now very early universe is the time when the universe was small the lake universe the time of day when a is very large was big enough this will become smaller than land but eventually will is big enough Lando will nominate and universal exponentially expand on the other hand through early times is when a small Monday is small it will be more important than land so very early times vacuum energy is not important very late times it dominates everything that's Hawaii when we make our observations we're in the forward in the process needing to say the universe is in the process now of making the transition from being matter dominated the cube here it's in the transition region with these 2 or more or less competing with each other so we're not yet scene Ginny wine exponential expansion Joylene still competition from this term over here even those forms bigger growth number of the borrowed from is a little bigger this was bigger but the competing but were beginning to see a transition from this behaving here To this behavior over here as curves that I've drawn repeatedly you look like fish or something which looks very much at early times like matter dominated but over the last won a total of 3 billion years we begin to see a deviation from it and deviations pointing in the direction Wiese cold accelerating its coal accelerating for the simple reason that aII increases exponentially and we calculate the acceleration that just the 2nd time derivative it's also increasing exponentially the derivatives of an exponential is just another exponential the secondary was just another exponential answer will be so the universe is not only expanding but expanding them X and exponential way but in an accelerated IndyCar could be accelerated without being exponential insolently about OK so what's the truth the truth is that observation at the present time confirmed acceleration more precision more precision we get the more it looks like it beginning to exponentially accelerated OK
many questions yet but say that FBl be positive that could be assessed it Rad stated so so they had to explain data all I know is you calculate the vacuum energy of quantum field it will be positive for both arms and negative mirrors so it's just a fact of a factor mathematics that be back in the energy for From HBO's arms is a half H borrow may go reach fluctuating motors miners bar of rich fermionic that the try and the question of where the vacuum energy cloned from were due on what is much a weird than having vacuum energy is having Novak is no known fury Our known theory that anyway consistent with the world as we know it which would predict 0 vacuum so when people talk about the mysterious dark energy what they should be setting is it's very mysterious because so little it on we can discuss in what numerically very small but it must be numerically very small and some sense of it took so long to discover it really not mean that I mean that is level would big after quarters an exponential expansion that we could see in his room Of course it would blow everything apart there would be a disaster for us but if it had any appreciable size that we were discovered the fact that we did discover but it was very hard to discover and it took enormous an enormously big telescopes seemed to the end of the universe in effect and that's an indication that some sentence of a small number of at a time game well this at the big ripped I never thought much about the big Gripper seemed to me movies these ideas which were oppressed like more than a physicist who with for the big grip as I understand it is or what happened if W was even more negative than minus wars and there's no known sensible fury with W was more negative than minus 1 hour nevertheless we could talk about it but I'm not sure what you have in mind as question it will be with us that I think the Represa marked W less than minus 1 of my or something like that led Lee said I'm getting my terminology right I never paid too much attention to it maybe it's worth paying attention to incidentally experiment focuses its focusing in more and more on W. Rickles minus 1 it's within the somewheres between minus 1 . 1 and 0 . . 19 and with the diminishing error bars but this Inc or it will never be a year Our high-precision number of pig bacon do who dig narrow it down more young it could be could it could and reason because they refer me on comes a court along with bows on the cancel each other exactly what I said what I chose my words I was thinking about exactly that I said no theory that agrees with everything we know about nature nor would not about nature's Fermi arms and embargoes on are not exactly match but it's not tainted 21 stepped between the calculated death 0 . Dia measure expected tend to the 120 I saw a lot now what is only once most of the constants of nature that we usually call kind of nature of Her breed of fundamental massive electron thought to be just a consequence of complicated crap there are these really fundamental constants of nature are see speed of light plant constant and the gravitational constant why do I say their fundamental because there's a sense in which there about universal things on down the missile and discuss a storm but what is universal about I'm not saying nothing in the world can move fast speed see no signal could grow faster than the speed of light so it really does Have a universe to it it's not their conditional on saying Well we're gonna be using oatmeal to send messages it the efforts it's the fundamental UK pass that's what about plants constant plants constant is also universal has to do with the uncertainty principle no matter what object is talking about doesn't matter the bowling ball or an electron uncertainty in position and uncertainty and momentum is always greater than or equal to plant constant period has a certain universal aspect to it on v Norton constant is also very universal again because the law of gravity all objects all exert forces between them gravitationally which are equal to the product of the masses the distance between them square and Newton's constant so the use of the word All in all of those 3 cases which an hour something deep and fundamental where there are other constants that we sometimes talk about the but say the ratio of electron well the electron masses of the proton it tho it is probably true that will protons and electrons have the same ratio as the zillions of particles lots of different particles the ratios of masses are not in a special way universal so we tend to think of G H Barnes sees very fundamental now G 8 see those current binge you can make an energy density you could make an energy you go 1st of all you could make a unit of energy call the plant energy the Plunkett a G corresponds to the energy of a man of about 10 to the minus 5 grams of Georgia macroscopic maps well microscopic microscopic may not like him elementary particle but like a little bit of dust little disaster for toe annihilate the energy that would be reaped leases the energy or the poor man's about a tank of gasoline or something like that all our energy there's a plant link plant point is very small Lincoln's upon a time there for you it units of links time and mess that you
could make a G each foreign state or another way of saying it is a unit of length from S and time that correspond to setting G H farm equaled 1 Micaela of light block length the and extent of the minus 33 centimeters you could also make a unit of volume of volume as 10 to the minus 19 90 centimeters and you have a unit of May has to have a unit of energy it's 1 plunk energy and then you have a unit of energy density 1 plant mass per cu plant claim that the natural unit Universal unit of the unit that and instantly how big is that rich city on the plan volume is tiny the plant commences pretty big that's a huge energy density vastly vastly bigger than anything we ever experienced on ordinary world but on the other hand what I say it's Hulu Jimmy huge by comparison with us ordinary creatures its v e unit of energy density it's the only unit of energy density that the other cars in very very basic physics how big it is a vacuum energy density the vacuum energy density is nowhere near as big as that it's about 123 orders of magnitude smaller OK so somewhere from somewhere unknown to us there is a kind kind energy density of the vacuum which is a 123 orders of magnitude smaller than what we might guests nobody knows how a calculated we might get off here is that it would be approximately 1 in natural units while we would never have guessed that because we never would've been here he gets it if you were true but but if you were to take during in the guests about what a set of laws of nature would produce it would be a 123 orders of magnitude bigger than what we see Sohail at for this course we will not maybe we will ask the question at least to the present time you won't ask the question where this fact in comes from but we should take note of that of of what is mysterious about was mysterious about not that it's there almost knocked that alone it's a lack of it which is mysterious OK young or under another example among the kids it would rise by there is that but what is it that that that was was most psychological and anything else Of at some time in the history of astronomy it was possible to lower the tragedy tactic let's say at the level of 10 to the minus a 100 And the miners rather big incidentally clear to you'd probably much too big but at some time in the history of astronomy 10 to the minus 100 could be discovered but not my time and mourned 101 was too small and astronomers did not discovered at the level of 10 to the minus a 100 seemed to be 0 at that level so they pushed ahead and they looked for it at the level of tend to the minus on under 1 still 0 and the miners 102 still 0 and the miners 103 and the the miners 120 still 0 and the miners 121 still 0 you got the feeling that maybe this thing really it 0 for reasons that we don't know this was a added to that had affected almost everybody in both physics astronomy astrophysics community Einstein himself in the world even think different ways but it was just the fact that was so small and each time a mother of best small was added to work the knowledge of it it was still 0 it just got people convinced there might be a 0 must be some reason Nova Logic go a crazy logic the cosmological constant seems to be exactly 0 it must be a consequence of the right theory of nature if we had the right period nature and of course everybody knows a right here choose a theory and therefore must be a concert consequence of string theory How we just explained it that that that meant both thing was really there really was there but just as we now have a hearing fog gravity inquiry mechanics and we know that the cosmological constant 0 then must predicted and predicted who we win was successful From the best laid plans of mice and men the turnout that worry that that energy chase that's measuring W. now you can obviously you can't army discover whether it will make some sudden lost any sudden change after trillion years the way to do that but you could try discover whether over the last hour billions 2 billion years of my changed by a smaller amount and that's equivalent the measuring W. Moore precisely so W was measured to about 10 per cent now and it's minus 1 to within 10 per cent that's evidence that at least over the relatively short turn a few billion years it hasn't changed much hasn't changed by more than a few cent dorm rooms and they will never be able to have Naylor completely make a difference in the value of this sort guaranteeing the 3rd the president does that explain why we have a very clear up for sale of would we see these things differently or the pretty similar attempts it doesn't depend much on their merger the 1st thing that we look the increase your details will be different general pattern would do the same despite that said do another Kaster for fund let's do it lender positive commercial my equations lost month and board perfect but still it case lender positive but instead equals 0 let's go OK Beagle plus 1 so this is their Oracle universe positively curve universe but with a positive cosmological constant or strive at work it's an interesting case is our work it out here is how we can
into into which the rough properties of it and then I'll give you the exact solution a dot on a square is equal to the energy density role and then if Keyes equal the plus 1 we get minus case over a squared that's our equation with Katie equal plus 1 what does not right remain role here I mean a pie ideal a pie 3 times world but at the same Islam that so here's our equation and let's see if we can make some sense out of the blue 1 way to make sense out of it the trial is to try and to see if it has the same structure as an equation where we may already have agreed to a familiarity with yes the stars and I'll show you what do you what you 1st multiplied by a square multiplied everything by squared away yet a crime if a . squared equal to starts right might minus land that a square I've transposed this the left side equals minus 1 we got back to a duo right yup OK now think of 80 the coordinator of a particle it half that has to be positive it is always positive but let's ignore that from moment and just think of it as the coordinator of a particle on a line we usually call expert with the score at 8 this would then be proportional to the kinetic energy of the particle 80 . square on the other hand is minus land a square could be like a potential energy potential energy for the potential energy be beat it would be a minus the constant Lander times a square Bellamy a potential energy a slightly unusual kind it would be negative because of a minus sign and it would increase with a square so did upside down parabola 8 not squared minus Let em when they squared equals minus 1 where say that the total energy can make of this fictitious particle the bubbly energy Connecticut plus potential is just equal to minus 1 so the vertical axis here is energy wise vertical axis energy because of potential energy a Maginot you had a particle moving in this kind of potential which had a total energy equal to minus 1 that's only hear how that particle moved well is pretty much only 1 kind emotion can have you from far away it will hit below over here and bounced back think I think of this as a high-heeled you will throw something up the hill it will get so far far and will go back down the hill or you could start in another way to think about it is started With total energy put minus 1 that's over here and let it go the fall back down but then you could say let's take a stake of falling down reverse it and think of a kind of balance will you go up come and go back he gets bigger and bigger you a start big shrinks comes to rest and then goes back and that's what this equation equations describing Our if we solve this equation basically it is the equation for a motion of a particle that comes up the hill and lacked Ariel What does a look-alike as a function of time if I plot kind this story in a this way they it starts out big in the past that negative pions shrinks and was back up another lawyer to so kind of that no kind of cosmology with the universe shrinks reaches some signs bounces and comes back out goalie Sobel exactly yes it's not hard to solve exactly on the value of the solution is you could work it out yourself check the ashamed but the shoes 1 0 Mosquero Lander times a hyper bike co-starring squared of Rwanda T. that's the exact solution but I'm just remind you what hyperbolic co-stars this In AT missed figure here is removed graph fiberboard course hyperbolic call sign is the same as the same has warned over square root of Lambda times he eat the square of land and sea plus each minors square of land a survivor Barco signed it's a symmetric function of time the same for negative time positive time symmetric it at a later time this piece of it is not important Tito minor squared away and the entity that goes 0 but this please exponentially increases exponentially increases on this side and on the opposite side it exponentially increases in the past if you want to draw this university won't picture of this universe most people would be inclined to draw following way let's put time off for now it is time going upward at time equals 0 the scale factor is as small as it will ever get so tired equals 0 the universe is a small sphere lumber being positives as a spherical geometry with drug that has a small circle over here as time goes forward the scale factor increases and increases exponentially generous increases increases like that but as time goes into that past this solution is simply reflected that a strange kind of uterus which exponentially increases in the future we learned that that a flat case also exponentially increased in the future of flat case had only the square root of land that it did not have this term here so very very late kind basically at late times they both just look exponentially expanding and they were very similar to each other overhaul geometry from negative in the infinite past the positive the test is usually described as a about about 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 1 in the uterus would be abounds we just wouldn't expand would could track wrote who we contracted bounce and expand this is called also de Sitter space now strangely flat case in this case are really secretly the same geometry of tried to explain to you that we get to it they really not different but there were real or pick some time to explore they look very different but not 0 any case of a case that now let's do 1 or their tastes let me think on negative cosmological constant let's try a case with negative cosmological constant old Camilla that when this is no negative let's described by putting a minus sign in front of your here In fact for simplicity USB really simple and just said the land for the absolute value of land the equal to 1 just to be really simple and has too many symbols he of call minus 1 cosmological constant that happens to be my 1 in some units are another minus 1 over a square virtually might scale squared mile KOA squared now cases 1 in this case another words if spaces positively this equation is nonsense from left inside a positive thing on the right hand side a negative thing if Keyes positive the answer is there is no solution our with the cosmological constant for the positive fraught with the negative cosmological constant for the positive for positively case but for negatively curved case theory the Syria figure our works like let's take minus 1 in which cases becomes plus 1 squared let's the same job under that we do His or equation let's converted to a simple mechanical system let's see what mechanical systems corresponds with positive cosmological constant corresponded to a potential energy which just don't wait off into the basement with 2 happens now With multiply by a square server we get a . cornered Norway squared the data equals minus a squared plus warning that multiply by a squared which transposed a square to the other side it becomes a dark squared plus a school arid if I Keppel land around it would multiply that public's now mosquito plus a squared equals 1 now what kind of system are we talking about this is chaotic energy and this is potential energy the potential energy is plus a square what kind of system has a potential energy which increases quadratic displacement harmonic oscillator right so this equation here there's actually energy conservation equations for energy equal to plus 1 for a harmonic oscillator within this case a unit spring constant the energy is potential energy total energy is equal the plus 1 what kind of emotion do you get In particular what can you get if you start at 8 0 0 it has always be positive so half of this really doesn't mean anything it is by definition positive but you started a cool 0 for the Big Bang is shoot the marble up hill what is it comes back and the numerous expands in crashes this is the case equals minus 1 this is would be open universe this is the opposite what we might have expected the Open University but with Kate with negative cosmological constant negative cosmological constant causes it instead of expanding exponentially just expands and lacking collapses a big crunch even the is open an infinite it still comes back and quench sorrow be thankful that we don't live in the universe with 2 large negative cosmological constant in fact be thankful that we don't live in numerous with 2 logic positive cosmological constant is the case would be deadly in 1 case the full would be so large outward there which is grab everything in the area are you hold on your wife and your husband's hand you wouldn't be able to overcome the outward flow of the was to begin positive it was negative and into a big you would just have a crunch OK that that that's the beauty of vacuum energy in a nutshell is somewhat cases if you're free look through them some of them make sent some of them don't make sense our but they're all have their own characteristic behavior in there it could always analyze by translating them into some sort of mechanical systems and thinking about the conservation of energy but that OK it is well best Lawrence center questions there are signs of these Einstein's equations applied to it I'm not sure that would mean field complex a radius of that but I know of and I don't know of any application of these ideas complex they were right yeah the other not for more or less accidently will be interesting cases like that armed but did not you aid to different fluids together it won't be true anymore there we don't have to different forwards or just say 1 range behave this way of arranged for his way but not a general fluid will not have that property Laura I'll tell you what it means however it means that the speed of sound In the material is constant and doesn't that doesn't depend on events but why that should be true natural for possible thing opted for more
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Titel Cosmology | Lecture 5
Serientitel Lecture Collection | Cosmology
Teil 5
Anzahl der Teile 9
Autor Susskind, Leonard
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DOI 10.5446/15061
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (February 11, 2013) After reviewing the cosmological equations of state, Leonard Susskind introduces the concept of vacuum energy. Vacuum energy is represented by the cosmological constant, and is also known as dark energy.

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