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# Cosmology | Lecture 5

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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
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
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
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