Dark Energy

hello today we're going to look at dark energy dark energy is a key explanation for accelerating universe expansion you can find experimental and observational evidence of dark energy if you type evidence for dark energy into any internet search engine a principal way of doing so is to observe supernovae in galaxies and the conclusion that is drawn is that those galaxies are accelerating away from us suggesting that the universe expansion is itself accelerating we can perhaps best start our story with Einstein who was trying to understand the nature of the universe and he was being told by the astronomers at the time that the universe was fixed and unchanging and that the galaxies in the universe these dots all represent galaxies in the universe were as it were suspended in space they had always been there they would always be there there was no start to the universe there will be no ending the galaxies simply are there there may be local movement the Sun goes around the galaxy the earth goes around the Sun but on a cosmic scale the universe just remains fixed and unchanging now this of course was an immediate problem for Einstein because he knew that the only long scale force that was operating that he was aware of was the gravitational force and that all of these galaxies would exert a gravitational force which was an attractive force on all the other galaxies and so what ought to happen is all those galaxies should be as it were attracting themselves and thus pulling all the galaxies in so the universe ought to be contracting but the astronomers at the time were quite adamant that the universe was fixed and unchanging so Einstein concluded that there must be another force which he called the cosmological constant which was equal and opposite to the force of gravity and which essentially maintained the galaxies in space in a fixed position now before Einstein got very far with this analysis a man called Hubble came along and showed that the universe was in fact expanding and you can see how he did that in my video on Hubble Big Bang and the age of the universe but let me for completeness just repeat the main points here the expansion of the universe can be thought of as in one dimension the expansion of a rubberband let's pin the rubberband at this end so this is fixed and then we'll have some coordinate points which we mark on the rubberband these are points 1 2 3 and this might be 4 and the scale factor that is the distance between any two adjacent points we give the letter a now obviously if we stretch the rubber band then the coordinate points will move and the spacing factor will also move so here is the rubberband and I've marked as you can see the coordinate points on it the distance between any two coordinate points to adjacent coordinate points is a but as the rubberband expands so the co-ordinate points move and the distance a increases it's always the same distance between the coordinate points but that distance between those points increases as the elastic band expands and essentially what we're saying is exactly the same thing that as the whoops as the universe expands so the distance between any two coordinate points also expands and that's the coordinate distance a so what we're essentially saying is that the distance between to coordinate points let's say we want to know the distance between coordinate point one and coordinate point four that distance is the difference in coordinate points four minus one would be three multiplied by a and that a we have to recognize is a function of time because a itself increases as the as equal the elastic band as the universe increases a will increase now we can perform a differential on this we can differentiate both sides with respect to time so we say that D D with respect to time which we usually call d dot d dot is the same as d d by dt d dot is well Delta X itself won't change the coordinate points will still be four and one and four minus one will still be three so Delta X remains unchanged but a of course does change with time and so you need to take the differential of a with respect to time but what is d dot what is DD by DT what is a distance divided by time that's simply velocity so the velocity of the coordinate point is equal to Delta X times a dot which is a function of time now we can always multiply anything by one so I'm going to multiply that by a divided by a a over a is 1 I've done no harm to that equation but you'll notice that Delta X times a is d so now we can say that V is equal to D times a dot over a I shall leave off the fact that these are time dependent because we've established that but we just need to bear that in mind and in my earlier video I pointed out that this term a dot over a is what's called Hubble's constant and so V equals Hubble's constant times D and that means that if you know the velocity of a receding galaxy you can calculate how far away it is it also implies that the further something is away from you the faster it's traveling one important point to make about Hubble's constant is that it is constant everywhere in space today but Hubble's constant changes with time so it's only a constant in space not a constant in time so now we have a picture of the universe where all the galaxies of the universe are moving away from one another there is no sense in which there is a center of the universe or indeed an age of the universe the universe is all space space itself is expanding and it is carrying the contents of space the galaxies with it and so everything is moving away from everything else that's rather light dots on a balloon you can take a balloon you can mark some dots on it and as the balloon expands you'll see that all the dots move away from all the other dots there's no sense that anywhere is the center where is the center of the surface of a balloon where is the center of the surface of a sphere it doesn't have one but you can regard you can simply take a coordinate position and say let us pick one particular point and let us call that the center so let's pick a particular galaxy and make that the center for our purposes it's not the center of the universe it's simply the center of the reference frame that we happen to choose and let's consider another galaxy let's say it's this one here and that is at a distance D at the moment from the one that we have assigned the coordinate position 0 2 and that galaxy will be moving away from this in a position which i'll marked with an X with a velocity V and we can say of course that V is equal to Hubble's constant times D that is the formula we just derived now the universe is generally regarded as being homogeneous and isotropic which basically means the same everywhere and the Newton's theorem tells us that there will be a gravitational force acting on this galaxy here and what Newton's theorem tells us is that the mass that will be acting on this galaxy here will be all the mass that is contained within a sphere of which this galaxy is at the edge and the coordinate point is at the center so the total mass of all the galaxies inside this sphere acts on this galaxy here which we'll say has a mass little m all the mass of galaxies outside that sphere will of course exert a gravitational attraction but Newton's theorem shows that all of the mass outside the sphere event effectively cancels out and the only relevant mass that will apply to this galaxy is the mass inside the sphere and so we get the formula the force the gravitational force acting on that galaxy will equal G which is the gravitational constant times Capital m which is the total mass of all the galaxies inside the sphere multiplied by little mass the mass of the galaxy itself divided by d squared and we've developed that formula in other videos we can also say that the potential energy of that galaxy is equal to minus G capital M little m divided by D now the total energy of this galaxy here was this galaxy here will be the to its kinetic energy plus its potential energy and that's obviously going to be the kinetic energy is half MV squared where m is the mass of the galaxy and V is its velocity as it expands as the universe expands the galaxy is traveling at velocity V plus the potential energy well the potential energy is minus G capital M little m over D and that combination of kinetic energy and potential energy is the total energy and that must remain a constant in time multiplying all terms by two we get that MV squared minus G capital M little m over D equals to K and then you'll see that you can divide through by M but two K over m is itself a constant two is a constant the mass of the galaxy remains constant so this is just another constant which I shall continue to call K and so we have derived the term that v squared minus G M over D is a constant this is the essentially the remains of the kinetic energy term this is the potential energy term and that becomes a constant now you'll recall that we earlier derived these two terms here that the distance is Delta X times a and the distance differentiated by time is Delta X times a dot and we're going to use those two terms to fit into the equation that we have just derived so just to remind you D is equal to Delta X the difference in coordinate points times the spacing factor a which itself is a function of time and velocity V is equals Delta X times the time differ of the spacing factor in other words the rate of change of the spacing factor with time and we're going to substitute D and V in this formula here and that will give us that V squared this term here is Delta x squared times a dot T all squared that's these squared minus 2 GM over D well D is Delta X a a function of T and that equals a constant K now mass is equal to the volume times the density density is defined of course as mass per unit volume so mass is simply volume times density and if we think about the expanding universe here is the picture we drew earlier here is the galaxy that expanding away is the center of the universe in terms of our frame of reference then as the universe expands the volume of this sphere will increase but the mass within it will not increase the number of galaxies remains the same and their mass remains the same so the volume will increase but the density will decrease and the mass will remain the same the volume of that sphere is obviously 4/3 PI D cubed that's just the volume of a sphere and the density is we're going to call that Rho but I'm going to put brackets T to remind us that the density will vary with time and now we can substitute this term here which is the mass for that M term there and so that gives us that Delta x squared a dot which is a function of T squared minus 2 G over Delta X a function of T multiplied by the mass which is all of this 4/3 PI D cubed Rho which is a function T is equal to K but D this D here is equal to Delta X times a which is a function of T so we can put instead of D there we can put Delta X cubed a t cubed that that term simply replaces the d cubed term there and then you will see that we've got a delta x times a in the denominator and a delta x cubed a cubed in the numerator so we can cancel out one of those and leave us with Delta x squared a T squared and I'll write that out so we can see what it is that becomes fill 2 x squared a dot function of T squared minus 2 G times 4/3 PI Delta x squared a T squared Rho of T equals I can just squeeze that on equals K now for an important thing to notice this term here has a delta x squared in it this term here has a delta x squared in it in order to be dimensionally consistent it must follow that K has got a Delta x2 squared term in it otherwise dimensionally it's not going to work so since K is itself embedded with a Delta x squared term we can actually cancel out the Delta X Squared's throughout it simply means we're left with a different constant but it's still a constant so let's just write that out that becomes a dot T squared - we'll multiply this now out you get 8 pi G over 3 times a T squared Rho T equals K and if we divide both sides now by a we get a dot T squared divided by a of T squared equals this comes on the other side so it now becomes plus 8 pi G over 3 the a T squared has been cancelled out because we've divided both sides by a T squared we now have a grow T by convention we say minus K over a T squared because we are dividing through by a squared and that is a simple version of what is called the Freedman robertson-walker formula and just to remind you that this can be traced back this term here it can be traced back essentially to the kinetic energy term this can be traced back essentially to the potential energy term and this is essentially the constant term now you can see something rather interesting from this equation firstly this term here the 8 pi 0 over 3 will always be a positive term because obviously these are all positive and Rho which is the density of the universe will always be positive so this is always positive if minus K is always positive if minus K is positive that means this whole term on the right hand side is positive which means that this term is always positive and that means that the universe will constantly expand forever and that's called an open universe if on the other hand this minus K term here can sometimes be negative there might come a point where the negative value of this term is bigger than the positive value of this term in which case this term would become negative and that means the universe would eventually stop growing and start contracting and that would be called a closed universe if on the other hand K is zero so that essentially kinetic energy and potential energy are balanced then that's what's called a flat universe it means that the universe will continue to expand but it will slow down and it will get to what's called an asymptotic value but only after an infinite amount of time this might be better shown by a diagram if we draw a graph and this is the universe size increasing and this is time increasing if you have an open universe then the universe expands and it just keeps on expanding it never stops if you have a closed universe then the universe expands but it's constantly being slowed by the gravitational forces until eventually the gravity stops the universe from expanding anymore and then causes it to collapse on itself until we get to this point here which is called the Big Crunch the whole universe crunches back into a single singularity and effectively the reverse of the Big Bang happens the middle way is called the flat universe and in those circumstances the universe expands but asymptotically towards a value that it never quite reaches it will reach that value but only after an infinite amount of time and so there is a maximum size of the universe that the universe will constantly be expanding to but never quite reaching that would be flat so this is open this is closed this is flat and before we move on it's also worth just noticing that the term here a dot squared over a squared well a dot over a is simply Hubble's constant that's how we defined it earlier in this video so a dot squared over a squared is Hubble Hubble's constant squared so you could replace this term by H squared if you wanted to now I want us to consider a cube of universe I won't draw the third dimension but it well I will live all is it's a cube essentially off side a which is the space dimension the spatial dimension between the coordinate points and we can and inside that cube all the galaxies have a total mass Capital m and we can say that the density of that cube will be the mass divided by the volume and the volume is obviously a cubed that's the volume of a cube and I'm now going to substitute this this term for row in the Freedman robertson-walker equation that I derived just a few moments ago so that will now look like a dot T squared squared divided by a T squared is equal to 8 pi G over 3 times Rho but for Rho I'm going to use M over a and I'll just put the Brack and the T in to remind us of that of course varies with time minus K divided by a T squared now it just so happens that the observational evidence is that the universe appears to be flat and that means that K equals 0 so this term if that is true that means that the universe will expand it will never contract again but it will expand to a particular size but it will only ever get that size after an infinite time and if that's the case then the Friedman robertson-walker equation simply becomes this term here and I want to know how a varies with time you know as what is the solution for a and the way to do that is to guess the way we're going to guess it is we're going to say let a which of course is a function of time equals some constant C this is not the speed of light this is just a constant times time to the power P and let's see if that guess will work in our formula well we first need to calculate the value of a dot because we're going to need that in this formula so a dot which is you differentiate this with respect to time you bring the P down this side and there becomes T into P minus 1 that's just basic calculus so now we've got values for a and a dot so what is a dot over a well that is PC T to the P minus 1 divided by C T to the power P the constants simply cancel out and this term here simply becomes T to the P minus 1 over T to the P is simply 1 over T so that becomes P divided by T and if we substitute this term P over T in this part of the formula here we've got to square it so that becomes we're going to take this formula here and simply insert P over T squared all squared that becomes that P squared over T squared which is the value of a dot squared over a squared is equal to this term here which I'm not going to change at all 8 pi G over 3 times M a t-cubed but a is C T to the P so a cubed is going to be C cubed T to the 3p now dimensionally we have a T squared term on that side of the equation and the only T term on the right-hand side of the equation is T to the 3p so T squared must if it's going to be dimensionally consistent equal T to the 3p and that means that P has to have a value of 2/3 and that means if we go back to our term that a is equal to C times T to the P that means we can now write that a equals C times T to the P which is T to the 2/3 so now we have found that the way in which the universe expands the scale factor of the expansion is some constant which we could work out but there's not another point the key thing is that it expands in proportion in proportion to the 2/3 power of the time so if we plot the value of a against the value of T time if it were a straight line if a equalled some constant times T it would look like that but actually it isn't that it's a 2/3 power which means that a is proportional to T to the 2/3 and that is the expansion rate for a matter-dominated universe but right back at the beginning beginning of time after the Big Bang and probably for about the first 10,000 years the universe would not have been matter-dominated it would have been radiation-dominated and why is that well when the Big Bang happened the assumption is that a huge amount of matter was created but all the matter that was created would have to have an equal amount of antimatter created that antimatter and matter would combine and annihilate and produce a huge amount of radiation in the form of photons surprisingly it also left a small surplus of matter which is essentially all the matter you see in the universe today how it did that we don't know strictly you would have thought that if the Big Bang had created equal amounts of matter and antimatter they would then have recombined produced a huge amount of radiation with no excess matter at all but somehow we managed to leave a small amount of matter and that's what you see in the universe but the consequence of the matter and antimatter colliding and annihilating meant that overwhelmingly in the universe at that time there would have been a huge amount of radiation in the form of photons so now let's consider our cube as we did for the matter-dominated universe when we consider the cube of galaxies now we're going to consider a cube of photons in the early universe when it was radiation dominated and once again the dimensions of this unit of this cube are a we know that the energy of a photon is Planck's constant times the frequency of the photon and you can also write that as Planck's constant times the speed of light that C is now the speed of light divided by the wavelength of the photon now as the universe expands in other words as this cube expands as a expands it will expand the contents with it and that means that as a expands so does lambda so does the wavelength of the photon that is extinct is itself expanding in space and consequently the energy of the photon which is HC over lambda since lambda is going to increase that means that the energy of the photon must decrease and that is exactly what has happened these high-energy photons possibly gamma rays created through the annihilation of matter and antimatter have been have had their wavelengths stretched as the universe has expanded and the wavelength is now in the microwave region and that's what we see in the sky it is the cosmic microwave background radiation and you can see that in all directions and the cosmic microwave background radiation is in fact they as it were the fossil the historic relic of the photons the high-energy photons at the very beginning of time after matter and antimatter had annihilated but to go back to this formula here we can say that the energy is proportional to 1 over lambda that's from this formula here H and C are both constant but lambda will vary as a the scaling factor of the universe expansion increases so that means that E is proportional to 1 over a and wherever you have a proportionality you can always say that e equals n over a where n is just some constant I've dreamt up it doesn't stand for anything it's just a letter that I've derived it's a constant to enable us to say that e equals n over a now what is the energy density well that will equal the energy divided by the volume here is the energy in over a and the volume is a cubed because that's the volume of this cube that we thought of in the first place so the energy density is equal to n some constant we don't need to worry about what that is the crucial thing is that it has an eighth of a fourth term in the denominator so grow that is the energy density for radiation-dominated universe is in over a 4 to the 4th and you'll recall that row for a matter-dominated universe was M divided by a cubed so for a matter-dominated universe the density was a function of a cubed 1 over a cubed for a radiation dominated universe the density is a function of 1 over a to the fourth now we can insert the radiation density into the formula that we derived earlier and instead of having the matter density which is this term here we're now going to substitute the radiation density so that means that we get a dot T squared over a T squared is equal to 8 pi G over 3 times the density which is n over a which is a function of T to the fourth power and once again I want to know how a expands with time with the matter-dominated universe remember it expands as T to the power 2/3 what does it what happens with radiation dominated universe well again we're going to guess and the guess is the same as last time we're going to assume that a is equal to C times T to the P where remember C is just a constant now it's not the speed of light and we already established last time that a dot squared over a squared is equal to P squared over T squared but that mathematic still remains from last time and that equals from this formula here 8 pi G over 3 times n over a to the fourth well that's simply going to be C to the fourth T to the 4 P and now you'll notice that there is a T squared term here and there's a T to the 4 P term here and those are the only two time terms in that equation so for dimensionality to be consistent they must be the same T squared must equal T to the 4 P and that obviously means that P is 1/2 so from this formula here a equals some constant we could work it out but there's not a lot of point because the key thing is that the expansion goes as the square root of time so if we plot density versus the scaling factor a we know that for a matter-dominated universe the graph is M over a cubed we know that for a radiation-dominated universe the graph is n which is a constant divided by a to the power 4 in other words the radiation-dominated density falls off much faster because it's an a to the fourth power then the radiative then the matter-dominated density which only falls off as a 1 over a cubed power we can also draw a universe expansion graph as we did before here is the universe increasing in size and here is time and what we say is that in the very early day of the universe the universe expansion was radiation-dominated and that means that there's a curve that looks like this this is where the scaling factor is equal to CT to the power 1/2 but that after about ten thousand years we'll put that here ten thousand years matter had been formed and it suddenly became matter that dominated the expansion of the universe because there comes a point this is the ten thousand year point where it is matter that's going to dominate because the radiation quality has fallen off so much that matter now takes over and so at this point the expansion of the universe becomes matter dominated and that increases as seed to the T to the 2/3 and as we've said it would reach an asymptotic limit at infinity so so far we would have to conclude that the universe expands under radiation dominance for about ten thousand years and it has then been expanding under matter dominance and it would if that was all there were - it slowed down because you'll notice that this curve shows that the expansion of the universe is slowing down as time increases so it would slow down and gradually get slower and slower and slower asymptotically reaching some fixed limit but I told you earlier that there is experimental evidence that the universe is accelerating in its expansion which means it's not going this way it's going that way and the question is why should that be and does anything that we've done so far give us any clue as to why that might be happening but before we do that I just want to take us on what I hope will be an interesting diversion I have pointed out that at the beginnings of the universe and the universe was radiation-dominated and that that radiation would have an energy which equals Planck's constant times the frequency now if you have high energy you also have high temperature its Boltzmann that created the or gave rise to the formula that says that the energy is equal to some constant which we call Boltzmann's constant times the temperature so the higher the energy the higher the temperature at around about 3,000 degrees Kelvin roughly atoms are what's called ionized that is to say that the electrons in those atoms have so much energy that they all disappear from the atom and the atom is simply left essentially as the nucleus so that means that in the very early stages of the universe when temperatures would certainly have been higher than 3000 degrees Kelvin there would have been no atoms they would just have been as it were hydrogen nuclei which are protons and free electrons and the consequence of that meant that the photons the huge number of photons that had been produced through the antimatter matter annihilation would not be able to get very far in the universe because they would constantly be interacting with electrons and swapping energy so photons would be hitting an electron and then being given off by that electron and consequently moving all around they wouldn't get anywhere they would just be moving around hitting electrons coming out in another direction the photons would have no way of traveling through the universe because they would be constantly interacting with electrons and consequently the universe in those days would be opaque you couldn't see through it because the elect because the photo couldn't get through it they're constantly interacting the temperature of the universe today is approximately three degrees Kelvin so at some point when the universe cooled it cooled enough for the electrons to be caught up by the protons to form hydrogen atoms and the minute that that happened then the photons would have a much easier route because although some of them would hit electrons within the hydrogen atoms most of them would travel straight through and at that point the universe became transparent you could actually see through it and what we're going to do is an interesting calculation based on what we have done so far to see when that might have happened now we say here that the energy is equal to boltzmann constant times the temperature but we've also shown earlier that it is that the energy is proportional to one over lambda and that is proportional to one over a and that means that T is proportional to one over a the temperature has fallen from 3,000 degrees Kelvin to three degrees Kelvin that's by a factor of 1,000 so a we can conclude must have increased by a factor of 1,000 so the scaling factor today we'll put a today divided by the scaling factor at the time that atoms were ionized is equal to 1,000 in other words the universe is a thousand times bigger than it was at the time when atoms were ionized but we can say that in a matter-dominated universe a today is C T to the 2/3 where T is the amount of time that the universe has been in a distance in other words it's t-today divided by C times T to the 2/3 when the universe was ionized so whatever time it was that the universe was ionized well the C's cancel and now we can as it were raised to the power 3 over 2 on both sides and that gives us that 1,000 times 3 to the 2 is equal to t-today divided by T at the time the universe was ionized rearranging that formula T at the time the universe was ionized is equal to T today which is approximately 10 billion years just call it 10 billion for ease divided by a thousand to the power 3 over 2 which is about 30 thousand and that gives you three hundred thousand years so what we've calculated is that the time since the Big Bang when the universe was ionized but just about to move to a point where the atoms would form was 300,000 years after the Big Bang but now let's get back to the main mission which is trying to find out something about this dark energy and just to recap we have derived a formula the Freedman robertson-walker formula which can be formulated a dot squared divided by action of a bracket there a dot squared divided by a squared is equal to 8 pi G over 3 times Rho and Rho will vary according to whether you are talking about a matter-dominated universe or a radiation-dominated universe Rho is essentially the density and what we've said is that for matter-dominated universe Rho is M a cubed and as a consequence of that the acceleration or the scaling factor of the universe varies according to T to the power 2/3 whereas for a radiation dominated universe Rho is equal to n over 8 of 4 and the scaling factor varies according to T to the power 1/2 that's the matter dominated section that's the radiation dominated section now we need to look at pressure and a little bit of thermodynamics first let's think about a matter dominated universe and we're going to take cube of that universe I don't call the third dimension and here are the galaxies inside that cube and the question is what pressure do those galaxies exert on the surface of the cube and the answer of course is 0 because the galaxies aren't bouncing backwards and forwards on the surfaces of the cube they are simply suspended in space as that cube expands as the universe expands so it will carry the galaxies with them but there is no pressure on the sides so for a matter-dominated universe the pressure on the sides of a cube is 0 but what of a radiation dominated universe let's suppose we think about a one-dimensional issue here a photon is traveling backwards and forwards between these two points it is traveling at speed C because it's a photon and the distance between these two points is a distance F what is the pressure on this surface here well we know that pressure is force over area and the force on this point here will be the rate of change of momentum DP by DT so let's suppose we have a photon of momentum P traveling in this direction hitting the wall and coming back again what is the change of momentum well it was traveling with P in this direction it's now traveling with P in this direction the total change of momentum is 2p so DP is 2p but what is DT well that's the time it takes for the photon to get all the way down to this point and come back and hit the wall again and what is that time we'll time will simply be distance divided by speed what's the distance well it travels L in that direction and L again so that's 2l divided by the speed which is C so the time is 2l sorry yes it's 2l over C and so our force on this point here is 2 PC over 2l of course the twos will immediately cancel out we have shown in previous videos that the energy of a photon is its momentum multiplied by its velocity so that PC term there is essentially its energy so now we can say that the force is equal to PC which is the energy divided by L and I over L is essentially a density term in one dimension but we've got to work in three dimensions so let's now take our cube of photons and they could be traveling of course in any direction but the components of the photons will be traveling either in that direction or in that direction for of course backwards or forward so it's side to side up and down or back and forth and if we consider the components of the photons that are moving in the side-to-side direction and hitting an area da on this wall here we've already shown that the force is the over L what is the pressure on this area a well that is going to be the force divided by the area which is d a so that is e over L which is the force times D a what is the length remember the length is essentially now the length of one side of the cube the length times da is simply the volume so now we've got energy divided by volume what is energy divided by volume that is the energy density so we've shown that for radiation dominated universe the pressure is equal to Rho but not quite because that presupposes that the components of the photons are all traveling in the side-to-side direction in fact there are three possible ways they can travel side to side up and down back and forth so only 1/3 of them only 1/3 of the components of the photon velocities will be in the side-to-side direction so strictly we have to say that the pressure on one side of the cube is going to be Rho over 3 and to recap for the matter-dominated universe we said the pressure was 0 because the galaxies do not bounce off the walls of the cube so we can now say that the pressure is equal to W times Rho where for a matter-dominated universe W is 0 and that gives you that the pressure is 0 and for a radiation dominated universe W is 1/3 and then you get pressure is Rho over 3 we can look at this thermodynamically we can take a cube of universe as we have before and we can allow that cube to expand I'm just going to go like this for the moment there is the expansion the expansion is by an additional volume called DV we're just going to assume that this is a box full of a gas this is not universe anymore we're just going to do a little bit of thermodynamics at this point this box is filled with a gas we allow the volume to increase and what we can say is that the work done is equal to the force times the distance and the force is always pressure times area and that means that the work done is the force which is pressure times area times distance area times distance is volume which is P DB so the work done is the pressure times the volume of the expansion but where does that work come from it has to come from the energy work is energy it has to come from the energy the existing energy the internal energy of the molecules and so the molecules lose energy and that means they lose temperature because energy and temperature are related so now we can say that the change in energy of these molecules is equal to minus P DV because as volume increases the energy decreases now we can also say that the total energy in the box is equal to the energy density Rho times the volume of the box density times volume becomes the total energy and calculus tells us that you can take this equation here and you can say that de equals Rho DV plus V D Rho but we know that de also equals minus P DV so that equals minus P TV rearranging you get that v-0 is equal to minus P plus Rho times DV or as you just bring the two DV terms together here and you take the one D Rho term there but we said earlier that we let P equals W times Rho so substituting that into this formula here you get V D Rho is equal to minus W Rho plus Rho DV and that means that V D Rho equals minus W plus 1 times Rho DV and if we rearrange that you get that D Rho by a Rho equals minus W plus 1 DV over V these are solved as log terms so now you can write that d log of Rho is equal to minus W plus 1 D log V and that means you can do a straight integral of that and get that log Rho is equal to minus W plus 1 log V and that means you can always take that and make that the power which means that log Rho is equal to the logarithm of V to the power minus W plus 1 and there's also a constant term in there so that means that since both sides are logarithms that value must equal the totality of that value which means that Rho is equal to some constant which I'll call C that's not the speed of light divided by V to the our W plus one because I've simply done this is one over V to the power W plus one now let's just think about our cube of space again with sides a the volume is a cubed substituting that into this formula here so we can say that Rho is equal to some constant divided by instead of V we're now going to have a cubed so that is a over three into W plus one what does that mean well when W is equal to zero that's a matter-dominated universe that gives us that Rho is equal to C over a cubed because W is zero and that's exactly what we determined before that the energy density increases as the foot the cube power of or rather the inverse of the cube power of a when w is equal to 1/3 which is the radiation dominated at universe Rho is equal to C over a to the fourth and that is exactly what we found before for radiation dominated universe so looking at it thermodynamically you get the same result so now we're going to insert this term for row in the Friedman Walcott Robertson formula and just to remind you what that was we have a dot squared over a squared is 8 pi G over 3 times low and for Rho you would either put the formula M over a cubed if it's a matter-dominated universe or in over a to the power 4 if it's a radiation dominated universe but we're now going to put 4 Rho so this now becomes 8 pi G over 3 for Rho we're going to have a constant divided by a to the 3w plus one and if W is zero then that is the formula for a matter-dominated universe and if W is one-third then that is the formula for a radiation dominated universe and we're almost there because now comes the exciting bit what happens if W equals minus one you'll observe that in those circumstances you get a to the power zero and a to the power 0 is simply 1 and that means that a dot squared over a squared is equal to 8 pi G over 3 times C because a this term here just becomes 1 if W is minus 1 and for C we have a special term Rho naught and that is essentially a constant none of those terms are going to vary and that is essentially the cosmological constant or we could write that a dot over a which is Hubble's constant is equal to the square root of 8 pi G Rho naught divided by 3 and that means if we take this term here in this term here and multiply both sides by a you get that a dot is equal to a into the square root of 8 pi G Rho naught over 3 now this you might recognize is the condition for an exponential where the differential of a value is equal to the value itself in other words if dy by DT is proportional to Y then that is an exponential equation and that means that you get that a is equal to some constant which we'll call C times the exponential of 8 pi G Rho naught over 3 times tea so it's the exponential of the square root of this term times time and that means that the scale factor a is exponentially increasing with time and if it's exponentially increasing the time it's accelerating with time and that is what we think the universe is doing and what is causing that acceleration well that's dark energy or if you like vacuum energy it's the energy that is inherent in all of space and it's causing the universe to accelerate because now if we draw the graphs that we drew before where we drew the density or the energy density against the value of a the universe expansion factor what we said was that radiation falls off as one over a to the power four we said that matter falls off at 1 over a to the power 3 and that therefore at this point which we said was about 10,000 years the density of matter dominates because the radiation density is fallen so low but we now have another density to put on here which is the dark energy density and that doesn't vary at all that's a constant and consequently there comes a point where that energy density is greater and has greater dominance than either the matter or the radiation densities and so if you then plot as we have done before universe expansion against time what you find is as we drew before you start off with radiation dominant and that's going up on the basis of T 2 power to after 10,000 years matter domination takes over and that would reach an asymptotic level going up at T to the power two-thirds but at some point the dark energy density takes over and that's constant and that creates acceleration and what happens then is that the universe expands in that way and it expands forever and there is a consequence of that we've just shown that a dot over a which is Hubble's constant is equal to the square root of 8 pi G times Rho naught the unchanging energy density associated with dark energy over 3 and we know that in Hubble's constant the velocity is equal to Hubble's constant times the distance away from us and that means you can simply say that the velocity is H which is the square root of 8 pi g rho naught over 3 times d now there will be a point out in space where d is such that the relative velocity is the speed of light C and that is calculated at somewhere along the rheine in the range of 10 to 12 billion light years so when a galaxy gets 10 to 12 billion light years away from us it is traveling as space expands it is being carried by space at the speed of light and there's nothing in special relativity that restricts that that's perfectly permissible two objects can't pass one another at a relative speed of more than the speed of light but there's nothing to stop space expanding at any rate it likes and carrying the galaxies with it so there will be galaxies that are going beyond the speed of light once relative to us one they got past this particular distance and if that's the case then light will no longer be able to reach us so those galaxies pass through what you might call a horizon and once they've gone through it as space expands you can see them anymore and so consequently as space accelerates in its expansion all the galaxies of the universe will eventually pass through this horizon they'll pass through the point where they are so far away that they will be traveling relative to us at more than the speed of light and we will no longer see them and consequently the fate for the earth were it to survive we are talking of course billions of years time when the Sun will no longer exist and neither will the earth but if it did then apart from our own local Milky Way and possibly some neighboring galaxies which are being held together by the strong gravitational attraction all the other galaxies of the universe will have passed out of view because they will be so far away that they will be traveling at more than the speed of light and that light from them will no longer reach us finally if you look at the relative amounts of energy density in the universe it is now thought that the energy associated with dark energy which is we call Rho naught divided by all the energy density in the universe is about 73% so 73% of the energy of the universe is dark energy and we don't actually know what it is if you look at the road the energy density of matter compared to the total energy in the universe that is about 27% and the energy density of radiation is now trivially small because all that high energy in the early stages of the universe has now because of the expanding universe been reduced to microwave radiation which very low energy and has virtually no contribution now to the overall energy density of the universe this matter density can be split into two parts there is the Rho associated with dark energy and the road sorry the Rho associated with dark matter on which I did a separate video just a little while ago looking at dark matter and dark matter accounts for about 23% and the visible matter that we can see accounts for about 4% so you can see that dark energy and we don't really know what that is accounts for 73% of all the energy in the universe dark matter and we don't really know what that is accounts for 23% of the energy in the universe and the visible matter that which we can see and we don't wholly understand that accounts for 4% of the universe so if you're studying physics there's a lot still to find out

Channel: DrPhysicsA

Views: 165,459

Rating: 4.9321337 out of 5

Keywords: dark, energy, Friedman, Robertson, Walker, equation, cosmology, universe, einstein, gravity, general, relativity, anti, matter, big, bang, radiation, density, hubble

Id: 9G5NAHsQkOA

Length: 69min 41sec (4181 seconds)

Published: Mon Jul 23 2012