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  • Okay, so we're gonna talk about the universe as it is now, not the early universe, but the universe.

  • That's something like 13 billion years old.

  • And it's doing something unusual when you look at distant Galaxies.

  • By that, I don't mean Galaxies nearby, our own Milky Way, like the Andromeda Galaxy's.

  • I'm talking about Galaxies that are tens of millions of light years away from us, and you look at how, then moving with respect to one another that generally moving away.

  • Now that's that's no unexpected, because we know the universe is expanding.

  • So if the universe is expanding, then it means the space between Galaxies is is is stretching.

  • But what's going on that's unusual is that it's not just moving apart.

  • It's moving apart, moving apart faster and faster, so the universe is actually accelerating.

  • So then you say, OK, well, fair enough.

  • The universe is accelerating.

  • What's weird about that?

  • What's funny about it is that if you just think of what's in the universe, we've got radiation and we've got matter, makeup us, and then we got doubt matter.

  • We would it actually expect that the gravitational pull from those objects would slow down that movement so that the universe is expanding because of the hot, big bang of the energy from the hot Big Bang.

  • But then gravity begins to pull things back, and we should expect the Galaxies to be slowing down.

  • But they're not the moving away faster on DSO.

  • It implies that one of the things that implies is that there's some form of energy density being pumped in that's causing it to expand its overcoming.

  • The natural pull of gravity is like an anti gravity.

  • Andi, this has been coin.

  • Does dark energy is that it's a bad name, really, But it's but it's on it.

  • Not only is doing this is dominating all the other energy contributions in the universe today.

  • In fact, if if you if if you add up all the types of energy that you could have, then you've basically got the radiation in the universe.

  • We have a matter content in the universe, and you've got this dark energy and you've got the possibility of the curvature of the universe acting like an energy source.

  • When you add all of these up to come to unity to in terms of these funny units, the dark energy is about 70%.

  • So 0.7 of this contributions from this weird stuff that we don't know, we don't understand the doubt matter, making up about 27% as well, so together than making up 97 98% of the of the energy density of the universe that we just don't understand what it is.

  • But today we're gonna talk about the dark energy.

  • So one of the things that I've been working on for the last few years is trying to understand the source of dark energy.

  • Now it could be that the source has always been there.

  • Andi.

  • It's been there throughout the history of the universe, and it's just that it's come to dominate everything today or in the recent past, by recent past.

  • By the way, I mean, within the last seven billion years or so, we're talking Cosmo, leave you here.

  • It could be that it's been constant.

  • That would be a term cut.

  • That would be a quantity that is known as a cosmological constant, and we can come back to that.

  • Another possibility is that it's something that has been evolving with time on that it had it was sub dominant early on, but perhaps it mimicked the matter and radiation and you notice it pretended it was like matter and radiation.

  • It followed the the density of matter and radiation dropping, dropping, dropping.

  • But for some reason, about seven billion years ago, it came to dominate everything and that those air colds generally coming to the name of quintessence theories where it's evolved with time.

  • Then there's possibilities that it's not due to some energy source it all that what we're seeing here isn't either.

  • Cosmological constant isn't some new form of energy, but what we're seeing is the first evidence that Einstein's theory of general relativity needs modify on.

  • We're seeing that it needs modifying on these large scales the size of the universe, the observable universe on these air called modified theories of gravity on DDE.

  • So these three possibilities are all being actively pursued, a za way of understanding the observation that the universe is accelerating today and I've been involved in a kind of all three of them trying to come up with models which try and mimic these.

  • So then it doesn't have to be dark energy because maybe everything doesn't need.

  • That's right.

  • If so, of course, when we say Einstein's got it wrong, it's a bit like he's got it wrong.

  • Perhaps on these largest scales, he's clearly brilliantly correct on solar system.

  • Everything within the solar system is beautifully explained by Einstein's theory of gravity.

  • In fact, to be honest, if I say it quietly, pretty much everything on even cosmological scale seems to be brilliantly explained by Einstein's theories of gravity.

  • So any modification would have to be kind of quite small.

  • But it could be that we just need these small modifications, in which case we don't need these new exotic forms of energy, energy density.

  • Thio explain it.

  • As you say it is due to some change in the gravity of the curvature of the universe that that that's leading to the observations That universe appears to be accelerating.

  • Why do you say Doc?

  • And he's no good.

  • What would you call up?

  • Well, I've just been thinking about that because I don't think dark energy is such a great name.

  • That's there's lots of things that a dark then It's not particularly that that's special.

  • Lots of things have got energy on this.

  • It's not really that that's special.

  • But the thing that makes dark energy the important quantities it is is that the pressure it has is negative.

  • It's got negative pressure, and that means it actually causes things toe push apart.

  • That's the way that you can imagine it.

  • So some of the discovered you can think of it.

  • It's been having a bit like attention on associated with it, and it's smooth.

  • That's the other thing about it.

  • It's It's a very smooth components through.

  • It's pretty uniform throughout the whole universe, so something like a smooth tension might be.

  • It might be a better thing toe have rather than rather just moved pension.

  • I guess the problem with that name is it doesn't it doesn't allude to the fact that it's completely backward.

  • Don't know.

  • It makes it sound like like we might know what we're talking about.

  • All that we know is that when you write down the equations, Einstein's equations on Dhe allow for something that can have a negative pressure.

  • It will do the job for you.

  • That's the clear.

  • So a cosmological constant will do that for you as well.

  • These concocted models that you have these contestants models, they will do it for you.

  • And then, of course you can.

  • You can also use your modify gravity, and the modifications will do it for you.

  • There is one other thing that will that can effectively do it for you, which is is less in favor but hasn't been completely ruled out.

  • And that is maybe the universe isn't a smooth on his uniform.

  • As we thought, maybe we knew of the re structure in the universe, right?

  • We know, Look at you and I were quite different.

  • Andi, Even though you're down the gym and I'm down the gym, we're still quite different.

  • And you So there are structures out there.

  • We know that there are that galaxy is a kind of clustered together on in filaments on those filaments kind of surround void, so voice where there is lower density than on the average.

  • But the thought is that as you get to bigger and bigger scales in the universe, that that the distribution of those voice becomes a bit more uniforms Andi there that their effect drops off, giving you this net homogeneity.

  • Things are very smooth, but maybe that's not the case.

  • you know, maybe actually, the universe is is homage in homage ing.

  • Yes, it's not homogeneous, nice and smooth and live scales.

  • And if it's in homogeneous, then and we happen to be living in a void in a region that's gotta lower density than on average than we would actually perceive the universe to be expanding at a different rate from outside on, it could be that that's what we're seeing here, that the effect of this acceleration is, and that's what an acceleration is.

  • You're seeing a different rate here, compared to here on the effects of the explosion is just that.

  • But because there are so many, that's probably not the case on.

  • The reason why it's not the case is is because of the the the fact that we can measure the different things in the universe and from it in first, something about the distribution of the matter so we can measure the microwave background radiation.

  • We can measure the distribution of the hot and cold spots in the microwave background.

  • We can measure the distribution of Galaxies and when you do in particular the microwave background on look at the distribution of the hot and cold spots.

  • It becomes clear that in order for this in homogeneous model type of models to work, then you need to be in a very special place in this void.

  • In fact, you need to be very close to the center of the void, and then you begin to ask, Well, why is it I would be very close to the center of a void could be anywhere in this void.

  • So there's some high degree of fine tuning going on in order for you to explain the apparent acceleration of the universe from the idea of these in homogeneous distribution of matter.

  • So it's kind of being it's under under threat a CZ a model, but it hasn't been ruled out yet, but it's certainly very fine tuned if if under the theory of I know there are always possible was cause if it is the cliche, dark energy is this, like some kind of particle or field that can be measured is that is that thing is the thing with mass or no mass has a core energy that can be measured in jewels.

  • Maura's this like it's dark and you just feel like I shall I get my cup of get my cup to show you the how much energy there is.

  • Okay, if you just hang on, I'll go get it.

  • What a shame has taken it.

  • I think I'm just taking it to university.

  • And she had a dark energy cut, but that is it for this cup here.

  • It's about this size this cup has in it about, I think, is a yack to Graham of Oh, 10 to the minus 24 jewels of energy of dark energy.

  • Because dark energy is smooth ride, it's throughout the whole universe.

  • So it is.

  • It has got an energy has got on that.

  • It's got an energy density and so you can work out how much energy is in a cup.

  • And I think it's about 10 to the minus 24 jewels, which is no, not a huge amount, but it's it's it's there.

  • So what makes it spent?

  • You said, What is it?

  • Well, we don't know what it is, and that's why we try lots of things that can and mimic its effect.

  • The the key ingredient isn't particularly that it's got an energy.

  • That's lots of things have got energy matter has got energy.

  • Radiation has got an energy associated with it.

  • That doesn't make it.

  • That doesn't make the end of the universe accelerate.

  • It actually slows down the universe.

  • But what The things that have energy also often have pressure.

  • And now we usually think of pressures of a positive quantity.

  • You think of particles banging against the surface of a container, creating a pressure pushing out.

  • And that's that's a positive quantity of radiation has in the in the early universe.

  • Relativistic red particles have a radiation pressure, which is 1/3 of the energy density had dark matter particles and the particles that you and I am made up off there called dust particles written.

  • That's the general name given to them.

  • They've got basically no pressure because they're hardly moving.

  • They're moving non relativistic Lee.

  • So you think of putting some doubt matter or some barriers that were made of in a in A in a tin on, they'll hardly hit the side of the tent because they're moving set so they don't create a pressure.

  • So that's got zero pressure or very close to zero.

  • What mix Dark Energy special is that the pressure, it has actually is negative on DSO.

  • It's you need.

  • You need something that can that can produce, have a positive energy density but have this negative pressure that's acting in the opposite direction to what you might imagine.

  • And there isn't any standard stuff.

  • Does that the stuff that you and I are made of, the particles that you know it just does not do that.

  • You need something more a bit more exotic on the took that the two favorite where's of doing it are basically to use what the particle cosmologist loves to use, which is a scale of field on a scale of field has both.

  • A It's a it's an object that's got a value everywhere on Dhe.

  • Its energy is made up basically of a kinetic energy.

  • So the kinetic energy of motion of that object on the potential energy that sort of telling you at any given point in space and time.

  • What's its potential associated with it now, if it turns out that potential dominates over the kinetic, so the kinetic energy of the field is Sasa letting around a small compared to its potential energy, then it will have a negative pressure, and it can cause a kind of a repulsive effect in terms of causing acting against gravity on dhe.

  • And so that's what a quintessence scenario will do for you.

  • You work in a regime where the potential dominance over the kinetic energy and it will give me this negative pressure.

  • It's what a cosmological constant does, in fact, in a cosmological constant.

  • It's a unique case where the energy density is exactly minus the pressure.

  • So they balanced that they're equal in magnitude, but opposite in sign, and that hasn't has the same effect.

  • The data suggests that the universe is perfectly consistent with the cosmological constant so that the acceleration of the universe has bean driven by something that looks like a cosmological constant.

  • In actual fact.

  • Just recently, there's a set of death have been coming out.

  • If you look at the plank on dhe microwave background data, if you look at people at the data coming from, people have looked at some supernova.

  • That, by the way, is the way in which we infer what the universe is doing.

  • You you will come back to that if you look at the that they're debtor they're actually finding that the pressure is actually less than the pressure is less than the energy dense, the minus.

  • Okay, What they're finding is that when I divide the pressure by the energy density for a cosmological constant, that number would be minus one.

  • If I do the same for ordinary matter, that number is positive.

  • So the pressure is 1/3 of the energy tested for radiation.

  • The pressure is zero for dust.

  • For a cosmological constant, the pressure is minus.

  • The energy densities of pressure divided by an agency is minus one.

  • For contestants scenario, you find that the pressure evolves down towards minus one.

  • It it begins to look eventually like a cosmological constant.

  • But there seems to be some tentative suggestions that, actually the pressure divided by the cut the energy does is less than minus one.

  • So that's lower than a cosmological constant would give.

  • That's a weird scenario on it.

  • It's probably not gonna last that I imagine the data will push everything back up towards the cosmological constant.

  • But at the moment there's a slight pressure toe in the data, suggesting it might be less than minus one.

  • That means that the universe is actually not just going to is going to end up with a big rip.

  • It is called a phantom scenario, and in the end there will be a future singularity where it will just tear apart because it's expanding so rapidly that first of all the Galaxies begin to move apart rapidly on.

  • Then the matter within the galaxy's begins to break up under the influence of the expansion of the universe.

  • So they all break up, and then eventually the fabric of space time itself will split.

  • That's the idea.

  • It's not good Zzyzx they're going to do much for, uh, no.

  • London will carry on.

  • I'll be no problem.

  • How far can he say, Yeah, Is that galaxy being pulled away from us or pushed away from us?

  • That's a good question.

  • The the dark energy is driving the expansion of the universe.

  • It's driving the relative size of the universe.

  • So this is so in that sense, I would say it.

  • Sze pulling is pulling the Galaxies apart.

  • You could also think of it is pushing them apart as well, though, but I I think I think I would no, actually, I think I think of it is pushing the Galaxies apart because to be pulling it apart, you need something outside of it to be doing that.

  • The dark energy is part of the overall on dhe system of space, time and matter and energy density.

  • And so it is integral to it.

  • So it's it's affecting the space time.

  • It's not outside of it, appalling the space time.

  • So I'd say it's its effects were pushing, pushing them apart through it's come through the effect it has on the overall size of the universe.

  • That's a good question.

  • I haven't really thought about it, so it's just kind of this insidious thing that's everywhere.

  • It's just making it be, yes, it's everywhere.

  • It's it's There's things with very little variation in it that no noticeable variation today.

  • It's in it so constant, like a cosmological constant would do that.

  • It's just the same everywhere.

  • What do you need to find?

  • What's the Large Hadron Collider?

  • Dark energy?

  • That's a very good question.

  • So the key thing I think you need in cosmology really is not a single is usually not a single smoking gun.

  • One of the benefits that you have in cosmology that you don't have necessarily the Large Hadron Collider in some sense, a complimentary in the Large Hadron Collider.

  • It's huge asset is the fact that it can produce the same situation.

  • Millions 6 600 million times a second collisions are carrying.

  • So 600 million times a second you're producing the conditions of the very early universe 10 to the minus 11 seconds after the Big Bang.

  • You can't do that in cosmology that it's gone, but what you can do is you can probe different epochs of the universe using different sets of data.

  • So, for example, the cosmic microwave background is really probing the universe, as it was 300,000 years after the Big Bang is telling you what the universe looked like as those photons decoupled from on the matter.

  • But on the other hand, if I look at, say, the distribution of Galaxies are of clusters of Galaxies, that's telling me about the universe much later on a few 1,000,000,000 years later.

  • If I look at the abundance of the primordial elements in the universe, that's telling me about the universe much earlier in a few minutes after the Big Bang so that there are these different epochs I can probe.

  • I can look, for example, of the distribution of the supernova in distant Galaxies.

  • That is giving me a snapshot of how the universe is expanding at different moments on by each of these that the distribution of the Galaxies, the distribution of the hot and cold spots in the microwave background, the relative distance between this distant supernovas.

  • They all depend upon how the universe is expanding, that that means they all depend upon the source of dark energy that you've got, Whether it's a source that's constant over time or whether it's the source, it's varying over time.

  • So what you really want to do is get as much data on these different eh pox as you can and see what models they're all consistent with.

  • And that's the that will be the key ingredient.

  • It won't be that the microwave background will pin it down for me or the clusters of Galaxies will pin it, and I will need all three, and I'll have to be able to bring them together and and find which of the if any of the models best fits the combined total that that'll be the way that we'll try and do it.

  • So currently, one of the key things that's out there, they're out.

  • There are specialised telescopes being built on DNA now working.

  • There's the dark energy survey, which you know nothing.

  • And we're part off on that.

  • You know that?

  • It says what is going to do on the 10 right?

  • It's going.

  • This is trying to probe for evidence of, you know, trying to find out what the dark energy is, and one of the ways it will is going to try and do it is by looking as deep as it can a distant supernova in distant Galaxies.

  • On why that could be really useful is that we've just said the universe is accelerating today, and we've just said What's the consequence of that?

  • We've said it's gonna rip everything apart, okay?

  • Eventually if it was to carry on.

  • But if I turned that if that if the universe always accelerated, then we would have never form structures right, the atoms in the universe would never have had a chance to pull together on the gravitational attraction because the acceleration would have pushed, pulled them apart, pushed them apart too quickly for gravity to pull them back.

  • So we know the universe can't have always accelerated.

  • So that means there's a knee pop.

  • There's a time in the in the universe where it went from a period of not accelerating toe accelerating.

  • And this is the part which the dark energy survey is going to be trying to probe, because it can look deep enough into the universe to look at the distribution of the supernova and Galaxies at that scale.

  • Now, if we can get enough information about that period, then what we will be able to do hopefully is start discriminating between the various models that try and explain dark energy like the cosmological constant, like a contestant's model like a modified gravity model.

  • Because they should all have slightly different ways in which we leave the decelerating period that's called the matter dominated era an end to the accelerating period.

  • So that's one way, for example, that we will try and test these dark energy if it exists.

  • Dark energy is everywhere.

  • You saying it's in this room, it's it.

  • It's in that cup.

  • So what is it about dark energy that makes you unable to detect it, measure or be able to prove it's here because it's hardly interact with anything in the sense of right on son.

  • It's a bit like gravity.

  • It's not like gravity in the sense gravity sucks.

  • And this isn't but gravity in this room is very difficult to pin down on dhe.

  • If you had very it's big influences on big scales with massive objects on dhe.

  • Dark energy is doing the same thing.

  • You know, just because you have put on a bit of weight over the last few months and you know you've grown, you can't put it down to affix accelerating universe.

  • Right?

  • Dark energy has absolutely no impact on us in our everyday lives, because the forces that are binding us together and much stronger than the foot than the dark energy which is tryingto pulls about the dark energy here is trying to pull us apart that that's true, but it's completely negligible compared to the forces within our bodies, which are keeping us together on the force of gravity, which is keeping us on the earth so it becomes very difficult to determine in the lab I am involved in an experiment, a proposed experiment, which is going to try and do that.

  • It's going to try and look for various forms of dark energy by which are called chameleon fields.

  • So this is one of these contestants type models which, whether dark energy can its contribution can change.

  • And this particular model is I'll just explain.

  • It briefly is very nice because the the effect of the chameleon field, which is going to be my dark energy field, depends upon its environment.

  • So in a in an environment where it's in the presence of lots and lots of matter, then it's a heavy field.

  • It doesn't move very much.

  • It doesn't do very much in the presence of very little matter, like on these very large scales thes cosmological scales.

  • The field is extremely light, and then it can.

  • Actually it's has a negative pressure associated with it, and it acts like a effective cosmological.

  • Constant drives the acceleration, so we've got a difference, right?

  • We've got if if I have a very light density of matter, it it's a very low density of matter.

  • The field is very light if I have a high density of matter.

  • The field is very massive and it reacts to the density.

  • So what?

  • This experiment that is that we're proposing with a colleague of mine that's a Nazi in Clare Burrage and then with Ed Heinz at Imperial is we're going to have If it works out, we're gonna have a comment, make a condensates.

  • A Bose Einstein condensate in the lab, in a in a chamber.

  • So you got this gas of in the of molecules in the in the chamber on, then we're going to separate them.

  • You can move him apart, and we're going to remember it.

  • If if this field is present, that we've said, it's everywhere, Okay, this chameleon field is every well, at the moment, it won't do anything special.

  • The to come and say it's a very similar.

  • It'll have the same value in each of the common sense.

  • But imagine now I bring a really massive source, something really massive close to one of the condensates.

  • So that means that this compensates is suddenly feeling a much bigger density.

  • It could be a heavy object.

  • Okay, it'll be that would be ideal.

  • If it was some very massive heavy object then then this bit of the bulls on.

  • Remember, I've split the bars.

  • Einstein.

  • Condensate.

  • Let me do it big.

  • Spit it in a big way here.

  • So there's one bit here that is now feeling this big sauce so it feels more massive.

  • Its density has increased, but this one is still no experience in that source.

  • But remember, the chameleon field does.

  • It reacts, Okay, It reacts to the local density, so it will feel much heavier here than it does here.

  • And so it will change its profile.

  • And what you can then do is you can quickly bring the source is back on the two values of the communion.

  • Feel kind of interact with one another and we'll see, like in like in what's known as an interferometer.

  • You can see interference effects from this chameleon and and that would be a way of testing for the chameleon model of back energy.

  • And that that is a lot.

  • But we can do in there.

  • We're going to try and do in the laboratory.

  • Talked about how there was this matter dominated period.

  • Where the where dark energy wouldn't have been.

  • Yes, President.

  • It could have been present.

  • Itjust mustn't be dominant.

  • Yes, Yes, yes, of course.

  • The tipping.

  • Quiet.

  • What would What would have reached what level?

  • Certainly that That's a good question.

  • Yeah, So when you've got and when you've got radiation and matter in the universe is expanding its contribution to the energy density drops.

  • That's really straightforward to understand.

  • But let's just remember what we mean by energy density.

  • We mean on the energy stored in a given volume.

  • So that's what that's what the density is.

  • It's per unit volume.

  • So if you imagine the volume is being our universe, okay?

  • And I have a given number of particles, nooses are matter.

  • Maybe I've got 10 particles.

  • Andi, Initially, the volume is maybe you know this big.

  • So I've got 10 particles in this volume.

  • So the energy is is this number of particles divided by this volume?

  • Maybe quite high.

  • Now I double the size of the box, so the number of particles has remained the same.

  • I've still got 10.

  • The energy is not changed.

  • I've got 10.

  • But the box has doubled in size, so its volume has gone up by a factor of two cubed eight.

  • So the densities dropped by a factor of eight.

  • And if I double it again, it goes down again by the Cube.

  • And so you can see that that matter.

  • In an expanding universe, its energy dense that inevitably drops OK, right.

  • So dark energy, because of its nature, is its energy density can remain constant all the way through.

  • On the models that I've were said that were putting forward in terms of these contestants models, they are such that they will act like radiation on matters, so they initially will drop.

  • But they'll have some feature about them in their potentials, which eventually means they come to Goa constant and they eventually come to dominate.

  • So there will always be a tipping point.

  • As you said, where the matter has to drop, radiation actually has to drop even faster radiations probably worth just mentioning.

  • So imagine I've got a photon of light and again I put it in this box, okay?

  • And I've got a photon of light in this box.

  • The universe expands, so the box doubles in size.

  • This for Tom?

  • It's structures because that's what the universe, it will stretch the its its wavelength will double.

  • Okay, where we know that the energy goes is one over the wavelength.

  • Energy and lights causes one over the wavelength, so if you double the size that the wavelength, it's energy has halved.

  • And so now you've got the energy has gone down by a factor of two from the radiation, but it's gone down by this factor of eight from the box.

  • So, in fact, it goes down by a factor of 16 to to the powerful, so the energy in the radiation drops even faster than the energy and matter.

  • So the early universe is one where you're initially dominated by radiation on matters sub dominant.

  • But the radiation density is dropping so rapidly that eventually there's a tipping point there.

  • That's a very famous e pop that's called matter radiation equality.

  • Where the match radiation keeps dropping, matters drops off less quickly.

  • But now, eventually, this.

  • There's another tipping point somewhere here, where the dark energy comes too dominant on dhe.

  • That's that, sir.

  • That's now that's no.

  • And as the universe gets bigger and bigger and the radiation and the matter has listen less sway, yeah, that's why it's gonna get that's right.

  • So unless this if it's if it's a pure cosmological constant.

  • That's will be the outcome.

  • The universe will just keep expanding.

  • It'll accelerating on goes into what's known as a distant to sitter expansions an exponential expansion on Yeah, the first of all that.

  • What will happen is the Galaxies will drift apart because they're being pushed apart.

  • So the first thing we will notice is that we'll see all the distant Galaxies just moving away from us and eventually won't see any.

  • There won't be any in our vicinity, so pretty sad.

  • But then, within our own galaxy, the stars will begin to move apart, and we'll see them drifting out and then within our own.

  • We will be here by then, of course, but within our own solar system, that will happen.

  • And then eventually they at the molecules and the atoms will themselves break up.

  • But most people, I think, believe that this is probably a transient feature and that what we're going to see is that there'll be some decay of if it's a field, a scale of field responsible, like quintessence that will decay just like the decay and inflation, which then allows the universe to reheat that there will be some decay will move back into something like a matter or a radiation dominated universe again.

  • Zero degrees.

  • You walk all the way around and you come back to 360 degrees with a straight string That's not quite right.

  • String sort of cuts out a bit of space, so when you walk around it, it makes your spares conical.

Okay, so we're gonna talk about the universe as it is now, not the early universe, but the universe.

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ダークエネルギー&ビッグリップ - 六十のシンボル (Dark Energy & The Big Rip - Sixty Symbols)

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    林宜悉 に公開 2021 年 01 月 14 日
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