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So we have seen there are two reasons to introduce a new form of energy called dark energy in the Universe. First, luminous and dark matter, as well as radiation, are not sufficient to ensure the energy density necessary for space to be flat. And we have seen that flatness is a prediction coming from the theory of inflation. Second, the expansion of Universe is presently accelerating. This was first observed in 1999. But we have also seen that matter and radiation will tend to slow down the expansion. So we need a new component that would explain this acceleration of the expansion.

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And we are going to see now that presumably, this new component, this unknown dark energy is provided for by what one calls vacuum energy, the energy of the quantum vacuum.

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So why do I say that vacuum energy is a likely candidate for dark energy? Well, we have encountered the energy of the vacuum in the context of inflation. Remember that inflation is this very rapid phase of expansion, actually so rapid that it keeps accelerating and becomes what we call exponential. And so you see that there’s a direct connection between the presence of this form of energy, which is the energy of the vacuum, and the acceleration of the expansion of the Universe. And so this is why we’re saying that probably history is repeating itself. And we might be on the verge of an inflationary epoch.

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Or at least, we might attribute the acceleration of the expansion of the Universe to this form of energy, which we call the energy of the vacuum.

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So what is exactly this quantum vacuum that I just mentioned? Well, for a physicist, the quantum vacuum is the fundamental state over which one builds the theory. To be a little more explicit, let me use a diagram that we physicists are using. This is an energy diagram. The fundamental state is at the bottom. So this is the foundation of the theory. And we build particle states over the vacuum. So that means if I introduce a particle of mass m, we know it has energy E equals mc squared. And so one particle state will be at an energy E equals mc squared above the fundamental state, above a vacuum. If I introduce two particles, energy, again, mc squared.

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So two mc squared for the two particles. And so we have a two-particle state which is at an energy level 2 mc squared, and so on. And so you see that when we introduce more and more particles, we’ll build a tower of states, which correspond to one particle, two particles, three particles, and so on. And all this tower of states is built over the foundation, which is the quantum vacuum. I understand this is a notion a little difficult to understand because we’re used to the fact that a vacuum is what remains where we have taken away everything.

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At least, it is difficult for us Westerners. In Eastern philosophies, people are used to this notion of a vacuum, which is a central notion. Let me just take an example which also will allow you to understand a little better what I mean by a quantum vacuum. I will take the example of a Zen garden, a Zen dry garden such as this one, where you have gravel. It’s gravel which has been combed by a rake. And so you can imagine that this gravel would be precisely the state of a quantum vacuum. And over this state, which you see as fluctuation– we’ll come back to that in a while. Over this vacuum, you are building structures– those would be particles.

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Those would be the matter in the universe, and so on– the structures of the different stones. And so you see that you can imagine a Zen garden such as this one as a sort of model of the Universe with a certain geometry, but also which is built over a quantum vacuum, that means a fundamental state of gravel with some fluctuations.

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But the quantum vacuum is not just a theoretical concept. It has also some experimental consequences. Let me illustrate that with what one calls the Casimir effect. Let us take two plates, two conducting plates of metal. Put them parallel to one another, completely in the vacuum. So there is no current running on the plates. And there is no gas. The two plates are just put in the vacuum parallel to one another. Now, we have seen that the vacuum is the location of fluctuations. Particles, anti-particles are created and annihilated. We are calling them quantum fluctuations. And so as in this diagram, the vacuum will be full of these fluctuations.

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Now, it turns out that the fluctuations in between the two plates are playing a specific role and will induce a force, an attractive force, which can be measured. And so you see that in a sense, the effect is contrary to what a gas would induce. Just imagine between the two plates you have put some gas, then there is a pressure of the gas, which would tend to repel the two plates. That would be due to the pressure of the gas, the positive pressure of the gas. In the case of the fluctuations coming from the vacuum, the effect is attraction. So we are talking about a negative pressure instead of a positive pressure of the gas.

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So you see that the effect of a quantum vacuum is somewhat opposed to the effect of a gas or more generally, to the effect of matter.

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Why does a vacuum energy accelerate the expansion of the Universe? So let me return and use the equivalence between what we call the acceleration and the gravitational field. Let me take the example of a lift. So on the left-hand side I have a freely falling lift– and we have seen that Einstein was establishing an equivalence between acceleration and gravitational field. But the fall of a lift is slowed down by the fact that it takes place above the sun. The sun is sending some radiation. Radiation of photon is exerting a positive pressure on the bottom of the lift as we have seen in the case of the radiometer. And so this is slowing down this fall.

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And similarly, radiation is slowing down the expansion of the Universe. On the right-hand side, I imagined a source of vacuum. Of course, that doesn’t exist. Vacuum is everywhere. But let me just imagine for the sake of the argument that there’s a source of vacuum. So that means that again, we’ll have fluctuations of the vacuum. And those fluctuations have an attractive effect. And so instead of slowing down the fall of the lift, it will, on the contrary, accelerate the fall of the lift. And so you see that in a sense, fluctuations of a vacuum have an accelerating effect. And it’s a similar effect that it has on the expansion of the Universe.

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So how is one going to identify the exact nature of dark energy and maybe prove that dark energy is vacuum energy? Well, astrophysicists have started a very ambitious programme to try to map the history of the Universe through large surveys of galaxies. So one is looking at the large portion of the sky with a very large number of galaxies. And one is trying to go as deeply as possible. That means to try to get back to very ancient times, very distant galaxies. And in this way, one hopes to be able to retrace the history of the Universe. And of course, the history of the Universe depends on the expansion.

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And because the dark energy precisely has strong effect on the expansion. By having a detailed history of the Universe through the different ages, one expects to be able to retrace the nature of dark energy. And so in this ambitious programme, let me pick two emblematic experiments. The first one is a US experiment. This is the LSST telescope. And it will be in Chile, in the Andes. This is a very large telescope, eight-meter telescope, which is precisely constructed in order to have very large surveys of galaxies and very deep surveys. The other experiment is a space mission, a European space mission of the European Space Agency. It’s called Euclid.

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And again, it will try to have pictures of large parts of the sky repeating over and over in order to have as deep as possible and as precise as possible maps of all these galaxies, again to trace the history of the Universe, and to try to identify the nature of dark energy. Both of these experiments will receive the first light in the early 2020’s. And so one expects that in the 2020’s to be able to have a much more precise idea of what is dark energy, and possibly show that this is the energy of the vacuum.

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So what is the value of the energy of the vacuum? Well, let us guess this value using a method that we learned from Galileo himself, which is called dimensional analysis. We are trying to compute the energy of the quantum vacuum. So that involves quantum physics. And we have seen that the characteristic constant of the quantum world is Planck constant, h. We’re of course in a gravitation context. And so we should use Newton’s constant, the constant of gravity. And we are using general relativity. And we have seen that the velocity of light, c, is playing a very important role.

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And so we should construct out of G, h, and c a quantity that has the unit of energy, which would be measured as an energy in our world. That quantity is called the Planck energy. And its exact value is 10 to the 19 GeV. And so that gives us a typical mass scale, typical energy scale, I should say, of the energy of vacuum. If we turn that into an energy density– remember that what is observed is 10 to the minus 26 kilogramme per cubic metres. Well, if we now use the Planck energy to compute this energy density, we find a number which is 120 orders of value too large.

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So that means it’s 10 to the 120 times what is observed in nature. So it means we miserably fail in our guess. And moreover, that means that we do not have in hand so far a theory of quantum gravity. So it means a theory that would encompass the quantum theory and the theory of gravity described by general relativity.

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To sum up, dark energy is most probably the energy of the vacuum, that fundamental state of the Universe which serves as a foundation for the construction of the material Universe. The vacuum energy already played a central role in the phase of inflation which followed the Big Bang, and so would play now a significant role in the recent acceleration of the expansion. And a large, experimental and observational programme has been set up worldwide in order to identify whether indeed dark energy is this energy of the vacuum.

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And if we want to compute this energy of the vacuum, a concept which lies at the interface between the theory of gravitation, general relativity, and the quantum theory, then we will have to reconcile these two theories, which are notoriously difficult to reconcile. That’s a task that might occupy theorists for the next years and maybe even the next decades.