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[Auto-generated transcript. Edits may have been applied for clarity.]
So if. Have to.
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Okay. So welcome back. So. So there been already a lot of questions about the early universe and how it began.
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So and Hardy is going to tell us about two possible windows into what might have happened in the early universe.
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And this is about cosmic strings and gravitational waves.
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So please read. Okay. Thanks very much for the introduction.
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So indeed, I'm going to describe cosmic strings and try to convince you that these, if they exist, which I hope they do,
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could give a fascinating window into the cosmological history of the universe, allowing us to learn new things.
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Beyond which you'll notice that Jerry has already described and potentially also learned about physics at extremely high energy scales,
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far beyond any that we could ever hope to probe into particle collider.
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Maybe I can also note that actually, there's a nice complementarity between what I'll say and what we'll be talked about
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in context of condensed matter in the next Saturday morning of theoretical physics.
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So the kind of physics I describe appears in many different places. Okay.
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Just to make sure we're all on the same page. Let me start by reviewing classical field theory.
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If I imagine that I've got some sort of mattress like structure with some masses connected by springs,
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then I can imagine that associated to each mass.
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There's some degrees of freedom in my theory, maybe just the position of the mass, maybe some other degrees of freedom.
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And there will also be interactions between the different masses owing to these springs.
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Now, if I take the distance between the different masses to be extremely small,
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or if I look at physics on extremely large distance scales compared to the spacing between the masses,
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then I can effectively describe my system as some continuous system,
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some continuous field where at each point in space I have some degrees of freedom.
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Actually, it's not at all obvious that this works.
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This is the magic of effective field theory that actually I can do this continuum description, but it turns out to be true that I can do this.
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Probably the most familiar example of a field theory that you will have all seen in great detail comes from electromagnetism.
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In this case, at each point in space I have some four vector, the electromagnetic four vector that I have called a hair,
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and this gives rise to the usual electric and magnetic fields that are very familiar in this kind of way.
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Actually, in this talk I'm going to focus on a simpler type of field theory,
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just a scalar field theory where instead of just having instead of having a full four vector worth of degrees of freedom at each point in space,
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I'm just going to have a single number. This might be a real number, in which case I'll talk about a real scalar field.
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It might be a complex number, in which case it's a complex scalar field. Unfortunately, due to time,
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I don't have the opportunity to explain why we believe that new scalar fields
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probably exist in the universe and probably exist at very high energy scales.
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Again, there was actually a very nice morning of theoretical physics about axioms that would
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be one of the canonical examples of new scalar fields that we believe may well exist.
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So that would be provide context. Of course, I'm happy to talk about this afterwards, but for the rest of the talk,
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just please take as a belief that we may well have these new scalar fields around us in the universe.
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I'm also going to work in the classical limits.
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Of course, as we have already heard this morning, we believe that the universe is fundamentally quantum,
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so that each degree of freedom here in my mattress should really be some quantum harmonic oscillator with corrections.
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And then as I take the continuum limit, I should have a degree, a quantum degree of freedom at each point in space.
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But as we know from electromagnetism, there are many scenarios in which it is useful to talk about a classic electric and magnetic field.
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We don't need to worry about the details of the individual photons that make up the field.
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And likewise, everything that I talk about today will have sufficiently large occupation numbers.
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If I can be a bit technical that we can treat the system as a classical field, which makes the physics much easier to analyse.
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Okay, that is the set up. But of course we want to talk about dynamics, about what happens with time.
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And again. Let me go back to a very simple system just to give an introduction.
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So if I think about just an individual one particle here represented by this.
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I saw this ball moving in some potential, which I'll call you.
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Of course, we all know that the total energy of the system is some kinetic energy plus the potential energy,
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and depending on how much energy the particle has, of course the system can have.
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The behaviour of the system can be different. If I have, say, a large amount of energy in the particle, then the particle can explore the full system.
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It can escape off to infinity. It's unbounded. This is for this particular example potential I've just drawn slightly randomly.
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If I have somewhat less energy than the particle can explore all of this region, including both of these minima.
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But if I go down to lower energy, then the particle will inevitably be trapped and either this minimum or this minimum.
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And just to reiterate, there's one degree of freedom in the system as I've drawn it, just the position of the particle.
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Let me go to the scalar field now. And just like each degree of freedom, the degree of freedom on the left hand side was moving in a potential.
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Now each degree of freedom, which is the value of the field at each point in space, will also be associated to a potential here.
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Because I have a continuous system, I talk about energy density rho rather than the total energy.
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Well, it's more useful to talk about the energy density, right?
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Rather than the total energy capital E, which is just the integral over space of the energy density.
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And the energy density takes a pretty similar form to on this side, we have the potential energy associated to the value of the field at each point.
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We also have some time derivatives of the fields.
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This is not as analogous to the kinetic energy of the particle, and we also have some spatial derivatives.
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Contribution to energy density. These, if you like, are associated to in the mattress description.
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The energy stored in the string in the springs between the different masses.
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Okay stream. You know them, but it will make it do with it.
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Let me start with a very simple potential. So just potential of this form here as a is some new energy scale which will actually be associated
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to the energy scale of the new physics or the scalar field that might arise in my theory,
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some coupling constant. And of course we all know what this potential energy looks like.
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Just looks like this. What is noteworthy is that this potential of this theory has a symmetry in particular is symmetric.
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Under the mapping, phi goes to minus phi. Of course the potential remains unchanged by this.
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And moreover, not not only is this a symmetry of the theory,
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but also it's a symmetry of the vacuum solution, by which I mean the lowest energy solution.
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Of course, just looking at the total energy density we have, that the energy density is minimised.
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When I minimise you and when I have no time or space derivatives in the field.
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So. As a result, if I imagine myself as an external observer to the system with some tile work, I can adjust the energy in the system,
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which equivalently, is adjusting the system from being at high temperature to being at low temperature.
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Then I get the following effects or dynamics.
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If I start off at high energy, high temperature way above this energy scale for the scalar field can explore all of its potential,
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a large part of its potential. Correspondingly, here the colours represent the different values of the scalar field.
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There's also enough energy that I can have sizeable derivatives both in time and space.
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So the field probably looks like this. Then, as I turn my dial and the energy density of the field goes down, well, the derivatives become smaller,
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but also the the value of the field at each point is in some sense channelled down to this one unique minimum.
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So what I get out at these low temperatures is really quite a boring system.
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Is just a scalar field. That's right. At its minimum, no derivatives and time and space.
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Much more interesting is if I make only a very minor change to my potential.
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Whereas before I had a plus sign here, now I just put in a minus sign.
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Fundamentally, we have no reason to think that this sign should be positive or negative.
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In this case, I still have my symmetry in the underlying theory. I can map phi to minus phi.
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The potential, which now looks like this is unchanged by this mapping.
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But if I look at the vacuum of the solution of the theory now the symmetry is spontaneously broken.
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Actually better I should say that the symmetry is hidden because it's still there is just not manifest in the same way.
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But everyone says I can follow this. I have to if I want to vacuum pick either this minimum or this minimum is just given by these values.
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And then when I apply, if I guessed minus phi, this vacuum will turn into this, not vacuum or vice versa.
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If I play the same game as before of starting off at high temperatures and lowering the temperature,
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this leads to much more interesting effects at high temperatures.
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Again, the field will explore only large parts of its potential, including both of the minima, so it's random.
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Has large derivatives, large gradients.
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But as I drop down to low temperatures, the field is going to have to pick one or other of the minima to go to.
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Now, locally, we expect the field to all go to the same minimum.
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If I'm sense, if I'm looking just at this small region in space, I can't have too large gradients.
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Otherwise that would cost too much energy.
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So I'm going to sit in this minimum, say, whereas if I'm looking over in this region of space, I might sit in this minimum.
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But there's no reason that the field over in this region, which is a long way away from this region, must be in the same minimum.
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Maybe it just happened to have a fluctuation in that direction over here, and the fluctuation over in that direction over there.
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So this leads to a much more interesting structure. And in fact,
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the structure that we end up with is in many ways similar to what would happen in
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this simple system where we have one spin degree of freedom at each point in space,
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which can either point up or down at high temperatures, the spins would be randomly up and down.
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But as I go to low temperatures, I end up with these kind of regions where the spin point up, these kind of regions where the spins will point down.
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These are called domains. Similarly, I'll call these regions of space where the scalar field has the sits in the same minimum domains.
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And between the different domains we have domain rules.
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So these domain wars, uh, interesting objects, they will actually be generalised to the strings that I'll talk about soon.
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If I imagine setting up my system so that I say sitting this minimum over here and space this minimum over here in space,
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then of course, the field cannot sharply jump from this minimum to this minimum.
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That would require an infinite gradient which would cost infinite energy.
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Instead, what I get is some sort of interpolating profile where the value of the scalar field as I go from here to here or here to here,
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takes this kind of form with the red region here corresponding to the domain.
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Well, there is an energy cost to this.
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Inevitably, if I have a domain wall, it costs potential energy because I have to have this region in space where I'm not sitting in one of the minima.
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It also cost gradients energy because I have to have the field varying in space.
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So you might say to me, well, this is not the vacuum solution, why would you care about it?
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And the reason I care is that actually, once I end up with this sort of configuration,
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it can be really hard for this to disappear into the true vacuum.
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If I imagine I have some finite size system like this, then actually the domain war would not be stable.
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There would be some small energy gain or energy decrease, I should say, in moving the domain will say slightly in this direction,
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which would slightly decrease the gradient sitting over here corresponding to this and the domain more could move to the edge of my fixed box,
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and at that point I would end up in the true vacuum.
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However, if the system is sufficiently large, if it's infinite, or if it's for all practical purposes,
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infinite, then there's no energy gain to moving the domain more one way or the other.
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So once I've got it there, I can't make it move to the edge of my system.
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You might also say maybe these quantum fluctuations that Joe talked about earlier could allow this domain wall to disappear.
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We heard that space time is actually quantum and is constantly fluctuating.
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So why doesn't this domain all just fluctuate away to nothing? And the answer is that for the domain wall to disappear in this way,
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I would need that the field in this entire region of space to simultaneously fluctuate over from this minimum into this minimum.
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This is exponentially unlikely to occur because fluctuations, quantum fluctuations or thermal fluctuations a small localised things.
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Whereas here I would need some coherent fluctuation to take me over the top of the barrier everywhere in space,
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or at least everywhere on this side of space. Okay, now turning to the early universe.
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Luckily, Joe gave a very nice introduction or summary of what we already know about the early universe.
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This is some sort of cartoon form. We live somewhere here where the temperature of the photons that make up the
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cosmic microwave background being about ten to the minus four electron volts.
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I'll always think in terms of natural units. I measure energies in terms of electron volts, or moves ten to the six electron volts,
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gives ten to the nine electron volts, and does a conversion into length scales.
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This is the rough comparison.
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Now, with the exception of inflation, which as we heard, leaves perturbations left over in the universe which we can detect and observe.
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Basically, all of the information that we have about the universe stems from this error here.
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And later of course, here. There's been lots of words and conjectures put for what might have happened earlier.
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But again, with the exception of inflation, this is all conjecture. Maybe just as a point of reference, the LHC energies correspond to about a TV.
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So that's sitting somewhere in this region. But of course,
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it's quite different physically to have two protons colliding with TV energies compared to the whole universe having the temperature of TV.
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So now let me suppose that I have my new scalar field,
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and let me assume that the scale of the scale of scalar associated the scalar field, which I'll call Fe,
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which is also the scale at which symmetry breaking happens, is somewhere in this region,
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say ten to the 16 GeV, maybe down to ten to the ten GV, something like that.
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Actually, again, there's good reasons to think that this may be the energy scale associated to new
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physics that could be associated to grand unified theories or similar things.
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But for now, let me just take this as a conjecture.
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Then provided the temperature of the universe did indeed reach values above alpha, and that inflation happened before this.
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As I evolve the universe forward in time, the universe gets cooler as we know, and we all go through this spontaneous,
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symmetry breaking process, and my configuration of the scalar field that I get out will look something like this.
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Like I had before with different domains in different patches of the universe.
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And actually my slightly fuzzy statement about regions of space being sufficiently
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far enough away from each other going into different vacua can now be made sharper.
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The largest distance over which any causality can occur over which signals to propagate,
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with the exception of inflation which happened earlier, is set by the Hubble parameter.
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So the Hubble distance is the inverse of the Hubble parameter, given by an expression of this form,
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where rho is the total energy density of the universe, which during radiation domination,
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which is everything from about here and earlier,
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earlier we think is given by this expression where t is the temperature of the universe and Planck is the Planck scale.
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That means that at these early times, the size of the Hubble horizon, which is the distance over which signals can propagate,
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over which information can be transferred, might be represented by this small Hubble patch.
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And then for an observer living in this region,
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there's no way that they could know anything about what has occurred outside their own Hubble horizon over here.
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So indeed, we have a sharp prediction that the spontaneous symmetry breaking should lead to the system
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being in different vacua in different parts of the universe that are causally disconnected.
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Now, let me run forward in time. The Hubble parameter decreases.
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The Hubble horizon grows. So that now is represented by this larger square.
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And at this stage you can see that for this region of space and for this domain surrounded by a domain wall,
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suddenly it will the system will realise if I can use very loose language,
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that actually this is a domain that is in some sense surrounded by a surrounded region of the opposite vacua.
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So then be tension on this domain wall which will cause it to shrink, disappear, so that subsequently this region will all end up in the same vacuum.
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On the other hand, there will still be domain walls separated by Hubble distances.
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For example, once all of this region turns blue. Still, there will be one domain over here and the different domain over here,
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which is separated by roughly the Hubble distance leading to a domain more that is actually roughly of Hubble size.
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And indeed, this pattern persists as time progresses.
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Domain moves get destroyed, but still successively more domain moves enter each other's horizons.
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What's this to do with strings? Well, actually, the physics of strings is very similar, but somewhat harder to visualise,
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or at least harder to draw on the presentation, which is why I started with domain walls.
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The only difference is that now I have my scalar field,
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and instead of it being a real scalar field with one real degree of freedom at each point in space,
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let me say that I have a complex scalar, so that each point in space I have a complex number.
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And let me assume that the potential associated to the scalar has this form where this is the absolute value of the complex number at each point.
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Now this potential looks something like this. Where in the vertical direction I have the value of the potential.
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With this direction, I have the real part of phi in the direction into the board.
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I have the imaginary part of phi. At early times, the scalar field can explore all of these minima.
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In fact, here we have a full circle.
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The minimum scalar field explains all of them, but as I go down in temperature, the scalar field will pick one of these particular minima to go to.
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Unlike the case of domain walls, there's no energy barrier between the different minima.
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There's a full continuous circle, but still I just end up with similar structures.
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The strings. So perhaps this is easiest to visualise first if I imagine a two dimensional space.
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So in this case, if I have my space being two dimensional, then I can have regions in space where as I do a loop in physical space,
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either on the board here or in the 2D plane of this picture.
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I also do a loop of the filled space. What does this mean?
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Well, it means that I end up with an object that, just like the domain more, cannot fluctuate away easily.
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I also cannot fluctuate fluctuate away this object easily.
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As I look further and further away from the centre of what will turn out to be the string.
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The field sits in this vacuum, but it is winding around 0 to 2 pi in the vacuum manifold, even infinitely far in space.
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This means that if I wanted to get rid of this string, I would need the fields all the way out here, way away from the string,
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to simultaneously fluctuate over the top of the potential such that I ended up in the same vacuum everywhere.
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If you like, there's some obstruction to this system relaxing down to the true vacuum,
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which would just consist of the scalar field being started a single point with no variation in time or space.
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It's sad, but okay. Now, of course, this situation can't persist arbitrarily close to the centre of the string, with the field sat in its vacuum.
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If I went infinitely close to the centre of the string, the gradients would become infinite.
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That would cost me infinite energy. I certainly don't have infinite energy in my system to start.
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And in fact, what happens is that when I get within a distance of one over half of the centre of the string,
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the scalar field gets forced back onto the top of its potential.
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So that characterises the really the string, like the string core or the centre of this string where the scalar field is forced.
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And if you want some more mathematical details, this is the typical form of the profile of a single string,
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where g is some function that varies between zero at the centre of the string and one at infinity,
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which corresponds to the field sitting in its vacuum manifold.
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One of the most important characteristics of these string like objects is their tension.
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By which I mean their energy per unit length. A straightforward calculation gives you that the tension, which is energy per unit length,
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is given by an expression roughly of this form pi times for skirt.
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Now this scale for can be extremely large. I was saying it could be something like ten to the 16 GV.
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That means that these strings carry an enormous amount of energy.
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They're extremely thin, but associated to the core of the string there's this huge energy density.
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Now let me look at the cosmological evolution. So a single long straight string is stable.
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There's no way of getting rid of it. Aside from these vanishingly rare properties, probability of having some fluctuation away.
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But in the early universe, what would actually happen is that first, I started off with some temperature that's very high.
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I would then call below the symmetry breaking phase transition.
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And I wouldn't just form one long straight string, I would form a complete network of strings.
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This is some picture from a numerical simulation. The strings would have complicated evolution.
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If I have small loops that are smaller than the Hubble horizon, these things can collapse.
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If I have two strings and a Hubble horizon, they can intersect, they can recombine, form new strings, and so on.
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But still there will be some evolution that continues.
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And in fact, just like the domain, walls persist with roughly one domain wall per Hubble patch surviving.
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Actually, in the case of strings, I also get roughly one string possible patch surviving.
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It might be, depending on the specific theory, that the strings are destroyed at some much later time.
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For example, in some axion theories, this happens when the temperature of the universe is about achieve.
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This is more model dependent. It could also be that the strings just survive indefinitely until the present day,
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so that we would still have strings out that with one in our observable universe.
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Of course, we don't just want to conjecture that these strings exist. We'd like to detect them and learn something.
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There's a few different ways that this can happen, but perhaps the most promising is to look for the dark matter.
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Look for dark matter that might be produced by the evolution of the string network.
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So during this evolution, strings are being destroyed. Some survived, but most are being destroyed.
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And in the process they release energy.
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In lots of sensible theories, this energy goes into dark matter and there's a huge energy density in the strings,
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which means actually I can produce the vast majority of my dark matter this way.
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Also because these strings have extremely high energies, potentially close to the Planck scale.
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They can source a sizeable amount of gravitational waves.
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These are ripples in space time that come about because the energy density of these objects are so high and they're fluctuating and moving.
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It could also be that if these strings survive, they can leave signals in the cosmic microwave background,
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which we could just look for directly by studying this background of photons extremely carefully.
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So actually, a sizeable part of my research these days is spent trying to understand the dynamics of these strings.
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Make predictions, for example, for the dark matter abundance that is produced.
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And you might think that this should be a straightforward task. I'm doing classical field theory, which is generally easier than quantum field theory.
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I've got a potential that is pretty straightforward,
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is the kind of potential that we just show our third year, fourth year undergraduates as the first example.
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So what is the problem? Why don't we just study the system directly?
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Well, really the issue is the combination of two factors. First of all, the dynamics of these strings are extremely nonlinear.
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They have complicated interactions. When the strings approach each other, the system is certainly far from any linear approximation.
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This means that analytic approaches will only get us so far,
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as anyone who works on basically anything involving partial differential equations will know.
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Nonlinear partial differential equations extremely complicated to solve.
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A natural other approach is just to do numerical simulations.
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Then the nonlinear behaviour is not a problem. But the real problem is that I have a huge scale separation in my problem.
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I'll say more about that in the next slide. But this in some sense is the real challenge.
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We haven't got good analytic approaches that really allow us as much precision as we want, and numerics are challenging.
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Let me say a little bit more about what I would do. What I do in simulations, in some sense we do the simplest possible thing.
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We take our complex scalar.
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We discretize again, we put it on some lattice, we get access to the biggest supercomputer that anyone will give us access to,
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and then we just solve the equations of motion of this complex scalar.
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If I start with sufficiently random initial conditions, the strings will automatically form.
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They will automatically evolve in the way that I've been describing, following the equations of motion.
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And I'll just be able to see what the system does. The problem comes about because in order to capture the string interactions,
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I need to make sure that I have at least a few lattice points per string core.
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Otherwise, if these two strings intersect each other, I'm not going to correctly resolve the dynamics of a system.
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So that's represented by this small circle.
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But at the same time, we know that the communication, the strings can communicate or interact over distances of order the Hubble distance.
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This is also the typical length of the strings that typical curvature.
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So my simulation I better make sure that I have at least a few Hubble patches within my grid, represented by this blue sky.
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Now the most we can do, even using the biggest computers we can get access to is evolve simulations, but something like 5000 cube grid points in them.
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This means that I'm limited to doing simulations where the ratio between the string thickness
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and the size of the Hubble parameter in cartoon form is less than the ratio of this big square,
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and the small circle is roughly less than, say, a thousand.
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The physical situation, meanwhile, is that this scale of separation is ten to the 30.
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And again, anyone who does simulations of any type will tell you that if I do simulations at this girl and I want to capture physics at this scale,
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things are not going to be safe. You certainly don't want to take at face value what you do in simulations here,
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and just blindly apply your results that you get out to the physical situation.
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I don't actually have any perfect solution to this. It's a challenge.
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What we do or what we attempt to do is do simulations as carefully as we possibly can in this regime,
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and try to make this huge extrapolation to the physical regime, at least being aware of the uncertainties.
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It's not perfect. I'll talk about some possible future improvements at the end.
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Unfortunately, physics is hard. This is the best we can do.
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Or at least the best we know how to do. Let me just briefly mention that there's one feature of the network, or the evolution of the strings,
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that gives us hope that such an extrapolation is possible and not completely out there.
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That's the existence of a so-called attractor solution.
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So regardless of the situation that I start my string network, often I actually get drawn into what is called an approximate scaling solution,
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where I have about one string length of one length of string, one Hubble length of string, Hubble patch.
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I can actually motivate this pretty quickly and in a kind of hand-wavy way.
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Imagine I start my string network off with way too much string. I have lots of strings per Hubble patch.
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These will very quickly interact with each other. They will annihilate away from loops that will shrink.
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And I'll get drawn down to this critical point where I have about the right amount of strings such that the right the strings are being destroyed,
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balances the rate at which the strings re-enter each other's Hubble horizons.
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On the other hand, if I have way too few strings per Hubble patch, basically nothing will happen.
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The system will just sit there, the strings won't be destroyed.
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The Hubble horizon will meanwhile grow, and I'll accumulate string within each Hubble horizon,
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which will again persist until I get to this so-called sort of critical point at which there's this balance between being stretched,
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strings being destroyed, and strings beginning to see each other across Hubble distances.
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If you like. In some sense, this is almost an instance of self-organized criticality.
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Okay, so that's the basic picture. We make use of this approximate scaling solution.
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We do careful extrapolations.
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You can argue that the energy emitted per Hubble time Hubble volume by the string network is given by an expression of this form.
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So we have this Fe which is the energy scale of the spontaneous symmetry breaking.
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It's this very large scale. And we have Hubble which enters because this is the typical length of the strings.
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And we try to make predictions for the dark matter abundance and for the amounts of that might be
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produced by these strings and the amount of gravitational waves that are produced through this evolution.
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Let me first talk a bit about the dark matter abundance. I'll start with an extremely simple formula.
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So the energy density in dark matter is given by the number density of dark matter times the mass of dark matter.
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Because the dark matter is non-relativistic today. Okay, this in some sense seems completely useless, but what is the point of it?
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Well, the energy density in dark matter is something that we can measure.
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We have no idea what the mass of dark matter is.
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It could vary anywhere between ten to the -20 electronvolts and say, ten to the ten GeV or even larger.
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This is many, many orders of magnitude. But still, with this combination of the total energy density, we can constrain, for example,
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by looking at the evolution of galaxies or the cosmic microwave background and similar things.
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Now, if we study the evolution of the string network in detail, the number density of dark matter is something that we can actually calculate.
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We can calculate how many dark matter particles are released by the string network.
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And if we've got this predicted or this calculated this observed,
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we can then make a prediction for the mass of dark matter and experimentalists like this.
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If I restrict myself just to axion dark matter, if you know what that is, that's fine.
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If you don't, that's fine as well. It's just some possible new scalar particle that might be dark matter.
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There's no reason to think that, well, aside from these type of calculations, the mass of the dark mass could be anywhere between ten to the -12.
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Even smaller, actually, somewhere up to this ten to the four electronvolts for this particular candidate.
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Experimentalists really don't like having to look over 20 orders of magnitude a mass for each of these different masses.
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They have to think of different experimental techniques to use here in the solid region.
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We've got existing experiments. These are some proposed new experiments, but still it's a major challenge.
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The benefit of our prediction is that we can highlight some particular range.
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In fact, for our latest calculations, we end up something like ten to the minus three electron volts,
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where if the scenario that I'm proposing is true,
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that stock match is produced by the string network, the dark matter mass should be given by this value.
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We can then go to experimentalists even in the basement of the Beecroft building.
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Tell them, please look in this mass range if they find something absolutely fantastic.
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If they don't find anything, then at least we have ruled out this entire scenario,
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teaching us something about the production, at least ruling out an entire class of production mechanisms for dark matter.
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Okay, so that's the. In some sense the aim for the dark matter abundance.
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The story for gravitational waves is pretty similar. Here I've plotted the sensitivity of proposed future experiments searching for dark matter.
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This is the energy that energy density and dark matter in the present day universe.
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This is the dark matter. Gravitational waves in the present day universe.
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This is the frequency of the gravitational waves.
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And these are different proposed observation techniques, as the iron is also being developed in the basement of the Beecroft building.
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If you get a chance to visit that area, it's very interesting.
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So what do our predictions look like? Well, these are our predictions.
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For what the gravitational wave spectrum coming from the string network should look like.
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Now there's a dependence on this scale for which is the energy density associated to the strings.
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If the strings have higher energy density, they're also thinner. They lead to more gravitational waves.
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And you can see that if all of these projections actually pan out and the
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experimentalists do incredible work and managed to reach these sensitivities,
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potentially we could be discovering string networks that arise from physics of energy scales of 10 to 14 GeV,
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possibly even as high as 10 to 15, ten to the 16 GV.
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This is almost absurdly high energy densities.
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And really, the reason that we actually have a chance of discovering or learning about this energy dense physics, these scales,
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is because the cosmic strings survive the evolution of the universe, because they're stable in the way that I described earlier.
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This is also, in some ways quite nicely complementary to collider searches.
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If you remember, collider searches are looking at energy scales of maybe TV,
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which is ten to the three GeV, possibly ten TV, maybe 100 TV in the future if we're really lucky.
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Whereas here we're looking at much, much higher energy scales. So in some sense we're attacking the problem from two directions.
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Okay. It's also actually quite nice that the shape of the spectrum looks like this.
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The fairly flat predictions.
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This means that if we do see this background of gravitational waves in, say, one experiment over here, we can then say, well,
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if this is being produced by cosmic strings in the early universe, we predict you should also see signal in these experiments over here.
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If they do, great.
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If they don't see anything, then we can be pretty sure that this is not cosmic strings that are producing the signal that we're seeing over here.
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The other really nice feature that I alluded to already is that these gravitational waves are produced in the very early universe.
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So now here on the top axis, I plotted the temperature of the universe when these string,
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when the gravitational waves that have the corresponding frequency are produced.
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Remember the limits of our current observations are somewhere around here.
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So actually, if we see a signal anywhere from here over to the right, at higher frequencies,
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we would be learning something or would be seeing a signal that is originating from the absurdly early universe again,
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way beyond anything that we have seen in any other way, with the possible exception of inflationary fluctuations.
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Perhaps in the best case scenario, if we see such a signal, we'd really be able to then study it in complete detail.
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Learn something about, again, physics that these kind of energy scales, which again, we would never directly be able to access.
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Okay, so let me just finish by talking a bit about work in progress.
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There's certainly, of course, a huge amount to be done on the experimental and observational side.
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The last two slides showed projections, but of course this takes incredible work for experimentalists to make these things actually happen.
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But there are still there's also things to be done from me, from my side on the theoretical and from other people's side on the theoretical side.
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The first thing is that we can improve our simulations.
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Some of you may well have been shouting in your head when I showed you this picture of a fixed lattice, especially if you are in numerical things.
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This is a very bad thing to do. We actually don't need resolution over all of our simulation.
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What we really care about simulating is the centre of the strings.
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So why don't we just put our lattice points at the centre of the strings when we need it.
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Then we can could access much bigger scale separations.
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It would make our huge extrapolation still bad, but less bad than it currently is.
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Indeed, this is something sensible to do. It's something that we, or in particular students and postdocs are working on at the moment.
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The real challenge here is that because we have such a huge extrapolation,
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any small systematic effects that we introduced by doing this meshing would extrapolate to huge problems and really change our predictions.
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So a lot of work is needed to really get reliable results.
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There's lots of other similar processes, actually in the early universe that could lead to gravitational wave signals.
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We could have phase transitions in which say that a similar scalar field starts at some false minimum,
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and then the true minimum by some sort of quantum tunnelling or thermal tunnelling.
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In this case, we would again get interesting to make. More structures which could collide, could produce gravitational waves.
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And lots of work is needed to really get accurate and reliable predictions for this system.
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Okay, I think that's pretty much all I want to say. So just to summarise, spontaneous symmetry breaking is already an interesting process.
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We have good reason to think that it might occur in the early universe, and if it does, then it often, not always,
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but in many cases that leads to these types of domain walls or strings, which collectively are called topological defects.
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The fantastic thing about these is that they persist from the early universe,
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so that we have access to the energy scales at which the spontaneous symmetry breaking happens.
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Okay, so I'm not entirely sure you can tell that.
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00:38:53,630 --> 00:39:04,300
This is better than this happening in an undergraduate lecture, I guess. As well as giving us access to the physics of extremely high energy scales.
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It also potentially allows us to learn something about the evolution of the universe at extremely early times,
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potentially with the proposed gravitational wave detectors,
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when the temperature of the universe was something like ten to the HGV, or modulo,
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with the usual definitions of times after the beginning of the universe.
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This is something like ten to the -22 seconds after the start of the universe,
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and there's a really extensive ongoing experimental and theoretical effort in this direction.
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Okay, so thank you.
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[Auto-generated transcript. Edits may have been applied for clarity.]
So if. Have to.
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Okay. So welcome back. So. So there been already a lot of questions about the early universe and how it began.
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So and Hardy is going to tell us about two possible windows into what might have happened in the early universe.
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And this is about cosmic strings and gravitational waves.
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So please read. Okay. Thanks very much for the introduction.
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So indeed, I'm going to describe cosmic strings and try to convince you that these, if they exist, which I hope they do,
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could give a fascinating window into the cosmological history of the universe, allowing us to learn new things.
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Beyond which you'll notice that Jerry has already described and potentially also learned about physics at extremely high energy scales,
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far beyond any that we could ever hope to probe into particle collider.
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Maybe I can also note that actually, there's a nice complementarity between what I'll say and what we'll be talked about
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in context of condensed matter in the next Saturday morning of theoretical physics.
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So the kind of physics I describe appears in many different places. Okay.
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Just to make sure we're all on the same page. Let me start by reviewing classical field theory.
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If I imagine that I've got some sort of mattress like structure with some masses connected by springs,
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then I can imagine that associated to each mass.
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There's some degrees of freedom in my theory, maybe just the position of the mass, maybe some other degrees of freedom.
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And there will also be interactions between the different masses owing to these springs.
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Now, if I take the distance between the different masses to be extremely small,
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or if I look at physics on extremely large distance scales compared to the spacing between the masses,
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then I can effectively describe my system as some continuous system,
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some continuous field where at each point in space I have some degrees of freedom.
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Actually, it's not at all obvious that this works.
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This is the magic of effective field theory that actually I can do this continuum description, but it turns out to be true that I can do this.
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Probably the most familiar example of a field theory that you will have all seen in great detail comes from electromagnetism.
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In this case, at each point in space I have some four vector, the electromagnetic four vector that I have called a hair,
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and this gives rise to the usual electric and magnetic fields that are very familiar in this kind of way.
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Actually, in this talk I'm going to focus on a simpler type of field theory,
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just a scalar field theory where instead of just having instead of having a full four vector worth of degrees of freedom at each point in space,
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I'm just going to have a single number. This might be a real number, in which case I'll talk about a real scalar field.
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It might be a complex number, in which case it's a complex scalar field. Unfortunately, due to time,
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I don't have the opportunity to explain why we believe that new scalar fields
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probably exist in the universe and probably exist at very high energy scales.
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Again, there was actually a very nice morning of theoretical physics about axioms that would
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be one of the canonical examples of new scalar fields that we believe may well exist.
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So that would be provide context. Of course, I'm happy to talk about this afterwards, but for the rest of the talk,
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just please take as a belief that we may well have these new scalar fields around us in the universe.
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I'm also going to work in the classical limits.
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Of course, as we have already heard this morning, we believe that the universe is fundamentally quantum,
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so that each degree of freedom here in my mattress should really be some quantum harmonic oscillator with corrections.
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And then as I take the continuum limit, I should have a degree, a quantum degree of freedom at each point in space.
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But as we know from electromagnetism, there are many scenarios in which it is useful to talk about a classic electric and magnetic field.
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We don't need to worry about the details of the individual photons that make up the field.
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And likewise, everything that I talk about today will have sufficiently large occupation numbers.
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If I can be a bit technical that we can treat the system as a classical field, which makes the physics much easier to analyse.
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Okay, that is the set up. But of course we want to talk about dynamics, about what happens with time.
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And again. Let me go back to a very simple system just to give an introduction.
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So if I think about just an individual one particle here represented by this.
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I saw this ball moving in some potential, which I'll call you.
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Of course, we all know that the total energy of the system is some kinetic energy plus the potential energy,
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and depending on how much energy the particle has, of course the system can have.
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The behaviour of the system can be different. If I have, say, a large amount of energy in the particle, then the particle can explore the full system.
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It can escape off to infinity. It's unbounded. This is for this particular example potential I've just drawn slightly randomly.
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If I have somewhat less energy than the particle can explore all of this region, including both of these minima.
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But if I go down to lower energy, then the particle will inevitably be trapped and either this minimum or this minimum.
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And just to reiterate, there's one degree of freedom in the system as I've drawn it, just the position of the particle.
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Let me go to the scalar field now. And just like each degree of freedom, the degree of freedom on the left hand side was moving in a potential.
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Now each degree of freedom, which is the value of the field at each point in space, will also be associated to a potential here.
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Because I have a continuous system, I talk about energy density rho rather than the total energy.
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Well, it's more useful to talk about the energy density, right?
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Rather than the total energy capital E, which is just the integral over space of the energy density.
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And the energy density takes a pretty similar form to on this side, we have the potential energy associated to the value of the field at each point.
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We also have some time derivatives of the fields.
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This is not as analogous to the kinetic energy of the particle, and we also have some spatial derivatives.
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Contribution to energy density. These, if you like, are associated to in the mattress description.
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The energy stored in the string in the springs between the different masses.
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Okay stream. You know them, but it will make it do with it.
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Let me start with a very simple potential. So just potential of this form here as a is some new energy scale which will actually be associated
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to the energy scale of the new physics or the scalar field that might arise in my theory,
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some coupling constant. And of course we all know what this potential energy looks like.
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Just looks like this. What is noteworthy is that this potential of this theory has a symmetry in particular is symmetric.
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Under the mapping, phi goes to minus phi. Of course the potential remains unchanged by this.
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And moreover, not not only is this a symmetry of the theory,
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but also it's a symmetry of the vacuum solution, by which I mean the lowest energy solution.
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Of course, just looking at the total energy density we have, that the energy density is minimised.
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When I minimise you and when I have no time or space derivatives in the field.
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So. As a result, if I imagine myself as an external observer to the system with some tile work, I can adjust the energy in the system,
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which equivalently, is adjusting the system from being at high temperature to being at low temperature.
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Then I get the following effects or dynamics.
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If I start off at high energy, high temperature way above this energy scale for the scalar field can explore all of its potential,
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a large part of its potential. Correspondingly, here the colours represent the different values of the scalar field.
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There's also enough energy that I can have sizeable derivatives both in time and space.
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So the field probably looks like this. Then, as I turn my dial and the energy density of the field goes down, well, the derivatives become smaller,
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but also the the value of the field at each point is in some sense channelled down to this one unique minimum.
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So what I get out at these low temperatures is really quite a boring system.
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Is just a scalar field. That's right. At its minimum, no derivatives and time and space.
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Much more interesting is if I make only a very minor change to my potential.
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Whereas before I had a plus sign here, now I just put in a minus sign.
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Fundamentally, we have no reason to think that this sign should be positive or negative.
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In this case, I still have my symmetry in the underlying theory. I can map phi to minus phi.
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The potential, which now looks like this is unchanged by this mapping.
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But if I look at the vacuum of the solution of the theory now the symmetry is spontaneously broken.
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Actually better I should say that the symmetry is hidden because it's still there is just not manifest in the same way.
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But everyone says I can follow this. I have to if I want to vacuum pick either this minimum or this minimum is just given by these values.
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And then when I apply, if I guessed minus phi, this vacuum will turn into this, not vacuum or vice versa.
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If I play the same game as before of starting off at high temperatures and lowering the temperature,
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this leads to much more interesting effects at high temperatures.
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Again, the field will explore only large parts of its potential, including both of the minima, so it's random.
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Has large derivatives, large gradients.
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But as I drop down to low temperatures, the field is going to have to pick one or other of the minima to go to.
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Now, locally, we expect the field to all go to the same minimum.
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If I'm sense, if I'm looking just at this small region in space, I can't have too large gradients.
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Otherwise that would cost too much energy.
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So I'm going to sit in this minimum, say, whereas if I'm looking over in this region of space, I might sit in this minimum.
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But there's no reason that the field over in this region, which is a long way away from this region, must be in the same minimum.
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Maybe it just happened to have a fluctuation in that direction over here, and the fluctuation over in that direction over there.
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So this leads to a much more interesting structure. And in fact,
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the structure that we end up with is in many ways similar to what would happen in
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this simple system where we have one spin degree of freedom at each point in space,
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which can either point up or down at high temperatures, the spins would be randomly up and down.
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But as I go to low temperatures, I end up with these kind of regions where the spin point up, these kind of regions where the spins will point down.
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These are called domains. Similarly, I'll call these regions of space where the scalar field has the sits in the same minimum domains.
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And between the different domains we have domain rules.
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So these domain wars, uh, interesting objects, they will actually be generalised to the strings that I'll talk about soon.
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If I imagine setting up my system so that I say sitting this minimum over here and space this minimum over here in space,
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then of course, the field cannot sharply jump from this minimum to this minimum.
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That would require an infinite gradient which would cost infinite energy.
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Instead, what I get is some sort of interpolating profile where the value of the scalar field as I go from here to here or here to here,
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takes this kind of form with the red region here corresponding to the domain.
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Well, there is an energy cost to this.
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Inevitably, if I have a domain wall, it costs potential energy because I have to have this region in space where I'm not sitting in one of the minima.
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It also cost gradients energy because I have to have the field varying in space.
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So you might say to me, well, this is not the vacuum solution, why would you care about it?
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And the reason I care is that actually, once I end up with this sort of configuration,
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it can be really hard for this to disappear into the true vacuum.
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If I imagine I have some finite size system like this, then actually the domain war would not be stable.
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There would be some small energy gain or energy decrease, I should say, in moving the domain will say slightly in this direction,
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which would slightly decrease the gradient sitting over here corresponding to this and the domain more could move to the edge of my fixed box,
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and at that point I would end up in the true vacuum.
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However, if the system is sufficiently large, if it's infinite, or if it's for all practical purposes,
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infinite, then there's no energy gain to moving the domain more one way or the other.
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So once I've got it there, I can't make it move to the edge of my system.
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You might also say maybe these quantum fluctuations that Joe talked about earlier could allow this domain wall to disappear.
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We heard that space time is actually quantum and is constantly fluctuating.
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So why doesn't this domain all just fluctuate away to nothing? And the answer is that for the domain wall to disappear in this way,
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I would need that the field in this entire region of space to simultaneously fluctuate over from this minimum into this minimum.
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This is exponentially unlikely to occur because fluctuations, quantum fluctuations or thermal fluctuations a small localised things.
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Whereas here I would need some coherent fluctuation to take me over the top of the barrier everywhere in space,
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or at least everywhere on this side of space. Okay, now turning to the early universe.
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Luckily, Joe gave a very nice introduction or summary of what we already know about the early universe.
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This is some sort of cartoon form. We live somewhere here where the temperature of the photons that make up the
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cosmic microwave background being about ten to the minus four electron volts.
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I'll always think in terms of natural units. I measure energies in terms of electron volts, or moves ten to the six electron volts,
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gives ten to the nine electron volts, and does a conversion into length scales.
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This is the rough comparison.
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Now, with the exception of inflation, which as we heard, leaves perturbations left over in the universe which we can detect and observe.
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Basically, all of the information that we have about the universe stems from this error here.
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And later of course, here. There's been lots of words and conjectures put for what might have happened earlier.
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But again, with the exception of inflation, this is all conjecture. Maybe just as a point of reference, the LHC energies correspond to about a TV.
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So that's sitting somewhere in this region. But of course,
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it's quite different physically to have two protons colliding with TV energies compared to the whole universe having the temperature of TV.
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So now let me suppose that I have my new scalar field,
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and let me assume that the scale of the scale of scalar associated the scalar field, which I'll call Fe,
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which is also the scale at which symmetry breaking happens, is somewhere in this region,
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say ten to the 16 GeV, maybe down to ten to the ten GV, something like that.
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Actually, again, there's good reasons to think that this may be the energy scale associated to new
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physics that could be associated to grand unified theories or similar things.
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But for now, let me just take this as a conjecture.
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Then provided the temperature of the universe did indeed reach values above alpha, and that inflation happened before this.
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As I evolve the universe forward in time, the universe gets cooler as we know, and we all go through this spontaneous,
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symmetry breaking process, and my configuration of the scalar field that I get out will look something like this.
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Like I had before with different domains in different patches of the universe.
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And actually my slightly fuzzy statement about regions of space being sufficiently
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far enough away from each other going into different vacua can now be made sharper.
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The largest distance over which any causality can occur over which signals to propagate,
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with the exception of inflation which happened earlier, is set by the Hubble parameter.
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So the Hubble distance is the inverse of the Hubble parameter, given by an expression of this form,
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where rho is the total energy density of the universe, which during radiation domination,
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which is everything from about here and earlier,
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earlier we think is given by this expression where t is the temperature of the universe and Planck is the Planck scale.
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That means that at these early times, the size of the Hubble horizon, which is the distance over which signals can propagate,
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over which information can be transferred, might be represented by this small Hubble patch.
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And then for an observer living in this region,
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there's no way that they could know anything about what has occurred outside their own Hubble horizon over here.
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So indeed, we have a sharp prediction that the spontaneous symmetry breaking should lead to the system
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being in different vacua in different parts of the universe that are causally disconnected.
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Now, let me run forward in time. The Hubble parameter decreases.
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The Hubble horizon grows. So that now is represented by this larger square.
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And at this stage you can see that for this region of space and for this domain surrounded by a domain wall,
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suddenly it will the system will realise if I can use very loose language,
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that actually this is a domain that is in some sense surrounded by a surrounded region of the opposite vacua.
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So then be tension on this domain wall which will cause it to shrink, disappear, so that subsequently this region will all end up in the same vacuum.
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On the other hand, there will still be domain walls separated by Hubble distances.
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For example, once all of this region turns blue. Still, there will be one domain over here and the different domain over here,
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which is separated by roughly the Hubble distance leading to a domain more that is actually roughly of Hubble size.
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And indeed, this pattern persists as time progresses.
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Domain moves get destroyed, but still successively more domain moves enter each other's horizons.
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What's this to do with strings? Well, actually, the physics of strings is very similar, but somewhat harder to visualise,
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or at least harder to draw on the presentation, which is why I started with domain walls.
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The only difference is that now I have my scalar field,
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and instead of it being a real scalar field with one real degree of freedom at each point in space,
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let me say that I have a complex scalar, so that each point in space I have a complex number.
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And let me assume that the potential associated to the scalar has this form where this is the absolute value of the complex number at each point.
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Now this potential looks something like this. Where in the vertical direction I have the value of the potential.
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With this direction, I have the real part of phi in the direction into the board.
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I have the imaginary part of phi. At early times, the scalar field can explore all of these minima.
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In fact, here we have a full circle.
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The minimum scalar field explains all of them, but as I go down in temperature, the scalar field will pick one of these particular minima to go to.
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Unlike the case of domain walls, there's no energy barrier between the different minima.
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There's a full continuous circle, but still I just end up with similar structures.
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The strings. So perhaps this is easiest to visualise first if I imagine a two dimensional space.
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So in this case, if I have my space being two dimensional, then I can have regions in space where as I do a loop in physical space,
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either on the board here or in the 2D plane of this picture.
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I also do a loop of the filled space. What does this mean?
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Well, it means that I end up with an object that, just like the domain more, cannot fluctuate away easily.
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I also cannot fluctuate fluctuate away this object easily.
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As I look further and further away from the centre of what will turn out to be the string.
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The field sits in this vacuum, but it is winding around 0 to 2 pi in the vacuum manifold, even infinitely far in space.
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This means that if I wanted to get rid of this string, I would need the fields all the way out here, way away from the string,
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to simultaneously fluctuate over the top of the potential such that I ended up in the same vacuum everywhere.
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If you like, there's some obstruction to this system relaxing down to the true vacuum,
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which would just consist of the scalar field being started a single point with no variation in time or space.
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It's sad, but okay. Now, of course, this situation can't persist arbitrarily close to the centre of the string, with the field sat in its vacuum.
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If I went infinitely close to the centre of the string, the gradients would become infinite.
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That would cost me infinite energy. I certainly don't have infinite energy in my system to start.
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And in fact, what happens is that when I get within a distance of one over half of the centre of the string,
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the scalar field gets forced back onto the top of its potential.
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So that characterises the really the string, like the string core or the centre of this string where the scalar field is forced.
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And if you want some more mathematical details, this is the typical form of the profile of a single string,
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where g is some function that varies between zero at the centre of the string and one at infinity,
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which corresponds to the field sitting in its vacuum manifold.
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One of the most important characteristics of these string like objects is their tension.
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By which I mean their energy per unit length. A straightforward calculation gives you that the tension, which is energy per unit length,
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is given by an expression roughly of this form pi times for skirt.
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Now this scale for can be extremely large. I was saying it could be something like ten to the 16 GV.
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That means that these strings carry an enormous amount of energy.
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They're extremely thin, but associated to the core of the string there's this huge energy density.
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Now let me look at the cosmological evolution. So a single long straight string is stable.
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There's no way of getting rid of it. Aside from these vanishingly rare properties, probability of having some fluctuation away.
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But in the early universe, what would actually happen is that first, I started off with some temperature that's very high.
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I would then call below the symmetry breaking phase transition.
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And I wouldn't just form one long straight string, I would form a complete network of strings.
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This is some picture from a numerical simulation. The strings would have complicated evolution.
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If I have small loops that are smaller than the Hubble horizon, these things can collapse.
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If I have two strings and a Hubble horizon, they can intersect, they can recombine, form new strings, and so on.
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But still there will be some evolution that continues.
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And in fact, just like the domain, walls persist with roughly one domain wall per Hubble patch surviving.
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Actually, in the case of strings, I also get roughly one string possible patch surviving.
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It might be, depending on the specific theory, that the strings are destroyed at some much later time.
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For example, in some axion theories, this happens when the temperature of the universe is about achieve.
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This is more model dependent. It could also be that the strings just survive indefinitely until the present day,
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so that we would still have strings out that with one in our observable universe.
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Of course, we don't just want to conjecture that these strings exist. We'd like to detect them and learn something.
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There's a few different ways that this can happen, but perhaps the most promising is to look for the dark matter.
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Look for dark matter that might be produced by the evolution of the string network.
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So during this evolution, strings are being destroyed. Some survived, but most are being destroyed.
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And in the process they release energy.
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In lots of sensible theories, this energy goes into dark matter and there's a huge energy density in the strings,
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which means actually I can produce the vast majority of my dark matter this way.
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Also because these strings have extremely high energies, potentially close to the Planck scale.
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They can source a sizeable amount of gravitational waves.
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These are ripples in space time that come about because the energy density of these objects are so high and they're fluctuating and moving.
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It could also be that if these strings survive, they can leave signals in the cosmic microwave background,
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which we could just look for directly by studying this background of photons extremely carefully.
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So actually, a sizeable part of my research these days is spent trying to understand the dynamics of these strings.
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Make predictions, for example, for the dark matter abundance that is produced.
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And you might think that this should be a straightforward task. I'm doing classical field theory, which is generally easier than quantum field theory.
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I've got a potential that is pretty straightforward,
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is the kind of potential that we just show our third year, fourth year undergraduates as the first example.
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So what is the problem? Why don't we just study the system directly?
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Well, really the issue is the combination of two factors. First of all, the dynamics of these strings are extremely nonlinear.
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They have complicated interactions. When the strings approach each other, the system is certainly far from any linear approximation.
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This means that analytic approaches will only get us so far,
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as anyone who works on basically anything involving partial differential equations will know.
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Nonlinear partial differential equations extremely complicated to solve.
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A natural other approach is just to do numerical simulations.
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Then the nonlinear behaviour is not a problem. But the real problem is that I have a huge scale separation in my problem.
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I'll say more about that in the next slide. But this in some sense is the real challenge.
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We haven't got good analytic approaches that really allow us as much precision as we want, and numerics are challenging.
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Let me say a little bit more about what I would do. What I do in simulations, in some sense we do the simplest possible thing.
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We take our complex scalar.
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We discretize again, we put it on some lattice, we get access to the biggest supercomputer that anyone will give us access to,
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and then we just solve the equations of motion of this complex scalar.
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If I start with sufficiently random initial conditions, the strings will automatically form.
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They will automatically evolve in the way that I've been describing, following the equations of motion.
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And I'll just be able to see what the system does. The problem comes about because in order to capture the string interactions,
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I need to make sure that I have at least a few lattice points per string core.
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Otherwise, if these two strings intersect each other, I'm not going to correctly resolve the dynamics of a system.
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So that's represented by this small circle.
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But at the same time, we know that the communication, the strings can communicate or interact over distances of order the Hubble distance.
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This is also the typical length of the strings that typical curvature.
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So my simulation I better make sure that I have at least a few Hubble patches within my grid, represented by this blue sky.
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Now the most we can do, even using the biggest computers we can get access to is evolve simulations, but something like 5000 cube grid points in them.
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This means that I'm limited to doing simulations where the ratio between the string thickness
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and the size of the Hubble parameter in cartoon form is less than the ratio of this big square,
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and the small circle is roughly less than, say, a thousand.
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The physical situation, meanwhile, is that this scale of separation is ten to the 30.
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And again, anyone who does simulations of any type will tell you that if I do simulations at this girl and I want to capture physics at this scale,
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things are not going to be safe. You certainly don't want to take at face value what you do in simulations here,
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and just blindly apply your results that you get out to the physical situation.
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I don't actually have any perfect solution to this. It's a challenge.
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What we do or what we attempt to do is do simulations as carefully as we possibly can in this regime,
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and try to make this huge extrapolation to the physical regime, at least being aware of the uncertainties.
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It's not perfect. I'll talk about some possible future improvements at the end.
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Unfortunately, physics is hard. This is the best we can do.
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Or at least the best we know how to do. Let me just briefly mention that there's one feature of the network, or the evolution of the strings,
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that gives us hope that such an extrapolation is possible and not completely out there.
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That's the existence of a so-called attractor solution.
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So regardless of the situation that I start my string network, often I actually get drawn into what is called an approximate scaling solution,
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where I have about one string length of one length of string, one Hubble length of string, Hubble patch.
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I can actually motivate this pretty quickly and in a kind of hand-wavy way.
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Imagine I start my string network off with way too much string. I have lots of strings per Hubble patch.
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These will very quickly interact with each other. They will annihilate away from loops that will shrink.
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And I'll get drawn down to this critical point where I have about the right amount of strings such that the right the strings are being destroyed,
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balances the rate at which the strings re-enter each other's Hubble horizons.
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On the other hand, if I have way too few strings per Hubble patch, basically nothing will happen.
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The system will just sit there, the strings won't be destroyed.
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The Hubble horizon will meanwhile grow, and I'll accumulate string within each Hubble horizon,
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which will again persist until I get to this so-called sort of critical point at which there's this balance between being stretched,
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strings being destroyed, and strings beginning to see each other across Hubble distances.
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If you like. In some sense, this is almost an instance of self-organized criticality.
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Okay, so that's the basic picture. We make use of this approximate scaling solution.
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We do careful extrapolations.
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You can argue that the energy emitted per Hubble time Hubble volume by the string network is given by an expression of this form.
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So we have this Fe which is the energy scale of the spontaneous symmetry breaking.
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It's this very large scale. And we have Hubble which enters because this is the typical length of the strings.
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And we try to make predictions for the dark matter abundance and for the amounts of that might be
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produced by these strings and the amount of gravitational waves that are produced through this evolution.
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Let me first talk a bit about the dark matter abundance. I'll start with an extremely simple formula.
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So the energy density in dark matter is given by the number density of dark matter times the mass of dark matter.
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Because the dark matter is non-relativistic today. Okay, this in some sense seems completely useless, but what is the point of it?
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Well, the energy density in dark matter is something that we can measure.
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We have no idea what the mass of dark matter is.
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It could vary anywhere between ten to the -20 electronvolts and say, ten to the ten GeV or even larger.
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This is many, many orders of magnitude. But still, with this combination of the total energy density, we can constrain, for example,
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by looking at the evolution of galaxies or the cosmic microwave background and similar things.
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Now, if we study the evolution of the string network in detail, the number density of dark matter is something that we can actually calculate.
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We can calculate how many dark matter particles are released by the string network.
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And if we've got this predicted or this calculated this observed,
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we can then make a prediction for the mass of dark matter and experimentalists like this.
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If I restrict myself just to axion dark matter, if you know what that is, that's fine.
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If you don't, that's fine as well. It's just some possible new scalar particle that might be dark matter.
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There's no reason to think that, well, aside from these type of calculations, the mass of the dark mass could be anywhere between ten to the -12.
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Even smaller, actually, somewhere up to this ten to the four electronvolts for this particular candidate.
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Experimentalists really don't like having to look over 20 orders of magnitude a mass for each of these different masses.
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They have to think of different experimental techniques to use here in the solid region.
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We've got existing experiments. These are some proposed new experiments, but still it's a major challenge.
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The benefit of our prediction is that we can highlight some particular range.
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In fact, for our latest calculations, we end up something like ten to the minus three electron volts,
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where if the scenario that I'm proposing is true,
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that stock match is produced by the string network, the dark matter mass should be given by this value.
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We can then go to experimentalists even in the basement of the Beecroft building.
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Tell them, please look in this mass range if they find something absolutely fantastic.
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If they don't find anything, then at least we have ruled out this entire scenario,
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teaching us something about the production, at least ruling out an entire class of production mechanisms for dark matter.
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Okay, so that's the. In some sense the aim for the dark matter abundance.
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The story for gravitational waves is pretty similar. Here I've plotted the sensitivity of proposed future experiments searching for dark matter.
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This is the energy that energy density and dark matter in the present day universe.
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This is the dark matter. Gravitational waves in the present day universe.
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This is the frequency of the gravitational waves.
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And these are different proposed observation techniques, as the iron is also being developed in the basement of the Beecroft building.
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If you get a chance to visit that area, it's very interesting.
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So what do our predictions look like? Well, these are our predictions.
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For what the gravitational wave spectrum coming from the string network should look like.
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Now there's a dependence on this scale for which is the energy density associated to the strings.
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If the strings have higher energy density, they're also thinner. They lead to more gravitational waves.
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And you can see that if all of these projections actually pan out and the
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experimentalists do incredible work and managed to reach these sensitivities,
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potentially we could be discovering string networks that arise from physics of energy scales of 10 to 14 GeV,
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possibly even as high as 10 to 15, ten to the 16 GV.
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This is almost absurdly high energy densities.
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And really, the reason that we actually have a chance of discovering or learning about this energy dense physics, these scales,
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is because the cosmic strings survive the evolution of the universe, because they're stable in the way that I described earlier.
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This is also, in some ways quite nicely complementary to collider searches.
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If you remember, collider searches are looking at energy scales of maybe TV,
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which is ten to the three GeV, possibly ten TV, maybe 100 TV in the future if we're really lucky.
366
00:34:47,300 --> 00:34:54,020
Whereas here we're looking at much, much higher energy scales. So in some sense we're attacking the problem from two directions.
367
00:34:54,440 --> 00:34:58,730
Okay. It's also actually quite nice that the shape of the spectrum looks like this.
368
00:34:59,060 --> 00:35:01,070
The fairly flat predictions.
369
00:35:01,490 --> 00:35:08,240
This means that if we do see this background of gravitational waves in, say, one experiment over here, we can then say, well,
370
00:35:08,240 --> 00:35:13,880
if this is being produced by cosmic strings in the early universe, we predict you should also see signal in these experiments over here.
371
00:35:14,300 --> 00:35:15,290
If they do, great.
372
00:35:15,590 --> 00:35:22,250
If they don't see anything, then we can be pretty sure that this is not cosmic strings that are producing the signal that we're seeing over here.
373
00:35:24,490 --> 00:35:30,280
The other really nice feature that I alluded to already is that these gravitational waves are produced in the very early universe.
374
00:35:31,210 --> 00:35:38,590
So now here on the top axis, I plotted the temperature of the universe when these string,
375
00:35:38,600 --> 00:35:42,160
when the gravitational waves that have the corresponding frequency are produced.
376
00:35:43,030 --> 00:35:47,710
Remember the limits of our current observations are somewhere around here.
377
00:35:48,250 --> 00:35:52,840
So actually, if we see a signal anywhere from here over to the right, at higher frequencies,
378
00:35:53,320 --> 00:35:59,770
we would be learning something or would be seeing a signal that is originating from the absurdly early universe again,
379
00:35:59,770 --> 00:36:04,570
way beyond anything that we have seen in any other way, with the possible exception of inflationary fluctuations.
380
00:36:07,410 --> 00:36:12,720
Perhaps in the best case scenario, if we see such a signal, we'd really be able to then study it in complete detail.
381
00:36:12,960 --> 00:36:19,260
Learn something about, again, physics that these kind of energy scales, which again, we would never directly be able to access.
382
00:36:21,230 --> 00:36:25,010
Okay, so let me just finish by talking a bit about work in progress.
383
00:36:25,340 --> 00:36:29,810
There's certainly, of course, a huge amount to be done on the experimental and observational side.
384
00:36:30,110 --> 00:36:36,920
The last two slides showed projections, but of course this takes incredible work for experimentalists to make these things actually happen.
385
00:36:37,730 --> 00:36:44,480
But there are still there's also things to be done from me, from my side on the theoretical and from other people's side on the theoretical side.
386
00:36:45,590 --> 00:36:48,440
The first thing is that we can improve our simulations.
387
00:36:49,310 --> 00:36:55,310
Some of you may well have been shouting in your head when I showed you this picture of a fixed lattice, especially if you are in numerical things.
388
00:36:55,610 --> 00:37:00,620
This is a very bad thing to do. We actually don't need resolution over all of our simulation.
389
00:37:00,800 --> 00:37:04,010
What we really care about simulating is the centre of the strings.
390
00:37:04,190 --> 00:37:07,700
So why don't we just put our lattice points at the centre of the strings when we need it.
391
00:37:08,090 --> 00:37:11,150
Then we can could access much bigger scale separations.
392
00:37:11,420 --> 00:37:15,740
It would make our huge extrapolation still bad, but less bad than it currently is.
393
00:37:16,730 --> 00:37:23,270
Indeed, this is something sensible to do. It's something that we, or in particular students and postdocs are working on at the moment.
394
00:37:24,170 --> 00:37:28,639
The real challenge here is that because we have such a huge extrapolation,
395
00:37:28,640 --> 00:37:36,560
any small systematic effects that we introduced by doing this meshing would extrapolate to huge problems and really change our predictions.
396
00:37:36,890 --> 00:37:40,220
So a lot of work is needed to really get reliable results.
397
00:37:41,360 --> 00:37:46,070
There's lots of other similar processes, actually in the early universe that could lead to gravitational wave signals.
398
00:37:46,520 --> 00:37:54,260
We could have phase transitions in which say that a similar scalar field starts at some false minimum,
399
00:37:54,860 --> 00:37:59,390
and then the true minimum by some sort of quantum tunnelling or thermal tunnelling.
400
00:38:00,260 --> 00:38:06,020
In this case, we would again get interesting to make. More structures which could collide, could produce gravitational waves.
401
00:38:06,290 --> 00:38:10,280
And lots of work is needed to really get accurate and reliable predictions for this system.
402
00:38:12,170 --> 00:38:21,250
Okay, I think that's pretty much all I want to say. So just to summarise, spontaneous symmetry breaking is already an interesting process.
403
00:38:21,350 --> 00:38:26,540
We have good reason to think that it might occur in the early universe, and if it does, then it often, not always,
404
00:38:26,540 --> 00:38:33,380
but in many cases that leads to these types of domain walls or strings, which collectively are called topological defects.
405
00:38:34,340 --> 00:38:37,370
The fantastic thing about these is that they persist from the early universe,
406
00:38:37,490 --> 00:38:42,080
so that we have access to the energy scales at which the spontaneous symmetry breaking happens.
407
00:38:44,000 --> 00:38:48,760
Okay, so I'm not entirely sure you can tell that.
408
00:38:53,630 --> 00:39:04,300
This is better than this happening in an undergraduate lecture, I guess. As well as giving us access to the physics of extremely high energy scales.
409
00:39:04,750 --> 00:39:10,330
It also potentially allows us to learn something about the evolution of the universe at extremely early times,
410
00:39:10,960 --> 00:39:14,020
potentially with the proposed gravitational wave detectors,
411
00:39:14,680 --> 00:39:19,510
when the temperature of the universe was something like ten to the HGV, or modulo,
412
00:39:19,820 --> 00:39:23,950
with the usual definitions of times after the beginning of the universe.
413
00:39:23,950 --> 00:39:28,120
This is something like ten to the -22 seconds after the start of the universe,
414
00:39:28,720 --> 00:39:33,040
and there's a really extensive ongoing experimental and theoretical effort in this direction.
415
00:39:34,450 --> 00:39:35,410
Okay, so thank you.
