Cosmic strings and gravitational waves from the early Universe

[Auto-generated transcript. Edits may have been applied for clarity.] So if. Have to. Okay. So welcome back. So. So there been already a lot of questions about the early universe and how it began. So and Hardy is going to tell us about two possible windows into what might have happened in the early universe. And this is about cosmic strings and gravitational waves. So please read. Okay. Thanks very much for the introduction.

So indeed, I'm going to describe cosmic strings and try to convince you that these, if they exist, which I hope they do, could give a fascinating window into the cosmological history of the universe, allowing us to learn new things. Beyond which you'll notice that Jerry has already described and potentially also learned about physics at extremely high energy scales, far beyond any that we could ever hope to probe into particle collider.

Maybe I can also note that actually, there's a nice complementarity between what I'll say and what we'll be talked about in context of condensed matter in the next Saturday morning of theoretical physics. So the kind of physics I describe appears in many different places. Okay.

Just to make sure we're all on the same page. Let me start by reviewing classical field theory. If I imagine that I've got some sort of mattress like structure with some masses connected by springs, then I can imagine that associated to each mass. There's some degrees of freedom in my theory, maybe just the position of the mass, maybe some other degrees of freedom. And there will also be interactions between the different masses owing to these springs.

Now, if I take the distance between the different masses to be extremely small, or if I look at physics on extremely large distance scales compared to the spacing between the masses, then I can effectively describe my system as some continuous system, some continuous field where at each point in space I have some degrees of freedom. Actually, it's not at all obvious that this works.

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. Probably the most familiar example of a field theory that you will have all seen in great detail comes from electromagnetism. In this case, at each point in space I have some four vector, the electromagnetic four vector that I have called a hair, and this gives rise to the usual electric and magnetic fields that are very familiar in this kind of way.

Actually, in this talk I'm going to focus on a simpler type of field theory, 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, 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.

It might be a complex number, in which case it's a complex scalar field. Unfortunately, due to time, I don't have the opportunity to explain why we believe that new scalar fields probably exist in the universe and probably exist at very high energy scales. Again, there was actually a very nice morning of theoretical physics about axioms that would be one of the canonical examples of new scalar fields that we believe may well exist.

So that would be provide context. Of course, I'm happy to talk about this afterwards, but for the rest of the talk, just please take as a belief that we may well have these new scalar fields around us in the universe. I'm also going to work in the classical limits. Of course, as we have already heard this morning, we believe that the universe is fundamentally quantum, so that each degree of freedom here in my mattress should really be some quantum harmonic oscillator with corrections.

And then as I take the continuum limit, I should have a degree, a quantum degree of freedom at each point in space. But as we know from electromagnetism, there are many scenarios in which it is useful to talk about a classic electric and magnetic field. We don't need to worry about the details of the individual photons that make up the field. And likewise, everything that I talk about today will have sufficiently large occupation numbers.

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.

Okay, that is the set up. But of course we want to talk about dynamics, about what happens with time. And again. Let me go back to a very simple system just to give an introduction. So if I think about just an individual one particle here represented by this. I saw this ball moving in some potential, which I'll call you.

Of course, we all know that the total energy of the system is some kinetic energy plus the potential energy, and depending on how much energy the particle has, of course the system can have. 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. It can escape off to infinity. It's unbounded. This is for this particular example potential I've just drawn slightly randomly.

If I have somewhat less energy than the particle can explore all of this region, including both of these minima. But if I go down to lower energy, then the particle will inevitably be trapped and either this minimum or this minimum. And just to reiterate, there's one degree of freedom in the system as I've drawn it, just the position of the particle. 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.

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. Because I have a continuous system, I talk about energy density rho rather than the total energy. Well, it's more useful to talk about the energy density, right? Rather than the total energy capital E, which is just the integral over space of the energy density.

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. We also have some time derivatives of the fields. This is not as analogous to the kinetic energy of the particle, and we also have some spatial derivatives. Contribution to energy density. These, if you like, are associated to in the mattress description. The energy stored in the string in the springs between the different masses.

Okay stream. You know them, but it will make it do with it. 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 to the energy scale of the new physics or the scalar field that might arise in my theory, some coupling constant. And of course we all know what this potential energy looks like.

Just looks like this. What is noteworthy is that this potential of this theory has a symmetry in particular is symmetric. Under the mapping, phi goes to minus phi. Of course the potential remains unchanged by this. And moreover, not not only is this a symmetry of the theory, but also it's a symmetry of the vacuum solution, by which I mean the lowest energy solution. Of course, just looking at the total energy density we have, that the energy density is minimised.

When I minimise you and when I have no time or space derivatives in the field. 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, which equivalently, is adjusting the system from being at high temperature to being at low temperature. Then I get the following effects or dynamics. If I start off at high energy, high temperature way above this energy scale for the scalar field can explore all of its potential,

a large part of its potential. Correspondingly, here the colours represent the different values of the scalar field. There's also enough energy that I can have sizeable derivatives both in time and space. 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, but also the the value of the field at each point is in some sense channelled down to this one unique minimum.

So what I get out at these low temperatures is really quite a boring system. Is just a scalar field. That's right. At its minimum, no derivatives and time and space. Much more interesting is if I make only a very minor change to my potential. Whereas before I had a plus sign here, now I just put in a minus sign. Fundamentally, we have no reason to think that this sign should be positive or negative. In this case, I still have my symmetry in the underlying theory. I can map phi to minus phi.

The potential, which now looks like this is unchanged by this mapping. But if I look at the vacuum of the solution of the theory now the symmetry is spontaneously broken. Actually better I should say that the symmetry is hidden because it's still there is just not manifest in the same way. 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.

And then when I apply, if I guessed minus phi, this vacuum will turn into this, not vacuum or vice versa. If I play the same game as before of starting off at high temperatures and lowering the temperature, this leads to much more interesting effects at high temperatures. Again, the field will explore only large parts of its potential, including both of the minima, so it's random. Has large derivatives, large gradients.

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. Now, locally, we expect the field to all go to the same minimum. If I'm sense, if I'm looking just at this small region in space, I can't have too large gradients. Otherwise that would cost too much energy. 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.

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. Maybe it just happened to have a fluctuation in that direction over here, and the fluctuation over in that direction over there.

So this leads to a much more interesting structure. And in fact, the structure that we end up with is in many ways similar to what would happen in this simple system where we have one spin degree of freedom at each point in space, which can either point up or down at high temperatures, the spins would be randomly up and down. 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.

These are called domains. Similarly, I'll call these regions of space where the scalar field has the sits in the same minimum domains. And between the different domains we have domain rules. So these domain wars, uh, interesting objects, they will actually be generalised to the strings that I'll talk about soon.

If I imagine setting up my system so that I say sitting this minimum over here and space this minimum over here in space, then of course, the field cannot sharply jump from this minimum to this minimum. That would require an infinite gradient which would cost infinite energy. 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, takes this kind of form with the red region here corresponding to the domain.

Well, there is an energy cost to this. 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. It also cost gradients energy because I have to have the field varying in space. So you might say to me, well, this is not the vacuum solution, why would you care about it?

And the reason I care is that actually, once I end up with this sort of configuration, it can be really hard for this to disappear into the true vacuum. If I imagine I have some finite size system like this, then actually the domain war would not be stable.

There would be some small energy gain or energy decrease, I should say, in moving the domain will say slightly in this direction, 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, and at that point I would end up in the true vacuum.

However, if the system is sufficiently large, if it's infinite, or if it's for all practical purposes, infinite, then there's no energy gain to moving the domain more one way or the other. So once I've got it there, I can't make it move to the edge of my system. You might also say maybe these quantum fluctuations that Joe talked about earlier could allow this domain wall to disappear. We heard that space time is actually quantum and is constantly fluctuating.

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, I would need that the field in this entire region of space to simultaneously fluctuate over from this minimum into this minimum. This is exponentially unlikely to occur because fluctuations, quantum fluctuations or thermal fluctuations a small localised things.

Whereas here I would need some coherent fluctuation to take me over the top of the barrier everywhere in space, or at least everywhere on this side of space. Okay, now turning to the early universe. Luckily, Joe gave a very nice introduction or summary of what we already know about the early universe.

This is some sort of cartoon form. We live somewhere here where the temperature of the photons that make up the cosmic microwave background being about ten to the minus four electron volts. 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, gives ten to the nine electron volts, and does a conversion into length scales. This is the rough comparison.

Now, with the exception of inflation, which as we heard, leaves perturbations left over in the universe which we can detect and observe. Basically, all of the information that we have about the universe stems from this error here. And later of course, here. There's been lots of words and conjectures put for what might have happened earlier. 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.

So that's sitting somewhere in this region. But of course, it's quite different physically to have two protons colliding with TV energies compared to the whole universe having the temperature of TV.

So now let me suppose that I have my new scalar field, and let me assume that the scale of the scale of scalar associated the scalar field, which I'll call Fe, which is also the scale at which symmetry breaking happens, is somewhere in this region, say ten to the 16 GeV, maybe down to ten to the ten GV, something like that.

Actually, again, there's good reasons to think that this may be the energy scale associated to new physics that could be associated to grand unified theories or similar things. But for now, let me just take this as a conjecture. Then provided the temperature of the universe did indeed reach values above alpha, and that inflation happened before this.

As I evolve the universe forward in time, the universe gets cooler as we know, and we all go through this spontaneous, symmetry breaking process, and my configuration of the scalar field that I get out will look something like this. Like I had before with different domains in different patches of the universe. And actually my slightly fuzzy statement about regions of space being sufficiently far enough away from each other going into different vacua can now be made sharper.

The largest distance over which any causality can occur over which signals to propagate, with the exception of inflation which happened earlier, is set by the Hubble parameter.

So the Hubble distance is the inverse of the Hubble parameter, given by an expression of this form, where rho is the total energy density of the universe, which during radiation domination, which is everything from about here and earlier, earlier we think is given by this expression where t is the temperature of the universe and Planck is the Planck scale.

That means that at these early times, the size of the Hubble horizon, which is the distance over which signals can propagate, over which information can be transferred, might be represented by this small Hubble patch. And then for an observer living in this region, there's no way that they could know anything about what has occurred outside their own Hubble horizon over here.

So indeed, we have a sharp prediction that the spontaneous symmetry breaking should lead to the system being in different vacua in different parts of the universe that are causally disconnected. Now, let me run forward in time. The Hubble parameter decreases. The Hubble horizon grows. So that now is represented by this larger square.

And at this stage you can see that for this region of space and for this domain surrounded by a domain wall, suddenly it will the system will realise if I can use very loose language, that actually this is a domain that is in some sense surrounded by a surrounded region of the opposite vacua. 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.

On the other hand, there will still be domain walls separated by Hubble distances. For example, once all of this region turns blue. Still, there will be one domain over here and the different domain over here, which is separated by roughly the Hubble distance leading to a domain more that is actually roughly of Hubble size. And indeed, this pattern persists as time progresses. Domain moves get destroyed, but still successively more domain moves enter each other's horizons.

What's this to do with strings? Well, actually, the physics of strings is very similar, but somewhat harder to visualise,

or at least harder to draw on the presentation, which is why I started with domain walls. The only difference is that now I have my scalar field, and instead of it being a real scalar field with one real degree of freedom at each point in space, let me say that I have a complex scalar, so that each point in space I have a complex number. 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.

Now this potential looks something like this. Where in the vertical direction I have the value of the potential. With this direction, I have the real part of phi in the direction into the board. I have the imaginary part of phi. At early times, the scalar field can explore all of these minima. In fact, here we have a full circle. 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.

Unlike the case of domain walls, there's no energy barrier between the different minima. There's a full continuous circle, but still I just end up with similar structures. The strings. So perhaps this is easiest to visualise first if I imagine a two dimensional space. 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, either on the board here or in the 2D plane of this picture.

I also do a loop of the filled space. What does this mean? Well, it means that I end up with an object that, just like the domain more, cannot fluctuate away easily. I also cannot fluctuate fluctuate away this object easily. As I look further and further away from the centre of what will turn out to be the string. The field sits in this vacuum, but it is winding around 0 to 2 pi in the vacuum manifold, even infinitely far in space.

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, to simultaneously fluctuate over the top of the potential such that I ended up in the same vacuum everywhere. If you like, there's some obstruction to this system relaxing down to the true vacuum, which would just consist of the scalar field being started a single point with no variation in time or space.

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. If I went infinitely close to the centre of the string, the gradients would become infinite. That would cost me infinite energy. I certainly don't have infinite energy in my system to start.

And in fact, what happens is that when I get within a distance of one over half of the centre of the string, the scalar field gets forced back onto the top of its potential. So that characterises the really the string, like the string core or the centre of this string where the scalar field is forced.

And if you want some more mathematical details, this is the typical form of the profile of a single string, where g is some function that varies between zero at the centre of the string and one at infinity, which corresponds to the field sitting in its vacuum manifold. One of the most important characteristics of these string like objects is their tension.

By which I mean their energy per unit length. A straightforward calculation gives you that the tension, which is energy per unit length, is given by an expression roughly of this form pi times for skirt. Now this scale for can be extremely large. I was saying it could be something like ten to the 16 GV. That means that these strings carry an enormous amount of energy. They're extremely thin, but associated to the core of the string there's this huge energy density.

Now let me look at the cosmological evolution. So a single long straight string is stable. There's no way of getting rid of it. Aside from these vanishingly rare properties, probability of having some fluctuation away. But in the early universe, what would actually happen is that first, I started off with some temperature that's very high. I would then call below the symmetry breaking phase transition. And I wouldn't just form one long straight string, I would form a complete network of strings.

This is some picture from a numerical simulation. The strings would have complicated evolution. If I have small loops that are smaller than the Hubble horizon, these things can collapse. If I have two strings and a Hubble horizon, they can intersect, they can recombine, form new strings, and so on. But still there will be some evolution that continues. And in fact, just like the domain, walls persist with roughly one domain wall per Hubble patch surviving.

Actually, in the case of strings, I also get roughly one string possible patch surviving. It might be, depending on the specific theory, that the strings are destroyed at some much later time. For example, in some axion theories, this happens when the temperature of the universe is about achieve. This is more model dependent. It could also be that the strings just survive indefinitely until the present day, so that we would still have strings out that with one in our observable universe.

Of course, we don't just want to conjecture that these strings exist. We'd like to detect them and learn something. There's a few different ways that this can happen, but perhaps the most promising is to look for the dark matter. Look for dark matter that might be produced by the evolution of the string network. So during this evolution, strings are being destroyed. Some survived, but most are being destroyed. And in the process they release energy.

In lots of sensible theories, this energy goes into dark matter and there's a huge energy density in the strings, which means actually I can produce the vast majority of my dark matter this way. Also because these strings have extremely high energies, potentially close to the Planck scale. They can source a sizeable amount of gravitational waves. 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.

It could also be that if these strings survive, they can leave signals in the cosmic microwave background, which we could just look for directly by studying this background of photons extremely carefully. So actually, a sizeable part of my research these days is spent trying to understand the dynamics of these strings.

Make predictions, for example, for the dark matter abundance that is produced. 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. I've got a potential that is pretty straightforward, is the kind of potential that we just show our third year, fourth year undergraduates as the first example. So what is the problem? Why don't we just study the system directly?

Well, really the issue is the combination of two factors. First of all, the dynamics of these strings are extremely nonlinear. They have complicated interactions. When the strings approach each other, the system is certainly far from any linear approximation. This means that analytic approaches will only get us so far, as anyone who works on basically anything involving partial differential equations will know. Nonlinear partial differential equations extremely complicated to solve.

A natural other approach is just to do numerical simulations. Then the nonlinear behaviour is not a problem. But the real problem is that I have a huge scale separation in my problem. I'll say more about that in the next slide. But this in some sense is the real challenge. We haven't got good analytic approaches that really allow us as much precision as we want, and numerics are challenging.

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. We take our complex scalar. We discretize again, we put it on some lattice, we get access to the biggest supercomputer that anyone will give us access to, and then we just solve the equations of motion of this complex scalar. If I start with sufficiently random initial conditions, the strings will automatically form.

They will automatically evolve in the way that I've been describing, following the equations of motion. And I'll just be able to see what the system does. The problem comes about because in order to capture the string interactions, I need to make sure that I have at least a few lattice points per string core. Otherwise, if these two strings intersect each other, I'm not going to correctly resolve the dynamics of a system. So that's represented by this small circle.

But at the same time, we know that the communication, the strings can communicate or interact over distances of order the Hubble distance. This is also the typical length of the strings that typical curvature. So my simulation I better make sure that I have at least a few Hubble patches within my grid, represented by this blue sky. 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.

This means that I'm limited to doing simulations where the ratio between the string thickness and the size of the Hubble parameter in cartoon form is less than the ratio of this big square, and the small circle is roughly less than, say, a thousand. The physical situation, meanwhile, is that this scale of separation is ten to the 30. 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,

things are not going to be safe. You certainly don't want to take at face value what you do in simulations here, and just blindly apply your results that you get out to the physical situation. I don't actually have any perfect solution to this. It's a challenge. What we do or what we attempt to do is do simulations as carefully as we possibly can in this regime, and try to make this huge extrapolation to the physical regime, at least being aware of the uncertainties.

It's not perfect. I'll talk about some possible future improvements at the end. Unfortunately, physics is hard. This is the best we can do. 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,

that gives us hope that such an extrapolation is possible and not completely out there. That's the existence of a so-called attractor solution. So regardless of the situation that I start my string network, often I actually get drawn into what is called an approximate scaling solution, where I have about one string length of one length of string, one Hubble length of string, Hubble patch. I can actually motivate this pretty quickly and in a kind of hand-wavy way.

Imagine I start my string network off with way too much string. I have lots of strings per Hubble patch. These will very quickly interact with each other. They will annihilate away from loops that will shrink. 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, balances the rate at which the strings re-enter each other's Hubble horizons.

On the other hand, if I have way too few strings per Hubble patch, basically nothing will happen. The system will just sit there, the strings won't be destroyed. The Hubble horizon will meanwhile grow, and I'll accumulate string within each Hubble horizon, which will again persist until I get to this so-called sort of critical point at which there's this balance between being stretched, strings being destroyed, and strings beginning to see each other across Hubble distances.

If you like. In some sense, this is almost an instance of self-organized criticality. Okay, so that's the basic picture. We make use of this approximate scaling solution. We do careful extrapolations. You can argue that the energy emitted per Hubble time Hubble volume by the string network is given by an expression of this form. So we have this Fe which is the energy scale of the spontaneous symmetry breaking.

It's this very large scale. And we have Hubble which enters because this is the typical length of the strings. And we try to make predictions for the dark matter abundance and for the amounts of that might be produced by these strings and the amount of gravitational waves that are produced through this evolution.

Let me first talk a bit about the dark matter abundance. I'll start with an extremely simple formula. So the energy density in dark matter is given by the number density of dark matter times the mass of dark matter. Because the dark matter is non-relativistic today. Okay, this in some sense seems completely useless, but what is the point of it? Well, the energy density in dark matter is something that we can measure. We have no idea what the mass of dark matter is.

It could vary anywhere between ten to the -20 electronvolts and say, ten to the ten GeV or even larger. This is many, many orders of magnitude. But still, with this combination of the total energy density, we can constrain, for example, by looking at the evolution of galaxies or the cosmic microwave background and similar things. 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.

We can calculate how many dark matter particles are released by the string network. And if we've got this predicted or this calculated this observed, we can then make a prediction for the mass of dark matter and experimentalists like this. If I restrict myself just to axion dark matter, if you know what that is, that's fine. If you don't, that's fine as well. It's just some possible new scalar particle that might be dark matter.

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. Even smaller, actually, somewhere up to this ten to the four electronvolts for this particular candidate. Experimentalists really don't like having to look over 20 orders of magnitude a mass for each of these different masses. They have to think of different experimental techniques to use here in the solid region.

We've got existing experiments. These are some proposed new experiments, but still it's a major challenge. The benefit of our prediction is that we can highlight some particular range. In fact, for our latest calculations, we end up something like ten to the minus three electron volts, where if the scenario that I'm proposing is true, that stock match is produced by the string network, the dark matter mass should be given by this value.

We can then go to experimentalists even in the basement of the Beecroft building. Tell them, please look in this mass range if they find something absolutely fantastic. If they don't find anything, then at least we have ruled out this entire scenario, teaching us something about the production, at least ruling out an entire class of production mechanisms for dark matter. Okay, so that's the. In some sense the aim for the dark matter abundance.

The story for gravitational waves is pretty similar. Here I've plotted the sensitivity of proposed future experiments searching for dark matter. This is the energy that energy density and dark matter in the present day universe. This is the dark matter. Gravitational waves in the present day universe. This is the frequency of the gravitational waves. And these are different proposed observation techniques, as the iron is also being developed in the basement of the Beecroft building.

If you get a chance to visit that area, it's very interesting. So what do our predictions look like? Well, these are our predictions. For what the gravitational wave spectrum coming from the string network should look like. Now there's a dependence on this scale for which is the energy density associated to the strings. If the strings have higher energy density, they're also thinner. They lead to more gravitational waves.

And you can see that if all of these projections actually pan out and the experimentalists do incredible work and managed to reach these sensitivities, potentially we could be discovering string networks that arise from physics of energy scales of 10 to 14 GeV, possibly even as high as 10 to 15, ten to the 16 GV. This is almost absurdly high energy densities.

And really, the reason that we actually have a chance of discovering or learning about this energy dense physics, these scales, is because the cosmic strings survive the evolution of the universe, because they're stable in the way that I described earlier. This is also, in some ways quite nicely complementary to collider searches.

If you remember, collider searches are looking at energy scales of maybe TV, which is ten to the three GeV, possibly ten TV, maybe 100 TV in the future if we're really lucky. Whereas here we're looking at much, much higher energy scales. So in some sense we're attacking the problem from two directions. Okay. It's also actually quite nice that the shape of the spectrum looks like this. The fairly flat predictions.

This means that if we do see this background of gravitational waves in, say, one experiment over here, we can then say, well, if this is being produced by cosmic strings in the early universe, we predict you should also see signal in these experiments over here. If they do, great. 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.

The other really nice feature that I alluded to already is that these gravitational waves are produced in the very early universe. So now here on the top axis, I plotted the temperature of the universe when these string, when the gravitational waves that have the corresponding frequency are produced. Remember the limits of our current observations are somewhere around here.

So actually, if we see a signal anywhere from here over to the right, at higher frequencies, we would be learning something or would be seeing a signal that is originating from the absurdly early universe again, way beyond anything that we have seen in any other way, with the possible exception of inflationary fluctuations. Perhaps in the best case scenario, if we see such a signal, we'd really be able to then study it in complete detail.

Learn something about, again, physics that these kind of energy scales, which again, we would never directly be able to access.

Okay, so let me just finish by talking a bit about work in progress. There's certainly, of course, a huge amount to be done on the experimental and observational side. The last two slides showed projections, but of course this takes incredible work for experimentalists to make these things actually happen. 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.

The first thing is that we can improve our simulations. 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. This is a very bad thing to do. We actually don't need resolution over all of our simulation. What we really care about simulating is the centre of the strings. So why don't we just put our lattice points at the centre of the strings when we need it.

Then we can could access much bigger scale separations. It would make our huge extrapolation still bad, but less bad than it currently is. Indeed, this is something sensible to do. It's something that we, or in particular students and postdocs are working on at the moment. The real challenge here is that because we have such a huge extrapolation, any small systematic effects that we introduced by doing this meshing would extrapolate to huge problems and really change our predictions.

So a lot of work is needed to really get reliable results. There's lots of other similar processes, actually in the early universe that could lead to gravitational wave signals. We could have phase transitions in which say that a similar scalar field starts at some false minimum, and then the true minimum by some sort of quantum tunnelling or thermal tunnelling. In this case, we would again get interesting to make. More structures which could collide, could produce gravitational waves.

And lots of work is needed to really get accurate and reliable predictions for this system. Okay, I think that's pretty much all I want to say. So just to summarise, spontaneous symmetry breaking is already an interesting process. We have good reason to think that it might occur in the early universe, and if it does, then it often, not always, but in many cases that leads to these types of domain walls or strings, which collectively are called topological defects.

The fantastic thing about these is that they persist from the early universe, so that we have access to the energy scales at which the spontaneous symmetry breaking happens. Okay, so I'm not entirely sure you can tell that. 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.

It also potentially allows us to learn something about the evolution of the universe at extremely early times, potentially with the proposed gravitational wave detectors, when the temperature of the universe was something like ten to the HGV, or modulo, with the usual definitions of times after the beginning of the universe. This is something like ten to the -22 seconds after the start of the universe, and there's a really extensive ongoing experimental and theoretical effort in this direction.

Okay, so thank you.