 I have been fortunate to have Thibaut as a collaborator and a friend for many years. So today I would like to talk about two topics which I know are close to his heart, which are black holes and cosmic strings. So I'm going to argue that loops of cosmic strings can be captured into black holes and then they can interact with black holes in a very interesting way. This is based on my work with Henrik Singh, Yuri Levine and Andrey Gruzinov. Okay, I don't know, maybe I can do it like that. So first let me briefly review some relevant properties of cosmic strings. Strings could be formed at symmetry breaking phase transitions in the early universe. And they are predicted in a wide class of elementary particle models. The strings are linear topological defects. They don't have ends. They can be either from closed loops or extend to infinity. The main parameter characterizing the strings is the mass per unit length, which will denote μ. And it's determined mostly by the symmetry breaking energy scale. A convenient dimensionless combination is gμ, where g is Newton's constant. And if the symmetry breaking scale varies between electroweak and grand unification, this parameter gμ varies in this huge range from 10 to the minus 34 to 10 to the minus 6. The strings have large tension which is equal to the mass per unit length, which is just a consequence of relativistic invariance. And because of this large tension, if you have a closed loop of string, it oscillates relativistically. Another important property of strings is that when two strings cross, they reconnect as shown here. So what do I press? That one? Okay. So when they cross, they reconnect at the crossing point, and this provides a mechanism for the formation of closed loops. The dynamics of strings is determined by the Numbugoto action, which is simply proportional to the world sheet area described by the string in space-time. The solution of corresponding equations of motion is very simple. It's given by this, where a and b are two arbitrary vector functions, arbitrary except that they satisfy these constraints. Sigma is a parameter along the string, and it varies between 0 and L, where L is what is called the invariant length. It is just the mass of the loop divided by mu. This solution describes a loop which oscillates periodically with a period L over 2. Now, if a loop of string runs into a black hole by some part of it, it will be captured like shown here, and we will be interested in the situation where the black hole is very small compared to the loop. So the size of the black hole is about gm, where m is its mass. So the loop is very large, but at the same time the black hole is much more massive than the loop. In this situation, you can think of this location where the black hole is as a point where the loop is pinned. So then you just have to solve the Numbugoto equations of motion with boundary conditions that there is one loop, one point on the loop which remains fixed, and then the solution depends on one rather than two arbitrary functions and is given by this. And the loop oscillation period is changed, now it is 2L rather than L over 2, but so far there is not much different from the free loop. Now, such an oscillating loop may self-intersect and then it will break into two loops and one part of one of them will fly away, but we experimented with thousands of kind of randomly formed loops and we discovered that in all of the cases that we looked at, after few reconnections you are left with a non-intersecting loop which is pinned to the black hole and keeps oscillating periodically. Okay, so now the fact that we now have just one vector function describing the dynamics of the loop allows us to think about it in the following way. We take this vector function and plot it. Its length is 2L where L is the invariant length and it is just a fixed curve in three-dimensional space. So its length is related to the length of the loop, it's twice the length of the loop, but the shape of this curve has little to do with the shape of the physical loop. But this is a very useful object nevertheless as we will see. This picture that I described where one point of the loop remains fixed is exact only in the limit when the size of the black hole ratio of R over L, R is the radius of the black hole, when this ratio goes to zero. When this ratio is finite there is still some energy and angular momentum exchange between the loop and the black hole and as a result this auxiliary curve as we call it is changing. But it's changing very slowly because the interaction is rather weak. It's suppressed by a power of R over L. So I will kind of describe exactly how this works. An important solution of stationary solution for a string in the gravitational field of a rotating black hole was found by Frolov and collaborators and this solution is very helpful for understanding what I'm going to tell you about. So the solution is the following. So this is a rotating black hole and its spin is perpendicular to the screen. The string is this red line, it extends to infinity along a straight line and as it comes towards the black hole it starts winding around the horizon. And this is a stationary solution. And then obviously this string will exert a torque on a black hole and this arm of the torque L is determined by the mass of the black hole which is this R which is GM and also on the angular velocity omega. So the torque is simply mu, the tension of the string times L and it's given by that. So this is if the string is in the equatorial plane of the rotating black hole. Now if the string is directed in some arbitrary direction specified by the unit vector N, then the torque is given by this. Okay, this is just a simple geometry. And then we add this extra term N cross N dot which is just allowing the because our strings are not stationary. We have a string that sticks out of a black hole at two points and then also these directions are gradually changing. So omega is the angular velocity of this vector in the direction of the string around the black hole and this term is added there so that the torque vanishes when the string is co-rotating with the black hole. Because when it rotates with the black hole there is no torque. So these vectors N1 and N2 vary on the time scale of the loop oscillation which is a water L, the length of the loop. And this is much greater than the characteristic time scale associated with the black hole. So in this sense these directions are, this situation is quasi-stationary so from which we conclude that we can use this expression which was obtained for a stationary solution but we still assume that it applies locally at any moment of time. We can use this formula. Okay, so now we can also calculate the rate of energy change of the black hole. Yeah, I should say that since the string exerts torque on the black hole the black hole exerts the same torque on the string and once you know the torque and angular velocity you can write the rate of energy change. This is the work done by the black hole on the string and this is non-zero even if the rotational velocity of the black hole is zero. In that case only that term contributes and the rate of energy change is given by this formula. And you can, well, omega is a water 1 over L so this is a small energy change in the sense that R over L squared appears and you can from this easily find that the loop loses all its energy in this way on this time scale L cubed over R squared and this is obviously much greater than L which is the one period of oscillation so it takes many oscillations until the loop loses its energy to this effect which we call horizon friction basically the energy is lost because these vectors move around along the horizon. Okay, in general, this is a general situation that the loop orbit evolves slowly compared to the loop oscillation period and this can be described mathematically as a continuous deformation of this auxiliary curve. So A of sigma is that auxiliary curve that I mentioned that describes the solution of a pinned loop and because of this effect of exchange of energy and angular momentum with the black hole, this auxiliary curve will be gradually deformed and this is what mathematicians call a geometric flow. In our case, I will not have time to give you the derivation but in our case this flow is described by this equation. V of sigma is the velocity of the point on this auxiliary curve at location sigma. Sigma labels locations along the curve and it is given in terms of this first and second derivatives of A of sigma. Okay, very simple equation. Now a special case of omega equal to zero attracted a lot of attention from mathematicians so then V is proportional to A double prime and this is basically the curvature of this curve and this even has a special name, curve shortening flow. So you can see that the directions of the flow are indicated here when it will be moving out at this point but inwards elsewhere and so the result will be that the curve will become more and more round and also it will be shrinking. Once it is round, it's obvious that it's going to be shrinking and mathematicians proved rigorously that the asymptotic state of this curve is that a small shrinking circle. Okay and the length of the curve is twice the length of the physical loop so this means that the loop loses its energy and eventually it is swallowed by the black hole but as I said the physical loop is different from this it's not circular at all so you can find that a circular auxiliary curve corresponds to a rotating double line so this is a black hole, the string comes out radially and then comes back along the same line and this double line rotates around the black hole in such a way that its tip rotates at the speed of light. So this configuration actually is a strong emitter of gravitational waves if you calculate using perturbation theory if you calculate the rate of gravitational radiation you find that it is logarithmically divergent but of course the actual rate will be finite for one thing is because the string is not infinitely thin and also this asymptotic configuration of double line is never physically rich so it will be close to double line. Okay, more interesting things occur if we allow omega rotation velocity of black hole to be non-zero so then this first term becomes important and you see each derivative with respect to sigma gives you a factor of 1 over L so the first term is dominant if omega L is much greater than 1 that is if the string is sufficiently long then the first term dominates and to see what effect it has let us consider again a circular auxiliary curve so suppose this curve is circular and suppose the omega angular velocity is points perpendicular to the screen in that case the velocity of deformation can be directed radially outward everywhere so the loop will be growing instead of shrinking and it will continue growing once you are in this configuration the loop will keep growing extracting energy from the black hole so where this energy comes from it comes from the rotational energy of the black hole so by solving this equation you can find that the length of the loop grows like square root of t and you may be wondering how important this effect would be if I for example have a string attached to a super heavy black hole in our galactic center and you find that it will be spun down completely during its lifetime if g mu is sufficiently large if it is greater than 10 to the minus 15 but 10 to the minus 15 is pretty small value it is well below the current upper bounds on this g mu so this effect can be quite significant in our actual universe another very interesting effect is very similar to super radiance which was first pointed out by Jakov Zildović Zildović pointed out that if you have a black hole a rotating black hole and you have a circular polarized electromagnetic wave this wave can be reflected from the black hole with amplification here a very similar effect if you have a helical wave traveling along the string around the string and hitting the black hole it also reflects with greater amplitude but the interesting thing here is that now this wave travels all the way around the loop and hits the black hole on the other side and it's amplified again so as a result even if you start with a tiny amplitude helical wave the amplitude will grow exponentially and the corresponding time scale is given by this so for maximally rotating black hole this omega r is a wider one and the time scale is comparable to the loop oscillation period which is much faster than the other time scale associated with this problem so one can imagine a scenario where this super radiance kind effect extracts rotational energy from the black hole then this wave becomes non-linear once it becomes non-linear it's easily imagined that self-intersections will result in loop formation and so the loops will be produced, will fly away and so this loop attached to the black hole can become a loop factory kind of by turning the rotational energy of the black hole into a number of closed loops okay, so these are the physical effects that can occur and now I would like to address the question how likely it is for a loop of string to actually be attached to a black hole so just a quick review of string evolution if strings were formed in the early universe numerous numerical simulations have shown that they were all in a scale invariant way so each horizon volume in the universe looks something like this with several long strings stretching across the horizon and a large number of closed loops which are barely visible here, they are shown in red and these loops have a wide distribution of sizes so the loops oscillate and decay by emitting gravitational waves as a side comment I will mention that these waves add up to a stochastic gravitational wave background and I enjoyed many discussions and writing papers with Tibor on the properties of this background in particular we found that this gravitational wave background is highly non-gaussian and includes large bursts of gravitational radiation but now we will not be worried about this background the average density of loops is proportional to g mu to the minus three-halves so the smaller g mu is, the larger is the density and this is easy to understand because if g mu is small the gravitational radiation is small and so the loop decays slower and the gravitational radiation therefore the number of loops is larger and therefore you could expect that the capture probability is higher for small values of g mu and this is indeed what one finds so we found that the probability of capture the most probable it's most probable for a loop to be captured is a supermassive black hole by a supermassive black hole simply because it has much bigger radius than smaller black holes and the probability that a supermassive black hole at the center of our galaxy has captured a loop is a water one if g mu satisfies this condition if g mu is less than ten to the minus eighteen and this I don't give the details of this calculation because you have to include the fact that the loops concentrated galactic halos and then the density increases towards the center so when you include all that you find this condition a very different scenario is obtained if the black holes are primordial primordial black holes are formed in the early universe they collapse of large density fluctuations and the fluctuations in the early universe can collapse only if they have a horizon size so you have this horizon volume and you have a fluctuation collapse into a black hole which has size comparable to the horizon and you see it is inevitable that this black hole will capture a few strings and then these strings will run from one black hole to another so the result is that you have a black hole string network which is an interesting object to study but it's hard to tell much about how it will evolve without an numerical simulation so now these simulations are underway so in conclusion I argued that string loops can be captured by black holes and moreover this is even likely if of course assuming that strings exist we know that black holes exist and this is likely for sufficiently small values of gμ once it happens this triggers a variety of physical effects such as black hole spin down super radiant amplification of waves gravitational wave emission and I should finally say that this is work in progress because first of all for all these effects we have not really included the effect of string reconnections and that needs numerical simulations to study also the evolution of black hole string network networks needs to be studied and there are other things as well this is all I had to say pleasure to be here happy birthday to you Bo in your equation of evolution of the formal shape of the string we did not include gravitational radiation back reaction because it's always negligible compared to the effect it's usually negligible but you are right that equation does not include gravitational back reaction maybe you can generalize it there is an interaction between string we see a string as a topological defect in spacetime the interaction is just like a casimir effect due to quantum fluctuation if you compute energy density without string and with the topological defect they are different and you can compute the interaction between the string it is not really like casimir effect casimir effect is due to quantum fluctuations and this one is not quantum fluctuation this is just a classical solution what I say is that there is an interaction due to quantum fluctuation between string did you take this into account in your simulation because I think it's important you mean interaction between string two strings, sure there is of course an interaction due to quantum effect but this would be negligible for the parameters that I am discussing here so you remember it could be very important if the topological defect is strong enough yes but we already have significant bounds coming actually from study of gravitational radiation from strings we have significant bounds on what this g mu is and it is pretty small so there is one question also on the chat you can go ahead and ask your question thank you very much if you can hear me I wanted to understand what is on the string tension or generally the string parameter space will you be able to set based on the various physical effects on supermassive buckles that you have in this case in particular would that exclude also the string parameter space thank you well at present the strongest bound on this g mu of strings comes from observations of millisecond pulsars it's conceivable that even stronger bounds come from interactions of strings with black holes but in order to really put reliable bounds we have to finish these numerical simulations so that because now some effects are not accounted for like crossing of the strings and so forth so at present we cannot improve on the millisecond bound okay here ah here when the string loop is captured by the supermassive black holes there will be a burst of gravitational waves is there a mean to distinguish the signal from the signal emitted by some emery or an ordinary black hole capture by the black hole well I'm not sure that the capture event will be very spectacular but when the string evolves and when it becomes close to a rotating double line the tip of this double line acts like a cusp on a loop so basically the tip is a strong emitter but of course you have to be lucky because if the tip rotates like in this plane you have to be lucky to be kind of looking in the right direction but there are many black holes in the universe so in principle we can observe these bursts of radiation from these rotating double lines but I don't think that the capture moment will be very important in terms of gravitational radiation okay so let's thank the speaker again okay thank you