 20-second delay, but that's fine, I'll let you know. Good, Nico, you can start. Okay, I think we are live. So welcome everyone to the Latin American webinar on physics. Today we are super happy to have Hardy Vermae from the National Institute of Chemical Physics and Biophysics in Tallinn, Estonia, where she's a researcher. And Hardy will talk today about probing primordial black holes with gravitational waves. So are you there, Hardy? Yes. So thanks a lot for accepting this invite. And please, to dance floor, it's all yours. Yes, thanks, and thanks for the invitation. So I said I will be discussing promoter black holes and promoter black holes have been a very active topic recently. So in a sense, primordial black holes are simply black holes that are formed in the early universe. There are several mechanisms by inflation, by collapse of cosmic strings in phase transitions. And today I will focus mainly on gravitational waves from promoter black hole mergers and gravitational waves that are induced by the formation mechanism that can also be used to probe promoter black holes in areas. So let's have a look at the possibilities for promoter black hole abundance. So the mass range can be quite large. So the lowest masses are actually of the order of grams, but such black holes evaporate. So, okay, I cannot show. So, and black holes that do not evaporate, that can be actually components of dark matter start from roughly dental minus 16 sort of masses. Before there are bones from evaporation that will affect, for example, CNB or produce cosmic ray background. Then there are various bone strains from microlensing that rule out relatively light promoter black hole and then there are other candidates from its intent to minus 11 up to 100 solar masses. So, and around the LIGO mass range, which is roughly one to 100 solar masses, LIGO gives the strongest constraint right now. And this is what I will be discussing in the second part of the talk. And above that you have constraints from accretion, so promoter black holes accrete and the infalling gas will will be hot and radiate and it can affect, for example, the CNB. And there is a wide range of dynamical constraints. For example, white binaries, if you have promoter black holes in dark matter halos, then they can disrupt binaries and just by the survival of white binaries, we can see, we can infer that there are no promoter black holes in this range. And so there are several other such constraints. So, and this is a brief overview of what we can have. So, and let's look at the main applications or phenomenological scenarios. The window for promoter black holes as dark matter is open for relatively light black holes, as you can see on this plot. Then there is the option of promoter black holes as the seeds for supermassive black holes. So far it's unknown how supermassive black holes are formed. You need intermediate black hole and one possibility of this is having an intermediate mass black hole as a promoter black hole that will then accrete and become supermassive black holes. And of course, the scenario that I will mostly discuss today is the promoter black hole scenario for the LIGO Virgo events. And yes, so let's move on. So the first thing that I will discuss is gravitational waves that are induced by promoter black hole formation. So if you have a large scalar perturbation, it will induce or it will source gravitational waves that can be picked up by the current or future gravitational wave experiments. So this is a rough, well, it's a plot for a power law spectrum. I would like to show, maybe I can do, no. So what you can see is that the scenario for light, promoter black hole dark matter is around 10 to minus what was this, 10 to minus 16 solar masses, which is about 10 to 14 in mega-parsecs. So, and the future experiments can test this scenario. While pulsar timing arrays can test scenarios that for heavier black holes that are about one solar mass or a bit heavier. And of course, so this clay line above is roughly the abundance that you need for the amplitude for the spectrum that you need to produce promoter black hole dark matter, all of it. So you need a peak that goes from the plunk, which we observe to roughly 10 to minus two, or 10 to minus one. And as you can see, there are some observations like mu distortions of the CMB that can also constrain as such peaks if they are not very narrow, if the slope is not very steep because you have to pass the copepharic pump. But, okay, I need to hurry up. So recently, Nanograv reported a potential signal for a gravitational wave background. So we should look at the first five bins. The last bins are probably just some random noise. And it's not clear that, so Nanograv is a pulsar timing array and they observed a residual noise in the timings of the pulsars. And they don't observe a monopolar or deep polar correlations, which means that it cannot be easily explained by astrophysical sources or clock, not just in clocks, but they also haven't found any quadropolar correlations, which means that correlations between, or negative correlations between the pulsars in with a 90 degree angle with respect to us, which would be a definite sign for a gravitational wave signal. So we cannot say that we have observed gravitational waves, but we can speculate. So what we did is we used two shapes for peaks in the promoted power spectrum. The first one is a broken power law, which shows up in most models of single field inflation that is used to produce promote the back holes. So it's a specific feature for when you have quasi-inflection point. And we choose the infrared tail to grow as K-4 and the UV tail should drop as K-4 into the minus 0.5. In this way, we get the end of inflation roughly at 50, 60 e-falls. And the second shape that we use is taken from a paper on hybrid inflation, which is a low normal function with the slight modification that it has exponentially we got off. And we also use a specific parameters for the shape of the peak. So let's move on. So this is a plot of the induced gravitational waves over the amplitude of the peak in the power spectrum, A squared. So since the gravitational waves are reduced at second order, then it's roughly quadratic in the, so it's roughly a square of the gravitational wave of the promote the power spectrum. So, and I also show the blue line here is the induced gravitational wave signal for a delta function. So, but in general, you can see that the height of the signal is roughly the same. So maybe if you have a very narrow spectrum that's close to the delta function and this is difficult to make, you can get a small peak. But if we can make a very rough estimate showing that we don't have, the non-rub signal doesn't predict enough power to have all dark matter in promote the back holes, which is what we need is roughly the amplitude has to be 0.02, which implies that the height is two times 10 to minus eight. But above you can see that the non-rub signal, at least in the one sigma region reaches up to 2.10 to minus nine. So it's a one order of magnitude lower. But let's look at some of the details because promote the back hole formation is exponentially sensitive to all the details in the form formation. So what we use is we assume critical collapse when following this paper by Young et al. And the mass of the promote the back hole that is forming is roughly the size of the horizon when the fluctuations enter times some factors. And if the density perturbation is below the threshold, you don't form anything. So what we assume here I give the parameters and we also use the non-linear equation between the relation between the density contrast and the curvature fluctuations, which is important, it will suppress the abundance a lot. And so another technical detail is the mass function. So what's what we obtain from by assuming Gaussian curvature perturbations. And as you can see, the promote the back holes, the sufficiently strong fluctuations that produce promote the back holes appear at the tight tail of the Gaussian distribution. So any modification to the Gaussian distribution can also affect the abundance a lot. And assuming the critical collapse what we obtain a mass function that in the infrared is a power law and in the ultraviolet in the large mass is cut off exponentially. So this is a feature for critical collapse. And so these are the results. As I said, the like of Virgo scenario for promote the back holes is outside of the range, at least this is what we find. And but we can explain, for example, the promote the back holes scenario for the supermassive black hole seeds, which is not in the middle of the range, but two sigma is good enough. And one more note that I want to make is that there are large theoretical uncertainties. For example, we use the, so our computation corresponds to the middle scenario. The middle, the yellow line for the critical collapse parameters. But as the formation depends or the critical threshold depends, for example, on the shape of the over density, then it can vary by several orders of magnitude. However, we find that this variation is not enough, it's not sufficiently large currently to bring the like scenario into the non-agrar range. But I also have to mention that there are a few other papers that say that you could have sort of mass promote the back holes, but this paper didn't consider the non-linear relation between the density and the curvature fluctuations, which can suppress the abundance a lot. And I think that's a good point. Corvacher fluctuations, which can suppress the abundance a lot. And if you change the cosmological background, for example, if the promote the back holes form in the meta-dominated period, then it's also possible to have heavy black holes consistent with the number of signal. And there was one more scenario that we didn't consider is when you have a very flat power spectrum, then it's possible to have promote the back holes in the low mass window. That's suitable for promote the back hole dark matter. So, a brief summary of this part is that we can explain promote the supermassive back hole seeds with non-linear signal, but it's not sufficiently large for the promote for the Ligo Virgo promote the back hole scenario. And of course, if you have a positive measurement of a gravitational wave background, then we have to have a better understanding of the uncertainties related to promote the back hole formation, because this is the mind, if you don't understand that, then it's difficult to make any definite conclusions. So, do you have any questions for this part? Well, you tell me we have the question at the very end, but I probably have some questions. Well, I have one. Can you write? Yes. So, you mentioned maybe anything that's true, that back hole school has been produced during a matter of an extended era. So, that's an important phenomenological difference between back holes being produced in already interrogation or matter. Well, the question is, is there a difference between back holes Well, yes, the collapse of density perturbation is different in the two eras. In a radiation-dominated epoch, all the fluctuations tend to be damped, but this is not the case in the matter-dominated era. So, it's easier to produce black holes and you need smaller peaks. Okay, okay. Okay, so, if there are no more questions, then I will move to the second part of the talk. So, promote the back hole mergers and the formation of promote the back hole binaries. I can distinguish two different scenarios. So, the first one is the early scenario where the promote the back holes form in the very early radiation-dominated universe, just by being close to each other and the coupling from expansion, so they start falling towards each other. And this is the mechanism that I will discuss today. And the second formation is so-called late formation where the binaries form in dark matter halos by close encounters. And the rates for these two processes can be very different. However, since promote the back holes have much smaller scale structure, so you form more small halos earlier on, then this can influence the late formation. So, it can enhance late formation and it can disrupt early binaries. So, the two numbers will be probably a bit more closer to each other on this slide. And the observed merger rate is about 10 cubic gigabytes per year. And if we compare this to the early mechanism, then it's clear that if it's true, then it's difficult to have all black holes as dark matter because we would see too many mergers. But this is what we will discuss in detail. So, the rough idea of computing the number of mergers, the merger rate at any time is that we have to take all the initial conditions in the early universe. So, it's a random field of stationary black holes that will decouple and form pairs. So, we just have to look at the initial density of promote the back hole pairs and look at how many of these pairs will form binaries that will merge today. So, the binaries that we can see with current experiments or see it all. The binaries that don't merge will not see. And of course, we also need to take into account that the binary that forms in the very early universe can be disrupted. It can interact with surrounding promote the back holes or surrounding matter and be perturbed. And we might not see it or it might... Well, it's very likely to merge at a different time. So, this is the initial setup that we have. So, in the early universe, it's just filled with promote the back holes that are at rest. And then at some point, they will decouple from the expansion. So, they have an initial combing distance and we also have to account for the closest black holes to the binary because if you have a nearby black hole, it can fall into the binary and disrupt it and we will not see the merger. So, we have to take this into account when we count the initial... the pairs that will form binaries that merge today. We assume a Poisson distribution. So, the black holes are distributed completely randomly. This is a good approximation, but it doesn't have to be true. We just... There are ways to generate non Poisson distributions. For example, by non Gaussianite. And then to study the initial binary, we can just solve its evolution. So, it will experience different forces. So, expansion which will become self-dominant to the self-gravity term, so the attraction. And then there is a tidal force, a tidal torque from surrounding black holes which prevents a head-on collision of these two black holes. So, they start... They will be very eccentric and they will start to orbit each other. So, these are some details of the formation which I don't want to discuss in too much detail. The essence is that we can compute the semi-major axis of the binary that's formed. From the initial distance, we can compute the angular momentum from the torque. And we can do this statistically by summing the torques and averaging over the torques of all surrounding promoter black holes. And also, matter perturbation. So, all of this will contribute to the angular momentum. And since we have the initial separation and we have the angular momentum, we can compute the collision time of the black holes. And one thing that I want to turn your attention to is that it depends on the seven power of the angular momentum, which is usually of the order of 10 to the minus 3. So, in any collision with surrounding black holes, the angular momentum can decrease by, let's say, it becomes 10 to minus 1. And then the collision time can increase by 10 orders of magnitude. So, it's really important that the initial eccentricity, which the binaries are very eccentric, they follow almost the line, is preserved. Otherwise, the initial binaries will not form or merge within the age of the universe. But once we know the distribution of distances of the semi-major axis, if we know the distribution of angular momentum, then we can compute the distribution of collision times, which means the merger rate. So, and we have performed the simulations. So, this is an initial setup that we put a binary by hand that has the correct distance to form a binary that will merge within the age of the universe. We looked at the very early universe, so we had only 70 black holes up to 380 kW of cosmic times, which means we roughly up to C and B. And we looked at simulations with the monochromatic mass distribution, where all the black holes have 13 solar masses. And the fraction of prominent black holes was 110 percent and 1 percent. And we run each simulation 3,000 times and look at the survival of the binary at the statistics of the angular momentum and the merger rate. And we also considered the extended mass functions where we had a number of 3, 10, and 30 solar mass black holes. And there is a typo here, so the fraction was 0.1 percent. But the fraction was 10 percent, not 100 percent. And the expansion from dark matter, we look at the expanding universe where the expansion, the dark matter component is accounted only by its contribution to the expansion. We don't have any other dynamics for dark matter. And we test the statistics of the analytic predictions. So this is a simulation where if you can follow, then I think it was this prominent back hole it will fall into the binary. And the binary is immediately perturbed. So if you look at this again, if you didn't see, so you have this black dot above the two red dots, which is the central binary which falls and the binary is disrupted. And just a second example of binaries where the binary forms, it has the behavior that we expect from our analytic estimates. But then it flies into the cluster and gets disrupted. So it slowly picks up a satellite and then it falls into this dense cluster and will be disrupted. And one more point that we want to make is that the rate shift at this point is relatively high, but you can already see lots of small clusters of black holes. So the structure of the structure formation, if prominent back hole starts immediately after meteredation equality and this can have a big impact on the survival of binaries because these binaries will find themselves, or most of the binaries at least, will find themselves in the small clusters and be disrupted by interactions with other black holes. But this is not the case when you have only a fraction of dark matter in prominent back holes because the noise introduced by black holes, prominent back holes is small. So we can distinguish two different types of disruption. The first is the disruption by the nearest neighbor. So when the closest black hole flies into the binary and we can estimate the rate shift when this happens by looking at when the closest binary is coupled when the closest prominent back hole is coupled to the binary and then we can estimate when it will fall into the binary or the rate shift when the closest approach happens. And from this we get the rough condition for the distance of the closest black hole to the binary. So it has to be farther than this y defined in this way. And the second source is disruption by clusters which I will discuss later. But in the simulation we can see that the disruption rate is quite high so we cannot neglect it. As for the implications for constraints or the theoretical predictions when the fraction is small then we can use, we can forget the disruption by clusters and compute only the fraction that the merger rate of initial binaries assuming that they will not be disrupted later during the revolution. So a few results of the simulation is that the first plot shows the distribution of angular momentum and we can see that analytic estimate matches numerical estimate very well. So the analytic estimate is the green curve and the yellow curve is the angular momentum distribution of binaries that were disrupted but not ionized. So that a black hole fell into the binary it could have formed a three-body system and then one of the black holes flew out of the system. And this is the distribution that you roughly get. Yes, so the second point is that the initial binaries are very eccentric so the angular momentum is roughly 10 to the minus 2 and the eccentricity is 1 minus j squared and the square root of it just if you want to think in terms of eccentricities. And the final point is that initial binaries are hard which means that the binding energy is much higher typically than the average kinetic energy of surrounding black holes implying that if you have a binary black hole encounter then this black hole cannot ionize the binary. So the binary can swap the black hole but usually what you get is a new binary. So let's look at some of the plots again. So the second plot shows the change in lifetime so we find that since we put in the central binary so that it should have the initial life merger time that coincides with H of the universe which is this tau zero we find a peak around one which corresponds to the peak of unperturbed binaries but we also see that when the binary is perturbed so this yellow region then the coalescence time is several orders of magnitude larger than the H of the universe as I said before so if you have this collection of binaries that you expect to merge and they meet with some other black holes then you will never see them again. And the last panel shows that the binding energy the ratio of the binding energies before and after a collision and what we can see is that the binding energy since the binaries are hard and hard binaries if they collide with other black holes they tend to get harder so the binding energy increases the binaries become more compact but the eccentricity decreases the angular momentum grows so the coalescence times become much larger so now since we have the results we can make predictions for the merger rate of binaries and in this plot we assume a log normal mass function and we can see the effect of inclusion of different factors for suppression so the blue line shows no suppression the yellow line accounts for the tidal torque from matter perturbations the green line accounts for the suppression so if we cut away the binaries that will interact with the closest black hole then this is the green line and finally if we account for the disruption in clusters we get the black line and it can be seen that the effect is to reduce the rate by several orders of magnitude when the fraction of promoter black holes is large and finally we can also estimate the merger rate of perturbed binaries and it turns out that it can be dominant when the fraction of promoter black holes is large but there are uncertainties in the computation so if we move on the merger rate contains several factors so the S in this formula is the suppression rate that can get contributions from several different sources like the nearest neighbors or disruption in clusters it also shows the dependence in time which can be used to discriminate between the promoter black hole scenario for LIGO and the astrophysical scenario for LIGO because as can be seen in the figure below the merger rate is expected to go down for astrophysical mergers so if we look at the high enough redshift but for promoter black holes it can go up and another effect is the modification of the distribution of angromenta or eccentricities by interactions with the surrounding matter that must be taken into account even if the fraction of promoter black holes is small so let me try now to estimate the suppression due to collisions of binaries and black holes in clusters so we will assume several things so first of all it's a good assumption that when perturbed when a binary collides with a promoter black hole that is supposed to merge within the age of the universe then it will not see the merger so we can just exclude this from the set of merging binaries the second assumption is that we can assume we should estimate this minimal size of the clusters in which binaries can be perturbed so since larger clusters form later they are less dense and they are less likely to disrupt binaries so we expect that there is a maximal cluster size above which binaries are unlikely to be disrupted but we assume that if within such a small cluster the binary will be disrupted with certainty and the last step would be to estimate the probability that the binary is inside such a cluster so if we know this probability then we can estimate the fraction of binaries that will not contribute the merger rate which will effectively give the suppression rate for the mergers and let's consider two time scales the first time scale is the time scale of collisions so we can take the cross-section for a collision that changes the angular momentum by a significant amount and since we know assuming that small clusters virialize fast we know the velocity, the dispersion, the density and we can estimate the time scale for collisions for a given cluster size and we find out that clusters that are smaller that contain less than 200,000 black holes assuming that all of the dark matter is in black holes then such clusters will always disrupt the binary so even relatively large clusters will disrupt the binaries there is also an assumption for the mass here which I don't remember so of course the number will change if it changes the promote the black hole mass and the second point is that if you have a larger cluster then such clusters are not stable so gravity structures are not stable and even if the collisions are infrequent then as we know from the dynamics of global clusters the density inside the core of such clusters can start to increase the binaries can also migrate toward the center of the cluster because binaries are heavier they are bound systems and they are heavier than individual black holes they can migrate into the core and since the core density is increasing then the black holes inside the core can become collisional meaning that the collisions in the core can have a time scale that is less than the age of the universe now and if we take this into account we get an even larger number for the minimal size of the cluster that doesn't disrupt binaries and so taking the last number we count the black holes that are not within clusters of the size determined by the core collapse and we can estimate the disruption or the suppression at the given rate shift and what we find is that for f equal 1 the suppression rate the suppression of the merger rate is roughly consistent with our simulations so even if our strategy was designed to overestimate the suppression rate we would assume that it should be not too far from the real suppression rate and so to go back then if we include this effect we get this black line so the final contribution is estimating the merger rate of per third binaries now since if we assume that the initial binary the initial pair would form a binary that coalesces within the age of the universe then if it's perturbed its coalescence time will be huge so if we want to estimate the contribution of perturbed binaries formed in the early universe we have to have a very compact distribution a very compact setup of three black holes and they will become bound before meteredation equality they will evolve and eventually one of the black holes will be ejected from this three-body system and as a result we have a binary the binary distribution of andromenta is roughly independent of the initial distribution and has a power law distribution which we estimate from our simulation and other work on three-body collisions so we get that this gamma the power law is roughly one or two so the distribution is either constant which is what we find or it is linear in chain which is the equilibrium distribution of andromenta and I will I don't have time to go into details which I didn't plan to do but also accounting for the average energy or the average semi-major axis of this of a binary is formed in this way we can estimate the merger rate and what we can see is that it has a different scaling with the fraction of promoter black holes it has a different scaling with the mass and it also has a different time dependence that these if this is a component that can dominate then the distribution or promoter black hole scenarios can be very different from what we would expect if the initial non-perturbed binaries contribute to that merger rate and the final point is that since these binaries don't have very high eccentricities and in fact the eccentricity distribution is close to the equilibrium one then further collisions in later halos will not increase the coalescence time but since hard binaries tend to get harder then the semi-major axis will decrease which tends to increase the merger rate so if you have lots of small scale structure that will perturb the binaries later then the merger rate can actually be higher than what we estimate here but we didn't take this into account so to yes so this is the one of the second to last slide so if you fit the mass function the log-normal mass function to the events found by Laigo and Virgo collaboration we can characterize the parameters of the distribution and we find that the mass, the central mass is roughly 20 total masses and the distribution is relatively narrow so it's 0.5 which means roughly one order of magnitude on either side the Cree line accounts for the constraints on premodal back holes and the points above the Cree line are excluded by several constraints in the one to one hundred mass region and we also find the constraints from Laigo since the scenario the suppression rate is expected to be expected to be conservative it's difficult to imagine how to suppress this or how to explain the constraint way further but of course there are several theoretical uncertainties associated to this process so but this is the result that we get and in the plot the test the Cree line denotes the region where we can explain the Laigo events but since we don't see many one hundred total mass black holes then the constraint is obtained by requiring that we don't see any any mergers so we account for the actual mass distribution that we obtain by Laigo we don't just fit the merger rate and this is a work in progress we also show the constraints or potential constraints from individual mergers on the left and on the right we show the constraints from a stochastic gravitational background this is something that I didn't discuss in the talk but when the binaries merge and they are let's say too far to pick up the signal for the individual merger then they will contribute to the gravitational way background and it's also possible to constrain from other black holes using this background and in fact I don't have a slide for this but using the stochastic gravitational way background from binaries since it depends on the rate of dependence it's also possible to look at the background and distinguish the promoter black hole scenarios and other physical scenarios because as I said in other physical scenarios the mergers at high redshifts are more infrequent and I will summarize so we find that the promoter black hole scenario for the Laigo Virgo events is possible roughly when the abundance is between 0.1 to 5% and one important point is that when the abundance is small then we don't have to account for the disruption and all this non-linear evolution in the late universe but if the abundance is relatively large then even for 5% this can affect for example the time dependence or the gravitational way backgrounds that allow what I want to say is the signature for separating the promoter and the other physical scenarios might change a bit if you have a larger fraction and this needs to be understood better and of course after our paper there has been more work so oppression has been shown to change the predictions and it can relax the constraints there have been studies of the small scale structure formation that is the clusters of black holes where they show that also since there is small scale structure that is enhanced you can have a larger late finite measure rate that can also contribute that can be picked up that can be large enough to be visible by Laigo and Virgo and this will also change the characteristics of the promoter black holes scenario and there have also been a few other studies so this is it thank you for listening and I will answer any questions if you have so thank you very much Hardy for the talk so I can see already a couple of questions from the YouTube chat so guys don't hesitate to send your questions to Hardy so there is one from Ivanova Ivanova is asking do you assume a specific pbh population size to obtain the rate of mergers in both the early and late phases sorry this is in chat I didn't understand okay the question is if you need to assume a specific pbh population size to obtain the rate of mergers do I assume a specific pbh population to obtain the rate of mergers we assume a low normal mass function in the numerical computations but in general it doesn't so it's possible to to to the estimates for any distribution of promoter black holes one thing that I didn't discuss ah okay I didn't discuss the effect of an extended mass function for example in the suppression factors but I don't have the question so I think the first slide of your second part we will talk about the pbh binary formation could you please go back to that slide yes do you mean the simulation no before that the very first one there is a lot of noise asking about this rate the rates are completely independent on the calculation right so this gives a rough estimate rough numbers for the upper bounds so that the largest numbers at least for the early one this is the large number and for the upper bound so I assume the def is one and the mass function is such that the mass function can be whatever actually and no suppression and so on and for the late one this is more like a typical number so it might be a bit larger but this is something that I haven't studied in detail so if you have lots of clustering then the late formation can be larger actually I was confused by this slide because you're saying that the formation rate for random like pbh moving randomly is higher than if pbh are already like in a halo and higher by five orders of magnitude the early one corresponds to when the pbh are at rest so they are at rest that the couple and then they fall into each other in the late case the four are in a halo the halos are much less dense than the early universe and also the pbh are moving so it's a bit more difficult to have closer encounters in the early universe if you have a pair so every nearby pair will interact and form a bound system so that's why the early rate is much larger or this is at least what they expect but of course the numbers in this slide did change when we discussed different suppression factors for example if you look at the rate here so this 10 to 5 corresponds to roughly the unsuppressed case so the actual rate is if you account for the suppression in halos then it's lower so that's not a question again by Ivanova asking if for this rate we assume that pbh are monochromatic not necessarily so in the simulation for monochromatic and non monochromatic cases but the non monochromatic just had 3-solar mass, 10-solar mass and 30-solar mass black hole so it's a toy model but our analytic estimates so for example this merger rate has our can be computed for arbitrary mass functions when we compute suppression in clusters then we assume monochromatic distribution and also this distribution of the merger rate of perturbed binaries we assume monochromatic distribution but so the effect if you have a non monochromatic distribution well there is one thing that I need to say before is that the distribution is relatively narrow what we observe the LIGO cannot correspond to very wide distribution so the monochromatic assumption is not that bad but anyway if you assume if you compute this three-body formation then the main effect that has to be accounted for that if you have a three-body system of different mass binaries then usually the lightest binaries is ejected so this population of binaries will likely contain the heavier tail of the mass distribution because all the lighter ones are ejected and also the lighter binaries are more easily disrupted okay thanks there's another question by Marco Sariela-Mides asking whether you're aware of a recent work by a team selling in which they used a Curta method for the pvh and found much more coalescence time that means far less stringent bound for fpvh and LIGO yes I am aware so the the metric that they assume describes promodal black holes with growing mass and such so if if you look at the behavior of matter around the black hole in the Curta metric then you see that there is a continuous flux of dust or whatever is surrounding the black holes into the black hole and I don't think this is a realistic description of accretion and then there is one more issue for me with this is that if you have all black holes all of dark matter is in black holes and the black hole accretes then it has to promote the black holes so the situation is not what they describe there is no uniform flow of dust into the black hole it's usually what happens even if you have lighter black holes or particles that you can have a flow into the black hole but then it's not completely spherical it's a metric but violent relaxation is the wrong word but most of the particles will not fall directly into the black hole they will form a cloud around the black hole and this also has been studied and it's easy to eat the cloud that has some velocity dispersion and if the particles have a large velocity they will it's difficult for the black holes to eat them so briefly I think they make a specific assumption for accretion which I think is not realistic but of course this deserves to be studied further okay great thanks maybe a last question we have been running for more than an hour so thanks Hardy thank you very much for the nice talk and a couple of comments or announcements better so next week we will have another webinar this time by Gino Itidori and in two weeks time we will have a chiptorn so we are super happy with this talk so please keep tuned and get connected next week for Gino's talk and thanks again Hardy thank you very much thank you see you guys next week bye