 We are live Cosimo Thank you for joining us. We apologize for this a little bit this delay We were having issues with Chinese internet and some firewalls My name is Alejandro, I'll be your host. I'm also having some issues with my video. So I believe you cannot see my face But but I'm here It is a great pleasure to introduce to Cosimo Bambi. He's an Italian relativistic and cosmologist Who is currently a professor at Fudan University in China Bambi's researcher interests are very broad, including strong fields, tests of general relativity, Modified theories of gravity, gravitational waves, quantum gravity, accretion disks around black holes Baryogenesis, galaxy cluster, and testing in general relativity He has more than hundreds of publications in all the topics I discussed and he has recently written a couple of books with Springle, about particle cosmology, general relativity, and accretion disks He got his bachelor from Florence University in 2003 And then he got his PhD at University of Ferrara under the supervision of Sasha Dolgov Then he became then he did a couple of postdocs At Wayne State University, at the Cowley Institute for the Physics and Mathematics of the Universe And then to at the Ludwig-Maxillian University of Munich before becoming a faculty and full professor At Fudan University in Shanghai in in China In 2016 he got the prestigious chair and now he is a she-sheeted junior Peri-professor at Fudan So it is very nice to have you here Cosimo We'll talk about today about testing General relativity with X-ray spectroscopy. So please could you please share your screen? I think I'm sharing the screen Can you let's no, I think we are seeing we are seeing you. Could you please try again? Do you see my screen? No, no, I think we saw it for for just one moment Yeah, now we see your screen so you can just make it like full Okay, can I start? Yes, so I'll just talk if if if we have any question or you can just go and then Okay, so Thank you for the invitation and sorry for the delay. You too is a technical issue. So The title of my talk is testing strong gravity using X-ray reflection spectroscopy And the idea is to test Generativity in the strong fuel regime So the project is something between physics and astrophysics in the sense We want to test the fundamental physics, but in the end the astrophysical part is very important So I start with motivation So generativity was proposed at the end of 1915 by Albert Einstein and despite some Preliminary uptent to test the theory. For example, like the observation of a light deflection by the sun Test of generativity start much later and this is because It is difficult to test gravity gravity is weak and you cannot test this in a laboratory So systematic test of generativity started in the sixties with experimenting the solar system and even seven in the seventies By observing a binary pulse So in the case of solar system experiment essentially you want to study the trajectory of planet or artificial satellite in the solar system Or you want to study the propagation of electromagnetic signal in the solar system In the case of a binary pulsar you study the orbital motion of this system A pulsar is a very compact star You can and you see A beam because Beam is along your line of sight and you can use this system as a very precise clock The point is that all these system the sense solar system and binary pulsar Are in the so-called weak field You can describe this system as a Newtonian system plus some correction And what you want to do now is to test the strong fuel regime And in this case you want to test the space time Near black holes So in this case, you don't have a Newtonian system plus some small correction. You have something completely different So this slide is just to remind the main properties of black hole in generativity So black hole in generativity are relatively simple object Because they are completely described by a few parameters the mass the spin and electric charge This is what this is A Consequence of the so-called no air theorem. There are some assumption, but more or less than me. They are RISMO And so basically black holes, yes, are completely described by just a few parameters And if you ignore the electric charge because for astrophysical system, this is usually irrelevant You have only the mass and the spin and The solution for mass spin black hole with mass spin is called the Kerr solution And generativity make a very clear prediction about the motion of particle around the black hole So what eventually want to check is the motion of particle massive particle or and massless particle around the black hole Now The fact that in generativity There are these black hole solution is not enough to say that this solution Well described the gravitational field around the astrophysical black hole And because I mean the This solution sort of ideal solution In the sense, they are stationary Asymptotically flat. There is an empty space time. So you can quantify possible deviation from the solution And it is easy to realize that Deviation from the solution when you consider the region close to the black hole are very small So for example, you can consider the fact that An astrophysical black hole should come from the gravitational collapse of a system And there may be some initial deviation from the Kerr metric. Well, these are Quickly radiated waves of emission of gravitational waves Then for example, you can consider The mass for example in the case of a binary system There is the companion star or the black hole can be surrounded by an accretion disk But these are very small effects. So for example, in the case of the accretion disk variation between the mass of the accretion disk of the black hole is typically something like 10 to minus 9 10 to minus 10 So it is very small and well The effect are extremely small and also impossible to measure And you can also consider for example, a net recharge and what typical astronomical system have in a binary charge, but this is extremely small and completely negligible for the metric around the black hole So in the end you expect from the theory that the Kerr metric will describe the gravitational field around an astrophysical black hole From the observational point of view, you have two main classes of astrophysical black hole stellar mass black hole They are the natural product of a collapse of heavy stars after the stars have exhausted all the nuclear fuel And you have a supermassive black hole at the center of the center of most galaxies Actually, you don't really know the exact origin of the system But I mean from observation you see that most galaxies have a supermassive black hole at the center and there may be also in a This third class intermediate mass black hole with a mass between the stellar mass and the supermassive one So this is an artistic picture of a stellar mass black hole with a companion star So on the right you have a companion star And then you see the accretion disk, which is yellow, red And at the center there is a black hole. So the black hole is just Stripping material from the companion star The gas Fall onto the gravitational potential of the black hole and you have the creation of this the creation of this accretion disk And what we want to do to test the prediction of generativity Is to verify the motion of the particle of the gas near the black hole And the propagation of the photons emitted by this particle and reaching the distant observer So this is a picture of the black hole, stellar mass black hole in our galaxy With a dynamical measurement of the mass In the sense that you can study the orbital motion of a companion star and you can measure the mass of a black hole There are about 20 systems Which is not much, but these are the known stellar mass black hole in the galaxy with a dynamical measurement of the mass And you expect on the other hand there are something like 10 to 8, 10 to 9 stellar mass black hole in the galaxy So in the end it is very challenging to Find this object And in our case we can test just a small number of objects in the end So from a certain point of view they are very special objects And this picture shows also the typical size of this system So on the top right corner There is the sun and mercury The radius of the sun is 0.7 million kilometers And the distance sun mercury is 50 million kilometers And you can see the size of this system. So there is this senior sex one, which is a binary of a stellar mass black hole with a blue giant And then there are other system I mean some some systems are larger some systems are smaller You cannot see here the black hole because the black hole is really tiny The radius is something like less than 100 kilometer But you can see the accretion disk because the accretion disk can be large for example in the case of grs 1915 It is very large And in our case we are interested in the radiation emitted from the very inner part of the accretion disk. So at Sun kilometers from the black hole And this is the case of The supermassive black hole at the center of our galaxy In this case the evidence comes from the study of orbital motion of individual stars This Plot is the result of something like 20 years of observation. You can study with Newtonian mechanics The motion of the star and in the end you can conclude that at the center there is a large mass in a small volume And so for this reason you say there is a black hole so Typically when we claim existence of black holes, for example the 20 stellar mass black hole in the galaxy or the supermassive black hole at the center of the galaxy or in nearby in nearby galaxies The My typical The argument is that there are some compact objects They are too compact. For example in the case of a stellar mass black hole They are too heavy To be a neutron star in the case of a supermassive black hole They are Too compact to be for example a cluster of neutron star. So there is no National explanation the framework of conventional physics. And so for this reason you say there is a black hole so what we want to do here is To check if the metric around this object is really described by Kerr's solution as expected in generativity And we can do this by studying the motion eventually the motion of the particle of the gas You know the Christian disc and the propagation of the photons And to do this we use this method Which is an x-ray reflections petroscopy So this is a cartoon of our system Uh, we have a black hole and we have a geometrically thin accretion disc The accretion disc emits as a black body locally and as a multi-temperature black body radially when integrated radially You have this corona which is some hot cloud Very close to the black hole even if you don't know the exact geometry of the corona So the thermal photon from the accretion disc can have inverse compound scattering of free electrons in the corona And the corona get this power law component And then the corona can also illuminate the disc we have here some absorption scattering fluorescence emission And we have this reflection component So what we want to do is to study the reflection component Because the other two components are not very informative the thermal component is just a multi-temperature black body spectrum Which is uh, I mean it is a very simple shape And not very informative A power law component. Well, it is just a power law. So there is no really any information about The metric the gravitational field around the core In the case of the reflection component There are some emission line and in particular the iron ke alpha line Which is a very narrow line in the rest frame of the gas But it is very Deformant that I started In when we we opt far from the from the black hole And this is because it is the result of a combination of all relativistic effect so Since you I mean from the observation of this Distorted line you can get information about the metric of a space time. This is the general idea This is a slide to show you some possible Corona model As I said, we know the corona is there because we observe the power spectrum. We observe the reflection component But we don't know the geometry of the corona It is close to the alcohol and it is compact But these are possible options. So you have for example a lamp post corona, which is uh, Just above the black hole We have this sandwich corona, which is some kind of atmosphere about above the a christian disc Or you have this kind of a spherical corona to roll the color For now, and these are just options and we don't know the exact configuration And well, this is another just to show you the The reflection spectrum the reflected component at the emission point where there are several emission lines So the basic idea is this one. So, uh, we know atomic physics So we can predict the reflection spectrum at the emission point We measure the reflection spectrum far from the source and the link between The connection spectrum at the emission point and the reflection spectrum far from the source is due to the Motion of the gas in the disc and the propagation of the photon from the point of emission to the point of detection so if I mean If I mean you you can predict the reflection spectrum at the emission point you measure the reflection spectrum far from the source And then you can verify if the motion of this particle is like the one you expect from generality This is the idea So, uh, to do this test we have a model Well, I before our model There was this uh, reexil, which is Currently the most advanced reflection model for the kermetric. So, um, this is Was developed something like five six years ago by Thomas Douser and avie Garcia And uh, it assume a kermetric This model Has two main block Silver and the right comb. Silver is a model for the reflection spectrum at the emission point So it depends only on atomic physics and right comb is a convolution model in which You assume the kermetric And require as input the reflection spectrum at the emission point And provide as output the reflection spectrum far from the source the one you observe So this is reexil And but people have used reexil in the past few years to measure black hole spin I mean, this is standard physics. You assume standard atomic physics. You assume generativity And you want to measure the black hole spin and also other parameters of of a system Typically, I mean concerning the black hole intrinsic parameter with the call. It is just the spin So this is uh, I have a few slides to show the impact of the main parameter Of the model on an iron line Well, a reflection spectrum You have to fit the old reflection spectrum, but The arrow line, I mean if you see the impact on the line It is easier to understand The impact of these parameters. So this is The arrow line for different value of inclination angle of a disc. So basically you change the Doppler boosting And you can see that for a high inclination angle, which means A jaune disc the The line Go to higher energies and this is because you can observe Radiation I mean the the energy photon come from the very inner part of a christian disc Where the motion of the gas is very high And here you have The impact on the line of the emissivity index. So you need the some emissivity profile For the calculation of a reflection spectrum These depend on the geometry of the corona and you don't know the geometry of the corona But typically if you consider a corona with arbitrary geometry in the sense you don't want to make an assumption About the geometry of the corona You assume a power law and this is the emissivity index in the power law So you you assume one over r to q And here you can see that q for three four five and the other number and As you increase q the contribution from photons closer to the recall becomes more important And this is The impact of a spin parameter on the arrow line. So the Main effect on the christian disc of a spin parameter is to change the inner edge of the disc because well because The innermost able you assume that the inner edge of the disc is at the innermost able circle orbit And the innermost able circle orbit depends on the black hole spin So if the spin increase the innermost the radius of the innermost able circle orbit decrease And so for this reason you can see photon. I mean they They The arrow line get a Low energy tail because you can see photon Very strongly redshifted because they come from the region very close to the black hole So these are the impact This is this is the impact of this spin parameter inclination angle and emissivity profile And If you assume Generativity, you can use this Technique to fit your data and measure black hole spin. So this is a table of spin measurement for stellar muscle color and well in the In the third line in the third column you have the measurement With this x-ray reflection spectroscopy Which is called also iron line method And well, you have something like 10 measurement Some black hole seems to be fast rotating some Not very fast rotating And here you have about 2030 spin measurement for supermass black hole And for the moment the x-ray reflection spectroscopy Is the only technique capable of measuring the spin of supermass black hole So what you want to do is This technique for testing the kinematic And what we have done is to Extend the rexil the new model is called rexi and k where and k stands for non-care And We maintain the silver because we don't want to make any different assumption I mean we we assume Atomic physics is the same And We are we have only difference in the metric of the spacetime So we have a different convolution model In which we do we don't assume the kinematic now there are Two natural approach to test the metric around the black hole And here I call this to approach top down and bottom up So in the top down is the idea is to compare the care solution of generativity With another black hole solution of another theory of gravity. So in this case you have just Two models generativity and the alternative gravity And what you can do is To implement these alternative gravity in our in the model And then you fit your data and you see if the data prefer The solution of the solution of generativity or the spacetime of the alternative gravity And This is I mean sort of very natural approach, but There are two problems first There are a large number of alternative theory. So That's why you should repeat your analysis for every theory Which is well very annoying in the end And then there is an even more practical problem is that typically you don't know the rotating black hole solution in alternative theory And this is because simply because it is difficult to find the rotating black hole solution But this is true even in generativity In generativity the Varshid solution for non-rotating black hole was found immediately after I mean the immediate after Einstein proposed the theory And on the other hand the care solution the solution for rotating black hole Was found more than 40 years after the non-rotating solution and this is just because it is Technically more difficult to find the rotating solution So for these two problems Most people working in the field of test of a care metric prefer this bottom-up approach Which remind the ppn formalism So in the case of a ppn formalism stand for parametrized post-itonian formalism This is was developed in the 60s To test This partial solution in the weak field limit in the solar system So in this case, you don't want to make any assumption about the metric of a spacetime So you write the most general Line element, which is static and spherically symmetric You employ m over r as your expansion parameter Where m is the mass of the sun and r is some radial coordinate This is a small parameter because in the case of the sun Is in any case less than 10 to minus six This is very very if you assume r Is the The the radius of the sun So this is a small parameter you can consider an expansion And you expand your metric And you parametrize your ignorance with some Free parameter here you see beta and gamma and When you in generativity the only A Spherical is symmetric vacuum solution Is this varshild metric and you know that when you write this varshild metric in isotropic coordinates beta and gamma are one And what the idea is to measure beta and gamma from observation And today we know that beta and gamma Are one at the level of 10 to minus 4 10 to minus 5 So eventually you can say that solar system with solar system experiment you can And confirm This varshild solution in the weak field limit with this precision So you would like to do something similar for the car metric Here there are some complication because well because You cannot use m over r as an expansion parameter because m over r is not a small Parameter any longer you want to probe with strong gravity region. So m over r is Back close to one And then if you deform The metric In the strong gravity region it is very easy to create some pathological space time I mean some space time with Naked singularities close time like curves. I mean something with very bad properties So there are a few proposals in the literature and Every proposal has its advantage and disadvantage For the moment We implemented this johnson metric which is one of the proposal in the literature Just to test The car metric with electromagnetic radiation. So for it is suitable for our case And in this case you have some deformation parameters here in this Version of a johnson metric you have epsilon 3 alpha 1 3 alpha 2 2 alpha 5 2 This deformation parameter you want to measure these deformation parameters and if This deformation parameter at zero you exactly recover the car metric And what you want to do is to get some constraint on these deformation parameters And I mean the approach is very similar to the ppm formal is for solar system experiment In the solar system experiment you have this ppm line element and you want to Measure bet and gamma And here you have this extra parameter and you want to see if these Deformation parameters are zero. So they are consistent With the car metric Here there is the impact of a deformation parameter on an arrow line For an inclination angle of 30 degree Here it is for 80 degree, but I mean of course every deformation parameter has Different properties a different impact on on the spectrum So you can imagine some deformation parameters are easier to constrain some deformation parameters are more challenging Probably some deformation parameters Don't have any effect on the reflection spectrum So We have this code This model reaction k We implemented this yarn symmetric And then we try to To fit Some x-ray data to get some constraint on these deformation parameters Now in this So here we have some sources for the moment. We started one h o 7 o 7 and arachelium 564 Which are two supermassive black hole And we have also result on archive of gx 339 Which is a stellar mass black hole And we have some work in preparation with other sources either supermassive black hole and stellar mass black hole So this is the first paper we wrote Uh one h o 7 o 7 is a supermassive black hole and we consider an xm newton observation of 2011 and the constraint on alpha 1 3 are In these two plot and they are to plot just because Well from this just this observation. We don't know which model is correct. So these two These two plot Have some slightly different assumption about the mission of the system And you have Along the x axis the spin parameter the y axis is for the deformation parameter alpha 1 3 You assume the other deformation parameter at zero just for simplicity For alpha 1 3 equals zero you recover the chermetic. So you want to see if your data are consistent with zero Then there is a gray region which is not studied because In that region there are pathological properties close them like curves make you singularities. I mean something you don't like And so you don't want to consider this that region in the In the calculation and the red green and blue curves correspond to the 68 90 and 99 percent confidence level curves for two parameters So this is in the case of xm newton data of 2011 of this source one h o 7 o 7 Here you have the same source but with different data From no star and swift. I mean you combine the data of no star and swift and you obtain this constraint and What we find for the moment is that these are consistent with the chermetic We studied arachaelian 564 with suzaku data And the constraint here seems to be stronger just because It seems that this source is a very fast rotating black hole And when you have a very fast rotating black hole, you can well the inner edge of the disk is very close to the black hole so you can better test the metric in the strong gravity region and also because If you deform car Well, if you deform a very fast rotating black hole Usually you obtain something that look like a slower rotating black hole So for this reason if you find something that look like a very fast rotating car black hole You can get a very strong constraint and this is the case of arachaelian 564 Then we have also this result from gx 39 from In observation of rxd But I mean you have some similar constraint and what you want to do is just to try to get a stronger and stronger constraint and to to see if you can I mean Just to to to see if This result are consistent with zero and to try to push. I mean your measurement I mean to To get a stronger constraint And this is the same gx 39. We run some mcmc simulation. And so we started the correlation among all parameters of the model And then we have this working preparation of with a gs 1354 of nuster data here the constraint seems to be very very good And this is because I mean the the inner edge of the disk Is very close to the black hole and at least for alpha 1 3 This permit us to put a very strong constraint However, I mean if you consider a different deformation parameter it is possible that I mean the constraint is not so good even if I mean in the end as you can imagine different Deformation parameter have a very different impact on the spectrum. So in this case alpha 1 3 with this gs 1354 we can obtain a very strong constraint So I arrived at my conclusion So this is just to summarize what we have done and what we plan to do. So what we have done in the past three years Is basically to construct this model, which is called the rex nk And we have started to Get some constraint from observation of specific sources Either a stellar mass and a supermass black hole So this is a sort of Our broad map for the next five years so first of all We want to improve the astrophysical model because Any astrophysical measurement is as good as the model And in our case there are many simplifications. So for example, the accretion disk is assumed to be geometrically thin And on the equatorial plane on the land you expect that as the mass accretion rate of the black hole increase The thickness of the disk increase and this can have some Introduce some systematic effect Then you have a simple I mean you describe the emissivity profile Of a reflection component with a power law or a broken power law. Even this for sure is an approximation In silver there are several approximations. For example, you have a fixed electron density And probably it is too low in comparison with the possible electron density in an accretion disk You ignore the temperature of Of a of a accretion disk. So basically silver is Mainly suitable for cold accretion disk. I mean there are several simplifications and you want to Remove this simplification to have a more sophisticated model just to Decrease the to reduce the systematic effect We plan to analyze analyze many more data There are many data already in archive in the sense just a previous observation and we have just to Download the data and fit the data with our model And try to get a stronger constraint And also, I mean at a certain point we plan also to Propose a new observation more suitable for this kind of test And then we want to test more deformation parameters and possible also specific alternative gravity For a moment, since we are just at the beginning we focus on this Johansson metric and in particular on this deformation parameter alpha one three and we are already Working on other deformation parameter for Johansson metric, but in general I mean you want to To have a wider idea about The impact of different deformation on your Reflection component and you can put different constraints on different deformation parameters As I said some deformation parameters probably are easier to constrain And other are more difficult and probably Some kind of deformation from care cannot be tested with this technique just because there are no specific signature On the reflection spectrum from certain deformation parameters And I think that is all I can stop here. Thank you. Thank you very much cosimo for this great webinar You can start with one question We have on our youtube channel from Probably pronounced this bad apologies for that shantanu disay And this person cosimo is asking I presume for the LIGO binary black hole detections Haven't most alternatives been ruled out or render really or render it unlikely Okay, so well the point is that electromagnetic test and Tested with gravitational waves are very different because in the case of electromagnetic test like X-ray reflection spectroscopy Actually you you test the motion of the particle in the accretion disk and The propagation of the photons from the point of emission to the point of detection In the case of gravitation waves you have to consider a specific theory because You need to calculate the gravitational wave in that specific theory. So in a certain sense, it is much more challenging to use gravitation wave for For Testing gr. I mean, I would say that the two approach are complementary because they they Test different part of the theory In the case of electromagnetic test you test the motion of particles And for example, you can have deviation from emotional particle. Even if you don't have any deviation from the gravitation wave signal and In the case of gravitation wave signal you test the Einstein equation And You may have deviation from the gravitation wave signal in the gravitation wave signal And you may not have deviation in the electromagnetic spectrum So basically they are two approaches and it is difficult to compare the two either you Consider a specific theory and then you can compare the two approaches. Otherwise typically they measure different things Okay, thank you um, okay, do we have any questions from our uh, low physics coordinators Yeah, I have a question for For costimo, but first of all it's very very nice your webinar costimo So my question was a little bit curious about this when you're in the part of the motivation You were talking about this corona model that you can have the corona on the vertical axis or in the accretion disc Do you know if this all these models are just kind of initial conditions for the accretion in the black hole or is Or do you expect that most of the black hole has this type of corona or you know My question is kind of how this corona get formed Yeah, uh, this depends on the physics of the accretion process, which is uh, uh Not clear in the sense. For example in the one possibility is that the corona is the base of the jet and in this case you have The corona only when there is the jet And this is an option or otherwise it is the atmosphere above the accretion disc The problem is that you don't really know well the physics of the accretion process You see this power law component which is Due to invest competence scattering. So you you expect there is this corona, but you don't know much You can learn more with the next generation of excel emission because in this case you will have also a Timing information of a signal and in this case you can start to figure out where exactly the corona is in the In the system corona and i mean black hole and accretion disc Okay, one more question. I'm sure more or less related with this that i'm asking just now What what happened in the case of the? Composition of the accretion disc in the sense of because most of the these accretion discs are coming from In some sense a part another star that is in a binary system with the black hole So if there is a strong effect because if the start the companion star is to all those two Would you say that more or less because of the determinization can kind of No, no, no it is important the The material of the companion star and Indeed when in the In the reflection model in rex and rex and k You have for example the metallicity of the of the accretion disc So if you have a high higher on abundance You may have a A stronger iron line which may be helpful for example Typically you expect all this system have more less A solar in iron abundance close to the one of the solar system because i mean more or less they are I mean you you observe this system now. So they are not very old or very young system. So typically you expect A typical metallicity of the solar system. However, yes, you may have some difference and this is important I mean you have to take into account the the possible difference of the composition of a star Okay, thank you Do we have any other question from our low physics coordinators? Because what it was i'll i'll start with my questions Okay, i'll go with one Could you please cause me a comment on the mcmc results you run because i believe it's like the first time you have done this type of analysis in your Research, uh, could you please see what was like the the generous here? What would you learn differently from the previous approaches? no, the problem here is just that I mean this model as many parameters and It is usual. It is it is difficult to find the actual minimum of the chi-square when you perform these chi-square statistics so it is So basically mcmc simulations are a sort of more sophisticated approach to To perform this kind of analysis because I mean the The pack the x-ray package we use with these relx and k is expect and I mean It is not very good to find The best the best fit and so on so with mcmc simulation what you want to do is had to study better the parameter degeneracy and the existence of many local minima and just to try to find the true minimum and Yes to have a A better idea of the actual uncertainty in your measurement. This is no actually there are no significant. I mean of course some parameters are Uh correlated, but I mean this is true in any system the problem is just that there are many parameters And mcmc simulation are probably it is a more suitable tool for this kind of test Okay, and then and then a little bit related to To the way you were like choosing sources and the parameter Could you please comment a little bit on the on why You have chosen those particular systems. I believe like as you said in there should be like in the in an archive Many observations x-ray observations, but it's like you have Chosen like particular sources and then you also in your webinar you mentioned like Maybe with those this type of analysis that your group Is performing you might be able to Propose observations that are more suitable. Could you please comment on what like Do we have like a selection bias here and what could be like a more? Yes. Yes. Okay. Yes, sir. Uh well, uh You would like to have a bright source because in this way you have a higher photo count And the statistic is better You want to have a simple source in the sense without many complications. For example in some cases you may have uh absorption from other system Like some cloud between the line of sight of you and the object uh, so Basically, we try to select simple sources particularly bright and also that Look like faster rotating care black hole This criteria actually is a little bit arbitrary just because We knew that for very fast. I mean if an object looks looks like a very fast rotating black hole and it is easier to Constrain this deformation parameter alpha one three. So for this reason For a moment our four sources are biased in the sense they're all fast rotating black hole but I mean Let us say this is a sort of a preliminary List of sources and yes, we want to explore also other Uh sources with the other features Okay, thank you. And my last question really quick is um So so in those analysis you you have at least four parameters I don't recall exactly which one impacts more The particular I don't line But then it is it seems that you just you are just using one of those parameters I understand like it's difficult to combine all of them. So it's it's good It's a good way like to go parameter right parameter. But why did you choose in particular alpha one three and not? Okay. Yes. There are four deformation parameters in these Johansson metric No, here I show you alpha one three because it's the one we started with And we have also uh data I mean we have already some Constraint on alpha two two The problem with the other deformation parameters is that Their impact on the reflection spectrum is very weak I see so basically especially Alpha five two probably you cannot constrain alpha alpha five two at all with the reflection spectrum So we knew this and okay, we started with other deformation parameters But I mean it makes sense, of course Even if we know that the impact this week to try but for a moment for alpha five two, we don't have any any study Okay So do we have any other question? Let me see the youtube channel. I believe no do we have any other question from the coordinators If not, like we basically finished on time just taking into account like our our shift at the beginning So thank you everybody for joining us today We hope to see you like in two weeks with our next webinar. So stay tuned. It could be Over our youtube channel follow us follow us on twitter or stay tuned on facebook to our post Or even in our in our web page to know what's coming up So thank you kosimo for this great webinar and thank you everybody for joining us today Okay, bye