 a few seconds and then, okay, wonderful. I think we are live. Welcome back everyone. Thank you for joining us for today's Love Physics webinar. My name is Alejandro and I'm going to be your host. Today we are presenting the relevance of being a small detecting fundamental fields with Lisa observations of extreme mass ration spirals of embryos by Andrea Mazzelli. Andrea received his bachelor masters and PhD from Lhasa Pienzo University of Rome. He then did postdocs at Pulse Mouth, Georgia Tech, Tubig & Centra, then Sapienza, before becoming an assistant professor at the Gran Sasso Science Institute in 2020. Andrea is an expert on a variety of things, but more particularly on strong gravity astrophysics phenomena starting in black holes and neutron stars interactions. He's interested in the theoretical modeling of these objects in general activity and extensions of it in the study of the dynamic and stability and their phenomena related to them. Today we will have a glance at one of his latest interests. So remember that you can ask questions over email through our YouTube channel or Twitter, Twitter or X. The questions will be read at the end of the talk. So without further ado, we will turn it over to Andrea. Thanks for joining us. Okay, thanks a lot, Alejandro for the very kind. First of all, for the invitation to this. I found this a super beautiful event and the organization is very wonderful. And thanks for the introduction. So what are we gonna talk today? We're talking about small objects and in the context of extreme astratious spirals. And I hope that at the end of the talk we'll be clear why I'm writing here the relevance of being small. And in particular, I'm talking about some future and these are sources that will be detected by hopefully by the Lisa Satellite, which is like a new generation of interferometers that will be launched in the 2037 hopefully and exploring a different kind of range of frequencies compared to what we have seen so far. So just to say that this is the work that I'm presenting here is a sort of line of research that we have launched with some collaborators and in particular with Thomas Sotiriou, a Diversion Nottingham, Susanna Barsanti, Andrew Sparce, Andrew Sposdoch with Thomas and Susanna with my PhD. And then Leonardo Gualtieri, Dan Pisa, Mattela Rocca PhD student and then Lorenzo Sperri and Nicola Franckini, Nicola is in Paris and Lorenzo is PhD at the AI. So I'll cover like this is like more or less some summary of the research line in the last two years which involves so many people. So let me give you a couple of points of motivation. So I think that it's fair to say that at least in the gravitational wave band after decades of being theoretically driven, we live in AIRA which is data driven. We have many facilities, I mean like LIGO, Virgo, Kagura. We recently got the news that India basically funded like LIGO India and that there is the event horizon telescope with fantastically, I mean new about like the last discovery of the confirmation actually of pulsar timing array but that's there are also probes in the electromagnetic sector like NISR and the future possibly is even brighter because there is Lisa in Europe and in US we are discussing about next generation of 3G the instant telescope computer explorer and then there are new observatories in the EM sector like ATIN. So all of these, I mean we'll produce a flood of data we know this and among the many spectacular things that we will be able to perform one of the probably the golden opportunities that we have is to test the natural comfort objects in particular black holes and natural stars and we can use them in order to search for new physics. So at least this is our role. Now there are two key points in the search of new physics that we can exploit. The first one is that in general when we talk about the new physics we tend to believe that this is connected to new fundamental fields that may couple with other forces in particular in my talk we'll talk about new fundamental fields in the gravity sector and then that we have a variety of theories that predict the structure and dynamical evolution of compact objects so we can use them in order to make predictions that will be tested by data. In my talk I will talk about a simple but very powerful like case which is a scalar fields. So extension of general activity in which there is an extra field which couple to gravity sector on top of black holes. And I'm using this because scalars and in particular like scalars are basically a ubiquitous in many extents from GRN of the standard models. So the interest of building models for this particular scenario is basically very wide that embeds a variety of communities. And I will use, I want to talk about how to test new fundamental fields and scalar fields through gravitational waves and in particular using asymmetric binaries. So what are asymmetric binaries? If you look at the catalog of events that we have observed so far through LIGO and Virgo I think that it's fair that we have spanned a relatively small interval of mass ratio, the mass ratio just the ratio between the masses of the binary components. And if I made it the correct number here like the largest mass difference that we have detect so far is one to 30, okay? But 3G detectors and in particular at least I will actually be able to observe and span a way wider mass range which can basically span mass ratio between 10 to minus six and 10 to minus seven. And these are very, very peculiar sources because the dynamics of these sources is mostly dictated by the mass ratio. And these sources feature a very long spiral with the number of cycles performed like in the band of the detector which grows as Q becomes smaller and smaller. So goes into 10 to minus six, 10 to minus seven. Now there are many reasons why these asymmetric binaries are particularly interesting in terms of fundamental physics but there are like three features that I want to highlight here and there are extremely promising in terms of their discovery potential. First of all, as I was mentioning here because since their dynamical evolution is dictated by the mass ratio they have a very, I would say super slow spiral and which means that we will be able to observe like for long periods, these asymmetric biaries performing tons and tons and it will be a bit more quantitative afterward tons of cycles in the band of Lisa. Then because the dynamical evolution of this binary is very rich and it presents several dynamical features that we can actually use in terms to improve parameter estimation because the variability in the dynamical evolution can help breaks the generous in the parameter estimation and then because there are some particular astrophysical setups that actually promote this binary to be golden sources for fundamental physics. And actually the point that I want to make here is that there are theoretical implication from the specific beyond GR models that we construct that actually promote asymmetric binaries to be such golden sources for fundamental physics. Any need, I mean there is a plethora of things that we can do with the asymmetric binaries and I actually will focus on this question here. So if we can actually use them in order to test the existence of fundamental fields coupled in the gravity sector in acting in the universe. So we mostly focus on extremist-rich spirals which are the extreme versions of the asymmetric binaries but some of the things that we are doing we are doing with the other collaborators actually hopeful we'll extend in the less extreme case. Now extremist-rich spiral are the prototype sources that we're gonna sell with Lisa and this is a pure new family of sources that is deemed to ground-based detector. The sources host a primary black hole, a massive black hole which is dubbed the primary with masses between 10 to four and 10 to eight solar masses and a secondary which I call the little guy which is of the order of the stellar mass such that if you cook the mass ratio between the primary and the secondary you have something between 10 to minus three and 10 to minus six. So very far from the one over 30 that we have seen so far. Okay, and this is a key point for our description. Now if you look on the bottom left panel basically what you see is the sensitivity curve of Lisa and on top of these you see some the strain emitted by some sources. For example, there is this yellow to dark red bands which are very massive sources evolving. Then you see like multi-band sources which actually are coming out from Lisa. Then you see the galactic binaries as stars and dot and then I don't have a pointer to point out but if you see right in the bucket of Lisa you see some this kind of red signals which are harmonics emitted by extreme mass ratio spiral. So this is the characteristic strain different harmonics emitted by extreme mass ratio spirals. And if you see they meet there if you look at the frequency band they are right in the bucket of the Lisa sensitivity curve. So millihertz band. And so they are completely obscure deem to ground-based detector. Okay. Now the richness of the evolution here on the left you see basically a sketch of what could be like the dynamic evolution in the last orbits of an extreme mass ratio spiral. So they feature non-equatorial orbits with large HN3 XN3 CT up to 0506. They may feature resonances but the key point is that if you take a standard one-year of observation that we could manage to perform with Lisa and the sources in this interval of time such sources will complete between 10 to four and 10 to five cycles before the plunge. Now this is an enormous amount of cycles of orbits. And as you say in English this is a blessing in these guys. It's a blessing because if you're able to track the dynamical evolution of these sources along all these cycles in principle you can have a very, very precise map of the space time which basically translates into very accurate parameter estimation. And indeed some of the parameter estimation for these sources are spectacular but is that these guys because you need to cook very accurate templates in order to follow and track all these cycles and to avoid miss even just a single of these cycles. Okay. And this is these building waveforms for these particular sources goes into the realm of a self-force formalism which is an approach that have been followed for now two decades, probably a bit more in order to develop a way for models in vacuum GR that will be tested with the least observation of Emrys. Now let me give you a sketch of how self-force is done or the basic ingredients in GR because this will be fruitful for like our description in beyond GR so with Scalafield. So since Emrys have this very small mass ratio basically they can be naturally studied in terms of perturbation theory where the perturbation parameter you can into it is the mass ratio of the binary which is a very small number. So what you do basically is you expand your metric around the background which is generated by the massive black hole. In this case, for example, the care solution basically you can order the solution you can basically cook up the Einstein equation and at the leading order basically will get this expression here which at the leading adiabatic order depending on whether your background is cottage, varsity, or care it will provide two famous sets of equation which are the reggiv wellerzirelli or the Tokersky equation depending on the particular background. Now you need the solutions of the age because you need the phase evolution which is determined by age I'll be more clear in the following but basically the game is that this 5T is basically the orbital phase which is of the if you wish of the second that are going around, the primary object And there is a way basically to classify different contribution to the phase evolution. So basically, at the linear, what I call here, adiabatic perturbation, basically the phase will just depend on this greenish term that I put here, which is called the adiabatic contribution, which scale as one over the mass ratio. So it's a very big number. And then you have what are called first post-adiabatic correction, which are order of one, okay? And basically if you talk with, or actually if you study a bit of that analysis, you see that if you want to track the extremist ratio spirals along all those large numbers of cycles that I was mentioning before, you need this blue first post-adiabatic correction in order to avoid missing cycles and to bias your parameter estimation. Now, the self-force program in GR is now completing this first post-adiabatic correction is I think incredibly tough job, but indeed it has taken more than two decades to compute like this blue sector here, okay? Let me give you why a bit of information on why we need this. If I actually expand a bit more quantitative like the metric tensor here, G around the background and now include also the second order contribution. Now, what I really want to do computing H and what I call the H2 here is basically compute the orbital trajectory of the secondary, okay? This D squared zeta, zeta is the trajectory. And you can actually recast these trajectory here in terms of the self-force components, this F1 and F2, where F1, F2 are what are called like the first and second order self-force term. And then there is a reminder of all mass ratio cubic terms, okay? Now, during the spiral, basically the binary will evolve on terms of a radiation reaction time scale, which is TRR, which basically goes on with this scale M over Q, okay? So if you basically compute the cumulative shift introduced by the self-force correction at the second order, so this F2, basically you see that this delta zeta, so the cumulative shift will be Q squared times T squared which is basically order Q0. Now, if you talk, if again, if you look at parameter estimation, basically the rule of thumb is that if you want to make a good, without being too precise, like parameter estimation, you need a cumulative shift which is less than one radian. So you need to include this self-force correction F2 in order to achieve basically this requirement. But this also tell you that basically high order terms, so what I put here in the reminder that would have been like F3 is actually subdominant because we contribute to this cumulative shift with powers of the mass ratio, which is then a very small number. And so you can safely neglect it, okay? Now the point is that completely this F2 is taking a very long time. Again, I said like more than 20 years and finally like I think last year, it was produced the first, what is called the first post-adiabatic waveform. So basically the one that includes this blueish stuff for two non-rotating black holes, okay? If I'm correct. So basically there is still a lot to do because I mean there has been to included the generic orbits and so on and so forth. Okay, and then again, let me just give you some practical idea of why how to use the perturbation. I'm giving you here a simple example for, I mean, maybe if you haven't seen it, if you want to perturb a partial black hole but the generalization to care is basically very easily. At the linear order, so what I call adiabatic, you can split the perturbation into polar and axial sector. And basically it's very well known from the fifth, the 60s and 70s by this beautiful paper by Reggio Wilhelm Zerilli. If you decouple your perturbation H, basically this you get like several polar component and the axial. This polar and axial just refer to the way that H, pole and H acts transform under party transformation but Reggio Wilhelm Zerilli basically found a particular gauge in which you can reduce the seven and three only to one degrees of freedom for the polar sector and one degrees of freedom for the axial sector. So if you cook up everything into your Aston equation with the source term and the source term is just a small particle going around the massive object. And if your background is not rotating, so it's Varship, you get one equation for the axial sector, one for the polar sector, which are this d r, d square r and d square zeta which are called the Reggio Wilhelm Zerilli equations and Zerilli equations. And then if you are actually perturbing as you should do in terms of realism like a Kerr-Black-Cole, you get through a miraculous decoupling I think as was called by Tchaikovsky. What is the Tchaikovsky equation which tells you the actually allows to compute the perturbation induced by the small particle on top of a Kerr-Black-Cole, OK? Now, why we need the perturbation? Because we need the perturbation to compute the gravitational fluxes, OK? So if you know, for example, zeta for a Kerr-Black-Cole, you can compute the energy fluxes and l, the angular momentum fluxes add infinity and add the horizon, OK? These are just proportional to the perturbation. And the fluxes drive the orbital evolution through the last equation that I put here. So the change in the distance between the secondary and the massive object, the radial distance and the change in the phase is basically driven by this e dot here. And once you know how r and phi change, then you can construct the tubularization and do whatever the parameter estimation has to do, OK? This is what is done in GR. And I put it here, I think, because it's useful for our discussion here beyond GR. OK, now, if I want to study basically extreme mass ratio spirals, and in particular, I try to see whether I can use them to test the, well, the existence of scalar fields, the first thing that I want to ask myself is, are extreme mass ratio spirals sensitive to new fields? And we can actually learn some lessons from the comparable mass systems, or not the asymmetric ones, the comparable mass that we have seen observed in LIG and Virgo. And we know that compact binaries can prove the existence of the new fields. And, pardon, one of the most famous examples is through the emission of, to the dipole emission. So basically, if you introduce a scalar field in the majority of cases, you will activate other channel of emission, and in particular, the dipole one, which is pre-Newtonian in jargon. In GR, basically, this term is not allowed. You start with a quadrupole term, but in other theories where there is a scalar field, basically, there is a pre-quadrupole component, which is the dipole one, OK? And you see in this plot here, I've shown constraints. And the first bar here, when it's written 5 minus 2, is actually constrained, obtained by the different catalogs of events observed by LIG and Virgo on this minus p1, so dipolar term here. So we know that actually we can test the existence of new fields because they will produce a detectable effect on the waveform. And in particular, in this case, I just saw this dipole emission minus 1 term. And we actually have already done it. We already constrained to gravitational wave observation, OK? These come from the spiral phase. But in the last years, I would say like five years, there have been a very active flow of efforts in order to go beyond the spiral description of beyond GR and also to study how the merger and the post merger that actually can be accessed only through numerical relativity beyond GR theories. And there is an active field of research which is actually trying to solve this problem, which is, as well, very, very tough. So this is for comparable mass binaries. So what about asymmetric binaries like extrema-stretchy spirals? Now, before going into this, let me give you a couple of slides. Let me make two steps back to trying to understand whether it's possible or not to test extrema-stretchy spirals, to use extrema-stretchy spirals to test black holes with the scalar fields. Now, I don't want to go too much in details because this is a gigantic field. But in general, what I would say that is not what I'm saying is we are looking for black hole solutions that are different from, that have signatures that allow to distinguish them from their GR counterparts. So basically, we are looking for solutions that have airs. Now, it's not very easy, in general, to cook black holes with airs. And we know from the 70s that there are basically nowhere theorems that, for example, in scalar tensor theories, even with self-interactions, basically provides stationary solutions which are the same as in GR. So GR and scalar tensor theory share the same solution. The theorems, for example, cooked by Stephen Hawking and then by Thomas Sottilio and Varego Farahoni, mostly assume a specific form of the action which provides these black holes, for example, having first derivative product or first derivative of the scalar field. But you can try to introduce more complex model. For example, like some years ago, there was a reviniscence of Ordeski theory, which is a big, broad class of models of scalar tensor theory, which basically embeds models which still provide second-order field sequestration. And even in this case, people have been found in particular subclass to find, like, nowhere solution. What I'm trying to say is that, in general, I mean, you can, even beyond GR, there are basically nowhere theorems which protect your black holes to acquire airs. So it's not super easy to have solutions which differs from general activity. A cracker actually can be used as test bed with astrophysical observation. However, I mean, there are, like, different loopholes. And because the perturbation of a black hole that even is similar to GR in a beyond GR model, this such perturbation can be different. There are notable exceptions as super radians. And then there are loopholes. And indeed, I want to talk to one of these loopholes because this connects to one of the most famous beyond GR models that we have now that have been deeply studied, which is basically this Einstein-Gauss-Bonnet theory, which is basically defined by these actions here. So the first term is just Einstein-Hilbert action here. And then if you see this model modify, like, general activity, because it has, like, a scalar field there, which, I mean, the first term is just the Kinect term. Plus, there is a minimal coupling between the scalar field phi and this G, which is the Gauss-Bonnet environment. Now, this G is just like a complex function of the Riemann tensor and the Ritchie tensor Ritchie scalar. So it's basically a combination of curvature tensor in an invariant way. And if you basically look like derive the field's equation, you get what it's there on the right. This box phi plus alpha Gb equals 0 and alpha is the coupling constant on the theory, which basically enters there in your action. Now, this is a very interesting model for several reasons. First of all, because introduce errors to black holes. And then this is because it's very difficult to constrain these with other classes of objects. First of all, because this capital G, which basically introduced the error, is proportional, as I was saying, as a product of the Riemann tensor. So basically, it's suppressed for weakly gravitating objects, for example, like normal stars. And then also because even for neutral stars, basically deviations from GR in this theory are very subdominant. So the only real object that we can use to test this theory beyond, like I guess, GR are just black holes. And also because if you take these alpha Gb phi capital G, this can be really see as a small coupling expansion of un-general theory. For example, like coming from models of quantum gravity, like string theory. Now, the key point of this theory here, however, is that, and this actually is a feature of a lot of beyond GR model, which have like non-gauge fields, is that there are, in order to have a solution which is stable. So basically, it does not explode in a particular regime. And in this case, at the horizon, you have to impose some regularity conditions. Here, for example, I have the values of the scalar field phi prime computed like solving the field sequestration for this theory. And actually, if you impose the regularity condition at the horizon, which basically means you find a way to avoid the scalar fields diverged at the horizon, so everything is regular, then you find that the scalar charge, which basically, roughly speaking, tells you the amount of air that the black-core air has basically scales in this way. So the scalar charge I will call here after, like d. And basically, this was scared by the coupling of the theory, this alpha gb, divided by m squared. This d is a dimensionless number, because alpha in this theory is a dimensionful coupling in geometrical units, basically, a dimension of length squared, or if you wish, mass squared. Now, this very little relation here, that the scalar charge, fixed by the regularity condition, is proportional to the coupling divided by the mass squared of the black-core air actually has very strong consequences. Because basically, if m, so the mass of the black-core, is the only relevant scale that you have in the game, then if the coupling is much smaller than m squared, basically the scalar charge is very small. So basically, what I'm saying is that if the coupling, which basically tells you the length scale of the new phasing happening here, is way smaller than the mass squared of the black-core, then the charge, so the difference from this black-core from the GR counterpart is extremely, extremely small. Okay, and this is actually the point if you want to test, like with this, with the Gauss-Bonnet, and with theories, which actually there are many, which feature basically the same kind of coupling that you have to accept that if there is a scale, like this alpha, which actually may be already be bounded or constrained by other astrophysical observation, then not all objects will actually be and though the same scalar charge, because this scalar charge will depend on the mass, so for example, more massive object will have very small mass scalar charge and lighter object will have a larger scalar charge. And indeed, now let me come back to the question of whether, like extreme mass-strategy spirals are actually sensitive to new physics through the scalar field and actually scalar fields, and actually after D, it may be extremely empty to answer no. First of all, again, because in many scalar tensor theory of gravities, black holes are actually protected by newer theorems, so they are the same in GR, and then because for any black holes, there are many theories like Gauss-Bonnet that have presented for which basically the new field coupled with what are called high curvature term, so these are basically proportional to product of the Riemann tensor and they feature dimensional couplings. So if I take these theories, basically the GR deviation as we've seen scale basically with the coupling or here a Britten generically the length scale divided by M to some powers, okay? And then if the length of the order of M basically the air is basically suppressed, okay? Now this happens precisely for all the binaries that we will observe with the future satellite, Lisa, because basically Lisa has roughly speak, I mean, we'll observe, there are let's say two big families of object that we will observe with Lisa, which are very massive binaries and then extremist stretch spiral and intermediate mass stretch spirals, but because of this suppression here that I'm showing for theories which coupled with higher curvature terms, the very massive guys that will be in comparable supermassive black holes or that will be at the center of asymmetric binaries will actually feature a negligible charge, okay? What I'm saying is that with Lisa, for example, we will observe the binary spiral of supermassive black hole with extremely large SNR with the order of 1000 or even more, okay? These, I mean, from outside could be seen as like general and golden sources to test like fundamental fields and deviation from GR, but then if you look at the theory that predict deviations, then you see that these objects for which there are two supermassive black holes actually have, for this particular systems, theories predict a very tiny amount of charge, so deviation from general activity, which is gonna be hardly beat by like the large SNR. So if you take Lisa, there is one particular families of system that is actually, instead, is prone to be used to test general activity and these are asymmetric binary and extreme mass ratio spiral. And this is because in extreme mass ratio spiral and the asymmetric binaries, you have a central black hole, which is still very massive. So for this primary massive black hole, the charge or if you wish to deviation from GR will still be very small, but there is the little guy. The little guy is a stellar mass object. The mass of this object is basically comparable to what we have observed so far in the LIGO band. And for this object, this suppression will not work, okay? And these, the stellar mass object will have like a non-negligible scar of charge, a non-negligible deviation from GR that will leave an imprint in the dynamical evolution of the binary. There is, of course, like some exception. One of these is in this red box. I mean, we can talk later if you're interested. And so this is basically why, I think, extreme mass ratio spiral are particularly interesting. Now, before I go now into the setup of this model that we have devised with Thomas and others, let me say that while for comparable mass systems, there is a clear way, and I show you like an example that I've been used with LIGO and with observation, to model wave force beyond general activity to test the existence of such fundamental fields. In asymmetric binary and extreme mass ratio spiral, the field is basically virgin. So besides this setup and this model that we have cooked with Thomas and others, they're basically nothing in the market. So there are no waveforms. And we are attempting a bit to bridge at least to fill the hole of this vacuum. And it is extremely difficult. And if you think that 2037, so when Lisa will fly, is very far in the future, just think that the self-force in vacuum GR have worked for 20 years and they still don't have full waveforms to be ready for that analysis. So it's actually very timely and we need to start working in order to produce waveform models, okay? Now, I want to tell you that if I then assume a particular model for my beyond GR models, there is a very nice way to produce with minimal cost model for extreme mass ratio spiral and asymmetric banners, which are almost ready to be used like for of course, mock parameter estimation and data challenges and blah, blah, blah. So my model is basically defined by this action here, which basically has three term. The first one is like Einstein, Albert Einstein action plus again, the kinetic term for the scalar field plus have these term here, which I'm calling no minimal coupling and these term here as in front like this dimension pool coupling, which I will assume in my model and then there is matter field, okay? Now, alpha for the geometric units are dimension of length to some powers of N, okay? Gauss-Bornet that I showed before is a particular example that belong to this class of general theories with N equal to, but I mean, I'm actually considering any theory in which the length, the alpha is basically dimension of length power of N, okay? And if you turn like these, this coupling in particle physics units, you basically see that this as negative mass dimension. So basically I'm working a sort of effective filter approach in which basically deviation from GR are actually suppressed by some particular energy scale, okay? Now, I have an isometric binaries for me, the primary black hole, the primary object is a black hole, okay? And S0 is protected by Noir theorem. So this will just give black hole as in GR. So if you want to introduce air, basically everything needs to come from this green arrow, which is the nominal coupling, okay? And actually this is a technicality with a document later, but in particular, if you're considering like shift symmetry, so for which like basically theories which are invariant, if you rescale pi by a constant term, basically you see that the only way to introduce air through SC is just through the Gauss-Bonne coupling, okay? So this is the only term that introduce a scalar air, okay? And for this model as we have seen before, the scalar charge is actually not an independent parameter, but depends on the coupling on the theory divided by the mass square, okay? Now, you can actually, if you are working this shift symmetric model, you can actually introduce more shift symmetric interaction in SC, the air will still be given at the linear order by like the Gauss-Bonne air, okay? And if you want to introduce other shift symmetric theory, there is a shift symmetric interaction, there is this paper by Thomas and Postdoc, make this a Ravani, that basically show that if the extra terms that are controlled by new couplings alpha, that I call alpha e, will actually correct the scalar charge by factor which are the old Gauss-Bonne times alpha, this new coupling. So we'll be further suppressed with respect to the linear term induced by the Gauss-Bonne coupling so they can be safely neglected, okay? All of this is for my primary. Then the secondary, the little guy, okay? So this is the mass, which there's more guy which is going around the massive black hole. And in this case, I mean, to model these, I'm using like what is called like the skeletonized approach, which was in basically develop long time ago by elderly and then there is this beautiful paper by Damour Esposito Farese, which basically provides the theory for like multi-scalars and gives a very nice definition of like this, this kind of skeletonized approach, which basically turns the matter action into a point particle action. So basically what you do is that you treat an extent, the extended body, which means my secondary, which is going around as a point particle, but the game is basically you replace the mass of the particle with a mass function. And this mass function now depends on the scalar field, okay? Which is given basically here by this m of five, okay? So this is basically the usual action for a point particle, but m is not constant, but depends on the scalar field. And then you can actually derive your equation and the scalar fields equation basically will be these and will be sourced basically by derivative of the scalar field mass function. Okay, now where are the simplification from this model because of the suppressions that we have discussed so far. So this suppression that comes because basically the air will always scale as the coupling divided by powers of the mass of the black hole. Basically, if you are assuming that your theories are continuously connected to generativity. So when you send your coupling to zero, alpha to zero, basically you recover GR solution. So Kerr black holes. Basically there is only one way to modify your metric and this is true this parameter zeta, which is alpha divided by the mass of the black hole, the big black hole, the primary to some powers of m. So this is proportional to the charge. And you can actually recast this parameter q in terms of zeta, in terms of the mass ratio. So basically if you multiply and divide by small m, which is the mass of the secondary object, basically this zeta then becomes qn the mass ratio times alpha divided by the mass of the small guy powers of m. Okay, now here I'm conservatively assuming that the ratio of alpha divided by m powers of m is of the order unit, which means that basically I'm assuming that the alpha as the length scale of a sterile mass object, which is particularly welfare because otherwise we would have seen deviations already in other astrophysical observation like live and Virgo just to tell you for example, this is the case for Gauss Bonnet for which actually alpha as dimension as the length of a sterile mass objects. This also means basically that since deviations from gr are just encoded in this zeta, which can be recast as the mass ratio, then actually put this guy here into my perturbative expansion for asymmetric binaries and still use one single parameter that I was using before in vacuum gr, which is the mass ratio, even to treat the deviation from generality. So any gr deviation will be controlled by this parameter, which is smaller than one because it's proportional to the mass ratio q actually powers of n, okay? And then because of this suppression, the air, the central black hole actually can be considered as care. Since as we discussed before in these theories like the charge or the deviations scale as zeta for the central bell hole where there is this capital M, all deviations from gr are basically negligible, okay? So basically then you can run all your perturbative framework and you can still use as a perturbative parameter your mass ratio with just enter now again into the perturbation induced by the small particle but also in the negation of gr and you can compute first and second order correction and in principle play the game that has been done in gr that are just for also for beyond gr model, okay? And then you get like at the linear order we are actually developing given the second cell force equation but for now let me just discuss about results of the linear level so what I call the adiabatic order. And basically if you may call this machinery, I mean I can actually answer question about technicality later but you see that the fields equation becomes this one. So you have like the Einstein equation but then the scholar fit the question there, okay? Now there are two points here. The first set of equation, the Einstein equation there are actually exactly the same adiabatic order. So the linear order that you would have in general activity. So basically on the left side, there is something that doesn't need to be solved because it has already been solved like so far you know the solution, you know the fluxes, you know everything. So the only things that you need to solve is the scholar field equation delta phi and the box phi. And basically the solution there besides this integral depends only on one single parameters, this D which is the scholar charge of the secondary object. So let me summarize in this framework in which I'm considering theories which actually cup in which like the scholar field couple couples with the curvature invariance sub all the deviation from GR of the central object are completely suppressed, okay? And indeed this reflects on the fact that the linear order, the Einstein equation are the same as in GR. And the only things that you need compute or to solve with the scholar field equation where the deviation from GR will appear in D which is the small charge of the secondary object which has a stellar mass size and so has a non-negligible charge. So all deviation from GR in the evolution of the system will actually be controlled by this D, okay? And then basically you can play again the game instead of having only this what I call E-gravitational which is the old gravitational flux is that again are the same in GR. Now you have also to add like the scholar field contribution. So what I called E dot scale which is the scholar field induced by the scholar flux induced by the scholar field and at this level the total energy emitted at the horizon and at infinity we just be the sum of these two contributions. So the old gravitational part and the new scholar part, okay? Now let me tell you as a general consequence is that when you actually add the fields what happens in general is that the binary accelerates because I mean you activate new fields new channels of emission. So the binary is losing more and more energy than I mean the usual GR case. So it's basically speeding up to the equal essence, okay? And then you can actually compute in these frameworks the frame of the waveform. So you can actually follow the same procedure that I said before. So compute the total energy flux which is given by the old GR plus now the correction given by the scholar field determine the dynamics and build the gravitation of a polarization. Now because of all the suppressions that we have seen before and this decoupling between basically the scholar field and the tensor sector basically if you look to this point one, two, three, four basically all of these is exactly the same as in general activity at the linear order. There is only one thing in these procedures that differs from things that have been done so far in GR which is in the computation of the fluxes because together to the well-known E dot GR you need to compute the square delta E dot and then everything is already ready in order to produce adiabatic waveforms for extreme stress spirals to be tested with the four parameter estimation, okay? And even the waveform polarization are actually the same because at the leading order the ison equation are exactly the same. So you have an universal family of waveforms that can be test against GR and actually depend on a single parameter which is the scholar charge. So in this sense also we have cook up a framework which is completely agnostic. If you realize basically we have progressively lost contact with the specific beyond GR model which was introduced by in the action because now everything is then into the scholar charge. I'll come back later on this point. Okay, I don't know how many minutes I have but I can actually now give you a practical example of how to use, okay, 10 minutes. Yes, some this framework basically to test general activity. So the first thing that you can do basically is in order to quantify the difference in dynamical evolution of an extreme stress spiral in GR and beyond GR is called what is called the defacing. Okay, so basically you compute the orbital phase in GR you compute the orbital phase evolution or gravitational phase evolution beyond GR and you see the difference in the two. Okay, now there is a rule of thumb which is identified by these horizontal line here and which basically tells you that everything so differences above that line in principle will allow to spot deviations from your vacuum GR model. Okay, what I'm putting here is basically this defacing for a prototype system that should be observed hopefully by Lisa with a system of 10 to 6, 10 solar masses and spinning black color at the center with spin parameter at 0.9. What I'm looking here is an observational wind over one year. So the zero here means on the x axis means the planche and each curve corresponds to a different values of the scholar charge. Let me repeat again of the secondary object. The primary massive black hole is basically uncharged. So it's like of the one is a curve black hole. Okay, and this is done. This horizontal line depends on the signal to noise ratio so the strength of the signal in your detector and basically you see that of course if you increase the charge the defacing increases and you get farther and farther from the horizontal line but basically this plus tells you that potentially Lisa will be able to the spot deviations in the evolution of extreme mass ratio spirals for charges of the order of 10 to minus three here this five, 10 to minus three. Okay, however, this defacing here basically doesn't take care of the correlations between the parameters is just the difference in the orbital phase. So in order to really understand whether you can actually disentangle this charge. So basically, because what I want to measure is the parameter that deviates from GRD that would be zero in general activity. I need to consider a most sophisticated approach. So a real parameter estimation that actually takes into account the correlation between the other parameters of the binary, the masses, the spin luminosity distance, inclination and all the other angles in the game. And indeed, with this, and in particular this plot here forecast basically the accuracy, this relative accuracy on the scalar charge that you will get on two prototype system with different signal to nurse ratio would consider like the SNR equal 30, like a pessimistic case. And I don't want to call like this optimistic, the purple line, but a regular case with the 150. And basically what you see is that the extreme mass ratio spirals will actually measure the scalar charge with the percent accuracy and better of course, depending on the value of the scalar charge. And in the bottom panel, basically what I plot here is basically the value of the measurement plus the three sigma and just to show that basically Lisa would be able to exclude the zero. So the GI hypothesis and more than three sigma for scalar charge of the order of 10 to minus one. Okay, now all of these basically tells you that this can be this framework that we have developed which is very general because it embeds a lot of theory can actually be used to measure to spot deviation from GR measuring the scalar charge agnostic only. Okay, so you don't have to do anything. You have your way from you measure a number which is the scalar charge. And then, but then the only things that you have you can say is the system that I have is different from GR with a certain level of confidence. However, you can actually trace back your scalar charge to an actual theory when you have a map between the scalar charge and the fundamental problem of the theory. And for these reasons, let me introduce again this Gauss-Bohm and gravity theory for which like this map has been computed by Emanuele Bertie, Felix, Julie, and the John Hopkins. So again, let me repeat some property of this theory. This is a theory which modifies like in the stronger but you're like the general activity and in our languages. So for alpha, which has dimension full couplings, this is length squared and deviation from GR in terms of our zeta parameter are controlled by this q square and then this alpha divided by the mass of the small guy. These f of five is a generic function of the scalar field. So indeed like Gauss-Bohm gravity is a family of theories depending on the way you choose the f of five. And then, as I was telling you, this Gauss-Bohm invariant is basically this combination of Ritchie scalar, Ritchie tensor and the Riemann tensor. Okay, now let me take this example for these two values of f of five, what is called the exponential coupling and the shift symmetric linear coupling, this E phi and the f of five. And basically you see here that basically Emanuele and Felix computed the map between D, the scalar charge and beta, beta just I renamed this alpha divided by the mass of the small guy. So basically there is a map between the scalar charge and alpha, the fundamental coupling of the theory. Okay, so the game here is that you can use the way from what I told you before, you can measure the scalar charge, which is a number which pop ups in your data, maybe different from zero, maybe not. And then you can actually translate the bound into actually a constraint or a bound on the fundamental coupling on the theory. Okay, I have very like probably in the last couple of slides here, just to tell you that this is an example on how actually to map the charge in the Gauss-Bohm coupling. And to do so, we've actually used the fast entry waveforms which is a pipeline that has been recently developed by Michael Katz and other people, I think at the Einstein Institute in Berlin. So let me stress you that the computing waveforms even just in GR for extremist spirals is not only difficult, extremely difficult and indeed it's an unsolvable problem, but also producing the end waveforms is extremely difficult because the longest spirals makes generating this waveform extremely difficult in terms of computational expenses or requests. So you need some particular breakthrough in this which is actually brought by fast entry waveforms which makes like neural network interpolation and other techniques to produce fast waveform models. And these after decades was only produced in 2021. And now is the standard in the LISA community. We have implemented our waveform in this pipeline here you see, for example, how to map the constraint that you get in the charge for different system. The labels here refer to different values of primary, the secondary, and actually you already have eccentric waveforms, the third number between parentheses here is eccentricity, 0.1, 0.4 of the system and how this turns into map of the Gauss-Bohm coupling. I don't want to get into details, we are coming out for a paper in the next three weeks. So, but I mean, I just want to tell you that we have an agnostic frame that actually can be turned into a fundamental physics tool to probe specific models, okay? And this is actually the status of every waveform models in which we are now trying to implement like care circular of equatorial waveforms beyond generativity with the scarlet field. Okay, let me tell you just last thing is that in principle everything that we've done so far actually has been done only for a massless field but with Susanna Barsanti actually led the work which is PhD student in Rome, we actually generalized this model in order to include massive scarlet fields which are actually defined by this action here. And these are a specific model with particular features which are new features which are excited like the massless case. And in particular one of these is actually that if you take a massive scarlet field like the scarlet flux that you get at infinity vanishes in some particular regions of the parameter space and in particular for frequencies smaller than smaller than the mass of the scarlet field, okay? Now, since the flux at infinity is always the dominant one compared to the horizon fluxes and basically drives the spiral, the question for like massive tier is massive scarlet field has always been whether like this suppression was actually too strong to actually forbid to use these systems in order to test generativity. So what we did again, we solve like basically these equations here, we compute like the scarlet field equation now depending also on the mass of the scarlet field we compute waveforms and then we got bounds which are depicted in these figures here, okay? So again, these are basically the counterparts of the bounds that I've showed you before. Again, this is the usual emery system with a 10 to six solar mass black hole at the center spin of 0.9 and then like different colors here refers to different colors refer to different values of the mass of the secondary from 1.4 solar mass to 15 solar masses, okay? And basically the message here you see that it's a bit of a pity I'd say that like the smallest cutest guy, this could be like an outro star basically it doesn't lead to any constraint because basically the constraint this black curves here are completely compatible with null hypothesis. So basically with the scalar charge being zero which is general activity, but you see actually that for other system you make a measurement and you actually you make a good measurement, okay? The right hand side plots show basically the marginalized posterior distribution for the charge and the mass of the scalar field, the charge of the secondary and the mass of the scalar field. And if you see basically in the majority of these you don't see that zero. The white region within like the color bands is the 90% confidence interval of the distribution and you see that basically the zero is in at least for the 10 and 15 solar mass secondary is basically excluded by this zero. So basically the measure of these is that Lisa could actually simultaneously constrain the scalar charge and the mass of the scalar fields to be different from zero with very good accuracy with observation of extremist species parallel. And this is basically was completely unproved before. Okay, this is my summary. So actually with this which is our extremist species parallel and I would say like asymmetric binary sensitive to new fields, the answer is yes. And actually for a period driven by data we can actually use theory to drive our way of modeling waveforms and the dynamical evolution of extremist species spirals because of this theory which actually predict the suppression of the air of overall the scalar charge as I said for very massive object. This leads at least to adiabatic order to very universal family of waveforms to be tested with GR and actually we have showed that Lisa will actually provide incredibly good constraints on the charge and actually also on the mass of the field in massive theories. There is basically tons of things to be done and completely unexplored which is computation of next to leading orders of force terms. So what I call like the post adiabatic order which are actually needed to make real parameter estimation, other fields vector fields and also correlation with other astrophysical effects also include generic orbits and in particular the effects of resonance. And I would say that this is all and thanks a lot for the attention. Thank you Andrea, wonderful webinar. I already received a question but I think you address it at the end of the talk. So let me just read the last part. So what are the main difference between considering massless or scalars with mass? If you can handle that. I can actually answer this more deeply. So one of these as I was mentioning is basically these vanishing of the flux for omega smaller than the mass of the fields. This can actually be recast in terms of like specific orbital radius. So basically this black point that I put here means that for any combination of mode there is a particular radius for which for orbital distance larger than this radius the scalar flux at the infinity just dies. And indeed in this plot here, if I remember you see the scalar flux and the continuous chain line is the massless case. So basically you see a continuous of the flux emitted but then for the other scales as you increase basically you see that for particular radii basically goes to zero. There is a second effect which is also very important which is the one which is called floating orbits. So basically during the orbital evolution you can have like the excitation of the quasi normal modes of the scalar field. And some of these in particular case in which basically when you get in resonance basically the amplitude of these excitation can be large enough to completely counterbalance the gravitational emission. So basically you have a cancellation between the scalar flux which gets amplified by this resonance and the gravitational flux. And so the binary just gets stuck. So basically what you have is what are called floating orbits for which basically the binary slows and slow downs until basically it halts completely. These are the two main effects. Okay, there is always a delay so maybe there is if there is a follow up question I'll let you know. Yeah, is there a question here from one of the coordinators in the call? Yeah, I have a question for Andrea. Very nice to talk. I was wondering when this object appears do you expect that there is kind of let's take on how to explain it like do you expect that this object are more frequent in the early stage of the universe or do you expect that there are kind of whatever there are these two objectives they could be the signal or I mean because of the big difference of masses or something like that, not because of the scalar field. I would be universal for all. I don't know actually how to answer to this question because I mean one of the weak point of all the embryo point of the program is that basically there are no rates. And so basically we can go for one to hundreds of events, detected rates. And basically it depends on the formation channel. So whether basically there is some sort of capture in vacuum or if there are dense environments with matters or other objects that actually push the secondary close to the black hole. What I'm saying is that I don't know whether they will appear but probably there is an action effect in the sense that we're gonna see this around probably zeta one two just because otherwise the SNR will be basically too low and this will just basically go below like the least as sensitivity curve. But in terms of where I mean in terms of time basically everything is very uncertain. Okay, thank you. There was actually a follow up question but then I would like to ask here. So there is one, I don't know if how crazy I know we don't have the rates Andrea but then in principle you can also have a triple hierarchical system. I saw I think it was a nature a couple of years ago where you have a super massive and then two small ones. Do you expect something interesting for this type of test if these two guys are, they carry some scalar field? There should be some. I think it would be super interesting. I mean already in GR because you have a lot of effects like a center of mass Doppler effects and whatever which are basically induced by the relative motion of the binary per se if you're meaning this in the field like of a biggest guy for example being the Christian disc. And I guess all of these will just be promoted in the context of like beyond your model. So these are incredibly interesting sources which are very difficult to model. Yeah, because of these rates maybe this is also valid to study and consider this is a scenario where this is actually and actually they evolve probably in this again in dirty environments like Christian disc. And I guess it's called like wet environments opposed to like the clean pure vacuum. And it could be that actually that in this particular system as I was saying actually the rate will actually become because some effects are actually pushed like the binaries to the center. So we'll actually be more in the regime observed by Lisa. Yeah, okay, yeah. Thank you. And then the follow up question was like for this slide I guess that you're showing is this is for ultra massive I guess it's ultra low massive scalar fields, right? This is very, very, yeah, yeah. All of these basically, well this one and the constraints here are 10 to minus 16 electron volt. And so these are ultra light fields. And also because bigger values actually would be forbidden by super radians basically would be excluded by super radians. So more massive fields were actually for this particular combination of 10 to six or massive will actually produce super radians. Okay, we'll see. Thank you. I'll let me see if there are more questions. Okay, we're past the hour. So I'm gonna try to also ask here and there is there a way for you to explain at least intuitively why more massive here? I mean, there are a couple of things I find here interesting at least like you see this this constraint that you're putting is in some sense changing rapidly with the mass, right? Like you go from the orange blood to like you just double the mass and then it's like way more constrained. But then also on the other hand, there's this competing effect which here I believe is very not dominant which is like now you have a more massive black hole where they actually the charge should be. Okay, great. This is a fantastic question. Fantastic question. So let me say this. So there is actually and this is actually something that we're trying to investigate. And first of all, the so these that you see here it also happens for like massless fields. Okay, so that in terms of constraints on the scalar charge you get better constraints in the less extreme cases. And I think that the explanation for this is basically there are multiple way up to see this. So if you fix like the mass of the central black hole if you increase the mass of the secondary, okay? The number of cycles becomes smaller. Okay, so you lose a bit of cycles but the SNR increases and also I would say like the flux increase increases because the flux scales as the mass ratio squared. So basically since pushing up like the mass of the secondary basically makes Q larger and larger. So there is a competition between these two which actually, I mean there is probably a suited spot for which at some point you cannot go too high in the mass of the secondary because then you lose too much orbits which we're trying to explore. But let me also make the points from these is that then when you, and this is actually why it makes extremely interesting when you try to map the charge to the coupling, okay? Then again, the lighter objects wins again. The lightest object wins again because basically the charge if you see for Gauss-Bonne here scale basically is alpha divided by the mass squared of a small object for Gauss-Bonne. So lighter is the secondary, biggest is gonna be like the this contribution. And indeed, let me see if I can, if you see actually the map here between the scalar charge and the secondary, let's take for example like the green, actually the red and yeah, even the, no. What is it? Yeah, okay, the green and the red, okay? Which are basically the only things that differs from them is like the mass of the secondary. The constraints on the scalar charge, scalar charge of the secondary are very similar. Okay, you see there. But then when you map then for Gauss-Bonne since the greenish guy is lighter, which means that the total scalar charge will be, or if you wish alpha will be like better because I mean of the suppression of the mass scale. So, and you see here, then the constraint on alpha will be way better than the red one. Okay, great, great, thank you. I have more questions, but I guess I'm gonna let the last one here that I got is what does your GR say about the structure of the secondary in terms of spin cell force and how these effects will compete with what you are describing? Yeah, so the spins basically enters into the post adiabatic term in the face the blueish one that I put somewhere here. And it's not clear actually whether it's gonna be measurable or not. We have done also some impure GR. We have done basically some works actually with the group in Rome, but it's basically could be like the effect to be too tiny to be measured. While let me say that the parameter of the spin of the primary will be measured with extraordinary accuracy. Okay, I think that in some cases with rapidly spinning black holes can be made the spin of the primary can be measured with like one part of 10 to five, 10 to six. So basically marvelous accuracy. If these are very, very fast spinning larger than 0.9, the spin of the secondary, which again enters like in the blueish thing here probably is not measurable. At least we haven't found a way to disentangle it from the other parameters. Other order effects, for example, like the quadruple of the secondary. So for example, if you have a neutral star and like you have like a spin induced quadruple moment we go on to post-post adiabatic correction. So basically it should be even in this case like subdominant even with these green and blue stuff. So should not contribute at all to the parameter estimation. Okay, wonderful. Thank you, Andrea. Thank you everybody for joining us. Let us see in the next webinar, which is going to be the 150th and we have a very, very another superb speaker for this occasion also. So thank you, Andrea. Thank you everyone for joining us today. All the best. See you soon. Bye-bye. Cheers.