## Final Report Summary - LISASIMICO (LISA gravitational wave astrophysics from simulations of inspirals of compact objects into massive black holes)

The main aim of the LISASIMICO project has been to contribute to the development of an emerging scientific area in the interface between astrophysics, cosmology, and fundamental physics: gravitational wave astronomy. The foundations of this new area lie in the General Theory of Relativity of Albert Einstein, which predicts that time-dependent distributions of mass / energy produce gravitational waves in the same way as moving charges generate electromagnetic waves. Then, by detecting and analysing the gravitational radiation coming from different astrophysical sources we can in principle develop a new kind of astronomy. The LISASIMICO project has focused on the low-frequency part of the gravitational-wave spectrum (around the milihertz) where there are plans for the construction of a space-based gravitational-wave observatory, laser interferometer space antenna (LISA); until recently a European Space Agency - National Aeronautics and Space Administration (ESA-NASA) mission and currently an ESA-only mission under the names eLISA /NGO). The main target of the project has been the study of one of the main sources of gravitational-waves that a detector like LISA is expected to detect: The inspiral, due to gravitational-wave emission, of binary systems whose components are a stellar-mass compact object (a white dwarf, or a neutron star, or a stellar black hole) and a (super)massive black hole sitting at a galactic centre (with a mass range between hundred thousand and ten million solar masses). Due to the extreme mass ratios involved they are usually known as extreme mass-ratio inspirals (EMRIs). They are long lasting sources and very relativistic since the stellar compact object can spend of the order of years in the strong field region of the supermassive black hole. During the last year before the system plunges, an EMRI system performs more than a hundred thousand cycles inside the LISA band, which means that the gravitational waves emitted by an EMRI contain very detailed information of the geometry of the supermassive black hole. Observations of EMRIs will provide invaluable information to understand a wealth of different subjects, from the dynamics of galactic nuclei to the geometry of black holes and the validity of general relativity, including constraints on galaxy formation models and measurements of cosmological parameters.

A large part of the efforts of the LISASIMICO project have been devoted to the modelling of the dynamics of EMRIs to produce theoretical gravitational waveforms that will be used to extract the EMRI signals, which will be buried in instrumental noise and the gravitational wave foreground, and to determine the physical parameters of the system. First of all, together with Priscil.la Canizares Martinez (a Doctor of Philosophy (PhD) student initially funded by the LISASIMICO project and later by an FPU fellowship of the Spanish Ministry of Science and Innovation) we have developed a new technique to model the evolution of ERMIs. In particular, it allows for a precise estimation of the self-force, a local force associated with the gravitational field created by the small compact object as it orbits the supermassive black hole, which is the responsible for the inspiral of the system. This technique, which we named the 'particle without particle' (PwP) formulation of the ERMI problem, is based on a division of the physical domain in which the stellar compact object moves around the supermassive black hole. This domain decomposition is such that the stellar compact object, described as a particle (we can neglect its internal structure due to extreme mass ratios involved), is always located at the interface between two subdomains. In this way, we have showed that we can get rid of the singularities associated with the particle description and have equations that have smooth solutions. Then, we have used pseudospectral collocation methods (PCM) for the numerical solution of the equations which then provide exponential convergence of the numerical computations. This makes our numerical computations the most accurate ones in the field for time-domain simulations. In this method, the stellar compact object, the particle, has been substituted by boundary conditions on the smooth solutions at each subdomain. We have developed a complex but well-structured numerical code using the C language and we have first computed the self-force for the case of circular orbits around a non-rotating black hole. We have shown that, to a high degree of precision, our results coincide with other computations in the literature. Afterwards, we extended the computational infrastructure to generalise the PwP method to the case of generic eccentric orbits by introducing a new technique of moving domains. The results of this pioneer work have been corroborated a posteriori by other authors. We have also tuned and improved the technique and numerical codes for efficiency and precision, establishing the PwP as an outstanding tool for self-force computations for EMRIs. In relation to this, during the LISASIMICO project we conducted an investigation of a controversy in the community on the possibility that time-domain computations of the self-force may be contaminated by junk solutions that are difficult to identify, which would mean that these computations may not be reliable. We found a flaw in the argumentation and concluded that these junk solutions are due to inconsistencies between the initial conditions and some of the equations and hence, they can always be safely avoided in numerical computations.

Apart from these high-precision simulations of ERMIs, we have developed a completely new scheme for the construction of approximate waveforms for EMRIs. The idea is that once we will have a complete knowledge of the gravitational self-force, we cannot integrate the full perturbative equations with the self-force to obtain the gravitational waveforms because that would be computationally very expensive. We rather need to have a phenomenological and efficient scheme fitted to the self-force results that provides these waveforms in a reasonable time. Although there are several such schemes, they are either not very precise or they only cover a sector of the configuration space and are not easy to generalise. Our scheme is a hybrid method that combines tools from different approximation techniques in general relativity:

(i) a multipolar, post-Minkowskian expansion for the far-zone metric perturbation (the gravitational waveforms) and for the local prescription of the self-force (both in the Lorenz/Harmonic gauge);

(ii) a post-Newtonian expansion for the computation of the multipole moments in terms of the trajectories; and

(iii) a Barker-Henderson (BH) perturbation theory expansion when treating the trajectories as a sequence of self-adjusting Kerr geodesics.

Then, the EMRI trajectory is made out of Kerr geodesic fragments joined via the method of osculating elements as dictated by the multipolar post-Minkowskian radiation-reaction prescription. Apart from the generation of waveforms, the local character in time of our multipolar post-Minkowskian self-force makes the scheme amenable to study the possible appearance of transient resonances in generic inspirals, an important question that we will study in the future.

Another line of work within the LISASIMICO project has been LISA phenomenology, that is, the exploration of different scenarios of ERMIs and the different science that we can extract from EMRI observations and the quality of this information. For instance, we have explored a system similar to EMRIs, in which the stellar compact object only does a few passages around the supermassive black hole before being lost due to scattering with other stellar compact objects. The interest for these systems, which were called extreme mass-ratio bursts (EMRBs), arises from the fact that there will be much more EMRBs than EMRIs although many of the EMRBs will not be detectable. We have analysed how relativistic effects in the modelling of these new systems can help in their detectability and the prospects to extract useful physical information (like the supermassive black hole mass and spin) from their observations with an observatory like LISA. Another important effort done in LISA phenomenology goes in the line of analysing the implications of LISA EMRI observations for fundamental physics, and more specifically the possibility of using EMRIs for testing the geometry of black holes and / or testing theories of gravity alternative to general relativity. First, we paved the way for this kind of studies by investigating the dynamics of perturbations of non-rotating black holes in Chern-Simons modified gravity (a possible alternative to general relativity motivated from high-energy physics). This is important since black hole perturbation theory is a key tool for modelling EMRIs. In this work we concluded that several formulations of four-dimensional Chern-Simons gravity are too restrictive in the sense that they are overconstrained gravitational theories. For example, they prevent most of the oscillations modes of black holes and hence they are not viable theories. Moreover, we also studied a class of theories of Chern-Simons modified gravity that do not suffer from this problem. This class of theories is known as dynamical Chern-Simons modified gravity, and the main difference with the non-viable theories is that the scalar field that introduces the Chern-Simons gravitational term is a dynamical field. We derived all the equations for the dynamics of perturbations of non-rotating black holes, analysed their structure and discussed how to solve them in comparison with what is done in general relativity. We have also developed different tools and computations to assess the ability of a gravitational-wave observatory like LISA to test this theory using EMRI observations. To that end, we have, for the first time, consistently computed the generation of gravitational waves from EMRIs in Chern-Simons modified gravity, showing that although point particles continue to follow geodesics in the modified theory, the background about which they inspiral is a modification to the Kerr metric, which imprints a CS correction on the gravitational waves emitted. We have also seen that Chern Simons modified gravitational waves are sufficiently different from the general relativistic expectation that they lead to significant dephasing after 3 weeks of evolution, but such dephasing will probably not prevent detection of these signals, but instead lead to a systematic error in the determination of parameters. One of the conclusions of this work is that one should be able to perform tests of Chern-Simons modified gravity with space-borne detectors that might be two orders of magnitude larger than current binary pulsar bounds. Finally, the researcher has done several other investigations on how to extend these studies to more general EMRI scenarios that can produce scientific results with impact in fundamental physics. A summary of these investigations have been published in an invited article in the journal Gravitational Waves Notes.

Apart from the work that fits with the main goals of the LISASIMICO project, we did other related research. First of all, given the importance of the LISA mission itself for this project and also given that the research group where the project is developed has a strong involvement in the precursor mission to LISA, LISA PathFinder (LPF; an ESA mission), the researcher has been involved in these activities, which include:

(i) the design and construction of the data management unit, the computer that controls the LISA technology package (LTP), a suite of experiments that will test the main technology for LISA, essentially the drag-free system that test the geodesic motion and measures the differential acceleration between two test masses inside the LPF mission;

(ii) the diagnostic subsystem, which involves thermal diagnostics, magnetic diagnostics, and a radiation monitor to measure the particles that go through the satellite and charge the test masses;

(iii) preparations for the data analysis for the LTP, participating in the development of the LTP data analysis tools. In this activity the researcher has involved a new PhD student: Nikolaos Karnesis (funded by the Catalan Government Agency, AGAUR).

Finally, the author has also done research in the modelling of another very important source of gravitational radiation for LISA (and also for ground detectors), namely the inspiral, merger and ringdown of massive black hole binaries. In particular the merger part requires solving the full non-linear Einstein equations, and this in turn requires the use of supercomputers. The researcher has been involved in the design of some of these simulations, in the comparison of results with approximation techniques like the post-Newtonian approximation theory, and in the constructions of waveform template banks for the NINJA collaboration, which aims at using results from numerical relativity to the analysis of data by the upcoming advanced ground-based observatories like AdvLIGO and AdvVIRGO. More recently, the author has started to get involved in research that consists in applying modelling techniques from general relativity (like numerical relativity or perturbation theory) to problems of current interest in high-energy physics, like the development and applications of the well-known AdS-CFT correspondence conjecture. The idea is to produce a synergy between two different communities (numerical relativity and high-energy physics) and the researcher is an author in a white paper that presents a roadmap for research in this new area.

A large part of the efforts of the LISASIMICO project have been devoted to the modelling of the dynamics of EMRIs to produce theoretical gravitational waveforms that will be used to extract the EMRI signals, which will be buried in instrumental noise and the gravitational wave foreground, and to determine the physical parameters of the system. First of all, together with Priscil.la Canizares Martinez (a Doctor of Philosophy (PhD) student initially funded by the LISASIMICO project and later by an FPU fellowship of the Spanish Ministry of Science and Innovation) we have developed a new technique to model the evolution of ERMIs. In particular, it allows for a precise estimation of the self-force, a local force associated with the gravitational field created by the small compact object as it orbits the supermassive black hole, which is the responsible for the inspiral of the system. This technique, which we named the 'particle without particle' (PwP) formulation of the ERMI problem, is based on a division of the physical domain in which the stellar compact object moves around the supermassive black hole. This domain decomposition is such that the stellar compact object, described as a particle (we can neglect its internal structure due to extreme mass ratios involved), is always located at the interface between two subdomains. In this way, we have showed that we can get rid of the singularities associated with the particle description and have equations that have smooth solutions. Then, we have used pseudospectral collocation methods (PCM) for the numerical solution of the equations which then provide exponential convergence of the numerical computations. This makes our numerical computations the most accurate ones in the field for time-domain simulations. In this method, the stellar compact object, the particle, has been substituted by boundary conditions on the smooth solutions at each subdomain. We have developed a complex but well-structured numerical code using the C language and we have first computed the self-force for the case of circular orbits around a non-rotating black hole. We have shown that, to a high degree of precision, our results coincide with other computations in the literature. Afterwards, we extended the computational infrastructure to generalise the PwP method to the case of generic eccentric orbits by introducing a new technique of moving domains. The results of this pioneer work have been corroborated a posteriori by other authors. We have also tuned and improved the technique and numerical codes for efficiency and precision, establishing the PwP as an outstanding tool for self-force computations for EMRIs. In relation to this, during the LISASIMICO project we conducted an investigation of a controversy in the community on the possibility that time-domain computations of the self-force may be contaminated by junk solutions that are difficult to identify, which would mean that these computations may not be reliable. We found a flaw in the argumentation and concluded that these junk solutions are due to inconsistencies between the initial conditions and some of the equations and hence, they can always be safely avoided in numerical computations.

Apart from these high-precision simulations of ERMIs, we have developed a completely new scheme for the construction of approximate waveforms for EMRIs. The idea is that once we will have a complete knowledge of the gravitational self-force, we cannot integrate the full perturbative equations with the self-force to obtain the gravitational waveforms because that would be computationally very expensive. We rather need to have a phenomenological and efficient scheme fitted to the self-force results that provides these waveforms in a reasonable time. Although there are several such schemes, they are either not very precise or they only cover a sector of the configuration space and are not easy to generalise. Our scheme is a hybrid method that combines tools from different approximation techniques in general relativity:

(i) a multipolar, post-Minkowskian expansion for the far-zone metric perturbation (the gravitational waveforms) and for the local prescription of the self-force (both in the Lorenz/Harmonic gauge);

(ii) a post-Newtonian expansion for the computation of the multipole moments in terms of the trajectories; and

(iii) a Barker-Henderson (BH) perturbation theory expansion when treating the trajectories as a sequence of self-adjusting Kerr geodesics.

Then, the EMRI trajectory is made out of Kerr geodesic fragments joined via the method of osculating elements as dictated by the multipolar post-Minkowskian radiation-reaction prescription. Apart from the generation of waveforms, the local character in time of our multipolar post-Minkowskian self-force makes the scheme amenable to study the possible appearance of transient resonances in generic inspirals, an important question that we will study in the future.

Another line of work within the LISASIMICO project has been LISA phenomenology, that is, the exploration of different scenarios of ERMIs and the different science that we can extract from EMRI observations and the quality of this information. For instance, we have explored a system similar to EMRIs, in which the stellar compact object only does a few passages around the supermassive black hole before being lost due to scattering with other stellar compact objects. The interest for these systems, which were called extreme mass-ratio bursts (EMRBs), arises from the fact that there will be much more EMRBs than EMRIs although many of the EMRBs will not be detectable. We have analysed how relativistic effects in the modelling of these new systems can help in their detectability and the prospects to extract useful physical information (like the supermassive black hole mass and spin) from their observations with an observatory like LISA. Another important effort done in LISA phenomenology goes in the line of analysing the implications of LISA EMRI observations for fundamental physics, and more specifically the possibility of using EMRIs for testing the geometry of black holes and / or testing theories of gravity alternative to general relativity. First, we paved the way for this kind of studies by investigating the dynamics of perturbations of non-rotating black holes in Chern-Simons modified gravity (a possible alternative to general relativity motivated from high-energy physics). This is important since black hole perturbation theory is a key tool for modelling EMRIs. In this work we concluded that several formulations of four-dimensional Chern-Simons gravity are too restrictive in the sense that they are overconstrained gravitational theories. For example, they prevent most of the oscillations modes of black holes and hence they are not viable theories. Moreover, we also studied a class of theories of Chern-Simons modified gravity that do not suffer from this problem. This class of theories is known as dynamical Chern-Simons modified gravity, and the main difference with the non-viable theories is that the scalar field that introduces the Chern-Simons gravitational term is a dynamical field. We derived all the equations for the dynamics of perturbations of non-rotating black holes, analysed their structure and discussed how to solve them in comparison with what is done in general relativity. We have also developed different tools and computations to assess the ability of a gravitational-wave observatory like LISA to test this theory using EMRI observations. To that end, we have, for the first time, consistently computed the generation of gravitational waves from EMRIs in Chern-Simons modified gravity, showing that although point particles continue to follow geodesics in the modified theory, the background about which they inspiral is a modification to the Kerr metric, which imprints a CS correction on the gravitational waves emitted. We have also seen that Chern Simons modified gravitational waves are sufficiently different from the general relativistic expectation that they lead to significant dephasing after 3 weeks of evolution, but such dephasing will probably not prevent detection of these signals, but instead lead to a systematic error in the determination of parameters. One of the conclusions of this work is that one should be able to perform tests of Chern-Simons modified gravity with space-borne detectors that might be two orders of magnitude larger than current binary pulsar bounds. Finally, the researcher has done several other investigations on how to extend these studies to more general EMRI scenarios that can produce scientific results with impact in fundamental physics. A summary of these investigations have been published in an invited article in the journal Gravitational Waves Notes.

Apart from the work that fits with the main goals of the LISASIMICO project, we did other related research. First of all, given the importance of the LISA mission itself for this project and also given that the research group where the project is developed has a strong involvement in the precursor mission to LISA, LISA PathFinder (LPF; an ESA mission), the researcher has been involved in these activities, which include:

(i) the design and construction of the data management unit, the computer that controls the LISA technology package (LTP), a suite of experiments that will test the main technology for LISA, essentially the drag-free system that test the geodesic motion and measures the differential acceleration between two test masses inside the LPF mission;

(ii) the diagnostic subsystem, which involves thermal diagnostics, magnetic diagnostics, and a radiation monitor to measure the particles that go through the satellite and charge the test masses;

(iii) preparations for the data analysis for the LTP, participating in the development of the LTP data analysis tools. In this activity the researcher has involved a new PhD student: Nikolaos Karnesis (funded by the Catalan Government Agency, AGAUR).

Finally, the author has also done research in the modelling of another very important source of gravitational radiation for LISA (and also for ground detectors), namely the inspiral, merger and ringdown of massive black hole binaries. In particular the merger part requires solving the full non-linear Einstein equations, and this in turn requires the use of supercomputers. The researcher has been involved in the design of some of these simulations, in the comparison of results with approximation techniques like the post-Newtonian approximation theory, and in the constructions of waveform template banks for the NINJA collaboration, which aims at using results from numerical relativity to the analysis of data by the upcoming advanced ground-based observatories like AdvLIGO and AdvVIRGO. More recently, the author has started to get involved in research that consists in applying modelling techniques from general relativity (like numerical relativity or perturbation theory) to problems of current interest in high-energy physics, like the development and applications of the well-known AdS-CFT correspondence conjecture. The idea is to produce a synergy between two different communities (numerical relativity and high-energy physics) and the researcher is an author in a white paper that presents a roadmap for research in this new area.