This page gives an overview of part of my research project. The context is explained in the lot of other sections under the ‘Research’ button. Summarizing my research line in one sentence is: ‘Testing strong gravity with compact objects’. This involves black holes and neutron stars and alternative theories of gravity mainly, but also black holes in high energy physics. Studying these objects in a more theoretical context leads to a better understanding of their mathematical properties, but is also related to string theory and condensed matter physics, thought the celebrated AdS/CFT duality. In this context, BH thermodynamics and AdS solution is of high importance. For instance, I have built an exact analogue in higher dimensions of a van der Waals BH (generalizing the result of this paper).
Dynamics in higher dimensions is also very relevant, because it can lead to mathematical simplifications allowing for analytic treatments of complex dynamical scenarios such as formation of accretion disk. These higher dimensional models provide toy model that should give qualitative hints for 4 dimensional scenarios. See my paper constructing exact solution of rotating thin shells in 5 dimensions. Here I will explain in more details only one aspect of my research, related to tidal dynamics.
The main framework of my research project has been introduced by myself with collaborators recently in a Newtonian context , and generalized to the relativistic case . The result extends the notion of Love numbers to ‘Love function of the frequency’, which we will call response function, since it encodes the matter response towards an external time dependent perturbation. A first theoretical success of this new approach is that it reproduces the known tidal coefficient in the appropriate limit (vanishing frequency) for both NS and BH. Furthermore, it contains the (known) absorption coefficient  and reproduces the previously known quadrupole beta function for BH . The innovative character of this proposal resides in the fact that it proposes a new paradigm in the way we think CO mode structure. Let us try to summarize its impact and consequences:
(i) It provides a new point of view on gravito-spectroscopy in the sense that it disentangles the central object’s internal dynamics from the gravitational one. Usually, gravito-spectra of CO are understood as QNM spectra of the underlying object. Here, the modes that appear are only due to the matter multipole modes (in the NS case, formally the extended source is modeled by a point multipole), which are then coupled to gravity.
(ii) As a byproduct, these new results define the relativistic counterparts of the Newtonian overlap integrals, which appear naturally as the width of the response function poles in the Newtonian case. The overlap integrals quantize the amount of energy transferred from tidal interactions to matter deformation. This is particularly relevant for relativistic extensions of Newtonian seismology .
(iii) This new perspective opens up many doors for modeling the mass multipole in a precise and controlled way. This new approach deserves to be given a chance to be developed. Applications and extensions are wide; some of them will be presented in this proposal.
Plan and methodology
This research project has many possible directions, let us summarize a few objectives.
(i) Demonstrate the applicability of the new technique on a concrete example. This can be achieved by performing explicit simulation of binary system evolution within the PN formalism, including the mode resonance predicted by the response function. Then systematize the computation of the response function parameters to realistic EOS.
(ii) Clarify and formalize the BH case, which is still plagued by an unclean treatment of the regularization scheme. Despite the fact that the result agrees with known results for the two first orders in frequency, the square term is plagued by an unknown extra scale. This is due to the use of two regularization schemes and will be cured by using a unique one, such as DR.
(iii) Waveform prediction and investigation of the imprint of the mode spectrum contained in the response function.
(iv) Include rotation. This is probably the most difficult step regarding the response function formalism. The case of rotation touches two aspects of this project. Indeed, a) it is a central ingredient for universal relations and b) it is likely the natural way of addressing the unsolved issue of relativistic Love numbers for rotating bodies.
(v) Explore the observational potential of universal relations, and extend these in some representative alternative theories.
An important expected result of the framework is to improve the semi-analytic predictions of gravitational waveforms, useful for the future gravitational wave detectors. This will be an extension of the binary motion modeling. In next stages, motion of BH-NS or BH-BH binaries may be envisaged, once the BH case is fully under control. On another hand, there are other candidates than GR for describing the gravitational dynamics of spacetime. Extending part of the approach to some representative of alternative theories is also one of the objectives. An expectation is to explore the following question: how can we distinguish GR from another model out of CO observations? A possible approach to the question consists in investigating universal relations of NS, since they allow to take away the uncertainty on the equation of state of the matter field. In the following, we give some more technical details on the methodology.
Realistic EOS and waveforms – During the last years I collected realistic EOS tables and implemented a thermodynamically consistent interpolation scheme . I am currently in touch with members of the Institute of Astronomy and Astrophysics of the Free University of Brussels (ULB), that develops nuclear energy density functionals for astrophysical applications such as supernovae cores or pulsars. These parametric EOS cover the physics from the crust to the core in a consistent way. This EOS model allows to investigate the influence of specific nuclear physics parameters on universal relations and the response function. The first step, after handling the equation of state is to build non rotating neutron star configurations. This is achieved by solving the TOV equation governing the hydrostatic equilibrium of a self gravitating fluid in GR. Then the slowly rotating approximation as described in the previous section can be applied. This should already give some insights on the relations between nuclear parameters and geometrical properties of the surrounding spacetime. Then the response function, as well as the first few modes and relativistic overlap integrals can be constructed. Once these data are collected, a model for the motion of stars in binary systems can be developed. The key point is the following: the response function has the same structure as in the non relativistic case (with other values of the coefficients). In the Newtonian case, the modes originate from the normal modes of the fluid displacement (difference between perturbed and unperturbed fluid location). This displacement is then expanded over its normal modes, and the amplitudes with which each normal mode appear form a simple forced harmonic oscillator equation. The response function in frequency domain is the superposition of the solutions to all these oscillators. This suggests to model the multipole dynamics in the relativistic case as a sum of harmonic oscillators – the amplitudes. An interaction potential can be derived from the effective action between the mass and the multipole, as already given in . The point here is to use the interaction term provided by the EFT and insert the gravitational field of a given multipolar order. The equation of motion can then be formulated in the harmonic approach to PN dynamics, where the Einstein equations their-selves are expanded in PN orders, in a suitable gauge. Finally the gravitational wave extraction, and waveform prediction is achieved by evaluating the energy balance equation far from the sources. The quadrupole formula allows to extract the amplitude of the gravitational waves emitted as a function of time, leading to the gravitational waveform .
Rotating Neutron Stars – Rotating compact objects can be either treated with full rotation, leading to systems of partial differential equations, or can be approximated using the slow rotation approximation scheme. This perturbative method where the perturbation parameter is the angular velocity has been first introduced by James Hartle in the late 60’s . This approximation leads to a tower of inhomogeneous linear ODE systems at each order, which are easier to handle technically than the original non perturbative equations. For the fully rotating case, there are already existing algorithms and publicly available codes for GR e.g. , while some further development need to be done for specific scenarios e.g. in the context of AMG. From the effective field theoretical point of view, addition of rotation – even slow rotation – needs to be taken into account by adding appropriate additional degrees of freedom. This will add contribution to the effective stress tensor. Under the slow rotation approximation, the resulting stress tensor will also be expanded in powers of the angular momentum. The procedure for the matching then needs to be carried out consistently. Indeed, here there are two small parameters: one is the rotational parameter, the other one is associated to the tidal perturbation. The perturbation scheme needs to accommodate both effects at the same time before any matching can be achieved. The perturbative approach seems to be the best choice because it allows to identify clearly the origin of each pieces entering the result, thanks to the small book-keeping parameters. For the black hole case, more progress can be expected. First, rotating black holes all belong to the Kerr family, where the so called Teukolsky formalism  to handle perturbations of rotating spacetime including sources is well knwon. The free Teukolsky equation is known to admit solutions in terms of series of hypergeometric functions , very similar to the case we already dealt with, so that our approach can be in principle applied to the sourced case, i.e. with the effective stress tensor from EFT.
Black holes and dimensional regularization – Finally, despite the exciting conclusions of previous work, the formalism needs to be improved in order to be more easily generalizable to more complicated situations like the non-perturbatively rotating case. So far, we use the Riesz Kernel regularization for the source and an analytic continuation for the homogeneous solutions of the perturbation equation. These two are actually similar to the dimensional regularization scheme (DR), which is confirmed by the fact that we recover the known beta function of the black hole quadrupole computed with DR. This suggest to apply DR directly to our model, which will be the first step towards a clear and clean treatment of the BH case. In the present status, the drawbacks are that two regularization scheme are used, leading to two renormalization scales in the final answer. For NS it is harmless because the final result for the response function is numerically insensitive to the value of one of the two renormalization scales. Instead for BH, the extra renormalization scale appears explicitly in the result, more precisely in the second order of the Taylor series over the frequency of the full response function .
The result is however quite encouraging because it contains the correct vanishing Love number at zeroth order in frequency, the correct dissipation coefficient of black holes to first order in frequency and reproduces the quadrupole beta function derived with dimensional regularization. This definitely calls for a consistent regularization scheme and dimensional regularization seems to be the best choice. Moreover, it turns out that this is precisely the technique mastered by the supervisor of this proposal (which was among the motivations for choosing the host institution). The same arguments applies for a clean treatment of rotating black holes, where the sourceless Teukolsky equations must be considered. Our model can in principle lead to the definition of Love Numbers for the Kerr black hole, which is presently unknown, even for slow rotations. Finally note that for the Kerr Love number, we likely don’t even need dimensional regularization since the zero frequency solutions are much better behaved.
Alternative theory of gravity
There are essentially two aspects concerned by this point. The first one is the existence of universal relations for non GR theories. As we already mentioned, one of the important consequence is that it allows to focus the gravitational dynamics. In particular, this opens the door to new ways of testing different AMG. This is of course a rough statement, and it depends on what one wants to measure or extract. The approach is the following: we first need the background NS in a given gravitational theory [42; 43]. A first step is to built the slowly rotating solution up to order at least 2 in the rotational parameter, as described in the previous sections. The first order in rotation gives the moment of inertia, while the second order provides the rotation induced mass quadrupole. Higher order give higher current or mass multipoles. These two (or more) quantities are then made dimensionless. Next the Love numbers have to be computed. The ‘procedure’ consists in building the general solution, without imposing specific boundary conditions in the exterior spacetime. The solution in the exterior consists in a combination of ‘regular’ and ‘irregular’ parts (at infinity). These are associated to the perturbation caused by an external perturbing field at infinity (irregular) and the response to this perturbation by the central object (regular). The ratio of the amplitude of these two pieces is proportional to the Love number for a given mode. The additional degrees of freedom (such as scalar or electromagnetic) also have to be matched since they can be excited by an external perturbation. This defines a “susceptibility” for them. The principle for constructing the response function goes along the same line, except that the perturbation is now time dependent, and the vacuum solution (at least in GR) is now formally expressed in terms of series of hypergeometric functions . For alternative theories with additional degrees of freedom, the representation of the solution has to be extended first. This is a non trivial task. Next the matching formula has to be derived; the degree of technical difficulties here is the same as in GR. Within the timeline of this project, the focus will be set on the invariant relations aspects, and some preliminary investigation will be started for the response function.
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