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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