The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.

The regularization and renormalization of the radiative mass-type quadrupole moment of inspiralling compact binaries (without spins) is investigated at the fourth post-Newtonian (4PN) approximation of general relativity. As clear from the conservative 4PN equations of motion, a dimensional regularization has to be implemented in order to properly treat the non-linear interactions experienced by gravitational waves during their propagation toward future null infinity. By implementing such procedure, we show that the poles coming from the source moment (computed in a companion paper) are exactly cancelled in the radiative moment, as expected for a physical quantity. We thus define and obtain a ``\textit{renormalized}'' source quadrupole, three-dimensional by nature, which is an important step towards the computation of the gravitational-wave flux with 4PN accuracy. Furthermore, we explicitly prove the equivalence between the dimensional regularization and the previously used Hadamard partie finie scheme up to the 3PN order.

With the aim of providing high accuracy post-Newtonian (PN) templates for the analysis of gravitational waves generated by compact binary systems, we complete the analytical derivation of the \textit{source type} mass quadrupole moment of compact binaries (without spins) at the fourth PN order of general relativity. Similarly to the case of the conservative 4PN equations of motion, we show that the quadrupole moment at that order contains a non-local (in time) contribution, arising from the tail-transported interaction entering the conservative part of the dynamics. Furthermore, we investigate the infra-red (IR) divergences of the quadrupole moment. In a previous work, this moment has been computed using a Hadamard partie finie procedure for the IR divergences, but the knowledge of the conservative equations of motion indicates that those divergences have to be dealt with by means of dimensional regularization. This work thus derives the difference between the two regularization schemes, which has to be added on top of the previous result. We show that unphysical IR poles start to appear at the 3PN order, and we determine all of these up to the 4PN order. In particular, the non-local tail term comes in along with a specific pole at the 4PN order. It will be proven in a companion paper that the poles in the source-type quadrupole are cancelled in the physical \textit{radiative type} quadrupole moment measured at future null infinity.

In the context of the recently proposed type-II minimally modified gravity theory, i.e. a metric theory of gravity with two local physical degrees of freedom that does not possess an Einstein frame, we study spherically symmetric vacuum solutions to explore the strong gravity regime. Despite the absence of extra degrees of freedom in the gravity sector, the vacuum solutions are locally different from the Schwarzschild or Schwarzschild-(A)dS metric in general and thus the Birkhoff theorem does not hold. The general solutions are parameterized by several free functions of time and admit regular trapping and event horizons. Depending on the choice of the free functions of time, the null convergence condition may be violated in vacuum. Even in the static limit, while the solutions in this limit reduce to the Schwarzschild or Schwarzschild-(A)dS solutions, the effective cosmological constant deduced from the solutions is in general different from the cosmological value that is determined by the action. Nonetheless, once a set of suitable asymptotic conditions is imposed so that the solutions represent compact objects in the corresponding cosmological setup, the standard Schwarzschild or Schwarzschild-(A)dS metric is recovered and the effective cosmological constant agrees with the value inferred from the action.