"Before we go into more details on how we explore ""beyond SM+GR"" physics in our project, we note that such a situation is not new; we have been in it twice in the 19th century when astronomers observed anomalies in the orbits of Uranus and Mercury. The former found an explanation in the form of ""dark matter"": a new planet, Neptune, was found based on theoretical predictions derived from Uranus' anomalies. Similarly, a planet ""Vulcan"" was conjectured to explain Mercury's orbital peculiarities, but Vulcan was never found. Instead, a new theory of gravity, namely Einstein's general relativity, provided the answer. Bearing in mind this lesson of history, we have explored both possibilities, dark matter and modified gravity. We find:
* Contrary to previous belief, boson fields accumulating around stars, do not trigger collapse to a black hole, but settle down into breathing configurations.
* Binary systems composed of compact, self gravitating boson fields, ""boson stars"", emit characteristic gravitational wave signals distinguishing them from black hole or neutron star binaries.
* Rotating boson stars made of scalar fields are not stable. In contrast, rotating boson stars composed of vector or ""Proca"" fields are stable.
* The presence of a background scalar field leaves characteristic imprints on the gravitational wave signal generated in the inspiral and merger of compact binaries.
* Black holes act as gravitational lenses, creating a characteristic, self-similar lensed image of objects behind them.
* Extended measurements of black-hole rotation rates will be able to identify or constrain the presence of dark-matter candidate fields such as dark photons or axion-like particles.
* The presence of a boson field around astrophysical X-ray sources leads to a shift in the quasi-periodic oscillations of the X-ray signal which can be detected with future missions such as LOFT.
* So-called black-hole mimickers generate waveform signals that may be detected with future space interferometers such as the ESA L-class mission LISA.
* We have found a long-lived (years or centuries) gravitational wave pattern arising in massive scalar-tensor theory of gravity. By directing gravitational wave searches at historic supernovae such as Kepler 1604, we can test these theories.
* The spontaneous scalarization of compact objects is a widespread phenomenon that occurs for neutron stars and black holes in various alternative theories of gravity.
* Contrary to our expectations, grazing collisions of black holes in more than four spacetime dimension can form dumbbell shaped configurations that ultimately lead to formation of a naked singularity. This behavior is not present in four spacetime dimensions suggesting the special nature of the four-dimensional world of our daily lives."
