Gravitational Waves

Immediately after the formulation of General Relativity (GR) it was realised in 1916 that the theory implied the existence of gravitational waves (GW). But, because of the weakness of the gravitational interactions, it took many years to develop the necessary technology for detecting them. Their existence was indirectly confirmed when the observations of the binary pulsar PSR 1913+16 in 1974 showed an orbital decay which matched Einstein's predictions of energy loss by gravitational radiation. The Nobel prize in physics was awarded to Hulse and Taylor for this discovery. The search for the necessary technology started in the sixties and culminated in this century with the construction of the two LIGO, one in Louisiana and the other in Washington state and of VIRGO near Pisa in Italy. On 11 February 2016 the LIGO/VIRGO collaboration announced the first observation of gravitational waves. The signal was named GW150914 because the waveform was observed on 14 September 2015, within just two days of when the advanced LIGO detectors started to collect data after their upgrade. It matched the prediction of GR for the inward spiral and merger of two black holes and subsequent ring-down of the resulting single black hole.

The LIGO and VIRGO detectors consist of a laser interferometer with two arms long 4 Km each in LIGO and 3 Km each in VIRGO. The arrival of a gravitational wave effectively changes the length of one or of both arms of the interferometer and this results in a measurable signal. This is a very tiny effect and, in order to observe it, one must be able to observe a stretch of the arms of the interferometer of less than a thousandth of the diameter of the proton! One must also be sure that this stretch is not due to other causes and this requires not only a high sensitivity but also the ability to isolate the system from other sources of stretches. Other laboratories for the detection of GW, as for instance LIGO in India, are been constructed and this gives the possibility of comparing the waveforms detected reaching higher reliability of the results obtained. In 2017 Weiss, Barish, and Thorne received the Nobel prize for decisive contributions to the LIGO detector and the observation of gravitational waves.

To be observed and analysed by the network of GW detectors, inspiralling compact binaries require high-accuracy templates that must be extracted from GR. For this one must solve the equations of motion that follow from the Einstein-Hilbert Lagrangian and from the Lagrangian describing the two compact objects interacting with gravity. The simplest version of this last Lagrangian is given by the world-line Lagrangian of a massive point-particle with spin for each of the two compact objects. The system constantly radiates gravitational waves which backreact on the orbiting bodies making the problem highly non-linear.

Few analytical approaches have been developed to solve the equations of motion derived from the previous Lagrangian. They are certainly accurate in the beginning stage of the inspiralling, but become inaccurate around the merging when the velocity of the two compact objects becomes of the order of the speed of light and the interaction becomes strong. Therefore, in order to get the waveform of the radiation, that is mostly emitted around the merging, one must also rely on the calculations performed in numerical relativity. On the other hand, in order to have a more direct understanding of what is happening in the inspiral process approaching the merging, few analytical approaches have been developed. One of them uses the post-Newtonian approximation consisting in an expansion for small velocity of the two compact objects. This expansion has been carried out up to 5PN order where there are still few disagreements among various calculations.

The other approach is the so-called Effective One Body (EOB) formalism introduced by Buonanno and Damour. It aims to describe all different phases of the two-body dynamics in a single analytical method and it leads to results faster than numerical relativity. The idea is that, instead of considering the real Hamiltonian describing the interaction between two compact objects with masses m1 and m2 that can, for instance in the PN expansion, be expressed as a sum of terms suppressed by higher and higher powers of the square of the speed of light c, one constructs an effective Hamiltonian for a particle with effective mass mu in an external metric that is fixed by requiring that the effective dynamics be the same as the original dynamics. In other words, it maps the more complicated general two-body problem to the simpler one of a test particle in an effective metric. EOB waveforms are an important class of inspiral-merger- ringdown waveforms models employed by LIGO/VIRGO searches.

Because of the recent developments in the study of scattering amplitudes many people have started to use the modern techniques from Quantum Field Theory to extract the physical properties of the two-body interaction in a simpler way and without needing to expand for small velocities. One performs the post-Minkowkian (PM) approximation expanding in powers of the Newton constant G keeping the fully relativistic expression without expanding for small velocities. The orders 1PM and 2PM are known since sometime and can be more easily derived in the probe approximation, where one studies the geodesic motion of one particle in the external field generated by the other particle. The new results are at 3PM where the probe approximation cannot fix everything and where radiation starts to appear. In the unbound problem where one considers the scattering rather than the merging of the two compact objects, one has computed the two-loop elastic scattering amplitude involving two scalar point-particles with different masses, describing the two compact objects, and one has extracted from it the classical quantities, obtained in the limit of the Planck scale h going to zero, as the eikonal or the radial action. From them one can then compute observables as the deflection angle.

At 3PM it has been shown that the sum of the conservative dynamics and of the effects due to radiation reaction has provided a deflection angle that is fully consistent and universal at high energy, as expected from the fact that, at high energy, the leading contribution comes from the massless particle with the highest spin, the graviton, while the contribution of all other massless particles with lower spin is subleading and can be neglected. Partial results are now also known at 4PM. What is known at 3PM and 4PM has been used in the EOB formalism to get a better agreement with the results obtained from numerical relativity.