The First Detection of Gravitational Waves

This article deals with the first detection of gravitational waves by the advanced Laser Interferometer Gravitational Wave Observatory (LIGO) detectors on 14 September 2015, where the signal was generated by two stellar mass black holes with masses 36 $ M_{\odot}$ and 29 $ M_{\odot}$ that merged to form a 62 $ M_{\odot}$ black hole, releasing 3 $M_{\odot}$ energy in gravitational waves, almost 1.3 billion years ago. We begin by providing a brief overview of gravitational waves, their sources and the gravitational wave detectors. We then describe in detail the first detection of gravitational waves from a binary black hole merger. We then comment on the electromagnetic follow up of the detection event with various telescopes. Finally, we conclude with the discussion on the tests of gravity and fundamental physics with the first gravitational wave detection event.

I.

AN OVERVIEW TO GRAVITATIONAL WAVES AND ITS SOURCES
Gravitational waves were detected for the first time on 14 September 2015, which is not only the landmark discovery on its own, but also opened up a completely new observational window through which to explore the universe. It is the upshot of the tireless efforts of thousands of physicists across the globe over several decades on numerous fronts: theoretical and experimental as well as computational. As was correctly pointed out by the Laser Interferometer Gravitational Wave Observatory (LIGO) spokesperson Gabriela Gonźales, "It takes a village to discover gravitational waves". This discovery, which entails measurement of displacements over distances as ridiculously small as a thousand times smaller than the diameter of the proton, is undoubtedly the epitome of human knowledge and endeavour.
In the year 1915 Einstein came up with the theory of general relativity, which is the best classical theory of gravity available to date. Einstein's theory has passed every experimental or observational test it has ever been subjected to. One of the predictions of general relativity was gravitational waves. In fact, Einstein himself came up with this prediction merely a year later in 1916. According to general relativity, gravity is described by the curvature of spacetime. Matter tells spacetime how to curve and spacetime tells test bodies how to move. Spacetime is dynamical unlike in the Newtonian view of the world. Gravitational waves are the tiny ripples on the fabric of spacetime which travel at the speed of light generated by the accelerating masses. When the spacetime metric g µν is written as g µν = η µν + h µν , where η µν is the flat metric expressed in Cartesian coordinates and h µν is the tiny perturbation around it and the vacuum Einstein equations are linearised in h µν , we get the wave equation h µν = 0. to the leading order, in the so-called TT gauge. Here is the flat space d´alembertian. Thus, it implies that the metric perturbation h µν represents the wave that travels at the speed of light and it is known as the gravitational wave [1, 2].
With the recent discovery of gravitational waves, Einstein's theory of general relativity has survived yet another round of intense scrutiny and passed an experimental test with the flying colours. Like electromagnetic waves, gravitational waves also admit two polarizations and are transverse to the direction of propagation. Consider a gravitational wave travelling along z -axis in the Cartesian coordinates in flat spacetime. If it is linearly polarized along xaxis, the polarization state is referred to as plus polarization. If it is linearly polarized along the direction oriented at an angle of 45 • to the x -axis in x -y plane, the polarization state is referred to as cross-polarization. The effect of the gravitational wave with plus polarization increases the proper lengths along x -axis and decreases the proper lengths along y-axis during the first half-cycle i.e., during the crest, whereas it decreases the proper lengths along the x -axis and increases the proper lengths along y-axis during the second half-cycle i.e., during the trough. For the cross polarization the same pattern is repeated but is rotated by 45 • in the x -y plane (see Figure 1). The strength of the gravitational wave, for instance with plus polarization, denoted by h, is a dimensionless number referred to as a gravitational wave strain and is defined as the difference of the fractional change in proper lengths along x -axis and y-axis, i.e., h = δLx Lx − δLy Ly . Gravitational wave strain is generally extremely small owing to the extreme weakness of gravity as compared to other forces. For instance, the peak gravitational wave strain during the first detection event was as small as 10 −21 . Thus, it is difficult to detect the gravitational waves. In fact, Einstein himself thought that the gravitational waves would never be detected. It took elaborate efforts over many decades to devise an experimental setup with immense sensitivity, namely the few-km-long Michaelson interferometer, that could directly detect the gravitational waves.
The process of generation of gravitational waves by the non-relativistic source can be understood by solving the linearized Einstein equation with source written in the Lorenz gauge, namely h µν = − 16πG c 4 T µν , where is again the d´alembertian,h µν is the trace reversed part of the metric perturbation, and T µν is the stress-tensor associated with the non-relativistic source. The solution of this equation is given byh ij = 2G c 4 rÏ ij (t r ) where r is the distance from the source and I µν is the mass quadrupole moment of the source computed at the retarded time t r = t − r c . This formula implies that the gravitational wave produced by the non-relativistic source is proportional to the second time derivative of its mass quadrupole moment. The rate of loss of energy P of the source via emission of gravitational radiation depends on the square of third time derivative of the quadrupole moment and is given by P = − G regime and linear approximation where the source exhibits strong gravitational field and motion of source is relativistic or where the higher order corrections are relevant for the description of the physical system, one has to resort to elaborate techniques such as black hole perturbation theory, numerical relativity and post-Newtonian theory. In fact, one of the main achievements of the discovery of gravitational waves is the possibility to compare the numerically calculated waveforms in the full nonlinear regime with the actual observations.
As a matter of fact, Einstein himself denied the existence of gravitational waves in full nonlinear theory. The first proper theoretical treatment was given by Trautman and Robinson [3,4].
There are many sources of astrophysical origin that emit gravitational radiation which can potentially be detected with the current detector sensitivities. These sources include black hole -black hole binary, black hole neutron star binary and neutron star neutron star binary systems, supernova explosions, rapidly rotating deformed neutron stars, inflation, phase transitions in early universe and dynamics of cosmic defects such as cosmic strings [5].
The most prominent sources for the ground-based gravitational wave interferometric detectors are the binary systems made up of compact objects such as black holes and neutron stars. In fact, the first detection of gravitational waves is associated with the binary black hole system. Compact objects go around each other in a quasi-circular shrinking orbit for a long time, emitting gravitational waves and eventually merge to form a single black hole.
The gravitational wave signal of neutron star binary is slightly different from that of the black hole binary during the orbital evolution due to the fact that the neutron stars are distorted due to the tidal interaction in the later stage when they are sufficiently close. This deformation, which depends on the equation of state of matter constituting the neutron star, is imprinted on the gravitational wave signal. The waveform during the merger phase is also quite different since the merger of neutron star gives rise to the hypermassive neutron star which radiates for some time before it collapses to form a black hole.
Another source of the gravitational radiation is supernovae. Massive stars, at the end of their life cycle when they run out of nuclear fuel ceasing the process of nuclear fusion and gravity takes over, die catastrophically. Their core undergoes a gravitational collapse, leading to the formation of a neutron star or a black hole, while the outer layers are blown apart constituting a supernova. If the collapse of the core is asymmetric it leads to the production of gravitational waves in the form of a burst. The waveform of the burst radiation is not very well known at present. If the supernovae take place in our galaxy or galaxies nearby we should be able to detect the burst signal.
Deformed rotating neutron stars with the mountain on their crust as high as a few cm emit continuous gravitational waves. The signal is almost a pure sinusoid at twice the frequency of the rotation of a neutron star in the source frame with a tiny negative frequency derivative. Although the signal is expected to be weaker than that due to compact binary coalescence, it lasts for a very long time. A continuous signal will allow us to constrain various properties of neutron stars such as nuclear matter equation of state. Although so far there is no detection, the spin-down limit has been beaten for Crab and Vela pulsars limiting the loss of rotational energy of pulsars due to the emission of gravitational waves to a small fraction .
Stochastic gravitational radiation is emitted by various physical processes that are inherently stochastic in nature or due to the incoherent admixture of various coherent signals that are weak enough and are not resolved individually. For instance, the inflation which is the phase of accelerated expansion in early universe at nearly constant Hubble rate gives rise to a stochastic gravitational wave background. The origin of gravitational radiation during the inflation is inherently quantum mechanical. Apart from inflation, various phase transitions in the early universe, oscillatory dynamics of cosmic defects such as cosmic strings are also expected to rise to such a signal. Stochastic gravitational wave background due to the unresolved binary black hole mergers could be detected in the upcoming years with ground based detectors.
Gravitational waves are also expected to surprise us, providing access to completely new objects and phenomena.

II. GRAVITATIONAL WAVE DETECTORS
The quest for the detection of gravitational waves started in the 1960s with the pioneer efforts of Weber. He tried to detect gravitational waves with heavy metal bars that had a size of around 1 m. A gravitational wave passing through the bar would excite it into oscillations due to its tidal effects which could in principle be measured. The bar detectors had low sensitivity essentially due to their small size. Moreover, sensitivity peaked sharply around the resonance making the detectors narrow band. In fact Weber claimed a detection, but his results could not be replicated by other groups [5]. The distance to the source was 1.3 billion light years [6]. The peak luminosity of the event was 3 × 10 56 erg/sec, making it the most luminous event exceeding the integrated luminosity of all the stars taken together in the observable universe.
Despite the effort to minimize the noise and enhance the sensitivity of the detector in the best possible way, the noise generally exceeds the signal in the detector output. The gravitational signal that is hidden in the noise must be excavated out using optimal data analysis techniques such as matched filtering where the expected signal is correlated with the detector output. Templates which represent all possible gravitational wave signals are used for this purpose. Thus, it is essential to have an accurate waveform a priori. In the case of binary black hole coalescence which is the most promising source of gravitational radiation for ground-based detectors and also the source of the signal detected by LIGO, it is essential to solve two-body problem in general relativity. It is a difficult problem unlike the Newtonian case where the exact solution is known. Various sophisticated techniques have been developed over any decades to deal with the two-body problem in general relativity.
Initially, two black holes orbiting one another in a quasi-circular orbit are far apart, the gravity is weak and the speeds are non-relativistic. The orbit slowly shrinks due to the emission of gravitational waves which steals the orbital energy. This phase is known as inspiral. Post-Newtonian corrections include various interesting effects such as spin-orbit coupling i.e., coupling between the spins of the black holes and the orbital angular momentum, and tail effects which account for the back-scattering of the gravitational radiation due to the spacetime curvature etc.
When the black holes come sufficiently close, the velocities are relativistic and gravity is strong, and post-Newtonian approximations becomes less reliable. Eventually, black holes plunge towards each other and collide at the velocity close to the speed of light. The black holes merge together to form a single remnant black hole. This phase is known as merger. A burst of radiation is given out during the merger with often the luminosity of the gravitational waves exceeding the integrated luminosity of all stars in the entire observable universe. As stated earlier, the peak luminosity in case of GW150914 was 3 × 10 56 erg/sec. Analytical perturbative and approximation techniques fail during the merger and one has to deal with the full non-linear dynamics of general relativity in its full glory using numerical relativity. Full-fledged numerical relativity simulations of the binary black hole merger were carried by many groups out after the initial breakthrough in the year 2005 in this field [7].
The final black hole that is formed as a result of merger is often in the excited state.  Figure   5.
The techniques of effective one body are also used to generate the waveform [8]. The post-Newtonian theory is extended into the strong field regime using the resummation techniques and it is complemented with the input from the numerical relativity simulations. It provides us a full waveforms for the binary black hole coalescence in the analytical form that are accurate and efficient. The detector output for GW150914 in the LIGO Livingston and Hanford detectors superimposed with the waveform of binary black hole coalescence predicted from general relativity using the techniques mentioned above are shown in 21 M . The signal-to-noise ratio was 13 [9]. There was yet another candidate with low statistical significance and hence it is not claimed to be the detection.

IV. ELECTROMAGNETIC FOLLOW-UP
The sky localization of gravitational wave event GW150914 was carried out from the information of the time delay of arrival of the signal between the two LIGO detectors and their directional sensitivities. The sky localisation of this event is depicted in Figure 6. The source could be located over a region in sky that is spread across the area as large as few 100 square degrees This is pretty large as compared to for instance the sky spanned by sun which is merely 0.5 100 square degrees. This makes the electromagnetic follow-up rather challenging. A lot of effort was put into search for an electromagnetic counterpart to the gravitational wave event by multiple ground based and space based telescopes all across the electromagnetic spectrum ranging from radio to gamma rays. However, no promising candidate was discovered [10]. This is in line with the expectation that the black hole coalescence events are not expected to harbour an electromagnetic counterpart.
The gravitational wave signal from the binary neutron star merger is expected to be observed in the future. The event of binary neutron star merger could be interesting from the point of view of joint electromagnetic-gravitational wave observation. A binary neutron star merger is believed to be the progenitor of a short gamma-ray burst, which is an energetic event. With the enhancement of sensitivity of the gravitational wave detectors and with multiple detectors going online in the near future, the sky localization of the gravitational wave event would improve significantly. This would make the electromagnetic follow-up significantly easier. Observation of short gamma-ray burst coincident with the gravitational wave detection of neutron star merger will allow us to confirm the alleged link between the two.

V. TESTS OF GENERAL RELATIVITY
All the tests of general relativity carried out so far which include solar system tests and tests concerning binary pulsars, deal with the regime where the gravitational fields are weak, velocities are small, and dynamics are quasi-static. For the first time, gravitational wave observation of binary black hole coalescence provided us with an opportunity to probe gravity and test general relativity in the strong field, large velocity and highly dynamical regime which was inaccessible before. Various tests to check the validity of general relativity were carried out as we describe below [11].
The best-fit general relativity waveform was subtracted from the detector output and it was checked whether the residual was consistent with pure noise or there was leftover power.
It was found that the residual was indeed consistent with Gaussian noise. General relativity was tested to 4% level using this test, in other words the correlation between the detector output and the waveform based on general relativity is greater than 96%.
The final mass and final dimensionless spin parameters of the remnant black hole, M f and a f , can be determined in two ways, by using the inspiral or low frequency part and by using the post-inspiral or high frequency part of the signal. In the case of GW150914 the critical frequency was 132 Hz. In order to determine the mass and spin of the remnant black hole input from the numerical relativity is required. Numerical relativity evolution starting from the information of the inspiral phase allows us to predict the final mass and spin of the remnant black hole assuming general relativity. The dark purple contour in Figure 7 confines the 90% confidence region in the M f −a f plane based on the prediction using inspiral part of at each order are fixed by general relativity. We modify those coefficients to mimic possible deviations from general relativity. Figure 8 shows upper bounds on the fractional deviations from the general relativistic predictions |δφ|, from gravitational wave and binary pulsar observations. The upper bound is quite strong at the leading order from the binary pulsar observations, but bounds are pretty weak at higher orders beyond 1 PN since the orbital velocity is small. Upper bounds imposed by gravitational waves observation at higher orders are significantly better since in the late inspiral regime we have access to higher velocities.
Gravitons are massless particles in general relativity and thus the gravitational waves travel at speed of light. At present we do not have access to sensible well-defined theory of massive gravity. However, one could deal with a hypothetical situation in which gravitons have non-zero fixed Compton length i.e., non-zero mass. Gravitational waves in that case would travel slower than speed of light, moreover, the speed will be different at different frequencies exhibiting non-trivial dispersion relation. The phase of the gravitational wave signal in such a case would deviate from the general relativistic prediction. This allows us to put constraints on the Compton wavelength. Newton's law of gravity in such a case will also get modified, which allows to put constraints from solar system tests of gravity. Figure 9 shows the probability distribution function for the Compton wavelength of graviton.
The lower bound on the Compton wavelength of graviton imposed by gravitational wave observation is a factor of 3 better than the one from solar system tests.
All studies carried out to detect deviation from general relativity indicate that GW150914, the process of binary black hole coalescence, was in accordance with the vacuum Einstein equation of general relativity and no statistically significant deviation was found within the limits such as sensitivity of the detectors and nature of event itself. In the future it would be possible to use better constraints with detectors with better sensitivities due to the availability of the network of detectors, with detections that exhibit higher signal to noise ratio and by combining multiple observations. We should be able to test the no hair theorem, the second law of black hole mechanics i.e., the area theorem, look for extra polarization modes, and so on. The golden era in the field of gravity, fundamental physics and astronomy has just begun with the first detection of gravitational waves (GW150914) and many more exciting developments and discoveries are ahead of us.