ON September 14, 2015, around 3:20 p.m. Indian Standard Time, ripples in space-time caused by the merger of two black holes hit the two Laser Interferometer Gravitational-wave Observatory (LIGO) centres in the United States. The signal was so strong that just by looking at the data one could “see” the signal. Yet, it took the collaborating scientific groups several weeks of analysis to confirm the detection and to extract the astrophysical information contained in the observed signal. In addition to confirming Einstein’s century-old prediction of the existence of gravitational waves (GWs), this detection marked the beginning of a new era in astronomy. The LIGO discovery was made possible by significant Indian contributions to theoretical source modelling and data analysis.

Indian scientists have made major contributions to GW physics over the past 25 years. In the late 1980s and early 1990s, Bala Iyer’s group at the Raman Research Institute (RRI), Bengaluru, in collaboration with a group of French scientists, pioneered the mathematical calculations used to model the GW signals expected from orbiting black holes and neutron stars. These calculations, using the so-called post-Newtonian methods, form the basis for computing the theoretical “templates” of the expected signals employed in GW detection.

Around the same period, in parallel, Sanjeev Dhurandhar’s group at the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, did foundational work on developing the data analysis techniques used to detect these weak signals buried deep in the detector noise. The present Indian GW research community has essentially grown out of research programmes these two groups carried out. Several researchers belonging to the next generations, including Sukanta Bose, A. Gopakumar, Sanjit Mitra, Rajesh Nayak, Archana Pai, B.S. Sathyaprakash, Anand Sengupta and the authors of this article, trained under these groups.

**Mergers of binaries**

A big discovery does not happen overnight and requires decades of preparation. A unique feature of GW astronomy is the rich interplay it demands between experiment, theory and data analysis. Historically, scientists had realised that mergers of binaries consisting of black holes or neutron stars were the best bet for the first detection of GWs. (A binary system is a system of two stellar objects—brown dwarfs, planets, neutron stars, black holes, galaxies or asteroids—which are close enough that their mutual gravitational interaction causes them to orbit each other around their common centre of mass.)

Experimental advances in the field were positively influenced by the theoretical progress made in solving the Einstein field equations for such systems to accurately compute the expected signal shapes, or “waveforms”. It was realised early on that if one could calculate the expected signal shapes, a powerful data analysis method called “matched filtering” could be employed for detection. This method involves comparing the data with theoretical templates of the expected signals making use of the mathematical operation called cross-correlation.

Immediately after the LIGO project was funded in the U.S., Kip Thorne at Caltech, California, convened an international meeting of experts in January 1994 to brainstorm the theoretical challenges the experiment posed. Bala Iyer was one of the participants at that meeting as was Dhurandhar and the French theoretical physicist Luc Blanchet. “During the meeting, Blanchet felt that the ‘post-Newtonian formalism’ he was developing with Thibault Damour could be generalised to the second post-Newtonian order and higher,” recalled Iyer.(Post-Newtonian expansions in general relativity are used to find an approximate solution to the Einstein field equations. The approximations are expanded in terms of a small parameter, *v*/
*c* (where *v* is the velocity of the astrophysical object and *c* is the speed of light), which express orders of deviations from Newton’s law of universal gravitation. Higher order terms can be added to increase accuracy. In the limit, when the small parameters approach zero, the post-Newtonian expansion reduces to Newton’s equations.)

Together with Blanchet, Damour and a few young colleagues, Iyer was involved in extending the theoretical formalism, now called the Blanchet-Damour-Iyer formalism, to higher orders, enabling the computation of theoretical waveforms to a very high accuracy. These waveforms constitute the theoretical inputs on which the present data analysis strategies rely.

The signal waveforms from binaries of black holes or neutron stars depend on the properties of these objects. For example, two neutron stars with masses comparable to that of the sun will orbit each other at very close separations (and hence high orbital frequencies) before they merge. Thus, they produce GW signals of very high frequencies (around 2,000 hertz) during the merger. On the other hand, heavy mass black holes such as the ones LIGO observed will merge at large orbital separations and hence low orbital frequencies (see Figure 1). Hence, their GW signals span only low frequencies (around 20-200 Hz), which are in the human audio frequency range.

Since one does not have prior knowledge of the properties of the signal that is buried in the data, each data segment has to be compared with signal waveforms corresponding to different values of the source parameters, such as the masses of the black holes. This calls for a clever optimisation: the “template bank” should contain a signal waveform that is close to the signal buried in the data; at the same time, the total number of templates in the bank has to be limited so that the data analysis is computationally tractable.

**Matched filtering technique**

The IUCAA scientists Dhurandhar and B.S. Sathyaprakash (who later went on to set up a leading research group at Cardiff University, Wales), along with a few young collaborators, laid the foundations for a method that makes use of a sophisticated geometric approach to tackle this problem. In the 1990s, the IUCAA became a major hub of activity in the then nascent field of GW data analysis. Dhurandhar and Sathyaprakash were among the first ones to think of using the matched filtering technique for GW data analysis and to implement it in a full-blown computer code.

A particularly intriguing contribution of Dhurandhar’s is the use of a mathematical technique called “stationary phase approximation” to compute the “Fourier transform” of the expected GW signals from binary systems. (The Fourier transform decomposes a signal into the frequencies that make it up, similar to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes—Wikipedia.) In private conversations, Dhurandhar confesses that this idea occurred to him in a rare moment of inspiration while he was jogging in a park in Cardiff. People in the field acknowledge this as Dhurandhar’s original contribution, though he never published a paper on it. Story has it that a reputed journal rejected his paper describing this method for the first time. However, this method was to become an essential tool for GW data analysis in the coming years.

Over the past decade, the Indian GW community has expanded to a number of educational and research institutions. Major contributions of Indian scientists include the development of techniques to “coherently” combine and analyse the data from multiple observatories, development of “hierarchical” search methods that allow them to progressively dig into the data, techniques to make “sky maps” of stochastic GWs similar to the sky maps of the cosmic microwave background, formulating methods to accurately test Einstein’s theory of general relativity using GW observations, modelling GW signals by combining post-Newtonian calculations with large-scale supercomputer simulations, and cross-correlation-based techniques to detect continuous GWs from spinning neutron stars.

The Indian participation in the LIGO experiment, under the umbrella of the Indian Initiative in Gravitational-wave Observations (IndIGO), as part of the worldwide LIGO Scientific Collaboration involves 61 scientists from nine institutions (see table). The IUCAA and the Bengaluru-based International Centre for Theoretical Sciences (ICTS) of the Tata Institute of Fundamental Research (TIFR), Mumbai, host LIGO Tier-3 grid computing centres. At the TIFR, a prototype detector is being built for training and research.

Indian groups have also contributed to the understanding of the response of the detector to the signals and terrestrial influences, to the method used to detect the signal, bounding the orbital eccentricity, to estimating the mass and spin of the final black hole and the energy and power radiated during merger, to confirming that the observed signal agrees with Einstein’s theory, and to the search for a possible electromagnetic counterpart using optical telescopes.

The shape of the observed signal allows scientists to identify the properties of its distant astrophysical source, such as the mass of the orbiting black holes and how fast they spin. In spite of having been formed only in 2013, the ICTS-TIFR group has already made significant contributions to the analysis of the data from the GW detection of September 2015. Making use of results from already available supercomputer simulations, this work has led to the accurate estimation of the mass and spin of the final black hole. The September 14 event has produced a new black hole that is approximately 62 times more massive than the sun and spinning at 67 per cent of the maximum possible spin rate for black holes according to the theory of general relativity (Figure 2). This is the most massive black hole in the stellar-mass range discovered so far, and one of the most accurate astronomical measurements of its kind.

**100 times more powerful**

Black hole mergers are the most powerful astronomical events in the universe after the Big Bang. Although simple analytical calculations provide order-of-magnitude estimates of the energy and power radiated during the merger, a faithful treatment of the systematic and statistical errors requires results from supercomputer simulations of the merger. In the September 14 event, scientists (with major contributions from the same Indian group) have estimated the peak power of the gravitational radiation to be as high as 3.6 × 10
^{49} watts. This makes this event at least 100 times more powerful than the brightest “gamma ray burst”, the most powerful astronomical phenomenon discovered so far. The peak power emission in this event (in the form of GWs) is larger than the average power emission from all the stars in the universe put together.

These powerful astronomical events allow scientists to test Einstein’s theory of general relativity in a hitherto unexplored regime. A new way of testing general relativity emerged out of a collaboration between the RRI group and other scientists, notably Sathyaprakash. They proposed a way of measuring the coefficients of the post-Newtonian expansion from the observed signal to see whether these agreed with the values predicted by the theory.

This test was performed on the recent LIGO observation, with contributions from groups at the Chennai Mathematical Institute, the Indian Institute of Science Education and Research (IISER) Thiruvananthapuram, and IISER Kolkata.

Yet another test proposed and implemented by the ICTS-TIFR group involved measuring the mass and spin of the final black hole from the “inspiral” part of the signal (produced by the motion of the black holes before they merge) and checking their consistency with the same parameters measured from the “post-merger” signal. Analyses performed on the September 14 event have revealed that the observations are fully consistent with the theory’s prediction within measurement uncertainties. These are some of the first tests of Einstein’s theory in the regime of extreme gravity and velocities.

At this point, it is important to mention the seminal work of C.V. Vishveshwara from the 1970s. Vishveshwara worked in Maryland, U.S., but later moved to the RRI. His study of how a black hole responds to an external perturbation led to the prediction of characteristic oscillation modes of black holes, called “quasi-normal modes”. It was later realised that a “perturbed” black hole formed by the merger of two black holes will settle into a stationary black hole by radiating the same type of quasi-normal modes. In the LIGO observation, the last part of the observed signal (referred to as the “ring down” of the black hole) is consistent with the presence of a quasi-normal mode inferred from theory.

The Indian involvement in experimental GW physics started only relatively recently, with the formation of the IndIGO consortium (see table), but the Indian scientific community has the relevant expertise in various aspects of the science of precision measurement. With the prospect of LIGO-India being established, the community aspires to play a leading role in this emerging research frontier.

*P. Ajith is a physicist at the International Centre for Theoretical Sciences (ICTS) of the Tata Institute of Fundamental Research. K.G. Arun is a physicist at the Chennai Mathematical Institute.*

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