Gravitational waves are produced by the merging of binary cosmic bodies such as neutron stars and black holes. Using the basis in Einstein’s theory of relativity, astronomers and scientists alike have tried to come up with efficient and accurate methods to detect and observe gravitational waves. This resulted in the development of detectors from resonant mass detectors to laser interferometers. Since the discovery of gravitational waves recently, it has become imperative to capitalize on the data and come up with templates that will make it easier to detect gravitational waves in the future. The dichotomy of detecting gravitational waves is that while they help in uncovering the mysteries of the universe that would otherwise been impossible with only electromagnetic waves, there is no way of knowing the details of their source. This paper explores how the waves from different combinations of neutron stars and black holes with varying masses, radii and orbital frequencies are analyzed to accurately categorize and profile their sources. Moreover, this paper intends to provide an insight in to the methods and theories that have or are being used, what they imply and their impact on future research.

Gravitational waves are ripples in the curvature of space-time travelling at the speed of light. It was Oliver Heaviside in 1893 who first made them analogous to the inverse-square law between gravity and electricity. The curvature of space-time is generally caused by the presence of a mass or in an astrophysical sense, a stellar mass. The greater the mass in a given volume, the greater the curvature of space-time at its boundary. As these masses move, the curvature changes to adjust for the movement and thus produces gravitational waves. In 1905, Henry Poincare compared them to accelerating electrical charges producing electromagnetic waves, but it wasn’t until 1915 when Einstein published his theory of general relativity that the concept of gravitational waves gained some traction. Poincare’s ideas implied that for gravitational waves to exist there would be no such thing as gravitational dipoles. Einstein on the other hand believed that there are three types of gravitational waves, namely longitudinal-longitudinal waves, longitudinal-transverse waves and transverse-transverse waves. Einstein and Eddington then studied the problem further and predicted that very small amounts of energy would be radiated by a spinning rod or a double star and this laid the foundation for future research into the topic. Einstein worked on a paper with Rosen where they came up with rigorous solutions for undulatory gravitational fields with cylindrical gravitational waves in Euclidean space. They used the approximate method of integration of gravitational equations of the general relativity theory to lead to the existence of gravitational waves. Einstein’s final result came about after the radiation in special relativity was analyzed using nonlinear theory and is represented by the leading order terms of the quadrupole formula for gravitational wave emissions. He started with the integration of gravitational equations of general relativity theory to provide the existence of gravitational waves defined as,

\begin{equation} R_{\mu\nu}-{1/2}g_{\mu\nu}R=-T_{\mu\nu},\nonumber \\ \end{equation}where \({g_{\mu\nu}}\) can be replaced by the equation

\begin{equation} g_{\mu\nu}=\delta_{\mu\nu}+\gamma_{\mu\nu},\nonumber \\ \end{equation}and

\begin{align} \delta_{\mu\nu} & =1 & \mu=\nu\notag \\ & =0 & \mu\neq\nu,\notag \\ \end{align}as theorized by (Einstein 1937).

An approximately small value of \({\gamma_{\mu\nu}}\) shows that the gravitational field is weak with its derivatives occurring in many high powers. These higher power derivatives can be neglected due to insignificance and the rest of the equation is represented as

which further expands into the relation

\begin{equation} \overline{\gamma}_{\alpha\alpha,\mu\nu}+\overline{\gamma}_{\mu\nu,\alpha\alpha}-\overline{\gamma}_{\mu\nu,\alpha\nu}-\overline{\gamma}_{\nu\alpha,\alpha\mu}=-2T_{\mu\nu},\\ \end{equation}as stated in (Einstein 1937).

The intention of those equations was to establish conditions under which the variables can exist as gravitational equations for empty space that can then be written as

thus providing the plane gravitational waves moving in a positive direction along the x-axis with the following conditions satisfied

\begin{align} \overline{\gamma}_{11}+i\overline{\gamma}_{14} & =0\notag \\ \overline{\gamma}_{41}+i\overline{\gamma}_{44} & =0\notag \\ \overline{\gamma}_{21}+i\overline{\gamma}_{24} & =0\notag \\ \overline{\gamma}_{31}+i\overline{\gamma}_{34} & =0,\notag \\ \end{align}The plane gravitational waves are then categorized into three types based on the values of \(\overline{\gamma}\). Pure longitudinal waves have only \(\overline{\gamma}_{11}\), \(\overline{\gamma}_{14}\) and \(\overline{\gamma}_{44}\) not equal to zero. Half longitudinal, half transverse waves have only \(\overline{\gamma}_{21}\) and \(\overline{\gamma}_{24}\) or only \(\overline{\gamma}_{31}\) and \(\overline{\gamma}_{34}\). The pure transverse waves have only \(\overline{\gamma}_{22}\), \(\overline{\gamma}_{23}\) and \(\overline{\gamma}_{33}\) that are not equal to zero. These categories of gravitational waves from Einstein’s theories laid a foundation that aided in the observation of gravitational waves many years later.

On September 14th, 2015, both twin Laser Interferometer Gravitational-wave Observatory (LIGO) detectors in Livingston, Louisiana and Hanford, Washington, detected the presence of gravitational waves emitted by a cataclysmic event in the distant universe. The LIGO scientists estimated that the black holes involved in the formation of gravitational waves were about 29 and 36 times the mass of the sun. This incident occurred 1.3 billion years in the past, and about 3 times the mass of the sun was converted in to gravitational waves in a very short amount of time. The power output of the binary merger between the black holes produced a power output equal to 50 times that of the visible universe.

A quadrupole formula was hence developed which defined the loss of orbital energy and angular momentum due to gravitational wave emission. The importance of the discovery of gravitational waves was that they contain information amongst themselves through their origin about the nature of gravity that could otherwise not be obtained and moreover their detection contains potential application for future research.

Gravitational waves reach earth continuously from a great distance. Their effect however is negligible. The gravitational waves produced by the merger of GW150914 travelled over a billion lightyears and its largest effect on earth was that it changed the length of one of the LIGO detectors by ten thousandth the width of a proton. LIGO states that any object with mass that accelerates produces gravitational waves even on as small of a magnitude as humans and vehicles; its just that they are too small to detect. The gravitational waves that are detected are normally a result of incredibly dense objects such as neutron stars spinning in the deep regions of space. The main sources of gravitational waves in space are mainly neutron stars, black holes and to a smaller extent supernovas. This is because all the events that cause the formation of neutron stars and black holes form a background in the audio frequency part of the spectrum. Previously there has been some evidence for the emission of gravitational waves like the Hulse-Taylor binary system which consisted of a pulsar. The pulsar had a 17 Hz radio emission in a 8 hour orbit and a companion neutron star. This led to astronomers finding more sources of gravitational waves.

Gravitational waves are similar to electromagnetic waves in many ways. They carry energy and angular momentum away from the source. Additionally, gravitational waves also carry off linear momentum whose ’kick’ in the region of approximately 4000 km/s can knock a coalescing black hole out of its host galaxy. (González 2007). Also gravitational waves show redshifts and blue shifts not only due to the relative velocities of the observer and the source but also because of changes in space-time and universal expansions.

All the sources can be generally classified into three distinct classes based on the signal that can be extracted. One of the categories is the burst sources consisting of the formation of neutron stars, black holes and supernovae. It is a short event with one or very few cycles of signal of broad bandwidth. Another category is the narrow-band sources consisting the rotation of non-axial symmetric stars such as pulsars and accreting neutron stars. These sources are much weaker than burst sources and are affected by the Doppler shift. The third category is the stochastic backgrounds which consist of weak periodic sources from galaxies or burst sources at very large distances. Binary neutron star systems are known to produce gravitational waves in all three classes. Also present are low-frequency sources which are normally the result of massive black holes. With masses ranging between \(10^{6}\textemdash{10^{9}}\) solar masses, waves are produced by low mass objects orbiting the aforementioned black holes. Neutron stars and black holes are known to have high spin rates and these spin rates give properties to the gravitational waves that are observed.

Bursts are transients of poorly known phase evolutions. The detectors that are setup to observe bursts are extremely sensitive to high signal-noise-ratios. For these waves to be detected, the merger or the supernova explosion that causes this needs to be asymmetric. Stochastic waves mainly arise from a superposition of sources according to (Riles 2013).Short period binary systems are the ones that produce an abundance of gravitational waves as long as their masses are greater than 0.6 solar masses, below which it becomes much harder to detect. The binary systems are quite similar until the collision, so there is an enveloping theory to accurately model them for future research.

Compact binary systems namely, neutron star/neutron star (NS/NS), neutron star/black hole (NS/BH) and black hole/black hole (BH/BH) are the most promising sources of gravitational waves in the universe. Under the effect of gravitational radiation, a neutron star binary (consisting of two neutron stars) with orbital periods generally around less than half a day, will spiral together and merge in less than a Hubble time. A Hubble time is the reciprocal of the Hubble parameter which is defined as the ratio between the size of the universe at a certain time to the size at a chosen time. Many x-ray binary systems show neutron stars spinning between 250 – 350 Hz. Their spin rate is determined by their mass accretion and angular momentum. The spiralling can take place over millenia as it is their mutual orbital energies that keep these binary systems from colliding. Over time these bodies loose the energy and move closer towards one another and subsequently revolve faster. Before the tidal disruption/coalescence stage begins, during the last few minutes of inspiral a strong gravitational wave signal is emitted. With LIGO and VIRGO detectors at optimum sensitivity, this is observable to a high degree of accuracy. In the past, there have been three occurrences of binary mergers, which when extrapolated to the rest of the universe, they have led to an estimated merger rate in the universe of about 102yr\({}^{-1}\)Gpc\({}^{-3}\). The binary merger of two black holes of 10 solar masses and higher can be detected upto distances with red shifts of about 2 – 3 based on the calculations of (Cutler 1994).

Stellar evolution studies mention that globular clusters are responsible for the formation of binary black holes which tend to coalesce without an electromagnetic signature under a Hubble time.

Binary systems while being the best source for gravitational waves have innately large uncertainties. One of these uncertainties are the coalescence rates of neutron stars and/or black holes. Data from LIGO suggests that the estimated merger rates of NS/NS binary systems range between \(2\times 10^{-4}\) –\(0.2\) per year, \(7\times 10^{-5}\) – \(0.1\) per year for NS/BH binary systems and \(2\times 10^{-4}\) – \(0.5\) per year for BH/BH mergers assuming they are within the boundary masses. A NS/NS binary is normally of the magnitude of 1.4-1.4 solar masses while a BH/BH binary is usually 106-106 solar masses.
Observers also have with them theoretically designed templates through previous experimentations and theories such as those by Einstein that they use to compare the current observed waveforms. A single detector provides enough output to determine the masses of the bodies in the binaries but unfortunately not their distance or their position in the sky.

Detectable gravitational waves have some very identifiable properties. The signal from gravitational waves is proportional to the size of the objects that are merging. They also have very low frequencies with similarly low graviton (\(\hbar\omega\)) energy. When passing through ordinary matter, only a negligible amount of gravitational wave gets scattered or absorbed, and pass unobstructed through dust clouds and stars. With that in mind detectors were created to analyze these waves.

Resonant mass detectors were first used in the 1960’s in experiments to detect gravitational waves. They could have been described as simple harmonic oscillators that were driven by gravitational waves. An example of a resonant mass detector is the mass quadrupole detector. By measuring the displacement amplitude and the power absorbed it provided important components of the Riemann tensor. These detectors were later improved into interferometric detectors by reducing the background noise and having a long-baseline broadband. Today the detectors are widely separated to distinguish the observed gravitational waves from environmental and background noise. Multiple detectors are needed to be setup because of the difficulty in detection gravitational waves. Each detector consists of mirrors with a distance of about 4 kilometers between the two sites. For increased sensitivity, each detector has a resonant optical cavity that multiplies the effect. The detectors two methods of searching that are namely the generic transient search and the binary coalescence search. The generic transient search does not use a specific waveform model but rather reconstructs the signal waveforms with common gravitational wave signal using a maximum likelihood method. The statistic is defined by the equation

where \({E_{c}}\) is the dimensionless coherent signal energy that is observed from the reconstructed waveforms, \({E_{n}}\) is the dimensionless residual noise.

Each event is then classified into search classes; class C1 being known population of noise transients, C3 being events with frequency that increases with time and C2 being all the remaining events.(Cutler 1994)

The binary coalescence search uses emissions from binary systems with individual masses lower than 99 solar masses. This model assumes that the spins of the binary system are aligned with orbital angular momentum and a template is made which covers an extremely large parameter space. It is safe to assume that the orbits are circular because the gravitational radiation reaction causes the eccentricity of the orbits to decrease during the spiral, according to equation

where \({P}\) is the orbital period and \({\epsilon}\) is the eccentricity.

And the bodies can be treated as structureless, spinning point masses because tidal interactions between them have been shown to be negligible. For the most frequent gravitational wave occurrences, the detectors calculate the strain amplitude with equation

where h(t) is a potential signal, and n(t) is the detector noise.

After convolving with Wiener’s optimal filter, \(\textit{s(t)}\to\int{(\omega(t-\tau)s(\tau)d\tau)}\), allows the measurement of signal to noise ratio S/N which is represented by the equation

where the denominator is the root-mean-square value of the numerator if the detector measured only noise.

In the absence of gravitational waves, the S/N[h] has a root mean square equal to 1. If the value is greater than or equal to 6.0 in each of the detectors then with almost certain confidence, a gravitational wave has been detected. This is done by comparing the value to around 10\({}^{15}\) templates of waveforms, with more being added every year. The signal however is limited to the distance at which it can measure. So a signal to noise ratio is created for the network of detectors defined by the equation,

where \(\rho_{a}\) is the signal to noise ratio in the ath detector.

The detection threshold is represented by \(\rho\) which are nearly equal to 8.5. The threshold is the point at which the most distant binary system can be observed by the detectors. This distance is usually 100 Mpc. Since the gravitational waves detected are determined mainly by the masses of the spinning bodies an equivalency is created for the combination of masses in the system. This equivalency includes the errors that can be created by the post-Newtonian effects. The equivalency is

defined as the chirp mass which in turn helps define each of the masses in the binary system.(Tutukov 1993)

The chirp mass in general relativity is the leading-order amplitude and frequency evolution of the gravitational wave signal that is emitted by a binary system. It helps in accurately determining the individual masses in the binary system.

The information collected however only represents a small portion of the data that needs to be measured because the detector will have noise with Gaussian and non-Gaussian components. The non-Gaussian noise is considered unimportant since most of the time it will not be detected simultaneously in the twin detectors. The remaining Gaussian part is defined by spectral density \({S_{n}(f)}\), where \({f}\) is the the frequency such that,

where \({S_{0}}=3\times 10^{-48}\)Hz\({}^{-1}\) and \({f_{0}}=70\)Hz.

The detector noise determines the strength of the signal that can be detected and the distance to which it can be observed. The previous equation can detect a NS/NS merger out to distances in the range of 1 Gpc. These detectors then use the Fourier transform of the Riemann tensor for further detection of the radiation, that will accurately identify the binary system. The Riemann tensor is a common method used to define the curvature of space-time and beneficial in the theories of general relativity.

The merger rates of neutron stars are usually estimated using binary pular statistics and supernova rates. Merger rates of black holes on the other hand correspond to the category of star clusters.
Theoretically there are two broad ways under which gravitational waves can be detected.

Classical physics involving stellar objects can be solved using classical or Newtonian gravitational two-body problems. This leads to an accurate model of a large range of astrophysical systems. Gravitational waves however are based on the theories of general relativity which is nonlinear so many systems are not symmetric and because it includes radiation, the bound solutions change due to angular momentum and energy. So it cannot be represented with a two-body solution. Hence these waves as a concept cannot exist in this system. In Newtonian gravity bodies follow closed elliptical orbits. A finite coupling coefficient by Einstein tries to express gravitational waves in Newtonian physics in the equation

\begin{equation} \textbf{T}=\frac{c^{4}}{8\pi G}\textbf{G},\nonumber \\ \end{equation}where T is the stress energy tensor, G is the Einstein curvature tensor, \({c}\) is the speed of light and \({G}\) is Newton’s gravitational constant.

This equation represents the high rigidity of space which is a good approximation of why the gravitational waves have small amplitude even with high energy density. Gravitational waves have a weak coupling to matter which further proves the massive elastic stiffness of space-time.

General relativity vastly differs from Newtonian physics. Unlike Newtonian gravity, in general relativity when the binary shrinks as the objects merge, the frequency and amplitude of the gravitational waves increase. The basis of the methods of the detection are based on the fact that waves are produced by a system of masses that are interacting with one another defined by the action principle,

\begin{equation} \delta I=\delta[-cm\int{ds}+W]=0,\\ \end{equation}where m is the rest mass and W is resultant of the motion of the mass reacting with another.(Weber 1960) The element ds is defined by

\begin{equation} ds^{2}=g_{\mu\nu}d\chi^{\mu}d\chi^{\nu},\\ \end{equation}which can further be approximated as the gravitational field equation,

\begin{equation} g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\nonumber \\ \end{equation}where \(\eta_{\mu\nu}\) is the metric of the flat background and \(h_{\mu\nu}\) is the perturbation on the background. The equations of general relativity then become a system of linear equations,

\begin{equation} \left(\nabla^{2}--\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\right)\nonumber \\ \end{equation}providing a three-dimensional wave equation with no \(z\)-component.(Ju 2000)

The formation of gravitational waves is an exciting event even in theory. The collision of two massive stellar objects such as stars or black holes to produce waves that are detected light-years away by instruments that were created on a hypothesis is extraordinary in itself. Moreover, the detection of gravitational waves provided the ability to view the universe in a completely different way. It potentially provides evidence to the fact that the universe expanded in a process known as cosmic inflation. Prior to their discovery the astronomical observations were restricted to electromagnetic waves and particle-like entities. These unfortunately have a limitation. They can be obscured or hidden behind other stellar objects that absorb these ’light’ waves. Gravitational waves on the other hand, do not. With one eye on the future, LIGO intends to detect five more black hole mergers in the next observing campaign and possibly 40 more binary star mergers each year. With these objectives come many related advancements in the technology. The signal-to-noise ratio upgrades are expected to double and improve detections by a factor of ten. A Laser Interferometer Space Antenna (LISA) is proposed to be placed in space with its main goal to detect gravitational waves. LISA will look to detect mergers a 1000 years before they merge that will allow astronomers to create classes of previously unknown sources. The information received from the amplitudes of the waves from a black hole merger make the measurement of distances more accurate.

Coalescing compact binaries are currently the best sources to detect and understand gravitational waves. Extrapolation from the data collected by the detectors shows that around several hundred binary neutron star mergers happen every year. Binary mergers hence are much better than stellar core collapses or pulsars and supernova as sources of gravitational waves.(González 2007)

The LIGO detectors have observed waves from the coalescence of a binary black hole system and the resultant waveforms match the theoretical templates that have been created and stored for such mergers. This observation shows the existence of binary black hole systems. The results can then be used to trace back to what binary systems led to the detection of the waves.

If the merging objects are black holes, they form a single perturbed black hole that radiates gravitational waves as a superposition of quasinormal ringdown modes according to (Abbott 2016). Eventually the oscillation gets damped after the merger and reaches constant frequency as the black hole reaches its final state. Black holes binaries also have a different phenomenon that help in their identification and detection of gravitational waves called the ’kick’. This recoil is due to the emission of anisotropic gravitational radiation and the loss of linear momentum as it is travels away. A big difference between neutron star mergers and black hole mergers are that black hole mergers are much rarer. Also currently with LIGO it is harder to measure black holes since they exist at much further distance and in some cases, their masses are too large for accurate detection. There will be more accurate data if and when LISA is activated. The present evidence though uses the eccentricity of the orbits and the asymmetric mass loss in the merger to extrapolate the findings. If tidal deformations or rotation-induced quadrupole moments are present in the analysis of companion stars, it becomes highly unlikely that it is a neutron star binary. This is consistent with the mass distribution normally present in neutron star binaries. Hence a system containing two neutron stars has relatively lower masses and an orbit that is inclined at nearly 90 degrees. The decay of the orbital period because of gravitational waves is measurable within 15 months. There are as many as five neutron star binary systems known in the universe so far, but only three have orbits tight enough to cause a merger within a Hubble time. Two of these neutron star binaries are located in the galactic field while the third is on the outskirts of the globular cluster. The two binary systems in the galactic field with merge with their companion stars because of the emission of gravitational waves that have been detected are in the right timescale for a merger. These properties also have an effect on the prediction of the rate of mergers between binary systems.(Burgay 2003)

If the mass of the objects after calculations turns out to be greater than 3 solar masses, then with high probability the binary system consists of