# Theory, observation and parameter estimation of gravitational wave radiators

## Abstract

One of the fascinating predictions of General Relativity is the existence of Gravitational Waves (GW). The linear approximation of Einstein Field Equations (EFE) resembles a wave equation and admits plane wave solutions which propagates at the speed of light. Very strong GWs are produced by catastrophic events like binary mergers which reach earth and are detected by ground based detectors like Advanced LIGO, VIRGO etc. In this article we study the gravitational radiation of a rotating four point-mass system, binary systems under the Quadrupole formalism of linearized gravity. This article also explains about various gravitational wave sources, detectors and estimation of various source parameters based on Maximum Likelihood estimation.

**Keywords**: gravitational waves, general relativity, parameter estimation, gravitational wave detectors

## INTRODUCTION

The assusmption that $|h_{\mu\nu}|\;<<1$ allows considering only first order in the perturbation. We can consider $h_{\mu\nu}$ as a symmetric tensor field propagating on a flat background spacetime. The resulting theory is called linearized theory. The linearized theory is invariant under finite Poincare transformations. To linear order in $h_{\mu\nu}$ the Riemanian tensor becomes

This linearized Riemanian tensor is invariant under the transformation ${h}_{\mu \nu}\u27f6{h}_{\mu \nu}-{\xi}_{\nu ,\mu}-{\xi}_{\mu ,\u0306}$ and which is greatly simplified if we introduce trace reverse tensor

where $h = \eta_{\alpha\beta}h^{\alpha\beta}$ and $\bar{h} = -h$. This along with the Lorentz gauge condition $\bar{h}^{\mu\nu}_{,\nu} = 0$ gives Weak Field Einstein equations

## TRANSVERSE TRACELESS GAUGE

As stated earlier we are interested in the study of GW outside the source. This is obtained by considering $T_{\mu\nu} = 0$ in the eqn(4), which gives

which also implies that GWs travel at the speed of light. We can now use Lorentz gauge along with the gauge freedom obtained by the infinitesimal coordinate transformtion $x^{\mu} \longrightarrow x^{'\mu} +\xi^{\mu}(x)$ with $|\partial_{\mu}\xi_{\nu}|<< 1$ in terms of $\bar{h}_{\mu\nu}$ gives

We can choose $\xi_{\mu}(x)$ such that it gives $\Box \xi_{\mu\nu} = 0$. This means we can choose $\xi_{\mu}$ so as to impose four conditions on $\bar{h}_{\mu\nu}$.

Therefore we have reduced the symmeric matrix $h_{\mu\nu}$ to just two degrees of freedom and it ensures $\bar{h}_{\mu\nu} = h_{\mu\nu}$. This metric in the transverse traceless gauge is denoted by $h^{TT}_{ij}$

$h_{+}$ and $h_{\times}$are the amplitudes of the two independent polarization of the waves. Their effect on a ring of test masses on a plane transverse to the direction of propagation of wave is shown in Fig. 1. Therefore the linearized theory also describes a classical gravitational field whose quantum description would be a massless spin -2 field . The metric can be written as

If we have a plane wave solution $h_{\mu\nu}(x)$ propagating in an arbitrary direction $(\hat{n})$ which is already in the Lorentz gauge, we can find the corresponding form in the TT gauge by using the operator $\Lambda_{ij,kl}$ called Lambda tensor, which is defined as

where $P_{ij}$ is defined as $P_{ij}(\hat{n}) = \delta _{ij} - n_{i}n_{j}$ which has a trace $P_{ii} = 2$ and has the following properties

Apart from having a simple form for the GWs in this gauge it also has some peculiar properties. If two masses are at rest (one at the origin and the other at $x^{1} = \xi$) before the arrival of the GWs they will remain at rest even after the arrival of the wave. This can proved from the Geodesic equation in the TT frame.

The last equation vanishes because in TT gauge $h_{00} = h_{0i} = 0$. Hence the particles remain at rest. However the proper distance oscillates as the wave passes, which can be calculated from the metric. Bernard.F.Schutz

Thus we can conclude that in TT gauge the coordinate position remains the same (initial separation $\xi$ remains constant) but the physical effect of the wave can be measured by calculating the *proper distance* or *proper time.*

## GENERATION OF GRAVITATIONAL WAVES

We found that in the Lorentz gauge, trace reversed metric perturbation is given by the relation

Its solution can be found by using retarded Greens function which represents the effect of signals of the source from the past light cone.

where $x^{i}$ is the position far away from the source where we are calculating the field and $R$ is the size of the source which is at the position $y^{i}$. We define $r^{2} = x^{i}x_{i}$ and we have $r^{2}>> y^{i}y_{i}$, therefore the source is far away from the field point so the contribution of $R$in the integral is dominated by $r$ which can be taken outside the integral. We consider a non-relativistic source, so R inside the time argument of the integral can be expanded as follows $t-R = t-r +n^{i}y_{i} + O(\frac1r)$ with $n^{i} = \frac{x^{i}}{r}$, $n^{i}n_{i}=1$.

The direction of the field point is given by the unit vector $n^{i}$. The Taylor expansion in time of the source with the above assumptions give

The above integral contain moments of components of stress-energy. Let denote the moments of density $T^{00}$ by $M$, moments of momentum $T^{0i}$ by $P$ and moments of stress $T^{ij}$ by $S$

The various moments of momentum $T^{0l}$ are

The various moments of momentum $T^{lm}$ are

From the energy momentum conservation $\partial _{\nu}T^{\mu\nu}=0$ we know that Mass, Momentum and angular momentum of the source is conserved.

Similarly,

Using the above relations we get following useful identities

Taking derivative of $\dot{M}^{jk}$ and using the symmetry $S^{ij} = S^{ji}$ gives the identity

Using the above relations we can show that

After applying TT gauge transformations we get a time-dependent part which is purely spatial, transverse and traceless. The leading term of the expression can be written as

where $Q^{ij} = M^{ij} - \frac13\delta ^{ij} M_{kk}$. Thus in the slow-motion limit the dominant source of radiation is the second time derivative of the second moment of mass density which is called the mass quadrupole moment.

## ENERGY CARRIED AWAY BY GRAVITATIONAL WAVES

Gravitational wave has physical effects on test particles, this means gravitational waves carry energy. Recalling the famous Feynman-Bondi sticky bead argument originally proposed by Richard Feynman that if a bead on a stick is placed transverse to the direction of propogation of the wave, the bead slides on the stick back and forth thereby heating the bead and the stick due to friction. This clearly implies that gravitational waves carry energy. The power emitted by a Gravitational source is given bySean.M.Carrol, et al

The minus sign represents the rate at which the energy is changing and the radiating sources will be losing energy.

## GRAVITATIONAL WAVES EMITTED BY A 4-POINT MASS SYSTEM

This system consists of four point masses, two are located in the $x^{1}-x^{2}$ plane and other two are located in the space diagonally opposite to each other making a polar angle $\theta$ w.r.t to $x^{3}$axis. The whole system rotates with angular velocity $\Omega$ with $x^{3}$as the axis of rotation. The corresponding energy density is

The various components of quadrupole moments are

Reduced Quadrupole Moment:

The components of reduced quadrupole moment tensor are $Q_{xx} = q_{xx} -\frac43MR^{2}$; $Q_{yy} = q_{yy} -\frac43MR^{2}$; $Q_{zz} = q_{zz} -\frac43MR^{2}$ other components are same as the quadrupole moments.

We shall compute the wave emerging in the $x^{3}$direction $\hat{n} \longrightarrow (0,0,1)$ for which the projector $P_{ij}$ becomes $P_{ij} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}$ and $\Lambda_{ij,kl}(\hat{n}) = P_{ik}P_{jl} - \frac12 P_{ij}P_{kl}$. This can used to project the quadrupole moments in the TT gauge. The components in the TT gauge are found to be

Thus the radiation emitted by this four point mass system is given by

In this direction ( $x^{3}$) the amplitude will be maximum when the two mass in the space makes an angle $\theta = \frac{\pi}{4}$ with the polar axis $x^{3}$

The Gravitational waves in this direction is generally right hand elliptically polarized.

The wave emerging in the $x^{1}$ direction $\hat{n} \longrightarrow (1,0,0)$ has a projector $P_{ij} = \begin{bmatrix}0&0&0\\0&1&0\\0&0&1\end{bmatrix}$ and the emitted radiation is

The wave emerging in the $x^{2}$ direction $\hat{n} \longrightarrow (0,1,0)$ has a projector $P_{ij} = \begin{bmatrix}1&0&0\\0&0&0\\0&0&1\end{bmatrix}$ and the emitted radiation is

## GRAVITATIONAL RADIATION EMITTED BY A BINARY

Let us consider two stars of mass $m_{1}$ and $m_{2}$ (assume those to be a point mass). Let $l_{0}$ be the orbital separation between them. The total mass given by $M = m_{1} + m_{2}$ and the reduced mass is given by $\mu = \frac{m_{1}m_{2}}{M}$. From Keplers law the orbital frequency is found to be $\Omega = \sqrt{ \frac{GM}{l_{0}^{3}} }$

Let $r_{1}$and $r_{2}$ be the distances from the centre of mass to the masses $m_{1}$ and $m_{2}$ respectively.

$r_{1} = \frac{m_{2}l_{0}}{M}$ and $r_{2} = \frac{m_{1}l_{0}}{M}$.

The stress energy tensor of the system is given by

Following the same procedure in the 4-Point mass system above, for a wave travelling in the $\hat{n} \longrightarrow (0,0,1)$ direction. The components of quadrupole moment in the TT gauge is found to be

The wave is circularly polarized and the radiation is emitted at twice the orbital frequency. The wave emitted from other directions are found to be linearly polarized.

This problem has an important significance. In 1975 R.A Hulse and J.H Taylor discovered a binary system PSR 1913+16 in which two neutron stars were neutron stars at very short distance from each other in a nearly circular orbit. The orbital period of the binary system decreases in time due to the emission of gravitational wave which is in good agreement with the predictions of General Relativity. This provided the first indirect evidence of gravitational wave. The decrease of orbital period due to GW emission (orbital evolution) brings the stars closer, the process becomes faster and they finally spiral down and coalesce.

The gravitational wave luminosity is found from the reduced quadrupole moment of the binary system which is

$L_{GW} = \frac{dE_{GW}}{dt} = \frac{32G^{4}\mu^{2}M^{3}}{5c^{5}l_{0}^{5}}$

Here we assume that the orbital parameters do not change significantly over the time interval taken to compute the average. This assumption is called adiabatic approximation. In this approximation the system has time to adjust the orbit to compensate the energy lost in gravitational waves with the change in orbital energy.

\frac{E}{dt} + L_{GW} = 0

where E is the orbital energy, which is given by $Orbital Energy = Kinetic + Potential$

The above equation gives the energy of the binary system. This can be used to prove how the decrease in period is related to the emission of gravitational waves. Differentiating eqn(44) w.r.t. time gives

We know that the orbital frequency is $\Omega ^{2} = \frac{GM}{l_{0}^{3}}$, taking logarithm of the on both sides gives $\frac{1}{\Omega} \frac{d\Omega}{dt} = -\frac{3}{2l_{0}} \frac{dl_{0}}{dt}$

The period of the orbit is given by $P = \frac{2\pi}{\Omega}$, which on differentiating with time gives the relation

Substituting equation (47) in (48) gives

This proves that the orbital period decreases due to the emission of gravitational waves.

## SOURCES OF GRAVITATIONAL RADIATION

## Continuous Sources

When a source emits gravitational waves continuously over a period of time longer than the observation time at a nearly constant frequency then we can say that it is a source of continuous gravitational waves. A neutron star with an axial-asymmetry or wobbling about their rotation axis can generate continuous gravitational waves. If the source is symmetric about the rotation axis it cannot emit gravitational waves. Even if the neutron star is axis-symmetric, it still can emit radiation if the axis of symmetry of the star is different from the axis of rotation. In continuous gravitational sources, like neutron stars, with time there will be a change in frequency due to loss of angular momentum in gravitational radiation. This is called spin down.

Another mechanism of generation of gravitational waves from rotating star is due to its fluid modes of oscillation. There are oscillatory fluid modes which moves retrograde with respect to stars rotation. In such a case positive amount of angular momentum will be carried away from the star. The pattern moving in the retrograde sense relative to the star will amplify the fluid mode due to decrease in angular momentum. This causes an instability known as Chandrasekhar-Friedman-Schutz instability or CFS instability. There are different modes in rotating star that causes this CFS instability which are damped out by viscous forces.

## Burst Sources

Coaslescing binaries are the sources of burst radiations. These are gravitational waves produced by binaries like Neutron stars, White dwarfs and Black holes. These binaries has been studied extensively in the literature and they are the promising sources for detection. Various binary coalescences in high frequency band has been detected by ground based detectors like LIGO. Two binaries collide and form a new compact object of the same time in which some of the mass will be emitted as gravitational radiation. If the coalescing binary consists of a Black Hole and a Neutron star the resulting object will be a Black Hole. The wave form modelling of these sources are very important for its detection. Non-relativistic binaries are modelled using Post-Newtonian theory. The life cycle of a binary has three phases- Inspiral, Merger and Ringdown. In inspiral phase the binaries orbit around their common centre of mass, the orbit shrinks due to loss of gravitational radiation and its period increases. The gravitational wave emission increases as the orbit shrinks and finally comes to a stable circular orbit which marks the end of inspiral phase and the transition to the merger phase where the gravitational wave emission peaks. Following the merger the final blackhole oscillates in shape between a distorted, elongated and a flattened spheroid. This oscillation dampens due to the emission of gravitstional waves. This phenomenon is called ringdown.

Another source of burst generation is the gravitational collapse of a star. Consider the collapse of a rotating stellar core of a star supported by electron degeneracy pressure to a neutron star. As the star collapses the angular momentum is conserved, so shape of the stellar core changes with the matter near the equator. This collapse continues until the stellar core reaches a nuclear density at which it is supported by neutron degeneracy pressure. The neutron star will be an axis-symmetric ellipsoid. An observer along the axis of the collapse does not see gravitational radiation but observer looking along the equator will see a linearly polarized gravitational wave.

## Stochastic Sources

The stochastic sources of gravitational waves arises from an extremely large number of weak, independent and unresolved gravitational wave sources. This gives an observational constraint to distinguish which gravitational waves comes from which sources. We can broadly classify such stochastic processes in two: Cosmological and Astrophysical Stochastic process. Whatever the nature of the statistical distribution of that process, the central limit theorem guarantees that the sum of the signals at any given time or frequency is a random variable drawn from a Gaussian distribution. This is the reason why we call it a stochastic process.

## GRAVITATIONAL WAVE DETECTORS

The first built gravitational wave detector is a resonant bar detector. These detectors are set into oscillations by gravitational waves of frequency that are close to the natural frequency of bar. There are several such detectors operating in the world, some are operated at very low temperature to reduce the thermal noise. Other type of detectors are Laser Interferometers - Advanced LIGO , VIRGO, TAMA, GEO600, INDIGO (planned) etc. These Interferometers already detected several binary coalescence events, they have sensitive in the frequency band 10Hz-1KHz. There are spaced based detectors like LISA which are sensitive in the band 1-100 mHz. Other experimets include pulsae timings which can observe gravitational waves in the frequency band nHz- mHz.

## Response of a Detector

The response of a detector can be written in terms of a Symmetric Trace Free tensor $D_{ij}$, its value depends on the type of detector. For a laser interferometer the form of the detector tensor is ]R. L. Forward, 1978

where $n_{1}$ and $n_{2}$ are the unit vectors along the direction of the interferometer arms. For a resonant bar the detector tensor has the form

$D_{ij} = n_{i}n_{j} - \frac13 \delta_{ij}$

The incoming gravitational wave can also be expressed in terms of a Symmetric Trace Free tensor. Then the response of a detector to the incomming gravitational wave can be written as

$h(t) = h_{+}(t)F_{+}(\theta,\phi) + h_{\times}F_{\times}(\theta,\phi)$

$e_{ij}^{+}(\hat{n}) = \hat{u_{i}}\hat{u_{j}}-\hat{v_{i}}\hat{v_{j}}$

$\hat{u}$ and $\hat{v}$ are unit vectors orthogonal to the direction of propagation of gravitational wave $\hat{n}$.

## Laser Interferometers

In an interferometer light from a laser after passing through a beam splitter (at the centre) is splitted, half of the light is sent down one arm and the other half is send to the other arm which is perpendicular. The beams sent to two arms have correlated phases. These beams after reflection from end masses are brought back for interference. The interference measures the change in length of two arms caused by the passage of gravitational wave,

The above relation shows that longer the arms of the intererometer, better the sensitivity. The interferometers suffers from different instrumental noises. The signal is extracted from such a noise by using a technique called matched filtering. To filter out ground vibrations, the optical components are suspended by two-three stage suspensions system which are made for the instrument to observe at lower frequencies. Thermal vibration of mirrors and suspension systems are another problem, which cannot be controlled by reducing the temperature because the heat generated from the laser will always remain. Thermal vibrations can be minimized by using high quality materials. Another source of noise is called the Photon shot noise which comes from the random fluctuations of intensity of the two beams. This can be reduced by increasing the number of photons, with more photons the power fluctuations will decrease.

## Laser Interferometer Space Antenna

Laser Interferometer Space Antenna (LISA) would be the first space based gravitational wave detector. It uses the same principles of laser interferometry. LISA has a constellation of three spacecraft arranged in the shape of an equilateral triangle. The distance between the spacecraft's are precisely measured using 2 W laser, the change in distance between spacecraft will be a sign of passing gravitational wave. The main challenge is that unlike ground based gravitational wave observatories the arm of LISA cannot be locked in a specific position, the distances between the spacecraft varies significantly over a period in orbit. This change must be tracked precisly and small thrusters should be used to adjusted the position. One of the main advantage of LISA is its long arm length (2.5 Million Kms), which helps in increasing the detection accuracy. LISA is sensitive to low frequency band of gravitational wave spectrum like Extreme-mass-ratio inspirals (EMRI), Supermassive blackhole binaries.

## Resonant Mass Detectors

The research for gravitational waves started with the resonant mass detectors or otherwise called as Weber bar because of its founder Joseph Weber. It consists of an aluminium cylinder of length two meters and 0.5 meter diameter and isolated from vibrations in a vaccum chamber. Today there are few such detectors kept in lower temperatures to reduce the noise from thermal effects. The principle of the weber bar is that the gravitational wave passing perpendicular to the axis of the detector produces tidal forces which will stretch and contract the length of the cylinder. This change in length is sensed by a piezo electric transducer wrapped around the detector and converts it into an electric signal. The weber bar is sensitive to a strain of $10^{-16}$.

## STATISTICAL THEORY OF SIGNAL DETECTION

The output of a Gravitational wave detector is a combination of signal and noise. The signal will be buried in a sea of noise, to extract such signals we need certain statistical methods. These are same methods employed in radio engineering. A function of time like record of strain in the detector is called a time-series, and a collection of such time series measured under similar condition is called an ensemble. A random function whose values are described only by means of a set of probability distributions referred to such an ensemble often goes under the name of stochastic process. A stochastic process is also called as random process. A particular member of the ensemble is called a realization of the stochastic process. In this chapter we will see the methods used to extract signals from the data.

## Stochastic Process

Let $T$ be a subset of real numbers $T \subset R$. A stochastic or a random process $x(t)$ is a family of random variables $x(t)$ labelled by the number $t \in T$. They have a joint n-dimensional cumulative distribution function F defined by

When T is a discreet set of points, $T = \{ t_{1},t_{2},...... \}$, the stochastic process is called a random sequence. The stochastic process $x(t)$ is Gaussian if the cumulative distribution function $F_{t_{1},......t_{n} }(x_{1},x_{2},.....x_{n})$ is Gaussian. The stochastic process is stationary if all the finite dimensional cumulative distribution functions $F_{t_{1},......t_{n} }(x_{1},x_{2},.....x_{n})$ defining the stochastic process remain the same if the set of points $T = \{ t_{1},t_{2},...... \}$ is shifted by an arbitrary constant $\tau$, i.e.) the probabilistic structure of the stochastic process is invariant under a shift of the parameter $t$

## Properties of stochastic process

The first order moment $\mu _{1}$ is called the mean value of the stochastic process $x(t)$ which is denoted by $m(t)$

The second-order moment, $\mu_{1,1}$is called the auto-correlation function, it is given as

## Testing of Hypothesis

The gravitational wave output is in the form of time series, the probability distribution describing the output of the detector varies in the presence and absence of signals. The output of the detector can be written in the form $x(t) = n(t) + h(t)$, where $n(t)$ and $h(t)$ are the gaussian stochastic noise and the signal we are looking respectively. Let $p_{0}(x)$ be the probability distribution when the signal is absent in the data and $p_{1}(x)$ be the distribution in the presence of the signal. Let's define the absence of signal as the null hypothesis $H_{0}$ and presence of signal as the alternate hypothesis $H_{1}$. Thereby we can partition the observation set into $R$ and $R'$. If the observed data is in $R$ we accept the null hypothesis otherwise we reject it. There are three very different approaches in hypothesis testing all leading to a famous test called likelihood ratio test. It is defined as $\Lambda(x) = \frac{p_{1}(x)}{p_{0}(x)}$ i.e.) it is the ratio of the probability distribution when the signal is present and when the signal is absent. We choose alternate hypothesis if the value of likelihood ratio $\Lambda(x)$ is greater than a threshold value.

## Bayesian approach

In this method we assign costs to our decisions, let $C=\left[\begin{array}{cc}{\textstyle {C}_{00}}& {\textstyle {C}_{01}}\\ {\textstyle {C}_{10}}& {\textstyle {C}_{11}}\end{array}\right]$