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Summer Research Fellowship Programme of India's Science Academies

Theory, observation and parameter estimation of gravitational wave radiators

Sree Kanth H

Department of Theoretical Physics, University of Madras, Chennai 25

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

According to General Relativity mass produces spacetime curvature. Non-uniform motion of mass is the sources of ripples in curved spacetime, which propogates at the speed of light. These propagating ripples in spacetime curvature are called Gravitational Waves. The weak coupling of GW to matter makes it so difficult to detect and is also what makes them astrophysically significant because they are least absorbed. Our primary focus is on time-varying weak gravitational fields propagating on empty spacetime ​Sathyaprakash, 2009​. The weakness of gravitational field is expressed by decomposing the metric into flat Minkowski metric plus a small perturbation. ​​

 gμν=ημν+hμν          hμν  <<1      μ,ν=0,1,2,3\displaystyle  g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\;\;\;\;\;\vert h_{\mu\nu}\vert\;<<1\;\;\;\mu,\nu=0,1,2,3

The assusmption that hμν  <<1|h_{\mu\nu}|\;<<1 allows considering only first order in the perturbation. We can consider hμνh_{\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 hμνh_{\mu\nu} the Riemanian tensor becomes

Rμνρσ=12(hμσ,ρν+hνρ,σμhνσ,ρμhμρ,σν)\displaystyle R_{\mu\nu\rho\sigma} =\frac12 (h_{\mu\sigma,\rho\nu} + h_{\nu\rho,\sigma\mu} - h_{\nu\sigma,\rho\mu} - h_{\mu\rho,\sigma\nu})

This linearized Riemanian tensor is invariant under the transformation hμνhμν-ξν,μ-ξμ,̆ and which is greatly simplified if we introduce trace reverse tensor

hˉμν=hμν12ημνh\displaystyle \bar{h}^{\mu\nu} = h^{\mu\nu} - \frac12\eta^{\mu\nu}h

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

hˉμν=16πGc4Tμν\displaystyle \Box\bar{h}_{\mu\nu} = - \frac{16\pi G}{c^{4}} T_{\mu\nu}

TRANSVERSE TRACELESS GAUGE

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

hˉμν=0\displaystyle \Box \bar{h}_{\mu\nu} = 0

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 xμxμ+ξμ(x)x^{\mu} \longrightarrow x^{'\mu} +\xi^{\mu}(x) with μξν<<1|\partial_{\mu}\xi_{\nu}|<< 1 in terms of hˉμν\bar{h}_{\mu\nu} gives

hˉμνhˉμν=hˉμν(μξν+νξμημνρξρ)\displaystyle \bar{h}_{\mu\nu} \longrightarrow \bar{h}^{'}_{\mu\nu} = \bar{h}_{\mu\nu} - (\partial _{\mu}\xi _{\nu} + \partial _{\nu}\xi _{\mu} - \eta_{\mu\nu}\partial _{\rho}\xi^{\rho})
 νhˉμν(νhˉμν)=νhˉμνξμ\displaystyle  \partial^{\nu}\bar{h}_{\mu\nu} \longrightarrow (\partial^{\nu}\bar{h}_{\mu\nu})^{'} = \partial^{\nu}\bar{h}_{\mu\nu} - \Box \xi _{\mu}

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

 h0μ=0 ,        hii=0 ,       jhij=0\displaystyle  h^{0\mu} = 0  ,         h^{i}_{i} = 0  ,        \partial ^{j} h_{ij} = 0

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

 hijTT(t,z)=[h+h×0h×h+000 0 ]Cos[ω(tz/c)]\displaystyle  h^{TT}_{ij}(t,z)=\begin{bmatrix} h_{+}&h_{\times}&0\\h_{\times}&-h_{+}&0\\0&0&  0   \end{bmatrix} Cos[\omega(t-z/c)]

​​​ h+h_{+} and h×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

 ds2=dt2+(1+hxxTT)dx2+(1hxxTT)dy2+2hxyTTdxdy+dz2\displaystyle  ds^{2} = -dt^{2} + (1+ h^{TT}_{xx})dx^{2} + (1- h^{TT}_{xx})dy^{2} + 2h^{TT}_{xy}dxdy + dz^{2}
polarizations.gif
    :  +and× polarizations of the  GW+ and \times  polarizations   of  the   GW

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

           Λij,kl(n^)=PikPjl12PijPkl\displaystyle               \Lambda_{ij,kl}(\hat{n}) = P_{ik}P_{jl} - \frac12 P_{ij}P_{kl}

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

    Λij,kl=Λkl,ij\displaystyle \Lambda _{ij,kl} = \Lambda _{kl,ij}
    Λij,kl=Λji,lk\displaystyle \Lambda _{ij,kl} = \Lambda _{ji,lk}
     niΛij,kl=njΛij,kl=nkΛij,kl=nlΛij,kl=0\displaystyle  n^{i}\Lambda _{ij,kl} = n^{j}\Lambda _{ij,kl} =n^{k}\Lambda _{ij,kl} =n^{l}\Lambda _{ij,kl} = 0
     Λij,mnΛmn,kl=Λij,kl\displaystyle  \Lambda _{ij,mn}\Lambda _{mn,kl} = \Lambda _{ij,kl}

    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 x1=ξx^{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.

    d2xidτ2  =Γνρidxνdτdxρdτ\displaystyle \frac{d^{2}x^{i}}{d\tau ^{2}}   = -\Gamma ^{i}_{\nu\rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}
    d2xidτ2  =Γ00idx0dτdx0dτ\displaystyle \frac{d^{2}x^{i}}{d\tau ^{2}}   = -\Gamma ^{i}_{00}\frac{dx^{0}}{d\tau}\frac{dx^{0}}{d\tau}
    Γ00i=12(2h0i,0h00,i)\displaystyle \Gamma ^{i}_{00} = \frac12(2h_{0i,0} - h_{00,i})

    The last equation vanishes because in TT gauge h00=h0i=0h_{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

    Δl=0ξds21/2=[1+12hxxTT]ξ\displaystyle \Delta l = \int_{0}^{\xi} |ds^{2}|^{1/2} = [1 + \frac12 h^{TT}_{xx}]\xi

    Thus we can conclude that in TT gauge the coordinate position remains the same (initial separation ξ\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

     hˉμν=16πGc4Tμν\displaystyle  \Box\bar{h}_{\mu\nu} = - \frac{16\pi G}{c^{4}} T_{\mu\nu}

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

     hˉμν(xi,t)=41RTμν(y,tR)d3y  ; R2=(xiyi)(xiyi)\displaystyle  \Box \bar{h}_{\mu\nu}(x^{i},t) = 4\int \frac1R T_{\mu\nu}(y,t-R)d^{3}y   ;  R^{2} = (x^{i}-y^{i})(x_{i} - y_{i})

    where xix^{i} is the position far away from the source where we are calculating the field and RR is the size of the source which is at the position yiy^{i}. We define r2=xixir^{2} = x^{i}x_{i} and we have r2>>yiyir^{2}>> y^{i}y_{i}, therefore the source is far away from the field point so the contribution of RRin the integral is dominated by rr 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 tR=tr+niyi+O(1r)t-R = t-r +n^{i}y_{i} + O(\frac1r) with ni=xirn^{i} = \frac{x^{i}}{r} , nini=1n^{i}n_{i}=1​.

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

     hˉμν=4r[Tμν(t,yi)+T,0μν(t,yi)njyj+12T,00μν(t,yi)njnkyjyk+16T,000μν(t,yi)njnknlyjykyl+...]d3y\displaystyle  \bar{h}^{\mu\nu} = \frac4r \int [T^{\mu\nu}(t^{'},y^{i}) + T^{\mu\nu}_{,0}(t^{'},y^{i})n^{j}y_{j}+\frac12 T^{\mu\nu}_{,00}(t^{'},y^{i})n^{j}n^{k}y_{j}y_{k} + \frac16 T^{\mu\nu}_{,000}(t^{'},y^{i})n^{j}n^{k}n^{l}y_{j}y_{k}y_{l} + ...]d^{3}y

    The above integral contain moments of components of stress-energy. Let denote the moments of density T00T^{00} by MM, moments of momentum T0iT^{0i} by PP and moments of stress TijT^{ij} by SS

    M=T00(t,yi)d3y\displaystyle M = \int T^{00}(t^{'},y^{i})d^{3}y
    Mj=T00(t,yi)yjd3y\displaystyle M^{j} = \int T^{00}(t^{'},y^{i})y^{j}d^{3}y
    Mjk=T00(t,yi)yjykd3y\displaystyle M^{jk} = \int T^{00}(t^{'},y^{i})y^{j}y^{k}d^{3}y
    Mjkl=T00(t,yi)yjykyld3y\displaystyle M^{jkl} = \int T^{00}(t^{'},y^{i})y^{j}y^{k}y^{l}d^{3}y

    The various moments of momentum T0lT^{0l} are

    Pl=T0l(t,yi)d3y\displaystyle P^{l} = \int T^{0l}(t^{'},y^{i})d^{3}y
    Plj=T0l(t,yi)yjd3y\displaystyle P^{lj} = \int T^{0l}(t^{'},y^{i})y^{j}d^{3}y
    Pljk=T0l(t,yi)yjykd3y\displaystyle P^{ljk} = \int T^{0l}(t^{'},y^{i})y^{j}y^{k}d^{3}y

    The various moments of momentum TlmT^{lm} are

    Slm=Tlm(t,yi)d3y\displaystyle S^{lm} = \int T^{lm}(t^{'},y^{i})d^{3}y
    Slmj=Tlm(t,yi)yjd3y\displaystyle S^{lmj} = \int T^{lm}(t^{'},y^{i})y^{j}d^{3}y

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

    0T00=lT0l\displaystyle \partial _{0}T^{00} = -\partial _{l}T^{0l}
     M˙=d3x0T00=d3xiT0i=dSiT0i=0\displaystyle  \dot{M} = \int d^{3}x \partial _{0}T^{00} = -\int d^{3}x \partial_{i}T^{0i} = -\int dS^{i} T^{0i} = 0

    Similarly,

    M˙i=d3xxi0T00=d3xxilT0l=d3xδliT0l=Pl\displaystyle \dot{M}^{i} = \int d^{3}x x^{i} \partial_{0}T^{00} = - \int d^{3}x x^{i} \partial_{l}T^0l = \int d^{3}x\delta^{i}_{l}T^{0l} = P^{l}

    Using the above relations we get following useful identities

    M˙=0 ;M˙k=Pk;  M˙jk=Pjk+Pkj;  M˙jkl=Pjkl+Pklj+Pljk\displaystyle \dot{M} = 0 \:; \dot{M}^{k}= P^{k} ; \;\dot{M}^{jk} = P^{jk} + P^{kj} ;\; \dot{M}^{jkl} = P^{jkl}+P^{klj}+P^{ljk}
     P˙j=0;    P˙jk=Sjk;    P˙jkl=Sjkl+Sjlk\displaystyle  \dot{P}^{j} = 0 ; \;\; \dot{P}^{jk} = S^{jk} ; \;\; \dot{P}^{jkl} = S^{jkl} + S^{jlk}

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

    Sjk=M¨jk2\displaystyle S^{jk} = \frac{\ddot{M}^{jk}}{2}

    Using the above relations we can show that

    hˉ00(t,xi)=4rM+4rPjnj+4rSjk(t)njnk+4rS˙jkl(t)njnknl+.....\displaystyle \bar{h}^{00}(t,x^{i}) = \frac4rM + \frac4rP^{j}n_{j} + \frac4r{S}^{jk}(t^{'})n_{j}n_{k} +\frac4r\dot{S}^{jkl}(t^{'})n_{j}n_{k}n_{l} + .....
     hˉ0j(t,xi)=4rPj+4rSjk(t)nk+4rS˙jkl(t)nknl+....\displaystyle  \bar{h}^{0j}(t,x^{i}) = \frac4r P^{j} + \frac4r S^{jk}(t^{'})n_{k} + \frac4r \dot{S}^{jkl}(t^{'})n_{k}n_{l} +....
     hˉjk(t,xi)=4rSjk(t)+4rS˙jkl(t)nl+...\displaystyle  \bar{h}^{jk}(t,x^{i}) = \frac4r S^{jk}(t^{'}) + \frac4r\dot{S}^{jkl}(t^{'})n_{l}+...

    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

    hˉijTT=2rQ¨ijTT(tr)\displaystyle \bar{h}^{TT}_{ij} = \frac2r \ddot{Q}^{TT}_{ij}(t-r)

    where Qij=Mij13δijMkkQ^{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 by​​Sean.M.Carrol, et al​

    P=G5c5  d3Qijdt3 d3Qijdt3 \displaystyle P = -\frac{G}{5c^{5}}\langle   \frac{d^{3}Q_{ij}}{dt^{3}}  \frac{d^{3}Q^{ij}}{dt^{3}}   \rangle

    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

    Untitled-6.jpeg
      Two point masses in x-y plane and two in the space

      This system consists of four point masses, two are located in the x1x2x^{1}-x^{2} plane and other two are located in the space diagonally opposite to each other making a polar angle θ\theta w.r.t to x3x^{3}axis. The whole system rotates with angular velocity Ω\Omega with x3x^{3}as the axis of rotation. The corresponding energy density is

       T00(t,X)= M[δ(x1RCosΩt)δ(x2RSinΩt)δ(x3)+δ(x1+RCosΩt)δ(x2+RSinΩt)δ(x3)+ δ(x1RSinθCosΩt)δ(x2RSinθSinΩt)δ(x3RCosθ)+  δ(x1+RSinθCosΩt)δ(x2+RSinθSinΩt)δ(x3+RCosθ)]\displaystyle  T^{00}(t,X) =  M[\delta (x^{1}-RCos\Omega t)\delta(x{2}-RSin\Omega t)\delta(x^{3}) \\+ \delta (x^{1}+ RCos\Omega t)\delta(x{2}+ RSin\Omega t)\delta(x^{3})\\+  \delta (x^{1}-RSin\theta Cos\Omega t)\delta(x{2}-RSin\theta Sin\Omega t)\delta(x^{3}-RCos\theta) \\+   \delta (x^{1}+RSin\theta Cos\Omega t)\delta(x{2}+RSin\theta Sin\Omega t)\delta(x^{3}+RCos\theta)]

      ​​The various components of quadrupole moments are

      qxx=MR2Cos2Ωt+MR2Sin2θCos2Ωt+Constant\displaystyle q_{xx} = MR^{2}Cos2\Omega t + MR^{2}Sin^{2}\theta Cos^{2}\Omega t + Constant
       qyy=MR2(1Cos2Ωt)+MR2Sin2θ(1Cos2Ωt)\displaystyle  q_{yy} = MR^{2}(1-Cos2\Omega t) + MR^{2}Sin^{2}\theta (1-Cos2\Omega t)
      qzz=MR2Cos2θ\displaystyle q_{zz} = MR^{2}Cos^{2}\theta
      qxy=qyx=MR2Sin2Ωt+MR2Sin2θSin2Ωt\displaystyle q_{xy} = q_{yx} = MR^{2}Sin2\Omega t + MR^{2}Sin^{2}\theta Sin2\Omega t
      qxz=qzx=MR2Sin2θCosΩt\displaystyle q_{xz} = q_{zx} = MR^{2}Sin2\theta Cos\Omega t
      qzy=qyz=MR2Sin2θSinΩt\displaystyle q_{zy} = q_{yz} = MR^2Sin2\theta Sin\Omega t

      Reduced Quadrupole Moment:

      Qij=qij13δijqkk\displaystyle Q_{ij} = q_{ij} - \frac13 \delta_{ij}q^{k}_{k}
      qkk=ηikηik=qxx+qyy+qzz\displaystyle q^{k}_{k} = \eta^{ik}\eta_{ik} = q_{xx} + q_{yy} + q_{zz}

      The components of reduced quadrupole moment tensor are Qxx=qxx43MR2Q_{xx} = q_{xx} -\frac43MR^{2}; Qyy=qyy43MR2Q_{yy} = q_{yy} -\frac43MR^{2} ; Qzz=qzz43MR2Q_{zz} = q_{zz} -\frac43MR^{2} other components are same as the quadrupole moments.

      We shall compute the wave emerging in the x3x^{3}direction n^(0,0,1)\hat{n} \longrightarrow (0,0,1) for which the projector PijP_{ij} becomes Pij=[100010000]P_{ij} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix} and  Λij,kl(n^)=PikPjl12PijPkl \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

      QijTT=[12(QxxQyy)Qxy0Qxy12(QyyQxx)0000]\displaystyle Q^{TT}_{ij}= \begin{bmatrix} \frac12(Q_{xx}-Q_{yy})&Q_{xy}&0\\Q_{xy}&\frac12(Q_{yy}-Q_{xx})&0\\0 &0&0 \end{bmatrix}

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

      hijTT=8GMR2Ω2c4z[(1+Sin2θ)Cos2Ωt(1+Sin2θ)Sin2Ωt0(1+Sin2θ)Sin2Ωt(1+Sin2θ)Cos2Ωt0000]\displaystyle h^{TT}_{ij} = -\frac{8GMR^{2}\Omega^{2}}{c^{4}z} \begin{bmatrix} (1+ Sin^{2}\theta)Cos2\Omega t & (1+ Sin^{2}\theta)Sin2\Omega t & 0 \\(1+ Sin^{2}\theta)Sin2\Omega t & (1+ Sin^{2}\theta)Cos2\Omega t & 0 \\0&0&0 \end{bmatrix}

      In this direction ( x3x^{3}) the amplitude will be maximum when the two mass in the space makes an angle θ=π4\theta = \frac{\pi}{4} with the polar axis x3x^{3}

      hijTT=8GMR2Ω2c4z[Cos2ΩtSin2Ωt0Sin2ΩtCos2Ωt0000]\displaystyle h^{TT}_{ij} = -\frac{8GMR^{2}\Omega^{2}}{c^{4}z} \begin{bmatrix} Cos2\Omega t &Sin2\Omega t & 0 \\Sin2\Omega t & Cos2\Omega t & 0 \\0&0&0 \end{bmatrix}

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

      The wave emerging in the x1x^{1} direction n^(1,0,0)\hat{n} \longrightarrow (1,0,0) has a projector Pij=[000010001]P_{ij} = \begin{bmatrix}0&0&0\\0&1&0\\0&0&1\end{bmatrix} and the emitted radiation is

      hijTT=4GMR2Ω2c4x[0000(1+Sin2θ)Cos2Ωt0.5Sin2θSinΩt00.5Sin2θSinΩt(1+Sin2θ)Cos2Ωt]\displaystyle h^{TT}_{ij} = -\frac{4GMR^{2}\Omega^{2}}{c^{4}x} \begin{bmatrix}0& 0& 0 \\0&- (1+ Sin^{2}\theta)Cos2\Omega t & 0.5Sin2\theta Sin\Omega t \\0&0.5Sin2\theta Sin\Omega t&(1+ Sin^{2}\theta)Cos2\Omega t \end{bmatrix}

      The wave emerging in the x2x^{2}  direction n^(0,1,0)\hat{n} \longrightarrow (0,1,0) has a projector Pij=[100000001]P_{ij} = \begin{bmatrix}1&0&0\\0&0&0\\0&0&1\end{bmatrix} and the emitted radiation is

      hijTT=4GMR2Ω2c4y[(1+Sin2θ)Cos2Ωt00.5Sin2θSinΩt0000.5Sin2θSinΩt0(1+Sin2θ)Cos2Ωt]\displaystyle h^{TT}_{ij} = -\frac{4GMR^{2}\Omega^{2}}{c^{4}y} \begin{bmatrix} (1+ Sin^{2}\theta)Cos2\Omega t& 0& 0.5Sin2\theta Sin\Omega t\\0&0 & 0\\0.5Sin2\theta Sin\Omega t&0&-(1+ Sin^{2}\theta)Cos2\Omega t \end{bmatrix}

      GRAVITATIONAL RADIATION EMITTED BY A BINARY

      Let us consider two stars of mass m1m_{1} and m2m_{2} (assume those to be a point mass). Let l0l_{0} be the orbital separation between them. The total mass given by M=m1+m2M = m_{1} + m_{2} and the reduced mass is given by μ=m1m2M\mu = \frac{m_{1}m_{2}}{M}. From Keplers law the orbital frequency is found to be Ω=GMl03\Omega = \sqrt{ \frac{GM}{l_{0}^{3}} }

      Let r1r_{1}and r2r_{2} be the distances from the centre of mass to the masses m1m_{1} and m2m_{2} respectively.

      ​​ r1=m2l0Mr_{1} = \frac{m_{2}l_{0}}{M} and r2=m1l0Mr_{2} = \frac{m_{1}l_{0}}{M}.

      The stress energy tensor of the system is given by

      T00=m1δ(xx1)δ(yy1)δ(z)+m2δ(xx2)δ(yy2)δ(z)\displaystyle T^{00} = m_{1}\delta (x-x_{1})\delta (y-y_{1})\delta(z) + m_{2}\delta (x-x_{2})\delta (y-y_{2})\delta(z)

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

      hijTT=4GμR2Ω2c4z[Cos2ΩtSin2Ωt0Sin2ΩtCos2Ωt0000]\displaystyle h^{TT}_{ij} = -\frac{4G\mu R^{2}\Omega^{2}}{c^{4}z} \begin{bmatrix} Cos2\Omega t &Sin2\Omega t & 0 \\Sin2\Omega t &- Cos2\Omega t & 0 \\0&0&0 \end{bmatrix}

      ​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

      LGW=dEGWdt=32G4μ2M35c5l05L_{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.

      dEdt+LGW=0\displaystyle \frac{dE}{dt} + L_{GW} = 0

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

      where E is the orbital energy, which is given by OrbitalEnergy=Kinetic+PotentialOrbital Energy = Kinetic + Potential

      Ek=12m1Ω2r12+12m2Ω2r22=GμM2l0\displaystyle E_{k} = \frac12 m_{1}\Omega^{2}r_{1}^{2} + \frac12 m_{2}\Omega^{2}r_{2}^{2} = \frac{G\mu M}{2l_{0 }}
      U=Gm1m2l0=GμMl0\displaystyle U = \frac{-Gm_{1}m_{2}}{l_{0}} = - \frac{G\mu M}{l_{0}}
      E=GμM2l0\displaystyle E = -\frac{G\mu M }{2l_{0}}

      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

      dEdt=12GμM2l02dl0dt\displaystyle \frac{dE}{dt} = \frac12 \frac{G\mu M}{2l_{0}^{2}} \frac{dl_{0}}{dt}

      We know that the orbital frequency is Ω2=GMl03\Omega ^{2} = \frac{GM}{l_{0}^{3}}, taking logarithm of the on both sides gives 1ΩdΩdt=32l0dl0dt\frac{1}{\Omega} \frac{d\Omega}{dt} = -\frac{3}{2l_{0}} \frac{dl_{0}}{dt}

      dEdt=23EΩdΩdt\displaystyle \frac{dE}{dt} = \frac23 \frac{E}{\Omega}\frac{d\Omega}{dt}

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

      1ΩdΩdt=1PdPdt\displaystyle \frac{1}{\Omega}\frac{d\Omega}{dt} = -\frac{1}{P} \frac{dP}{dt}

      Substituting equation (47) in (48) gives

      dPdt=32PELGW\displaystyle \frac{dP}{dt} = -\frac32 \frac{P}{E} L_{GW}

      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.

      phases.png
        Coalescing binaries

        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.

        sens.png
          Detectors and sources

          Response of a Detector

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

          Dij=n1in1jn2in2jD_{ij} = n_{1i}n_{1j}-n_{2i}n_{2j}

          where n1n_{1} and n2n_{2} are the unit vectors along the direction of the interferometer arms. For a resonant bar the detector tensor has the form

          Dij=ninj13δijD_{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+(θ,ϕ)+h×F×(θ,ϕ)h(t) = h_{+}(t)F_{+}(\theta,\phi) + h_{\times}F_{\times}(\theta,\phi)

          F+(θ,ϕ)F_{+}(\theta,\phi) and F×(θ,ϕ)F_{\times}(\theta,\phi) are antenna pattern functions which depends on the direction n^=(θ,ϕ)\hat{n}=(\theta,\phi). The patterns functions are smooth functions of position of source in the sky ​Y. Giirsel, et al, 1989​ . The antenna pattern functions are defined as F(n^)=Dijeij++Dijeij×F(\hat{n}) = D^{ij}e_{ij}^{+} + D^{ij}e_{ij}^{\times}, where

          eij+(n^)=ui^uj^vi^vj^e_{ij}^{+}(\hat{n}) = \hat{u_{i}}\hat{u_{j}}-\hat{v_{i}}\hat{v_{j}}

          eij×(n^)=ui^vj^+vi^uj^e_{ij}^{\times}(\hat{n}) = \hat{u_{i}}\hat{v_{j}}+\hat{v_{i}}\hat{u_{j}}

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

          Δl=0ξds21/2=[1+12hxxTT]ξ\displaystyle \Delta l = \int_{0}^{\xi} |ds^{2}|^{1/2} = [1 + \frac12 h^{TT}_{xx}]\xi

          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.

          laserint.png
            Laser Interferometer

            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 101610^{-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 TT be a subset of real numbers TRT \subset R. A stochastic or a random process x(t)x(t) is a family of random variables x(t)x(t) labelled by the number tTt \in T. They have a joint n-dimensional cumulative distribution function F defined by

            Ft1,......tn(x1,x2,.....xn)=P(x(t1)x1,x2,.....xtnxn)F_{t_{1},......t_{n} }(x_{1},x_{2},.....x_{n}) = P(x(t_{1}) \leq x_{1},x_{2},.....x_{t_{n}} \leq x_{n})

            When T is a discreet set of points, T ={ t1,t2,...... }T  = \{  t_{1},t_{2},......  \}, the stochastic process is called a random sequence. The stochastic process x(t)x(t) is Gaussian if the cumulative distribution function Ft1,......tn(x1,x2,.....xn)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 Ft1,......tn(x1,x2,.....xn)F_{t_{1},......t_{n} }(x_{1},x_{2},.....x_{n}) defining the stochastic process remain the same if the set of points T ={ t1,t2,...... }T  = \{  t_{1},t_{2},......  \} is shifted by an arbitrary constant τ\tau, i.e.) the probabilistic structure of the stochastic process is invariant under a shift of the parameter tt

            Ft1+τ,t2+τ......tn+τ(x1,x2,.....xn)=Ft1,......tn(x1,x2,.....xn)F_{t_{1}+ \tau ,t_{2}+\tau......t_{n} +\tau}(x_{1},x_{2},.....x_{n}) = F_{t_{1},......t_{n} }(x_{1},x_{2},.....x_{n})

            Properties of stochastic process

            The first order moment μ1\mu _{1} is called the mean value of the stochastic process x(t)x(t)​ which is denoted by m(t)m(t)

            m(t)=E{x(t)}m(t) = E\{ x(t) \}

            The second-order moment, μ1,1\mu_{1,1}is called the auto-correlation function, it is given as

            K(t,s)=E{ x(t)x(s)}K(t,s) = E \{  x(t)x(s) \}

            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)x(t) = n(t) + h(t), where n(t)n(t) and h(t)h(t) are the gaussian stochastic noise and the signal we are looking respectively. Let p0(x)p_{0}(x) be the probability distribution when the signal is absent in the data and p1(x)p_{1}(x) be the distribution in the presence of the signal. Let's define the absence of signal as the null hypothesis H0H_{0} and presence of signal as the alternate hypothesis H1H_{1}. Thereby we can partition the observation set into RR and RR'. If the observed data is in RR 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 Λ(x)=p1(x)p0(x)\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 Λ(x)\Lambda(x) is greater than a threshold value.

            Bayesian approach

            In this method we assign costs to our decisions, let C=[C00C01C10C11]