LISA is a space-borne, laser-interferometric gravitational-wave detector currently under study by the European Space Agency. We give a brief introduction about the main features of the detector, concentrating on its one-year orbital motion around the Sun. We compute how the amplitude as well as the phase of a gravitational wave are modulated due to this motion by transforming an arbitrary gravitational-wave signal in a reference frame that is rigidly fixed to the arms of the detector. To see how LISA works the detector response to a gravitational wave which is purely monochromatic in the barycentric frame will be discussed. A brief review of the theory of parameter estimation, based on the work of Finn and Cutler, will be given. Following this theory the detection of a gravitational-wave signal buried in detector noise was simulated numerically. We interpret the results of this simulation to determine the angular resolution of LISA

Danzmann, K.

Jennrich, O.

Peterseim, M.

English

MPG.PuRe

Accuracy of parameter estimation of gravitational waves with LISA

LISA is a space-borne, laser-interferometric gravitational-wave detector currently under study by the European Space Agency. We give a brief introduction about the main features of the detector, concentrating on its one-year orbital motion around the Sun. We compute how the amplitude as well as the phase of a gravitational wave are modulated due to this motion by transforming an arbitrary gravitational-wave signal in a reference frame that is rigidly fixed to the arms of the detector. To see how LISA works the detector response to a gravitational wave which is purely monochromatic in the barycentric frame will be discussed.

A brief review of the theory of parameter estimation, based on the work of Finn and Cutler, will be given. Following this theory the detection of a gravitational-wave signal buried in detector noise was simulated numerically. We interpret the results of this simulation to determine the angular resolution of LISA

Peterseim, Michael

Jennrich, Oliver

Danzmann, Karsten

Max Planck Society eDoc Server

Class. Quantum Grav.13 (1996) A279–A284. Printed in the UKAccuracy of parameter estimation of gravitational waveswith LISAMichael Peterseim, Oliver Jennrich and Karsten DanzmannUniversiẗat Hannover, Appelstrasse 2, D-30167 Hannover, GermanyAbstract. LISA is a space-borne, laser-interferometric gravitational-wave detector currentlyunder study by the European Space Agency. We give a brief introduction about the main featuresof the detector, concentrating on its one-year orbital motion around the Sun. We compute howthe amplitude as well as the phase of a gravitational wave are modulated due to this motion bytransforming an arbitrary gravitational-wave signal in a reference frame that is rigidly fixed tothe arms of the detector. To see how LISA works the detector response to a gravitational wavewhich is purely monochromatic in the barycentric frame will be discussed.A brief review of the theory of parameter estimation, based on the work of Finn and Cutler,will be given. Following this theory the detection of a gravitational-wave signal buried in detectornoise was simulated numerically. We interpret the results of this simulation to determine theangular resolution of LISA.PACS number: 0480N1. IntroductionThe LISA mission consists of six spacecraft forming a laser interferometer in a plane inclined60◦ with respect to the ecliptic and along a circle with a radius of 3× 109 m [1]. Twospacecraft will be placed at each of three points on this circle forming an equilateral trianglewith baselines of 5× 109 m which rotates clockwise around its centre, as viewed from theSun, with a period of one year. The complete constellation describes an approximatelycircular orbit at a distance ofR = 1 AU from the Sun and trailing the Earth in its orbit by20◦. As a single, nonmoving detector reveals no information about the directional parametersof the source of the gravitational wave, all the information about the source parameters iscontained in the variation of the detector response that results from the described motion [5].Firstly, the detected amplitude of the gravitational wave is modulated, as the detector’ssensitivity pattern is not isotropic, so the detected amplitude depends on the location of thesource relative to the detector and this location changes due to the rotation of the detectorformation around its centre.Secondly, the detector is moving relative to the source due to the periodic motion ofits centre along its orbit around the Sun, Doppler-shifting the frequency of the gravitationalwave. This results in a phase modulation of the detector output.The amplitude modulation as well as the phase modulation will spread a sharply peakedmonochromatic signal into a set of sidebands separated from the carrier by integer multiplesof the fundamental frequency,(1 year)−1. In the following sections both modulations willbe calculated and a review of some standard techniques of parameter estimation will begiven. Finally, these techniques are applied to the simulated data for a linearly polarized,monochromatic wave at a frequency of 3 mHz and interpreted to give information aboutthe potential accuracy of the estimation of the angular parameters of the source.0264-9381/96/SA0279+06$19.50c© 1996 IOP Publishing Ltd A279A280 M Peterseim et al2. The amplitude modulationThe amplitude modulation can be calculated by transforming the metric tensorsh× := h× 0 1 01 0 00 0 0 and h+ := h+ 1 0 00 −1 00 0 0 (1)which are defined in the source frame, i.e. a system with itsx-axis in thex–y-plane ofthe barycentric frame, itsz-axis pointing towards the Sun and the source at its origin. Thetransformation is split into one transforming the source system into the barycentric systemand another one from the barycentric frame into the detector frame, which is rigidly fixedto the interferometer arms.Let θ andφ be the Euler angles that define the source position in the barycentric frame,with its x–y-plane in the ecliptic (cf figure 1). The transformation into the source systemis composed of two rotations. The first, realized by the rotation matrix1, turns they-axisof the barycentric frame on the projection of the line connecting the Sun and the source onthe ecliptic, that is counterclockwise through an angleφ − 90◦ around thez-axis,a1 := sinφ − cosφ 0cosφ sinφ 00 0 1 . (2)Figure 1. Orientation of the source in the barycentric frame.A second rotationb1 turns the system counterclockwise around the newx-axis by180◦ − θ . With T1 := b1a1 the matrixh+ of (1) is transformed from the source systeminto the barycentric frame byh+ → T t1h+T1. (3)The following angles are used to calculate the transformation into the detector system:ψa := 2πtTψb := 13π ψc := −2πtT+ α (4)where 13π is the angle of LISA with respect to the ecliptic [1] andα is the phase betweenLISA’s motion around the Sun and the motion around its centre of mass.Now a rotation matrixa2 turns the frame of reference in the barycentric systemcounterclockwise around thez-axis byψa, so that the newy-axis points towards LISA.Parameter estimation of gravitational waves with LISA A281Then b2 turns it byψb out of the ecliptic. Finally,c2 turns it clockwise around the newz-axis byψc. A vector is transformed from the barycentric into the detector system byT2 = c2b2a2. Therefore the matrixh+ is transformed ash+ → T2T t1︸ ︷︷ ︸=:Th+T1T t2. (5)3. The phase modulationThe translatory motion of the detector relative to the source leads to a phase modulation ofthe gravitational-wave signal. This modulation can be easily calculated with the so-calledbarycentric transform between time of arrival at the Solar System and time at the detector[2]. In the former system, which can be considered to be a convenient inertial frame, thesignal is not modulated at timescales short or comparable to an Earth orbit and thereforeis of fixed frequency. Letsd and sb be the signals at the detector and at the barycentre,respectively, then by definitionsd(td) = sb(tb[td, θ, φ]), (6)where(θ, φ) is the angular position of the source (cf figure 1). The relation between thetwo time variablestb, td is given bytb[td, θ, φ] = td + En(θ, φ)Ed(td)c(7)with En being a unit vector pointing towards the source andEd a vector connecting LISA andthe solar system, so it has a length ofR = 1 AU:En = cosφ sinθsinφ sinθcosθ Ed = R sin(2πt/T )cos(2πt/T )0 . (8)Therefore the relation between the two signalssd andsb as functions of time issd(td) = sb(td + R sinθccos(2πtT− φ)). (9)So if the signal in the inertial frame is purely sinusoidal of frequencyfGW, in the detectorresponse it appears assd(td) = sin(2πfGWtb)= sin(2πfGWtd + 2πfGWR sinθccos(2πtT− φ)︸ ︷︷ ︸8(t)), (10)including a phase modulation8(t) with a modulation indexm ofm = 2πfGWR sinθc≈ π sinθ(fGW1 mHz). (11)A282 M Peterseim et al4. The LISA response to a gravitational waveA gravitational wave which is purely sinusoidal in the barycentric frame causes a strainperceived by the detector given byH = T0 0 0 00 hpl hkr 00 −hkr hpl 00 0 0 0 T t exp{i[2πfGWt − EkGWEx +8(t)]} (12)with the phase modulation8(t) of equation (10) and the modulated amplitudeT (h+ + h×)T t (cf equation (5)). To see how the gravitational-wave detector works, recallthat general relativity predicts that a ray of light connects a set of points by an interval ofzero ords2 = 0. (13)For simplicity, one of the detector arms is chosen to give thex-axis of the detector frame,while the other arm points to the ‘first quadrant’ at an angle ofα = 60◦ to thex-axis. Theabove equation then becomes0 = ds2= gµν dxµ dxν= −c2 dt2 + (1 +H11(t, Ex)) dx2 + (1 +H22(t, Ex)) dy2 +H12(t, Ex) dx dy+H21(t, Ex) dy dx= −c2 dt2 + [(1 +H11(s)) cos2 α + (1 +H22(s)) sin2 α+ (H12(s)+H21(s)) sinα cosα] ds2 (14)where the parameters is the parametrization of the trajectory andα equals 60◦ for one armof the interferometer and 0◦ for the other. This means that the gravitational wave modulatesthe light travel timeτtt between two neighbouring points of fixed coordinate separationwhich can be calculated by integrating the square root of the above equationcτtt∫0dt =L∫0√1 +H11(s) cos2 α +H22(s) sin2 α + 12(H12(s)+H21(s)) sin 2α ds−0∫L√1 + · · · ds,where L is the interferometer arm length. Assuming that the metric perturbation isapproximately constant during the time any given wavefront is present in the apparatus(i.e. fGWτtt  1), and expanding the root, leads to a difference in light travel time for theinterferometer arms of1τtt = 2L8c√3[(H22 −H11)√3 +H12 +H21]and, takingλ to be the laser wavelength, a corresponding measurable phase difference of18 = 2Lπ4λ√3[(H22 −H11)√3 +H12 +H21] (15)whereH still has the same time dependence as in (12). Due to the motion of the detectora signal that is monochromatic in the source system will now be spread into a set ofParameter estimation of gravitational waves with LISA A283sidebands in the detector system that contains all the information about the parameters ofthe gravitational wave. The resulting line spectra for two source locations are presentedin figures 2 and 3. The frequency of the gravitational wave in the example given in thefigures is 3 mHz, and the source locations are(θ, φ) = (π/2, 0) or (θ, φ) = (π/4, 0)respectively.-20 -10 0 10 20∆f [year-1]0.000.100.200.300.400.50Figure 2. Sideband structure of the detector re-sponse to a monochromatic, pluspolarized gravita-tional wave at 3 mHz, corresponding to a modula-tion index ofm ≈ 10 (cf equation (11)). The sourceis located at(θ, φ) = (π/2, 0).-20 -10 0 10 20∆f [year-1]0.000.100.200.300.400.50Figure 3. Sideband structure of the detector re-sponse to a monochromatic, pluspolarized gravita-tional wave at 3 mHz, corresponding to a modula-tion index ofm ≈ 10 (cf equation (11)). The sourceis located at(θ, φ) = (π/4, 0).5. Review of parameter estimationThe problem of measurement is to determine the values of some or all parameters of thesignal [3]. It will be shown how accurately that can be done. Consider a streams(t) thatrepresents the pure detector outputh( Eµ), parametrized by several unknown parametersµicollectively denoted asEµ = (µ1 = θ, µ2 = φ, . . .) plus additional noisen(t). Now one hasto find a probability density functionP( Eµ, s) for the parametrizationEµ that characterizesthe detector outputh( Eµ). Assuming thatn(t) is a Gaussian process with zero mean,A284 M Peterseim et alcharacterized by the one-sided power spectral densitySn(f ), it can be shown [4] thatP( Eµ, s) ∼ exp〈s, h( Eµ)〉where the symmetric inner product is defined as〈s, h〉 = 2∞∫0s(f )h∗(f )+ h(f )s∗(f )Sn(f )df.From this definition it follows that, for a waveformh( Eµ), the signal-to-noise ratio is givenapproximately bySN[h( Eµ)] = 〈h( Eµ), h( Eµ)〉RMS(〈h( Eµ), n〉) =√〈h( Eµ), h( Eµ)〉.The error in measurement is taken to be the width of the probability density functionP( Eµ, s)for the measured valuêEµ, i.e. the variance–covariance matrix6ij =∫(µi − µ̂i)(µj − µ̂j )P ( Eµ, s)dnµ.There are several choices of data-processing algorithms to findÊµ; one is the so-called Bayesestimator ÊµBayes=∫ EµiP ( Eµ, s)dnµ.6. ResultsIn table 1 RMS errors are presented as square roots of6θθ and6φφ that have been calculatedfor a monochromatic, linearly plus-polarized wave at 3 mHz with constant amplitude overone year of observation. Only values ofθ < π/2 andφ = 0 appear in that table, becausethe errors are symmetrical inπ/2 ± θ and in φ. For a signal-to-noise ratio of 100 theposition accuracy comes out as 1–10 mrad, depending on the value ofθ .Table 1. The RMS errors for the angular parametersθ and φ (cf figure 1 for φ = 0 and asignal-to-noise ratio of 115, that put a constraint on the angular position accuracy of 10−6 sr).Location (rad) δθ (mrad) δφ (mrad) d = sinθ δθ δφ (mrad2)θ = 0.10π 1.99 5.58 3.10θ = 0.15π 2.33 3.65 3.87θ = 0.20π 2.79 2.84 4.64θ = 0.25π 3.27 2.43 5.45θ = 0.30π 3.92 2.08 6.47θ = 0.35π 5.52 1.88 9.06θ = 0.40π 6.81 1.69 10.82θ = 0.45π 9.27 1.59 14.38θ = 0.50π 10.94 1.57 16.55References[1] Danzmann Ket al 1993 Proposal for a laser-interferometric gravitational wave detector in spaceReport MPQ177 Max-Planck-Institut f̈ur Quantenoptik[2] Schutz B F 1991The Detection of Gravitational Wavesed D Blair pp 406–52[3] Cutler C and Flanagan E 1994Phys. Rev.D 49 2658[4] Finn L S 1992Phys. Rev.D 46 5236[5] Saulson P R 1994Fundamentals of Interferometric Wave Detectors(Singapore: World Scientific)

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Accuracy of parameter estimation of gravitational waves with LISA

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