LISA data analysis: Doppler demodulation

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1 INSTITUTE OF HYSICSUBLISHING Class. Quantum Grav. 20 (2003) S163 S170 CLASSICAL ANDQUANTUM GRAVITY II: S (03) LISA data analysis: Doppler demodulation Neil J Cornish 1 and Shane L Larson 2 1 Department o hysics, Montana State University, Bozeman, MT 59717, USA 2 Space Radiation Laboratory, Caliornia Institute o Technology, asadena, CA 91125, USA Received 18 September 2002 ublished 28 April 2003 Online at stacks.iop.org/cqg/20/s163 Abstract The orbital motion o the laser intererometer space antenna (LISA) produces amplitude, phase and requency modulations o a gravitational wave signal. The modulations have the eect o spreading a monochromatic gravitational wave signal across a range o requencies. The modulations encode useul inormation about the source location and orientation, but they also have the deleterious eect o spreading a signal across a wide bandwidth, thereby reducing the strength o the signal relative to the instrument noise. We describe a simple method or removing the dominant, Doppler component o the signal modulation. The demodulation reassembles the power rom amonochromatic source into a narrow spike and provides a quick way to determine the sky locations and requencies o the brightest gravitational wave sources. ACS numbers: Nn, Ym (Some igures in this article are in colour only in the electronic version) 1. Introduction This paper introduces a quick and dirty method or making a irst pass at the laser intererometer space antenna (LISA) data analysis problem. LISA will be sensitive to gravitational wave sources in the requency range Hz, which nicely complements the requency range covered by the ground-based detectors ( Hz). The data analysis challenges posed by LISA are very dierent rom those encountered with the ground-based detectors. Unlike the situation aced by the ground-based observatories, most o the sources that LISA hopes to detect have well-understood gravitational waveorms indeed many o them will be monochromatic and unchanging over the lie o the experiment. The diiculty comes when we include LISA s orbital motion and the modulations this introduces into the signal. Since LISA is designed to orbit at 1 AU, the modulations introduce sidebands at multiples o the orbital requency o m = 1/year. The approach we investigate here works by correcting or the dominant, Doppler modulation. This has the eect o re-assembling most o the Fourier power o a monochromatic /03/ $ IO ublishing Ltd rinted in the UK S163

2 S164 NJCornishand S L Larson source into a single spike. Since the correctiondepends on where the source is located on the sky, we are able to locate a source and its requency. The Doppler demodulation approach is amiliar to radio astronomers and has been discussed in relation to gravitational wave astronomy [1]. Doppler demodulation orms an integral part o ground-based gravitational wave searches or continuous gravitational wave sources such as pulsars [2, 3]. However, the unique orbital motion o the LISA observatory and the dierent requency range that it covers demand a separate treatment o the Doppler demodulation approach to LISA data analysis. 2. Signal modulation LISA will output a time series describing the strain in the detector as a unction o time, h(t). The strain will be a combination o the signal s(t) and noise n(t) in the detector: h(t) = s(t) + n(t). Because o LISA s orbital motion, a source that is monochromatic at the Sun s barycentre will be spread over a range o requencies. A monochromatic gravitational wave with requency and amplitudes h + and h in the plus and cross polarizations will produce the response [4] s(t) = A(t) cos (t) (1) where (t) = [2π t + ϕ 0 + φ D (t) + φ (t)]. (2) The amplitude modulation A(t), requency modulation φ D (t) and phase modulation φ (t) are given by A(t) = [(h + F + (t)) 2 + (h F (t)) 2 ] 1/2 (3) φ D (t) = 2π R c sin θ s cos(2π m t φ s ) (4) ( h F ) (t) φ (t) = arctan. (5) h + F + (t) Here F + (t) and F (t) are the detector antenna patterns in barycentric coordinates [4]. Each o the modulations is periodic in the orbital requency m = 1/year. The angles θ s and φ s give the sky location o the source, ϕ 0 is the phase o the wave and R = 1AUistheEarth Sun distance. The requency modulation, which is caused by the time-dependent Doppler shiting o the gravitational wave in the rest rame o the detector, is the dominant eect or requencies above Hz [5]. Ignoring the amplitude and phase modulations or now, pure Doppler modulation takes the orm s(t) = A cos[2π t + β cos(2π m t φ s ) + ϕ 0 ] ( ) =R A J n (β) e 2πi( +mn)t e iϕ 0 e in(π/2 φ s ). (6) n= Here J n is a Bessel unction o the irst kind o order n and β is themodulation index β = 2π R c sin θ s. (7) The bandwidth o the signal deined to be the requency interval that contains 98% o the total power is given by B = 2(1+β) m. (8) Sources in the equatorial plane have bandwidths ranging rom B = Hz at = 10 3 Hz to B = Hz at = 10 2 Hz.

3 LISA data analysis: Doppler demodulation S Demodulation I LISA were at rest with respect to the sky, each monochromatic source would produce a spike in the power spectrum o s(t). While this would make data analysis very easy, it would also severely limit the science that could be done as it would be impossible to determine the source location or orientation. Since LISA will move with respect to the sky, each source will have its own unique modulation pattern, and this pattern can be used to ix its location and orientation. On the other hand, the modulation makes the data analysis more complicated as the Fourier power o each source is spread over a wide bandwidth B. Oneapproach to the data analysis problem is to demodulate the signal, thereby re-assembling all the power o a given source at one requency. Since each source has a unique modulation, each source will also require a unique demodulation. Demodulating a particular source is not diicult i one happens to know its requency, location, orientation and orbital phase. One could imagine searching through this parameter space and looking or spikes in the power spectrum corresponding to sources that have been properly demodulated. A more practical approach is to ocus on the Doppler modulation as it causes the largest spreading o the Fourier power. Consider the Doppler modulated phase [ (t) = 2π t + R ] c sin θ s cos(2π m t φ s ) + ϕ 0. (9) We seek a new time coordinate t in which this phase is stationary d = 2π = d dt dt dt dt. (10) Thus, t ( t = 1 v ) c sin θ s sin(2π m t φ s ) dt. (11) Working to irst order in v/c we have t = t R c sin θ s cos(2π m t φ s ). (12) Taking the data stream rom the detector over a one year period and perorming a ast Fourier transorm allows us to write s(t) = a n e 2πimnt. (13) n erorming the coordinate transormation (12) wearrive at the new Fourier expansion s(t ) = c k e 2πi mkt, (14) k where the Fourier coeicients c k are given by c k = ( ) R a n J n k 2πn m c sin θ s e i(n k)( φ s π/2). (15) n Since the modulation has a limited bandwidth, an excellent approximation to c k is given by where c k k+l n=k l a n J n k (α) e i(n k)( φ s π/2), (16) α = 2πk m R c sin θ s (17)

4 S166 NJCornishand S L Larson Figure 1. ower spectra, (), beore (dashed line) and ater (solid line) Doppler demodulation. and l = 1+[α]. (18) The square brackets imply that we have taken the integer part o α. Consider the pure Doppler modulation described in (6) orthesimple case where = q m,andqis an integer (the non-integer case is more involved, but the idea is the same). The source has Fourier components a n = A e iϕ 0 J n q (β) e i(n q)(π/2 φ s ), (19) so that c k A e iϕ 0 e i(q k)( φ s π/2) J n q (β)j n k (α). (20) n Using the Neumann addition theorem, J n q (β)j n k (α) = J k q (α β), (21) n and the act that α β,weind c k A e iϕ 0 δ kq. (22) In other words, the Doppler demodulation procedure re-assembles all the power at the barycentre requency = q m. The demodulation is less eective when applied to a real LISA source because it does not correct orthe amplitude and phase modulations. Our Fourier space approach is very eicient i one is interested in a limited requency range. It does not oer any great saving over a direct implementation in the time domain i one wants to consider all requencies at once. Hellings [6] has recently implemented the Doppler demodulation procedure in the time domain and ound similar results to ours. To illustrate how the Doppler demodulation works, we consider monochromatic, circular Newtonian binaries as described in [4]. Figure 1 shows the power spectrum o s(t) beore and ater Doppler demodulation or a source with = Hz, θ s = and φ s = We see that most o the power is collected into a spike o width 3 m about the barycentre requency.

5 LISA data analysis: Doppler demodulation S Figure 2. ower spectra beore (dashed line) and ater (solid line) Doppler demodulation. The upper graph shows the demodulation when the detector is made stationary with respect to the higher-requency source, while the lower graph shows the same or the lower-requency source Figure 3. ower spectra beore (dashed line) and ater (solid line) Doppler demodulation. The upper graph shows the demodulation when the detector is made stationary with respect to source A, while the lower graph shows the same or source B. Table 1. Description o sources A and B. (Hz) θ s φ s A B Next we consider an example wherethereare two sources that have nearly the same requency but, nevertheless, do not have overlapping bandwidths. The sources have = Hz, ( θ s, φ s ) = ( , ) and = Hz, ( θ s, φ s ) = ( , ). The higher-requencysource has an amplitude times larger than the lower-requency source. The eect o demodulating each source is shown in igure 2. The perormance o the demodulation procedure is considerably less impressive when the two sources have overlapping bandwidths. Consider sources A and B described in table 1. The amplitude o source B is 1.59 times larger than the amplitude o source A. Figure 3 shows

6 S168 NJCornishand S L Larson Figure 4. Determining the sky location o a source with ( θ s, φ s ) = ( , ). The sky map uses the Mollweide projection in ecliptic coordinates with (π/2, 0) at the centre o the map. what happens when we demodulate sources A and B in turn. Because the two sources overlap, their signals interere with each other and the demodulation is able to detect only the stronger o the two sources Angular resolution The Doppler demodulation technique can be used to ind the sky location o a source by successively demodulating each point on the sky. In practice, we use the HEALIX [7] hierarchical, equal area pixelization scheme toprovide a inite number o sky locations to demodulate. We search or the maximum power contained in three adjacent requency bins and record this value or each sky pixel. Rather than ind the maxima across all requencies, we produce separate sky maps or requency intervals o width 4π R c m.there is little point in trying to use smaller requency intervals due to the intererence eect discussed earlier. We begin by considering an isolated source (i.e., no other sources with overlapping bandwidths) with = Hz and ( θ s, φ s ) = ( , ). Usingsky pixels o angular size 0.92,wearrive at the graph shown in igure 4. The demodulation code produced the best it values o = Hz and ( θ s, φ s ) = (1.253, 5.04) and ( θ s, φ s ) = (1.889, 5.04). The degenerate it or the sky location illustrates one o the drawbacks o the Doppler demodulation method it is unable to distinguish between sources above or below the equator. The errors in the source location ( θ s = 2.1, φ s = 6.3 ) are larger than the pixel size and can be attributed to the amplitude and phase modulations which we have not corrected or. It should be noted that the error in the source location is not caused by instrument noise since we are working in the large signal-to-noise limit (n(t) = 0). Indeed, instrument noise has little eect on the demodulation procedure since the noise is incoherent. The noise gets moved about in requency space, but there is little power accumulation at any one requency. When noise was added to the previous example at a signal-to-noise ratio o 5, the requency determination and φ s determination were unaected, while the error in θ s increased to θ s = 3.4. Finally, we illustrate how the source intererence phenomena discussed earlier limit our ability to locate sources that overlap in requency. Figures 5 and 6 show how the demodulation procedure is able to locate sources A and B when one o the sources is turned o. Figure 7 shows what happens when both sources are present. We see that source B shows up clearly, while source A iswashedout.

7 LISA data analysis: Doppler demodulation S169 Figure 5. Determining the sky location o source A without source B. Figure 6. Determining the sky location o source B without source A. Figure 7. Determining the sky location o sources A and B together. 4. Discussion The Doppler demodulation procedure provides a quick way o inding the requencies and sky locations o the brightest sources detected by LISA. The method has a number o limitations, the most serious being its inability to locate more than one source per bandwidth and its

8 S170 NJCornishand S L Larson inability to determine i a source is in the northern or southern hemisphere. Despite these limitations, the Doppler demodulation procedure will be a useul tool in the LISA data analysis arsenal. Acknowledgments It is a pleasure to thank Tom rince and the members o the Montana Gravitational Wave Astronomy Group Bill Hisock, Ron Hellings and Matt Benacquista or many stimulating discussions. The work o NJC was supported by the NASA ESCoR programme through Cooperative Agreement NCC The work o SLL was supported by LISA contract number O Reerences [1] Livas J C 1987 hd Thesis Massachusetts Institute o Technology [2] Brady R, Creighton T, Cutler C and Schutz B F 1998 hys. Rev. D [3] Dhurandhar S V and Vecchio A 2001 hys. Rev. D [4] Cutler C 1998 hys. Rev. D [5] Cornish N J and Larson S 2002 Lisa data analysis: II. Source identiication and subtraction (in preparation) [6] Hellings R W 2002 in preparation [7] Gorski K M, Wandelt B D, Hivon E, Hansen F K and Banday A J 1999 reprint astro-ph/ (web page: