The cross-correlation search for a hot spot of gravitational waves
Sanjeev Dhurandhar, Hideyuki Tagoshi, Yuta Okada, Nobuyuki Kanda, Hirotaka Takahashi
aa r X i v : . [ g r- q c ] M a y The cross-correlation search for a hot spot of gravitational waves
Sanjeev Dhurandhar, Hideyuki Tagoshi, Yuta Okada, Nobuyuki Kanda, and Hirotaka Takahashi
4, 5 Inter-University Centre for Astronomy and Astrophysics,Post Bag 4, Ganeshkhind, Pune 411007, India Department of Earth and Space Science, Graduate School of Science,Osaka University, Toyonaka, Osaka 560-0043, Japan Department of Physics, Graduate School of Science, Osaka City University,Sugimoto 3-3-138, Sumiyoshi-ku, Osaka 558-8585, Japan Department of Humanities, Yamanashi Eiwa College,888, Yokone, Kofu, Yamanashi 400-8555, Japan Earthquake Research Institute, University of Tokyo, Bunkyo-Ku, Tokyo 113-0032, Japan (Dated: November 14, 2018, ver.3.11)The cross-correlation search has been previously applied to map the gravitational wave (GW)stochastic background in the sky and also to target GW from rotating neutron stars/pulsars. Herewe investigate how the cross-correlation method can be used to target a small region in the skyspanning at most a few pixels, where a pixel in the sky is determined by the diffraction limit whichdepends on the (i) baseline joining a pair of detectors and (ii) detector bandwidth. Here as one ofthe promising targets, we consider the Virgo cluster - a ”hot spot” spanning few pixels - which couldcontain, as estimates suggest ∼ neutron stars, of which a small fraction would continuously emitGW in the bandwidth of the detectors. For the detector baselines, we consider advanced detectorpairs among LCGT, LIGO, Virgo, ET etc. Our results show that sufficient signal to noise can beaccumulated with integration times of the order of a year. The results improve for the multibaselinesearch. This analysis could as well be applied to other likely hot spots in the sky and other possiblepairs of detectors. PACS numbers: 95.85.Sz,04.80.Nn,07.05.Kf,95.55.Ym
I. INTRODUCTION
An enigmatic prediction of Einstein’s general theoryof relativity are gravitational waves (GW). With the ob-served decay in the orbit of the Hulse-Taylor binary pul-sar agreeing within a fraction of a percent with the theo-retically computed decay from Einstein’s theory, the exis-tence of GW was firmly established. Currently there is aworldwide effort to detect GW with the operating inter-ferometric gravitational wave observatories, the LIGO,Virgo, GEO and TAMA [1]. Now the advanced detec-tors being constructed include the upgraded LIGO andVirgo, the LCGT of Japan, LIGO-Australia and futurepossibilities such as Einstein Telescope (ET) [2].Different types of GW sources have been predicted andmay be directly observed by these advanced detectors inthe near future (see [3] and references therein for recentreviews). In this paper we will address the problem ofthe targeted search of stochastic GW from a small regionin the sky, typically of linear size of a few degrees (fewpixels - a pixel determined by the diffraction limit) - a”hot spot” - where there is likely to be an abundance ofindependent, unresolved GW sources continuously pro-ducing a relatively large stochastic background. Such ascenario seems feasible for the Virgo cluster, which couldcontain about 10 neutron stars, the current estimatebeing 10 − per galaxy. Out of these neutron starsa small fraction of them could be rotating sufficientlyrapidly emitting GW in the advanced detector bandwidthof several 100 Hz to about 1 kHz. These could produce a reasonable signal-to-noise ratio (SNR) with an integra-tion time of the order of an year. Thus, the GW sourceconsists of spinning asymmetric neutron stars whose am-plitudes and phases are randomly distributed. We willbe thus dealing with a localized stochastic GW source.This is only one type of GW source, but there could becontributions from other sources such as supernovae withasymmetric core collapse, binary black hole mergers, low-mass X-ray binaries and hydrodynamical instabilities inneutron stars, or even GWs from astrophysical objectsthat we never knew existed. These will only in general(statistically) add to the SNR. The detectors we considerfor this paper are advanced detectors such as the LIGO,Virgo, LCGT, ET etc. which are expected to have suffi-cient sensitivity for detecting a hot spot.The appropriate method for observing such a sourceis the cross-correlation method described in [4] (hence-forth referred to as paper I), which is also generallyknown as the radiometric method. The idea is to cross-correlate data streams from two detectors with an ap-propriate time-delay, namely, the time-delay between ar-rival times of a GW wavefront from a specific direction ˆΩ . This choice of time-delay allows the sampling of thesame wavefront. As the detector baseline rotates with theearth, the time-delay between the data streams changesduring the course of the day. The statistic targets a patch(pixel) in the sky around ˆΩ its size being determined bythe diffraction limit, namely, the inverse of the band-width divided by the light travel time along the baseline.This statistic in fact is a point estimate of the signal re-ceived from the given direction ˆΩ and is most appropriatefor observing a hot spot and could be made optimal by‘masking’ the rest of the sky if the hot spot emits a strongsignal.The GW strain amplitude for a rotating neutron staris proportional to the square of the frequency [5], h ∼ π α Gc εIR f , (1)where α ∼ < G is the Newton’sgravitational constant, c the speed of light, ε is the el-lipticity of the neutron star, I the moment of inertia, R the distance to the source and f the GW frequency.Since the cross-correlation statistic is quadratic in thestrain amplitude, it scales as the fourth power of the fre-quency and therefore the main contribution to the SNRwill tend to come from high frequency sources assumingthat they are relatively abundant in the high frequencyregime. Thus it is the population of millisecond neutronstars that we must primarily consider. We then estimatethe millisecond neutron star population from the astro-physical information that is available and show that onecan get an acceptable SNR, ρ ∼
1, for an integration oftime of about an year. Using multiple baselines improvesthe SNR further. We find that among the current or nearfuture baselines, the baseline of the two LIGOs and thebaseline of LIGO Livingston and a LIGO like detector atAIGO site stand out - they give dominant contributionto the SNR.In section II, we give a brief description of the cross-correlation method and the statistic and then derive anexpression for the optimal SNR. In section III, we stateour results and discuss them in light of the astrophysicalscenarios that are possible and the sensitivities of thefuture advanced detectors such as the ET.
II. THE CROSS-CORRELATION STATISTICFOR TARGETING A HOT SPOT
We refer to paper I for the detailed arguments involvedin defining the cross-correlation statistic. Here we onlyfurnish the salient steps. Since here we are interested inobserving a hot spot, we will restrict our discussion to apoint source. The full statistic, which we denote by S , isa weighted sum of elementary pieces ∆ S k , k = 1 , , ...n defined over a time-segments t k − ∆ t/ ≤ t ≤ t k +∆ t/ k . The full observation time is T = n ∆ t . The ∆ t is so chosen that it is much larger than thepossible time-delay between the detectors (which mustbe less than about 40 ms for ground-based detectors)and much less than the time required for the orientationof the detectors to change appreciably and also on thetimescale in which the noise is stationary. Current valuesof ∆ t used in LSC data analysis vary from 32 to 192seconds. Let us consider a pair of detectors labeled by I = 1 ,
2, then the data in the I th detector is given by x I ( t ) = h I ( t ) + n I ( t ), the signal h I ( t ) is added to thenoise n I ( t ) in the I th detector. For a point source in the direction ˆΩ , the ∆ S k also becomes a function of ˆΩ . Itcan be expressed easily in the Fourier domain,∆ S k ( ˆΩ ) = Z ∞−∞ df e x ∗ ( t k ; f ) e x ( t k ; f ) e Q ( t k , f, ˆΩ ) , (2)where the e x ∗ I ( t k ; f ) are short term Fourier transforms(SFT) defined only over the interval ∆ t around t k ,namely, e x I ( t k ; f ) := Z t k +∆ t/ t k − ∆ t/ dt ′ x I ( t ′ ) e − πift ′ . (3)The Q ( t k , f, ˆΩ ) is a filter function chosen so that it op-timizes the filter output. It also depends on the powerspectrum of the GW source and the power spectral den-sities of the noises in each of the detectors. As discussedin paper I, in the general case it is a far more complicatedobject - a functional - but for the case of a point source,it reduces to a function of the direction ˆΩ . Even thenit remains a functional of the signal power spectral den-sity and the noise power spectral density (PSD). With aslight abuse of notation we still write it as a function of f . The ∆ S k are random variables because of the noiseand for different k we take them to be uncorrelated. Themean and the variance of ∆ S k are denoted respectivelyby µ k = h ∆ S k i and σ k = h ∆ S k i − h ∆ S k i . It has beenshown in paper I that the linear combination that yieldsthe maximum SNR is: S = P nk =1 µ k σ − k ∆ S k P nk =1 µ k σ − k , (4) ρ = ( n X k =1 µ k /σ k ) , (5)where ρ is the SNR. The sum over k can be convertedinto an integral over t and henceforth in this article wedrop the suffix k and replace t k by just t . This helps toavoid clutter without jeopardizing clarity.We now turn to the noise and signal PSDs in termsof which the SNR can be finally expressed. The signalcross-correlation in the two detectors in the limit of largetime segment can be written as: h e h ∗ ( t, f ) e h ( t, f ′ ) i = δ ( f − f ′ ) H ( f ) γ ( t, f, ˆΩ ) , (6)where γ ( t, f, ˆΩ ) is the so called directed overlap reductionfunction analogous to the one defined in [6] for the fullsky, and given in the case of the point source by, γ ( t, f, ˆΩ ) = Γ( t, ˆΩ ) e πif ˆΩ · ∆x ( t ) /c , (7)Γ( ˆΩ , t ) = F +1 ( t, ˆΩ ) F +2 ( ˆΩ , t )+ F × ( ˆΩ , t ) F × ( ˆΩ , t ) , (8)and where the ∆x ( t ) is the vector joining detector 1 todetector 2 and rotates with the Earth tracing out a cone.The F + I , F × I , I = 1 ,
2, are the antenna pattern func-tions for the two detectors and for the two polarizations.As mentioned in paper I the directed overlap reductionfunction has a bandwidth of about 750 Hz as comparedto the few tens of Hz for the overlap reduction functionfound by integrating over the full sky. This is the mainadvantage of this method in which the sensitive regionof the detector bandwidth is sampled by the statistic.Further the quantity f H ( f ) is essentially the flux perunit frequency per unit solid angle. For the noise, wetake the noise in the two detectors to be uncorrelated, h n ( t ) n ( t ′ ) i = 0, and the one-sided noise PSD in eachdetector I is given through the defining equation, h e n ∗ I ( t ; f ) e n I ( t ; f ′ ) i = 12 δ ( f − f ′ ) P I ( t ; | f | ) . (9)We also assume h h I ( f ) n J ( f ′ ) i = 0 , I, J = 1 ,
2, that isthe signal and noise are uncorrelated. We are now readyto write down the optimal filter. In paper I it has beenshown that the optimal filter for a given time segmentlabeled by t and for a point source in the direction ˆΩ isgiven by, Q ( t, f, ˆΩ ) = λ ( t ) H ( f ) γ ∗ ( t, f, ˆΩ ) P ( t ; | f | ) P ( t ; | f | ) , (10)where λ ( t ) is a normalization constant, which in any casecancels out in the SNR. The SNR ρ is given in terms of µ ( t ) and σ ( t ) which are the mean and standard deviationrespectively of ∆ S ( t ). To keep the expressions simple weassume that the noise in the detectors is stationary. Thiscertainly will not be the case, but since we are only inter-ested in order of magnitude results, the assumption is notunjustified. Then P I becomes a function of f only. Alsowe consider a band-width f ≤ f ≤ f for evaluating theSNR; the lower limit f is determined by the seismic cut-off, while the upper limit f is decided by the GW sourcesabove which we do not expect significant contribution tothe SNR. Given this, the relevant quantities can be bestexpressed in terms of the following two averages: h H i BW = 2∆ f Z f f df H ( f ) P ( f ) P ( f ) , (11) h Γ i ( ˆΩ ) = 1 T Z T Γ ( ˆΩ , t ) dt , (12)where ∆ f = f − f . The first is the noise weighted av-erage of the signal H ( f ), the suffix BW denotes band-width, while the second is the time average of the squareddirected overlap reduction function taken over one side-real day. It is a function of sky position of the source.But since the azimuth is averaged over 2 π , it is just afunction of the declination of the source. Then in termsof these averages we have, µ ( t ) = λ (∆ t ∆ f ) h H i BW Γ ( ˆΩ , t ) , (13) σ ( t ) = 12 λ (∆ t ∆ f ) h H i / Γ( ˆΩ , t ) . (14) h Γ i /
21 day
LIGO-H Virgo LCGT AIGOLIGO-L 0.387 0.288 0.224 0.452LIGO-H − − − − − − are given for the Virgo cluster whose dec-lination is ∼ +12 . ◦ (the RA is irrelevant since we take aone day average). LIGO-L stands for LIGO-Livingston andLIGO-H for LIGO-Hanford. Then using the continuous limit of Eq. (5), we may writethe SNR ρ in terms of the averages as follows: ρ = " t Z T dt µ ( t ) σ ( t ) , = 2 ( T ∆ f ) h H i / h Γ i /
21 day . (15)We now use this expression to compute the SNR for thecontinuous wave sources from the Virgo cluster. We ob-serve that the SNR scales as √ T .To fix ideas we can look at a simplified situation ofidentical detectors with white noise P I ( f ) = P in thefrequency range f ≤ f ≤ f and P I = ∞ otherwise.Similarly we may consider flat signal spectrum H ( f ) = H , then the SNR simplifies to: ρ = 2 [ T ∆ f ] H P h Γ i /
21 day . (16)The values of h Γ i /
21 day for various combinations of de-tector baselines are given in Table I for the Virgo clusterwhich has a declination ∼ +12 . ◦ . III. RESULTS AND DISCUSSIONA. Pulsar population and distribution
We consider gravitational waves from rotating neutronstars in the Virgo cluster. One important parameterof this source is the population of such neutron stars.The number of Galactic neutron stars is estimated tobe 10 − since the birth rate is about 10 − / yr andthe age of the Galactic disk is about 10 yr. What ismore important in our case is the number of Galacticneutron stars whose rotation period is of the order ofmilliseconds. From the survey of radio pulsars in ourGalactic disk, the population of millisecond pulsars is es-timated to be at least 40000 [7–9] which implies a birthrate of 2 . × − / yr. This is consistent with other stud-ies of the millisecond pulsar population by Ferrario andWickramasinghe (3 . × − / yr) [10] and by Story et al.(4 − × − / yr) [11]. From the recent observation of FIG. 1: The distribution of observed radio pulsars. The hor-izontal axis is log ( f r ) where f r is the rotational frequencyof pulsars. The histogram is the observed number. The solidline is the two component Gaussian model of the distribution. gamma rays with the Fermi satellite [12], the popula-tion of millisecond pulsars in our Galactic globular clus-ter is estimated to be 2600-4700 which is one order lowerthan millisecond pulsars in the Galactic disk. Althoughthere might be significant population of millisecond pul-sars which do not emit radio waves, X and gamma raysnow, since the life times of millisecond pulsars are be-lieved to be long ( ∼ yr) [13], we do not expect a largepopulation of such millisecond pulsars to exist. Thus weadopt 40000 as a typical number of neutron stars pergalaxy whose rotation period is of the order of millisec-onds.A catalog of radio pulsars is given in the ATNF pulsardatabase [14]. The distribution of observed radio pulsarsis given in Fig.1. We find that the distribution naturallyfalls into two regions separated by 50Hz. In each region,the distribution is approximately Gaussian as seen fromthe figure. This means that the distributions in eachregion may be approximated as log-normal distributionsgiven by: P (log f r ) d (log f r ) = 1 √ πσ e − (log fr − log µ σ d (log f r ) , (for f r > , (17) P (log f r ) d (log f r ) = 1 √ πσ e − (log fr − log µ σ d (log f r ) , (for f r < , (18)where µ = 219Hz, σ = 0 . µ = 1 . σ = 0 . f r = f / f is the gravitational wavefrequency). P and P are normalized to unity when in-tegrated from f r = 0 to infinity. We assume a similar bi-modal form of distribution of neutron stars in the Virgo cluster. We assume that the total number of neutronstars in our Galaxy is 10 for f r < f r > galaxies inthe Virgo cluster, total number of neutron stars in Virgocluster is N low ∼ for f r < N high ∼ × for f r > N ( f ) df =( N high P (log f r ) + N low P (log f r )) df r f r ln 10 . (19)Since the length of data of one time-segment, ∆ t is atmost 10 seconds, the frequency resolution is larger than10 − Hz. The frequency bandwidth can be taken as 10 Hz. Thus the number of frequency bins is 10 . Since thenumber of pulsars with f >
100 Hz is 10 , the numberof pulsars in each frequency bin is about 10. In the lowfrequency regime this number is much larger. Thus, it isnot possible to resolve the signal from each pulsar, whichconfirms the stochastic nature of the Virgo cluster hotspot. B. Signal-to-noise ratio
The spectral density of gravitational radiation fromneutron stars in Virgo cluster, H ( f ) is given as H ( f )= h h i N ( f )= (cid:20) . × − (cid:16) ε − (cid:17) (cid:18) I . × gcm (cid:19)(cid:21) ×h α i f N ( f ) , (20)where h α i represents the average with respect to theinclination angle and the polarization angle. Assuminguniform distribution of the sources over the angles, wehave h α i = 0 .
4. We also used the distance R = 16 . H ( f )is compared with the noise power spectral density, it isconvenient to define an effective source power, H eff ( f )by, H ( f ) = 8 T obs h Γ i f H ( f ) , (21)where T obs is the observation time.Then the signal-to-noise ratio is given by, ρ = "Z f f dff H ( f ) P ( f ) P ( f ) / . (22)The noise power spectral density of various advanced de-tectors including Einstein Telescope, as well as H eff ( f )are plotted in Fig.2. Here, we assume T obs = 1yr and h Γ i /
21 day = 0 .
2. In this plot, H eff ( f ) is plotted for ε = 10 − , − , − . Although in these plots, we in-clude the contribution from low frequency neutron starswith f r < FIG. 2: One-sided noise power spectral density of LCGT, ad-vanced LIGO, advanced Virgo, and Einstein Telescope (ET-B). LCGT noise curve is ”variable RSE in broadband mode”(VRSE(B)) [15]. Advanced LIGO noise curve is ”Zero Det,High Power” taken from [16]. Advanced Virgo noise curveis take from the Virgo website[17]. The Einstein Telescopenoise curve is called ”ET-B” [18]. The effective source power H / ( f ) is also plotted. In this plot, we assume T obs = 1yrand h Γ i /
21 day = 0 . We now consider the quantity h H i / ∆ f / . We findthat, h H i / ∆ f / ∝ (cid:16) ε − (cid:17) (cid:18) I . × gcm (cid:19) (cid:18) N msp × (cid:19) , (23)where N msp is a numbers of millisecond pulsars. Thesignal-to-noise ratio with 1 year observation is written as ρ = ρ s T obs h Γ i /
21 day . ! (cid:16) ε − (cid:17) × (cid:18) I . × gcm (cid:19) (cid:18) N msp × (cid:19) . (24)We now compute the observation time required to achieve ρ = 3, which we denote by T ρ =3obs . We choose this value ofSNR because the noise in the statistic S ( ˆΩ ), as arguedin paper I, is distributed as a Gaussian with mean µ S and standard deviation σ S . This is the consequence ofthe generalized central limit theorem. The ρ is µ S /σ S .When no signal is present, we have, µ S = 0. Thus when we take ρ >
3, there is more than 99.7% chance that thenoise is not masquerading as the signal. We then havethe following result: T obs = T ρ =3obs h Γ i /
21 day . ! − (cid:16) ε − (cid:17) − × (cid:18) I . × gcm (cid:19) − (cid:18) N msp × (cid:19) − (cid:16) ρ (cid:17) . (25)The values of ρ and T ρ =3obs are given in Tables II andIII for h Γ i /
21 day = 0 .
2. These tables along with Eq.(24)and Eq.(25) can be used to obtain the ρ and the T obs forany other value of h Γ i /
21 day or equivalently for any othersky location, not just the Virgo cluster. From the tables,we find that in the case of ε = 10 − we can achieve ρ = 3in about 3 months with advanced LIGO noise PSD. Foradvanced Virgo and LCGT, it takes about 1.5 year toachieve it. For Einstein Telescope, it will be quite easyto observe it. However, the results strongly depend onthe value of ǫ . If ε = 10 − , it will become difficult toobserve the Virgo cluster hot spot with advanced LIGO,advanced Virgo and LCGT. Only Einstein Telescope willbe able to detect it. Note that these are order of mag-nitude results where we have assumed a typical value of h Γ i /
21 day ∼ . ρ and T ρ =3obs ob-tained for various detector combinations of advanced de-tectors, are given for the specific source location of theVirgo cluster. Since h Γ i /
21 day is roughly factor of 2 largerthan 0.2 for the LIGOs and AIGO network, the values of ρ and T ρ =3obs are improved significantly for these base-lines. For example, the ρ = 3 is achieved in 26 daysby LIGO-L and LIGO-H, and in 19 days by LIGO-L andAIGO. ρ LIGO Virgo LCGT ET-BLIGO 5.8 2.5 3.0 52Virgo − − − − − − . × TABLE II: The signal-to-noise ratio ρ which can be ob-tained with 1 year observation time for each combinationof the detectors’ noise PSD assuming h Γ i = 0 . ε = 10 − . The results improve if we employ several baselines of anetwork of detectors. A full treatment of multi-baselinegravitational wave radiometry has been given in [19]. Theresults of this paper can be easily applied to the case ofthe hotspot where the source consists of a single pixel orat most a few pixels. In this case the beam matrix fora single baseline essentially consists of a single diagonal T ρ =3obs LIGO Virgo LCGT ET-BLIGO 0.26 yr 1.5 yr 1.0 yr 1.2 dayVirgo − − − − − − . × secTABLE III: Observation time T ρ =3obs required to achieve ρ =3 for each combination of the noise PSD and assuming h Γ i = 0 . ε = 10 − . ρ LIGO-H Virgo LCGT AIGOLIGO-L 11.3 3.54 3.36 13.2LIGO-H − − − − − − ρ which can be ob-tained with 1 year observation time for each combination ofthe detectors’ noise PSD and h Γ i in Table I. ε = 10 − is assumed. Noise PSD of AIGO is assumed to be the sameas that of LIGO noise PSD. term for a single pixel or in case of few pixels, a smallblock diagonal matrix having dominant diagonal terms.The ρ which we have defined above is then just the SNRobtained for the log likelihood statistic λ defined in thatpaper. We also deduce from further results of that paperon sensitivity that approximately in our case, ρ = X I ρ I , (26)where ρ network is the SNR for the network and the index I runs over all the baselines of the network. Similarly,it is easy from the foregoing to deduce that the obser-vation times to reach an SNR of 3, namely T ρ =3obs , addharmonically; more specifically we have,1 T ρ =3obs = X I T ρ I =3obs , (27)where now the T ρ =3obs denotes the time of observation re- T ρ =3obs [day] LIGO-H Virgo LCGT AIGOLIGO-L 25.8 262 291 18.9LIGO-H −
474 319 39.5Virgo − −
907 266LCGT − − − T ρ =3obs required to achieve ρ = 3for each combination of the noise PSD and h Γ i in TableI. ε = 10 − is assumed. Noise PSD of AIGO is assumed tobe the same as that of LIGO noise PSD. Detector combination ρ T ρ =3obs [day]L-H-V 12.1 22.4L-H-J 12.2 22.1L-H-A 19.6 8.55L-V-J 5.24 120L-V-A 14.1 16.5L-J-A 14.1 16.4H-V-J 4.57 157L-V-A 10.1 32.1L-J-A 10.4 30.3V-J-A 5.54 107L-H-V-J 13.1 19.1L-H-V-A 20.4 7.90L-H-J-A 20.5 7.81L-V-J-A 15.1 14.4H-V-J-A 11.5 25.0L-H-V-J-A 21.4 7.21TABLE VI: The signal-to-noise ratio ρ which can be ob-tained with 1 year observation time and the observation timerequired to achieve ρ = 3 by more than 2 detectors. Theseare derived from Eqs. (26) and (27) and Tables IV and V.L: LIGO-Livingston, H: LIGO-Hanford, V: Virgo, J: LCGTin Japan, A: a detector with LIGO’s noise PSD at the AIGOsite in Australia. quired for the network and T ρ I =3obs denotes the observationtime required for the baseline I to reach the SNR of 3.We can now apply these results to various networks.The results are given in Table VI. We first consider the3 detectors, LIGO-Virgo (LHV) network. Just compar-ing tables IV and VI, the ρ goes up from 11.3 for twoLIGOs to 12.1 for L-H-V network which is about 7 %increase. Note that one must here take into account 3baselines L-H, L-V and H-V. The T ρ =3obs comes down from25.8 days for the two LIGOs to 22.4 day for the L-H-Vnetwork which is a decrease of 13 %. If one considers thetwo LIGOs along with the LCGT the improvement is al-most similar to Virgo case, that is, the observation timecomes down to 22.1 day. The improvement of addingother baselines to the L-H baseline is marginal becausethe LH contribution is dominant. Note however that aninteresting improvement is obtained if one considers adetector at AIGO site assuming same noise PSD as theLIGOs. In such a L-H-A network, L-A contribution be-comes dominant because of largest h Γ i in Table I,and ρ goes up to 19.6 and T ρ =3obs comes down to 8.5days. L-V-A and L-J-A networks are similar and givessecond largest value of ρ among 3 detector networks.They are better than L-H-V and L-H-J cases.In case of a 4 or 5 detector network, we can have furtherimprovement, but the effect is not so large since the L-H-A contribution dominates ρ . For the 4 detector case,L-H-J-A network gives the largest value of ρ = 20 . ρ = 21 . T ρ =3obs = 7 .
21 days.
IV. SUMMARY
In this article we address the question of observinga hotspot of stochastic GW using the cross-correlationstatistic. The idea is to restrict the statistic to a singleor few pixels in the sky and target possible point stochas-tic sources. A possible source which we pick is the Virgocluster which could be a rich bed of rotating neutron starscontaining an estimated number of 10 . Out of these therotating neutron stars emitting GW which fall into thebandwidth of the advanced detectors are primarily themillisecond neutron stars. We assume that the distribu-tion of such neutron stars follows a bimodal distributionsimilar to that of the radio millisecond pulsars observedin our galaxy. We then see that with advanced detec-tors the observation time required to accumulate SNR ∼ ε ∼ − . Several baselines havebeen considered as well as multiple baselines correspond-ing to networks of detectors. In these calculations, thebaselines that stand out are the two LIGO detectors andthe LIGO Livingston and a LIGO like detector at theAIGO site in Australia. These baselines have the bestsensitivity, because for these baselines, the detectors arealmost co-aligned. In such cases, the observation timerequired to achieve SNR ∼ ε = 10 − . The future proposed Einstein Telescopecan easily detect the hotspot. In fact, Einstein Telescope would be the only detector which can observe the Virgocluster hotspot if ε ∼ − . In that case, only one Ein-stein Telescope will be sufficient to detect the Virgo clus-ter by cross-correlating with other detectors like LIGOs,Virgo and LCGT.Besides the Virgo cluster, there could be other candi-dates for hotspots such as the Andromeda galaxy or ourown galactic centre. Although in these cases, the numberof sources contributing to the GW background may besmaller than the Virgo cluster, their distances are muchsmaller, which makes up for the overall strength of thestochastic sources. Acknowledgments
S. Dhurandhar thanks S. Bose and S. Mitra for usefuldiscussions on multiple baselines. We thank F. Taka-hara and S.J. Tanaka for useful discussions on the pop-ulation of neutron stars. S. Dhurandhar also acknowl-edges the DST and JSPS Indo-Japan international coop-erative programme for scientists and engineers for sup-porting visits to Osaka City University, Japan and Os-aka University, Japan. H. Tagoshi, N. Kanda and H.Takahashi thank JSPS and DST under the same Indo-Japan programme for their visit to IUCAA, Pune, India.H.Tagoshi’s work was also supported in part by a MonbuKagakusho Grant-in-aid for Scientific Research of Japan(No. 20540271). H.Takahashi’s work was also supportedin part by a Monbu Kagakusho Grant-in-aid for ScientificResearch of Japan (No. 23740207). [1] Abramovici, A., Althouse, W.E., Drever, R.W.P.,G¨ u rsel,Y., Kanwamura, S., Raab, F.J. Shoemaker, D.,Sievers, L., Spero, R.E., Thorne, K.S., Vogt, R.E., Weiss,R., Whitcomb, S.E., and Zucker, Z.E., 1992, Sciences,256, 325; Bradaschia, C., et al., 1990, Nucl. Instum.Methods Phys. Res. A, 289, 518;Danzmann K et al.,1995, in First Edoardo Amaldi Conference on Gravita-tional Wave Experi- ments, Ed. E Coccia, G Pizzella, FRonga, Singapore: World Scientific; Tsubono, K., 1995, ,”300-m Laser Interferometer Gravitational Wave Detec-tor (TAMA300) in Japan”, in Gravitational Wave Ex-periments, Ed., Coccia, E. Pizzella, G., Ronga, F., WorldScientific Singapore.[2] Kuroda, K., 2006, Class. Quantum Grav., 23, S215.[3] K. S. Thorne, in
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