Detecting the anisotropic astrophysical gravitational wave background in the presence of shot noise through cross-correlations
DDetecting the anisotropic astrophysical gravitational wave background in the presence of shot noisethrough cross-correlations
David Alonso, ∗ Giulia Cusin,
1, 2, † Pedro G. Ferreira, ‡ and Cyril Pitrou § Astrophysics, University of Oxford, DWB, Keble Road, Oxford OX1 3RH, UK Universit´e de Gen`eve, D´epartement de Physique Th´eorique and Centre for Astroparticle Physics,24 quai Ernest-Ansermet, CH-1211 Gen`eve 4, Switzerland Institut d’Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France. (Dated: Received February 10, 2020; published – 00, 0000)The spatial and temporal discreteness of gravitational wave sources leads to shot noise that may, in someregimes, swamp any attempts at measuring the anisotropy of the gravitational wave background. Cross-correlating a gravitational wave background map with a sufficiently dense galaxy survey can alleviate this issue,and potentially recover some of the underlying properties of the gravitational wave background. We quan-tify the shot noise level and we explicitly show that cross-correlating the gravitational wave background and agalaxy catalog improves the chances of a first detection of the background anisotropy with a gravitational waveobservatory operating in the frequency range ( , ) , given sufficient sensitivity. I. INTRODUCTION
The detection of gravitational waves is now in full flowheralding a new era in gravitational physics. One next frontieris the measurement and characterization of the gravitationalwave background, a smooth but structured bath of gravita-tional radiation which may have come from the primordialUniverse, but also from the plethora of gravitational wave sig-nals emitted by different astrophysical sources from the begin-ning of stellar activity until today [1]. One hopes that a cleanmeasurement of this background will shed light on the physicsof the early universe as well as the astrophysical properties ofastrophysical sources (e.g. population of compact binaries).The astrophysical gravitational wave background (AGWB),i.e. the background generated by gravitational events at latecosmic times, is quantifiable through its isotropic energy den-sity level and through the spatial angular power spectrum en-coding its anisotropy. Existing data already place bounds onboth the isotropic and anisotropic components [e.g. see 2–6].A detection of the isotropic level could come as early as 2020[7].The AGWB in the LIGO band is mostly generated by a su-perposition of discrete events – binary mergers, cataclysmicgravitational events, etc – and, as such can be thought of asequence of random processes. One starts off with the spa-tial distribution of the underlying density field, which can bemodelled as a continuous random field (e.g. a realizationof a multivariate Gaussian distribution on sufficiently largescales). Sources of gravitational waves will arise from a dis-crete sampling of this underlying density field; the simplestapproach is to assume that it is a spatial Poisson process wherethe variance is set by the local number density of GW sourceswhich is a relatively complicated function of the underlyingdensity field. Finally, the events that lead to gravitational ∗ Author list is alphabetised.; [email protected] † [email protected] ‡ [email protected] § [email protected] waves will often also be discrete in time, leading to a thirdlayer of stochasticity.The discrete nature of the processes underlying the AGWBlead to a source of noise which is familiar from the analy-sis of galaxy surveys – Poisson or shot noise. If the discretepart of the process is sufficiently sparse (i.e. the number den-sity of sources or the rate of gravitational wave events is suffi-ciently small) shot noise may dominate, making it impossibleto characterise the anisotropy of the AGWB (i.e. the underly-ing smooth field). A clear derivation of the problem from firstprinciples can be found in [8].It has been suggested that it could be possible to sidestepthe shot noise problem by cross-correlating a GW map witha dense galaxy sample tracing the same large-scale structure[9]. In this situation, the shot noise level of the cross-spectrumis primarily driven by the density of the much denser galaxysurvey (although the GW shot noise will still be a significantcontribution to the signal to noise of the cross-correlation).In this paper we explore this claim and quantify how muchone can alleviate the shot-noise problem in measurements ofthe anisotropy of the AGWB with current gravitational waveexperiments.This paper is structured as follows. In Section II we presenta succint synopsis of the anisotropy of the AGWB and ofits cross-correlation with the galaxy distribution. In Sec-tion III we discuss shot noise (both spatial and pop corn)and we sketch the argument of why cross-correlating a mapof the AGWB with a galaxy survey may mitigate the shot-noise problem. In Section IV we compute signal-to-noise ra-tio (SNR) for auto-correlation and cross-correlations and weshow that the SNR for cross-correlation is significantly largerthan the auto-correlation one and has a mild dependence onthe cut-off chosen to filter out resolvable GW sources. Wealso show that most of the SNR comes from z < .
5, and thatthe result depends very mildly on the number density of galax-ies. The SNR is also enhanced when considering the likelyhigher rate of neutron star mergers on top of black hole merg-ers, hence this observable might be a realistic and promisingtarget for present and future galaxy surveys.We must clarify from the outset, that we only intend to a r X i v : . [ a s t r o - ph . C O ] F e b explore the shot-noise problem here, in isolation from othersources of noise. Therefore, all results reported below mustbe understood as forecasts for a perfect experiment with no in-strumental noise. They therefore represent the best-case sce-nario for the detectability of the AGWB in the presence ofspatial and temporal shot noise. II. AGWB AND GALAXY ANGULAR POWER SPECTRA
The isotropic AGWB signal can be characterized by the en-ergy density in gravity waves, ρ GW , per logarithmic frequencyinterval in units of the critical density, ρ c , averaged over di-rections: ¯ Ω GW ≡ d ρ GW / d ln f / ρ c where f is frequency. It canalso be written as the sum of contributions from sources lo-cated at all the (comoving) distances r in the form ¯ Ω GW ( f ) ≡ π (cid:82) ∂ r ¯ Ω GW ( f , r ) d r . Each astrophysical model predicts afunctional dependence for ∂ r ¯ Ω GW ( f , r ) where we define ∂ r ¯ Ω GW = f πρ c a (cid:90) d L GW ¯ n G ( L GW , r ) L GW , (1)¯ n G ( L GW , r ) is the average physical number of galaxies at dis-tance r with gravitational wave luminosity L GW . We use here,for definiteness, the reference astrophysical model of [9].Of interest in this paper is the anisotropy of the AGWB,which can be modelled as δ Ω GW ( e ) = (cid:90) d r ∂ r ¯ Ω GW δ G ( r e ) , (2)when sub-leading contributions from peculiar velocities andmetric perturbations are ignored [10]. The line of sight direc-tion is given by the unit vector e , and we assume an inhomoge-neous distribution of galaxies hosting the gravitational wavesources, characterized by a density n G = ¯ n G ( r )( + δ G ( r )) ,where ¯ n G ( r ) = (cid:82) ¯ n G ( L GW , r ) d L GW is the mean density ofgalaxies in the Universe at comoving distance r .In the Limber approximation, the general expression ofthe angular power spectrum of the anisotropies, C GW (cid:96) of theAGWB reduces to [11, 12] C GW (cid:96) ( f ) (cid:39) (cid:0) (cid:96) + (cid:1) − (cid:90) d k P G ( k ) (cid:12)(cid:12) ∂ r ¯ Ω GW ( f , r (cid:96) ) (cid:12)(cid:12) , (3)where (cid:96) is the multipole in the spherical harmonic expansion, P G ( k ) is the galaxy power spectrum, and r (cid:96) = ( (cid:96) + / ) / k .Thus a measurement of C GW (cid:96) is sensitive to the shape of ∂ r ¯ Ω and of P G ( k ) . For details and derivations see [9, 11–13].Consider now a direct measurement of the galaxy distribu-tion, and let us construct a weighted average of the galaxyoverdensity of objects along the line of sight by ∆ G ( e ) = (cid:90) d rW ( r ) δ G ( r e ) , (4) We assume that all galaxies are observed, hence W ( r ) is not a selectionfunction but rather a weight used to combine the distance dependent over-densities δ G ( r ) . It can be considered as an artificial selection function W ( r ) / [ r a ¯ n G ( r )] . where the weight function W ( r ) is normalized so (cid:82) W ( r ) d r =
1. The auto-correlation and cross-correlation with AGWB inthe Limber approximation are given by C ∆ (cid:96) (cid:39) (cid:0) (cid:96) + (cid:1) − (cid:90) d k P G ( k ) | W ( r (cid:96) ) | , (5)and C GW , ∆ (cid:96) ( f ) (cid:39) (cid:0) (cid:96) + (cid:1) − (cid:90) d k P G ( k ) W ( r (cid:96) ) ∂ r ¯ Ω GW ( f , r (cid:96) ) . (6)As we can see, and very much along the lines of what isdone in large-scale structure studies [e.g. 14, 15], we havea full set of spectra and cross spectra, Eqns. (3), (5) and(6) which characterize the statistical properties of the data [ δ Ω GW ( e ) , ∆ G ( e )] . Measuring these spectra can give us awealth of information about the underlying processes that leadto the generation of gravitational waves in the late Universe,see [9, 16]. III. SPATIAL AND POP CORN SHOT NOISE
In this section we derive how shot noise arises and how itaffects auto and cross-correlations. We distinguish betweenthe shot noise arising from the discreteness of gravitationalwave sources in space and due to their Poisson nature in thetime domain. This will be useful in our estimate of the SNRin the next section. We note that, while in the previous sec-tion, we have presented auto and cross-correlations in termsof angular power spectra, our discussion here will be in termsof real-space correlations.
A. The shot noise between two Poisson processes
Consider two discrete sets of points, a and b . In a givenpixel p there are N a , bp points of each type, of which N cp arecommon to both sets. We will write the ensemble averageof each quantity as (cid:104) N xp (cid:105) ≡ N x . In a given pixel, let us write N xp = N cp + N x − cp . Assuming Poisson statistics, the first twomoments of the distribution are: (cid:104) N xp (cid:105) = N x , (cid:104) ( N xp ) (cid:105) − (cid:104) N xp (cid:105) = N x . (7)The covariance between a and b is therefore: (cid:104) N ap N bp (cid:105) − (cid:104) N ap (cid:105)(cid:104) N bp (cid:105) = (cid:104) ( N cp ) + N cp N a − cp + N cp N b − cp + N a − cp N b − cp (cid:105) − N a N b ≡ ( N c ) + N c + N c N a − c + N c N b − c + N a − c N b − c − ( N c + N a − c )( N c + N b − c )= N c , (8)where in the second line we have used the fact that N c , N a − c and N b − c are all uncorrelated. We thus see that thecross-variance of two Poisson samples is equal to the num-ber of events in the intersections of the two samples, i.e.Cov( N ap , N bp )= N c . We will use this result in the next sections. B. AGWB-galaxy count cross-correlations and shot noise
The gravitational wave density fluctuation δ Ω GW , p in apixel p is given by the cumulative flux of all gravitationalwave sources along the line of sight p . Let us discretize thisline of sight into intervals of comoving distance r . Discretiz-ing also the range of GW luminosities, L GW , and ignoringmetric perturbations and peculiar velocities, we can write Ω GW , p = f ρ c θ p ∑ r ∑ L GW N L GW r , p L GW π ( + z ) r , (9)where N L GW r , p is the number of sources in pixel p , in the radialbin r and in the luminosity bin L GW , and where θ p is thearea of the pixel. In the continuum limit, taking the ensembleaverage, and writing (cid:68) N L GW r , p (cid:69) = a r θ p d r d L GW ¯ n G ( L GW , r ) , (10)we find (cid:10) Ω GW , p (cid:11) = (cid:90) d r (cid:90) d L GW f a L GW π ρ c ¯ n G ( L GW , r ) , (11)to recover the integral over the radial coordinate of Eq. (1).On the other hand, the weighted galaxy number per solidangle along pixel p , ∆ G p , is simply given by1 + ∆ G p = ∑ r W ( r ) θ p r a ¯ n G ( r ) N G r , p , (12)where N G r , p is the number of galaxies in pixel ( r , p ) ,whose av-erage is (cid:10) N G r , p (cid:11) = a r θ p d r ¯ n G ( r ) , (13)and by construction we have (cid:104) + ∆ G p (cid:105) = ∑ r W ( r ) d r = N G r , p and N L GW r , p ,we can now compute the variance of the different auto- andcross-correlations [17]. AGWB auto-correlation: (cid:10) Ω GW , p Ω GW , p (cid:48) (cid:11) − (cid:10) Ω GW , p (cid:11) (cid:10) Ω GW , p (cid:48) (cid:11) = δ pp (cid:48) ∑ L GW ∑ r (cid:32) f L GW π ( + z ) r ρ c θ p (cid:33) (cid:68) N L GW r , p (cid:69) = δ pp (cid:48) θ p (cid:90) d rr a ¯ n G (cid:0) ∂ r ¯ Ω GW (cid:1) , (14)where, in the last line, we have taken the continuum limit andwe have assumed that all galaxies have the same GW lumi-nosity, i.e. ¯ n G ( L GW , r ) = δ ( L GW − L ) ¯ n G ( r ) . Number counts auto-correlation: (cid:10) ∆ p ∆ p (cid:48) (cid:11) − (cid:10) ∆ p (cid:11) (cid:10) ∆ p (cid:48) (cid:11) = δ pp (cid:48) ∑ r (cid:34) W ( r ) θ p r a ¯ n G ( r ) (cid:35) (cid:10) N G r , p (cid:11) = δ pp (cid:48) θ p (cid:90) d rr a ¯ n G W ( r ) (15) AGWB - number counts cross-correlation: (cid:10) Ω GW , p ∆ p (cid:48) (cid:11) − (cid:10) Ω GW , p (cid:11) (cid:10) ∆ p (cid:48) (cid:11) = δ pp (cid:48) θ p (cid:90) d rr a ¯ n G W ( r ) ∂ r ¯ Ω GW , (16)where we have assumed a monochromatic GW luminos-ity function, and that all galaxies emit GWs (and thereforeCov ( N L GW p , r , N G p , r ) = (cid:104) N L GW r , p (cid:105) as shown in the previous sub-section III A).We observe that the integral in Eq. (14) diverges at thelower limit, when r =
0, hence the contribution of Poissonnoise of the AGWB auto-correlation depends on the cut-offused to regularize it. The reason for this divergence is that,for fixed luminosities, the flux of nearby sources increaseslike ∼ r − , and therefore the very few closest sources endup dominating the total GW intensity across the sky. Froman observational point of view, the physical quantity whichsets the cut-off is the observed flux: sources with a fluxabove a given threshold can be resolved and filtered out of thedata. Given that the flux per unit frequency Φ from a sourcein z is related to the luminosity per unit frequency through Φ ( f ) = ( + z ) L GW / ( π d L ) , we have that a lower bound on Φ is translated into a lower bound in redshift and an upperbound in luminosity . Assuming that all galaxies have thesame associated luminosity, the cut-off on flux directly trans-lates into a lower cut-off in redshift (or analogously in r ). C. Pop-corn shot noise
So far we have only considered the effect of the spatial dis-creteness of the sources of gravitational waves. In the fre-quency band of terrestrial interferometers, e.g. the LIGO-Virgo frequency band, the dominant contribution to the back-ground comes from the merging phase of the evolution ofsolar-mass compact objects. The signal is “pop-corn”-like:events are separated in time and with almost no temporal over-lap. Thus, there is a second shot-noise component due to thefact that events are discrete in time, and only some of themwill contribute to the GW intensity mapped in a given timeperiod. In this paper we focus on the contribution to the back-ground coming from mergers of binary black hole systems.To compute this pop-corn shot noise we need to use thefact that the number of galaxies is a Poisson variable andeach galaxy has a given (small) probability β T of containinga merger during the observation time T , with a Poisson distri-bution. We then use properties of compound Poisson distribu-tions, see e.g. Ref. [8]. The only difference brought by thispop-corn noise on the results of III B is that the variance ofthe AGWB auto-correlation gets a correction prefactor of theform ( + / β T ) , but the variance of cross-correlation (and More precisely, it defines the region of integration in the plane ( z , L GW ) . galaxy auto-correlation) remain the same .The value of β T can be estimated β T = Ta ¯ n G × d N d t d V , (17)where d N / d V / d t is the merger rate per units of observedtime and volume. Consider the upper bound for the mergerrate [18] d N d t d V < d N d t m d V ∼
100 Gpc − yr − , (18)where t m is the comoving time of the source, i.e. t = ( + z ) t m > t m . Then using a constant comoving galaxy density a ¯ n G ∼ . − , we find β T / T < − / yr. It follows thatin the Hz band the shot noise of the AGWB auto-correlation(dominated by pop corn shot noise) is enhanced typically bya factor 10 with respect to the shot noise in the mHz band(which is purely of spatial type). On the other hand, theshot-noise level of the cross-correlation stays the same overthe whole frequency range and no enhancement due to thestochasticity in time of sources is present. This highlights thepower of cross-correlation. We stress that this is just an orderof magnitude estimate. To derive more accurate predictionsfor SNR in the next section, we will need to keep track of allthe redshift factors in Eq. (17). IV. RESULTS
Before we embark on assessing the impact of cross-correlations, we note that the weight function, W ( r ) , shouldbe chosen so as to maximize the SNR of cross-correlation.This can be done as long as radial information (i.e. accurateredshifts) are available for all galaxies in the survey we cross-correlate with, which we will assume here. As detailed inAppendix A, the optimal weights can be derived in terms of aWiener filter, finding the result W ( r ) ≡ π∂ r ¯ Ω GW ¯ Ω GW . (19)Physically this means that we approximately weight all galax-ies by a 1 / r factor, hence mimicking the properties of a back-ground mapped in intensity. In principle this means that the Let us denote N GW = ∑ Ni y i the total number of GW events in a pixel. N is the number of galaxies in that same pixel which follows a Poisson distri-bution of average (cid:104) N (cid:105) , and the y i also follow a Poisson statistics of average (cid:104) y i (cid:105) = β T due to the pop-corn nature of GW events. The compound statis-tics is found by averaging first over the statistics of the y i at fixed N and thenover the statistics of N . One finds easily Cov ( N GW , N GW ) = (cid:104) N (cid:105) ( β T + β T ) and Cov ( N GW , N ) = (cid:104) N (cid:105) β T . To compute the modification brought by thepop-corn nature (due to small values of β T ) we must form the ratio of theseexpressions with their asymptotic behaviour when β T → ∞ . Hence we find forthe auto-correlation of N GW a modification factor ( β T + β T ) / β T = + β − T ,whereas the modification factor for cross-correlation is trivially β T / β T = In [19] it is also found that the inferred merger rate is consistent (at the 68%confidence level) with being uniform in a comoving volume and source frametime. spatial shot noises of auto and cross-correlations (Eqs. (14-16)) have exactly the same expressions (up to normalisationfactors 4 π / ¯ Ω GW ). In detail this is not exactly the case sincethe full expressions for galaxy numbers and for the GW back-ground also involve sub dominant metric and velocity contri-butions, as well as the dominant galaxy overdensity term ineqs. (4-6). See [9, 11–13, 20] for details. This implies thatthe optimal weight function found from the Wiener filter mustdiffer slightly from (19).We can now estimate the SNR of the cross-correlation inthe Hz (LIGO-Virgo) frequency band. We assume that shotnoise is the only noise component, i.e. we assume an ideal ex-periment with no instrumental noise. The SNR of the AGWBauto-correlation is given by (cid:18) SN (cid:19) = ∑ (cid:96) ( (cid:96) + ) (cid:18) C GW (cid:96) C GW (cid:96) + N GW (cid:96) (cid:19) , (20)while the one of the cross-correlation is given by (cid:18) SN (cid:19) , ∆ = ∑ (cid:96) ( (cid:96) + )( C GW , ∆ (cid:96) ) ( C GW , ∆ (cid:96) + N GW , ∆ (cid:96) ) + ( C GW (cid:96) + N GW (cid:96) )( C ∆ (cid:96) + N ∆ (cid:96) ) . (21)The noise power spectrum N (cid:96) is in fact given by the constantsmultiplying δ pp (cid:48) / θ p in Eqs. (14-16), as found from the dis-crete to continuous rule δ pp (cid:48) / θ p → δ ( e − e (cid:48) ) .In both (20) and (21) the dominant contribution to thedenominator comes from the variance of AGWB auto-correlation N GW (cid:96) due to the large pop-corn shot noise. Hencethe SNR of cross-correlation will be typically enhanced withrespect to the AGWB auto-correlation one. Having chosenthe optimal weight (19), it is very easy to obtain analyticapproximations. We first use that C ∆ (cid:96) = ( π / ¯ Ω GW ) C GW , ∆ (cid:96) =( π / ¯ Ω GW ) C GW (cid:96) (this is only approximate when includingthe subdominant metric contributions). Furthermore, we alsofind that all spatial shot noises are similarly related by fac-tors ( π / ¯ Ω GW ) . Including the pop corn shot noise in GWauto-correlations, we then have N ∆ (cid:96) = ( π / ¯ Ω GW ) N GW , ∆ (cid:96) =( π / ¯ Ω GW ) N GW (cid:96) / ( + β − T ) . For (cid:96) (cid:29) C (cid:96) scale roughlyas 1 / ( (cid:96) + / ) , as a consequence of the Limber expressions(3)-(6) with large kernels. Using that β T (cid:28)
1, the cumulativeSNR of the auto-correlation scales as (cid:18) SN (cid:19) GW ( (cid:96) max ) ∼ β T α cut (cid:112) ln (cid:96) max , (22) The optimal full sky C (cid:96) estimator for two observables x and y is ˆ C xy (cid:96) = ∑ m a x (cid:63)(cid:96) m a y (cid:96) m / ( (cid:96) + ) . Its variance is easily deduced from the assumed Gaus-sianity of the a x , y (cid:96) m , and it allows to deduce the SNR from a Fisher matrixanalysis. The prefactor 2 ( (cid:96) + ) for the auto-correlation SNR (instead of theusual cosmic variance ( (cid:96) + ) /
2) is due to the fact that the signal, which isthe amplitude of the GW background, appears quadratically in the observ-ables (the C (cid:96) ).
10 10010 - - ℓ max S / N AutoCross z < < - -
110 Cut - off ( Mpc ) S / N ℓ≤
100 Auto ℓ≤
10 Auto ℓ≤
100 Cross ℓ≤
10 Cross - - - - - - - - a n G ( Mpc - ) S / N ℓ≤
100 Auto ℓ≤
10 Auto ℓ≤
100 Cross ℓ≤
10 Cross - - - - - - β T S / N ℓ≤
100 Auto ℓ≤
10 Auto ℓ≤
100 Cross ℓ≤
10 Cross
FIG. 1.
Top left : cumulative SNR for AGWB auto-correlation (black dashed line) and cross-correlation (black cont. lines) with galaxy numbercounts using the optimal galaxy weight (and its restrictions to redshift bins in colors).
Top right : dependence on the cut-off for two differentmaximum multipoles ( (cid:96) max =
10 in thick lines and (cid:96) max =
100 in thin lines). Auto-correlations are in dashed lines, and cross-correlations withthe optimal galaxy weight are in continuous lines.
Bottom left : same curves varying instead the galaxy number density a ¯ n G . Bottom right: same curves varying instead the pop corn enhancement factor β T . When not varied, the cut-off distance is 60 Mpc, the comoving galaxydensity is 0 . − , and the pop corn enhancement factor is β T = − . where we defined the cut-off dependent quantity α cut ≡ ( (cid:96) + / ) C ∆ (cid:96) = / N ∆ (cid:96) , which is approximately constant for low (cid:96) .A quick estimate for this coefficient is α cut (cid:39) a ¯ n G r cut × (cid:82) P G ( k ) d k , which is independent of the details of ∂ r ¯ Ω GW . Forthe SNR of the cross-correlation, one has (cid:18) SN (cid:19) GW , ∆ ( (cid:96) max ) ∼ (cid:112) β T α cut (cid:96) max , (23)where we used the scalings Eqs. (5) and (6). For an orderof magnitude estimate, let us consider a cut-off at 60 Mpcfor which α cut ∼ × . Then assuming integration time ofone year and using for the value of β T = its upper boundfound in Sec. III C, we have ( S / N ) GW ( (cid:96) max ) ∼ . √ ln (cid:96) max and ( S / N ) GW , ∆ ( (cid:96) max ) ∼ √ . (cid:96) max . The SNR up to a given (cid:96) max , when using either the auto-correlation or the cross-correlations, is presented in the top left panel of Fig. 1. Weobserve that the behaviour with (cid:96) max is well described by theanalytical scalings we have found. Note that our analysis dif-fers significantly from Ref. [17], where the constraints derivedon posterior distributions are only cosmic variance limited.The dependence on the cut-off distance used when com-puting the GW pop corn shot noise is also illustrated in the top right panel. For these plots we have integrated the sig-nal over the frequency range 10Hz < f < N (cid:96) are inde-pendent of (cid:96) . This is why the turning point for the total SNRup to (cid:96) max moves toward higher values cut-off distances as weincrease (cid:96) max .In the bottom left panel we have shown the effect of re-ducing the number density of galaxies, and it is clear thatthe SNR is rather insensitive to its precise value as long as a ¯ n G > − , which is comparable with current spectroscopicsurveys.Finally, the dependence on the enhancement factor β T isillustrated in the bottom right panel. In this work we havestudied only the contribution to the background coming frommergers of black holes in the Hz band. Another importantbackground component in this band is given by merger ofbinary neutron star systems, see e.g. [9]. The merger rateof neutron stars is expected to be much higher (a factor 10)than the one of black holes [21], the current upper limit ford N / d t / d V being 2810 Gpc − yr − . Hence the pop corn shotnoise will affect in a less severe way this background com-ponent as we expect β T to be typically larger by an order ofmagnitude.We find that most of the signal of the auto-correlationcomes from low redshift ( z < . V. CONCLUSION
The shot noise due the pop corn nature of GW sources in theHz band is not a fundamental limitation that prevents one fromgetting information about the GW background anisotropies.Restricting to the background component coming from merg-ers of binary black hole systems, we have considered that shotnoise and cosmic variance are the only noise components. Inthat idealized case, the SNR of the cross-correlation with agalaxy catalog is found to be much higher than that of theauto-correlation. The SNR of the cross-correlation is of order ∼
10 for large ( (cid:38) (cid:96) max for realistic galaxy number den-sities ( a ¯ n G ∼ − Mpc − ). Moreover, we have shown thatmost of the signal comes from low redshift z < .
5, indicatingthat present galaxy catalogs can already be used to constructthese cross-correlations.While this analysis has shown that there is some promisein this method, it is useful to take a more conservative viewof its feasibility with up and coming data, and what we maylearn from such an observation. Currently, it is envisaged that all events out to approximately 1 Gpc will be resolved (of or-der 10 in total). If we take this to be the effective cutoffwe see that the SNR can be appreciable if we are able to re-solve the map down to approximately 1 ◦ , i.e. (cid:96) max ∼ z ∼ . β T by one order of magnitude (i.e. β T (cid:39) − for oneyear of observation). Also, β T ∝ T , and therefore the SNRwill keep on improving as more data is collected. The rateof improvement will ∝ √ T and ∝ T for the cross-correlationand auto-correlation respectively. For very long total observa-tion time (such that β T reaches 10 − ), the significance of bothobservables becomes comparable.It is worth noting that, although it is always possible toavoid the offset in the AGWB power spectrum caused by shotnoise, commonly called the “noise bias”, by using only cross-correlation between different data splits [26] (a technique thatis extensively used in the CMB community to avoid compli-cated instrumental noise biases), this will not mitigate in anyway the impact of shot noise on the variance of the estimatedpower spectrum (which will be the dominant contribution).Cross-correlating with a denser sample that traces the sameunderlying structure, on the other hand, does lead to a signif-icant mitigating factor (see Eqs. 20 and 21). This has beenused in large-scale structure surveys to study the clustering ofsparse samples, such as damped Lyman- α systems [27, 28].Most importantly, we must emphasize the fact that our anal-ysis has not accounted for any form of instrumental noise. Fora realistic instrument, it is in principle not clear what the beststrategy would be to carry out this cross correlation. One pos-sibilities would be to use only resolved events. In this casethe signal would be dominated by radiometer noise, and thedetection would be limited by the small number of resolvableevents. The second possibility would be to search for a back-ground of unresolved events. In this case, the detection islikely to be limited by detector noise and the poor angular res-olution of Earth-based facilities. This translates into an loweffective (cid:96) max , and from the top left panel of Figure 1 we inferthat the associated SNR would probably remain below unity.LIGO-Virgo is expected to detect the isotropic componentof the background with the design sensitivity [7]. Giventhat the typical amplitude of the AGWB anisotropies aresuppressed by a factor of at least 10 − with respect to themonopole, this means that an improvement in design sensi-tivity of at least a factor 10 is necessary to get a detection(on the angular scales accessible given the diffraction patternof the observatories). Einstein Telescope is expected to reachthis sensitivity threshold [29]. However, an improved sensi-tivity also implies that the catalogue of resolvable sources onecan detect becomes much more complete and deep in redshiftand one expects to have a broad redshift cover (up to z ∼ − intrinsic (irreducible)background. The shot noise in the LISA band will thereforeonly be due to the discreteness in space of the GW sources,and will be a subdominant contribution to the total error budget (see also [16]).In summary, we have found that there are no intrinsic (i.e.shot-noise-like) noise components that constitute a funda-mental barrier to obtaining information from the anisotropiesin the AGWB in the Hz band, and that cross-correlating witha galaxy survey traceing the same underlying structures is apromising method to get a first detection of the anisotropies.This result holds in idealized case without instrumental noise,and a future work will be dedicated to applying this analysisto realistic GW detector networks and galaxy surveys. Acknowledgements —
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Appendix A: Optimal weights
Consider a vector of N measurements x = ( x , ..., x N ) thatwe want to linearly combine to find the best estimate of agiven quantity y . Assuming Gaussian statistics, the probabil-ity for y given x is − p ( y | x ) = z T C − zz z − x T C − xx x , (A1)where we have defined the vector z = ( y , x , ..., x N ) , and C ab isthe covariance matrix between a and b ( C ab ≡ (cid:10) ab T (cid:11) ). There-fore C zz is C zz = (cid:18) C yy C Txy C xy C xx (cid:19) . (A2)The maximum-likelihood estimator for y can be found bysolving the equation − ∂ y log p ( y | x ) =
0. After a little alge-bra, this estimator isˆ y = w T x ≡ C Txy C − xx x . (A3)The linear coefficients w are the so-called Wiener filter . In our case, y is the gravitational wave background in agiven pixel Ω GW , p , and x is a vector of number count measure-ments along that pixel’s line of sight N r , p . Assuming Poissonerror bars, the different covariance elements are C rr (cid:48) ≡ Cov ( N r , p , N r (cid:48) , p ) = δ rr (cid:48) d r r a ¯ n G ( r ) θ p , (A4) C r Ω ≡ Cov ( N r , p , Ω GW , p ) = d r ∂ r ¯ Ω GW . (A5)Therefore the Wiener filter weights of the sum (12) are W ( r ) r a ¯ n G ( r ) ∝ ∂ r ¯ Ω GW r a ¯ n G ( r ) . (A6)Using these weights allows to build the most likely GW back-ground in a given pixel only from its galaxy number mea-surement binned in redshifts. Hence it is the good variableto use for cross-correlating with the directly measured GWbackground of that pixel so as to constrain its global ampli-tude. Once properly normalized, the weights (A6) lead to theweight function (19).Equivalently, the weights can be obtained from a least-squares analysis of the variables N r , p Ω GW , p . This al-lows to build an estimator for the GW amplitude A p = π Ω GW , p / ¯ Ω GW asˆ A p ≡ ∑ r (cid:0) ∑ r (cid:48) C r (cid:48) Ω C − r (cid:48) r (cid:1) N r , p Ω GW , p ∑ rr (cid:48) C r (cid:48) Ω C − r (cid:48) r C r Ω , (A7)where C rr (cid:48) ≡ Cov ( N r , p Ω GW , p , N r (cid:48) , p Ω GW , p ) = C rr (cid:48) C ΩΩ + C r Ω C r (cid:48) Ω . Assuming that this covariance is dominated by thepop-corn noise of the GW background, we can then approxi-mate C rr (cid:48) (cid:39) C rr (cid:48) C ΩΩ , from which we infer again that the opti-mal weights are ∝ ∑ rr
0. After a little alge-bra, this estimator isˆ y = w T x ≡ C Txy C − xx x . (A3)The linear coefficients w are the so-called Wiener filter . In our case, y is the gravitational wave background in agiven pixel Ω GW , p , and x is a vector of number count measure-ments along that pixel’s line of sight N r , p . Assuming Poissonerror bars, the different covariance elements are C rr (cid:48) ≡ Cov ( N r , p , N r (cid:48) , p ) = δ rr (cid:48) d r r a ¯ n G ( r ) θ p , (A4) C r Ω ≡ Cov ( N r , p , Ω GW , p ) = d r ∂ r ¯ Ω GW . (A5)Therefore the Wiener filter weights of the sum (12) are W ( r ) r a ¯ n G ( r ) ∝ ∂ r ¯ Ω GW r a ¯ n G ( r ) . (A6)Using these weights allows to build the most likely GW back-ground in a given pixel only from its galaxy number mea-surement binned in redshifts. Hence it is the good variableto use for cross-correlating with the directly measured GWbackground of that pixel so as to constrain its global ampli-tude. Once properly normalized, the weights (A6) lead to theweight function (19).Equivalently, the weights can be obtained from a least-squares analysis of the variables N r , p Ω GW , p . This al-lows to build an estimator for the GW amplitude A p = π Ω GW , p / ¯ Ω GW asˆ A p ≡ ∑ r (cid:0) ∑ r (cid:48) C r (cid:48) Ω C − r (cid:48) r (cid:1) N r , p Ω GW , p ∑ rr (cid:48) C r (cid:48) Ω C − r (cid:48) r C r Ω , (A7)where C rr (cid:48) ≡ Cov ( N r , p Ω GW , p , N r (cid:48) , p Ω GW , p ) = C rr (cid:48) C ΩΩ + C r Ω C r (cid:48) Ω . Assuming that this covariance is dominated by thepop-corn noise of the GW background, we can then approxi-mate C rr (cid:48) (cid:39) C rr (cid:48) C ΩΩ , from which we infer again that the opti-mal weights are ∝ ∑ rr (cid:48) C rr
0. After a little alge-bra, this estimator isˆ y = w T x ≡ C Txy C − xx x . (A3)The linear coefficients w are the so-called Wiener filter . In our case, y is the gravitational wave background in agiven pixel Ω GW , p , and x is a vector of number count measure-ments along that pixel’s line of sight N r , p . Assuming Poissonerror bars, the different covariance elements are C rr (cid:48) ≡ Cov ( N r , p , N r (cid:48) , p ) = δ rr (cid:48) d r r a ¯ n G ( r ) θ p , (A4) C r Ω ≡ Cov ( N r , p , Ω GW , p ) = d r ∂ r ¯ Ω GW . (A5)Therefore the Wiener filter weights of the sum (12) are W ( r ) r a ¯ n G ( r ) ∝ ∂ r ¯ Ω GW r a ¯ n G ( r ) . (A6)Using these weights allows to build the most likely GW back-ground in a given pixel only from its galaxy number mea-surement binned in redshifts. Hence it is the good variableto use for cross-correlating with the directly measured GWbackground of that pixel so as to constrain its global ampli-tude. Once properly normalized, the weights (A6) lead to theweight function (19).Equivalently, the weights can be obtained from a least-squares analysis of the variables N r , p Ω GW , p . This al-lows to build an estimator for the GW amplitude A p = π Ω GW , p / ¯ Ω GW asˆ A p ≡ ∑ r (cid:0) ∑ r (cid:48) C r (cid:48) Ω C − r (cid:48) r (cid:1) N r , p Ω GW , p ∑ rr (cid:48) C r (cid:48) Ω C − r (cid:48) r C r Ω , (A7)where C rr (cid:48) ≡ Cov ( N r , p Ω GW , p , N r (cid:48) , p Ω GW , p ) = C rr (cid:48) C ΩΩ + C r Ω C r (cid:48) Ω . Assuming that this covariance is dominated by thepop-corn noise of the GW background, we can then approxi-mate C rr (cid:48) (cid:39) C rr (cid:48) C ΩΩ , from which we infer again that the opti-mal weights are ∝ ∑ rr (cid:48) C rr (cid:48) Ω C − rr
0. After a little alge-bra, this estimator isˆ y = w T x ≡ C Txy C − xx x . (A3)The linear coefficients w are the so-called Wiener filter . In our case, y is the gravitational wave background in agiven pixel Ω GW , p , and x is a vector of number count measure-ments along that pixel’s line of sight N r , p . Assuming Poissonerror bars, the different covariance elements are C rr (cid:48) ≡ Cov ( N r , p , N r (cid:48) , p ) = δ rr (cid:48) d r r a ¯ n G ( r ) θ p , (A4) C r Ω ≡ Cov ( N r , p , Ω GW , p ) = d r ∂ r ¯ Ω GW . (A5)Therefore the Wiener filter weights of the sum (12) are W ( r ) r a ¯ n G ( r ) ∝ ∂ r ¯ Ω GW r a ¯ n G ( r ) . (A6)Using these weights allows to build the most likely GW back-ground in a given pixel only from its galaxy number mea-surement binned in redshifts. Hence it is the good variableto use for cross-correlating with the directly measured GWbackground of that pixel so as to constrain its global ampli-tude. Once properly normalized, the weights (A6) lead to theweight function (19).Equivalently, the weights can be obtained from a least-squares analysis of the variables N r , p Ω GW , p . This al-lows to build an estimator for the GW amplitude A p = π Ω GW , p / ¯ Ω GW asˆ A p ≡ ∑ r (cid:0) ∑ r (cid:48) C r (cid:48) Ω C − r (cid:48) r (cid:1) N r , p Ω GW , p ∑ rr (cid:48) C r (cid:48) Ω C − r (cid:48) r C r Ω , (A7)where C rr (cid:48) ≡ Cov ( N r , p Ω GW , p , N r (cid:48) , p Ω GW , p ) = C rr (cid:48) C ΩΩ + C r Ω C r (cid:48) Ω . Assuming that this covariance is dominated by thepop-corn noise of the GW background, we can then approxi-mate C rr (cid:48) (cid:39) C rr (cid:48) C ΩΩ , from which we infer again that the opti-mal weights are ∝ ∑ rr (cid:48) C rr (cid:48) Ω C − rr (cid:48) rr