Low latency search for Gravitational waves from BH-NS binaries in coincidence with Short Gamma Ray Bursts
LLow latency search for Gravitational waves from BH-NS binaries in coincidence with Short GammaRay Bursts
Andrea Maselli and Valeria Ferrari
Dipartimento di Fisica, Universit`a di Roma “La Sapienza” & Sezione, INFN Roma1, P.A. Moro 5, 00185, Roma, Italy. ∗ (Dated: October 18, 2018)We propose a procedure to be used in the search for gravitational waves from black hole-neutron star coalesc-ing binaries, in coincidence with short gamma-ray bursts. It is based on two recently proposed semi-analyticfits, one reproducing the mass of the remnant disk surrounding the black hole which forms after the merging asa function of some binary parameters, the second relating the neutron star compactness, i.e. the ratio of massand radius, with its tidal deformability. Using a Fisher matrix analysis and the two fits, we assign a probabilitythat the emitted gravitational signal is associated to the formation of an accreting disk massive enough to supplythe energy needed to power a short gamma ray burst. This information can be used in low-latency data analysisto restrict the parameter space searching for gravitational wave signals in coincidence with short gamma-raybursts, and to gain information on the dynamics of the coalescing system and on the internal structure of thecomponents. In addition, when the binary parameters will be measured with high accuracy, it will be possibleto use this information to trigger the search for o ff -axis gamma-ray bursts afterglows. Keywords: gravitational waves, neutron stars, short gamma-ray bursts
Introduction. — The advanced gravitational wave detectorsLIGO and Virgo (to hereafter AdvLIGO / Virgo) are expectedto detect signals emitted by coalescencing compact binaries,formed by neutron stars (NS) and / or black holes (BH) [1].These catastrophic events have an electromagnetic counter-part. For instance, the coalescence of NS-NS and BH-NS bi-naries has been proposed as a candidate for the central engineof short Gamma Ray Bursts (SGRB), provided the stellar-mass BH which forms after merging is surrounded by a hotand su ffi ciently massive accreting disk, but this model needsto be validated (see for instance [2] and references therein).Since the electromagnetic emission is produced at large dis-tance from the central engine, it does not give strong infor-mation on the source. In addition, the emission is beamed,and consequently these events may not be detected if one islooking in the wrong direction. Conversely, the gravitationalwave (GW) emission is not beamed, and exhibits a character-istic waveform (the chirp) which should allow a non ambigu-ous identification of the source. GRBs are characterized by aprompt emission, which lasts a few seconds, and an afterglow,whose duration ranges from hours to days.Thus, gravitational wave detection may be used to trig-ger the afterglow search of GRBs which have not been de-tected by the on-axis prompt observation, and to validate the“jet-model” of SGRB. Or, in alternative, the observation of aSGRB may be used as a trigger to search for a coincident GWsignal. Indeed, this kind of search has already been done inthe data of LIGO and Virgo [3, 4].Since not all coalescences of compact bodies produce ablack hole with an accreting disk su ffi ciently massive to powera SGRB, we need to devise a strategy to extract those havingthe largest probability to produce a SGRB. This is one of thepurposes of this paper. In a recent paper of the LIGO-Virgocollaboration [5] a plausible observing schedule has been in-dicated, according to which within this decade the advanced ∗ [email protected] detectors, operating under appropriate conditions, will be ableto determine the sky location of a source within 5 and 20 deg .Given the cost of spanning this quite large region of sky tosearch for a coincident SGRB with electromagnetic detectors,indications on whether a detected signal is likely to be associ-ated with a SGRB is a valuable information.The procedure we propose has several applications. It canbe used in the data analysis of future detectors i) to gain in-formation on the range of parameters which is more useful tospan in the low latency search for GWs emitted by BH-NSsources [6], ii) for an externally triggered search for GW co-alescence signals, following GRB observations [3, 4], and iii)when the binary parameters will be measured with su ffi cientaccuracy and in a su ffi ciently short time to allow for an elec-tromagnetic follow-up, to search for o ff -axis GRB afterglows.Although our method is devised for BH-NS coalescing bina-ries, it will also be applicable to NS-NS binaries, when a reli-able and suitable fit for the mass of the accretion disk whichforms around the black hole produced in the coalescence willbe provided by numerical studies of such systems (see below).A large number of numerical studies of BH-NS coales-cence, have allowed to derive two interesting fits. The first[7] gives the mass of the accretion disk, M rem , as a functionof the the NS compactness C = M NS / R NS , where M NS and R NS are the NS mass and radius, the dimensionless BH spin, χ BH ∈ [ − , q = M BH / M NS : M rem M b NS = K (3 q ) / (1 − C ) − K q C R ISCO . (1)Here M bNS is the NS baryonic mass which, following [8], weassume to be 10% larger than the NS gravitational mass; R ISCO is the radius of the innermost, stable circular orbit for a Kerrblack hole: R ISCO M BH = + Z − sign( χ BH ) (cid:112) (3 − Z )(3 + Z + Z ) , (2)where Z = + (1 − χ ) / (cid:104) (1 + χ BH ) / + (1 − χ BH ) / (cid:105) and Z = (3 χ BH + Z ) / [9]. The two coe ffi cients K = . ± a r X i v : . [ g r- q c ] F e b .
011 and K = . ± .
007 have been derived [7] througha least-square fit of the results of fully relativistic numericalsimulations [10–13]. M rem is a key parameter in our study. Neutrino-antineutrinoannihilation processes extract energy from the disk [14], andseveral studies have shown that this process could supply theenergy required to ignite a short gamma-ray burst, if M rem ∈ (0 . ÷ . M (cid:12) [15–17]. In the following we shall assume asa threshold for SGRB formation M rem = . M (cid:12) . The resultswe will show do not change if we choose M rem = . M (cid:12) .The second fit [18] is a universal relation between the NScompactness C and the tidal deformability λ = − Q i j / C i j ,where Q i j is the star traceless quadrupole tensor, and C i j = e α (0) e β ( i ) e γ (0) e δ ( j ) R αβγδ is the tidal tensor, i.e. the Riemann tensorprojected onto the parallel transported tetrad attached to thestar e α ( µ ) : C = . − . × − ln ¯ λ + . × − (ln ¯ λ ) , (3)where ¯ λ = λ / M . This fit is found to reproduce the valuesof the star compactness with accuracy greater 3%, for a largeclass of equations of state (EoS). Hereafter, we shall denoteby C λ the NS compactness obtained from this fit.Let us now assume that the gravitational wave signal emit-ted in a BH-NS coalescence is detected; a suitable data anal-ysis allows to find the values of the symmetric mass-ratio ν = ( M NS M BH ) / ( M NS + M BH ) and of the chirp mass M = ν / ( M NS + M BH ), from which the mass ratio q can be derived,and of the black hole spin χ BH , with the corresponding er-rors. Knowing q ± σ q and χ BH ± σ χ BH , using the fit (1) wecan trace the plot of Fig. 1 in the q − C plane, for an assigneddisk mass threshold, say M rem = . M (cid:12) . This plot allows toidentify the parameter region where a SGRB may occur, i.e.the region M rem (cid:38) . M (cid:12) (below the fit curve in the figure),and the forbidden region above the fit ( M rem (cid:46) . M (cid:12) ). Inaddition, we identify four points X , . . . X , which are the in-tersection between the contour lines for χ BH ± σ χ BH and thehorizontal lines q ± σ q . Let us indicate as C , . . . , C the cor-responding values of the neutron star compactness. Since thefit (1) is monotonically decreasing, C < C < C < C . Atthis stage we still cannot say whether the detected binary fallsin the region allowed for the formation of a SGRB or not.In order to get this information, we need to evaluate C . Asdiscussed in [19–24], Advanced LIGO / Virgo are expected tomeasure the gravitational wave phase with an accuracy su ffi -cient to estimate the NS tidal deformability λ . Thus, usingthe fit (3), the NS compactness C λ and the corresponding un-certainty σ C λ = σ + (cid:88) i , j ∂ C λ ∂ p i ∂ C λ ∂ p j Cov( p i , p j ) (4)can be inferred. In (4) p i = { λ , M NS } , and Cov( p i , p j ) is theircovariance. As shown in [18], σ fit = . C is the largest rela-tive discrepancy between the value of C obtained from the fitand the value computed solving the equations of stellar per-turbations, for a set of EoS covering a large range of sti ff ness. q + q q q B H B H B H + B H X X X X M rem . . M M rem & . M C q FIG. 1. Contour plot of the fit (1) in the q - C plane, for M NS = . M (cid:12) , χ BH = . M rem = . M (cid:12) . The fit separates the region allowedfor SGRB ignition (below the fit curve) from the forbidden region(above the fit). Given the measured values of q ± σ q and χ BH ± σ χ BH ,a detected signal can correspond to a NS with compactness C whichfalls in the blue, green or yellow region. Since C also comes with anerror σ C , in order to infer if it can be associated with a SGRB, weneed to evaluate the probability P( C ≤ C ) and P( C ≤ C ) (see text). Knowing the parameters and their uncertainties, the prob-ability that a SGRB is associated to the detected coalescencecan now be evaluated.We assume that ( q , C λ , χ BH ) are described by a multivariateGaussian distribution: P ( q , C λ , χ BH ) = π ) / | Σ | / exp (cid:34) − ∆ T Σ − ∆ (cid:35) , (5)where ∆ = ( (cid:126) x − (cid:126)µ ), (cid:126)µ = ( q , C λ , χ BH ), Σ is the covariancematrix. Then, we define the maximum and minimum proba-bility that the binary coalescence produces an accretion diskwith mass over the threshold, ¯ M rem , asP MAX ( M rem (cid:38) ¯ M rem ) ≡ P( C λ ≤ C ) , (6)P MIN ( M rem (cid:38) ¯ M rem ) ≡ P( C λ ≤ C ) , where P( C λ ≤ C i ) is the cumulative distribution of Eq. (5),which gives the probability that the measured compactness C λ , estimated through the fit (3), is smaller than an assignedvalue C i .As an illustrative example, we now evaluate the probabil-ity that a given BH-NS coalescing binary produces a SGRB,assuming a set of equations of state for the NS matter andevaluating the uncertainties on the relevant parameters usinga Fisher matrix approach. Evaluation of the uncertainties on the binary parameters. —The accuracy with which future interferometers will measurea set of binary parameters θ , is estimated by comparing thegravity-wave data-stream with a set of theoretical templates.For strong enough signals, θ are expected to have a Gaussiandistribution centered around the true values , with covariancematrix Cov ab = ( Γ − ) ab , Γ ab = (cid:32) ∂ h ∂θ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ h ∂θ b (cid:33) . (7)being Γ the Fisher information matrix [25]. ( ·|· ) is the innerproduct between two GW templates ˜ h ( f ) and ˜ g ( f ):( h | g ) = (cid:90) f max f min ˜ h ( f )˜ g (cid:63) ( f ) + ˜ h (cid:63) ( f )˜ g ( f ) S n ( f ) d f , (8) (cid:63) denotes complex-conjugation, and S n ( f ) is the noise spec-tral density of the considered detector. To model the waveformwe use the TaylorF2 approximant in the frequency domain,assuming the stationary phase approximation [26]: h ( f ) = A ( f ) e i ψ ( f ) = (cid:114) M / π / d f − / e i ψ ( f ) . (9)where d is the source distance. The post-Newtonian expansionof the phase includes spin-orbit and tidal corrections. It canbe written as ψ ( f ) = ψ PP + ψ T , where the point-particle termis ψ PP ( f ) = π t c − φ c − π + M π f ) − / (cid:88) i = e i ( m π f ) i / , (10)and t c and φ c are the time and the phase at coalescence. Thecoe ffi cient e i are listed in [27, 28]. The tidal contribution ψ T is given by [20, 29] ψ T ( f ) = − Λ ν m x / (cid:34) + x − π x / ++ (cid:32) + (cid:33) x − π x / (cid:35) , (11)where x = ( m π f ) / and Λ is the averaged tidal deformability,which for BH-NS binaries reads [30]: Λ = λ + q )26 .We consider non rotating NSs, as this is believed to be areliable approximation of real astrophysical systems [31, 32].The GW waveform is therefore described in terms of the fol-lowing set of parameters, θ = (ln A , t c , φ c , ln M , ln ν, Λ , β ),where β is the 2 PN spin-orbit contribution in ψ PP : β = χ BH M m + ν ˆ L · ˆ S BH , (12)where ˆ L , ˆ S NH are the unit vectors in the direction of the or-bital angular momentum and of the spin, respectively. Wechoose the BH spin aligned with the orbital angular mo-mentum. Moreover, since χ BH ≤ | | , β (cid:46) .
4; thereforewe consider the following prior probability distribution on β : p (0) ( β ) ∝ exp (cid:104) − ( β/ . (cid:105) . Thus, we need to compute a 7 × A is uncorrelated with the other variables, we performderivatives only with respect to the remaining six parameters θ = ( t c , φ c , ln M , ln ν, Λ , β ). In our analysis we consider bothsecond (AdvLIGO / Virgo) and third generation (Einstein Tele-scope, ET, [33]) detectors. For AdvLIGO / Virgo we use the
ZERO DET high P noise spectral density of AdvLIGO [34],in the frequency ranges [20 Hz , f ISCO ]; for the Einstein Tele-scope we use the analytic fit of the sensitivity curve providedin [35], in the range [10 Hz , f ISCO ]. f ISCO is the frequency ofthe Kerr ISCO including corrections due to NS self-force [36]: f ISCO = M BH m π Ω ( χ BH ) (cid:104) + ν c GSF ( χ BH ) (cid:105) , (13)with Ω ( χ BH ) = sign( χ BH ) M / / ( r / + χ BH ).We model the NS structure by means of piecewise poly-tropes, [24]. The core EoS changes with an overall pressureshift p , specified at the density ρ = . × g cm − .Once the adiabatic index Γ core is fixed, increasing p producesa family of neutron stars with growing radius for a given mass.Choosing Γ core = p = (10 . , . , . , . )g / cm we obtain four EoS, , H , HB and B , which denote verysti ff , sti ff , moderately sti ff and soft EoS, respectively. Thestellar parameters for M NS = (1 . , . M (cid:12) , are shown in Ta-ble I. EoS M NS ( M (cid:12) ) C λ (km ) M NS ( M (cid:12) ) C λ (km ) H HB B C = M NS / R NS , and the tidal deformability λ . Numerical results. —Following the strategy previously out-lined, we compute the minimum and maximum probabilities(6) that the coalescence of a BH-NS system produces a rem-nant disk with mass above a threshold ¯ M rem , for the NS mod-els listed in Table I and di ff erent values of the mass ratio q .The results are given in Table II, for q = q =
7, blackhole spin χ BH = (0 . , . , . M NS = (1 . , . M (cid:12) , anddisk mass thresholds ¯ M rem = . M (cid:12) .For AdvLIGO / Virgo we put the source at a distance of 100Mpc. For ET the binary is at 1 Gpc. In this case the signalmust be suitably redshifted [23, 37], and we have assumedthat z is known with a fiducial error of the order of 10% [38].The first clear result is that as the BH spin approaches thehighest value we consider, χ BH = .
9, and for low mass ra-tio q =
3, the probability that a BH-NS coalescence producesa disk with mass above the threshold is insensitive to the NSinternal composition, and it approaches unity for all consid-ered configurations. These would be good candidates for GRBproduction. For the highest mass ratio we consider, q = ffi ciently massive disk depends onthe NS mass and EoS, and on the detector. In particular, itdecreases as the EoS softens, and as the NS mass increases.This is a general trend, observed also for smaller values of χ BH . However, when χ BH = . (cid:38)
50% .Let us now consider the results for χ BH = .
2. If the NSmass is 1 . M (cid:12) the probability that a detected GW signal froma BH-NS coalescence is associated to the formation of a blackhole with a disk of mass above threshold is (cid:38)
50% for both q = d =
100 Mpc AdV q = d = q = d =
100 Mpc AdV q = d = M NS = . M (cid:12) χ BH χ BH χ BH χ BH EOS C λ . . . . . . . . . . . . H HB B M NS = . M (cid:12) H HB B MIN ,P MAX ] that the coalescence of a BH-NS binary produces a disk mass larger than ¯ M rem = . M (cid:12) for AdLIGO / Virgo (AdV) and for the Einstein telescope (ET), for binaries with q = q =
7, NS masses (1.2,1.35) M (cid:12) , and BH spin χ BH = (0 . , . , . d =
100 Mpc for advanced detectors, and d = C λ is estimated throughout the universal relation (3). AdvLIGO / Virgo and ET, provided q =
3. For larger NS mass,this remains true only if the NS equation of state is sti ff ( or H ). High values of q are disfavoured.When the black hole spin has an intermediate value, say χ BH = .
5, Table II shows that, the NS compactness playsa key role in the identification of good candidates for GRBproduction, for both detectors. Again large values of the massratio yield small probabilities.The range of compactness shown in Table II includes neu-tron stars with radius ranging within ∼ [10 ,
15] km. Fromthe table it is also clear that if we choose a compactnesssmaller than the minimum value, the probability of generat-ing a SGBR increases, and the inverse is true if we considercompactness larger than our maximum.
Concluding remarks. —The method developed in this paper can be used in severaldi ff erent ways. In the future, gravitational wave detectors areexpected to reach a sensitivity su ffi cient to extract the param-eters on which our analysis is based, i.e. chirp mass, mass ra-tio, source distance, spin and tidal deformability. We can alsoexpect that the steady improvement of the e ffi ciency of com-putational facilities experienced in recent years will continue,reducing the time needed to obtain these parameters from adetected signal. Moreover, the higher sensitivity will allow todetect sources in a much larger volume space, thus increas-ing the detection rates. In this perspective, the method weenvisage in this paper will be useful to trigger the electromag-netic follow-up of a GW detection, searching for the afterglowemission of a SGRBs.Waiting for the future, the method we propose can be usedin the data analysis of advanced detectors as follows: • Table II indicates the systems which are more likely toproduce accretion disks su ffi ciently massive to generatea SGRB. The table can be enriched including more NSequations of state or more binary parameters; however,it already contains a clear information on which is the range of parameters to be used in the GW data anal-ysis, if the goal is to search for BH-NS signals whichmay be associated to a GRB. For instance, Table II sug-gests that searching for mass ratio smaller than, or equalto, 3-4, and values of the black hole angular momen-tum larger than 0.5-0.6 would allow to save time andcomputational resources in low latency search. In ad-dition, it would allow to gain sensitivity in externallytriggered searches performed in time coincidence withshort GRBs observed by gamma-ray satellites. • If a SGRB is observed su ffi ciently close to us in theelectromagnetic waveband, the parameters of the GWsignal detected in coincidence would allow to set athreshold on the mass of the accretion disk. If the GWsignal comes, say, from a system with a BH with spin χ BH = .
5, mass ratio q =
7, and neutron star mass M NS = . M (cid:12) , from Table II, equations of state softerthan the EoS would be disfavoured. Thus, we wouldbe able to shed light on the dynamics of the binary sys-tem, on its parameters and on the internal structure ofits components. We would enter into the realm of grav-itational wave astronomy.Finally, it is worth stressing that as soon as the fit (1) will beextended to NS-NS coalescing binaries, this information willbe easily implemented in our approach. Being the rate of NS-NS coalescence higher than that of BH-NS, our approach willacquire more significance, and will be a very useful tool tostudy these systems. Acknowledgements. —We would like to thank M. Branchesifor her comments on the paper, which helped us to clarify sev-eral issues. We would like to thank L. Pagano, R. Amato andM. Muccino for useful discussions and comments. A. M. issupported by a ”Virgo EGO Scientific Forum” (VESF) grant.Partial support comes from the CompStar network, COST Ac-tion MP1304. [1] , . [2] William H. Lee and Enrico Ramirez-Ruiz, New J. Phys. , 17(2007).[3] J. Abadie et al. [LIGO Scientific Collaboration], Astrophys. J. (2012) 2.[4] J. Abadie et al. [LIGO Scientific Collaboration], Astrophys. J. (2012) 12.[5] J. Aasi et al., arXiv:1304.0670 (2013).[6] J. Abadie et al. [LIGO Scientific Collaboration], A&A A155 (2012).[7] Francois Foucart, Phys. Rev.
D 86 , 124007 (2012).[8] B. Giacomazzo, R. Perna, L. Rezzolla, E. Troja and D. Lazzati,The Astrophysical Journal Letters, :L18, (2013).[9] J. M. Bardeem, W. H. Press and S. A. Teukolsky, Astrophys. J. , 347 (1972).[10] K. Kyutoku, H. Okawa, M. Shibata, K. Taniguchi, Phys. Rev.D , 064018 (2011).[11] Z.B. Etienne, Y.T. Liu, S.L. Shapiro and T.W. Baumgarte, Phys.Rev. D , 044024 (2009).[12] F. Foucart, M.D. Duez, L.E. Kidder, M.A.Scheel, B. Szilagyi,S.A. Teukolsky, Phys. Rev. D , 044015 (2012).[13] F. Foucart, M.D. Duez, L.E. Kidder, S.A. Teukolsky, Phys. Rev.D , 024005 (2011).[14] T. Piran, Rev. Mod. Phys. , 1143 (2004).[15] S. Setiawan, M. Ru ff ert and H. -T. Janka, Mon. Not. Roy. As-tron. Soc. , 753 (2004).[16] T. D. Matteo, R. Perna and R. Narayan, Astrophys. J. , 706(2002).[17] R. Popham, S. E. Woosley and C. Fryer, Astrophys. J. , 356(1999).[18] A. Maselli, V. Cardoso, V. Ferrari, L. Gualtieri and P. Pani,Phys. Rev. D , 023007 (2013). [19] W. Del Pozzo, T. G. F. Li, M. Agathos, C. V. D. Broeck andS. Vitale, Phys. Rev. Lett. , 071101 (2013).[20] T. Damour, A. Nagar, L. Villain, Phys. Rev. D
85, 123007(2012).[21] J. Read, et al., Phys. Rev. D
88, 044042 (2013).[22] F. Pannarale, L. Rezzolla, F. Ohme, J. S. Read, Phys. Rev. D D
88, 104040(2013).[24] J. Read, et al., Phys. Rev. D
79, 124033 (2009).[25] E. Poisson and C. M. Will, Phys. Rev. D , 2 (1995).[26] T. Damour, B. Iyer, B. Sathyaprakash, Phys. Rev. D , 061501 (2004).[28] L. Blanchet, T. Damour, G. Esposito Farese and B. R. Iyer,Phys. Rev. Lett. , 091011 (2004).[29] J. Vines, E.E. Flanagan, T. Hinderer, Phys. Rev. D
83, 084051(2011).[30] E.E. Flanagan, T. Hinderer, Phys. Rev. D
77, 021502 (2008).[31] L. Bildsten and C. Cutler, Astroph. J , 175 (1992).[32] C. S. Kochanek, Astroph. J , 234 (1992).[33] [34] D. Shoemaker, https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=2974 .[35] B.S. Sathyaprakash and B.F. Schultz, Living Rev. Relativity ,2 (2009).[36] Marc Favata, Phys. Rev. D , 024028 (2011).[37] C. Cutler and E. E. Flanagan, Phys. Rev. D (1994) 2658.[38] C. Messenger and J. Read, Phys. Rev. Lett.108