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Possible Strong Gravitational Wave Sources for the LISA Antenna José C.N. de Araujo, Oswaldo D. Miranda1 and Odylio D. Aguiar Divisaõ de Astrof´ısica - Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas 1758, Saõ José dos Campos, 12227-010 SP, Brazil [email protected], [email protected], [email protected] 1 0 0 ABSTRACT 2 n a Recently Fuller & Shi proposed that the gravitational collapse of supermassive J objects (M & 104M⊙) could be a cosmological source of γ-ray bursts (GRBs). The 2 major advantage of their model is that supermassive object collapses are far more 1 energetic than solar mass-scale compact mergers. Also, in their proposal the seeds of v 2 supermassive black holes (SMBHs) thus formed could give rise to the SMBHs observed 2 at the center of many galaxies. We argue here that, besides the generation of GRBs, 0 1 there could well occur a strong generation of gravitational waves (GWs) during the 0 formation of SMBHs. As a result, the rate of such GW bursts could be as high as the 1 0 rate of GRBs in the model by Fuller & Shi. In this case, the detection of GRBs and / bursts of GWs should occur with a small time difference. We also argue that the GWs h p produced by the SMBHs studied here could bedetected when the Laser Interferometric - o Space Antenna (LISA) becomes operative. r t s a Subject headings: gravitational waves – supermassive black holes : v i X r 1. Introduction a The Laser Interferometric Space Antenna (LISA) is designed to detect low frequency gravitational waves in the frequency range 10−4 −1 Hz, which cannot be detected on the Earth becauseof seismic noise. Alot of very interesting astrophysical phenomenaarebelieved togenerate GWs in this frequency band: the formation of supermassive black holes (SMBHs), SMBH-SMBH binary coalescence, compact stars orbiting SMBHs in galactic nuclei, pairs of close white dwarfs, pairs of neutron stars, neutron star and black hole binaries, pairs of contact normal stars, normal star and white dwarf binaries, and pairs of stellar black holes. We are particularly concerned here with SMBHs, which are believed to be present in galactic nuclei (Blandford 1999). Lynden-Bell (1969) originally proposed that active galaxies harbor a 1 Present address: Department of Physics, Washington University, Campus Box 1105, One Brookings Drive, St. Louis - MO 63130-4899 - USA. – 2 – SMBH engine fed by accretion and there is now solid observational evidence for this (Richstone et al. 1998), although there remain some unanswered questions related to their formation. Several interestingpapersstudythemassfunctionofSMBHsingalaxies (Franceschini, Vercellone&Fabian 1998; Salucci et al. 1999), using different combinations of optical, infrared, radio and X-ray data. SMBHs could form through the dynamical evolution of dense star cluster objects; by the merging of SMBHs of smaller masses and by the viscous evolution and collapse of self-gravitating gaseous objects (e.g., supermassive stars). Quinlan & Shapiro (1990) assumed the existence of a densestarclusterinagalactic nucleusandfollowed thebuild-upof100M⊙ orlargerseedblackholes by collisions. Another possibility is that ∼ 106M⊙ SMBHs form by coherent collapse in galactic nuclei before most of the bulge gas turns into stars (Silk & Rees 1998; Haehnelt, Natarajan & Rees 1998). Other interesting studies concerning SMBH formation are discussed by Rees (1997, 1998); Haehnelt & Rees (1993); Haehnelt (1994); Eisenstein & Loeb (1995); Umemura et al. (1993) and Fuller & Shi (1998; hearafter FS). SMBHs may produce a strong GW signal during their formation, which could be detectable by LISA even at cosmological distances. Since most galaxies could harbor SMBHs it is argued that the number of events expected could be several per year or even per day. It is worth studying whether other astrophysical phenomena related to the formation of such putative SMBHs, such as the emission of electromagnetic radiation and neutrinos, could help constrain the SMBH production rate and formation epoch. For example, γ-ray could be related to the production of GWs since the formation of SMBHs may be a very energetic phenomenon. In particular GBRs have been puzzling astrophysicists, because of the enormous electromagnetic energy produced, ∼ 1051 − 1052 ergs, the spatial isotropy (which suggests that the sources are cosmological), and the event rate of several sources per day. Recently FS (see also Shi & Fuller 1998; Abazajian, Fuller & shi 1999) proposed that the gravitational collapse of supermassive objects (M & 104M⊙), either as relativistic star clusters or as a single supermassive star could account for cosmological GRBs. These authors also proposed thatsuchsupermassiveobjectsshouldproduceneutrinoemission,buttheydidnotconsiderwhether such γ-ray and neutrino sources could be also strong GW sources. Since the FS model involves the formation of a SMBH it is hard to avoid GWs being also produced. The paper is organized as follows: §2 deals with the GWs generated by GRB SMBHs and §3 presents the discussion and conclusions. 2. Gravitational Waves from GRB SMBHs This paper extends the study by FS, which considers whether the collapse of supermassive objects could account for cosmological GRBs. We argue that such a source of γ-rays could also be a strong source of GWs. Then we propose an independent way to check FS model through GW – 3 – astronomy. FS define a supermassive object in terms of a star or star cluster that undergoes the general relativistic Feynman-Chandrasekhar instability during its evolution. Supermassive objects with M & 5×104M⊙ could leave black hole remnants of M & 103M⊙. To account for the observed rate of GRBs the supermassive object collapses should amount to several per day. Each collapse probablyleadstoablackholeremnant,soitishardtoavoidtheconclusionthatGWsaregenerated with the same frequency. If other processes of SMBH formation do not involve GRB events, GW production rate could well be even higher. If all supermassive objects form and collapse at a redshift z, as assumed by FS, the event rate is dr ρ F(1+z)3 R ≃ 4πr2a3 b , (1) BH zdt M 0 where r is the Friedman-Robertson-Walker comoving coordinate of the supermassive object, a is z scale factor of the Universe at redshift z, t the age of the Universe, ρ is the present value of the 0 b baryonic density, F is the fraction of baryons incorporated in supermassive objects, M is the mass of theinitial hydrostatic supermassivestar, taken tobeM = 10M , whereM isthemass of the BH BH resulting SMBH (FS; Shi & Fuller 1998; Abazajian, Fuller & Shi 1999). This rate can be rewritten as R ≃ 4πr2cn , (2) BH BH where n is the number density of SMBHs, given by BH ρ F b n = . (3) BH M Equation(2)isimplicitintheequationsderivedbyCarr(1980)inastudyconcerningthegeneration of GWs from SMBHs. The GW amplitude associated with the formation of each SMBH is (Thorne 1987) 1/2 −1 15 GM M r h = ε BH ≃ 7.4×10−20ε1/2 BH 0 , (4) BH (cid:18)2π (cid:19) c2 r0 (cid:18) M⊙ (cid:19)(cid:18)1Mpc(cid:19) where ε is the efficiency of generation of GWs. The collapse to a black hole produces a signal with frequency ν = 1 c3(1+z)−1 ≃ 1.3×104Hz M⊙ (1+z)−1. (5) obs 5πM G (cid:18)M (cid:19) BH BH – 4 – The ensemble of SMBHs formed should produce a background of GWs with amplitude 1 h2 = h2 dR (6) BG νobs Z BH BH (de Araujo, Miranda & Aguiar 2000; Miranda, de Araujo & Aguiar 2000), where dR is the BH differential SMBHformation rate. IftheSMBHs areassumedto have thesamemass andformation redshift, as in the FS model, we have 4πR2cn 1/2 BH h = h . (7) BG BH (cid:18) ν (cid:19) obs This equation can be written as 1/2 τ h = h , (8) BG BH (cid:18)∆t(cid:19) 0 (cf. Carr 1980) where the subscript zero indicates a present day value, τ is the duration of each 0 burst and ∆t is the interval between bursts. Unlike Carr, we assume that the above equation 0 holds only for (τ/∆t) & 1. These time scales are 0 1 τ ≃ (9) 0 ν , obs and 1 ∆t ≃ . (10) 0 R BH The ratio τ 4πR2cn BH ≃ , (11) (cid:18)∆t(cid:19) ν 0 obs is called duty-cycle and can be interpreted as the number of overlapping bursts. If the bursts overlap, (τ/∆t) is greater than 1 and thus h > h ; on the other hand, if 0 BG BH (τ/∆t) is less than 1, they do not overlap and the GW background is not continuous, but consists 0 of a sequence of spaced bursts with a mean separation ∼ ∆t (see Ferrari, Matarrese & Schneider 0 1999, who consider the case where a non-continuous background also appears). The cosmological model considered here has a density parameter Ω = Ω = 0.1 and Hubble 0 b constant H0 = 50kms−1Mpc−1. For aSMBH formed at redshiftz ≃ 3 with mass 107M⊙, theGWs – 5 – would be detected at frequency ν ≃ 3.3×10−4Hz, so the characteristic duration of the burst is obs τ ≃ 3×103 s. If ∆t ≃ 1/R = 1 day−1, as observed for GRBs, we obtain 4.0×10−2 for the 0 0 BH duty-cycle. In this case, a population of SMBHs formed at z ≃ 3 with mass 107M⊙ cannot produce a background and one will observe a burst a day with duration τ , amplitude h and frequency 0 BH ν . obs The results are summarized in Fig. 1 which shows the duty-cycle (τ /∆t ) as a function of the 0 0 mass of the SMBHs, for the formation redshift range z = 1−5. We also present, for comparison, results for R ∼10 day−1. BH The energy density of the GWs can be written in units of the critical density as 1 dρ GW Ω = , (12) GW ρ dlogν c obs where ρ = 3H2/8πG. Equivalently c ν 4π2 Ω = obsF = ν2 h2 . (13) GW c3ρ ν 3H2 obs BH c Assuming a maximum efficiency for the generation of GWs (ε ≃ 7×10−4; Stark & Piran 1986) during the collapse of an object to a black hole, one has Ω < 10−6 for the redshifts and masses GW studied here. In Fig. 2 we present the amplitude h as a function of the observed frequency (ν ) for BH obs different values of ε, SMBH mass and formation redshift. We also present the LISA sensitivity (h ) s for a signal-to-noise ratio of 1 for burst sources. For example, hBH > hs for MBH = 106M⊙ and ε > 10−5. Thus, even for low GW efficiency the signal produced by these SMBHs could be detected by LISA. 3. Discussion and Conclusions The results presented here were obtained for an open Universe model with Ω = 0.1 and b H = 50km s−1Mpc−1. We also assume the same scenario as FS, with all the SMBHs forming 0 at the same redshift. For a given event rate, and for a given range of mass, we first calculate the duty-cycle to see whether the GWs produced by the ensemble of SMBHs generate a stochastic background. For an event rate exceeding 1−10 day−1 we find that the bursts do not overlap and so they do not produce a continuous stochastic background. In particular, a stochastic background could occur for black holes with MBH ∼ 107M⊙ only if the event rate exceeded 30 per day−1. In this case we would have τ /∆t > 1 and the GWs of different seeds could overlap producing a 0 0 background with amplitude given by equation (7). SMBHs formed with masses < 106M⊙ could – 6 – produce a GW background for the same event rate only if they formed at z >5. The major advantage of the FS scenario, as a cosmological source of γ-ray emission, is its enormous energy reservoir; the gravitational binding energy is Eg ∼ 1054(MBH/M⊙) erg. Another advantage of this scenario is related to the angular scale of the sources. Although tremendous energy is deposited into the fireball (∼ 1052 ergs during the collapse to a black hole of 106M⊙), the distortion produced in the cosmic background radiation through the scattering of hot electrons (Sunyaev-Zeldovich effect) occurs on a very small angular scale (θ . 10−10 arc seconds) and is therefore undetectable. In the FS model a potential problem, as a GRB source, is related to the “baryon-loading” 2, that is, the confinement of the electron/positron/photon fireball by the baryons which could carry energy of it in the form of kinetic energy, thus diminishing the amount of energetic photons (the gamma ones). This suggests that the region at several Schwarzschild radii from the supermassive star core should have extremely low baryon density. There are, at least two ways to avoid the excessive baryon-loading: rotation of the star producing the flattened collapse or the collapse of a dense star cluster instead of a single object. This could result in a different event rate for the GRBs and the GW bursts, not all GW bursts being related to GRBs in the present scenario since the baryons could block the γ−ray. Even if the GRBs and GW bursts have completely different event rates, either because the source of GWs does not produce GRB at all or because the gamma radiation is blocked, it would be possible to verify the FS scenario by looking for GRBs once GW bursts associated with SMBH formation are observed and identified. There will be a time interval between the GRB and the GW burst because the types of radiation are generated in different ways. The generation of the GRB depends on a series of physical processes after the collapse of the core, e.g., the generation of the fireball to accelerate the matter to the ultra-relativistic regime when the kinetic energy in the fireball could be converted to γ-rays. The GWs, on the other hand, are mainly produced when the SMBH is formed, through the excitation of its quasi-normal modes. A detailed modeling is required however to evaluate the time interval between the GRB and the GW burst. Using the LISA observatory to detect GW bursts related to the SMBHs formation, one could find their GW amplitudes, the characteristic frequencies and also the formation rate of SMBHs. If we also find the redshift associated with the events (by observing in the electromagnetic window) we will be able to obtain the SMBH masses and the GW efficiency using the model proposed here. By comparing the SMBH formation GW event rates with the GRB rates one could also infer what fraction of an ensemble of SMBHs had conditions to generate GRBs and to impose constraints on the FS scenario. Then in the present study we are proposing an independent way to check FS model through GW astronomy. 2 There are many papers in the literature discussing aspects related to the injection of energy (including the baryon-loading problem) associated with GRBs. In particular, we refer the reader for the papers of Shemi & Piran 1990; Kobayashi, Piran & Sari 1999 and Fuller, Pruet & Abazajian 2000. – 7 – The Brazilian agency FAPESP partially supports this work (JCNA under grants 97/06024-4 and 97/13720-7; ODM under grant 98/13735-7). ODA thanks CNPq (Brazil) for financial support (grant300619/92-8). WewouldliketothankDr. RobinStebbinsandProf. Peter Benderforkindly providing us with the LISA sensitivity curve. Finally, we would like to thank an anonymous referee for his (her) criticisms and suggestions which greatly improved the present version of our paper. REFERENCES Abazajian, K., Fuller, G.M. & Shi, X. 1999, in Activity in Galaxies and Related Phenomena, proceedings of IAU symp.194, eds. Y. Terzian, E. Khachikian, D. Weedman (San Francisco: ASP), p. 235 Blandford, R.D. 1999, in Galaxy Dynamics, ASP Conference Series vol. 182, eds. D.R. Merritt, M. Valluri, J.A. 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Israel (Cambridge: Cambridge University Press), p. 331 Umemura, M., Loeb, A. & Turner, E.L. 1993, ApJ, 419, 459 This preprintwas prepared with the AASLATEX macros v5.0. – 9 – (cid:3) (cid:11)D(cid:12)(cid:3))XOOHU(cid:3)(cid:9)(cid:3)6KL(cid:3)UDWH (cid:11)E(cid:12)(cid:3)(cid:20)(cid:19)(cid:3)[(cid:3))XOOHU(cid:3)(cid:9)(cid:3)6KL(cid:3)UDWH (cid:26) ](cid:3) (cid:3)(cid:20) ](cid:3) (cid:3)(cid:24) (cid:20)(cid:19) (cid:12) ~ 0 (cid:11)D(cid:12) (cid:11)E(cid:12) (cid:3)(cid:11) (cid:25) (cid:3) +(cid:20)(cid:19) % 0 (cid:24) (cid:20)(cid:19) ](cid:3) (cid:20) ](cid:3) (cid:3)(cid:24) (cid:16)(cid:23) (cid:16)(cid:22) (cid:16)(cid:21) (cid:16)(cid:20) (cid:20) (cid:20)(cid:19) (cid:20)(cid:19) (cid:20)(cid:19) (cid:20)(cid:19) t D (cid:3)(cid:18)(cid:3) W (cid:19) (cid:19) Fig. 1.— Duty-cycle versus the mass of SMBHs for the formation redshift range z = 1 − 5. The results are presented for 1 and 10 events day−1. The cosmological model considered has Ω = Ω = 0.1 and H = 50km s−1Mpc−1. 0 b 0 – 10 – (cid:3) (cid:16)(cid:20)(cid:26) (cid:20)(cid:19) e (cid:3)(cid:16)(cid:3)(cid:22) (cid:11)D(cid:12)(cid:3) (cid:3) (cid:3)(cid:20)(cid:19) (cid:26) e (cid:3)(cid:16)(cid:3)(cid:24) (cid:20)(cid:19) 0 ~ (cid:11)E(cid:12)(cid:3) (cid:3) (cid:3)(cid:20)(cid:19) (cid:16)(cid:20)(cid:27) (cid:20)(cid:19) (cid:11)D(cid:12) (cid:25) (cid:16)(cid:20)(cid:28) (cid:20)(cid:19) 0 ](cid:3) (cid:3)(cid:20) ~ (cid:20)(cid:19) K (cid:3) (cid:24) (cid:11)E(cid:12) (cid:20)(cid:19) 0 ~ (cid:16)(cid:21)(cid:19) ](cid:3) (cid:3)(cid:24) (cid:20)(cid:19) ](cid:3) (cid:3)(cid:20) (cid:16)(cid:21)(cid:20) (cid:20)(cid:19) /,6$(cid:3)6HQVLWLYLW\(cid:3)(cid:11)K (cid:12) 6 ](cid:3) (cid:3)(cid:24) (cid:16)(cid:21)(cid:21) (cid:20)(cid:19) (cid:16)(cid:23) (cid:16)(cid:22) (cid:16)(cid:21) (cid:16)(cid:20) (cid:20)(cid:19) (cid:20)(cid:19) (cid:20)(cid:19) (cid:20)(cid:19) n (cid:3)(cid:11)+](cid:12) REV Fig. 2.— Dimensionless amplitude h as a function of observed frequency for ε = 10−5 and BH ε = 10−3 for the burst of GWs for MBH = 105, 106and107M⊙ at redshifts z = 1 − 5. The LISA sensitivity for burst sources (h ) is also plotted. The cosmological model considered has S Ω = Ω = 0.1 and H = 50km s−1Mpc−1. 0 b 0