The Minimum and Maximum Gravitational-Wave Background from Supermassive Binary Black Holes
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MNRAS 000,1{9 (2018) Preprint 22 October 2018 Compiled using MNRAS LATEX style file v3.0 The minimum and maximum gravitational-wave background from supermassive binary black holes Xing-Jiang Zhu1;2?, Weiguang Cui3 and Eric Thrane1;2 1School of Physics and Astronomy, Monash University, Clayton, Vic 3800, Australia 2OzGrav: The Australian Research Council Centre of Excellence for Gravitational Wave Discovery 3Departamento de F´ısica Te´orica, M´odulo 15, Facultad de Ciencias, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Accepted XXX. Received YYY; in original form ZZZ ABSTRACT The gravitational-wave background from supermassive binary black holes (SMBBHs) has yet to be detected. This has led to speculations as to whether current pulsar timing array limits are in tension with theoretical predictions. In this paper, we use electromagnetic observations to constrain the SMBBH background from above and below. To derive the maximum amplitude of the background, we suppose that equal- mass SMBBH mergers fully account for the local black hole number density. This yields −15 a maximum characteristic signal amplitude at a period of one year Ayr < 2:4 × 10 , which is comparable to the pulsar timing limits. To derive the minimum amplitude requires an electromagnetic observation of an SMBBH. While a number of candidates have been put forward, there are no universally-accepted electromagnetic detections in the nanohertz band. We show the candidate 3C 66B implies a lower limit, which is inconsistent with limits from pulsar timing, casting doubt on its binary nature. −17 Alternatively, if the parameters of OJ 287 are known accurately, then Ayr > 6:1×10 at 95% confidence level. If one of the current candidates can be established as a bona fide SMBBH, it will immediately imply an astrophysically interesting lower limit. Key words: gravitational waves { galaxies: evolution { quasars: individual(OJ 287) { galaxies: individual(3C 66B) { black hole physics { pulsars: general 1 INTRODUCTION Begelman et al. 1980). Assuming that binaries are circular with the orbital evolution driven by GW emission, the char- The detection of gravitational waves (GWs) from several acteristic amplitude h of such a background signal can be stellar-mass compact binary merger events (Abbott et al. c well described by a power-law spectrum h ∼ f −2/3 with 2016a, 2017a,b) by ground-based laser interferometers Ad- c f being GW frequency (Phinney 2001); the amplitude at vanced LIGO (Aasi et al. 2015) and Advanced Virgo (Acer- f = 1 yr−1, denoted by A , can be conveniently used for nese et al. 2015) heralded a new era of GW astronomy. In yr comparing steadily-improving experimental limits and dif- parallel to ground-based detectors that are monitoring the ferent model predictions. Prior to the formation of timing audio band (∼ 10 − 103 Hz) of the GW spectrum, decades arrays, Kaspi et al.(1994) used long-term observations of of efforts have gone into opening the nanohertz frequency arXiv:1806.02346v3 [astro-ph.GA] 19 Oct 2018 two pulsars to constrain A ≤ 2:6 × 10−14 with 95% confi- (10−9 − 10−6 Hz) window (Sazhin 1978; Detweiler 1979; yr dence (the same confidence level applied for limits quoted Hellings & Downs 1983; Jenet et al. 2005; Hobbs et al. 2010; below). Jenet et al.(2006) reduced this limit to 1:1 × 10−14 Manchester 2013). These experiments, called pulsar timing using a prototype data set of the Parkes Pulsar Timing Ar- arrays (PTAs; Foster & Backer 1990), use a number of ul- ray (PPTA; Manchester et al. 2013). Since the establish- tra stable millisecond pulsars collectively as a Galactic-scale ment of three major PTAs, including PPTA, NANOGrav GW detector. (McLaughlin 2013) and the European PTA (EPTA; Kramer Previous PTA searches (e.g., Yardley et al. 2011; van & Champion 2013), the constraints have been improved by Haasteren et al. 2011; Demorest et al. 2013) have primar- an order of magnitude over the last decade. Currently the ily targeted a GW background (GWB) formed by a cosmic best published upper limit on A is 1 × 10−15 by the PPTA population of supermassive binary black holes (SMBBHs; yr (Shannon et al. 2015) with comparable results from the other two PTAs (Lentati et al. 2015; Arzoumanian et al. 2018). The three PTAs have joined together to form the Interna- ? E-mail: [email protected] © 2018 The Authors 2 Zhu, Cui & Thrane tional Pulsar Timing Array (IPTA; Verbiest et al. 2016), Second, we present a novel Bayesian framework to infer aiming at a more sensitive data set. the SMBBH merger rate based on a gold-plated detection of Over the past two decades, theoretical predictions of a single system. The derived merger rate can be combined Ayr have also evolved. Rajagopal & Romani(1995) consid- with the system chirp mass to compute the GWB signal am- ered several mechanisms that can drive SMBBHs into the plitude using the practical theorem of Phinney(2001). We GW emission regime and obtained an estimate of Ayr = consider several SMBBH candidates with inferred masses − 2:2 × 10 16. Jaffe & Backer(2003) used galaxy merger rate and orbital periods, and derive lower bounds on Ayr. estimates and the scaling relation between black hole mass This paper is organized as follows. In Section2, we re- and the spheroid mass of its host galaxy to compute the view the formalism of Phinney(2001) and provide two useful GWB spectrum; they confirmed the power-law relation and equations for quick computation of Ayr. In Section3, we de- −15 found Ayr ∼ 10 . Subsequently, Wyithe & Loeb(2003) em- rive the maximum signal amplitude from several black hole ployed a comprehensive set of semi-analytical models of dark mass functions. In Section4 we present the Bayesian frame- matter halo mergers and the scaling relation between black work for inferring the SMBBH merger rate from a single sys- hole mass and the velocity dispersion of its host galaxy; they tem. We apply this method to several SMBBH candidates found that the GWB is dominated by sources at redshifts to derive plausible lower bounds of Ayr. Finally, Section5 z . 2. Most of recent studies generated consistent results contains summary and discussions. −15 with median values at Ayr ≈ 1 × 10 (Sesana et al. 2008; Sesana 2013; Ravi et al. 2014), whereas some suggested that the signal may be a factor of two stronger (Kulier et al. 2015) 2 THE FORMALISM or even a factor of four stronger (McWilliams et al. 2014). Theoretical prediction of the GWB is essential to the in- In this section we present a phenomenological model for the terpretation of current PTA upper limits. In Shannon et al. GWB formed by a population of SMBBHs in circular orbits. (2015), theoretical models that predict typical signals of We start from the calculation of ΩGW¹ f º { the GW energy −15 Ayr & 1 × 10 were ruled out with high (& 90%) confi- density per logarithmic frequency interval at observed fre- dence. It was further suggested that the orbital evolution of quency f , divided by the critical energy density required to 2 2/ SMBBHs is either too fast (e.g., accelerated by interaction close the Universe today ρc = 3H0 c 8πG. Here H0 is the with ambient stars and/or gas) or stalled. Such a tension be- Hubble constant. Assuming a homogeneous and isotropic tween models and observations can be eliminated using dif- Universe, it is straightforward to compute this dimension- ferent black hole-host scaling relations (Sesana et al. 2016). less function as (see Phinney 2001, for details): Recently, Middleton et al.(2018) quantified this problem ¹ 1 1 N¹zº dEGW within a Bayesian framework by comparing a wide range of ΩGW¹ f º = dz; (1) ρc ¹1 + zº d ln fr models with the PPTA limit and found that only the most 0 fr=f ¹1+zº optimistic scenarios are disfavoured. where N¹zº is the spatial number density of GW events at Furthermore, understanding uncertainties in Ayr pre- redshift z; the ¹1+zº factor accounts for the redshifting of GW dictions is critical to the evaluation of future detection energy; fr = f ¹1+ zº is the GW frequency in the source's cos- prospects. For example, in Taylor et al.(2016), the model mic rest frame, and d fr¹dEGW/d frº is the total energy emitted of Sesana(2013) was used in combination with PTA up- in GWs within the frequency interval from fr to fr + d fr. per limits to calculate the time to detection of the GWB. There are two other quantities that are commonly Based on simple statistical estimates, they suggested ∼ 80% used for the characterization of a GWB, namely, the one- of detection probability within the next ten years for a large sided spectral density Sh¹ f º and the characteristic amplitude and expanding timing array. Needless to say, this state- hc¹ f º. They are related to ΩGW¹ f º by (Maggiore 2000): ment hinges on accurate predictions of the minimum GWB. 3H2 Dvorkin & Barausse(2017) attempted to make such a pre- 2 0 −2 h ¹ f º = f Sh¹ f º = f ΩGW¹ f º: (2) diction within a semi-analytical galaxy formation model by c 2π2 artificially stalling all SMBBHs at the orbital separation In the Newtonian limit, the GW energy spectrum for from which GW can drive binaries to merge within a Hubble an inspiralling circular binary of component masses m and time; they suggested that a PTA based on the Square Kilo- 1 m is given by (Thorne 1987): metre Array (SKA; Lazio 2013) is capable of making a detec- 2 / tion in this least favourable scenario. Recently, Bonetti et al. dE ¹πGº2/3 M5 3 GW = c f −1/3; (3) (2018) and Ryu et al.(2018) showed that this scenario might d f 3 be too pessimistic because triple/multiple interactions can 3/5 drive a considerable fraction of stalling binaries to merge; where Mc is the chirp mass defined as Mc = Mη , with 2 both suggested that the GWB is unlikely to be lower than M = m1+m2 being the total mass and η = m1m2/M being the −16 Ayr ∼ 10 .