arXiv:0809.3412v2 [astro-ph] 25 Mar 2009 o.Nt .Ato.Soc. Astron. R. Not. Mon. .Sesana A. Timing Pulsar with Arrays observations hole and black systems massive binary resolvable from waves Gravitational eevd— Received asv lc oe(B)bnr ytm ihmse in masses with systems binary range (MBH) the hole black Massive INTRODUCTION 1 yte&Le 03 eaae l 04 eaae l 2005). al. et Sesana 2003; 2004, band Backer al. frequency et & The Sesana 2003, Jaffe Loeb 1994; & Haehnelt Wyithe e.g., (see, frequencies aesucso rvttoa ae Gs at (GWs) waves gravitational of sources date ae nefrmtrSaeAntenna Space Interferometer Laser e en eeoe yEAadNS.Teobservational The NASA. and 10 interferome ESA window by laser developed wave being gravitational ter space-borne a 1998),
O fteple n h O codn oagvnmodel, given actual a the to of according TOA fit the the (TOA) and of pulses arrival the residuals of of timing TOA time The the pulses. produc- in radio 1983), of signature al. characteristic et (e.g. a Earth Bertotti the ing the 1979, to on Detweiler pulsar GWs 1978, a Sazhin from of signals effect radio radio-telescope, the of Parkes propagation exploit PTAs the 2008). e.g. (PTAs; Manchester Arrays Timing sar c 1 2 3 etrfrGaiainlWv hsc,TePnslai St Pennsylvania The Physics, Wave Gravitational for Center colo hsc n srnm,Uiest fBirmingham, of University Astronomy, and Physics of School et fAtooy nvriyo ihgn n ro,M 48 MI Arbor, Ann Michigan, of University Astronomy, of Dept. 00RAS 0000 ∼ − 9 10 Hz 4 1 .Vecchio A. , − − 10 10 ∼ 10 − 10 6 M zi led cesbewt Pul- with accessible already is Hz 000 − ⊙ 5 r mns h rmr candi- primary the amongst are Hz 0–0 00)Pitd2 coe 08(NL (MN 2018 October 22 Printed (0000) 000–000 , − inlrdainfo asv lc oebnr ytm ihmasses with obs systems to binary means hole only black massive the from relevant provide pla the evolutio radiation currently or into tional (PTAs) and insight on-going Arrays gain currently assembly to Timing are and the Pulsar surveys objects of these of of components number demographics a key and are structures holes black Massive ABSTRACT oe,w siaeta naeaea es n eovbesuc p source resolvable one least at average range con on the the in black that explore residuals massive estimate specific and we the redshift, phys on model, and main Depending masses observations. the as Arrays of black Timing such distribution massive sources, of the the range investigate of wide t we a of scenarios, consider level We assembly resolved. the individually exceeds be significantly and/or could that co close radiation sufficiently that gravitational Sources background Arrays. ate Timing stochastic Pulsar a upcoming produces with population cosmic whole u yteLsrItreoee pc nen ( future Antenna function Space distribution Interferometer hole Laser black the massive by naturally would the out sy observations of These unique rare. end these be high-mass detect to likely plausibly are can events the (SKA), Array Kilometre Square e words: Key general zwl epoe ythe by probed be will Hz 1 2 n .Volonteri M. and ( LISA edre al. et Bender , ∼ H mHz - nHz lc oepyis rvttoa ae omlg:ter pulsa – theory cosmology: – waves gravitational physics, hole black ∼ - t nvriy nvriyPr,P 60,USA 16802, PA Park, University University, ate 5 dbso,Briga,B52T UK 2TT, B15 Birmingham, Edgbaston, − 0,USA 109, 0n.Pla iigAry,adi atclrtefuture the particular in and Arrays, Timing Pulsar ns. 50 3 fteesresadtepandSur ioee Array Kilometer Square planned the and (SKA; surveys NanoGrav these and of 2008), al. et (Janssen 1 Array European Timing 2005). the al. Pulsar 2008), et (Manchester Jenet PTA Parkes 1983, complete Downs The & in Helling effects, (e.g. GWs unmodelled cluding about information physical the carry 1 of number a address and population, this to access direct l 07 rdc h xsec falrenme fmas- frequency of the number in large emitting between a et (MBHB) range Yoo of binaries 2007, hole existence 2003, black al. the sive Loeb et predict & Malbon 2007) 2006, Wyithe al. Zentner 2003; & Madau Koushiappas & Haardt Volonteri, sensitivity. r xetdt mrv osdrbyo h capabilities the on considerably improve to expected are http://arecibo.cac.cornell.edu/arecibo-staging/nano oua cnro fMHfrainadeouin(e.g. evolution and formation MBH of scenarios Popular www.skatelescope.org ∼ A T E 10 tl l v2.2) file style X LISA − 9 zad10 and Hz ). ilpoueamjrla in leap major a produce will ) − 6 eune o Pulsar for sequences ebcgon and background he opeeto the on complement z Tscngain can PTAs Hz. ndt rb the probe to nned hsclprocesses. physical l edetectable be uld tm,although stems, oepopulation hole clparameters ical oue timing roduces uvy carried surveys ∼ > asv gener- massive regravita- erve 10 fcosmic of n oebinary hole 7 M grav/ ⊙ The . rs: - 2 A. Sesana et al. unanswered questions in astrophysics (such as the assem- (http://www.g-vo.org/Millennium) and populate them with bly of galaxies and dynamical processes in galactic nuclei), central MBHs according to different prescriptions (Tremaine by detecting gravitational radiation of two forms: (i) the et al. 2002, Mclure et al. 2006, Lauer et al. 2007, Tundo et al. stochastic GW background produced by the incoherent su- 2007). The Millennium simulation (Springel et al. 2005) cov- 3 3 perposition of radiation from the whole cosmic population of ers a comoving volume of (500/h100 ) Mpc (h100 = H0/100 MBHBs and (ii) GWs from individual sources that are suf- km s−1 Mpc−1 is the normalized Hubble parameter), ensur- ficiently bright (and therefore massive and/or close) so that ing a number of massive nearby binaries adequate to con- the gravitational signal stands above the root-mean-square struct the necessary distribution. For each model we com- (rms) value of the background. Both classes of signals are pute the stochastic background, the expected distribution of of great interest, and the focused effort on PTAs could lead bright individual sources and the value of the characteristic to the discovery of systems difficult to detect with other timing residual δtgw, see Equation (20), for an observation techniques. time T . The signal-to-noise ratio at which a source can then The possible level of the GW background, and the con- be observed scales as SNR≈ δtgw/δtrms where δtrms is the sequences for observations have been explored by several root-mean-square level of the timing residuals noise, both authors (see e.g. Rajagopal & Romani 1995; Phinney 2001, coming from the receiver and the GW stochastic background Jaffe & Backer 2003; Jenet et al. 2005; Jenet et al. 2006; contribution. In the following we summarise our main re- Sesana et al. 2008). Recently, Sesana Vecchio & Colacino sults: (2008, hereinafter PaperI) studied in details the properties (i) The number of detectable individual sources for dif- of such a signal and the astrophysical information encoded ferent thresholds of the effective induced timing residuals into it, for a comprehensive range of MBHB formation mod- δt is shown in Table 1. Depending on the specific MBH els. As shown in PaperI, there is over a factor of 10 uncer- gw population model, we estimate that on average at least one tainty in the characteristic amplitude of the MBHB gener- resolvable source produces timing residuals in the range ated background in the PTA frequency window. However, ∼ 5 − 50ns. Future PTAs, and in particular SKA, can plau- the most optimistic estimates yield an amplitude just a fac- sibly detect these unique systems; the detection is however tor ≈ 3 below the upper-bound placed using current data by no means guaranteed, events will be rare and just above (Jenet et al. 2006), and near-term future observations could the detection threshold. either detect such a stochastic signal or start ruling out se- (ii) As expected, the brightest signals come from very lected MBHB population scenarios. Based on our current 8 massive systems with M > 5 × 10 M⊙. Here M = astrophysical understanding of the formation and evolution M 3/5M 3/5/(M + M )1/5 is the chirp mass of the binary of MBHBs and the estimates of the sensitivity of SKA, one 1 2 1 2 and M > M are the two black hole masses. Most of could argue that this instrument guarantees the detection of 1 2 − − the resolvable sources are located at relatively high redshift this signal in the frequency range 3 × 10 9 Hz − 5 × 10 8 Hz (0.2
c 0000 RAS, MNRAS 000, 000–000 Massive black holes and pulsar timing 3 the paper: the expected timing residuals from the estimated region of the parameter space when one deals with individ- population of MBHBs, including detection rates for current ual sources. and future PTAs. We also provide a comparison between the To avoid this problem, in this paper we generate distri- stochastic background computed according to the prescrip- butions of coalescing MBHBs using the galaxy haloes merger tions considered here and the results of PaperI. Summary rates derived from the on-line Millennium run database. and conclusions are given in Section 5. The Millennium simulation (Springel et al. 2005) covers a 3 3 volume of (500/h100 ) Mpc and is the ideal tool to con- struct a statistically representative distribution of massive 2 THE MASSIVE BLACK HOLE BINARY low/medium–redshift sources. In fact, the typical ensemble POPULATION of events available to construct the mass function of coa- lescing binaries is ∼ 100 times larger than in a typical EPS- In this section we introduce the population synthesis model based merger tree realization. As a first step we compile cat- adopted to estimate the number, and astrophysical param- alogues of galaxy mergers from the semi-analytical model of eters of MBHBs that emit GWs in the frequency region Bertone et al. (2007) applied to the Millennium run. probed by PTAs. The two fundamental ingredients to com- pute the merger rate of MBHBs are (i) the merger history of galaxy haloes, and (ii) the MBH population associated 2.2 Populating galaxies with massive black holes to those haloes. We discuss them in turn. In building and We need to associate to each merging galaxy in our catalogue evolving the population of sources we follow exactly the a central MBH, according to some sensible prescription. The same method as in PaperI, to which we refer the reader for Bertone et al. 2007 catalogue contains many properties of further details, with one important difference: the galactic the merging galaxies, including the bulge mass M , and haloes merger rates are derived using the data provided by bulge the bulge rest frame magnitude M both of the progenitors the Millennium simulation, and not EPS-based models. We V and of the merger remnant. This is all we need in order will justify this choice in the next sub-Section, but note that to populate a galaxy with a central MBH. The process is the two methods yield (within the statistical error) the same twofold. results. This is a result that is important in itself and has far reaching consequences (outside the specific issues related (i) We populate the coalescing galaxies with MBHs ac- to PTAs). cording to four different MBH-host prescriptions:
• MBH − Mbulge in the version given by Tundo et al. 2.1 From merger trees to the Millennium (2007, “Tu” models, see Table 1): database M M BH = 8.31 + 1.12 log bulge , (1) In PaperI we used models based on the Extended Press & M⊙ 1011 M⊙ Schechter (EPS) formalism (Press & Schechter 1974, Lacey & Cole 1993, Sheth & Tormen 1999), that trace the hier- with a dispersion ∆ = 0.33 dex. archical assembly of dark matter haloes through a Monte- • MBH −Mbulge, with a redshift dependence in the ver- Carlo approach. Although EPS–based models tend to over- sion given by Mclure et al. (2006, “Mc” models, see Table predict the bright end of the quasar luminosity function at 1): z < 1 (e.g. Marulli et al. 2006), we showed that EPS halo M BH = 2.07 log(1 + z) − 3.09, (2) merger rates at low redshift are consistent with observations Mbulge of close galaxy pairs (Lin et al. 2004, Bell et al. 2006, De Pro- pris et al. 2007). In this paper we focus on MBHBs, whose with a redshift dependent dispersion ∆ = 0.125z + 0.25 GWs induce timing residuals above the stochastic signal dex (see Figure 3 of Mclure et al., 2006). from the whole population, and are therefore detectable as • MBH −MV as given by Lauer et al. (2007, “La” mod- individual sources. The population of low/medium-redshift els, see Table 1): and high-mass sources will particularly impact on the re- M M + 22 BH = 8.67 − 1.32 V , (3) sults. At low redshift, the EPS-based merger tree outputs M⊙ 2.5 need to be handled with care. In models such as those consid- with dispersion ∆ = 0.35 dex. ered in PaperI, each realization of the Universe is obtained • M − σ as given by Tremaine et al. (2002, “Tr” by reconstructing the merger history of about 200 dark mat- BH models, see Table 1): ter haloes, see Volonteri et al. (2003) for details. The out- puts of the models are a list of coalescences labelled by MBH MBH = 8.13 + 4.02 logσ200, (4) masses (for a given recipe that associates a MBH mass and M⊙ to a given dark matter halo) and redshift. These events are where σ200 is the velocity dispersion in units of 200 km then properly weighted over the observable volume shell at − s 1, and the assumed dispersion of the relation is ∆ = 0.3 each redshift to obtain the distribution d3N/dMdzdt (see dex. σ is obtained applying the Faber-Jackson (Faber & PaperI), that is the coalescence rate (the number of coales- Jackson 1976) relation in the form reported by Lauer et cences N per time interval dt) in the chirp mass and redshift al. 2007 to the values of MV obtained by the catalogue. interval [M, M + dM] and [z,z + dz], respectively. The re- sulting distribution is reasonably smooth over most of the To each merging system we assign MBH masses according (M,z) plane, but small number statistics becomes impor- to equations (1)-(4) so that we have the masses of the two 9 tant at z < 0.5 and M > 10 M⊙, which is an important MBH progenitors, M1 and M2. For each prescription we also
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MBH − Mbulge MBH − Mbulge MBH − MV MBH − σ Tundo et (2007) Mclure et al. (2006) Lauer et al. (2007) Tremaine et al. (2002)
Single BH accretion Tu-SA Mc-SA La-SA Tr-SA 6.9 (2.7) 6.6 (2.5) 8.1 (3.0) 6.2 (2.5) 1.5 (1.1) 1.0 (0.8) 1.7 (1.2) 0.8 (0.8) 0.2 (0.4) 0.02 (0.1) 0.5 (0.6) 0.01 (0.1) 0.04 (0.2) 0.002 (0.04) 0.1 (0.2) 0.002 (0.04)
Double BH accretion Tu-DA Mc-DA La-DA Tr-DA 8.3 (2.9) 7.3 (2.7) 9.6 (3.2) 7.0 (2.8) 2.2 (1.4) 1.6 (1.1) 2.6 (1.5) 1.2 (1.0) 0.6 (0.7) 0.2 (0.4) 0.8 (0.8) 0.06 (0.2) 0.2 (0.4) 0.03 (0.2) 0.3 (0.5) 0.007 (0.1)
No accretion (before merger) Tu-NA Mc-NA La-NA Tr-NA 6.4 (2.5) 6.0 (2.4) 6.8 (2.7) 6.0 (2.5) 1.3 (1.0) 0.5 (0.6) 1.5 (1.1) 0.5 (0.6) 0.1 (0.3) 0.07 (0.1) 0.1 (0.3) 0.003 (0.05) 0.02 (0.1) 0.001 (0.03) 0.02 (0.1) − (−)
Table 1. The table summarises the 12 models of assembly of massive black hole binary populations considered in the paper – ”Tu”, ”Mc”, ”La” and ”Tr” identify the MBH-host relation; “SA”, “DA”, and “NA” label the accretion mode; full details on the models are given in Section 2 – and the total number of individually resolvable systems N(δtgw ) for selected values of the characteristic timing residuals (for each model, from top to bottom δtgw = 1, 10, 50 and 100 ns considering an integration time of 5 yrs). The values in the table are the sample mean and standard deviation within brackets computed over the 1000 Monte-Carlo realisations for each model.
M calculate the mass of the MBH remnant, Mr, using the same β = r − 1 . (6) equations. In all cases (1)-(4), the remnant mass is Mr > M1 + M2 M1 + M2, consistent with the fact that MBHs are expected We label this accretion mode as “DA” (double BH accre- to grow predominantly by accretion. We also emphasize that tion, see Table 1). the observed scatter is included in each relation, according to the observational evidence that similar bulges may host The ”SA” and the ”DA” modes are to be expected in gas- significantly different MBHs. poor mergers, especially in non-equal-mass mergers, where (ii) For each MBH-host relation we consider three differ- the dynamical evolution of the binary is much slower (e.g., ent accretion scenarios: Yu 2002) than the infall timescale of the gas (e.g., Cox et al. 2008). In a stellar environment the orbital decay of MBHBs • The masses of the coalescing MBHs are M1 and M2. is expected to be much longer than in a gaseous environment That is, either no accretion occurs, and the merger rem- (e.g., Sesana et al. 2007, Dotti et al. 2006). nant, M1 + M2 < Mr, sits below the predicted mass, or The MBHB models that we consider here relay on two accretion is triggered after the MBHB coalescence. We la- assumptions: all bulges host a MBH, and a MBHB always bel this accretion mode as “NA” (no accretion, see Table coalesces following the hosts’ merger. Regarding the first 1). assumption, dynamical processes such as gravitational re- Post-coalescence accretion is expected for gas-rich coil and triple MBH interactions may deplete bulges from mergers, where MBH pairing and coalescence is believed their central MBH. However, if one compares the mass func- to occur on very short timescales (Mayer et al. 2007). If tion of coalescing binaries obtained here to the results of we are to assume that during a galaxy merger the MBH EPS merger tree models, where both gravitational recoil and remnant is always brought on the correct correlation with triple interactions are consistently taken into account, one its host, by the combination of merging and accretion, two finds that the two distributions (shown in the upper–left additional routes are possible. panel of figure 1) are in excellent agreement, within the sta- • Accretion is triggered before the MBHB coalescence tistical uncertainties. This is because triple interactions are and only the more massive MBH (M1) accretes mass; in likely to eject the lighter MBH from the host, leaving behind this case the masses of the coalescing MBHs are αM1 and a massive binary in the vast majority of the cases (Volon- M2, where teri Haardt & Madau 2003, Hoffman & Loeb 2007). Grav- itational recoil, on the other hand, may be effective in ex- Mr − M2 α = − 1 . (5) pelling light MBHs from protogalaxies at high redshift, but M 1 has probably a negligible impact on the population of MBHs We label this accretion mode as “SA” (single BH accre- in the mass range of interest for PTA observations (Volonteri tion, see Table 1). 2007). The assumption that MBHBs coalesce within an Hub- • Accretion is triggered before the MBHB coalescence ble time following the hosts’ merger is justified by several and both MBHs are allowed to accrete the same fractional recent studies of MBHB dynamics. The stellar distribution amount of mass; in this case the masses of the coalescing interacting with the binary may be efficiently repopulated MBHs are βM1 and βM2, where as a consequence of non-axisymmetric or triaxial galaxy po-
c 0000 RAS, MNRAS 000, 000–000 Massive black holes and pulsar timing 5 tentials (Merritt & Poon 2004, Berzcik et al. 2006), or by massive perturbers (Perets & Alexander 2008); moreover, if one considers post-Newtonian corrections to the binary evo- lution and the effects of eccentricity, one finds that the coa- lescence timescale is significantly reduced (Berentzen et al. 2008). If the binary evolution is gas–driven, typical harden- ing timescales are expected to be shorter than 108 yr (Escala et al. 2005, Dotti et al. 2006). In summary, we build a total of 12 models (4 MBH-host prescriptions × 3 accretion modes). Hereinafter, we will re- fer to each model using the labels associated to the prescrip- tions employed in it. As an example, the model based on the MBH − σ relation (”Tr”) with single BH accretion prescrip- tion (”SA”) will be referred to as Tr-SA, see Table 1 for a summary.
2.3 Computing the coalescence distributions Assigning a MBH to each galaxy, we obtain a list of coa- lescences (labelled by MBH masses and redshift); the same output quantity given by the EPS-based merger trees, used as a starting point in PaperI. Each event in the list can be now properly weighted over the observable volume shell at Figure 1. Mass function of coalescing MBHBs according to each redshift to obtain the distribution d3N/dMdzdt along MBH–host relations reported by several authors. The thick his- the lines described in PaperI. tograms refer to the model labeled at the top of the panel and described in the text. Error bars are calculated assuming a Pois- A further technical detail has to be considered to son error in the number count of events from the coalescence cat- smooth the coalescence distributions. The Millennium sim- alogues that contribute to the chirp mass interval. In the top-left ulation provides better statistics for constructing the mass panel the thin histogram is the coalescing MBHB mass function function of merging objects, but the redshift sampling is predicted by the VHMhopk model studied in PaperI. rather poor. In fact, the Millennium database consists of 63 snapshots of the whole simulation. The most recent ones are taken at z = 0, 0.02, 0.04, 0.064 and 0.089; we need, at least at low redshift, to spread the events over a continuum in z, to obtain a sensible distance distribution of the GW sources. To this aim, we decouple the (M,z) dependence in the dif- ferential mass function at z < 0.3 and re-write d3N/dMdzdt in the following form: The M distributions of coalescing binaries are shown in d3N 0 d3N figure 1 for all the MBH-host prescriptions assuming accre- = Ψ(z) × dz = Ψ(z) ×F(M,t) (7) tion on M1 only (models Tr-SA, Tu-SA, Mc-SA, La-SA ). dMdzdt Z0.3 dMdzdt The top-left panel also shows the distribution obtained by a By doing so we are redistributing according to a given reference EPS merger tree model (VHMhopk; Volonteri, Sal- function Ψ(z) the average mass function obtained at z < vaterra & Haardt 2006) used in PaperI. The agreement with 0.3. This is justified since, as expected, the mass func- 8 the MBH − σ prescription (”Tr”) is good for M > 10 M⊙, tion does not show any significant evolution for redshifts although the statistical errors are large due to low number below 0.3. The dependence on z should be of the form 8 2 statistics. The discrepancy for M < 10 M⊙ is due both to Ψ(z) ∝ nG × dVC /dz, where nG is the galaxy/MBH num- the resolution limit of the Millennium simulation and to the ber density and dVC /dz is the differential comoving volume fact that we relate MBHs to bulges. However, the fact that shell. At such small redshifts the impact of merger activity 8 our sample may be incomplete for M < 10 M⊙ has little (if on galaxy/MBH number density is negligible (of order of any) impact on the results of this paper: the MBHB popu- 0.1/Gyr, e.g., Wake et al. 2008; White et al. 2007; Masjedi lation observable with PTAs is by far dominated by sources et al. 2006; Bell et al. 2006); therefore, we assume nG to 8 with M > 10 M⊙, as we have shown in PaperI, and even be constant. On the other hand the Universe can be consid- more so when we consider systems that can be individu- ered Euclidean, so that the differential volume shell is just 2 ally resolved. We will further discuss this point in Section proportional to z . We then obtain the coalescing MBHB 4. Note that, as discussed by, e.g., Lauer et al. 2007 and distribution in the form Tundo et al. 2007, the high mass end of the population de- d3N rived using the M − σ relation (”Tr” models) drops very = C × z2 ×F(M,t), (8) BH dMdzdt steeply. The drop is much faster than in all the other cases, where C is a normalization factor set by the condition that is for distributions inferred from the MBH − Mbulge or the MBH − MV relations (”Tu”, ”Mc”, ”La” models). This 0 0 d3N is because σ seems to converge to a plateau in the limit of dM dz = dM dzC×z2×F(M,t).(9) very massive galaxies (Lauer et al. 2007). Z Z0.3 dMdzdt Z Z0.3
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5/3 −1 3 TIMING RESIDUALS FROM RESOLVED M D ≃ 25.7 MASSIVE BLACK HOLE BINARIES 9 10 M⊙ 100 Mpc The search for GWs using timing data exploits the effect of f −1/3 × ns (17) gravitational radiation on the propagation of the radio waves 5 × 10−8 Hz from one (or more) pulsar(s). The characteristic signature is an overall scale factor that sets the size of the residuals; D of GWs on the time of arrival (TOA) of radio pulses (e.g. is the luminosity distance to the GW source. The expression Sazhin 1978, Detweiler 1979, Bertotti et al. 1983) is a linear for r(p)(t) can be simply obtained from Equations (15), (16) combination of the two independent GW polarisations. In and (17) by shifting the time t → t − d(1 − cos µ)/c, see practice, the analysis consists in computing the difference equation (12) and (13). The two functions a and a× are between the expected and actual TOA of pulses; the timing + the ”antenna beam patterns” that depend on the source residuals contain information on all the effects that have not location in the sky and the polarisation of the wave. been included in the fit, including GWs. In this section we MBHBs observable with PTAs produce a quasi- summarise the observed signal produced by GWs, following monochromatic signal – the frequency change is ≃ 3 × closely Jenet et al. (2004), to which we refer the reader for −2 9 5/3 −8 11/3 −1 10 (M/10 M⊙) (f/5 × 10 Hz) nHz yr – of further details. known form (though unknown parameters). The optimal The observed timing residual generated by a GW source data analysis approach to search for these signals is by means described by the independent polarisation amplitudes h × +, of the well known technique of matched-filtering. The data is set can be represented as: 1 r(t)= (1 + cos µ) [r (t) cos(2ψ)+ r×(t) sin(2ψ) , ] (10) 2 + δt(t)= r(t)+ δtn(t) (18) where t is the time at the receiver, µ is the opening angle where r(t) is the contribution from GWs, and δtn(t) ac- between the GW source and the pulsar relative to Earth and counts for the fluctuations due to noise; the latter contribu- ψ is the GW polarisation angle. The two functions r+,×(t) tion is the superposition of the intrinsic noise in the measure- are defined as ments and the GW stochastic background from the whole (e) (p) population of MBHBs. The angle-averaged optimal signal- r ×(t) = r (t) − r (t) , (11) +, +,× +,× to-noise ratio at which a signal from a MBHB radiating at t (GW) frequency ≈ f can be detected using a single pulsar r(e) (t) = dt′h(e) (t′) , (12) +,× +,× is Z0 t 2 (p) ′ (p) ′ d 2 δtgw(f) r+,×(t) = dt h+,× t − (1 − cos µ) . (13) hρ i = . (19) c δtrms(f) Z0 h i
(e) (p) In the previous expression δtrms(f) is the root-mean-square Note that r+,×(t) and r+,×(t) have the same functional form, and result from the integration of the time evolution of value of the noise level δtn at frequency f, h.i represents the the polarisation amplitudes at different times, with a delay average over the source position in the sky and orientation of ≃ 3.3 × 103 (d/1 kpc)(1 − cos µ) yr, where d is the distance the orbital plane, and δtgw(f) is the characteristic amplitude of the pulsar from the Earth. For GWs propagating exactly of the timing residual over the observation time T defined along the Earth-pulsar direction, there is no effect on the as: TOAs (r(t) = 0 for cos µ = ±1). 8 δtgw(f)= α(f) fT , (20) From now on we will concentrate on the timing residu- 15 p als produced by binary systems in circular orbit. We model where the numerical pre-factor comes from the angle average gravitational radiation at the leading quadrupole Newto- of the amplitude of the signal: nian order, that is fully justified by the fact that binaries 2 2 2 2 2 8 in the mass and frequency range considered here are far a (1 + cos ι) + 4a× cos ι = . (21) + 15 from the final merger; in fact, the time to coalescence is 9 −5/3 −8 −8/3 Equation 19 is appropriate to describe observations us- ≃ 615(M/10 M⊙) (f/5 × 10 Hz) yr. The timing residuals (10) can be written as (Jenet et al, 2004): ing a single pulsar. In reality one can take advantage of the several pulsars that are continuously monitored to increase r(t)= r(e)(t) − r(p)(t) (14) the signal-to-noise ratio, and therefore the confidence of de- tection: adding coherently the residuals from several pul- where sars – currently the Parkes PTA contains about 20 pulsars, (e) 2 r (t)= α(t) a+(1 + cos ι) cos Φ(t) + 2a× cos ι sin Φ(t) (15) and more are expected to available with future instruments – yields an increase in signal-to-noise ratio proportional to In the previous expressions ι is the source inclination angle, the square-root of the number of pulsars used in the observa- Φ(t) the GW phase related to the frequency f(t) (twice the tions. We will use the characteristic amplitude of the resid- orbital frequency) by uals δtgw to quantify the strength of a GW signal in PTA t observations; δtgw can be used to compute in a straightfor- ′ ′ Φ(t) = 2π f(t )dt , (16) ward way the signal-to-noise ratio, as a function of the noise Z level and number of pulsars in the array (all of which are and quantities that do not depend on the astrophysical model), M5/3 and therefore asses the probability of detection of sources in α[f(t)] = [πf(t)]−1/3 D the context of a given MBHB assembly scenario.
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4 RESULTS For each of the twelve models considered in Section 2 – the models are the result of four MBH-host galaxy prescriptions and three different accretion scenarios – that encompass a very broad range of MBHB’s assembly scenarios, we com- pute the number of sources that are potentially resolvable individually and several statistical properties of the popu- lation, such as the redshift, chip-mass and frequency distri- butions, by means of Monte-Carlo realisations of the whole population of MBHBs according to a given model. Before describing the results, we provide details about each step in the computation of the relevant quantities. The distribution given by equation (8) is straightfor- 3 wardly converted into d N/dzdMdlnfr, i.e. the comoving number of binaries emitting in a given logarithmic (rest- frame) frequency interval with chirp mass and redshift in the range [M, M + dM] and [z,z + dz], along the lines de- scribed in Section 3 of PaperI. As in PaperI, we then assume that in the frequency range of interest (f > 3 × 10−9 Hz) the binary evolution is driven by GW emission only. This is a reasonable assumption, since the coalescence timescale 6 for these systems is typically ∼<10 yr; any other putative mechanism (i.e. star ejection, gas torques) of angular mo- Figure 2. Summary of the properties of the population of mas- mentum removal must have an enormous efficiency to com- sive black hole binary systems – according to model Tu-SA – pete with radiation reaction on such short timescales. For that generate gravitational waves in the frequency window cov- ered by Pulsar Timing Arrays. Top-left panel: characteristic am- each of the twelve models we estimate the GW stochastic plitude of the timing residuals δtgw (equation( 20)) as a function background (and the corresponding rms value of the tim- of frequency; the asterisks are the residuals generated by individ- ing residuals as a function of frequency) generated by the ual sources and the solid line is the estimated level of the GW sources following the scheme described in Section 4 of Pa- stochastic background. Top-right panel: δtgw as a function of the perI. Finally we generate distribution of bright, individually number N(δtgw ) of total (dotted line) and individually resolvable resolvable sources by running 1000 Monte-Carlo realisations (solid line) sources, see equation 22. Bottom-left panel: Distri- of the whole population of MBHBs and by selecting only bution of the number of total (dotted lines) and resolvable (solid those sources whose characteristic timing residuals, equation lines) sources per logarithmic frequency interval dN(δtgw )/d log f ( 20), exceed the stochastic background level. We note that as a function of the GW frequency for different values of δtgw : the result of which and how many sources raise above the av- from top to bottom 1, 10 and 100 ns, respectively. Bottom-right panel: distribution of the total (dotted lines) and individually erage noise level depends on the duration of the observation resolvable (solid lines) number of sources per logarithmic chirp T , for two reasons: (i) T affects the size of the observational mass interval dN(δtgw )/d log M as a function of chirp mass for window in frequency space, in particular the minimum fre- different values of δtgw : from top to bottom 1, 10 and 100 ns, quency 1/T that can be reached, and (ii) the background respectively. An observation time of 5 years is assumed. level decreases as the observation time increases (as the size of the frequency resolution bin ∆f = 1/T decreases), en- hancing the number of individually resolvable sources. For parameter); the values quoted in the next section and sum- the results that are described here and summarised in table marised in table 1 refer to the sample mean computed over 1, we set T = 5 yrs; increasing the data span to 10 years, the set of Monte-Carlo realisations and the sample standard the background level would be slightly lower, and the ad- deviation. Figure 4 quantifies the typical 1-σ error in our ditional resolved sources would be barely brighter than the estimate of the number of sources. background. The statistics of bright, well resolvable sources is basically unaffected. We can then cast the results in terms of the cumulative number of resolvable sources as a function 4.1 Single resolvable binaries of the timing residuals, according to: The large number of Monte-Carlo realisations allows us ∞ dN to study the details of the properties of the individual N(δt )= δt′ , (22) gw ′ gw sources in a statistical sense. We concentrate in particular Zδt δtgw gw on the physical properties of the population, such as the ex- where the integral is restricted to the sources that produce pected number of sources per logarithmic frequency interval residuals above the rms level of the stochastic background; dN(δtgw)/d log f, chirp mass range dN(δtgw)/d log M and if we do not consider this additional constraint, then, for redshift dN(δtgw)/dz, and the observable parameters, such any given δtgw , N(δtgw) simply returns the total number of as the timing residuals produced by each system and the sources in the Monte-Carlo realisation above that particular overall expected number of resolvable MBHBs at a given residual threshold. level of timing residual noise. A summary of the typical Each Monte-Carlo realisation clearly yields a different range of information that can be extracted from the sim- value for N(δtgw) (or its distribution according to a given ulations is shown in figure 2 for the specific model Tu-SA.
c 0000 RAS, MNRAS 000, 000–000 8 A. Sesana et al.
Figure 3. Left panel: The effect of the MBH-host galaxy relation, assuming that accretion always takes place for a single black hole before merger (”SA” models), on the number of observable systems. The plot shows the number of total (thin lines) and resolvable (thick lines) sources N(δtgw ) as a function of δtgw , see equation 22. Four different MBH merger scenarios are considered: Tu-SA (solid line), Tr-SA (long–dashed lines), Mc-SA (short–dashed lines) and La-SA (dotted lines), see Section 2 for a description of the models. Right panel: The effect of the MBH accretion model on the number of individually observable systems. The plot shows the number of resolvable sources only, N(δtgw ) as a function of δtgw . As reference for the MBH-host galaxy relation, models ”La” (thick lines) and ”Tr” (thin lines) are considered. The line style is as follow: model La-SA and Tr-SA (solid lines), La-DA and Tr-DA (short–dashed lines), and La-NA and Tr-NA (long–dashed lines). The duration of the observation is set to T =5 yr
The top-left panel shows the induced residuals of each sin- quency at which the background level equals the selected gle source compared to the level produced by the stochas- value of δtgw. At higher frequencies the number of sources tic background from the whole population; the plot clearly drastically drops because the number of emitting binaries is shows the importance of taking into account the additional a quite steep function of frequency, N(f) ∝ f −8/3; the de- ”noise contribution” from the brightness of the GW sky in crease at lower frequencies is because most of the emitters considering the detectability of resolvable systems. There actually contribute to the background and are not individ- are many sources inducing residuals above, say the 5 ns ually resolved (as clearly shown by the dotted lines). The level, however most of them contribute to the build-up of qualitative behaviour of the results obtained using different the background, and are not individually resolvable. The astrophysical models is very similar to the one described expected number of bright resolvable MBHBs at frequencies in figure 2, with differences that affect only the numerical < 10−7Hz, and at a timing level > 1ns, is typically around values of the different quantities. ten, with fainter sources resolvable at higher frequencies. The central question that we want to address in this The top-right panel shows the mean number of individual paper is what is the expected number of individually resolv- sources detectable as a function of δt from 1000 Monte- gw able sources that produce an effective timing residual above Carlo realizations of the emitting population. In this partic- a given value, as a function of different models of MBHB ular case a sensitivity of ≈ 10 ns is required to resolve an formation and evolution. We summarise these results in fig- individual source; for a timing precision of 100 ns there is ures 3 and 4, and in Table 1, where we show the mean total a 5% chances to observe a particularly bright source. Note number of individual sources that exceed a given level of that at the 1 ns level, there are ∼ 100 MBHBs contribut- timing residual (as defined by equation (22)), as a function ing to the signal, however 90% of them contribute to the of the timing residual. The qualitative behaviour of the re- background and only about 10 sources are individually re- sults is similar for all the scenarios, but the actual numbers solvable. In the two bottom panels of figure 2 we plot the vary significantly. In fact, both MBH-host relations and ac- frequency and chirp mass distributions of resolvable sources cretion prescriptions have a strong impact on the statistics for selected values of the minimum detectable residual am- of the bright sources and, consequently, on their detection. plitude δt . Not surprisingly, the chirp mass of observable gw We analyse the effect of the black holes populations and of systems decreases for smaller values of the considered resid- the accretion in turn. The left panel of figure 3 shows re- ual threshold, however even for an rms level of 1 ns all the 8 sults where the accretion prescription is the same (”SA”), systems are characterised by M > 10 M⊙. The frequency but the underlying MBH-host relation changes; on the other distribution shows instead a peak corresponding to the fre- hand, in the right panel, we select two MBH-host relations,
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Figure 4. The number of individually resolvable sources for se- Figure 6. Distribution of the number of resolvable sources as lected MBHBs assembly models. The plots show N(δtgw ) as a a function of the gravitational wave frequency. The plots show function of δtgw , see equations 22 and 20, for a typical model dN(δtgw )/d log f for different values of the characteristic ampli- (Tu-SA, top panel), and, in the lower panel, the two models that tude of the timing residuals (from right to left δtgw = 1, 10, 50 yield the largest (La-DA: top curve) and smallest (Tr-NA: bottom and 100 ns). Each panel refers to a different MBH-host relation, curve) number of sources. The solid lines represent the mean value while SA mode is considered (see labels in each panel). Chirp of N(δtgw ) and the shaded area the 1−σ region as computed from mass and redshift distributions for the same models are given in 1000 Monte-Carlo realisations of each MBHB population, see also figures 7 and 8. Table 1.
Figure 7. Chirp mass distribution of the number of resolvable Figure 5. Top panel Cumulative mean number of resolvable sources. The plots show dN(δtgw )/d log M for the same charac- sources N(δtgw ) as a function of the characteristic timing resid- teristic amplitude of the timing residuals and models as in fig- 9 ual δtgw for different mass cuts: solid line: M > 10 M⊙; dashed ures 6 and 8. 8 9 7 line: 10 M⊙ < M < 10 M⊙; dotted line 10 M⊙ < M < 8 10 M⊙.Bottom panel: same as top panel but for different red- shift intervals: solid line z < 0.1; dashed line 0.1
c 0000 RAS, MNRAS 000, 000–000 10 A. Sesana et al.
Figure 8. Redshift distribution of the resolvable sources. The Figure 9. The characteristic amplitude of the GW stochastic plots show dN(δtgw )/d log z for the same characteristic amplitude background from the population of MBHB systems. In each panel of the timing residuals and models as in figures 6 and 7. the thin lines identify the estimated level of the stochastic back- ground assuming “SA” (solid line), “DA” (dashed line) and “NA” (dotted line) accretion modes. The total GW amplitude from a but change the accretion history. The left panel shows that single Monte–Carlo realisation of the signal corresponding to the assuming a sensitivity threshold of 30 ns, one expects to ob- “SA” accretion mode is also shown as thick solid line. serve of the order of one source in the La-SA model, while there is only a probability ≈ 5% for the Tr-SA model. If we maintain the same MBH-host relation and we consider different accretion scenarios for e.g. the Lauer et al. popula- tion, the mean number of expected sources varies by a fac- tor of ≈ 5 between 0.3 (La-NA) and 1.5 (La-DA), see right 9 panel. Figure 3 shows that in the most pessimistic case – Tr- 10 M⊙ produce indeed larger timing residuals, but are also NA model, that has the sharpest cut–off of the MBHB mass much rarer). Interestingly, the vast majority of detectable < < function bright end – a precision of ≈ 5ns should guarantee a sources are at redshift 0.1 ∼ z ∼ 1, which shows that PTAs positive detection; in the optimistic La-DA case, a precision could probe the medium-redshift Universe, and are unlikely of ≈ 50ns should be sufficient. In turn the timing precision to discover nearby sources. The reason is simply that, at required for positive detection is in the range 5 − 50ns, that least at small redshift, the Universe volume increases as z3. is basically consistent with a factor ≈ 10 uncertainty in the A summary of the properties of individual re- background level estimated in PaperI. solvable sources, dN(δtgw)/d log f, dN(δtgw)/d log M and The typical spread around the mean values obtained in dN(δtgw)/d log z, is given in figures 6, 7 and 8, respectively, the Monte-Calro realisations is shown in figure 4 for selected for all the four MBHB population models considered here, models. When N(δtgw) ≫ 1 the 1-σ range is roughly the with accretion limited to a single black hole prior to merger Poisson error around the mean (reflecting the uncorrelated considering different residual thresholds δtgw = 1, 10, 50, 100 nature of sources in each Monte Carlo realisation). When ns. All the models show the same qualitative features, as we N(δtgw) < 1 it can be interpreted as the probability to find have highlighted before. The frequency distribution shown a single source above the considered δtgw value if the actual in figure 6 was discussed above and is the same for all the MBHB population in the Universe follows the prescription models. The distribution of the detectable sources as a func- 8 given by the considered model; in this case a non-detection tion of chirp mass (figure 7) peaks at ≈ 3 × 10 M⊙ for all is trivially consistent with the model predictions. Table 1 models assuming δtgw = 1 ns. Increasing δtgw, the distribu- provides a summary of the results for the 12 models. tion peak shifts towards higher M and the mean number It is also interesting to investigate the mass-redshift dis- of events is strongly model dependent for δtgw > 10 ns, cf. tribution of the detectable sources. Figure 5 shows the ex- the values in Table 1. The redshift distributions (figure 8) pected number of detectable individual sources (as a func- consistently show a broad peak in the range 0.2
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8 particle in the Millennium simulation is 8.66/h100 ×10 M⊙, allowing the reconstruction of haloes with minimum mass of 10 the order of ≈ 5 × 10 M⊙. Assuming a barionic fraction of 0.1, the simulation is then incomplete for barionic struc- 9 tures less massive than ≈ 5 × 10 M⊙. We checked this by plotting the mass function of barionic structures and find- 9 ing a sudden drop below 10 M⊙. It is then inevitable that in the results derived from the Millennium Simulation most 6 of the MBHs with mass below a few×10 M⊙ are missing. Since many of these MBHs are expected to merge with more massive ones during cosmic history, the (spurious) lack of MBHs in this mass rage explains the flattening of the mass function dN˙ M /d log M shown in the top-left panel of figure 1. All the backgrounds are rather similar at f > 10−7 Hz because all the MBH prescriptions adopted lead to similar 8 MBH mass functions at MBH < 10 M⊙ (this fact is inde- pendent of the incompleteness issue). This means that the slope of the background on the right of the knee has a well defined dependence upon the adopted MBH population: the more pronounced is the high mass tail of the MBH mass function, the steeper is the high frequency end of the GW background. In turn, models constructed using the Millen- Figure 10. The characteristic amplitude of the GW stochastic nium simulation confirm the findings of PaperI. background compared to the estimate given in PaperI. The thick dashed line is the stochastic background predicted by the Tr-SA model; the dotted lines bound the background levels computed for 5 SUMMARY AND CONCLUSIONS all the models investigated in this paper. For comparison, the solid thick line is the background predicted by the VHMhopk model We have investigated the ability of Pulsar Timing Arrays and the shaded area the range of uncertainty of the strength of (PTAs) to resolve individual massive black hole binary sys- the signal, as reported in PaperI (see Sections 4 and 5 of PaperI tems by detecting gravitational radiation produced during for details). the in-spiral phase through its effect on the residuals of the time of arrival of signals from radio pulsars. We have con- sidered a broad range of assembly scenarios, using the data 4.2 Stochastic background of the Millennium simulation to evaluate the galactic haloes As a sanity check, we compare the stochastic backgrounds merger rates, and a total of twelve different models that derived according to the MBH populations inferred using control the relations between the mass of the central black the Millennium Simulation to the predictions of EPS–based holes and the galactic haloes, and the evolution of the black models reported in PaperI (the reader is referred to PaperI hole masses through accretion. These models therefore cover for the technical details). In figure 9 we show Monte Carlo qualitatively (and to large extent quantitatively) the whole generated signals for each model; in each panel we plot the spectrum of MBH assembly scenarios currently considered. stochastic levels according to the three different accretion Regardless of the model, we estimate that at least one re- modes discussed in Section 2. Both the accretion prescrip- solvable source is expected to produce timing residuals in tion and the adopted MBH-host correlation influence the the range ∼ 5 − 50 ns, and therefore future PTAs, and in level of the background. If accretion occurs onto the rem- particular the Square Kilometre Array may be able to ob- nant (i.e. after coalescence, ”NA” models), the predicted serve these systems. A whistle-stop summary of the mod- characteristic amplitude of the GW background can be up els and results is contained in Table 1. The total number to a factor of 3 lower with respect to models in which both of visible events clearly depend on the sensitivity of PTAs the MBHs accrete before the final coalescence (”DA” mod- and on the astrophysical scenario. As expected, the bright- els); on the other hand, MBH − σ (”Tr”) models predict est sources (for PTAs) are very massive binaries with chirp 8 lower backgrounds compared to MBH −Mbulge (”Tu”, ”Mc”) mass M > 5×10 M⊙. However, (initially) quite surprisingly and MBH − MV (”La”) models. A comparison of these re- most of the resolvable sources are located at relatively high sults with those presented in PaperI is given in figure 10. At redshift (z > 0.2). In conjunction with the observation of 10−8 Hz the models studied here cover a characteristic am- the stochastic GW background from the whole population plitude range consistent with the uncertainty estimated in of MBHBs, the identification of individual MBHBs could PaperI. The Tr-SA model predicts a stochastic background provide new constraints on the populations of these objects that agrees with the typical EPS–model within 30% for and the relevant physical processes. f < 10−7Hz. The high frequency end is instead steeper; this As a by-product of the analysis, we have also estimated effect is caused by incompleteness in the low mass end of the the level of the GW stochastic background produced by MBH population. As shown in figure 1, the mass function of the different models, finding good agreement with the es- coalescing MBHB derived from the Millennium simulation timates derived using merger tree realisations based on the is not consistent with the one obtained using the EPS for- Extended Press & Schechter formalism considered in PaperI. 8 malism for M < 10 M⊙. The weight of a single dark matter Such agreement provides a further validation of the results
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