ON THE COMPLEMENTARITY OF PULSAR TIMING AND SPACE LASER INTERFEROMETRY FOR THE INDIVIDUAL DETECTION OF SUPERMASSIVE BLACK HOLE BINARIES

In the frequency band between 10⁻⁹ Hz and some fraction of 10⁻⁶ Hz, PTA offers the unique chance to observe gravitational radiation. Beyond the observation of a stochastic background, the challenge of observing a single SMBHB by PTA has recently received growing attention. Simulations concur in predicting a scenario wherein some SMBHBs stick out of the stochastic signal floor of unresolved SMBHBs. The observation of an individual SMBHB provides opportunities for new measurements in general relativity and new perspectives for the scientific community.

The state of the art on timing residual precision¹ from current PTAs² lies in the 100–50 ns domain, while improvement by SKA³ down to 10–1 ns is expected (Liu et al. 2011).

Investigations⁴ on individual sources aim to recover physical parameters such as the spins and the masses of, and the distance to, an SMBHB (Sesana & Vecchio 2010) using the gravitational wave front curvature (Deng & Finn 2011), or the Pulsar term⁵ (Corbin & Cornish 2010; Lee et al. 2011; Mingarelli et al. 2012). From these studies it emerges that the distance to a pulsar is important for individual SMBHBs, unlike statistical background observation (Jenet et al. 2004; Sesana et al. 2009; Finn & Lommen 2010). Finn & Lommen (2010) and Pitkin (2012) analyzed bursts coming from different sources including individual SMBHBs.

Burt et al. (2011) suggest refraining from searching for new pulsars, unless close to an existing cluster of good pulsars; instead, they recommend the allocation of more observation time to already low-noise pulsars.

Turning to specific observation targets, searches have so far produced negative results: Jenet et al. (2004) found no evidence of the emission of gravitational waves by a supposed SMBHB in 3C 66B; neither did Lommen & Backer (2001), who were seeking evidence for an SMBHB in Sgr A*.

Finally, Yardley et al. (2010) describe the observations used to produce the sensitivity curves for the Parkes radio telescope and propose a method for detecting significant sinusoids in PTA.

Gravitational wave detection in space was first proposed by means of a small-sized interferometer on board a single satellite (Grassi-Strini et al. 1979), before shaping into a triangular satellite configuration (Bertotti 1984; Faller et al. 1985, 1989). The LISA pathfinder (Antonucci et al. 2012) is deemed an important step toward technological maturity.

NGO⁶ (ESA 2011), which is derived from the previous LISA proposal,⁷ is a space project designed to measure gravitational radiation over a broadband at low frequencies, where the universe is richly populated by strong sources of gravitational waves, including SMBHBs. NGO plans to trace the formation, growth, and merger history of supermassive black holes (SMBHs) during different epochs, measuring spin and masses, with an unprecedented precision, often where the universe is blind with our current electromagnetic techniques (ESA 2011). In fundamental physics, different tests on general relativity, including the no-hair theorem and the dynamics in strong field, and on alternative theories will be feasible with SLI. NGO (Amaro-Seoane et al. 2012a, 2012b) implies a shift to higher frequencies of the sensitivity band, as compared to LISA.

Pitkin et al. (2008) first proposed sequential detection, but it appeared necessary to improve and update their initial work for several reasons. We begin by ascertaining that the total SMBHB mass is generally larger than the 50 million solar masses considered by Pitkin et al. (2008). New analyses are carried out herein. First, we examine various astrophysical scenarios combining SMBH–host relations and accretion processes, and dry and wet mergers. Second, we span a large range of residuals (50–1 ns). Third, we estimate the number of events and the probability of sequential detection, building our investigation upon the recent statistical findings of individual detection by PTA–SKA, which were not available at the time of the work by Pitkin et al. (2008). Fourth, we present the signal-to-noise ratio (S/N) of SLI for the sources concerned.

Section 2 is devoted to the computation of transit times (how long it takes for a binary to switch bands) from the PTA–SKA to the SLI band for a range of SMBHB masses. In the same section, the ringing frequencies are computed for the (stationary and rotating) SMBHB remnants.

Section 3 is the core of the paper, in which we study the impact of astrophysical, observational, and detection constraints separately, one step after the other, steadily building up our analysis. Details of the models are described by Sesana et al. (2009), to which the reader is referred for further information. SMBH-host relation models are then combined with accretion prescription models, and different values of timing residuals are considered. We determine first the maximum number of sequential detections within a given time to merger, assuming that SLI would catch all the sources that were previously observed by PTA–SKA. We then consider the impact of the SLI low cutoff frequency and of the spin of the remnants. Then for NGO, we compute the S/N at different values of the redshift z, and we finally adapt our previous estimates for sequential detection.

Section 4 sums up the conclusions; the Appendix attempts to sketch some of the operational scenarios that may lie ahead.

We refer to the total mass of the binary M, normalized to 1 + z, where z is the redshift (Hughes 2002). The mass enters in the orbit evolution equation with the timescale Gm/c³, the timescale being redshifted. The consequence⁸ is that a binary at z = 1.5 and of mass 2 × 10⁸ M_☉ is equivalent to a binary at z = 0.2 and of mass 4.17 × 10⁸ M_☉. Herein, given the range of z, the factor (1 + z) introduces an uncertainty of less than 2.25 on the mass determination.

We use the terms of observation and detection when referring to PTA–SKA and SLI, respectively, while sequential detection implies both observation by PTA–SKA and detection by SLI. Finally, the term "event" refers mostly to astrophysical phenomena, or it is used whenever a specific labeling is not intended.

Concerning individual observation solely by PTA, Sesana & Vecchio (2010) estimated that only a handful of observable binaries exist, including those at the 5 ns effective noise level; the number of resolvable systems quickly drops if the timing precision degrades to, e.g., 50 ns. This result confirms that of Sesana et al. (2009), who analyzed a wide range of population models and indicated 5–15 individual sources having residuals larger than the stochastic background.¹¹ The residuals are between 2 and 60 ns in the frequency range 2 × 10⁻⁸–10⁻⁷ Hz. Further, they identified most of the individually resolvable SMBHB sources as having a mass larger than 0.5 × 10⁹ M_☉ and lying at a redshift0.2 < z < 1.5.

These findings are based on a statistical sample of merging massive galaxies generated from the online Millennium database¹² built by Springel et al. (2005). The Millennium simulation covers a comoving volume of (500/h₁₀₀)³ Mpc³ (h₁₀₀ = H₀/100 km s⁻¹ Mpc⁻¹ is the normalized Hubble parameter). Sesana and coworkers populated the merging galaxies with central SMBHs according to different models. For each model, the expected distribution of bright individual sources and the associated timing residuals were computed.

The following procedure has been adopted. Out of the Millennium database, for every cell of the (m₁, m₂, z) distribution, we determine the radiating frequency of those SMBHBs which are (1) in an orbital phase at most time t_r before merger; (2) radiating at a frequency below f_p; and (3) generating a timing residual larger than a given threshold. We compute the number of SMBHBs per unit time and per frequency bin (of size equal to 1/t_r) such that ${\cal N}(f) = \int 1/t_{r} d{\cal N}(f)/df$. Afterward, we integrate $d^4{\cal N}/dm_1 dm_2 dz dt$ along t_r and obtain the number of sources for a given bin. Finally, the results are summed up for all bins, masses, and redshifts. The outcome is the number of sources, individually observable by PTA–SKA, merging in time t_r.

This observation probability depends on the details of the merging SMBHB population, and specifically on the number of coalescing binaries and on their mass. Dependence on other intrinsic binary parameters such as spin and eccentricity may be negligible, though recent studies show that 1.5 post-Newtonian terms, including spin-orbits effects, may be detected (Mingarelli et al. 2012). Indeed, such systems, in the PTA–SKA band, will be far enough from merger for spin-orbit and spin–spin terms not to affect our analysis significantly, but will be close enough to merger, such that the residual eccentricity is likely to be smaller than 0.1.

Since binaries producing timing residuals in this frequency range are assumed to be extremely massive systems, the rates will depend on the high-mass end of the SMBH mass function. This is an important point since this is precisely where the two most popular massive black hole (MBH) mass predictors give inconsistent results. MBH masses inferred by the M–σ relation (Tremaine et al. 2002; Gultekin et al. 2009; Graham et al. 2011) can indeed be up to an order of magnitude lower with respect to their M-bulge inferred counterparts (Häring & Rix 2004; Tundo et al. 2007; Lauer et al. 2007; Gultekin et al. 2009).

We have tested nine different models, implementing three different SMBH–host relations and three different accretion prescriptions. On this basis, we have built our catalogue of merging SMBHs starting from the Millennium Run. Nine cases result from the cross-combinations.

The SMBH–host relations explored, including intrinsic scattering (Sesana et al. 2009), are configured in three types.

• A.  
  M–σ relation according to Tremaine et al. (2002). Following their relation, SMBH ∝σ⁴ implies that SMBHs with masses larger than 10⁹ M_☉ are extremely rare.
• B.  
  M–σ relation according to Gultekin et al. (2009). This relation shows a steeper dependence on σ, SMBH ∝σ^4.24; the intrinsic scattering is larger than in Tremaine et al. (2002), predicting more massive binaries.
• C.  
  M-bulge relation, again according to Gultekin et al. (2009). The M-bulge relation generally predicts higher SMBH masses for a given galaxy host; see, e.g., Häring & Rix (2004), Lauer et al. (2007), and Tundo et al. (2007). In this case, we have SMBH up to 10¹⁰ M_☉.

Accretion prescriptions (Sesana et al. 2009) are grouped in three classes.

• a.  
  Accretion occurs onto the SMBH's remnant, meaning that the two merging SMBHs are undermassive with respect to the selected SMBH–host relation.
• b.  
  Accretion occurs onto the primary SMBH before coalescence, meaning that the total mass of the two merging systems follows the SMBH–host relation, but the mass ratio is usually quite high (implying a weaker gravitational wave signal).
• c.  
  Accretion occurs onto both SMBHs before coalescence, meaning that the total mass of the two merging systems follows the SMBH–host relation, and the mass ratio of the merging SMBH is closer to unity (the most favorable situation for gravitational wave detection).

The combination of the above assumptions encompasses a large range of SMBH mass functions, bracketing the observations. The most pessimistic model—Aa—predicts that hardly any SMBHBs with a total mass larger than 10⁹ M_☉ will coalesce, while the most optimistic model—Cc—delivers coalescences with a total mass even larger than 10¹⁰ M_☉.

Twenty-seven cases are shown; see Table 1. We consider only SMBHBs that are individually observable by PTA–SKA, generating a timing residual above a given threshold (50, 5, 1 ns), and merging in less than 10 years. For these sources, we provide the total number of observations in 10 years in n_{10, 10} units (we consider f_p = 8 × 10⁻⁷ Hz only). The resolvable SMBHBs are mostly situated at a redshift 0.2 < z < 1.5.

The paucity of the rates in Table 1 is far from promising. Further, there is a spread of seven orders of magnitude between the most pessimistic model combined with a 50 ns timing precision and the most optimistic model combined with a 1 ns timing precision. For a 10 year survey of SMBHBs merging within 10 years, the Cc scenario predicts 4.7 × 10⁻² observations at 1 ns, whereas the Aa scenario predicts 3.5 × 10⁻⁹ at 50 ns. We emphasize that the values in Table 1 include those SMBHBs which would not enter any SLI bandwidth, as they are too massive.

But a different reading of the results is possible, as outlined in the abstract. Indeed, these numbers may be interpreted as ideal upper limits for sequential detection, when assuming (1) an SLI detector with a low cutoff coincident to the high cutoff frequency of PTA–SKA, i.e., f_s = f_p; (2) a noiseless interferometer; under these two conditions, no source would remain undetected by SLI; and finally (3) PTA (or SKA) and SLI operating simultaneously for the same survey duration.

Further, the rates in Table 1 are approximately proportional to the time to merger of the SMBHB, and are obviously proportional to the duration of the survey, so we may switch to $N_{y^{-2}}$ units. Thus, when referring to the conditions (1–3) above, the probability of sequential detection at 1 ns will never be better than 4.7 × 10⁻⁴ yr⁻², i.e., per year to merger and per year of survey (Cc), or 3.3 × 10⁻⁶ yr⁻² (Aa), since a factor of 100 stands between the two units.

A third presentation of the results may be proposed, when searching for the ideally minimum number of years necessary for a single sequential detection. The number of years will be given by $\sqrt{\vphantom{A^A}\smash{\hbox{$1/N_{y^{-2}}$}}}$. At 1 ns, for the Cc scenario we get 46 years versus 550 years for Aa.

This is an extremely optimistic, i.e., fairly unrealistic, scenario for sequential detection, as it implies hypotheses (1) and (2) above, (3) pulsar timing and SLI surveys lasting 46 or even 550 years, and (4) a precision of 1 ns (conversely, the number of years may be less, when rates include inspirals with masses lower than 1 × 10⁸ M_☉; provided that PTA–SKA is sensitive to lighter masses, the latter have a longer inspiral time). A space mission lasting for decades or centuries is out of the question, but such a constraint does not have not to be fulfilled literally. Indeed, if the mass of the SMBHB can be evidenced from pulsar timing, it is possible to predict when the SMBHB would be visible by an SLI. It would then be the task of the space agencies to launch a mission in time for this rendezvous, a few months or a few centuries after observation by radio astronomy.

It is worth pointing out that the spread among different model predictions decreases as timing precision improves. From Table 1, we conclude that only the precision of 1 ns timing residuals deserves to be retained for further analysis, as the other precisions are coupled to even more negligible chances.

A caveat is that the population of the merging SMBHBs is constructed out of the Millennium Run. Thus, the total coalescence rate is fixed to that predicted by the Millennium. The actual coalescence rate in the universe is poorly constrained, and it is mostly determined in the high-mass, low-redshift range (relevant to this study), by counting galaxy pairs (Patton et al. 2002; Lin et al. 2004, 2008, 2010; Bell et al. 2006; Xu et al. 2012). The rates predicted by pair counting are consistent with those predicted by semi-analytic merger trees (Volonteri et al. 2003), and by the Millennium Run within a factor of a few (Sesana et al. 2008, 2009). The numbers are, therefore, uncertain by a factor of a few due to different estimates in the merger rate.

Further, the true SMBHB population might not be perfectly described by current models or might come from a completely unexplored physical mechanism (Volonteri 2011).

We now wish to compute how the values in Table 1 are affected by introducing an SLI low cutoff frequency f_s. The answer depends heavily upon the ring-down frequency of the merging system, Equation (3). Here, we consider three different remnant spins (a = 0, 0.9, 0.99) and two different low cutoff frequencies for SLI (f_s = 1 or 3 × 10⁻⁵ Hz). We infer the number of possible sequential detections, assuming 1 ns residual precision. We obtain the values reported in Tables 2 and 3. They are meant as optimal sequential detection rates; the noise characteristics of a given SLI have been ignored. Further, for each column it is assumed that all remnants acquire the same final spin, i.e., the coalescences produce remnants solely of spin a = 0, or a = 0.9, or a = 0.99. Larger rates are associated with higher values of spin.

The optimal sequential detection rates of Tables 2 and 3 are, obviously, lower than the rates of Table 1, because most SMBHs have a ring-down frequency below the f_s frequency cutoff. Numbers, in n_{10, 10} units, are in the range of 10⁻⁴ to 10⁻², if an SLI low-frequency cutoff at 10 μHz is assumed.

The chances of detection critically drop by about two orders of magnitude when the cutoff is shifted even slightly to higher frequencies, from 10 to 30 μHz. Optimal sequential detection rates are in the range of 10⁻⁶ to 10⁻⁴ for an SLI cutoff frequency at 3 × 10⁻⁵ Hz. Further, we have tested that the rates are exactly zero in our models for a cutoff at 10⁻⁴ Hz. The spread in the rate values due to different spins lowers with a decreasing cutoff frequency.

Again, the values may be read differently in $N_{y^{-2}}$ units, obtained by dividing the values in Tables 2 and 3 by a factor of 100. Finally, a single detection occurs every 79 years for the Cc scenario, every 587 years for the Aa one, both for a cutoff at 10 μHz and spin of 0.9, while for a cutoff at 30 μHz and a spin of 0.9, 542 years are required for the Cc scenario, and 1187 years for the Aa one. The preceding values no longer imply condition (1) in the previous section, but the other conditions hold, i.e., (2) noiseless SLI, (3) pulsar timing and SLI surveys lasting 79 or even 1187 years, or alternatively the possibility of launching an SLI mission at any time within a given period, (4) 1 ns timing precision, and (5) all remnants possessing a spin of 0.9.

It is legitimate to ask what the rates would be for scenarios displaying multiple spins. Lousto et al. (2010a, 2010b, 2010c) show an analytic distribution for dry mergers following the Kumaraswamy (1980) functional form, peaked at a = 0.75. For wet mergers, Dotti et al. (2010) provide a statistical distribution, peaked at a = 0.89. The distribution of cases between dry and wet mergers is not known.

We now turn to a specific configuration, namely NGO, and attempt to make a more realistic estimate by considering the contribution of noise. Computations relative to other SLI projects will be carried out when such projects will have been planned by space agencies.

For the computation of the S/N for NGO¹³ (Jennrich et al. 2011), we stick to the mass range of 2 × 10⁸–2 × 10⁹ M_☉, the dimensionless spin Kerr parameter (a = 0, 0.9, 0.99), and introduce four distances (z = 0.2, 0.3, 1, 1.5); see Table 4. Recently, progress has been made on the computation of S/Ns from inspiral to ring-down (Santamaría et al. 2009, 2010; Ajith et al. 2011). Thus, the fifth column in Table 4 shows the S/N for phenomenological wave forms, computed with the PhenomC model (Santamaría et al. 2010). For comparison and for the sole inspiral phase, the fourth column displays the S/N, computed using a post-Newtonian approximation (Poisson & Will 1995).

As expected, the computation of S/N for NGO shows that part of the sources that have come all the way from the PTA–SKA bandwidth to that of NGO are not necessarily detected by the interferometer, due to the presence of noise. The heaviest SMBHB remnants at z = 0.3 are not detected if they are associated with a low spin, and for any spin value for z > 0.5; see Table 4. NGO is optimized for lower mass SMBHBs and this explains the low values of S/N for the sources considered herein.

The lowest frequency requirement on NGO (ESA 2011) is 10⁻⁴ Hz. However, the goal¹⁴ of NGO is to reach 3 × 10⁻⁵ Hz (ESA 2011); further, Jennrich et al. (2011) show that the sensitivity curve stretches below this frequency, down to 10⁻⁵ Hz. This sensitivity curve has been used for computation of the S/N; see Table 4.

To estimate the rates, we must look for a commensurate answer to the spread of several orders of magnitude, due to different astrophysical scenarios (SMBH–host relation, accretion process; for the remnant, fast or slow spin, spin distribution vis-à-vis dry and/or wet mergers) and to the paucity of the events. Further, other uncertainties play an important role. Thus, a coarse estimate on the rates is adequate. The other major uncertainties are the following.

In Section 3.2, we showed that a difference of 20 μHz in the SLI low-frequency cutoff determines a difference in the rates (both n_{10, 10} and $N_{y^{-2}}$ units) of two orders of magnitude. Moreover, we noted that for a cutoff at 10⁻⁴ Hz the chances of sequential detection are null for our simulations. Will NGO perform according to its requirements or its goals? The chances of sequential detection may be meager or simply nonexistent.

The timeline scenario is uncertain at both ends. When will the ambitious 1 ns precision be achieved? Will NGO fly or will there be another SLI-type mission? And when? And of what duration? Let us assume that both conditions, i.e., the residual precision requirement and an SLI launch, will be met in the second half of the next decade. Then, five years may well represent the maximum value for t_r that we can observe. When the latter is coupled to, e.g., a three year SLI survey, this implies that the rates in n_{10, 10} units must be divided approximatively by a factor (10/5) × (10/3) = 6.6.

Finally, the interferometer noise will further reduce the rates. Altogether, it seems judicious to apply between one and two orders of magnitude of reduction to the rates of Table 2, if NGO goals on sensitivity are met and a cutoff at 10⁻⁵ Hz is assumed.