Insights on the astrophysics of supermassive black hole binaries from pulsar timing observations
aa r X i v : . [ a s t r o - ph . C O ] J u l Insights on the astrophysics of supermassive blackhole binaries from pulsar timing observations
A Sesana Max-Planck-Institut f¨ur Gravitationsphysik, Albert Einstein Institut, Am M¨ulenber1, 14476 Golm, Germany
Abstract.
Pulsar timing arrays (PTAs) are designed to detect the predictedgravitational wave (GW) background produced by a cosmological population ofsupermassive black hole (SMBH) binaries. In this contribution I review the physics ofsuch GW background, highlighting its dependence on the overall binary population,the relation between SMBHs and their hosts, and their coupling with the stellar andgaseous environment. The latter is particularly relevant when it drives the binaries toextreme eccentricities ( e > .
1. introduction
The pulsar timing arrays (PTAs) described in this volume, provide a unique opportunityto obtain the very first low-frequency gravitational wave (GW) detection. The EuropeanPulsar Timing Array (EPTA) [1], the North American Nanohertz Observatory forGravitational Waves (NANOGrav) [2], and the Parkes Pulsar Timing Array (PPTA)[3], joining together in the International Pulsar Timing Array (IPTA) [4], are constantlyimproving their sensitivity in the frequency range of ∼ − − − Hz. Inspirallingsupermassive black hole (SMBH) binaries populating merging galaxies throughout theUniverse are expected to generate the dominant signal in this frequency band [5, 6, 7, 8].Despite the fact that theoretical models of galaxy formation in the standardhierarchical framework predict a large population of SMBH binaries forming duringgalaxy mergers, to date there is only circumstantial observational evidence of theirexistence. Less than 20 SMBH pairs with separations of ∼
10 pc to ∼
10 kpc are knownto date (see [9], for a comprehensive review). At smaller separation, only a handfulof candidate sub-parsec bound Keplerian SMBH binaries have been identified, basedon peculiar broad emission line shifts [10, 11]; however, alternative explanations to thebinary hypothesis exist [9], and unquestionable observational evidence is still missing.If as abundant as predicted, SMBH binaries are expected to form a low frequencybackground of gravitational waves (GWs) with a typical strain amplitude A ∼ − at a frequency f = 1 / yr [7, 8, 12, 13], with a considerable uncertainty of ≈ ‡ .The aforementioned studies indicate that such a signal is expected to be dominatedby a handful of sources, some of which might be individually resolvable. On onehand, the unresolved background provides innovative ways to test fundamental physicsand alternative theories of gravity; on the other hand, electromagnetic counterparts toindividually resolvable sources can be searched for with a number of facilities openingnew avenues toward a multimessenger based understanding of these fascinating systemsand their hosts. These themes are not included in this paper, but are covered in K.J. Leeand T. Tanaka & Z. Haiman contributions to the present special issue. Here I providea general overview of the predicted GW signal as a whole, discussing uncertaintiesin normalization and spectral shape stemming from the underlying properties of theemitting binaries. I will be generally concerned with the level of the background , withoutentering into its peculiar properties in terms of non-Gaussianity and resolvability [14, 12],nor in issues related to detection, which are treated in the contributions by X. Siemensand collaborators, J. Ellis and N. Cornish & A. Sesana.The paper is organized as follows. In Section 2, the concept of GW backgroundis introduced, and the relevant ingredients that enter its computation are identified.The main focus of Section 3 is on the overall cosmological population of SMBH binaries(namely their number and typical masses) and on the information that can be extractedby a putative PTA observation. Section 4 is devoted to the coupled dynamical evolutionof SMBH binaries and their star/gas rich environment. This coupling has importantconsequences on the source frequency distribution and their eccentricity, which leavesimportant signatures in the signal. A summary of the main results is given in Section5. Throughout the paper a concordance Λ–CDM universe with Ω M = 0 .
27, Ω λ = 0 . h = 0 . G = c = 1.
2. General model of the GW background
Consider a cosmological population of merging SMBH binaries. Each merging pairis characterized by the masses of the two holes M > M , defining the massratio q = M /M . Without making any restrictive assumption about the physicalmechanism driving the binary semimajor axis and eccentricity evolution, we can writethe characteristic amplitude h c of the GW signal generated by such population as: h c ( f ) = Z ∞ dz Z ∞ dM Z dq d NdzdM dqdt r dt r d ln f K ,r × h ( f K ,r ) ∞ X n =1 g [ n, e ( f K ,r )]( n/ δ (cid:20) f − nf K ,r z (cid:21) . (1) ‡ In astronomy, the notation dex is commonly used for the log unit; therefore 0.5dex= 10 . . Here, h ( f K ,r ) is the strain emitted by a circular binary at a Keplerian rest framefrequency f K ,r , averaged over source orientations h ( f K ,r ) = r M / D (2 πf K ,r ) / , (2)where we have introduced the chirp mass M = ( M M ) / / ( M + M ) / , and thecomoving distance to the source D . The function g ( n, e ) [15] accounts for the fact thatthe binary radiates GWs in the whole spectrum of harmonics f r,n = nf K ,r ( n = 1 , , ... ),and is given by, e.g., equations (5)-(7) in [16]. The δ function, ensures that each harmonic n contributes to the signal at an observed frequency f = nf K ,r / (1 + z ), where thefactor 1 + z is given by the cosmological redshift. d N/ ( dzdM dqdt r ) is the differentialcosmological coalescence rate (number of coalescences per year) of SMBH binaries perunit redshift z , primary mass M , and mass ratio q , and dt r /d ln f K ,r is the time spentby the binary at each logarithmic frequency interval. These two latter terms, takentogether, simply give the instantaneous population of comoving systems orbiting at agiven logarithmic Keplerian frequency interval per unit redshift, mass and mass ratio. Inthe case of circular GW driven binaries, g ( n, e ) = δ n , dt/d ln f is given by the standardquadrupole formula, and equation (1) reduces to the usual form h c ( f ) = 4 f − / π / Z Z dzd M d ndzd M z ) / M / , (3)where we have introduced the differential merger remnant density (i.e. number ofmergers remnants per co moving volume) d n/ ( dzd M ) (see [17, 18] for details). Inthis case, h c ∝ f − / ; it is therefore customary to write the characteristic amplitude inthe form h c = A ( f / yr − ) − / , where A is the amplitude of the signal at the referencefrequency f = 1yr − . Observational limits on the GW background are usually given interms of A .Equation (1), together with a prescription for the eccentricity distribution of theemitting SMBH binaries as a function of the frequency, namely e ( M , q, f K , r ), providesthe most general description of the GW background generated by a population of SMBHbinaries. The signal depends on three distinctive terms:(i) the cosmological coalescence rate of SMBH binaries in the Universe, d N/ ( dzdM dqdt r );(ii) the specific frequency evolution of each binary, dt r /d ln f K ,r ;(iii) the eccentricity evolution of the systems, which determines the emitted spectrumfor a given binary Keplerian frequency.In the following section, we will examine the impact of the items listed above on the GWsignal; on the other hand we will highlight the enormous potential of PTA observations inimproving our understanding of the global population of SMBH binaries in our Universeand of their dynamical evolution.
3. Spectral normalization: cosmological SMBH binary coalescence rate
As written in equation (1), the GW strain amplitude is proportional to the square root ofthe cosmic coalescence rate of SMBH binaries, and it is sensitive to the mass distributionof those binaries. The SMBH binary coalescence rate therefore sets the normalization of the detectable signal. This, in practice, depends on four ingredients: (i) the galaxymerger rate; (ii) the relation between SMBHs and their hosts, (iii) the efficiency of SMBHcoalescence following galaxy mergers and (iv) when and how accretion is triggered duringa merger event.
Galaxy merger rate.
Despite the number of observations of massive galaxies atrelatively low redshift, their merger rate is not very well constrained, and is one of themajor factors of uncertainties in the calculation of the signal. As detailed in [18], onepossible observationally based way to estimate the galaxy differential merger rate is thefollowing: d n G dzdM dq = φ ( M, z ) M ln 10 F ( z, M, q ) τ ( z, M, q ) dt r dz . (4)Here, φ ( M, z ) = ( dn/d log M ) z is the galaxy mass function measured at redshift z ; F ( M, q, z ) = ( df /dq ) M,z is the differential fraction of galaxies with mass M at redshift z paired with a secondary galaxy having a mass ratio in the range [ q, q + δq ], and τ ( z, M, q )is the typical merger timescale for a galaxy pair with a given M and q at a given z . φ and F can be directly measured from observations, whereas τ can be inferred bydetailed numerical simulations of galaxy mergers. All these quantities are known atbest to within a factor of 2 , implying that the galaxy merger rate can be estimatedwithin an accuracy of a factor of a few. Alternatively, the galaxy merger rate can beestimated from large N-body simulations of structure formation. Here the problem isthat only few of those are available to date (see, e.g., the Millennium run [19]), and itis therefore difficult to extract sensible errorbars on the numbers. SMBH-host relations.
The more massive the SMBH, the stronger the emittedGW signal. Observationally, SMBHs correlate both with the velocity dispersion, σ ,and the mass, M bulge , of the host galaxy bulge [20, 21]. However, those correlationscome in different flavors, and are constantly re-calibrated to include new availabledata (see, e.g., [22, 23, 24, 25, 26]). Most noticeably, the discovery of SMBHs with M > M ⊙ in two brightest cluster galaxies (BCGs) [27] resulted in a recentupward revision to the established SMBH-host relations by a factor 0.2-0.3dex [25, 26].Moreover, these relations have a significant intrinsic scatter ( ≈ Efficiency of SMBH coalescence.
Even if galaxies merge, the two SMBHs have tomake their way to the center of the merger remnant, form a Keplerian binary, and getrid of their energy and angular momentum to enter the efficient GW emission stage.In most of the models underlying current analysis efforts [8, 13, 12], the coalescenceefficiency is taken into account through the estimate of the Chandrasekhar dynamicalfriction timescale. If this timescale is longer than the Hubble time, the secondarySMBH never makes it to the center of the merger remnant. Otherwise, a Keplerianbinary forms, and its subsequent evolution is assumed to occur quickly on cosmologicaltimescales (e.g., . z occurs at z − ∆ z , where ∆ z is given by the the dynamical friction timescale). Accretion.
A further factor of uncertainty is related to accretion onto the SMBHs.During galaxy mergers, large amounts of gas are subject to dynamical instabilities andare prone to fall towards the minimum of the evolving potential well [32], eventuallytriggering accretion that increases the mass of the SMBHs. Whether this occurs before,during or after the Keplerian binary stage has an effect onto the effective mass and massratio of the GW emitting systems. Quantitative estimations by [14] showed that thiscan have a factor of ≈ The first, obvious payoff of a PTA detection is the direct confirmation of the existenceof a vast population of sub-pc (to be precise, sub-0 . . z <
1, but uncertainties in allsuch measurements are reflected into a large range of predicted signals. The differencebetween the top-left and the top-right panel is given by the recent upgrades in theSMBH mass-host relation [25, 26] to include the overmassive black holes measured inBCGs [27]. The range of expected signal is boosted by a factor of two, with the 99 . Figure 1.
Characteristic amplitude of the GW signal. Shaded areas represent the68%, 95% and 99 .
7% confidence levels given by our models. In each panel, the blackasterisk marks the best current limit from [33]. Shaded areas in the upper left panelrefer to the 95% confidence level given by [13] (red) and the uncertainty range estimatedby [8]. See main text for discussion of the individual panels. which is A ≈ × − . In the lower panels, we consider two subset of the modelsfeaturing these upgraded relations: (i) those in which accretion does not occur priorto binary coalescence, and (ii) those in which accretion precedes the formation of thebinary, and is more prominent on the secondary SMBH [35]. In the latter case, binariesobserved by PTA are way more massive, and with a larger mass ratio, implying a muchlarger (by almost a factor of three) signal. If, for instance, a signal with amplitude A ∼ × − at a frequency of 1yr − is detected, this would support a picture in whichSMBHs accrete copious amount of gas before forming a binary, since SMBHs that do notdo so, thus correlating with the merger progenitors, are unable to produce such a strongGW background. However, the more likely A ∼ × − region, can be the result ofseveral combinations of the parameters defining the merging SMBH binary population,and detailed information about each single ingredients will be hard to disentangle.
4. Spectral shape: environment coupling and eccentricity evolution
SMBH binaries evolve in a complex, dense astrophysical environment. Forming aftergalaxy mergers, they sit at the center of the stellar bulge of the remnant, and theyare possibly surrounded by massive gas inflows triggered by dynamical instabilitiesrelated to the strong variations of the gravitational potential during the merger episode.Accordingly, two major routes for the SMBH binary dynamical evolution have beenexplored in the literature: (i) gas driven binaries, and (ii) stellar driven binaries. Adetailed description of both scenarios is beyond the scope of this contribution; here weconsider simple evolutionary routes and assess their impact on the GW signal.
Let restrict ourselves to circular binaries first. Whatever is the driving dynamicalmechanism, the emitted GW is always given by equation (2). What is different isthe time spent by the binary at a given frequency, enclosed in the dt r /d ln f r term. Inthe GW driven case this is simply dt r d ln f r = 564 π / M − / f − / r . (5)Combining equations (5) and (2) yields a contribution to h c ∝ M / q / f − / . Therefore,integrating over the coalescence rate, the standard f − / power law follows.A background of stars scattering off the binary, drives its semimajor axis evolutionaccording to the equation [36] dadt = a Gρσ H, (6)where ρ is the density of the background stars, σ is the stellar velocity dispersion and H is a numerical coefficient of order 15. A problem with equation (6) is that theSMBH binary efficiently ejects stars from the galaxy core, and the subsequent evolutionrelies on the pace at which they diffuse into the so called binary loss cone . As shownby [18], substituting ρ i at the binary influence radius ( r i ≈ GM/σ ) in equation (6)corresponds to ’full loss cone at the influence radius’, which has to be expected in acomplex triaxial environment of a merger remnant, as corroborated by recent numericalsimulations [28, 29]. If we consider, for simplicity, an isothermal sphere, we substitute ρ i in equation (6), and we assume M BH ∝ σ , we get that in the stellar driven case dt/d ln f ∝ f / M / , which yields to a contribution of the single binary to the GWbackground of the form h c ∝ M qf .In the case of circumbinary disks, things are even more subtle, and the detailedevolution of the system depends on the complicated and uncertain dissipative physicsof the disk itself. Here we consider the simple case of a coplanar prograde disk, with acentral cavity maintained by the torque exerted by the binary onto the disk. No mass isallowed to flow through the cavity and the mass accumulates at its edge. This scenarioadmits a selfconsistent, non stationary solution that was worked out by [37]. In thiscase, the binary evolution rate can be approximated as [37, 38] dadt = 2 ˙ Mµ ( aa ) / . (7)Here, ˙ M is the mass accretion rate at the outer edge of the disk, a is the semimajoraxis at which the mass of the unperturbed disk equals the mass of the secondary blackhole, and µ is the reduced mass of the binary. Considering a standard geometricallythin, optically thick disk model [39], one finds dt/d ln f ∝ f − / M / , which yield to acontribution of the single binary to the GW background of the form h c ∝ M / q / f / .Compared to the GW driven case, ( da/dt ) GW ∝ a − , equations (6) and (7) havea very different (milder and positive) a dependence. Therefore, equating equations (6)and (7) to ( da/dt ) GW gives the transition frequency between the external environmentdriven and the GW driven regimes: f star / GW ≈ × − M − / q − / Hz f gas / GW ≈ × − M − / q − / Hz , (8)where M = M/ M ⊙ . We therefore see that if the signal is dominated by 10 M ⊙ SMBH binaries, then the transition frequency is located around 10 − Hz.
The eccentricity evolution of the binary has a major impact on the GW backgroundthrough the function g ( n, e ) [15]. The net effect of a large eccentricity is to move powerfrom the second to higher harmonics. However, since the energy carried by the waveis proportional to f h , shifting the emission to higher harmonics effectively removes power at low frequencies [40], without a significant enhancement (just marginal) of h at higher frequencies. Therefore, generally speaking, highly eccentric binaries pose athreat to PTA GW detection.It is well known that GW emission efficiently circularizes binaries, however thingscan be drastically different in the star and gas dominated stages. If binaries getvery eccentric in those phases, they can retain substantial eccentricity even duringthe GW dominated inspiral relevant to PTA observations, beyond the decouplingfrequencies given by equation (8). The eccentricity evolution in stellar environmentshas been tackled by several authors by means of full N-body simulations. Despite thelimited number of particles ( N < ), resulting in very noisy behavior for the binaryeccentricity, clear trend have been tracked. In general, equal mass, circular binariestend to stay circular or experience a mild eccentricity increase [41], while binaries thatform already eccentric, or with q ≪ § , e , which is often found to be larger than 0.6 in numerical studies [29].Large e implies that systems emitting in the nHz regime can be highly eccentric, causinga significant suppression of the GW signal, as we will see in the next section. In thecircumbinary disk scenario, excitation of eccentricity has been seen in several simulations[45, 46]. In particular, the existence of a limiting eccentricity has been studied in [47]through a suite of high resolution smoothed particle hydrodynamics simulations, in thecase of massive selfgravitating disks. They find a critical value e crit ≈ . − .
8. Theauthors presented an analytical model that agrees with their simulations, predicting thelimiting eccentricity to be: e crit = 0 . p ln( δ − .
65) + 0 .
19, where δ ≈ R c /a is theradius of the disk central cavity in units of the binary semimajor axis. Therefore, alsoin gaseous rich environments, eccentric binaries might be the norm (even though theextreme eccentricities ( e > .
9) that might be reached in the stellar driven case areunlikely).
The main effect of the binary-environment coupling is to suppress the low frequencysignal [48, 40, 49], as shown in figure 2. The energy of the SMBH binaries is transferredto the environment instead of going into GWs, the binary evolution is faster, andconsequently there are less systems emitting at each frequency. The transition frequency(equation (8)) is around 10 − Hz (corresponding to 30 yr timescale), but can still have asignificant impact on the detection. As depicted in figure 2, PTA limits on the observedbackground are usually established at very low frequency (given by the timespan ofthe observations) and then extrapolated at 1yr − assuming an f − / power law. Onemight therefore think that, just by keeping observing, PTAs will eventually hit thelow frequency background. If the spectral shape changes or, even worse, if there is aturnover frequency, this will not be the case. In figure 2 we examine one specific SMBHpopulation model, but we vary the environmental coupling. If we assume roughly thecurrent sensitivity, observing for 8 more years will eventually lead to a detection ifSMBH binaries are circular and GW driven . It might take just a couple of years moreif the system are driven by stellar scattering, but it might take some extra 10 yearsmore if all the systems are surrounded by massive circumbinary disks (this clearlydepends on the detailed physics of the disk-binary interaction, we just show here aselected case for the sake of the discussion). The situation gets even worse if binariesare eccentric. In particular, we consider the case where SMBH binaries are stellardriven, and all have an eccentricity e = 0 . § The moment of formation, or pairing, is defined here as the transition point between the earlydynamical friction driven stage and the late three body ejection driven stage in the binary evolution.This occurs when the mass in stars enclosed in the binary semimajor axis a is of the order of M , whichcorresponds to a ≈ − ∼ M ⊙ systems of interest here. More details can be found in[44]. Figure 2.
Influence of the binary-environment coupling on the GW signal. The blackdotted line is the standard f − / spectrum for a population of circular GW drivensystems. Red lines are for star driven binaries with eccentricity of 0 (solid) and 0.7(long-dashed) at pairing; the green dot-dashed line is for circular gas-driven binaries.A sketch of the current PTA sensitivity is given by the solid blue line, which is thenextrapolated to the limit at 1 yr − . Also shown in blue are extrapolation of thecurrent sensitivity to include 8 and 30 more years of observations (here we assumeno improvement in the timing of the pulsars, the mild improvement in the sensitivityfloor is given by the T / gain that comes from the longer integration time), as wellas the sensitivity given by putative arrays with 4 and 6 time better timing precision.We stress that the sensitivity curves are sketchy and only illustrative, but capture thetrends relevant to the discussion in the text. spectrum at f < × − Hz. With the current timing precision, 30 more years would beneeded to detect such signal. It is therefore extremely important for PTAs to constantlyimprove their intrinsic sensitivity, by reducing timing noise or adding new pulsars, toavoid unpleasant surprises related to SMBH binary dynamics. It is also clear that thedetermination of the GW background spectral slope carries a lot of information aboutthe dynamics of SMBH binaries. A well defined turnover frequency around 10 − Hzwill be the distinctive signature that strong coupling with the environment is the norm,1whereas a plateau might be indicative of a population of highly eccentric systems. PTAdetection will therefore provide important information about the dynamics of individualSMBH binaries, not only about the statistics of their collective population.
5. Conclusions
Pulsar timing arrays are achieving sensitivities that might allow the detection of thepredicted GW background produced by a cosmological population of SMBH binaries.Beyond the obvious excitement of a direct GW observation, the detection of suchsignal, together with the determination of its amplitude and spectral slope, will providean enormous wealth of information about these fascinating astrophysical systems, inparticular:(i) it will give direct unquestionable evidence of the existence of a large population ofsub-parsec SMBH binaries, proving another crucial prediction of the hierarchicalmodel of structure formation;(ii) it will demonstrate that the ’final parsec problem’ is solved by nature;(iii) it will provide important information about the global properties of the SMBHbinary population, giving, for example, insights about the relation between SMBHbinaries and their hosts;(iv) it will inform us about the dynamics of SMBH binaries and their stellarand/or gaseous environment, possibly constraining the efficiency of their mutualinteraction;(v) it will tell us if very eccentric SMBH binaries are the norm.Identification and sky localization of individual sources (not treated here, see T. Tanaka& Z. Haiman contribution to this issue), will add further items to this list, makingmultimessenger studies of SMBH binaries and their hosts possible. Pulsar timing arraysare not mere gravitational wave detectors, but also groundbreaking astrophysical probesthat will shed new light on some of the fundamental, yet most elusive objects of ourUniverse: supermassive black hole binaries.
Acknowledgments
A.S. acknowledges the DFG grant SFB/TR 7 Gravitational Wave Astronomy and byDLR (Deutsches Zentrum fur Luft- und Raumfahrt).
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