An estimate of the gravitational-wave background from the observed cosmological distribution of quasars
Nicolas Sanchis-Gual,1 Vicent Quilis,2, 3 and José A. Font2, 3 1Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal 2Departament d’Astronomia i Astrofísica, Universitat de València, c/ Dr. Moliner 50, 46100, Burjassot (València), Spain 3Observatori Astronòmic, Universitat de València, c/ Catedrático José Beltrán 2, 46980, Paterna (València), Spain We study the gravitational-wave background from the observed cosmological quasar distribu- tion. Using the DR9Q quasar catalogue from the ninth data release of the Sloan Digital Sky Survey (SDSS), we create a complete, statistically consistent sample of quasars from z = 0.3 to 5.4. Employing the spectroscopic information from the catalogue we estimate the masses of the supermassive black holes hosted by the quasars in the sample, resulting in a log-normal distribu- 8.32±0.33 tion of mean 10 M . The computation of the individual gravitational-wave strains relies on specific functional forms derived from simulations of gravitational collapse and mergers of massive black hole binaries. The background gravitational-wave emission is assembled by adding up the individual signals from each quasar modelled as plane waves whose interference can be constructive or destructive depending on the quasar evolutionary state. Our results indicate that the estimated gravitational-wave background discussed in this work could only be marginally detectable by LISA. This conclusion might change if more complete quasar catalogs than that provided by the SDSS were available.
I. INTRODUCTION merge to reach the observed final masses [24–27]. The 6 direct dynamical formation of MBH with M ≥ 10 M Besides the extraordinary relevance of the landmark or the mergers of binaries of less massive such objects, discoveries of Advanced LIGO and Advanced Virgo, the may produce large-amplitude gravitational waves within detection of gravitational waves from binary black hole reach of the LISA interferometer [28–30]. Even if the as- (BBH) mergers and binary neutron stars (BNS) merg- sociation of SMBH birth from gravitational collapse can- ers [1–5] has updated previous estimates of the stochas- not be connected with every instance of active quasars, tic gravitational-wave backgrounds generated by these such model allows to study the generation of an stochas- sources [6–8]. Such backgrounds are a consequence of tic background by transient processes. However, no de- the superposition of the gravitational waves produced tectable gravitational radiation would be produced if by a large number of unresolved, independent sources, SMBHs were instead the result of mass accretion onto much too weak to be individually detected with cur- significantly less massive seed black holes [31–34]. rent technology. In addition to those originating from Given the uncertainties in the actual processes op- BBH and BNS mergers, stochastic backgrounds of as- erating during the formation of quasars, we assume in trophysical origin exist for other types of sources, both this work that the dynamics of the quasar phase is vio- transient, as supernovae or mergers of white dwarf bina- lent enough to produce low-frequency gravitational-wave ries, and continuous, as spinning neutron stars or mag- bursts and ask ourselves if the combined signals from netars (see [9, 10] and references therein). Moreover, the observed quasar distribution might contribute to a gravitational-wave backgrounds in the nHz regime from new gravitational-wave background. For this purpose arXiv:2106.10039v1 [gr-qc] 18 Jun 2021 supermassive black hole binaries (SMBHBs) have been we use the DR9Q quasar catalogue from the Sloan Dig- extensively studied by the Pulsar Timing Array (PTA) ital Sky Survey [35] to establish a mock quasar dis- community [12–19]. tribution and compute the integrated amplitude of the In this work we estimate the relevance of the observed gravitational-wave strain. This computation relies on cosmological distribution of quasars from redshift z = 0.3 specific functional forms for the strain derived from sim- to 5.4 as a possible stochastic gravitational-wave back- ulations of both gravitational collapse [36] and massive ground. The intense sustained activity of quasars during BBH mergers [37], which we use to produce the stochastic their ∼ 108 yr lifetime (see [20] for current observational gravitational-wave background from the observed quasar estimates) is linked to violent processes associated with sample. We note that results on gravitational waves from the presence of supermassive black holes (SMBHs) in the catalogues have been reported before (e.g. [38, 39]) but 6 10 10 − 10 M mass range at their cores [21–23]. Pro- those works have not considered the observed quasar dis- posed mechanisms for quasar formation typically involve tribution. the gravitational collapse of Population III supermassive This paper is organized as follows. In Section II we stars (SMS) or other large distributions of gas clouds to describe the methodology we follow to obtain our mock yield massive black holes (MBH) that can subsequently sample of quasars and the total gravitational-wave emis- 2
sion. Section III presents our results for two different 10 models. We close with Section IV which presents our 10 conclusions.
109 ]
II. METHODOLOGY sun
A. Quasar sample 108
The ninth data release of the Sloan Digital Sky Sur- 7 vey [35] includes a quasar catalogue (DR9Q) with 87,822 SMBH mass [M 10 objects. This catalogue covers several patches of the sky but is far from providing a full celestial sample and gives −1 an average abundance of 27 QSO deg . Despite the 106 sparseness of the observed catalogue, a mock quasar cat- 23456789 alogue covering the whole sky can be created. To do radial distance from Earth [Gpc] so, we first fit the masses of the SMBH hosted by the quasars to a log-normal distribution (top panel of Fig. 2; the corresponding histogram of redshifts is plotted in the bottom panel). Quasars produce a characteristic elec- tromagnetic emission due to accretion processes of mat- ter onto their central SMBH. Their emission-line spectra present a great number of events in forbidden transitions. Such transitions include the Lyman alpha forest, but also C-III, C-IV, and Mg-II emission lines. The last two can be used to estimate the BH mass [40] by relating the BH mass to the emission-line bolometric luminosity, L, and its fullwidth at half maximum of either transition, 0.60±0.16 FIG. 1: Top panel: SMBH mass distribution of 81% of the 5 LC-IV MBH(C-IV) = 4.5 × 10 42 −1 quasars from the SDSS III DR9Q catalog. Bottom panel: 10 erg s Sky positions of the 70,964 quasars whose SMBH mass was 2 FWHMC-IV derived using C-IV and/or Mg-II emission line spectrometry. × M , (1) 1000 km s−1 0.57±0.12 6 LMg-II MBH(Mg-II) = 2.9 × 10 1042 erg s−1 • First, we produce a random sample of masses (see 2 middle panel of Fig. 2) following the redshift dis- FWHMMg-II tribution of the real catalogue (see bottom panel of × −1 M . (2) 1000 km s Fig. 2). We estimate the masses of the BHs associated with the • Next, we distribute the quasars in three redshift 87,822 quasars of the SDSS DR9Q catalog using Eqs. (1)- bins. Using the Landy & Szalay [41] estimator, (2), where the luminosities are computed combining sev- the correlation length and slope of the DR5Q cata- eral spectrometric parameters provided by the catalogue log sample [42] has been obtained in three studies, for each quasar. When both emission lines are present we for the low-redshift range 0.3 ≤ z ≤ 2.2 [43], the take the arithmetic mean of the two masses. Quasars are intermediate-redshift range 2.2 ≤ z ≤ 2.8 [44], and discarded if none of the lines are present. We are able the high-redshift range 2.9 ≤ z ≤ 5.4 [45]. For sim- to compute the BH mass for a total amount of 70,964 plicity, we associate three different mean redshifts quasars, which represents 81% of the total DR9Q cata- z = {1.80, 2.44, 3.29} of the previous ranges. Fol- 106M log. BH masses span the interval between and lowing the redshift distribution shown in the bot- 1010M . In Fig. 1 we plot the distribution of the SMBH tom panel of Fig. 2 we distribute the proportional masses as a function of the radial distance to Earth and number of quasars to each redshift range, yield- the sky positions of the quasars to illustrate both prop- z ing Nquasars = {243081, 658064, 213854} quasars in erties of the sample. The resulting distribution of masses each bin. can be well described by a normal distribution with a 8.32±0.33 6 mean value of M = 10 M where the error stands The final result is a sample containing ∼ 1, 115 × 10 for 1 σ deviation (see top panel of Fig. 2). QSOs between z = 0.3 to 5.4 and spanning all 4π sr of the Once we have identified the mass distribution of our sky. This sample is statistically comparable to the obser- sample of quasars, we proceed as follows: vational catalogue in the sense that the two-point corre- 3
Νquasars B. Gravitational-wave signal 500
400 The crucial ansatz is this work is that every observed quasar is emitting gravitational waves as a result of the 300 violent processes occurring at their cores. Over cos- mological time scales, the formation of quasars can be 200 approximated as an instantaneous process. Therefore, 100 the observation of a quasar by means of any electro- magnetic signal directly implies that the emission of its M log10 gravitational radiation is also maximum, both in fre- 6 7 8 9 10 11 M☉ quency and strain. Under this assumption, all quasars Νquasars 25 000 detected at a given redshift should be emitting the max- imum of their gravitational-wave signals. Quasars may 20 000 be active electromagnetically during different phases of AGN activity, e.g. through accretion processes. How- 15 000 ever, these timescales are much longer than those of the gravitational-wave channel. We use this notion to un- 10 000 derpin our simplified model in which the bulk of the gravitational-wave emission is either produced during the 5000 gravitational collapse of a SMS or in the late inspiral and M 0 log10 merger of two MBHs. A more complex and realistic de- 6 7 8 9 10 11 M☉ scription should possibly include other AGN-related pro- Ν quasars cess and their gravitational-wave outputs besides the two 500 main channels considered in this paper. Therefore, the 400 results drawn from our models could be interpreted as a lower constraint to the total gravitational-wave back- 300 ground produced by the quasar distribution. We assume that gravitational collapse is likely play- 200 ing a major role on the dynamics and we model the gravitational-wave strain and frequency using results 100 from the numerical relativity literature on direct grav- z itational collapse of supermassive stars [36, 49–53]. Un- 0 1 2 3 4 5 der this assumption, the dominant contribution to the gravitational-wave strain and frequency is given by the FIG. 2: Top panel: Histogram of the observed (DR9Q) SMBH l = m = 2 quasinormal-mode (QNM) ringdown of the mass. Middle panel: Histogram of the complete mock QSO catalogue. Bottom panel: Histogram of redshifts. SMBH 6 −2 −1 10 M fQNM = 2 × 10 (1 + z) [Hz], (4) M 2 1/2 −2 1/2 ∆EGW/Mc 10 [Hz] lation function of both distributions is the same. How- −19 hQNM = 6 × 10 −4 ever, as the SDSS catalogue is not complete, the mock 10 0.5 fQNM 1/2 catalogue generated underestimates the total number of M 1 Gpc quasars. × 6 , (5) 10 M dL In order to fix the properties of each object in the new −4 2 sample, we need to establish a cosmology. We adopt a where ∆EGW . 10 Mc is the total radiated en- ΛCDM model with cosmological parameters H0 = 70 ergy. We include the factor = 0.1 in Eq. (5), ab- −1 −1 km s Mpc , Ωm = 0.3, and ΩΛ = 0.7 [46]. In our sent in the work of Saijo [50], to accommodate new re- approach we also need to know the luminosity distance, sults from recent simulations in which the amplitude is dL, of each quasar. For a given cosmology, this quantity smaller [53, 54]. Note that, in this case, the emission of is given by (see e.g. [47, 48]): gravitational radiation is a short transient process (the collapse of SMS and SMBH birth) and cannot be associ- ated with every stage of quasar activity. While limited by c(1 + z) Z z dx the duration of the burst, this scenario will show that we dL = p (3) can model a stationary stochastic background from tran- H0 (1 + x)3Ω + Ω 0 m Λ sient gravitational waves from quasars. This approach would be a simplification of a more realistic scenario in where c is the speed of light. which the gravitational radiation from the evolution of 4
SMS before undergoing gravitational collapse were con- 3e-16 sidered. In addition, we also consider the potential con- 2e-16 QNM model tribution of MBH mergers to the gravitational-wave sig- 1e-16 (t)
nal [37]. For a MBH binary, assuming that the orbit T 0 is quasi-stationary until the separation reaches the in- h -1e-16 nermost stable circular orbit (ISCO), the gravitational- -2e-16 wave amplitude and maximum frequency at the ISCO are given by 3e-12 2e-12 Binary model 1/2 −5 −1 M1 1e-12 fbin = 4.4 × 10 (1 + z) 1 + M2 (t)
T 0
8 h 10 M -1e-12 × [Hz], (6) M1 -2e-12 5/3 2/3 -3e-12 −17 Mchirp f(1 + z) 0 1e+07 2e+07 3e+07 hbin = 3.5 × 10 8 −7 Time (s) 10 M 10 [Hz] 1 Gpc × (1 + z), (7) FIG. 3: Top panel: total gravitational-wave strain for quasar dL formation from direct gravitational collapse of SMS. Bottom panel: total gravitational-wave strain for quasar formation −1/3 3/5 where Mchirp = [M1M2(M1 + M2) ] . For simplic- from MBH binary mergers after a year of evolution of the ity we consider one single merger of equal-mass compo- sources. nents, namely M1 = M2 = 0.5 MBH. This seems a con- servative assumption considering that cosmological sim- ulations show that SMBHs are formed by a combination study we compute the strain produced by two different of accretion and an important number of merger events models: QNM ringdown and binary BH merger. We as- involving similar objects [55]. The evolution of the MBH sume that the emission from different quasars starts at orbit is not accounted for in our model, and the MBH different times, and the total strain reads: gravitational-wave sources are always active during the simulated observation period. Therefore, we ignore the N X previous contribution of the earlier inspiral stages by tak- hT(tobs) = H(t − tk) hk(tobs − tk) , (9) ing the peak emission and the maximum frequency as an k=1 upper limit for the MBHs. We ignore the contribution of the earlier inspiral stages by taking the peak emission where tobs is the observation time of the gravitational- and the maximum frequency as an upper limit for the wave detector, N is the total number of quasars in our MBH mergers. sample, H(t − tk) is the unit step function and tk is the The gravitational-wave output from each quasar k is time at which the gravitational-wave signal from the k-th modeled as a plane wave quasar is observed for the first time by the gravitational- wave detector. However, to avoid an unphysical non- i(ωkt+φk) hk(t) = h e , (8) stationarity in the binary model since the signals do not decay and the total strain would increase linearly with where ωk is the frequency and φk is the phase. The time, we compute the gravitational-wave strain only after frequency is assumed to be either complex, to account all sources are emitting. In this case, we let the sources for the exponential damping of the QNM signal, or real, evolve during a year and then start the observation pe- when dealing with the (continuous) inspiral signal of the riod. MBH binary, respectively. The real part of ω is com- We further assume one year of observation time, tobs = 6 puted from Eqs. (4) and (6) and the imaginary part is [0 : 32 × 10 ] s with a time resolution of ∆tmin = 100 s. extracted from [56]. Finally, the strain amplitude h is This choice is mainly computationally motivated since evaluated from Eqs. (5) and (7). we need to resolve frequencies that span from 10−6 Hz to 10−4 Hz. Therefore, the computation requires high time resolution but also a long observation period. One year III. RESULTS offers a good time window estimate for the frequencies in this range. Considering the mock catalogue described in Sec- In Fig. 3 we plot the resulting gravitational wave- tion II A and assigning a gravitational-wave signal to form obtained from (9). This figure compares the each object of this sample according to the method de- gravitational-wave output associated with the two most scribed in the previous section, the total gravitational- favored mechanisms for quasar formation, namely grav- wave strain produced by the complete distribution of itational collapse of SMS (blue curve) and mergers of quasars is the sum of all interfering plane waves. In our MBH binaries (red curve). 5
-16 10 these spectra for the two mechanisms. This figure also includes the sensitivity curve of the LISA interferometer -18 and the Galactic confusion noise component, obtained 10 from the analytic fits in [57]. Our results indicate that the amplitude of the estimated gravitational-wave back- -20 10 ground signal hardly falls within the sensitivity range of the LISA interferometer for the binary model but not for -22 the QNM model. We note however that our estimation 10 could be regarded as a lower bound given the fact that the SDSS catalogue is not complete. -24 10 Characteristic strain Binary model QNM model -26 LISA IV. CONCLUSIONS 10 Gal. noise -6 -5 -4 10 10 10 In this work we have studied the gravitational-wave frequency [Hz] background produced by the observed cosmological quasar distribution. To do so we have built a mock sam- FIG. 4: PSD for the two mechanisms. The design sensitiv- ple of quasars covering the whole sky from the mass and ity of LISA and the Galactic confusion noise component are redshift distributions of a real, but limited, sample ex- shown for reference [57]. tracted from the Sloan Digital Sky Survey DR9Q cata- log [35]. We have assumed that each quasar has emitted a gravitational-wave signal that reaches the Earth when a gravitational-wave detector is observing during a one- year period. Our approach has assumed that the SMBH In the binary merger case we let the source evolve at the center of a quasar has formed through two differ- during a one-year span prior to the observation time. ent mechanisms, (a) the gravitational collapse of a SMS The amplitude grows with time in this phase because and (b) the merger of two massive BHs. By computing individual quasars start to emit gravitational waves at the total contribution of all quasars from our mock sam- different times: their contributions add up to the to- ple we have provided estimates of the GW backgrounds tal strain and the signals do not decay. At the begin- produced in the two physical mechanisms investigated ning of the observation time all quasars are active and here. the amplitude is quasi-stationary (see bottom panel of While in this work we have only considered two pos- Fig. 3). On the other hand, since the amplitude of the sible quasar formation channels, this is still an open is- QNM ringing signal decays exponentially with time, the sue undergoing a deep debate. One possible manner to emission is a short transient. However, each quasar is ac- form SMBHs may be through multiple mergers of inter- tive at a different time. The top panel of Fig. 3 shows a mediate mass BHs with masses between 10 and 100 M total gravitational-wave strain with approximately con- from Pop III stars (see for instance [58]). However, this stant amplitude from this type of transient signal. mechanism could be not efficient enough, failing to form Finally, Fig. 3 shows that the amplitude of the sig- the very SMBHs in quasars, as discussed by [59]. The nal is about three orders of magnitude larger in the case other channel, commonly accepted to form the SMBHs of MBH mergers than for direct SMBH formation from present in quasars, is by mergers of black holes of similar the gravitational collapse of SMS. The frequency range masses. In this case, depending on the model, the num- −4 is shown up to ∼ 3 × 10 Hz, which is the maximum ber of mergers between black holes with masses larger 6 frequency of our sample, obtained from the mass dis- than 10 M , would range between 1 and 10 [60]. More- tribution and Eqs. (4) and (6). Furthermore, the total over, observational evidences seem to point that the pres- amplitude in the binary case is larger because the grav- ence of SMBHs in quasars could be well explained by itational wave of each quasar does not decay with time, a simple scenario of only a few galaxy mergers hosting since the frequency is real. In this case, a beating pattern SMBHs [61]. Other processes related to the evolution of in observed. The frequency distribution of our sample of AGN, such as accretion, could be considered as potential −4.78 quasars is peaked around a mean frequency of 10 Hz sources of gravitational waves. The inherent complexity and a standard deviation of σ = 0.332. Therefore, most of these process is out of the scope of the simple mod- of the quasars are emitting in a small frequency range, els employed in this paper, whose main goal has been creating the interference phenomena. On the other hand, to provide a lower estimate of total gravitational-wave for the QNM model the beating is also present, with a signal produced by the quasar distribution. −4.57 mean frequency and the standard deviation, 10 Hz As a consequence of our findings, the non-detection and σ = 0.330. of the cosmological gravitational-wave background pro- By performing a Fourier transform of the waveforms posed in this work would also be a relevant result. Thus, we obtain the corresponding power spectra. Fig. 4 shows the eventual absence of signal from the cosmological dis- 6 tribution of QSOs would imply that the process of forma- tions (CIDMA) through the Portuguese Foundation for tion and evolution of SMBHs within galaxies would be Science and Technology (FCT - Fundação para a Ciên- dominated by a quiescent form of accretion rather than cia e a Tecnologia), references UIDB/04106/2020 and by any violent phase of activity. UIDP/04106/2020 and by national funds (OE), through FCT, I.P., in the scope of the framework contract fore- seen in the numbers 4, 5 and 6 of the article 23, of Acknowledgments the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from We thank the referee for his/her constructive criticism the projects PTDC/FIS-OUT/28407/2017, CERN/FIS- and useful suggestions and Julien Baur for his early work PAR/0027/2019 and PTDC/FIS-AST/3041/2020. This on this project during a research internship at the Univer- work has further been supported by the Euro- sity of Valencia. Work supported by the Spanish Agen- pean Union’s Horizon 2020 research and innova- cia Estatal de Investigación (grants PID2019-107427GB- tion (RISE) programme H2020-MSCA-RISE-2017 Grant C33, PGC2018-095984-B-I00), by the Generalitat Valen- No. FunFiCO-777740. The authors would like to ac- ciana (grant PROMETEO/2019/071), by the Center for knowledge networking support by the COST Action Research and Development in Mathematics and Applica- CA16104.
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