Searching for primordial black holes with stochastic gravitational-wave background in the space-based detector frequency band
SSearching for primordial black holes with stochastic gravitational-wave background inthe space-based detector frequency band
Yi-Fan Wang,
1, 2, 3, ∗ Qing-Guo Huang,
4, 5, 6, 7
Tjonnie G.F. Li, and Shihong Liao Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut), D-30167 Hannover, Germany Leibniz Universit¨at Hannover, D-30167 Hannover, Germany Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, 88 South University Avenue, 225009, Yangzhou, China Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University, 36 Lushan Lu, 410081, Changsha, China Key Laboratory for Computational Astrophysics, National Astronomical Observatories,Chinese Academy of Sciences, Beijing 100012, China
Assuming that primordial black holes compose a fraction of dark matter, some of them mayaccumulate at the center of galaxy and perform a prograde or retrograde orbit against the gravitypointing towards the center exerted by the central massive black hole. If the mass of primordialblack holes is of the order of stellar mass or smaller, such extreme mass ratio inspirals can emitgravitational waves and form a background due to incoherent superposition of all the contributions ofthe Universe. We investigate the stochastic gravitational-wave background energy density spectrafrom the directional source, the primordial black holes surrounding Sagittarius A ∗ of the MilkyWay, and the isotropic extragalactic total contribution, respectively. As will be shown, the resultantstochastic gravitational-wave background energy density shows different spectrum features suchas the peak positions in the frequency domain for the above two kinds of sources. Detection ofstochastic gravitational-wave background with such a feature may provide evidence for the existenceof primordial black holes. Conversely, a null searching result can put constraints on the abundanceof primordial black holes in dark matter. I. INTRODUCTION
The recent direct detections of gravitational waves(GWs) by the LIGO and Virgo collaborations open aunique window to observe black holes (BHs) [1–8]. Theevent rate of binary BH merger at local Universe is es-timated to be 53 . +58 . − . Gpc − yr − from the detections[9]. Among the GW events, the relatively large mass ofthe first detection ( ∼ M (cid:12) ), GW150914, has stimulateddiscussions that the binary BHs of GW150914 could beof primordial origin, instead of products of stellar evolu-tion [10–13]. Ref. [12] shows that the binary stellar-massprimordial BHs coalescence scenario can give the correctorder of magnitude of event rate, if the abundance ofprimordial BHs in dark matter is ∼ − .Primordial BHs are a long hypothesized candidate fordark matter [14–17]. Assuming all the primordial BHshave the same mass, a variety of observations from as-tronomy and cosmology have given constraints on theprimordial BH abundance in dark matter, for example,gravitational lensing of stars and quasars, dynamics ofdwarf galaxies, large scale structure formation and accre-tion effects on the cosmic microwave background (CMB)(see Refs. [18–21] and references therein). The possibil-ity that all the dark matter are primordial BHs with the ∗ [email protected] same mass has been ruled out given all the constraintsaforementioned (but see, e.g., Ref. [22]). Nevertheless, itis still interesting to consider the scenario where primor-dial BHs compose a part of dark matter and propose newmethods to seek for evidence of primordial BHs or con-strain their abundance in dark matter, especially lever-aging the newly opened GW window [23, 24].In this work, we investigate the scenario in which pri-mordial BHs constitute a fraction of dark matter in thegalactic center. Astrophysical observations (see, e.g.,Refs. [25, 26] for a review) indicate that massive BHs withmass 10 M (cid:12) − M (cid:12) are ubiquitous and reside at thecenter of almost every massive galaxy. If some fractionof dark matter is composited by primordial BHs, theyshould perform a prograde or retrograde orbit againstthe gravity pointing towards the galactic center exertedby the central massive BH, and such a system becomesthe so-called extreme mass ratio inspiral (EMRI) systemwhose mass ratio is usually larger than 10 [27]. EMRIsare one of the important scientific targets of space-basedGW detector, such as Laser Interferometer Space An-tenna (LISA) which is anticipated to be launched in the2030s . Once detected, the GW signals from EMRIs canprovide valuable information such as event rate estima-tion [27, 28] and tests of general relativity [29, 30]. a r X i v : . [ a s t r o - ph . C O ] A p r The focus of our work is the stochastic gravitational-wave background (SGWB) energy density spectrum fromthe EMRI system consisting of a massive BH at the galac-tic center and a subsolar mass primordial BH. SGWB isan incoherent superposition of numerous GWs, includ-ing those too weak to be detected individually [31–35] orhaving an intrinsic stochastic nature, such as the primor-dial GW which is generated by quantum fluctuations inthe early Universe. The SGWB from binary stellar-massprimordial BH coalescence is calculated by Refs. [36–39]and, in particular, Ref. [37] shows that the null result ofSGWB in the LIGO frequency band ( ∼ [10 , , M (cid:12) . In the LISA frequency band, Refs. [40, 41]have considered the stochastic background from subsolarmass primordial BHs inspiraling to Sagittarius A ∗ , i.e.,the central massive BH of the Milky Way. Our work willexpand the study of Refs. [40, 41] in the following twoaspects. First, we calculate the SGWB energy densityspectrum in the frequency domain, i.e., Ω GW ( ν ) fromthe primordial BHs surrounding Sagittarius A ∗ . Second,we investigate the complete SGWB contributions fromextragalactic sources by modeling the event rate of pri-mordial BH EMRIs throughout the cosmic redshift.The rest of the paper is arranged as follows. In Sec-tion II we model the primordial BHs density profilearound a massive BH, and apply this relation to theSagittarius A ∗ in the Milky Way to derive the SGWBenergy density spectrum. We proceed to model the num-ber density of massive BHs at different redshift epochsand calculate the SGWB spectrum from extragalacticsources in Section III. We forecast the ability of LISA fordetecting the SGWB signal or, if there is a null result,constraining the abundance of primordial BHs in darkmatter in Section IV. The results show that LISA canprobe the existence of primordial BHs with mass range[10 − , M (cid:12) and constrain the abundance of primordialBH with 1 M (cid:12) to be 10 − in the optimal case where thedark-matter spike scenario with a steeper initial powerindex γ = 2 is valid. The main uncertainty is subjectto the value of the dark-matter initial power index γ .We summarize the conclusions in Section V. Throughoutthis work we assume the mass distribution of primordialBHs is a delta function due to the uncertainty of the pri-mordial BH population. Therefore the results should beseen as being from the primordial BHs with a represen-tative mass. Actually, as will be shown in the following,the mass of primordial BHs only serves as a scaling factorfor the amplitude of the resulting SGWB spectra and theshape of spectra only depends on the mass of the centralmassive BHs. II. THE STOCHASTICGRAVITATIONAL-WAVE BACKGROUND FROMSAGITTARIUS A ∗ A. Primordial Black Holes Density Profile
To model the event rate of primordial BH EMRIs, wefirst infer the primordial BH mass density at the galacticcenter. Since we expect that primordial BHs compose apart of dark matter, it is natural to use the dark-matterdensity profile to characterize the primordial BH massdensity around the central massive BH.For an initial dark-matter density profile with the fol-lowing power law form ρ ( r ) = ρ (cid:16) r r (cid:17) γ , (1)where ρ and r are halo parameters and to be deter-mined, γ is the power index, r is the radius of dark mat-ter, Ref. [42] suggests that the adiabatic growth of thecentral massive BH can enhance the surrounding dark-matter density at galactic center and form a spike dis-tribution, i.e., the halo will end up with the followingdensity [42, 43] ρ sp ( r ) = ρ R (cid:18) − R s r (cid:19) (cid:18) R sp r (cid:19) γ sp , (2)where the power index is enhanced from the initialvalue by γ sp = (9 − γ ) / (4 − γ ), the halo parameter ρ R = ρ ( R sp /r ) − γ with R s being the Schwarzschildradius of the central massive BH, R sp ( γ, M MBH ) = α γ r ( M MBH / ( ρ r )) / (3 − γ ) is the radius to which thedark-matter spike extends, α γ is derived numerically fordifferent γ in Ref. [42]. For an initial Navarro-Frenk-White (NFW) profile [44] with γ = 1, the final spike hasa index γ sp = 7 /
3, thus significantly boosting the innerprofile around the central massive BH.To connect ρ and r with the massive BH’s property,we employ the relation among the dark-matter halo virialmass M vir , the concentration parameter c con ≡ r vir /r where r vir is the halo virial radius, and the mass of mas-sive BH M MBH . The relation of c con − M vir for NFWprofile is given by Ref. [45]log c con = a + b log M vir h − M (cid:12) , (3)where a = 0 .
520 + 0 .
385 exp( − . z . ) and b = − .
101 + 0 . z are numerical factors at redshift z . Theparametrized formula Eq. (3) is obtained by numericallyfitting to a suite of N-body simulations for NFW pro-file with Planck , σ in the spheroidal regionand the total mass of the host galaxy, indicating a coevo-lution history with the whole galaxy. By employing theobservational M MBH − σ relation and using the quasar lu-minosity function to link σ with the halo mass, Ref. [47]gives a parametrized relation between the massive BH’smass M MBH and the host dark-matter halo’s virial mass M vir ,log (cid:18) M MBH h − M (cid:12) (cid:19) = ( − . ± .
33) + (1 . ± . × log (cid:20) β H ( z ) (cid:18) M vir h − M (cid:12) (cid:19)(cid:21) . (4)Here H ( z ) = H (cid:112) Ω m (1 + z ) + Ω Λ is the Hubble pa-rameter at redshift z , Ω m and Ω Λ are the matter anddark energy fractional densities, respectively. β is a ra-tio between the dark-matter halo’s circular velocity andvirial velocity, whose value is of order unity. Eq. (3) to-gether with Eq. (4) can fix the corresponding coefficientsof NFW density profile and NFW induced spike profilegiven the mass of massive BH. − − − − r [pc]10 ρ [ M (cid:12) M p c − ] cutoff frequency γ =2 spike γ =1 spikeNFW profile FIG. 1: Dark matter density profile around a massiveBH with mass 4 × M (cid:12) . The blue and orange solidcurves show the spike profile for power index γ = 2 and γ = 1, respectively. As a comparison, the NFWdark-matter profile is also plotted. The dark-matterdensity profile at galactic center is boosted significantlyfor the γ = 1 spike profile compared with the NFWprofile. The γ = 2 spike has even larger density than γ = 1 by three orders of magnitude. The verticaldashed line represents the orbital radius where the GWfrequency is 10 − Hz which is the lower cutoff frequencyof the LISA sensitive band.Fig. 1 shows the dark-matter spike profile around amassive BH with mass 4 × M (cid:12) . Given the uncertaintyof the initial profile power index, we choose two valuesfor γ , γ = 1 for NFW profile and γ = 2 representinga steeper profile for an optimistic result. We observethat the dark-matter density near the galactic center isboosted significantly for the γ = 1 spike profile comparedwith the NFW profile. The γ = 2 spike has even largerdensity than γ = 1 by three orders of magnitude. In Fig. 1, the vertical dashed line represents the radiuswhere the emitted GW frequency is 10 − Hz where wehave assumed primordial BHs perform circular motion.Therefore, the region of interest for the LISA frequencyband is between 4 R s as indicated by Eq. (2), and thedashed line. B. Stochastic Gravitational-Wave BackgroundSpectrum
The fractional energy density spectrum of SGWB isdefined as Ω GW = νρ c dρ GW dν , (5)where dρ GW is the gravitational-wave energy density inthe frequency band [ ν, ν + dν ], ρ c = 3 H c / πG is thecritical energy to close the Universe, G is the gravita-tional constant, c is the speed of light.The local energy density is related to the GW energyflux by ρ GW ( ν ) = F GW ( ν ) /c . F GW ( ν ) from SagittariusA ∗ can be derived by taking the integral of the density ofprimordial BH EMRI with respect to the orbital radius r , F GW ( ν ) = (cid:90) f PBH ρ DM ( r ; M MBH ) m PBH dE/dt ( r ; M tot , η ) d L r dr, (6)where d L is the luminosity distance from sources toGW detectors, ρ DM ( r ; M MBH ) is the dark-matter den-sity at radius r to the center given M MBH , m PBH and f PBH are the primordial BH mass and abundance, η ≡ m PBH M MBH / ( m PBH + M MBH ) is the symmetric mass ra-tio, M tot ≡ m PBH + M MBH is the total mass of the primor-dial BH EMRI system. To the leading order, GW powercan be calculated by the quadrupole formula [48, 49], dEdt ( r ; M tot , η ) = 325 G c η (cid:18) M tot r (cid:19) . (7)The radius r can be equivalently replaced by the GWfrequency ν under the assumption of Keplerian motion, r ( ν, M tot ) = (cid:20) ( GM tot ) / π (1 + z ) ν (cid:21) , (8)where the factor of (1 + z ) accounts for the cosmologicalexpansion and the GW frequency ν is twice of the orbitalfrequency. From Eq. (8) a differential relation can bederived , ddν = − π z ( GM tot ) / r / ddr . (9)Applying Eq. (9) to Eq. (6) and using the condition M MBH (cid:29) m PBH , the differential GW power from Sagit-tarius A ∗ is dF GW dν ( ν ) = 64 π
15 (1 + z ) G / c f PBH ρ DM m PBH M / r / d L , (10)Therefore the SGWB energy density spectrum isΩ SgrA ∗ GW ( ν ) = νρ c πG / c f PBH m PBH M / ρ DM r / d L . (11)The mass of Sagittarius A ∗ located at the center ofthe Milky Way is ∼ × M (cid:12) and the distance d L is ∼ m PBH = 1 M (cid:12) and f PBH =1 × − . Note that the term m PBH f PBH in Eq. (11)only serves as an overall scaling factor. We showcasetwo choices of the initial dark-matter profile power index, γ = 1 and 2, respectively. As a comparison, the LISAsensitivity curve of SGWB for the six links, 5 millionkm arm length configuration and 5 year mission duration(the optimal case) is plotted [50, 51].The figure shows that the amplitude of SGWB with γ = 2 is larger than that with γ = 1 by three orders ofmagnitude, inheriting from the three order of magnitudedifference of the dark-matter halo density as shown inFig. 1. Another feature is that the SGWB peaks at 3 × − Hz, which is determined by the mass of the centralmassive BH given the dark-matter spike distribution.It should also be noted that the SGWB from Sagittar-ius A ∗ is a directional signal, but the sensitivity curveis for isotropic SGWB. Therefore the comparison onlyserves as an order of magnitude estimation due to thelack of a sensitivity curve of space-based detectors fora directional background, but note that the techniqueof GW radiometer and the sensitivity for a directionalstochastic background for LIGO have been proposed andcalculated by Refs. [52]. The search for persistent GWsignals from pointlike sources has been performed by Ad-vanced [53, 54] and Initial LIGO and Virgo [55]. Es-pecially, using the data from the first and second ob-servational runs of Advanced LIGO and Virgo, the up-per limits for GW strain amplitude have been given forthree sources with targeting directions: Scorpius X-1, Su-pernova 1987 A and the Galactic Center, respectively.Refs. [56, 57] have considered the anisotropic SGWBsearch with space-based detectors. However, a special-ized investigation for directional SGWB from primordialBH EMRIs at the Galactic center has not been achievedin the context of space-based GW detectors. We thusleave the relevant study as a future work. III. THE STOCHASTICGRAVITATIONAL-WAVE BACKGROUND FROMEXTRAGALACTIC SOURCESA. Massive Black Hole Population
To model the massive BH population throughout thecosmic history, we employ the dark-matter halo massfunction and the M MBH − M vir relation which is char-acterized by Eq. (4). We choose the halo mass functionin Ref. [58] which calibrates with a suite of cosmologi-cal N-body simulations and takes the finite box size and FIG. 2: The SGWB energy density spectra ofprimordial BH EMRIs surrounding Sagittarius A ∗ , i.e.,the massive BH at the center of Milky Way. The massof primordial BH is chosen to be m PBH = 1 M (cid:12) and theabundance in dark matter is f PBH = 10 − . For theinitial dark-matter profile power index, γ = 2 and γ = 1are chosen for illustration. Both SGWB results peak atthe frequency 3 × − Hz which is determined by themass of the massive BH. The amplitude with γ = 2 islarger than γ = 1 by three orders of magnitude,inheriting from the three order of magnitude differenceof the dark-matter halo density. The LISA sensitivitycurve for detecting isotropic SGWB is also plotted for aqualitative comparison since the SGWB fromSagittarius A ∗ is a directional signal. The sensitivitycurve is for the optimal configuration of LISA which hassix links with 5 million km arm length and a 5 yearmission duration.the cosmic variance effects into account. In the actualpractice this halo mass function is generated through in-voking the Reed07 model in the python package hmf (anacronym for “halo mass function”) [59], where the cosmo-logical parameters are set to be the value of the
Planck satellite’s 2018 results [60].
Reed07 model is shown tobe valid up to redshift z ∼
30 and for halos with masses10 − h − M (cid:12) [58], which is sufficient for our purpose.Fig. 3 shows our results of the number density dn/d ln M MBH for massive BHs in different redshifts.Since astronomical observations have confirmed the ex-istence of massive BHs with ∼ M (cid:12) (for example seeRef. [61]), we will consider the SGWB spectrum from themassive BHs in the mass range [10 , ∼ ] h − M (cid:12) as afiducial model. The upper mass limit is determined bythe condition that the emitted GWs are in the LISA sen-sitive band and ∼ M (cid:12) is sufficient for this purpose. B. Stochastic Gravitational-Wave BackgroundSpectrum
The complete SGWB contribution can be obtained bytaking the sum from the EMRIs of all extragalactic mas-FIG. 3: The solid curves represent the number densityof massive BH at different redshifts which is derivedfrom
Reed07 dark-matter halo mass function. We willconsider the SGWB spectrum from the massive BHs inthe mass range [10 , ∼ ] h − M (cid:12) as a fiducial model.sive BHsΩ GW ( ν ) = νρ c πc (cid:90) (cid:90) dF GW dν dndM MBH dM MBH χ dχ, (12)where dn/dM MBH is the number density of massive BHsand χ is the sources’ comoving distance. CombiningEq. (10) and Eq. (12) yieldsΩ GW ( ν ) = f PBH m PBH νρ c π G / c (13) × (cid:90) dz (1 + z ) H ( z ) (cid:90) ρ DM M / r / dndM MBH dM MBH . As a fiducial model, we consider the sources in the red-shift range [0 , m PBH = 1 M (cid:12) and f PBH = 10 − , the results ofextragalactic SGWB with γ = 2 and γ = 1 are shown inFig. 4. The results from Sagittarius A ∗ are also replot-ted for comparison. The extragalactic SGWB peaks at ahigher frequency ( ∼ × − Hz), due to the contribu-tions from BHs less massive than Sagittarius A ∗ .Note that the extragalactic SGWB is an isotropic sig-nal, and thus can be compared with the LISA sensitivitycurve directly. It shows that the amplitude of extragalac-tic SGWB for γ = 2 already reaches the detectable zone,which means that the future LISA searching results canprobe the abundance of 1 M (cid:12) primordial BHs to be as lowas ∼ − for the γ = 2 case. The amplitude of γ = 1SGWB is smaller by three orders of magnitude due to asmaller dark-matter density profile.The main uncertainty comes from the dark-matter ini-tial profile power index. Future astrophysical progress onunderstanding the dark-matter profile at galactic centercan shed light on a more robust prediction on the pri-mordial BH EMRIs event rate. Conversely, future GWsearch with space-based detectors can also be beneficialfor study for the dark-matter profile. FIG. 4: The SGWB spectra from primordial BH EMRIssurrounding Sagittarius A ∗ (“Sgr A ∗ ”) and theextragalactic massive BHs (“ExtGal”) in the redshiftrange [0 , M (cid:12) and the abundance in dark matter is 10 − .The dark-matter initial profile power index is chosen tobe γ = 1 and γ = 2, respectively. The γ = 2extragalactic SGWB already reaches the detectablezone of LISA. C. Density Enhancement due toGravitational-Wave Dissipation
Another consideration is that the orbit of inspiralingprimordial BHs will gradually decay due to GW dissi-pation. As a consequence, the primordial BH densityprofile gets further concentrated. To quantify this effect,we notice that astrophysical spectroscopy surveys haveconfirmed that the first galaxies form at redshift z (cid:38) ∼
13 Gyr from now) [62]. Therefore we make the extremeassumption that all the primordial BHs EMRIs start todecay at z = 10, thus the decay duration equals to ∼ t , thefinal orbit radius r f and the initial radius r i is given byRefs. [48, 49] as follows∆ t = 5256 c G ηM tot ( r i − r f ) . (14)Let ∆ t = 13 Gyr and employ the mass conservationrelation ρ f ( r f ) = r i r f ρ i ( r i ) , (15)the enhanced density profile ρ f can be determined fromthe initial dark-matter profile ρ i which is assumed to thespike distribution. We have numerically verified that the mass loss due to GW dissi-pation is negligible by comparing the GW energy with the massenergy of primordial BHs.
For comparison, we also consider a less optimistic casewhere the orbital decay duration ∆ t is an average of a flatdistribution which ranges from zero to 13 Gyr, i.e., theprimordial BH final orbital radius after GW dissipation r f can be determined by r i with the following expression r f = (cid:90)
13 Gyr0 (cid:18) r i − G c ηM tot ∆ t (cid:19) / d∆ t (cid:44)
13 Gyr . (16)However, we find numerically that the relative differenceof the resulting SGWB spectra between the most opti-mistic case (i.e., ∆ t =13 Gyr) and the less-optimistic caseis negligibly small and at most O (10%); therefore we willonly consider the most optimistic case in the following asa representative of the GW dissipation effect.Choosing γ = 2, the fiducial SGWB result withoutdensity enhancement for m PBH = 1 M (cid:12) , f PBH = 10 − and the SGWB spectra with density enhancement for m PBH = 1 M (cid:12) , − M (cid:12) and 10 − M (cid:12) , respectively, areshown in Fig. 5. For the enhanced SGWB results, the pri-mordial BH abundance is modified accordingly to keep m PBH f PBH = 10 − . Since GW dissipation makes pri-mordial BHs get concentrated and closer to the center,the amplitude of SGWB spectra gets boosted and the fre-quency of the peak changes to ∼ − Hz. As the mass ofprimordial BHs decreases, the amplitude boost becomesmore significant. The amplitude for m PBH = 10 − M (cid:12) islarger than the fiducial result by one order of magnitude.This can be expected from Eq. (14) that a smaller valueof symmetric mass ratio η leads to a more significant or-bital decay.FIG. 5: A comparison between the SGWB spectra withand without primordial BH density enhancement effectdue to GW dissipation. The fiducial model is calculatedfrom the γ = 2 dark-matter spike distribution for m PBH = 1 M (cid:12) , f PBH = 10 − . The three dashed lines arefrom the γ = 2 enhanced dark-matter spike distributiondue to GW dissipation. The mass and abundanceparameters are so chosen to keep m PBH f PBH = 10 − . D. Modeling Systematics of the Massive BlackHole Population
To investigate the systematics on modeling the ex-tragalactic massive BH population, we vary the red-shift integral limits, the lower mass of massive BHsand the massive BH population model, respectively, andplot the corresponding SGWB spectra. The results areshown in Fig. 6. All SGWB spectra are calculated from γ = 2 dark-matter spike distribution and f PBH = 10 − , m PBH = 1 M (cid:12) .
1. The Redshift Limits of the Integral
The left column of Fig. 6 shows the SGWB componentcontributions from different redshift ranges, [0 , , [1 , , , z ∈ [3 ,
5] is subdominant andaccounts for at most 10% of the fiducial result. We thusconclude that the choice of 5 for the redshift upper limitis sufficient to capture the dominant contribution.
2. The Lower Mass Cutoff of the Massive Black Holes
As mentioned, we chose 10 h − M (cid:12) to be the mini-mum mass of massive BHs to account for the existence ofsuch massive BHs from astronomical observations. Nev-ertheless we change this value to 10 h − M (cid:12) to inves-tigate the contribution from different mass ranges. Theresult is shown in the middle column of Fig. 6. Com-pared with the fiducial result whose lower mass cutoffis 10 h − M (cid:12) , the result shows that the SGWB con-tributed by [10 , ] h − M (cid:12) massive BHs is mainly inthe relative high frequency range [ ∼ − , ∼ − ] Hz.Therefore the shape of the SGWB spectrum, especiallythe frequency of the peak, can provide information of theunderlying massive BH population.
3. The Population Model of Massive Black Holes
A third investigation is to substitute the massive BHpopulation model derived from the
Reed07 dark-matterhalo mass function to that proposed by Refs .[63–66] ac-counting for the massive BHs formed from population IIIstar seeds. The number density of massive BHs of thismodel is dnd log M MBH = 0 . (cid:18) M MBH × M (cid:12) (cid:19) − . Mpc − (17)The right column of Fig. 6 presents the results. It showsthat the amplitude of the peak is one order of magnitudesmaller than the fiducial model. This is because the num-ber density of massive BHs with mass [10 , ] h − M (cid:12) from Eq. (17) is less than that from the Reed07 derivedpopulation.The above investigations give a quantitative measure-ment about the effect on SGWB spectra from differentmodeling systematics of extragalactic massive BH, whichcan be improved in the future once a better understand-ing of the population of massive BHs is obtained. In addi-tion, the future SGWB search with space-based detectorscan serve as a tool to probe the population informationof massive BHs.
IV. CONSTRAINTS ON PRIMORDIAL BLACKHOLE ABUNDANCE
By comparing the SGWB spectrum with the LISA sen-sitivity and applying the conditionΩ GW ( ν ; m PBH , f
PBH ) ≤ Ω LISA sensitivityGW , (18)the upper limit on primordial BH abundance f maxPBH can beobtained for a null searching result. In Fig. 7, we plot theresults for different primordial BH masses from the mod-els of γ = 1, γ = 2, with and without the enhancementdue to GW dissipation, respectively. As a comparison,we also plot the current constraint from the microlensingsearch collaborations OGLE [67] and EROS [68].It shows that LISA can probe the primordial BH abun-dance in a large range of masses, [10 − , M (cid:12) for γ = 2and [10 − , M (cid:12) for γ = 1, respectively. We do notconsider more massive primordial BHs because it maybreak the condition of extreme mass ratio. For primor-dial BHs with 1 M (cid:12) , LISA can constrain the abundanceto be ∼ − ( γ = 1) or ∼ − ( γ = 2). The enhance-ment effect due to GW dissipation can further improvethe constraint at the lower end of the mass range. Thiswould be the most sensitive method proposed and quan-tified by now for detecting or constraining subsolar massprimordial BHs. V. CONCLUSIONS
In this work we have investigated the SGWB from pri-mordial BH EMRIs surrounding Sagittarius A ∗ and theextragalactic massive BHs, respectively, expanding theprevious work of Ref. [40, 41] by including the complete extragalactic contribution. After modeling the eventrate, the SGWB energy density spectra are calculated.The contributions from Sagittarius A ∗ and extragalacticmassive BHs peak at different characteristic frequencies.Future space-based GW detectors such as LISA may uti-lize this feature to give decisive evidence about the exis-tence of primordial BH dark matter at the galactic center.Finally, LISA can also constrain the abundance of pri-mordial BH in dark matter if there will be a null SGWBsearching result. For a NFW induced dark-matter spikeprofile ( γ = 1), LISA can exclude the existence of 1 M (cid:12) primordial BH with any abundance greater than 10 − ofdark matter. A steeper dark-matter profile power index γ = 2 can make the constraint even tighter by three or-ders of magnitude. This renders GWs in the space-basedfrequency band a powerful tool to hunting for subsolarmass primordial BHs. However, modeling uncertaintiesexist from the dark-matter spike profile power index andthe extragalactic massive BHs population as quantified inSection III. Future astrophysical progress on understand-ing these modeling systematics can help further improvethe ability of the GW window to search for primordialBHs. In addition, as a future work, we will apply thespecialized technique of the GW radiometer on the LISAdetector to investigate the detection ability for the direc-tional EMRI background from Sagittarius A ∗ . It wouldalso be interesting to study whether the stochastic back-ground is continuous or popcorn [69, 70], thereby ap-plying different optimal detection strategies accordingly[71]. ACKNOWLEDGMENTS
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