Gravitational Waves from Primordial Black Holes and New Weak Scale Phenomena
GGravitational Waves from Primordial Black Holes and New Weak Scale Phenomena
Hooman Davoudiasl ∗ and Pier Paolo Giardino † Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA
We entertain the possibility that primordial black holes of mass ∼ (10 –10 ) g, withSchwarzschild radii of O (cm), constitute ∼
10% or more of cosmic dark matter, as allowed byvarious constraints. These black holes would typically originate from cosmological eras correspond-ing to temperatures O (10 − O (10) Mpc encompassing the LocalSupercluster of galaxies. The proposed Einstein Telescope would further expand the reach for thesesignals. A positive signal could be further corroborated by the discovery of new particles in the O (10 − The presence of cosmic dark matter (DM) is firmly es-tablished by various cosmological and astronomical ob-servations [1]. However, all existing evidence for DM isfrom its gravitational effects. While it is widely believedthat DM has non-gravitational interactions that governedits production in the early Universe, all attempts to un-cover those interactions have been unsuccessful. Thissituation motivates one to entertain the possibility thatDM is of a purely gravitational nature. In particular,if DM is composed of primordial black holes (PBHs) [2–5], formed via gravitational collapse of primordial matteraround over density perturbations in the early Universe,it may only manifest itself through its gravitational ef-fects.The above PBH scenario removes the need to postulatenew particles and interactions associated with DM, whichis often invoked as strong motivation to search for physicsbeyond the Standard Model. This intriguing possibility isquite constrained by various observations [6–9] over mostof the viable parameter space. However, some parts ofthe parameter space allow for PBHs to be a significantcomponent of DM. In fact, allowing for deviations froma monochromatic spectrum, which is expected to be thecase in realistic scenarios [8], some narrow ranges of pa-rameters could possibly allow for all DM to be composedof PBHs.The primordial nature of the DM black holes implies aninteresting correspondence between the masses of PBHsand the era in which they were produced. That is, sincePBHs are assumed to be formed by the collapse of mat-ter and energy over a Hubble volume, the mass M PBH of a PBH is a measure of the horizon size, and hencethe temperature of the Universe at the time of the PBHformation. ∗ email: [email protected] † email: [email protected] PBH masses that could potentially originate fromfirst order phase transitions [10] at temperatures T ∼O (10 − T can be associated with extensions of the electroweaksector and the Higgs potential and may also lead to longwavelength primordial gravitational waves, which maybeen within the reach of future space-based observatories[11]. Those extensions may play a role in electroweakbaryogenesis and also provide new possibilities for mi-croscopic DM candidates, with PBHs comprising a sub-dominant, but potentially significant, DM population.The above range of T roughly corresponds to [10]10 g < ∼ M PBH < ∼ g . (1)Current bounds, including the recent micro-lensingsearches from observations of the Andromeda galaxy bythe Subaru Hyper Suprime-Cam [9], still allow about5 −
10% of DM to be comprised of PBHs, for masses in theabove range. With the assumption of a distribution for M PBH , it might be possible that a somewhat larger frac-tion of DM is made up of PBHs over the range (1). TheSchwarzschild radius R Sch of a black hole scales linearlywith its mass M BH and in the range (1) above, the cor-responding Schwarzschild radii are R Sch ∼ . −
10 cm.In this work, we will consider values of M PBH in therange (1) and explore the possibility that a neutron star(NS) or an astrophysical black hole (ABH) in our galac-tic neighborhood may have captured a PBH of suchmasses in an orbit around them. As we will show, thegravitational wave signals from these “David & Goliath(D&G)” binary systems can be detectable at AdvancedLIGO (aLIGO) or Advanced Virgo (AdV), at their de-sign sensitivity up to distances of ∼
10 Mpc, covering Note that, in our version of the confrontation, Goliath fares farbetter than in the original story. a r X i v : . [ g r- q c ] M a r he Local Supercluster of galaxies. The envisioned fu-ture Einstein Telescope could possibly extend the reachfor the parameters considered here to O (50) Mpc, goingbeyond the Local Supercluster.We point out that there can be two possible forma-tion mechanisms for D&G binaries: (a) through radiativecapture; see for example Refs. [12–14] and (b) throughadiabatic contraction and dynamical friction. The firstpossibility has been examined extensively in the contextof solar mass black holes and could in principle be appli-cable here. We suggest the second possibility based onproposed constraints for PBHs, where one estimates thelikelihood that a PBH be captured during star formationand later end up within the compact remnant, such asa white dwarf, destroying it [6]. It seems plausible thatone may also use this process to form D&G binaries thatwill later coalesce and yield our signal. We will focus onthe first mechanism (a), however possibility (b) may alsoresult in viable candidates. Hence, our estimate for therate of D&G inspiral events could be considered conser-vative in this sense. Without a more dedicated analysis- which is outside the scope of this work - it may notbe possible to determine which of the (a) or (b) optionsyield the dominant rate and what a reliable estimate ofthat rate would be.As we will discuss in the appendix, formation of D&Gbinaries via radiative capture is likely a rare occurrenceand our estimated rate might be ∼ − per year orless. However, our proposed signals could be detectedusing the existing aLIGO/AdV facilities and do not re-quire dedicated new experiments. In light of the above,and given the major impact of a potential PBH discoveryon our understanding of the Universe, an examination ofour proposal appears worth while, even if PBHs consti-tute only a subdominant contribution to DM. Let us begin with some general information about theastrophysical objects of interest. The known NS andABH populations have masses M NS ∼ − M (cid:12) and M ABH > ∼ M (cid:12) respectively, where M (cid:12) ≈ × gis the solar mass. For concreteness, in what follows wewill choose M NS = 1 . M (cid:12) and M ABH = 10 M (cid:12) , (2)as our reference values, however recent gravitational waveobservations [17] suggest that values of M ABH ∼ M (cid:12) are not necessarily uncommon. We note that the nearestknown NS and ABH are at distances d NS ∼ . d ABH ∼ See Refs. [15, 16] for recent works that examine whether theobservation of gravitational waves [17] from the merger of blackholes with ∼
30 solar masses corresponds to detection of DMcomposed of PBHs. Ref. [18] examined solar mass PBH binariesand Ref. [19] considered sub-lunar mass PBH binaries. For otherpossible signals of PBHs see Refs. [20, 21]. to optical observations. In principle, there could be iso-lated compact stellar objects that do not emit detectableoptical signals and may be closer to the Solar System. Inany event, we will use d > ∼ O (1) eccentricities e .The orbits get circularized as the binary evolves, how-ever for very hierarchic mass ratios the rate at whichthe eccentricity decreases de/dt ∝ − M PBH /M ABH [23] isslow and the eccentricity may still be sizable at the finalmerger. Hence, the circular orbit approximation may notbe very accurate for the systems we focus on. One of themain consequences of having e ∼ n = 2 harmonic and has significant com-ponents from higher harmonics [24, 25]. These effects donot, by and large, change the orders of magnitude for ourestimates.In order to estimate the proposed signal strengths, wewill need to make sure that parameters of the orbits weexamine can yield valid results. In this regard, we needto know the last stable orbit (LSO) for our systems. Ac-cording to the results in Ref. [26], for a test particle goingaround a black hole of mass M in an orbit with eccen-tricity e , the radius of the LSO is given by r LSO = G N c (6 + 2 e ) M e , (4)where G N = 6 . × − cm g − s − is Newton’s constantand c = 3 . × cm/s is the speed of light. Fora circular orbit with e = 0 we get the familiar result r LSO = 3 R Sch and for e = 1 we find r LSO = 2 R Sch .Hence, as long as we choose r > R Sch , we can assumestable orbits in our analysis. For simplicity, we will use r LSO = 3 R Sch for both the NS and ABH cases. Theresults of Ref. [27] suggest that this would also be a goodestimate for the NS case.Gravitational waves cause oscillations in the local met-ric as they travel through spacetime. These oscillationsgive rise to strain, i.e. variations in physical length scales,the size of whose amplitude we denote by h . Measure-ment of strain is the basis of gravitational wave detection.In the following, non-relativistic speeds and orbits largecompared to radii of the compact stellar objects are as-sumed. The simple formalism that we will use sufficesto get reasonable order of magnitude estimates. See e.g. Ref. [28] for an accessible presentation and Ref. [29] fora detailed exposition to the relevant subjects.2or a binary system, with component masses M and M , in a circular orbit of size r at a distance of d fromthe observer, we have h = 4 G N c M M r d . (5)The frequency of the corresponding gravitationalwaves are then given by f = 1 π (cid:20) G N ( M + M ) r (cid:21) / . (6)The radiation of gravitational waves by the binary systemcauses the decay of its orbital radius r to a smaller radius r f after a time [23]∆ t f ( r ) = 5 c G N (cid:34) r − r f M M ( M + M ) (cid:35) . (7)Of particular interest is the time ∆ t LSO , which we obtainfrom Eq. (7), required for the system to evolve to the LSOat r f = r LSO .For concreteness, we will consider f ∗ = 150 Hz as atypical value where aLIGO/AdV reach for gravitationalwaves is optimal. Our estimates do not sensitively de-pend on the exact value of f ∗ near our reference value.Using our reference values in Eq. (2), Eq. (6) yields theradius r ∗ corresponding to f ∗ r ∗ ≈
97 km (NS) and r ∗ ≈
182 km (ABH) . (8)Note that for the “D&G” binaries of interest here, wehave M PBH (cid:28) M (cid:12) and hence the frequency f ∗ of thewaves is independent of M PBH , to a very good approxi-mation. We see that for the above choice of parameters, r ∗ is well above the radius of the NS, about 10 km, andthe implied value of r LSO from Eq. (4).The decay time ∆ t LSO versus M PBH is plotted in Fig.1,for M NS = 1 . M (cid:12) , M ABH = 10 M (cid:12) , and f ∗ = 150 Hz.We see that 4 × s < ∼ ∆ t LSO < ∼ s. We will choosethe“observation time” t obs = ∆ t LSO , (9)which we will assume over the parameter space of ouranalysis.In Fig.2, we have plotted the expected size of thestrain signal hN − / √ t obs , with t obs = N t coh , ver-sus M PBH for M NS = 1 . M (cid:12) and distance from Earth5 kpc ≤ d ≤ t coh is the time scale overwhich the signal can be coherently observed. The valueof r ∗ has been chosen from Eq. (8) corresponding to theNS case. The horizontal dotted, dashed, and dot-dashedlines mark the projected AdV, aLIGO, and the proposedEinstein Telescope (ET) [30] sensitivities at f = f ∗ , in1 / √ Hz, of approximately 5 × − , 4 × − [31], and4 × − [32], respectively. We have used t coh = 2000 s PBH mass (cid:72) g (cid:76) (cid:68) t L S O (cid:72) s (cid:76) FIG. 1: Time, in seconds, required for the binary with M NS =1 . M (cid:12) (dahed) and M ABH = 10 M (cid:12) (solid) to evolve from anorbit where it emits gravitational waves at f ∗ = 150 Hz tothe last stable orbit given by r = 3 R Sch (assumed for boththe NS and ABH cases, using the corresponding mass). (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) PBH mass (cid:72) g (cid:76) S t r a i n (cid:72) H z (cid:45) (cid:144) (cid:76) AdV aLIGO ET
FIG. 2: Gravitational wave strain signal, in Hz − / , fora PBH-NS binary system, as a function of M PBH , with M NS = 1 . M (cid:12) and r = r ∗ , corresponding to a frequencyof f ∗ = 150 Hz. Different shades of red from darker tolighter correspond to the distance d intervals, (5 ,
50) kpc,(50 , . ,
5) Mpc, and (5 ,
50) Mpc. An observa-tion time of t obs = N t coh has been assumed, using Eq. (9)and t coh = 2000 s. The horizontal dotted, dashed, and dot-dashed lines represent the expected final design sensitivitiesfor AdV, aLIGO, and ET, respectively. (see for example Ref. [33]) in obtaining the results inFig.2. We see that for most of the range of M PBH con-sidered here, the entire Milky Way ( d < ∼
50 kpc) is withinthe reach of aLIGO/AdV.We note that the rate of the frequency increase for thesystems we examine is intrinsically quite slow, and onecould also focus the search on O (2000) known “pulsars”in our Galaxy whose optical signals determine their po-sitions in the sky. This feature allows one to accountfor signal modulation due to the motion of the observerwith respect to the barycenter of the Solar System, which3 .5 1.0 5.0 10.0 50.0 100.010 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) PBH mass (cid:72) g (cid:76) S t r a i n (cid:72) H z (cid:45) (cid:144) (cid:76) AdV aLIGO ET
FIG. 3: Gravitational wave strain signal, in Hz − / , fora PBH-ABH binary system, as a function of M PBH , with M ABH = 10 M (cid:12) and r = r ∗ , corresponding to a frequencyof f ∗ = 150 Hz. Different shades of blue from darker tolighter correspond to the distance d intervals, (5 ,
50) kpc,(50 , . ,
5) Mpc, and (5 ,
50) Mpc. An observa-tion time of t obs = N t coh has been assumed, using Eq. (9)and t coh = 2000 s. The horizontal dotted, dashed, and dot-dashed lines represent the expected final design sensitivitiesfor AdV, aLIGO, and ET, respectively. may lead to t obs = t coh , enhancing the reach for GalacticNS-PBH systems.The values of hN − / √ t obs versus M PBH are given inFig.3, for the ABH case is Eqs.(2) and (8), where we haveagain assumed t coh = 2000 s. Our results in Fig.3 suggestthat for M PBH ≈ g, aLIGO/AdV can be sensitiveto the gravitational wave signals of a PBH-ABH binaryout to distances of O (10) Mpc, while ET can probe d < ∼
50 Mpc, beyond our Local Supercluster.Note that our signal will not be mistaken for that of asmall planet or asteroid captured around an NS or ABH.This is because our gravitational wave signals are ob-tained for r ∗ ∼
100 km. This should be compared to themuch larger radius of the Earth R ⊕ ∼ M ⊕ ∼ × g is in the M PBH range of our pro-posal. In any event, a compact star will tidally destroya terrestrial scale rocky object, well before reaching anorbit comparable to its size.In conclusion, we illustrated, as a proof of principle,that if a primordial black hole of mass ∼ –10 gis captured by a neutron star or an astrophysical blackhole in our galactic neighborhood, gravitational wave sig-nals of their “D&G” confrontation could be detected byaLIGO/AdV or the proposed Einstein Telescope. Cur-rent constraints allow these primordial black holes toconstitute a significant fraction of cosmic dark matter.Although the signals we consider might be rare, theirdiscovery could shed light on early Universe phase tran-sitions in the visible and hidden sectors relevant to weakscale phenomena. As such, we may also expect that oursignals may be accompanied by discovery of new states ∼ −
100 GeV and also long wavelength primordialgravitational waves from the phase transition era. There-fore, we believe that searching for these signals in theexisting and future data is well motivated.The observation of gravitational waves by LIGO hasopened an exciting new front in the exploration of theCosmos. We hope that our work would further expandthe range of questions that could potentially be examinedat this front.We thank Scott Hughes for very helpful commentsand constructive criticism regarding our proposal andTongyan Lin for useful discussions. This work is sup-ported by the United States Department of Energy underGrant Contract de-sc0012704.
Appendix
Here, we provide an order of magnitude estimate forthe rate of D&G binary signal. As discussed before, thebinaries may form either in the process of star forma-tion, via the capture of a PBH by a massive star whoseremnant later forms a binary with the PBH, or throughradiative capture. Here, we focus on the second possibil-ity, and estimate the rate for an ABH to capture a PBHthrough gravitational radiation; the realistic rate may po-tentially be larger. Also, there is some contribution fromNS-PBH binaries that could add to the expected signalrate. In any event, given the multitude of contributingfactors, the following should be viewed as a rough guide.Following the discussions in Refs. [13, 14], let η ≡ M PBH
M/M , where M is the mass of the NS or ABHand M tot ≡ M PBH + M . The maximum impact parame-ter b that leads to the formation of the binary, assuminga relative velocity of w , is given by b max = (cid:18) π (cid:19) / M tot η / w / G N c − . (10)We will choose M PBH ∼ g, since it offers the far-thest reach in our range of PBH masses in (1) as seenfrom Fig.3, and set M = M ABH ∼ M (cid:12) . The re-sults of Ref. [34] suggest that within the inner 100 pcof the Milky Way, one could have a DM content of ∼ × M (cid:12) , though this quantity has large uncertain-ties. Hence, asuming some enhancement of DM densitytowards smaller radii, we can reasonably assume thatthe DM mass contained within the central 10 pc of theGalaxy is ∼ M (cid:12) . The simulations of Ref. [35] also im-ply that ∼ ABHs of mass 10 M (cid:12) could be containedwithin the same radius. Hence, the contribution of DM(including a sub-dominant PBH population) and ABHscan be comparable and of order 10 M (cid:12) . Assuming thatthe total mass within 10 pc of the center of the Galaxyis ∼ few × M (cid:12) , we find that w ∼
30 – 40 km/s canbe a fair estimate.4or the above set of parameters, one finds the crosssection σ ABH ∼ πb ∼ km . Assuming that thePBHs are distributed around the value chosen here, wefind a PBH number density of n PBH ∼ − km − .We may then estimate the capture rate for D&G bina-ries of interest, near the core of the Milky Way, as R ∼ σ ABH n PBH w N
ABH ∼ − yr − . Here, N ABH ∼ isthe number of ABHs within the inner ∼
10 pc of theGalaxy. Given our results in Fig.3, we may assume thatfor the chosen parameters aLIGO/AdV could be sensitiveto sources ∼
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