Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves
KKEK-TH-2260KEK-Cosmo-0263CTPU-PTC-20-22
Solar-Mass Primordial Black Holes ExplainNANOGrav Hint of Gravitational Waves
Kazunori Kohri a,b,c and Takahiro Terada da Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan b The Graduate University for Advanced Studies (SOKENDAI),1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan c Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa 277-8583, Japan d Center for Theoretical Physics of the Universe,Institute for Basic Science (IBS), Daejeon, 34126, Korea
Abstract
The NANOGrav collaboration for the pulsar timing array (PTA) observation re-cently announced evidence of an isotropic stochastic process, which may be the firstdetection of the stochastic gravitational-wave (GW) background. We discuss the possi-bility that the signal is caused by the second-order GWs associated with the formationof solar-mass primordial black holes (PBHs). This possibility can be tested by futureinterferometer-type GW observations targeting the stochastic GWs from merger eventsof solar-mass PBHs as well as by updates of PTA observations. a r X i v : . [ a s t r o - ph . C O ] O c t Introduction
Gravitational-wave (GW) astronomy started with the successful observations of GWs frommerger events of binary black holes by LIGO/Virgo collaborations [1]. GWs are also avaluable probe for the early Universe cosmology and particle physics. In particular, interestsin primordial black holes (PBHs) [2–4] were reactivated after the first detection of GWs [5–7]. In the PBH scenario, GWs can be emitted not only from the merger of binary PBHs butalso from the enhanced curvature perturbations that form PBHs [8–10]. This is due to thescalar-tensor mode couplings appearing at the second-order of the cosmological perturbationtheory [11–16]. It is interesting that we can indirectly probe physics of inflation by probingthe primordial scalar (curvature/density) perturbations inferred from the second-order GWsand PBH abundances [17–22].Recently, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav)released its 12.5-year pulsar timing array (PTA) data [23]. They search for an isotropicstochastic GW background by analyzing the cross-power spectrum of pulsar timing residu-als. They reported evidence of a stochastic common-spectrum process, parametrized as apower-law. The significance of the quadrupole nature in the overlap reduction function is notconclusive, whereas the monopole and dipole are relatively disfavored. This implies that theNANOGrav collaboration might have detected an astrophysical or cosmological stochasticGW background.It should be noted that the NANOGrav 12.5-yr signal is in tension with their previous11-yr result [24] as well as with Parkes PTA (PPTA) [25] (see Ref. [26] for the NANOGrav11-yr constraints on PBHs and also Ref. [27] related particularly to European PTA (EPTA)constraints [28]). According to the NANOGrav collaboration [23], the previous analysesmay have put too stringent constraints. It is important to carefully check their results withmore data or by reanalyses of the previous constraints. It is also crucial to establish thequadrupole (Hellings-Downs [29]) nature of the GWs before one claims the detection of theGW signals.Assuming the observed stochastic process is due to the detection of stochastic GW back-ground, the NANOGrav paper [23] studied the possibility that the GWs are produced fromsupermassive black hole merger events (e.g., see Ref. [30]). Other possibilities for the sourcesof GWs include cosmic strings [31–33], the PBH formation [34, 35], and a phase transitionof a dark (hidden) sector [36, 37].In this paper, we discuss the possibility that the putative GW signal is the second-order GWs induced by the curvature perturbations that produced solar-mass PBHs. Themain difference from Refs. [34, 35] is the mass range of the dominant PBH component.1ef. [34] concluded that the solar-mass PBHs abundance must be negligible and also that thesupermassive black holes may be responsible for the NANOGrav signal. Ref. [35] considered awide spectrum of the curvature perturbations and studied the possibility that the dark matterabundance is explained by O (10 − ) solar mass PBHs and a subdominant abundance of thesolar-mass PBHs explain the NANOGrav signal. Further comparisons with Refs. [34, 35]are made in Section 5. We compare the second-order GWs and the NANOGrav result inSection 2 and interpret it in terms of PBH parameters in Section 3. Then, we discuss futuretests of the scenario by measuring the stochastic GW background from mergers of solar-massPBHs in Section 4. After the discussion in Section 5, we conclude in Section 6. We adoptthe natural unit (cid:126) = c = 8 πG = 1. NANOGrav measures the strain of the GWs which is assumed to be of the power-law typein the relevant range of the analysis, h ( f ) = A GWB (cid:18) ff yr (cid:19) α , (1)where f is the frequency, f yr = 3 . × − Hz, A SGW is the amplitude, and α is the slope.More directly, they measure the timing-residual cross-power spectral density, whose slope isparametrized as − γ = 2 α −
3. They report preferred ranges of the parameter space spannedby A GWB and γ .These parameters are related to the energy-density fraction parameter Ω GW ( f ) = ρ GW ( f ) /ρ total in the following way, where ρ total is the total energy density of the Universe and the GWenergy density is given by ρ GW = (cid:82) d ln f ρ GW ( f ): [24]Ω GW ( f ) = 2 π f H A (cid:18) ff yr (cid:19) − γ , (2)where H ≡ h km/s/Mpc is the current Hubble parameter.In this paper, we discuss the possibility to explain the putative signal by the secondary,curvature-induced GWs produced at the formation of O (1) M (cid:12) PBHs. For such PBHs, itturns out that f (cid:38) f yr does not contribute significantly, and so we consider the frequencyrange 2 . × − Hz ≤ f ≤ . × − Hz [23, 31], which corresponds to the orange contourof figure 1 of Ref. [23].The current strength of the second-order, curvature-induced GWs is given by Ω GW ( f ) =2 Ω GW,c ( f ), where D = ( g ∗ ( T ) /g ∗ , )( g ∗ ,s, /g ∗ ,s ( T )) / Ω r is the dilution factor after the matter-radiation equality time with Ω r being the radiation fraction , and Ω GW,c ( f ) is the asymptoticvalue of Ω GW ( f ) well after the production of the GWs but before the equality time. This isgiven byΩ GW,c ( f ) = 112 (cid:18) f πaH (cid:19) (cid:90) ∞ d t (cid:90) − d s (cid:20) t ( t + 2)( s − t + s + 1)( t − s + 1) (cid:21) × I ( t, s, kη c ) P ζ (cid:18) ( t + s + 1) f π (cid:19) P ζ (cid:18) ( t − s + 1) f π (cid:19) , (3)where aH is the conformal Hubble parameter evaluated at the conformal time η c , P ζ ( k ) isthe dimensionless power spectrum of the primordial curvature perturbations, and I ( t, s, kη c )is the oscillation average of the kernel function, whose analytic formula has been derived inRefs. [39, 40]. For the recent discussions on gauge (in)dependence, see Refs. [41–48].We assume the log-normal power spectrum P ζ ( k ) = A s √ πσ exp (cid:32) − (ln k/k ∗ ) σ (cid:33) , (4)on top of the quasi-scale-invariant power spectrum measured at the cosmic-microwave-background (CMB) scale, where k = 2 πf is the wave number, A s is the amplitude, σ is the variance, and ln k ∗ is the average. As a natural setup, we take σ = 1 throughout thepaper and treat A s and k ∗ as free parameters. These can be translated to the GW parameters A GWB and γ and to the PBH parameters f PBH and M PBH , which are defined below. In thecase of the log-normal power spectrum, the full (approximate) analytic formula of Ω
GW,c ( f )is available [49] although we compute it numerically with the aid of an extrapolation intothe IR tail using the formula of Ref. [50].An example of the spectrum of the second-order GWs is shown as the thick black line inFig. 1. Also shown are power-law lines whose amplitude and slope correspond to points onthe contours of the NANOGrav favored region on the ( A GWB , γ )-plane (the green contoursin Fig. 2). The blue and cyan lines correspond to points on the upper half of 1 σ and 2 σ contour, while the orange and yellow lines correspond to points on the lower half of 1 σ and2 σ contour, respectively. The shaded regions are the constraints from the previous PTAobservations: EPTA [28], NANOGrav 11-yr [24], and PPTA [25]. The pink line at the For simplicity, we assume the Standard Model degrees of freedom and that neutrinos are massless. g ∗ ( T )and g ∗ ,s ( T ) are the effective relativistic degrees of freedom for the energy density and the entropy density,respectively [38]. These are evaluated at the horizon entry of the corresponding mode, while the quantitieswith the subscript 0 are evaluated at the present time. (cid:1)(cid:2) × (cid:3)(cid:4) - (cid:1)(cid:2) (cid:3)(cid:2) × (cid:3)(cid:4) - (cid:3) (cid:1)(cid:2) × (cid:3)(cid:4) - (cid:3) (cid:3)(cid:2) × (cid:3)(cid:4) - (cid:4) (cid:1)(cid:2) × (cid:3)(cid:4) - (cid:1)(cid:1) (cid:3)(cid:2) × (cid:3)(cid:4) - (cid:1)(cid:2) (cid:1)(cid:2) × (cid:3)(cid:4) - (cid:1)(cid:2) (cid:3)(cid:2) × (cid:3)(cid:4) - (cid:3) (cid:1)(cid:2) × (cid:3)(cid:4) - (cid:3) NANOGrav 11PPTAEPTA SKA
Figure 1: Example of the spectrum of the second-order GWs induced by the curvatureperturbations that produced PBHs of M PBH = 1 M (cid:12) and f PBH = 1 × − (thick black line).The power-law lines in the interval 2 . × − Hz ≤ f ≤ . × − Hz are also shown thatcorrespond to a rough visual guide of the NANOGrav signal range. The amplitudes andslopes of blue (cyan) and orange (yellow) lines are on the upper and lower 1 σ (2 σ ) contoursof the NANOGrav signal, respectively. The previous PTA constraints are shown by shadedregions: EPTA [28], NANOGrav 11-yr [24], and PPTA [25]. The pink line at the bottomright is the prospective constraint of SKA [51].As shown in the figure, there is tension between the NANOGrav 12.5-yr result and theexisting PTA constraints. According to the NANOGrav collaboration, the previous analysestend to underestimate the simulated GW signals, so the constraints were overestimated [23].An ongoing joint investigation among the PTA datasets implies a similar tendency for dataother than those of NANOGrav 11-yr [23]. Therefore, we do not worry too much about thetension between these preexisting PTA constraints and our explanation for the NANOGrav12.5-yr hint of the GWs in the following analyses. The relation between the second-order GWs and the properties of PBHs are as follows. TheGWs are induced by the enhanced curvature perturbations, which also produce PBHs. Theenergy density fraction β of the PBHs at the formation time, which also has the meaning4f the formation probability of a PBH in a given Hubble patch, is calculated in the Press-Schechter formalism [52] as β = (cid:90) ∞ δ c d δ (cid:112) πσ exp (cid:18) − δ σ (cid:19) (cid:39)
12 Erfc (cid:32) δ c (cid:112) σ (cid:33) , (5)where we have assumed that the primordial curvature perturbations have the Gaussianstatistics, δ c is the critical value of the coarse-grained density perturbations that producesa PBH [58–64], for which we take δ c = 0 .
42 [64, 65] , Erfc is the complementary errorfunction, and the variance σ of the coarse-grained density perturbations is defined as σ ( k ) = 1681 (cid:90) ∞−∞ d ln x w ( x ) x P ζ ( xk ) , (6)where w ( x ) is the window function, which we take as the modified Gaussian function w ( x ) =exp( − x / f PBH = ρ PBH /ρ CDM . This is related to β as follows, f PBH = (cid:90) d ln M Ω m Ω CDM g ∗ ( T ) g ∗ ( T eq ) g ∗ ,s ( T eq ) g ∗ ,s ( T ) TT eq (cid:15)β, (7)where the subscript m and eq denote the non-relativistic matter and the equality time, thetemperature T is evaluated at the horizon entry of the corresponding mode k , and (cid:15) denotesthe fraction of the horizon mass that goes into the PBH, which we take (cid:15) = 3 − / [4]. Moredetailed explanation for PBH formation and parameter dependencies can be found, e.g., inRefs. [69, 70] and in reviews [71–76].We relate k and the horizon mass in the standard way, i.e., using the Friedmann equation.Note, however, that there is a discrepancy between the average PBH mass M PBH and a naive For simplicity, we adopt the Press-Schechter formalism in this paper. However, we would like the readersto refer to Refs. [53–57] for more rigorous treatments. For the modified Gaussian window function, it is stated that δ c = 0 .
18 in Table 1 of Ref. [66] withouta detailed derivation. This may apparently be at odds with a naive expectation that δ c should be higherthan in the case of other window functions for the window-function dependence to be suppressed since themodified Gaussian window function enhances the value of σ . For this reason, we take δ c = 0 .
42 as the valueused more frequently in the literature. k ∗ because of two reasons: the peak position of σ ( k ) is smallerthan k ∗ , and each PBH mass is (cid:15) times smaller than the corresponding horizon mass. Theseshifts of peak positions were discussed, e.g., in Ref. [77] and recently emphasized again [67].Concretely, the relation among the wave number k ∗ , the corresponding frequency f ∗ = k ∗ / (2 π ), the corresponding horizon mass M , and the average PBH mass M PBH is as follows: M PBH . M (cid:12) (cid:39) M . M (cid:12) (cid:39) (cid:18) k ∗ . × Mpc − (cid:19) − (cid:39) (cid:18) f ∗ . × − Hz (cid:19) − . (8)We vary the scalar amplitude in the range 0 . ≤ A s ≤ .
040 and the average PBHmass in the range 0 . ≤ M PBH /M (cid:12) ≤
5. The resultant Ω GW h is fitted by a power-law linein the aforementioned range 2 . × − Hz ≤ f ≤ . × − Hz to extract the amplitudeof the GW strain A GWB and the slope γ . Note that A GWB ∝ A s , but it also depends on k * (or M PBH ) since the pivot scale is fixed to f yr (see eq. (1)). The result is shown in Fig. 2.From the figure, we see that a large fraction of the scanned parameter space can explain theNANOGrav signal. - - - -
040 and 0 . ≤ M PBH /M (cid:12) ≤ A s corresponds to a larger A GWB , and a larger M PBH corresponds to a larger γ . The thin red lines correspond to f PBH = 10 − , 10 − , 10 − , and10 − from top to bottom. The 1 σ and 2 σ NANOGrav contours are also shown.The scanned parameter range for A s corresponds to that of the PBH abundance f PBH as shown in Fig. 3. The upper and lower ends correspond to M PBH = 0 . M (cid:12) and 5 M (cid:12) ,6espectively. - - - - Figure 3: Relation between the scalar amplitude A s and the PBH abundance f PBH for M PBH /M (cid:12) = 0 . M PBH , f PBH ), which are shown as the green contours inFig. 4. The non-smoothness of the contours largely originates from the non-smoothness ofthe original NANOGrav contours. The uncertainty of extracting the data from the originalcontours is magnified in this figure compared to Fig. 2. Therefore, the 1 σ and 2 σ boundaryhas an uncertainty of very roughly an order of magnitude.Fig. 4 shows that the PBH mass should be around a solar mass to explain the NANOGravsignal. Also, it shows that f PBH close to unity is disfavored, but f PBH ∼ . σ contour depending on the value of M PBH .A part of such regions is excluded by existing constraints shown by shaded regions atthe top of the figure. These include the microlensing constraints by EROS/MACHO col-laborations [78, 79], the caustic crossing constraint [80], Advanced LIGO constraints on thesubsolar mass range (individual events [81] and superposition of events [82, 83]), and theconstraints due to photo-emission during gas accretion onto PBHs [84–86]. There are manysubdominant but independent and complementary constraints around this mass range (seeRef. [75]). There is also the LIGO/Virgo constraints on supersolar mass range [87, 88].Ref. [88] implies a substantial dependence on the width of the mass function, so we do notinclude it in Fig. 4. 7 .2 0.5 1 2 510 - - - C M B d i s k acc r e t i o n E R O S / M A CH O a L I G O O caustic crossing S G W B Figure 4: NANOGrav contours (green) on the plane of the average PBH mass M PBH and thePBH abundance f PBH . The dark shaded regions at the top are constraints from EROS-2 [78]and MACHO [79] (brown), caustic crossing [80] (purple), Advanced LIGO O2 (subsolar massrange) [81] (gray), Advanced LIGO non-detection of the stochastic GW background [82, 83](cyan), and the E -mode polarization of the CMB due to the disk-shaped gas accretion [84](blue). The solar-mass PBH possibility for NANOGrav can be tested by the detection of stochasticGW background from the superposition of binary solar-mass PBH merger events. The GWspectrum is obtained asΩ mergerGW ( f ) = f H (cid:90) f cut f − d z R ( z )(1 + z ) H ( z ) d E GW d f s , (9)where f cut is the UV cutoff frequency (see Refs. [89, 90] for the IR cutoff frequency), f s isthe frequency at the source frame (i.e., without the redshift factor), z is the redshift, R isthe comoving merger rate, and E GW is the energy of the GWs at the source frame. Theexpressions of f cut , R , and d E GW / d f s are found in Appendices B and C of Ref. [77]. See alsoRefs. [7, 74, 82, 91, 92] for more details.The result is shown in Fig. 5 as the black lines where M PBH = 1 M (cid:12) and f PBH = 10 − (solid), 10 − (dashed), 10 − (dotted), and 10 − (dot-dashed). Various prospective constraints8 (cid:2) - (cid:1) (cid:2)(cid:3)(cid:2)(cid:1) (cid:1)(cid:2)(cid:2)(cid:1)(cid:2) - (cid:2)(cid:3) (cid:1)(cid:2) - (cid:2)(cid:4) (cid:1)(cid:2) - (cid:2)(cid:5) (cid:1)(cid:2) - (cid:2)(cid:2) (cid:1)(cid:2) - (cid:6) (cid:1)(cid:2) - (cid:3) SKA LISATianQin BBO DECIGOAION km AEGDE ET+2CEHLVIKaLIGO O2
Our results depend on various assumptions. Some of them have been already stated, but weemphasize them again. First, we do not consider the effect of the critical collapse [73, 118–120] since it occurs only when the spherical symmetry is precisely respected. It is clear thatthe rare high-peak has approximately the spherical shape [121], but the spherical symmetrymust be realized to high precision for the critical collapse to happen [58]. On the other hand,Refs. [34, 35] include the effect of the critical collapse. It will be interesting to compare ourresults with an analysis including the critical collapse effect using a consistent parameterset [67]. In our preliminary study, we found a qualitatively similar feature that f PBH tendsto become larger than those reported in Refs. [34, 35].Second, we have chosen the modified Gaussian window function, whose width is twice aslarge as the standard Gaussian window function. This boosts the value of f PBH for a givenvalue of A s . This may be the largest difference compared to Refs. [34, 35] in which muchsmaller f PBH were reported.Third, we have not taken into account the nonlinear relation between the primordialcurvature perturbations and the density perturbations (see Refs. [53, 122]). This inevitablyleads to non-Gaussianity of the density perturbations [122]. Also, the inclusion of the intrinsicnon-Gaussianity of the primordial curvature perturbations significantly affects f PBH [123,124]. It also affects the second-order GWs [125–129].Fourth, we have not included the transfer function of the curvature perturbations in thedefinition of σ . This is preferred in Ref. [66]. If we include the transfer function, however, σ will reduce by “several” percent. This reduces f PBH non-negligibly.It is also worth mentioning that we have not taken into account the softening of theequation-of-state during the phase transition/crossover of quantum chromodynamics (QCD).See Refs. [35, 130, 131] for its enhancement effect on the PBH abundance f PBH for a givenscalar amplitude A s . Depending on the boost factor, this may realize a better fit for theNANOGrav signal simultaneously with stronger and more easily detectable GWs from merg-ers of the solar-mass binary PBHs. The softening also slightly affects the spectrum of thesecond-order GWs [132].We discussed a possible detection of the PBHs with the masses of O (1) M (cid:12) only by afuture interferometer-type GW observations in Section 4. Complementarily, however, we canalso measure such PBHs by the future optical/IR telescopes through microlensing events,e.g., Subaru HSC towards M31 for 10 year observations [133] or by the future precise CMBobservations of E - and B -mode polarization due to photon emission from an accretion diskaround a PBH, e.g., by LiteBIRD [134] or CMB-S4 [135].10 Conclusion
In this paper, we have interpreted the recently reported NANOGrav 12.5-yr excess of thetiming-residual cross-power spectral density in the low-frequency part as a stochastic GWbackground. We conclude that, under our assumptions, the second-order GWs induced bythe curvature perturbations that produced a substantial amount of O (1) solar-mass PBHscan explain the NANOGrav stochastic GW signal. In particular, the abundance of thePBHs can be sufficiently large so that future GW observations can test this possibility bymeasuring the stochastic GW background produced by mergers of the solar-mass PBHs.This is nontrivial since the suitable scalar amplitude A s could a priori produce too manyPBHs that are excluded by existing observational constraints or too few PBHs that do notlead to the detectable stochastic GW background from merger events. Similarly, for a given f PBH , the second-order GWs could be too strong or weak. Since the relation between A s and f PBH depends crucially on the ambiguity for the choice of the windows function as discussedin the previous section, a further study to refine the PBH formation criterion is necessary.
Note Added
Taking into account the uncertainties of PBH abundance calculations, i.e., the differentchoices of the window function, the value of δ c (see footnote 3), etc., our results are largelyconsistent with those of Ref. [35] [136]. The difference from Ref. [34] is also discussed inRef. [34]. Acknowledgments
K.K. thanks Misao Sasaki and Shi Pi for useful discussions. We thank Valerio De Luca,Gabriele Franciolini, and Antonio Riotto for useful discussions. This work was supported byJSPS KAKENHI Grant Number JP17H01131 (K.K.), MEXT KAKENHI Grant NumbersJP19H05114 (K.K.) and JP20H04750 (K.K.), and IBS under the project code, IBS-R018-D1(T.T.).
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