Formation of supermassive primordial black holes by Affleck-Dine mechanism
IIPMU 19-0093
Prepared for submission to JCAP
Formation of supermassiveprimordial black holes byAffleck-Dine mechanism
Masahiro Kawasaki a,b
Kai Murai a,b a Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan b Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo,Kashiwa 277-8583, Japan
Abstract.
We study the supermassive black holes (SMBHs) observed in the galactic centers.Although the origin of SMBHs has not been well understood yet, previous studies suggest thatseed black holes (BHs) with masses − M (cid:12) exist at a high redshift ( z ∼ ). We examinewhether primordial black holes (PBHs) produced by inhomogeneous baryogenesis can explainthose seed black holes. The inhomogeneous baryogenesis is realized in the modified Affleck-Dinemechanism. In this scenario, there is no stringent constraint from CMB µ -distortion in contrastto the scenario where Gaussian fluctuations collapse into PBHs. It is found that the model canaccount for the origin of the seed BHs of the SMBHs. Keywords: physics of the early universe a r X i v : . [ a s t r o - ph . C O ] J u l ontents There are many evidences that most galaxies have supermassive black holes (SMBHs) withmasses − . M (cid:12) in their centers [1–3]. However, the origin of SMBHs has not been wellunderstood yet. It is conventionally argued that stellar black holes (BHs) with masses O (10) M (cid:12) produced by stellar collapses grow to SMBHs by accretion and mergers. But observations of highredshift QSOs indicate that SMBHs already exist at z > [e.g. [4]] and hence it is in disputewhether stellar BHs have time to grow to SMBHs.Another candidate for SMBHs are primordial black holes (PBHs) which are produced fromlarge density fluctuations in the early universe [5–7]. Such large density fluctuations can beproduced by inflation [8–10]. PBHs have been attracting great interest because they can bedark matter of the universe or they are a good candidate for BHs that cause gravitational waveevents discovered by LIGO [11–25]. In Refs. [26, 27], it is shown that BHs exsiting at z ∼ can be seeds for SMBHs if they have masses around − M (cid:12) . If PBHs are formed at thecosmic temperature ∼ MeV, they have masses ∼ − M (cid:12) and account for those seed BHs forSMBHs. In fact an inflation model producing such SMBHs was proposed in Ref. [28]. However,the scenario for producing supermassive PBHs has a difficulty. PBH formation requires largedensity fluctuations on small scales, which leads to the CMB spectral distortion due to the Silkdamping [29, 30]. In fact the µ -distortion of CMB gives a stringent constraint on the amplitude– 1 –f the power spectrum of the curvature perturbations, P ζ (cid:46) − for k ∼ − Mpc, fromwhich PBHs with masses between × M (cid:12) and × M (cid:12) are excluded [31].However, this constraint is obtained assuming that the fluctuations are nearly Gaussian andmodels for PBH formation based on highly non-Gaussian fluctuations can evade it. One exampleis the PBH formation using the inhomogeneous Affleck-Dine baryogenesis [32–35]. (For anotherexample, see [36].) Recently, Hasegawa and one of authors [37, 38] showed that the inhomo-geneous Affleck-Dine baryogenesis is realized in the framework of the minimal supersymmetricstandard model (MSSM) and it produce high baryoin bubbles (HBBs) which is regions with highbaryon-to-entropy ratio η b . Those HBBs collapse into PBHs with masses (cid:38) M (cid:12) that explainthe LIGO gravitational wave events.In this paper, we examine whether the inhomogeneous Affleck-Dine baryogenesis can ac-count for SMBHs. It is found that the PBHs formed in the inhomogenous Affleck-Dine mechanismcan be the seed BHs of the SMBHs in the galactic centers. By varying the model parameters, wecan easily control the mass distribution of the PBHs. The scenario of Affleck-Dine mechanismdepends on the SUSY breaking scheme. We mainly discuss the gravity-mediated SUSY breakingscenario, where the baryon asymmetry in the HBBs become the massive baryons after the QCDtransition and generate the density contrast. Finally, we make some comments on the gauge-mediated SUSY breaking scenario, where the AD field fragments into the stable Q-balls. In thisscenario, the Q-balls can contribute to both the seed PBHs and the dark matter abundance.This paper is organized as follows. In Sec. 2, we review the outline of the inhomogeneousAffleck-Dine baryogenesis and evaluate the distribution of the HBBs. In Sec. 3, we explain howthe HBBs evolve and gravitationally collapse into the PBHs. We show the conditions for theseed BHs of the SMBHs and estimate the abundance of the PBHs in Sec. 4. Finally, we concludein Sec. 5. In this section, we briefly review the generation of inhomogeneous baryon asymmetry and highbaryon bubbles (HBBs) based on [37, 38].
In the previous works [37, 38], it was shown that HBBs are produced in the modified version ofthe AD baryogenesis in MSSM. In this model, we modify the conventional AD baryogenesis bymaking two assumptions:(i) During inflation, the AD field has a positive Hubble induced mass, while it has a negativeone after inflation. – 2 –ii) Just after inflation, the thermal potential for the AD field overcomes the negative Hubbleinduced mass around the origin.These assumptions are easily satisfied by appropriate ( and natural ) choice of the model param-eters. Under these assumptions, the potential for the AD field φ = ϕe iθ is written as V ( φ ) = (cid:16) m φ + c I H (cid:17) | φ | + V NR , (during inflation) (cid:16) m φ − c M H (cid:17) | φ | + V NR + V T ( φ ) , (after inflation) (2.1)where c I , c M are dimensionless positive constants, m φ is the soft SUSY breaking mass for theAD field, and V NR represents the non-renormalizable contribution given by V NR = (cid:32) λa M m / φ n nM n − + h . c . (cid:33) + λ | φ | n − M n − , (2.2)where λ, a M are dimensionless constants. The integer n ( ≥ ) is determined by specifying theMSSM flat direction. V T is the thermal potential for the AD field induced by the thermalizeddecay product of the inflaton and is written as V T ( φ ) = (cid:40) c T | φ | , | φ | (cid:46) T,c T ln (cid:16) | φ | T (cid:17) , | φ | (cid:38) T, (2.3)where c , c are O (1) parameters relevant to the coupling between the AD field and the thermalbath. Let us describe the dynamics of the AD field in this scenario. During inflation, the potential ofthe AD field has the minimum at ϕ = 0 due to the positive Hubble induced mass. Furthermore,when c I < , the AD field acquires quantum fluctuations around it. Therefore, the AD fieldcoarse-grained over the Hubble scale stochastically fluctuates and takes a different value in eachHubble patch shown in Fig. 1.After inflation, by the assumption (ii), the negative Hubble induced mass and the thermalpotential bring on the multi-vacuum structure with the vacuum “A” at ϕ = 0 and the vacuum“B” at ϕ (cid:54) = 0 (see Fig. 1). Then, the AD field rolls down the potential to one of the two vacuadepending on values of the AD field at the end of inflation t e . If ϕ ( t e ) in some patch is smallerthan the maximal point between the two vacua ϕ c ( t e ) , the AD field rolls down to the vacuum Asoon after inflation Thus, in this case almost no baryon number is produced, i.e. η ( A ) b (cid:39) . (2.4)On the other hand, in the patches satisfying ϕ ( t e ) > ϕ c ( t e ) , the AD field rolls down to thevacuum B. The vacuum B vanishes later due to the thermal potential or soft SUSY breaking– 3 – B B V ( ) V ( ) P ( N e , ) c c Figure 1 . The schematic view of the dynamics of the AD field during and after inflation.
Upper side :During inflation, the coarse-grained AD field diffuses in the complex plain and takes different values indifferent Hubble patches.
Lower side : Just after inflation, the multi-vacuum structure appears due tothe thermal potential and the negative Hubble induced mass. In the patches with | φ | > φ c , the AD fieldrolls down to the vacuum B. On the other hand, the patches with | φ | < φ c , the AD field rolls down tothe vacuum A. mass term and the conventional AD baryogenesis takes place. Therefore, the AD field begins tooscillate around the origin at H ( t ) (cid:39) H osc , which produces the baryon asymmetry given by η ( B ) b (cid:39) (cid:15) T R m / H (cid:18) ϕ osc M Pl (cid:19) , (2.5) (cid:15) = (cid:114) cn − q b | a M | sin( nθ + arg( a M ))3 (cid:16) n − n − + 1 (cid:17) , (2.6)where the subscript “osc” represents the values evaluated at the onset of the oscillation, T R isthe reheating temperature, q b is the baryon charge of the AD field, and θ is the initial phase ofthe AD field. In this way, the inhomogeneous AD baryogenesis takes place and the HBBs withlarge baryon asymmetry are formed.Here, we derive the condition for our assumption (ii). When the decay products are ther-malized instantaneously, the cosmic temperature is given by T inst ( t ) (cid:39) (cid:0) T R H ( t ) M Pl (cid:1) / . (2.7)With use of Eq. (2.7), the assumption (ii) is satisfied if ∆ ≡ T R M Pl H ( t e ) (cid:38) , (2.8)– 4 –here we set c M , c ∼ for simplicity. The critical point just after inflation ϕ c ( t e ) can be writtenas ϕ c ( t e ) ≡ ϕ c = ∆ / H ( t e ) , (2.9)where we also set the O (1) model parameters to unity. Next, we analytically evaluate the volume fraction of the HBBs. The evolution of the coarse-grained AD field during inflation is described by the Langevin equation including the Gaussiannoise [39–41] and the probability distribution function of the AD field with respect to e -foldingnumber N ( ∝ ln a [ a : scale factor ]) , P ( N, φ ) follows the Fokker-Planck equation: ∂P ( N, φ ) ∂N = (cid:88) i =1 , ∂∂φ i (cid:20) ∂V ( φ ) ∂φ i P ( N, φ )3 H + H π ∂P ( N, φ ) ∂φ i (cid:21) , (2.10)where ( φ , φ ) = ( (cid:60) [ φ ] , (cid:61) [ φ ]) . The first term in the RHS is classically induced by the poten-tial and the second term represents the quantum fluctuations. Assuming the initial condition P (0 , φ ) = δ (0) and the constant Hubble parameter during inflation H ( t ≤ t e ) = H I , we obtain P ( N, φ ) = 12 πσ ( N ) e − ϕ σ N ) , (2.11) σ ( N ) = (cid:18) H I π (cid:19) − e − c (cid:48) I N c (cid:48) I , (2.12)where we have used V ( φ ) (cid:39) c I H I φ and defined c (cid:48) I ≡ (2 / c I . The phase of the AD field θ israndom unless large CP violation terms such as Hubble induced A-terms are introduced.As discussed above, the patches where the AD field rolls down to the vacuum B cause theAD baryogenesis and form HBBs. Therefore, at the e -folding number N , the physical volume ofthe regions which would later become HBBs, V B ( N ) is evaluated as V B ( N ) = V ( N ) (cid:90) ϕ>ϕ c P ( N, φ )d φ ≡ V ( N ) f B ( N ) . (2.13)Here, we represent the physical volume of the Universe at N as V ( N ) ∼ r H e N , where r H ∼ H − I is the Hubble radius during inflation. The volume fraction of the regions which would laterbecome HBBs, f B ( N ) , is evaluated as f B ( N ) = (cid:90) π d θ (cid:90) ∞ ϕ c d ϕϕ e − ϕ σ N ) πσ ( N ) = e − π σ N ) , (2.14)where ˜ σ ( N ) ≡ (cid:16) − e − c (cid:48) I N (cid:17) /c (cid:48) I . We show the evolutions of f B ( N ) in Fig. 2 (left panel).– 5 – �� �� �� �� �� ���� - �� �� - �� �� - �� �� - �� � �� �� �� �� �� ���� - �� �� - �� �� - �� �� - �� Figure 2 . The evolutions of the volume fraction of HBBs f B ( N ) (left) and the production rate of HBBsat β B ( N ) (right) for various c (cid:48) I . ∆ is adjusted for each plot so that f B ( N = 60) takes the same value. The creation rate of the regions which would later become HBBs is obtained by differentiating V B ( N ) with respect to N as d V B ( N )d N = 3 V B + V ( N ) (cid:90) ϕ>ϕ c P ( N, φ )d N d φ. (2.15)The first term in the RHS represents the growth of the existing regions which would later becomeHBBs due to the cosmic expansion and the second term represents the creation of such regions.Therefore, the fraction of the HBBs with the scale exiting the horizon at N is evaluated at theend of inflation N e as β B ( N ; N e ) = 1 V ( N e ) · e N − N e ) (cid:20) V ( N ) (cid:90) ϕ>ϕ c d P ( N, φ )d N d φ (cid:21) = dd N f B ( N ) = ( πc (cid:48) I ) ∆sinh (cid:0) c (cid:48) I N/ (cid:1) f B ( N ) . (2.16)We show the evolutions of β B ( N ) in Fig. 2 (right panel).The scale of the HBBs k can be expressed in terms of the e -folding number N when thescale k exits the horizon as k ( N ) = k ∗ e N − N CMB , (2.17)where k ∗ is the CMB pivot scale and N CMB is the e -folding number when the pivot scale exitsthe horizon. The horizon mass when the scale k re-enter the horizon is evaluated as M H (cid:39) . M (cid:12) (cid:16) g ∗ . (cid:17) − / (cid:18) k Mpc − (cid:19) − , (2.18)where M (cid:12) is the solar mass and g ∗ is the effective degree of freedom of relativistic particles.Equivalently, the e -folding number N is evaluated in terms of M H as N ( M H ) (cid:39) −
12 ln M H M (cid:12) + 21 . N CMB , (2.19)– 6 –here we used g ∗ = 10 . and k ∗ = 0 .
002 Mpc − . It is known that the typical inflationarymodels suggest that N e − N CMB ∼ − . (2.20)On the other hand, to solve the horizon and flatness problem of the Big Bang cosmology, thetotal number of e -foldings of the inflation era should be N e (cid:38) . (2.21)In the rest of this paper, we fix N e = 60 and N CMB = 10 . It is also convenient to relate thecosmic temperature T to the horizon mass as T ( M H ) = 434MeV (cid:18) M H M (cid:12) (cid:19) − / . (2.22) In this section, we discuss the evolution of the HBBs and their gravitational collapse into PBHs.
Just after inflation, the energy density inside and outside the HBBs are almost the same becauseof the energy conservation of the AD field and the domination of the oscillation energy of theinflaton. After the AD field decays, the quarks carry the produced baryon asymmetry. As long asthe quarks remain relativistic, the density fluctuations are not generated. After the QCD phasetransition, the baryon number is carried by massive baryons (protons and neutrons). Sincethe energy of the baryons behaves as non-relativistic matter, their energy density is given by ρ (cid:39) n b m b ( m b : the nucleon mass). Thus, the density contrast between inside and outside theHBBs is written as δ ≡ ρ in − ρ out ρ out (cid:39) n in b m b ( π / g ∗ T (cid:39) . η in b (cid:18) T (cid:19) − θ ( T QCD − T ) , (3.1)where T QCD is the cosmic temperature at the QCD phase transition and θ ( x ) is the Heavisidetheta function. Here, we have used m b (cid:39) MeV.Here, we make three comments. First, HBBs are considered as top-hat type baryonicisocurvature fluctuations. Such small-scale isocurvature fluctuations are hardly constrained bythe CMB observations. Although isocurvature fluctuations induce adiabatic ones after theQCD phase transition, the produced perturbations are highly non-Gaussian. In addition, HBBsare presumed to be rare objects and adiabatic perturbations averaged over the whole universe Large isocurvature baryonic perturbations are constrained from the big-bang nucleosynthesis if they areGaussian [42]. – 7 –re small. Therefore, this model does not suffer from the stringent constraints from CMB µ -distortion and the PTA experiments [43–45]. Second, the baryon asymmetry η in b is different fromHBB to HBB depending on the initial phase of the AD field θ , which is efficiently random dueto the flatness of the phase direction. Although this effect does not bring a substantial changeto the following discussion, we assume that all HBBs have the baryon asymmetry for simplicity.Actually this assumption can be realized by introducing the Hubble induced A-term. Third,we implicitly assumed that the AD field decays into the quarks above. However, it is knownthat the coherent oscillation of the AD field usually fragments to the localized lumps called Q-balls [46–48]. In this paper, we mainly focus on the gravity-mediated SUSY breaking scenario,where the produced Q-balls are unstable and decay into the quarks. On the other hand, in thegauge-mediated SUSY breaking scenario, the produced Q-balls are stable and can make densityfluctuations which become PBHs later. We will make a comment on this case later. From Eq.(3.1), the HBBs become over-dense after the QCD transition. If the density contrastis large enough, the overdense regions collapse into PBHs after they re-enter the horizon. In theradiation-dominated era, the threshold value of the density contrast for the PBH formation isroughly estimated as δ c (cid:39) w with w ≡ p/ρ [6], which we adopt in this paper. Notice that ourmodel is hardly sensitive to the choice of estimation of δ c unlike the case of the Gaussian densityperturbations.In the present case w is written in terms of δ ( T ) as w ( T ) = p in ρ in (cid:39) p out ρ in = 13 11 + δ ( T ) . (3.2)Thus, the condition for the PBH formation δ ( T ) > δ c ( T ) (cid:39) w ( T ) is given by δ ( T ) (cid:38)
13 11 + δ ( T ) ⇐⇒ δ ( T ) (cid:38) . , (3.3)and this gives the upper bound of the temperature at the horizon re-entry for the PBH formation.The critical temperature for the PBH formation T c is obtained from Eq.(3.1): T c (cid:39) Min (cid:2) η in b MeV , T
QCD (cid:3) , (3.4)which leads to the lower bound of the horizon mass for the PBH formation, M c (cid:39) Max (cid:34) . (cid:0) η in b (cid:1) − M (cid:12) , . M (cid:12) (cid:18) T QCD (cid:19) − (cid:35) . (3.5)Here we used Eq.(2.22). Since the mass of the formed PBH is comparable with the horizon massat the horizon re-entry, M PBH ∼ M H , the mass distribution of the PBHs is given as β PBH ( M PBH ) = 12 β B ( N ( M PBH )) θ ( M PBH − M c ) , (3.6)– 8 –here β PBH is the volume fraction of the HBBs which become the PBHs with a mass M PBH atthe horizon re-entry over logarithmic mass interval d (ln M PBH ) . The relation between N and M H , Eq.(2.19) introduces the factor / in Eq.(3.6). In this section, we investigate the possibility that the PBHs in our scenario account for theSMBHs. First, let us summarize the conditions for the seed BHs of the SMBHs. As supportedby observations of high red-shift ( z ∼ ) SMBHs, the formation of galaxies is considered tobe preceded by the formation of seed BHs [49]. Therefore, we assume that the number densityof SMBHs is equal to that of galaxies. The number density of galaxies N gal is not well-known.Here, we take it as N gal = (10 − − − ) Mpc − . (4.1)This is consistent with the estimated values in Refs. [50, 51]. In Ref. [27], it is shown in thenumerical simulation that PBHs with masses around (10 − ) M (cid:12) subsequently grow up to M (cid:12) . Since PBHs heavier than (10 − ) M (cid:12) are also supposed to become SMBHs [52], weassume that PBHs heavier than a certain boundary value M b are seed BHs of SMBHs. Here, wetake M b as M b = (10 − ) M (cid:12) , (4.2)following Ref. [35].Next, let us estimate the abundance of the PBHs. The present abundance of the PBHswith mass M PBH over logarithmic mass interval d(ln M PBH ) is given by Ω PBH ( M PBH )Ω c (cid:39) ρ PBH ρ m (cid:12)(cid:12)(cid:12)(cid:12) eq Ω m Ω c = Ω m Ω c T ( M PBH ) T eq β PBH ( M PBH ) (cid:39) (cid:18) β PBH ( M PBH )1 . × − (cid:19) (cid:18) Ω c h . (cid:19) − (cid:18) M PBH M (cid:12) (cid:19) − / , (4.3)where Ω c and Ω m are the present density parameters of dark matter and matter, respectively. h is the present Hubble parameter in units of km/sec/Mpc. T ( M PBH ) and T eq are respectivelythe temperatures at the formation of PBHs with a mass M PBH and at the matter-radiationequality. The number density of PBHs over d(ln M PBH ) is written as d N PBH d(ln M PBH ) ( M PBH ) = Ω
PBH ( M PBH ) ρ crit M PBH (4.4) (cid:39) . × Mpc − (cid:18) β PBH ( M PBH )1 . × − (cid:19) (cid:18) M PBH M (cid:12) (cid:19) − / , (4.5)where ρ crit ≡ H M is the critical density. The relation which N gal and M b should satisfy iswritten as N gal = (cid:90) ∞ ln M b d(ln M PBH ) d N PBH d ln( M PBH ) . (4.6)– 9 – ��� �� � �� � �� � �� � �� - �� �� - �� �� - �� �� - � �� - � �� - � �� - � �� � �� � �� � �� � �� � �� - �� �� - �� �� - � �� - � �� - � �� - � �� - � Figure 3 . The PBH abundance. The left panel is the case with M c = 10 M (cid:12) and the right panel is thecase with M c = 10 M (cid:12) . In each panel, three lines correspond to N gal = 0 . , . , . Mpc − from topto bottom if M c ≥ M b . ���� �� � �� � �� � �� � ������������������ Figure 4 . The number density of PBHs assuming M c ≥ M b . Each plot is normalized by its maximumvalue. If M c < M b , the cutoff below which there is no PBH appears at M PBH = M b . In Fig. 3, we show the abundance of the PBHs with the parameters satisfying Eq. (4.6). Althoughthe number density of galaxies and the masses of the seed BHs are not well-known and we putsome assumptions on them in the above discussion, the mass distribution of the PBHs can beeasily modified in our model by varying the model parameters ( c I , ∆ , η in b ) .The HBBs with masses less than M c do not collapse into PBHs but they can contribute to– 10 –he baryon asymmetry of the Universe. In order to evaluate this contribution, we introduce thequantity η Bb = (cid:0) f B ( N e ) − f B ( N ( M c )) (cid:1) η in b , (4.7)which represents the averaged baryon asymmetry. Those baryons are highly inhomogeneous andnucleosynthesis in the HBBs are different from the standard BBN because of high baryon densityenvironment, which predicts quite different abundances for light elements. Therefore, in ordernot to spoil the success of the standard BBN, we should require η Bb (cid:28) η ob b ∼ − where η ob b isthe observed baryon asymmetry. The parameters in Fig. 3, where η Bb /η ob b (cid:46) − , satisfy thisrequirement. In Fig. 4, we show the number density spectrum of the PBHs. It can be seen thatthey have similar shapes with sharp peaks at M c independent of the choice of the parameters.Since the HBBs can account only for the small fraction of the observed baryon asymmtery, weneed another baryogenesis mechanism. The simplest possibility is to utilize another AD field (flatdirection) with negative Hubble mass during and after inflation, which leads to the conventionalAffleck-Dine baryogenesis.Finally, we comment on the gauge-mediated SUSY breaking scenario. As mentioned inthe previous section, in this scenario, the coherent oscillation of the AD field in the HBBs canfragment into stable Q-balls. Then, the density contrast between inside and outside the HBBsgrows as ∝ T − and the HBBs which re-enter the horizon sufficiently later collapse into PBHs.On the other hand, the residual Q-balls in the small HBBs which re-enter the horizon earlierand hence do not collapse into PBHs survive until now and they contribute to the current darkmatter abundance. Interestingly, for the appropriate parameters c (cid:48) I (cid:46) . , the PBHs originatedfrom HBBs become the seed BHs of the SMBHs and at the same time the surviving Q-ball canaccount for the whole dark matter. In Fig. 5, we show the abundance of PBHs with c (cid:48) I = 0 . .In the case of M c = 10 M (cid:12) and N gal = 0 . Mpc − in Fig. 5, the surviving Q-balls account foralmost all the dark matter. In this paper, we have examined whether the HBBs produced in the modified AD mechanismproposed in Ref. [37] account for the seed PBHs of the SMBHs in the galactic centers. Themodified AD mechanism in MSSM realizes inhomogeneous baryogenesis by taking into accountthe Hubble induced mass and the finite temperature effect, which leads to the formation ofhigh baryon density regions called HBBs. We have mainly studied the gravity-mediated SUSYbreaking scenario. In this scenario, after the QCD phase transition, the massive baryons insidethe HBBs behave as non-relativistic matter and the HBBs have larger densities than the outsideof the HBBs. Because the density perturbations produced by the HBBs are highly non-Gaussian– 11 – ��� �� � �� � �� � �� � �� - �� �� - �� �� - �� �� - � �� - � �� - � �� - � �� � �� � �� � �� � �� � �� - �� �� - �� �� - � �� - � �� - � �� - � �� - � Figure 5 . The PBH abundance. The left panel is the case with M c = 10 M (cid:12) and the right panel is thecase with M c = 10 M (cid:12) . In each panel, three lines correspond to N gal = 0 . , . , . Mpc − from topto bottom if M c ≥ M b . and HBBs are rare objects, the produced perturbations hardly constrained by the CMB µ -distortion or the PTA experiments.If the HBBs have sufficiently large densities when they re-enter the horizon, the HBBsgravitationally collapse into PBHs. The density contrasts of the HBBs increase as the cosmictemperature deceases. Therefore, only HBBs with large masses, which re-enter the horizon atlate epochs, can form PBHs. Thus, the condition for the PBH formation introduces a lowercut-off M c on the mass of the produced PBHs. The mass distribution of the PBHs includingthe cut-off depends on the parameters of inflation and the potential for the AD field. We haveshown that the PBHs produced by this mechanism can have a reasonable number density andmasses as the seed BHs of the SMBHs for appropriate values of the model parameters.The small HBBs which do not collapse into PBHs contribute to the baryon asymmetry ofthe universe. However, since the produced baryons are highly inhomogeneous, nucleosynthesisproceeds quite differently from the standard BBN. Thus, the success of the BBN requires thatthe baryon asymmetry produced through HBBs should be much smaller than the observed one.So, we need another mechanism or another AD field to produce the observed baryon asymmetry.In the gauge-mediated SUSY breaking scenario, the stable Q-balls can contribute to boththe PBHs and the dark matter abundance. In our model, the PBHs produced from the Q-ballscan have the reasonable number density and masses as the seed BHs of the SMBHs and, at thesame time, the residual Q-balls can constitute the whole dark matter abundance.Although the required mass distribution of the seed BHs of the SMBHs has some uncer-tainties associated with the number density of the galaxies and the mass growth of the SMBHs,the validity of these result are hardly affected because the mass distribution of the PBHs in ourscenario can be easily controlled by varying the parameters.– 12 – cknowledgments This work was supported by JSPS KAKENHI Grant Nos. 17H01131 (M.K.) and 17K05434(M.K.), MEXT KAKENHI Grant No. 15H05889 (M.K.), World Premier International ResearchCenter Initiative (WPI Initiative), MEXT, Japan (M.K., K.M.), and Program of Excellence inPhoton Science (K.M.).
References [1] J. Kormendy and D. Richstone,
Inward bound: The Search for supermassive black holes in galacticnuclei , Ann. Rev. Astron. Astrophys. (1995) 581.[2] J. Magorrian et al., The Demography of massive dark objects in galaxy centers , Astron. J. (1998) 2285, [ astro-ph/9708072 ].[3] D. Richstone et al.,
Supermassive black holes and the evolution of galaxies , Nature (1998)A14–A19, [ astro-ph/9810378 ].[4] Y. Matsuoka et al.,
Subaru high-z exploration of low-luminosity quasars (SHELLQs). I. Discoveryof 15 quasars and bright galaxies at 5.7 < z < 6.9 , Astrophys. J. (2016) 26, [ ].[5] S. Hawking,
Gravitationally collapsed objects of very low mass , Mon. Not. Roy. Astron. Soc. (1971) 75.[6] B. J. Carr and S. W. Hawking,
Black holes in the early Universe , Mon. Not. Roy. Astron. Soc. (1974) 399–415.[7] B. J. Carr,
The Primordial black hole mass spectrum , Astrophys. J. (1975) 1–19.[8] J. Garcia-Bellido, A. D. Linde and D. Wands,
Density perturbations and black hole formation inhybrid inflation , Phys. Rev.
D54 (1996) 6040–6058, [ astro-ph/9605094 ].[9] J. Yokoyama,
Formation of MACHO primordial black holes in inflationary cosmology , Astron.Astrophys. (1997) 673, [ astro-ph/9509027 ].[10] M. Kawasaki, N. Sugiyama and T. Yanagida,
Primordial black hole formation in a double inflationmodel in supergravity , Phys. Rev.
D57 (1998) 6050–6056, [ hep-ph/9710259 ].[11] D. Blais, C. Kiefer and D. Polarski,
Can primordial black holes be a significant part of darkmatter? , Phys. Lett.
B535 (2002) 11–16, [ astro-ph/0203520 ].[12] N. Afshordi, P. McDonald and D. N. Spergel,
Primordial black holes as dark matter: The Powerspectrum and evaporation of early structures , Astrophys. J. (2003) L71–L74,[ astro-ph/0302035 ].[13] P. H. Frampton, M. Kawasaki, F. Takahashi and T. T. Yanagida,
Primordial Black Holes as AllDark Matter , JCAP (2010) 023, [ ].[14] S. Bird, I. Cholis, J. B. Muñoz, Y. Ali-Haïmoud, M. Kamionkowski, E. D. Kovetz et al.,
Did LIGOdetect dark matter? , Phys. Rev. Lett. (2016) 201301, [ ].[15] A. Kashlinsky,
LIGO gravitational wave detection, primordial black holes and the near-IR cosmicinfrared background anisotropies , Astrophys. J. (2016) L25, [ ].[16] M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,
Primordial Black Hole Scenario for theGravitational-Wave Event GW150914 , Phys. Rev. Lett. (2016) 061101, [ ].[17] B. Carr, F. Kuhnel and M. Sandstad,
Primordial Black Holes as Dark Matter , Phys. Rev.
D94 (2016) 083504, [ ]. – 13 –
18] M. Kawasaki, A. Kusenko, Y. Tada and T. T. Yanagida,
Primordial black holes as dark matter insupergravity inflation models , Phys. Rev.
D94 (2016) 083523, [ ].[19] S. Clesse and J. García-Bellido,
The clustering of massive Primordial Black Holes as Dark Matter:measuring their mass distribution with Advanced LIGO , Phys. Dark Univ. (2017) 142–147,[ ].[20] K. Inomata, M. Kawasaki, K. Mukaida, Y. Tada and T. T. Yanagida, Inflationary primordial blackholes for the LIGO gravitational wave events and pulsar timing array experiments , Phys. Rev.
D95 (2017) 123510, [ ].[21] K. Inomata, M. Kawasaki, K. Mukaida and T. T. Yanagida,
Double inflation as a single origin ofprimordial black holes for all dark matter and LIGO observations , Phys. Rev.
D97 (2018) 043514,[ ].[22] G. Ballesteros and M. Taoso,
Primordial black hole dark matter from single field inflation , Phys.Rev.
D97 (2018) 023501, [ ].[23] Y. N. Eroshenko,
Gravitational waves from primordial black holes collisions in binary systems , J.Phys. Conf. Ser. (2018) 012010, [ ].[24] K. Ando, K. Inomata, M. Kawasaki, K. Mukaida and T. T. Yanagida,
Primordial black holes forthe LIGO events in the axionlike curvaton model , Phys. Rev.
D97 (2018) 123512, [ ].[25] K. Ando, M. Kawasaki and H. Nakatsuka,
Formation of primordial black holes in an axionlikecurvaton model , Phys. Rev.
D98 (2018) 083508, [ ].[26] R. Bean and J. Magueijo,
Could supermassive black holes be quintessential primordial black holes? , Phys. Rev.
D66 (2002) 063505, [ astro-ph/0204486 ].[27] Y. Rosas-Guevara et al.,
Supermassive black holes in the EAGLE Universe. Revealing theobservables of their growth , Monthly Notices of the Royal Astronomical Society (07, 2016)190–205, [ ].[28] M. Kawasaki, A. Kusenko and T. T. Yanagida,
Primordial seeds of supermassive black holes , Phys.Lett.
B711 (2012) 1–5, [ ].[29] W. Hu, D. Scott and J. Silk,
Power spectrum constraints from spectral distortions in the cosmicmicrowave background , Astrophys. J. (1994) L5–L8, [ astro-ph/9402045 ].[30] J. Chluba and R. A. Sunyaev,
The evolution of CMB spectral distortions in the early Universe , Mon. Not. Roy. Astron. Soc. (2012) 1294–1314, [ ].[31] K. Kohri, T. Nakama and T. Suyama,
Testing scenarios of primordial black holes being the seeds ofsupermassive black holes by ultracompact minihalos and CMB µ -distortions , Phys. Rev.
D90 (2014) 083514, [ ].[32] A. Dolgov and J. Silk,
Baryon isocurvature fluctuations at small scales and baryonic dark matter , Phys. Rev.
D47 (1993) 4244–4255.[33] A. D. Dolgov, M. Kawasaki and N. Kevlishvili,
Inhomogeneous baryogenesis, cosmic antimatter,and dark matter , Nucl. Phys.
B807 (2009) 229–250, [ ].[34] S. Blinnikov, A. Dolgov, N. K. Porayko and K. Postnov,
Solving puzzles of GW150914 byprimordial black holes , JCAP (2016) 036, [ ].[35] A. D. Dolgov and S. Porey,
Massive primordial black holes in contemporary universe , .[36] T. Nakama, T. Suyama and J. Yokoyama, Supermassive black holes formed by direct collapse ofinflationary perturbations , Phys. Rev.
D94 (2016) 103522, [ ]. – 14 –
37] F. Hasegawa and M. Kawasaki,
Cogenesis of LIGO Primordial Black Holes and Dark Matter , Phys. Rev.
D98 (2018) 043514, [ ].[38] F. Hasegawa and M. Kawasaki,
Primordial Black Holes from Affleck-Dine Mechanism , JCAP (2019) 027, [ ].[39] A. Vilenkin and L. H. Ford,
Gravitational Effects upon Cosmological Phase Transitions , Phys. Rev.
D26 (1982) 1231.[40] A. A. Starobinsky,
Dynamics of Phase Transition in the New Inflationary Universe Scenario andGeneration of Perturbations , Phys. Lett. (1982) 175–178.[41] A. D. Linde,
Scalar Field Fluctuations in Expanding Universe and the New Inflationary UniverseScenario , Phys. Lett. (1982) 335–339.[42] K. Inomata, M. Kawasaki, A. Kusenko and L. Yang,
Big Bang Nucleosynthesis Constraint onBaryonic Isocurvature Perturbations , JCAP (2018) 003, [ ].[43]
NANOGrav collaboration, Z. Arzoumanian et al.,
The NANOGrav Nine-year Data Set: Limitson the Isotropic Stochastic Gravitational Wave Background , Astrophys. J. (2016) 13,[ ].[44] L. Lentati et al.,
European Pulsar Timing Array Limits On An Isotropic StochasticGravitational-Wave Background , Mon. Not. Roy. Astron. Soc. (2015) 2576–2598,[ ].[45] R. M. Shannon et al.,
Gravitational waves from binary supermassive black holes missing in pulsarobservations , Science (2015) 1522–1525, [ ].[46] A. Kusenko and M. E. Shaposhnikov,
Supersymmetric Q balls as dark matter , Phys. Lett.
B418 (1998) 46–54, [ hep-ph/9709492 ].[47] K. Enqvist and J. McDonald,
Q balls and baryogenesis in the MSSM , Phys. Lett.
B425 (1998)309–321, [ hep-ph/9711514 ].[48] S. Kasuya and M. Kawasaki,
Q ball formation through Affleck-Dine mechanism , Phys. Rev.
D61 (2000) 041301, [ hep-ph/9909509 ].[49] R. C. E. van den Bosch, K. Gebhardt, K. Gultekin, G. van de Ven, A. van der Wel and J. L.Walsh,
An Over-Massive Black Hole in the Compact Lenticular Galaxy NGC1277 , Nature (2012) 729–731, [ ].[50] C. J. Conselice et al.,
The Evolution of Galaxy Number Density at z < 8 and its Implications , TheAstrophysical Journal (2016) 83, [ ].[51] H.-A. Shinkai, N. Kanda and T. Ebisuzaki,
Gravitational waves from merging intermediate-massblack holes : II Event rates at ground-based detectors , Astrophys. J. (2017) 276, [ ].[52] A. D. Dolgov,
Massive and supermassive black holes in the contemporary and early Universe andproblems in cosmology and astrophysics , Usp. Fiz. Nauk (2018) 121–142, [ ].].