Primordial black holes from long-range scalar forces and scalar radiative cooling
IIPMU20-0092
Primordial black holes from long-range scalar forces and scalar radiative cooling
Marcos M. Flores and Alexander Kusenko
1, 2 Department of Physics and Astronomy, University of California, Los AngelesLos Angeles, California, 90095-1547, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIASThe University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: September 7, 2020)We describe a new scenario for the formation of primordial black holes (PBHs). In the earlyuniverse, the long-range forces mediated by the scalar fields can lead to formation of halos of heavyparticles even during the radiation-dominated era. The same interactions result in the emission ofscalar radiation from the motion and close encounters of particles in such halos. Radiative coolingdue the scalar radiation allows the halos to collapse to black holes. We illustrate this scenario on asimple model with fermions interacting via the Yukawa forces. The abundance and the mass functionof PBHs are suitable to account for all dark matter, or for some gravitational waves events detectedby LIGO. The model relates the mass of the dark-sector particles to the masses and abundance ofdark matter PBHs in a way that can explain why the dark matter and the ordinary matter havesimilar mass densities. The model also predicts a small contribution to the number of effective lightdegrees of freedom, which can help reconcile different measurements of the Hubble constant.
Primordial black holes (PBHs) formed in the earlyUniverse can account for all or part of dark matter [1–32]. Furthermore, PBHs can seed supermassive blackholes [33–35], can play a role in the synthesis of heavyelements [36–38], and can be responsible for some ofthe gravitational wave events detected by LIGO [39–44].High energy density in the early universe facilitates for-mation of PBHs in the presence of large perturbationsfrom inflation (e.g., [5–7]) or from the scalar field dy-namics [22–24, 29]. The scalar forces can generate insta-bilities [4, 45] leading to PBHs [4, 22–24, 29]. However,in this class of scenarios, PBHs can only form from rare,overdense, spherical halos, while the rest of the halosvirialize and remain mechanically stable until the decayof their constituent particles, Q-balls or oscillons [29].Scalar force instability can lead to a growth of structuresand formation of halos of interacting particles even dur-ing the radiation dominated era [46–50], and it was con-jectured that such early growth of structure could pro-duce PBHs [49]; but, unlike the scalar field fragmentationscenarios [22–24, 29], the growth of structure in the mat-ter composed of elementary particles leads to virializedhalos, not PBHs [29, 46–48, 50].We describe a new scenario for PBH formation, whichis simple and generic: in its minimal realization it in-volves only one species of heavy particles interacting viathe Yukawa forces mediated by a scalar field. The samelong-range scalar interactions that cause the formationof halos during the radiation dominated era [46–51] al-low for emission of scalar waves, which drain energy fromthe virialized halos and facilitate their collapse to PBHs.Let us consider a fermion ψ interacting with a scalarfield χ : L ⊃ m χ χ + m ψ ¯ ψψ − yχ ¯ ψψ + ... (1)We assume that the universe was radiation dominated at temperatures T > m ψ , and that the ψ particles hadequilibrium density. We will also assume that the par-ticle number is preserved by an approximate symmetry,and we allow an asymmetry η ψ = ( n ψ − n ¯ ψ ) /s = 0 todevelop, in analogy with the baryon asymmetry η B , asin the asymmetric dark matter models [52, 53]. We willassume that the χ field is either massless or very light, m χ (cid:28) m ψ /M P , and that the ψ particles are either sta-ble or have a total decay width Γ ψ (cid:28) m ψ /M P , where M P = M P lanck / √ π ≈ × GeV is the reducedPlanck mass, so that there is a cosmological epoch dur-ing which the ψ particles are nonrelativistic, decoupledfrom equilibrium, and they interact with each other viaan attractive long-range force mediated by the χ fieldand described by the potential V ( r ) = y r e − m χ r . (2)During the radiation dominated era, gravitational in-teractions are not sufficient to allow for a linear growthof structures. However, scalar forces are usually (and,possibly, always [54–57]) stronger than gravity, β ≡ y ( M P /m ψ ) (cid:29)
1, and such forces can cause the fluctu-ations in the ψ particle number to grow even in the ra-diation dominated era [46–50]. We note that the scalarforces couple not to the mass density, but to the ψ num-ber density, and the halos of ψ particles grow in the oth-erwise uniform background of radiation as a form of anisocurvature perturbation.The adiabatic density perturbations δ ( x, t ) = δρ/ρ grow only logarithmically during the radiation dominatedera. However, the presence of a long-range “fifth force”stronger than gravity causes the fluctuations ∆( x, t ) =∆ n ψ /n ψ for an out-of-equilibrium population of heavy,nonrelativistic ψ particles to grow rapidly, as long as ψ is decoupled from radiation, so that the pressure can be a r X i v : . [ a s t r o - ph . C O ] S e p neglected.For the model of Eq. (1), if the mean free path of χ particles in a halo of ψ particles is longer than the size ofthe halo, the halo is not subject to radiative pressure dueto the χ radiation. The temperature at which it is true forthe Hubble size halos, and the structures start growing,is T g ∼ m/ [ln( y M P /m )]. This temperature is also closeto the temperature T f at which the annihilation reactions¯ ψψ → χχ freeze out, which, for y ∼ η ψ (cid:28) T f ∼ m/
36 [58].In Fourier space, the growth of these perturbationsbelow T f is described by the system of coupled equa-tions [46–50]¨ δ k + 1 t ˙ δ k − t (Ω r δ k + Ω m ∆ k ) = 0 (3)¨∆ k + 1 t ˙∆ k − t [Ω r δ k + Ω m (1 + β )∆ k ] = 0 , (4)where Ω r = ρ r / ( ρ r + ρ ψ ) and Ω ψ = ρ ψ / ( ρ r + ρ ψ )are the radiation and matter fractions, respectively, andΩ r + Ω m = 1. Assuming that only radiation and ψ parti-cles are present, and anticipating that all the ψ particleswill end up in PBHs, which also scale as matter, the timedependence of these fractions before the matter-radiationequality, t < t eq is given by Ω r = [1 + p t/t eq ] − andΩ m = [1 + p t eq /t ] − . In the limit β (cid:29)
1, the pertur-bations grow fast, so that one can assume Ω ≈ const,and∆ k ( a ) ≈ ∆ k, in (cid:18) tt in (cid:19) p/ , p = r
32 (1 + β )Ω ψ (5)For p (cid:29)
1, the time scale τ ∆ ≡ ∆ k / ( d ∆ k /dt ) is shorterthan the Hubble time, which implies a very rapid struc-ture formation. Thus, in the limit of a strong Yukawaforce, the structures form almost instantaneously on allscales up to the horizon size as soon as the ψ parti-cles decouple. This process was studied in the past,but the fate of the nonlinear structures was not eluci-dated. In Ref. [49], it was conjectured that the struc-tures could form black holes, but it was later realizedthat, instead, these structures remain as virialized darkmatter clumps [50]. In the absence of energy dissipation,the latter conclusion is correct because virialization putsan end to any further contraction of halos, unless energyand angular momentum can be transferred out of thecontracting halo.However, the same long-range forces that cause thegrowth of structure in the ψ -particle fluid also causeany particles moving with an acceleration to emit scalarwaves, which can dissipate energy from a halo. This isthe key element of PBH formation in the system of mat-ter particles interacting by long-range attractive forces.A virialized halo of N particles interacting by scalarYukawa forces has the potential energy E ∼ y N R , where R is the characteristic size of the halo. Each particleis a source of a scalar field which can be thought of asclassical and long-range on the length scales shorter than m − χ . A collection of N particles moving inside the halocan radiate scalar waves in several ways.First, if the motion is coherent, a dipole moment ro-tating with a frequency ω can produce a dipole radiation P coh ∝ y N . However, for a system of N identical par-ticles, the dipole moment about the center of mass isidentically zero because the charge is proportional to themass, and the first moment of the mass distribution iszero (by the definition of the center of mass). The coher-ent quadrupole radiation is possible, but it is suppressed.Second, if each particle is treated as an incoherentsource of radiation, the radiated power is proportionalto the square of the orbital acceleration a = ω R , where ω can be different for different particles. The radiatedpower P incoh ∝ y ω R N scales as the first power of thenumber of particles. This is the correct picture of radia-tive energy losses in the limit of relatively low numberdensity of particles.Third, there is scalar bremsstrahlung radiation simi-lar to free-free emission of photons from plasma [59, 60].Unlike the usual plasma with two charges of parti-cles, our system has identical particles, so the lead-ing bremsstrahlung radiation in two-particle collisions isquadrupole, not dipole, and it is similar to e − e compo-nent of the free-free emission from plasma [59, 60].Finally, if the contracting halo becomes opaque, theradiation is trapped, and the halo turns into a fireball oftemperature T halo . This happens when the collapse timescale τ coll = R ( t ) / ( dR/dt ) is shorter than the diffusiontime scale for χ radiation τ diff ∼ R /λ χ , where λ χ is themean free path of the χ particle in the halo. The meanfree path of the radiated χ particles is λ χ = 4 π/y T halo .When the χ radiation is trapped, the cooling proceedsfrom the surface; it can be approximated by the black-body radiation with the power P surf ∼ πR T . Theenergy transfer inside the halo can proceed either by dif-fusion or by convection, and the latter dominates. Forlarge β (cid:29)
1, the scalar force gradients (which exceedthe gravitational accelerations) overwhelm the viscosity,leading to very large Rayleigh numbers and fast convec-tion time scales. The time scale for convective trans-port is τ conv ∼ η/ ( Rρg ), where g = y N/mR , and η ∼ T is viscosity [61], leading to the Rayleigh num-ber Ra ∼ y N RT /m (cid:29)
1, which indicates the halois highly convective, and convection dominates the heattransport from the core to the surface.Each of these mechanisms can reduce the energy of thehalo on some characteristic time scale. The energy losstime scale is given by τ = EdE/dt = EP incoh + P ff + P surf ... , (6)where E ∼ y q N R , (7) P incoh ∼ y q N m R , (8) P ff ∼ y q N T eff m R ln (cid:18) T eff m (cid:19) (9) ∼ y q N m R ln (cid:18) N y q mR (cid:19) , (10) P surf ∼ πR T = 4 π y q N R , (11)where T eff ∼ y N/R is the energy per particle beforethe radiation is trapped, while T halo ∼ √ yN /R is thetemperature of trapped radiation after thermalization,as discussed below. Here q = 1 for a single ψ particle,while a clump of particles in orbital motion can have q (cid:29)
1. The particle mass includes the finite-temperaturecorrections, m = m ψ + ( y/ T [62].When the particle density is very low, the incoherentemission (8) is the dominant channel for the energy loss.However, when the mean separation between particlesis smaller than the radiation length, the radiation fromthe neighboring particles can interfere, and Eq.(8) is notapplicable. However, since the structure we consider ex-ists on a broad range of scales, small clumps rotating inthe larger halo can radiate as “particles” in Eq.(8) with q (cid:29)
1. In the absence of N-body simulations, we cannotreliably count on this dissipation channel. Therefore, wewill base the discussion on the bremsstrahlung emission(10), yielding a conservative estimate, which can only behelped by any additional dissipation.A halo of size R can lose energy and contract to ablack hole at temperature T as long as τ ( R ) < M P /T .Since the time scale is an increasing function of the halosize, the halos with smaller R , for which τ < M P /T ,collapse first. Those halos for which τ ( R ) > M P /T may never collapse if the formation of PBH from smallerhalos eliminates the long-range scalar forces.Initially, the halo of size R has a potential energy ∼ y N /R i , and it initially radiates with the power P ff ,Eq. (10). As the halo contracts, the number density in-creases and the χ radiation is trapped forming a fireball oftemperature T halo that can be estimated from energy con-servation: − y N /R i = − y N /R ( t )+(4 π/ R ( t ) T .This implies the halo temperature T halo ∼ √ yN /R ( t ).The solution for the size of the halo determined by dE/dt = P surf , which implies R ( t ) = R (0)(1 − t/τ surf ). Asthe halo starts to shrink, the characteristic time scale τ decreases, leading to even faster energy dissipation. Thissignals collapse of the halo to a PBH.At high densities, the ψ particles can form bound stateswith discrete quantum levels, and the emission picturechanges to that which is similar to hot gas emitting pho-tons. The viscosity and the ram pressure of such a gas of “atoms” can speed up the process of collapse into a blackhole.This very simplified thermal history involves twostages: the initial cooling by bremsstrahlung, until theradiation is trapped, and the following cooling from thesurface of a hot fireball. The bremsstrahlung time scale τ ff is the longer of the two, and it serves as the bottlenecklimiting the collapse of the larges halos.We find that for a wide range of parameters and y (cid:38) − , the radiative cooling time scale in either the high-density, low-density, or intermediate-density regimes issmaller than the Hubble time. Therefore, the col-lapse of a halo to a black hole is possible and it pro-ceeds unimpeded as the radius decreases and reaches theSchwarzschild radius.Formation of black holes halts further structure evolu-tion because, in accordance with the no-hair theorems,black holes do not carry global charges and do not feelthe long-range forces due to scalar interactions of parti-cles that fell into the black holes. The strong long-rangeforces are likely to cause all or most of the ψ particlesto end up in PBH. The cosmological PBH abundance isthen equal to the ψ particle abundance, and their fractionat present time is related to the baryon density: f PBH = Ω
PBH Ω DM = 0 . m ψ m p η ψ η B = (cid:16) m ψ (cid:17) (cid:16) η ψ − (cid:17) . (12)Therefore, our scenario has the same potential to explainthe closeness of Ω DM and Ω B , as the models with asym-metric particle dark matter [52, 53]. The asymmetry η ψ can arise from the same process that produces the baryonasymmetry of the universe.Let us now estimate the mass function of PBHs start-ing with the smallest masses. The limit N > ( M P /m ) can be derived by requiring that, as R approaches theSchwarzschild radius R S = mN/M P , the halo is stilllarger than the Compton wavelength of the ψ particle. Itis unlikely that a black hole would form from a halo withfewer particles than N min = ( M P /m ) . For fermions ψ ,one also needs to require that, as the Fermi degeneracy isreached in the course of a collapsing halo, the Fermi en-ergy be small compared to the potential energy y N/R as R → R S . This condition turns our to be less constrainingthan the quantum condition N > N min . We note thatthe Chandrasekhar limit of
N > ( M P /m ) derived forthe gravitational potential is effectively weakened hereby a factor ( m/yM P ) (cid:28)
1. A naive lower limit onthe mass of a halo that can form a PBH could be setas
M > mN min = M P /m = 5 × − M (cid:12) (1 GeV /m ).However, it is unlikely that a black hole could form closeto the quantum uncertainty limit. Viscous friction, tidalfriction, and gravitational mergers cause multiple neigh-boring halos to merge and form a single black hole, henceincreasing the minimal size. We parameterize the mini-mal PBH mass in the form M min = ζN min m = 10 − M (cid:12) (cid:18) ζ (cid:19) (cid:18) m (cid:19) . (13)Here ζ = F visc F mergers , where F visc is the effect of viscousfriction and tidal effects that could lead to merger ofneighboring dense halos into one, and F mergers representsthe effects of gravitational merger of black holes. Theexact values of these factors require detailed analysis andnumerical simulations. We assume that the viscosity andthe gravitational tidal forces act at least on the lengthscales of the order of (10 − R , in a volume thatencompasses more than 10 halos, so that F visc (cid:38) , F mergers (cid:38) , leading to ζ ∼ , which we will use asan illustrative value.Since the PBH formation is rapid and takes about oneHubble time, the mass function of PBHs should representa snapshot of the structure in the ψ fluid at the time offormation. In the absence of N-body simulations, the de-tails of the ψ halo structure formation are not known,but the structure can be described approximately. Sincethe collapsing halos are formed from the growth of per-turbations followed by a short history of mergers, theresulting PBH mass function can be approximated by aPress–Schechter function: M dN h dM = 1 √ π n ψ (cid:18) MM ∗ (cid:19) / e − M/M ∗ . (14)The characteristic mass M ∗ is set by the largest size R ∗ for which the emission time scale τ ( R ∗ ) in Eq.(6)is smaller than the Hubble time. For the relevantrange of parameters, the main emission channels arebremsstrahlung ( τ ∼ τ ff ) followed by the radiative cool-ing from the surface ( τ ∼ τ surf ). Since τ ff > τ surf , it is thebremsstrahlung time scale τ ff that determines whether ornot a given halo has time to collapse before the smallerhalos become black holes and terminate the action of thelong-range forces. Solving for the size τ ff ( R ∗ ) = t H , wefind the characteristic mass M ∗ = 4 π mn ψ R ∗ , R ∗ ’ × η ψ M P y g / ∗ m ! / . (15)We can parameterize M ∗ in the form: M ∗ = 4 . × η ψ M P y g / ∗ m ! / (16)= 1 . × − M (cid:12) (cid:16) η ψ − (cid:17) / × (17) × (cid:18) m ψ (cid:19) / (cid:18) y × − (cid:19) / . (18)The resulting mass function is shown in Fig. 1 for ourmodel with m ψ = 5 GeV, η ψ = 10 − . The χ particle mass m χ must be small enough to allowfor the long-range forces. If m χ > T f /M P , the long-range force cuts off at distances R ∼ /m χ , resulting inthe upper limit on the size of he characteristic scale inthe Press-Schechter function, R ∗ < /m χ .The radiative cooling of a collapsing halo is a complexdynamical problem. We have neglected the spatial den-sity and temperature distributions and the existence ofsmaller halos inside larger halos, as well as screening ofthe long-range forces by the finite density and temper-ature corrections to the scalar mass [63, 64], which inturn depend on the density distribution. These effectscan be studied in numerical N-body simulations. If thecollapse is delayed by some dynamics not captured byour discussion, the delay allows larger structures to formand collapse, extending the mass function toward largermasses. - - - - FIG. 1. The mass functions of PBH (line labeled “DM”) canaccount for all dark matter if the asymmetry in the dark sectoris the same as the baryon asymmetry, η ψ ∼ η B ∼ − , in amodel with m ψ = 5 GeV and y = 5 × − . The PBHs are inthe mass range of interest to LIGO, Virgo, and KAGRA (linelabeled “GW”) for m ψ = 5 MeV, y = 1 . × − , η ψ = 10 − .The constraints are from Refs. [65–71]. Our scenario can be realized in a variety of models withdifferent degrees of complexity in the dark sector. Thesimplest model described by the Lagrangian (1) is partic-ularly appealing. Let us assume that the asymmetry inthe dark sector is similar to the baryon asymmetry of theuniverse as in popular models of asymmetric dark mat-ter [52, 53]. Then the abundance of PBH (12) is just rightto explain all dark matter for m ψ = 5 GeV. The resultingmass function of PBHs, shown in Fig. 1 by a solid linelabeled “DM”, is consistent with all present observationsand can account for all dark matter.We note that, if m ψ (cid:29) ∼ ∼ N eff = 3 .
05. If one assumes that the dark sector, com-prising ψ and χ particles, had the same temperature asthe visible sector at T ∼ m ψ , one can estimate the con-tribution ∆ N eff of the light χ particles to radiation. Inthe dark sector, the number of effective light degrees offreedom goes from g = 1 + (7 / × g = 1. Thiscontributes to the measured value of N eff [72–74]:∆ N eff = 14( g /g ∗ ( T d )) / ≈ . − . , (19)where the model-dependent temperature for decouplingbetween the visible and the dark sectors is taken to bein the range T d = 1 −
100 GeV. The value ∆ N eff ∼ . N eff = 0 . − . m ψ = 5 MeV, y = 1 . × − , η ψ = 10 − , the resulting mass functionextends to M ∗ (cid:38) M (cid:12) , with a sufficient abundance toexplain some of the events reported by LIGO [39].In summary, we have presented a novel scenario forthe formation of primordial black holes. The scalar fieldsthat mediate long-range attractive forces enable both theclustering of heavy particles and the radiative cooling byemission of scalar waves. The cooling facilitates collapseof the halos into black holes, which can account for alldark matter. In the example using decoupled fermionsinteracting by the Yukawa forces, the resulting PBH darkmatter density is related to the particle mass and cannaturally explain the dark matter abundance.We thank K. Petraki, J. Rubio, M. Sasaki, V. Takhis-tov, and E. Vitagliano for helpful discussions. This workwas supported by the U.S. Department of Energy (DOE)Grant No. DE-SC0009937. A.K. was also supported bythe World Premier International Research Center Initia-tive (WPI), MEXT, Japan. [1] Y. B. Zel’dovich and I. D. Novikov, Sov. Astron. , 602(1967).[2] S. Hawking, Mon. Not. Roy. Astron. Soc. , 75 (1971).[3] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron.Soc. , 399 (1974).[4] M. Khlopov, B. Malomed, and I. Zeldovich, Mon. Not.Roy. Astron. Soc. , 575 (1985). [5] J. Yokoyama, Astron. Astrophys. , 673 (1997),arXiv:astro-ph/9509027.[6] J. Garcia-Bellido, A. D. Linde, and D. Wands, Phys.Rev. D54 , 6040 (1996), arXiv:astro-ph/9605094 [astro-ph].[7] M. Kawasaki, N. Sugiyama, and T. Yanagida, Phys. Rev.
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