THE

PRIMORDIAL BLACK HOLE

MASS RANGE

Paul H. Frampton

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REFERENCES

P.H. Frampton,The Primordial Black Hole Mass RangearXiv:1511.xxxxx [hep-ph]

P.H. Frampton,Searching for Dark Matter Constituents withMany Solar Masses.arXiv:1510.00400[hep-ph]

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Introduction

According to global analyses of the cosmologi-cal parameters one quarter, or slightly more, ofthe energy of the universe is in the form of darkmatter whose constituent is the subject of thepresent paper. Recently it has been proposedthat the dark matter constituents are blackholes with masses many times the mass of theSun. In a galaxy like the Milky Way, the pro-posal is that residing in the galaxy are betweenten million and ten billion black holes withmasses between one hundred and one hundredthousand solar masses.

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Black holes in this range of masses are com-monly known as Intermediate Mass Black Hole(IMBHs) since they lie above the masses ofstellar-mass black holes and below the massesof the supermassive black holes. It has longbeen mysterious why there is a mass gap be-tween stellar-mass and supermassive black holes.If the proposed solution of the dark matterproblem is correct, it will answer this old ques-tion.

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There is irrefutable evidence for stellar-massblack holes from observations of X-ray bina-ries. Such systems were first emphasized byZeldovich then further studied in by Trimbleand Thorne. All the known stellar-mass blackholes are members of X-ray binaries. The firstwas discovered over fifty years ago in 1964 inCygnus X-1 and many stellar-mass black holeshave since been discovered from studies of X-ray binaries, with masses in a range between5M� and 100M�, where the first-discoveredCygnus X-1 is at about 15M�.

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There is irrefutable observational evidence alsofor supermassive black holes from the obser-vations of fast-moving stars around them andsuch stars being swallowed or torn apart bythe strong gravitational field. The first discov-ered SMBH was naturally the one, Sag A∗, atthe core of the Milky Way which was discov-ered in 1974 and has mass MSagA∗ ∼ 4.1 ×106M�. SMBHs discovered at galactic coresinclude those for galaxies named M31, NGC4889,among many others. The SMBH at the coreof the nearby Andromeda galaxy (M31) hasmass M = 2 × 108M�, fifty times MSagA∗.The most massive core SMBH so far observedis for NGC4889 withM ∼ 2.1×109M�. Somegalaxies contain two SMBHs in a binary, be-lieved to be the result of a galaxy merger. Quasarscontain black holes with even higher masses upto at least 4× 1010M�.

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We note historically that dark matter was firstdiscovered by Fritz Zwicky in 1933 in the ComaCluster, and its presence in galaxies was demon-strated convincingly by Vera Rubin in the 1960sand 1970s from the rotation curves of manygalaxies. Rubin has more recently made a pre-scient remark about not liking a universe filledwith a new kind of elementary particle and weshall return to this, with the full quote, at theend of our final discussion.

Regarding the PBH mass range, the purposeof the present article is to convince the readerthat the possible PBH masses extend upwardsto many solar masses and above, far beyondwhat was was thought possible not many yearsago when ignorance about PBHs with manysolar masses probably prevented the MACHOand EROS Collaborations from discovering allthe dark matter.

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The plan of the present talk is

Section 2 : we review the original implementa-tion a la Carr and Hawking of PBH formation.

Section 3 : we discuss parametric resonance inhybrid inflation which can produce PBHs witharbitrary mass.

Section 4 : possible implications are discussedespecially for dark matter but also for galactic-core supermassive black holes and unassoci-ated black holes.

Section 5 : final discussion.

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2. PBHs a la Carr and Hawking

If all black holes were formed by gravitationalcollapse then black holes with MBH � M�would be impossible because stars powered bynuclear fusion cannot be far below M = M�.It was first suggested by Zel’dovich and byHawking that black holes can be produced inthe early stages of the cosmological expansion.

Such PBHs are of special interest for severalreasons. Firstly, they are the only type of blackhole which can be so light, down to 1012kg ∼10−18M�, that Hawking radiation might con-ceivably be detected ∗. Secondly, PBHs in theintermediate-mass region 100M� ≤MIMBH ≤106M� can provide the galactic dark matter.Thirdly, supermassive PBHs with MSMBH ≥106M� can play a role at galactic centers andprovide some of the cluster dark matter.

∗We shall, however, confirm at the end of this Section that such detection is impracticable.

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The mechanism of PBH formation involves largefluctuations or inhomogeneities. Carr and Hawk-ing argued that we know there are fluctuationsin the universe in order to seed structure for-mation and there must similarly be fluctua-tions in the early universe. Provided the radi-ation is compressed to a high enough density,meaning to a radius as small as its Schwarzschildradius, a PBH will form. Because the densityin the early universe is extremely high, it isvery likely that PBHs will be created. Thetwo necessities are high density which is guar-anteed and large inhomogeneities.

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During radiation domination

a(t) ∝ t1/2 (1)

and

ργ ∝ a(t)−4 ∝ t−2 (2)

Ignoring factors O(1), as we shall do through-out this talk, and bearing in mind that theradius of a black hole is

rBH ∼

(MBH

M2Planck

)(3)

with

MPlanck ∼ 1019GeV ∼ 10−8kg ∼ 10−38M�(4)

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Using the Planck density ρPlanck

ρPlanck ≡ (MPlanck)4 ∼ (10−5g)(10−33cm)−3

(5)the density of a general black hole ρBH(MBH)is

ρBH(MBH) ∼

(MBH

r3BH

)= ρPlanck

(MPlanck

MBH

)2

(6)which means that for a solar-mass black hole

ρBH(M�) ∼ 1018ρH2O (7)

while for a billion solar mass black hole

ρBH(109M�) ∼ ρH2O. (8)

and above this mass the density falls as M−2BH .

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The mass of the Carr-Hawking PBH is derivedby combining results. We see that MPBHgrows linearly with time and using Planckianunits or Solar units we find respectively

MPBH ∼(

t

10−43sec

)MPlanck ∼

(t

1sec

)105M�

(9)which implies, if we insist on PBH formationbefore the electroweak phase transition, t <10−12s, that

MPBH < 10−7M� (10)

Such an upper bound explains historically whythe MACHO searches around 2000, inspired bythe 1986 suggestion of Paczynski, lacked moti-vation to pursue searching beyond 100M� be-cause it was thought incorrectly at that timethat PBHs were too light.

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It was known correctly that the results of grav-itational collapse of normal stars, or even largeearly stars, were below 100M�. Supermassiveblack holes with M > 106M� such as SagA∗

in the Milky Way were beginning to be dis-covered in galactic centers but their origin atthat time was mysterious. We shall discuss thisagain later in the talk.

Hawking radiation implies that the lifetime fora black hole evaporating in vacuo is given bythe cubic formula

τBH ∼(MBH

M�

)3

× 1064years (11)

so that to survive for the age 1010 years of theuniverse, there is a lower bound on MPBHgiving as the full range of Carr-Hawking PBHs:

10−18M� < MPBH < 10−7M� (12)

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The lowest mass PBH which survives to thepresent time has the extraordinary density ρ ∼1058ρH2O. It has the radius of a proton andthe mass of ten thousand aircraft carriers

The radiation domination ends at t ∼ 47ky ∼1012secwhich permits a PBH with mass 1017M�.This has Schwarzschild radius ∼ 104pc, Hawk-ing temperature ∼ 6 × 10−25K, and density∼ 10−16g/cm3. Such a possible primordialsuper-duper-massive black hole would be a hun-dred times the mass of the Virgo cluster andone millionth the total mass of the visible uni-verse. Such an object might be unassociatedwith any galaxy or cluster of galaxies.

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The Hawking temperature TH(MBH) of a blackhole is

TH(MBH) = 6× 10−8K

(M�MBH

)(13)

which would be above the CMB temperature,and hence there would be outgoing radiationfor all of the cases with MBH < 2×10−8M�.Hypothetically, if the dark matter halo weremade entirely of the brightest possible (in termsof Hawking radiation) 10−18M� PBHs, the ex-pected distance to the nearest PBH would beabout 107 km. Although the PBH tempera-ture is ∼ 6 × 1010K, the inverse square lawrenders the intensity of Hawking radiation toosmall, by many orders of magnitude, to allowits detection by any foreseeable apparatus onEarth.

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3. Parametric Resonancein Hybrid Inflation

The original Carr-Hawking mechanism producesPBHs with masses in the range up to 10−7M�.In this Section we shall exhibit formation ofPBHs by a different mechanism. As discussed,PBH formation requires very large inhomogeneities.Here we shall merely illustrate how to produceinhomogeneities which are exponentially large.

In a single inflation, no exceptionally large den-sity perturbation is expected. Therefore we usetwo-stage hybrid inflation with respective fieldscalled inflaton and waterfall. The idea of para-metric resonance is that after the first inflationmutual couplings of the inflaton and waterfallfields cause both to oscillate wildly and pro-duce perturbations which grow exponentially.

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The secondary (waterfall) inflation then stretchesfurther these inhomogeneities, enabling pro-duction of PBHs with arbitrarily high mass.The specific model provides an existence theo-rem to confirm that arbitrary mass PBHs canbe produced. The resulting mass function isspiked, but it is possible that other PBH pro-duction mechanisms can produce a smoothermass function, as deserves further study.

We use a supergravity framework, defining byS the inflaton superfield and by Ψ, Ψ the wa-terfall superfields. The superpotential is

W = S

(µ2 +

(ΨΨ)2

M2

)(14)

in which µ is the inflation scale and M is acut-off.

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The Kahler potential is

K = |S|2 + |Ψ|2 + |Ψ|2 (15)

and thence the potential is

V (σ, ψ)∼

(1 +

σ4

8+ψ2

2

)(−µ2 +

ψ4

4M2

)2

+σ2ψ6

16M4(16)

where we have defined ψ = 2<(Ψ) and σ =√2<(S) with < ≡ real part.

Stationarizing gives vacua at σ = 0 and ψ =2√µM . For the case σ >

õM/2 there is a

σ-dependent minimum for ψ at

ψ0 ∼(

2√3

)(µM

σ

). (17)

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Because ψ has a large mass, it rolls to ψ0 andintegrating it out results in the potential

V (σ) =µ4

(1 +

σ4

8− 2

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µ2M2

σ4

)

=µ4 +µ4

8

(σ4 − σ4

0

(σ0

σ

)4)

(18)

in which σ0 =√

2/338(µM)

14. So long as the

first term is largest, the inflaton slow rolls.

After this inflation, the σ and ψ fields oscillate,decaying into their quanta via their self andmutual couplings. Specific modes of σ and ψare amplified by parametric resonance.

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We may write the equation of motion for aFourier mode σk as the vanishing of

σ′′k+3Hσ

′k+

[k2

a2+ m2

σ + 3m2σ

ψ√µM

cos(mσt)

]σk

(19)

where we defined mσ =√

8µ3/M . and ψ isthe amplitude of ψ oscillations.

This recognized to be of Mathieu type withthe required exponentially-growing solutions.Numerical solution shows that the peak wavenumber kpeak is approximately linear in mσ.The resultant PBH mass, the horizon masswhen the fluctuations re-enter the horizon, isapproximately

MPBH ∼ 1.4× 1013M�

(kpeak

Mpc−1

)−2

(20)

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Explicit plots have been exhibited for the casesMPBH = 10−8M�, 10−7M� and 105M� butit was checked that the parameters can be cho-sen to produce arbitrary PBH mass.

In this production mechanism based on hybridinflation with parametric resonance, the massfunction is sharply spiked at a specific massregion. Whether such a mass function is ageneral feature of PBH formation, or is onlya property of this specific mechanism, meritsfurther study.

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4. Dark Matter andSupermassive Black Holes

In Section 2 we discussed the method of pro-ducing PBHs proposed by Carr and Hawk-ing. Insisting that the production take placebefore the electroweak phase transition, andbearing in mind the survival to the age of theuniverse from Hawking radiation, led us to arange of possible PBH masses from 10−18M�to 10−7M�.

In Section 3, using a different production mech-anism based on parametric resonance in hybridinflation this was augmented to a much biggermass range

10−18M� < MPBH < 1017M� (21)

which adds to Carr-Hawking, inter alia, Pri-mordial Intermediate-Mass Black Holes (PIMBHs)in the range

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102M� < MPIMBH < 106M� (22)

and Primordial Supermassive Black Holes (PSMBHs)in the range

106M� < MPSMBH < 1017M�. (23)

where we have truncated the upper end at 1017M�as the heaviest conceivable black hole likely toexist in the Universe.

For dark matter in galaxies, PIMBHs are im-portant, where the upper end may be trun-cated at 105M� to stay well away from galacticdisk instability. For supermassive black holesin galactic cores, PSMBHs are natural candi-dates, as they are also for a part of the darkmatter in clusters.

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Dark Matter in Galaxies

The dark matter in the Milky Way fills out anapproximately spherical halo somewhat largerin radius than the disk occupied by the lumi-nous stars. Numerical simulations of structureformation suggest a profile of the dark matterof the NFW type. The NFW profile is fullyindependent of the mass of the dark matterconstituent.

Our discussion focuses on galaxies like the MilkyWay and restricted the mass range for the ap-propriate dark matter to only three orders ofmagnitude

102M� < M < 105M� (24)

We shall not repeat the arguments here, justto say that the constituents are Primordial In-termediate Mass Black Holes, PIMBHs.

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Given a total dark halo mass of 1012M�, thenumber N of PIMBHs is between ten million(107) and ten billion (1010) Assuming the darkhalo has radius R of a hundred thousand (105)light years the mean separation L of PIMBHscan be estimated by

L ∼(R

N

)(25)

which translates to

100ly < L < 1000ly (26)

which is also an estimate of the distance of thenearest PIMBH to the Earth.

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It may be surprising that as many as 107 ≤N ≤ 1010 intermediate-mass black holes in theMilky Way have remained undetected. Theycould have been detected more than a decadeago had the MACHO Collaboration persistedin its microlensing experiment at Mount StromloObservatory in Australia. We shall return tothis point in our final discussion.

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Dark Matter in Clusters

The first discovery of dark matter by Zwickywas in the Coma cluster which is a large clusterat 99 Mpc containing over a thousand galaxiesand with total mass estimated at 6×1014M�.A nearer cluster at 16.5 Mpc is the Virgo clus-ter with over two thousand galaxies and whosemass ∼ 1015M� is also dominated by darkmatter, as well as a small amount of X-rayemitting gas. A proof of the existence (if morewere needed) of cluster dark matter was pro-vided by the Bullet cluster collision where thedistinct behaviors of the X-ray emitting gaswhich collides, and the dark matter which doesnot collide, was clearly observable.

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Since there is not the same disk stability limitas for galaxies, the constituents of the clusterdark matter can involve also PSMBHs up tomuch higher masses. In the Universe, we mayspeculate here that there may be unassociatedPBHs with any mass up to 1017M� driftingoutside of any galaxy or cluster of galaxies.

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Supermassive Black Holes atGalactic Centres

As mentioned in the Introduction, in the MilkyWay there is SMBH, SagA∗, with massMSagA∗ ∼4× 106M�. Other galaxies have SMBHs withmasses ranging up to 2.1 × 109M� (for thegalaxy NGC4889). Only a tiny fraction ofgalaxies have been studied, so the range ofgalaxies’ core SMBHs is likely broader.

A black hole with the mass of SagA∗ woulddisrupt the disk dynamics were it out in thespiral arms but at, or near to, the center ofmass it is more stable. SagA∗ is far too mas-sive to have been the result of a gravitationalcollapse, and if we take the view that all blackholes either are the result of gravitational col-lapse or are primordial then the galaxies’ coreSMBHs must be primordial. This offers a newexplanation of their origin.

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Galaxy formation

If our discussion is correct, it provides a cleartime-ordering for galaxy formation that the darkmatter precedes star formation by half a billionyears. Let us consider the history of the MilkyWay.

The constituents of the Milky Way’s dark mat-ter halo, PIMBHs, were produced in the eraof radiation domination which ended at timet ∼ 47ky (red shift Z ∼ 4760). Only muchlater, after 560 million years (Z ∼ 8), did starformation begin in the Milky Way.

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In this version of cosmic history, much of thelarge-scale structure formation including of galax-ies such as the Milky Way progresses duringthe half billion years represented by the redshifts 4760 > Z > 8. This stage impor-tantly involves only dark matter. Baryonicastrophysical objects like the Solar System ap-pear only when Z < 8 and are demoted to anafterthought with respect to the Milky Way’sformation.

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5. Discussion

Such a bold solution of the dark matter prob-lem cries out for experimental verification. Threemethods have been discussed: wide binaries,distortion of the CMB, and microlensing. Ofthese, microlensing seems the most direct andthe most promising.

Microlensing experiments were carried out bythe MACHO and EROS Collaborations severalyears ago. At that time, it was believed thatPBH masses were below 10−7M� by virtue ofthe Carr-Hawking mechanism. Heavier blackholes could, it was then believed, arise onlyfrom gravitational collapse of normal stars, orheavier early stars, and would have mass below100M�.

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For this reason, there was no motivation to sus-pect that there might be MACHOs which ledto higher-longevity microlensing events. Thelongevity, t, of an event is

t = 0.2yrs

(MPBH

M�

)12

(27)

which assumes a transit velocity 200km/s. Sub-situting our extended PBH masses, one findsapproximately t ∼ 6, 20, 60 years forMPBH ∼103, 104, 105M� respectively, and searching forlight curves with these higher values of t couldbe very rewarding.

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Our understanding is that the original tele-scope used by the MACHO Collaboration atthe Mount Stromlo Observatory in Australiawas accidentally destroyed by fire, and thatsome other appropriate telescopes are presentlybeing used to search for extasolar planets, ofwhich two thousand are already known.

It is seriously hoped that MACHO searcheswill resume and focus on greater longevity mi-crolensing events. Some encouragement canbe derived from this, written this month bya member of the original MACHO Collabora-tion:

There is no known problem with searchingfor events of greater longevity than thosediscovered in 2000; only the longevity of thepeople!

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That being written, convincing observationsshowing only a fraction of the light curves couldsuffice? If so, only a fraction of the e.g. sixyears, corresponding to PIMBHs with one thou-sand solar masses, could well be enough to con-firm the theory.

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Finally, going back to the 2010 Vera Rubinquote mentioned in the Introduction, it is

”If I could have my pick, I would like tolearn that Newton’s laws must be modifiedin order to correctly describe gravitationalinteractions at large distances. That’s moreappealing than a universe filled with a newkind of sub-nuclear particle.”

If our solution for the dark matter problem iscorrect, Rubin’s preference for no new elemen-tary particle filling the Universe would be vin-dicated, because for dark matter microscopicparticles become irrelevant. Regarding New-ton’s law of gravity, it would not need modifi-cation beyond general relativity theory whichis needed for the black holes. In this sense,Rubin did not need to pick either alternativeto explain dark matter.

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