primordial black holes in cosmology -...

Primordial Black Holes in Cosmology

Lecture 3 : Constraints on their existence

Massimo Ricotti (University of Maryland, USA)

Institute of Cosmos Sciences, University of Barcelona 23/10/2017

Astrophysical Constraints

Institute of Cosmos Sciences, University of Barcelona

• Microlensing

• Macho, EROS, etc

• UCMHs: lensing, astrometry, DM annihilation

• Dynamical constraints

• Power spectrum

• Dynamical heating (halo binaries and dwarfs)

• Gravitational waves from binary PBHs

• Capture

• Primordial binaries

• Effects on CMB, reionization and detection in X-ray, Radio (next lecture)

WIMPS DM


Constraints on evaporating PBHsCarr, Bernard J. Springer Proc.Phys. 170 (2016) 23-31 arXiv:1402.1437 [gr-qc]


Constraints on non-evaporating PBHsand/or radiation or by merging with other PBHs. WhileHawking radiation is completely negligible forintermediate-mass PBHs, their growth can be very impor-tant in the matter-dominated epoch [47,179,180]. Forinstance, it has been conjectured that PBHs with a massof 102–104M⊙ could provide seeds for the supermassiveblack holes of up to 1010M⊙ in the centers of galaxies[181]. However, this involves a growth of many orders ofmagnitude and careful numerical integration is required tostudy this, allowing for the dilution of the PBHs due tocosmic expansion and the merger of the smaller onesoriginating from critical collapse. The clustering of PBHswill also have significant effects on their merger rates[55,182,183]. In particular, Chisholm [175] showed thatthe clustering would produce an inherent isocurvatureperturbation and used this to constrain the viability ofPBHs as dark matter. Later he studied the effect ofclustering on mergers [184] and found that these coulddominate over evaporation, causing PBHs with mass below1015 g to combine and form heavier long-lived black holesrather than evaporating. So far, no compelling study of thiseffect has been carried out for a realistic mass spectrum, sowe will not include it in our discussion below.

V. SUMMARY OF CONSTRAINTS ONMONOCHROMATIC NONEVAPORATED

BLACK HOLES

We now review the various constraints associated withPBHs which are too large to have evaporated yet, updatingthe equivalent discussion which appeared in Carr et al. [11].All the limits assume that PBHs cluster in the Galactic haloin the same way as other forms of CDM. In this case, thefraction fðMÞ of the halo in PBHs is related to β0ðMÞ byEq. (8). Our limits on fðMÞ are summarized in Fig. 3,which is an updated version of Fig. 8 of Ref. [11]. A list ofapproximate formulas for these limits is given in Table I.Both Fig. 3 and Table I are intended merely as an overviewand are not exact. A more precise discussion can be foundin the original references. Many of the constraints dependon other physical parameters, not shown explicitly. Ingeneral, we show only the most stringent constraints ineach mass range, although constraints are sometimesomitted when they are contentious. Further details of theselimits and similar figures can be found in other papers:for example, Table 1 of Josan, Green, and Malik [45], Fig. 4of Mack, Ostriker, and Ricotti [185], Fig. 9 of Ricotti,Ostriker, and Mack [15], Fig. 1 of Capela, Pshirkov, andTinyakov [36] and Fig. 1 of Clesse and Garcia-Bellido[186]. We group the limits by type and discuss those withineach type in order of increasing mass. Since we are alsointerested in the mass ranges for which the dark-matterfraction is small, where possible we express each limit interms of an analytic function fmaxðMÞ over some massrange. We do not cover Planck-mass relics, since the only

constraint on these is that they must have less than theCDM density, but we do discuss them further in Sec. VI.

A. Evaporation constraints

A PBH of initial mass M will evaporate through theemission of Hawking radiation on a time scale τ ∝ M3

which is less than the present age of the Universe forM lessthan M# ≈ 5 × 1014 g [35]. PBHs with M > M# could stillbe relevant to the dark-matter problem, although there is astrong constraint on fðM#Þ from observations of theextragalactic γ-ray background [4]. Those in the narrowband M# < M < 1.005M# have not yet completed theirevaporation but their current mass is below the massMq ≈ 0.4M# at which quark and gluon jets are emitted.For M > Mc, there is no jet emission.For M > 2M#, one can neglect the change of mass

altogether and the time-integrated spectrum dNγ=dE ofphotons from each PBH is just obtained by multiplying theinstantaneous spectrum d _Nγ=dE by the age of the Universet0. From Ref. [11] this gives

FIG. 3. Constraints on fðMÞ for a variety of evaporation(magenta), dynamical (red), lensing (cyan), large-scale structure(green) and accretion (orange) effects associated with PBHs. Theeffects are extragalactic γ-rays from evaporation (EG) [11],femtolensing of γ-ray bursts (F) [187], white-dwarf explosions(WD) [188], neutron-star capture constraints (NS) [36], Keplermicrolensing of stars (K) [189], MACHO/EROS/OGLE micro-lensing of stars [27] and quasar microlensing (broken line) [190](ML), survival of a star cluster in Eridanus II (E) [191], wide binarydisruption (WB) [37], dynamical friction on halo objects (DF) [33],millilensing of quasars (mLQ) [32], generation of large-scalestructure through Poisson fluctuations (LSS) [14] and accretioneffects (WMAP and FIRAS) [15]. Only the strongest constraint isusually included in each mass range, but the accretion limits areshown with broken lines since they are highly model dependent.Where a constraint depends on some extra parameter which is notwell known, we use a typical value. Most constraints cut off at highM due to the incredulity limit. See the original references for moreaccurate forms of these constraints.

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FIG. 3. Constraints on fðMÞ for a variety of evaporation (magenta), dynamical (red), lensing (cyan), large-scale structure (green) and accretion (orange) effects associated with PBHs. The effects are extragalactic γ-rays from evaporation (EG) [11], femtolensing of γ-ray bursts (F) [187], white-dwarf explosions (WD) [188], neutron-star capture constraints (NS) [36], Kepler microlensing of stars (K) [189], MACHO/EROS/OGLE microlensing of stars [27] and quasar microlensing (broken line) [190] (ML), survival of a star cluster in Eridanus II (E) [191], wide binary disruption (WB) [37], dynamical friction on halo objects (DF) [33], millilensing of quasars (mLQ) [32], generation of large-scale structure through Poisson fluctuations (LSS) [14] and accretion effects (WMAP and FIRAS) [15]. Only the strongest constraint is usually included in each mass range, but the accretion limits are shown with broken lines since they are highly model dependent. Where a constraint depends on some extra parameter which is not well known, we use a typical value. Most constraints cut off at high M due to the incredulity limit. See the original references for more accurate forms of these constraints.


Microlensing

•Initial MACHO results: observed 17 events and claimed that these were consistent with compact objects of M ∼ 0.5M⊙ contributing 20% of the halo mass (Alcock et al. 2000)

•Later these lensing events revisited by EROS, OGLE, etc and attributed to self-lensing or clumping in the halo.

•Limits shown in the figure are from null detections


Refs: MACHO collaboration [e.g., Alcock et al. (1998, 2000,

2001); Hamadache et al. 2006]; EROS collaboration; Lacey &

Ostriker 85; Moore 93; Carr 94; Afshordi, McDonald & Spergel

03; Yoo et al. 04

Physics Coll. Virginia Tech, 02-08-2008 – p.10/37


From Mack, Ostriker, Ricotti (2006)

Matter power spectrum at small scales

N. Afshordi ,P. McDonald and D. N. Spergel (2003)

Poisson statistics: P(k)=kn with n=0 sigma=M(n+3)/6=M1/2


Ly-alpha forest, flux power spectrum (1D)


Dynamical constraints

• Unbinding of soft binaries in galactic halo

• Disk heating/stability

• Stability of tidal streams

• Heating of stars in dwarfs (ultra-faint dwarfs)


T. D. Brandt, Astrophys. J. 824, L31 (2016).


GW constraints

• Primordial-binaries: (Nakamura et al.,1997, Sasaki et al. 2016)

• Binary formation by capture in mini halos (Bird et al. 2016, Kovetz 2017)


(Sasaki et al. 2016)


(Kovetz 2017)


14

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FIG. 6. Merger rate of PBH binaries if they make up all ofthe dark matter, and provided PBH binaries are not signifi-cantly perturbed between formation and merger (solid line).Superimposed are the upper limits from LIGO given in TableI and described in the main text.

also strongly constrains masses M 10 M�, and deferthis detailed analysis to the LIGO collaboration, updat-ing that carried out in Ref. [39] with the S2 run. Wesummarize our estimated limits in Table I.

We show these limits in Fig. 6, alongside the PBH bi-nary merger rate if they make all of the dark matter, andif PBH binaries are not significantly perturbed betweenformation and merger. We see that the latter largely

exceeds the estimated upper limits, by 3 to 4 orders ofmagnitude, depending on the mass. This indicates thatLIGO could rule out PBHs as the dominant dark mat-ter component, and set stringent upper limits to theirabundance.

To estimate these potential limits, we solve for themaximum PBH fraction for which the merger rate is be-low the LIGO upper limits. Note, that the merger rate isnot linear in f , nor a simple power law through all rangeof f , so these limits must be computed numerically. Weshow the result in Fig. 7, alongside other existing boundsin that mass range. We see that LIGO O1 may limitPBHs to be no more than a percent of the dark mat-ter for M ⇠ 10 � 300 M�. If confirmed with numericalcomputations, these would become the strongest existingbounds in that mass range.

VI. DISCUSSION AND CONCLUSIONS

NSTT [38] pointed out long ago that PBHs wouldform binaries in the early Universe, as a consequence ofthe chance proximity of PBH pairs, and estimated theirmerger rate at the present time. Following the first de-tection of a binary-black-hole merger [5], Sasaki et al. [9]updated this calculation to 30 M� PBHs, and general-ized it to an arbitrary PBH abundance. They focused onthe case where PBHs are a very subdominant fraction ofthe dark matter, as was implied by the stringent CMBspectral distortions bounds at the time [23], since then

micro-lensing wide binariesultra-faint dwarfs

potential limits from LIGO O1 run

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CMB anisotropies

FIG. 7. Potential upper bounds on the fraction of dark matterin PBHs as a function of their mass, derived in this paper (redarrows), and assuming a narrow PBH mass function. Thesebounds need to be confirmed by numerical simulations. Forcomparison we also show the microlensing limits from theEROS [21] (purple) and MACHO [20] (blue) collaborations(see Ref. [74] for caveats and Ref. [32] for a discussion ofuncertainties), limits from wide Galactic binaries [22], ultra-faint dwarf galaxies [25], and CMB anisotropies [24].

revised and significantly alleviated [24] (see also [33]).

In this paper, we have, first of all, made several im-provements to the calculation of NSST, and accuratelycomputed the distribution of orbital parameters of PBHbinaries forming in the early Universe. Specifically,we have computed the exact probability distribution ofinitial angular momentum for a close pair torqued byall other PBHs, and have accounted for the tidal fieldof standard adiabatic density perturbations, dominantwhen PBHs make a small fraction of the dark matter.

Our second and most important addition was to checkthoroughly whether the highly eccentric orbits of PBHbinaries merging today can get significantly disturbedbetween formation and merger. To do so, we have esti-mated the characteristic properties of the first non-linearstructures, and as a consequence their e↵ects on the or-bital parameters of PBH binaries. We found that PBHbinaries merging today are essentially unscathed by tidaltorques and encounters with other PBHs. This robust-ness stems from the fact that these binaries typically formdeep inside the radiation era and are very tight. We havealso estimated the e↵ect of baryon accretion to be muchweaker than previous estimates [43], but potentially im-portant if unknown numerical prefactors happen to belarge.

Thirdly, we have revisited the calculation of Ref. [8]for the merger rate of PBH binaries forming in present-day halos through gravitational recombination. We haveexplicitly accounted for the previously neglected Pois-son fluctuations resulting from the granularity of PBHdark matter. This shot noise greatly enhances the vari-ance of density perturbations on small scales, and haspronounced e↵ects on the properties of low-mass halos.

Ali-Haimoud et al. 2017

Effects of broad mass spectrum of PBHs

the dark matter at M ∼ 30M⊙ of the kind displayed in theleft panel of Fig. 1. As observations of the dwarf galaxyand wide binaries improve, this gap may be filled and eventhe present ones shrink it according to Ref. [38]. However,a monochromatic mass function is not very physical. Amodel-independent way of assessing the more realisticextended-mass-function case is to consider where the differ-ent constraints cross. For ρ¼0.1M⊙ pc−3, σ¼5 kms−1 (redsolid curve), which is also the line chosen in Ref. [191], theEridanus II and microlensing constraints cross atM ∼ 10M⊙and f ≈ 0.4. This means that 40% of the dark matter can becontained in PBHs with M < 10M⊙, thereby evading the

microlensing bounds, and another 40% in PBHs withM > 10M⊙, thereby evading the Eridanus II constraints.Hence the Eridanus II and microlensing constraints togetherexclude PBHs from having more than 80% of the darkmatter in this intermediate-mass range. The slightly lessrestrictive Eridanus II constraint with ρ ¼ 0.1M⊙ pc−3,σ ¼ 10 km s−1 (red dashed line) crosses the microlensingconstraints at M ∼ 20M⊙ and f ≈ 0.5, marginally allowingthe dark matter to be in PBHs in this range. However, in thiscase the extended mass function has to be perfectly tuned tofit beneath the bounds, which is unlikely. On the other hand,for ρ¼0.01M⊙ pc−3, σ ¼ 5 km s−1 (red dot-dashed curve)

1 5 10 50 100 500 10000.0

0.2

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EROS

Eridanus II

WB

1015 1016 1017 10180.0

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extragalacticGRBs

GRBfemtolensing

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Relicsdominate

atevaporation

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Kepler

GRBfemtolensing

Whitedwarf

capture Neutron starcapture

FIG. 4. Four windows in which PBHs could conceivably provide the dark-matter density. Upper left panel: (A) Intermediate-massblack holes. The constraints in this mass range are EROS and MACHO microlensing bounds [27] (in blue), dynamical constraints (inred) from the lifetime of the central star cluster in the Eridanus II dwarf galaxy [191], as well as dynamical constraints (in green) from theexistence of wide-binary star systems [37]. Upper right panel: (B) Sublunar black holes. In this case the constraints (in blue) are again thefemtolensing of GRBs from [187], while the limits from neutron-star capture (in green) are taken from [36]. The red-shaded regiondenotes microlensing constraints from the Kepler survey [189], while the red-shaded region on the left shows constraints from white-dwarf explosions [188]. Lower left panel: (C) Subatomic black holes. The constraints here (red-shaded region) stem from nondetectionsof extragalactic γ rays that would be observable from the evaporation of PBHs of these masses [11,35] and (in blue) femtolensing of γ-ray bursts (GRBs) taken from Fermi data [187]. Lower right panel: (D) Planck-mass relics from PBH evaporations. This shows the massrange of the initial PBHs if they derive from inflation [62] but there are no observational constraints on such relics. Details on all theseregimes and the meaning of the constraints can be found in the subsections on the respective scenarios.

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inflation model from Ref. [5] with the broadest DHF(centered at the same mass) which satisfies the ultrafaintdwarf disruption constraint and the narrowest DHF whichsatisfies the EROS-2 microlensing constraint. The axion-curvaton DHF is significantly broader than the broadestDHF which satisfies the ultrafaint dwarf disruption con-straint and significantly narrower than the narrowest DHFwhich satisfies the EROS-2 microlensing constraint. It istherefore clearly excluded by both constraints.

IV. DISCUSSION

Microlensing surveys [15,28] constrain the halo fractionof MACHOs with 10−7<M=M⊙<10, while dynamicalheating constraints are sensitive to M=M⊙≳10 [16].Together they exclude MACHOs with 10−7<M=M⊙<105 and a delta function mass function from making up allof the dark matter. However Refs. [5,8] have pointed outthat MACHOs with an extended mass function, as expectedfor PBHs formed from the collapse of large inflationarydensity perturbations [9,10], might be compatible withthese constraints. Furthermore interest in PBHs withM ∼ 10 M⊙ [11–13] has recently been stimulated by thediscovery of gravitational waves from massive black holemergers by LIGO [14].We have explicitly calculated the microlensing and

dynamical constraints for the DHFs found for PBHsproduced by two inflation models in Ref. [5] and alsofor a variable width DHF which replicates their shape.The DHFs studied in Ref. [5] both produce Nexp > 3

microlensing events in EROS-2 and also cause excessivedynamical heating of ultrafaint dwarf galaxies. In generalwe find that the dynamical constraints place a (central massdependent) upper limit on the width of the DHF (i.e. widedistributions are excluded), while the microlensing con-straints place a (central mass dependent) lower limit on thewidth of the DHF (i.e. narrow distributions are excluded).These constraints overlap and there are no parameter valueswhich satisfy both the microlensing limt and the weakestultrafaint dwarf heating limit.We have not proved that there is no extended DHF, with

all of the dark matter in compact objects in a single massrange, which can satisfy both the microlensing anddynamical constraints. However we have shown that

(i) to ascertain whether an extended DHF satisfies themicrolensing and dynamical constraints it is neces-sary to recalculate the limits for the specific massfunction, rather than using the limits derived for adelta function mass function;

(ii) generic DHFs, which replicate the PBH distributionsproduced by inflation models, cannot simultane-ously satisfy the EROS-2 microlensing constraintand also the weakest ultrafaint dwarf heating limit.

ACKNOWLEDGMENTS

A.M. G. acknowledges support from STFC GrantNo. ST/L000393/1 and is grateful to Marit Sandstad foruseful comments that have improved the presentation of themanuscript.

FIG. 4. The differential PBH halo fraction, df=dM, for theaxion-curvaton (dotted red line) inflation model from Ref. [5]compared with the broadest differential halo fraction, centered atthe same mass, which satisfies the ultrafaint dwarf disruptionconstraint (solid green) and the narrowest differential halofraction which satisfies the EROS-2 microlensing constraint(dashed black).

FIG. 3. Constraints on the width, σ, of the DHF functionalform, Eq. (10), as a function of the central mass Mc. Parametervalues in the red hatched area in the bottom left produce Nexp ≥ 3

microlensing events in the EROS-2 survey and are excluded at95% confidence. The blue hatched area in the top right isexcluded by the heating of ultrafaint dwarfs. The constraintfrom the disruption of the star cluster in Eri II is tighter andexcludes a large region of parameter space.

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FIG. 3. Constraints on the width, σ, of the DHF functional form, Eq. (10), as a function of the central mass Mc. Parameter values in the red hatched area in the bottom left produce Nexp ≥ 3 microlensing events in the EROS-2 survey and are excluded at 95% confidence. The blue hatched area in the top right is excluded by the heating of ultrafaint dwarfs. The constraint from the disruption of the star cluster in Eri II is tighter and excludes a large region of parameter space.

inflation model from Ref. [5] with the broadest DHF(centered at the same mass) which satisfies the ultrafaintdwarf disruption constraint and the narrowest DHF whichsatisfies the EROS-2 microlensing constraint. The axion-curvaton DHF is significantly broader than the broadestDHF which satisfies the ultrafaint dwarf disruption con-straint and significantly narrower than the narrowest DHFwhich satisfies the EROS-2 microlensing constraint. It istherefore clearly excluded by both constraints.

IV. DISCUSSION

Microlensing surveys [15,28] constrain the halo fractionof MACHOs with 10−7<M=M⊙<10, while dynamicalheating constraints are sensitive to M=M⊙≳10 [16].Together they exclude MACHOs with 10−7<M=M⊙<105 and a delta function mass function from making up allof the dark matter. However Refs. [5,8] have pointed outthat MACHOs with an extended mass function, as expectedfor PBHs formed from the collapse of large inflationarydensity perturbations [9,10], might be compatible withthese constraints. Furthermore interest in PBHs withM ∼ 10 M⊙ [11–13] has recently been stimulated by thediscovery of gravitational waves from massive black holemergers by LIGO [14].We have explicitly calculated the microlensing and

dynamical constraints for the DHFs found for PBHsproduced by two inflation models in Ref. [5] and alsofor a variable width DHF which replicates their shape.The DHFs studied in Ref. [5] both produce Nexp > 3

microlensing events in EROS-2 and also cause excessivedynamical heating of ultrafaint dwarf galaxies. In generalwe find that the dynamical constraints place a (central massdependent) upper limit on the width of the DHF (i.e. widedistributions are excluded), while the microlensing con-straints place a (central mass dependent) lower limit on thewidth of the DHF (i.e. narrow distributions are excluded).These constraints overlap and there are no parameter valueswhich satisfy both the microlensing limt and the weakestultrafaint dwarf heating limit.We have not proved that there is no extended DHF, with

all of the dark matter in compact objects in a single massrange, which can satisfy both the microlensing anddynamical constraints. However we have shown that

(i) to ascertain whether an extended DHF satisfies themicrolensing and dynamical constraints it is neces-sary to recalculate the limits for the specific massfunction, rather than using the limits derived for adelta function mass function;

(ii) generic DHFs, which replicate the PBH distributionsproduced by inflation models, cannot simultane-ously satisfy the EROS-2 microlensing constraintand also the weakest ultrafaint dwarf heating limit.

ACKNOWLEDGMENTS

A.M. G. acknowledges support from STFC GrantNo. ST/L000393/1 and is grateful to Marit Sandstad foruseful comments that have improved the presentation of themanuscript.

FIG. 4. The differential PBH halo fraction, df=dM, for theaxion-curvaton (dotted red line) inflation model from Ref. [5]compared with the broadest differential halo fraction, centered atthe same mass, which satisfies the ultrafaint dwarf disruptionconstraint (solid green) and the narrowest differential halofraction which satisfies the EROS-2 microlensing constraint(dashed black).

FIG. 3. Constraints on the width, σ, of the DHF functionalform, Eq. (10), as a function of the central mass Mc. Parametervalues in the red hatched area in the bottom left produce Nexp ≥ 3

microlensing events in the EROS-2 survey and are excluded at95% confidence. The blue hatched area in the top right isexcluded by the heating of ultrafaint dwarfs. The constraintfrom the disruption of the star cluster in Eri II is tighter andexcludes a large region of parameter space.

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Accretion constraints

• Effects on the CMB anisotropies (Ricotti et al. 2007, Ali-Halımoud & Kamionkowski 2017, Horowitz 2016, Poulin et al. 2017, Bellomo et al. 2017, Bernal et al 2017)

• Moving PBHs in galactic center (Gaggero et al. 2017)


Moving PBHs in Galactic center(Gaggero et al. 2017)


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