Constraints on Primordial Black Holes with Extended Mass Functions

0.55 MB·

by  Florian Kuhnel

Constraints on Primordial Black Holes with Extended Mass Functions
Florian Ku¨hnel1,2,∗ and Katherine Freese3,1,†
1The Oskar Klein Centre for Cosmoparticle Physics,
Department of Physics, Stockholm University, AlbaNova University Center,
Roslagstullsbacken 21, SE–10691 Stockholm, Sweden
2Department of Physics, School of Engineering Sciences,
KTH Royal Institute of Technology, AlbaNova University Center,
Roslagstullsbacken 21, SE–10691 Stockholm, Sweden
3Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
(Dated: Thursday 23rd February, 2017, 1:43am)
Constraints on primordial black holes in the range 10−18M to 103M are reevaluated for a
7 (cid:12) (cid:12)
general class of extended mass functions. Whereas previous work has assumed that PBHs are
1
produced with one single mass, instead there is expected to be a range of masses even in the
0
case of production from a single mechanism; constraints therefore change from previous literature.
2
Although tightly constrained in the majority of cases, it is shown that, even under conservative
b assumptions, primordial black holes in the mass range 10−10M to 10−8M could still constitute
(cid:12) (cid:12)
e
theentiretyofthedarkmatter. Thisstressesboththeimportanceforacomprehensivereevaluation
F
of all respective constraints that have previously been evaluated only for a monochromatic mass

2 function, and the need to obtain more constraints in the allowed mass range.
2

]
O Introduction—Primordial black holes (PBHs) are black any monochromatic scenario of 100% PBH dark matter
C holes produced in the early Universe, and have received is strictly excluded. This statement crucially relies on
considerable attraction since they were proposed more thevalidityoftheconstraintsusedinthiswork. However,
.
h
than four decades ago [1, 2]. In principle, they can span someofthesehavebeendisputed. Forinstance,thosede-
p
a huge range of mass scales—from as low as the Planck riving from neutron-star capture of black holes [28] are
-
o mass to many orders of magnitude above the solar mass. challenged due to uncertainties in the amount of dark
r They are unique probes of the amplitude of the density matter inside globular clusters (cf. Ref. [29, 67]). Fur-
t
s fluctuation on the very small scales of their production. thermore, the bounds from PBH accretion [38–40] rely
a
WiththemilestonediscoveryoftheLIGOandVIRGO on highly model-dependent and relatively insecure as-
[
collaboration of black-hole binary mergers [3, 4], the in- sumptions (spherically-symmetric Bondi accretion etc.).

2 terest in the question of whether PBHs could constitute Also, they depend on the so-called duty-cycle parame-
v the dark matter (DM) [5] has recently been revived [6– ter whose exact value varies significantly in the relevant
3
20]. Depending on the mass scale(s) involved, the black literature. Hence, monochromatic scenarios of 100% of
2
holes potentially cause large observable effects. For the PBH dark matter might still be allowed. In this arti-
2
7 case where all PBHs have one single mass (monochro- cle, however, our point is to investigate whether—even
0 matic mass function) somewhere in the range 10−18M if these stringent constraints (to which we will refer as
(cid:12)
. to103M ,Figure1summarisesthestrongestconstraints ’conservative’) are taken at face value—extended mass
1 (cid:12)
0 from previous literature at each value of possible PBH functions still allow for the possibility that the entirety
7 mass. The figure caption lists the physical effects and of the dark matter consists of PBHs.
1 provides respective references. In this Letter we reconsider the bounds on primordial
v: Most of the constraints derived in the literature, in- black-holedarkmatterintherange10−18M(cid:12) to103M(cid:12)
i cluding those in Fig. 1, are subject to the (at best for a wide class of extended mass functions. All realistic
X
over-simplifying)assumptionthatPBHformationoccurs cases with extended mass spectra require a rederivation
r mono-chromatically,ie.atoneparticularmassscaleonly, of the constraints and an integration of these over the
a
despite the fact that PBH mass spectra are generically whole mass range; this will be done in the present work.
extended due to the nature of the gravitational collapse Primordial Black-Hole Formation—There is a plethora
leading to their formation [21–27]. This incorrect as- of scenarios which lead to the formation of PBHs. All of
sumption can lead to large errors in the prediction of these require a mechanism to generate large overdensi-
the PBH abundance (cf. Ref. [27]). It should be stressed ties. In many scenarios, these overdensities are of infla-
that, given the current constraints displayed in Fig. 1, tionaryorigin[41–43]. Afterreenteringthehorizon,they
collapse if they are larger than a (medium-dependent)
threshold, where the case of radiation domination is the
one most often considered in the literature. Many more
∗Electronicaddress: fl[email protected] possibilitiesforPBHformationexist,suchasthosewhere
†Electronicaddress: [email protected] the source of the inhomogeneities are first-order phase
2
out recently by Green [20], a (quasi) log-normal PBH
1
EG mass function with derivative
0.500 K
(cid:34) (cid:35)
F dn ≡N exp −(logM/Mf)2 , (1)
NS ML dM 2σ2
0.100 f
WD mLQ
0.050
fitsaverylargeclassofinflationaryPBHmodelsreason-
f PT
τ ably well (cf. Refs. [61–63] for the derivation and use of
log-normal mass functions).[68] Above, N is a normali-
0.010
sation constant chosen such that the integral of dn/dM
0.005
over all masses is equal to one. As a function of M, the
r.h.s. of Eq. (1) is peaked at M ; its width is controlled
f
byσ . Inpractice, foreachpairofparameters(σ , M ),
0.001 f f f
10-16 10-12 10-8 10-4 10 102 a given set of constraints needs to be evaluated on the
whole mass range, which then yields a limit on the PBH
M/M
⊙��� content.
FIG.1: Summaryofpreviousliteraturefortheidealisedcase The observational constraints which are included in
in which the entire PBH dark matter consists of PBHs of a ouranalysisareallofthosestatedinthecaptionofFig.1,
single mass M (mono-chromatic mass function): Constraint adaptedfortheextendedmassfunctionEq.(1)whichwe
“curtain” on the dark-matter fraction f ≡ ρ /ρ for a
PBH DM use throughout the rest of our paper.
varietyofeffectsassociatedwithPBHsofmassM inunitsof
solar mass M . Only strongest constraints are included. We
(cid:12) Neutron-Star Capture: Primordial black holes cap-
show constraints from extragalactic γ-rays from evaporation
turedbyneutronstars(NS)genericallyleadtotherapid
(EG) [31], femtolensing of γ-ray bursts (F) [32], white-dwarf
destruction of the NS. Hence, the observations of neu-
explosions(WD)[33], neutron-starcapture(NS)[28], Kepler
tron stars pose constraints on the PBH abundance. For
microlensing of stars (K) [11], MACHO/EROS/OGLE mi-
a mono-chromatic mass spectrum, it has been argued in
crolensing of stars (ML) [34] and quasar microlensing (ML)
[35], millilensing of quasars (mLQ) [36], pulsar timing (PT) Ref. [28] on the basis of a sufficiently large neutron-star
(SKAforecast)[37],andaccretioneffectsontheopticalthick- survival probability that the constraint can be phrased
ness(τ)[38]. Moredetailscanbefoundin[17]. Whereasthese as
constraints are only valid for the case of a mono-chromatic
mass function, instead all realistic cases with extended mass 1≥ft F , (2)
NS 0
spectra require a rederivation of those constraints and an in-
tegrationoftheseoverthewholemassrange;thiswillbedone with the PBH dark-matter fraction f ≡ ρ /ρ .
PBH DM
in the present work. Here, t is the age of the star. The capture rate F
NS 0
inside of a globular cluster (with core dark-matter den-
sity ρ ) reads
DM,GC
transitions [6], bubble collisions [44, 45], the collapse of
√ ρ 2G M R
cosmic strings [46, 47], necklaces [48] or domain walls F = 6π DM,GC N NS NS ×
[49]. 0 M v¯(1−2GNMNS/RNS)
As mentioned above, in general, one encounters ex- (cid:20) (cid:18) (cid:19)(cid:21)
3E
tended PBH mass spectra rather than PBHs of a sin- × 1−exp − loss , (3)
Mv¯2
gle mass. The reason is two-fold: On the one hand,
the initial spectrum of overdensities is already extended with G being Newton’s constant, v¯ is the dispersion
N
in essentially all of the models mentioned above. On of the assumed Maxwellian velocity distribution of the
the other hand, even if the initial density spectrum was PBHs, R is the radius of the neutron star, and M
NS NS
mono-chromatic, the phenomenon of critical collapse [7] itsmass. Finally,theaverageenergylossE isapprox-
loss
will inevitably lead to a PBH mass spectrum which is imately given by
spread out, shifted towards lower masses and lowered,
leading to potentially large effects (cf. Ref. [27]). 58.8G2M2M
E (cid:39) N NS . (4)
loss R2
Bounds for Extended Mass Functions—The derivation NS
of bounds for extended mass functions is strictly speak- Inthecaseofanextendedmassdistribution,thebound
ing always subject to a specific model, such as the given in Eq. (2) changes. Let I ∈ N denote the num-
bin
axion-curvatonmodel[50–52],thehybrid-inflationmodel ber of bins {[M , M ], i ≤ I }. Then, for a given
i i+1 bin
[53,54], ortherunning-massmodel[55–60], justtomen- distribution specified by some dn/dM, we have
tionafew(cf.Ref.[17]foranextensiveoverview). There-
fcoornec,luitsiosenesmisnhtohpiseleressgatrod.draHwowaenvyerm, oitdeils-ininddeepeedndweenltl 1(cid:38)(cid:88)Ibin tNSfF0(i)(cid:90) Mi+1dM ddMn , (5)
possible at an approximate level. As has been pointed i=1 Mi
3
where F(i) is evaluated on a mass within each bin, and
the bin n0umber I should be chosen such that, to the 2.0 Log10(f)
bin
-0.2
precisionsought,itdoesnotmatterwhereexactlyineach -0.4-0.6 0
bin the quantity F0 is evaluated. For the evaluations 1.5 -0.8
-1
of the constraint we use the parameter values M =
NS -1
1.4M , R = 12km, ρ = 2 × 103GeVcm−3, -2
(cid:12) NS DM,GC σf -3
t =1010yr, and v¯=7kms−1, as given in Ref. [28]. 1.0 -4
DM -5 -3
-6
Pulsar Timing: Ref.[37]arguesthattheabundanceof
0.5 -5
1–1000M PBHs might be considerably constrained via
(cid:12)
the non-detection of a third-order Shapiro time delay as
the PBHs move around the Galactic halo. They present
-15 -10 -5 0
resultsofarespectiveMonte-Carlosimulationleadingto
a forecast for the Square Kilometre Array (SKA) which Log10(Mf/M⊙)���
approximately follows an f ∼ M1/3 scaling. More pre-
cisely, and adopted to an extended PBH spectrum, the FIG.2: MaximumPBHdark-matterfractionf ≡ρPBH/ρDM
respective constraint might be written in the form as a function of σf and Mf for the extended mass func-
tion specified in Eq. (1). The region enclosed by the red-
1(cid:38)(cid:88)Ibin (cid:32) M˜i (cid:33)1/3 f(cid:90) Mi+1dM dn , (6) dpaosshsiebdilitcyonotfou1r00i%n tphreimmoriddidallebloafckt-hheolepldotarkin-mdiactatteers (tsheee
1M dM the legend). The various contours are at log (f) =
i=1 (cid:12) Mi 10
0(red-dashed line),−0.2,−0.4,−0.6,−0.8,−1,−2,...,−6.
where M˜ is some mass within each bin. As before, I
i bin
should be chosen sufficiently large. smallestoneineachbin(wheretheconstraintisweakest).
Thismayoverestimatetheconstraint,butforthesmooth
AccretionEffects: AsfirstanalysedbyCarr[64],inthe mass functions given by Eq. (1) and the constraint used,
period after decoupling, PBH accretion and emission of by making the bins small enough, the error can be made
radiation might have a strong effect on the thermal his- arbitrarilysmall,andinparticularmuchsmallerthanthe
toryoftheUniverse. Ricottietal.[38]haveanalysedthis errorofusingEq.(1)insteadoftheactualmassfunction
possibility in detail. In particular, it has been discussed of a given model.[69]
that one associated effect of PBH is that they increase
the optical thickness τ →τ +(∆τ) which leads to Results—Figure 2 shows our main results for the con-
PBH
straints on the amount of PBH dark matter in the
1(cid:38)(cid:88)Ibin 17.4(cid:32) M˜i (cid:33)2 f(cid:90) Mi+1dM dn , (7) (Hσefr,e,Mthf)e-pvlaalnuee oufsiMng ,thaet dwihsticrihbuthtiiosndigsitvreibnubtiyonEqp.ea(k1s).,
1M dM f
i=1 (cid:12) Mi lies between 10−16M(cid:12) and 103M(cid:12), and its width σf ∈
[0.2, 2]. The region enclosed by the red-dashed contour
which is valid in the mass range [30M(cid:12), 103M(cid:12)]. in the middle of the plot around 10−9M indicates the
(cid:12)
In Eq. (7), we utilise the latest Planck constraint
possibilityforaPBHdark-matterfractionof100%. This
(∆τ) <0.012(95%C.L.) as in Ref. [65].
PBH is not possible outside this narrow region, and excludes
too narrow mass function (σ (cid:46) 0.4), which in particular
Other Bounds: Forthebounds from extragalactic γ-rays
applies to the monocromatic case. Hence almost all of
from evaporation, white-dwarf explosions, and lensing,
the parameter space does not allow for 100% PBH dark
we use a similar method to the one in Ref. [17]. This al-
matter,giventhevalidityoftheverystringentboundsin
lowstoapproximatelydetermine,fromconstraintscalcu-
themassrangeunderinvestigation. Asmentionedinthe
latedassumingadelta-functionhalofraction,whetheran
Introduction, it should be stressed that several of these
extendedmassfunctionisallowed. Thismethodsutilises
constraints are based on rather uncertain assumptions
binning of the relevant mass range. Specifically, a given
(suchasthosederivingfromaccretion, neutron-starcap-
constraintf isfirstdividedintolocallymonotonicpieces.
c ture, or ultra-faint dwarfs), and might be weakened sig-
In each of these pieces one starts with the bin, say i,
nificantly once a more elaborated treatment of those has
where the constraint is smallest, and integrates dn/dM
beenperformed. However, evenifallthementionedcon-
in this bin in order to obtain the fraction f . Then one
i straints are taken at face value, in the middle of the M
goestothenextbin,integratingover[M , M ]inorder f
i i+1 axis,thereremainsaregion,andthereforeaclassofmod-
to obtain f , and so on. In the original formulation of
i+1 elswithanecessarilyextendedmassfunction,whichstill
Ref. [17], each f is then compared to the largest value
j allows for PBHs to constitute all of the dark matter.[70]
of f in this bin (where the constraint is weakest). Re-
c
cently, Green [20] pointed out that there may be cases Summary & Outlook—In this Letter we have presented
where this procedure underestimates the constraint. the results of our systematic investigation of constraints
In this work, in order to avoid such a possible issue, for a wide class of extended mass function in the mass
instead of using the largest value of f , we will use the range 10−18M to 103M . For these results, which are
c (cid:12) (cid:12)
4
visualised in Fig. 2, very restrictive constraints on and Acknowledgments
forecasts for the allowed abundance of PBHs (as sum-
marised in Fig. 1) have been used. Hence, Fig. 2 pro-
It is a pleasure to thank Alexander Dolgov, Benjamin
vides an approximate lower bound on the allowed PBH
Horowitz, Alexander Kashlinsky, and Peter Klimai for
dark-matter fraction.
helpfulcomments. K.F.acknowledgessupportfromDoE
We confirm the results of Ref. [17] that there still is a grant DE-SC0007859 at the University of Michigan as
window in the mass range 10−10M(cid:12) to 10−8M(cid:12) which wellassupportfromtheMichiganCenterforTheoretical
can accommodate for 100% PBH dark matter. Apart Physics. K.F. and F.K. acknowledge support by the
from the possibility of Planck-mass relics, to pose new Vetenskapsr˚adet (Swedish Research Council) through
constraints in the mentioned mass window seems to be contract No. 638-2013-8993 and the Oskar Klein Centre
crucial for providing an answer to the question whether for Cosmoparticle Physics.
primordial black holes can constitute the entirety of the
dark matter.
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