Planck star phenomenology

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Planck star phenomenology

A. Barrau, Carlo Rovelli

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Planck

star

phenomenology

Aurélien Barrau

a,

∗

,

Carlo Rovelli

b,c

a_{LaboratoiredePhysiqueSubatomiqueetdeCosmologie,UniversitéGrenoble-Alpes,CNRS–IN2P3,53,avenuedesMartyrs,38026Grenoblecedex,France} b_{AixMarseilleUniversité,CNRS,CPT,UMR7332,13288Marseille,France}

c_{UniversitédeToulon,CNRS,CPT,UMR7332,83957LaGarde,France}

a

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t

i

c

l

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f

o

a

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Articlehistory:

Received 13 October 2014

Received in revised form 7 November 2014 Accepted 12 November 2014

Available online 15 November 2014 Editor: S. Dodelson

It is possible that black holes hide a core of Planckian density,sustained by quantum-gravitational pressure.Asablackholeevaporates,thecorerememberstheinitialmassandthefinalexplosionoccurs atmacroscopicscale.Weinvestigatepossiblephenomenologicalconsequencesofthisidea.Underseveral roughassumptions,weestimatethatuptoseveralshortgamma-rayburstsperday,around10 MeV,with isotropicdistribution,canbeexpectedcomingfromaregionofafewhundredlightyearsaroundus.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

1. Themodel

Recently, a newpossible consequenceof quantumgravity has beensuggested[1].Theideaisgroundedinarobustresultofloop cosmology[2]:whenmatterreachesPlanckdensity,quantum grav-ity generates pressure sufficient to counterbalance weight. For a blackhole,thisimpliesthat matter’scollapsecan bestopped be-forethecentralsingularityisformed:theeventhorizonisreplaced bya“trapping”horizon[3]whichresemblesthestandardhorizon locally,butfromwhichmattercaneventuallybounceout.Because ofthehugetimedilationinsidethegravitationalpotentialwellof thestar,thebounceisseeninextremeslowmotionfromthe out-side,appearing asanearly stationaryblackhole.The core,called “Planckstar”,retainsmemoryoftheinitialcollapsedmassmi

(be-cause there is no reason for the metric of the core to be fully determinedby the area of theexternal evaporating horizon) and thefinal explodingobject dependsonmi andismuchlargerthan

Planckian[1].Theprocess isillustratedbythe conformaldiagram inFig. 1.

Inparticular,primordialblackholes explodingtodaymay pro-ducea distinctive signal. The observability ofa quantum gravita-tionalphenomenon is madepossible by theamplification dueto thelargeratiooftheblackholelifetime(HubbletimetH)overthe

Plancktime[4].

Ifthisscenarioisrealized innature,canthe finalexplosionof aprimordialPlanckstarbe observed?Thisisthequestion we in-vestigatehere.

*

Corresponding author.

E-mailaddresses:[email protected](A. Barrau), [email protected]

(C. Rovelli).

Fig. 1. Penrose

diagram of a collapsing star. The dotted line is the external boundary

of the star. The shaded area is the region where quantum gravity plays an important role. The dark line represents the two trapping horizons: the external evaporating one, and the internal expanding one. The lowest light-line is where the horizon of the black hole would be without evaporation. P is

where the explosion happens.

The thin arrows indicate the Hawking radiation. The thick arrow is the signal stud-ied in this paper.

2. Dynamics

As a first step, we evaluate the energy of the particles emit-tedbytheexplosionofa primordialPlanckstar.Letmf

=

ami be

thefinalmassreachedbytheblackholebeforethedissipation of the horizon(atthe point P inFig. 1). In[1],an argument based on informationconservationwas given, pointingtothe preferred value

http://dx.doi.org/10.1016/j.physletb.2014.11.020

0370-2693/©2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by

406 A. Barrau, C. Rovelli / Physics Letters B 739 (2014) 405–409

a

∼

√

1

2

,

(2.1)

where mi is the initial mass. This value contradicts the

expec-tation from the semiclassical approximation; it follows from the hypothesis that the black hole information paradoxes might be resolved andfirewalls avoided if the semiclassical approximation breaksdownearlierthannaivelyexpected,asaconsequenceofthe strongquantumgravitationaleffectsinthe coreandthefact that theyalter theeffectivecausalstructureofthe evolvingspacetime (the point P isin thecausal futureof thequantum gravitational region).In[5,6]theextremeoppositecasederivedfromthesame ideas isexplored. As shownby Hawking, non-rotatinguncharged blackholesemit particleswithenergyintheinterval

(

E

,

E

+

dE

)

atarate[7] d2N dEdt

=

Γ

s h



exp



8

π

GmE

¯

hc3



− (−

1

)

2s



−1 (2.2)

perstateofangularmomentumandspin s.Theabsorption coeffi-cient

Γ

s,thatistheprobabilitythattheparticlewouldbeabsorbed

ifit wereincidentinthisstateontheblackhole,isafunctionof

E,m and s.Byintegratingthisexpression itisstraightforwardto showthatthemasslossrateisgivenby

dm dt

= −

f

(

m

)

m2

,

(2.3)

where f

(

m

)

accountsforthe degreesoffreedom ofeachemitted particle.Astime goeson,themassdecreases,thetemperature in-creasesandnewtypesofparticlesbecome“available”totheblack hole.Aboveeachthreshold, f

(

m

)

isgivenapproximatelyby[8]

f

(

m

)

≈ (

7

.

8

α

s=1/2

+

3

.

1

α

s=1

)

×

1024g3s−1

,

(2.4) where

α

s=1/2 and

α

s=1 arethenumberofdegreesoffreedom (in-cluding spin, charge and color) of the emitted particles. If f

(

m

)

isassumedtobeconstant,e.g. f

(

m

)

=

f

(

mi

)

,theinitialandfinal

massesofa blackhole reaching its final stage today,that is ina Hubbletime tH,are easy to calculate.Inthe Planckstar

hypoth-esis, the final stage is reached when m

=

mf



mPl. Integrating

Eq.(2.3)leadsto mi

=



3tHf

(

mi

)

1

−

a3



1 3

.

(2.5)

Inpractice,toaccountforthesmoothevolutionof f

(

m

)

when thedifferentdegreesoffreedomopen up,anumericalintegration hastobecarriedout.Thisleads,fora

=

1

/

√

2,to

mi

≈

6

.

1

×

1014g

,

(2.6)

and

mf

≈

4

.

3

×

1014g

.

(2.7)

Thevalueofmi isveryclosetotheusual valuem∗ corresponding

toblackholesrequiringjusttheageoftheUniversetofully evapo-rate.Thiswasexpectedastheprocessisexplosive.Thedetailsmay changedepending ontheexactshape chosenfor f

(

m

)

.In princi-ple, the accurate value of mi should be altered by the fact that

the Hawkingformula usedto calculateit is purely semi-classical whereastheendoftheprocessisstronglydominatedbyquantum gravity effects. In practice this has no effect as the evaporation time isvery strongly dominatedby the firststages.For example, increasing by afactorof 100themass lossrateinthe last tenth ofthemassintervalspannedbytheevaporatingblackholewould changeitslifetimebylessthan0.1%.

The value of the radius when m reaches mf is rf

≈

6

.

4

×

10−14cm.Thesizeoftheblackholeistheonlyscaleinthe prob-lemandthereforefixestheenergyoftheemittedparticlesinthis last stage. We assume that all fundamental particles are emitted withthesameenergytakenat

Eburst

=

hc

/(

2rf

)

≈

3

.

9 GeV

.

(2.8)

Of course, a more reliable model would be desirable but this is the mostnaturalhypothesisatthisstage. Infuture works, differ-ent hypothesis will be considered. However, as the spectrum of gamma-raysisverywidelyspreadevenformonochromaticjetsof quarks,itcanbesafelyassumedthatsomebroadeningofthe emit-tedspectrumwouldnotqualitativelychangetheresult.

3. Singleeventdetection

We now study the signal that evaporating Planck star would produce.Fromthephenomenologicalviewpoint,itisnaturalto fo-cusonemittedgamma-rays:chargedparticlesundergoadiffusion process in the stochastic magnetic fields and cannot be used to identify asingle eventwhereas neutrinosare hard todetect. The importantfactisthatmostoftheemittedgammasarenotemitted attheenergyEburst.Onlythosedirectlyemittedwillhavethis

en-ergy. Butassuming that thebranching ratiosarecontrolled, asin theHawkingprocess,bytheinternaldegreesoffreedom,this rep-resentsonlyasmallfraction(1

/

34oftheemittedparticles).Most gamma-rayswillcomefromthedecayofhadronsproducedinthe jetsof quarks,notably fromneutralpions. Eburst is alreadymuch

smaller than the Planck scale butthe mean emission ofemitted photonsisevensmaller.

Tosimulatethisprocess,wehaveusedthe“LundMonteCarlo” PYTHIA code (with some scaling approximations due to the un-usually low energy required forthis analysis). It contains theory and models fora number ofphysics aspects, including hard and softinteractions,partondistributions,initial- andfinal-stateparton showers, multiple interactions, fragmentation and decay. PYTHIA allowedustogeneratethemeanspectrumexpectedforsecondary gamma-rays emittedby a Planckstarreaching theendofitslife. The mainpoint tonoticeisthat themeanenergyis oftheorder of 0

.

03

×

Eburst,that is inthe tens of MeV rangerather than in

theGeVrange.Inaddition,themultiplicityisquitehighataround 10photonsperqq jet.

¯

Fig. 2shows themeanspectrumofphotons resultingfrom105jetsof3.9GeVuu quarks.

¯

It is straightforward to estimate the total numberof particles emittedmf

/

Eburstandthenthenumberofphotons



Nburst



emit-ted during the burst. As fora blackhole radiatingby the Hawk-ing mechanism, we assume that theparticles emitted during the bursts(thatisthosewithm

<

Eburst)areemittedproportionallyto

theirnumberofinternaldegreesoffreedom:gravityisdemocratic. Thespectrumresultingfromtheemittedu,d,c,s quarks(t and b are too heavy),gluons andphotonsis shownin Fig. 3. The little peakontherightcorrespondstodirectlyemittedphotonsthatare clearly sub-dominant.Byalsotakingintoaccount theemissionof neutrinosandleptonsofallthreefamilies(leadingtovirtually no gamma-rays andthereforebeinghereapure missingenergy),we obtain



Nburst



≈

4

.

7

×

1038.

Thequestionofthemaximumdistanceatwhichasingleburst can be detected naturally arises. Ifone requiresto measure Nmes

photonsinadetectorofsurface S,thisissimplygivenby

Rdet

=



S



Nburst



4

π

Nmes

.

(3.1)

If we set, e.g., Nmes

≈

10 photons in a 1 m2 detector, this

Fig. 2. Gamma-ray spectrum resulting from 105_{uu jets (linear scales).¯}

Fig. 3. Full spectrum of gamma-rays emitted by a decaying Planck star (log scales).

detection ofexplodingPlanckstarsislocal.Themaximumdistance atwhichsuchaneventcanbedetectedisjustafewtenstimesthe distancetotheneareststar.Thisisatinygalacticpatcharoundus. Thisalreadyhasaninteresting consequence.Exceptifthereis asignificantdarkmatter clumpwithin thissmallsphere–which isunlikely, thesignalisexpectedtobeisotropic asthehalois ho-mogeneousatthisscale.Thisisvery differentfrommostgalactic signalsusuallypeakedeitherinthegalacticcenterdirection,orthe directionofthe motionof thesolar systemfordirected searches fordarkmatter.ThismeansthatthePlanckstarsignalcouldmimic acosmologicalorigin.

Let us call the local density of dark matter

ρ

DM

∗

≈

0

.

3

±

0

.

1GeV

/

cm3 [9].IfPlanckstarsreachingmf weretosaturatethe

darkmatterbound,andiftheyclusterasordinarycolddark mat-ter,theirnumberwithinthedetectablehorizonwouldbe

Nmax_det

=

4

πρ

DM ∗ 3mf



S



Nburst



4

π

Nmes



3 2

≈

3

.

8

×

1022

.

(3.2)

However,astheirhistoryisnotthatdifferentfromtheoneof stan-dardprimordialblackholes withthesameinitialmass(of course the end of their lives is different but the total energy emitted remains the same) the usual constraint

Ω

PBH

<

10−8 _{for initial} massesaround1015_{g basicallyholds[10].Thisleadsto}

Ndet

<

4

πρ

DM ∗

Ω

PBH 3mf



S



Nburst



4

π

Nmes



3 2

≈

3

.

8

×

1014

.

(3.3)

Thisnumberisstillquitehigh.Itshowsthattheindividual detec-tionisfarfrombeing,inprinciple,outofreach.

One can then estimate the number ofevents that can be ex-pectedforagivenobservationtime



t.ThiscorrespondstoPlanck stars that havemasses betweenmf andm

(

t

)

at thebeginning

oftheobservationtime,withinthevolume R

<

Rdet.Inthiscase,

m

(

t

)

issimply

m

(

t

)

=



m3_f

+

3 f

(

m

)

t



13

_.

_(3.4)

Thisnumbern

(

t

)

is(estimatedforaunitvolume)givenby

n

(

t

)

=

m

(t)

mf

dn

dmdm

,

(3.5)

wheredn

/

dm isthedifferentialmassspectrumofPlanckstars to-day,stillperunitvolume.Importantly,theshapeofthismass spec-trumintheinterestingregionismostlyindependentoftheinitial shape.Thisisexactlytrueonlyinthelimitm

3 f

(

m

)

tH and

con-stitutes arough approximation here. To getorders ofmagnitude, we however assume this to be correct. In this case, due to the dynamicsoftheevaporation,dn

/

dm

∝

m2_.Thiscanbe straightfor-wardlyseenbywritingdn

/

dm

=

dn

/

dmi

×

dmi

/

dm,wheredn

/

dmi

istheinitialmassspectrumanddmi

/

dm

=

m2

(

3 f

(

m

)

t

+

m3

)

−2/3.

If primordial black holes leading to Planck stars are formed throughakindofphasetransitionintheearlyUniverse,theirmass spectrumcanbeverynarrow.Inthemostextremeoptimisticcase, thiswouldcoverexactly therangeofmassesreaching mf in the

observationtimewindow



t.Inthatcase,thenumberofobserved explosionswouldbe Nexpl

=

Ndet.Thisisobviouslyunrealistic.On

the other extreme, one can assume a very wide mass spectrum. Asthe primordialcosmologicalpowerspectrum P

(

k

)

∝

kn isnow known to be red

(

n

<

1

)

whereas it wouldhave hadto be blue toproduceasizableamountofprimordialblackholesbystandard processes,theusualhistoricalspectrum

dn dmi

=

αm

−1− 1+3w 1+w i

,

(3.6)

408 A. Barrau, C. Rovelli / Physics Letters B 739 (2014) 405–409

canbeexactlycalculatedinsomemodels,forexamplein inflation-aryscenarioswithadistinguishedscale[11].Itwillbeabsorbedin themorephenomenologicallyrelevant parameter

Ω

PBHusedlater oninthisarticle.

Thisleads,assuming thattheformation occurredinthe radia-tiondominatedera,toacontemporaryspectrumreading

dn dm

∼

α

m−52

_Θ(

_m

−

_m∗

₎

+

_m− 9 2 ∗ m2

Θ(

m_∗

−

m

)



,

(3.7)

wherem_∗ istheinitialmassofablackholeexplodingtoday. Thenumberofexpected“events”during anobservingtime



t

isgivenby N

(

t

)

=



m(t) mf dn dmdm



mmax mf dn dmdm

Ω

PBHNmax_det

Ω

sr

,

(3.8)

where mmax is the maximum mass up to which we assume the

massspectrumtobegivenby Eq.(3.7) and

Ω

sr isthesolid angle

acceptanceoftheconsidered detector.Here,thevalue of

Ω

PBH is noteasy toconstrain. An upperlimit canbetakenconservatively at 10−8. To fix orders of magnitude, if we set mmax

=

m∗ anda densityof a few percents ofthe maximumallowed density, that is

Ω

PBH

∼

10−10_{,thisleadstoone eventperday.Detectionisnot} hopeless.

4. Veryshortgamma-rayburstsanddiffuseemission

Could it be that those events havealready been detected?In particular:could theybe associatedwithsome gamma-raybursts (GRBs)?There are two main classes of GRBs:the long ones and theshort ones. The longones are quitewell understood andare associatedwiththedeathsofmassivestarsandhavenolinkwith thisstudy.Our modelforthe endofPlanck starsdoesnot allow for the calculation of anykind of light curve. This would be far beyondthecurrentdevelopment. But,obviously,thetime-scaleis expectedtobeshort.

IfPlanckstarexplosionsaretobeassociatedwithsome ofthe measured GRBs, this would obviously be with short gamma-ray bursts (SGRBs) [12]. This could make sense for several reasons. Firstly,SGRBsare thelesswellunderstood.Inparticular, the red-shiftsarenotmeasuredforalargefractionofthem,whichmeans thattheycould,inprinciple,beoflocalorigin.Secondly,SGRBsare known to havea harder spectrum and some of them do indeed reach the energies that we have estimatedin this work. Thirdly, a sub-classofSGRB,theveryshortgammaraybursts(VSGRBs),do exhibitanevenharderspectrumandcanbeassumedtooriginate fromadifferent mechanismastheSGRB time distributionseems tobebimodal[13].

Nothingcanbeconcludedatthisstagebutitisworthnoticing that thereare indicationsthatsome VSGRBsare compatiblewith aPlanckstarorigin. Thefactthattheir angulardistributionfavors a cosmologicaloriginisinfact alsofullycompatible withalocal bubbleorigin.

Anotherimportant question to address is the possible diffuse emissionfromPlanckstars.Notthesingleeventdetectionbutthe integrated signal over huge distances. In that case, no time sig-naturecan be expectedbutthe globalspectrumcan retain some specificcharacteristics.Twoeffectsarecompeting.Letusconsider a Planck star exploding at a redshift z. Because, to be detected now,ithasexplodedinthepast,itmeansthattheamountof cos-mictimerequiredforittoreachitsfinalmasswassmallerthanin ourvicinity.Itsinitialandfinalmasseswerethereforesmaller.This meansthatthemeanenergyofemitted particleswashigher (and

the totalamount smaller) than forPlanck stars explodingwithin

Fig. 4. Full

spectrum of gamma-rays emitted by a decaying Planck star at

z=3 (log scales).

theGalaxy.Butjustbecauseoftheredshift,themeasuredenergy is smaller than the emitted one by a factor

(

1

+

z

)

. In practice, thissecondeffectslightlydominatesoverthefirstone.For exam-ple, aPlanck starexplodingat z

=

3 emits photonswitha mean energyhigherthanaround usby afactor1.9. Thisenergyis then redshifted by afactor4.Fig. 4 showsthesamespectrum thanin Fig. 3 butfora z

=

3 Planckstar(it isnot justarescaling ofthe previousoneastheenergyofjetschanges).

When considering a diffuse signal withvery small absorption effects, like gamma-rays in this range, the integration effect is drastic.Foreachshell,thenumberofdetectedphotonsisinversely proportional to the squared shell distance because of the solid angle effect.But the numberof sources per shell isproportional to thesquareddistance.The twoeffectscompensate (modulothe slightenergyvariationmentioned intheprevious paragraph)and each shell contributes the same. The neat flux is therefore fully determined by the cutoff. Weleave thedetailedstudy forfuture worksas,inthiscase,itwouldbemandatorytotakeintoaccount not only the explosion spectrum but also the last stages of the Hawkingspectrumduetotheevaporationbeforetheexplosion.It can howeverbe easily concluded that thesignal can in principal be detected as the mean densityrequired to produced a sizable amountofgamma-raysisverysmall.

5. Conclusionandprospects

We haveshown that thedetection ofindividual explosions of Planck stars isnot impossibleand we haveestablished themain spectral characteristicofthe signal.Quantum gravitymightshow upinthetensofMeVrange.Wehaveestimatedtheorderof mag-nitudes for the expected frequency of events: in some cases, it might not be small. Several approximations aboutthe dynamics of theevaporationcan be improved.The details ofthe explosion remaintobeinvestigated.Theshapeofthediffuseintegrated sig-nal, andmore specificallyits potential specificsignature allowing to distinguish itfrom standard primordialblackholes, requires a full numerical analysis. It could also be interesting to investigate the emission of charged articles, in particular positrons and an-tiprotons (some interesting threshold effects couldbe expected). Aswellknown,themuchsmallerhorizoncanbecompensatedby thelargegalacticconfinementeffect.

Acknowledgement

A.B.wouldliketothankWeiXiang.

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