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Constraints on cosmic strings due to

arXiv:hep-ph/9306221v1 3 Jun 1993Constraints on cosmic strings due toblack holes formed from collapsed cosmic string loopsR. R. Caldwell† and Evalyn Gates†‡†NASA/Fermilab Astrophysics CenterFermi National Accelerator LaboratoryP.O.

Box 500Batavia, Illinois 60510-0500‡University of Chicago5640 S. Ellis AvenueChicago, Illinois 60637May, 1993ABSTRACTThe cosmological features of primordial black holes formed from collapsed cosmicstring loops are studied. Observational restrictions on a population of primordialblack holes are used to restrict f, the fraction of cosmic string loops which col-lapse to form black holes, and µ, the cosmic string mass-per-unit-length.

Usinga realistic model of cosmic strings, we find the strongest restriction on the para-meters f and µ is due to the energy density in 100MeV photons radiated by theblack holes. We also find that inert black hole remnants cannot serve as the darkmatter.

If earlier, crude estimates of f are reliable, our results severely restrict µ,and therefore limit the viability of the cosmic string large-scale structure scenario.0

I. IntroductionThe cosmic string scenario for the formation of large scale structure hasmany observable features. Primarily, cosmic strings may serve to produce per-turbations to the cosmological fluid of the necessary magnitude and distributionto seed the formation of galaxies and clusters, as observed today.

Cosmic stringsleave an observational signature through these perturbations, as well as throughthe emission of gravitational radiation. Broadly, then, there are two areas of cos-mic string research.

These are studies of the large-scale structure produced bycosmic strings, and tests of the compatibility of cosmic strings with cosmologicalobservations. Such tests focus, for example, on the anisotropies produced by cos-mic strings in the microwave background and the noise in pulsar timing due tothe cosmic string stochastic gravitational wave background.

Ultimately, the testof compatibility results in a restriction on µ, the mass-per-unit-length and solefree parameter in the cosmic string model. In this report, we will examine therestrictions on black holes formed from collapsed cosmic string loops.It is well known that a sufficiently smooth, circular cosmic string loop maycollapse to form a black hole [1,2,3,4,5].

During the evolution of a network ofcosmic strings, some cosmic string loops may collapse to form black holes. In thiscase, the observational restrictions on a population of primordial black holes maybe used to restrict such a cosmic string scenario.The study of primordial black holes has been vigorously carried out in, forexample, [6,7,8,9,10,11,12].

We will take advantage of this work in applying con-straints to a population of black holes formed from collapsed cosmic string loops.In turn, we will place restrictions on the cosmic string network. In this paper,we will find observational restrictions on the cosmic string scenario from cosmicstring loops which collapse to form black holes.The organization of this paper is as follows.

In section II we will summarizeprevious efforts to estimate the fraction f of cosmic string loops which collapse toform black holes. In section III we will present the models of cosmic strings andblack hole evaporation used to calculate the energy density in black holes andblack hole radiation.

In section IV we will present the observational constraintson a population of black holes formed from collapsed cosmic string loops. We willconclude in section V with a restriction on the parameters f and µ.II.

Collapse of Cosmic String Loopsto form Black HolesA cosmic string which contracts under its own tension to a size smaller thanits Schwarzschild radius will form a black hole. In this section, we will conduct abrief review of the analysis of this phenomena.

We will present a naive estimateof the probability that a realistic cosmic string loop will collapse to form a blackhole. While no conclusive work has been carried out to determine this fraction,our naive estimate will serve as a rough guide for the cosmological analysis in thesucceding sections.1

The phenomena by which a cosmic string loop collapses to form a black holemay be best understood by examining a simple case. We will consider a perfectlycircular, planar cosmic string loop of mass m. The string equations of motiondictate that such a loop will expand and contract under its own tension, witha maximum radius Rmax = m/2πµ.

When it contracts under its own tensionto within its Schwarzschild radius RS = 2Gm = 4πGµR, it will form a blackhole. (We use units ¯h = c = 1 and G = m−2planck.) A loop may never contract towithin its Schwarzschild radius, however, if it is sufficiently non-circular.

As well,a loop with a Schwarzschild radius comparable to its thickness may dissipate, byradiating the quanta trapped in the string, before a black hole may form. Thus,not all cosmic string loops collapse to form black holes; in fact we expect only asmall fraction to do so.We are interested in determining which loops formed by a realistic cosmicstring network collapse to form black holes.

Simplifying this problem, we ask whatfraction f of cosmic string loops collapse to form black holes. We will ultimatelyfind that observational restrictions on black holes formed from collapsed cosmicstring loops will depend linearly on this fraction f. In this study we will not beable to conclusively determine f. After reviewing past work on the properties andbehavior of realistic cosmic string loops, however, we will be able to determinethe relevant properties that effect this fraction.

In addition, previous attemptsto determine this fraction f, along with reasonable assumptions about the looppopulation, will be shown to indicate a rough value for f. If these estimates arereasonable, we may be able to place severe restrictions on cosmic string scenarios.The initial investigation of black holes formed from cosmic string loops wascarried out by Hawking [1]. Considering a scenario where the loop would have asimilar probability of collapsing to form a black hole in each oscillation period,he proposed that the fraction f is a function of the mass per unit length of theloop and the number of kinks (n) on a string loop.

His expression for f, however,depends exponentially on n. Numerical simulations of cosmic string evolutionsuggest that a reasonable range for n is given by n ∼2−5 [13], which correspondsto, for µ = 10−6, f ∼1−10−36. Thus, unless a more accurate determination of nis achieved, this approach does not provide a conclusive estimate for f that maybe useful for constraining cosmic string scenarios.A numerical analysis of several families of parameterized loop configurationswas carried out by Polnarev and Zembowicz [4].

They examined the fraction ofparameter space for which Burden and Kibble-Turok families of loops collapseto form black holes. These families model loops which contain cusps, and maynot necessarily be representative of realistic loops.

Depending on the measureassigned to the configuration parameter space, they found the fraction may lie inthe range f ∼10−9 −10−15. For loops with few kinks, this range of values seemsto be roughly in accord with Hawking’s estimates.There are two major features of a cosmic string loop which may determinewhether the loop will collapse to form a black hole.

These are, roughly, the large2

(underlying loop configuration or shape) and small (kinks and cusps) scale fea-tures of the loop. One may attempt to determine the fraction of loops collapsingto black holes by asking the following questions: (i) what fraction of realistic loopspossess an underlying configuration which would lead to the formation of a blackhole, and (ii) what fraction of these loops possess kinks and cusps (fluctuations)small enough that a black hole may still form.

Previous numerical simulations ofcosmic string evolution [13,14,15,16] provide some insight into the relevant issues.The study of the effects of the gravitational back-reaction on the evolutionof cosmic string loops indicates that the gravitational back-reaction will set theminimum scale of structures on long strings. These structures are the predeces-sors of parent loops which are chopped offthe long strings.

The parent loopsthen undergo fragmentation and rapidly evolve towards simple, non-intersectingconfigurations, containing on the order of 2 −5 kinks. Quashnock and Spergel[17] found that the kinks and small scale structure on string loops rapidly decay,and the loops then oscillate in a self-similar manner.

Cusps, however, are not sup-pressed by the gravitational back-reaction and persist throughout the evolutionof the loop. Thus, except for configurations that contain cusps, the underlyingshape of a realistic loop is dominated by low mode or long wavelength oscillations.The work of Garriga and Vilenkin [18] considered nucleated cosmic stringloops, which may collapse to form black holes.

The behavior of classical fluc-tuations on the loops were examined, and it was shown that while transverseperturbations maintain constant amplitude as the cosmic string loop contracts,radial perturbations shrink by a factor of the perturbation mode number. Sincethe loops are dominated by low mode oscillations, the maximum allowed ampli-tude of a perturbation that is consistent with collapse to a black hole is of orderRs.

That is, the maximum tolerable fluctuation which will not prevent the for-mation of a black hole contains only ∼Gµ of the total loop energy. However,while we expect most of the daughter (non-self-intersecting) loops to be relativelysimple, we currently have no information regarding the frequency or distributionof fluctuations and loop configurations.We may nevertheless attempt to estimate the fraction f using Hawking’sapproach, incorporating the work of [13] on the properties of realistic cosmicstring loops.

We consider the properties of stable, non-self-intersecting loops, andassume that the number of kinks on a loop is the dominant factor in determiningwhether such a loop will collapse to form a black hole. Note that since theseloops oscillate in a self-similar manner, we are concerned only with whether agiven configuration will immediately collapse to form a black hole.

We do notintegrate this probability over the number of oscillations in the black hole lifetime,as Hawking did in his original calculation. Thus, we then integrate Hawking’sexpression for the number of loops which collapse immediately to form blackholes over the distribution of the number of kinks on stable daughter loops.

Thisintegral is dominated by the contribution from loops with two kinks. We findf ∼10−12.3

This estimate is roughly in the same range as that given in the work by Zembowiczand Polnarev. We must stress that although this is just a rough estimate, we mayuse this fraction as a guide for our study of the observational constraints in thefollowing section.A careful determination of f will be necessary to conclusively evaluate theobservational constraints on cosmic strings.

Such a study will be the focus of afuture work [19]. In the meantime, we will adopt the working hypothesis thata fraction f of realistic loops are smooth enough at the time they are choppedoffthe cosmic string network that they may immediately collapse to form blackholes.

We may now proceed to evaluate the observational constraints on blackholes formed from collapsed cosmic string loops.III. Production and Evolution of Black Holesfrom the Collapse of Cosmic String LoopsThe properties of a population of black holes formed from collapsed cosmicstring loops are well specified by the properties of the cosmic string networkand by the properties of quantum mechanical evaporation by a black hole.

Inthis section we will first present the model of cosmic strings used in this study,focusing on those aspects relevant to the population of black holes producedfrom collapsed cosmic string loops. Second, we will describe how the quantummechanical decay of black holes is incorporated into the cosmic string scenario.Third, we will outline the calculation of the physical properties of the populationof black holes necessary to make contact with cosmological observations, andsubsequently restrict the cosmic string model.We use the “one-scale” model of kinky cosmic strings.

The properties of thismodel have been well described by [20]. We will repeat the necessary elements ofthis model.i.

The background cosmology is a spatially flat FRW spacetime with scale factora(t) ∝t1/2 in the radiation dominated era, and a(t) ∝t2/3 in the matterdominated era.ii. For simplicity, we will calculate physical quantities in a fiducial, physicalvolume V (t) = a3(t)r3 where r is an arbitrary coordinate length.iii.

We define loops to be closed cosmic strings formed, with an initial size L(t) =αl(t), through the intercommutation of long strings, where l(t) is the horizonradius. All other cosmic string is contained in long strings.iv.

Loops are considered to be non-self-intersecting.One may argue that anewly formed loop may self-intersect and fragment at a rate proportional tothe loop oscillation frequency, producing smaller loops. This rate at whicha loop self-intersects, however, is much faster than both the rate at whicha loop radiates gravitational waves and the expansion rate.

Thus, we arejustified in assuming that a loop fragments rapidly; we may consider thatL(t) represents the size of the final, non-self-intersecting loops.4

v. The rate of loop formation is [21]dNloopdt= 4Aα t−4V (t)(III.1)where A ≈10 gives the number of long, horizon-length cosmic strings presentin a horizon volume, as determined by numerical simulations [14,15]. Thevalue α ≈10−4 is given by the observational bounds on cosmic string gravi-tational radiation [21].We may now obtain the rate of black hole formation from collapsed cosmicstring loops.

Applying our hypothesis regarding the formation of black holes fromcosmic string loops to equation III.1, we finddNbhdt= f dNloopdt. (III.2)This equation states that Nbh black holes of mass m(t) = αµl(t) were formedduring the time interval t to t + dt.This equation is valid for times t ≫ti,where ti ∼α−1µ−3/2tplanck gives the time at which the Schwarzschild radius ofa newly formed loop is comparable to the thickness of the cosmic string.

Thus,the properties of the cosmic string network determine the initial properties of thepopulation of black holes.The cosmological evolution of a black hole is dominated by quantum mechan-ical emission of a spectrum of particles [2]. This radiation, which has been wellinvestigated, is the most important cosmological aspect of a black hole.

Thus,in order to follow the evolution of the black holes formed from collapsed cosmicstring loops we need the black hole decay rate. For a black hole of mass m, thedecay into massless particles is given by [22]d2mdωdt =XiΓ(m, ω, s)ωe8πmω ± 1(III.3)where the sum is over all particle species i.

The ± refers to boson or fermionstatistical weights, and Γ(m, ω, s) is a dimensionless function of the black holemass, the radiated particle spin s, and frequency ω. We will be interested pri-marily in the emission of photons, for which Γ(m, ω, s) = 64m4ω4/9.

ExaminingIII.3, we see that a black hole emits a burst of thermal radiation, characterizedby the black hole temperature, which is inversely proportional to the black holemass. This emission will continue until the black hole has completely evaporatedaway, or, as has been suggested [11,23,24], an inert Planck-mass object remains.In such a case, the black hole evaporation rate will truncate when m ∼mplanck.Adding the black hole decay rate into our model, therefore, we have completelyspecified the cosmological evolution of a population of black holes.5

We may now use the expressions for the rate of black hole formation and therate of black hole evaporation to calculate the energy density produced in blackholes and black hole radiation. The fraction of critical energy density in blackholes isΩbh(t) =1ρcrit(t)1V (t)Z ttidt′ dNbhdt′ m(t′, t).

(III.4)The limits of integration are from the time ti when the cosmic string loops mayfirst collapse to form black holes, to the present time t. Here, the function m(t′, t)gives the mass of a black hole formed at time t′ at a later time t. This function maybe obtained by integrating the black hole evaporation rate III.3 over frequencyand time, applying suitable boundary conditions. Examining III.4, the energydensity in black holes at time t is dominated by those surviving black holeswhich formed earliest, as the rate of black hole production is a rapidly decreasingfunction of time.

The fraction of critical energy density in black hole radiation,in a logarithmic frequency interval, isdΩbhrad(t)d ln ω=1ρcrit(t)1V (t)Z ttidt′ dNbhdt′Z τ(t′)+t′t′dt′′ ω(t′′)d2mdω(t′′)dt′′ . (III.5)Here, τ(t′) is the lifetime of a black hole, and ω(t′′) = ωa(t)/a(t′′) gives therelationship between the frequency as emitted at time t′′, ω(t′′), and the frequencyobserved at time t. This power spectrum is dominated by the contribution fromblack holes evaporating at the present time.

These expressions, III.4 and III.5have been integrated numerically; the results will be presented in section IV.We will be interested, as well, in constructing the function β(m) in order toevaluate the observational constraints on this population of black holes formedfrom collapsed cosmic string loops. This function represents the fraction of criticalenergy density in black holes formed during the time interval t to t + dt.β(m) =1ρcrit(t)m(t)V (t) dNbh=256π3Aµf(III.6)(Here, we have used ρcrit(t) = 3t−2/32π.) It is not surprising that this function isreally a constant.

The gross features of the cosmic string network scale with thehorizon radius: the gross features of the population of black holes formed fromcollapsed cosmic string loops scale with the horizon radius. It is important to notethat equation III.6 describes a population of black holes different from the blackholes described by the function β(m)horizon found in the primordial black holeliterature (for example, see [10]).

There, β(m)horizon represents the fraction ofcritical energy density in black holes which enter the horizon in the time intervalt to t + dt. Such a black hole will be much smaller than the horizon radius, as6

are the black holes formed from collapsed cosmic string loops, at the later time(2αµ)−1t. Therefore, to relate equation III.6 to the function β(m)horizon foundin the literature, we writeβ(m) =β(m)horizona(t/2αµ)a(t)≈(2αµ)−1/2β(m)horizon.

(III.7)Here, we have simply accounted for the growth in the black hole energy densityover the background radiation energy density from the time the black hole entersthe horizon to the time that a black hole of the same mass would be formed froma collapsed cosmic string loop. The function β will be used in section IV, as hasbeen used in [10,12], to evaluate the restrictions on black holes.We have apparently neglected to consider the loops formed along with thelong strings at the time of the cosmological phase transition.

These loops, ashas been recently shown in [18], may be smoothed by the friction with the cos-mological fluid [25]. The loops most smoothed by the friction, however, haveSchwarzschild radii smaller than the string thickness; these loops will not col-lapse to form black holes.

The remaining, unsmoothed loops, which are largerthan the horizon radius at time ti, behave simply as long strings. Thus, we arguethat we have considered all loops which may collapse to form black holes, andmay now proceed to evaluate the observational constraints on black holes.IV.

Observational Constraints on Black Holesfrom Collapsed Cosmic String LoopsThe numerous observational restrictions on a population of primordial blackholes are a direct consequence of the richness of the physics of black hole evapo-ration. Through quantum mechanical decay, a black hole will radiate all particlespecies.

Primordial black holes may be observed then through the emitted par-ticle spectra. Consequently, the observation of spectra produced by such blackholes may serve to indicate exotic events which may have taken place in the earlyuniverse.

Figuratively, the cosmic string energy invested in black holes in theearly universe provides a return with observational consequences today. In thefollowing paragraphs we will present the observational constraints on black holesformed by collapsed cosmic string loops.

We will begin by evaluating equationsIII.4-5 for constraints on the energy density in black hole photon radiation andremnants. Next, we will use equations III.6-7 to evaluate more observational con-straints.

These constraints will be expressed in terms of a restriction on f, usingthe preferred value µ = 10−6. We will conclude this section with an interpretationof the observational constraints on the cosmic string parameters.The strongest constraint on this population of black holes formed from col-lapsed cosmic string loops is due to the γ ray flux observed at 100MeV [26,10,27].We require that the fraction of critical energy density in photons emitted by7

black holes, with energy in a logarithmic interval at 100MeV , be less thanΩγ = 10−8h−2. Integrating equation III.5, we finddΩbhγ(t0)d ln ω|ω=100MeV = 109h−2f ≤10−8h−2→f ≤10−17.

(IV.1)Black holes of mass m ∼1016g (with a lifetime ∼1017s) which evaporate todayserve as the dominant source of photons at this energy.In the work of both Hawking [1] and Polnarev and Zembowicz [4] the cos-mological constraints on cosmic strings due to the γ rays emitted by black holesformed from collapsed cosmic string loops were evaluated.These calculationsused a very rough cosmic string model. Our work improves upon their resultsby implementing a realistic cosmic string model, as we take advantage of resultsfrom numerical simulations to determine the average number of long strings in ahorizon volume and the size of newly formed loops.

The improved limits are dueto our improved model.Further observational constraints on primordial black holes, which are typ-ically stated as a restriction on the energy density in black holes in a particularmass range, may be easily evaluated analytically using β. We have taken advan-tage of the literature [10,12], in which the restrictions on black holes are statedin terms of β(m)horizon, which we may simply convert into β according to equa-tion III.7.

In the following table, then, we list the observational constraint, therestriction on β, and the resultant limit on f, the fraction of collapsing cosmicstring loops.Observational Restrictions on fdiffuse γ ray backgroundβ(m15) ≤10−21f ≤10−17interstellar e+ backgroundβ(m15) ≤10−21f ≤10−17interstellar ¯p backgroundβ(m15) ≤10−21f ≤10−17interstellar e−backgroundβ(m15) ≤10−20f ≤10−16photodissociation of d by photonsβ(m10) ≤10−16f ≤10−14distortion of CMBRβ(m13) ≤10−15f ≤10−13photon-to-baryon ratioβ(m13) ≤10−14f ≤10−12n/p by nucleonsβ(m10) ≤10−11f ≤10−9entropy productionβ(m11) ≤10−3f ≤10−1remnants overclose universeβ(m−2) ≤10−18 f ≤10−158

In this table, mX indicates black holes formed with mass 10Xg. The first fourconstraints are taken from [10].The limit due to the observed diffuse γ raybackground at 100MeV is the strongest constraint.

Uncertainties in the clusteringof black holes and the diffusion of charged particles within the galaxy may weakenthe constraints on the interstellar e± and ¯p backgrounds. The next five constraintsare taken from [12].

These constraints focus primarily on the nucleosynthesisrestrictions on black hole radiation. None of these nucleosynthesis limits are verystrong.The constraint on black hole remnants requires that the remnants not over-close the universe.

However, it has also been suggested [11,24] that inert, Planckmass black hole remnants may provide a substantal fraction of the dark matter.Integrating equation III.4, we findΩbhr(t0) = 1015f ≤1→f ≤10−15. (IV.2)Black holes of mass ∼10−2g, the first black holes formed from collapsed cosmicstring loops at the time ti, serve as the dominant source of remnants.

This limitis a conservative upper bound on f, as the energy density in remnants dependscritically on this inital time ti. Although we have stated earlier that black holesmay form only at times t ≫ti, when the cosmic string loop Schwarschild radiusis much greater than the string thickness, we have actually included black holesformed starting at time ti.

Therefore, this limit on f may weaken somewhatdepending on the detailed behavior of the collapse of a thick cosmic string toform a black hole. Then, closure density in remnants requires f ≈10−15, whichis in disagreement with the γ ray background limit, f ≤10−17.

Therefore, therestrictions on a population of black holes formed from collapsed cosmic stringloops indicate that these black hole remnants cannot serve as the dark matter.We may now interpret the restrictions on the population of black holes formedfrom collapsed cosmic string loops in terms of restrictions on cosmic strings. Theγ ray background limit requiresf ≤10−17forµ = 10−6.

(IV.3)This equation gives the strongest restriction on cosmic strings due to black holesformed from collapsed cosmic string loops. If the rough estimates of the magni-tude of f ∼10−12 are reliable, then our results would indicate that µ ≤10−11 isnecessary for compatibility with the γ ray background.

In this case, we could ruleout the cosmic string scenario of large scale structure formation, which demandsµ ∼10−6. We are not confident in these crude estimates of f, however, as wehave indicated in section II.

Clearly, a more detailed investigation is necessary todefinitively determine f [19].9

V. ConclusionIn this work we have analyzed the restrictions on black holes formed fromcollapsed cosmic string loops.We have found that the requirement that thephoton flux due to evaporating black holes does not exceed the observed γ raybackground flux serves as the strongest restriction on such black holes formedfrom cosmic string loops. Using a realistic model of cosmic strings, we find thatthis observation requires f ≤10−17 for µ = 10−6.

Thus, a fraction of no morethan 10−17 of newly formed cosmic string loops may collapse to form black holesin order that µ = 10−6 remains compatible with observation. This restriction alsoprecludes black hole remnants from serving as the dark matter.

We plan to studyin greater detail the fraction f of loops which collapse to form black holes [19].If a lower bound f ≥10−16 is found, the γ ray background limit on evaporatingblack holes would serve as the strongest observational bound on cosmic strings,and would rule out the cosmic string scenario of large-scale structure formation.AcknolwedgementsWe would like to thank James E. Lidsey and Jean Quashnock for usefulconversations. The work of RRC and EG was supported in part by the DOE(at Chicago and Fermilab) and the NASA through grant # NAGW-2381 (atFermilab).10

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