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This content was downloaded from IP address 130.64.25.60 on 13/07/2019 at 23:53 JCAP11(2018)008 hysics P le c ic t ar strop A https://doi.org/10.1088/1475-7516/2018/11/008 and Andrei Gruzinov b Yuri Levin a osmology and and osmology C 1808.00670 Cosmic strings, domain walls, monopoles, GR black holes, gravitational waves Cosmic strings and primordial black holes (PBHs) commonly and naturally form rnal of rnal ou An IOP and SISSA journal An IOP and 2018 IOP Publishing Ltd and Sissa Medialab Department of Physics and574 Astronomy, Tufts Boston University, Avenue, Medford, MADepartment 02155, of U.S.A. Physics, Columbia538 University, West 120th street, NewDepartment York, of NY Physics, 10027, New U.S.A. 726 York University, Broadway, New York, NY 10003, U.S.A. E-mail: , [email protected] , [email protected] [email protected] b c a c Alexander Vilenkin, Cosmic strings and primordialholes black J Received September 10, 2018 Accepted October 10, 2018 Published November 7, 2018 Abstract. in many scenarios describingand the PBHs are early present, universe.time their of interaction Here leads their we to formation, showleading a the that to range PBHs if of the get interesting both formation attachedloop consequences. of to cosmic production At black-hole-string the strings the networks strings ingive and and a rise commonly influence range to to their evolution, the of smallblack suppression nets redshifts. holes of made in of Subsequently,properties the several reconnections of network black within the as holesof strings. the and well gravitational connecting network as waves. The strings. the This netsThe stability leads The oscillate strings to of number and can potentially of the shrink keepgenerally observable applicable. nets exponentially PBHs string-driven Alternatively, from mergers due depend heavy PBHs of galactic toof on can PBHs. halos, the drag galaxies. the low-tension making emission topological strings thesupermassive into The the black current superconducting centers holes. bounds strings on can PBHs appear not asKeywords: radio filaments pointing/ towards sources, gravitational waves / theory ArXiv ePrint: JCAP11(2018)008 = 2 the defects are called N – 1 – 2 strings attached to each monopole [5]. In the case of ≥ In this paper we discuss what happens when both strings and PBHs are present in the A seemingly unrelated direction of cosmological research is the formation and evolution N 3.1 The minimal3.2 scenario4 Black hole3.36 detachment Persistent networks: the maximal scenario8 2.1 PBH2 formation 2.2 String evolution and3 capture universe. We find that theyThe generically evolution form and a observational network effects whereand of PBHs strings vice are may connected versa. then by be strings. enhanced In significantly in particular, affected some by scenarios. observational PBHs boundsconnected Another by interesting on effect strings. the is string the increased energy merger scale rate are for PBHs strongly “necklaces”, with monopoles andA antimonopoles discovery alternating of like cosmicnew beads strings observational along would bounds the open on string strings a may [6]. window rule into out physics classes of of very particle high physics energies, models. and 1 Introduction Primordial black holes (PBHs)played an could important be role formedThey in could the early serve as evolution in seeds of for theand the supermassive black radiation could universe holes (see observed era account at refs. for and thesuggested [1,2] centers that merging could of for most PBHs BH recent have galaxies could reviews). binaries constitute currently a substantial observed part, byof if LIGO. not topological It all, defects, has of inparticle dark also particular physics matter. been cosmic models strings.review). and Some can Strings models are produce predict predictedwith a more in complicated variety defects a of — wide astrophysical monopoles class effects connecting of by (see strings, [3,4] for a 3 Evolution of BH-string4 network 4 Black holes with loops, and5 nets9 Gravitational radiation from nets 6 Constraints on persistent networks 7 Conclusions 11 12 13 Contents 1 Introduction1 2 Black hole formation and string capture2 JCAP11(2018)008 and (2.1) (2.2) (2.3) . For . The f 11 M . With t 1. Then − 3

1 on such / 1 10 M  − ∼ ∼ λ λ δ . 100 λ 16 ∼ is − − f f where PBHs may and the comoving d t 10 10

of equal matter and M − ∼ M 6 eq 12 t − M 10 form at 10 − , which gives M 3 3 ∼ and − 10 λ

, 10 , respectively. In order to account 2 8 is ∼ M / , ∼ 1 − λ 2 s, 10 /  M − 10 M

1 f eq M  10 ∼ M M f M 5 eq λ t − t −  , and the probability of having  15 λ 10 5 − 2 f and − ∼ ∼ 1 and [14] – 2 – 10 16 Gt 10 and their average separation is f

− ∼ ∼ 3 ∼ M GM 10 CDM − f does not change with time and is therefore a useful , so black holes of mass Mn ρ − M 10 ∼ λt rms M δ 13 ∼ CDM f ∼ t/G f t ∼ − ρ is the horizon mass at the time ∼ M f M 10 f n

J ∼ M M λ 17 . The corresponding range of 3 10 − ∼ Mpc 1 /G . 0 eq t ∼ 1. On the other hand, the rms fluctuation on scales accessible to CMB and ∼ ∼ eq M 1, eq.) (2.3 then gives The density fluctuation required for a horizon-size region to collapse to a black hole is Let us now indicate some cosmologically interesting values of the parameters Suppose the fraction of horizon regions that turn into BHs at time We now mention some other mechanisms of PBH formation. High-energy vacuum bub- δρ/ρ ∼ ≡ during the radiation era is . There are only two observationally allowed windows for the mass M cold dark matter (CDM) mass density at that time is large-scale structure observations is δ with their Schwarzschild radius comparable to the cosmological horizon. the BH density at formation is radiation densities. The quantity characteristic of PBHs. 2 Black hole formation2.1 and string capture PBH formation PBHs can be formedcollapse by of a primordial variety of overdensitiescould mechanisms. during originate The the from most radiation quantum fluctuations widely erat in discussed [7,8]. the scenario inflationary The is epoch. initial the The overdensities Jeans mass at time scales is negligibly small.enhanced Hence one in has the to range assumesimplicity of that that scales the the fluctuation corresponding fluctuation amplitudethe to spectrum is PBH has same strongly a formation. time narrow and Heresome peak, have inflationary we so nearly models shall that assume (e.g., the all [9–11]). for same PBHs form mass. at about Such fluctuation spectra naturally arise in so the fraction of dark matter in the form of BHs is where λ account for all dark matter [12, 13]: bles may nucleate andsizes. expand during After inflation inflation, ends, resultingto the in bubbles BHs a collapse much very toform wide smaller form spectrum with BHs than of their [15, the bubble Schwarzschild16 ].is radius horizon, Small PBH comparable bubbles but formation to collapse by starting thelarge collapse horizon. walls with of form a A vacuum BHs closely domain at certain related the walls critical scenario horizon [15, size scale.17–20]. the PBHs Once could BHs again, also sufficiently be formed by collapse of cosmic PBHs to serve asPBH seeds density of supermassive BHs, we need f for LIGO observations, we need JCAP11(2018)008 . 1 t 1 . 0 (2.4) (2.5) ∼ s is the , where l p 44 m −  10 and the string η 2 ∼ η p s. 1. The strings are t ∼ 4 − . Long strings move µ  t 10 2 ) ∼  p t , where l p t 1 η/m ( − ) ∼ Gµ ( Gµ contains a few long strings stretching ∼ t s t . , 11 − l . Gµ 10

Γ is l M . – 3 – ∼ l . τ Gµ M 2, so they typically have a chance to intersect with other . 0 ∼ on the string mass parameter]: [27 v 2 , is due to CMB observations [29]. 7 . − . The strings carry a magnetic flux of a heavy gauge field (which acquires 10 M µ . = due to the symmetry breaking). When two strings cross, they reconnect in such and Gµ 50 is a numerical coefficient. Gravitational waves emitted by loops over the cosmic GeV is the Planck mass. The mass per unit length of string is T η η ∼ 19 ∼ 10 Numerical simulations of string evolution indicate that strings evolve in a scale-invariant The lifetime of a loop of initial length PBH formation can be enhanced if the universe went through some episodes of reduced In what follows we shall assume that PBHs are formed on the horizon scale. We shall Any strings that formed prior to inflation will be inflated away, so that the distances ∼ We do not expectThis any bound phase applies transitions only to later strings than formed as the a QCD result transition of gauge at symmetry breaking. Global strings, resulting 1 2 p strings in a Hubble time. Loops are chopped off the long strings with a typical size from breaking ofmost a of global the symmetry, energysuch emit going strings, a into radiation negligible of fraction massless of Goldstone their bosons. energy The in strongest observational gravitational bound waves, on with Planck time. Note thatvalues this of is earlier than PBH formation for all astrophysically interesting manner [24–26]. A Hubble-size volume at any time and may only account for PBHs with across the volume andat a mildly relativistic large speeds, number of closed loopsMuch smaller of loops length are also producedthe in speed localized of regions light. where The theand loops string gradually oscillate velocity periodically, approaches shrink lose and their disappear.both energy by necessary gravitational Long radiation, to string sustain reconnections and the chopping scale-invariant nature off of of loops evolution. are where Γ a mass history add up toamplitude a of stochastic this gravitational background wavecan is background. impose not Requiring in an that conflict upper the with bound predicted the millisecond pulsar observations, one pressure. This couldreheating happen after at inflation [see, cosmologicalmay e.g., phase be [23] transitions strongly and or reducedthe references horizon. and during therein]. the a This typical In period mechanism, black this however, of is hole case slow likely size the to may Jeans operate be mass only significantly incomment smaller on the than the very alternative early possibility universe at the2.2 end of the next String subsection. evolutionCosmic and strings capture are characterized by the energy scale of symmetry breaking, a way that theof direction string of gravity is the set magnetic by flux the is dimensionless preserved parameter along the strings. The strength between them will become much greater than the present horizon and we will not be able to string loops [17, 21, 22],produced but in this this requires the way loops is to not be cosmologically nearly interesting. circular and the PBH density m tension is formed at a symmetry breaking phase transition at JCAP11(2018)008 /G f (2.6) t . The . Oth- f t λ  2 2 / ) 1 f M − λ ∼ GM/t , the probability for f steps is exponentially t 1 ∼ − t λ . To see what effect this s . Hence the average length t  λ replaced by ( . At n 1 which is destined to become  t ∼ λ f ∼ ∼ t δ . f n 2  f t is the BH density at formation and GM – 4 – f  n . In fact, there will be very few strings much longer ∼ 10 long strings to be captured in each BH. In other /λ n ∼ f t ∼ l 10 long strings. These strings suddenly find themselves partially ∼ 20 long strings coming out of the BH, with half of them carrying flux ∼ 20 strings attached to the BH, they will frequently intersect and reconnect. ∼ , it finds itself surrounded by a domain of significantly lower density. At that point an Long strings have the shape of random walks of step Suppose now that PBHs are formed at some time We finally comment on the scenarios where BHs have size much smaller than the horizon f t ∼ a string to encounterof a string BH connecting two at BHs each is step along its length is than that (since the probability of not encountering a BH for This is equivalent to theerwise, above the scenario evolution with of the the parameter network is similar to that3 with BHs formed at Evolution the of horizon BH-string scale. 3.1 network The minimalStrings scenario emanating from any giventime BH scale. typically change With their shape and direction on a Hubble When intersecting strings haveleave opposite a directions string of segment thestring with magnetic pieces flux, both move such away. itsloops. reconnections The ends segment Eventually attached will what toreduces oscillate, remains the self-intersect the of BH, and number the while chop of segmentis off the strings will when some reconnected attached be a closed long to swallowed string by the moving the BH through BH. by the This two. horizon process volume Another intersects possibility two to of consider the strings attached words, there will be in and the other half out of the BH. engulfed by a BH. Thus, we expect see any strings within thetransition observable after universe. inflation will On evolve theMany as other described inflationary hand, above models and strings based will formedof survive on at superstring-inspired till a grand models the phase predict unified present epoch. stringe.g., theories formation [30, of at31] particle the very and physicsquite end and references generic. or a therein). after number inflation Hence (see, coexistence of stringswill and have PBHs on appears strings,a to consider be BH. an While overdensethe region this background with region radiation is densitya bigger and somewhat the than denser stringst homogeneous the evolve universe. horizon, just its When asapparent evolution horizon the they forms, is would region encompassing in unremarkable: comesbecomes more a a or within both BH. less region the The the of size horizonso entire of it region, at the so typically region the at contains region’s that time interior is comparable to the cosmological horizon, suppressed). The average initial separation of BHs connected by a string is is their mass, the initial density of BHs attached to the network is then at formation. StringsBH will happens be to captured be inbe within this a attached case Schwarzschild to radius only more from when than a two the string, strings. and center it of If is a unlikely newly for formed a BH to evolution of this BH-string network will be discussed in the next section. JCAP11(2018)008 ) at is (3.5) (3.3) (3.1) (3.2) (3.4) (3.6) h t t ( d 10 strings the strings ∼ h ). We shall refer t t > t ( f can be found from ∼ t 1, the distance travelled  is also t . This length becomes comparable , Gµ/λ 2 2 / /λ − 1 f ∼ 3 t t f t . / − t ) t 2 h t t ∝ . . The typical BH velocity at time v ( ∼ 3 2 ∝ ) l Gµ µ / µ t t ) ) 1 f ) t ( t t t ∼ ∼ ( ( 3 3 ( l 10 / t − 3 1 ) a µn a 3 ∼ . Hence we can write , then − ) ) 3 t – 5 – t ) λ λ / ∼ ( ( t 2 s ( f ) t µ/M s ∼ t ρ the network consists of BHs separated by distances (  ( ρ ) ∝ ∝ s t h ∝ . ∼ ρ ( t ) ) ) per BH in the network at time h t ) t d Gµ v t t t ( ( ( ∼ ( d n l , so BHs are nearly stationary. Then at E t h t  , and it follows from eq. (3.4) that the BH separation at time , so the energy density in long strings is 3 h /λ t v f t . Thus, at ∼ h t h ) does not change with time, t t is the scale factor. The total string length decreases in the same way, with ∼ ( l of the network length still survives. If we assume that the loops are chopped 2 the BH separation is much larger than the horizon. So the string dynamics / , all BHs are attached to the string network. At later times, only a fraction 2 1 / per volume f f t 1 t t t ) ∝ ∼ ∼ /t ∼ ) f t t t t 1, whichever is smaller. If ( ( a and connected by more or less straight strings. The subsequent evolution depends on ∼ BHs are pulled by the strings with a force At According to the minimal scenario, the number density of BHs in the network is The average length of string At ∼ There is no significant stretching of strings by the expansion during the radiation era. h ) v 3 t t ( the average BH velocity ∼ that time is also to the horizon at This shows that and the distance between them is off completely at random andfraction that of this BHs process that is unaffected are by still the attached presence to of the BHs, network then at the time The total energy of long strings in a large comoving volume decreases as where the remaining length going into closed loops. f to this as “theof minimal BHs scenario”. remaining in (In the the network may next actually subsection be we significantly shall larger.) argue that the fraction or by BHs in a Hubble time is to the BH. Onceresult again, in if the the formationgo of flux down a directions as sub-horizon in thestrings attached the number in segment. attached of this strings attached The way.BHs strings are rate We that decreases, opposite, of shall remain but these argue, this connected still processes however, will to that most will the at BHs string any may network. timeis lose largely there all unaffected is their by afirst BHs, nonzero recall except density that in of a theof string rare length network horizon without regions BHs that exhibits do a contain scaling BHs. evolution, Let with us JCAP11(2018)008 . It (3.8) (3.9) (3.7) , the 2 d (3.10) (3.11) t ) t > t µ/M ( ∼ ). For t ( d is independent of the . d h ∼ t t > t ). λ s. . At later times, the scale of the

. 1. The BHs are still moving non- BH M f , M t ρ 2 1 3 .  / − 2 , where ∼ M 1 µt ) / ∗ 3 − 1 s t / ) λ ρ t. ∼ 1 t Gµ 3 ) ∼ t h ( / t ) ∼ 2 5 t t ( − − ) ( t ) – 6 – v ( ∼ 10 Gµ/λ d ) ( ∼ Gµ t ∼ ( ( ) the BHs become energetically subdominant compared t d ∼ ( ∼ µ d M d ). The network configuration in this regime is changing d d t t v ( ∼ v t > t ∗ t . t 1, so BHs become relativistic right after . eq ∼ ∼ are occasionally formed by self-intersection of long Brownian strings. Such loops h < t t v Suppose now there is a BH attached to the string and let us assume for d ∼ , t 5 h 4 t will be comparable to the distance traveled by the BHs: , then ) at t ( d ∼ Gµ > λ The BHs start moving relativistically at The key assumption in the above scenario is that in the early stages of evolution the loops The distance travelled by a non-relativistic BH in a Hubble time is If Large loops of size We assume that 5 4 BHs move a distance largerstring than intersections, their as current a separation result of inin which the the some network. BHs network This will disconnect, should and lead the to new BH separation either reconnect to theloop network formation or is fragment subdominant. into smaller loops. Simulations indicate that this mechanism of relativistically, but they canand no longer reconnect, be chopping regarded off as loops stationary. and The finite strings BH-string will now nets cross of size At this time the network reaches equipartition, However, the equipartition betweencreasingly strings large and Lorentz BHs factors.large would degree Accelerating require of to the coherence ultra-relativisticgenerally BHs in velocities take move the place. would with direction require Instead, in- of at to a the strings. tension force Note acting that on the a evolution BH, of and the does not infinite network at on the Hubble timescale are chopped off atsection random, we regardless shall of argue whether thatso or chopping our off not a estimates the loop can loop with be contains a a regarded BH BH. as may3.2 In be lower the bounds considerably more next on difficult, the Black number hole ofLoops detachment BHs are in chopped the offopposite network. directions. the long stringssimplicity that by there collision is of only one wiggles other traveling string along attached to the that string BH. in We shall assume also that becomes The BH velocity at that time is where we have used eq. (3.6) for network is initial BH density (which is characterized by the parameter connecting BHs become nearlyexpansion. straight, and The at BH later separation times in they this are regime simply is stretched by the JCAP11(2018)008 6 1 1. t . 0 6, so  , then ∼ t t ∼ 10 √ is its subgroup. 10 strings within H ∼ N along the strings. Now , which decreases much t /N ∼ 1 and some later time − t 1 t can be expected to disconnect ∝ 1 ) t t ( f 6. This corresponds to a solid angle (1) monopoles. With is a semisimple group and O π/ G ∼ – 7 – , where 2 Z × ). This has the solution t H ( f 1 6 strings attached to it. These strings have the same direction of ) is the fraction of BHs remaining in the network at time → t − ( assumed in the minimal scenario. e ∼ f 2 / ∼ 1 U(1) . We can expect that the strings will be cleared of monopoles in this range, so the ) − t . If t t × 1 1 . t N 0 4 ∝ e H ( ∼ ) ). Hence, a BH attached to two strings at 10 t f → ( 1 f ∼ G t/t 1 t ln( N e ∼ A loop that includes the BH can only be formed if the two strings emanating from the Simulations indicate that strings in the network consist of nearly straight segments This is of course only a rough estimate, but it does suggestWe that note BH also survival that in strings of the a special kind, called necklaces, are more likely to remain ∼ “Dangerous” string crossings that can result in BH detachment are likely to occur close to the BH, e.g., t 6 N string orientation will not reverse. within a distance there should be at least the magnetic flux, so their reconnections do not change the number of the attached strings. hole cross one another.a It stationary can be BH shown is thatstring simply is a reflected pinned wave traveling by at along a theangle a fixed BH from straight position). (approximating string one the So towards another, if BHa waves the collision as traveling two along becomes a strings the come very point out strings likely where of if are the the the not BH angle likely atseparated to between a by the collide. substantial kinks two However, [32]. stringsare is The rather sufficiently kinks mild, small. travel along sothere the the are strings string also at some direction the large varies speed kinks, very of which little are light. from separated Most one by kinks distances segment to another. But the BH is sothe massive strings. that it This is is not a moved reasonable much assumption (in at a early Hubble stages time) of by the the evolution. forces exerted by consider a stringdirection attached of to the a string does BH.change not of As change much kinks direction —reconnection travel will until a of along occur large two the kink strings about arrives. becomes string once likely Such towards is sudden, in the large a BH, Hubble the time. Suppose the angle at which Then we expect the strings toNow, be the in number a “dangerous” of configuration Hubble about every times 10 between Hubble some times. initial time slower than Then monopoles carrying theand magnetic their flux flux of is U(1)gets squeezed are attached into formed to strings in two at strings. theby the the first (The second string.) phase magnetic phase transition, The charge transition,generic monopoles of so33]. [ the will that then monopole The each be evolution is monopole that of like twice relativistic necklaces beads the string has on flux motion been the carried causes studiedto strings. the in the monopoles These refs. speed defects to [34, of move are35 ], light.efficient, along quite with the As and the strings a conclusion the at result, speeds monopole-antimonopoleMonopoles separation close annihilation therefore between on have the them little stringformed, on is effect it rather a on captures long the some stringblack string monopoles hole dynamics and is size and antimonopoles at comparable evolution. together formationto to and with the When the horizon, the the a each monopole strings. horizon. string BH separation is Since on is likely to the the contribute strings are both comparable network may be morereliable determination likely of than the theof fraction naive of the surviving string-BH analysis BHs network. in would the require a preceding numerical section simulation attached indicates. to BHs A thantransitions “ordinary” strings. Consider a sequence of symmetry breaking phase is at we must have the Hubble volume, we can expect the total magnetic charge of the BH to be 2 JCAP11(2018)008 is t (3.13) (3.19) (3.20) (3.12) (3.14) (3.15) (3.16) (3.18) (3.17) , when ∗ t ∼ , but at later t d t at ∼ t M ∼ the strings are stretched by the . d t h . . 4 . f . ). It becomes Gµ. f 3 − t d t 2 t 3 , / 1 ( / v 3 / 1 h , f 1 t / 2 − v . t 2 ) 2 ) ∼ t > t − µd t / ) − 3 t f λ ∼ f ) ) ( t t Gµ λ ∼ t Gµ 1 ( d n 2 3 ( ∼ f 3 t 2 / 3 t / ) − Gµ is t 2 h / ( 1 ) ( t 2 λ 9 d − ) – 8 – / ∼ − ∼ λ 2 Gµ ∼ λ ) µ/M ) ( − t is ( t 3 ∼ ( λ ( ∼ / r l µ/M t > t ) h net 2 n ρ ( ∼ t t ∼ ρ λ ( d d d ∼ dom v t ∼ t h ∗ v t . h ) at t , the average length of strings can be found from eqs. (3.1),3.5): ( t t ( ∼ d  ) and t ∼ ) ( t 2 ( − is the radiation density. The network dominates the energy density of d 1 µd − ) the BH separation becomes comparable to the horizon. The average string 1. Assuming this is the case, at ∼ 2 f t Gt 3 net  ( / ρ 2 . It becomes h 2 − ∼ is v t λ ) , r ∗ ρ ∼ M There is a range of possible evolution scenarios between the minimal and maximal mod- The typical BH velocity at time As long as BHs move non-relativistically, their displacement during a Hubble time In this scenario, the BH number density in the network is h t µ/M ∼ t > t ( times multiple string crossings in theduction of vibrating closed BH-string loops network could and result smallin in nets. the minimal copious Then scenario. pro- the Numerical subsequent evolution simulations will could be be necessary similar to to explore that these possibilities. els. For example, the network may evolve as in the maximal scenario until At early times, when where the universe at length at that time is also so the average BH separation is At For most of the astrophysicallywe interesting have parameter values thatexpansion we and mentioned become in nearly section straight. II ∼ The BH velocity at that time is At this point BHs startat moving relativistically. The energy density of the BH-string network 3.3 Persistent networks:Considering the the maximal possibility scenario BHs that are the detached from minimal the network,that scenario we a shall may substantial now overestimate discuss fraction the thethe of opposite maximal ease BHs limiting scenario. case, remain at assuming attached which to the network. We shall refer to this as The average kinetic energyµd of BHs at JCAP11(2018)008 , assuming that l 10 loops of length ) is from eq. (3.9). t ∼ = 2 ( , with about one loop d d T t ∼ t . In a BH-string network, the 3 2 are connected by 2 strings. The ), where t t − ( 3 d ∼ ), so a string segment attached to a BH t ( d – 9 – in a Hubble volume t would then oscillate with a period l We did find some examples of pinned loops that did not self-intersect, 7 the BH size is much smaller than d ) formed per Hubble time in a volume t ( d . A stable binary: two black holes with charges 2 and . In the minimal model the loop production resumes at t > t h ∼ are produced per Hubble time t t At PBHs with attached string loops can be of astrophysical interest. For example, if the Another interesting object is a small net consisting of a few BHs connected by strings. Another example of a simple net is a triangular configuration with three BHs connected 1 ∼ . This is in contrast to string loops without BHs, where simple families of loop solutions are non-intersecting t 7 0 can be approximately described asA a segment segment with of both invariant ends length pinned at a fixed point in space. PBHs were captured intothe halos Milky of Way galaxies and theytectable. therefore would make An drag exploration the the of gravitational loops theto waves attached dynamics BHs produced and to will by gravitational-wave them signatures be the of into published loops loops elsewhere. attached more de- The simplest netfigure1. is a In binary anbreak system up oscillating if consisting binary the of stringstwo the two have strings strings opposite should BHs flux be will connected directions.opposite opposite, frequently by For magnetic but “ordinary” intersect two for charges. strings, and necklaces strings, the In the the fluxes the see fluxes in latter binary can the case will be theby the binary three same is if “ordinary” long-lived. the strings;straight, BHs see assuming have figure2. that If theother the BH on strings a motion timescale are is longer nearly than non-relativistic the straight, and light-crossing they time that will for the the remain triangle’s BHs side. accelerate The each strings will Apart from closed stringoscillating loops, string segments the attached networks to will BHs yield and other small gravitational BH-string wave nets. sources — it does not self-intersect.number of We, such together pinned withof segment Elena them configurations self-intersect. Murchikova, and have found, studiedbut somewhat numerically it surprisingly, that a is most farafter from several clear self-intersections whetherintersecting these and configuration. are pinching This generic. off would It endow the is a “looplets”, PBH possible with will that an a evolve oscillating generic into string pinned a “hair”. loop, non-self- of size in a substantial portion of the parameter space. loop production is initially theat same as in the standard scenario, but then it ceases completely connections are indestructible by thegravitational internal waves. dynamics of the binary. The binary shrinks by emitting 4 Black holes with loops,The and spectrum nets of stochasticdetermined gravitational by radiation the produced by sizerather oscillating different distribution loops from of that of in loops.∼ string the is standard In cosmic the string presence scenario, where of PBHs this distribution is Figure 1 JCAP11(2018)008 2 are connected by 2 pairs − 2 and − – 10 – 2 by two pairs of strings; see figure3. This configuration is topologically − . A stable triangle: three black holes with charges 4, . An oscillating triangle: three non-relativistically moving black holes connected by ordinary , but gravitational waves are still generated by the vibrating BH-string network. The A triplet net can also be formed for necklace-type strings, with three BHs having mag- In the maximal scenario, the rate of loop formation remains strongly suppressed even at d Figure 3 netic charges 2,holes 2 of and charge -4. In this triple the BH with charge 4t is > connected t to two black strings. If thechaotic oscillatory sides movement. remain Itsides perfectly also support shrinks straight, by large-amplitude emitting they waves, the gravitationallead never strings waves. to self-intersect from However, the different if and triangle sides the the destruction. will stringy likely triangle . reconnect undergoes which would strings. The connections areof topologically the protected triple. and The are triple indestructible shrinks by by the emitting internal gravitational dynamics waves. then never self-intersect.off However, BHs if [34], waves the on strings strings may are reconnect amplified and by the multiple triangle reflections may be destroyed. protected, in a sense thateven the if internal the dynamics tension of the waves are triple will excited not on change the its strings configuration stretched between the black holes. Figure 2 JCAP11(2018)008 R .A (5.1) (5.5) (5.2) (5.4) (5.3) µ E 1 − ) . GM sec ( 2 in the minimal scenario. 2 ) − λ .  from the cluster is . Gµ  (  , this implies that the BHs are 10 r R − 10 sec Gµ × − H Gµ 10 ∼ − t 17 Gµ ) 10 10 for GM Schwarzschild radii. This condition is   ∼  connected by strings with tension

2 d GM 5 r t ( M  M / M Gµ GµR/r. ) = 10  ∼ − µR Gµ 15 µ ( 10 2 RM – 11 – 10 ) ¨  Q/r r . The characteristic angular frequency of oscillations ∼ Λ G Gµ 2

( ∼ ) ∼ µR M ∼ h ∼ Gµ ( v/R 2 100 /G 2 ∼ ∼ ω Mv ω 2 GM/ r M ∼ 2 ∼ h ¨ Q . This means that the string tension is stronger than gravitational GW 2 ∼ − ) τ and and later times. 2 d t . This means that BHs have more mass than strings and they move non- MR dE/dt GM/R ( µR ∼  Q  8 Gµ interaction between BHs. Numerically, for M relativistically. This condition is satisfied at further away from each other than about 10 satisfied at Obviously, the expression above is a rough estimate and the precise answer depends Let us estimate the power radiated in gravitational waves. The quadrupole moment of A similar estimate was obtained in a different context in ref. [36]. Gravitational radiation from relativis- 2. 1. 8 tically moving masses connected by strings was studied in refs. [37, 38]. The characteristic strain of a gravitational wave at distance on the initial geometrybecomes of greater than the the system. stringto tension, chaotic Eventually either motion the due that to gravitational wouldgravitational the force randomly interaction overall bring between shrinking dominates, two of the the blacka the BHs shrinking holes net, series to accelerates or of close due and mergers. separation.of one Remarkably, certain When obtains the BH mass a timescale scale in merger, are eq. or merging (5.4) at seemingly the implies present that epoch: only BHs One can find the energy loss rate from the emission of gravitational waves: gravitational wave spectrum differsonly substantially consider between some different aspects scenarios. of Here gravitational we radiation shall by small5 string-BH nets. Gravitational radiation fromLet’s nets consider a smallspecial net case of is black a holes tripletnon-relativistic, of of the black mass strings holes, segments connectedintersect remain by each straight, straight and other, strings. since the Sobe the long triangle the sides as can characteristic of the size remain motion a of stable is triangle the never for net. many We dynamical shall timescales. work in Let the following approximations: Therefore, the energytimescale of the net and its size decay approximately exponentially, on the the net is is JCAP11(2018)008 (6.2) (6.1) (6.3) (6.4) , cm Hz (5.6) 2 4 / / 1 3  

10 . . s M − 0 f M / t Gµ t at which the force of gravitational 10  . 0 3 f

Gµ / 1 3 M   / M − 12 2 3

λ 100 km λ / M 2 2 22 M . − − 12  2 10 λ / 10 1 12 ∼ 01 ) . − ∼ – 12 – 0 eq 2 0 2 0 10 z /M t d ∼ 2 0 / 4 1 / ) µd Gµ 3 ( ) eq Gµ < t ∼ ∼ f s Gµ 0 t 0 ( ( ρ ρ v 3 / 3 1 c − GM λ 1, we obtain the constraint ∼ ∼ 0 < 0 f . The energy density of strings relative to the background is d 12 − 10 λ/ ≡ is the Hubble time and Λ is the number of e-foldings by which the net needs to 12 H t λ For BHs to play a role in structure formation, we require that their string-induced A number of questions regarding small BH-string nets look interesting but are left Requiring that this is velocities are sufficiently small: where unanswered: does thetheir chaotic nature GW of spectrum? netresulting in dynamics GW In significantly bursts affect particular,torques or/and their BH due how lifetime mergers? to common and What theof are is string string the close tension? attached typical to BH encounters Howescape spin BHs? of common through resulting Close are from BHs the to the non-intersecting BHlength in the trajectories over horizon. points a cosmological for of timescales? net, How loops loop important attachment, is some this string mechanism length in may 6 depleting the loop’s Constraints on persistentSuppose networks now that BHs remain connectedat to present strings is until the then present time. The BH separation where shrink before theselected BHs by physically the merge. agescale of In is the other within Universe words, the andspace-based mass a detectors. the range certain string This detectable argument, mass tension,of however, by and scale may the the be for is net current too realistic may naturally ground-based simplistic: produce parametersand detectors the bursts this this and chaotic of may dynamics future gravitational cause wavesthe a when net the substantial possesses BHs variation a pass inattraction characteristic close the between oscillation to the merger frequency each BHs timescale. other, is It comparable is to worth string noting tension: that For realistic values ofLISA the or BH Pulsar Timing mass Arraybackground and bands. that string is The produced tension, spectral by shape an thisat ensemble of this frequency of the frequency. shrinking could stochastic nets gravitational-wave well may have lie an observable within feature JCAP11(2018)008 . 16 − (6.5) ] 10 . ]. Gµ SPIRE IN ][ arXiv:1801.05235 is the current age of the H t , the density and velocity con- 15 . − 2 / 10 arXiv:0912.5297 1 , where (2018) 063001[  2 ∼ Primordial black holes — perspectives in New cosmological constraints on −

λ ) 35 M M  and /Gµ 3 / , the velocity constraint requires 10

1 12 , respectively. Note that even if PBHs are cap-

− λ M 26 – 13 – M (2010) 104019[ 19 − 13 6 − )(10 − 10 10

10 10 . M ∼ D 81 ∼ M 100 Gµ Class. Quant. Grav. M Gµ < , M/ ( and H t 23 Phys. Rev. , − ∼ 10 . GW τ Gµ ]. SPIRE IN For supermassive BHs with One of the main observational signatures of cosmic strings is the gravitational wave For PBH dark matter with primordial black holes gravitational wave astronomy [ [1] B.J. Carr, K. Kohri, Y. Sendouda and J.[2] Yokoyama, M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, This gives the constraint Universe. If such objectsthis would exist give and most are valuableinvestigate observational insights detected signatures for by of particle LIGO, the physics model,to Pulsar and the be Timing astrophysics properties explored Arrays, alike. of in nets or To greater with further LISA, black detail. holes need Acknowledgments We are grateful tosions Ken Olum, and Jose comments Blanco-Pillado, onScience and the Foundation. Elena manuscript. Murchikova for The useful work discus- of A.V.References was supported by the National (GW) background produced byproduction oscillating by the string network loops. isaddition, significantly modified, the In resulting network the in provides a presence anotherand modified source of strings. GW of PBHs, spectrum. GW The the In —on nets loop small a lose oscillating their timescale nets energy of of black by holes gravitational radiation and shrink exponentially tured in galaxies, they mayPBHs avoid capture attached in to much strings smaller halos, effectivelythe due act absence to as of the warm pull overly dark of dense the matter; cores strings. they at may galactic also centers help7 (predicted to by explain the standard Conclusions ΛCDM). In this workboth we present. introduced a Thean new interaction infinite between class string the of network twostrings models with is cross BHs in dramatic and at and whichhave reconnect, the results cosmic discussed and nodes. in strings two some the limiting In andBHs fraction formation scenarios, the that PBHs of of providing remain course are in BHs upper of therequire is the and infinite numerical following lower detached network. simulations. bounds evolution, from A more on the definitive the network. analysis density of network of We evolution will straints require If this is satisfied, BHsthey would could be be captured observed by as galaxies.possibility lines If of that the radio the strings emission are radio attachedto to superconducting filament a BHs [39], observed [40]. cosmic near string We note the [41]. the central intriguing BH in our Galaxy may be due JCAP11(2018)008 ] A (2017) D 94 121 ]. Phys. (2018) Astron. , 12 , (1993) 3265 SPIRE Class. Quant. D 97 (2016) 064 (1987) 1131 IN , JCAP Phys. Rev. 02 , ]. , Cambridge , D 47 arXiv:1706.03746 (1985) 2398 Erratum ibid. D 35 ]. 55 SPIRE JCAP Phys. Rev. Proc. Roy. Soc. Lond. , IN , (1989) 237[ , ][ Primordial black holes as all Phys. Rev. SPIRE Mon. Not. Roy. Astron. Soc. ]. , , Phys. Rev. IN , ][ B 231 Double inflation as a single origin (2017) 063507[ (2016) 061101[ ]. SPIRE ]. IN 117 Phys. Rev. Lett. ][ ]. Primordial black hole scenario for the D 96 , ]. SPIRE SPIRE IN Primordial structure of massive black hole IN Phys. Lett. ][ astro-ph/9605094 , ][ SPIRE Density perturbations and black hole formation in SPIRE ]. IN ]. ]. IN ][ – 14 – Phys. Rev. astro-ph/0401532 (1967) 602. Primordial black holes from inflaton and spectator field , SPIRE SPIRE Black holes and the multiverse SPIRE Primordial black holes as dark matter Primordial black hole and wormhole formation by domain ]. IN 10 Phys. Rev. Lett. IN IN Beads on strings Cosmic strings and superstrings arXiv:1001.2308 ][ The hypothesis of cores retarded during expansion and the ][ ][ (1996) 6040[ ]. Cosmic strings and other topological defects ]. Formation of primordial black holes by cosmic strings Evolution of cosmic networks SPIRE Black holes in the early universe Spherical domain wall collapse in a dust universe ]. IN (2005) 265[ 150914, arXiv:1612.03753 Black holes from nucleating strings D 54 astro-ph/9509027 SPIRE ]. ][ Primordial black hole formation by vacuum bubbles SPIRE IN 23 arXiv:1411.7479 ]. IN ][ Sov. Astron. SPIRE (2010) 023[ , IN SPIRE 04 ][ IN SPIRE ][ Phys. Rev. (2017) 050[ arXiv:0911.1345 IN arXiv:1603.08338 (1997) 673 [ Black holes from cosmic strings , arXiv:1607.06077 Formation of MACHO primordial black holes in inflationary cosmology 04 JCAP , (1991) 1106[ 318 (2015) 155003[ Astropart. Phys. ]. ]. ]. arXiv:1711.06129 , JCAP 32 D 43 , arXiv:1710.02865 (2010) 623[ (1974) 399 [ SPIRE SPIRE SPIRE IN IN IN hep-ph/9208212 arXiv:1512.01819 466 [ [ hot cosmological model 168 hybrid inflation Astrophys. dark matter [ perturbations in a matter-dominated era University Press, Cambridge, U.K., (2000) [ (2016) 083504[ of primordial black holes043514[ for all dark matter and LIGO observations gravitational-wave event GW [ Rev. clusters walls Grav. (2018) 059901][ [ 044[ [5] T. Vachaspati and A. Vilenkin, [9] J. Garc´ıa-Bellido,A.D. Linde and D. Wands, [6] M. Hindmarsh and T.W.B. Kibble, [7] Ya.B. Zeldovich and I.D. Novikov, [8] B.J. Carr and S.W. Hawking, [4] E.J. Copeland and T.W.B. Kibble, [3] A. Vilenkin and E.P.S. Shellard, [10] J. Yokoyama, [11] P.H. Frampton, M. Kawasaki, F. Takahashi and T.T. Yanagida, [23] B. Carr, T. Tenkanen and V. Vaskonen, [12] B. Carr, F. Kuhnel and M. Sandstad, [13] K. Inomata, M. Kawasaki, K. Mukaida and T.T. Yanagida, [14] M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, [18] M. Yu. Khlopov, S.G. Rubin and A.S. Sakharov, [21] S.W. Hawking, [22] A. Polnarev and R. Zembowicz, [19] N. Tanahashi and C.-M. Yoo, [20] H. Deng, J. Garriga and A. Vilenkin, [17] J. Garriga and A. Vilenkin, [16] H. Deng and A. Vilenkin, [15] J. Garriga, A. Vilenkin and J. Zhang, JCAP11(2018)008 , , ]. ]. ] Phys. ]. , 05 D 92 SPIRE SPIRE IN B 595 IN ]. SPIRE ][ ][ JCAP Int. J. Mod. IN Phys. Rev. Lett. , , (1989) 2776 ][ , SPIRE ]. 63 Phys. Rev. IN ]. arXiv:1303.5085 (2015) 094002 , ][ Nucl. Phys. , SPIRE IN SPIRE ]. G 42 IN ][ astro-ph/9904315 arXiv:1709.02434 Superconducting cosmic strings (2014) A25[ arXiv:1101.5173 Phys. Rev. Lett. J. Phys. ]. , , 571 (1985) 557[ results. XXV. Searches for cosmic arXiv:0903.4686 ]. SPIRE (2018) 392[ ]. 2013 New limits on cosmic strings from IN B 249 arXiv:1711.04190 Large parallel cosmic string simulations: new Cosmic string loop shapes How generic is cosmic string formation in (1999) 083001[ hep-ph/0308134 ][ SPIRE ]. (2011)[ 083514 SPIRE IN B 778 Planck IN A nonthermal radio filament connected to the galactic – 15 – ][ ]. D 60 Dynamics of cosmic necklaces ]. ][ Gravitational waves from broken cosmic strings: the (2009) 123519[ D 83 SPIRE ]. Astron. Astrophys. Inflation, string theory and cosmic strings Monopole annihilation in cosmic necklaces Monopole-anti-monopole bound states as a source of IN (2017) L23[ Nucl. Phys. , Monopoles on strings SPIRE , ][ SPIRE IN Cosmic string evolution D 79 IN (2003) 103514[ 850 Phys. Lett. SPIRE ][ , ]. ][ Cosmic string evolution: a numerical simulation Phys. Rev. IN Stochastic gravitational waves from cosmic string loops in scaling arXiv:1412.0579 , Gravitational radiation from monopoles connected by strings Phys. Rev. , D 68 gr-qc/9612008 SPIRE ]. arXiv:1709.03845 IN Phys. Rev. , ][ (1986) 944[ SPIRE arXiv:1508.02693 IN Phys. Rev. , (2015) 1530010[ Astrophys. J. Lett. astro-ph/0005411 arXiv:0707.3460 (1997) 6054[ D 34 Superconducting strings , (2017) 027[ collaboration, P.A.R. Ade et al., ]. ]. 12 D 24 D 55 (1990) 119[ SPIRE SPIRE IN IN arXiv:1506.02022 [ 64 results on loop production Phys. Rev. black hole? gravitational wave observation bursts and the beads JCAP Planck (2001) 402[ Rev. strings and other topological defects [ [ (2010) 014[ ultrahigh-energy cosmic rays SUSY GUTs Phys. (2015) 063528[ [24] D.P. Bennett and F.R. Bouchet, [25] B. Allen and E.P.S. Shellard, [26] J.J. Blanco-Pillado, K.D. Olum and B. Shlaer, [27] J.J. Blanco-Pillado, K.D. Olum and X. Siemens, [28] C. Ringeval and T. Suyama, [38] L. Leblond, B. Shlaer and X. Siemens, [39] E. Witten, [40] E.M. Chudnovsky, G.B. Field, D.N. Spergel and A. Vilenkin, [41] M.R. Morris, J.-H. Zhao and W.M. Goss, [29] [35] J.J. Blanco-Pillado and K.D. Olum, [30] R. Jeannerot, J. Rocher and M. Sakellariadou, [33] T.W.B. Kibble and T. Vachaspati, [34] X. Siemens, X. Martin and K.D. Olum, [37] X. Martin and A. Vilenkin, [36] J.J. Blanco-Pillado and K.D. Olum, [31] D.F. Chernoff and S.-H. Henry Tye, [32] J.J. Blanco-Pillado, K.D. Olum and B. Shlaer,