VOLUME 76, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 1996

elkstherg

erue

Absence of Open Strings in a Lattice-Free Simulation of Cosmic String Formation

Julian BorrillBlackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom

and Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755*(Received 10 November 1995; revised manuscript received 22 February 1996)

Lattice-based string formation algorithms can, at least in principle, be reduced to the study of thstatistics of the corresponding aperiodic random walk. Since in three or more dimensions such waare transient, this approach necessarily generates a population of open strings. To investigate wheopen strings are an artifact of the lattice we develop an alternative lattice-free simulation of strinformation. Replacing the lattice with a graph generated by a minimal dynamical model of a first-ordphase transition we obtain results consistent with the hypothesis that the energy density in string is dto a scale-invariant Brownian distribution of closed loops alone. [S0031-9007(96)00118-4]

PACS numbers: 98.80.Cq, 11.27.+d

i

ea

et

ho

g

s

e

dto

edtedldofce,uchtticekenofenich

m

tshis5],alyld

theseareeactinoric

cesn

isg,tof

s

Cosmic strings are the longest standing candidatea source of primordial cosmological perturbations arisifrom particle physics beyond the standard model. Fdescribed by Kibble almost two decades ago [1] thhave yet to be demonstrated to be incompatible wany of the subsequent cosmological observations, frthe anisotropies in the cosmic microwave backgroundthe distribution of luminous matter on the largest scalThe field theoretic requirements of the basic modelsimple: the spontaneous breaking of some symmeof our theory yielding a degenerate vacuum manifowhose first homotopy group is nontrivial. The single frparameter in the theory is then the energy scale oftypically grand unified theory, symmetry breaking. Thcomplexity of the ensuing cosmology (in part responsibfor the model’s longevity) lies in the need to track thnonlinear evolution of the string-bearing field through tsymmetry breaking phase transition and on to the epof any observable consequences.

The starting point for any analysis of string cosmolois in the statistics of the strings’ initial distributionUnless terminating at a pair of monopoles or spatsingularities, all strings must be either closed loopsopen (and hence infinite). Here we are interested ininitial distribution of strings both between and within eacof these populations. We can trivially define a fractio0 # fc # 1 of the string energy density to be in closeloops, leavingfo ­ 1 2 fc in open string; this is thequantity of most interest in this Letter. Further, it is eato show, under the assumption that the string loopsBrownian and scale invariant, that the number densityloops with lengths in the rangefl, l 1 dld must go as [2]

dn ~ l25y2 dl . (1)

Numerical simulations of string formation assigphases taken from theS1 manifold of vacuum states tothe vertices of some three-dimensional lattice. (Nothat by “lattice” we mean a regular array of connect

0031-9007y96y76(18)y3255(4)$10.00

forngrsteyithomtos.retrylde

he,eleeech

y.ialorthehnd

yareof

n

ted

vertices, while the oxymoron “random lattice” is termea graph.) Along the lattice edges the field is takenfollow the shortest path on the manifold—the so-callgeodesic rule. Each face of the lattice is then inspecto determine whether, in traversing its boundary, the fiecovers the entire vacuum manifold. If so then a zerothe field is necessarily present somewhere within the faand a string segment is located passing though it. Sa segment connects the centers of the adjacent lavolume elements associated with the face, and tatogether all the segments join to form a populationcomplete connected strings. If the lattice is periodic thall the strings are closed loops; otherwise strings whintersect the lattice boundaries simply terminate there.

The original simulation [2] took phases at random frothe minimal three-point discretization ofS1 on a cubiclattice, resolving the ambiguity when four string segmenmet in a single cube by chance. Later refinements to tapproach included the use of a tetrahedral lattice [3–a continuous vacuum manifold [3,6], and a dynamicallocation of the field phases [7,8]. However, the kecommon feature of all these simulations is that the fieis set at the vertices of a lattice, and hence thatstrings form along the edges of its dual. In every casuch simulations do indeed generate strings whichBrownian and scale invariant, with the majority of thstring energy density being in open strings. The exfraction depends on the details of the simulation, andparticular on the geometry of the lattice and its dual. Fexample, the simple cubic lattice with its simple cubdual givesfo , 0.8 [2], while the tetrahedral lattice withtetrakaidekahedral dual givesfo , 0.63 [4,5].

Since such simulations construct strings as sequenof vertices on a lattice, it is not surprising that they caalso be represented as random walks. Since therean excluded volume around the path of a single stringiving a directional bias away from its origin, we mighnaively expect the ensemble to have the statisticsa self-avoiding random walk. However, if the string

© 1996 The American Physical Society 3255

VOLUME 76, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 1996

oid

ig

l

t

ed

ha

hdoaus

n

r

n

nTl

ow

ee

nsheis-s,

asithnu-h to

dallald

-ts.hen

ingofthe

icseyoptthegylyn.on

it,y,fultheee

n6a

re

erert

tersheer

are dense the associated ensemble of excluded voluremoves this bias, and the statistics turn out to be thof a Brownian random walk instead. Moreover, itknown that in three or more dimensions an aperiorandom walk on a lattice cannot be recurrent [9], andhas a nonzero probability of not returning to the origgiving a nonzero fraction of open string. Exploitinthis representation, analyses of the statistics of Brownrandom walks on the appropriate lattices give simiresults, withfo $ 0.6 [3,10].

Since open strings appear to be an inevitable conquence of the lattice-based formation algorithms, it is dsirable to consider formulations which do not involvelattice, and use them to test the alternative hypothesisthe string energy density may be accounted for solelyterms of a scale-invariant Brownian population of closloops. The results of one such algorithm are presentethis Letter.

Algorithm.—The use of a lattice in string formationsimulations allows us to know in advance where tstrings may be found, and hence to bypass the dynics of the phase transition. Without a lattice we mutherefore include the dynamics explicitly, and for comptational tractability we adopt the simplest possible dynaical model of a first-order phase transition. Sphericasymmetric bubbles of the true vacuum (with phases csen at random from the minimally discretized manifolare nucleated at random space-time events in a perifalse vacuum background. They are then taken to expuniformly at the speed of light for the duration of the simlation, colliding with one another, trapping regions of falvacuum, and generating strings.

The exterior (i.e., outside of any other bubble, aso in the false vacuum) intersection of the surfacesany three bubbles of differing phases generates a stsegment with open ends. As the bubbles continueexpand this segment traces the locus of intersectiontheir surfaces, lengthening, and, except in the case whthe three bubbles are the same size, bending (cf. lattbased simulations where the bubbles are always the ssize, and the string segments are necessarily straigAs the bubbles continue to expand they eventually ma fourth bubble. One and only one of the collisiobetween this bubble and each pair of the original tripalso generates a string segment. The final four-bubintersection event then sees the joining of one of the eof each segment to form a new extended segment.exterior intersection of the surfaces of any four bubbcan therefore also be viewed as the meeting of four thrbubble intersection loci, and hence of the joining of eithzero or two string segments.

The key to this algorithm is the location of the pointsintersection of all bubble triplets and quadruplets. Hoever, since the bubbles are spheres of known centerspredictably time-dependent radii these points of interstion can be determined analytically. Having calculat

3256

messe

sicson,

ianar

se-e-ahatindin

em-stu-m-llyo-)dicnd-e

dofingtoof

ereice-ameht).

eets

letbledshe

esee-er

f-

andc-d

the location of all the three- and four-bubble intersectioin the simulation space-time we can then construct tcorresponding string distribution and determine its stattics. Taking the four-bubble intersection points as nodeand their four constituent three-bubble intersection lociconnections, it is clear that we are generating a graph wfourfold connectivity. Note also that the precise locatioof the strings on this graph is determined by the particlar phase realization, enabling us to use the same grapgenerate many different string sets.

This algorithm is the natural extension of that employein two dimensions to assess the effect of bubble wspeed on vortex formation [11]. Its three-dimensionrealization, including the formation of both strings anmonopoles, is described in full detail elsewhere [12].

Implementation.—This algorithm is clearly much morecomplex (and computationally intensive) than its latticebased counterparts; indeed the limiting factor in iapplication will time and again prove to be its CPU time

In a finite simulation we can never hope to represent tlongest strings—for a string bearing field with correlatiolength j in a box of side L any string longer thanOsL2yjd is likely to cross the boundary. If we leavethe boundaries open, we can estimate the open strenergy density by seeing whether the number densityboundary crossings reaches a nonzero asymptote asbox size increases [2]. If instead we impose periodboundaries, forcing all strings to be loops, we can uthe theoretical scarcity of long closed loops to identifany excess as representing the open strings. We adthe second approach for a number of reasons. Sincehypothesis under consideration is that the string enerdensity is due to loops alone, we are looking for exactthe excess that might occur in a periodic simulatioMoreover, the open boundary approach requires runsa wide range of box sizes to test for an asymptotic limand hence falls foul of our CPU-time constraints. Finallthe requirement that all strings be closed provides a usecheck that the phase transition has completed and thatbubbles have indeed filled the entire simulation volumby the end of the run, as well as an overall test of throbustness of the code.

Since we wish to calculate many-bubble intersectiopoints, periodicity is implemented by incorporating 2copies of the simulation volume to surround it. Forsimulation of box sizeL and durationT we then consideronly those bubbles withinT of the central volume, andonly those intersections that occur inside it. We therefohave to balance the wish to maximize the ratioLyT so asto minimize the periodic copies, and the increased numbof bubbles necessary to fill a larger volume in a shorttime. Again the final constraint is CPU time. Note thathe simulation size and time are dimensionless parameand that the physical scale of the system is set by tbubble nucleation rate, and hence by varying the numbof bubbles nucleated.

VOLUME 76, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 1996

s

i

oa

o

i

i

t

igh-n toical

yndheea

ictan

gh-asall

ns

he

le-

alanlypsalestin

bye

findgs.rkheyot.at

We determine the graphs associated with five simutions each starting with 5000 bubbles in a periodic boxsize53 with a simulation run time of 1 in units of the boxlength, each graph requiring approximately 250 h of CPtime on a twin-processor Sun SPARC 10. We then genate the associated string populations from 1000 differerandom phase realizations in each case, giving someNs ,300 000 strings. Figure 1 shows the normalized numbdensityNsld of all these strings binned by length againthe lengthl.

We can immediately see that there are three phapresent: (i) A low-end tail population of loops withl , lo,with lo , e0.5 here. Clearly the loop distribution cannofollow Eq. (1) down to arbitrary short lengths, since thwould generate an infinite energy density. This populatihas never been explicitly identified before, since latticbased simulations automatically impose a low-end cutof the order of a few lattice spacings; it is worth noting thin this simulation the range of loop lengths covers mothan 6 orders of magnitude. (ii) A population of scaleinvariant Brownian loops with number density falling al25y2. (iii) A high-end tail population of loops withl1 , l , lmax, with l1 , e3.5 here (note thate3.5 , 52,as expected). The truncation of the longest strings (blong finite closed loops and infinite open strings) by thperiodic boundaries generates an excess of strings lonthanl1, so that in this region the loop number density falmore slowly, asl2a with a , 25y2.

The final feature to note is that since we have a fintotal number of loopsNs in our simulation there is aminimum measurable nonzero normalized bin numbdensity of1yNs, clearly identifiable as the extreme highend plateau in Fig. 1. As the string number densapproaches this the relative error in the measurements

FIG. 1. A log-log plot of the normalized number densityNsldof strings binned by length against their lengthl. The solidline is the simulation data, the dashed line is the best fit toBrownian regimeNsld ­ 0.67l22.5, and the dotted line is thebest fit to the periodicity-distorted regimeNsld ­ 0.082l21.9.

la-of

Uer-nt

ert

ses

tsone-fft

re-s

theger

ls

te

er-tycan

he

be seen to increase dramatically. These extreme hend points then correspond to those bins which happecontain one or more strings, despite having a theoretdnsld , 1yNs.

This is in many ways a familiar picture, quantitativelgiving the predicted behavior of the midrange loops aat least qualitatively the deviations at either extreme. Tfirst indication of significant differences comes with thvalue of a. Analytic estimates for random walks onperiodic cubic lattice givea ­ 1 [10], whereas we finda , 1.9. The fact that our long loop distribution fallsoff much faster than a random walk model would predindicates that the amount of long string is much less thin such models.

The periodicity of our simulation forces any open strinback into the box, where it would appear in the higend tail. We can now estimate its energy densitythe difference between the observed energy density instrings longer thanl1, Eobssl $ l1d, and the theoreticalenergy density in loops longer thanl1 alone,

Ethl sl $ l1d ­ k

Z `

l1

l23y2dl

­ 2 kl21y21 . (2)

To determine the normalizationk we can use the con-vergence of the observed and theoretical loop distributioin the regionlo # l # l1, giving

k ­Eobsslo # l # l1d

2sl21y2o 2 l

21y21 d

. (3)

The remarkable result when we do so for each of tfive runs is that

fo ­Eobssl $ l1d 2 Eth

l sl $ l1dEobssl $ 0d

­ 0.0062 (4)

(with a variance of 0.0050) compared to the0.66 0.8 wemight expect. We can account forall the energy densityin the strings as being due to a single population of scainvariant Brownian loops.

In conclusion, we have used a minimal dynamicmodel of a first-order phase transition to generateinitial string distribution whose statistics are significantdifferent from the standard lore. We find that the looshort enough to be unaffected by the box size are scinvariant and Brownian (except at the very shortelengths) as before, but that now the energy densitythe longer loops can also be accounted for simplythis distribution continued out to infinity. The absencof an energy excess in the long loops means that weno need to include a second population of open strinMoreover, we can readily see why all lattice-based wonecessarily generates open strings, whether or not tshould be present, while graph-based work need nIntriguingly, however, recent lattice simulations show th

3257

VOLUME 76, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 29 APRIL 1996

i

t

hi

h

x

.

.

it is also possible to achieve a very significant reductin the fraction of open strings either by increasing tvariance of the sizes of the initial phase domains [1or by decreasing the index of the power spectrum ofstring-bearing field [14,15].

Our simulations are certainly constrained by their CPtime demands. The range of the central, scale-invarBrownian loop, distribution is narrower than we miglike. However, this is an extremely difficult problemto address; the number of bubbles required to fillsimulation increases with the volumeL3, and the numberof four nodes to calculate then increases asN4. Sincethe runs presented here are already taking aroundh, even an order of magnitude increase inL2 would beprohibitively expensive.

We can expect the consequences of our resultsstring cosmologies to be profound. Simulations of strievolution with all the open string removed suggest tsuch distributions may reach a different scaling solutfrom the usual one [16]. Furthermore, current modelsstring-induced density perturbations focus on the waof long strings, and if we remove the open strinsuch models will require a much higher string numbdensity to have sufficient large loops to generaterequired perturbations. However, this number densityindependently bounded from above by the millisecopulsar limits on gravitational wave production in thdecay of small loops. Whether cosmic string scenarcan be made to fit these new constraints remains an oquestion.

The author acknowledges the support of NSF GrNo. PHY-9453431, and wishes to thank Andy Albrec

3258

onhe3]

the

U-iantht

he

250

forngat

onof

kesgerthe

isndeiospen

antt,

Ed Copeland, Pedro Ferreira, Mark Hindmarsh, TomKibble, James Robinson, Paul Shellard, Karl Strobl, AleVilenkin, and Andrew Yates for useful discussions.

*Present address.[1] T. W. B. Kibble, J. Phys. A9, 1387 (1976).[2] T. Vachaspati and A. Vilenkin, Phys. Rev. D30, 2036

(1984).[3] R. J. Scherrer and J. A. Frieman, Phys. Rev. D33, 3556

(1986).[4] R. M. Bradley, P. N. Strenski, and J. -M. Debierre, Phys

Rev. A 45, 8513 (1992).[5] M. Hindmarsh and K. Strobl, Nucl. Phys.B437, 471

(1995).[6] R. Leese and T. Prokopec, Phys. Rev. D44, 3749 (1991).[7] H. Hodges, Phys. Rev. D39, 3557 (1989).[8] P. Shellard (private communication).[9] F. Spitzer,Principles of Random Walk(D. Van Nostrand,

New York, 1964).[10] D. Austin, E. J. Copeland, and R. Rivers, Phys. Rev. D49,

4089 (1994).[11] J. Borrill, T. W. B. Kibble, T. Vachaspati, and A. Vilenkin,

Phys. Rev. D52, 1934 (1995).[12] J. Borrill (to be published).[13] A. Yates and T. W. B. Kibble, Imperial College Report

No. IMPERIALyTPy94-95y49.[14] Alex Vilenkin (private communication).[15] J. Robinson and A. Yates, Imperial College Report No

ImperialyTPy95-96y27.[16] Pedro Ferreira (private communication).