Indications of de Sitter spacetime from classical sequential growth dynamics of causal setsMaqbool AhmedCentreforAdvancedMathematicsandPhysics,CampusofCollegeofE&ME,NationalUniversityofSciencesandTechnology,PeshawarRoad,Rawalpindi,46000,PakistanDavid RideoutPerimeterInstituteforTheoreticalPhysics,31CarolineStreetNorth,Waterloo,OntarioN2L2Y5,Canada(Received30November2009;published22April2010)A large class of the dynamical laws for causal sets described by a classical process of sequential growthyieldsacyclicuniverse, whosecyclesof expansionandcontractionarepunctuatedbysingleoriginelementsofthecausalset.Wepresentevidencethattheeffectivedynamicsoftheimmediatefutureofone of these origin elements, within the context of the sequential growth dynamics, yields an initial periodof de Sitter-like exponential expansion, and argue that the resulting picture has many attractive features asa model of the early universe, with the potential to solve some of the standard model puzzles without anyne-tuning.DOI: 10.1103/PhysRevD.81.083528 PACSnumbers: 98.80.Bp, 02.10.Ox, 02.50.Ga, 04.60.mI. INTRODUCTIONCosmologyis the natural playgroundfor theories ofquantumgravityfor manyreasons. Themost obviousisthe fact that all quantum gravity theories are formulated atsuch high energies (or small distances) that it is practicallyimpossibleforanearthbasedlaboratoryoracceleratortotestthem. However, the early universe canprovidesuch alaboratory. In addition, cosmology is one of the mostnaturalapplicationsofgeneralrelativity,whichisexactlywhatatheoryofquantumgravityhopestounifywithquantum theory. On the other hand, the standard model ofcosmology [1,2] also suffers from some problems/puzzles[3]thatwarrantextensionstoit.Mostofthesehavetheirorigins in our lack of understanding of the physics of veryhighenergies andthus a successful theoryof quantumgravity should be able to address these problems.Quantumgravityhasrecentlybeguntoshedsomelightonthepuzzles of cosmology, suchas resolutionof thecosmic singularity [46], providing some hints as to alter-nativestoination[7], andprovidingpotential explana-tions of density perturbations [8]. Additionally theprediction of a nonzero cosmological constant arises natu-rally from the discreteness expected from quantum gravity[9]. Thus quantumgravity is beginning to showsomepromise in resolving some of the paradoxes of the standardmodel. However, thereisfarfromanyconsensusinthisregard, so cosmologists generally look for alternativeexplanations.The most common path taken is an extension via particlephysics that introduces a scalar eldwithveryspecialproperties in the early universe. If the potential of the eldsatises certain conditions it can be arranged that theuniverseundergoesrapidexpansioninadeSitter-likephase.Thisrapidexpansion,calledtheinationaryeraorination,getsridofmanyofthestandardmodelpuzzles/problemssuchastheso-calledhorizonproblem,theatness puzzle, the monopole problem, etc. For a morecomplete discussion see Refs. [10,11]. Despite the fact thatthis scenario has problems of its own, as has beenpointedoutbymanyauthors[7,12],theresultingfeaturesareveryattractiveandhardtoignore.Ontheotherhand,thelackofanycompetingmodel leavesascienticvoidthat needs to be lled if the case for or against ination hasto bedecided.Fortunately,causalsettheoryhasreachedastageinitsevolutionwheresomepredictionsabout cosmologyhavecome out [9,13]. Based on a mixture of classical dynam-ics (referred to as the classical sequential growth [14] orCSGmodels)andsomeexpectationsaboutquantumdy-namics, causal set theory predicts uctuations in thecosmological term around a mean value. If this mean valueis taken to be zero, the uctuations are of the right magni-tudetoexplainthepresentobservations.Computersimu-lations of the behavior of the universe with such auctuating cosmological termhaveshownthat the energydensity in follows the total energy density of the universeand is roughly of the same order. This is the rst time thatsuchtestablepredictionshavecomeoutofafundamentaltheory of quantum gravity.Causal set theory is also different frommost otherquantumgravity theories in the sense that it assumesfundamental discreteness. There have been many argu-mentsfordiscretenessat themost basiclevel, but all ofthemhaveeitherbeenmerelyphilosophical, orregardedonlyasmethodsofregulatingtheinnitiesandsingular-itiesinparticlephysicsandgeneral relativity. Intheab-sence of real predictions it has always been difcult to seeif these considerations have more than a philosophicalvalue.Predictionaboutuctuationsin,however,drawslifedirectlyfromfundamental discreteness, andit isbutnatural to wonder if causal set theory has more to say aboutcosmology. Inthis paper wewill present evidence thatPHYSICALREVIEWD81,083528(2010)1550-7998,2010,81(8),083528(11) 083528-1 2010 TheAmericanPhysicalSocietymanyof theCSGmodelsproduceadeSitter-likeearlyuniverse, and thus may prove helpful towards solvinganother puzzlewhyistheuniversesolargewhenit isnot so old?Ageneralquestionwhichnaturallyarisesincausalsettheory is whether causal sets which are well approximatedbycontinuaarisedynamically.Ithasbeenshownthatthesequential growth models possess continuumlimits, asN -and -0, howevertheresultingcontinualooknothinglikespacetimemanifoldsofdimension>1[15].However, it may still be the case that something resemblinga spacetime arises at nite . We consider this latterquestion here.This paper is organized as follows. In Sec. II we brieydescribe that portion of causal set theory which is relevantto the current work. In Sec. III we describe the behavior oftheoriginarypercolation dynamics,whicharisesasaneffective dynamics of the early universe of CSGmodels.TheninSec. IVwe compute the spacetimevolume ofAlexandrovneighborhoods(causaldiamonds)indeSitter space of arbitrary (integer) dimension. In Sec. V wedescribe the particular simulation we perform, with resultsin Sec. VI, and wrap up with some concluding remarks inSec.VII.II. CAUSAL SETSAcausal set, or causet for short, isalocallynitepartially ordered set, whose elements can be thought of asirreducible1atoms of spacetime. A partially ordered setCconsistsofagroundset,whichonegenerallylabelswithintegersfrom0toN 1(Ncanbeinnite), alongwithabinaryrelation -whichis irreexive(xx) andtransitive (x - y - z =x - z). Local niteness is the con-dition that every order interval (or simply interval)jx, y| y}x - y - z \x, z Chas nite cardinality.A. KinematicsThe connection to macroscopic spacetime arises via thenotionofasprinkling,inwhichoneselectseventsofaspacetime at random by a Poisson process, identies themwith causal set elements, and then deduces a partial order-ingamongtheelementsfromthecausal structureofthespacetime. One regards a continuum spacetime as being agoodapproximationtoanunderlyingcausal set if thatcausal set islikelytohavearisenfromasprinklingintothat spacetime. Foranextensivereviewofthecausal setprogram, see [1619].The connection between familiar concepts from contin-uum geometry and their discrete counterparts on the causalsetisthedomainofcausalsetkinematics.Wewillmakeextensive useof two resultsin this regard.Therst is statedas adenitionat this stageof thetheorys development, andis implicit inthedescriptioninthe previous paragraphof thecorrespondenceof thecausalsetwiththecontinuum.Inorderforacausalsettobe likely to arise from a sprinkling, it must be the case thatthe number of elements sprinkled into any region of space-time with volume Vis Poisson distributed, with a mean ofV. This Poisson uctuation in the correspondence betweenspacetime volume and number of elements plays a crucialroleinthepredictionof auctuatingcosmological con-stant [9].The second result relates to the correspondence betweenthe length of chains and proper time. A chain is a subset ofa causal set for which each pair of elements is related. Thelength L of a chain is the number of elements in the chainminus 1. In Minkowski space of any dimension, it has beenproven that the length of the longest chain between any pairof elements is proportional to the proper time between theeventsatwhichtheyaresprinkled,inthelimitofinnitesprinklingdensity[20,21]. In[22] theproportionalityisclaimedtoholdfor anyspacetime. Following[21], wedene m to be the constant of proportionality, so thatr mL. (2.1)B. DynamicsThere are a number of approaches toconstructingadynamical law for causal sets. Perhaps the most developedto date is the classical sequential growth model [14,23,24]mentioned in the Introduction. It describes the causal set asgrowing via a sort of cosmological accretion process, inwhichelements of thecauset ariseoneat atime, eachselectingsomesubsetofthecausalsettobeitspast.Theprocess of growth in the model is stochastic; each newbornelementselectsaprecursorsetatrandom,withproba-bilitieswhichsatisfyadiscreteanalogofgeneralcovari-anceandacausalityconditionakintothatusedtoderivetheBellinequalities.Thisrandomnessisregardedasfun-damental, andyet purelyclassical innature, becauseitdoes notallowfor any quantum interferenceamongalter-nativeoutcomes. Giventheclassicalnatureoftheproba-bilitydistribution,thedynamicsisincomplete,butcanbeseenasasteppingstonetowardformulatingafullyquan-tumprocess, whichcouldthenberegardedasageneral-ization of classical probability theory. Although thedynamicallygeneratedcausal sets donot leadtoorderswhich are readily approximated by smooth spacetimemanifolds, they do have a number of striking cosmologicalfeatures, which we explore further in this paper.The sequential growthdynamics is describedtotakeplace instages, thoughit is important toemphasizethat the discrete general covariance condition enforcesthat this orderinginwhichthecauset elements ariseis1Itmaybenecessarytoaddmatterdegreesoffreedomtothecauset elements, and at some stage it may be important tocoarsegrain thecauset sothat asingleelement maystandin for many, but for the moment we can think of the elements asnotcontaininganyinternalinformation.MAQBOOLAHMED ANDDAVIDRIDEOUT PHYSICALREVIEWD81,083528(2010)083528-2pure gaugeit has no effect on the probability of form-ing a particular (order equivalence class of) causal set. Atstage n the causal set has n elements so far, and the taskis to select a precursor set for the new element which arisesinthisstage. TheprobabilitiesoftheCSGmodel derivefroma sequence of non-negative couplingconstants(in), n 0. With these weights, the probability for select-ing a precursor set Sis proportional toi}S}.2Thus theprobability to choose a particular set SisPr(S) i}S}

n|0(n|)i|.Onceaprecursorischosen, tobetothepast ofthenewelement, all therelations impliedbytransitivityarein-cludedaswell. Thusit isthepast closureofSwhichforms the past ofthe newly generated element.The particular sequence in in, for a single non-negativereal number i, gives risetoadynamics calledtransitive percolation [14]. This sequence plays an impor-tant role, as we will see in a moment. The rule for decidingwhichelementstoselectforthepastofanewelementisparticularly simple for transitive percolation. The newbornelement simply considers each already existing element inturn, and selects it to be to its past with a xed probability i,(1 +i). It then adds to its past every element whichprecedesanyof theoriginally selectedelements,to main-tain transitivity.C. Cosmic renormalizationConsideranelementofacausalset,calledinthecom-binatorics literatureapost, whichis relatedtoeveryotherelement ofthecausal set. Thiswouldresembleaninitial or nal singularity of a universe, in that the entiretyof the universe is causally related to it. Now for any nite, it has been proven that a (an innite) causet generated bytransitive percolation almost surely contains an innitenumberofposts[25].Itisthelargescalebehavioroftheuniverse subsequent toone of these posts whichis thesubject of this paper. Wewill present evidencethat theperiod immediately following a post is one of rapid expan-sion of spacetime volume with respect to proper time. Thusat the largest scales the causal sets generated by transitivepercolationresembleabouncinguniverse,whichperiodi-cally undergoes collapse down to a nal singularity and anensuing reexpansion.It has further beenshownthat a large class of CSGmodels,whichincludesthesequence in (n, ln(n))nforn > 0, n >r2,3 also lead to causets which almost surelycontain an innite number of posts [26]. Now the presenceof postsinthedynamical model suggestsaninterestingpossibility, as described in [5], that the dynamics followinga post can be regarded as growing an entirely new universe,except with coupling constants which are renormalizedwithrespect tothoseof theprevious era. Muchis nowknown about the ow of the coupling constants (in) underthiscosmicrenormalization[27,28], inparticular, thatthe transitive percolation dynamics family in informs aunique attractive xed point. The sequence in in,n! hasalsobeenstudiedinsomedetail[5,29].Thereitisshownthattheregionimmediatelysubsequenttoapostbehavesliketransitivepercolation, withaparameteriwhichgetsdriven toward zero under the cosmic renormalization [i -i,N_for N elements to the past of the current eras post (ororigin element)].III. ORIGINARY PERCOLATION AND RANDOMTREESNote that the effective dynamics following a post comeswith a caveat: each element is required to be related to thepost, by denition. Therefore we nd an originary dynam-ics, for which the possibility of being born unrelated to anyotherelementisexcluded,andallremainingprobabilitiesarenormalizedcorrespondingly.Thustheprobabilitiesofanoriginarydynamicsareequal tothoseof anordinaryCSGmodel, conditionedontheevent that thenewbornelement connects to at leastone other element. (The orig-inary dynamics is in fact one of the general class ofsolutions tothe covariance andcausalityconditions onsequential growth, described in [30].)Originarypercolationistheoriginaryversionoftran-sitive percolation. As mentioned in Sec. II B, at each stageof the growth process, the newborn element considers eachexistingelementxinturn, andselectsxtobeinitspastwithprobability.Inordertomaintaintransitivityoftheorder, if it choosesxfor part of itspast, it includes allancestorsofxaswell. Intheevent that noelement xisselectedinthisprocess, it simplytriesagain,soastomaintain the condition of originarity. At stage n (meaningthat there are nelements currently in the causet), theprobabilitytoselectaparticularsubsetSofexistingele-ments isPr(S) }S}qn}S}1 qn,whereq 1 ,andthefactor1,(1 qn) accountsfortheoriginarycondition,whichexcludesthepossibilityofnot connecting to anything (which occurs with probabilityqn). Once a set S is chosen, the past closure of S becomesthe past of the newborn element x.A. Random tree eraFor small valuesof , theearlyuniverseof originarypercolation, by which we mean the structure of that portion2Thedenitionofprecursorsetusedherediffersfromthatin[14,23]. Thereonlyprecursor sets whichcontainedtheir ownpast wereincluded. Thisdenitionallowsfor amuchsimplerexpressionofthetransitionprobabilities(e.g.asgivenin[15]),whichbetterrevealsthephysicalmeaningofthein.INDICATIONSOFDESITTERSPACETIMEFROM. . . PHYSICALREVIEWD81,083528(2010)083528-3of the causal set whichis bornshortlyafter the originelement, formsarandomtree, withhighprobability. Tosee why this occurs, consider the probability that theselectedprecursor set contains m elementsis givenbyPr(}S} m) nm_ _}S}qn}S}1 qn.For small this becomes vanishingly small for any m >0.However, thecasem 0isexcludedbytheoriginaritycondition, whileitremainstrue that, aslong as N, which does not occur in(3.1).Therandomtreeera willcontinueuntil n 1,.Afterthis stage anewborn element becomes aslikely to choosemore than one parent as not. Thus we expect that the Nofthe random tree era is 1,. As far as the level populationdiscussiongoes, thisisnot theendof thestory, asit ispossible for sucha nontree element toselect all itsparentsfromelementsofthetreeatanearlyleveli, andthus itself to join an early level, say one much earlier thanthe maximum of (3.1), which is ln(1,). Thus (3.1)providesonlyalower boundonthecardinalityof earlylevels.ItisinterestingtonotethatGerhardtHooftpredictedalmost this exactscenario in 1978, cf. Fig. 10 of[19].B. Originary percolationTo get a better handle on the initial rate of expansion, weperformsimulationsofthefull originarypercolationdy-namics.ThistaskisgreatlysimpliedthroughtheuseoftheCAUSALSETStoolkit withinCACTUSframework[32].All we needtodois write a thorn (module) whichcountsthenumberofelementsineachlevel, andcountsthenumberofelementsandlongestchainsineachorderinterval, as explainedinSec. V. Theabilitytogeneratecausal setsviatheoriginarypercolationdynamicsisal-ready provided within thetoolkit.Asanillustration, weshowinFig. 2asmall examplecausal set generated by originary percolation with N 16, 0.2. Thepast of anyelement istheset of elementswhich can be reached fromit by traversing the lines(links) downward. The origin element/post is at thebottom. Theredsquaresareelementswhicharepart ofthe tree era.Results for originary percolation at 0.001, N 11 585, are depicted in Fig. 3. In addition to the cardinalityof each level mentioned above, we compute the cardinalityof a foliation of the causal set byinextendible anti-chains.Thisisamoreappropriateanaloguetothe(edge-less) spatial hypersurfaces of general relativity. An 0 500 1000 1500 2000 2500 051015202530NttFIG. 1(coloronline). MeanpopulationoflevelsforanN 16 384element randomtree. Themeananditserror aremea-suredfrom100samplesoftherandomtreeprocess.Thettingfunctionproportional toaPoissondistributionisshown. HereA 16 687 85and 9.306 0.022.MAQBOOLAHMED ANDDAVIDRIDEOUT PHYSICALREVIEWD81,083528(2010)083528-4antichainisasubset ofthecausal set whichcontainsnorelations. An inextendible antichain is one which is maxi-mal in the sense that no elements can be added to it whileremaininganantichain, i.e. everyother element of thecausal set istothefutureorpast ofoneofitselements.The inextendible antichains we employ here are dened asfollows. Theleveliasdenedaboveformsanantichain,but in general it will not be inextendible. We can extend itbyadjoiningthemaximal elements(oneswhichhavenoelementstotheirfuture)ofthat portionofthecausal setwhichisunrelatedtoanyelementofleveli. Itiseasytoshow that this will always yield an inextendible antichain.Note that all of the subcausal set which is unrelated to thelevel i antichainlives inlevels 1,thentheinitial treesitsintheveryearlypart ofthepercolatedcausalset. Theexponentialexpansionextendswell beyondthetreeera, andthustheinitial exponentialgrowthof(3.1)isindeedonlyaprecursortoanensuingexponential growth involving a much larger portion of thecausal set.Beforeclosingthissection, itisimportanttonotethatthefutureofeveryelement ofapercolatedcausal set isitselfaninstanceoforiginarypercolation. Thisissimplybecausepercolationiscompletelyhomogeneousthefu-ture of an element is the same (in probability) as that of anyother element. However, bydiscussingthefutureof anelement x, oneisconditioningoneachelement beingtothe future of x, which is exactly the condition of originarypercolation. Thus originary percolation describes a homo-geneous universe, for which the future of every element isexponentiallyexpanding. Thissoundsalot likedeSitterspace.IV. VOLUME OF AN ALEXANDROVNEIGHBORHOOD IN DE SITTER SPACETIMEFor aspacetimerespectingthecosmological princi-ple, an exponential expansion means the de Sitter space-time. If the universe is described by something like a causalset, the early universe region that we consider is veryyoung. Itdoesnotlooklikeaspacetimeyetinthesensethat it does not render itself easily to many of the familiarconcepts of the continuum. This is particularly clear if oneconsiders, for example, the randomtree era. It is notpossibletodenethenotionofspacelikedistanceinarandomtreeas notwoelements haveacommonfuture[33].Similarly,itisdifculttoseewhatcurvaturewouldmeaninthiscase. Ontheotherhand, thenotionsofthelength of the longest chain between two causal set elements(L),whichisproportionaltothepropertimebetweenthetwo events (r), and the number of causal set elements NFIG. 3 (color online). Mean spatial volume of ten originarypercolatedcausal sets withN 11 585, 0.001. Shownisthe cardinality of level i in red, the cardinality of an inextendibleantichaincontaining level iin green,thenumberofelementsoftheinitial treeat level i inblue, andthenumber of maximalelementsoftheinitialtree,atleveli,inmagenta.FIG. 2(coloronline). Asamplecausalsetgeneratedbyorig-inarypercolationwithN 16, 0.2.Elementsoftheinitialtreeareshownbyredsquares.3It turns out that this inextendible antichain is equivalent to theone which arises by taking the maximal elements of those whoselevelis i.INDICATIONSOFDESITTERSPACETIMEFROM. . . PHYSICALREVIEWD81,083528(2010)083528-5which are causally between two given elements,4which isproportional tothevolumeof theAlexandrovneighbor-hood5Vformed by the two elements, is still dened. Wetry to see if L and N follow the same relationship as r andVin D+1-dimensional deSitter spacetime.We useJs2 Ji2+e2i,(Jr2+r2J2D)as the D+1-dimensional de Sitter metric [34], where istheradiusof curvatureandall other symbolshavetheirusual meaning. As we have spherical symmetry in this casewe can represent the Alexandrovneighborhoodof twoevents in i and r space as sketched in Fig. 4. The spacetimevolume of this region can be written asV_0i1JieDi,_ro0JrrD1_JD+_i20JieDi,_r|0JrrD1_JD. (4.1)As the light cones in de Sitter space follow_ r ei,andwechooseoutgoingroandingoingr|radial coordinatessuch that ro(0) r|(0) R, we can write ro R +(1 ei,) and r| R +(ei,1). Using these we canwrite (4.1) asV CDD__0i1Ji_R +ei,1_D+_i20Ji_R ei,+1_D_, (4.2)where CD is the volume of a D-dimensional unit ball. Usingi1 ln+R, i2 lnR, and i1+i2 r, we can sim-plify (4.2) toV CDD+1_ln cosh2_ r2_+

D|1(1)|+1|D|_ ___1 +tanh_ r2__|+_1 tanh_r2__|2__(4.3)for D odd andV CDD+1_r+

D|1(1)||D|_ ___1 +tanh_ r2__|

_1 tanh_ r2__|__(4.4)forDeven.OneobviouscaseofinterestisD 3.Usingtheabove-mentionedexpressionsandthefact that C34r,3, itturns out thatV4r34_lncosh2_r2_tanh2_ r2__for afour-dimensional deSitter spacetime. It shouldbenoted that VrD+1forr1. For thefour-dimensional deSitter spaceVr24r4+O(r5) for r ).R rttt12FIG. 4 (color online). An Alexandrov neighborhood (orderinterval)indeSitterspace.FIG. 5(color online). The set of all pairs (L, N) for eachrelatedpairof elements, inthreecausal sets. Eachcauset wasgeneratedwiththesamevalueof 0.025butthreedifferentvaluesof N 500, 1500, and2500. (Tomaketheguresizemanageable, weplot onlyevery4thpoint for N 500, every10thforN 1500,andevery 200thpointforN 2500.)Notethat the smaller data sets are a subset of the larger ones, and thatatsomepointthemaximum N(L) nolongerincreaseswith N.4For an order interval jx, y|, we dene N }z}x - z - y} +1, where } } indicatesset cardinality. The +1allowsN Lforanintervalwhichisachain.5The Alexandrov neighborhood of two events is the overlap ofthepastofthefuturemosteventwiththefutureoftheother.MAQBOOLAHMED ANDDAVIDRIDEOUT PHYSICALREVIEWD81,083528(2010)083528-6V. SIMULATION DETAILSWe want to compare the relationship between Vand rgivenbyEqs.(4.3)and(4.4)withthatproducedbyorig-inary percolation between N and L. In a given simulationwegenerateacausalsetviaoriginarypercolation,withagiven number of elements N and the percolation parameter j0, 1|. Wethencalculatethelengths of thelongestchains L between all pairs of elements and the correspond-ingnumberofelementsNthatareconnectedtobothofthese elements andlie causallybetweenthem. For thisexercise the typical values of Nlie between1000and50 000andof between0.0001and0.03. Theprimarycomputational constraint is run time, as nding the lengthof the longest chain in an interval involves an O(N2)algorithm, and there are O(N2) intervals to check.For a given causal set, we collect a large number of pairsof numbers (L, N), one for every related pair of elementsinthecauset.The setofsuch pairs forthreecausalsets isplotted in Fig. 5. We wish to compare these data points withthefunctional forms (4.3) and (4.4), for some value ofthedimension D. If these causal sets are exactly represented byde Sitter space, i.e. if they arose from a Poisson sprinklingofaregionofdeSitterofspatialdimensionD, thenonewould expect the data points to be scattered about the curve(4.3)or(4.4),withPoissonuctuations.Thereareindica-tions that, in spacetime dimensions larger than 3, theuctuations in the length of the longest chain in a sprinkledinterval of Minkowski spacegrowsonlylogarithmicallywithL[35], soonemight guessthat wewouldseedatapoints distributed roughly uniformly above the curve (4.3)or (4.4). However, for reasons we do not fully understand,itturnsoutthatthedatapointsallseemtofallbelowthecurve (4.3) or (4.4), such that the maximum value of N fora given L, for an appropriate range of values of L, gives anexcellent match to one of the functions (4.3) or (4.4).It is important to notice that almost all of the physics inthis scenario is dictated by the choice of , as long as N >1,. This can be easily seen from Fig. 5, by observing thatthedatapointsfor smaller Nareeffectivelyasubset ofthose for a larger value of N. Notice, in particular, that themaximumvaluesofNfortheN 1500causetarethesame as those for the N 2500 causet. Thus, as long as Nis large enough to capture the relevant region of exponen-tial expansion, increasing Nfurther will have no effect ontheresultsof interest.6Inparticular, thismeansthat thedimensionDwhichgivesthebest t, for example, willonly depend on . If N is too small, on the other hand, thenthe causal set is not large enough to sample the region ofinterest, and we will get poor results. This is manifested inFig.5bythefactthatthemaximumNforN 500aresubstantially smaller than thosefor thelarger causets.The reader may be concerned that we use the maximumNfor a given value of L, rather than the mean. This is anindication that the percolated causal set is not exactlymanifoldlike. However this is not toosurprising, as wealreadyknowthattheCSGmodelsdonothavenontrivialspacetimes as their continuum limits [15]. Another indica-tionthat theseare not quite manifoldlikeis that at thesmallest scalestheyaretrees, asexplainedinSec. III A,andthusonedimensional (becausetheshortest intervalswill alwaysbechains). Thisfailureof themeantogivegood results may be expected, in that it gets contributionsfromall sortsof intervals, includingonesthat might beclosetoaboundary,suchthattheyhavesmallNforlargeL. Inasenseweareconsideringonlyintervalsasmeasured by observers which are stationary in the cosmicrest frame, sothat theycanget themost elementsfor agiven proper time separation.SinceeachcausalsetonlyprovidesasinglemaximumNfor each L, we repeat the computation for a number ofcausal sets, and from these compute a mean maximum Nwithits error. We thent eachsuchdata set withtheexpressionsgiveninEqs.(4.3)and(4.4),withrreplacedby L,m. and m are used as tting parameters. For D 3the tting expression looks likeV4r34_ln cosh2_r2_tanh2_r2__. (5.1)As mentioned above, at the smallest scales L the causal setbehaveslikeatreeandistherefore, inthesenseoforderintervals, 0 +1 dimensional. At the largest scales theintervals seetheinfraredcutoff N, andthereforearenot expected to give good results. We thus only t our datawithin a range of L values, as shown in Table I.Furthermore, since the error bars are muchsmaller forthesmallintervalsthanforthelargeones,ttingdirectlytotheformsabovewouldstronglyfavorthesmallscales,and tend to ignore the data for larger scales. We handle thisbytting(thelogofthemaximumN)tothelogofthefunctions above such as (5.1), which has the effect of ttingto the relative error in the maximum N.At no point have we ever mentioned any number for thedimension,inexpressingthedynamics.Thuswehavenoidea what dimension of de Sitter space to expect from ourresults.Wethereforetourdatatoevery (spatial)dimen-sion, usually from 1 to 9, and take the one which ts best.VI. RESULTSFigure6shows atypical behavior of the plot of themaximumnumber of elements inanorder interval andthe corresponding longest chains, for N 15 000 and 0.001. Interestinglyenough, thebest t wasachievedbythe function for 3 +1 dimensional de Sitter space, which is6Itistruethatoriginarypercolationforxed andinnite Ncontains every nite partial order as a suborder. Thus somewherein that innite causet is an interval with height L and cardinalityarbitrarilylarge. However, wedonotsendN -forxed,we are only interested in the early universe of originarypercolation, withNnolargerthansay1,3. InsucharegimethemaximumNiseffectivelyindependentofN.INDICATIONSOFDESITTERSPACETIMEFROM. . . PHYSICALREVIEWD81,083528(2010)083528-7shown in black. The best ts for the two neighboringdimensions are alsoshown, togive some indicationoftherobustnessof the dimension measurement.Theresultsfor all our runs, for avarietyof valuesof, are summarized in Table I. All ts are performedwith the GNUPLOT t function. For each value of we have considered, Table I provides the value of Nwe have used, the best t spacetime dimension, thebest t values for and m with their errors,

LjN,LV(L)|2,(NL2)a2N,L_(whereNListhenumber of data points t), the range of L values we t, andalso the number of causal sets generated. All reportederrors are as given byGNUPLOT.AsdiscussedinSec.II A,propertimesareexpectedtoberelatedtolengthof thelongest chainby(2.1). If weassume that the largest intervals of our causal sets doindeedbehavelikeintervalsof deSitter space, thenthets of Table I serve as an alternate measurement of m, in deSitterspacetimeofthreeandfourdimensions. Itisinter-esting to see that the values come out comparable to thosefor Minkowski space, whichfall between1.77and2.62[21]. Themeasurements indicatethat wecangrowauniverse which isroughly 2m 36 elements across.Figure6containstheresultsfromourlargest dataset(largest number of causal sets generatedwiththosepa-rameters). The plot for our smallest value of is shown inFig.7.TheretherangeofLvaluesavailableforthetissmaller, becauseoneneedsaverylargecausal settogetlargechainswithsuchasmall.Thecurvefor3 +1deSittercontinuestomakeanexcellent t, andbetterthancurves for different dimensions of de Sitter space. Figure 8portrays another example of the ts, this time in log scale.Our nal example, Fig. 9, comes from a larger value for .Here the universe is quite small, with a radius of curvatureofjust 4infundamentalunits.Thebesttdimensionisonly 2 +1 for this tiny universe.Beforeconcluding, wereturnattentiontotheissueoftting the mean vs the maximumN. In Fig. 10 we directlycompare the two on the same data set, generated from fourN 15 000 0.0008causal sets. It is clear that theFIG. 6 (color online). A plot of the maximum values of NasafunctionofL for 0.001andN 15 000,alongwithbesttcurvesfordeSitterspaceofthreedifferentdimensions.Theverticallineontheleftmarkstheendofthetreeera,whiletheone on the right separates the points that see the niteness ofthe causal set. The overall best t is achieved from the curve for3 +1dimensions,andisshowninblack.FIG. 7 (color online). A plot of the maximum values of Nasa function of L for 0.0001 and N 50 000, along with besttcurvesfordeSitterspaceofthreedifferentdimensions.Theoverall best t is achieved from the curve for 3 +1 dimensions.TABLEI. Fittingparametersforsomevaluesof. N D+1 m FittingrangeinL Numberofruns0.0001 50 000 4 8.7 1.8 2.105 0.028 2.32 925 60.0002 30 000 4 6.81 0.72 1.926 0.023 3.07 827 30.0005 20 000 4 7.81 0.57 1.787 0.022 5.59 732 100.0008 15 000 4 6.86 0.22 1.749 0.019 4.69 635 40.001 15 000 4 6.20 0.12 1.710 0.013 4.97 735 230.003 15 000 4 3.73 0.013 1.483 0.009 5.12 6100 200.005 15 000 4 3.097 0.009 1.388 0.009 4.69 5150 200.01 2000 3 4.086 0.028 1.136 0.006 2.75 539 500.03 1000 3 2.331 0.011 1.046 0.006 0.663 553 5MAQBOOLAHMED ANDDAVIDRIDEOUT PHYSICALREVIEWD81,083528(2010)083528-8maximums t the 3 +1 de Sitter form quite well, while themeanvalues donot. We omit the errors inthe means,becausetheyareextremelysmall forthesmall intervals,and strongly bias the ts away from the data at larger L. Inany eventthetispoorforthemeans,forany dimension(though again 3 +1 ts the best there as well).VII. SUMMARYAND CONCLUSIONSAfter motivating the study of originary percolation as anappropriate dynamical model for the early universe ofcauset set theory, at least withinthecontext of classicalsequentialgrowthmodels,weexploredanumberofindi-cations that it yields an exponentially expanding universe.Inparticular, for