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On recent constraints for the minimum scale of a small compact universe with three-torus topology Lee, Henry 1993

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ON RECENT CONSTRAINTS FOR THE MINIMUM SCALEOF A SMALL COMPACT UNIVERSE WITHTHREE-TORUS TOPOLOGYByHenry LeeB. Sc. (Physics) Simon Fraser University, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Henry Lee, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of PhysicsThe University of British Columbia6224 Agricultural RoadVancouver, B.C., CanadaV6T IZIDate:^Oci-oll&L /5, 0q3AbstractWe have embarked upon simple tests to gauge the validity of assumptions made by Fangand Liu in their assessment of opposite quasar pairs as a probe of global topology inthe universe. Constraints for the scale of a three-torus (T-3) universe obtained fromsearches of opposite pairs of quasars by Fang and Liu have been claimed as the strongestto date. However, two assumptions involved are shown to be invalid: quasar imagesare not distributed uniformly across the sky and the observer and the object within thefundamental cell are not coincident. So, detected images of the object are not locatedback-to-back.We perform two numerical calculations in a simulated survey. A three-dimensionalcomputational lattice with unit volumes was arranged as the model for a small universewith three-torus topology. The first calculation sets coincident observer and object po-sitions within the unit cell while the second calculation places the observer randomlywithin the unit cell. The computational lattice was surveyed with two by two degreesquare beams for images of the object and the number of opposite pairs of images wascounted.Our results show that opposite pairs of quasars are detected infrequently and thatthe Fang and Liu limit is overestimated by a factor of ten in probability. To obtain their90% confidence in pair detection probability, their limit underestimates the number oftoroidal diameters by a factor of two. Consequently, their claimed lower limit of 200Mpc for the minimum scale of a three-torus universe is reduced by two to 100 Mpc. Theabsence of many opposite quasar pairs does not constrain the minimum scale of a toroidaluniverse and any search for opposite quasar pairs is not a useful method to investigatethe existence or constrain the size of a possible toroidal universe.Table of ContentsAbstract^ iiTable of Contents^ iiiList of Tables^ vList of Figures^ viAcknowledgements^ viii12IntroductionTopology of the Universe142.1 Curvature, Geometry and Topology ^ 42.1.1^Spatial Homogeneity and Isotropy 42.1.2^Friedmann-Lemaitre-Robertson-Walker Spacetime ^ 72.1.3^Topology and Manifolds ^ 102.2 Considerations from Quantum Gravity and Inflation ^ 132.3 Small Universes and Three-Torus Topology ^ 143 Observations and Astrophysical Considerations 203.1 Possible Observational Evidence for Multiply Connected Topology . . . 203.1.1^Lower Limits from Surveys of Galaxies and Clusters of Galaxies 203.1.2^Other Observations ^ 211114 Constraints for the Minimum Scale of A Small Toroidal Universe^244.1 Quasars as Probes at Intermediate Redshifts  ^254.2 Constraints from Opposite Pairs of Quasars : Fang and Liu Limit . .^265 Simulations and Discussion^ 285.1 Computational Survey  285.2 Computational Algorithm ^  295.3 Results  ^305.3.1 The QS0 Luminosity Function and Comoving Space Density .^335.4 Implications for the Cosmic Microwave Background (CMB) ^ 375.4.1 Three-Torus Small Universe Models from CMB Anisotropy . .^386 Conclusions and Final Remarks^ 41 •Bibliography^ 43A Listing of Computational Program^ 45ivList of Tables2.1 Relationships between curvature and geometry: each row lists a differentuniverse for a given curvature constant, k, the mass density, S2, the result-ing geometry and curvature type. The sum of interior angles in a triangleand the area of a circle and the eventual fate for each universe are alsogiven.   92.2 Typical global topologies for FLRW universes: The first row lists the threepossible FLRW geometries. A global topology of the universe is given foreach kind of open or closed universe.   125.1 The number of toroidal diameters, the mean angular separation betweenQS0s from z = 0 to z = 2, the scale of a three-torus unit cell and numberof distinct QS0s about z = 2 are given. The number of distinct QS0sis calculated from the integrated QSO luminosity function in the narrowredshift interval about z = 2 given by equation (5.2). As usual, the Hubbleparameter is given by H0 = 100h0 km s Mpc-1   35VList of Figures2.1 Unlikely and likely isotropies in a homogeneous universe: (a) if the curva-ture was found to be maximum in one direction and zero in the orthogonaldirection, it is more likely that we would live on a cylindrical surface of cir-cular cross-section than oval cross section; (b) if the curvature was foundto be isotropic, it is more likely that we would live on the surface of asphere than on a surface of an ellipsoid.   62.2 Identification of the fundamental cube volume element for the simplestthree-torus topology: each pair of opposite faces is identified without ro-tation   152.3 Two-dimensional representation of the covering space for the three-torus:the unit cell with equal side lengths, a, is highlighted with a thick border.The asterisk symbol denotes the observer while the small filled ellipseat the centre of the unit cell denotes the object whose images are beingobserved. The observer looks out into the covering space within theirparticle horizon to constant redshift z. Survey beams originating from theobserver typically do not detect many opposite twin images of the object. 172.4 This is how Einstein sitting in his living room would appear if his livingroom was the fundamental volume in a three-torus universe. Even thoughthere is an infinite array of images within the "lattice," the three-torussmall universe is finite in size   18vi5.1 The probability of detecting at least one opposite image pair, Pi (N), isplotted as a function of the toroidal diameter, N. The star indicates theFang and Liu limit, the set of open circles is the result of a simulationwith coincident observer and object positions in the unit cell and the setof filled circles is the result of a full calculation with the observer positionaveraged over the unit cell. The dotted line is a probability curve for aPoisson distribution of images.  viiAcknowledgementsI would like to thank the following people; without them, there would have been littlejoy in the pursuit of knowledge• Mark Halpern, for supervising me through this thesis project, for his endless pa-tience and understanding, and valuable discussions in cosmology and answeringquestions from his vast knowledge of physics and astrophysics,• Geoff Hayward, for introducing me to this thesis project, our various discussions onsmall universe models and the possible observational consequences of these modelsand his guidance towards the completion of our project,• Ed Wishnow, for continually asking questions, for conversations and discussionsregarding cosmology, physics and sports, and providing a valuable perspective onmatters of academic research,• Jeff and Holly: a man is of great wealth to have such friends,• and to a number of my colleagues in 311 (Dan, Scott, Martin, Roger, Sebastian,Glenn, Bill and Dominique) who provided rousing discourse and unique views inmatters of research, the fine art of diplomacy in politics, playoff hockey, "softwareacquisitions," and "life after the thesis."viiiChapter 1IntroductionIn ancient times, people wondered about the universe and especially spent time wonderingabout the shape of their local universe: the earth. Accepted history shows that earlyGreek astronomy from 300 B.C. to approximately 100 A.D. made great strides towardsour understanding of relationships between the sun, the earth and the moon. Aristotle(384-322 B.C.) argued that the earth had a spherical shape because (i) he observed thatthe shadow of the earth on the moon was curved during lunar eclipses and (ii) northboundtravellers saw more of the northern sky exposed while more southern stars seemed todisappear below the horizon. Knowing that the earth was spherical in shape meant thatthe radius of the earth could be measured. Eratosthenes 200 B.C.) estimated theradius of the earth to within 20% of the correct value by noting that an obelisk in Syeneat high noon at the summer solstice cast a shadow whereas an obelisk many miles away atAlexandria cast none. Almost two thousand years later, mathematician Fredrick Gaussfound an angle excess over 180° when he measured the interior angles of a large geodetictriangle formed by three mountain peaks in Germany. This allowed Gauss to make anindependent estimate of the radius of the earth to within five percent of the current value.The shape and curvature of the earth could be measured without departing the surfaceof the earth.Today cosmologists ponder a similar question about much larger surroundings: whatis the "shape" or topology of the entire universe? To answer this question, one notes that[Fang & Mo 198713] the local spacetime curvature of the universe may be obtained from1Chapter 1. Introduction^ 2astronomical observations. We hypothesize that the spacetime curvature is the samethroughout the entire universe. This follows from the cosmological principle which statesthat no observer in the universe occupies a preferred position. If a special place existed inthe universe, an observer could describe physics particular only to their neighbourhood.However, neither the metric which describes the local structure of spacetime nor theEinstein field equations describing the dynamics of the universe determine the topologyof the universe. Depending on the spacetime curvature, there may be many differenttopologies assigned for each metric. The problem is how can one uniquely specify aparticular topology that is consistent within our current understanding of cosmology.All our observations are made within the universe we occupy; one can not escape or beoutside of the universe.Much work and effort has gone into the determination of curvature and geometry forwhich there are several tests including number counts in deep surveys and the angularsizes of objects as a function of increasing redshift. There is a similar quest for such testsfor the topology. There has been some progress in understanding the mathematics andtopological structure of three-dimensional manifolds with the assigned local geometry.However, it has only been recently that additional work has discussed the possibility ofdetecting topology in the universe. Even so, the study of topology in the universe is notdrawing much attention from most cosmologists today, primarily because of the lack offirm observational predictions.In subsequent chapters, the utility of the three-torus universe model is examined aswell as recent efforts which purportedly constrain the scale for such a universe. Theconnection between geometry and topology is explored in Chapter 2. The spatiallyhomogeneous Friedmann world model is used as the starting point for a discussion oftopology which then leads to the examination of the three-torus model. Chapter 3 dealswith observations which could be explained by a small universe model with three-torusChapter 1. Introduction^ 3topology. Recent constraints by Fang and Liu (1988) on the minimum scale for a three-torus universe are discussed in Chapter 4. In Chapter 5, the numerical algorithm andthe results of two simulations are discussed. We show that the technique used by Fangand Liu to obtain their constraints to be weak and that looking for opposite pairs ofquasar images is useless for constraining the minimum scale of a three-torus universe.Brief remarks are made concerning constraints on the scale of three-torus small universemodels from a consideration of the COBE detection of microwave background anisotropy.Specifically, the latest reports which claim to rule out three-torus models from the COBEdata are discussed.Chapter 2Topology of the Universe2.1 Curvature, Geometry and TopologyThe relationship between spacetime curvature, the geometry and the topology in theuniverse is examined. We consider the following question : if the universe is curved insome global sense and the topology can be determined, can we measure or deduce theradius of curvature for the universe?The principle of homogeneity and isotropy is discussed, leading up to Friedmannuniverses and possible topologies which admit the Friedmann world models. Finally, thesmall universe model will be discussed; in particular, the three-torus topology for thesimplest small universe model will be explored.2.1.1 Spatial Homogeneity and IsotropyFrom our limited vantage point, observations on the largest scales indicate that theuniverse appears to be homogeneous and isotropic [Ellis Sz Harrison 1974]. However,we occupy a small section of spacetime and our observations are limited to whateverinformation is available on our past lightcone. These limitations force us to inquirewhether or not any observations of symmetry are the result of our special location in theuniverse. Any symmetry observed on small scales could also exist on larger scales to givehomogeneity and isotropy.We consider two key questions. First, what isotropies does a spatially homogeneous4Chapter 2. Topology of the Universe^ 5universe allow? Second, is the universe spatially homogeneous if a certain kind of isotropyis allowed to exist everywhere?We address the first question. Spatial homogeneity does not automatically implyisotropy because any inhomogeneous universe may bear some special location for whichan observer sees isotropy in his neighbourhood. The cosmological principle mentioned inChapter 1 is used to solve this problem. Two examples help illustrate how the principleworks [Ellis & Harrison 1974].We imagine that we occupy some small region or neighbourhood of space on somelarge surface and that we are capable of measuring the curvature of space in every direc-tion from our region. In the first example, we suppose that the measured curvature isanisotropic; that is, the curvature is maximum in one direction and zero in the orthog-onal direction. There is reflection symmetry in the direction of maximum curvature. Itis very unlikely that we live at a line of reflection symmetry, R, on a cylindrical surfaceof oval cross-section (Figure 2.1a). It is more likely that we live on a cylindrical surfaceof circular cross-section with reflection symmetry at all points on the surface. In thesecond example, one supposes that the measured curvature is now isotropic. We couldbe living at one of the endpoints, I, of isotropic curvature of an ellipsoid but these wouldbe special locations (Figure 2.1b). According to the cosmological principle, it would beassumed that we reside on a spherical surface with isotropic curvature at every point onthe surface. All directions are the same and there is rotational symmetry.To address the question of whether or not the universe is spatially homogeneous,there are three basic isotropies which admit spatial homogeneity: spherical symmetry,cylindrical symmetry and discrete symmetry.A universe with spherical symmetry is everywhere isotropic and spatially homoge-neous. The Friedmann spacetime metric describes such a universe and is discussed inSection 2.1.2 below.Chapter 2. Topology of the Universe^ 6Figure 2.1: Unlikely and likely isotropies in a homogeneous universe: (a) if the curvaturewas found to be maximum in one direction and zero in the orthogonal direction, it ismore likely that we would live on a cylindrical surface of circular cross-section than ovalcross section; (b) if the curvature was found to be isotropic, it is more likely that wewould live on the surface of a sphere than on a surface of an ellipsoid.Chapter 2. Topology of the Universe^ 7The universe is invariant under rotations about some preferred axis uncle' cylindri-cal symmetry. This symmetry occurs in six spatially homogeneous models with localspacetime symmetry.A discrete reflection symmetry occurs for three preferred axes and a prescribed ori-gin of some coordinate system. All Bianchi class I homogeneous (anisotropic) modelssatisfy this symmetry [Ryan & Shepley 1975]. There exist five additional homogeneouscosmologies which satisfy a reflection symmetry only through three preferred axes.We do not directly observe a specific spatial isotropy. Rather, this is deduced fromthe cosmological principle and any symmetries observed in our past lightcone. Additionaldata is sought to confirm which homogeneous model is admitted by the particular isotropyinferred. The cosmological principle helps generate symmetry on larger scales from thatobserved on small scales. The principle is not useful if there is no symmetry observedlocally.2.1.2 Friedmann-Lemaitre-Robertson-Walker SpacetimeThe standard cosmological model of the universe relies on the concepts of homogeneityand isotropy [Peebles 1993]. On sufficiently large scales 100 Mpc), the spatial dis-tribution of galaxies is considered smooth and homogeneous. Also, recent observations[Smoot et al. 1992] have shown that the cosmic microwave background is quite isotropicover the entire sky. There is no reason why this cosmic background would not appearisotropic to any other observer in the universe in accordance with the cosmological princi-ple. Since the observed universe appears approximately homogeneous and isotropic, it isreasonable to assume that spacetime on horizon scales is also homogeneous and isotropic.For these reasons, homogeneous and isotropic universe models with constant curvaturegeometry are emphasized.Chapter 2. Topology of the Universe^ 8The most general metric describing homogeneous isotropic spaces is the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. The local metric structure of spacetime ischaracterized by the line element. The line element in an homogeneous and isotropicFLRW space is given byds2 = —c2dt2 a(t)2(dr2 f(r)2(d02 + sin20 d02)), (2.1)where r is the spatial radial distance coordinate, 0 and 0 are angular coordinates, anda(t) is the scale factor which describes the expansion of the universe at time t. The scalefactor also sets the length scale in the spatial coordinate. The function f(r) has thefollowing valuessin r, k = +1r,^k = 0sinh r, k = —1where k is the curvature constant which determines the geometry of the spatial sections.The spatial sections are also known as spacelike slices, spacelike cross-sections or spatialhypersurfaces at constant time.The curvature constant can be determined from measurements of the density of mat-ter, p, in the universe and the Hubble rate of expansion, Ho, at the present epoch,H2k = —a(t)2(CI — 1),c2where St = P/ Pc is the density parameter and pc =(2.2)3H2 is the critical mass density87rGneeded to close the universe [Kolb & Turner 1987, equation 3.16]. Throughout the entirediscussion, the Hubble parameter is taken to be Ho = 1001/0 km s-1 Mpc-1. Mass densitymeasurements and constraints come from various sources: nucleosynthesis of the lightelements, the stellar composition in galaxies, luminous matter found in the intergalacticmedium, velocity dispersion of galaxies within clusters of galaxies and dynamics on scalesf(r) =Chapter 2. Topology of the Universe^ 9from several to tens of megaparsecs. The Hubble parameter can be obtained in principleby measuring the velocities of nearby galaxies whose radial distances are independentlydetermined by some well-established calibration method.The spatial curvature and the geometry are related in a specific but limited way.While the curvature may determine the future evolution of the universe, we do not knowwhether the extent of the universe is finite or infinite. Table 2.1 lists relationships betweenseveral parameters and properties of simple polygons for universes with homogeneous ge-ometries. For each value of the curvature constant, the corresponding density parameter,k CZ Geometry Curvature Sum of InteriorAngles in AArea ofCircleFuture of Universe+1 L_ > 1 Elliptic positive > 180° < 77-2 eventual recollapse0 = 1 Euclidean flat = 180° = 7r2 expansion rate --4 0-1 < 1 Hyperbolic negative < 180° > 7r2 indefinite expansionTable 2.1: Relationships between curvature and geometry: each row lists a differentuniverse for a given curvature constant, k, the mass density, Q, the resulting geometryand curvature type. The sum of interior angles in a triangle and the area of a circle andthe eventual fate for each universe are also given.geometry and curvature are given. The sum of the interior angles in a triangle and thearea of a circle in the corresponding universes are also given. The last column lists theeventual fate for each world model. Note that the zero curvature flat universe presentsthe "boundary" case where the universal expansion gradually slows down and eventuallystops after an infinite amount of time. On the other hand, a positive curvature universeeventually collapses after a finite amount of time while a negative curvature universeexpands indefinitely without bound.Chapter 2. Topology of the Universe2.1.3 Topology and ManifoldsOur discussion of topology and geometry begins with some definitions [Weeks 1985].Topology and geometry are distinguished in the following way. The topology or "rubber-sheet geometry" of a surface is the study of those properties or features of a surfacewhich remain unaffected by continuous transformations or deformations. Bending andfolding are permitted deformation processes while tearing and reattaching parts of asurface are not permitted. Surfaces with nonzero curvature cannot be spread flat (asimple deformation) without tearing or cutting. For example, a spherical cap cannot bespread flat without rips in the cap. So, another way of stating what topology is that itis the mathematical study of how lines, surfaces, etc. connect together with no regardto the particular geometry. The geometry of a surface consists of those properties thatchange under simple deformations.The difference in the use of terms "local" and "global" with respect to topologyand geometry can best be explained as follows. A two-dimensional manifold or surfaceis a space with the local topology of a plane. A three-dimensional manifold or three-manifold (analogous to a surface) is a space with the local topology of "ordinary" three-dimensional space. Three examples help illustrate the discussion. A flat two-dimensionaltorus obtained by abstract gluing of the opposite sides of a square and a doughnut surfacehave the same global topology but different local geometry. As a second example, thesame two-torus and a plane have different global topologies but have the same localgeometry. Finally, the surface of a sphere (the two-sphere) and the surface of an irregularsolid blob with no holes have the same global topology. As we shall see, a three-torus hasthe same local geometry as three-dimensional Euclidean space but the global topologycan be quite different. More features of toroidal topologies are discussed below.In principle, the topology of the universe may be obtained simply by examining whatChapter 2. Topology of the Universe^ 11kinds of three-dimensional manifolds admit one of the three homogeneous geometrieslisted in Table 2.1. A three-dimensional manifold can be produced when the surfaces ofa given fundamental polyhedron are mathematically identified in some chosen manner.Such a manifold may be used as a model for the topology of the universe. It is muchsimpler to study topological properties of a manifold if the manifold is smoothly deformeduntil the curvature is constant throughout the entire manifold.Homogeneity and isotropy require constant curvature but allows the topology to be‘'open-ended." Since the spacetime metric does not determine the topology, there is thepossibility of universes with different global topologies and yet the same local spacetimestructure. We give an example of universes whose spatial sections have flat geometry.Spatial sections in a flat universe admit the infinite three-dimensional Euclidean geome-try. Geodesics between two points are unique and do not ever intersect; so, the topologyis simply connected. However, points in this universe may also be identified to give a"periodic" space which still admits a Euclidean geometry:, y, z) (x + las, y + may, z naz) (2.3)where ax, ay, a, are constants for all integers /, m, n. Here, the geodesics between twopoints are not unique; there is more than one way of connecting two points in t he space-time. The spatial section thus obtained has the multiply connected topology of a three-torus. Since the three-torus model is one of the simplest models for multiply connectedtopology [Sokolov & Shvartsman 1974, Gott 1980, Weeks 1985, Ellis & Schreiber 1986],this model is discussed in some detail below in Section 2.3.Table 2.2 (see ref. [Weeks 1985]) lists typical three-dimensional manifolds for differentkinds of universes. The following topologies in Table 2.2 are compatible with FLRWuniverses: three-sphere (S3), ordinary Euclidean space (P), hyperbolic space (H3) andthe three-torus (T3). The Seifert-Weber dodecahedral space in Table 2.2 consists of aChapter 2. Topology of the Universe^ 12dodecahedron whose opposite sides are identified with an added three-tenths clockwiserotation. The S3, E3, H3 topologies are all simply-connected while the three-torus andthe Seifert-Weber space are multiply-connected. Weeks cautions that even if our universeconnected with itself in some way as in the three-torus, this would not necessarily meanthat our universe curves around in some four dimensional space.Elliptic Euclidean Hyperbolicclosed 3-sphere(S3)3-torus(T3)Seifert-Weberdodecahedral spaceopen none ordinary Eucl.space (E3)hyperbolicspace (H3)Table 2.2: Typical global topologies for FLRW universes: The first row lists the threepossible FLRW geometries. A global topology of the universe is given for each kind ofopen or closed universe.We should mention that there are a large number of different topologies for eachvalue of the curvature constant. Aside from those mentioned in Table 2.2, there is awide variety of topological structure associated with different kinds of unit cells from anynumber of possible identifications used to identify the opposite faces of the fundamentalpolyhedra (unit cells). Other examples of closed three-dimensional manifolds which givemultiply connected topology are discussed in an excellent general discussion of topologywritten by W.P. Thurston and J.R. Weeks in Sci. Am. July 1984, pg. 108.Recent mathematical studies have shown that most finite topological models of spaceare three-dimensional manifolds with locally hyperbolic geometry [Hayward 1992]. AsTable 2.1 shows, hyperbolic models with 1-2 < 1 is an open finite universe that is foreverexpanding. Unfortunately, open universe models contrast with the widely popular infla-tionary model [Guth 1981] which predicts CZ = 1. However, since the abundance of lightChapter 2. Topology of the Universe^ 13elements from primordial nucleosynthesis strongly constrains the baryon content of theuniverse (gb 0.1), the inflationary model requires nonbaryonic matter which has yetto be discovered.2.2 Considerations from Quantum Gravity and InflationAdditional theoretical considerations could also describe how the global topology is as-cribed from initial quantum processes in the very early universe. In the Planck era whenthe universe was less than 10' seconds old, the structure of spacetime must be explainedby a quantum theory. Gravity and the spacetime manifold are incorporated into a singlephysical entity in the very early universe. The quantization of gravity is required: thelarge scale description of gravity and the quantum mechanical description of spacetimeare the same. Currently, there is no successful model or theory for quantum gravity.The topology is "frozen" into the universe after the Planck epoch since general relativ-ity preserves the topology through evolution of the Einstein field equations. The globaltopology of spacetime is what remains after the quantum gravity-spacetime symmetry isbroken. Present observations of the topology are the same for any observer at any giventime since the Planck epoch.One can consistently include the inflationary scenario within a small universe model[Barrow & Tipler 1985, Ellis & Schreiber 1986]. The original inflation model [Guth 1981]and its descendants have been invoked to deal with the flatness and the horizon problems[Peebles 1993] which plagued the Big Bang world model for a time. If inflation occursin a small universe, homogeneity and isotropy could be guaranteed to occur on scalesmuch larger than the identification scale. The scale of the horizon at decoupling wouldbe several times larger than the identification scale so that in pre-inflationary times,the universe would really be very small. More importantly, Morrow-Jones and WittChapter 2. Topology of the Universe^ 14[Morrow-Jones Si Witt 1993] have shown that inflation is not excluded from a universesolely on global topological arguments.2.3 Small Universes and Three-Torus TopologyOne possibility currently attracting attention is that the universe might be spatially finitewithout boundary and small enough in size that photons could have circumnavigated theentire universe at least once in the time between the initial singularity and the currentepoch. Models with such a feature are called small universe models. The spatial sectionsin these models are finite and the present scale of a small universe is smaller than thepresent size of the horizon.Many would consider a small universe as a universe with high spatial curvature. Theradius of curvature of the universe [Kolb & Turner 1987, equation 3.18] can be obtainedwith equation (2.2) and one obtains'1Rcurv a(t)1k1-1/2 cH0 (2.4)_A universe with a small radius of curvature (large spatial curvature) requires a massdensity much larger than the critical density. Recent observations and estimates of themass density indicate that we do not live in such a universe. Here, we consider smalluniverse models with zero spatial curvature.An example of a flat small universe model is obtained by taking a cubic fundamentalvolume element and topologically identifying the opposite faces of the cube. The cube isa spatial section of flat FLRW cosmology. Figure 2.2 shows how the identification takesplace: face 1 is identified with its opposite face 2, face 3 with 4, and face 5 with 6; eachpair of faces is simply identified without rotation. The resulting small universe has aspatial section with the multiply connected topology of a three-torus (T-3).'The Gaussian curvature is given by 1/R urv = I a(t)2Chapter 2. Topology of the Universe^ 15Figure 2.2: Identification of the fundamental cube volume element for the simplestthree-torus topology: each pair of opposite faces is identified without rotation.The three-torus is a homogeneous three-manifold because every point in the manifoldhas the same local geometry of Euclidean space. The average length scale for the three-torus model, a, may be defined by the volume of the fundamental element, a' = axayaz.This small universe model is considered "compact" because the volume of the three-torusunit cell is finite. The identifications of the cube shown above give the fundamental unitcell or compact three-manifold. There should be no confusion between the cosmologicalexpansion factor, a(t) shown in equation (2.1), and the identification scale of the unitcell, a, defined above.We consider what an observer would measure along the surface of the manifold withan example. One imagines forming a tube from a rectangular piece of paper by tapingthe top and bottom edges together. An observer on the surface of the tube walkingoff the top edge automatically re-enters from the bottom. This observer would measurezero curvature on the surface of the tube and can travel infinitely far in a straight linewhile visiting the same regions on the surface repeatedly. An identification scale mayChapter 2. Topology of the Universe^ 16be detected for which repetitions occur periodically in the axial direction. If the tworemaining edges of the tube are taped, the observations are very much the same exceptthat the identification scale in the orthogonal direction may not be the same as thatfound earlier. The resulting object is a doughnut-like surface with zero thickness; this isthe two-torus manifold. However, the identification scale would be the same everywhereon the surface if this thought experiment was performed with a square piece of paper.This identification can be mathematically performed in all three dimensions to give thethree-torus topology described above.To examine what one sees in a three-torus, it is useful to develop the following picture.To an observer, a three-torus will appear as a room whose walls, ceiling and floor arecovered with mirrors. If the room only contains the observer, the observer sees imagesof himself in the three-torus in every direction except that there is no mirror reversalof images. An object leaving the cube on one side immediately re-enters the cube fromthe corresponding spot on the opposite side. The universal covering space appears asan infinite rectangular lattice of images which extends in all directions because any lineof sight is propagated indefinitely across the three-torus. The three-torus remains finite,however, because the infinite array of images originate from the same object.The universal covering space is obtained by indefinite repetition of the unit cell orfundamental volume. Figure 2.3 shows a two dimensional representation of the universalcovering space for a three-torus topology. The identification scale, a, is much smallerthan the horizon. An observer sees the portion of the covering space which, at that time,lies within his particle horizon out to constant redshift z. A thick border highlights theunit cell with equal side lengths a. The observer is shown by the asterisk symbol whilethe object being imaged is indicated by the small filled ellipse at the center of the cell.Survey beams emerging from the observer shown in the vertical direction detect oppositeimage pairs of the observer. Each member of an opposite image pair is at the samea6(ponstantChapter 2. Topology of the Universe^ 17Figure 2.3: Two-dimensional representation of the covering space for the three-torus: theunit cell with equal side lengths, a, is highlighted with a thick border. The asterisk symboldenotes the observer while the small filled ellipse at the centre of the unit cell denotesthe object whose images are being observed. The observer looks out into the coveringspace within their particle horizon to constant redshift z. Survey beams originating fromthe observer typically do not detect many opposite twin images of the object.Chapter 2. Topology of the Universe^ 18distance from the observer. The remaining pair of opposite survey beams shown is moretypical in that very few opposite twin images of the object are detected.A three-dimensional view of this covering space is shown in Figure 2.4 taken fromthe cover of Weeks (1985). We see that the unit cell in the immediate foreground isFigure 2.4: This is how Einstein sitting in his living room would appear if his living roomwas the fundamental volume in a three-torus universe. Even though there is an infinitearray of images within the "lattice," the three-torus small universe is finite in size.Albert Einstein's living room. Einstein is sitting in the chair situated approximately atthe centre of the room and a table in the far left corner is the second object in the room.Just as the labelling indicated in Figure 2.2, we can explain the images in Figure 2.4 interms of abstract gluing. The left wall is glued to the right wall, the front wall is gluedto the back wall and the floor is glued to the ceiling.Chapter 2. Topology of the Universe^ 19Within the three-torus, physical processes evolve just as they would in ordinary Eu-clidean space. The processes of star formation and galaxy formation, for example, wouldoccur no differently than in a simply connected universe. If the matter distribution isperfectly homogeneous in the unit cell, the resulting small compact universe is physicallyindistinguishable from an ordinary simply connected universe with flat HAW cosmol-ogy. If the matter distribution in the fundamental cell is not perfectly homogeneous,then observers see inhomogeneities periodically tiled across the sky on scales larger thanthat of the cell. On scales much larger than that of the fundamental cell, the periodictiling smooths out to apparent homogeneity.Chapter 3Observations and Astrophysical Considerations3.1 Possible Observational Evidence for Multiply Connected TopologyIn principle, the topology of the universe could be determined from observations ofgalaxy and quasar distributions at intermediate to high redshifts. The largest struc-tures observed constrain the minimum small universe identification scale, a, which ismuch smaller than the present Hubble scale, given by LH = CI10-1 3000ho-1 Mpc. Fur-ther structure beyond the minimum scale results from the multiple imaging of objectswithin the fundamental volume.3.1.1 Lower Limits from Surveys of Galaxies and Clusters of GalaxiesA number of recent observations do not rule out the existence of a small universe. Sokolovand Shvartsman (1974) obtained a 280ho-' Mpc from Abell cluster richness data. Froma map of galaxy counts based on the Shane-Wirtanen sample (1967, Soniera kr, Peebles1978), Gott (1980) estimated a > 2004-1 Mpc from the absence of multiple images ofgalaxy clusters in the Serpens-Virgo supercluster. These observations showed that amultiply connected topology for the universe could not be excluded.Stronger lower limits could be obtained with current and future large deep galaxy sur-veys. Some data suggests that the distribution of galaxies appears smoother on lengthscales larger than a few hundred megaparsecs while other data exhibits clustering struc-ture on comparable scales. There is some optimism that the next round of deep surveys20Chapter 3. Observations and Astrophysical Considerations^ 21will solve this puzzle.3.1.2 Other ObservationsSmall universe models have been invoked to explain the apparent periodic peaks in thequasar redshift distribution [Fang 1990], apparent periodicities in pencil beam galaxysurveys [Broadhurst et al. 1990, Hayward & Twamley 1990] and isotropy in the cosmicmicrowave background [Ellis & Schreiber 1986, Fang & Mo 1987a]. Additional observa-tions which may be explained by multiply connected topology include neighbouring pairsof quasars, open-ended clustering scales and apparent quasar-galaxy associations. Inorder to assess the plausibility of these explanations, it is important to check carefullyconstraints observational data place on these nonstandard models [Fang k Mo 1987b].One finds a small probability of observing many close neighbouring quasar pairs withvarying redshifts that are seen coincidentally in a chance projection. A multiply con-nected topology can arrange images of different redshifts to appear in the same directionfor the observer. In that case, the objects which produce the images of close quasar pairswould be placed fairly close to the observer in the unit cell.Periodic perturbations in the matter distribution could occur within the fundamen-tal volume in a multiply connected topology over length scales corresponding to co-moving lengths at early quasar epochs. These perturbations could cause periodicitiesin the redshift distribution of emission and absorption lines from quasars [Fang 1990,Fang Si Sato 1985].The upper limit on galaxy clustering length scales in a simply connected topology canbe as large as the horizon scale. The latest observations show that the largest clusteringlengths are at the extent of the galaxy surveys. In a multiply connected topology, themaximum clustering scale is bound by the size of the small universe. Future surveysshould set better limits on clustering scales, especially where the zero-point clusteringChapter 3. Observations and Astrophysical Considerations^ 22length scale occurs.Density fluctuations in the cosmic microwave background (CMB) on large scales ina multiply connected topology were shown [Fang Mo 1987a] to be smaller than thosein a simply connected topology. This may be explained by the absence of correlationson scales larger than the size of a small compact universe. Fang and Mo claim thatthis result is independent of the specific type of multiply connected topology and onlyrequires that the repetition scale of a small universe be smaller than the present horizonlength scale. Further work in the CMB needs to be done for setting upper bounds on theCMB anisotropy in a possible small universe. More discussion on the impact of CMBanisotropy on small universe models is found in Section 5.4.Quasar-galaxy associations have proven difficult to explain in the past. High red-shift quasars are found in close proximity to lower redshift galaxies to form apparentlyphysically related associations [Arp 1987]. A multiply connected topology can explainquasar-galaxy associations as objects which form from the same clump of matter. Imagesof these objects are seen projected in the same direction. It is entirely possible that thequasar which formed at an earlier epoch evolved into the galaxy seen to be aligned withthe quasar itself. Since quasars have short lifespans 10" years), their remnants may beassociated with galaxies in the surrounding neighbourhood. The low redshift galaxy andthe high redshift quasar observed in these associations may be explained in this manner.This may also help explain the evolution of quasars near galaxies.However, gravitational lensing has been invoked as a more popular explanation forthe common origin of close quasar-galaxy associations [Schneider et al. 1992] withoutresorting to multiply connected topology. A foreground galaxy or cluster of galaxies actsas a gravitational lens for a high redshift background quasar, thus producing the closeassociation between a low redshift galaxy and an image of a high redshift quasar. A faintquasar when lensed could appear as luminous as the foreground galaxy.Chapter 3. Observations and Astrophysical Considerations^ 23Finally, if the small universe scale, a, is found to be smaller than the horizon scale, thenwe could detect all the luminous and dark matter in the universe. Current observationsshow that matter continues to congregate into large structures at increasing length scales.In this small universe model, we would have a possible upper limit on the length scaleup to which matter can assemble itself into structures. There would be little troubleexplaining the now discrepant mass density measurements on small (few Mpc) and largelength (tens to hundreds Mpc) scales because all the matter in the universe would becontained within the fundamental volume. This would prove very useful in resolving anydoubts on 52 by observations of images of galaxies and clusters of galaxies.Chapter 4Constraints for the Minimum Scale of A Small Toroidal UniverseAs a topological model of the universe, the three-torus topology is the simplest to test fora possible minimum scale and observational consequences may be predicted. We saw inChapter 3 how constraints on the minimum size of a three-torus universe were obtainedfrom surveys of galaxies and clusters of galaxies. Here we discuss recent attempts toconstrain the minimum size of a small three-torus universe with quasars.Demianski and Lapucha (1987) looked at catalogs of quasars for quasar pairs inopposite directions and did not find any significant excess of opposite pairs compared tothe number expected for a random distribution. Effects such as the gravitational lensingof images by clusters of galaxies, random velocities of objects being imaged and the finitelifetime of quasars were considered. Both gravitational lensing and random velocitieseffects produce an angular offset in the opposite images of a quasar. The finite lifetime ofquasars indicates that an opposite pair of images is observed if both images have similarredshifts to some given tolerance representing the interval of lookback time for which thequasars remain visible.Fagundes and Wichoski (1987) attempted to determine the three principal axes of apossible three-torus unit cell by searching for opposite pairs of quasars from the Hewittand Burbidge (1987) catalog of quasars. Fang and Liu (1988) tried constraining the min-imum scale of a three-torus universe based on Fagundes and Wichoski data. Since effortswere made to constrain some minimum physical size for such a nonstandard universe, wefirst explain why quasars have been used to probe the universe for the topology.24Chapter 4. Constraints for the Minimum Scale of A Small Toroidal Universe^254.1 Quasars as Probes at Intermediate RedshiftsIt is interesting to examine why quasars have become such useful objects to probe theuniverse for evidence of topology at early times. We know that there are a large numberof quasars and that they are unusually luminous if their redshifts are believed to becosmological. No nuclear processes can explain the large luminosities emanating fromthese objects in a small physical volume. It is believed that quasars are powered by theaccretion of matter onto a supermassive black hole 108 solar masses) in the centresof host galaxies. A discussion of the physics of quasar structure is found in Weedman(1986).For a zero curvature FLRW universe, the elapsed Hubble time, tH, at redshift z isgiven by [Kolb & Turner 1987, §3.2]2tH = 3H(1 + z)-3/2.^ (4.1)0The Hubble time is equal to the present value for the age of the universe when the redshiftis zero. It is known that the comoving space density of luminous quasars has a relativelyflat peak near a redshift of two [Hartwick Schade 1990]. From equation (4.1) the peakin the space density occurs at a time when the universe was merely 19% of its present age.That epoch where the space density of quasars is a maximum gives clues to conditionswhich led to the formation of quasars and possibly galaxies.Classical cosmological tests have yet to be applied successfully to quasars because theevolution for these objects is still not well understood. Nevertheless, quasar number countsurveys show that the universe is mostly isotropic on large scales and spatial clusteringof quasars show that the universe is homogeneous on large scales as well. Althoughthe space density of bright galaxies exceeds that of quasars for z < 2, there are morebright quasars at z 2 than at present. A large number of luminous quasars at suchlarge lookback times serve as possible probes of the spatial topology. Since the topologyChapter 4. Constraints for the Minimum Scale of A Small Toroidal Universe^26remains unchanged from the Planck epoch, a large number of bright quasars provides anexcellent set of "bright field markers" for possible observational signatures for a multiplyconnected topology. Further discussion on the spatial density of quasars is found inSection 5.3.1.4.2 Constraints from Opposite Pairs of Quasars : Fang and Liu LimitFang and Liu argued that if the spatial sections of the universe have three-torus topology,images of distant quasars should have exact twin images on the opposite side of the sky.Since these images are expected to be equidistant from the observer, any evolution ofthe quasars occurring since the time the light was emitted to the time the images areobserved can be ignored. Fagundes and Wichoski searched the Hewitt and Burbidgecatalog of quasi-stellar objects (QS0s) for opposite twin images. Excluded were QS0swith redshifts smaller than one and galaxies due to possible confusion in identifyingimages from a large population of sources at low to moderate redshifts. Fang and Liuexamined the Fagundes and Wichoski QS0 subsample for the frequency with whichopposite quasar pairs occurred to constrain the minimum size of a three-torus universe.A list of 47 pairs was obtained where the QSO images in opposite directions werefound to within a two degree square and to have the same redshift to within five percent.Further observations revealed spectral differences in 41 pairs, leaving six candidate pairs.However, the Hewitt and Burbidge QS0 catalog is not complete which makes a directstatistical analysis difficult. On the other hand, opposite regions of the sky have beenexamined in the 47 fields reported by Fagundes and Wichoski. This apparent low rate ofobserving pair candidates in those fields suggests that the expected likelihood of detectingpaired QS0 images in a given set of opposite two degree patches of the sky is less than90%. Fang and Liu estimate that this condition is met when the number of unit cellsChapter 4. Constraints for the Minimum Scale of A Small Toroidal Universe^27within the survey limit is no more than 90% of the number of two degree square patches(a = 2°) in the sky. That is,4steradians- N3 < 0.93^—^(a. w/180)2or N < 13,^(4.2)where N is the survey limit expressed in terms of unit cell or toroidal diameters. Thepresent radial distance r(z) to an object observed at redshift z for the zero curvatureFLRW model is given by2cr(z) = -170(1 — (1+ z)-1/2). (4.3)The Fagundes and Wichoski list contains QS0 images with redshift limit of z 2. Withequation (4.3) above, the upper limit on N in equation (4.2) corresponds to a lower limiton the toroidal diameter, a:r(z)a >N = 2001q,1 Mpc.This is widely quoted as the most stringent limit on a available.Two assumptions were made in the derivation of equation 4.4: (i) QS0 images aredistributed uniformly between patches of the sky along different lines of sight and (ii)QS0 images are all oppositely paired. We find that these assumptions are invalid. First,images of a particular QS0 in a three-torus topology will not be distributed uniformlywith respect to different lines of sight, but are actually clustered along lines of sightparallel to the axes of the unit cell. This slightly increases the chance of observing noimages at all in a given survey. Second, and more crucially, when the observer andobject are not coincident within the unit cell, QSO images of the object generally do nothave twin images on the opposite side of the sky. Figure 2.3 shows that all images ofthe observer are always found oppositely paired but that images of the object are not.Details of results from our calculations bear our these objections.(4.4)Chapter 5Simulations and Discussion5.1 Computational SurveyWe have performed simulations to test quantitatively the effect of Fang and Liu's arbi-trary assumptions on the probability of finding opposite QSO image pairs in a three-torusuniverse. The basic elments of our computations are described as follows. A repeatingthree-dimensional lattice is constructed with an object at the origin and one observerplaced in the cubic unit cell. The placement of the object at the origin does not violatethe cosmological principle because the flat three-torus is homogeneous. Every integer ver-tex corresponds to an image of the object. A line-of-sight direction is chosen randomlyand the number of detected QS0 images is calculated as the number of integer verticesfalling within some specified angle of the line of sight and closer than N lattice spacingsto the observer. The specified survey angle with halfwidth, 0, is chosen to have the samecoverage area as a two degree square to match the Fagundes and Wichoski report; thatis, the solid angle for survey angle, 0, is the same as the solid angle for a two degreesquare:C2(2° square) = St(circ.beam)(2 7180)2 = R-020 = 1.13°.^ (5.1)For every detected vertex a search is made for a vertex in the opposite direction at thesame distance from the observer. Results are averaged after searching a large number of28Chapter 5. Simulations and Discussion^ 29line-of-sight directions.5.2 Computational AlgorithmThe algorithm used in our two calculations was generally the same with exception tothe manner in which the initial observer position was set. In the first simulation, theobserver was set to one constant position in the unit cell while in the second simulation,the observer was set to 100 random positions within the cell. Presented below are moredetails of the algorithm executed in these computations.The survey depth was assumed to be at maximum redshift z = 2. The cell diameterwas set to an initial value N = 8 which was subsequently incremented by one to N = 30.From equation (4.4), these limits for N corresponded to unit cells with scales a = 317h0-1-Mpc and a = 84.5110-1 Mpc, respectively. These limits contained a reasonable range ofvalues within current constraints for minimum cell size a.For a given value of N, one of 500 survey directions, g, was set in a random directionand intersected the sphere of diameter N. The integer vertex, V, nearest to the intersec-tion in the computational lattice was identified. Two checks were performed to ensureV remained inside the sphere and contained inside the survey beam with halfwidth(equation 5.1). If the checks were successful, an opposite beam, —g, surveyed the an-tipodal direction for the nearest opposite integer vertex 0 with the same redshift fromthe observer as vertex V. The six closest integer vertices neighbouring 0 were found.Redshifts were determined for the opposite vertex 0 and its six closest vertex neighbours.From this group of seven vertices, the vertex whose redshift difference compared to theredshift for V from the observer was the smallest and below five percent was labelled 0'.If 0' was found to be within beam the "image" or vertex pair V — 0' was labelledas an opposite pair.Chapter 5. Simulations and Discussion^ 30The next positive vertex along the original beam C, in the direction from the sphereintersection towards the observer was found and the process of checking the new vertexand finding its opposite vertex was renewed. If all the vertices along Ci were exhausted, anew survey beam direction, -C,+1 was set randomly to recommence the search for oppositevertex pairs. Whereupon searches in all survey directions were completed (+Ci : i1, 500), the number of opposite image pairs for the particular cell diameter, N, wascounted. The cell diameter was then incremented by one to restart the algorithm. Theresults over all survey directions were averaged giving the average number of detectedimages and average number of detected opposite image pairs as a function of N. Resultsare also averaged over all random observer positions in the second simulation. Finally,the probability of finding at least one image pair for each N was calculated. The entireprogram listing used for these calculations is included in Appendix A.5.3 ResultsTypically, the first simulation for constant observer position took less than ten seconds ofreal time to run. The second (full) simulation with random observer position then tookapproximately eight to twelve minutes of real time per run, depending on the computeruser load. Over the course of the simulations, we examined what effect the relativeobserver and object positions within the unit cell would have on the probability, Pi (N),of detecting a minimum of one opposite QS0 image pair as a function of the surveylimit expressed in number of cell diameters, N. With the object located at the origin,probability curves are plotted in figure 5.1 for an observer coincident with the object(open circles) and for observer positions averaged over many random positions in theunit cell (filled circles). A Poisson probability curve shown by the dotted line is plottedfor comparison. The Poisson distribution is given as follows: the probability of detectingChapter 5. Simulations and Discussion^ 31Figure 5.1: The probability of detecting at least one opposite image pair, Pi (N), isplotted as a function of the toroidal diameter, N. The star indicates the Fang and Liulimit, the set of open circles is the result of a simulation with coincident observer andobject positions in the unit cell and the set of filled circles is the result of a full calculationwith the observer position averaged over the unit cell. The dotted line is a probabilitycurve for a Poisson distribution of images.Chapter 5. Simulations and Discussion^ 32a minimum of one QS0 image in a random distribution is 1 — er, where r is the ratio ofthe number of QSO images in the sky to the number of independent survey directions.The set of points for coincident observer and object positions (open circles) are inagreement with the Poisson distribution even though the distribution of images is notpurely random. All QS0 images in this case have opposite twin images at the sameredshift from the observer. However, observations indicate that the sky is definitelynot Poisson in nature; the distribution of galaxies and quasars, for example, is highlystructured, at least in our local neighbourhood. Rather, the impression that the skyappears Poisson is obtained from the random nature of the survey technique in a highlystructured universe (see Figure 2.3). A random distribution of images tends to be more"clumped" in places than a uniform distribution; that is, one can find coherent line-like or void-like structures more easily in a random distribution. In this way, the surveytechnique which relies on scanning through a number of different random directions in theorganized lattice can simulate observations which appear to look at a Poisson distributionof sources in the sky.The pair probability for coincident observer and object positions within the unit cellshows that the Fang and Liu 90% detection limit (shown as a star symbol in Figure 5.1)is overestimated by approximately 50 percent at N = 13. Our simulation obtains a 90%detection level at N = 18 which is almost a factor of two higher than the Fang and Liulimit.An estimation of the pair detection probability is shown by the filled circles fromthe second simulation for which the observer position has been averaged over the unitcell. As expected, the QS0 pair detection probability is much smaller than the firstsimulation. Compared to the calculated pair detection probability, the Fang and Liulimit is overestimated by a factor of ten at N = 13.From Figure 5.1, results from the second calculation give a 90% detection at N 24.5Chapter 5. Simulations and Discussion^ 33which is almost a factor of two larger than N = 13 for the Fang and Liu limit. Thisreduces the reported Fang and Liu limit on a by a factor of about two to a 1004-1Mpc. This corresponds to requiring about six times as many objects within the givensurvey volume; almost a factor of ten in number count is necessary.Note that as the typical survey beam fills an increasing volume of a small universefor increasing N, the pair detection probability for the second simulation (filled circles)exhibits a slower rise than the calculated probability from the first survey (open circles).One expects to find many opposite images of the object if that object is located inclose proximity to the observer in our three-torus model. The randomization of observerposition in the cell samples a greater number of possible observer object configurationsbut the number of exact opposite image pair configurations decreases, thereby lesseningthe chance of finding opposite images. Few images of objects located close to the observerare found and hence, very few opposite pairs of images of these objects are observed. Theordered distribution of images on the sky for a three-torus universe and the random natureof the survey technique combine to give a low likelihood of finding exact opposite imagesof the object.5.3.1 The QS0 Luminosity Function and Comoving Space DensityWe can further understand limits placed on the minimum scale of a small universe bychecking current observational data concerning the comoving space density of quasars atthe appropriate redshift survey depth. One needs an understanding of how the luminosityof quasars and the comoving space density of quasars vary as a function of redshift.The quasar luminosity function, (I)(M, z), at absolute magnitude M and redshift zrefers to the number of QS0s per unit volume per unit magnitude interval (Gpc-3 mag')in a specified redshift range. The luminosity function was derived from QS0 observationsin several major surveys, a list of which is compiled by Hartwick and Schade (1990). WeChapter 5. Simulations and Discussion^ 34consider the luminosity function in the redshift range 1.9 < z < 2.2 because the peak inthe space density of quasars is known to occur at a redshift of about two. The integratedluminosity function (ILF) gives the spatial density of QS0s in Gpc-3 for all QS0s brighterthan absolute magnitude -26 in the blue (A = 440 nm). The ILF is given byT -26L (M) dM = 333 +59 Gpc-3. (5.2)For our purposes here, the quoted error is ignored below.Table 5.1 below lists N, the number of cell diameters in a "sphere" of "radius" z = 2;cesep, the mean angular separation between QS0s from z = 0 to z = 2; the distance scale,a, which is calculated from Equation (4.4) and finally, the number of distinct QS0sobservable observed at redshift z = 2. Since the luminosity function was given in termsof ho = 0.5, the identification scale was calculated with this Hubble parameter and thensubsequently converted to ho = 1. All references to the Hubble parameter below aremade to Ito = 1.Equation (4.2) was used to calculate the mean angular separation between QS0s,asep (fullwidth, in degrees), from z = 0 and z = 2180 ( 31/2asep^-) •^ (5.3)N3This condition is derived within the assumption of one object (for which quasar imagesare seen) per unit cell and that the number of unit cells within a sphere of radius z = 2is about the same as the number of a„p degree square patches. A specified characteristicangle, 0, can be optimized for a survey experiment. When 0 is smaller than asep (pencilbeam survey), one would not expect to see too many QS0 images. If 0 is larger thanasep, one would expect to see many but not all of the QSO images; if 0 is much larger,however, then one would just about see every quasar image in the universe.The observed number of distinct QS0s at z = 2 is related to the scale of the unit cell.The product of the ILF with the comoving spatial volume of the unit cell, a3, gives theChapter 5. Simulations and Discussion^ 35N cesep(°) a (Mpc) No. of DistinctQS0s about z = 2ho = 0.5 ho = 18 4.39 634.0 317.0 84.99 3.68 563.6 281.8 59.610 3.14 507.2 253.6 43.411 2.72 461.1 230.6 32.612 2.39 422.7 211.4 25.213 2.12 390.2 195.1 19.814 1.89 362.3 181.2 15.815 1.71 338.1 169.1 12.916 1.55 317.0 158.5 10.617 1.42 298.3 149.2 8.818 1.30 281.8 140.9 7.519 1.20 266.9 133.5 6.320 1.11 253.6 126.8 5.521 1.03 241.5 120.8 4.722 0.96 230.5 115.3 4.124.4 0.82 208.0 104.0 327.9 0.67 181.8 90.9 235.2 0.48 144.3 72.2 1Table 5.1: The number of toroidal diameters, the mean angular separation between QS0sfrom z = 0 to z = 2, the scale of a three-torus unit cell and number of distinct QS0sabout z = 2 are given. The number of distinct QS0s is calculated from the integratedQS0 luminosity function in the narrow redshift interval about z = 2 given by equation(5.2). As usual, the Hubble parameter is given by 1/0 = 1001/0 km s Mpc'.Chapter 5. Simulations and Discussion^ 36number of distinct QS0s found within that volume at redshift z = 2. We can approachthis another way: suppose now that an observer does see a given number, n, of distinctquasars, all of which are near or at redshift two. The corresponding scale for the unitcell, a(n), can be calculateda(n)3 = a(1)3 • n Or a(n) a(1) = (5.4)where a(1) is the size of the unit cell for one distinct quasar at z = 2.We see from Table 5.1 that N = 13 or a 2004-1 Mpc corresponds to finding 20distinct QS0s per cell at redshift two. What can we say though about the total numberof quasars surveyed? The Hewitt and Burbidge survey lists about 3600 QSO objects andit is known to be incomplete. If this survey only sampled two percent of the total actualQS0 population, the new "limit" N = 24.5 corresponds to three distinct QS0s per unitcell volume at redshift two. Since one need only detect four distinct QS0s to provide astronger limit, this technique of surveying for opposite QSO images is not useful sincewe see that there is a large number of spectrally distinct QS0s for different redshifts.It is easy to imagine detecting a large number of quasar images at the survey depthof redshift two. However, it does not seem obvious or trivial to devise a method whicheasily selects those pairs of quasar images which are opposite from the observer to withina required small redshift tolerance. Nevertheless, an improved method of constrainingthe size of a small universe would be to count the number of distinct quasars in a quasarcatalog up to redshift two. The number of distinct quasars could then be used withequation (5.4) to calculate the minimum size of the three-torus unit cell. For example,if n = 500 distinct quasars were found in some catalog, this would correspond to a unitcell with scale a(500) = 57344 Mpc and N = 4.4 cell diameters out to redshift two.From Table 5.1, this new "limit" (N = 24.5; anew^100ho-1 Mpc) approaches thecell diameter for which a given two degree fullwidth square beam survey encompassesChapter 5. Simulations and Discussion^ 37the entire small universe at redshift z = 2. An observer will see any object in every twodegree square surveyed on the sky at z = 2. Every image detected in this manner wouldhave an opposite image.A number of weaknesses with the Fang and Liu method have been discussed. Asystematic count of distinct quasars from a QS0 catalog out to redshift two would seta better constraint on the minimum size of a three-torus unit cell. If the 90% detectionlimit from the full calculation (shown by the set of filled circles in Figure 5.1) was true,and the unit cell scale was 100ho-1 Mpc, a two degree square beam survey would fillmost of the volume of a small universe at redshift two. Any image detected would bepaired with an opposite image. This would render the Fang and Liu technique of findingopposite pairs useless because we would observe every image in the small universe.Even if the 100h0-1 Mpc lower limit was determined to a high degree of confidence,this would be ruled out by observations from redshift surveys. If a 100h0-1- Mpc wasadopted, the "new" lower limit to the three-torus scale obtained would be ruled outby the largest galaxy supercluster observed in the first Center for Astrophysics redshiftsurvey [Geller & Huchra 1989]. Dubbed the "Great Wall," this structure was found tobe a sheet of clusters of galaxies approximately 1704' Mpc long, 601/0-1 Mpc wide and54-1 Mpc thick. The new limit on the minimum three-torus scale is not valid becausethe supercluster is larger than 100h0-1 Mpc. These results demonstrate that a search foropposite QS0 pairs is not a good method for obtaining a limit on the size of our universe.5.4 Implications for the Cosmic Microwave Background (CMB)We consider what the small universe model suggests for the spatial distribution of thecosmic microwave background. The presently observed angle in the sky, Ocell, whichcorresponds to the present small universe length scale in the Fang and Liu limit at theChapter 5. Simulations and Discussion^ 38decoupling epoch (Zdecf_`-2 1000) 1S given by°cell^200rh(0-zdlecM) pc^2°. (5.5)The present size of the horizon at decoupling is given by [Kolb & Turner 1987, equation9.140]^od„ = 0 . 8 70Q1 /2 (zdec/11 00 )- 1/2^1° . (5.6)Angular scales smaller than 1° correspond to comoving length scales that are smaller thanthe horizon at decoupling while large angular scales 1°) correspond to length scaleslarger than the horizon at decoupling. Structure seen in the CMB on large angular scalesby the DMR instrument 7°) on the COBE satellite [Smoot et al. 1992] is larger thana unit cell at the decoupling last scattering surface. However, the optical depth at thatsurface (8z 0.1zd„) is only a few percent of the size of the unit cell and one subsequentlysees different sections of the cell in different directions. A non-negligible thickness of thelast scattering surface could severely smear the distribution of temperature fluctuationsin the three-torus universe.An apparent spatial pattern in the CMB anisotropy on large scales may be possiblein a toroidal topology [Hayward 1993]. In principle, one could assume some prescribedstandard or non-standard temperature distribution on the last scattering surface andthen proceed to "tile" this surface with unit cells. By further smoothing of the hot andcold temperature "spots" or surfaces inside the fundamental volume, one can hope toreproduce the temperature fluctuations and the correlation function measured by theDMR instrument.5.4.1 Three-Torus Small Universe Models from CMB AnisotropyRecent papers have discussed new constraints on the spatial topology from temperaturefluctuations in the cosmic microwave background. Starobinsky (1993) and Stevens, ScottChapter 5. Simulations and Discussion^ 39and Silk (SSS) (1993) independently claim that the small universe model is no longerinteresting.With no assumptions made on the initial perturbation spectrum, Starobinsky cal-culated temperature fluctuations inside a three-torus universe and found that the lowerlimit for a three-torus universe was a 225044 Mpc or 75 percent of the horizonscale, LH. The SSS calculations for large scale CMB temperature fluctuations in a smalluniverse with three-torus topology contained the following assumptions: (i) an initialn = 1 scale-invariant' inflationary power spectrum of density perturbations, (ii) pertur-bations that were integral multiples of the cutoff wavelength, (iii) equal length sides andorthogonal axes for their three-torus unit cell, (iv) the temperature fluctuation ampli-tude was matched to 1 = 20 and (v) a negligible thickness for the last scattering surface.Their calculations showed that the quadrupole moment was suppressed for "small" scales(a < LH) and when the 1 > 2 results were compared to the COBE observations alongwith a Monte Carlo test, they showed that the minimum scale was a = 24001/0-1 Mpc or80 percent of the horizon scale. Thus, SSS claimed that a "small" universe model is nolonger interesting.However, we emphasize that it is the particular three-torus model along with theassumptions used by SSS in their calculation which is no longer interesting. Althoughthe COBE detection is consistent with inflation, there is yet no firm observational orexperimental evidence of the scalar field supposedly responsible for inflation. It is possiblethat some other non scale-invariant fluctuation perturbation spectrum could result inthe observed temperature fluctuations. As well, the simplest three-torus model usedincorporates equal side lengths and orthogonal axes for the unit cell. Since there is noa priori justification of why the simplest three-torus model was used, other three-torusmodels with different side lengths and non-orthogonal axes (abstract gluing of sides with'Such a spectrum invokes an equal fluctuation amplitude for each length scale entering the horizon.Chapter 5. Simulations and Discussion^ 40rotation) seem just as likely.The SSS calculations show that the low order multipoles are suppressed for smallthree-torus scales. One could implement a simple temperature distribution within anassumed three-torus unit cell of reasonably small length scale which could in principlereproduce the necessary observed quadrupole moment. For example, the following tem-perature distribution could be arranged: the top and bottom third of a unit cell areset to "cold" (2.73°K + 30°1tK) while the middle third volume of the unit cell could beset to "hot" (2.73°K — 30°,11K). One would invoke some way of smoothing the hot andcold boundaries in making "warm and fuzzy" intersecting surfaces. If one allowed fora non-zero thickness of the last scattering surface, the intersecting surfaces could smearout appreciably and could possibly take up a large volume of the toroidal cell at thoseplaces where the "sphere" of the last scattering surface crosses the toroidal cells. Ourpoint here is that there could be a completely different set of initial conditions for a smalluniverse model for which the predicted CMB fluctuations could remain consistent withobservations.The following questions are raised. Can limits on the observed CMB anisotropycompletely rule out all three-torus small universe models? Could these limits also ruleout all small universe models (eg. those with hyperbolic topology)? Or can we showthat these small universe models remain consistent with observed anisotropy limits givensome nonstandard initial conditions within the unit cell? In this way, one may be ableto show that a subset of small universe models, however unlikely, are not impossible.Chapter 6Conclusions and Final RemarksWe have performed simple calculations to test the validity of assumptions made by Fangand Liu in their attempt to constrain the minimum scale for a three-torus universe. Wehave found that the minimum scale derived by Fang and Liu from the Fagundes andWichoski sample has been overestimated by approximately a factor of two from 200 Mpcdown to 100 Mpc. This corresponds to an underestimate by a factor of about six inthe number of images for any given object allowed per unit volume. In many areas ofcosmology, order-of-magnitude estimates are often sufficient. However, a factor of twocan be critical when it comes to explaining periodicities on scales of hundreds of Mpcor observed structure in the CMB anisotropy on degree scales with a small universecosmology. Even if there was some confidence with the 100 Mpc limit derived from theFang and Liu method, that limit is immediately ruled out by the Great Wall superclusterobservation.While a number of large scale observations may be explained with a multiply con-nected universe, there has been a lack of significant data which conclusively supports orrules out the three-torus topology or any other topological model as a likely model for asmall universe. The largest structures are observed to have sizes of the extent of latestsurveys which pushes the minimum size of a three-torus universe close to a significantfraction of the horizon length. Further investigations into the topology of the universecould emphasize more complete search techniques. It is possible that a complete samplingof a relatively small patch of sky (eg. one degree square survey beam) in a narrow redshift41Chapter 6. Conclusions and Final Remarks^ 42interval (z 2) might reveal a coherent pattern of images which is predicted by somechoice of topology. One might also imagine some nonstandard initial conditions withinthe unit cell which could produce a distribution of temperature fluctuations consistentwith the CUBE detection.The small universe is attractive because observations such as spatial isotropy onthe largest scales may be explained. The difficulty lies in obtaining firm evidence forproving or disproving this model. The latest mathematical research in this area appearsto favour topological manifolds with hyperbolic geometry, but no significant researchinto reproducing large scale structure has yet been done with hyperbolic topologies.Additional work in quantum cosmology, the mathematics of topology and observationalcosmology could provide a better handle on the topology of the universe and whether ornot the topology is multiply connected.Bibliography[Arp 1987][Barrow Sz Tipler 1985][Broadhurst et al. 1990][Demiariski & Lapucha 1987][Ellis & Harrison 1974][Ellis Si Schreiber 1986][Fagundes & Wichoski 1987][Fang 1990][Fang & Liu 1988][Fang Si Mo 1987a][Fang & Mo 1987b][Fang Si Sato 1985][Geller Huchra 1989][Gott 1980][Guth 1981][Hartwick Schade 1990]H. Arp, Quasars, Redshifts and Controversies (Interstel-lar Media), 1987.J. Barrow and F. Tipler, Mon. Not. Royal Astr. Soc. 216,395 (1985).T.J. Broadhurst, R.S. Ellis, D.C. Koo and A.S. Szalay,Nature 343, 726 (1990).M. Demiariski and M. Lapucha, Mon. Not. Royal Astr.Soc. 224, 527 (1987).G.F.R. Ellis and E.R. Harrison, Comm. Astrophys. SpacePhys. 6, 23 (1974).G.F.R. Ellis and G. Schreiber, Phys. Lett. A115, 97(1986).H.V. Fagundes and U.F. Wichoski, Ap. J. 322, L5 (1987).L.Z. Fang, Astr. Astrophys. 239, 24 (1990).L.Z. Fang and Y.L. Liu, Mod. Phys. Lett. A3, 1221(1988).L.Z. Fang and H.J. Mo, Mod. Phys. Lett. A2, 229 (1987).L.Z. Fang and H.J. Mo, in Observational Cosmology, eds.A. Hewitt et al. (IAU 124), 461 (1987).L.Z. Fang and H. Sato, Gen. Rel. Gray. 17, 1117 (1985).M.J. Geller and J.P. Huchra, Science 246, 897 (1989).J.R. Gott, Mon. Not. Royal Astr. Soc. 193, 153 (1980).A. Guth, Phys. Rev. D 23, 347 (1981); see also additionalreferences in Kolb and Turner [Kolb & Turner 1987].F.D.A. Hartwick and D. Schade, Annu. Rev. Astron. As-trophys. 28, 437 (1990).43Bibliography[Hayward 1992][Hayward 1993][Hayward & Twamley 1990][Hewitt Sz Burbidge 1987][Kolb & Turner 1987][Morrow-Jones & Witt 1993][Peebles 1993][Ryan & Shepley 1975][Schneider et al. 1992][Shane Si Wirtanen 1967][Smoot et al. 1992][Sokolov Si Shvartsman 1974][Starobinsky 1993][Stevens et al. 1993][Weedman 1986][Weeks 1985]44G. Hayward, talk presented at American MathematicalSociety meeting, Los Angeles (1992).G. Hayward, private communication.G. Hayward and J. Twamley, Phys. Lett. A149, 84(1990).A. Hewitt and G. Burbidge, Ap. J. Supp. 63, No. 1(1987).E. Kolb and M. Turner, The Early Universe (Addison-Wesley), 1991.J. Morrow-Jones and D. Witt, Phys. Rev. D 48, 2516(1993).P.J.E. Peebles, Principles of Physical Cosmology (Prince-ton University Press), 1993.M.P. Ryan and C.S. Shepley, Homogeneous RelativisticCosmologies (Princeton University Press), 1975.P. Schneider, J. Ehlers and E.E. Falco, GravitationalLenses (Springer-Verlag), 1992.C.D. Shane and C.A. Wirtanen, Pub. Lick Obs. 22, part 1(1967); R. Soniera and P.J.E. Peebles, Astr. J. 83, 845(1978).G.F. Smoot, et. al., Ap. J. 396, Li (1992).D.D. Sokolov and V.F. Shvartsman, Soy. Phys.-J.E.T.P.39, 196 (1974).A.A. Starobinsky, J.E.T.P. Lett. 57, 622 (1993).D. Stevens, D. Scott and J. Silk, Phys. Rev. Lett. 71, 20(1993).D.W. Weedman, Quasar Astronomy, (Cambridge Uni-versity Press), Chapter 9, 1986.J.R. Weeks, The Shape of Space, Monographs and Text-books in Pure and Applied Mathematics; 96, MarcelDekker, Inc., 1985.Appendix AListing of Computational ProgramThe following is a listing of the Fortran77 program used for the computations discussedin Chapter 5.C This program searches for back to back images in the sky forC a three-torus universe. The unit cell is assumed to have limitsC (-0.5, 0.5) in each of the x, y and z directions. If an objectC is located at the origin (0,0,0) in a cubic unit cell in this geometry,C images of the object will appear at all integer vertices of aC rectangular three-dimensional lattice. This program placesC an observer at location (bl,b2,b3) in the unit cell and searchesC along a random or chosen line of sight for images of the object.C Only the nearest vertices to the line of sight are searched.C Various useful comments are included.C Created by Mark Halpern : 6 January 1993C Modified by Henry Leeimplicit nonereal hO, c, pi, x, cosx, omega, om, avg, start, ran3, dummyreal b(3), b1, b2, b3, r, gam, phi, g, alf, bet, x, y, z, vlenreal dp, x2, y2, z2, v12, dp2(0:6)real frac(0:6), temp(0:6), ztest, holdinteger iavg, ibegin, k, 1, ir, idir, ndet, npair, ix, iy, izinteger ndt, npt, ix2, iy2, iz2, ixopp, iyopp, izoppinteger n, loop, i, ind, ixtemp, iytemp, iztemp, jinteger ilast, iptr, ifirst, imin, nd(11,8:30), np(11,8:30)integer totobj(8:30), totprs(8:30)character*20 nameC Fang and Liu use Hubble parameter of 50.h0 = 50.c = 3.E5pi = 4.*atan(1.)C These are for counting total detected objects and pairs.C A circle of halfwidth 1.128 degrees contains 4 square degrees in area.C So, we use a two by two degree square beam.C We calculate cos(1.128 deg) = 0.999806 to check whether or not any vertexC found is within the survey beam; this is essentially a dot product.45Appendix A. Listing of Computational Program^ 46X = 1.128COSX = COS(pi*X/180.)omega = 0.25*pi*(2.*x*pi/180.)**2om = (3. / omega)**(1/3)C If desired, an output file is openedC WRITE (6,1100)C1100 FORMAT(' Type an output file name.')C READ (5,1101) NAMEC1101 FORMAT(A20)C Two output files are opened : one called 'zz' and one called 'percent'C unit 3 (zz) : number of images and pairs detected for each RC unit 99 (percent) : chance of finding at least 1 opp pair as function of RC NOTE : R in this program corresponds to N in equation (4.2)name = 'zz'open(unit=3, file=name, status='new')open(unit=99, file='percent', status='new')write(99,989)^989^format('# R^% chance of at least 1 pair obs')iavg = 100avg = float(iavg)start = ran3(-371)C For coincident observer position, iavg = 1C For 100 random observer positions, iavg = 100WRITE (3,7000), iavg7000^FORMAT(' PROBE3.FOR : avg over ,'i3,' position(s) for observer')do 9999 ibegin = 1,iavgC The following sets the position of the observer within the unit cell.C One chooses either the the random number generator or by settingC the position coordinates to constant numbers.do 111 k = 1,3do 222 1 = 1,int(100.*start)222^dummy = ran3(1)ccc print*, 'k dummy : ',k,dummy111^b(k) = dummyb1 = b(1) - .5b2 = b(2) - .5b3 = b(3) - .5c b1 = 0.3c b2 = 0.3c b3 = 0.3print*, 'ibegin^Obs Posn : ',ibegin,b1,b2,b3c Probe for varying sizes of the small universe : R = toroidal diameterAppendix A. Listing of Computational Program^ 47do 888 ir = 8,30R = real(ir)c Probe in 500 random directions for a given size of the celldo 899 idir = 1,500c Each direction possibly finds new pairs; reset counters.ndet = 0npair = 0C Use RAN3 to generate line of sight:GAM = RAN3(1)PHI = 2.*3.14159*RAN3(1)IF (GAM .GT. 1.)^GAM = 1.G = SQRT(1.-GAM*GAM)ALF = G* COS(PHI)BET = G* SIN(PHI)WRITE (6,1006), ALF, BET, GAM1006^FORMAT(3F10.5)C Find intersection of beam with sphere of "radius" R:X = B1 + ALF*RY = B2 + BET*RZ = B3 + GAM*RC Find nearest integer vertex:IX = NINT(X)IY = NINT(Y)IZ = NINT(Z)C WRITE (6,1), IX,IY,IZ1^FORMAT(3I5)2100^CONTINUEC Check if this vertex is still within sphere:VLEN = SQRT( (IX-B1)**2 + (IY-B2)**2 + (IZ-B3)**2 )IF (VLEN .GT. R) GO TO 3000IF (VLEN .EQ. 0) GO TO 5000C Check dot product: is vertex within 2 degrees of beam?DP = ABS( ALF*(IX-B1) + BET*(IY-B2) + GAM*(IZ-B3) )/VLENIF (DP .LT. COSX) GO TO 3000C Image found. Count, then check opposite direction.NDET = NDET + 1NDT = NDT + 1Appendix A. Listing of Computational Program^ 48C Opposite point, on other side of the observer:X2 = B1 - VLEN*ALFY2 = B2 - VLEN*BETZ2 = B3 - VLEN*GAMC Nearest integer vertex in the opposite direction:IX2 = NINT(X2)IY2 = NINT(Y2)IZ2 = NINT(Z2)ixopp = ix2iyopp = iy2izopp = iz2C Check if nearest opposite integer vertex in cone:VL2 = SORT( (IX2-B1)**2 + (IY2-B2)**2 + (IZ2-B3)**2 )DP2(0) = ABS(ALF*(IX2-B1) + BET*(IY2-B2) + GAM*(IZ2-B3))/VL2C Check for distanceztest = abs(v12 - vlen)/vlenC Checking the six nearest integer vertices to opp vertex, 0C Check nearest opp integer vertex, 0 and 0 +/- 100, 010, 001 in vertices.C For arrays frac and dp2: indices 1,2 correspond to +/- x,C indices 3,4 to +/- y and indices 5,6 to +/- zdo 990 n = 0, 3loop = 0if (n .eq. 0) thenv12 = sqrt((ixopp-b1)**2 + (iyopp-b2)**2 + (izopp-b3)**2)dp2(0) = abs(alf*(ixopp-b1)+bet*(iyopp-b2)+gam*(izopp-b3))/v12frac(0) = abs(v12 - vlen) / vlenelsedo 991 i = -1, 1, 2ind = 2*n + loop - 1if (n .eq. 1) thenixtemp = ixopp + iv12 = sqrt((ixtemp-b1)**2 + (iyopp-b2)**2 + (izopp-b3)**2)dp2(ind)=abs(alf*(ixtemp-b1)+bet*(iyopp-b2)+gam*(izopp-b3))/v12elseif (n .eq. 2) theniytemp = iyopp + iv12 = sqrt((ixopp-b1)**2 + (iytemp-b2)**2 + (izopp-b3)**2)dp2(ind)=abs(alf*(ixopp-b1)+bet*(iytemp-b2)+gam*(izopp-b3))/v12elseif (n .eq. 3) theniztemp = izopp + iv12 = sqrt((ixopp-b1)**2 + (iyopp-b2)**2 + (iztemp-b3)**2)dp2(ind)=abs(alf*(ixopp-b1)+bet*(iyopp-b2)+gam*(iztemp-b3))/v12Appendix A. Listing of Computational Program^ 49endiffrac(ind) = abs(v12 - vlen) / vlenloop = loop + 1991^continueendif990^continuec Sort list of fractional z tolerances in ascending order;c smallest tolerance in 0th element of array temp(j)do 444 j = 0,6444^temp(j) = frac(j)ilast = 6do 1000 j = 0,5iptr = jifirst = j + 1do 555 k = ifirst,ilast555^if (temp(k) .1t. temp(iptr)) iptr = khold = temp(j)temp(j) = temp(iptr)temp(iptr) = hold1000 continuec imin is index that points to vertex correspondingc to smallest redshift tolerance below 5%imin = 0777^if (temp(0) .ne. frac(imin)) thenimin = imin + 1go to 777endifc write(6,487) (temp(j),j=0,6)c write(6,489) imin,frac(imin),dp2(imin)c write(6,485) ix,iy,iz,ixopp,iyopp,izoppc write(6,486) (j,j=0,6)c write(6,488) (frac(j),j=0,6)c write(6,490) (dp2(j),j=0,6)485^format('Forw Vertex ',3i3,' Opp Vertex ',3i3)486^format('index ',7i3)487^format('temp array 0-6:',7f8.3)488^formaWfrac array 0-6:',7f8.3)489^format('imin = ',i2,3x,'frac(imin) = ',f8.3,3x,& 'dp2(imin) = ',f8.3)490^formaWdp2 array 0-6:',7f9.6)c Is the chosen vertex within given 5% redshift tolerance AND still in beam?CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC THE FOLLOWING if-then CONDITION IS IMPORTANTCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCAppendix A. Listing of Computational Program^ 50if ((dp2(imin) .ge. cosx) .and. (frac(imin) .1e. 0.05)) thennpair = npair + 1npt = npt + 1elseif ((dp2(0) .ge. cosx) .and. (frac(0) .1e. 0.05)) thennpair = npair + 1npt = npt + 1endifC Done with this vertex. On to the next positive vertex, inwards.3000^CONTINUEC Modified to search towards origin along longest unit vector:IF (ABS(GAM).GT.ABS(ALF) .AND. ABS(GAM).GT.ABS(BET)) THENIZ = IZ-GAM/ABS(GAM)IF (IZ .EQ. 0) GO TO 5000C New IX, IY at this IZ:X = (ALF / GAM) * (IZ - B3) + B1Y = (BET / GAM) * (IZ - B3) + B2IX = NINT(X)IY = NINT(Y)ELSE IF (ABS(ALF).GT.ABS(GAM) .AND. ABS(ALF).GT.ABS(BET)) THENIX = IX -ALF/ABS(ALF)IF (IX .EQ.0) GO TO 5000C New IY and IZ at this IX:Y = (BET/ALF) * (IX-B1) + B2Z = (GAM/ALF) * (IX-B1) + B3IY = NINT(Y)IZ = NINT(Z)ELSEIY = IY - BET/ABS(BET)IF ( IY .EQ.0 ) GO TO 5000C New IX and IZ at this IYX = (ALF/BET) * (IY-B2) + B1Z = (GAM/BET) * (IY-B2) + B3IX = NINT(X)IZ = NINT(Z)END IFC Return to calculate image location.GO TO 2100C Exit or choose new direction.5000 CONTINUEC Tally up the number of opposite image pairs for each survey direction:C WRITE(6,1005), NDET,NPAIR1005^FORMAT(' NDET',I5,' NPAIR',I5)IF (NPAIR.GT .10) NPAIR = 10Appendix A. Listing of Computational Program^ 51IF (NDET.GT .10)^NDET = 10np(npair+1, ir) = np(npair+1, ir) + 1nd(ndet+1, ir) = nd(ndet+1, ir) + 1899^continueC Sum up the number of opposite image pairs for each R:totobj(ir) = totobj(ir) + ndttotprs(ir) = totprs(ir) + nptc About to increment R by one; so, reset some countersndt = 0npt = 0888^continue9999^continueWRITE (3,12)12^FORMAT('^))c Averaging ... and print output to filesc unit 3 (zz) : number of images and pairs detected for each Rc unit 99 (percent) : chance of finding at least 1 opp pair as function of Rc NOTE : R in this program corresponds to N in equation (4.2)do 999 ir = 8,30write(3,11) ir,& nint(float(totobj(ir))/avg), nint(float(totprs(ir))/avg)11^format(' Sphere Radius = ',i3,' Tot # Objs, Tot # Pairs = ',2i5)write(3,1010), ( nint(float(nd(j,ir))/avg),j=1,11 )write(3,1012), ( nint(float(np(j,ir))/avg),j=1,11 )1010^FORMAT('Objects: ',11I5)1012^FORMAT(' Pairs: ',11I5)write(99,895) ir, (500.-float(np(1,ir))/avg)/500.895^format(",i3,10x,f10.5)write(3,12)999^continueWRITE(3,1011)(N,N=1,10)1011^FORMAT('^N= 0',10I5)write(3,12)ENDC end main programFUNCTION NINT(X)NINT = INT( X + 0.5* X/( ABS(X) + 0.0000001) )RETURNAppendix A. Listing of Computational Program^ 52ENDREAL FUNCTION RAN3(IDUM)C (from Numerical Recipes page 199)C RAN3 returns a uniform random deviate between 0.0 and 1.0.C The integer IDUM is set to any negative value to initializeC or reinitialize the sequence.REAL FACINTEGER IDUM, MBIG, MSEED, MZ, MJ, MK, I, II, K, INEXT, INEXTPPARAMETER (MBIG=1000000000,MSEED=161803398,MZ=0,FAC=1./MBIG)DIMENSION MA(55)DATA IFF /0/IF (IDUM .LT. 0 .0R. IFF .EQ. 0) THENIFF = 1MJ = MSEED - IABS(IDUM)NJ = MOD(MJ,MBIG)MA(55) = MJMK = 1DO 11 I = 1,54II = MOD(21*I, 55)MA(II) = MKMK = MJ - MKIF (MK .LT. MZ) MK = MK + MBIGMJ = MA(II)11^CONTINUEDO 13 K = 1,4DO 12 I = 1,55MA(I) = MA(I) - MA(1 + MOD(I+30, 55))IF (MA(I) .LT. MZ) MA(I) = MA(I) + MBIG12^CONTINUE13^CONTINUEINEXT = 0INEXTP = 31IDUM = 1ENDIFINEXT = INEXT + 1IF (INEXT .EQ. 56) INEXT = 1INEXTP = INEXTP + 1IF (INEXTP .EQ. 56) INEXTP = 1NJ = MA(INEXT) - MA(INEXTP)IF (MJ .LT. MZ) MJ = NJ + MBIGMA(INEXT) = MJRAN3 = MJ*FACRETURNEND

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