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Gravitational effects of cosmic strings Vollick, Daniel 1992

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GRAVITATIONAL EFFECTS OF COSMIC STRINGSByDaniel VollickH. B. Sc. (Applied Mathematics) University of Western OntarioM. Sc. (Physics) University of VictoriaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Daniel Vollick, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Pksc.SThe University of British ColumbiaVancouver, CanadaDate 4pril (92.DE-6 (2188)AbstractCosmic strings are line-like distributions of energy which may have formed in the earlyuniverse. Recent simulations of the evolution of cosmic string networks show that stringshave a significant amount of small scale structure (i.e. wiggles). In this thesis we willexamine some of the gravitational effects produced by wiggly cosmic strings.One intriguing possibility is that cosmic strings produced the density perturbationswhich triggered the formation of galaxies and large scale structure. Calculations by otherswhich neglect the wiggles suggest that this scenario is ruled out. Using the Zel’dovichapproximation and taking the wiggles into account I found that the wakes produced bythe strings in cold dark matter can account for the observed large scale structure of theuniverse. Using the Gilbert equation I found that the size of the structure producedin hot dark matter is smaller but comparable to the size of the observed large scalestructure. I also found that the wiggles can fragment the wakes into pieces which havethe mass of a galaxy.One of the interesting observational effects of cosmic strings is the production ofdouble images through gravitational lensing. The gravitational field of a wiggly stringdiffers from the field of a straight string in that the curvature tensor outside a straightstring is zero, whereas the curvature tensor diverges as one approaches a wiggly string.Could this make a significant observational difference? To examine this I calculated theeffect of a wave pulse on the images of objects located behind the string and found thatdifferences exist, but are small.11Table of ContentsAbstract iiList of Tables vList of Figures viAcknowledgement xi1 Introduction to Cosmic Strings 11.1 Introduction 11.2 Formation of Cosmic Strings 51.3 Equations of Motion and Gravitational Fields of Cosmic Strings 81.4 Numerical Simulations 162 Gravitational Lensing Properties of Strings 292.1 Introduction 292.2 Straight Cosmic Strings . . . 292.3 Wiggly Cosmic Strings 332.4 Double Images 462.5 Conclusion. 673 Galaxy Formation in Cold Dark Matter 703.1 Introduction 703.2 Introduction to String Seeded Galaxy Formation 721113.3 The Zel’dovich Approximation and Straight Strings 743.4 Velocity Perturbations 1 833.5 Velocity Perturbations 2 913.6 The Accretion Wake 943.7 Fragmentation of the Wake 983.8 Conclusion 1054 Galaxy Formation in Hot Dark Matter 1074.1 Introduction 1074.2 The Gilbert Equation 1074.3 The Accretion Wake 1144.4 Wake Fragmentation 1214.5 Conclusion 1275 Conclusion 129Appendices 133A 133B 137C 139Bibliography 142ivList of TablesVList of Figures1.1 The form of the effective potential for temperatures above the criticaltemperature 61.2 The form of the effective potential for temperatures below the criticaltemperature 71.3 A box of size dH/2 taken from a Bennett and Bouchet[7] simulation in theradiation dominated era. Notice that the loops produced by the networkare very small and that the long strings are quite wiggly 181.4 This figure shows the string network relaxing to a scaling solution fromdifferent initial densities in the Bennett and Bouchet[7] simulations . . 201.5 The fractal dimension (d) of the strings in terms of Hubble sized units inthe Bennett and Bouchet[16J simulations. The slope of the graph is (d-1) 211.6 A box of size dH taken from a Bennett and Bouchet[7j simulation in thematter dominated era 231.7 The initial conditions for the Allen and Shellard[59J simulation 241.8 This figure shows the string network relaxing towards a scaling solutionfor different initial string densities in the Allen and Shellard[59] simulations 251.9 A box of size dH taken from an Allen and Shellard[59] simulation inthe radiation dominated era 261.10 The fractal dimension (d) of the strings in terms of Hubble sized units inthe Allen and Shellard[59J simulations. The slope of the graph is (d-1). 27vi2.11 Double images produced by a straight string. The string is located at thevertex of the shaded wedge. The source of light is denoted by S and theobserver is denoted by 0 312.12 The motion of a particle as it crosses the wedge. Figure a shows two chartswhich cover the manifold. Figure b shows a particle path getting rotatedas it crosses the wedge (see text for discussion) 322.13 The path of an object with & = 0.05 rad and w = 0 rad. The arrows givethe direction of motion of the image in time 442.14 The path of an object with 0 = 0.5 rad and w = 0.5 rad. The arrows givethe direction of motion of the image in time 452.15 The motion of the double images for f=e2,g=0 projected onto a plane aunit distance away. The string is situated along the vertical axis and thewave is propagating in the positive x direction. Its amplitude is lOx i0and its displacement is out of the page. The initial position of the imagesis marked by a small solid circle. The position of the images at t=0 andt=±2 are also marked. The arrows give the direction of motion of theimages in time. The units of time are related to the units of distance. Forexample if the units of distance are light years then the units of time areyears 56vii2.16 The motion of the double images for f=O, g=e_u2 projected onto a plane aunit distance away. The string is situated along the vertical axis and thewave is propagating in the positive x direction. Its amplitude is lOx iOand its displacement is in the positive Z direction. The initial position ofthe images is marked by a small solid circle. The position of the images att=O and t=±2 are also marked. The arrows give the direction of motion ofthe images in time. The units of time are related to the units of distance.For example if the units of distance are light years then the units of timeare years 582.17 The redshift of the right image as a function of time for f=e2,g=O. Theunits of time are related to the units of distance. For example if the unitsof distance are light years then the units of time are years 612.18 The redshift of the left image as a function of time for f=O, g=e_u2. Theunits of time are related to the units of distance. For example if the unitsof distance are light years then the units of time are years 622.19 The redshift of the right image as a function of time for f=O, g=e_L2. Theunits of time are related to the units of distance. For example if the unitsof distance are light years then the units of time are years 632.20 The initial grid of stars for the f wave. Initially the wave pulse is outside thefield of view so that the string is straight and lies on the x axis. Columnsof stars labeled with the same letters (i.e. b b’) are images of the same stars 64viii2.21 The same grid of stars as in figure 2.19 with the f wave centered at x=0.The displacement of the string is out of the page and the amplitude of thepulse is 10 on the scale used. The pulse is propagating in the positive xdirection. Columns of stars labeled with the same letters (i.e. a a’) areimages of the same stars 652.22 The initial grid of stars for the g wave. Initially the wave pulse is outsidethe field of view so that the string is straight and lies on the x axis.Columns of stars labeled with the same letters (i.e. b b’) are images of thesame stars 662.23 The same grid of stars as in figure 2.21 but with the g wave as shown. Thewave pulse is propagating in the positive x direction. Columns of starslabeled with the same letters (i.e. a a’) are images of the same stars. . .. 683.24 The flow of cold collisionless matter around an infinitely long straightstring (in the rest frame of the string). As the two streams of matter flowpast the string they merge to form a wedge of overdensity (op/p = 1)behind the string. The string is located at the vertex of the angular wedgewhich is cut out of the space 733.25 The triangular surface formed by a triangular wave pulse propagating inthe positive x direction. The velocity of the string is in the positive ydirection 99ix3.26 A cross section of the turnaround surfaces surrounding the surface produced by the triangular wave pulse with = 0.15, h=,= = 1,and a = iü. The outer surface is a turnaround surface. Particles outside this surface are flowing away from the surface produced by the wave.Particles inside the outer surface and outside the inner surface are headinginwards. The inner surface is generated by the jump discontinuity in thevelocity as one crosses the string. Particles inside this surface are given animpulse outwards and are hence moving in the outward direction 1024.27 The fraction (f) of matter in the universe accreted by the wakes plottedagainst the time () at which the wakes formed. For the upper curveI6 = 4 and h=1. For the lower curve I6 = 4 and h = 1174.28 F(a) plotted against a/dH 125xAcknowledgementI would like thank Bill Unruh for helpful and illuminating discussions.xiChapter 1Introduction to Cosmic Strings1.1 IntroductionIn this thesis we examine the gravitational lensing properties of cosmic strings and the rolethat strings may have played in the formation of galaxies. In particular we will argue thatthe small scale structure on cosmic strings can make them much more efficient generatorsof large scale structure in the universe than has hitherto been claimed.In this introductory chapter we will first give a brief overview of cosmic strings anddiscuss why they are of interest. We will then examine in more detail the formation,evolution, and gravitational fields of cosmic strings. We begin with a brief discussion ofcosmic strings.Cosmic strings are line-like distributions of energy which may have formed in the earlyuniverse. Since strings cannot have free ends they are either infinitely long or form closedloops. Cosmic strings can carry waves, which are referred to as small scale structure ifthey are much smaller than the Hubble radius. Since the tension in the string is equalto its mass density these waves propagate at the speed of light.Cosmic strings are of interest because if their energy density is sufficiently large theymay have produced the density perturbations in the matter of the universe which evolvedinto galaxies. To be able to calculate the density perturbations produced by the stringnetwork, we need to know the initial configuration of the string network and how itsubsequently evolves. It is believed that the string network forms as a random Brownian1Chapter 1. Introduction to Cosmic Strings 2walk with most of the string existing in the form of infinitely long strings (referred to aslong strings). The remaining string takes the form of closed loops. Nilmerical simulationsperformed by Vachaspati aild Vilenkin[641 support this belief. The subsequent evolutionof the string network is a very complicated process. Not only do we have to calculatehow individual strings behave iii an expanding universe but we have to take into accountwhat happens when strings cross. Calculations indicate (Shellard[50]) that when stringscross they intercommute (i.e. exchange partners). Therefore if a long string crosses itselfit produces a ioop of string. Similarly if a loop of string intersects a long string the loopwill reconnect onto the long string. Therefore it is not surprising to find that, to getan accurate picture of how the string network evolves, the calculations have to be donenumerically.The basic ideas on how the string network evolves have undergone major changes iiithe last few years. Originally it was believed that the strings were essentially straight onscales of order of the Hubble radius and that the loops produced by the string networkhad a size comparable to the Hubble radills. This scenario was in agreement with theearly simulations of string evolution performed by Albrecht and Turok[4]. The densityperturbations produced by the strings in this scenario have been extensively studied (forexample see Stebbins et. al.[53J). The long strings and loops produce density perturbations in different ways. The long strings give particles a velocity impulse towards thesurface traced out by the string, whereas loops of string attract matter like a point mass(for particles which are far from the loop). It was found that the density perturbationsproduced by the loops were more important than the density perturbations produced bythe long strings.Recent simulations performed by Bennett and Bouchet[7, 8, 9, 10, 11, 12, 13] and byAllen and Shellard[6, 48] show that this original scenario of cosmic string evolution isincorrect. They find that the long strings are very wiggly and that the loops producedChapter 1. Introduction to Cosmic Strings 3by the string network are very small. Since the loops are small they will not be veryimportant in galaxy formation [8, 53] and it will therefore be the long strings that producethe important density perturbations. We will therefore only consider the perturbationsproduced by the long strings.The gravitational field produced by a wiggly string differs significantly from the fieldproduced by a straight string. The space-time outside an infinitely long straight stringis conical. Since a conical space-time is locally fiat any particle initially at rest withrespect to the string will remain at rest. In other words the string does not attract thesurrounding matter. Even though the space-time outside the string is locally fiat it isnot globally fiat. Because of the curvature concentrated at the string, the global metricis equivalent to a fiat space-time minus an angular wedge. Particle trajectories which areinitially parallel and which pass on opposite sides of the string will be focused towardseach other. This focusing is the only gravitational effect produced by an infinitely longstraight string. On the other hand, the space-time outside a wiggly string is not locallyfiat. Particles initially at rest will be attracted towards the string. This attractionproduced by the wiggles will also enhance the focusing of particle trajectories. Thus,the gravitational effects produced by a wiggly string may differ significantly from thoseproduced by a straight string. This is why it is important to examine the gravitationaleffects produced by the wiggles.A substantial portion of this thesis deals with galaxy formation in the ‘new scenario’of cosmic string evolution. We will find that the density perturbations produced bywiggly cosmic strings can be much larger than the perturbations produced by straightstrings. We will also find that strings may be able to account for observations[24] whichindicate that galaxies may lie on the surface of bubbles whose interiors are low densityvoids.The density perturbations produced by the string network depend on the linear massChapter 1. Introduction to Cosmic Strings 4density (ii) of the strings. We now consider two physical effects produced by cosmicstrings which can be used to constrain the value of , for any strings which exist. Thefirst effect involves the production of gravitational radiation by the string network. Asthe string network evolves the rapid motion of the long strings and ioops produces gravitational radiation which will still exist today. This gravitational radiation can producestochastic variations in the timing of a millisecond pulsar. The amount of radiation produced by the string network will depend on and on how the strings evolve. If we assumethat the Bennett and Bouchet simulations accurately describe the evolution of the stringnetwork we can calculate the amount of radiation produced by the string network asan increasing function of i. Timing of the millisecond pulsar by Taylor[54J shows nostochastic variations down to a certain accuracy. This measurement in conjunction withthe simulations of Bennett and Bouchet places an upper limit on the value of. As isshown by Bouchet and Bennett [17] this constraint can be written in terms of the dimensionless quantity Gt/c2 as, Gu/c2 < 4 x 10—6. This constraint will be used throughoutthe calculations on string induced galaxy formation in chapters 3 and 4.Cosmic strings will also produce anisotropies in the microwave background radiation.These anisotropies have been calculated by Bouchet et.al.[18] based on the results fromthe Bennett and Bouchet simulations. The observed upper bound on the anisotropy ofthe microwave background will then put an upper bound on i. This upper bound turnsout to be G/c2 < 5 x 10—6, which is almost identical to the upper bound from the timingof the millisecond pulsar.In the old picture of cosmic string evolution, these values of are too small for thelong strings to create the density fluctuations required to give the large scale structure.However, we will find that the wiggles on the long strings significantly lower the value ofi required for the strings to produce large scale structure. In fact we will find that thevalue of ji required to produce large scale structure, similar to that observed, is close toChapter 1. Introduction to Cosmic Strings 5the upper bound on deduced from the timing of the millisecond pulsar.Another physical effect of the wiggles could be to change the gravitational lensingproperties of cosmic strings. Cosmic strings can produce double images of objects whichlie behind the string. This effect is of interest because it can be used it to search for cosmicstrings. We will focus our attention on the effect the wiggles have on the gravitationallensing properties of a string.This concludes our brief overview on cosmic strings. In the rest of this chapter wewill discuss the formation, evolution and, gravitational fields of strings in more detail.1.2 Formation of Cosmic StringsIn this section we discuss the formation and initial configuration of the string network.We begin by discussing the formation of cosmic strings within the context of a simplemodel.Cosmic strings are believed to form during a phase transition in the matter fieldsin the early universe[35, 36, 37, 39, 40, 67, 81]. The simplest kind of string arises froma phase transition in which a complex scalar field forms the order parameter (phasetransitions in a real scalar field lead to domain walls). At temperatures above the criticaltemperature (Ta) the effective potential, V($), for the scalar field will have the formshown in figure 1.1. Hence for T > T the vacuum expectation value of is zero (i.e.>=0). For temperatures below the critical temperature the effective potential willhave the form shown in figure 1.2. The field will settle down so that II2 i. We willassume that the field at each point can be taken also to have a definite value for thephase 0 (perhaps because of decoherence due to interactions with the environment), sothat the expectation value of will be(1)Chapter 1. Introduction to Cosmic Strings 6V (q5)I4IFigure 1.1: The form of the effective potential for temperatures above the critical temperature.Chapter 1. Introduction to Cosmic Strings 7V(çb)Figure 1.2: The form of the effective potential for temperatures below the critical tern-perature.Chapter 1. Introduction to Cosmic Strings 8where is the value of at its minima and 0 is that phase.As the universe expands after the big bang it cools and its temperature eventuallydrops below T. When this occurs can be taken to have the above non-zero expectationvalue. Due to thermal fluctuations during formation and perhaps due to causality in theexpanding universe, 0 cannot be the same everywhere. After the phase transition it mayhappen that as one moves around a closed loop in space, 0 varies by 2n7r. For nO wemust have < >= 0 somewhere within the ioop. The field will attempt to make theregion where = 0 as small as possible, although there will be an equilibrium thicknesswhere the gradient energy due to (V)2 will balance the energy due V(qS = 0). The linealong which = 0 can clearly have no ends, and thus must either be infinitely long orform a closed ioop. The energy per unit length will be given by the balance between theenergy and V() and is typically assumed to be on the order of 1021kg/m3,fromgrand unified theories.In the next section we will discuss the equations of motion of cosmic strings and thegravitational fields they produce.1.3 Equations of Motion and Gravitational Fields of Cosmic StringsIn the previous section we examined the initial state of the string network. In this sectionwe will examine the equations of motion for strings and the gravitational field producedby strings. The results of this chapter will be repeated used throughout chapters 2, 3and, 4.We begin by calculating the equations of motion for cosmic strings. The motion of astring in Minkowski space generates a two dimensional surface which can be parameterized by two variables T and ci. The action for a cosmic string is taken to bes = _f/_g(2)dTdu (2)Chapter 1. Introduction to Cosmic Strings 9where g(2) is the determinant of the induced metric on the surface and ji is the linearmass density of the string. This action is just the area swept out by the string and isreferred to as the Nambu action.To justify the use of this action consider the action for a continuous mass distribution,s = _fpi/i—2dVdt (3)Integrating over the cross section of of the string gives3_f2ddt (4)where dt? is an infinitesimal length along the string and p is the linear mass density ofthe string. Since the energy of the string is proportional to its length (this is set by themicrophysics of the scalar field), [L will be constant. The proportionality between theenergy of a string and its length also implies that the tension in the string is equal to itsenergy density. Now consider the energy-momentum tensor of this string. It will be ofthe formp0000—p 0 0TIw=00000000since the tension equals the energy density. An energy-momentum tensor of this formis invariant under boosts along the string. This implies that only the component of thestring velocity orthogonal to the string is physically meaningful. Thus ± in 4 shouldChapter 1. Introduction to Cosmic Strings 10be replaced by i. The action then becomes(5)It is shown by Goddard et. al. [30] that the above action is equivalent to the Nambilaction, which justifies our use of the Nambu action.We now calculate the induced metric on the surface traced out by the string. Thisresult will then be used to calculate the equations of motion for a string in a flat background space-time. We will take T to be a timelike coordinate and o to be a spacelikecoordinate. The vectors and form a basis for the tangent space (T(S)) of thestring world-sheet (S). Any vector v’ E T(S) can therefore be written as9x’ 9x’2 (6)Nowv11 — 2 -- 2& 8x’ôx” A2—g- ig=where ab is the induced metric on the surface (S) and is given byaxI1 axvgab=g---5-- (7)with ‘ = T and 2 = u. From gab given in (7) we can calculate g(2) and then find theequations of motion.We now consider the motion of a cosmic string in a Minkowski space. The determinantg(2) is given byg(2) = (th)2x’— ( . x’)2 (8)whereOx11 8x11x1=— x11=—. (9)OcrVarying the action (2) gives the equations of motionChapter 1. Introduction to Cosmic Strings 11—(± X)X’+_[)X— ( x’)±’— 0 (10)8T [(i. — ±2z’2] J [(± x’)2 —By performing a coordinate transformation on the surface it is always possible [32] tofind a gauge in which(11)If we write x’ = (t, (t, u)) the equations of motion become the familiar wave equation,(12)with the constraints(13)The equations of motion tell us that a cosmic string can carry waves which propagatewith the speed of light and the constraint i• §‘ = 0 tells us that the velocity of the stringis perpendicular to the string.Let us now examine the gravitational field produced by a cosmic string. The firstquantity we need is the energy-momentum tensor of a string, which is given by{76]T— 2 {/EL) 0 0(/’L)14—— 0x )} (where L is the matter Lagrangian density, g’ is the four-dimensional metric and g is itsdeterminant. Writing the action asS =— f[f _g(2)4(x — (r, ))drdu]d4x (15)and substituting the Lagrangian L = —i f _g(2)4(xI — x(r, u))drdu into (14) givesT = f da{±v — x’z’v]3(— f(t, a)) . (16)in a fiat background space-time.Chapter 1. Introduction to Cosmic Strings 12The metric in the weak field limit can be written as = + h, where isgiven by[76]h1v4Gfdx’’J (17)‘TR is the retarded time and S = Substituting equation(16) into equation(17)gives=4Gf du[X — xx _±2 (18)R-R•xwhere 1? = — (a, ‘TR) and all quantities in the integral are evaluated at the retardedtime TR.We now calculate the metric, in the weak field limit, for a straight string lying onthe x-axis[701. The more interesting case of a string carrying waves propagating in onedirection will then be considered. For a straight string the position vector of the stringis given by(o,t)=(t,r,0,0). (19)The energy-momentum tensor is then given by10000 —1 0 0TIW = ,u6(y)5(z)00000000This energy-momentum tensor corresponds to a perfect fluid with P = —p, P, = P = 0.The field h,1 is given by00000000= h00100001Chapter 1. Introduction to Cosmic Strings 13whereii = 8Gdn(r/r0) , (20)r2 = x2 + y2, aild r0 (which is of order of the radius of the string) is a constant ofintegration. The line element is therefore given byds2 = —dt2 + dx2 + (1 — h)(dy2 + dz2) (21)The line element can be brought into Lorentzian form by transforming to coordinates(r’,8’) defined by(r’)2 = (1 + 8Gt)(1 — h)r2 (22)(23)In these coordinates the line element takes the formds2 = —dt2 + dx2 + dr’2 + r’2d0’ (24)where 0 < ‘ < (1 — 4G[t)27r. This is the metric of a Minkowski space minus a wedge.Hence the space-time about an infinitely long thin straight cosmic string is conical witha deficit angle (z) given by= 8rG (25)As a more complex case let us consider a string carrying waves propagating in onedirection only (this will be used later in the thesis). We could describe these waves inthe gauge we have been using but it will be more convenient to describe them byt=r , y=f(a+T) , z=g(a±r) (26)where f and g are arbitrary functions of the same argument (i.e. either u + T or a — T).It is easy to show (26) satisfies the equations of motion (10). These solutions representwaves propagating along a string which is situated on the x axis. It is important to noteChapter 1. Introduction to Cosmic Strings 14that the sum of two waves of the form (26) propagating in opposite directions is not, ingeneral, a solution to the equations of motion. For definiteness we will take f and g tobe functions of u=x-t from now on.We now calculate the gravitational field of a string carrying waves of the form (26).The energy-momentum tensor for these solutions has been found by Vachaspati[62] andis given by1 + f’2 + g’2 f’2 + g’2 —f’ —gI #9 I #9 If2+g —1+f+g—f —gT’ = 1u5(y — f)6(z— g)—f’ —f’ 0 0—9’ —g’ 0 0where f’ = In the weak field limit the metric is—[1 + h(f2 + g’2)] h(f’2 + g’2) —hf’ —hg’h(f’2 + g’2) 1 — h(f’2 + g’2) hf’ hg’g, =—hf’ hf 1—h 0—hg’ hg’ 0 1—hwhereh = 4Gt1n{p’[(y— f)2 + (z — g)21} (27)and Po is a positive constant of integration. For f=g=0 this metric reduces to the metricof a straight string, as it should.The above expression gives the linearized (bi << 1) solution. Surprisingly, an exactsolution to the field equations for a wave pulse travelling down the string has been foundby Garfinkle[27]. Since this solution will be used in chapter 2 we will discuss it here. Webegin by writing the linearized metric in coordinates (ll,x,p,) defined byu=x—t (28)Chapter 1. Introduction to Cosmic Strings 15pcos=y—f(u) (29)psin8=z—g(n) . (30)The metric then becomes= + (31)Powhere i/f, is given byf’2+g’ f’2+g’—f’ —g’f’2+g’ —f’2g f’ g’17w- =—1 0—g’ g 0 —1Garfinkle finds an exact solution by writing the exact metric as= ‘j + B(p)M (32)and substituting it into the vacuum field equations. These equations reduce to oneequation for B(p),d1 p dB033dpl—Bdp —The general solution to this equation isB(p) =1- (P)-8G (34)P0where Po and t are arbitrary constants (P0 > 0). Thus if is identified as the linear massdensity of the string Garfinkle’s solution reduces to the Vachaspati metric in the weakfield limit provided thate<<—-<<e (35)PoFor G 106, it is clear that the radius of a cosmic string is much larger than e*(note that e_106 is much smaller than the Plank length) and that e7 is an extremelyChapter 1. Introduction to Cosmic Strings 16large number (larger than any physical scale). Thus the weak field limit is valid in anyregion of interest exterior to the string.The metric can be written in a simpler form by transforming to coordinates definedbyv= —u+x+f’(u)pcosO+g’(u)psin84G 1r=cp0 pa(O<)1c— 1—4GIn these coordinates the line element is given byds2 =2dudv+dr+rq5F( ,r,)du (36)whereT ‘ 2 ‘ 2F = _2po(__)a[f (u) cos cq + g (u) sin c] + f (u) + g (u) . (37)PoGarfinkle’s solution will be used in chapter 2 to calculate the light deflection producedby a string carrying a wave pulse.In the next section we discuss the numerical simulations of cosmic strings which havebeen performed by three different groups.1.4 Numerical SimulationsTo be able to examine the density perturbations produced by the string network we needto know how the string network evolves. In particular we will need to know the averageinterstring separation, the velocity of the strings and the fraction of energy of the stringsthat resides in the small scale structure.Sillce the evolution of the string network, including intercommuting, is too complicated to solve analytically numerical solutions must be used. The numerical evolutionof cosmic strings has been studied by three groups, Albrecht and Turok [2, 3, 4, 56],Chapter 1. Introduction to Cosmic Strings 17Bennett and Bouchet [7, 8, 9, 10, 11, 12, 13, 16] aild, Allen and Shellard[6, 48]. Althoughthe Bennett and Bouchet simulations are similar to the Allen and Shellard simulationsthey both disagree in some aspects from the Albrecht and Turok simulations. The firsttwo are believed to be more accurate so they will be discussed first.We begin with a discussion of the Bennett and Bouchet simulations. They set up theinitial conditions for their simulations using the procedure of Vachaspati and Vilenkin[64]. This consists of dividing a large cube into smaller cubes and randomly assigninga phase to each of the vertices. If the phase rotates through 2ir as we go around theface of a small cube we say that a string passes through that face. This procedureproduces a network of random walks of persistence length o in the large cube. Periodicboundary are then imposed at the surface of the large cube. The physical motivation forthis procedure is that the strings are expected to form with the scalar field having somecoherence length. If the small cell size is larger than the coherence length, the phases atthe various vertices should be random. However, the formation of the strings occurredat very early times and the simulations are relevant for late times. The string networkat these late times is thus determined by the string dynamics rather than any remnantof the formation process. The hope is that the initial conditions actually used will havelittle effect on the actual string distribution after a short evolution time.The Vachaspati-Vilenkin procedure does not involve any length scale. This meansthat the ratio of to the Hubble radius is a free parameter in the simulation. Changingthe initial size of the Hubble radius (keeping fixed) is equivalent to changing the initialstring density.We now consider the results of their simulations. Figure 1.3 shows a box of size dH/2(dH is the distance to the Hubble radius) that has been taken out of their radiationsimulation. The scale factor has increased by a factor of 4.93 at this time. From figure1.3 one can see that although the long strings keep more or less the same direction upCDclCDc÷I-,-. N•CDCDO CD-.CD cCD C (i);1j,-iCDI-o CDCCDC)I- CDCDQi—CDe4 I 0I I’ 00Chapter 1. Introduction to Cosmic Strings 19to scales ‘ di/2 they are very wiggly on small scales 4L). One can also see that theloops produced in the simulation are very small. In fact most of the loops are too smallto be resolved because there is a small scale ioop cutoff in the simulations.The smallness of the loops also means that they are probably not very important inthe formation of galaxies and large scale structure [8].The small scale structure on the long strings is produced when strings cross andintercommute. This process produces four kinks, a right and left propagating kink oneach string. Bennett and Bouchet show that the kink amplitude decays as a2<)>_1due to the expansion of the universe. In the radiation dominated universe they find that> 0.43. Therefore in the radiation dominated universe the kink amplitude decaysasa014. The half-life of the kink is then 140 expansion times in a radiation dominateduniverse. We can conclude that the expansion of the universe is not very effective atsmoothing out the kinks.Kinks can also be removed from the long strings by loop production if sections of thestring with a high kink density have a high probability of self intersecting. This effectdoes occur, but from their simulations it can be seen that it is not effective at keepingthe kink density low. There is also back reaction from gravitational radiation but thishas no effect on scales present in the simulations [16]. Thus kinks are important and thenumerical codes must evolve these discontinuities accurately over long periods of time.One important conclusion that can be drawn from the Bennett and Bouchet simulations is that the long string density approaches what is called a scaling density. That isthe energy density of the long strings decays like radiation (i.e. as ) in the radiationdominated era and like matter (i.e. as in the matter dominated era. Figure 1.4 showsthe relaxation of the string energy density to the scaling solution value for different initialstring densities. It can he seen thatChapter 1. Introduction to Cosmic Strings 20I I I I I I I I I II I I i. i I I10 20 30Figure 1.4: This figure shows the string network relaxing to a scaling solution fromdifferent initial densities in the Bennett and Bouchet[7] simulationspSaT/)i=pLsdH/p 52± 10 (38)which is equivalent topLSt2/IL 13 ± 2.5. (39)Since we are interested in the wiggles on the strings, we now consider the fractaldimension of the strings in the Bennett and Bouchet simulations. The fractal dimensionof a curve (d) is defined throughLccR” (40)where L is the distance along the curve between two points on the curve and R is thestraight line distance between the points (the fractal dimension of a straight line is 1whereas the fractal dimension of a Brownian curve is 2). If the string network reachesa scaling solution the properties of the string network are independent of time if all theChapter 1. Introduction to Cosmic Strings 2121.50Figure 1.5: The fractal dimension (d) of the strings in terms of Hubble sized units in theBennett and Bouchet[16j simulations. The slope of the graph is (d-1).distances are rescaled by the Hubble radius. We therefore expectL RdH(t) (41)where d will depend only on the size scale we are considering and not the time. Figurel.5shows logio(L/R) vs Log1o(R/H(t)). The slope of the curve is (d-1). On large scales wehave d=2 just like a random walk. On small scales d1.12 and thus the strings are notsmooth on these smaller scales. The persistence length is defined as the length scale atwhich the fractal dimension changes from 2 to 1.12. It is also verified that ocH(t)as expected for a scaling solution. Thus the persistence length moves to larger physicalscales as t increases. It is found that in the radiation dominated era‘-‘-‘ dH. In theradiation dominated era it is also found that the energy in the small scale structure isabout 45 percent the the total string energy. This percentage remains approximatelyconstant as the string network evolves.—4 —3—2—1 0Log10( R/H )Chapter 1. Introduction to Cosmic Strings 22This behaviour changes in the Bennett and Bouchet simulations for the evolution ofthe string network in the matter dominated era. Figure 1.6 shows a box of size dH takenfrom a matter era simulation. It is easily seen that the matter era strings have less smallscale structure and thus produce fewer ioops than the radiation era strings. In the matterdominated era Bennett and Bouchet find that <v2 > 0.37, which means that the kinkamplitude decays as a’26. Thus the kinks decay faster in the matter dominated erathan in the radiation dominated era. They also find that the energy in the small scalestructure is about 28 percent of the total string energy in the matter dominated era. Asin the radiation dominated era this percentage remains approximately constant as thestring network evolves.The results of the Allen and Shellard simulations are similar. These simulations arefor strings in the radiation dominated era. To produce the initial conditions for thesimulation they use a modified Vachaspati-Vilenkin method. They also allow non-zeroinitial velocities to be assigned to all the points on the string. Since the average velocityof the strings in flat space-time is they either assign this velocity or zero velocity toeach point on the string. The velocity is chosen to rotate as one moves along the stringwith a coherence length of (the coherence length of the long strings). Figure 1.7 showsan example of the initial conditions for the string network. They find that both initialconditions (i.e. zero and non zero initial string velocities) lead to similar results at latetimes. This gives us some confidence that the behaviour of the string network at latetimes does not depend strongly on the ad hoc initial conditions.Figure 1.8 shows the results of several runs. The final string density approachespLSt2/I = 16 ± 4 at late times. This agrees with the Bennett and Bouchet simulationswhich find pLst2/,u = 13 ±2.5 at late times. Figure 1.9 shows a box of volume (dH)3after 4 expansion times. This is visually similar to results from the Bennett and Bouchetsimulations.Chapter 1. Introduction to Cosmic Strings 23Figure 1.6: A box of size dH taken from a Bennett and Bouchet[7J simulation in thematter dominated era.Chapter 1. Introduction to Cosmic Strings 24Figure 1.7: The initial conditions for the Allen and Shellard[59] simulationWe now consider the fractal dimension of the strings in the Allen and Shellard simulations. Figure 1.10 shows the fractal dimension of the strings in their simulation. Theydistinguish 4 regions.1) small scale: d 12) intermediate fractal: d increases from 1 to 1.253) transition region: d increases from 1.25 to 24) Brownian region: d 2They also find an interesting result for the velocity of the strings which may helpto explain why the large loops produced by the network quickly break up into smallerloops. On small scales they find < v > 0.62c. On length scales of order they find<v > 0.15c. With coherent velocities of only 0.15 large loops will collapse into verysmall volumes producing many tiny ioops.Chapter 1. Introduction to Cosmic Strings 25Physical time tFigure 1.8: This figure shows the string network relaxing towards a scaling solution fordifferent initial string densities in the Allen and Shellard[59] simulations.3025201510___I500 2 4 6 8Chapter 1. Introduction to Cosmic Strings 26Figure 1.9: A box of size taken from an Allen and Shellard{59] simulation in theradiation dominated era.Chapter 1. Introduction to Cosmic Strings 275 I I I I I) I I I I I I IJ .1 I I I I I IIIO/III I I I I4 70A :I2 : 0, —- 0000__V . _aQR0 /ooov_p9d1ø1, //1 I IlIIII1• 11.11.111 I II’IIIIII •IIIIIIII Iio io2 i1 1o io1 to2E/tFigure 1.10: The fractal dimension (d) of the strings in terms of Hubble sized units inthe Allen and Shellard[59] simulations. The slope of the graph is (d-1).Finally we discuss the simulations performed by Albecht and Turok. In the Albrechtand Turok simulations the long strings do not have much small scale structure and thestring network produces large stable loops. They also find that pLSt2/ = 50 ± 25, whichis larger than the value obtained by Bennett and Bouchet and by Allen and Shellard.According to Bennett and Bouchet, and Allen and Shellard the discrepancies betweentheir simulations and the Albrecht and Turok simulations arise because the latter simulations smooth out the kinks on the strings rapidly. This rapid smoothing of the kinksreduces the amount of small scale structure on the long strings. It will also cause theloops which are produced by the network to be smoother and to break up into fewerstable loops than they should. Thus one would expect Albrecht and Turok to have toomany large loops in their simulations, which they do. These large loops are more efficientthan smaller loops at transferring energy back to the long strings. Albrecht and Turokfind that about 50 percent of the energy lost to loops gets returned to the long stringChapter 1. Introduction to Cosmic Strings 28network whereas Bennett and Bouchet find that only about 10 percent of the energy getsreturned. In fact Bennett and Bouchet get results similar to Albrecht and Turok if theyincrease the size of their small scale cutoff (i.e. reduce resolution). This may explain whythe simulations of Albrecht and Turok differ from the simulations performed by Bennettand Bouchet and by Allen and Shellard. We will use the results of the Bennett andBouchet and the Allen and Shellard simulations.To conclude this section let us briefly summarize the important properties of thestring network which come from the simulations performed by Bennett and Bouchet andby Allen and Shellard.1) The density of long strings reaches a scaling solution2) The loops produced by the string network are very small and the long strings are quitewiggly.3) The small scale structure on the strings contains about 45 percent of the total stringenergy in the radiation dominated era and about 28 percent in the matter dominatedera.4) In the radiation dominated era the average separation between the long strings isabout one third of the Hubble radius and the coherent velocities of the long strings isonly about 0.15c.Chapter 2Gravitational Lensing Properties of Strings2.1 IntroductionIn the previous chapter we discussed how cosmic strings may have formed during a phasetransition in the early universe. We then discussed the evolution of the string networkand the gravitational field of strings. In this section we will examine the gravitationallensing properties of wiggly cosmic strings.One possible way to detect cosmic strings is through gravitational lensing. If a cosmicstring lies between us and a distant object it may produce double images of that object.The lensing properties of straight cosmic strings have been investigated by Vilenkin[69]and by Gott [32]. As was discussed in chapter 1, cosmic strings are actually quite wiggly.Therefore to get a realistic idea of the lensing properties of strings we need to calculatethe gravitational lensing properties of wiggly cosmic strings. The research I carried outon this problem, in collaboration with Bill Unruh, will be discussed after we briefly reviewthe gravitational lensing properties of straight cosmic strings.2.2 Straight Cosmic StringsIn this section we examine the gravitational lensing produced by a straight string. Thespace-time outside an infinitely long straight string is conical. This means that the spaceorthogonal to the string can be represented as a plane with an angular wedge cut out.Consider a straight string (at the vertex of the wedge), an observer (0) and a source of29Chapter 2. Gravitational Lensing Properties of Strings 30light (S) (eg. a quasar) as shown in figure 2.11. For simplicity we take the string to beorthogonal to the plane of the page. The shaded region in the figure is the wedge thathas been cut out of the space. The opposite sides of the wedge are to be identified. Tosee what happens to particle trajectories that cross the wedge cover the manifold withtwo charts as shown in figure 2.12a. Coordinates on charti can be taken to be the polarcoordinates (r,8) with 0 r < oc and 0 <8 < 2r — Chart2 straddles the wedge andcan be described by the polar coordinates (r’,8’). In the overlap region (A) let r = r’and 8 = 8’. In the overlap region (B) let r = r’ and 8 = 8’ — . Thus all particletrajectories get rotated through an angle LI as shown in figure 2.12b. From figure 2.11we can conclude that the observer will see double images of any source located behindthe string, within an angle of (A4 is the angular deficit=8irGi/c2).The angular separation of the images (6) is approximately given by£(d + £)‘ (42)where £ is the distance from the string to the source and d is the distance from the stringto the observer. For d- £ and Gji/c2 = 10—6 we find that 6 5 arcsec. It is easy tosee that there is no focusing of the light rays so that both images will have the samebrightness.The expected number (N) of double quasars produced by strings will be approximatelygiven byAQ3VqN (43)where L\ is the solid angle subtended by a strip of width N is the number ofstrings between us and the quasars and Vq is the angular number density of quasars. Thesolid angle subtended by a strip of width is The angular density of quasarsis 240 sr1 [69]. We also expect N8 1 since there are only a few strings per Hubbleregion in which sourcesproduce double imagesIdentifyFigure 2.11: Double images produced by a straight string. The string is located at thevertex of the shaded wedge. The source of light is denoted by S and the observer isdenoted by 0.Chapter 2. Gravitational Lensing Properties of Strings 31S — — —— j1- 0S—— —light rays0Chapter 2. Gravitational Lensing Properties of Strings 32chartichart2(a)(b)Figure 2.12: The motion of a particle as it crosses the wedge. Figure a shows two chartswhich cover the manifold. Figure b shows a particle path getting rotated as it crossesthe wedge (see text for discussion).particleoutgoing particleChapter 2. Gravitational Lensing Properties of Strings 33volume. Thus for ZM = 2, N 1, 1q 240sr’ and G = f6 x 10-6 we find thatN 12116 x i0 . (44)Even for 116 = 4 (which is the largest value consistent with the timing of the millisecondpulsar) we find hat N is only about 0.05. Thus we can conclude that N ‘1 and that itis not too likely that the strings will produce any double images of quasars. In fact noneof the known quasar pairs are good candidates for string lensing.Since galaxies are much more numerous than quasars it may be more profitable tolook for string lensed galaxies. Due to the large number density of galaxies we need tobe able to determine the spectra of galaxy pairs to be sure they are images of the samegalaxy. The angular number density of galaxies for which we can obtain a spectrum is3 x 106sr’ [69]. Therefore we would expect.- 150116 pairs of string lensed galaxies.For 116 = 4 we therefore expect 600 lensed galaxy pairs. For a straight string, thesedouble galaxies would line up in a straight line.In the next section we examine the gravitational lensing properties of wiggly cosmicstrings. In particular we will examine the effect of a propagating wave pulse on the doubleimages of objects located behind the string. This is of interest since recent simulationsof the evolution of cosmic strings show that they are very wiggly.2.3 Wiggly Cosmic StringsIn this section I present the research I carried out in collaboration with Bill TJnruh onthe gravitational lensing properties of curved cosmic strings[74, 75]. This investigation isimportant since, as discussed in chapter 1, recent simulations of the evolution of cosmicstring networks show that strings have significant small scale structure on scales fromthe Hubble radius down to the limit of resolution of the simulation. This investigation isparticularly relevant because of the non trivial curvature near the string. The curvatureChapter 2. Gravitational Lensing Properties of Strings 34tensor diverges as one approaches the string[27]. Clarke, Ellis and Vickers[23] have raisedthe concern that this could completely alter the gravitational lensing properties of strings.Therefore if we are to search for cosmic strings via their gravitational lensing propertieswe mllst investigate the lensing produced by curved strings.We will first calculate, to linear order in perturbation theory, the effect of a sharplypeaked wave pulse on the image of a object located infinitely far behind the string.We will show that this naive pertllrbation expansion may break down for objects whichappear close to the string. Since it is exactly these objects which are of interest we willcalculate the effect of the pulse on their images numerically. To give an idea of how agaussian wave pulse effects the stars behind it we produce a picture of the images of arectangular grid of stars which are located infinitely far behind the string. We will useboth the linearized and exact solutions for the gravitational field of a pulse, as discussedin chapter 1, for these calculations.We now calculate, to first order in perturbation theory, the effect of a sharply peakedwave pulse on the image of an object located infinitely far behind the string. Recall fromthe introduction that a solution to the string equations of motion ist=r , x=u , y=f(a+T) , (45)where f and g are arbitrary functions. These solutions represent waves propagating inone direction on a string which is situated on the x axis. We will examine solutions withf and g functions of ‘a r — u only.The metric, in the weak field limit, for a string carrying waves of the form (45) is (seeChapter 2. Gravitational Lensing Properties of Strings 35section 1.3)—[1 + h(f’2 + g’2)] h(f’2 + g’2) —hf’ —hg’h(f’2 + g’2) 1 — h(f’2 + g’2) hf’ hg’g1111 =—hf’ hf’ (1— h) 0—hg’ hg’ 0 (1 — h)where= df(:), (46)h = ln[p2{(y- f)2 + (z - g)2}] , (47)(48)and Po is a constant of integration.To calculate the location of the images, we examine the null geodesics in the abovespace-time. Initially we will do so only to lowest order in the size of the pulse, i.e. tofirst order in f and g. Terms which are independent of ci (such as those proportional tof or f2) do not occur. Terms independent of f (or g) simply change the uniform (conical)deflection by the string and are of no interest. Thus the lowest order term which wouldgive results of interest is proportional to cf. Since we have restricted our attentionto terms which are linear in f and g, the effects of the two polarizations add linearly.Therefore we can solve for one polarization and later add the solution for the other. Wetherefore take g=0. We will calculate the deflection of light for a short wave pulse (ie. fis non zero only on a small interval in u).The above metric is just the linearized metric. Unfortunately we will not be able todirectly use this to examine the effect of such pulses on the most interesting cases, thosewhere the object lies behind the string so as to produce ‘double images’. We will findthat the deflection produced by a pulse is proportional to-f f(u)dn where r1 is thechapter 2. Gravitational Lensing Properties of Strings 36distance of the photon from the string when it is influenced by the pulse and f(u) is thewave pulse (u=x-ct). Since r scales as ,u for light rays which produce double imagesthe naive effect of the pulse will scale as . This implies that the effect of such pulseson the double images must be calculated more carefully, which will be done later in thischapter.Before we begin the calculation of the light deflection it will be useful to simplify themetric by transforming to coordinates (u,v,r,6), defined byx = l(n + v)2 (49)y = pcos8 + f(n)z = psinOwhere gQu) is taken to be zero, p is given byp = r[1 — (1 — lnr/ro)] (50)and. r2 = x2 + y2. In the new coordinate system, (u,v,r,O), the metric is given by0 (1+)f’cosO —r(1-i-—)f’sinO0 0 0=(1+ft)f’cos6 0 1 0—r(1+—)f’sinO 0 0 (1—c)r2whereh=cdnr/ro (51)to lowest order in a (i.e. it).The equations of motion for particles follows from the geodesic actionS = fLds (52)Chapter 2. Gravitational Lensing Properties of Strings 37where L is a Lagrangian given bydx dxLiL=g-—--— . (53)and s is an affine parameter. The motion of the particle is then given by the Euler-Lagrange equationsd1OL—o 54dsO± 0x12 —For g given by (51) the equations of motion (to lowest order in f with r0 = 1)(55)d2v off’ dr 2 dr dO dO 2—2f sinO———orf cosO(—) =0 (o6)+ (1+ )f” cosO()2 - (1- )r()2 0 (57)d20 1 h o ,, du 2 2d6dr—-(1. + + —)f sin&(-) + = 0 (58)where s is some affine parameter. From (55) we see that we can take s=u by a suitablechoice of the affine parameter.We now examine the deflection of light in the plane orthogonal to the string. Integrating equation (58) givesr2=+fr(1++)fu’sinOdu (59)where £ is a constant. Let H(u) = fr(1 + + )f” sinOdu and r = . It can easily beshown that equation(57) becomesd2w 1 dHdw 1 h(1— ci)w +w2ldndO — w22(1 + -)f cos8 = 0 (60)To order f=0 we have the well-known solution,w = /3sin[TZ c(O—6)] (61)Chapter 2. Gravitational Lensing Properties of Strings 38where, to this order, 6 is the outgoing angle (measured in the plane orthogonal to thestring, relative to the (x,y) plane) of the photon at r = oo. This will also be the outgoingangle when we include terms linear in f, since we will take f=0 when the photon is atinfinity.To write equation (60) in terms of 0 we need to express f” in terms of 0. To lowestorder f” is given bydf£2,84 sin2 (sin2f) (62)where =—(0— 6). Substituting this into equation(60) gives+ (1- )w = /3(1+ ft):in@ - 4. (63)The general solution to this equation isw(0) = /3 sill[\/1— c(8— 6)] + w(0) (64)where w(O) is the particular solution, which satisfies the initial conditions w(6) =w,(6)=O.The incoming photon at r = will have an angle (0) given by the solution of/3 sin[1-c(O - 6)] + w(O) = 0 (65)for near +6. To find 0 set 0 = 6+ + y, where -y is small. Solving for ‘y gives667— (The angle 7 then represents the additional deflection which the ray of interest suffers.To calculate-y we need the particular solution to eqllation(63). It can be written asw(0)= j’ G(0,00)[1+ - (67)Chapter 2. Gravitational Lensing Properties of Strings 39where G(8, 0) is the Green’s function which satisfiesd2G-(0,0w) + (1 — a)G(8,0o) = 6(0 — 0) (68)and the boundary conditions G(8 = 6) = G’(O = 6) = 0. The Green’s function whichsatisfies these conditions is1 0 0<0G(0,00)=____( L_sin[/1—o(0—0o)] 0>00Thus w(0) is given by/32 (1 + ) sin( — 0 ) d dfw(0)= f 2 sinth0 0 0 sin[/1 — c(0 — 0o)J-[sin2o-jd0o (69)where h0 = cdn[r(0o)J and =— c(0o— 6). Letw(0) = f k(8, 0o)(sin20L)do (70)where k(0, 6) = (1 + h0/2) sin(co — 0) sin[/1 — c(0 — 0)]/ sin cf. To simplify thisexpression we integrate by parts twice and take f(6)= () = 0. This gives/32 dk 0 d dkw(0)= —{f(0o)sinqo——j (71)Now let f(u) be sharply peaked enough about u = u1 (O = OQai)) so that 0(n) isessentially constant over the support of f(u). Then w(O) can be approximated by= / [sin2 01) ff(0o)dOo (72)where we have taken f(O) = 0. It can be shown that_[sin2çidO1)]= —sin(20i —36). (73)Substitutingof(0o)d0o=_ff(n)dn (74)Chapter 2. Gravitational Lensing Properties of Strings 40equations(72) and (73) into equation(66) gives(75)The deflection of light is therefore given by=+4Gtf f(u)dusin(201 — 36)/; /;. (76)From this we see that the pulse produces a deflection in the plane orthogonal to thestring which drops off as (where r is the distance of the photon from the string whenit is influenced by the pulse). It is also clear that this additional deflection vanishes ifthe total area of the wave pulse is zero.So far we have calculated the deflection in the plane orthogonal to the string. To findthe deflection in the direction along the string consider the equationd2x 1 , cosO dr dr dii dii 2+ cfr—— rcosO(j) ] = 0 (77)which can be found from equations(49), (50) and (56). To the correct order this can bewritten as+ ‘234fsinqcos(ii — 2b) = 0. (78)This equation has the particular solutionx(O) = _c 3f G(O,Oo)-sing5ocos(8o— 26)d80 (79)where the Green’s function G(O, O) is given by0G(O,&0)=gr2dO rr2dO n nJ ff)OzzOo U>U10Chapter 2. Gravitational Lensing Properties of Strings 41and satisfies G(6, O) = G’(6, O) = 0. Integrating (79) by parts and taking the pulse tobe peaked givesx(O) = f f(Oo)do{O1) sin1 cos(Oj — 26) + G(8, i) cos(2e1 — 36)} (80)Now project the photon’s motion onto the (x,y) plane. The change in the angle,6w,(measured relative to the y-axis) is given by£/3cosw6w = Ax[sinw — ]cosw (81)cos 6where=- J f(u)du cos(2&i - 36) (82)and w is the initial angle (this is found from cosw= B) with A=(1,,0) andB=(0,-1,0) ). We can see that the along the string vanishes if the total area of the pulseis zero.There is a simple expression for the magnitude of the deflection produced by the wavepulse which we will now discuss. The order of magnitude of the maximum deflection,both in the plane orthogonal to the string and in the direction along the string, is givenby where z\ is the angular deficit and= f f(n)dn (83)is the solid angle subtended by the pulse as seen by an observer situated at a distance r1from the pulse.We now examine the time evolution of the image of an object located infinitely farbehind the string, as seen by an observer located at (xo, r0, Oo). This information can beobtained by examining the null geodesics which come in from infinity (at an angle 0) andare received by the observer. At large negative t the wave pulse is at a large negative xand is propagating in the positive x direction. A photon coming in from infinity at thisChapter 2. Gravitational Lensing Properties of Strings 42time will move in a flat geometry until its x coordinate is the same as the x coordinate ofthe front of the wave pulse. The photon then briefly passes through a region of nonflatgeometry, after which it once again moves in a flat geometry. Photons received beforethe pulse passes the x coordinate of the observer will not undergo a deflection. Sincew(8) = 0 after the wave pulse effects the photon we only need to consider photonsreceived when wp(80) = 0.Consider the path of the photon projected onto the plane orthogonal to the string.The angle (8) that the photon comes in from infinity at is given by(84)where32w(O) =— 2 j f(zt)dusin(28 — 36) . (85)2r1The angle (A) between photon’s path and the line joining the observer and string is givenbyA =— c(Oo — 6) . (86)The null geodesics which come from a fixed object at infinity will all have the same 6 butwill in general have different 6. Therefore the observer will see an object with a given 6at the angleA = — c(8 8) + r + rosinow() (87)which contains the time dependent termNow consider the motion of the photon projected onto the (x,y) plane. In this planelet the photon come in from infinity with an angle w relative to the y-axis and be receivedby the observer with an angle w + 6w. It can be shown that the change in this angle (6w)is given by£,Bcosw6w=—x[sinw— }cosw (88)cos 8Chapter 2. Gravitational Lensing Properties of Strings 43where x is given by (82).We now calculate the relationship between £ and /3. The relationship, to order a =f = 0, between £ and /3 can be found by setting ds2 = 0. This gives(89)or£/3=±4/1+c0551 (90)V 1 — cos5sinwThis allows us to eliminate £/3 in favor of w and S in expressions (87) and (88).So far we have discussed the position of the image as a function of the angles 0 andw. We now calculate how the image changes in time (or u). The motion of the photon interms of u, to lowest order, is given by setting a= f = 0 in equation (59) and integrating.The result iscot q5 = cot(80 — ö) +£2/3u0 (91)where u0 is the value of u when the photon is received by the observer (i is the angleof the photon when it is influenced by the wave pulse). We have also taken u1 = 0(i.e. the pulse is peaked at u=0). Since n0 =— t0 and x0 is fixed, equation (91)gives as a function of t (the time the photon is emitted by the observer). Thereforeequations (87), (88) and (91) together give the time evolution of the images of objectslocated infinitely far behind the string. The image of the object, as seen by the observer,will move along a closed path in space. Two such paths, projected onto the plane, aunit distance from the observer and orthogonal to the unperturbed position of the objectare shown in figures 2.11 and 2.13. In these figures the x axis is a line on the planeorthogonal to the unperturbed position of the object (the origin), which is parallel tothe (x,y) plane. The path in figure 2.14 is not closed because we have not included themotion of the object when the photons are received in the nonflat geometry, since thispart of the motion depends on the exact shape of the wave pulse.Chapter 2. Gravitational Lensing Properties of Strings 44Figure 2.13: The path of an object with 0 = 0.05 rad and w = 0 rad. The arrows givethe direction of motion of the image in time.Chapter 2. Gravitational Lensing Properties of Strings 45Figure 2.14: The path of an object with 0 = 0.5 rad and w 0.5 rad. The arrows givethe direction of motion of the image in time.0.228-1.64yx2-0.724Chapter 2. Gravitational Lensing Properties of Strings 46This concludes the calculations using a first order perturbation expansion in t. Asmentioned at the beginning of this section this expansion may be suspect for objectswhich produce double images. For the remainder of this chapter we focus on objectswhich produce double images.2.4 Double ImagesIn this section we calculate the effect of a gaussian wave pulse on both the observedpositions and redshifts of the double images of an object using the exact metric recentlydiscovered by Garfinkle[26}. We use this to ensure that the naive behaviour is properlyhandled. We also calculate the effect of the wave pulse on the images of a rectangulargrid of stars located infinitely far behind the string.To begin these calculations we will now set up the equations of motion for photons.We will also calculate the light deflection for photons which have large impact parameters.This deflection will then be compared to the deflection produced by a point mass movingwith a luminal velocity.We begin by recalling from section 1.3 (expression(36)) that the metric for a wavepropagating along the string can be written asds2 = 2dudv + dr2 +r2dq + F(u, r, /i)du2 (92)wherer ‘I .‘ 2 / 2F= —2ro(——)[f (u)cosc+g (n)sinçj+f(u) +g(u) . (93)ar0Chapter 2. Gravitational Lensing Properties of Strings 47These coordinates are related to the coordinates of the weak field limit (x,p,6) viav = —u +x + f’Qu)pcosO + g’(’u)psinOr = crpc(94)9=. (°)1where p2= (y — f)2 + (z — g)2.Now consider the motion of particles in this space-time. The Lagrangian for particlemotion isL = 2 + 2 +r2b + F(u, r, )n2 (95)The Lagrangian and hence the equations of motion are quite insensitive to the value ofr0 since only factors of appear in equation(93). We will therefore take r0 = 1.The equations of motion for particles in this space-time ared[dv+F’1 — 1OF(du’2dsLds dsi— 20u”dsJ— 0ds2— (96)— r() = —()(°1)[f”(u) cosc + g”(u) sin(r2)= ([f”(u) sill oq — g”(u) coscj()2where s is an affine parameter. From (96) we can choose u=s.For light rays we also have the first integral+ )2 + r2()2 + FQu,r,b) = 0 (97)which follows from ds2 = 0.An important symmetry of these equations for g(u)=0 is the discrete symmetry—. That is, if [v(u),r(u),g(u)J is a solution so is [v(u),r(u),-g(u)}. This symmetry willbe used later.Chapter 2. Gravitational Lensing Properties of Strings 48It is also important to note that (r,) are measured from the string not from the xaxis. Therefore as the wave passes by, the (r,) coordinates shift relative to the (x,y,z)coordinates. To get an idea of the magnitude of the effect of the wave pulse on the lightrays consider the equations of motion in a coordinate system which does not get shiftedby the wave. Define the coordinates Y and Z byY=rcos+f(u) , Z=rsin+g(u) (98)Note that Y and Z are not the same as the (y,z) coordinates introduced earlier whendiscussing the weak field limit The (Y,Z) coordinates defined by (98) are coordinates inflat space-time minus a wedge when f=g=0.The equations of motion for Y and Z are= f”[l— ()‘cos(c — 1)j— ()1g” sin( — 1)çb ()Z” = — (ry—l cos(c— l)$] + ()‘f” sin(c — 1)For (o — 1)lnr << 1 and (Q — 1) << 1 these equations can be written as—— 1)f” mr— (c — 1)g (100)Z —(c—1)g 1nr+(c—1)fHence for f” and g” not too large the motion in (Y,Z) is nearly a straight line. A similarargument holds in the x direction. We therefore do not expect the deflection to be largeeven though the curvature diverges as r — 0.We will now examine the light deflection when the impact parameter of the photon isvery large compared to the amplitude and width of the pulse. We take the pulse to havesupport on a finite interval. Under these conditions the (Y,Z) coordinates of the photonwill change little relative to its distance from the string during the time it is influencedby the pulse. We therefore write Y = Yo + 6Y and Z = Z0 + 6Z where (Y0,Z0) is theposition of the photon when it begins to be influenced by the pulse and (bY, 6Z) is theChapter 2. Gravitational Lensing Properties of Strings 49variation of the photon position during the interaction. For simplicity we will take g=0.Expanding equation(99) to second order in and with g=0 gives= —(cr— 1)f” lnr0 — f”[coso(bY— f) + sinb06Z]—ffl{cos2o[6Z2— (6Y— f)2] — 2sin2q5o(6Y — f)6Z}= (c— 1)f”o + -f”[coso6Z — sin— (102)— f)2 — Z9 — 2cos2o(Y — f)6Z}where = tan’() is the angle of the photon at the time it begins to be influenced bythe pulse. To obtain eqs.(101) and (102) we have neglected higher order terms in (c— 1).It can easily be shown that the light deflection produced by the higher order terms will bemuch smaller than the deflection produced by the first order terms if I(c — 1)lnro << 1and ( — 1) << 1. These conditions will be satisfied in any region of interest outsidethe string.We now consider the light deflection produced by the wave pulse. As we shall showbelow the deflection is related to= f ÔY”du = J Z”du (103)where the limits of integration extend from a time before the interaction with the wavepulse begins to a time after it ends. Thus we wish to calculate Y’ and ZiZ’.To find an expression for 5Y” and t5Z” we use the method of successive approximations. To lowest order (i.e. f=g=0) the path of the photon is given bybY =— VyU(104)= A— Vznwhere (‘vy, VZ) and (Ar, Az) are integration constants. Recall that u=x-t. This meansthat Vy $ unless the initial motion of the photon in the x direction is zero. Substituting (104) into th right hand side of (101) and (102) gives the next correction to the photonChapter 2. Gravitational Lensing Properties of Strings 50motion. It is easily seen that if we neglect terms of order - which are nonlinear in fthen /Y’ aild zZ’ involve terms proportional to f f”du, J’ ‘uf”du, f ff”du and fu2f”d’u.The first integral is zero since f’ is zero at the end points of integration. By integratingby parts the second integral can also be seen to be zero. Thus the only contribution tothe light deflection will come from the terms proportional to f ffdu =— f f’du and tof uf”du, 2f fdu. We find that=‘)cosof f’2du ‘ ffdu[(v—v)cos2o—2vvvzsin2o] (105)= _( ‘)sinoff’2du( ‘)Jfdn[(v —v)sin2o+2vyvzcos2oj(106)To get the next order correction we would substitute the first order correction intothe equations of motion. This only gives terms which fall off faster than and termsproportional to (c — 1)2 (i.e. G2t). These terms are not of interest as they are smallerthan terms already retained so they will not be considered. Therefore /Y’ and /.Z’ willbe given by (105) and (106). For the remainder of this section we will refer to Y’ =and Z’ = as the velocity of the photon in the (Y,Z) plane.We now calculate the deflection () in the plane orthogonal to the string fromthe expressions for /.Y’ and JZ’. Let us denote the magnitude of the velocity changeorthogonal to the initial velocity by tv and the magnitude of the initial velocity in the(Y,Z) plane by VyZ. Since the deflection of the photon is small the deflection angle willbe approximately equal to . The deflection of the photon due to the wave pulse isthen given byc23yro ft2dusin(cbo — ) — 4Gy f fdusin(2o — 36) (107)where ,i3yz = vyz/c and 6 is the incident angle of the photon at r = cc. This deflectiondoes not include the deflection produced by the conical geometry surrounding the straightChapter 2. Gravitational Lensing Properties of Strings 51part of the string. To get the total light deflection we add to (107) the deflection producedby a straight string (i.e. The last term in (107) is the expression derived for thelight deflection in the previous section. The first term in (107) is proportional to f’2 andhence was not found in the previous section where we neglected nonlinear terms in f. Itis this term, however, that will dominate the deflection at large r0, even for small f. For fsmall enough there will exist a region for which the second term in (107) dominates butas r0 increases the first term will eventually come to dominate the deflection.We now compare the deflection produced by a wave pulse of energy E to the deflectionof a point mass of energy E and velocity v —÷ c. To lowest order the motion of the photonis given by= sin(_ ) (108)where b is the impact parameter of the photon relative to the x-axis. The first term in(107), including the g wave, becomes4G,ub r‘2 ‘22 2J[f +g ]du . (109)CThis is identical to the deflection produced by a point particle with finite energy movingat a velocity v —* c if we take the energy of the particle to beE = f[f12 + g’2]du (110)Thus the deflection of light at large distances from the wave pulse is identical to thedeflection produced by a moving point mass with velocity v —f c and energy E given by(110). Since T°° for the string is given by[62]T°°=(y — f)b(z — g)[1 + f’2 + g’2] (111)this is also the energy we would associate with the wave pulse from the energy-momentumtensor.Chapter 2. Gravitational Lensing Properties of Strings 52We now calculate the effect of a gaussian wave pulse on the double images of an objectlocated infinitely far behind the string. We take the following scenario to perform thecalculations. To calculate the effect of a wave pulse on the double images we took thefollowing scenario. The observer is located at x=0, ç& 0 at a distance of io units (eg.light years) from the string. The string lies along the x axis and the source is locatedat x=0, q = at an infinite distance behind the string (we actually worked with thecoordinate 8 to avoid the wedge). In the conical geometry of the string without a wave,the source is located behind the string relative to the observer. The wave pulse was takento be a gaussian packet with unit half-width and unit height, i.e. f = e2 or g = e2.To find the photon path coming in from the source at infinity and reaching the observerat time u, we emit photons at time u from the observer backwards in time and check tosee if the photon goes to the source. By varying the “initial” data (two angles describedbelow) at the observers position, the photon path connecting the source and observercan be found. We did not calculate the position of the image for photons received whenthe observer is in the nonflat space-time produced by the wave pulse. The deflection forthese photons will be exceedingly small since their distance to the string is quite largewhen they are influenced by the wave pulse. For u < —10 or u > 10, the space-timearound the string is flat, and we can solve for the motion of the photon analytically.For the time period —10 < u < 10 we solved the equations of motion numerically usingan adaptive fifth-order Runge-Kutta routine. The complete solution was obtained bymatching the analytic solutions to the numerical solution. To determine the positionof the image we take the components of the tangent vector of the photon path, at theobserver, in the (x,Y) plane and (Y,Z) plane and find the angle these vectors make withthe y-axis. These are also the two angles used for initial data as described above. Theimage is then projected onto an (x,Z) plane a unit distance away along the Y-axis.To carry the photon from the observer to u=-10 we start with the equations of motionChapter 2. Gravitational Lensing Properties of Strings 53with f=g=O.2_O r2=l (112)where 1 is a constant of integration. Combining these two equations givesd2r 12 (113)This equation has the first integraldr j2(—)2+-=E (114)where E is a positive integration constant. This can be integrated givingr =4+ E(u - u) (115)where ü is a constant of integration. If the photon is received at zt0 by the observerlocated at r0 then__________________________r=+E[_uo+4_]2. (116)At r = r0 we want < 0. This means that we must choose the minus sign in the ±above. We can now integrate (112) to obtain= o + tan’[(n - UO)--+ tan’[[-(117)Equations(116) and (117) allow us to carry the photon to the interaction region in the(r,) plane.To carry the photon in the x direction consider equation(97) with F=0.+2+r22)=0. (118)Hence 7) = — E. This implies thatdx E—1dtl-i-E (119)Chapter 2. Gravitational Lensing Properties of Strings 54Since E is a constant of the motion we can carry the photon to the interaction region viav = v0 — E(u — (120)where v0 = (xo + t0) = t0.We now calculate the initial and outgoing angles. For F=0 we introduce the followingcoordinatesn=x—tv = (x + t)(121)y = r cosz=rsinWe are interested in the angle (7) of the projection of the photon path in the (y,z) planewith the y-axis and the angle () of the projection of the photon path in the (x,y) planewith the y-axis.To find y we calculate the angle between the two vectorsA=(0,,.) =(0,-1,0) . (122)The angle -y is given by4cos = = — du (123)AI(BIAt the observer = 0 and thus cos = —. Since i = — we havecos’y= iji—. (124)For small -y we then have12Er0This is the initial angle in the (y,z) plane. The outgoing angle at r — oc is identical withas r —* cc.Chapter 2. Gravitational Lensing Properties of Strings 55We now calculate the angle . In the (x,y) plane we have2=2+1=1-E (126)d’u duNow is the angle between the vectors and where Ô is given by(127)du duThuscos = — = — du (128)BIICIwhich can be written ascos — rsin5cos — du du (129)— E)2 + [ cos — r sin 5}2For r >> 1 we have At the observer = 0 and we therefore havecoso±2lE (130)Now let E = 1 + zE. We findc +/E (131)Since from (119) we have that = —AE we must choose the negative sign in the +above. The angle at cc (D) can be found from (129). It is given by+2/cos qcosAoc, = (132)/(i —E)+4EcosThe term rsinç has been neglected since rsin = sinçb— 0 as r— 0. This is allthe information we need to calculate the motion and angles for f=g=0.We now use the above results coupled with numerical integration to calculate themotion of the double images of an object located behind the string. Figure (2.15) showsthe motion of the double images for f = e_u2, g=0. Note that in figures 2.15, 2.16, 2.20,Chapter 2. Gravitational Lensing Properties of Strings 56Figure 2.15: The motion of the double images for f=e2, g=O projected onto a planea unit distance away. The string is situated along the vertical axis and the wave ispropagating in the positive x direction. Its amplitude is lOx 1O and its displacement isout of the page. The initial position of the images is marked by a small solid circle. Theposition of the images at t=0 and t=±2 are also marked. The arrows give the directionof motion of the images in time. The units of time are related to the units of distance.For example if the units of distance are light years then the units of time are years.ziOa20 10’x10-10-20Chapter 2. Gravitational Lensing Properties of Strings 572.21, 2.22 and 2.23 the x axis is the vertical axis (figures 2.17, 2.18 and 2.19 are redshiftplots). The arrows give the direction of the motion of the image in time. The initialposition of the image is on the Z-axis at about 13 x iO. This point is marked by asmall solid circle. The position of the images at times t=O and t=+2 are also marked. Attime t=O the gaussian wave pulse would appear centered at x=O. The units of time arerelated to the units of distance. For example if the units of distance are light years thenthe units of time are years. Note that the motion is symmetric for the two images. Thisis a consequence of the symmetry—>— of the equations of motion, and the geometryof the source-string-observer.Figure 2.16 shows the motion of the double images for f=O, g = e_U2. Notice thatthe right image disappears behind the wave pulse and then reappears at a later time. Itcan be seen from the figures that the motion of the image is symmetric about the Z axisfor the g wave but not for the f wave. This behavior for the motion of the images in theplane orthogonal to the string can be seen from equation(99). The light deflection in theplane orthogonal to the string is proportional to zZ’. For the g wave (i.e. f=O) we seethat Z” depends only on r and hence the deflection in the plane orthogonal to the stringwill be symmetric about the Z axis. For the f wave Z” depends only on b and hence themotion in the plane orthogonal to the string will not be symmetric about the Z axis. Asimilar argument holds in the x direction. In these figures the maximum amplitude ofthe wave pulse is 10 x i0.From the figures we see that the maximum angle of deflection is of the same orderas the amplitude of the wave. For the case examined in the previous section (ie. f smalland the impact parameter not too small) the deflection angle scaled as , where d is thedistance to the string (i.e. d scales as the impact parameter for a fixed source at infinity)and L is the size of the wave (i.e. f fdu L2). Since is the angle subtended by (orthe apparent size of) the pulse we see that the ratio of the maximum deflection angle toChapter 2. Gravitational Lensing Properties of Strings 58Figure 2.16: The motion of the double images for f=O, g=e_u2 projected onto a planea unit distance away. The string is situated along the vertical axis and the wave ispropagating in the positive x direction. Its amplitude is lOx io and its displacement isin the positive Z direction. The initial position of the images is marked by a small solidcircle. The position of the images at t=O and t=*2 are also marked. The arrows givethe direction of motion of the images in time. The units of time are related to the unitsof distance. For example if the units of distance are light years then the units of timeare years.1050-5z15x 1O-10-15Chapter 2. Gravitational Lensing Properties of Strings 59the apparent size of the pulse decreases as . To see how the deflection angle varies withdistance in the case of double images we repeated the above computations varying d (dis the distance from the observer to the string) from 102 to i0. We define the deflectionorthogonal to the string to be the maximum deflection of the image in the directionorthogonal to the string. The deflection parallel to the string is defined analogously. Forboth waves the magnitudes of these two deflections are of the same order in angular size.First consider the f wave. In the region 102 d io both deflections (i.e. paralleland perpendicular to the string) are nearly constant and the paths traced out by theimages are similar in shape. Since the angular size of the wave pulse scales as theratio of the deflection size to the angular size of the wave pulse grows with d in thisregion. For iO < d i0 the angular size of the deflections begin to decrease and forio < d < iO the deflection varies as . The ratio of the deflection size to the angularsize of the wave pulse remains approximately constant in this region. In this region regionthe paths traced out by the images are also similar in shape. From our analysis of thelight deflection at large distances we know that the deflection will continue to decreaseas out to d —> oc.Now consider the deflection produced by the g wave. In this case we have to considerthe two images separately. For the left image the deflections are approximately constantfor 102 < d io and the paths have similar shapes. In the region iO < d 106 thedeflections begin to decrease and for 106 d 10 the defiections decrease as “.-‘ . Inthe latter region the path shapes are again similar. Once again from our analysis of thelight deflection at large distances we know that the deflection will continue to decreaseas ‘- out to d — cc. The behavior of the right image is the same as the left imageexcept that the deflection grows slowly with distance in the region 102 d iOn.Thus if we are close enough to the string the deflection remains constant or grows withincreasing distance the ratio of the deflection to the apparent size of the pulse increasesChapter 2. Gravitational Lensing Properties of Strings 60as d increases. If we are far enough from the string the deflection and the ratio ofthe deflection to the apparent size of the pulse is approximately constant.To calculate the redshift caused by the wave pulse we emit two photons, at slightlydifferent times, backwards in time from the observer and find the time between thereception of the photons at the source. As before we analytically carry the photons tou=-10 and numerically integrate to u=10. At this point the photons are in flat space-timeand it is then easy to find the time separation between the reception of the photons atthe source. As expected the redshift is of order t (see the expression for the temperaturediscontinuity caused by a wave in ref [62]). The redshifts for the f and g waves shown infigures 2.17, 2.18 and 2.19. As before the units of time in these figures are related to theunits of distance since we have set c=1. For example if the unit of distance is light-yearsthen the unit of time is years.Lastly we calculate the effect of a gaussian wave pulse on a rectangular grid of starswhich are located infinitely far behind the string. To do this we sent out a grid of photonsfrom the observer towards the string and calculated the outgoing angles for the grid. Wethen used linear interpolation to find the observed positions of the fixed background stars.We checked the accuracy of the interpolation on a portion of the stars and found thatthe actual position of the star is within the cross plotted for the star position.Now consider the effect of the f wave on the grid of stars. Figure 2.20 shows the initialgrid used for the f wave. Initially the wave is out of the field of view so that the stringis straight and is located on the x axis (the vertical axis). Columns of stars labeled withthe same letters (eg. b b’) are images of the same stars. Figure 2.21 shows this grid withthe f wave centered at x=0 and its displacement is out of the page. The double imagesproduced by the string are roughly but not quite perpendicular to the x axis.We now consider the effect of the g wave on the grid of stars. Figure 2.22 shows theinitial grid used for the g wave. In Figure 2.23 we have plotted the star positions andChapter 2. Gravitational Lensing Properties of Strings 61Figure 2.17: The redshift of the right image as a function of time for f=e_uZ, g=O. Theunits of time are related to the units of distance. For example if the units of distance arelight years then the units of time are years.t12-15 -10 -5-2-4-6Chapter 2. Gravitational Lensing Properties of Strings 62Figure 2.18: The redshift of the left image as a function of time for f=O, g=e_u2. Theunits of time are related to the units of distance. For example if the units of distance arelight years then the units of time are years.2—1-5Chapter 2. Gravitational Lensing Properties of Strings 63I I IFigure 2.19: The redshift of the right image as a function of time for f=O, g=e2. Theunits of time are related to the units of distance. For example if the units of distance arelight years then the units of time are years.zio-3x210-15-10 -5 5 t0—1-2-3Chapter 2. Gravitational Lensing Properties of Strings 64•4 + + +., +., ++ + + ++ + + ++ + + ++ + + ++ + ., ++ + + ++ + + ++ + + ++ + + +• + + ++ + + +4-10 + +• + + ++ 4 + +• + + ++ ÷ + ++ + + ++ + + +• + + ++ + + ++ + + ++ + + +•ê 4 + 44 1 + ++ + + ++ + + ++ + + •f4 + 4 ++ + + +• + + +4 + + ++ + 4 +4 + + ++ + + 4+ 4 4 ++ + + ++ 4 4 4• + +-.5+ + + ++ + + ++ + + +4 + + ++ + + ++ + 4 ++ + + ++ + + ++ + + ++ + + ++ + + ++ + + +10+ 4 + •b •c ÷d4 + + + + 44 + 4 + + 4+ + + + + ++ 4 + + 4 ++ + + + + ++ 4 + + 44 + + + $ +• + 4 + + ++ + + + + ++ + + + + +4 4 + + 4 ++ 4 + 4 •+ + + + + $0+ + 4 + + ++ + + + + +4 + + + 4 +4 4 + + + 4+ + 4- 4 4 -4 + + + 4 ++ + + + 4 4+ + + 4 + 44 + + 4 + ++ + + + 4 ++ + 4 + + 4- -.10+ + 4 •l + +db+ c+ d+ ++ 4 + +4 + + +4 + + ++ + 4 ++ + 4 ++ + + ++ 4 + ++ 4 + +4 + + ++ + 4 +4 + 4 4+ + + ++ 4 + +4 + 4 ++ + + 4+ + + ++ + + ++ + + ++ 4 + ++ + 4 ++ 4 + ++ + + ++ 4 + ++ 4 + +,b+ C d++ + 4 + 4 + + 4 + 4+ 4 + + 4 + + + + ++ + + + + + + + + 4+ + 4 + + + + + + +4 + + 4 + + + + + ++ + + 4 4 4 + + + +4 + + 4 + 4 4 + 4 ++ + + 4 + 4 + + 4 ++ 4 + + + + + 4 + 4+ + + + 4 + + + + ++ + + + + + 4 + + ++ + 4 4 4 4 + + 4 ++ 4 4 4 4 4 4 + 4 +4 +5+ + + + + 1O. •+ + 4 + 4 + 4 4 + ++ + + + + + + 4 + ++ + 4 4 4 + + + + 4+ + + + + 4 + + + ++ + + 4 4 + + + 4 4+ + 4 4 + + + 4 4 4+ 4 + + $ + + 4 + ++ + + + + + + + + ++ + + + + + + + 4 +4 + + + + + 4 + + ++ + + + + + + + + ++ 4 + 4 + + + 4 4 4Figure 2.20: The initial grid of stars for the f wave. Initially the wave pulse is outsidethe field of view so that the string is straight and lies on the x axis. Columns of starslabeled with the same letters (i.e. b b’) are images of the same stars.Chapter 2. Gravitational Lensing Properties of Strings 65+ .,+5+ ++ 4+ ++ ++ 0b+ +++.4 4 + ++ + + 4 + + + ++ + + + + + + 4a + + + +e + + + ++ 4 + + + + + 4 ++ + + + + + + + ++ + + + + + + + 4.+ + + + + + + + +4 + + + + + + + 4.+ + + +.4 + + + ++ .4.4 4 + 4 + + ++ + + + + + + + ++ + 4 + 4+ + + 4 4. ++ + + + ++ • + + 4+ + + + 4+ .4 + 4 ++ 4 + .4 4+ + + + ++ + + + 44 + + + 4+ + + 4 ++ + + + +4 + + + +* 5* t 4. ++ + + + + 4 + + + ++ + + + + + + + + +4 + + + + + + + 4 .44 . + 4. + + .4 + 4 +4 + 4 .4 4. .4 + 4- + ++ + + 4- + .4 + .4 + ++ .4 + 4. .4 + .4 .4 + ++ + + + + ++ + + 4 .4 + + .4 + ++ + + + 4 + + .4 + .4+ + + + + +.4 .4 4- .4++ :10+ 4- 4 ++ + + .4+ .4 + 44. 4 4 4+ + .4 ++ .4 .4 +.4 4 + +4 .4 4 4+- + + 4.4 + + .4+ 4- + .4.4 ÷ .4 .4 + .4 +Figure 2.21: The same grid of stars as in figure 2.20 with the f wave centered at x=0.The displacement of the string is out of the page and the amplitude of the pulse is 10on the scale used. The pulse is propagating in the positive x direction. Columns of starslabeled with the same letters (i.e. a a’) are images of the same stars.xc+ d+ .4 4 .4 +4 + 10.4 + .4 4 +.4 .4.4 .4 4 + .4.4.4 .4.4+ + + +.4+ + a+ ÷+ e+ +.4.4 4. + .4 ++ + + .4 .4+ + + + .4+ .4 .4 + 44. .4 .4 .4 + .4 .4+ .4 + 4 .4 + + .4.4 + .4 .4.4 + .4 .4 4 + .4 ++ + +4 + + + 4 4- 4 .4 .4.4 -1 +.4 .4 + .4 + 4 .4 .4 .4 + +.4.4 + + + .4 + + .4 I. 4+ .4 -4 4 + + 4 .4 + 4 + ++ + + + + + 4 + + .4 + 4.4 + 4 .4 .4 .4 + + + + + ++004.44.4+.4.44..4++ + + .4 -4 .44 .4 4 .4 4 4 .4.4 + .4 .4 4 .4 +4 4 + + + .4 .44 .4 .4 .4 .4 + +.4 + .4 +-4 + +.4 +.4 .4 .4 + .4.4 4 .4 + 4 + ++ 4 4 .4 .4 .4 .4+ 4- + + + + ++ 4 + .4 •-+.4.4 .4 .4 .4.4 .4 4i *+ .4.4 +4 4.4 +.4 .44 .4.4 +4 44.4.4 4+ ++ ++ +4 .44 44 4.4 4+ ++ 44 4+ +4.4z4 .4+ 4+ 44 .4+45+ 4.4 4+ 4+ +4 ++ -14• + + + +.4 + ++ a b’+ c d: e# +a’ ÷1! +c’ +d’ +V +Chapter 2. Gravitational Lensing Properties of Strings 66Figure 2.22: The initial grid of stars for the g wave. Initially the wave pulse is outsidethe field of view so that the string is straight and lies on the x axis. Columns of starslabeled with the same letters (i.e. b b’) are images of the same stars.a’ bt cS+ + ++ + +• + ++ + ++ • A• + ++ + ++ + ++ + ++ + 2+ + ++ + ++ + ++ + ++ + axat bt c+ + + + t + + + + + + + + ÷ + + +4 + + + + + + + + + + + + + + + + + + +4 + + + 4 4 + + + + + + + + t 4 + + + +4 + + 4 + + + + + t + + + + + + + + + +t t + + + + + + + + 4 + + t 4 + + + + ++ + + + + + + + + + + + + + + + + + + +4 + + + + + + + + + +., + 4 t + + + + +4 4 4 + + + 4 4 + 4 4 4 4 4 4 4 + 4 4 +4 4 4 + + + 4 + 4 4 + 4 4 + 4 4 4 4 4 4t 4 4 + 4 + + + 4 4 4 + + + 4 + + + 4 +t 4 + 4 + 4 4 + 4 4 + 4 + 4 + + + + 4 +4 4 4 4 + + 4 + 4 4 4 + t + t 4 4 4 4 44 4 4 4 + + 4 4 4 4 4 4 4 + 4 + + + 4 +4 4 t + 4 4 + + 4 4 + + 4 + + 4 4 + 4 44 4 + 4 + 4 4 4 4 4 4 4 4 + 4 4 4 + 4 44 4 4 4 + 4 4 4 + 4 4 4 4 4 4 4 4 + 4 +4 + to4 t ++ + +4 + 4-24 4 44 + +4 4 ++ 4 +-44 4 +4 4 44 + 44 4 +44-6a’t L’ c’z2 4 +4 4 4 6+ 4 Ut 12 ‘ 14’ 164 4 4 4 + 4 4 + 4 + 4 + 4 4 4 4 4 + + +4 4 + 4 4 + 4 + 4 t 4 4 4 4 4 + 4 4 44 4 4 4 4 + + 4 4 4 + 4 4 4 4 4 4 4 +t + 4 4 + 4 4 4 4 + 4 4 4 + 4 4 4 + 44 + 4 + 4 4 4 4 4 + 4 4 4 4 + 4 4 4 + +4 4 + 4 4 + 4 4 4 4 4 4 + + 4 4 + 4 4 44 + + 4 + 4 4 4 + + + 4 + 4 + + + + + +t + + + 4 4 + + + + + 4 4 t + + 4 4 +4 + + + + + 4 + + + 4 4 + + + + + + 4 +t + + + 4 + 4 + + + + + + + + + + 4 + +4 + + 4 4 + + + + + ÷ + + + + + + + + +4 + 4 + ÷ + + + + 4 ÷ + 4 + 4 4 + 4 + +4 + + + + 4 4 + + + + + + + t_ + + + 4 ÷: 6: : : : : : : : : : : : : : : : : : :Chapter 2. Gravitational Lensing Properties of Strings 67the position of the string. The observed position of the string is determined by findingthe photon paths which leave the string and reach the observer. We find that the shapeof the string as seen by the observer is the same as the wave pulse shape g = e_u2. Onceagain the line joining the double images is roughly but not quite orthogonal to the string.2.5 ConclusionIn this chapter we numerically examined the effect of a wave pulse on both the observedposition and redshift of double images. We took the wave pulse to be the gaussian e’2and found that the deflection for both f and g waves, in the region d > 102, remainsapproximately constant or slowly increases close to the string and decreases as furtherfrom the string. In the region where the deflection remains constant or slowly increasesthe ratio of the deflection to the apparent size of the pulse increases as d increases. In theregion where the deflection decreases as the ratio of the deflection to the apparent sizeof the pulse is approximately constant. We also calculated perturbatively the deflectionfor small pulses (ie. we neglected nonlinear terms in f) and for impact parameters whichare not too small (ie. not of order i or smaller). We found that the magnitude of themaximum defection is LbZ where Zq is the angular deficit of the string and z2 is thesolid angle subtended by the pulse as seen by the photon when it is influenced by thewave. If we denote the size of the pulse by L the maximum deflection angle is of orderThis means that the ratio of the maximum deflection angle to the apparent size ofthe pulse decreases as . Therefore nonlinear effects of the wave pulse coupled with asmall impact parameters change the results qualitatively. We also calculated the redshiftcaused by the wave pulse and found that it is of order ji as expected.We then calculated the effect of the gaussian wave pulse on a grid of stars locatedinfinitely far behind the string. It was found that the line joining the double images is++ + ++ + ++ + -2’+ + ++ + ++ + ++ + ++ +4+ + ++ + ++ + +• + ++ +* +•a +b ec +d + + + +.t + + ++ + + +0 + + • + + + ++ * + + + + + + + + ++ + ++ + ++ + + + ++ ++ 4f + + + + + ++ ++ + ++ + + + ++++ + + + + + + +++ + ** + + + 4+ + + + + + + + ++ + + + + + + + ++ + + + + + + + ++ +g • + + + ++ + + + + + + ++ + + + + + + ++ + • + + + + +++ 12 ++ + ++ + + ++ ++ + ++ 4 + ++ + 4 ++ + + + ++ +++g’+ + + ++ 4 ++ + ++ + + +4 ++ ++ + ++ + ++4 ++ ++ + ++ ++ ++ ++ + + + ++++++ ++ ++ + ++ ++++ +++ + ++ + ++ •+ +1 + + ++ + + ++ +++ +4 + ++ + + ++ ++ ++ ++ ++ + +++ ++ +0 +++ + ++ +‘h +1 +j ed’+ ++ ++ + ++Chapter 2. Gravitational Lensing Properties of Strings 688 DC6+ + 4+ + ++ + ++ + +++ 4+ + ++ + ++ + ++ + ++ ++ ++ + ++ + ++ + ++ +• + * a+ b Ce d++ + + + + + + ++ + + + a’ + + ++ + 4 + + + + ++ + + 4 + 4 + ++ + + + + 11+ + ++ + + + + + + ++ + + + + + + ++ + 4 + + + c, ++ + + + + + 4 ++ + + + + + + ++ + + + + + + 4+ + + + + + + +• + + 4 + + + •+ + + + + + + ++++++÷+4+++4+++++++4++++++++ + + ++ + + + + + ++ + + + + + ++ + 4 + + + ++ + + + ++ ++ + + + ++ ++ + + + 4 + j’+ + 4 + 4+ ++ + + ++ + ++ + + ++j1t 44 + + ++ +++ + + + 0++ ++ + + + h +++ + ++ +4 ++ + ++ h 1+ 1++ ++ 4+ ++ ++ 4+ ++ ++ ++ ++ +* 4++dz-8Figure 2.23: The same grid of stars as in figure 2.22 but with the g wave as shown. Thewave pulse is propagating in the positive x direction. Columns of stars labeled with thesame letters (i.e. a a’) are images of the same stars.Chapter 2. Gravitational Lensing Properties of Strings 69roughly but not quite orthogonal to the string. We also find that the deflection producedby a wave pulse remains small even if the photon gets arbitrarily close to the string. Thisanswers the concern raised by Clarke, Ellis and Vickers about the structure of the doubleimages produced by a curved string. If we find a series of double images produced by awiggly string they will be roughly but not exactly orthogonal to the string. The redshiftsof the double images can also differ by an amount of order G[t.For most of the calculations we took d = units. A reasonable choice of unitswould be 1 unit=.O1 Mpc. This would put the string at a distance of 102 Mpc. In theseunits the half-width and height of the wave are .01 Mpc. On this scale the wave pulsewould appear stationary, so we would not see the images move.Chapter 3Galaxy Formation in Cold Dark Matter3.1 IntroductionOne of the most important problems of modern cosmology is to explain how galaxies andlarge scale structure formed. In this chapter we examine string seeded galaxy formationin cold dark matter. We begin with a discussion on dark matter and galaxy formation.Observations of the dynamics of galaxy clusters and observations of the rotation curvesof spiral galaxies indicate that a large fraction 90%) of the matter in these systems isnot visible. This invisible matter is called dark matter (see Trimble [55j for an excellentreview of dark matter). This dark matter is usually classified into two categories,1) cold dark matter if the thermal velocities of the particles are very small near teq (thetime at which the matter and radiation densities are equal)or2) hot dark matter if the thermal velocities of the particles are large near teq.Both forms of dark matter are assumed to be very weakly interacting. If dark matter makes up 90% of the matter in the universe it will obviously be important inunderstanding galaxy formation.It is also important to note that the perturbations in dark matter can begin to growbefore the perturbations in the baryonic matter. Prior to recombination the baryonsare strongly coupled to the background photons. This coupling inhibits the growth ofdensity perturbations in the baryons. Therefore density perturbations in the baryons70Chapter 3. Galaxy Formation in Cold Dark Matter 71cannot begin to grow until after recombination. It can be shown that [44j the darkmatter perturbations begin to grow when the universe switches from being radiationdominated to being matter dominated. This switch occurs well before reconibination.Therefore by the time the baryonic perturbations can grow there may be significantdensity perturbations in the dark matter. Regions of high dark matter density will attractthe baryonic matter producing density perturbations in the baryons. We assume thatgalaxies will then form in these regions. Since most of the calculations in this chapter dealwith dark matter and most of the observations refer to galaxies (i.e. luminous matter)we need to make some assumption on how the distributions of dark and luminous matterare related. We will assume that the distribution of the luminous matter traces thedistribution of the dark matter.To calculate the density perturbations produced by cosmic strings we need to makesome assumption on the nature of the dark matter. In this chapter we will take the darkmatter to be cold. Thus we will be consider string induced density perturbations in auniverse which contains cold dark matter, radiation and of course some baryons. In thenext chapter we will examine string induced density perturbations in hot dark matter.In the next section we will discuss how the ideas on string seeded galaxy formationhave changed over the last few years. We will then discuss the density perturbationsproduced by straight strings in a fiat background space-time. In section 3.3 we discussthe Zel’dovich approximation and apply it to the velocity impulse produced by straightstrings. In sections 3.4 and 3.5 we calculate the velocity impulse produced by wigglystrings and in section 3.6 we examine the accretion wakes produced by wiggly strings.We will find that wiggly strings can produce structure which is similar to the observedlarge scale strllcture. In section 3.7 we examine the effect of the wiggles on the accretionwake produced by the string. It will he shown that wave pulses propagating along thestring can fragment the wake into pieces which have the mass of a galaxy.Chapter 3. Galaxy Formation in Cold Dark Matter 723.2 Introduction to String Seeded Galaxy FormationIn this section we discuss how the ideas on string seeded galaxy formation have changedover the last few years. Initially[14, 19, 42, 47, 49, 51, 53, 58, 67, 69] it was assumedthat there was essentially no structure on scales smaller than the Hubble radius. Thiswas assumed because wiggles on a string which are smaller than the Hubble radius getstretched as the universe expands. This tends to straighten the strings. Since the stringsare fairly straight on scales of order of the Hubble radius we expect the loops producedby the network to be very large. In this old scenario of string evolution it was theselarge loops that would produce the density perturbations which subsequently evolvedinto galaxies. Early numerical simulations performed by Albrecht and Turok [4] seemedto confirm this scenario.Recent simulations [7, 8, 9, 10, 11, 12, 13, 6, 48] of the evolution of cosmic stringevolution show that this old picture is incorrect. In these new simulations the longstrings are quite wiggly on small scales and the loops produced by the string networkare very small. The size of the loops is actually related to the size of the wiggles on thelong strings. Since the long strings are wiggly on small scales we expect that the stringswill cross themselves on small scales and produce small loops. Even if a large loop isproduced it will be very wiggly and will likely fragment into many tiny loops.The early simulations agreed with the old scenario because they smoothed out thesmall scale structure rapidly. Without the small scale structure these simulations wouldinevitably produce large loops.The switch from the old scenario to the new scenario means that most of the earlyideas on string seeded galaxy formation are incorrect. We therefore need to re-examinethe scenario of string seeded galaxy formation. As noted above one of the main differencesbetween the old and the new scenario is the size of the loops produced by the stringChapter 3. Galaxy Formation in Cold Dark Matter 73Figure 3.24: The flow of cold collisionless matter around an infinitely long straight string(in the rest frame of the string). As the two streams of matter flow past the string theymerge to form a wedge of overdensity (6p/p = 1) behind the string. The string is locatedat the vertex of the angular wedge which is cut out of the space.network. It has been shown (Bennett and Bouchet[8}, Stebbins[52]) that the loops are sosmall that they will probably produce only subgalactic sized perturbations. Thus in thenew scenario the long wiggly strings will be the source of the dominant perturbations.The velocity and density perturbations produced by these wiggly strings is the topic ofthis and the next chapter.Before tackling the perturbations produced by wiggly strings it will be instructive tobriefly discuss the perturbations produced by straight strings in a flat background spacetime [33, 51, 53, 63]. Consider the effect of a straight string moving through a mediumof cold collisionless matter in a flat background space-time. As the matter flows by thestring the two streams of matter on opposite sides of the string merge forming a wedgewedgeof overdensityIdentifytterChapter 3. Galaxy Formation in Cold Dark Matter 74of overdensity = 1 behind the string (see figure 3.24). The opening angle of the wedgeis 8rGji (c=1). For Gp 10 and v c the string leaves a nearly plane wake ofoverdensity behind it.In the next section we examine the Zel’dovich approximation and apply it to thevelocity impulse produced by a straight string. Throughout this and the next chapterwe will take the cosmological constant to equal zero.3.3 The Zel’dovich Approximation and Straight StringsIn this section we will review the Zel’dovich approximation for the growth of density andvelocity perturbations [80, 15] and apply it to the velocity impulse produced by a movingstraight string. These results will later be compared, to the perturbations produced bywiggly strings. We will take the universe to contain both matter and radiation.To begin the Zel’dovich approximation we write the trajectory of a particle of colddark matter in, Eularian coordinates, as= [+(t)J (133)where t is some initial time, are the comoving coordinates of the matter and is theperturbation to the Hubble expansion. The Zel’dovich approximation consists in findingan expression for and relating the velocity and density perturbations to it.To find a differential equation for ‘i we first pick an origin for our coordinate system.The Newtonian approximation will be valid as long as the particle velocities do notapproach the speed of light and as long as we deal with particles which are well withinthe Hubble radius. For particles which satisfy these conditions we have= (134)Chapter 3. Galaxy Formation in Cold Dark Matter 75where is the gravitational potential which satisfiesV4(i, t) = 4’lrG[pb(t) + 3Pb(t) + (5PmQT, t)} (135)pb(t) is the total background density (i.e. matter plus radiation), Pb(t) is the backgroundpressure and Pm(’, t) is the density perturbation in the matter.To calculate the density perturbation transform to the comoving coordinates. Thistransformation is well behaved as long as (i) is a single valued function. When )becomes a multiple valued function we say that shell crossing has occurred. From now onwe shall only consider regions in which shell crossing has not occurred. In the comovingcoordinates the matter density (p)) is constant. Transforming back to the coordinatesgives(c)Pm UI (136)where DetJ is the determinant of the Jacobian of the transformation from to Thematter density is then given bya(t)3 (c)pm(t)= det[6ijP;]1. (137)For <<1 we can expand the determinant to linear order in to getpm(,t) p(t)[1 - q. t)j (138)where p(t) is the background matter density. Therefore the density perturbation tolinear order is given by6pm(,t) —p(t).. (t) . (139)We now return to the calculation of V2. Substituting (139) into (135)t) = 4G[p&(t) + 3Pb(t) - p(t)V t)] . (140)Chapter 3. Galaxy Formation in Cold Dark Matter 76The solution to this equation ist) = G[(p6(t) + 3Pb(t))- 3p(t)t)}. (141)Here we have assumed that ‘. x = 0 (i.e. that the perturbations do not producevorticity in the cold dark matter). To justify this assumption substitute (133) into (134)and take the curl with respect to This gives2àV x + x = 0 . (142)To linear order x = Eq.(142) then becomes822—( x‘) + x ‘) = 0 . (143)Hence if q x and q x are initially zero then x will be zero at all times. It willbe shown later that the perturbations produced by cosmic strings do satisfy x =x = 0, so that we are justified in taking x = 0.The equation of motion for can be obtained by substituting (141) and (133) into(134). This gives02 aD[ + 2— — 4rGp(t)j = 0 . (144)The solution to this equation is well known[44 76] and for an Einstein-de Sitter spacetime (ie. Qo = 1) is given byt) = ()D1t)+D2(t) (145)whereD1(t) = 1 + a(t) (146)D9(t)=[1+ a(t)J ln[’- 3[1 + a(t)]4 (147)Chapter 3. Galaxy Formation in Cold Dark Matter 77and and are arbitrary functions of For a(t) we use the scale factor for a universecontaining both radiation and matter and take a(teq)=1. The first term in (145) is referredto as the growth solution and the second term is referred to as the decay solution. Wefirst consider wakes which form at teq.As we shall see below the effect of the cosmic string can be approximated by a purevelocity perturbation. That is as the string passes by a given particle we can approximateits effect on the particle as a velocity perturbation at time teq. We will find that it will bea reasonable approximation to give every particle its velocity perturbation at the sametime. Hence the initial conditions for (144) areteq) =0, (teq)=(148)where ziT(4) is the velocity impulse produced by the string.For t >> teq the decay solution becomes negligible and the solution to (144) satisfyingthe initial conditions (148) is0.63teqL\fki(t) . (149)For i 1, O.63teq(1 + zeq) where (1 + zeq) = a(to) and t0 is the present time.The perturbation is given byt) O.63teqa(t)q . (150)As we shall see the velocity perturbations produced by the string satisfy• ((qj) = 0(except Ofl the surface swept out by the string) so that to linear order the string producesno density perturbations. The density perturbations will appear only when the aboveapproximation breaks down, namely when 1.We now calculate the velocity perturbations generated by a straight string[53]. Thegravitational field produced by the string at distances which are much smaller thanChapter 3. Galaxy Formation in Cold Dark Matter 78the Hubble radius will be closely approximated by the gravitational field of a stringin Minkowski space. To calculate the effect of the string on its surroundings we willconsider a straight string moving in a medium of cold collisionless matter. As the stringsweeps by a given particle it will attract that particle towards the surface generatedby the motion of the string. For small G1.i the velocity of the particle will always benonrelativistic. Since the influence of the the string is largest when it is closest to theparticle, it will be a reasonable approximation to give to each particle its impulse at sometime close to when the string passed by. Since the velocity of the string is— O.15c (Allenand Shellard [5, 48]) and the motion of the particles is nonrelativistic it will also be areasonable approximation to give to each particle its impulse at the same initial timeteq. The problem has now been reduced to one of essentially planar geometry, for whichthe Zel’dovich approximation turns out to give the exact results up to the time of shellcrossing (i.e. when fluid particles pass through each other).The equations of motion for a particle ared2x’ dx dx+ F--—-—-— = 0 (151)where is the connection on the space-time and s is the arc length along the particletrajectory. For particles with nonrelativistic velocities the equations of motion in theweak field ared2xz ah0 1 ãh00 (152)The total impulse given to the particle, taken to be at a fixed position, is then1 008hvi=_f_idt+fa-0dt . (153)We will take the string to lie along the x axis at t=0 and have a velocity = 13c3. Sincethe problem has planar symmetry we only need to calculate The first term in (153)Chapter 3. Galaxy Formation in Cold Dark Matter 79involves h03 evaluated at +oo, which is zero. Therefore1 °c-.= f Vh0dt . (154)We now digress for a moment to justify the assumption x ‘I’ = 0, which was madeto obtain (141). It was shown that if q x and x are initially zero then Xwill equal zero at all times. From (148) we see that qx ‘i = 0 (since L = 0) and that= x if(4). Since zSi is given by (154) we will have V.x = = 0and hence ‘. x ‘.1’ will be zero at all times.We now resume our calculation of From expression (18)hoo=4Gtf du (155)- Ix — x(u, TR) — [x — x(u, TR)] . x(, TR)where TR = t — — (a, rR) is the retarded time. The differential of the retarded time,drR, and the differential of the time, dt, are related bydt = [1— [—f(u,rR)J •(u,TR)]d. (156)x — x(a, ‘TR)IThus Jv can be written as= 2G[L I dt I dcrV “ “ . (157)J—c,o J—oo— x(o,t)jThis expression is valid for a straight or wiggly string. For a straight string (u, T) =71ai +/3CT3. Substituting this into the above expression for zv and integrating gives= —4irGp’y3c .sgn(z). (158)To get an idea of the magnitude of /v take Gu = 106 and ‘y/3 = 1. Then10c, which is certainly nonrelativistic. It is important to notice that Iv2iis independent of z.We now examine the wake produced by the string. The velocity perturbations produced by the string create two surfaces of interest. The first surface, known as theChapter 3. Galaxy Formation in Cold Dark Matter 80turnaround surface, consists of those particles whose velocity in the z-direction, at thepresent time, is zero. The particles which define this surface are in the process of turningaround from the Hubble flow and heading back towards the plane traced out by thestring. Later we will use this surface to define the amount of matter accreted by thewake. To find this surface we look for the coordinates which satisfy= a(t)[q + (t)] + a(t)(t) =0. (159)For ‘i(’ t) given in (149) this becomesq = 2() 1.3teq(1 + zeq)vzi(2 . (160)For the velocity perturbation of a straight string qz is given byq ±5.0teq(1 + zeq)7c . (161)We take teq = 3 X 10’°h4,1 + Zeq = 2.5 x 104h2Q0 and = I6 x 10. Thecoordinate distance to the turnaround surface is then given byq +1.2 x1047/916hMpc. (162)The physical distance to this surface (z = (1 + zeq)(qz + IJ)) is 1.57/3/16oMpc. Thesurface density contained within the turnabout surfaces is2(1 + zeq)qzpo 1.7 x10’27/16hM®/Mpc (163)whereP0 G20 (164)irand t is the present time.The other surface of interest is the surface which has turned around and reachedz=0 at the present time. This surface is defined by z = (1 + zeq)(qz + ‘Ps) = 0. TheChapter 3. Galaxy Formation in Cold Dark Matter 81coordinate distance to this surface is one-half of the coordinate distance to the turnaroundsurface. Therefore the surface density of the matter that has fallen back onto the wakeis a 8.7 x10’y3i62M0/ pc.So far we have discussed wakes which are produced by straight strings at teq. Thestring network also produces wakes at other times and we need to decide which of thesewakes produces the large scale structure. I will show below that the wakes which formvery early accrete almost all of the matter in the universe and that wakes which formvery late accrete only a small amount of the matter. Thus the last wakes which accretealmost all of the matter will accrete wakes produced earlier and will not be accreted bywakes which form later. Therefore it is reasonable to assume that it is the last wakeswhich accrete almost all of the matter in the universe which set the size of the large scalestructure.To determine which wakes produce the large scale structure we need to calculate howthe fraction of the matter (fa) accreted by the wakes and the present interwake separation(z) depends on the time of wake formation (ti). I will define the amount of matter accreted by the wakes to be the amount of matter contained within the turnaround surfaces.I will take the ratio of the present coordinate thickness of the wake to the coordinateinterwake separation as an estimate of the fraction of the matter in the universe thathas been accreted by the wakes. To simplify the calculation I will assume that = 1.This is a reasonable assumption since observations indicate that 0.2‘‘ 1 and sincethe inflationary scenario predicts that = 1. I will also assume that the universe ismatter dominated from teq to the present, i.e. that t is later than teq. Corrections to thisapproximation will be discussed in section 3.6 when we consider the wakes produced bywiggly strings. It is shown that these corrections are small and we will therefore neglectthem here. From equation (161) we have q t(1 + z) t. Since the coordinate interstring separation (Ac) is o t the present interwake separation (A) is Ac(1 + z) t.Chapter 3. Galaxy Formation in Cold Dark Matter 82We also have fa q/c t. Since fa t we see that very early wakes accrete amuch larger fraction of the matter than wakes formed at late times.We can now calculate fa and for wakes formed at t from fa o t and thevalues of fa and z at teq. From (30) the coordinate thickness of wakes formed at teq is2.4 x107/3I6hMpc and their coordinate interwake separation is 2.1 x 104hMpc(i.e. 1/3 of the Hubble radius). Therefore the wakes formed by straight strings at teqhave accreted 1 0t6/37h2percent of the matter in the universe. As discussed inthe introduction simulations of the evolution of cosmic string networks indicate that0.15. Therefore wakes formed at teq have accreted 17i6h2percent of the matter inthe universe. The present interwake separation for wakes formed at teq is 5.3h2Mpc.Using these values of fa and at teq together with fa t and t gives5.3()h2Mpctq (165)fa 0.17it6()_*h2The last wakes which accrete almost all of the matter (say fa=0.8) then form at,u6h2( )rt (166)Substituting this into the expression for L in (165) gives2.4/h’Mpc (167)This is the size of the structure produced by the string network. It will be shown insection 3.6 that L is not too sensitive to the value of fa (0.8 in the above) chosen. If z isto match the observed large scale structure (25-5Oh1Mpc), /16 has to satisfy I6 100.But this value of /16 is much larger than the constraint (/16 4) imposed by the timingof the millisecond pulsar. Thus straight strings would be ruled out as the origin of thelarge scale structure.Chapter 3. Galaxy Formation in Cold Dark Matter 83So far we have calculated the value of P6 required for the strings to produce theobserved large scale structure for o = 1. If o 1, I found that the required value ofP6 does not change significantly.The above calculations dealt with galaxy formation induced by strings which whereessentially straight (i.e. in the old scenario of string seeded galaxy formation). Sincerecent simulations (see introduction) show that strings are very wiggly we need to reexamine string induced galaxy formation. In the following sections in this chapter Ipresent the research that I carried out on the wakes produced by wiggly cosmic strings incold dark matter[73j. I found that the size of the structure produced by wiggly strings iscomparable to the size of the observed large scale structure for P6 4. Thus the wiggleson the strings save the scenario of string seeded structure formation. I also found thatthe small scale structure on the strings can fragment the wake into pieces which have themass of a galaxy.3.4 Velocity Perturbations 1In the previous section we examined the wakes produced by straight cosmic strings.Since strings are actually quite wiggly we need to examine the wakes produced by wigglystrings. As a first step towards this goal we will now calculate the velocity perturbationsproduced by a string carrying travelling waves of the formtT x=u y=f(u±T) z=g(u+r) (168)(see equation(26) in chapter 1). The velocity perturbations produced by strings carryingwaves propagating in both directions will be examined in the next section.Solutions of the form (168) are not actually valid in an expanding universe. To seethis let y=f be the displacement of the string from its equilibrium position. In a radiationChapter 3. Galaxy Formation in Cold Dark Matter 84dominated universe with If’I lf << 1 the string satisfies the equation of motion[67jf+f—f”=0 (169)where r is the conformal time defined by T cx t and a(T) cx T. This equation has theplane wave solutionf(x,r) = sin[k(x - r)] (170)where x is a comoving coordinate and ) = is the comoving wavelength. Thus thephysical amplitude of the wave is constant but the physical wavelength grows with time.For particles which are well within the horizon it will he a reasonable approximation toneglect the stretching of the string.To calculate the velocity impulse we will use the metric for a string in fiat space-timeand take f and g to be the waveforms on the string as the string passes the particle.For this metric to be a good approximation we require that the particle be much closerto the string than the horizon and that the amplitude of the waves on the string bemuch smaller than the horizon. To simplify the calculations we will also assume that theamplitude of the waves are much smaller than the coordinate distance from the stringto the particles of interest (i.e. particles on the turnaround surfaces). As before we willtake the string to pass through the x axis at t=0 and to have the velocity = /3c3. Asin the case of a straight string we are interested in The z-component of the totalimpulse given to a particle, taken to be at rest, isah 1 006hvz_fdt+Jdt . (171)The first term is just h03 evaluated at the end points of integration. Since h03 = (1—r_8)g’ the impulse given to the particle in this coordinate system will oscillate withno matter how far away the string is. But since the curvature tensor behaves as[26jwe can see that this is just a coordinate effect. Thus there will exist a coordinateChapter 3. Galaxy Formation in Cold Dark Matter 85system (S) which in the vicinity of the particle becomes Minkowski when the string isfar away. It is in this coordinate system that we need to calculate v1. In S the velocityperturbation, will be essentially unchanged if the amplitude of the infinite wavetrain is set to zero at very large distances. For the truncated wave in the coordinatesystem we have used the particle will begin and end up in a coordinate system which canbe made locally Minkowski by a simple coordinate transformation. This transformationwill change the velocity by an amount of order p2, which we can neglect. Therefore toorder p the velocity change in the coordinates we have used with h03 = 0 at t = ±oogives the correct impulse. Therefore1 coãh (172)To calculate we need h00 for a moving string. The metric given in (27) is for astring at rest on the x axis. The h00 component of the metric for a string moving withvelocity vj is found by performing a Lorentz transformation on the metric given in (27).This gives—4Gp72ln{[7(y— act)— f]2 + (z — g)2}[(g’) + (/3— f’)2] (173)wheref = f[x — 7(ct — /3y)] , g = g[x — 7(ct — /ly)] . (174)The integration constant (P0) in (27) has been dropped since it does not appear inSubstituting this into (172) gives (see appendix C)‘2 ‘2= -4Gp’y2L [((y/3) f)2 g)2]dt. (175)Now consider particles with x=y=0. Particles with x 0, y 0 can be consideredby shifting the wave up or down the string. As stated before we are only interested inChapter 3. Galaxy Formation in Cold Dark Matter 86particles with z >> f, g. The velocity impulse then becomes‘2 ‘2= _4G72zj [(7/3ct)+zj dt. (176)To proceed with the calculation of /v we write g(—7ct) and f(—7ct) asg(—7ct)=gi7Cktdk, f(—ct)=fkei7tdk. (177)Substituting (177) into (176) and integrating gives= —47c[/3+2i L kfkeHdk_ L 1 kk’(gkgI +fkfk’ )H (k+;’) dkdk’]sgn(z)(178)The first term in the bracket is the velocity change produced by a straight string while thelast two terms are the velocity change produced by the waves on the string. An importantproperty of the last term is its dependence on . The reason for this dependence on /3is that as /3 gets smaller the string spends more time in the vicinity of the particle andhence gives the particle a larger impulse.It will be instructive to consider /v for a monochromatic wave on the string. Takeg=asin[ko(x—7(ct—/ly))] . (179)and f=O. The Fourier transform of g isg = -[5(k + k0) — 5(k — k0)] . (180)At x=y=0 the velocity impulse is given by= -4r7[/3 + (f)2(i + e)]c sgn(z). (181)where ‘.o= .For zJ > )‘, is essentially independent of z.An important property of to notice is that even in the limit a —* 0 the wave willstill have an effect if () remains non-zero. Therefore small scale structure on any scaleChapter 3. Galaxy Formation in Cold Dark Matter 87can influence the velocity perturbations produced by the string. This could be importantfor predictions of density perturbations which come from numerical simulations of cosmicstring evolution. In these simulations there is a lower limit of resolution for the small scalestructure. Therefore if there is significant small scale structure on scales smaller than thelimit of resolution the predictions of density perturbations from these simulations will betoo small.To get an idea of the effect of the sine wave on we can take the result from theBennett and Bouchet[7, 8, 9, 10] simulation that in the radiation dominated era about50 percent of the energy of the string resides in the small scale structure. The energyof a moving string can be found from its rest energy (i.e. the energy in the rest frameof the string) and momentum via a Lorentz transformation. Actually the energy andmomentum of a segment of string will not in general transform as a four-vector if f isnon-zero at the endpoints (due to the lack of absolute simultaneity). If we are interestedin the energy of a length Ax of string this effect will be small if is small. Thereforethe energy of a length Ax of string with << 1 isXO+a, I IE =7i[Ax+ f [(f )2 + (g )2 - f ]dx] (182)where f’ and g’ are the wave forms in the rest frame of the string. For f=0 and g givenby (180) we findE 7tAx[1 + 22()2] (183)for Ax >> ).. Therefore the ratio of the energy in the waves (Em) to the energy of thestring is‘,2(a2—V 184E — 1+27r2()Here we have defined the energy in the waves as the total energy of the string segmentminus the energy of a straight piece of string connecting the endpoints of the segment.Chapter 3. Galaxy Formation in Cold Dark Matter 88From (184) we find21r2()2 = E/E (185)1—E/ESubstituting this into (181) gives= -4K7{/3+J(1 + e)}c sgn(z) . (186)For waves whose wavelength is much smaller than the z coordinate of interest and for=wefindG1i 1= —4ivy—[8 + ä]c sgn(z) . (187)For = 0.15 the second term (due to the wiggles) is 44 times larger than the firstterm (due to a straight string). If there is significant small scale structure on scalessmaller than the resolution limit of the simulation the effect of the wave on the velocityperturbation will be larger.In the early universe we expect there to be a network of strings. We are interestedin the average perturbations by this string network. VTe therefore consider the averageimpulse generated by a statistical ensemble of strings. The ensemble average of f’ (t) andg’(t) will be taken to satisfy=< g’(t)> = 0 (188)We now calculate the average velocity perturbation produced by the string network.Taking the ensemble average of (176) givesGi 2 °°_____________<v >= —4—7 ze I dt . (189)c2 [(7vt)2 + z2}where we have taken each string to have the same velocity. This gives the average velocityperturbation at some z distance away from the strings.‘We now relate the velocity perturbation to the ratio of the energy in the waves (Em)to the total energy of the string (E). The energy in expression (184) is the energy of aChapter 3. Galaxy Formation in Cold Dark Matter 89length Zx of string. If we consider a length L.x for each member of the ensemble theaverage energy will beXO+V,<E >= 7[x + j (<f 2 > + <g2 >)dx] . (190)We now consider ensembles which in addition to satisfying (188) also satisfy the requirement that < f’2 > + < > be independent of x (and hence independent of t). Forsuch ensembles we have(191)The average energy in the waves is then<E >= y[< f’2 > + <g’ >]/x . (192)Substituting (191) and (192) into (189) gives<v >= —47c[ + iEW/E )jsgn(z) . (193)The above expression for < > is identical to expression (186) for /v if lzI > ).We now calculate the velocity perturbation, to lowest order, produced by waves propagating in opposite directions. To begin with consider solutions to the equations ofmotion (10) of the form= [t,a,f(x,t),g(x,t)] (194)withfl , , << 1 . (195)To linear order in f and g the equations of motion become linear wave equations. Thusfor f and g satisfying (195) the general solution will be= [t, a, f(x — t) + f(x + t), gu(x — t) + gv(x + t)] (196)Chapter 3. Galaxy Formation in Cold Dark Matter 90where f, ft,, g, g, are arbitrary functions of their respective arguments. This solutionsatisfies the equations of motion up to second order in f and g. Hence the next ordercorrection to (196) is cubic in f and g.We now calculate the energy momentum tensor. To second order in f and g the energymomentum-tensor can be written asTIW= T + T + + (197)where1 0000 —1 0 0T=j6(y—f)6(z—g)00000000(f’2 j (2 ((2 (2“ôtzJ ‘8u) Ou 9u(Of2 + (.N2 (Of2 (8g2 —“8u1 “&ui “8u1 ‘ &u .9u_f! _1!. 0 08u 8u_thL— 0 08n 8u12(2 _(2_(2 2I“8vJ “Dv! Dv Dv_(&‘2 — (Q2 (f2 (2 2LL“Dv! “Dv! “Dv) “Dv) Dv Dv1!L— 0 0Dv Dv_!L 0 0Dv Dvand00 0 000 0 0= —2p,S(y— f)5(z— g)0 0 21(-’ j 1[(th“DuI”Dv Du”8v)’\Dv”Du0 0“DuI”Dv) “DvJ”thi) ‘DuPDvChapter 3. Galaxy Formation in Cold Dark Matter 91Thus and are the energy-momentum tensors for the u and v waves respectivelyand is the interaction energy-momentum tensor.We now show that in some cases the velocity perturbations produced by waves propagating in opposite directions is the sum of the velocity perturbations produced by eachof the waves separately. If the waves propagating in opposite directions have differentpolarizations then only T = will be nonzero. This component is not ilsed in thecalculation of the velocity perturbation produced by the string. Therefore if the wavespropagating in opposite directions have different polarizations the total velocity impulse,to lowest order, will be the sum of the velocity perturbations produced by the individualwaves.Now consider an ensemble of strings in which the waves propagating in oppositedirections are uncorrelated and hence satisfy<((9fU)(öfV) >< (0fU)(V) >< (0U)(6fV) >< (8U)(V)>= 0 . (198)It is easy to see from (17) that for y2 + z2 >> f2 + g2 (i.e. the particle is much furtherfrom the string than the amplitude of the wave) the average impulse given to the particlecan be written as<v >=< > + <v> (199)where < > is the average impulse produced by the u wave and < > is theaverage impulse prodilced by the v wave. Thus on average, to lowest order, the impulseproduced by the two waves adds linearly. This result will he used in section 3.7 when wedisdllss how the small scale structure on the strings can fragment the wake.3.5 Velocity Perturbations 2In the last section we calculated the velocity perturbation produced by waves propagatingalong the string in one direction. We also considered the velocity perturbation, to lowestChapter 3. Galaxy Formation in Cold Dark Matter 92order in E/E, produced by waves propagating in opposite directions. In this section weexamine the velocity perturbations generated by a string which is described in the gaugegiven in (11). In this gauge we can have waves propagating in both directions.We begin by writing the position of the string asf(u,t) = cu + Ilct3 + h(u,t) (200)where represents the waves propagating along the string and c is a constant. For (u, t)to satisfy the constraints (11) we must have2+ 2c1h+ 2 + 2ah + + c_22 = 1 (201)andvh + + . = 0 . (202)We want to be able to write the velocity perturbation in terms of E/E, as we did inthe last section. To be able to do this we will now derive a relation between E/E and. We define the energy in the waves as we did previously for the travelling waves. Theenergy of a length /.o of string isE = . (203)The lellgth /u corresponds to a length Zx via(204)For a long segment of string with h not too large the last term in (204) will be negligible.The energy of a straight string with length Lx is Wyzx z’yLa. The energy in thewaves is therefore given by(205)Thus c can be written as= 7’(l — E/E) . (206)Chapter 3. Galaxy Formation in Cold Dark Matter 93This result will be used near the end of this section to write the velocity perturbation interms of E/E.As in the last section we are interested in the average velocity perturbation producedby the string network. We will therefore consider an ensemble of strings which satisfieshj <<Zta. >< > 0 (207)where Zia is the z coordinate of the turnaround surface. We also take each string to havethe same E/E and the same velocity. This means that each string has the same o.To calculate the average velocity perturbation produced by this ensemble take theensemble average of (157). For x=y=0 this gives<v >= _2zf duf dt C+ < 3 . (208)_ (2g + /32ct+ z2)rBy integrating by parts twice and using the equations of motion (12) we getc2i’2I dul dt =1 dal dt +J_ J,0 [2g + /32ct+ z2] J_ Jo [22 /32ct+ z2](Co coo— c--[E.I do- I dt°t . (209)J_ J [2o- + /32ct+ z2]rIf we require that the ensemble satisfy <• >= <• ‘ >= 0 then < > canbe written as<v >= _2zf do-f dt + [<2> +c2 <‘2>] (210)—cc—cc [a2o- + /32ct+ z2]We also require that <h > +c2 <h’2> be independent of a and t. We then have<v >= _4[2 + (c_2 < > + < >)j sgn(z) . (211)Taking the ensemble average of (201) givesc_2<i >+<‘2>=(1_c2_2) . (212)Chapter 3. Galaxy Formation in Cold Dark Matter 94Hence </.v > can then be written as<v >= -4K7[+ (Eç,)]sgn(z)c. (213)Expression (213) is the velocity perturbation we will use when calculating the accretionwake produced by a cosmic string.3.6 The Accretion WakeIn this section we examine the accretion wakes produced by strings with small scalestructure.We first consider wakes formed at teq. The average distance to the turnaround surfaceis given by<qz > 1.3teq(1 + zeq) <LV> (214)±1.2 x 107[ + ( ç,El6h_2MPC . (215)We now take = 0.15 and E/E = 0.4 (average of E/E for radiation and matterdominated eras). The coordinate distance to the turnaround surface isq 4.5 x 104i6h2Mpc. (216)The physical distance to the surface is 5.7,tt6oMpc and the surface density containedwithin the turnaround surfaces is6.4 x10’2hM®/Mpc. (217)So far we have discussed wakes which are produced at teq. Again, as in the straightstring analysis, the string network produces wakes at other times and we again needto decide which of these wakes produces the large scale structure. As before (section3.3) I will show below that the wakes which form very early accrete all of the matterChapter 3. Galaxy Formation in Cold Dark Matter 95in the universe and that wakes which form very late accrete only a small amount of thematter. Thus the last wakes which accrete almost all of the matter will accrete wakesproduced earlier but will not be accreted by wakes which form later. Therefore it is againreasonable to assume that it is the last wakes which accrete almost all of the matter inthe universe which set the size of the large scale structure.To determine which wakes produce the large scale structure we again need to calculatehow the fraction of matter(fa) accreted by the wakes and the present interwake separation() depends on the time of wake formation (ti). As before I will take the ratio of thepresent coordinate thickness of the wake to the coordinate interwake separation as anestimate of the fraction of the matter in the universe that has been accreted by thewakes. To simplify the calculations I will again assume that = 1 and will again usethe approximation that the universe is matter dominated from t to the present time.From equation (214) we have q cx t(1 + z) cx t. Since the coordinate interstringseparation () is cx t the present interwake separation () is c(1 + z) cx t. Wealso have fa cx cx t. Since fa cx t we see that early wakes accrete a muchlarger fraction of the matter than wakes produced at late times.As before we can calculate fa and z from fa cx t, cx t7 and from the valuesof fa and L\ at teq. This assumes again that the universe is exactly matter dominatedfrom teq to the present. The correction due to the finite transition time from radiation tomatter dominance will be discussed below, and again is small. From (216) the coordinatethickness of the wake is 9.0 x104t6h2Mpc. Since the coordinate interwake separationis 2.1 x 104hMpc these wakes have accreted 4206h2 percent of the matter inthe universe. Since the present interwake separation is zi 5.3h2Mpc we will have5.3()h_2Mpc(218)fa 4.2it6()_h2Chapter 3. Galaxy Formation in Cold Dark Matter 96The last wakes which accrete almost all of the matter (say fa = 0.8) then form at(6g)teq (219)Substituting this into the expression for in (218) givesc l24/Jh1Mpc (220)The coefficient (i.e. 12) is 5 times larger than for straight strings. This implies thatwiggly strings produce much larger structure than straight strings. If we take z to bethe size of the observed large scale structure (25-50h’Mpc), we see that that 6The wiggles therefore allow us to create large scale structure similar to that observedeven for values of p6 which obey the experimental constraints.It is important to see how sensitive Li is to the value of fa chosen (above we tookfa0.8). By eliminating tilteq in equations (218) it can be seen that Li cc i//j. Forexample Li increases by about 40 percent if fa is lowered from 0.8 to 0.4. Thus Li is onlymoderately sensitive to the value of fa chosen to represent the large scale structure seentoday.In the above calculations we used the approximation that the universe was matterdominated from teq to the present. It is not. There is a finite time during which theradiation still plays an important role in determining the expansion rate of the universe.We now discuss corrections to this approximation. From (219) we see that the wakeswhich produce the large scale structure form atti 12([L6hteq (221)For [‘6 4 and h 1 we see that ti > teq. Before calculating the corrections tothe approximation let us see how these corrections will likely behave. The calculationsfor the wakes which formed at teq were carried out by evolving the velocity perturbations produced by the strings at teq using the exact linear equations (i.e. the Zel’dovichChapter 3. Galaxy Formation in Cold Dark Matter 97approximation). To approximate the values of z and fa for wakes formed at t we usedthe values for z and fa for wakes formed at teq and then used the matter dominatedapproximation fa O( t and L\ cc t. To calculate the corrections we first calculatedfa and Z for wakes formed at t2 using the exact linear equations (as we did for wakesformed at teq). The corrections to the matter dominated approximation were then foundby taking the ratio of the exact (to linear order) values of fa and to the values foundby using the matter dominated approximation. There will therefore be no correction tothe matter dominated approximation for t = teq (since this was calculated using theexact linear equations). As t increases the corrections to the approximation will initiallyincrease. Eventually t will be well within the matter dominated era and the correctionswill then stop increasing (i.e. fa really will scale as t, etc.). For i6 = 4 and h=,equation (219) was found to give t l2t. For these wakes I have calculated that theapproximation that the universe is matter dominated underestimates fa by a factor of1.2 and L by a factor of 1.3. For 6 = 4 and h=1 equation (219) gives t lOOt.For these times, the approximation underestimates fa and L by similar amounts (i.e.ti = l2teq is within the matter dominated region and the corrections do not change significantly for larger ti). Since simulations indicate that E/E changes from about 0.4in the transition era to about 0.3 in the matter dominated era we have also taken thisinto account. The net effect of both of these corrections is to increase the size of thelarge scale structure produced by the strings by a factor of about 1.3. Thus the size ofthe interwake separation increases from 12/h’Mpc to 16’hMpc. We cantherefore conclude that wiggly strings can produce structure whose size is similar to thatobserved (25-50hMpc) if /6 4, which is just consistent with the constraint ‘ 4imposed by the timing of the millisecond pulsar.Recent surveys[1, 21, 22, 25] indicate that regions of size 45h’Mpc may havecoherent streaming velocities of up to io km/s. The local group of galaxies is alsoChapter 3. Galaxy Formation in Cold Dark Matter 98known to be moving with respect to the microwave background radiation with a velocityof 600km/s. It is therefore of interest to calculate the peculiar velocities generatedby cosmic strings. The peculiar velocity in the Zel’dovich approximation for structureformed at teq is given by= (1 + zeq) 0.25(1 + zeq)4 (222)for (1 + zeq) >> 1 (recall that we have used a(t) for a universe containing both matterand radiation). From equation(213) we can expect peculiar velocities of magnitudev 1507[/3 + (c/E]6hokm/s. (223)For E/E = 0.4 and /9 = 0.15560jih2okm/s . (224)We expect these velocities to be coherent over some fraction of the interwake separation. If we consider wakes with an interwake separation of Lh’ we find v3.41O6km/sr.,. 75t6km/s (for L = 45). For an average interwake spacing of 20h’we find v -. 170i6km/s which is close to the observed value for I6 = 4, although thescale over which this velocity is coherent is smaller than observed.3.7 Fragmentation of the WakeWe now consider the effect of the small scale structure on the fragmentation of a wakeformed at teq. Consider a wave pulse propagating along the string. As the wave propagates it will form a surface if the string velocity is non-zero (see figure 3.25). As we willsee below, the surrounding matter will be attracted to this surface. The effect of thiswave pulse will then be to generate a tube-like overdensity within the accretion wake ofthe string.Chapter 3. Galaxy Formation in Cold Dark Matter 99yxFigure 3.25: The triangular surface formed by a triangular wave pulse propagating in thepositive x direction. The velocity of the string is in the positive y direction.triangular surface1Chapter 3. Galaxy Formation in Cold Dark Matter 100We now calculate the velocity impulse produced by a wave pulse propagating alonga cosmic string. The velocity impulse generated by a moving string (f=0) can be foundfrom1 °°.Vh00dt . (225)Differentiating (173) and integrating by parts taking g=0 at t = +00 gives (see appendixC)00 (z— )[/32 ‘2]= 42GJ2(y — )2 + (z— g)2dt (226)= —47Gr g’ — g) + 7g’2(y — Vt) dt (227)- 7(y—vt) +(z—g)and,= —/3’yL\v . (228)Since the velocity of the wave pulse in the (x,y) plane is (c/7, v) we see that the impulsegiven to the particles is a right angles to the tube generated by the wave pulse and istowards the tube.Since the particles are attracted to the surface produced by the wave pulse there willexist a turnaround surface surrounding it. Particles on this surface are in the process ofturning around from the Hubble flow and heading back towards the surface produced bythe string. Because of the symmetry we only need to examine the cross section of theturnaround surface on the plane x = —/3’yy. Let IIS denote the rectangular coordinateson this plane by (l,z). To find the turnaround surface we look for particles whose (x,y)velocity orthogonal to the tube is equal to zero. This condition is v, = /37v. Substituting= (1 + zeq)[+ 2I’] (229)into v = f3v and using = —3-’yv and q = —137q9 gives= qy + 1.3teq(1 + zeq)vy = 0 . (230)Chapter 3. Galaxy Formation in Cold Dark Matter 101This defines the turnaround surface surrounding the surface produced by the wave.To calculate the velocity impulse produced by the wave pulse we take the wave pulse,in the rest frame of the string, to be the triangular wave0 u—aa+u —a<u<0g(u)=--a—u 0<u<a0 u>aThe amplitude of the wave pulse will be written as a = odH 6.4 x 102th4 2pc (dHis the Hubble radius at time teq). The ratio of the wave amplitude to the Hubble radiusat time teq is then given by a. We also write L\v = —4Gi3(q, q, c, 3). Equation(230)then becomesqy = . (231)Transforming g(u) into a frame in which the string is moving with a velocity 3 andintegrating (215) gives (see appendix C)= —4Gix{sgn[y - 7(z + x- a)][tan’(1 r) - tan’()]+ (232)sgri[y + ,87(z— x— a)j[tan1(1 @Zr) — tan’()}This together with (231) defines the turnaround surface.The solutions to (231) for the triangular wave pulse form two surfaces. For c << 1(II6 ‘-‘ 1) the outer surface is the usual turnaround surface. The inner surface canbe divided into two subsurfaces. For z > 0 the velocity changes sign discontinuouslyacross the surface. This part of the surface is generated by the discontinuous changein the velocity perturbation as we cross the string. For z < 0 the velocity changessign continuously. These two surfaces are shown in Figure 3.26 for !3 = .15, c = i0,= 1, h=, and o = . In the interior of the inner surface the 1-component of theChapter 3. Galaxy Formation in Cold Dark Matter 102Figure 3.26: A cross section of the turnaround surfaces surrounding the surface producedby the triangular wave pulse with 3 = 0.15, h=,= , = 1, and c = ion. Theouter surface is a turnaround surface. Particles outside this surface are flowing away fromthe surface produced by the wave. Particles inside the outer surface and outside the innersurface are heading inwards. The inner surface is generated by the jump discontinuity inthe velocity as one crosses the string. Particles inside this surface are given an impulseoutwards and are hence moving in the outward direction.1zChapter 3. Galaxy Formation in Cold Dark Matter 103velocity is outwards. In the region between the inner and outer surfaces the 1-componentof the velocity is directed inwards. In the region exterior to the outer surface the 1-component of the velocity is outwards. Linear perturbation theory will break downnear the surface of discontinuity since shell crossing will have occurred if the matter iscollisionless. If the matter is collisional a shock wave will form which will prevent shellcrossing. For the collisionless matter we are considering the discontinuity in the velocitystays at the same comoving coordinates as time evolves. This follows from the fact thatthe equations describing the evolution of perturbations involve only time derivatives notspatial derivatives.We now consider how the dimensions of the turnaround surface depend on, 7,h, and, . Numerically we find that dimensions of the outer turnaround surface scale as7gh-’. This behavior can easily be seen for = 0, q >> q, a. Equation(231) becomesqy = 76.1ith_2sgn(y)[tan_1(64X 10h — z) + tan’()] (233)Iqj4.9 x 10 6h2 (234)qywhich has the scaling stated above for 3 = 0.We also need to know how the turnaround surface changes as the shape of the triangular wave is changed. If we denote the height of the wave by a and its base half-widthby w, we find that the dimensions of the outer turnaround surface increase as (a/w) .So as the wave pulse becomes more peaked the turnaround surface becomes larger. Thisbehavior of the turnaround surface can be seen in the following way. The impulse givento a particle close to the triangular wave pulse, but not to close to the kinks, will be thesame as the impulse produced by an infinitely long string. This impulse is proportional to/3-y where ,6j is the component of the string velocity which is orthogonal to the string.Chapter 3. Galaxy Formation in Cold Dark Matter 104A simple calculation for the triangular wave pulse shows that the orthogonal velocity forthe sides of the triangle is1- 722 (235)for w/a << 1, where /3 is the velocity of the string as a whole. Thus the impulsecx ‘y(). Therefore the velocity impulse grows as a/w for particles near thestring. For /3 = 0 and for >> a, qI we see from (234) that the distance to theturnaround surface goes as the square root of the impulse. Hence the result that thedimensions of the turnaround surface go as (a/w) 4 is not surprising. Since we expect themajority of waves on the string to be not too sharply peaked we will take a/w=1 for thesubsequent calculations.Now consider what happens when there is more than one wave pulse propagating onthe string. If we have two waves on the string following each other too closely the velocity perturbations between the two surfaces formed by these waves will tend to canceland the outer turnaround surface will surround the surfaces produced by both waves.If we have two wave pulses propagating in opposite directions on the string we expectthe intersection of the two waves to be the center of a region of large mass accretion.From (199) we see that for an ensemble of strings the average of the total velocity perturbation, to lowest order in E/E, will be the sum of the average velocity perturbationsproduced by the oppositely propagating waves. We have also found that the velocityperturbation produced by oppositely polarized waves propagating in opposite directionsis, to lowest order, the sum of the perturbations produced by the individual waves. Wetherefore expect there to be a turnaround surface surrounding the intersection regionwhich contains a coordinate volume of order The masscontained in this volume is -‘ 10b0()47/4h_1OM). For (j r) y4h’o > 1this is of order of the mass of a galaxy. For = 102, I-6 = 4, ‘y = 1, and 20h’ = 1 theChapter 3. Galaxy Formation in Cold Dark Matter 105mass is 2.5 x 10’2M®. Hence the intersection region of the two wave pulses can accretea galactic mass. Since wakes which form earlier accrete most of the mass they may alsoassist in fragmenting the matter accreted by later wakes.3.8 ConclusionIn this chapter we examined the velocity and density perturbations produced by a cosmicstring with small scale structure moving through a medium of cold collisionless matter.In this concluding section we will summarize the new results found in this chapter andcompare them to the observations.We found that the average velocity perturbation generated by an ensemble of stringsis given by<v >= -4[ + (Eç/EJsgn(z)c (236)where E/E is the ratio of the energy of the waves on the strings to the total energy ofthe strings. For E/E = 0.4, ,8 = 0.15 and an average interwake separation of Sh’Mpcthe wakes produced by the strings have accreted 1.21o6 percent of the matter inthe universe. Wakes which accrete 80 percent of the matter in the universe have anaverage interwake spacing of 16’7hMpc. Since the underdensity scales as anyvoid with a significant underdensity will have a similar average size. We assume that thedistribution of luminous matter mirrors the distribution of the cold collisionless matterso that the wakes will contain the same fraction of luminous matter as cold collisionlessmatter.A recent survey by the CfA indicates that galaxies may lie on the surfaces of bubblelike structures which contain voids of size 25 — 50h’Mpc and density 20 percent ofthe mean density. Thus for 16 ‘ 3 strings may be able to account for the observed largeChapter 3. Galaxy Formation in Cold Dark Matter 106scale structure (although it difficult to estimate the uncertainty in the calculated size ofthe structure produced by the strings).It has also been observed that large regions 45hMpc may be streaming coherently with peak velocities of up to lO3km/s. Strings can produce peculiar velocities ofmagnitude 34,10[L6km/s over distances less than For S=45, v 75ji6km/swhich is a bit small. For S=20, v -‘ 170i6km/s which is close to the observed value forI6 = 4, but the distance over which these velocities are coherent is only 10hMpc.We also find that wave packets propagating on the string will produce tube-like overdensities within the accretion wake of the string. For a triangular wave pulse the coordinate cross section of this tube has dimensions 847h’pc, where c is theratio of the amplitude of the pulse to the horizon at the time of formation of the wake.We also find that the region of collision between two wave pulses propagating in oppositedirections can accrete Thus structure on the strings canfragment the wake into galaxy mass objects.Chapter 4Galaxy Formation in Hot Dark Matter4.1 IntroductionIn the previous chapter we reviewed the problem of galaxy formation and examined stringinduced density perturbations in cold dark matter. Since the nature of the dark matteris unknown at present we also want to consider string induced density perturbations inhot dark matter. In this chapter we will examine the density perturbations produced bystrings in hot dark matter (HDM). The HDM will be taken to be composed of massiveneutrinos with a mass lOOh2ev. This mass is just sufficient to close the universe (ie.= 1). The main difference between hot and cold dark matter is that the hot darkmatter particles have significant random thermal velocities while the cold dark matterparticles do not. This random thermal velocity of the hot dark matter particles will tendto erase density perturbations.In this chapter we derive the Gilbert equation which describes the evolution of densityperturbations in hot dark matter. We then examine the wakes produced by wiggly stringsand the fragmentation of these wakes. Throughout this chapter we choose our units sothat Boltzrnann’s constant, k=l.4.2 The Gilbert EquationIn this section the Gilbert equation[14, 29, 20, 45] and the equation which describes theevolution of the peculiar velocity field will be derived. The derivation of the Gilbert107Chapter 4. Galaxy Formation in Hot Dark Matter 108equation will closely follow reference [45].We begin by discussing the mechanics of a particle in an expanding universe. TheNewtonian approximation will be sufficient as long as the particle velocities are nonrelativistic and as long as the particles of interest are well within the Hubble radius. TheLagrangian for such particles isL = m2- m,t) (237)where in is the mass of the particle and t) is the Newtonian potential which satisfies= 4rGp(,t) . (238)It will be convenient to transform to coordinates () defined by = a(t) and toperform the canonical transformation(239)where L’ = rriaal2.The coordinates follow the isotropic Hubble flow and hence cffferfrom the comoving coordinates ( used in the Zel’dovich approximation which follow thecold dark matter particles. The new Lagrangian isL = ma2L2— mff(,t) (240)where cff(x, t) satisfiesVcbff(,t) = 4rGa2[p(,t)— pb(t)] = 4irGa2pb(t)b(f,t) (241)pb(t) is the unperturbed background density and t) is the density contrast which isdefined by—p(, t) — pb(t) 6p(, t)242\X— pb(t)— Pb(t)Chapter 4. Galaxy Formation in Hot Dark Matter 109To calculate the equations of motion we first need the canonical momentum (pj, whichis given by= ma2= maipec (243)where pec = a is the peculiar velocity. The equations of motion for j5 arep=—mVxeff(x,t) (244)We now consider the equation which describes the evolution of the neutrino phasespace density. Since neutrinos are weakly interacting their phase space density f( t)will satisfy the collisionless Boltzrnann equation,+ ma- mVeff pf =0. (245)We now assume that the string produces a small perturbation (fi) to the homogeneousbackground (fo). We can therefore writef(,j5,t)= fo() +f1(,t) (246)where Ifil << fol, fo is given byfo(p) = (e + 1) (247)and T is the neutrino temperature (Ta is a constant since T x ). Since the source foreff is op it is already a small quantity. The background density pb(t) and the densityperturbation Op(, t) are related to fo() andf1(,t) throughpb(t)=(22rn)3 f fo()d3p (248)andop= (2 )3ffl(xTht)dP . (249)Chapter 4. Galaxy Formation in Hot Dark Matter 110To linear order the collisionless Boltzmann equation becomes+ — mV€ff = 0 . (250)The above partial differential equation can be written as an integral equation if we goto Fourier transform space. The advantage an integral equation over a partial differentialequation is that it is much simpler to handle numerically. To begin with we write fi,g5eff and 5p in terms of their Fourier transforms f, and 5, i.e.f1(,t) = 1 ffi(,t)ed3k (251)(2K)2q(,t) = 1 f(,t)ecl3k (252)(27r)2and1f(,t)ed3k. (253)(2r)2Substituting (251-253) into (250) gives++ 4jma2pfo(p) = 0. (254)At this point it is useful to change time variables from t to , where is defined byd = dt/a2( ) . (255)In terms of equation(254) becomes= _4 maej6. fo(p) . (256)Integrating this from some initial time to time gives) = - 4iGm() fa’),(257)Chapter 4. Galaxy Formation in Hot Dark Matter 111where f (p) = From this equation we can find the Gilbert equation and theequation describing the evolution of the peculiar velocity field.To obtain the Gilbert equation we integrate (257) over j5 and divide by f fo(p)d3p.This gives= -- ‘)]d’ (258)whereI[k(-= J (259)To evaluate I[k( —‘ )j we follow reference [45] and use the approximationfo() 3e (260)where1 0o p2(l+ep)dP (261)The value of is chosen so that the total number of neutrinos is unchanged by theapproximation, i.e. f fo(p)d3p= feP1d3p. This approximation overestimates thenumber of low momentum neutrinos and underestimates the number of high momentumneutrinos. Thus in using this approximation we will slightly overestimate (by less than20 percent) the amount of matter accreted by the wakes. With this approximationI[k(— ‘)J can be evaluated analytically and is found to be (see appendix B)“1 8iriii3(Ta)I[k —= m [1 +c2k(— 1)2]2 (262where c = Substituting this into (258) gives— ffi(, 8Gm3(Ta) a(’)(-( ,)— ffo(p)d3p + J [1+2k(-’) ()(263)Chapter 4. Galaxy Formation in Hot Dark Matter 112Before writing the Gilbert equation in its final form we change time variables fromto a dimensionless conformal time (T) defined bydr=- (264)4awhere Heq is the Rubble parameter at teq. We also scale a(t) so that a(teq)=1. Thistransformation will greatly simplify the Gilbert equation and the equation for the peculiarvelocity field. The relation between and r is given by(265)‘1eqand a(T) is given by (see appendix A)a() = 4(r2 + r) . (266)As we will see later the first term in (263) will be of the formHeqw(— i) 267)where w depends on A but not . Substituting the above into (263) gives,r)= [1+k2g(r,)]+6f [1 +k2g(T’)2j2(’) (268)whereg(r, T’) = ln(1 + ) — ln(1 + (269)and ==This is the Gilbert equation for T). As stated above thederivation of the Gilbert equation closely followed reference [45].We now derive an equation for the peculiar velocity field in a way which I have notseen in the literature. The mean streaming velocity, , is defined by1 f15*fdSp (270)ma ffd3pChapter 4. Galaxy Formation in Hot Dark Matter 113The equation for the Fourier transform of is then obtained by multiplying (257) byintegrating over and dividing by ma f fo(p)d3p. The Fourier transform of thussatisfies= ffi( - (271)where= (272)A simple calculation shows that (see appendix B)fi[k,,’] = . (273)We now decompose into a component parallel to 1 (v11) and a component perpendicularto k (vj). These components are defined by, vi(k,) = i(k,) k and ‘j = — vjk.These quantities satisfy the following equations= f( + (274)and— f{- (lv. jf)e()dpv(k,)— 3 . (275)mak f fo(p)d pBy comparing (274) and (258) we see that— i a6(k,)=— (276As is usual decays with time and will therefore be ignored. Thus equation(276) isthe required equation describing the time evolution of the peculiar velocities. For theremainder of this chapter I present the research I carried out on the density perturbationsproduced by wiggly strings in hot dark matter[72].Chapter 4. Galaxy Formation in Hot Dark Matter 1144.3 The Accretion WakeIn this section the thickness of the nonlinear region of the accretion wake and the fractionof matter in the universe accreted by the wakes will be calculated. We also calculate thepeculiar velocity fields and the height of the turnaround surface. These results willthen be compared to the relevant observations. Some numbers that will be useful insubsequent calculations are H1 4.4 x 10°hs and 1+ Zeq 2.9 x 104h2 (see appendixA). We also have[45, 76] Teq 5.8 x l04h2Kelvin. (the neutrino temperature at teq) andVeq = 0.05.We begin this section by setting up the Gilbert equation. The (average) velocityperturbation (v) produced by the string at distances which are much larger than thesize of the small scale structure is given by=--y[ + (Eç,)lC (277)where E is the energy in the waves on the string and E is the total energy of the string(see expression (213) in chapter 3). Since we need the velocity perturbation at all pointsin space we consider the case when the size of the small scale structure goes to zero. Inthis case expression (277) is valid everywhere.To set up the Gilbert equation we need fi which we will now calculate. Since theeffect of the string is approximated as a pure velocity perturbation we have= fo( 55) (278)where 5j5 is the change in the canonical momentum produced by the string. Expanding(278) to first order in 5j5we find thatfi(,) = PPf1() = mPzaivf(p)sgn(z) (279)Chapter 4. Galaxy Formation in Hot Dark Matter 115where zv is the magnitude of the velocity change given in (277), a2 a(t) and t is thetime at which the wake formed. Taking the Fourier transform of fi gives)6(k) . (280)kpFrom f we can calculate the first term in the Gilbert equation(i.e. equation (263)).It is given by— (281)[1 +a2k(—Hence w in (267) is given by4aiAv6(k)(k) (282)WHeqNow define a reduced density contrast (6) byr) 4ai6vR(kT)(k)6(k) (283)HeqThe reduced density contrast then satisfiesg(T,)+6I g(T,T’)(k,T’)d6(k,r) = [1 + kg(T, )2j2 JT2 [1 + kg(T, r’ )2}2 TI . (284)(recall —_____*— mHeqWe now consider solutions to equation (284). This is a Volterra integral equation ofthe second kind and can be solved numerically using the trapezoidal rule. For cik <<Twe find that the solution to (284) can be approximated byA 26R(kZ,T) B4+ck4T (285)where A and B are functions of (the value of T when the wake forms). For T = q wefind that A0.21 and B0.72. Substituting (285) into (283) gives4fia/v_________Ho(k)b(ky)B4AT2 . (286)eqChapter 4. Galaxy Formation in Hot Dark Matter 116So far we have calculated the Fourier transform (6) of the density contrast. We areactually interested in 6(f, T) which is given byb(f. T) = 1 j,r)ed3k . (287)(2r)2The inverse Fourier transform of (286) can be found analytically and 6(, T) is given by2AuaiZv_1 . 2b(z, i= B4Heqe srn(crz + (288)where=We now examine the wake produced by the string. The nonlinear region of the wakewill be defined as the region in which 6 1. The boundary of this region is then foundby setting 6 = 1. For 16 = 4 and < h < 1 we find that the height of the nonlinearregion varies little with r for 0.1 Ti 0.5 (0.2teq ‘ ti ‘ 7teq) (recall that I6 is definedby = jt6 x 10 and h is the ration of Hubbies constant to 100 Km/s/Mpc). We willtake the ratio of the coordinate thickness (see below for exact definition) of the nonlinearregion to the interwake separation at T as an estimate of the fraction (f) of HDM in theuniverse that has been accreted by the wakes. In figure 4.27 we plot f vs Ti. For the uppercurve /6 = 4 and h=1 and for the lower curve 6 = 4 and h = . As before we will assumethat it is the last wakes which accrete almost all of the matter in the universe which setthe size of the large scale structure. From fig 4.27 we see that the wakes of interest formbetween Ti ‘ 0.5 for h= and between 0.5 Ti 0.8 for h=1. As we can see fromfigure 4.27 the wakes which accrete 80 percent of the matter in the universe formed at0.33 2.6teq for h= and at Ti 0.51 2 7teq for h=1. The interwake separation forthese wakes is 35Mpc for h= and l3Mpc for h=1. For h= the wakes which formbetween 0.3k Ti 0.5 have an interwake separation of 30-5OMpc. For h=1 the wakeswhich form between 0.5’’ T 0.8 have an interwake separation of 13-2OMpc. Thusthe size of the structure prodilced by the strings is smaller but comparable to the sizeChapter 4. Galaxy Formation in Hot Dark Matter 1170.80.6f0.40.20Figure 4.27: The fraction (f) of matter in the universe accreted by the wakes plottedagainst the time (ri) at which the wakes formed. For the upper curve t6 = 4 and h=1.For the lower curve /‘6 = 4 and h =Chapter 4. Galaxy Formation in Hot Dark Matter 118of the observed large scale structure. As with cold dark matter it is difficult to estimatethe uncertainty in the above numbers.We now examine in detail the wakes formed at 2teq. From the previous paragraphwe see that for these wakes to be of interest we must have h . For ti = 2.3teq we findthat A 0.22, B 0.82 anda2=1.6. Substituting the values for the various quantities in(288) gives6(z, r) = 4.3 x 108vesin[zI + ]ii6r2 (289)where u pc’h4. In the Bennet and Bouchet[8, 9] simulations E/E 0.5 in theradiation dominated era and 0.3 in the matter dominated era (see section 1.3 fordetails). We will therefore take E/E = 0.4 for t teq. The Allen and Shellard[6, 48]simulations also show that the coherent string velocities are about 0.15c. From 1 + Zeq =4(r2 + r) we find that r 85h. Substituting the above results into (289) gives6(z,ro) 4.4e1sin[az + ]t6h2 . (290)The maximum value of 6(z, r0) occurs at z=0. Hence for any nonlinear region (i.e. 6 1)to exist today we must have 6(0, To) 1. If 6(0, r0) 1 we see that 6 must satisfy/J6 0.32h2 . (291)This restriction on P6 is for wakes which form at ti = 2teq. I find that this restriction onP6 is is not sensitive to r for 0.2 < r < 1. This means that the above restriction on I6is essentially independent of over the range of T we are interested in.We now calculate the height of the nonlinear region. To do this we set 6 = 1 and solvefor z. The coordinate height of the wake will be denoted by z. For h = and bt6 = 4we find that z, llOOpc. The approximate physical height is 1100(1 + z) 8Mpc.To estimate the fraction of matter accreted by the wakes we do not want to use thepresent coordinate thickness of the wake (2z) since the matter presently at z has fallenChapter 4. Galaxy Formation in Hot Dark Matter 119towards the wake from an initially larger z coordinate. It is the initial z coordinate (z)of the matter presently at z that we need to use to estimate the fraction of matteraccreted by the wakes. To calculate z we first need to calculate the peculiar velocityfield produced by the string.To begin the calculation of the peculiar velocity field we change variables from to Tin (276). This gives) — iHeq 86(k, ‘r) -.r) = — k— Vf_k2 OTk . (292)Therefore(, r) = iHeq J ã, r)ed3k . (293)Substituting the expression for 6(, r) from (286) into (293)and integrating gives=— Avar [1— e cos(uz)j sgn(z)6rn/s (294)—4.6 x 1O°[i — e cos(z)} sgn(z)t6hrn/s . (295)We now return to the calculation of z. To calculate the distance (Zz) that the fluidhas fallen towards the wake we integrate (294). If uz > 1, which will be the case here,we can neglect the e1-cos(oIzI) term in (294). Integrating (294) from r to TO with<<T0 gives______2HeqB4r0 . (296)For ti 2.3teqAz 2.5 x lO3pc. (297)We can therefore write z=z+Az.We take the ratio of 2z (thickness of the wake) to the coordinate interwake separationas an estimate of the fraction of the matter in the universe which has been accreted byChapter 4. Galaxy Formation in Hot Dark Matter 120the wakes. Since the coordinate interwake separation is 6.6 x lO3pc these wakes haveaccreted all of the matter in the universe.Another surface of interest is called the turnaround surface. It consists of thoseparticles which are in the process of turning around from the Hubble expansion andheading back toward the wake. To find the turnaround surface set = 0 in the zdirection. This gives= —H’(1 + zeq)1v (298)where H0 is the value of the Hubble constant at the present time. Using H’ to andto 2 x 10’7hs givesz 155[16h2[1— e1cos(z)]pc . (299)For h = and [16=4 we find z 2.6 x lO3pc, which is about twice zn,. For a solution toexist to equation (299) we find that [16h2 must satisfy [16h2 ‘ 0.3. For 155t6h2>>the solution to (299) can be approximated by z 155[16h2.In the last few paragraphs we have taken h. We now resume our discussion forii < 1. Since expression (291) is not sensitive to for 0.2< r 1 we can concludethat if the universe is dominated by HDM and if strings are relevant for galaxy formationthen 116 is must satisfy0.3h2 [L 4. (300)The upper limit comes from the timing of the millisecond pulsar (see section 1.1) and thelower limit is from the above analysis. For h = we see that [16 must lie in the narrow< <range 1 116 4.re are now in a position to compare the peculiar velocities predicted by this scenariowith the observed large scale peculiar velocities. From expression (296) we see thatthe peculiar velocity field is zero at the wake center and tends to increase as we moveChapter 4. Galaxy Formation in Hot Dark Matter 121outwards. The maximum peculiar velocity of galaxies will therefore occur near the edgeof the nonlinear region. For 1 <6h2 4 I find that the velocity at the edge of the wakeis about 300 — 400p6hkm/s. For 6h 2 this is close to the magnitude of the observedlarge scale streaming velocities, but the distance over which this velocity is coherent issmaller than that observed.We expect galaxies to begin to form when 6 ‘ 1. Thus the time (TNL) at whichgalaxies begin to form can he estimated as the time at which 6(z = 0) goes nonlinear.Setting 6(0, T) = 1 for ti = 2.3teq gives TNL This means that 1 + ZNL 3.1[L6h2Wakes which formed at T 2Teq 4.5teq go nonlinear first at l+ZNL 3.3t6h2 Overthe range 0.2 r 1, l+ZNL 3,u6h2. Observations indicate that galaxy formationoccurred for 1+z’3. This is consistent with ,u6h2 = 1. If h 1 and 1u 4 the wakes gononlinear much earlier.4.4 Wake FragmentationIn this section we consider the effect of the small scale structure on the wake producedby a string. It was shown in chapter 3 that for CDM (cold dark matter) the smallscale structure can fragment the wake into galaxy mass objects. In HDM neutrino freestreaming tends to erase density perturbations on small scales so that the perturbationsproduced by the small scale structure may be erased.We now consider how the total density perturbation can be decomposed into theperturbation produced by the string without a wave pulse plus the perturbation dueto the wave pulse. The velocity perturbation produced by the string will be the sumof the velocity perturbation produced by the string without the pulse plus the velocityperturbation due to the pulse. That is, we can write the total velocity impulse as= /v + (/v — /v) (301)Chapter 4. Galaxy Formation in Hot Dark Matter 122where /v is the velocity perturbation produced by the string without the wave pulseand (v—is the velocity perturbation due to the pulse. From the Gilbert equationwe can see that the density perturbation depends linearly upon the velocity perturbation. Therefore to linear order the density perturbation will also be equal to the densityperturbation produced by the string without the pulse plus the perturbation due to thepulse. Hence if the pulse is to have a significant effect on the wake it needs to producesignificant (i.e. nonlinear) density perturbations. We can therefore confine our analysisto the density perturbations produced by the pulse alone.We now discuss a simplification in the velocity perturbation which will simplify thesubsequent calculations. The velocity perturbations produced by the pulse depend on thevelocity of the string (3). The perturbations produced by the pulse with string velocity= 0 and ,6 = .15 are essentially identical except very near the string. Even near thestring they do not differ greatly. Therefore we will take 41 = 0 in the expression for thevelocity perturbation produced by the pulse to greatly simplify the calculations.We now need to choose a specific form for the wave pulse. Since the calculations canbe done analytically for a triangular wave pulse we takeo u<—aa+u —a<u<agQu) =a—u O<u<a0 u>0where u=x-t. The velocity perturbation produced by this wave pulse is calculated inAppendix C. Taking 41 = 0 in expressions (370), (376), and (377) giveszvx = 0L\v = —8G[tan(-) + tan()] . (302)Zv = _4G/1ln[2( )21Chapter 4. Galaxy Formation in Hot Dark Matter 123The next step in calculating the density perturbation is to compute the Fourier transforms (ak) of /vk. They are defined by= 1Jvi()ed3x (303)(27r)2and are given by= 0167rG k/k r1— ej6(k) , (304)= (2ir)4 k+k L— 16irGi [l—e’]— (2r) k+kNext we need the Fourier transform of fi(, , It is given by16rGtma (1 — —ika\eJiO) = -___________(2k k+k f0(p)6(k) (305)pThe first term in the Gilbert equation (263) is then given by16irGtia(1 — e’) ( —— 1 6(k) [1 +c2k(— .)2]2 (306)(27r)kHence w in (267) is given by16\/Gpia(1 — e_ikL)w =— ö(k) (307)HeqkzTo simplify the Gilbert equation we write it in terms of a reduced density contrast (bR)which is defined byik a),T) =— 16Gtia(1 — e (k)R(k, k, r) (308)HeqkzIn terms of6R(k, k, ‘r) the Gilbert equation becomesg(r,r) T g(T,T)= [1 + kg(T,r’)2]+ 6f [1 + (309)where k2 = k2 ± k2yz y z•Chapter 4. Galaxy Formation in Hot Dark Matter 124Now consider solutions to the Gilbert equation. Once again the solution for r >>is given byR(k, k,= B4 + ckzT(310)The density contrast, 6(, T), is given byT) — j’ (1— e) ei(kyl2) dk dk (311— (2’7r)HeqO j k + (1 + k)2 Y zwhere r = = 1/u. The k integral can be done givingb( T) = _8iAar [ (1— C )e1om(O + bO)ejdk (312)‘./HeqQ J_ kb sin 20where b = (u + k and 0 is defined by cos20 =We now want to see if any nonlinearities are produced by the wave pulse. Sinceall particles are attracted to a small region centered on (y,z)=(0,a/2) we will calculate6o(r) = 6(y = 0,z = a/2,r).— 16AGtar2 j°° sin(ka/2) sin06oiT)_ 1 . dk 313J— k( + k4) sin 20= 16AGwar2 sin[uJ du (314)HeqB4 JO u(1--u4)cos0where to obtain the last expression we have taken u = k/uk. Define1 ça sin[-u] duF(a)=—I 3 . (315)2Jo z(1+u4)cosOFor t = 2.3teq we have6o(ro) 1.5F(a)t6h2. (316)For the wave pulse to produce a nonlinear region we require that 50(TO) 1. For u6h2 = 1we find that 5 = 1 at F(a) 0.67. In figure 4.28 we plot F(a) vs a/dH (dH is the distanceto the Hubble radius at tq). From figure 4.28 we see that F(a)—’0.67 for a 0.O5dH. HenceChapter 4. Galaxy Formation in Hot Dark Matter 125F(a)0 0.05 0.1 0.15 0.2 0.25 0.3Figure 4.28: F(a) plotted against a/dH.Chapter 4. Galaxy Formation in Hot Dark Matter 126triangular pulses with a << O.O5dH cannot produce nonlinear density perturbations.Pulses with aZ O.O5dH can produce nonlinear density perturbations.For pulses which may be able to seed nonlinear structure it is important to have anestimate of the size of this structure. This can be estimated by considering 6(y = 0, z, r).From (312)6(y = 0 z, r) 16AGwar2oo sin[u] cosR/cT(z — a/2)u] du (317)HeqB4 Jo u(1+u4)T cosI find (numerically) that 6 is approximately constant for z<<a. For a o1 0.O4dH, 6begins to decrease as z —* a and then decreases rapidly for zZ a. Thus the extent of thenonlinear region will be of order the pulse size, which will contain a significant amountof matter if a’ 0.O5dH.Above we considered a triangular wave whose base length was twice its height. We alsoneed to consider triangular waves with different dimensions. We will therefore consider atriangular wave pulse with amplitude (a) and a base half-width (w). The impulse givento a particle close to the triangular wave pulse, but not to close to the kinks, will be thesame as the impulse produced by an infinitely long string. This impulse is proportionalto 16’y where,6j is the component of the string velocity (due to the wave pulse) which isorthogonal to the string. A simple calculation shows that /3’yj cx a/w for w/a<.<1 (seesection 3.7). Thus the velocity impulse and hence 6 grows as the pulse becomes morepeaked. Therefore very peaked small pulses may be able to produce nonlinear densityperturbations. But such pulses are expected to be rare. Hence we can conclude that smallscale structure whose size is << .O5dH cannot fragment the wake. Pulses whose size is‘ .O5dH may produce nonlinear regions of similar size. Pulses which produce significantdensity perturbations (ie. with 6 of order unity) will also cause the perturbations tobecome nonlinear earlier and hence allow galaxies to form before 1+z 3i6h2.Chapter 4. Galaxy Formation in Hot Dark Matter 1274.5 ConclusionIn this chapter we examined the velocity and density perturbations produced by cosmicstrings in hot dark matter.We assumed that it is the last wakes which accrete almost all of the matter in theuniverse which set the size of the large scale structure. We found that for h= and u6 = 4(the largest value of ,u6 consistent with observations) these wakes form between 2teqand 6.tq and have a present interwake separation of 30-5OMpc. For h=1 and 6 = 4we find that these wakes form between 6.teq andl9.5teq and have a present interwakeseparation of 13-2OMpc. This is smaller but comparable to the size of the observedlarge scale structure (25-50h’Mpc) It is difficult to estimate the uncertainties in thesenumbers but we can conclude that this scenario for galaxy formation is promising.We also found that for strings to produce any nonlinear density perturbatiolls t6 mustsatisfy [16 . O.3h2. For wakes which formed at 2teq with h= and i = 4 the physicalheight of the nonlinear region is 5Mpc.The string also produces a peculiar velocity field which increases as we move outwardfrom the wake center. For 1 < 1t6h2 4 the peculiar velocity at the edge of the wake is300-400 ht6km/s. For h[16 2 the magnitude of this velocity is close to the magnitudeof the observed large scale peculiar velocities, but the distance over which this velocityis coherent is smaller than observed. For wakes formed at 2teq the redshift at whichnonlinearities set in is (1 + ZNL) 3.1[16h2 Wakes which formed at 4.5teq go nonlinearfirst at a redshift of c 3.3[16h2 Over the range of r of interest l+ZNL 31i6h2. This canbe taken as an estimate of the redshift at which galaxies formed. Observations indicatethat galaxy formation occurred for 1+z 3. This is consistent with [16h2 = 1.We also find that for h= small scale structllre whose size is << 0.O5dH (dH is thedistance to the Hubble radius at ‘ 2teq) cannot produce nonlinear regions and hencechapter 4. Galaxy Formation in Hot Dark Matter 128cannot fragment the wake. Waves whose size is Z O.OödH may be able to producenonlinear regions whose extent is of order of the size of the wave. For h=1 the criticalsize for the waves is O.OldH (dH is the distance to the horizon at 7teq).Chapter 5ConclusionIn this concluding chapter we will summarize the new results obtained in this thesis anddiscuss their implications. The main objective of this thesis was to examine the effectof the wiggles, which propagate along cosmic strings, on various physically importanteffects produced by the strings. In particular, we examined the effect of these wiggles onthe density perturbations and light deflection produced by cosmic strings.In chapter 2, I investigated the gravitational lensing properties of wiggly cosmicstrings. This investigation is of interest because we may be able to detect cosmic stringsthrough the gravitational lensing they produce. As one approaches a wiggly string thecurvature tensor diverges. Clarke, Ellis and, Vickers [23] have raised the concern that thiscould completely alter the gravitational lensing properties of the string, in comparisonto a straight string.I found that the divergence of the space-time curvature has little effect on the structureof the double images. That is, the position of the double images is close to what onewould expect based on the lensing properties of straight strings.In chapter 3 the density and velocity perturbations produced by strings in cold darkmatter were calculated using the Zel’dovich approximation and results from recent simulations of cosmic string evolution.I found that the average velocity perturbation produced by a string carrying waves129Chapter 5. Conclusion 130propagating in both directions is given by<v >= 4K7[ +(Eç1)]C (318)where E/E is the ratio of the energy in the wiggles to the total energy of the string.This velocity is directed towards the surface traced out by the string. The first term in(318) is the velocity perturbation produced by a straight string. The second term is thevelocity perturbation produced by the wiggles on the string. Numerical simulations ofthe evolution of the string network indicate that 3 0.15 and 0.3‘ E/E ‘‘ 0.5. Forthese values of 3 and E/E the second term in (318) is much larger than the first term.As the strings moves through the matter in the universe they generate wakes. Thefraction of matter 111 the universe accreted by these wake structures depends on the timethey form. Wakes formed very early accrete all of the matter in the universe. Wakeswhich form later accrete less of the matter in the universe. I assume that it is the lastwakes which accrete almost all of the matter in the universe which set the size of thelarge scale structure. This is a reasonable assumption because wakes which form earlierwill be accreted by the wakes we are considering. Wakes which form after the wakes weare considering will not accrete much matter. The average separation of the wakes alsodepends on the time they formed. We can therefore write the fraction of matter accretedby the wakes (fa) as a function of the interwake separation (Sh’). I found the relationbetween fa and S to be,fa (319)where t6 = i/106 (recall that t6 < 4). The last wakes which accrete almost all ofthe matter in the universe will have fa -‘ 0.8 (the actual value chosen for fa is not tooimportant). Therefore the size of the scale produced by the strings isS i—’ 16,/iMpc . (320)Chapter 5. Conclusion 131Thus we expect the string network to produce a network of sheetlike overdensities. Theregions between the sheets will be low density voids. The average size of the voids willbe 16/7Mpc.We now see how this compares to the observed large scale structure. Recent observations indicate that galaxies lie on the surface of bubble-like structures of size 25-50h’Mpc. The interors of these bubbles are voids with a density of only 20% of themean density. From (320) we see that S ‘—‘ 32Mpc for ,u6 4. This means that theaverage interwake separation is 32h1Mpc. This compares favorably with the size ofthe observed large scale structure (25-50h’).I also calculated the large scale peculiar velocity fields produced the the strings. Ifound that the strings can produce peculiar velocities with magnitudes similar to thoseobserved, hut the distance over which this velocity is coherent seems to be, on the basisof crude estimates, significantly smaller than observed.Lastly I considered the effect of the small scale structure on the wakes produced bythe strings. I found that a wave pulse propagating down the string would generate atubelike overdensity within the accretion wake. If we have two waves propagating on thestring in opposite directions the tubelike overdensities they produce will intersect. Thisregion of intersection will be a region of large mass accretion. I showed that this regioncould easily accrete the mass of a galaxy. Hence the small scale structure on the stringscan fragment the wake into galaxy mass objects.In chapter 4 the wakes produced by the string network in hot dark matter (neutrinos)were examined. I found that the size of the structure produced in hot dark matteris smaller but comparable to the size of the structure in cold dark matter. This isa consequence of neutrino free streaming which tends to erase density perturbations.Because of this free streaming the wakes in hot dark matter accrete less matter than thewakes in cold dark matter. This means that the last wakes which accrete almost all of theChapter 5. Conclusion 132matter in the universe form earlier in hot dark matter than in cold dark matter. Thusthe size of the structure will be smaller in hot dark matter than in cold dark matter.I also examined the effect of the small scale structure on the wakes produced by thestrings in hot dark matter. I found that for h= waves with a size greater than about0.05 times the Hubble radius may be able to fragment the wake. Waves which are muchsmaller than this will not be able to fragment the wake. This result is not surprisingif one considers the effect of neutrino free streaming. Small waves will produce smalldensity perturbations which will be erased by neutrino free streaming. Larger waves willproduce density perturbations which will be large enough not to be significantly effectedby free streaming.In conclusion we can say that the cosmic string scenario for structure formationlooks promising in both cold and hot dark matter, although the parameter u is tightlyconstrained if strings are to play this role.Appendix AIn this appendix we review some of the equations of elementary cosmology.For a homogeneous isotropic universe the line element can be written as[76]ds2 = —dt2 + a(t)2[1 dr2 + d2 +r2sind] (321)For simplicity we will only consider universes with k=O (i.e. 12=1). Substituting theabove line element into the Einstein field equations— = 87rGT, (322)givesa2 = pa (323)where p is the total density. For a universe filled with an ideal fluid and a null fluid(radiation) p is given byp(t) = + (324)where aeq is the scale factor at time t, when the matter and radiation have the samedensity. p and Pq are the matter and radiation densities at teq. Scaling a(t) so thata(teq)=1 and substituting (324) into (323) gives•28’wGr1+aa— Peq[ 2At teq the Hubble parameter is given by H = lpr Thus(326)133Appendix A. 134Integrating equation (326) and taking a(t=0)=0 gives(1+a) -2(1+a)4+ (327)A simpler expression for the ‘time’ evolution of the scale factor can be found by transformillg to the dimensionless conformal time (T) defined bydT= Heq dt (328)4iJa(t)Integrating the above and taking t(r=0)=0 gives16V’1 1t = [3 + —r2j (329)eqIn terms of the variable r equation (326) becomes=41+a (330)Integrating and taking a(r=0)=0 givesa(r) = 4(r2 + T) (331)Setting a=1 and solving for r gives Teq 0.207.We now calculate the values of some useful quantities. From (327) we findHeqteq = (1/ — 1) 0.552 (332)From (326) we havea2 i He (333)which givesHeq= at) (334)We now need to calculate a(to)=1+zeq (where t0 is the present time). Since p =p(l + zeq) and p = Pq(1 + zeq)4 we have1+Zeq (335)p;Appendix A. 135For t2 = 1 we have= 3H(to)x 1026hkg/m3 (336)The density of radiation is given by[76jp(t) = [1+ fl(±)]pem(t) (337)where n is the number of neutrino species and pem(t) is the density of the backgroundelectromagnetic radiation. For n=3, pr(t) 1.68pern(t). The present density of thebackground radiation is pm 4.5 x 1031kg/m. Thus p 7.6 x 1031kg/m andl+Zeq 2.5 x 104h2. Substituting H0=lOOh km/s/Mpc and a 2.5 x 104h2 into (334)givesHeq 1.81 X 10’h4s’ (338)and = 5.5 x 10hs. teq is then 3.0 x 100h4s. The physical distance to thehorizon is given byD(t) = a(t)cfa(t) (339)Transforming to T givesD(r) = 4a(T)c f dr (340)ThusD(r) = 16c(3+ T2) (341)The horizon distance at Teq S dH(Teq) 627h4pc. In the matter dominated era (T>> 1)D16\,c3= 3t (342)and in the radiation dominated era (r << 1)D16c2= 2t (343)Appendix A. 136The Hubble constant isH== Heq 2T+1 (344a 16/Qr2+r)In the matter dominated era (r>> 1)H Heq 2 (345)8/r3 3tand in the radiation dominated era (T>> 1)HHeq (346— 16/T2 2tAppendix BIn this appendix we evaluate the integralsI[k(-= I (347)andI,, ‘)= f (348)Let ) lie along the z axis. I is then given by2- oo kI= f f f cosOf(p)exp[—i--cosO( —)]p2sin8dpdOdçb (349)Now let u = cosO.I = 2 L’1 f uf(p) exp[—in( — ‘)]p2dpdn (350)= 4Ki [b cos b — sin b]p2d (351)where b=— a’). Changing variables to b givesI=4ik3( ‘)3 j f(b)[bcosb — sin b]db (352)Using the approximation fo() 3e9 and integrating givesI[k(— a’)] = 8ii(Ta)3[1 +2k(— !)2}2 (353)where c =Now H11 = ft. Ic is given byH11=(354)137Appendix B. 138im 8 r k•15—-— J (355)Thereforeim 8H11 = —.-I[k(—(356)Now consider-i(x=f()(xpjfo(p)e m (357)It is easily seen that x fi = 0, so thatim0= —--I[k(— ‘)] (358)Appendix CIn this appendix we calculate the velocity perturbations and /v produced bya triangular wave pulse. From (154) we have1 cocOhoo=— / ãdt (359)where= —4Gi72[g’ + (3— f)2] ln{[7(y — 3t) — f]2 + (z— g)2}1 (360)andf=f[x—-y(t--/y)j , g= [x—’y(t—/3y)] (361)We will take the triangular wave to be in the (x,z) plane. Thus f=0. Substituting (360)and (361) into (359) gives for Z\v2 °° (z — g)g’(2 + g’2)= 4G7 J72(y — 3t)2 + (z— g)2 — g g ln[7(y — j3t)2 + (z — g)2]dt (362)By integrating by parts and taking g=g’=O at t = +oo it can be shown that/ g g ln[72(y — t)2 + (z — g)2]dt =— / g’27 — t) — g’(z — dt (363)J-oo y2(y — 3t)2 + (z — g)2Combining (363) and (362) gives/3g’(z— g) + 7g2(y—/3t)dt (364)= 4Gt72L 72(y — t)2 + (z — g)2For /v we getI II= 42J[2(— t) — 7g(z— g)](2 +‘2)g g— 3t)2 + (z — g)2(365)139Appendix C. 140Substituting expression (363) into (365) gives= —4G[00 (72/32 + g’2)(y — 9t) — /337g’(z — (366)i-cc— /3t)2 + (z — g)2Nowcc j32y(y—3t)f__cc 2(Y_i3t)2+(Z_g)2dt = f°00 I{ln[’(y — /3t)2 + (z — g)2]}dt_____________(367)(z—g) dt/3If g has support on a finite interval in t then the first integral on the RHS of (367) iszero. Using this and = —79’ givescc /32’y(— /3t) cc /3yg’(z—g) dt (368)— /3t)2 + (z — g)2dt = fcc 72(y — /3t)2 + (z — g)2Substituting this into (366) gives“OO= —47Gi / /3g’(z — g) + ‘yg2( — /3t)dt (369)[y(y — /3t)2 + (z — g)2]By comparing this to the expression for /v3, one can readily see that= —/37v (370)For Lv we getcoo (z_g)(/32+g12 d (371)= —472GJ 72(y — /3t)2We take the triangular wave pulse in the rest frame of the string to be0[a+ —a<u<0[a—u0 uaAppendix C. 141where u=zx-t. In the frame of reference in which the string is moving with velocity , gis given byo tti1gl=a_x+7(t_/3Y)tltt2(g={=a+x—7(t—y) t2<tt3o tt3 Jwheret1 = /3y + ‘y’(x — a) (372)t2 = y +1x (373)andt3 = y +71(x + a) (374)Thust1 y23(y—3t) dt + ft2 (1+713)(y—1 t)+7/33z gl)= —472G{f_7(y—t)+z ti 72(y—t)+(z g1) dt____________________ __________(375)+ft3+ f0° 72(y—t) dt2 2(y—/3i)+(z—gl t3 (y_8t)_z 1Since g and g are linear functions of t these integrals can be found in any decent tableof integrals. 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