aa r X i v : . [ h e p - t h ] M a y TOPICS IN INFLATIONARY COSMOLOGY AND ASTROPHYSICSbyMatthew M. GlenzA Dissertation Submitted inPartial Fulfillment of theRequirements for the Degree ofDoctor of Philosophyin PhysicsatThe University of Wisconsin-MilwaukeeDecember 2008OPICS IN INFLATIONARY COSMOLOGY AND ASTROPHYSICSbyMatthew M. GlenzA Dissertation Submitted inPartial Fulfillment of theRequirements for the Degree ofDoctor of Philosophyin PhysicsatThe University of Wisconsin-MilwaukeeDecember 2008Major Professor DateGraduate School Approval DateiiBSTRACTTOPICS IN INFLATIONARY COSMOLOGY AND ASTROPHYSICSbyMatthew M. GlenzThe University of Wisconsin-Milwaukee, 2008Under the Supervision of Distinguished Professor Leonard ParkerWe introduce a general way of modeling inflation in a framework that is indepen-dent of the exact nature of the inflationary potential. Because of the choice of ourinitial conditions and the continuity of the scale factor in its first two derivatives, weobtain non-divergent results without the need of any renormalization beyond what isrequired in Minkowski space. In particular, we assume asymptotically flat initial andfinal values of our scale factor that lead to an unambiguous measure of the number ofparticles created versus frequency. We find exact solutions to the evolution equationfor inflaton perturbations when their effective mass is zero and approximate solutionswhen their effective mass is non-zero. We obtain results for the scale invariance ofthe inflaton spectrum and the size of density perturbations. Finally, we show that asubstantial contribution to reheating occurs due to gravitational particle productionduring the exit from the inflationary stage of the universe.The second part of this dissertation deals with a post-Minkowski approximation toa binary point mass system with helical symmetry. Numerical solutions for particlesof unequal masses are examined in detail for two types of Fokker actions, and thesesolutions are compared with predictions from the full theory of General Relativity andiiiith post-Newtonian approximations. Analytic solutions are derived for the ExtremeMass Ratio case.The third part of this dissertation discusses the detection sensitivity of the Ice-Cube Neutrino Telescope for observing interactions involving TeV-scale black holesproduced by an incoming high-energy cosmic neutrino colliding with a parton in theAntarctic ice of the south pole. Parton Distribution Functions and the black holeinteraction cross section are computed numerically. Our computation shows that Ice-Cube could detect such black hole events at the 5-sigma level for a ten-dimensionalPlanck mass of 1.3 TeV.Major Professor Dateiv able of Contents
List of Figures ixList of Tables xAcknowledgments xi1 Introduction 1Part I - New Aspects of Inflaton Fluctuations 72 Inflationary Cosmology 8
Part II - Binary System of Compact Masses 1084 Unequal Mass Binary Solution in a Post-Minkowski Approximation109
Part III - Production and Decay of Small Black Holes at the TeV Scale1365 TeV-Scale Black Hole Production at the South Pole 137 vii ist of Figures C . . . . . . . . . . 683.7 Particle Production in the Massless Case. . . . . . . . . . . . . . . . . 743.8 Particle Production in the Effective- k Approximation. . . . . . . . . . 783.9 Particle Production in the Dominant Term Approximation. . . . . . . 803.10 Non-Zero Mass, Negligible with Respect to H. . . . . . . . . . . . . . 813.11 Massless Dispersion Spectrum. . . . . . . . . . . . . . . . . . . . . . . 843.12 Massive Dispersion Spectrum. . . . . . . . . . . . . . . . . . . . . . . 883.13 Inflaton Spectrum Characterized in Terms of Inflaton Mass. . . . . . 903.14 Modes Exiting the Hubble Radius. . . . . . . . . . . . . . . . . . . . 1004.1 Binary in Circular Motion. . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Retarded Angle ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3 Parametrization-Invariant (PN) Omega versus Velocity. . . . . . . . . 1164.4 Parametrization-Invariant (SPN) Omega versus Velocity. . . . . . . . 117viii.5 Parametrization-Invariant (PN) Energy versus Omega. . . . . . . . . 1184.6 Parametrization-Invariant (SPN) Energy versus Omega. . . . . . . . . 1194.7 Parametrization-Invariant (PN) Angular Momentum versus Omega. . 1214.8 Parametrization-Invariant (SPN) Angular Momentum versus Omega. 1224.9 Full Affine Case of Omega versus Velocity. . . . . . . . . . . . . . . . 1234.10 Truncated Affine Case of Omega versus Velocity. . . . . . . . . . . . . 1244.11 Full Affine Case of Energy versus Omega. . . . . . . . . . . . . . . . . 1254.12 Full Affine Case of Angular Momentum versus Omega. . . . . . . . . 1264.13 Truncated Affine Case of Energy versus Omega. . . . . . . . . . . . . 1274.14 Truncated Affine Case of Angular Momentum versus Omega. . . . . . 1285.1 Parton Distribution Functions: Lower Momentum Transfer. . . . . . . 1435.2 Parton Distribution Function: Higher Momentum Transfer. . . . . . . 1445.3 An Accurate Fit for a Small Range of Data. . . . . . . . . . . . . . . 1475.4 A Reasonable Fit for a Large Range Data. . . . . . . . . . . . . . . . 1485.5 Cross Section: New and Old Slicing. . . . . . . . . . . . . . . . . . . 1535.6 Cross Section: Varying Semi-Classical Regime. . . . . . . . . . . . . . 1545.7 IceCube and LHC Discovery Reach. . . . . . . . . . . . . . . . . . . . 159A.1 Velocity versus Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168A.2 External Force versus Time. . . . . . . . . . . . . . . . . . . . . . . . 169A.3 Energy versus Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171ix ist of Tables δ H for V = m φ . . . . . . . . . . . . . . . . . . . . 995.1 Probability of Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.2 Number of Signal Events. . . . . . . . . . . . . . . . . . . . . . . . . 1585.3 10-Dimensional Planck Mass Sensitivity. . . . . . . . . . . . . . . . . 158xCKNOWLEDGMENTSI wish to thank my advisor, Distinguished Professor Leonard Parker, for suggestingPart I of this dissertation. I appreciate his patience, his trust, and his guidance.Without his pioneering work on gravitational particle production, this dissertationwould not have been possible.I am also thankful for my other collaborators on Parts II and III, K¯oji Ury¯u andLuis Anchordoqui. K¯oji graciously let me contribute to his research, even though hecould have calculated my results faster by himself. I appreciate Luis’s generosity andhis sincere desire to see me succeed in physics and life.I am grateful for the support of the Lynde and Harry Bradley Foundation, andfor the support of the National Space Grant College and Fellowship Program and theWisconsin Space Grant Consortium.My wife, Alyson, sacrificed her own scholarships so that I might attend the Uni-versity of Wisconsin—Milwaukee. Thank you.xi Chapter 1Introduction
This dissertation is an exploration of space on scales that are small (quantum fluctua-tions, TeV-scale black holes, vacuum particle creation); scales that are big (anisotropiesin the Cosmic Microwave Background, seeding of large-scale structure, the Hubbleradius); and scales that are in between (Extreme Mass Ratio binary black holes,temperatures associated with horizons, Innermost Stable Circular Orbits). The firstpart of this dissertation is an outgrowth of methods developed by my thesis advisorin the following works [1, 2, 3, 4]. These methods are applicable to the creation ofquantized perturbations of the inflaton field, which is the topic we explore in Part I.The new results that appear in this dissertation are based primarily on the work ofthree papers. The first of these papers, “Study of the Spectrum of Inflaton Pertur-bations,” examines an exact calculation of the evolution of quantum fluctuations andthe subsequent particle creation in a model of the early expansion of the universethat is relevant to a wide range of inflationary potentials consistent with observationsand that does not depend on renormalization in curved spacetime [5]. The secondof these papers, “Circular solution of two unequal mass particles in Post-Minkowskiapproximation,” computes numerically a set of solutions to a helically symmetricbinary system of point masses in a particular approximation to General Relativityand presents analytical formulas for the limit that the mass of the lighter particle isnegligible with respect to that of the more massive particle [6]. The third of thesepapers, “Black Holes at the IceCube neutrino telescope,” calculates the experimentalsensitivity for observing TeV-scale black holes produced by a gravitational interactionbetween a cosmic neutrino and an elementary particle within the atomic nuclei of icemolecules [7]. This dissertation is divided into three main parts corresponding tothese three papers.In Part I, “New Aspects of Inflaton Fluctuations,” we begin with a brief summaryof early universe cosmology. Two of the most important cosmological theories ofthe twentieth century are the Big Bang theory and the theory of Inflation. The BigBang theory supposes that our universe was once much smaller and much hotter thatit is today. It explains the expansion of the universe, the presence of the CosmicMicrowave Background Radiation, and the primordial abundances of light elements.Cosmological Inflation supposes that the early universe underwent an extremely largeincrease in size in a very small amount of time. This explains why the density of ouruniverse today is so close to the critical density that separates a universe that expandsforever from one that eventually recollapses, it explains the near homogeneity andisotropy of the universe, and it explains why we don’t observe magnetic monopoles.Most importantly of all, however, inflation explains the origins of those anisotrophiesthat do exist in our universe. Although a key ingredient of the Big Bang theory is ahigh energy density in the early universe and a correspondingly high temperature, theclassical theory of inflation predicts an extreme cooling of the universe as it expands—much like the air in a piston cools as it expands to do work on its surroundings.We consider Reheating, and specifically the energy density of particles created byan expanding universe, as a means of preserving both theories without sacrificingany of their successes. We give a general overview of the amplification of quantumfluctuations into large-scale density perturbations during inflation, and we describesome of the ways of relating theoretical predictions to observations. We then list someof the observational findings of experiments.We continue with the details of the method we use to model inflation. Instead ofspecifying an inflationary potential, as is usually done, we specify directly the changein the scale factor, which is a measure of the size of the universe, versus time. Weconsider a scale factor that accommodates several parameters, but its most importantfeatures are that it asymptotically approaches a constant values at early times, thatit approaches a different constant value at late times, and that its first two derivativeswith respect to time are continuous. The asymptotically flat regions of our scale fac-tor allow us to associate our model with Minkowski spacetime at early and late times.Identification with a Minkowski vacuum at early times leads us to initial conditionsthat contain no infrared divergences, and comparison with a Minkowski spacetime atlate times leads us to an unambiguous measure of the frequency-dependent densityof particles created by the expansion of the universe. That our scale factor is contin-uous up to its second derivative with respect to time ensures we have no ultravioletdivergences, in addition to the prevention of infrared divergencies mentioned before.We choose for our scale factor a composite of three segments. The initial and finalsegments are each associated with a particular form of asymptotically flat scale factorwith different choices of parameters. The middle segment of the scale factor, wheremost of the expansion takes place, is a region that grows exponentially with respect toproper time. Such an exponential growth is indicated by experimental observations.We solve for the matching conditions necessary to maintain the desired continuityof our composite scale factor. For each of our scale factor segments we have exactsolution to the evolution equation for fluctuations of a massless, minimally-coupledscalar field. We also describe two different approximations to the case of a constantmass. We match up our solutions to the evolution equation at the interfaces betweenthe segments of our composite scale factor, and at late times we are able to determinethe particle production due to the expansion of the universe. From here we discussthe dispersion spectrum. We note the scale-invariance of the scalar index, providedthe requirement is met that each mode be converted into a curvature perturbation ata time related to when it crosses the Hubble radius, and that all modes not be con-verted at once after the end of inflation. Using a hybrid combination of our methodwith the slow roll approximation, we describe a way of calculating the density pertur-bations produced by inflation. Finally, we show how Reheating, or a return to the hotBig Bang conditions after the end of inflation, can accompany inflation. We discusspossible consequences of Reheating and its relationship to constraints on predictionsfor exotic particles and high energy physics.In Part II, “Binary System of Compact Masses,” we examine a post-Minkowski ap-proximation to a helically symmetric binary system of point masses. The helical sym-metry is maintained through the presence of half-advanced and half-retarded fields.The equations of motion are given for one of two Fokker actions— parametrization-invariant and affine— by Friedman and Ury¯u in [8], and from their results we calculatenumerically the solutions in the case of unequal masses. We also derive analytical for-mulas for the Extreme Mass Ratio limit where the ratio of the smaller mass divided bythe larger mass goes to zero. This limit would be applicable to the inspiral of a solar-mass black hole into a billion-solar-mass black hole, such as is predicted to exist atthe centers of many galaxies. For both the numerical computations and the analyticequations, we plot three graphs: the angular momentum versus the velocity of thelighter particle, the unit energy of the lighter particle versus the angular momentum,and the unit angular momentum of the lighter particle versus the angular momentum.These plots are given for four mass ratios and for both types of Fokker action. For theparametrization-invariant case we include one of two different correction terms thatgenerates solutions that agree with the first post-Newtonian approximation, and wedemonstrate this in the Extreme Mass Ratio limit. We discuss the locations of Inner-most Stable Circular Orbits, and we compare the predictions of this post-Minkowskiapproximation with both those of the post-Newtonian approximation and those ofthe full theory of General Relativity.In Part III, “Production and Decay of Small Black Holes at the TeV-Scale,” weinvestigate the possibility of using the IceCube Neutrino Telescope to detect TeV-scale black holes. In the physics of the Standard Model, it is not impossible thata cosmic neutrino could come close enough to an elementary particle in the cubickilometer of ice in the IceCube experiment to form a black hole. Such interactionsinvolving gravity, however, are so much less likely than interactions involving the weakforce, that IceCube would never differentiate their signal from the background noise ofweak-interaction event rates. Many theories of physics beyond the Standard Model,such as string theory, require additional dimensions of spacetime beyond the 3+1dimensions of our common experience. These additional dimensions might not havebeen noticed before if they were compactified, or curled up, with a simple examplebeing the topology of a higher-dimensional torus. At the compactification scales,then, gravity would be much stronger than in a 3+1-dimensional theory, whereasat macroscopic scales gravity would appear to be much weaker than the strong andelectroweak forces. In addition, if only gravitons propagated into the compactifieddimensions, then the scale of compactification could be anything small enough notto conflict with observations. On distances smaller than this scale, gravity wouldgrow stronger with decreasing separation faster than an inverse-square law wouldpredict. If the strength of gravity were equal to the strength of the electromagneticforce around energies of roughly one TeV, or 10 − meters, the scale at which theelectromagnetic and weak forces unify into the electroweak force, then gravity couldbe sufficiently strong that the IceCube detector could observe the production of TeV-scale black holes in the interactions between cosmic neutrinos and partons, whichare the fundamental particles— both quarks and gluons— that are found withinnucleons in atoms. For the high energies of interest for this experiment, the nucleonscannot be treated as single particles, which is why we treat them as collections ofpartons. At any moment, a parton can have an energy ranging from nothing to theentire rest mass energy of the nucleon, and parton distribution functions describethe probabilities of finding each parton with a given energy. We develop simple fitsto a specific model of the parton distribution function, and with this informationwe are able to numerically integrate an expression giving us the cross section forthe gravitational interaction. The black holes formed by these interactions woulddecay almost immediately via Hawking radiation, or particles produced by the strongcurvature of spacetime outside of black holes. The Cherenkov light of these eventscould be measured by the photomultiplier tubes of IceCube, and signals could bepicked out from the background event rate by searching for muon-daughter particleswith less than 20% of the total energy, which is sufficiently unlikely in Standard Modelphysics that we would be able to discern TeV-scale black hole events from interactionsthrough the weak force. We find that the IceCube detector could measure TeV-scaleblack holes at a statistically significant 5 σ excess for a 10-dimensional Planck scaleof 1.3 TeV.The relationship between space at the smallest and largest scales is, perhaps,nowhere so evident as the inflation of quantum fluctuations from below the Plancklength to sizes beyond our observable universe in what follows: Part I - New Aspectsof Inflaton Fluctuations. Part I:New Aspects ofInflaton Fluctuations
Chapter 2Inflationary Cosmology
At the beginning of the twentieth century, most scientists believed that the universewas infinite and eternal. Such a situation is not compatible with cosmology governedby the theory of General Relativity, which predicts that a static universe would beunstable to perturbations. From this it follows that our expanding universe startedfrom a singularity of infinite density and temperature. This Big Bang theory of theuniverse successfully explains several observational phenomena. One of these is theexpansion of the universe and Olber’s paradox, which asks— if the universe is infi-nite, then why do we not observe stars in every direction; why do we see dark spacebetween stars? With help from Hubble, Einstein and others came to realize that theuniverse is not only expanding, but it must also have a finite age. Thus, not all ofthe light from stars in the universe has had time to reach us, and for distant starsthis light is redshifted by the expansion of the universe. Another question resolvedby the Big Bang theory is that of the primordial abundances of the light elements:hydrogen, deuterium, tritium, helium-3, helium-4, and lithium. Stars convert hydro-gen to heavier elements through nuclear fusion, but the light elements are found indefinite ratios in galactic dust thought never to have been part of any star. Thisis explained by looking back to the high temperatures and pressures of the universewhen it was much more dense, shortly after the Big Bang. The universe was hotterthan any star, and a series of calculations involving the thermal-equilibrium ratio ofprotons to neutrons, the ratio of baryons to photons, the half-life for a free neutron,and the cross section for neutrons to become bound in nuclei [9, 10]; predicts ratios ofprimordial abundances of the light elements that agree very well with observations.A final success of the Big Bang theory is the explanation of the observed CosmicMicrowave Background Radiation (CMBR) at a temperature of approximately 2.7Kelvin. This was first discovered by Penzias and Wilson in 1965 while they wereworking at Bell Labs, and for this discovery they were awarded a Nobel Prize in1978. This background noise is the red-shifted relic of the early universe’s radiationdominance. Although the Big Bang theory explained some questions about our uni-verse, Cosmological Inflation was necessary to explain other observed properties ofour universe.Inflation was originally conceived to explain three primary phenomena. The firstof these was the flatness problem. The density of our universe is surprisingly closeto the critical density needed to close the universe, above which a closed universewould eventually re-collapse into a Big Crunch and below which an open universewould expand forever— neglecting acceleration caused by the presence of dark en-ergy. Surprisingly close, because unless our universe’s density is precisely equal tothe critical density— and there is no reason to assume it must be— the ratio betweenthe two drifts rapidly away from 1 in a Big-Bang-only universe. Inflation solves thisproblem by very rapidly driving this ratio exceedingly close to 1 during a short periodof enormous growth of the universe. The second argument for inflation is that all theCMBR is, to excellent approximation of within about one part in ten thousand, inthermal equilibrium. Just as the resolution to Olber’s paradox involves light taking afinite time to reach the Earth, so does this present a problem for early-universe light,emanating from different directions, that is just now reaching us. In a Big-Bang-onlymodel, widely separated regions of the currently observable universe weren’t previ-ously in causal contact, and that they should be in thermal equilibrium now is a0mystery. This problem is resolved by explaining how the space in minute regions ofour universe that were once in thermal contact expanded sufficiently rapidly duringinflation to remove the different parts of the equilibrated sections to causally discon-nected parts of the universe: the space between points within equilibrated regionsof the universe grew much faster than signals could travel across the distance be-tween those points. Thus, the CMBR reaching the Earth today, even from differentdirections, has come from regions of the universe that were previously in thermalequilibrium. The third issue that motivated inflation is the observed absence of mag-netic monopoles, which may have been created in the very early universe. Inflationresolves this by showing how monopoles could be inflated away with the expansion ofspace such that— unless monopoles were produced after inflation— on average thereshouldn’t be any monopole close enough to us to detect after inflation.Inflation has come up with an unforseen prediction that has since turned outto be more important than any of the historical justifications for its existence: thecreation of fluctuations during inflation that lead to the anisotropies of our present-day universe. For NASA-COBE’s (Cosmic Background Explorer) 1989 detection ofthese anisotropies in the CMBR, Mather and Smoot were awarded a Nobel Prize in2006. In the most widely used models of inflation, this expansion is driven by theinflaton field, which is a scalar quantum field, and the perturbations of the inflatonfield seed galaxy formation and are responsible for large-scale structure of our universetoday.
In units of c = ~ = 1 Einstein’s equation is [11, 12] G ab ≡ R ab − Rg ab = 8 πG T ab . (2.1)1On large enough scales, our universe appears to be of a fairly uniform density in alldirections. If the Earth is not in a privileged position in the universe, this impliesthat the universe is homogeneous and isotropic. Following the example of [12, 13], ifwe assume no distinction between the spatial directions, we can write the Friedmann-Robertson-Walker-Lemaˆıtre (FRWL) metric as ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θ dφ ) (cid:21) , (2.2)where a ( t ) is the scale factor that relates the chosen coordinate scale to the propertime t , and the variable k describes the topology of the universe: k > k = 0 corresponds to zero intrinsic curvature (flatuniverse), and k < g ab = − a ( t ) − kr a ( t ) r
00 0 0 a ( t ) r sin θ , (2.3) g ab = − − kr a ( t ) a ( t ) − r −
00 0 0 a ( t ) − r − sin − θ . (2.4)In this section, only, we will not use the Einstein summation convention. In the basisof { t, r, θ, φ } , the Christoffel symbols are given byΓ cab = X d (cid:20) g cd ( ∂ a g bd + ∂ b g ad − ∂ d g ab ) (cid:21) , (2.5)where ∇ a V c = ∂ a V c + Γ cab V b and ∇ a W c = ∂ a W c − Γ bac W b , with ∂ a the covariantderivative operator of the flat metric [14]. For the metric given by Eq. (2.2), we see2that g cd = δ dc g cc and g cd = δ dc g cc , where δ cd is the Kronecker delta, so we haveΓ cab = 12 g cc (cid:0) δ bc ∂ a g cc + δ ac ∂ b g cc − δ ab ∂ c g aa (cid:1) . (2.6)In the set of coordinates defined by { t, r, θ, φ } , we consider the four cases of a = b = c , a = b = c , a = b = c , and a = b = c (each of the indices is different in this last case)to get a = b = c : Γ ccc = 12 g cc ∂ c g cc , (2.7) a = b = c : Γ caa = − g cc ∂ c g aa , (2.8) a = b = c : Γ cca = 12 g cc ∂ a g cc , (2.9) a = b = c : Γ cab = 0 . (2.10)The non-zero derivatives are ∂ t g rr , ∂ t g θθ , ∂ t g φφ , ∂ r g rr , ∂ r g θθ , ∂ r g φφ , and ∂ θ g φφ . Thus,the non-vanishing Christoffel symbols are Γ trr , Γ tθθ , Γ tφφ , Γ rrr , Γ rθθ , Γ rφφ , Γ rrt = Γ rtr ,Γ θφφ , Γ θθt = Γ θtθ , Γ θθr = Γ θrθ , Γ φφt = Γ φtφ , Γ φφr = Γ φrφ , and Γ φφθ = Γ φθφ .When we write the Ricci tensor as [12] R ab = X c ( ∂ c Γ cab − ∂ a Γ ccb ) + X c,d (cid:0) Γ dab Γ ccd − Γ dcb Γ cda (cid:1) , (2.11)we find, using an underline to indicate terms that cancel, using an overline to indicateterms to be consolidated, and using a = a ( t ), ˙ a = da/dt , and ¨ a = d ˙ a/dt , that R tt = − ∂ t (cid:0) Γ rrt + Γ θθt + Γ φφt (cid:1) − (cid:0) Γ rrt Γ rrt + Γ θθt Γ θθt + Γ φφt Γ φφt (cid:1) = − (cid:20)(cid:18) ¨ aa − ˙ a a (cid:19) + (cid:18) ¨ aa − ˙ a a (cid:19) + (cid:18) ¨ aa − ˙ a a (cid:19)(cid:21) − "(cid:18) ˙ aa (cid:19) + (cid:18) ˙ aa (cid:19) + (cid:18) ˙ aa (cid:19) = − aa , (2.12)3 R rr = (cid:0) ∂ t Γ trr + ∂ r Γ rrr (cid:1) − (cid:0) ∂ r Γ rrr + ∂ r Γ θθr + ∂ r Γ φφr (cid:1) + h Γ trr (cid:0) Γ rrt + Γ θθt + Γ φφt (cid:1) + Γ rrr (cid:0) Γ rrr + Γ θθr + Γ φφr (cid:1) − (cid:0) Γ rrr Γ rrr + Γ θθr Γ θθr + Γ φφr Γ φφr + 2Γ rtr Γ trr (cid:1) i = (cid:18) a ¨ a + ˙ a − kr (cid:19) + (cid:18) r + 1 r (cid:19) + h a ˙ a − kr (cid:18) − ˙ aa + ˙ aa + ˙ aa (cid:19) + kr − kr (cid:18) r + 1 r (cid:19) − (cid:18) r + 1 r (cid:19) i = a − kr (cid:18) ¨ aa + 2 ˙ a a + 2 ka (cid:19) , (2.13) R θθ = (cid:0) ∂ t Γ tθθ + ∂ r Γ rθθ − ∂ θ Γ φφθ (cid:1) + h Γ tθθ (cid:16) Γ rrt + Γ θθt + Γ φφt (cid:17) +Γ rθθ (cid:16) Γ rrr + Γ θθr + Γ φφr (cid:17) − (cid:16) Γ φφθ Γ φφθ + 2Γ tθθ Γ θtθ + 2Γ rθθ Γ θrθ (cid:17) i = (cid:18) r (cid:8) a ¨ a + ˙ a (cid:9) − (cid:8) − kr (cid:9) + (cid:26) θ sin θ (cid:27)(cid:19) + h r a ˙ a (cid:18) ˙ aa − ˙ aa + ˙ aa (cid:19) − r (cid:8) − kr (cid:9) (cid:18) kr − kr − r + 1 r (cid:19) − cos θ sin θ i = a r (cid:18) ¨ aa + 2 ˙ a a + 2 ka (cid:19) . (2.14) R φφ = (cid:0) ∂ t Γ tφφ + ∂ r Γ rφφ + ∂ θ Γ θφφ (cid:1) + h Γ tφφ (cid:16) Γ rrt + Γ θθt + Γ φφt (cid:17) + (cid:16) Γ θφφ Γ φφθ (cid:17) +Γ rφφ (cid:16) Γ rrr + Γ θθr + Γ φφr (cid:17) − (cid:16) tφφ Γ φtφ + 2Γ rφφ Γ φrφ + 2Γ θφφ Γ φθφ (cid:17) i = (cid:0)(cid:8) a ¨ a + ˙ a (cid:9) r sin θ − (cid:8) − kr (cid:9) sin θ + (cid:8) sin θ − cos θ (cid:9)(cid:1) + h a ˙ ar sin θ (cid:18) ˙ aa + ˙ aa − ˙ aa (cid:19) − (cid:0) − cos θ (cid:1) − (cid:8) r − kr (cid:9) sin θ (cid:18) kr − kr + 1 r − r (cid:19) = a r sin θ (cid:18) ¨ aa + 2 ˙ a a + 2 ka (cid:19) . (2.15)The Ricci Scalar Curvature is R ≡ X ab g ab R ab g tt R tt + g rr R rr + g θθ R θθ + g φφ R φφ = 6 (cid:18) ¨ aa + ˙ a a + ka (cid:19) . (2.16)The most general stress tensor associated with homogeneity and isotropy is that of aperfect fluid [12], given by T ab = ρU a U b + P ( g ab + U a U b ) , (2.17)where ρ is the energy-density, P is the pressure, and in these coordinates U a =( − , , ,
0) is the four-velocity of a comoving observer, and U b = X a g ab U a . (2.18)The time-time components of the Einstein Equation, Eq. (2.1), give us the Friedmannequation: G tt = − aa − (cid:20) (cid:18) ¨ aa + ˙ a a + ka (cid:19)(cid:21) ( −
1) = 3 ˙ aa + 3 ka = 8 πG ρ, (2.19)or, H ( t ) = 8 πG ρ − ka , (2.20)where the Hubble constant is defined by H ( t ) ≡ d a ( t ) /d ta ( t ) . (2.21)Any same space-space components of the Einstein equation, for which we will use r - r ,give us the Raychaudhuri equation: G rr = g rr (cid:18) ¨ aa + 2 ˙ a a + 2 ka (cid:19) − (cid:20) (cid:18) ¨ aa + ˙ a a + ka (cid:19)(cid:21) g rr = 8 πG P g rr , (2.22)5or, 2 ¨ aa + ˙ a a + ka = − πG P, (2.23)which, when we use H = H ( t ) and ˙ H = d H/d t = a − ¨ a − a − ˙ a , can be written2 ˙ H + 3 H + ka = − πG P, (2.24)which we rewrite, using Eq. (2.20), as either˙ H = − πG ( ρ + P ) + ka , (2.25)or as the Raychaudhuri equation, which is˙ H + H = − πG ρ + 3 P ) . (2.26)We get the continuity equation by taking the time derivative of Eq. (2.20) and theninserting Eq. (2.25) to find8 πG ρ = 2 H ˙ H = 2 H (cid:20) − πG ( ρ + P ) + ka (cid:21) , (2.27)which becomes ˙ ρ = − H ( ρ + P ) + 3 H πG ka . (2.28)In a flat universe, where k/a can be neglected and the metric can be written as ds = − dt + a ( t ) ( dx + dy + dz ), the continuity equation becomes˙ ρ = − H ( ρ + P ) . (2.29)A simpler way of deriving this equation would be to use conservation of energy in a6comoving reference frame to show, in units where E = mc = m , that d (cid:18) EV (cid:19) = − MV dV − PV dV, (2.30)where M = ρV and V ∝ a . If there were no pressure, as is the case for what is re-ferred to as dust, then in the coordinates { t, x, y, z } this would reduce to conservationof a density current:0 = X a [ ∇ a ( ρU a )]= X a [ U a ∂ a ρ + ρ ∇ a U a ]= U t ∂ t ρ + ρ (cid:0) U t Γ xxt + U t Γ zzt + U t Γ zzt (cid:1) = − ∂ t ρ − Hρ. (2.31)For dust, which is the term for matter that satisfies P = 0, such as cold dark matterand— to good approximation— galaxies, we can solve the differential equation˙ ρρ = − aa , (2.32)by integrating both sides with respect to time to getln ρ ∝ − a, (2.33)or ρ ∝ a − . (2.34)We combine this with Eq. (2.20) to get˙ a a ∝ a − , (2.35)7which leads to ˙ a ∝ a − / , (2.36)and (with k = 0) a dust ( t ) ∝ t / . (2.37)We refer to this as a matter-dominated universe. For the case of a radiation-dominateduniverse, where radiation obeys the equation of state P = 13 ρ, (2.38)we would have ρ ∝ a − , (2.39)˙ a ∝ a − , (2.40)and (with k = 0) a radiation ( t ) ∝ t / . (2.41)In the next section we will show that a slowly-changing scalar field displaced from itsminimum potential energy obeys the equation of state P ≃ − ρ, (2.42)for which we have from Eq. (2.25) ˙ H inflation ≃ . (2.43)We discuss inflation in more detail in the next section, but first we mention thatwith a time-invariant Hubble constant, we would have (in a flat universe) a de Sittermetric given by ds = − dt + e Ht ( dx + dy + dz ) . (2.44)8Whether k = 0 in Eq. (2.20), or not, we may define a critical density that wouldproduce an equivalent Hubble constant if k were 0. This we define as ρ c = 3 H πG . (2.45)We define the density parameter asΩ ≡ ρρ c = πG (cid:0) H + ka (cid:1) H πG = 1 + ka ( t ) H ( t ) , (2.46)where a in a flat universe ( k = 0), we would have Ω = 1. One of the primarymotivations for inflation was reconciling observations that in our universe Ω ≃ H ∝ t − when k ≃ rad = 1 + ka ( t ) H ( t ) = 1 + ˜ k t, (2.47)Ω mat = 1 + ka ( t ) H ( t ) = 1 + ˜ k t / , (2.48)where ˜ k ∝ k . The Big Bang theory predicts— based on the presence of the ap-proximately 2 .
7K CMBR and the relationship between the current matter densityand Hubble constant— that our universe was radiation-dominated until it was about300,000 years old and has been roughly matter-dominated (neglecting any recent ac-celeration of the universe due to dark energy) since then. Thus, Ω in our universeshould diverge rapidly from 1, unless the value of k was very nearly zero at earlytimes in our universe. One mechanism for driving Ω close to 1 is inflation. When a ( t ) = e Ht and H = constant, we haveΩ infl = 1 + ka ( t ) H ( t ) = 1 + kH − e − Ht . (2.49)9Inflation very rapidly drives the value of Ω towards 1. With enough inflation, aninitial value of Ω that may have differed from 1 by orders of magnitude, could havebeen driven close enough to 1 that it would still be approximately equal to 1 in ouruniverse today. For the rest of this dissertation we will assume that the universe isflat, in the sense that we will take the curvature constant k to be zero. From now onwe will not make use of this variable and will reserve k for other quantities, namelythe Fourier mode-number. For the rest of this dissertation, we will adopt the Einstein summation convention.The Lagrangian density of a scalar field with metric signature of +2 is [15, 16, 17] L = 12 | g | / ( − g ab ∂ a φ∂ b φ − m φ − ξRφ ) , (2.50)where g ≡ det( g ab ). A massless ( m = 0), uncoupled ( ξ = 0) field with a φ -dependentpotential, where the potential may incorporate a non-zero scalar field mass, becomes L = − | g | / g ab ∂ a φ∂ b φ − | g | / V ( φ ) . (2.51)The origin of this potential depends on the various models being considered, butthe main prerequisites are that φ initially be displaced from the true minimum ofthe potential, and that some portion of the slope of the potential must be relativelyflat with respect to changes in φ during the slow roll approximation, for which seeSec. 2.3.1. If we were to retain the Ricci curvature scalar in Eq. (2.50), then thevariation of the action would lead to the Einstein Eq. (2.1) in the calculation below[16][17, pp. 491-505]. The action is [15] S = Z d x ′ L = Z d x ′ (cid:20) | g | / ( − g a ′ b ′ ∂ a ′ φ∂ b ′ φ − V ) (cid:21) , (2.52)0and the stress-energy tensor is [15] T ab = 2 | g | / δ S δg ab . (2.53)Using the identities [16] δg ab = − g ac g bd δg cd , (2.54) δ | g | / = 12 | g | / g ab δg ab , (2.55)leads to T ab = 2 | g ( x ) | / δ R d x ′ [ | g ( x ′ ) | / ( − g a ′ b ′ ( x ′ ) ∂ a ′ φ∂ b ′ φ − V )] δg ab ( x )= δ R d x ′ [ | g ( x ′ ) | / ( − g a ′ b ′ ( x ′ ) ∂ a ′ φ∂ b ′ φ − V )] | g ( x ) | / δg ab ( x )= Z d x ′ | g ( x ′ ) | / | g ( x ) | / δg a ′ b ′ ( x ′ ) δg ab ( x ) (cid:20) g a ′ b ′ ( x ′ )( − ∂ c φ∂ c φ − V ) + ∂ a ′ φ∂ b ′ φ (cid:21) . (2.56)Finally, using the delta function identity [16] δg a ′ b ′ ( x ′ ) δg ab ( x ) = g a ′ a g b ′ b δ ( x ′ , x ) , (2.57)the stress tensor is T ab = g ab (cid:18) − ∂ c φ∂ c φ − V (cid:19) + ∂ a φ∂ b φ, (2.58)and T a b = g a b (cid:18) − ∂ c φ∂ c φ − V (cid:19) + ∂ a φ∂ b φ. (2.59)1The spatial slicing and coordinate threading of time is chosen such that φ = φ ( t ). Inabsence of perturbations, space-time is homogeneous and isotropic: T a b = g a b (cid:18)
12 ˙ φ − V (cid:19) − δ a δ b ˙ φ , (2.60)where a dot represents derivatives with respect to time. Because of homogeneity andisotropy, the stress tensor is described by a perfect fluid, T ab = − ρ P P
00 0 0 P , (2.61)where ρ is the energy density and P is the pressure. It is now possible to solve forthe energy density and pressure: the energy density is equal to minus the time-timecomponent of the stress tensor; and the pressure is equal to any of the three diagonalspace-space components of the stress tensor [10]. ρ = − T = − (cid:20)(cid:18)
12 ˙ φ − V (cid:19) − ˙ φ (cid:21) = 12 ˙ φ + V ( φ ) , (2.62) P = T = T = T = 12 ˙ φ − V ( φ ) . (2.63)The Friedmann equation, H = 8 πG ρ, (2.64)and the continuity equation, ˙ ρ = − H ( ρ + P ) , (2.65)become H = 8 πG (cid:18)
12 ˙ φ + V ( φ ) (cid:19) , (2.66)2and ˙ φ ¨ φ + ˙ V ( φ ) = − H ˙ φ , (2.67)¨ φ + dV /dtdφ/dt = − H ˙ φ, (2.68)¨ φ + V ′ = − H ˙ φ, (2.69)where a dot represents a derivative with respect to time and a prime represents aderivative with respect to φ . The curvature term in the Friedmann equation is hereset to zero. Whether or not this is precisely the case, soon after inflation begins thecurvature of the universe will become negligible. Well after inflation has begun, the scalar field can be treated as a homogeneous,isotropic classical field with the fluctuations consisting of quantum perturbations. In-flation smooths out all other perturbations to the point that quantum fluctuationsare all that remain. For models of inflation driven by a single scalar field, pertur-bations can be expressed as time-dependent, location-dependent fluctuations on ahomogeneous, time-dependent background: φ ( ~x, t ) = φ ( t ) + δφ ( ~x, t ) . (2.70)The Euler-Lagrange equation, ∂ φ L − ∂ a (cid:20) ∂ L ∂ ( ∂ a φ ) (cid:21) = 0 , (2.71)with Eqs. (2.50) and (2.51), becomes − √− g V ′ ( φ ) + 12 ∂ a (cid:0) √− g g ab ∂ b φ (cid:1) + 12 ∂ b (cid:0) √− g g ab ∂ a φ (cid:1) = 0 , (2.72)3or 1 √− g ∂ a (cid:0) √− g g ab ∂ b φ (cid:1) − V ′ ( φ ) = 0 , (2.73)which is equivalent to [15, p. 38][17, p. 542] (cid:3) φ − V ′ ( φ ) = 0 . (2.74)If we perturb this with Eq. (2.70), then we get (cid:3) ( φ + δφ ) − V ′ ( φ + δφ ) = 0 . (2.75)To first order in δφ , we write this as (cid:3) φ + (cid:3) δφ − [ V ′ ( φ ) + δφV ′′ ( φ )] = 0 , (2.76)and we then use Eq. (2.74) to show (cid:3) δφ − δφV ′′ ( φ ) = 0 . (2.77)We can see from Eq. 2.50) that for a free field we may make the association V ′′ ( φ ) = m + ξR, (2.78)where m is the scalar mass, ξ is the coupling constant, and R is the Ricci scalarcurvature.The perturbation of Eq. (2.70) expanded in terms of creation and annihilationoperators is [16] δφ = (volume) − / X ~k [ a ~k g k ( t ) e i~k · ~x + H.C. ] . (2.79)where volume = [ L a ( t )] , (2.80)4which is the physical length found from multiplying the coordinate length times thescale factor. The time dependent part of the fluctuations is ψ k , where ψ k ≡ a ( t ) − g k , (2.81)and | δφ k | = L − | δψ k | . (2.82)The solution thus far has periodic boundary conditions, but in the limit that L → ∞ ,a volume even as large as the observable universe will not be affected by this choiceof boundary conditions. Combining the metric ds = − dt + a ( t ) ( dx + dy + dz ); (2.83)where g ab = − a ( t ) a ( t )
00 0 0 a ( t ) , (2.84) g ab = − a ( t ) − a ( t ) −
00 0 0 a ( t ) − , (2.85)and p | g | = p | [ − a ( t ) ][ a ( t ) ][ a ( t ) ] | = a ( t ) , (2.86)with the massless, uncoupled scalar field equation [15] (cid:3) δφ − δφV ′′ ( φ ) = 1 | g | / ∂ a ( | g | / g ab ∂ b δφ ) − δφV ′′ ( φ ) = 0 , (2.87)5yields0 = a ( t ) − ∂ t (cid:2) a ( t ) ( − ∂ t δφ (cid:2) + a ( t ) − ∂ i (cid:2) a ( t ) (cid:0) a ( t ) − (cid:1) ∂ i δφ (cid:3) − δφV ′′ ( φ )= ∂ t δφ + 3 H ( t ) ∂ t δφ − a ( t ) − ∂ i ∂ i δφ + δφV ′′ ( φ ) , (2.88)where H ≡ da/dta . (2.89)With the spatial dependence given by Eq. (2.79), the evolution equation for mode- k becomes ∂ t δφ + 3 H ( t ) ∂ t δφ + k a ( t ) δφ + δφV ′′ ( φ ) = 0 . (2.90)Using the scale factor associated with the de Sitter universe given by Eq. (2.44), a = e Ht , (2.91)and assuming a constant value of V ′′ ( φ ) to simplify the calculation, leads to an evo-lution equation for mode- k of ∂ t δφ + 3 H∂ t δφ + k e Ht δφ + δφV ′′ = 0 . (2.92)Combining this with Eq. (2.79) leads to0 = (cid:16) (cid:20) H e − Ht g k − He − Ht ∂ t g k + e − Ht ∂ t g k (cid:21) +3 H (cid:20) − He − Ht g k + e − Ht ∂ t g k (cid:21) + h k e − Ht g k i + V ′′ h e − Ht g k i (cid:17) = e − Ht ∂ t g k + k e − Ht g k − H e − Ht g k + V ′′ e − Ht g k . (2.93)Using the change of variables, u ≡ − kH e − Ht , (2.94)6which is k times the conformal time, we then have ∂ t = dudt ddu = ke − Ht ∂ u , (2.95)and ∂ t = ke − Ht ∂ u ke − Ht ∂ u = ke − Ht ∂ u [ − Hu∂ u ]= − kHe − Ht ∂ u − ukHe − Ht ∂ u , (2.96)so the evolution equation Eq. (2.93) for mode- k in terms of u is0 = e − Ht (cid:2) − kHe − Ht ∂ u − ukHe − Ht ∂ u (cid:3) g k + k e − Ht g k − H e − Ht g k + V ′′ e − Ht g k = H e − Ht (cid:26) − kH e − Ht ∂ u g k − u kH e − Ht ∂ u g k + k H e − Ht g k − g k + V ′′ H g k (cid:27) = u ∂ u g k + u∂ u g k + (cid:20) u − (cid:18) − V ′′ H (cid:19)(cid:21) g k . (2.97)Eq. (2.97) is Bessel’s equation. The most general solution for a given k -component, g k , is [18] g k ( t ) = 12 p π/H (cid:26) c H (1) q − V ′′ H ( u ) + c H (2) q − V ′′ H ( u ) (cid:27) . (2.98)We then have ψ k ( t ) = a ( t ) − p π/H (cid:26) c H (1) q − V ′′ H ( u ) + c H (2) q − V ′′ H ( u ) (cid:27) , (2.99)but for the k = 0 mode of the massless, minimally coupled case in a purely deSitter universe, a universe that has an infinite history and future that is at all timesdescribed by the metric of Eq. (2.44), see also Refs. [19, 20].For sufficiently large k -modes the solution should be asymptotically insensitive tothe de Sitter curvature, as this corresponds to very small wavelengths. On a verysmall scale that locally appears nearly flat, the curvature becomes negligible. For7these large k -modes, the solution we expect— due to the rapid attenuation of matterand radiation in a de Sitter universe— is that of the positive frequency WKB vacuumsolution [16] ψ k ( t ) ∼ p ω k ( t ) a ( t ) e − i R ω k ( t ′ ) dt ′ = 1 p − a ( t ) Hu e − iu , (2.100)where the frequency is ω k ( t ) ≡ s k a ( t ) + m . (2.101)See also Sec. 3.2. To match our constants, c and c , when k → ∞ , we use the largeargument expansion of the Hankel functions [21] H (1) ν ( z ) ∼ p / ( πz ) e i ( z − νπ − π ) H (2) ν ( z ) ∼ p / ( πz ) e − i ( z − νπ − π ) , (2.102)which means that, to within a phase, ψ k ( t ) = 1 p − a ( t ) Hu (cid:8) c e iu + c e − iu (cid:9) . (2.103)To match to the positive-frequency, vacuum solution given by Eq. (2.100) we choose[16, 18] lim k →∞ c ( k ) ∼ , lim k →∞ c ( k ) ∼ . (2.104)The de Sitter metric and the physical volume are symmetric under the transforma-tion [16] t → t + t and ~x → e − Ht ~x. (2.105)8The Killing vector generating this isometry, [22] ξ = 1 , ξ i = − Hx i . (2.106)corresponds to conservation of energy. Since the vacuum fluctuations can be expectedto share this symmetry of space-time, provided— as will be explained in Sec. 3.4.1—there is an infinite expansion and the universe is de Sitter in the infinite past andinfinite future, the variable u is thus invariant under t → t + t and ~k → ~ke Ht . (2.107)Then, with k ′ ≡ ke Ht , ψ k ′ ( t + t ) = ψ k ( t ) (2.108)requires c ( k ′ ) = c ( k ) and c ( k ′ ) = c ( k ) . (2.109)Thus, because t is arbitrary, we have [23], ψ k ( t ) = 12 a ( t ) − / p π/H H (2) q − V ′′ H ( u ) . (2.110)We note for future reference that changing the sign of the argument in Eq. (2.98)also yields a linearly independent solution to Eq. (2.97) under the transformation u → ˜ u = − u , because the Hankel functions of the first and second kind form anorthogonal and complete set. The coefficients c ( k ) and c ( k ) will, in general, changeunder the transformation u → ˜ u , but the procedure outlined above for finding thesecoefficients in the k → ∞ limit, leads to c ( k ) = − i and c ( k ) = 0. A simpler wayof seeing this, once we have Eq. (2.110), is to change the sign of H . Although wewill later take H to be real and positive, we have not yet made this assumption, sochanging the sign of H should leave Eq. (2.110) intact in the flat-space limit of k → ∞ ,9where again a mode should not see the curvature of space. Using Eq. (2.102), we seethat this large argument limit of the Hankel functions takes— to within a phase— H (2) v ( z ) → − iH (1) v ( − z ). In this section we will focus on defining the slow roll approximation, the slow rollparameters, the number of e-folds, the curvature perturbation, the spectrum of cur-vature perturbations, and the spectral index.In the slow roll approximation [24, 25, 26, 27, 28, 29, 30, 31, 32]˙ φ ≪ V ( φ ) (2.111)and | ¨ φ | ≪ | V ′ | . (2.112)This means Eqs. (2.66) and (2.69) become H ≃ πG V ( φ ) (2.113)and ˙ φ ≃ − V ′ H . (2.114)These conditions ensure that P ≃ − ρ , which is the property of a space-time domi-nated by a cosmological constant, or de Sitter space; and that the kinetic term doesnot grow appreciably since the potential is assumed to be flat and H is large. Duringinflation, the slow roll parameters must satisfy [33] ǫ ≪ | η | ≪ , (2.115)0where the slow roll parameters are defined by [33] ǫ ≡ πG (cid:18) V ′ V (cid:19) ≃ − ˙ HH , (2.116) η ≡ πG (cid:18) V ′′ V (cid:19) , (2.117)Using the slow-roll equations (2.113) and (2.114), we can express the number ofe-folds of inflation as [34] N e ≡ ln (cid:20) a ( t final ) a ( t initial ) (cid:21) = Z t final t initial H dt ≃ πG Z φ initial φ final V ( φ ) V ′ ( φ ) dφ. (2.118)We define a mode to be crossing the Hubble radius when the mode’s wavelength, a ( t ) /k , is the same size as the Hubble radius, H − , which would be the horizon sizein a purely de Sitter universe. During inflation, when the scale factor is growingexponentially and k and H are both constant, a mode exits the Hubble radius when k/ [ a ( t ) H ] = 1. After inflation, when the scale factor is given by either a radiation-dominated a ( t ) ∝ t / growth or by a matter-dominated a ( t ) ∝ t / growth, where forboth cases H ∝ t − , then k/ [ a ( t ) H ( t )] = 1 defines the time when a mode re-entersthe Hubble radius.We can apply the small argument limit of the Hankel functions [21, Eq. 9.1.9], (cid:12)(cid:12) H (1) v ( z ) (cid:12)(cid:12) ≃ (cid:12)(cid:12) H (2) v ( z ) (cid:12)(cid:12) ≃ (cid:18) Γ( v ) π (cid:19) (cid:18) | z | (cid:19) − v , (2.119)when the real part of the parameter v is positive and non-zero, to Eq. (2.110), to get | ψ k | ≃ π H a ( t ) − (cid:18) Γ( v ) π (cid:19) (cid:18) ka ( t ) H (cid:19) − v . (2.120)In the massless, minimally-coupled case, v = 3 /
2, and we find | ψ k | ≃ π H a ( t ) − (cid:18) √ π/ π (cid:19) (cid:18) k a ( t ) H (cid:19) − . (2.121)1Late enough into inflation for a given mode to be well outside the Hubble radius, wethen have | ψ k | ≃ H k , (2.122)which is approximately half the value of | ψ k | at the time it exits the Hubble radius—see Sec. 3.5. Although this perturbation of the inflaton field is not a gauge-invariantquantity, there is a gauge-invariant quantity, a curvature perturbation that we call R k , that is approximately conserved outside of the Hubble radius, and we can use itto relate the inflaton fluctuations to density perturbations at the time of re-entry asfollows: [9, 10, 24, 27, 29, 34, 35, 36, 37] δφ k ˙ φ H ≃ R k, exit ≃ R k, re − entry ∝ δ k ≡ δρ k ρ , (2.123)where for re-entry into a matter-dominated universe δ k ≃ R k , and for re-entry into aradiation-dominated universe δ k ≃ R k . The value of δφ k is usually taken (neglectingthe coordinate length L ) to be the unrenormalized value H /k obtained at the timeof exiting the Hubble radius. The justification for using an unrenormalized value of δφ k , when it is well known that the Bunch-Davies state given by Eq. (2.110) leads to adivergent δφ when summed over all modes, is usually given as implicit large and smallcutoff frequencies. It is often assumed that the infrared and ultraviolet divergencescome from infrared and ultraviolet frequencies that do not affect the treatment ofmodes exiting the Hubble radius during inflation. Parker [38], however, has shownthat the divergences affect every mode, and that neglecting a proper renormalizationdrastically alters the results that are obtained.We use the definition of a spectrum given by Liddle and Lyth [34]: P f ( k ) ≡ (cid:18) L π (cid:19) πk h| f k | i . (2.124)Thus, under the standard assumption that it is not necessary to renormalize the2inflaton fluctuations as they are exiting the Hubble radius, we could show P δ ∝ P R = (cid:18) H ˙ φ (cid:19) P δφ = (cid:18) H ˙ φ (cid:19) (cid:18) H π (cid:19) , (2.125)from the super-Hubble radius behavior given by Eq. (2.122). The renormalization of[38], however, changes this: the renormalized spectrum of inflaton perturbations, atthe time of exiting the Hubble radius when k/ [ a ( t ) H ] = 1, is P δφ = (cid:18) H π (cid:19) (cid:18) π (cid:12)(cid:12) H (1) n (1) (cid:12)(cid:12) − m H + m H + m H + 2( m H + 1) / (cid:19) , (2.126)where m H ≡ m/H and n ≡ p / − m H . The renormalized inflaton fluctuationdepends critically on the mass and when the magnitude of the fluctuation is evaluated.In the massless case, (cid:12)(cid:12)(cid:12) H (1) n (1) (cid:12)(cid:12)(cid:12) = 4 /π , and the renormalized fluctuation is preciselyzero. Well outside the horizon, the renormalized P δφ also goes to zero, but this isperhaps not a problem, as R k is the conserved quantity, not δφ k , and the value of R k given by Eq (2.123) is typically evaluated at the time a mode crosses the Hubbleradius. Thus, renormalization has the potential to greatly alter the character of thespectrum of perturbations.The scalar spectral index, n s , is a measure of how the magnitude of density per-turbations changes with scale. A value of n s = 1 indicates scale-invariance. A valueless than one is called a red-tilted spectrum, and a value greater than one is called ablue-tilted spectrum. It is defined as n s ( k ) − ≡ d ln P R d ln k , (2.127)where the value of n s is given for a specific value of k , called the pivot value, which isnormally either of k = 0 .
05 Mpc − [39] or k = 0 .
002 Mpc − [40], relative to the value ofthe scale factor fixed to be such that a ( t now ) = 1. There is little running, or change in n s ( k ) with changing scales, so the choice of k pivot is somewhat arbitrary. We can relate3the scalar spectral index to the slow roll parameters given in Eqs. (2.116) and (2.117)[34, 41]. Because the curvature perturbations are evaluated at the time of Hubbleradius crossing, when k = a ( t ) H ≃ He Ht , we see that with a nearly constant valueof H during inflation d ln k = d [ln( H ) + Ht ] ≃ H dt . This leads to, with Eq. (2.114)rewritten as dt = − H/V ′ dφ , dd ln k ≃ − V ′ H ddφ ≃ − πG V ′ V ddφ . (2.128)Again using Eq. (2.114), the spectrum of curvature perturbations given by Eq. (2.125)becomes P R = (cid:18) H V ′ (cid:19) (cid:18) H π (cid:19) . (2.129)With Eq. (2.113), this becomes P R = (cid:18) πG VV ′ (cid:19) πG V π = (8 πG ) π V V ′ , (2.130)where the observed value of P R is typically listed for the specific value of k =0 .
002 Mpc − , which is different from the value of k used with the scalar spectral index[39]. In [40], the pivot scale for the spectrum of curvature perturbations is chosen tobe k = 0 .
02 Mpc − , as this is a scale that puts tighter constraints on the magnitude ofthe curvature perturbation spectrum for a wider array of model assumptions. Withinthe assumptions of various models, there is still a relatively scale-invariant spectrumof curvature perturbations.With Eq. (2.128), Liddle and Lyth find d ln P R d ln k ≃ − πG V ′ V ddφ ln (cid:18) (8 πG ) π V V ′ (cid:19) ≃ − πG V ′ V ddφ (3 ln V − V ′ ) ≃ − πG V ′ V (cid:18) V ′ V − V ′′ V ′ (cid:19) ≃ − πG (cid:18) V ′ V (cid:19) + 2 18 πG (cid:18) V ′′ V (cid:19) ≃ − ǫ + 2 η, (2.131)where the slow roll parameters are given by Eqs. (2.116) and (2.117). Thus, n s − − ǫ + 2 η. (2.132)See also the end of Sec. 3.5 for a slightly different derivation.Finally, we note that when we define the mass by m ≡ d V /dφ , we find that m H ≡ m/H = p m /H = p V ′′ / (8 πG V ) = p η, (2.133)and thus the effective inflaton mass is related to the Hubble constant during inflationthrough the slow roll parameter η . The Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) data measures ascalar spectral index of n s (0 . / Mpc) ≃ .
96 [42]. The Sloan Digital Sky Survey(SDSS) measures a scalar spectral index of n s (0 . / Mpc) ≃ .
95 [43]. Because theWMAP experiment measures fluctuations in the CMBR, while the SDSS observes thelocations of galaxies and large-scale structure in our universe, there is good, indepen-dent accord for the red-tilted spectral index measured by these different approaches.The Five-Year WMAP data finds a curvature perturbation spectrum of P R (0 . /M pc ) ≃ . × − . (2.134)What follows in this section, where we apply these observations to two particularmodels, is based upon work done by [44]. The first model we consider, the quadraticchaotic inflationary potential [45, 46], is in good agreement with the Three-Year5WMAP data [47]. The second model, a type of Coleman-Weinberg model [29, 48], isin good agreement with the Five-Year WMAP data [40, 49].The quadratic chaotic inflationary potential is given by [45, 46] V ( φ ) = 12 m φ . (2.135)In Fig. 2.1, we plot the potential given by Eq. (2.135) versus φ . In chaotic inflation, the V [ φ ] / m φ Figure 2.1: Quadratic Chaotic Potential.value of φ is initially perturbed away from the minimum and rolls slowly— providedthe slope of the potential is sufficiently gradual— down the potential to the minimumat φ = 0. To be contrasted with chaotic inflation is new inflation, in which φ beginsnear the maximum value of the potential located at φ = 0 and rolls slowly to aminimum of the potential [50]. The Coleman-Weinberg potential, which was actuallyone of the earlier models considered for an inflationary potential that did not involve6tunneling through a potential barrier and its associated problems with bubbles ofinflation not coalescing, is an example of new inflation.The one-loop Coleman-Weinberg potential is given in the zero-temperature limitby [29, 48] V ( φ, T ) = 12 Bσ + Bφ (cid:20) ln( φ /σ ) − (cid:21) . (2.136)In Fig. 2.2, we plot a dimensionless potential V ( φ ) / ( Bσ ) versus the dimensionless V [ φ ] / ( B σ ) φ / σ Figure 2.2: Coleman-Weinberg Potential.parameter φ/σ . In the low temperature limit, the stable minima of the potential arelocated at φ = ± σ . At the beginning of inflation φ ≃
0, where the slow roll conditionsare satisfied, and φ rolls to either of two (in the low temperature limit) stable minima.Classically, inflation is a period of super-cooling, so the low-temperature limit shouldbe justified, but see also Sec. 3.7.For the quadratic chaotic inflationary potential, the slow roll parameters of Eqs. (2.116)7and (2.117) are equal to each other, and we have ǫ = η = 14 πGφ . (2.137)From Eq. (2.132) and the Five-Year WMAP spectral index of n s − ≃ − .
04, wefind n s − − ǫ + 2 η = − η ≃ − . , (2.138)or ǫ = η ≃ . . (2.139)From Eq. (2.133), we have mH ≃ √ . ≃ . . (2.140)From Eqs. (2.137) and (2.139), 14 πGφ ≃ . , (2.141)or φ cmb ≃ G − / √ . π , (2.142)where φ cmb corresponds roughly to that range of φ at which the modes observed byWMAP were exiting the Hubble radius during inflation. Using Eq. (2.118), we findthe number of e-folds before the end of inflation at which these modes were exitingthe Hubble radius: N e ≃ πG Z φ cmb VV ′ dφ ≃ πG Z φ cmb φ dφ ≃ πG φ ≃ . ≃ . (2.143)For the value of m H ≃ . P δφ ≃ (cid:18) H π (cid:19) (cid:18) . − . . (cid:19) ≃ (cid:18) H π (cid:19) (0 . ≃ . H . (2.144)Using the relation given in Eq. (2.125) and the slow roll approximation given inEq. (2.114), we have P R = (cid:18) H ˙ φ (cid:19) P δφ ≃ (cid:18) H m φ (cid:19) . H ≃ m H − φ . H ≃ . H φ , (2.145)then, as a rough estimate of the general order of magnitude, we use φ cmb to get P R ≃ . H . π ( G − / ) ≃ (cid:18) HG − / (cid:19) . (2.146)We can equate this with the amplitude of the spectrum found in the Five-Year WMAPdata to write 37 (cid:18) HG − / (cid:19) ≃ . × − , (2.147)and HG − / ≃ × − . (2.148)Using the Planck scale, G − / ≃ . × GeV, finally we have H ≃ × GeV , (2.149)which can be seen as an upper limit on H near the beginning of inflation, around thetime the modes observed by WMAP were exiting the Hubble radius; as φ rolls downthe potential towards zero, the size of H decreases.For the Coleman-Weinberg potential given by Eq. (2.136), we have V ′ = 4 Bφ ln φ σ , (2.150)9 V ′′ = 12 Bφ (cid:18)
23 + ln φ σ (cid:19) . (2.151)The slow roll parameters are ǫ = 116 πG Bφ ln φ σ Bσ + Bφ (cid:2) ln( φ /σ ) − (cid:3) ! = ( G − / ) πσ r ln r + r (cid:2) ln( r ) − (cid:3) ! , (2.152) η = 18 πG Bφ (cid:16) + ln φ σ (cid:17) Bσ + Bφ (cid:2) ln( φ /σ ) − (cid:3) = ( G − / ) πσ r (cid:0) + ln r (cid:1) + r (cid:2) ln( r ) − (cid:3) ! , (2.153)where r ≡ φ/σ . We assume the values given by [9, p. 292] of σ ≃ × GeV ,B ≃ − . (2.154)With those values and G − / ≃ . × GeV, taking r ≪ ǫ ≃ (cid:0) . × (cid:1) r (cid:0) ln r (cid:1) , (2.155) η ≃ (cid:0) . × (cid:1) r ln r , (2.156)and we find in the limit φ ≪ σ , that ǫ ≪ η . Using the WMAP value of 0 .
96 for thespectral index, this leads to − ǫ + 2 η ≃ η ≃ − .
04, or η ≃ − . . (2.157)Then we have m H ≃ − .
06, or m H ≃ . i ≃ i/ . (2.158)0An imaginary physical mass could lead to tachyonic behavior [51], however in thiscase, recall we are dealing with an effective mass. To find r , which we assume to bemuch less than one, we combine Eqs. (2.156) and (2.157) to get r ≃ ± . × − . (2.159)With Eq. (2.118), we have N e ≃ π σ ( G − / ) Z . × − + r (cid:2) ln( r ) − (cid:3) r ln r ! dr ≃ . (2.160)Finally, Eqs. (2.114), (2.125), (2.126), (2.134), and (2.158) lead us to2 . × − ≃ (cid:18) H ˙ φ (cid:19) (cid:18) H π (cid:19) (0 . ≃ H π (4 Bφ ln φ σ ) ! (0 . . (2.161)Then, using the values given in Eqs. (2.154) and (2.159), we have H = 4 . × GeV . (2.162)This value of H listed here for the Coleman-Weinberg potential can be compared withthat found in Eq. (2.149) to see how discrepancies can arise when choosing betweendifferent models consistent with observations.The usual method of describing inflation by first specifying a potential and thencalculating observable quantities is thus in some ways not very constraining in its pre-dictions for the early universe. In the next chapter we will discuss a means of modelinginflation in a potential-independent way by specifying the evolution of a scale factorconsistent with inflation instead of attempting to discern between individual modelsof potentials consistent with inflation.1 Chapter 3Spectrum of Inflaton Fluctuations
In [38], Parker showed how to renormalize fluctuations in the inflaton field in curvedspacetime using adiabatic regularization, for which see also [52, 53, 54]. Other papers[55, 56] have since found similar disagreement with the standard treatment of thedispersion. The technique used in [38] has been shown to give the same results inhomogeneous and isotropic universes as other methods of renormalization, such aspoint-splitting, and to be related to the Hadamard condition in curved space time[5, 57, 58, 59, 60], which states that the two-point function h | φ ( x ) , φ ( x ′ ) | i , in thelimit x ′ → x takes the form of a Hadamard Solution [15, 59] S ( x, x ′ ) = ∆ / π (cid:18) σ + v ln σ + w (cid:19) , (3.1)where σ is the proper distance of interval of spacetime between x and x ′ , ∆ ≡− det[ ∂ a ∂ b σ ][ g ( x ) g ( x ′ )] − / and reduces to [ − g ( x )] − / as x ′ → x , and v ≡ ∞ X l =0 v l σ k , (3.2) w ≡ ∞ X l =0 w l σ k . (3.3)2As an additional check on adiabatic regularization, we examine the spectrum of in-flaton perturbations in spacetimes that asymptotically approach Minkowski space atearly and late times. This is a method introduced and used in Parker’s analysis ofparticle creation by an expanding universe [1, 2, 3], and it requires no renormalizationbeyond that already known in Minkowski space. To make use of Minkowski spacein the analysis of the spectrum of inflaton perturbations coming from inflation, weinvestigate a scale factor, which is a measure of the size of the universe, that is com-posed of different scale factor segments joined together, similar to the treatments of[61, 62]. We first tried evolving forward the inflaton perturbations using a fourthorder Runge-Kutta numerical integration routine in C++ code, but we realized thatwe would need to use greater precision for our computation. We decided instead touse an analytical calculation by matching known solutions to the evolution equationat the boundary conditions joining the different segments of the scale factor. Ourcalculations were performed using 500 digit precision in Mathematica. We consider the metric ds = dt − a ( t ) (cid:2) ( dx ) + ( dy ) + ( dz ) (cid:3) . (3.4)The time t will run continuously from −∞ to ∞ . The scale factor a ( t ) will becomposed of three segments. Our scale factor will generally be C , i.e., a continuousfunction with continuous first and second derivatives everywhere, including at thejoining points between segments. Briefly, we will consider scale factors that are only C or C at the joining points. The initial and final segments are asymptoticallyMinkowskian in the distant past and future, respectively. The middle segment is anexponential expansion with respect to the time t . We choose specific forms for a ( t )in these segments that have exact solutions of the evolution equations for inflaton3quantum fluctuations of zero effective mass.Fig. 3.1 shows an example our composite scale factor plotted versus dimensionless a ( t ) H infl t a t , τ ’ f t , τ i Figure 3.1: Composite Scale Factor.time. This illustrative example summarizes our notation using a moderate expansionof ∼ a ( t ), is continuous, as are ˙ a ( t ) and ¨ a ( t ). In this case,the parameters for the initial asymptotically flat segment are a i = 1, a i = 2, and s i = 1. The free parameters of the final asymptotically flat segment are a f = 9 and a f = 6. Both asymptotically flat scale factors are given by different parameter choicesof Eq. (3.6) with the parameter b in both cases equal to zero. The asymptotically flatscale factor of the initial region joins the exponentially expanding scale factor of themiddle region at a time t in t -time and τ i in τ -time. The exponentially expandingscale factor of the middle region joins the asymptotically flat scale factor of the finalregion at a time t in t -time and τ ′ f in τ ′ -time of the final segment, where a prime isused to distinguish between the τ -times of the initial and final segments.4The equation for the middle (inflationary) segment of our composite scale factoris given in terms of proper time by a ( t ) = a ( t ) e H infl ( t − t ) , (3.5)where H infl is the constant value of H ( t ) ≡ a − da/dt during the exponential expansionof the middle segment.We define the quantity of Eq. (2.118), N e ≡ ln ( a f /a i ), in terms of a ( t initial ) = a i and a ( t final ) = a f . When there is a long period of exponential growth, N e is essentiallythe number of e-foldings of inflation. Typically, N e will be about 60. Within the finalasymptotically flat scale factor, the ratio of a f to a f determines how gradually theexponential expansion transitions to the asymptotically flat late-time region. (Forexample, this ratio might be 1 e-fold, which we would consider to be relatively gradual,or it might be 1.0001, which we would consider to be relatively abrupt.) The initial and final asymptotically flat regions permit us to unambiguously interpretour results for free fields without having to perform any renormalization in curvedspacetime. The final asymptotically flat region will not significantly affect the resultobtained for the spectrum of inflaton perturbations created by the inflationary seg-ment of the expansion. The initial asymptotically flat region should have a negligibleeffect on the spectrum resulting from a long period of inflation, although we do findremnants of the early initial conditions in the late-time inflaton dispersion spectrum,which we will discuss in Sec. 3.4.1.We base each asymptotic segment on a scale factor of the form, a ( t ( τ )) = (cid:8) a + e τ/s [( a − a )( e τ/s + 1) + b ]( e τ/s + 1) − (cid:9) , (3.6)5where τ is related to the proper time t by dτ ≡ a ( t ) − dt. (3.7)See Fig. 3.2. This figure shows the asymptotically flat scale factor, a ( t ( τ )), and the a ( τ ) , s H ( τ ) τ /s a sH Figure 3.2: Asymptotically Flat Scale Factor.associated dimensionless Hubble parameter, sH ( t ( τ )) = sa − da/dt = sa − da/dτ , ofEq. (3.6) with a = 1, a = 2, b = 0, and s = 1. Note in the graph that the maximumof H occurs at a value of a ( t ( τ )) closer to a than to a . In both the case where a ≫ a and the case where a ≃ a , H max occurs at a value of the scale factor where a ( t ( τ )) ≃ a .The form of the scale factor in Eq. (3.6) is based on the form of the index ofrefraction used by Epstein to model the scattering of radio waves in the upper atmo-sphere and by Eckart to model the potential energy in one-dimensional scattering in6quantum mechanics [63, 64]. It was first used in the cosmological context by Parker[4, 65, 66] to model a ( t ). As can be seen from Fig. 3.2, this scale factor approachesthe constant a at early times and the constant a at late times, and the constant s determines roughly the interval of τ -time for a ( t ) to go from a to a . A sufficientlylarge magnitude of b would produce a bump or valley in a ( t ), but unless otherwisenoted, we will take the value of b to be zero. The parameters a , a , b , and s aredifferent in the initial and final asymptotically flat segments. Where confusion wouldarise we will include subscripts i in the initial set of parameters and f in the final setof parameters. With our choices of a ( t ) in the three segments, we are able to join them so that a ( t )and its first and second derivatives with respect to time are everywhere continuous.This requires that we join the exponentially expanding segment, in which H ( t ) hasthe constant value H infl , to the initial and final segments at the times when H ( t ) isan extremum. This is a maximum value, when b = 0, and we equate this maximumvalue of H ( t ) with H infl . A simple power law form of the scale factor, such as thatof a radiation-dominated universe, could not be used to simultaneously maintain thecontinuity of the scale factor and its first and second derivatives when matched directlyto the inflationary segment of exponential expansion. An application of these methodsof matching continuously to C for the radiation reaction of the electromagnetic forceis given in the Appendix A. With b i = 0 and b f = 0, we then find the following expressions. The time τ i at whichthe first segment joins to the exponential segment is τ i = s i ln (cid:18) a i − a i + C i a i (cid:19) . (3.8)7The constant a ( t ) in Eq. (3.5) is a ( t ) = (cid:18) − a i − a i + C i (cid:19) / . (3.9)Because the maximum value of H ( t ) in the first segment must equal H infl , we findthat H infl = " / ( − a i + a i ) a i (11 a i − a i + C i ) s i × (cid:0) − a i − a i + C i (cid:1) / × (cid:0) a i − a i + C i (cid:1) , (3.10)where C i ≡ q a i + 46 a i a i + 9 a i . (3.11)Once we choose values for a f and a f , the remaining constants are determined tohave the following values: s f = " / (cid:0) − a f + a f (cid:1) a f (cid:0) a f − a f + C f (cid:1) H infl × (cid:0) − a f − a f + C f (cid:1) / × (cid:0) a f − a f + C f (cid:1) . (3.12)We denote the parameter τ of Eq. (3.6) as τ ′ in the final segment. At the time τ ′ f when the exponential segment joins to the final segment, we find that τ ′ f = s f ln a f − a f + C f a f ! . (3.13)The corresponding proper time t at which the exponential segment joins to the finalsegment is t = 14 H infl ln (cid:18) − a f − a f + C f − a i − a i + C i (cid:19) + t , (3.14)8where C f ≡ q a f + 46 a f a f + 9 a f . (3.15)See Fig. 3.1 for a schematic diagram of how we match our segments of the scalefactor together.Fig. 3.3 shows an example of our composite scale factor and a particular dimension- a ( t ) , ψ k ( t ) √ — s i H infl t a Re Im Abs Figure 3.3: Matching Boundary Conditions.less solution to the evolution equation, where both are plotted versus dimensionlesstime. This example shows our composite scale factor over a moderate expansion of ∼ a ( t ), is continuous, as are ˙ a ( t ) and ¨ a ( t ). The parametersfor the first asymptotically flat segment are a i = 1, a i = 2, and s i = 1. The freeparameters of the end asymptotically flat segment are a f = 9 and a f = 6. Wechoose b i = b f = 0. We plot the k = 2 Fourier mode of √ s i ψ k alongside the scalefactor to show how this representative evolution solution changes with respect to the9scale factor. The real part of √ s i ψ k , “Re,” the imaginary part of √ s i ψ k , “Im,” andthe magnitude of √ s i ψ k , “Abs,” are all plotted. We have checked our method against known mathematical theorems. One such theo-rem is that in an oscillator with a changing frequency, the quantity
E/ω is conservedif the changes in frequency are made continuously in all derivatives with respect totime; however, if any of the derivatives of the frequency with respect to time arediscontinuous, then this introduces changes to the conserved quantity of order N ,where the N -th derivative is the first discontinuous derivative [67]. It is also shownby [68] that for adiabatic changes, the changes to the conserved quantity fall off withincreasing frequency faster than any power of the frequency. We find in this conservedquantity a close analogy with the average number of particles created per mode forhigh-energy particles, which are those particles whose wavelengths have not yet exitedthe Hubble radius before the end of inflation. It is found in Ref. [1], that when thescale factor is changed adiabatically, the amount of particle production falls off withfrequency faster than any power of the frequency. The dependence of high-frequencyparticle production upon the continuity of the scale factor is also noted in [69]. Thescale factor must maintain continuity in the zeroth, first, and second derivatives toavoid an ultraviolet divergence in the energy density. This is the reason why wechoose matching conditions that are continuous in a ( t ), H ( t ), and ˙ H ( t ). We couldin principle maintain continuity in higher derivatives of our composite scale factor,as well, which would further reduce the amount of high-energy particle production.This further reduction in the high-energy particles would not appreciably improveupon any of our qualitative or quantitative results. The need for C matching con-ditions when trying to calculate a finite energy density was previously realized by[61]. In the work of [62, 70] upon the creation of gravitons during inflation, thescale factor is not C , and both authors adopt a UV-cutoff frequency. The author of0[62] recognizes the dependence of high-energy particle production upon the transitionfrom de Sitter space to a radiation dominated universe, and he attributes the entireamount of high-energy particle production to the instantaneous change in the Ricciscalar curvature given by Eq. (2.16) from 12 H during inflation to 0 in a radiationdominated universe. In [3], Parker has shown that massless gravitons satisfying aconformally invariant spin-2 field would not be produced for any a ( t ). However, anEinstein graviton that instead satisfied a weak field approximation such as Eq. (4.1),which in vacuum would lead to (cid:3) ¯ h ab = 0, is not conformally invariant. (We use herethe definition ¯ h ab ≡ h ab − hη ab , and we work in the Lorentz gauge where ¯ h ab,β = 0,which means ¯ h ,β = − h ,β = 0 [17].) This is analogous to a massless, minimally-coupledKlein-Gordon field equation of the form of Eq. (2.87), except for the two polarizations( h + and h × ) of gravitational waves [71, 72, 73, 74]. This means that for quanta of thislinear field, we would expect the same results for average number of quanta createdper mode for each polarization; therefore, | β k | = 2 | β k | . Consider an inflaton field composed of a spatially homogeneous term plus a first orderperturbation, φ ( ~x, t ) = φ (0) ( t ) + δφ ( ~x, t ) . (3.16)We investigate, in units of ~ = c = 1, a minimally-coupled scalar field that obeysEq. (2.90), which we will refer to as the evolution equation: ∂ t δφ + 3 H∂ t δφ − a − ( t ) X i =1 ∂ i δφ + m ( φ (0) ) δφ = 0 . (3.17)The mass term is related to the inflationary potential by m ( φ (0) ) = d Vd ( φ (0) ) . (3.18)1For simplicity, we take m ( φ (0) ) as a constant, m . This is an effective mass, andfrom now on m will refer only to this effective mass, which may or may not be thesame as the mass of the scalar field, which we will call m scalar . In Eq. (2.78), we showhow m could incorporate a scalar coupling to the background curvature. In whatfollows, we will assume the minimally coupled case of ξ = 0, even though the m termcould include a non-zero coupling term if the curvature were also constant. (In theasymptotically flat segments of our composite scale factor the Ricci scalar curvatureis not a constant.) We note that the massless, conformally-coupled case of m scalar = 0and ξ = 1 / g ab ( x ) → ˜ g ab ( x ) = Ω( x ) g ab ( x ) , (3.19) φ ( x ) → ˜ φ ( x ) = Ω( x ) const φ ( x ) , (3.20)where Ω( x ) is a continuous, finite, real, scalar function; in the conformally-invariantcase, no particle production occurs [1, 2, 3, 15, 16].The quantized field δφ can be written in terms of the early time creation andannihilation operators, A † ~k and A ~k , as δφ = X ~k (cid:16) A ~k f ~k + A † ~k f ∗ ~k (cid:17) , (3.21)where f ~k = V − e i~k · ~x ψ k ( t ( τ )) . (3.22)We are imposing periodic boundary conditions upon a cubic coordinate volume, V = L . In the continuum limit L would go to infinity. The function ψ k ( t ) satisfies ∂ t ψ k ( t ) + 3 H∂ t ψ k ( t ) + k a ( t ) ψ k ( t ) + m ψ k ( t ) = 0 , (3.23)2where k = 2 πn/L , with n an integer. Because the creation and annihilation operatorsin Eq. (3.21) correspond to particles at early times, we require that ψ k satisfies theearly-time positive frequency conditionlim τ →−∞ ψ k ( t ( τ )) ∼ p a i ω i ( k ) e − ia i ω i ( k ) τ , (3.24)where ω i ( k ) ≡ p ( k/a i ) + m .At late times, this solution will have the asymptotic formlim τ ′ →∞ ψ k ( t ( τ ′ )) ∼ p a f ω f ( k ) h α k e − ia f ω f ( k ) τ ′ + β k e ia f ω f ( k ) τ ′ i , (3.25)where ω f ( k ) ≡ p ( k/a f ) + m . Consider a spacetime composed of three segments of the scale factor, a ( t ), in a homo-geneous background metric given by Eq. (3.4). For an example, see Figs. 3.1 and 3.3.The first and second segments are joined at the time t , and the second and thirdsegments are joined at the time t .The quantities ψ k and dψ k /dt are continuous across the joining regions given acontinuity of the scale factor of at least C . Using Eq. (3.47), it is possible to showthe conservation of the Wronskian. Multiplying Eq. (3.47) by its conjugate leads to d ψ k ( t ) ∗ dτ ψ k ( t ) = d ψ k ( t ) dτ ψ k ( t ) ∗ . (3.26)Integrating by parts shows (cid:20) ψ k ( t ) dψ k ( t ) ∗ dτ − ψ k ( t ) ∗ dψ k ( t ) dτ (cid:21) boundary = 0 . (3.27)3Since the boundary conditions are arbitrary, it follows with Eq. (3.7) that the Wron-skian, a ( t ) (cid:20) ψ k ( t ) dψ k ( t ) ∗ dt − ψ k ( t ) ∗ dψ k ( t ) dt (cid:21) , (3.28)is a constant. Using Eq. (3.24), we see that this constant is just i ; and using Eq. (3.25),we see that iα k α k ∗ − iβ k β k ∗ = i , or [1] | α k | − | β k | = 1 . (3.29)We have two linearly independent solutions to the evolution equation in both thesecond segment, with solutions h ( t ) and h ( t ); and the third segment, with solutions g ( t ) and g ( t ); for a total of four separate functions. These functions are multipliedby constant coefficients that we must determine. During the second segment, from t to t , we have: ψ k ( t ) = Ah ( t ) + Bh ( t ) , (3.30) ψ ′ k ( t ) = Ah ′ ( t ) + Bh ′ ( t ) . For t > t , we have: ψ k ( t ) = Cg ( t ) + Dg ( t ) , (3.31) ψ ′ k ( t ) = Cg ′ ( t ) + Dg ′ ( t ) . If we require that ψ k ( t ) and ψ ′ k ( t ) be continuous at t and t . This imposes 4 matchingconditions: Ah ( t ) + Bh ( t ) = ψ k ( t ) , (3.32) Ah ′ ( t ) + Bh ′ ( t ) = ψ ′ k ( t ) ,Cg ( t ) + Dg ( t ) = Ah ( t ) + Bh ( t ) , Cg ′ ( t ) + Dg ′ ( t ) = Ah ′ ( t ) + Bh ′ ( t ) . Given the values of ψ k and ψ ′ k , and the matching conditions Ah ( t ) + Bh ( t ) = ψ k ( t ) = ψ k , (3.33) Ah ′ ( t ) + Bh ′ ( t ) = ψ ′ k ( t ) = ψ ′ k ,Cg ( t ) + Dg ( t ) = Ah ( t ) + Bh ( t ) ,Cg ′ ( t ) + Dg ′ ( t ) = Ah ′ ( t ) + Bh ′ ( t ) , we wish to calculate the constant coefficients C and D in terms of the functions h ( t ), h ( t ), g ( t ), and g ( t ); and the values of ψ k , ψ ′ k , t , and t . (Here a prime denotesderivative with respect to t .) Rearranging the first two matching conditions leads to B = (cid:20) ψ k − Ah h (cid:21) t = t , (3.34) A = (cid:20) ψ ′ k − Bh ′ h ′ (cid:21) t = t . Combining these two equations leads to A = (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t ,B = (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t . (3.35)At the time t we have: ψ k ( t ) = Ah ( t ) + Bh ( t )= n (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t h ( t )+ (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t h ( t ) o , (3.36)5and ψ ′ k ( t ) = Ah ′ ( t ) + Bh ′ ( t )= n (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t h ′ ( t )+ (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t h ′ ( t ) o . (3.37)Let us also define ψ k ≡ ψ k ( t ) and ψ ′ k ≡ ψ ′ k ( t ). In terms of ψ k and ψ ′ k the lasttwo boundary conditions in Eq. (3.33) become C = (cid:18) ψ ′ k g − ψ k g ′ g ′ g − g g ′ (cid:19) t = t ,D = (cid:18) ψ ′ k g − ψ k g ′ g ′ g − g g ′ (cid:19) t = t . (3.38)Substituting for ψ k and ψ ′ k yields C = (cid:18) [ Ah ′ + Bh ′ ] g − [ Ah + Bh ] g ′ g ′ g − g g ′ (cid:19) t = t ,D = (cid:18) [ Ah ′ + Bh ′ ] g − [ Ah + Bh ] g ′ g ′ g − g g ′ (cid:19) t = t . (3.39)Finally, expressing A and B in terms of the given values of ψ k and ψ ′ k specified at t leads to C = 1( g ′ g − g g ′ ) t = t × n (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t ( h ′ g − h g ′ ) t = t + (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t ( h ′ g − h g ′ ) t = t o , (3.40)and D = 1( g ′ g − g g ′ ) t = t × n (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t ( h ′ g − h g ′ ) t = t + (cid:20) ψ ′ k h − ψ k h ′ h ′ h − h h ′ (cid:21) t = t ( h ′ g − h g ′ ) t = t o , (3.41)which are the combined joining conditions for ψ k and ψ ′ k .We find ψ k and ψ ′ k from the solution to the evolution equation in the initialasymptotically flat segment of the scale factor. In the massless case, this solution isgiven by Eq. (3.43). The functions h ( t ) and h ( t ) are to be related to the evolutionequation solutions in the inflationary middle segment of the scale factor. Comparingthis with Eqs. (3.45) and (3.48) shows A = E ( k ) and B = F ( k ). Similarly, thefunctions g ( t ) and g ( t ) are to be related to to the evolution equation solutions inthe final asymptotically flat segment of the scale factor, and we will later make theidentification C = N ( k ) and D = N ( k ), where the coefficients N ( k ) and N ( k ) aredefined through their use in Eq. (3.46). We will first consider the case, m = 0. Rewriting the evolution equation, Eq. (3.23),in terms of τ instead of t leads to d ψ k dτ = − k a ψ k . (3.42)For the first segment of our composite scale factor, the solution of (3.42) havingpositive frequency form (3.24) at early times is the hypergeometric function [4, 16,65, 66] ψ k ( t ( τ )) = 1 √ a i k e − ika i τ F ( − ika i s i + ika i s i , − ika i s i − ika i s i ; 1 − ika i s i ; − e τsi ) , (3.43)7where F ( a, b ; c ; d ) is the hypergeometric function as defined in [21, see 15.1.1]: F ( a, b ; c ; z ) = Γ( c )Γ( a )Γ( b ) ∞ X n =0 Γ( a + n )Γ( b + n )Γ( c + n ) z n n ! . (3.44)For the exponentially expanding segment of the scale factor in the massless case( V ′′ = 0 in Eq. (2.99) above) ψ k ( t ) = a ( t ) − (cid:20) E ( k ) H (1) (cid:18) ka ( t ) H infl (cid:19) + F ( k ) H (2) (cid:18) ka ( t ) H infl (cid:19)(cid:21) , (3.45)where H (1) and H (2) are the Hankel functions of the first and second kind. The vari-ables t and τ are related by Eq. (3.7). The coefficients E ( k ) and F ( k ) are determinedby the matching conditions of the first joining point at t = t . We note that the finiteperiod of exponential inflation lacks the full symmetries of a de Sitter universe. Inthe pure de Sitter case, as shown in [20], the k = 0 mode has to be chosen in a specialway to avoid infrared divergences. For our a ( t ), infrared divergences do not arise (seeSec. 3.2.2).For the final segment of our composite scale factor, the solution of the evolutionequation (3.42) is a linear combination of hypergeometric functions [4, 16, 65, 66]: ψ k ( t ( τ ′ )) = N ( k ) e − ika f τ ′ F ( − ika f s f + ika f s f , − ika f s f − ika f s f ; 1 − ika f s f ; − e τ ′ sf )+ N ( k ) e ika f τ ′ F ( ika f s f + ika f s f ,ika f s − ika f s f ; 1 + 2 ika f s f ; − e τ ′ sf ) , (3.46)where the coefficients N ( k ) and N ( k ) are determined by the matching conditionsof the second joining point at t = t . An example of the evolution for a particularmode is plotted for a specific choice of parameters using our composite scale factorin Figs. 3.3 and 3.4.Fig. 3.4 shows a dimensionless solution to the massless evolution equation, where8 -0.5 0 0.5 0 1 2 3 ψ k ( t ) √ — s i H infl t Real Part Imag. Part Magnitude
Figure 3.4: A Dimensionless Solution to the Evolution Equation.the k = 2 Fourier mode is plotted versus dimensionless time for the same compositescale factor used in Fig. 3.1. The real part of √ s i ψ k , the imaginary part of √ s i ψ k ,and the magnitude of √ s i ψ k are all plotted.With joining conditions for the segments of the scale factor, the derived solution tothe evolution equation can be matched up with the known solution for the exponentialexpansion of an inflationary segment by matching δφ k ( t ) and its time derivative acrossthe boundary conditions. See Figure 3.5 for the evolution of modes in the middle ofa long inflationary period for the massless case. The time t is taken to be zero when k = a ( t ) H (when the plotted mode exits the Hubble radius) and depends on themode number k . Multiplied by k /H and plotted against this mode-dependenttime, all of the different fluctuation modes align along the same curve in this graph.This shows, in the massless case, the scale-invariance of the spectrum for those modesthat exit the Hubble radius during a period of constant H ( t ).9 -10-5 0 5 10 15 20 25-15 -10 -5 0 5 10 15 L n [ | δ φ k ( t ) | k / H ] H t
Figure 3.5: Scale-Invariance of Inflaton Perturbations in Continuum Limit.
In the case of a massive scalar field, the evolution equation, Eq. (3.23), can be writtenin terms of τ as d ψ k dτ = − ( k a + m a ) ψ k . (3.47)For the middle, inflationary segment of our scale factor, our solution given by Eq. (2.99)is ψ k ( t ) = a ( t ) − (cid:20) E ( k ) H (1) √ − m H (cid:18) ka ( t ) H infl (cid:19) + F ( k ) H (2) √ − m H (cid:18) ka ( t ) H infl (cid:19)(cid:21) , (3.48)where we define m H in terms of the effective mass by m H ≡ mH infl . (3.49)We know the solution to the evolution equation for the region of the scale factor0given by Eq. (3.5) exactly, but we do not have an analytic solution for an asymptot-ically flat segment of our scale factor except for the trivial case of a constant scalefactor. We instead use one of two different approximations that we find reduce tothe same numerical solutions in their mutual realms of applicability: the effective-kapproach and the dominant-term approach. In the first of these approximations, the effective-k approach, we choose our initialand final asymptotically flat segments of the scale factor such that a i ≃ a i and a f ≃ a f . The middle segment of our scale factor, under these conditions, is thuswhere almost all of the change in the scale factor occurs, and we make use of ourexact solution in this region. In the beginning and final asymptotically flat segmentswe make the transformation k → k eff , where k eff is an effective k defined in the initialregion as k i eff ≡ q k + m a i , (3.50)and in the final region by k f eff ≡ q k + m a f . (3.51)In the limit that a = a in a given segment, the approximation becomes exact andreduces to the known Minkowski flat space solution of ψ k ( t ( τ )) = 1 √ a ω h α k e − ia ωτ + β k e ia ωτ i , (3.52)where ω is given by ω ≡ r k a + m . (3.53)The closer the ratio a /a comes to unity in an asymptotically flat segment of the scalefactor, the more trustworthy the effective-k approach becomes. If the two parametersare precisely equal, however, then the scale factor becomes a constant in time and1derivatives of the scale factor are equal to zero. In such a case where a = a , wecannot join to the inflationary middle segment continuously in any derivatives of thescale factor. When a f ≃ a f in the end segment of our composite scale factor, weobserve ultraviolet particle production due to the rapid breaking, or deceleration, ofthe scale factor’s expansion. This is true regardless of effective mass, because this“extended” region of particle production occurs where the mass is negligible and( k/a ( t )) ≫ m . The Effective- k Approach works very well— especially for the case where the finalasymptotically flat scale factor is parameterized such that a f ≃ a f . The Effective- k Approach need not be as accurate when a f ≪ a f , and for this situation we introducean alternate massive approximation, that of the Dominant Term Approach. In thiscase we introduce a new asymptotically flat scale factor that yields an exact solutionin the limit that k →
0. For a fixed mass, this approximation becomes exceedinglyclose to the exact solution whenever | m | ≫ k/a ( t ). In the Dominant Term Approach,when k/a ≫ | m | , we use the asymptotically flat scale factor given above along withthe massless solution; and when | m | ≫ k/a ( t ), we use a new asymptotically flat scalefactor and its associated zeroth Fourier mode solution. These two solutions can bematched up for the case of modes in the intermediary- q region, where we would usethe massless solution for the initial asymptotically flat scale factor and the massivesolution for the final asymptotically flat scale factor. The Dominant Term Approachis suspect at the interface between the small- and intermediary- q behaviors and at theinterface between the intermediary- and large- q behaviors, where the justification forneglecting either the m -term or the k/a -term is weakest. Depending upon which termis neglected, however, this method provides tight upper and lower limits on the averageparticle production per mode even at these interfaces. When an abrupt transitionfrom the exponential inflation of the middle scale factor segment to the asymptotically2flat final scale factor segment is taken to make a fair comparison, the DominantTerm Approach is in excellent agreement with the Effective- k Approach— even atthe interfaces of q ≃ q ≃ exp( − N e ). When the final transition between thesecond and third scale factor segments is not taken to be abrupt, the upper- and lower-limits place the results of the Dominant Term Approach very close to the Effective- k Approach— even at the interfaces— and they differ only in their descriptions of thelarge- q behavior. This is because the Effective- k Approach requires an abrupt end toinflation and is not a contradiction between the two approaches, but rather is a resultof the previously mentioned fact that an abrupt transition at the end of inflationproduces a high-energy region of residual particle production.
Inflaton Field of Fixed Mass and Zeroth Fourier Mode
In units of ~ = c = 1, the perturbations to the inflaton field satisfy the evolutionequation for mode- k ¨ δφ k + 3 H ( t ) ˙ δφ k + k a ( t ) δφ k + m δφ k = 0; (3.54)where a dot represents a derivative with respect to the proper time; where a ( t ) isthe scale factor; where H ( t ) ≡ ˙ a ( t ) /a ( t ) is the Hubble constant, which may varywith time; and where m is taken to be a constant effective inflaton mass, which isequal to the square root of the second derivative of the inflationary potential withrespect to the homogeneous, background part of the inflaton field. With a change ofvariables from the proper time, t , to a new time variable that satisfies the relationship dτ ≡ a ( t ) − dt ; and examining the zeroth Fourier mode, where k = 0, which can infact can be taken to be approximately correct whenever k/a ( t ) ≪ m , the evolutionequation becomes d δφ dτ = − m a ( τ ) δφ . (3.55)3Using an analysis patterned after that which Epstein used to model the scatteringof radio waves off the ionosphere [63] and that which Eckart used to model potentialenergy in one-dimensional scattering in quantum mechanics [64], we define a scalefactor that is asymptotically flat in both the past- and future-time infinities as a ( τ ) = n a + e τ/s [( a − a )( e τ/s + 1) + b ]( e τ/s + 1) − o . (3.56)The form of this scale factor is modeled after the scale factor first introduced byParker [4, 16, 65, 66] which has four adjustable parameters a , a , s , and b that allowone to approximate a wide range of possible scale factors a ( τ ). The field equation,Eq. (3.55), with this scale factor, a ( τ ), has exact solutions in terms of hypergeometricfunctions [63, 64]. With this scale factor, Eq. (3.55) becomes d δφ dτ = − m (cid:8) a + e τ/s [( a − a )( e τ/s + 1) + b ]( e τ/s + 1) − (cid:9) δφ . (3.57)A change of variables to u ≡ e τ/s leads to d δφ d ( s ln u ) = − m (cid:8) a + u [( a − a )( u + 1) + b ]( u + 1) − (cid:9) δφ . (3.58)With the chain rule, we use d δφ d ( s ln u ) = 1 s (cid:18) d ln udu (cid:19) − ddu "(cid:18) d ln udu (cid:19) − ddu δφ = us ddu (cid:20) u ddu δφ (cid:21) = u s d du δφ + us ddu δφ (3.59)to write, with a prime denoting a derivative with respect to the variable u , δφ ′′ + δφ ′ u + s m u (cid:8) a + u [( a − a )( u + 1) + b ]( u + 1) − (cid:9) δφ = 0 . (3.60)4Without having yet made any assumption as to the reality of τ /s , the variable u mayrange from −∞ to + ∞ on the complex plane. Portions of this evolution equationcan be seen to become infinite at u = 0 and u = −
1. For the case of u = 0, wherethe evolution equation becomes δφ ′′ + δφ ′ u + s m u a δφ = 0 , (3.61)we use the chain rule to change variables to v = ln u , where ∂ u = u − ∂ v , to get e − v ∂ v (cid:0) e − v ∂ v δφ (cid:1) + e − v ∂ v δφ + e − v s m a δφ = 0 , (3.62)which simplifies to ∂ v δφ = − s m a δφ , (3.63)the solution of which is, δφ = e ± isma v = u ± isma . (3.64)For the case of u = −
1, where the evolution equation becomes δφ ′′ − δφ ′ + s m (cid:8) a − b ( u + 1) − (cid:9) δφ = 0 , (3.65)we test the analog of the solution found in Eq. (3.64) to look for a solution of theform δφ = ( u + 1) x , (3.66)and insert this into the evolution equation for the case of u = − x ( x − u + 1) x − − x ( u + 1) x − + s m a ( u + 1) x − s m b ( u + 1) x − = 0 . (3.67)5Because ( u + 1) = 0, the factors with the lowest exponential power of ( u + 1) x − dominate this equation, and at the point of u = − x ( x − u + 1) x − = s m b ( u + 1) x − , (3.68)or x ( x −
1) = s m b, (3.69)with solutions x ± = 1 ± √ s m b , (3.70)so at u = − δφ = ( u + 1) x ± . (3.71)A second order differential equation has at most two distinct solutions; therefore, ourtest has found all the solutions for the case of u = −
1. To write the u = 0 case in anequivalent form, we define p ≡ isma , (3.72)such that for the u = 0 case δφ = u ± p , (3.73)and define for later use p ≡ isma . (3.74)To find the general solution of δφ ( u ), we write δφ = (1 + u ) x − u − p f [ u ] , (3.75)6where the function f [ u ] is defined by this equation. We insert this expression for δφ back into Eq. (3.60) to get0 = (cid:0) (1 + u ) x − u − p f [ u ] (cid:1) ′′ + ((1 + u ) x − u − p f [ u ]) ′ u (3.76)+ s m u (cid:8) a + u [( a − a )( u + 1) + b ]( u + 1) − (cid:9) (1 + u ) x − u − p f [ u ] , which, with s m a , = − p , and s m b = x − x + = x − − x − , becomes0 = x − ( x − − u ) x − − u − p f [ u ] − p x − (1 + u ) x − − u − p − f [ u ] + x − (1 + u ) x − − u − p f ′ [ u ] − p x − (1 + u ) x − − u − p − f [ u ] + p ( p + 1)(1 + u ) x − u − p − f [ u ] − p (1 + u ) x − u − p − f ′ [ u ]+ x − (1 + u ) x − − u − p f ′ [ u ] − p (1 + u ) x − u − p − f ′ [ u ] + (1 + u ) x − u − p f ′′ [ u ]+ x − (1 + u ) x − − u − p f [ u ] − p (1 + u ) x − u − p − f [ u ] + (1 + u ) x − u − p f ′ [ u ] u + 1 u (cid:8) − p + u [( − p + p )( u + 1) + x − x + ]( u + 1) − (cid:9) (1 + u ) x − u − p f [ u ] , (3.77)multiplying by (1 + u ) − x − +1 u p +1 produces0 = x − ( x − − u ) − uf [ u ] − p x − f [ u ] + x − uf ′ [ u ] − p x − f [ u ] + p ( p + 1)(1 + u ) u − f [ u ] − p (1 + u ) f ′ [ u ]+ x − uf ′ [ u ] − p (1 + u ) f ′ [ u ] + (1 + u ) uf ′′ [ u ]+ x − uf [ u ] − p (1 + u ) uf [ u ] + (1 + u ) uf ′ [ u ] u + u + 1 u (cid:8) − p + u [( − p + p )( u + 1) + x − x + ]( u + 1) − (cid:9) f [ u ] , (3.78)which can be simplified to0 = u ( u + 1) f ′′ + [2 x − u − p (1 + u ) + (1 + u )] f ′ (3.79)+ h x − ( x − − u ) − u − p x − + x − + [ p ( p + 1) − p ](1 + u ) u − + u + 1 u (cid:8) − p + u [( − p + p )( u + 1) + x − x + ]( u + 1) − (cid:9) i f, u ( u + 1) f ′′ + [2 x − u − p (1 + u ) + (1 + u )] f ′ (3.80)+ h ([ x − − x − ] u + x − x + )( u + 1) − + ( p − p ) u − +( − p x − + p − p + p − p + x − ) i f, then to 0 = u ( u + 1) f ′′ + [2 x − u − p (1 + u ) + (1 + u )] f ′ (3.81)+ (cid:0) − p x − + p − p + x − (cid:1) f, and finally to 0 = u ( u + 1) f ′′ + [(2 x − − p + 1) u + (1 − p )] f ′ (3.82)+( x − − p + p )( x − − p − p ) f. This is a hypergeometric equation and can be solved in terms of the hypergeometricfunction f = F ( x − − p + p , x − − p − p ; 1 − p ; − u ), using the notation of [21]. Joining Scale Factors Continuously to Second Derivative
To achieve a finite energy density we must maintain the continuity of the compositescale factor to C at the matching points of the individual scale factor segments.Sec. 3.3.1 discusses further the need for C joining conditions. See Figure 3.6 foran example of the asymptotically flat scale factor described in the previous sectionjoined to a region of inflation where the scale factor grows exponentially with respectto proper time. This graph shows how an asymptotically flat region could be joinedonto the beginning or end of an exponential region.To join these different scale factors continuously to the second derivative, wenote that an exponentially growing scale factor, of the form a ( t ) = a exp( Ht ),8 L n [ a ( t )] H infl t Asymptotically Flat Exponential
Figure 3.6: Joining Segments of Scale Factor Continuously to C .9has a time-independent Hubble constant. To find a point in the asymptoticallyflat scale factor described above where ˙ H = 0, we must find a local extremum of H ( t ). When b = 0, there is a unique maximum value of H ( t ). In a simpler scalefactor of the form a ( t ) ∝ t n , which describes a radiation- or matter-dominated uni-verse, no such point would exist. Using the relationship dτ ≡ a ( t ) − dt , the Hub-ble constant is H ( t ) ≡ a ( t ) − ( da/dt ) = a ( τ ) − ( da/dτ ), and its time-derivative is˙ H ( t ) = a ( τ ) − ∂ τ [ a ( τ ) − ∂ τ a ( τ )] = a ( τ ) − ∂ τ a ( τ ) − a ( τ ) − [ ∂ τ a ( τ )] . This is zerowhen ( d a ( τ ) /dτ ) = 4 a ( τ ) − ( da/dτ ) ; in other words, when " (cid:8) − (1 + e − τ/s ) e τ/s (cid:9) (1 + e − τ/s ) − ( a − a ) e − τ/s s [ a + (1 + e − τ/s ) − ( a − a )] (3.83)+ " − (1 + e − τ/s ) − ( a − a ) e − τ/s s [ a + (1 + e − τ/s ) − ( a − a )] = 4 (cid:2) a + (1 + e − τ/s ) − ( a − a ) (cid:3) − " (1 + e − τ/s ) − ( a − a ) e − τ/s s [ a + (1 + e − τ/s ) − ( a − a )] , where the parameter b in Eq. (3.60) has been taken to be zero so that there might bea unique maximum value of the Hubble constant. To simplify this, we multiply bothsides of the equation by 12 s a ( τ ) ( a − a ) − (1 + e − τ/s ) e τ/s to get (cid:0)(cid:8) e τ/s − e − τ/s ) e τ/s (cid:9) (cid:2) a (1 + e − τ/s ) + ( a − a ) (cid:3)(cid:1) + (cid:18) −
53 ( a − a ) e τ/s (cid:19) = (cid:18)
43 ( a − a ) e τ/s (cid:19) , (3.84)which can be expressed as2 a e τ/s + ( a − a ) e τ/s − a = 0 . (3.85)This is a quadratic equation with two roots for e τ/s . The ratio τ /s is now taken tobe real, which means e τ/s is non-negative; this leaves only the positive root solution0of e τ/s = a − a + √ a + 14 a a + a a . (3.86)Once that is found, the C matching conditions for τ , a ( τ ), and H are τ = s ln (cid:20) a − a + √ a + 14 a a + a a (cid:21) , (3.87) a ( τ ) = (cid:18) a (5 a − a + √ a + 14 a a + a ) a + 3 a + √ a + 14 a a + a (cid:19) , (3.88) H = √ − a + a )3 a (5 a − a + √ a + 14 a a + a ) s ! × ( a − a + p a + 14 a a + a ) × q − a − a + p a + 14 a a + a . (3.89) At late times, our solution to the evolution equation will have the asymptotic formgiven by Eq. (3.25). The early- and late-time vacua are related through a Bugoli-ubov Transformation [1] (alternately Romanized in the literature from the Cyrillic asBugolubov or Bugolyubov or Bogoliubov), where the early-time creation and anni-hilation operators ( A † ~k and A ~k ) are related to the late-time creation and annihilationoperators ( a † ~k and a ~k ) through a ~k = α k A ~k + β ∗ k A † ~k , (3.90)where α k and β k are the Bugoliubov coefficients given by Eq. (3.25) and satisfyingEq. (3.29). Because our scale factor is asymptotically Minkowskian, the meaning ofparticles at early and late times has no ambiguity. At late times, the number operatoris h N ~k i t →∞ = h | a † ~k a ~k | i = | β k | , (3.91)1where | i is the state annihilated by the early-time annihilation operators A ~k . Forthe rest of this chapter, the notation | δφ k | is defined as h | δφ k δφ k | i = | f ~k | . Inthe continuum limit, this reduces to (2 π ) − | ψ k | . Thus, | β k | is the average numberof particles in mode- ~k created by the expansion of the scale factor from a state thatinitially has no particles [1, 3]. In the absence of units, the magnitudes of k , a , H , and m have no inherent sig-nificance. The ratio of the Hubble radius, H − , to wavelength, a/k , however, doeshave significance. This combination of k/aH is what we call q when we take theparticular values of a = a f and H = H infl . The other relevant dimensionless ratiosare m H ≡ m/H infl and N e . Transformations that simultaneously leave the values of k/ ( a ( t ) H ( t )) and m H intact do not change the arguments of any of the evolution so-lutions used in our composite scale factor. See Eq. (3.48) for the inflationary middlesegment of our composite scale factor. For an asymptotically flat scale factor of eitherthe form described by Eq. (3.6) or the form described by Eq. (3.56), no matter howwe scale a = a ( τ /s, a , a ), the ratio a /a remains a constant; furthermore, whenkeeping the particular value of τ /s fixed, H ∝ / ( sa ) ∝ / ( sa ). For example,if we multiply k by a constant and multiply a ( t ) by that same constant, we don’tchange the wavelength of our mode. If we don’t alter H , this rescaling won’t change | β q | . When b = 0, we see that this transformation is k → k ∗ xa → a ∗ xa → a ∗ xs → s ∗ x − . (3.92)2For a second example, rescaling k , H infl , and m by the same factor is equivalent to k → k ∗ ys → s ∗ y − m → m ∗ y. (3.93)This second example won’t change the average number of particles created per mode,either. We note that in the massless case the coefficient 1 / √ a i k from Eq. (3.43)may change in invariant transformations, but | β q | does not change because Eqs. (3.94)and (3.95) contain factors that compensate for the change in N . The same is true inthe massive case under the transformation k/a ( t ) → p ( k/a ( t )) + m . In the massless case we find the following:
For our choice of the final asymptotically flat segment given by Eq. (3.6), where weuse Eq. (3.46) to define our functions g ( t ) and g ( t ) in terms of the relationship ψ k ( t ) = N g ( t ( τ )) + N g ( t ( τ )), we find the coefficients α k and β k of Eq. (3.25) fromthe large argument asymptotic forms [4, 16, 65, 66, 21]. With b f = 0, c ≡ iks f a f ,and c ≡ iks f a f , we have α k = q ka f (cid:20) C Γ(1 − c )Γ( − c )Γ(1 − c − c )Γ( − c − c ) + D Γ(1 + 2 c )Γ( − c )Γ(1 + c − c )Γ( c − c ) (cid:21) , (3.94)and β k = q ka f (cid:20) C Γ(1 − c )Γ(2 c )Γ(1 − c + c )Γ( − c + c ) + D Γ(1 + 2 c )Γ(2 c )Γ(1 + c + c )Γ( c + c ) (cid:21) . (3.95)Recall that C and D and the functions g ( t ) and g ( t ) were defined in Sec. 3.2.1. Auseful check of our method is the test of whether Eq. (3.29) is validated, which wefind to be true in all our numerical calculations.The variable | β k | is the average number of particles created in the mode k, as3measured at late times, from the expansion of the scale factor through N e number of e-folds, starting from a universe that is initially in a vacuum state that is asymptoticallyMinkowskian. We use the dimensionless variable q ≡ ka f H infl , (3.96)where k is the wave number, a f is the asymptotically flat late-time scale factor, and H infl is the constant value of ( ˙ a ( t ) /a ( t ))— where the dot represents a derivative withrespect to proper time— during the exponential expansion of the middle segment. Weexpress our results using q instead of the wave number, k , because we find that | β q | is an invariant quantity (see Fig. 3.7), whereas | β k | depends on the arbitrary value ofthe scale factor. By | β q | , we refer to the average number of particles created in themode given by k = q H infl a f . See the end of Sec. 3.4.3 for a discussion of invarianttransformations.We define three regions of q . Values of q . exp( − N e ) are in the small- q region.Values of exp( − N e ) . q . q region. Values of 1 . q arein the large- q region.Fig. 3.7 shows the average late time particle number per mode ( | β q | ) versus q = k/ ( a f H infl ) for 60 e-folds of inflation. Two cases are plotted for the masslesscase based on the behavior at the matching conditions: the scale factor continuousin 0th, 1st, and 2nd derivatives ( C ); and the scale factor continuous in 0th and 1stderivatives ( C ). Note that in the C case, | β q | transitions from a q − dependenceat the end of the intermediary- q region all the way to a q − dependence, temporarilyparallel to the C large- q regime, before settling down into its ultraviolet q − behav-ior. For the wiggles near the transition from the small- q region to the intermediary- q region at q = e − N e , compare with the graph of the dispersion spectrum in Fig. 3.11.When a ( t ) is C or C , i.e. when H infl is continuous, we find numerically that the4 -20 0 20 40 60 -30 -20 -10 0 10 L og [ | β | ] Log [q ] C C Figure 3.7: Particle Production in the Massless Case.5particle production per mode in the small- q region, ( q . e − N e ), is β q = sinh[ N e ] . (3.97)We also find this to be the case, analytically, by taking the limit k →
0. Thisanalytical limit can be seen as follows. Eq. (3.42), in the k → dψ k ( τ ) /dτ is constant. From Eq. (3.24), we see that at early times ψ k ( τ ) = 1 / √ ka i and dψ k ( τ ) /dτ = − i p ka i / → k → dψ k ( τ ) /dτ is bothconstant and zero, so must ψ k ( τ ) be constant. Matching ψ k ( τ ) and dψ k ( τ ) /dτ withthe late-time conditions— which do not make any assumptions about the changingscale factor before the late-time asymptotically flat region of spacetime is reached—leads to two boundary conditions:1 / p ka i = ( α k + β k ) / p ka f , (3.98) − i p ka i / − iα k + iβ k ) (cid:18)q ka f / (cid:19) . (3.99)This leads to α k + β k = e N e , (3.100) α k − β k = e − N e . (3.101)The solution to this is α k = cosh N e , (3.102) β k = sinh N e . (3.103)In the limit of k →
0, both coefficients happen to be real, and we can see thatEq. (3.29) is naturally satisfied. Although this result was derived in the k → k/ ( a i H infl ) ≪
1. This small- q limit holdsfor arbitrary expansions, besides those described by our parameterized composite6scale factor, provided they initiate from a Minkowski vacuum state. We find that therequirement for an alternative to the Bunch-Davies state for the k = 0 mode in deSitter space would be a consequence of taking N e → ∞ in this analytical limit.For at least a moderate number of e-folds, this simplifies to | β q | ≃ e N e . (3.104)The dependence in the intermediary- q region ( e − N e . q .
1) for the C or C massless case is | β q | ≃ q − . (3.105)When N e is finite, with our composite scale factor there are no infrared diver-gences. For infinite inflation, where N e → ∞ , we find the infrared divergences of ade Sitter universe. This problem is resolved for a true de Sitter universe in [20]. Ourcomposite scale factor is different from a purely de Sitter universe in that our initialconditions are specified by our initial asymptotically flat region of the scale factor.Discontinuities in the derivatives of the scale factor at the matching points in-troduce additional particle production for modes in the large- q (or q &
1) region.For the C case, where the scale factor and H = ˙ a ( t ) /a ( t ) are both continuous, thelarge- q region goes like | β q | = n q − . (3.106)For the C case, where the scale factor and H = ˙ a ( t ) /a ( t ) and ˙ H ( t ) are all continuous,the large- q region goes like | β q | = n q − . (3.107)Here n and n are constant coefficients, with n ≃ n ≃ O (1 /
4) for a gradual endto inflation. For a sufficiently abrupt end to inflation, n and n can be made to bearbitrarily large. See Sec. 3.7.In the C case, H ( t ) is not continuous, and we find quite a different behavior.7The evolution equation, Eq. (3.23), may be written [3] d ψ k ( t ) dt + " k a ( t ) + m − (cid:18) ˙ a ( t ) a ( t ) (cid:19) −
32 ¨ a ( t ) a ( t ) ψ k ( t ) = 0 . (3.108)At the discontinuity in ˙ a ( t ), if we express the jump as a step function, then the form of¨ a ( t ) picks up a delta-function contribution. Thus, there is a finite jump in dψ k ( t ) /dt across the discontinuity. The Wronskian is still conserved. In the C case, | β q | isproportional to q − in the small- and large- q regions, and it is proportional to q − in the intermediary- q region. A C scenario would suffer from both infrared andultraviolet divergences, hence we will not consider it further.For a non-composite scale factor composed of one asymptotically flat scale factordefined by Eq. (3.6), at large values of q the value of | β q | falls off faster than anypower of q , and in terms of k we have: [4, 16, 65, 66] | β k | = sin (cid:0) [1 − √ k s b ] (cid:1) + sinh [ πks ( a − a )]sinh [ πks ( a + a )] − sinh [ πks ( a − a )] . (3.109)In the limit that k → k → | β k | = sinh [ N e ], where in this case N e is ln ( a /a ). This is the same small- q limit for the average number of particles created per mode as we found above inEq. (3.97). The analog of the intermediary- q region extends over a range of ln q equal to 2 N e , as opposed to N e for the particle production associated with our com-posite scale factor. Thus, a graph of the average number of particles created permode for a single asymptotically flat scale factor would look similar to Fig 3.7, exceptthe region analogous to the intermediary- q region would be twice as long and wouldhave half the slope relative to a scale factor dominated by an exponential expansion. In the massive case we find the following:
In Fig. 3.8, the dependence of particle production ( | β q | ) on mass is shown for an8 -40-20 0 20 40 60 80 -40 -30 -20 -10 0 10 L og [ | β | ] Log [q ] m H =0 m H =Sqrt[0.1] m H =1 Figure 3.8: Particle Production in the Effective- k Approximation.9expansion of 60 e-folds. The beginning and end segments are defined by a i = a i (1 +10 − ), a f = a i e , and a f = 0 . a f . The massless case can be compared withthe plot in Fig. 3.7 which is continuous up to the second derivative of the scale factorto see that the two graphs are the same for q .
1. In this graph, however, there isan “extended” region of | β q | ∝ q − shortly after q ≃ q ≃ before the ultraviolet behavior of | β q | ∝ q − is seen. The term “extended” is definedin Sec. 3.7. This is due to particle creation caused by the rapid transition from theinflationary region to the asymptotically flat scale factor. The two approximations,the effective-k approach and the dominant-term approach, give the same results withthis particular parameterization of inflation. Both of the massive cases shown hereproduce more red-shifted particles of low momentum than the massless case. Thecase of m H = 1 /
10 produces many more low momentum particles than the case of m H = 1. See also Figs. 3.9 and 3.13.In Fig. 3.9, the dependence of particle production ( | β q | ) on mass is shown for anexpansion of 60 e-folds. This graph is different from Fig. 3.8 in that the transitionfrom exponential expansion to the final asymptotic segment of the scale factor ismore gradual, happening over about an e-fold. Thus, we use the dominant-termapproximation. The effective-k approach, in spite of the gradual transition to anasymptotically flat scale factor, overlaps with the dominant-term approach in thisgraph except very close to q = 1. For values of q .
1, this graph is identical to thatof Fig. 3.8.
In Fig. 3.10, particle production as a function of q is plotted for 60 e-folds for boththe massless case and the case of m = 10 − H infl , labeled as m << H . This graphwas made using the dominant-term approximation. The effective-k approach wouldoverlap on this graph except very near to q = m H = 10 − . It is always the casethat ( k/a ( t )) ≫ m for q > m H and in this region the plot of m H = 10 − overlaps0 -40-20 0 20 40 60 80 -40 -30 -20 -10 0 10 L og [ | β | ] Log [q ] m H =0 m H =Sqrt[0.1] m H =1 Figure 3.9: Particle Production in the Dominant Term Approximation.1 -40-20 0 20 40 60 80 -40 -30 -20 -10 0 10 L og [ | β | ] Log [q ] m< 1, the value of | β q | approaches the constant (1 / q N e . In the region of m H exp( − N e ) < q < m H , wehave ( k/a ( t )) ≫ m in the initial asymptotically flat region and ( k/a ( t )) ≪ m inthe final asymptotically flat region. Between q ≃ m H exp( − N e ) and q ≃ exp( − N e ),we see | β q | ∝ q − ; and between q ≃ exp( − N e ) and q ≃ m H , we see | β q | ∝ q − .In light of these characteristics, a comparison of Eqs. (3.115) and (3.116) can be madewith consideration to where ( k/a ( t )) ≫ m and to where ( k/a ( t )) ≪ m . Such ananalysis shows that in the tiny mass limit of m H ≪ 1, the dispersion spectrum reducesto the massless dispersion spectrum. The tiny mass limit bridges the transition fromthe massless case to the case of small, non-negligible m H such as m H = 0 . 01, and thedispersion spectra as a function of q for all cases changes continuously when goingfrom massless to tiny mass to small mass. This is a successful check on our method. The dispersion spectrum is [16, 75] h | δφ | i = 12( a f L ) X k " | β k | p ( k/a f ) + m . (3.110)We will first consider the massless case where m = 0. See below in Sec. 3.4.3for the massive case. We subtract off the late-time Minkowski vacuum contribution,which is that part of the unrenormalized dispersion which would be present in aMinkowski vacuum without any particles ( | β k | = 0 for all k ), to get the dispersion h | δφ | i = 12( a f L ) X k | β k | p ( k/a f ) = 1 a f L X k | β k | k , (3.111)3which in the continuum limit becomes h | δφ | i = 1 a f (2 π ) Z ∞ | β k | k d k. (3.112)Spherical symmetry, where d k = 4 πk dk , gives us h | δφ | i = 12 π a f Z ∞ k | β k | dk. (3.113)With k = q a f H infl and dk = dq a f H infl , we have h | δφ | i = a f H π a f Z ∞ q | β q | dq = H π Z ∞ q | β q | dq . (3.114)In the massless case, the dispersion spectrum amplitude is thus Z ≡ q | β q | H π . (3.115)We plot Z/H in Fig. 3.11. We see that in both the case where a ( t ), ˙ a ( t ), and¨ a ( t ) are all continuous; and the case where a ( t ) and ˙ a ( t ) are continuous; h | δφ | i isfinite without the need for any renormalization beyond subtracting off the Minkowskivacuum terms. When none of the derivatives of the scale factor is continuous, thenthe dispersion spectrum does not converge.Fig. 3.11 shows the dispersion spectrum Z/H given by Eq. (3.115) for ourcomposite scale factor continuous in a ( t ), ˙ a ( t ), and ¨ a ( t ) over an expansion of 60 e-folds. The y-axis, Z/H , is shown multiplied by a factor of e − N e ; and the x-axis, q , isshown multiplied by a factor of e N e . When using this scaling, the region plotted in thisgraph would look identical for an expansion of 10 e-folds, and it would look identicalfor an expansion of 80 e-folds. In the case of a i = a i + w , where w ≡ − a i ,we see marked peaks in the dispersion spectrum. When we change the parameters4 Z e - N e / H q e N e a =a +w a =10a Figure 3.11: Massless Dispersion Spectrum.5in the initial asymptotically flat region to a i = 10 a i , these peaks are damped asshown. The ending conditions of the final asymptotically flat segment do not affectthese peaks.A calculation of the dispersion spectrum in the massive case leads to an equationanalogous to Eq. (3.115): Z ≡ q | β q | H π q m H q . (3.116) We take for our initial conditions a quantum state to be asymptotic at early timesto that of a Minkowski vacuum spacetime for all modes. This is a consequence ofour asymptotically flat scale factor and our assumption that no particles are initiallypresent. It is more common in the literature to take instead the Bunch-Davies statefor quantum fluctuations, that is to assume a de Sitter spacetime. As pointed out by[62], this leads to an infrared divergence of the two-point function, where the two-pointfunction is another name for our dispersion spectrum, and the cause of this divergenceis correctly diagnosed as being due to the choice of initial conditions in [70]. Both ofthe authors of [62, 70] handle these infrared divergences with a cutoff frequency thatomits modes that are currently outside the Hubble radius of our observable universe.The use of de Sitter initial conditions is equivalent to supposing an inflationaryperiod that extends over an infinite number of e-folds, or N e → ∞ . If we assumea finite N e , and if we assume that in the future our universe will be approximatelymatter-dominated for all times, which means neglecting any dark energy or cosmo-logical constant, then eventually every mode that exited the Hubble radius duringinflation would eventually re-enter the Hubble radius of our universe after inflation ifit has not already done so.Both Figs. 3.11 and 3.12 show additional peaks after the primary peak, where theprimary peak roughly indicates the interface between small- q and intermediary- q behavior. These minor peaks are caused by phase differences between modes with6similar wavelengths as they exit the Hubble radius near the beginning of inflation.The modes that exit the Hubble radius with a large amplitude— either a positive realamplitude, a negative real amplitude, a positive imaginary amplitude, or a negativeimaginary amplitude— quickly have this large amplitude translated into a near con-stant value outside of the Hubble radius. Those modes that exit the Hubble radiuswith relatively small amplitudes are frozen into evolutions of relatively small mag-nitudes outside of the Hubble radius; these relatively low-amplitude modes have arelatively high change in amplitude with respect to time, but this initial excess inthe derivative of the amplitude with respect to time is rapidly redshifted away dur-ing inflation. With an abrupt transition from an asymptotically Minkowski vacuumto an exponential inflation of the scale factor, by which we mean that a ( t ) ≃ a i ,where a ( t ) is the scale factor at the transition from the initial asymptotically flatsegment to the exponentially growing segment of inflation, and where a i is the scalefactor at early times, we see that the minor peaks are more pronounced. With a moregradual transition from the initial asymptotically flat segment of the scale factor toinflation (when a ( t ) ≃ . a i ), these minor peaks are damped out. If these modeswere observable in our universe, that is if they have already re-entered our Hubbleradius, their measurement might tell us something about initial conditions before thebeginning of inflation: whether there had been a phase transition from the very earlyuniverse to inflation, how rapidly the very early universe had been expanding (orcontracting) relative to the expansion of inflation, and what the dominant contribu-tion to the evolution of our universe might have been before the start of inflation.Because measuring the contribution of these minor wiggles to the scale dependence oflarge-scale structure would be experimentally challenging (if not impossible), this isin some sense speculation, but that does not change the fact that the two dispersionspectra shown in Fig. 3.11 are different, and this difference— if observed— would tellus about our pre-inflationary universe.7 Consider a quantum fluctuation of the particular mode that, at the beginning ofinflation, has a wavelength equal to the Planck length. By the time this wavelengthhas been stretched to the point that the mode is exiting the Hubble radius, it willhave a wavelength the size of the Hubble radius. For this to happen, the scale factormust increase by a factor of H − /ℓ Planck .With ~ = c = 1, the Planck length is ℓ Planck = √ G = 8 × − (GeV) − . Usingthe value of H infl = 7 × GeV given in Eq.(2.149), we find H − /ℓ Planck ≃ × ,which corresponds to a mode exiting the Hubble radius ln(10 ) ≃ 12 e-folds afterthe start of inflation. All higher frequency modes, that is for q & e − N e +12 , will haveoriginated from trans-Planckian modes during inflation. With N e = 60 e-folds ofinflation, if we use the estimate of the number of e-folds before the end of inflationin which the observable modes of the CMB are exiting the Hubble radius given byEq. (2.143) (50 e-folds) or by Eq. (3.144) (53 e-folds), then it might be possible toobserve the difference in amplitudes between those modes that were initially super-Planckian quantum fluctuations and those that were initially sub-Planckian quantumfluctuations before the start of inflation. With either a smaller value of the Hubbleconstant during inflation or with a larger number of total e-folds of inflation, there-entry of the first trans-Planckian modes back into our Hubble radius after inflationcould be postponed to epochs of our universe much later than recombination. Fig. 3.12 shows a comparison of the dispersion spectrum ( Z/H ) given by Eq. (3.116)and normalized to 1 for our composite scale factor continuous in a ( t ), ˙ a ( t ), and ¨ a ( t )over an expansion of 60 e-folds for various masses. The values of Z/H were dividedby the maximum value of the primary peak for each located at q ≃ exp( − N e ). Tonormalize these peaks, Z/H was divided by the following factors: 1 . × for the8 Z / H N o r m a li ze d t o q e N e m H =0 m H =Sqrt[0.1] m H =1 Figure 3.12: Massive Dispersion Spectrum.9massless case, 2 . × for m H = 0 . 1, and 2 . × for m H = 1.The dispersion spectrum is plotted for three different cases of m H in Fig. 3.12.The effective-k approach is useful for this approximation. This approach demandsthat in the initial asymptotically flat segment of the scale factor, a ( t ) must always beapproximately equal to a i . Because a ( t ) ≃ a if either a ≃ a or a ≫ a , however,this approach can be used with a wide range of initial conditions. Specifically, when a ≫ a in Eq. (3.6), we have a ( t ) ≃ (7 / (1 / . Although we are not at the momentconsidering the case of a ≫ a in Eq. (3.56), for comparison we note that it wouldlead to a ( t ) ≃ (1 / . In both cases a ( t ) ≃ . a . We have found that, even inthe massive case, the observed humps are dependent only upon the initial conditions.In the region shown in this figure, the graph would not be significantly altered byusing C joining conditions instead of our C matching conditions. The effective-kapproximation plotted on this graph would overlap with the exact solution, if anexact solution were available. The shapes of the curves are fixed above a moderatenumber of e-folds. We define the variable J such that the maximum value of Z/H for the major peak, which is the peak located nearest to q = e − N e , is J e ( P − N e in the massless case and is J e ( P − N e in the massive case. Then, the normalizationfactor scales like e ( P − N e in the massless case, as can be seen from Eq. (3.115); andthe normalization factor scales like e ( P − N e in the massive case, as can be seen fromEq. (3.116), where we define the exponent P in the following way: | β q | ≃ q − P (3.117)in the region of intermediary- q ( e − N e . q . | β q | ≃ e P N e (3.118)in the small- q region ( q . e − N e ). The exponent P is well described by a q -independent value in the case of m = 0 and in the case of 0 . . m H . / P = 2, so the height of the major peak in the graph of themassless case in Fig. 3.12 grows with an increasing number of e-folds as e N e , while thewidths of the peaks narrow with an increasing number of e-folds as e − N e . The areaunder an individual peak in the massless graph therefore does not change appreciablywhen changing the number of e-folds of expansion, provided there are at least a fewe-folds of inflation. For the massive cases, we see that P = 2 . m H = 0 . P = 2 . m H = 1.Fig. 3.13 shows the dependence of the variable P , as defined in Eq. (3.117), upon P m H Figure 3.13: Inflaton Spectrum Characterized in Terms of Inflaton Mass. m H = m/H infl . The calculated data points shown lie on the curve P = p − m H .Outside of the region plotted, however, P does not have a constant, q -independentvalue. For m H > . 5, the argument, p (9 / − m H , of the Hankel functions becomesimaginary, and | β q | oscillates with changing q . For an example of a non-zero massmuch smaller than H infl , see Fig. 3.10.1Table 3.1: Approximation h | δφ | i /H . . m H . . J e ( P − N e + R e − Ne dq J q − P m H = 0 J + R e − Ne dq J q − P Table 3.2: Configuration Space Dispersion h | δφ | i /H . . m H . . (cid:0) − P + − P − P e ( P − N e (cid:1) Jm H = 0 (cid:0) + N e (cid:1) J We wish now to approximate the dependence of the configuration space dispersion h | δφ | i /H with regard to the number of e-folds. In our approximation we neglectthe minor peaks; we assume that the major peak is located exactly at q = e − N e , that Z/H increases linearly with q up to the major peak, and that Z/H decreases as( q e N e ) − P in the massive case— or as ( q e N e ) − P in the massless case— until theonset of large- q behavior at q = 1, which effectively serves as a cut-off point. Themaximum of the major peak is given by height = J e ( P − N e , in the massive case; andheight = J e ( P − N e , in the massless case. We find J ≃ . 01 for all three cases. In thissimple approximation, the configuration space dispersion is given by TABLES 3.1and 3.2. The small mass limit of P → P = 2, both reduceto the same limit of | δφ | ≃ H infl r N e + 12 . (3.119)For further discussion of the small mass limit reducing to the massless dispersionspectrum, see Fig. 3.10. We define a given mode of δφ k to be crossing the Hubble radius when k/ ( a ( t ) H ( t )) =1. We define a mode of k to be inside the Hubble radius when k > a ( t ) H ( t ), andwe define a mode of k to be outside the Hubble radius when k < a ( t ) H ( t ). Modesin the intermediary- q range exit during inflation to eventually re-enter the Hubble2radius at some time after inflation has ended, provided any cosmological constant ordark energy can be taken to be negligible. Using our composite scale factor, we notethat after a few e-folds of inflation, the quantum perturbations that are exiting theHubble radius are found numerically to satisfy | ψ k | = H k D ( m H ) , (3.120)where | ψ k | is the time-dependent part of | δφ k | , as given by Eqs. (3.21) and (3.22).The variable D ( m H ) ≃ (1 + m H ) is a constant of order 1 that we have evaluatednumerically to be D ( m H = 0) = 1 . ,D ( m H = √ . 1) = 1 . ,D ( m H = 1) = 1 . . (3.121)Thus, our spectrum of | δφ k | , if evaluated at the time of exiting the Hubble radius,is scale-invariant, regardless of effective mass. By Eqs. (2.124) and (2.125) we have P R ∝ k | δφ k | . (3.122)The scalar spectral index given by Eq.(2.127) is n s = 1 + d ln P R d ln k . (3.123)We see that when taken at the time of crossing the Hubble radius, the spectrum,which is proportional to k | δφ k | , has no k-dependence because we have shown thespectrum is proportional to k H /D ( m H ). Evaluating the scalar spectral index atthe time of exiting the Hubble radius thus leads to n s = 1, which can be used as thedefinition of a scale-invariant spectrum.3The modes that exit the Hubble radius at the very beginning of inflation, however,along with those that exit the Hubble radius before reaching the middle segment ofour composite scale factor where a ( t ) begins to grow exponentially with respect to t ,are not described by Eq. (3.120). These modes in the small- q region are not scale-invariant; therefore, well after the end of inflation, long-wavelength modes that arenot scale-invariant would eventually re-enter the Hubble radius of a matter-dominateduniverse. If the total number of e-folds of inflation is sufficiently small, it would bepossible to observe a transition from the scale-invariance to a scale-dependence oflarge-scale structure. See Figs. 3.7, 3.8, and 3.9. Because the small- q modes of largeenough wavelength exit the Hubble radius before evolving away from the early-timeconditions specified by Eq. (3.24), we would expect a massless inflaton to generatea spectral index of n s = 3 in the small- q region, and we would expect a massiveinflaton to generate a spectral index of n s = 4 in the small- q region. If scale-invariance continued indefinitely for large wavelength modes, the dispersion would beinfrared divergent, so this eventual end to scale-invariance is not an artifact of ourinitial conditions. The modes responsible for the galaxy-size structure of today leftthe Hubble radius approximately 45 e-folds before the end of inflation [9, p. 285], soif N e were not too much larger than this, we would expect it to be possible to measurethe end of scale-invariance in our observable universe.In Fig. 3.14 we plot six scenarios depicting the behavior of | ψ k | after exiting theHubble radius. The first two cases, A and B, are for m H = 0. In both of these cases,an expansion of 20 total e-folds is plotted. In case A, there is a gradual end to inflationspanning one e-fold; and in case B, there is an abrupt end to inflation. Because themode has exited the Hubble radius, neither of these end conditions changes | δφ k | ,and the two lines overlap. Here, and in general for the massless case, | δφ k | reaches aconstant value a few e-folds after exiting the Hubble radius, and this constant valueis close to the value at the time of exit. In the massless case, we find a scale-invariantspectrum even when the spectrum is defined in terms of the value of | δφ k | at the4end of inflation. This can be found by noting that at the time a particular mode iscrossing the Hubble radius, the value of | ψ k | given by Eq. (3.45) is approaching aconstant value as the argument k/ [ a ( t ) H infl ] becomes much less than 1.For cases labeled C, D, E, and F; we use m H = √ . 1. In the massive cases, | ψ k | never reaches a constant value, although it changes much more slowly after exiting theHubble radius. Cases C and D are the massive analogs of cases A and B, respectively.In case E, we end inflation gradually over the length of one e-fold, starting just as ourspecific mode crosses the Hubble radius. In case F, we end inflation abruptly just asour specific mode crosses the Hubble radius.From [21], { Eq. 9.1.9 } , we see that in the small argument limit of the Hankelfunctions (cid:12)(cid:12) H (1) v ( z ) (cid:12)(cid:12) ≃ (cid:12)(cid:12) H (2) v ( z ) (cid:12)(cid:12) ≃ (cid:18) Γ( v ) π (cid:19) (cid:18) z (cid:19) − v , (3.124)when the real part of the parameter v is positive and non-zero. In Eq. (3.48), we findnumerically that E ( k ) ∼ − ( i/ p π/H infl and F ( k ) ∼ q ,which are the modes that exit during the exponential expansion of our composite scalefactor. Long after these modes have exited the Hubble radius, when k/ [ a ( t ) H infl ] ≪ 1, we expect Eq. (3.48) to approach | ψ k | ≃ a − (cid:12)(cid:12)(cid:12) H (1) v ( z ) (cid:12)(cid:12)(cid:12) ∝ a − z − v , where z = k/ [ a ( t ) H infl ] and v = p (9 / − m H . Using this small argument approximation withEqs. (3.122) and (3.123) leads to d ln P R d ln k = 3 − q − m H , (3.125)under the assumption that we evaluate | δφ k | at late times, in which case n s = 4 − q − m H . (3.126)In determining the spectrum, | δφ k | is often evaluated at the time when a modeexits the Hubble radius [34, 10]. In this case, we would get n s = 1, exactly. The5WMAP results [42] find n s ≃ . 96, which would suggest a value of m H ≃ i/ H infl , provided the end of inflationis the appropriate time to evaluate P R . This value of m H ≃ i/ m scalar would leadto tachyonic behavior [51], but here m is an effective mass, so this need not be aproblem.A different method of calculating Eq. (3.126), involves combining Eq. (3.117) and q ≡ k/a f H infl with Eq. (3.132). To renormalize Eq. (3.132) would involve droppingthe Minkowski vacuum term such that (1 + 2 | β k | ) → | β k | ; although, in the caseof the intermediary- q modes, the term to be subtracted off is already negligiblecompared with the particle number per mode. To simplify the massive case, we treat | m | ≫ k/a ( t ) in the intermediary- q region of modes that exit the Hubble radiusduring inflation. Then Eqs. (3.122) and (3.123), together with the relationship givenin Fig. 3.13 of P = p − m H , or P = 2 in the massless case, give us the same resultas in Eq. (3.126).If H infl were not constant, but were slowly decreasing during inflation, then wewould find a red-tilted spectrum. We could incorporate this effect into our exactcalculation by taking the adiabatic approach and using the value of H infl ( t ) for ourfirst matching conditions and the value of H infl ( t ) for our second joining. CombiningEqs. (2.125), (3.21), (3.22), (3.120), and (3.123), we find n s = 1 + dd ln k ln (cid:18) H infl ( t ) ˙ φ (cid:19) , (3.127)which, with Eq. (2.114), becomes n s = 1 + dd ln k (6 ln[ H infl ( t )] − V ′ ]) , (3.128)where a dot denotes a derivative with respect to time, and a prime denotes a derivative6with respect to φ . Then, using d/d ln k = H − d/dt , we have n s = 1 + 6 ˙ H infl ( t ) H − V ′ V ′ H , (3.129)which, through the chain rule and with Eq. (2.114), ˙ V ′ = ˙ φV ′′ = − V ′ V ′′ / (3 H infl ( t )), so that, finally, with Eq. (2.113), we have n s = 1 − − ˙ H infl ( t ) H infl ( t ) + 2 18 πG (cid:18) V ′′ V (cid:19) , (3.130)which we write in terms of Eqs. (2.116) and (2.117) to get n s = 1 − ǫ + 2 η, (3.131)which is equivalent to Eq. (2.132) first shown by [41]. Fig. 3.14 shows that the maximum difference between the late-time values of | δφ k | in all six of the cases plotted is about 40%. We conclude that when m H ≪ 1, thevalue of | δφ k | at late times is a reasonably good indicator of the value of | δφ k | at Hubble radius exit. For the rest of this section we will adopt the assumptionthat the late time value of | δφ k | is indicative of the value of | δφ k | at the time ofexiting the Hubble radius. This assumption allows us to extrapolate our method oflate-time renormalization in Minkowski space to a time of curved spacetime in lieuof applying a more rigorous analysis that would require a more complex method ofcurved spacetime renormalization such as in [38].The final conditions do not affect the value of | δφ k | much once a given mode hascrossed the Hubble radius. Thus, we could end inflation just after a mode has exitedthe Hubble radius to find that the value of | δφ k | will be very close to its late-time7value. At late times, Eqs. (3.22) and (3.25) show that the time averaged expectationvalue— h| δφ k | i = 12 L a f ω f (cid:0) | α k | + | β k | (cid:1) = 12 L a f ω f (cid:0) | β k | (cid:1) . (3.132)This value of h| δφ k | i obtained from Eqs. (3.21), (3.22), and (3.120), however, isun-renormalized. To use the renormalized values, we take (1 + 2 | β q | ) → | β q | .Although [9, p. 285] identifies the scale factor, a gal = a f e − , as the one in whichthe k-modes responsible (by seeding the density perturbations) for the formation ofgalaxies are exiting the Hubble radius; we note that when m H ≪ | δφ k | after a mode crosses the Hubble radius, and thus our subsequentmethod is widely applicable to the range of intermediary- q modes. In our assumptiondescribed above, a mode defined by q = 1 at late times is an excellent indication ofthe state of any mode just after crossing the Hubble radius when m H ≪ 1. We canassume for the moment that inflation ends abruptly just as the mode k = a f H infl exits the Hubble radius. This abrupt ending does not change | β q | for the q = 1mode, because we have found that the ending conditions do not affect modes of q . 1. In this case, | δφ q | isn’t changing from its value at Hubble radius crossing(or is roughly equal to the late-time value it would have reached a few e-folds aftercrossing the Hubble radius), the late-time value of | β q | for the q = 1 mode isn’tchanging (because there is no more inflation and the mode q = 1 is insensitive toother factors), and the scale factor isn’t changing; therefore the renormalized value of | δφ q | is not changing. This argument wouldn’t hold for modes of large- q , becausethey are sensitive to the time-derivatives of the scale factor, but we find that thelate-time dispersion spectra for the mode q = 1 is a good approximation to therenormalized value of | δφ q | at the time any mode exits the Hubble radius. Foran analysis of the instantaneous renormalized value of δφ k that does not rely on a8late-time argument, see [38].We next consider the curvature perturbation given by Eq. (2.123) and defined atthe time of Hubble radius crossing as R k = − H ˙ φ δφ k . (3.133)The variable ˙ φ is the rate of change of the homogeneous background scalar field.The quantum perturbations we have considered so far, δφ k , are assumed to be muchsmaller in magnitude than the zeroth-order field. So far our method in this chapter has not been linked to any particular potential ormodel of inflation. In what comes next, we choose a simple potential, which is foundto be in good agreement with the 3-Year WMAP data [47], and we use a hybridcombination of our method and the slow roll approximation. The remainder of thissection is intended to be of a more speculative nature than the rest of this dissertation.For our example, we use the Linde quadratic chaotic-inflation potential [45, 46] V = 12 m φ . (3.134)From Eqs. (2.113) and (2.114), the two slow roll conditions are H ≃ πG V, (3.135)and ˙ φ ≃ − dV /dφ H . (3.136)9Table 3.3: Comparison of δ H for V = m φ H = 10 GeV H = 10 GeV H = 10 GeV m H = 0 . . × − . × − . × m H = 0 . 01 2 . × − . × − . × − m H = 0 . . × − . × − . × − m H = 0 . 25 8 . × − . × − . × − m H = 1 1 . × − . × − . × − We combine these two slow roll equations with the potential specified in Eq. (3.134)to find ˙ φ ≃ − m r 23 1 √ πG . (3.137)We rewrite this as ˙ φ ≃ − H m H r 23 1 / √ πGH infl . (3.138)In our notation, with δφ k taken from Eq. (3.116), R k = − H infl − H m H q 23 1 / √ πGH infl vuuut ( q → | β q | → ) H π r m H ( q → ) ; (3.139)therefore, with 1 / √ πG ≃ . × GeV, R k = 14 π s m H p m H (cid:18) H infl . × GeV (cid:19) . (3.140)The magnitude of the curvature perturbation has been shown to be a conservedquantity outside of the Hubble radius [35, 36], and the curvature perturbation can berelated to the amplitude of density perturbations at the time of re-entry, when onceagain k/ [ a ( t ) H ( t )] = 1. In a matter-dominated universe this relationship is [34] δρ k ρ ≡ δ k = 25 R k . (3.141)See TABLE 3.3 for sample values of δ H , the density contrast defined in [34], atthe time of re-entry into the Hubble radius and for the potential given by Eq. (3.134).00 In Fig. 3.14, the value of | ψ k | , in units of H /k and for the mode k = ( a ( t ) e ) H infl , -5 0 5 10 5 10 15 20 L og [ | ψ ( t ) | ] H infl t A B C D E F Figure 3.14: Modes Exiting the Hubble Radius.is plotted versus dimensionless time H infl t . The graph shows the relative constancy of | δφ k | for modes that have exited the Hubble radius during inflation. For additionaldiscussion of this graph and the difference between cases A-F, see Sec. 3.5.The inflaton perturbations only approach a true constant well outside the Hubbleradius for the massless case. During an exponential expansion in the massive case, m H ≫ k/ [ a ( t ) H infl ], and we may rewrite Eq. (3.23) as ∂ t ψ k ( t ) + 3 H∂ t ψ k ( t ) + m ψ k ( t ) = 0 . (3.142)01The two linearly independent solutions to this are ψ ± ∝ exp (cid:20) − 12 (3 ± P ) H infl t (cid:21) , (3.143)where P is defined as in Fig 3.13. In the small mass limit of P → 3, well outside theHubble radius one of these linearly independent solutions approaches a constant valuewith respect to t , while the other solution decays exponentially. With m H of order 1,both linearly independent solutions decay exponentially outside the Hubble radius.Although the magnitude of these massive perturbations are constant when the scalefactor is constant in time, the rates of their decay well outside the Hubble radiusdepends on how the scale factor is changing. For our composite scale factor with anabrupt end to inflation, where almost all of the expansion occurs in the exponentiallygrowing segment of our scale factor, there is more small- q particle production thanin our composite scale factor with a relatively gradual end to inflation, where moreof the total expansion of the scale factor takes place in the final asymptotically flatsegment of our scale factor. We turn now to tracing a particular mode as it exits theHubble radius until it re-enters our observable universe.Consider, as an example, the k -modes responsible for large-scale structure for-mation. As an approximation, take the following three epochs to be simultaneous:recombination (the time light was emitted from the surface of last scattering), thetransition from a radiation-dominated universe to a matter-dominated universe, andthe re-entry of the modes that would provide the density perturbations to seed galax-ies. Furthermore, also as an approximation, assume a transition to a radiation-dominated universe, where a ( t ) = Ct / , immediately after the end of inflation suchthat H ( t ) is continuous. Call the time of the end of inflation t f , and call the timeof re-entry and recombination t r . Turner and Kolb give the temperature of inflationand the temperature at recombination as 10 GeV and 1 eV, respectively [9]. In aradiation-dominated universe, the energy density— neglecting particle production—02is related to the scale factor as ρ rad ∝ a ( t ) − , and the temperature is related tothe energy density as T ∝ ρ / , so the temperature is related to the scale factor as T ∝ a ( t ) − after radiation and matter have decoupled and are no longer in ther-mal equilibrium. Thus we know that a ( t r ) = 10 a ( t f ). The radiation-dominatedscale factor then gives us ( t r /t f ) / = 10 , or t r = 10 t f . The hubble constantin the radiation-dominated universe is H ( t ) = a ( t ) − da ( t ) /dt = (2 t ) − . Because H ( t f ) = H infl , we have H ( t r ) = 10 − H infl . When a mode exits the Hubble radiusduring inflation, we have k/ ( a ( t ) H ( t )) = k/ ( a ( t f ) e − K e H infl ), where the variable K e isthe number of e-folds before the end of inflation at which a mode exits the Hubbleradius. When our example mode re-enters the Hubble radius after inflation, we have k/ ( a ( t ) H ( t )) = k/ ( a ( t r ) H ( t r )) = 1. By equating the relations for exit and re-entry,we have k/ ( a ( t f ) e − K e H infl ) = k/ (10 a ( t f )10 − H infl ), or e − K e = 10 − . This meansin our approximation K e ≃ 53 e folds . (3.144)Turner and Kolb find, with a more detailed calculation, a value of 45 for this number[9, p. 285]. The simplification of treating recombination, matter-radiation equality,and galaxy seeding as concurrent is a relatively useful approximation. The radiation-dominated universe transitions to a matter-dominated universe at a temperatureroughly one order of magnitude higher than the temperature of recombination, whichmeans the mode that will later re-enter the Hubble radius at the time of radiation-matter equality exits the Hubble radius during inflation roughly 2 e-folds later thanthe mode that will later re-enter the Hubble radius at recombination. The exact rela-tionship between these two events with the Hubble crossing for the modes responsiblefor seeding galaxy formation depends on the nature of dark matter: the current sizeof galaxies does not lead to a simple estimate of their size in the past, because theirsize does not scale with the size of the universe once they have become gravitation-ally bound. Baryonic matter will clump to structure initiated by cold dark matter,but not until after recombination, when radiation pressure overcomes gravitational03attraction; dark matter will start clumping earlier than this, at the epoch when itdecouples from the dominant radiation background [9, 10, 34]. The approximationof an immediate transition from inflation to a radiation-dominated universe is lesscertain, as the validity of this approach could vary based on the specific inflationarypotential being considered. Our analysis of this particle creation reveals a mechanism for Reheating, which isa return to the temperatures and densities that are responsible for the successes ofthe Big Bang model. We find that the energy density present after inflation dependson how abrupt the transition is from the inflationary middle segment of exponentialgrowth to the final asymptotically flat region of the scale factor. Our scale factor can be made to be continuous to the scale factor and two of itsderivatives, but no more, so we see additional particle production caused by discon-tinuities of higher derivatives. When we maintain continuity of a ( t ), ˙ a ( t ), and ¨ a ( t ),the particle number is proportional to q − for large- q . The energy of a particle ofmode- k at late times is ω f = q ( k/a f ) + m . The energy per mode in the large- q regime is then proportional to ( k/a f ) q − = q − H infl .With a gradual transition between segments of a ( t ), the large- q behavior in which | β q | falls off as q − starts around q ≃ 1. With an arbitrarily abrupt transition fromthe end of inflation to our final asymptotically flat scale factor, however, this transitioncan be prolonged to an arbitrarily high value of q , which we denote by q − off . Wefind empirically that q − off ≃ a f / ( a f − a f ). We define the region between 1 . q . q − off as the “extended” region. In the “extended” region the fall off of | β q | ∝ q − is extended from q ≃ q , such as the value of q ≃ shown in04Fig. 3.8, in which a f − a f ≪ a f as a ( t ) makes a rapid transition to flatness. Thisextension is caused by the production of particles of higher momenta by the rapidchange in H ( t ) after inflation. When the transition of a ( t ) is gradual, one finds, asin Fig. 3.9, that beyond q ≃ 1, the quantity | β q | falls off more rapidly, eventuallygoing as q − if the function a ( t ) is C . With sufficient extension, the particle numberper mode in the “extended” region is proportional to q − , regardless of the valueof P in the intermediary- q region, so for both the massless and massive cases thecontribution to the total energy density is dominated by these “extended” modes, andwe neglect both the red-shifted modes and the ultraviolet modes. When a ( t ) is C andthere exists a significant “extended” region, the contribution to the energy densityfrom values of q > q − off is negligible. The energy density associated with the“extended” region, which dominates the total energy density when a f − a f ≪ a f ,is h E i ≃ πa f ) Z a f H infl q − off a f H infl ka f | β q | d k = Z q − off q | β q | H π dq = Z q − off q H π dq . (3.145)When q − off ≫ 1, we find for the energy density h E i ≃ H (cid:16) a f a f − a f (cid:17) π . (3.146)Because q − off ≃ ( a f / [ a f − a f ]), we can see that an abrupt end to inflation canlead to energy densities large enough to produce reheating. For particle productionas the cause of reheating, see also [61].05 In units of ~ = c = k B = 1, the temperature is T = ( h E i /σ ) / , where σ is theStefan-Boltzmann constant. Then the energy density attributable to an abrupt endto inflation given by Eq. (3.146) leads to an effective temperature of T ≃ s a f / √ σa f − a f H infl √ π . (3.147)This approximation holds for any relatively abrupt transition and does not dependon any discontinuities of the scale factor.In an expansion governed by the asymptotically flat scale factor of Eq. (3.6) withno exponential middle segment, the large- k behavior— in both the massless case andthe effective- k approach— follows a thermal spectrum given by [4, 16, 65, 66] T = 14 πsa < a . (3.148)When a ≃ a , we use H max ≃ − a a a s , (3.149)to show that in our notation this is equivalent to T ≃ H infl π (1 − a f a f ) (3.150)for a single asymptotically flat scale factor with a ≃ a . In the large- q regime ofour composite scale factor with a f ≃ a f , we would expect to find the temperatureapproaching this same value, regardless of mass, of P , and of the number of e-folds;but only if we were able to maintain continuity with the previous segments of thescale factor across an infinite number of derivatives.With a gradual transition between segments of a ( t ), the large- q behavior in which | β q | falls off as q − starts around q ≃ 1. For such a gradual transition, we find a06late-time temperature— which is red-shifted after the end of inflation by the expan-sion of the final asymptotically flat segment of the scale factor— that is comparableto the Gibbons-Hawking temperature of H/ (2 π ) [76].It is tempting to imagine the temperature varying continuously from the Gibbons-Hawking temperature describing a de Sitter state— or from an approximate Gibbons-Hawking temperature associated with the approximate de Sitter state in our case—to the near Gibbons-Hawking temperature equivalent at late times in our asymptot-ically flat space, but this is perhaps unwarranted. At late times, the average numberof particles created per mode from an early-time vacuum is well defined. This is notnecessarily the case during inflation, when a choice must be made whether to make ameasurement rapidly or slowly. If the measurement were made quickly, then by thetime-energy uncertainty relationship, particles would be created through the act ofmeasurement; if the measurement were made slowly, then the size of the scale factorwould change appreciably during the measurement process, which could change theoutcome [1]. Just as an observer accelerating through a Minkowski vacuum mea-sures particles [16, 77, 78, 79, 80, 81], so would a temperature-measuring device beexcited in de Sitter space; however, unlike a thermal bath in flat spacetime, a mov-ing observer in de Sitter space would register no red-shifting in any direction. Infact, the authors of [82] find that for a massless, minimally-coupled scalar field inde Sitter space, no particles would be produced, and the associated effective tem-perature from these particles would be zero. With our composite scale factor, andusing our late-time evaluation method alone, it is difficult to say whether particles arepresent during the exponential expansion, or whether they are created by the chang-ing Hubble constant at the end of inflation. It is likely that during the expansion,the long-wavelength modes that have exited the Hubble radius correspond to real,low-energy particles, while the high-frequency modes that have not left the Hubbleradius correspond to virtual particles whose promotion to real particles depends uponthe future evolution of the universe— such as our matching conditions— but to say07conclusively whether particles exist during inflation would require a quantum fieldrenormalization in curved spacetime, such as the adiabatic method given by [38].By showing that the particle production of certain predicted particle species wouldcause conditions incompatible with observations in our universe, high-energy particlephysics may be able to constrain the amount of reheating. Because we have shownhow reheating— subject to ending conditions— is general to large- H infl inflationarymodels, this can similarly be used to place model-dependent constraints on predictionsfor new particles, such as theorized supersymmetric partners of observed particles,under particular values of H infl . In one such analysis [83], if the gravitino ˜ G is thelightest supersymmetric particle (LSP), then this constrains the maximum reheatingtemperature to be less than 10 GeV. If the ˜ G is not the LSP, and if its mass mightbe expected to be ∼ 100 GeV, then the maximum reheating temperature may still beless than or about 10 GeV [84]. Another example of a constraint on reheating is forthe particle creation of scalar moduli, which may be present in supersymmetry andstring theories: if the magnitude of the effective mass of the moduli field is less than H infl , then the upper limit on the reheating temperature could be as low as 100 GeV[85]. This constraining works both ways. If evidence were found for the existence ofsuch a reheating-constraining particle, this could eliminate those models of inflationthat predict a large, nearly constant value of H infl along with a rapid end to inflation.Those models that would be in agreement with such a low reheating temperaturewould be those with either a relatively small value of H infl , or those with a finalperiod of inflation at which the inflationary potential has reached a near-minimumvalue, but at which it remains the dominant influence on the evolution of the scalefactor, so that the initial high-energy particle production is greatly red-shifted andso that any unwanted relic particles are sufficiently attenuated such that they do notinterfere with later early-universe processes, such as Big Bang Nucleosynthesis.08 Part II:Binary System ofCompact Masses Chapter 4Unequal Mass Binary Solution in aPost-Minkowski Approximation In [8], Friedman and Ury¯u investigate a particular system of binary point massesthat acquires a helical symmetry by taking the half-advanced plus half-retarded fieldsfrom the linearized Einstein equation. This time-invariant system in the co-rotatingreference frame provides for an action at a distance theory, as has been previouslydiscussed by [86, 87, 88]. It allows for a single action integral that depends on thedynamical variables and trajectories of each particle, without requiring a descriptionof the force field acting on the particles. Such an action is called a Fokker action[8]. The Fokker action is not a true action, as the variation of the Fokker actionintegral depends on the boundary conditions and it involves integrals over each pointmass’s parameter time. When, however, a limit is taken after the variation of theFokker action, in which its endpoints are taken at times of −∞ and + ∞ , the variationyields the correct equations of motion. The conserved energy and angular momentumassociated with the Fokker action remain finite, even though energy and angularmomentum of the field are infinite due to radiation from the system occurring overan infinite amount of time.In the post-Minkowski (PM) approximation, the metric is assumed to be flat with10small perturbations of the form g ab = η ab + h ab , where to linear order h ab is the half-advanced plus half-retarded field of each particle. Unlike the post-Newtonian (PN)approximation, however, v/c ≪ c = G = 1. Friedman and Ury¯u note that in zeroth order PMapproximation T ab = ρu a u b = 0, and particles travel on flat space geodesics. A naivefirst order perturbation would then lead to δT ab = δρu a u a + ρδu a u b + ρu a δu b = δρu a u a ,which, because u a is the unperturbed straight-line motion, does not allow for boundorbits. In [8], this is avoided by considering a parameterized family of solutions to T ab ( s ) = ρ ( s ) u a ( s ) u b ( s ) + p ( s )[ g ab ( s ) + u a ( s ) u b ( s )] that corresponds to flat space for s = 0. In a radiation gauge, − G (1) ab ≡ (cid:3) ( h ab − η ab ) h = − πT (1) ab , (4.1)where h is the trace h aa , the first-order stress-tensor is constructed from the first-order u a , from the first-order ρ , and from the flat-space metric. In the binary solution, tofirst order the motion of each mass is given by the linear field of the other, and theself-force serves only to renormalize the mass as a self-energy. Furthermore, Friedmanand Ury¯u note of their post-Minkowski solution that it is correct to Newtonian order(0PN), the radiation field of the linearized metric is correct to 2.5PN, and a correctionterm to the equations of motion is necessary to have the orbits agree with the 1PNsolutions. For the case of the electromagnetic force, a specific example, in whichthe self force and radiation reaction are calculated, is given in greater detail in theAppendix A. For the case of gravity, instead of photons the radiation takes the formof gravitational waves. The measurement of the energy loss due to this radiation in aparticular binary system which contained a pulsar earned Hulse and Taylor a NobelPrize in 1993.Although in linearized gravity non-linear terms are dropped that are of the samePN-order as linear terms that are kept, which means the next highest PM-order will11have terms of equal magnitude to those used at linear PM-order, the post-Minkowskiapproximation may be helpful in evaluating solutions involving the full Einstein equa-tions in General Relativity that use helically symmetric initial data sets. Such initialconditions neglect the radial velocities associated with the radiation-reaction force,but a second-order post-Minkowski framework might lead to a better understandingof requirements for initial data in full-GR simulations.Fig. 4.1 shows the two point masses, m and ¯ m , with respective velocities v and m (cid:143) v (cid:143) v ma (cid:143) a Figure 4.1: Binary in Circular Motion.¯ v . The radial parameters can be expressed as a ≡ v/ Ω and ¯ a ≡ ¯ v/ Ω, where Ω is theangular velocity shared by both point masses. Accounting for relativistic velocities,12the radial parameter is not equal to the 1 / (2 π ) times the circumference observed inthe particle’s co-moving frame. The position vectors are x a = tt a + a̟ a and ¯ x a =¯ tt a + ¯ a̟ a . The trajectory of m is tangent to the helical Killing vector k a = t a + Ω a ˆ φ a ,and the trajectory of ¯ m is tangent to the helical Killing vector ¯ k a = ¯ t a + Ω¯ a ˆ φ a , where γ ≡ dt/dτ .In Fig. 4.2 the Law of Cosines relates t = a + ¯ a − a ¯ a cos( π − ϕ ), or ( ϕ/ Ω) = j mm (cid:143) a (cid:143)= v (cid:143)€€€€W v €€€€W a =j€€€€W t = Figure 4.2: Retarded Angle ϕ .( v/ Ω) + (¯ v/ Ω) + 2( v/ Ω)(¯ v/ Ω) cos( ϕ ), so that the retarded angle, which is equal inmagnitude to the angle associated with the advanced position, is given by the positiveroot of the transcendental equation ϕ = v + ¯ v + 2 v ¯ v cos ϕ .13From two types of Fokker action, a parametrization invariant action with a post-Newtonian correction and an affinely parametrized action, the equations of motionand expressions for conserved energy and angular momentum are derived following thevariational calculation of Ref. [90]. In the Affine case, we parameterize the trajectoriesusing the perturbed flat-space metric as ( η ab + h ab ) ˙ x a ˙ x b = − η ab +¯ h ab ) ˙¯ x a ˙¯ x b = − x ( τ ) and¯ x (¯ τ ). This leads to γ = (1 − v − h ab k a k b ) − / and ¯ γ = (1 − ¯ v − ¯ h ab ¯ k a ¯ k b ) − / . In theparameter-invariant case, we parameterize the trajectories using the flat-space metricas η ab ˙ x a ˙ x b = − η ab ˙¯ x a ˙¯ x b = − 1. This leads to γ = (1 − v ) − / and ¯ γ = (1 − ¯ v ) − / .The affine parameterization is characterized by the following: the parameter times ofgeodesics are the proper times of the perturbed metric; the PM-form of the geodesicequation ( η ab + h ab )¨ x b + C abc ˙ x b ˙ x c applies, where C abc ≡ (1 / ∇ b h ac + ∇ c h ba − ∇ a h bc );and, finally, the 4-velocity is orthogonal to the 4-acceleration, or U a ∇ a U b = 0, thatis the particles travel along geodesics. The linear post-Minkowski approximation isnot at this point accurate to 1PN order, but Friedman and Ury¯u give two differentadjustments to the parametrization-invariant case: the simplest correction consistentwith 1PN (called PN where confusion will not arise) and a correction that is bothparameterization-invariant and special-relativistically covariant (SPN), where resultsare given in [8] for the deDonder gauge. They show also that for both of the Fokkeractions the form of the first law of thermodynamics dE = Ω dL holds, and this lawcan be used to check for the presence of an Innermost Stable Circular Orbit (ISCO).We find a solution describing a helically symmetric circular orbit in the post-Minkowski approximation (with post-Newtonian corrections) that is analogous to thecircular solution of two charges obtained by Schild for the electromagnetic interaction[91]. In [6] we report results supplementing those of [8]: numerically computed solu-tion sequences for unequal mass particles, and analytic formulas in the extreme massratio limit. The latter results agree with the first post-Newtonian (1PN) formulas;hence a consistency of our model is confirmed in this limit.14We present a set of formulas governing the helically symmetric circular orbits oftwo point particles, { m, v } and { ¯ m, ¯ v } , and derive analytic expressions in the extrememass ratio limit q ≡ m/ ¯ m → 0. The set of algebraic equations is solved numericallyfor a fixed binary separation to specify each circular orbit. The result for the unequalmass binary orbit is presented in Sec. 4.1.We compute the solution to the equation of motion numerically for three massratios: q=1.0, q=0.1, and q=0.001. We solve the equation of motion for each massratio in the PM+PN model, the PM+SPN model, and the affine model. We alsocalculate the solution for the q → We discuss solutions to the post-Minkowski approximation in the case of parametrization-invariant plus 1PN correction terms, and then we discuss solutions in the affine case.For the analytical solution in the q → Parameter Invariant Circular Solution We first list the result from [8] for the parametrization invariant model with 1PNcorrection terms. After integration, the equations of motion for particles m and ¯ m are written in terms of the velocities, v and ¯ v , of particles m and ¯ m , which are relatedto the orbital radius by a ≡ v/ Ω and ¯ a ≡ ¯ v/ Ω, through the equations: − mγ v Ω = − m ¯ mγ ¯ γ Ω [ F ( ϕ, v, ¯ v ) + ( m + ¯ m )Ω F I ( ϕ, v, ¯ v, γ, ¯ γ ) ] , (4.2) − ¯ m ¯ γ ¯ v Ω = − m ¯ mγ ¯ γ Ω [ ¯ F ( ϕ, v, ¯ v ) + ( m + ¯ m )Ω ¯ F I ( ϕ, v, ¯ v, γ, ¯ γ ) ] . (4.3)15As shown below, { ϕ, v, ¯ v, γ, ¯ γ } are not independent. The functions F ( ϕ, ¯ v, v ) =¯ F ( ϕ, v, ¯ v ) are the post-Minkowski terms, while F I ( ϕ, ¯ v, v, ¯ γ, γ ) = ¯ F I ( ϕ, v, ¯ v, γ, ¯ γ )is either of two alternative 1PN correction terms that agree at 1PN order: F I = F PN ( ϕ, v, ¯ v, γ, ¯ γ ) derived from a non-relativistic correction, or F I = F SPN ( ϕ, v, ¯ v, γ, ¯ γ )derived from a special relativistically invariant correction. These are F ( ϕ, v, ¯ v ) ≡ − ϕ + v ¯ v sin ϕ ) n (1 + v ¯ v cos ϕ )¯ v × ( ϕ cos ϕ − v sin ϕ ) + 12 v (1 − ¯ v )( ϕ + v ¯ v sin ϕ ) − 12 [¯ v sin ϕ ( ϕ + v ¯ v sin ϕ ) + (1 + v ¯ v cos ϕ )( v + ¯ v cos ϕ ) − v − v ( ϕ + v ¯ v sin ϕ ) ]Φ( ϕ, v, ¯ v ) o , (4.4) F PN ( ϕ, v, ¯ v, γ, ¯ γ ) ≡ − γ ¯ γ ( v + ¯ v ) (cid:20) γ v ( v + ¯ v ) (cid:21) , (4.5) F SPN ( ϕ, v, ¯ v, γ, ¯ γ ) ≡ − γ ¯ γ ) / ϕ + v ¯ v sin ϕ ) n γ v + ¯ v sin ϕϕ + v ¯ v sin ϕ + (1 + v ¯ v cos ϕ ) ( v + ¯ v cos ϕ )( ϕ + v ¯ v sin ϕ ) o . (4.6)The function Φ( ϕ, v, ¯ v ) is defined byΦ( ϕ, v, ¯ v ) ≡ (1 + v ¯ v cos ϕ ) − (1 − v )(1 − ¯ v ) ϕ + v ¯ v sin ϕ . (4.7)For the parametrization invariant models, γ and ¯ γ are derived from a flat-spacenormalization of the four-velocity, γ = (1 − v ) − , ¯ γ = (1 − ¯ v ) − . (4.8)The retarded angle ϕ is the positive root of ϕ = v + ¯ v + 2 v ¯ v cos ϕ .In Fig. 4.3 the angular velocity, in dimensionless form Ω M , is plotted againstthe velocity of the lighter particle for 3 mass ratios and the q → -6 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 Ω M V q=1.0 q=0.1 q=0.001q=0.0 Figure 4.3: Parametrization-Invariant (PN) Omega versus Velocity.17for q → q = 0 . 001 overlap each other in the plot. The inflectiondisplayed in the logarithmic plot changes near the cutoff velocity for the small massratio cases. In Fig. 4.4 the angular velocity, in dimensionless form Ω M , is plotted -6 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 Ω M V q=1.0 q=0.1 q=0.001q=0.0 Figure 4.4: Parametrization-Invariant (SPN) Omega versus Velocity.against the velocity of the lighter particle for 3 mass ratios and the q → q → q = 0 . 001 overlap each other in the plot. The inflectiondisplayed in the logarithmic plot changes near the cutoff velocity for the small massratio cases.In this notation Kepler’s Law, ( T period ) = 4 π a / ¯ m , may be written as ( ¯ m Ω) =( ¯ m/a ) [6]. In this form it may be compared with Eq. (4.2), when written as,(Ω ¯ m ) = (cid:16) ¯ ma (cid:17) (cid:8) v ¯ γ [ F + ( m + ¯ m )Ω F I ] (cid:9) . (4.9)18To see how this post-Minkowski approximation is related to Newtonian gravity in thenon-relativistic ( v << 1) limit, see Sec. 4.2. Parameter Invariant Energy and Angular Momentum In Figs. 4.5 (PN) and 4.6 (SPN), the unit energy of the lighter particle, in dimen- ( E - M b a r) / m Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.5: Parametrization-Invariant (PN) Energy versus Omega.sionless form ˆ E/m , where ˆ E = E − ¯ m , is plotted against Ω M . In the limit that theparticle approaches becoming unbound, v → 0, the unit energy of the lighter massapproaches 1, its rest mass energy. As it becomes more tightly bound, its energydecreases below the rest mass energy it would have in flat space.The conserved energy and angular momentum for the parametrization invariantmodel are written E = E PM + e I , and L = L PM + ℓ I , (4.10)19 ( E - M b a r) / m Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.6: Parametrization-Invariant (SPN) Energy versus Omega.20where E PM and L PM are the post-Minkowski terms E PM = mγ + ¯ m ¯ γ (4.11) L PM = 2 m ¯ mγ ¯ γ Φ( ϕ, v, ¯ v ) , (4.12)and e I and ℓ I are the parametrization invariant 1PN corrections e I = e PN and ℓ I = ℓ PN ,or those of the special relativistically invariant model e I = e SPN and ℓ I = ℓ SPN givenby e PN = 12 Ω ℓ PN , (4.13) e SPN = 12 Ω ℓ SPN , (4.14) ℓ PN = − m ¯ m ( m + ¯ m )Ω γ ¯ γ ( v + ¯ v ) , (4.15) ℓ SPN = − m ¯ m ( m + ¯ m )Ω( γ ¯ γ ) / ϕ + v ¯ v sin ϕ ) . (4.16)In Figs. 4.7 (PN) and 4.8 (SPN); angular momentum, in dimensionless form J/ ( mM ), where M is the total mass of both particles and m is the mass of the lighterparticle having velocity, v; is plotted against Ω M for 3 mass ratios and the q → v for each mass ratio, beyondwhich there are no further solutions, there is an Innermost Circular Orbit (ICO).One possible explanation for the termination of solutions can be found by looking atEq. (4.9), which may be written as a quadratic equation in terms of ( ¯ m Ω). Beyondthe maximum value of v , the solutions for ( ¯ m Ω) become imaginary.21 0 0.05 0.1 0.15 J / ( m M ) Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.7: Parametrization-Invariant (PN) Angular Momentum versus Omega.22 0 0.05 0.1 0.15 J / ( m M ) Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.8: Parametrization-Invariant (SPN) Angular Momentum versus Omega.23 Affine Circular Solution In Fig. 4.9 The angular velocity, in dimensionless form Ω M , where M ≡ ( m + ¯ m ), -6 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 0.5 Ω M V q=1.0 q=0.1 q=0.001q=0.0 Figure 4.9: Full Affine Case of Omega versus Velocity.is plotted against the velocity of the lighter particle for 4 mass ratios in the affinemodel. The behavior for solutions existing beyond v isco , the velocity at which theminimum energy and angular momentum occur, is most prominently displayed forthe q=1.0 case. In the q = 1 case the ISCO occurs at v ∼ − mγ v Ω = − m ¯ mγ ¯ γ Ω F A ( ϕ, v, ¯ v ) , (4.17) − ¯ m ¯ γ ¯ v Ω = − m ¯ mγ ¯ γ Ω ¯ F A ( ϕ, v, ¯ v ) , (4.18)24 -6 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 Ω M V q=1.0 q=0.1 q=0.001q=0.0 Figure 4.10: Truncated Affine Case of Omega versus Velocity.25where the function F A ( ϕ, ¯ v, v ) = ¯ F A ( ϕ, v, ¯ v ) is defined as F A ( ϕ, v, ¯ v ) ≡ − ϕ + v ¯ v sin ϕ ) n (1 + v ¯ v cos ϕ )¯ v ( ϕ cos ϕ − v sin ϕ )+ 12 v (1 − ¯ v )( ϕ + v ¯ v sin ϕ ) − 12 [¯ v sin ϕ ( ϕ + v ¯ v sin ϕ )+(1 + v ¯ v cos ϕ )( v + ¯ v cos ϕ )]Φ( ϕ, v, ¯ v ) o . (4.19)For the affinely parametrized world line, γ and ¯ γ satisfy − γ (1 − v ) + 4 ¯ mγ ¯ γ Ω Φ( ϕ, v, ¯ v ) = − , (4.20) − ¯ γ (1 − ¯ v ) + 4 mγ ¯ γ Ω Φ( ϕ, v, ¯ v ) = − . (4.21) Affine Energy and Angular Momentum In Fig. 4.11 the lighter particle’s unit energy per mass, in dimensionless form ˆ E/m , ( E - M b a r) / m Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.11: Full Affine Case of Energy versus Omega.26where ˆ E = E − ¯ m , is plotted against Ω M for the affinely-parameterized case. InFig. 4.12 angular momentum, in dimensionless form J/ ( mM ), where M is the total 0 0.05 0.1 0.15 J / ( m M ) Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.12: Full Affine Case of Angular Momentum versus Omega.mass of both particles and m is the mass of the lighter particle having velocity, v ; isplotted against Ω M for 3 mass ratios and the q → E = mγ + ¯ m ¯ γ + 4 m ¯ mγ ¯ γ Ω Φ( ϕ, v, ¯ v ) , = mγ (1 − v ) + ¯ m ¯ γ (1 − ¯ v ) − m ¯ mγ ¯ γ Ω Φ( ϕ, v, ¯ v ) , (4.22) L = 2 m ¯ mγ ¯ γ Φ( ϕ, v, ¯ v ) , = 2 m ¯ mγ ¯ γ (1 + v ¯ v cos ϕ ) − (1 − v )(1 − ¯ v ) ϕ + v ¯ v sin ϕ , (4.23)27where the form of Φ( ϕ, v, ¯ v ) is the same as that of the parametrization invariant model(4.7). Using Eq.(4.20) and (4.21), the energy can be rewritten E = 12 mγ + 12 mγ (1 − v ) + 12 ¯ m ¯ γ + 12 ¯ m ¯ γ (1 − ¯ v ) . (4.24)This can be compared with Eq. (4.11), noting the different definitions of γ in theparametrization-invariant and affine models.Figs. 4.13 and 4.14 show the same data as Figs. 4.11 and 4.12, respectively, except ( E - M b a r) / m Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.13: Truncated Affine Case of Energy versus Omega.that only the solutions where 0 ≤ v ≤ v isco are plotted.In the affine case, for any mass ratio q ∈ [0 , 1] we find a simultaneous minima inthe energy and angular momentum which corresponds to the ISCO. The values of thenormalized angular velocity, angular momentum, and energy that occur at the ISCOin the affine model vary monotonically from q=1 to 1=0. With q ranging from 1 to28 0 0.01 0.02 0.03 0.04 0.05 0.06 J / ( m M ) Ω M q=1.0 q=0.1 q=0.001q=0.0 Figure 4.14: Truncated Affine Case of Angular Momentum versus Omega.290, Ω M decreases from ≈ . ≈ . L/ ( mM ) increases from ≈ . ≈ . E/m decreases from ≈ . ≈ . Numerical Solutions for Unequal Mass Circular Orbit A circular solution is calculated from algebraic equations given in Eqs. (4.2) and (4.3)for the parametrization invariant model or Eqs. (4.17) and (4.18) for the afinelyparametrized model. One method of solving for a fixed ratio q = m/ ¯ m is— (1)assume a ratio of velocities v/ ¯ v and determine the corresponding mass ratio from theequations of motion, then (2) change the velocity ratio to adjust the value of the massratio to a fixed value (using the bisection method, for example). The mass ratio, q , canbe determined by multiplying both sides of either Eqs. (4.2) and (4.3) or Eqs. (4.17)and (4.18) by ¯ v ¯ γ / ( vγ ) and then dividing both sides of the first equation listed ineither of these pairs of equations with the second equation to yield an expresion forthe mass ratio q .Another method involves solving the relationship M Ω[ q, v, ¯ v ]= M Ω[(1 /q ) , ¯ v, v ] byvarying the parameters v and ¯ v . For a description of an efficient means of samplinga parameter space and fine-tuning the optimal result, see Sec. 5.2.1. For the Extreme Mass Ratio, the mass of the lighter particle is negligible relativeto that of the more massive particle. This would be appropriate for a test massorbiting in the spherically symmetric gravitation field of a much more massive ob-ject. Whereas the signal from a merger of identical black holes, each with a masson the order of one solar mass, would fall within the sensitivity of LIGO’s (Laser In-terferometer Gravity-wave Observatory) frequency band; the inspiral of a solar-massblack hole into a billion-solar-mass black hole, such as those predicted to be at thecenters of many galaxies, would fall into the most sensitive frequency band of LISA30(Laser Interferometer Space Antenna) [92]. The Extreme Mass Ratio is an excellentapproximation to this latter scenario of a mass ratio on the order of 10 − . Extreme Mass Ratio Limit The extreme mass ratio limit q ≡ m/ ¯ m → v → v → 0, we may assume that v and ¯ m remain finite. Consequently,we have ¯ γ → ϕ → v , and ¯ m → M , where M ≡ m + ¯ m is the total mass. With v and Ω regarded as independent variables, Eq. (4.2) is a quadratic equation for Ω M ,whose q = 0 form is F I (Ω M ) + F (Ω M ) − v = 0 , (4.25)with physical solution Ω M = 12 F I (cid:16) − F + p F + 4 F I v (cid:17) . (4.26)The functions F (the post-Minkowski term), F I = F PN (the simplest 1PN correction),and F I = F SPN (the special-relativistically covariant 1PN correction) for q = 0 become F ( ϕ, v, ¯ v ) = 1 − v v (1 − v ) , (4.27) F PN ( ϕ, v, ¯ v, γ, ¯ γ ) = − v (cid:18) − v (cid:19) , (4.28) F SPN ( ϕ, v, ¯ v, γ, ¯ γ ) = − (1 − v ) / v (cid:18) − v (cid:19) , (4.29)where Φ has the form Φ( ϕ, v, ¯ v ) = 1 + v v . (4.30)Without the 1PN correction, the parametrization-invariant post-Minkowski model isgiven by setting F I = 0, and therefore Ω M = v/F . In the q → M = v (1 − v )1 − v . (4.31)31 Parameter Invariant Solution Sequence in q → Limit In the q → b E PM ≡ E PM − ¯ m , and taking the limit ¯ v → m → M , wehave b E PM m = (1 − v ) / , (4.32) L PM mM = 1 + v v (1 − v ) / , (4.33) e PN m = 12 ℓ PN mM Ω M, (4.34) e SPN m = 12 ℓ SPN mM Ω M, (4.35) ℓ PN mM = − (1 − v ) / v Ω M, (4.36) ℓ SPN mM = − (1 − v ) / v Ω M. (4.37)In [8], it is shown that the first law of thermodynamics that relates the changes inthe conserved energy and the angular momentum, dE = Ω dL , is satisfied by binarysolutions derived from the parametrization invariant Fokker action. This relationis used to cross check both the analytic formula in the q → d b E/dv = Ω dL/dv , where b E ≡ b E PM + e I .In the parametrization invariant post-Minkowski model without a 1PN correc-tion, the normalized angular velocity, Ω M , is defined in an interval 0 ≤ v < / √ q → 0, and Ω M becomes infinite at v = 1 / √ 3. With the 1PN correction F I = F PN ,the range of finite Ω M is approximately 0 ≤ v . . F I = F SPN , it is 0 ≤ v . . M = 0 . q = 0 ISCO are Ω M = 0 . 124 and0 . q → M .In Eq. (4.26), an expansion of Ω M in the small v limit becomes Ω M = v + 3 v + O ( v ) for both PN and SPN models, and this is inverted to write v in terms of smallΩ M as v = (Ω M ) / − Ω M + O ((Ω M ) / ) . (4.38)Substituting this into the energy and angular momentum formulas, the leading twoterms agree with the post-Newtonian formulas (see e.g. [93]) up to the 1PN order forthe extreme mass ratio q → b Em = 1 − 12 (Ω M ) / + 38 (Ω M ) / + O ((Ω M ) ) , (4.39) LmM = 1(Ω M ) / (cid:20) M ) / + O ((Ω M ) / ) (cid:21) . (4.40) Affine Solution Sequence in q → Limit Eq.(4.17) implies Ω ¯ m = v ¯ γ − F A ( ϕ, v, ¯ v ) − , (4.41)where ¯ γ is evaluated from Eqs. (4.20) and (4.21).33In the limit of q → v → F A ( ϕ, v, ¯ v ) = 1 − v v . (4.42)From Eq. (4.17) and (4.20), we have γ = (cid:18) − v − v − v (cid:19) / , (4.43)while in Eq. (4.21), taking ¯ v → m → γ → 1. As a result we have inthe extreme mass ratio, Ω M = v − v . (4.44)In the q → b E ≡ E − ¯ m , and the angular momentum become b Em = (1 − v )[(1 − v )(1 − v − v )] / , (4.45) LmM = 1 + v v (cid:18) − v − v − v (cid:19) / . (4.46)The first law δE = Ω δL is also satisfied for the affinely parametrized model, andhence one can cross check formulas in the q → d b E/dv =Ω dL/dv . Although the lighter particle’s normalized angular velocity, Ω M , is finitein an interval v ∈ [0 , γ as well as conserved quantities E and L become infinite at v = p √ − ≈ . M = (cid:0) √ − (cid:1) / / (3 − √ ≈ . . In this interval, v ∈ [0 , p √ − v = p (1 + 2 / − / ) / ≈ . , which corresponds toΩ M = (cid:0) / − / (cid:1) / √ − / + 2 / ) ≈ . . (4.47)34The Schwarzschild ISCO occurs at Ω M = 6 ( − / √ ≃ . Radial Parameter in q → Limit With the definition a = v/ Ω, we can write a/M = v/ ( M Ω), where M = m + ¯ m in the q → M = ¯ m . Then we insert into Eqs. (4.31) (0PN parametrization-invariant without 1PN correction term), (4.26) (PN and SPN cases), or (4.44) (0PNaffine case) the maximum velocity (parametrization-invariant) or the ISCO velocity(affine). These cutoff velocities are— 1 / √ . . . M − , these minimum radial parameters are as follows: 0 (PM), 2 . 67 (PN andSPN), and 3 . 24 (Affine). Note that the ISCO of the affine case occurs at v = 0 . a/M ≃ . M that is predicted by the full theory of general relativity. In the caseof the PM+1PN correction term model, and in the affine case without a correction,the minimum radial parameter occurs on the order of the 2 M event horizon for aSchwarzschild black hole. For comparison with our post-Minkowski analysis, we list the 1PN equations of [93],where Blanchet’s ν is our q (1 + q ) − . In our notation, EM = − q ( q + 1) ( M Ω) / (cid:20) − (cid:18) 34 + 112 q (1 + q ) (cid:19) ( M Ω) / (cid:21) , (4.48)35or, ˆ Em = 1 − 12 1 q + 1 ( M Ω) / (cid:20) − (cid:18) 34 + 112 q (1 + q ) (cid:19) ( M Ω) / (cid:21) ; (4.49)and LM = q (1 + q ) M Ω) / (cid:20) (cid:18) 32 + 16 q (1 + q ) (cid:19) ( M Ω) / (cid:21) , (4.50)or LmM = 11 + q M Ω) / (cid:20) (cid:18) 32 + 16 q (1 + q ) (cid:19) ( M Ω) / (cid:21) . (4.51)Thus we show explicitly in the limit q → ϕ to the next or-der in the velocities, v and ¯ v , as ϕ ≈ ( v + ¯ v )(1 − v ¯ v/ q = 0 case.36 Part III:Production and Decayof Small Black Holesat the TeV Scale Chapter 5TeV-Scale Black Hole Productionat the South Pole We discuss the possibility of observing TeV-scale black holes produced at theIceCube Neutrino Telescope [7]. After giving a brief summary of the IceCube exper-iment, we explain what TeV-scale black holes are. We then examine a gravitationalinteraction between a neutrino and a nucleon. Because a nucleon is not a pointparticle, we rely on the parton model, which describes the nucleon as a collectionof quarks and gluons. Following this, we describe our method for modeling PartonDistribution Functions (PDFs), we evaluate the cross section for the interaction ofneutrino+nucleon → black hole, and then we calculate IceCube’s detection sensitivityfor observing TeV-scale black holes. In the Standard Model a neutrino can interact with a nucleon through both chargecurrent (CC) interactions and neutral current (NC) interactions [94, 95, 96]. In aCC interaction, a neutrino (anti-neutrino) interacts with a quark to become a lepton38(anti-lepton), conserving electron-, muon-, and tau-lepton number. In this interactiona W + ( W − ) particle is exchanged with a down-quark (up-quark), which becomes anup-quark (down-quark). In a NC interaction, a neutrino exchanges a Z with a quarkand neither the neutrino nor the quark changes flavor.Because neutrinos experience only gravity and the weak force, they may travel as-tronomical distances without interactions. Thus, they preserve information about theenvironment in which they were produced. The corollary to this is that a sufficientlylarge detector must be used to observe these cosmic neutrinos here on Earth.The IceCube Neutrino Telescope is composed of approximately one cubic kilometerof Antarctic ice ranging from 1400 meters in depth to 2400 meters in depth below thesurface near the Amundsen-Scott Station located at the geographic South pole [97].IceCube is already taking data, and it is scheduled to be fully operational by 2009-2010. At that time, it will consist of 80 strings, each a kilometer long, of 60 evenlyspaced PhotoMultiplier Tubes (PMT) each, for a total of 4800 PMT. The stringsare 125 meters apart, and each interior string will be surrounded by six equidistantneighbors [98].When high energy charged particles move faster than the local speed of lightthrough the ultraclear Antarctic ice, in which the absorption length of the relevantwavelengths is greater than 100 meters [99], they emit Cherenkov radiation. Thisradiation, within a range that includes visible light and some UV light, can be detectedby the PMT used in IceCube, and the time at which this happens— including the timefor the signal to register— can be recorded within an accuracy of a few nanoseconds[100]. The paths of these charged particles may be dominated by jets from a highenergy muon or tau. They may also be diffused throughout a shower. With sufficientdata, these paths can be used to reconstruct the particle interactions that have takenplace. This requires the energy of the incident neutrino to be greater than 100 GeV.In the case of a series of interactions caused by a single incident particle, the totalenergy— provided that it is contained within the volume of IceCube and is less than3910 GeV so that it does not saturate the detector— can be measured [99].When measuring neutrino interactions, one must contend with a background eventrate of charged particles, such as muons produced by cosmic rays hitting the atmo-sphere [101]. Examining upward going tracks, or particles that have passed througha significant fraction of the Earth, effectively restricts the progenitor particle of aninteraction to a neutrino, which, because it is only weakly interacting, is able to eas-ily penetrate the Earth, whereas charged particles are not. A horizontally travelingneutrino passing through the center of IceCube travels through 150 kilometers of theEarth [102]. There is also a background trigger rate for IceCube’s PMT of less thanone kilohertz [97, 99]. That is, in the absence of a signal a PMT will discharge on av-erage no more than once every millisecond. This is not a problem, because the transittime across IceCube for those particles that produce Cherenkov radiation is on theorder of a few microseconds and the PMT recording time is accurate to within a fewnanoseconds. Although the volume of IceCube is one cubic kilometer, the effectivevolume for detecting neutrinos is larger, because muons may be produced outside theIceCube volume and still travel inside to be measured [103].Another useful veto is the IceTop surface array of 160 Cherenkov detectors of 2.7meter diameter tanks of ice spread out over one square kilometer of area [104]. IceTophelps reject background events and is also useful for calibration.Amanda, the prototype of IceCube that proved the viability of detecting neutrinosin polar ice caps, is still running. Because its volume overlaps with the volume ofIceCube, it can either contribute to IceCube’s sensitivity, or it can serve as a checkon IceCube detections, depending upon whether data from the two experiments isexamined collectively or independently [100].40 In the standard model (SM), gravity is by far the weakest of the four fundamentalforces. It has been theorized that this weakness is due to the presence of extradimensions beyond the 4 familiar dimensions of our spacetime [105]. If gravitonspropagated into the extra dimensions while SM fields were confined to our brane of3+1 dimensions, then gravity thus diluted would appear much weaker than the otherforces. In this case, gravity might become much stronger at small distances than a4-dimensional theory would predict.We will investigate the possibility that the distance at which gravity and theelectromagnetic force have the same strength is at ∼ − m, the distance at whichthe electromagnetic and weak forces unify as the electro-weak force. This wouldmean that for the small distances at which gravity matched the electro-weak force instrength, there would be a fundamental D -dimensional Planck mass of about 1 TeV,in which case our 4-dimensional Planck mass would just be an effective Planck massover macroscopic dimensions.The strength of TeV-scale gravity at small distances could potentially make iteasier for interacting particles to form black holes. This can be qualitatively un-derstood via Gauss’s Law [106, 107]. The surface area of a sphere in D dimen-sions, where there is 1 time dimension and D − r D − . The magnitude of a D -dimensional Newtonian gravitational forceacting between two masses would be proportional to M m G D , where G D is the D -dimensional gravitational constant. Spread evenly over the surface area of asphere, this force would be proportional to M m G D r − D . The value of the po-tential energy at a separation r between the two masses would be proportional to M m G D r − D . Taking M as the primary mass and m as a test mass, then using anon-relativistic argument to relate the maximum kinetic energy of a test mass movingnear the speed of light ( mc / 2) to the potential energy, places an event horizon at r ∝ ( M G D /c ) (1 / [ D − . The dimensionality of G D is length D − mass − time − . The41 D -dimensional Planck mass, M D , is then proportional to ( ~ D − c − D G D − ) (1 / [ D − .In units of ~ = c = 1, then G D ∝ M − DD , and the event horizon would be r ∝ ( M G D ) (1 / [ D − ∝ ( M M − DD ) (1 / [ D − ∝ (1 /M D )( M /M D ) (1 / [ D − .It has been suggested that the Large Hadron Collider (LHC) could easily producesuch black holes in this scenario [108, 109, 110]. If the LHC would be powerful enoughto detect this sort of black hole interaction, then cosmic rays would also be energeticenough to produce this interaction. In particular, we will discuss the possibility thatneutrinos produce black holes in the ice of the south pole and can be detected by theIceCube Neutrino Telescope.To model the gravitational interaction between a neutrino, which is a point par-ticle, and a nucleon, which is an object of finite extent and which has an internalstructure attributable to constituent point particles, we turn to Parton DistributionFunctions. In high energy interactions between a neutrino and a nucleon, the neutrino interactsprimarily with a single parton, a quark or a gluon. For these collisions, the protonand neutron are not just an up-up-down and an up-down-down, but are composed ofthese and other, virtual particles that are continually created and annihilated throughthe time-energy uncertainty relationship.Similarly, for low energy interactions, a nucleon acts as a single particle of restmass energy m N in its rest frame. For high energy interactions between a neutrinoand a parton, in the nucleon’s rest frame the parton will have have some fractionof the total energy rest-mass of the nucleon. This fraction is denoted as x , where x ranges from 0 to 1, or from none of the nucleon’s total energy to all of it [111].A Parton Distribution Function (PDF) describes the probability of finding a givenparton— up ( u ), anti-up (¯ u ), down ( d ), anti-down ( ¯ d ), strange ( s ), anti-strange (¯ s ),charm ( c ), anti-charm (¯ c ), bottom ( b ), anti-bottom (¯ b ), or gluon ( g )— with a fraction42 x of the total rest energy of m N . The contributions from the PDFs for the super-massive top and anti-top within the nucleons at rest are negligible, and we neglectthem. Thus, the probability that an i th species of parton exists with a fractionalenergy between x and x is P = Z x x f i ( x, Q ) dx. (5.1)The variable Q is the momentum transfer, where we choose Q ≃ r − s [112], and thePDFs are somewhat insensitive to changing Q [113]. We thus use Q = min { r − s , 10 TeV } . (5.2)The PDFs cannot be calculated analytically from first principles in the StandardModel. They must be fitted to experimental data. We use the CTEQ6D PDFs [114].The largest uncertainty in the PDFs exists for large- x gluons, where f ( x, Q ) gluon maybe off by more than a factor of 2 [115]. At small x , where the PDFs are muchmore certain, the gluon quickly comes to dominate the neutrino-parton interactionsthrough its high probability of being available for a collision.Fig. 5.1 plots log( x ) versus log( xf i ( x, Q )) for a representative quark, the up, andfor a gluon, both of which for the relatively low Q of 10 GeV, or Q = 100 (GeV) [116]. Fig. 5.2 plots log( x ) versus log( xf i ( x, Q )) for a representative quark, the up,and for a gluon, both of which for the relatively high Q of 10 TeV, or Q = 100,000,000(GeV) [116]. The variable Q changes by six orders of magnitude between these twocases, but the PDFs shown only change by about an order of magnitude. For x lessthan about 10 − , the graphs of the PDFs are nearly linear in these log-log plots. Forthis reason we use different models of these PDFs for small x and large x . We alsouse a different modeling of PDFs for the ranges Q > 10 TeV, 10 TeV > Q > > Q > 100 GeV, 100 GeV > Q > 10 GeV, 10 GeV > Q > > Q . These different regimes of PDFs lead to the almost imperceptible bulge43Figure 5.1: Parton Distribution Functions: Lower Momentum Transfer.44Figure 5.2: Parton Distribution Function: Higher Momentum Transfer.45between E ν = 10 GeV and E ν = 10 GeV in Fig. 5.5.For these different regions of x and Q , we make use of simple approximations tothe PDFs by fitting the CTEQ6D data to a form of f i ( x, Q ) = Ax n , (5.3)where, for example, in the small- x and large- Q regime, n ∼ − . A ∼ . A ∼ . Ax n is a good approximation that is simple to usewhen we integrate the cross section for a black hole interaction.What follows is a brief description of our numerical method. To accurately fit boththe variables A and n , we refine our best guess and also sample the two dimensionalparameter space. At a fixed value of Q and given a two dimensional array relating f i ( x ) to x , we start with a reasonable guess for the variables A and n and a reasonablevalue for our step variable. At each iteration, we compare our previous lowest resultfor the sum of the squares of the difference between the given data points and Ax n forall the points in the array with new values of the variables A and n . We try alteringour current best values of A and n by increasing or decreasing one or the other orboth in tandem or opposition for a total of 8 different combinations. If one of thesecombinations results in a better fit, then we store these new values of A and n as ournew current best values, and we retain the new sum of the squares of the differencebetween the given data points and the new Ax n as the new best target, and then werepeat the eight combinations. If we do not find a better fit, then we decrease the sizeof our step variable and repeat the above algorithm. If we reach a sufficiently smallstep variable, we do not yet give up: there are local quasi-minima in the parameterspace that are not good fits, such as A = 0 and n ≪ − 1. We instead pick a newvalue of our step variable that is large enough to jump to unexplored, and potentiallyrewarding, areas of the parameter space. To prevent getting stuck with the same46poor choice iteration after iteration, we choose a random number between 0 and areasonable maximum for our new step variable. At this point it is better to choosea step variable that is too big, or big enough to jump to a new region of parameterspace, than too small to be effective. The subsequent shrinking of the step variablewill take care of any initial excess. After a set number of loops of this entire processwhere the best fit remains unchanged, we take the resulting best values of A and x as our best fit.A simpler method than the one just described involves simultaneously changingboth variables (or more, if a problem requires sampling a higher dimensional param-eter space) with different random steps weighted towards zero. This method wasnot used for the project described here, but I have used it elsewhere with success.Each parameter is changed by a different step value, and each step value involvestwo layers of randomness. The first layer determines the order of magnitude of thestep, with a sizable probability it will be insignificantly small or zero, and the secondlayer determines the coefficient and sign associated with the order of magnitude. Thissimpler method effectively encapsulates the whole of the important parts of the abovemethod in very few lines of code and is faster at searching for and homing in on thebest solution.To check our results, we plot our best Ax n against the CTEQ6D PDF data toensure our data is a good fit. For an appropriately nearby value of Q , the benefit tousing a simple function over an array of data is that we can calculate f i ( x ) for anygiven x, and we avoid both having to maintain in our program memory a complicatedseries of PDF arrays and having to interpolate between data points in these arrays.Our fits are an excellent approximation to the PDFs being used and are certainly wellwithin the uncertainty of the PDFs, themselves.For examples of these PDF fits for the u -quark, see Figs. 5.3 and 5.4. The first ofthese shows why the PDF fits are broken up across the range of the parton momentumfraction, x , and the resulting excellent fit. The second of these figures shows how the47 10 20 30 40 50 -6 -5.5 -5 x f( x , Q ) Log x CTEQ6D up up Fit Figure 5.3: An Accurate Fit for a Small Range of Data.48 x f( x , Q ) Log x CTEQ6D up up Fit Figure 5.4: A Reasonable Fit for a Large Range Data.49PDF fit can drift from the exact PDF when overextending the fit over too large arange of x . This may also result from overextending a particular fit over too widea range of Q . In both of these figures a value of Q = 10 (GeV) is used for theCTEQ6D PDFs. Our fit in Fig. 5.3 is given by Eq. (5.3) with A = 0 . n = − . A = − . n = − . In its simplest form, calculating the cross section for the interaction of neutrino +nucleon → Black Hole involves using the Thorne Hoop Conjecture [117] and checkingto see if the neutrino and parton come close enough together to be within the radiusof the Schwarzschild black hole that would be formed from their combined center-of-mass energy. At this stage of our simple approximation of checking to see if the impactparameter, b , is smaller than the Schwarzschild radius, r s , for our cross section, wewould simply have the area πr s of a disk.To find the center-of-mass energy, E CM , we use the conservation of relativistic4-momentum, P total a = P νa + P parton a . We define our metric as η ab = diag( − , , , P ν a = ( E ν , p ν,x , p ν,y , p ν,z ) and P parton a = ( xm N , , , x is the fraction of the total rest-mass energy of the nucleon presentin the parton at the time of the interaction. Squaring the 4-momentum, [118] P total a P total a = ( P parton a + P νa )( P parton a + P νa )= P parton a P parton a + 2 P parton a P νa + P νa P νa , (5.4)and using P a P a = − E CM , (5.5)we have − E CM = − x m N + 2( − xm N E ν + 0 · ~p ν ) − m ν . (5.6)50Because we are interested in energies where E ν ≫ m N and m ν , we find E CM = 2 xm N E ν . (5.7)We denote this quantity by ˆ s ≡ xm N E ν . (5.8)In terms of the variables √ ˆ s , the neutrino-parton center-of-mass energy; D , thetotal number of dimensions of spacetime; and M D , the D -dimensional Planck scale;we express the Schwarzschild radius as [106, 107] r s ( √ ˆ s, D, M D ) = 1 M D " √ ˆ sM D D − " D − π ( D − / Γ( D − ) D − D − . (5.9)From here on we will work within the assumption of string theory that D = 10, andwe will have M D = M , which we will eventually take to be near 1 TeV [119]. In 10dimensions, we then have r s ( √ ˆ s, M ) = 1 M " √ ˆ sM π / Γ(9 / / (5.10)for the Schwarzschild radius.The actual radius of the black hole will differ from the Schwarzschild radius r s ,due to factors such as angular momentum and the geometry of spacetime, and wewill call this corrected cross sectional area F πr s , where the variable F is a prefactorused to correct for differences from an exact Schwarzschild metric. We define theinelasticity as [120] y ≡ M BH √ ˆ s , (5.11)which is a measure of how much of the center-of-mass energy is available to the blackhole for Hawking radiation [121, 122, 123]. The energy difference, the deficit betweenthe final mass of the black hole after its ring-down phase [124, 125, 126, 127] and51the center-of-mass energy initially present in the collision, is carried off via incomingshock wave multipole moments radiating gravitational waves [128, 129, 130, 131, 132].The inelasticity y depends on the impact parameter b , and we define z ≡ bb max , (5.12)where b max = √ F r s . The values of F and y ( z ) calculated depend upon the slicing ofspacetime used to determine whether or not an apparent horizon is present. In thework of [133, 134, 135], it is found for D = 10 that F = 1 . 819 and we approximate theirfindings for the inelasticity as y ( z ) = 0 . − . z . In the later work of [136], in whicha slicing on the future light cone is used, it is found for D = 10 that F = 3 . 09 andwe approximate their findings for the inelasticity as y ( z ) = 0 . − . z + 0 . z .We will refer to these two different slicings as the “old slice” and the “new slice,”respectively.The prefactor F and the inelasticity y ( z ) were derived using classical generalrelativity. Since we don’t yet have a quantum theory of gravity, we need to make surewe stay within a semi-classical regime. We expect a thermal distribution of Hawkingradiation [137, 138, 139] for M BH ≥ x min M , (5.13)where x min = 3 ensures a well-defined resonance not dominated by the 3-brane tension[120, 140], and thus M BH ≥ i ofparticles of spin s with initial total energy between ω and ω + dω is [141]˙ N i dω = σ s ( ω )Ω D − ω D − ( D − π ) D − (cid:2) e ω/T − ( − s (cid:3) − , (5.14)where T = D − π r s (5.15)52is the instantaneous Hawking temperature,Ω D − = 2 π ( D − / Γ[( D − / 2] (5.16)is the volume of a unit ( D − σ s ( ω ) is the greybody factor that accountsfor the backscattering of part of the outgoing radiation into the black hole [142]. Notethat a rough estimate of the instantaneous Hawking temperature can be found fromthe first law of black hole thermodynamics (which is analogous to the combined firstand second law of thermodynamics): T = dE/dS ≃ ( dA/dM ) − [107]. CombiningEqs. (5.8), (5.11), and (5.13) shows that χ ≡ ( x min M ) m N E ν y ( z ) ≤ x, (5.17)where to find the cross section we integrate the PDFs over the parton momentumfraction x and use χ as our lower limit of integration.In addition to integrating the PDFs over the parton momentum fraction, we alsointegrate over z for an impact parameter-weighted average over parton cross sections.The area of a thin ring of inner radius z and thickness of dz is proportional to zdz .We multiply this by a factor of 2, so that when we integrate R zdz alone, we geta factor of 1; therefore, if y ( z ) did not depend on z , this weighted average could beneglected. Because the value of y ( z ) does in fact depend on z , the weighted averageensures we use the correct lower limit of integration, χ , when integrating over theparton momentum fraction, x .The final expression for the νN → BH cross section is [143] σ = Z z dz Z X dx F πr s ( √ ˆ s, M ) X i f i ( x, Q ) , (5.18)where i labels parton species, and the f i ( x, Q ) are PDFs.Fig. 5.5 shows log( σ ) plotted versus log( E ν ) for both the case of apparent horizons53Figure 5.5: Cross Section: New and Old Slicing.54 L og σ ( pb ) Log E ν (GeV) x min =1 x min =3 x min =5 Figure 5.6: Cross Section: Varying Semi-Classical Regime.55on the “old slice” (dot-dash line) and the “new slice” (solid line). The cross sectionis given in units of picobarns (pb) and the energy of the incoming neutrino is givenin units of GeV. We use x min = 3 and M = 1 TeV.Fig. 5.6 shows log( σ ) plotted versus log( E ν ) on a log-log scale for different valuesof x min using the new slicing. The cross section is given in units of picobarns (pb)and the energy of the incoming neutrino is given in units of GeV, where M = 1 TeVand Q = min { r − s , 10 TeV } .The cross sections were integrated with a variable step size with respect to theparton momentum function x . The dominant contribution from the PDFs comesfrom the small- x region, which is only probed when the lower limit of integration χ is sufficiently small. This happens with large enough values of the incoming neutrinoenergy E ν . We keep the step variable of integration smaller than χ/ 100 for x < − and equal to 1 / 100 for x > − . This gives us excellent accuracy and a fast numericalcalculation of the cross section. One of the major outstanding questions that IceCube is hoped to be able to answeris— what is the flux rate of cosmic neutrinos? A good estimate involves a consid-eration of the number of neutrinos expected to be created in association with theobserved flux of charged cosmic ray particles: this is the Waxman-Bahcall (WB) flux[144] of φ ν ≃ . × − ( E ν / GeV) − GeV − cm − s − sr − , (5.19)including all species of neutrinos. Another estimated flux assumes that extragalacticcosmic rays dominate the spectrum at energies above ∼ . GeV and that additionalneutrinos are to be expected from sources opaque to ultra-high energy cosmic rays;56Table 5.1: Probability of Signal. M BH P sig M M M M M φ ν ≃ . × − ( E ν / GeV) − . GeV − cm − s − sr − , (5.20)including all species of neutrinos.To confirm the existence of black hole interactions amidst the background noise ofstandard model (SM) interactions, we pick out a signal that has a high likelihood forthe relatively democratic Hawking radiation and a low likelihood for charge current(CC) interactions: we search for soft muons, or muons with less than 20% of theincident neutrino energy. In SM CC interactions, a produced muon will generallycarry away at least 80% of the incident energy. We only consider interactions withat least 4 secondary particles, where at least one of them is a muon [108]. The crosssection for the SM CC interaction producing a soft muon is [7] σ y> . ≃ . E ν / GeV) . pb . (5.21)For incident neutrino energies larger than 10 GeV, the background number of SMCC interactions meeting these criteria for the AARGHW flux, which produces moreevents than the WB flux, is 10 events over the 15 year lifetime of IceCube. For E ν > GeV, the expected event rate for the SM CC interaction over IceCube’slifetime is less than 1 event.The probability that a black hole interaction produces the criteria we proposeto search for depends on the mass of the black hole formed [7]. See Table 5.1 for57some values of the signal probability versus the size of the black hole created froma neutrino-parton interaction. In this probability we neglect the gravitons radiatedinto the bulk of the compactified dimensions, but these are thought to carry awayless than 15% of the radiated energy when D = 10 [146, 147, 148].With the probability of signal given as a function of black hole mass, we need away to determine the expected number of TeV-scale black holes formed within a givenmass range. We do this by dividing the expected number of black holes produced intobins at 0 . M mass intervals. We vary our value of x min , and repeat our calculationfor the expected number of black holes created at IceCube. For example, shoulda rate of 235 TeV-scale black holes be created at IceCube for x min = 3 . 1, and 246created for x min = 3 . 0, then we could assign 11 black holes to the M BH = 3 . x min ≥ 3, we get our cumulative totals.We will integrate, with respect to energy, the neutrino flux over the 15 year lifetimeof the IceCube experiment, or T ≃ . × seconds. At the energies of interest theEarth is opaque to neutrinos. Hence, we will only consider neutrinos passing downthrough the Antarctic ice, and we will only accept measurements from this half of theavailable directions, which makes for 2 π steradians of solid angle for observation. Thebackground rate of non-neutrino events at such high energies is entirely negligible.IceCube’s effective volume is 1 km [104], which at a density of 900 kg/m means thenumber of nucleons available for neutrino interaction targets is n T ≃ . × . Ourupper limit of integration is an energy of 10 GeV, because beyond this the IceCubedetector will be saturated and unable to resolve all the details of the interaction [99].The total number of black hole signal events over the life of the IceCube experiment58Table 5.2: Number of Signal Events. x min N BH [WB] N BH [AARGHW]3 43 (19) 69 (30)4 34 (15) 43 (19)5 27 (12) 28 (12)6 22 (9) 20 (9)Table 5.3: 10-Dimensional Planck Mass Sensitivity. x min M / TeV [WB] M / TeV [AARGHW]3 1.5 (1.2) 1.5 (1.2)5 1.3 (1.1) 1.3 (1.1)7 1.2 (1.0) 1.2 (1.0)9 1.1 (1.0) 1.1 (0.9)is N sig = 2 π n T T Z dE ν σ ( E ν ) φ ν ( E ν ) P sig . (5.22)In Table 5.2 we calculate the expected number of black hole signals over thelifetime of IceCube. With a lower limit of integration of 10 GeV, we fix M = 1TeV, but we allow x min to vary. We compare the number of events for the WB fluxto the AARGHW flux. For each flux, we have calculated the number of events usingboth the “new slice,” which is given without parentheses; and the “old slice,” whichis given inside parentheses.In Table 5.3 we calculate the maximum 10-dimensional Planck mass for which wewould expect be able to observe the interaction at the 3 σ level. With a lower limitof integration of 10 GeV, and for differing values of x min , we find the correspondingvalue of M . We do this for both the WB flux and the AARGHW flux. For eachflux, we have calculated the number of events using both the “new slice,” which isgiven without parentheses; and the “old slice,” which is given inside parentheses.In Fig. 5.7 we plot the TeV-scale discovery reach for both IceCube and the LargeHadron Collider (LHC) [120], assuming a cumulative integrated luminosity of 1 ab − over the life of the collider. We calculate the maximum value of M that could beobserved at the 5 σ level versus x min , and we use a lower limit on the energy integral59Figure 5.7: IceCube and LHC Discovery Reach.60of 10 GeV. We plot the IceCube discovery reach only in the semi-classical regime of x min ≥ 3; however, the LHC could potentially be focused on superstring resonances[149, 150, 151], and could thus be able to probe the quantum regime [7].61 Chapter 6Conclusion Using the de Sitter Bunch-Davies state for modes of intermediary- q and large- q isvalid in the exponentially growing region of the scale factor, but imposing the Bunch-Davies state on modes of small- q leads to infrared divergences in the dispersionspectrum. Maintaining continuity of the scale factor to C is necessary to preventultraviolet divergences of the energy density of particles created during inflation. Theasymptotically Minkowskian regions of our composite scale factor do not affect thenear scale-invariance of the intermediate- q region of particle production, but it doesallow for an unambiguous interpretation of the number of particles produced versus q , and it allows for flat-space renormalization. An asymptotically flat scale factorsegment may be joined continuously to C with an exponentially growing segmentof scale factor, whereas a simple power law such as a ( t ) ∝ t n may not. Both of ourmassive approximations are trustworthy in their respective regimes: little growth ofthe composite scale factor outside of the exponentially growing region for the effective- k approach, and with modes not at the interface between the small- and intermediate- q behavior and not at the interface between the intermediate- and large- q behaviorfor the dominant-term approach. In our model, the average number of particlescreated per mode can be characterized in terms of three parameters: the number ofe-folds, N e ; the ratio of the mass to the Hubble constant during inflation, m H ; and62the dimensionless mode number, q . We find a scale-invariant spectrum when H infl and m H are both constant, provided modes are converted individually into curvatureperturbations soon after exiting the Hubble radius. The spectral index can be shiftedtowards a blue spectrum if all the curvature perturbations are created around thesame time or at a time after the end of inflation. The spectral index can be shiftedtowards a red spectrum by taking into consideration a changing value of H infl or ˙ φ .We find that an abrupt end to inflation leads to a boosted production of high-energyparticles and an associated high temperature. If monopoles, or certain other exoticparticles, were found to be created copiously at low temperatures— at the LHC, forinstance— it could place rigorous constraints on the characteristics of inflation.The predicted energy and angular momentum in the post-Minkowski approxi-mation for our binary point mass system with helical symmetry agrees to first post-Newtonian order in the case of parametrization-invariant action plus either of the 1PNcorrection terms. With q → 0, we can make a comparison with the Schwarzschildsolution of General Relativity. Here, both the affine case and the parametrization-invariant with a 1PN correction term have an Innermost Circular Orbit at about 3 M ,which is outside the event horizon of GR located at 2 M . Only the affine case has anInnermost Stable Circular Orbit, and it occurs at ∼ . M , which is outside of theISCO predicted by GR located at 6 M . These discrepancies may be due to the lin-ear order of the post-Minkowski approximation, or they may be due to the radiationbeing pumped into the binary system by the half-advanced plus half-retarded helicalsymmetry. A form of the first law of thermodynamics dE/dv = Ω dL/dv is satisfied,and this serves as a useful check on the analytical and numerical results.With a flux of cosmic neutrinos at the Waxman-Bahcall rate, over its 15 yearlifespan the IceCube Neutrino Telescope could detect TeV-scale black holes at the5 σ level up to a maximum 10-dimensional Planck mass of 1.3 TeV. Our analysisshows that PDFs can be approximated well by fits to x f ( x ) = A x n , provided therange of the parton momentum fraction, x , for each fit is restricted to a few decades63of variation on a log scale. The fitting of the parameters A and n can best beaccomplished by simultaneously varying each, and by sampling a large enough areaof parameter space to ensure a false minimum deviation is avoided. The integrationinvolved in calculating the cross section of the gravitational interaction between aparton and a cosmic neutrino is most efficiently carried out with a variable step sizeof integration. Values of the parton momentum fraction closest to the lower limit ofintegration dominate the cross section, so care must be taken to use a small enoughstep size in this range so that these values are not over-weighted in the cross section.A convenient way of associating events with a given value of M BH is to recalculate thenumber of lifetime events for different values of x min , and then subtract the differencebetween the events from incremental values of x min into bins.In the three parts of this dissertation, we have focused on the topics in inflationarycosmology and astrophysics described in three papers: [5, 6, 7].64 Appendix AComposite History of an ExactReaction-Force Solution This Appendix is motivated by and based on the work of [152]. What follows isan application of the more general techniques presented in Sec. 3.1.2 for matchingcontinuously to second derivative in what could be taken as either a scale factor onthe one hand or as a particle’s velocity on the other. It is hoped that this exampleserves to illustrate some aspects of the self force and radiation reaction mentioned inChapter 4. We begin with a charged particle that in its rest frame emits radiationwhen accelerated as given by the Larmor formula (in Gaussian units) of P = 23 e c ˙ v , (A.1)which leads to, in addition to the external force, a radiation-reaction force of the form[153, 154, 155] ~F applied = m ˙ ~v − e c ¨ ~v, (A.2)65as perceived by the particle in its momentarily-comoving rest frame. In this example,we will consider only rectilinear motion, so we rewrite this as F applied = m ˙ v − mτ ¨ v, (A.3)where τ ≡ e mc . (A.4)We thus define the reaction force, or self force, as F self ≡ − mτ ¨ v. (A.5)For constant acceleration, we have v c ( t ) = a c ( t − t ) , (A.6)˙ v c ( t ) = a c , (A.7)¨ v c ( t ) = 0 , (A.8) F c self ( t ) = 0 , (A.9) F c applied ( t ) = ma c , (A.10) P c radiated ( t ) = mτ a c . (A.11)We then introduce a two similar velocity histories given by a hyperbolic tangent inanalog with Sec. 3.1.2, v i ( t ) = v i + ∆ i tanh t − t i s i , (A.12) v f ( t ) = v f + ∆ f tanh t − t f s f , (A.13)where ∆ i is twice the difference between early- and late-time velocities for the firstvelocity history, t i is the time at which v i ( t ) = v i , and s i is a throttling parameter66that decreases the change in velocity with respect to time as it increases in magnitude;and where ∆ f is twice the difference between early- and late-time velocities for thefinal velocity history, t f is the time at which v f ( t ) = v f , and s f is a throttlingparameter that decreases the change in velocity with respect to time as it increasesin magnitude. We will take both s i and s f to be ≥ 0. Then we have v i ( t ) = v i + ∆ i tanh t − t i s i , (A.14)˙ v i ( t ) = ∆ i s i (cid:18) − tanh t − t i s i (cid:19) , (A.15)¨ v i ( t ) = 2 ∆ i s i (cid:18) tanh t − t i s i (cid:19) (cid:20)(cid:18) tanh t − t i s i (cid:19) − (cid:21) , (A.16) F i self ( t ) = − mτ ∆ i s i (cid:18) tanh t − t i s i (cid:19) (cid:20)(cid:18) tanh t − t i s i (cid:19) − (cid:21) , (A.17) F i applied ( t ) = m ∆ i s i (cid:18) − tanh t − t i s i (cid:19) − mτ ∆ i s i (cid:18) tanh t − t i s i (cid:19) (cid:20)(cid:18) tanh t − t i s i (cid:19) − (cid:21) , (A.18) P i radiated ( t ) = mτ (cid:20) ∆ i s i (cid:18) − tanh t − t i s i (cid:19)(cid:21) , (A.19)and v f ( t ) = v f + ∆ f tanh t − t f s f , (A.20)˙ v f ( t ) = ∆ f s f (cid:18) − tanh t − t f s f (cid:19) , (A.21)¨ v f ( t ) = 2 ∆ f s f (cid:18) tanh t − t f s f (cid:19) (cid:20)(cid:18) tanh t − t f s f (cid:19) − (cid:21) , (A.22) F f self ( t ) = − mτ ∆ f s f (cid:18) tanh t − t f s f (cid:19) (cid:20)(cid:18) tanh t − t f s f (cid:19) − (cid:21) , (A.23) F f applied ( t ) = m ∆ f s f (cid:18) − tanh t − t f s f (cid:19) − mτ ∆ f s f (cid:18) tanh t − t f s f (cid:19) (cid:20)(cid:18) tanh t − t f s f (cid:19) − (cid:21) , (A.24) P f radiated ( t ) = mτ (cid:20) ∆ f s f (cid:18) − tanh t − t f s f (cid:19)(cid:21) . (A.25)67At times t i and t f , respectively, we have v i ( t i ) = v i , (A.26)˙ v i ( t i ) = ∆ i s i , (A.27)¨ v i ( t i ) = 0 , (A.28) F i self ( t i ) = 0 , (A.29) F i applied ( t i ) = m ∆ i s i , (A.30) P i radiated ( t ) = mτ (cid:18) ∆ i s i (cid:19) , (A.31)and v f ( t f ) = v f , (A.32)˙ v f ( t f ) = ∆ f s f , (A.33)¨ v f ( t f ) = 0 , (A.34) F f self ( t f ) = 0 , (A.35) F f applied ( t f ) = m ∆ f s f , (A.36) P f radiated ( t ) = mτ (cid:18) ∆ f s f (cid:19) . (A.37)We then specify a composite velocity history by matching the velocity histories of v i ( t ) to v c ( t ) to v f ( t ). We can maintain C joining conditions— meaning the velocity,acceleration, and radiation-reaction force are all kept continuous— by joining theinitial segment to the start of a region of constant acceleration at t = t i , and byjoining the final segment to the end of a region of constant acceleration at t = t f .See Fig. A.1, where we plot a dimensionless example of a composite velocity where∆ i = ∆ f = s i = s f = a c = 1. In this example we take t i = 0, t f = 10, v i = 1, and v f = 11.Maintaining continuity of the velocity history up to its second derivative imposes,68 v ( t ) t Figure A.1: Velocity versus Time.69in addition to the two conditions of matching times, the following boundary conditions∆ i s i = a c = ∆ f s f , (A.38) v i = lim t →−∞ [ v ( t )] + ∆ i , (A.39) v f = lim t → + ∞ [ v ( t )] − ∆ f . (A.40)We find that t = t i − v i /a c , and the duration of constant acceleration is t a ≡ t f − t i .See Fig. A.2, where we plot a dimensionless example of the applied force necessary -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -5 0 5 10 15 F a pp li e d ( t ) t τ =1.0 τ =0.1 Figure A.2: External Force versus Time.to maintain the motion of the particle shown in Fig. A.1 for two different dimensionlessvalues of τ . In the small- τ limit, we get a Newtonian 2nd Law of F = ma . In thelarge- τ limit, we note some peculiarities of the self-force. To initiate the acceleration,a force must initially be applied opposite to the direction of motion— this is to be70compared with the pre-acceleration found for radiation-reaction forces that eliminatesrunaway-acceleration solutions. To end the period of constant acceleration, the forcemust be increased in the direction of motion. As will be shown below, this additionalwork is needed to compensate for the energy dissipated by the radiation emitted.The total change in kinetic energy of the particle is∆ KE = 12 m ((cid:18) lim t → + ∞ [ v ( t )] (cid:19) − (cid:18) lim t →−∞ [ v ( t )] (cid:19) ) = 12 m (cid:8) ( v i + a c t a + ∆ f ) − ( v i − ∆ i ) (cid:9) = m a c v i ( t a + s i + s f ) + 12 ma c (cid:0) t a + 2 t a s f + s f − s i (cid:1) . (A.41)The total power radiated is P radiated total = (cid:18)Z t i −∞ P i dt (cid:19) + (cid:18)Z t f t i P c dt (cid:19) + Z + ∞ t f P f dt ! = 23 mτ ∆ i s i + mτ a c t a + 23 mτ ∆ i s i = mτ a c (cid:18) s i + t a + 23 s f (cid:19) . (A.42)The total work done on the particle by the external force is W total = (cid:18)Z t i −∞ F i applied v i ( t ) dt (cid:19) + (cid:18)Z t f t i F c applied v c ( t ) dt (cid:19) + Z + ∞ t f F f applied v f ( t ) dt ! = (cid:18) ma c (cid:20) v i ( s i − τ ) + s i a c (cid:26) τ − s i (cid:27)(cid:21)(cid:19) + (cid:18) m a c v i t a + 12 m a c t a (cid:19) + (cid:18) ma c (cid:20) v i ( s f + τ ) + a c (cid:18) s f + s f t a + 23 s f τ + t a τ (cid:19)(cid:21)(cid:19) = ma c v i ( s i + t a + s f ) + 12 ma c (cid:0) t a + 2 t a s f + s f − s i (cid:1) + mτ a c (cid:18) s i + t a + 23 s f (cid:19) . (A.43)71We find that W total − P radiated − ∆ KE = 0 , (A.44)and thus energy is conserved at early and late times. See Fig. A.3, for the case of -80-60-40-20 0 20 40 60 80 100 -5 0 5 10 15 E n e r gy t -Radiated -Kinetic +Work =Total Figure A.3: Energy versus Time.energy conservation between early and late times. The velocity history is given inFig. A.1, and we choose τ = 1. The energy deficit that develops is primarily dueto the energy dissipated through the emitted radiation during the phase of constantacceleration. This negative energy must be balanced by an additional amount of workapplied to the particle to end the acceleration. If additional energy is not provided tothe system, Wiseman has proven that the kinetic energy of the particle decreases tocompensate [152]. In the limit of s i → s i → 0, we see that the work associatedwith overcoming the reaction force at the initial and final joining points is − m a c v i τ and m a c v f τ , respectively. Because in this velocity history ¨ v = 0 if t = t i and72 t = t f , and because W self = R F self ( t ) v ( t ) dt , we see that in the instantaneous limit, F self ( t ) = ma c τ [ δ ( t − t f ) − δ ( t − t i )], where δ ( t ) is the Dirac delta-function.73 Bibliography [1] L. Parker, The creation of particles by the expanding universe , Ph.D. thesis (XeroxUniversity Microfilms, Ann Arbor, Michigan, No. 73-31244), Harvard University(1966).[2] L. Parker, Phys. Rev. Lett. , 562 (1968).[3] L. Parker, Phys Rev. , 1057 (1969).[4] L. Parker, Nature , 20 (1976).[5] M.M. Glenz and L. Parker, “Study of the Spectrum of Inflaton Perturbations,”to be submitted.[6] M.M. Glenz and K. Ury¯u, Phys. Rev. D 76 , 027501 (2007).[7] L.A. Anchordoqui, M.M. Glenz, and L. Parker, Phys. Rev. D 75 , 024011 (2007).[8] J.L. Friedman and K. Ury¯u, Phys. Rev. D 73 , 104039 (2006).[9] E.W. Kolb and M.S. Turner, The Early Universe , (Perseus Publishing, Cam-bridge, MA, 1994).[10] S. Dodelson, Modern Cosmology , (Academic Press, Boston, 2003).[11] A. Einstein, Preuss. Akad. Wiss. Berlin, Sitzber., 844-847 (1915).[12] R.M. Wald, General Relativity , (The University of Chicago Press, Chicago, 1984).74[13] H.C. Ohanian and R. Ruffini, Gravitation and Spacetime , Second Edition, (W.W.Norton & Company, New York, 1994).[14] L.P. Hughston and K.P. Tod, An Introduction to General Relativity , (CambridgeUniversity Press, Cambridge, England, 1990).[15] N.D Birrell and P.C.W. Davies, Quantum Fields in Curved Space , (CambridgeUniversity Press, New York, 1982).[16] L. Parker and D.J. Toms, Principles and Applications of Quantum Field Theoryin Curved Spacetime , (Cambridge University Press, 2009), to appear.[17] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation , (Freeman, New York,1973).[18] A. Vilenkin and L.H. Ford, Phys. Rev. D 26 , 1231 (1982).[19] B. Allen, Phys. Rev. D 32 , 3136 (1985).[20] B. Allen and A. Folacci, Phys. Rev. D 35 , 3771 (1987).[21] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions , (NationalBureau of Standards, Washington, D.C., 1972).[22] N. Deruelle, J. Katz, and J-P. Uzan, Class. Quantum Grav. , 421 (1997).[23] T.S. Bunch and P.C.W. Davies, Proc. Roy. Soc. London A 360 , 117 (1978).[24] A.H. Guth, Phys. Rev. D 23 , 347, (1981).[25] K. Sato, Phys. Lett. B 99 , 66, (1981).[26] A.A. Starobinsky, Phys. Lett. B 91 , 99 (1980).[27] A.A. Starobinsky, Phys. Lett. B 117 , 175 (1982).[28] A.H. Guth and S.-Y. Pi, Phys. Rev. Lett. , 1110 (1982).75[29] J.M. Bardeen, P.J. Steinhardt, and M.S. Turner, Phys. Rev. D 28 , 679 (1983);[30] G. Boerner, The Early Universe , (Springer, Berlin, 1988).[31] R. Brout, F. Englert, and E. Gunzig, Ann. of Phys. , 78 (1978).[32] V.F. Mukhanov, H.A. Feldman, and R.H. Brandenberger, Phys. Rept. , 203(1992).[33] H.V. Peiris et al , Astrophys. J. Suppl. , 213 (2003).[34] A.R. Liddle and D.H. Lyth, Cosmological inflation and large-scale structure ,(Cambridge University Press, New York, 2000).[35] J.M. Bardeen, Phys. Rev. D 22 , 1882 (1980).[36] M. Sasaki, Prog. Theor. Phys. , 1036 (1986).[37] S.W. Hawking, Phys. Lett. B 115 , 295 (1982).[38] L. Parker, arXiv:hep-th/0702216v2 (2007).[39] D.N. Spergel et al , ApJS , 175 (2003).[40] E. Komatsu et al , arXiv:astro-ph/0803.0547v1 (2008).[41] A.R. Liddle and D.H. Lyth, Phys. Lett. B 291 , 391 (1992).[42] J. Dunkley et al , arXiv:astro-ph/0803.0586v1 (2008).[43] M. Tegmark et al , Phys. Rev. D 74 , 123507 (2006).[44] M.M. Glenz, X. Huang, and L. Parker, in preparation (2007).[45] A.D. Linde, Phys. Lett. B 108 , 389 (1982).[46] S. Habib et al , Phys. Rev. D 71 , 043518 (2005).[47] D.N. Spergel et al , ApJS , 377 (2007).76[48] S. Coleman and E. Weinberg, Phys. Rev. D 7 , 1888 (1973).[49] Q. Shafi and V.N. Senoguz, Phys. Rev. D 73 , 127301 (2006).[50] D.H. Lyth and A. Riotto, Phys. Rept. , 1 (1999).[51] P.R. Anderson et al , Phys. Rev. D 62 , 124019 (2000).[52] L. Parker and S.A. Fulling, Phys. Rev. D 9 , 341 (1974).[53] S.A. Fulling, L. Parker, and B.L. Hu, Phys. Rev. D 10 , 3905 (1974).[54] P.R. Anderson and L. Parker, Phys. Rev. D 36 , 2963 (1987).[55] F. Finelli et al , arXiv:0707.1416v1 (2007).[56] I. Agull´o et al , arXiv:0806.0034v1 (2008).[57] N.D. Birrell, Proc. Roy. Soc. (London) A 361 , 315 (1978).[58] C. L¨uders and J.E. Roberts, Commun. Math. Phys. , 29 (1990).[59] K. Pirk, Phys. Rev. D 48 , 3779 (1993).[60] W. Junker and E. Schrohe, Annales Poincare Phys. Theor. , 1113 (2002).[61] L.H. Ford, Phys. Rev. D 35 , 2955 (1987).[62] B. Allen, Phys. Rev. D 37 , 2078 (1988).[63] P.J. Epstein, Proc. Nat. Acad. Sciences (US) , 627 (1930).[64] C. Eckart, Phys. Rev. , 1303 (1930).[65] L. Parker, “The Production of Elementary Particles in Strong GravitationalFields,” in Asymptotic Structure of Space-Time , edited by F.P. Esposito and L.Witten, (Plenum Press, New York), 107 (1977).77[66] L. Parker, “Quantized Fields and Particle Creation in Curved Spacetime,” 66pages in Relativity, Fields, Strings and Gravity: The Second Latin American Sym-posium on Relativity and Gravitation (SILARG 2) , editor C. Aragone. (Universi-dad Simon Bolivar, Caracas, 1975).[67] R.M. Kulsrud, Phys. Rev. , 205 (1957).[68] J.E. Littlewood, Annals of Physics (New York) , 233 (1963).[69] D.J.H. Chung, E.W. Kolb, and A. Riotto, Phys. Rev. D 59 , 023501 (1999).[70] U.A. Yajnik, Phys. Lett. B 234 , 271 (1990).[71] E.M. Lifshitz, Zh. Eksp. Teor. Fiz. , 587 (1946).[72] L.P. Grishchuk, Zh. Eksp. Teor. Fiz. , 825 (1974).[73] L.P. Grischuk, Sov. Phys.—JETP , 409 (1975).[74] L.H. Ford and L. Parker, Phys. Rev. D 16 , 245 (1977).[75] L. Parker, “Time’s Arrow and the Strength of Inflation,” talk presented at theOrigins of Time’s Arrow conference at the New York Academy of Sciences, October15-16 (2007).[76] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 , 2738 (1977).[77] W. Rindler, Am. J. of Phys. , 1174 (1966).[78] S.A. Fulling, Ph.D thesis (unpublished), Princeton University (1972).[79] S.A. Fulling, Phys. Rev. D 7 , 2850 (1973).[80] W.G. Unruh, Phys. Rev. D 14 , 870 (1976).[81] P.C.W. Davies, J. Phys. A 8 , 609 (1975).78[82] P.R. Anderson, C. Molina-Paris, and E. Mottola, Phys. Rev. D 72 , 043515(2005).[83] J. Pradler and F.D. Steffen, Phys. Lett. B 648 , 224 (2007).[84] R.H. Cyburt et al , Phys. Rev. D 67 , 103521 (2003).[85] G.F. Giudice, I. Tkachev, and A. Riotto, J. High Energy Phys. , 9908 (1999).[86] A.D. Fokker, Zeits. f. Physik , 386 (1929).[87] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. , 157, (1945).[88] J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. , 425, (1949).[89] T. Ledvinka, G. Sch¨afer, and J. Biˇc´ak, Phys. Rev. Lett. , 251101 (2008).[90] J.W. Dettman and A. Schild, Phys. Rev. , 1057 (1954).[91] A. Schild, Phys. Rev. , 2762 (1963).[92] LISA Mission Science Office , LISA − LIST − RP − D 65 , 124009 (2002).[94] L.A. Anchordoqui et al , Annals Phys. , 145 (2004).[95] F. Mandl and G. Shaw, Quantum Field Theory , Revised Edition, (John Wiley& Sons, New York, 2005).[96] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nu-clei, and Particles , Second Edition, (John Wiley & Sons, New York, 1985).[97] A. Karle et al , Nucl. Phys. Proc. Suppl. , 388 (2003).[98] L.A. Anchordoqui et al , Phys. Rev. D 74 , 125021 (2006).79[99] F. Halzen, Eur. Phys J. C 46 , 669 (2006).[100] IceCube Collaboration , Preliminary Design Document,(http://icecube.wisc.edu).[101] P. Lipari, Astropart. Phys. , 195 (1993).[102] J. Alvarez-Mu˜niz et al , Phys. Rev. D 65 , 124015 (2002).[103] M. Kowalski, A. Ringwald, and H. Tu, Phys. Lett. B 529 , 1 (2002).[104] L.A. Anchordoqui and F. Halzen, Annals Phys. , 2660 (2006).[105] N. Arkani-Hamed, S. Dimopoulos, and G.R. Dvali, Phys. Lett. B 429 , 263(1998).[106] R.C. Myers and M.J. Perry, Ann. Phys. , 304 (1986).[107] P.C. Argyres, S. Dimopoulos, and J. March-Russell, Phys Lett. B 441 , 96(1998).[108] S. Dimopoulos and G. Landsberg, Phys. Rev. Lett. , 161602 (2001).[109] S.B. Giddings and S.D. Thomas, Phys. Rev. D 65 , 056010 (2002).[110] A. Ringwald and H. Tu, Phys. Lett. B 525 , 135 (2002).[111] F. Halzen and A.D. Martin, Quarks And Leptons: An Introductory Course InModern Particle Physics , (John Wiley & Sons, New York, 1984).[112] R. Emparan, M. Masip, and R. Rattazzi, Phys Rev. D 65 , 064023 (2002).[113] L.A. Anchordoqui et al , Phys. Rev. D 65 , 124027 (2002).[114] J. Pumplin et al J. High Energy Phys. , 012 (2002).[115] D. Stump et al J. High Energy Phys. , 046 (2003).80[116] Graph created from CTEQ data (http://durpdg.dur.ac.uk/hepdata/pdf3.html).[117] K.S. Thorne, in Magic Without Magic: John Archibald Wheeler , edited byJ. Klauder (Freeman, San Francisco, 1972) p. 231.[118] B.F. Schutz, A First Course in General Relativity , (Cambridge University Press,New York, 2004).[119] I. Antoniadis et al , Phys Lett B 436 , 257 (1998).[120] L.A. Anchordoqui et at , Phys. Lett. B 594 , 363 (2004).[121] S.W. Hawking, Nature (London) , 30 (1974).[122] S.W. Hawking, Commun. Math. Phys. , 199 (1975).[123] S.W. Hawking, Commun. Math. Phys. , 206(E) (1975).[124] V.P. Frolov and D. Stojkovic, Phys. Rev. D 67 , 084004 (2003).[125] V.P. Frolov and D. Stojkovic, Phys. Rev. D 68 , 064011 (2003).[126] V.P. Frolov, D.V. Fursaev, and D. Stojkovic, J. High Energy Phys. , 057(2004).[127] V.P. Frolov, D.V. Fursaev, and D. Stojkovic, Classical Quantum Gravity ,3483 (2004).[128] P.C. Aichelburg and R.U. Sexl, Gen. Relative. Gravit. , 303 (1971).[129] R. Penrose, unpublished.[130] P.D. D’Eath and P.N. Payne, Phys. Rev. D 46 , 658 (1992).[131] P.D. D’Eath and P.N. Payne, Phys. Rev. D 46 , 675 (1992).[132] P.D. D’Eath and P.N. Payne, Phys. Rev. D 46 , 694 (1992).81[133] H. Yoshino and Y. Nambu, Phys. Rev. D 66 , 065004 (2002).[134] H. Yoshino and Y. Nambu, Phys. Rev. D 67 , 024009 (2003).[135] D.M. Eardley and S.B. Giddings, Phys. Rev. D 66 , 044011 (2002).[136] H. Yoshino and V.S. Rychkov, Phys. Rev. D 71 , 104028 (2005).[137] L. Parker, Phys. Rev. D 12 , 1519 (1975).[138] R.M. Wald, Commun. Math. Phys. , 9 (1975).[139] S.W. Hawking, Phys. Rev. D 14 , 2460 (1976).[140] J. Preskill et al , Phys. Lett. A 6 , 2353 (1991).[141] T. Han, G.D. Kribs, and B. McElrath, Phys. Rev. Lett. , 031601 (2003).[142] D.N. Page, Phys. Rev. D 13 , 198 (1976).[143] L.A. Anchordoqui et al , Phys. Rev. D 68 , 104025 (2003).[144] E. Waxman and J.N. Bahcall, Phys. Rev. D 59 , 023002 (1998).[145] M. Ahlers et al , Phys. Rev. D 72 , 023001 (2005).[146] V. Cardoso, M. Cavaglia, and L. Gualtieri, Phys. Rev. Lett. , 071301 (2006).[147] V. Cardoso, M. Cavaglia, and L. Gualtieri, Phys. Rev. Lett. , 219902(E)(2006).[148] V. Cardoso, M. Cavaglia, and L. Gualtieri, J. High Energy Phys. , 021 (2006).[149] L.A. Anchordoqui et al , Phys Rev. Lett. , 171603 (2008).[150] L.A. Anchordoqui et al , arXiv:hep-ph/0804.2013 (2008).[151] L.A. Anchordoqui et al , arXiv:hep-ph/0808.0497 (2008).[152] A.G. Wiseman, unpublished (2008).82[153] H.A. Lorentz, The Theory of Electrons and its Applications to the Phenomenaof Light and Heat , Second Edition, (G.E. Stechert & Co., New York, 1916).[154] E. Poisson, arXiv:gr-qc/9912045v1 (1999).[155] J.D. Jackson, Classical Electrodynamics , Third Edition, (John Wiley & Sons,New York, 1999).83CURRICULUM VITAE Matthew GlenzEDUCATION Ph.D., Physics University of Wisconsin—Milwaukee Dec. 2008B.S., Physics Iowa State University, Honors May 6, 2000Studied Abroad at Lancaster University, England 1998-1999 EMPLOYMENT Research Assistant University of Wisconsin—Milwaukee 2006-2008Teaching Assistant University of Wisconsin—Milwaukee 2004-2006Technical Services Epic Systems Corporation, Madison, WI 2001-2004Support Technician Gundersen-Lutheran Hospital, LaCrosse, WI 2000-2001Research Aide U.S. Dept. of Energy, Iowa State University 1997-1998Head Cook/Supervisor Boy Scout Camp Decorah, Holmen, WI 1997Nature Counselor Boy Scout Camp Decorah, Holmen, WI 1996Scout Craft Director Boy Scout Camp Decorah, Holmen, WI 1995 AWARDS American Physical Society Travel Grant 2008Papastamatiou Scholarship 2008NASA / Wisconsin Space Grant Consortium Fellowship 2007-2008Bradley Fellowship, Lynde and Harry Bradley Foundation 2006-2008UWM Chancellor’s Fellowship 2004-2008ISU Foreign Language Student of the Year 1998ISU Dedicated Service Award 1997National Merit Scholarship 1996 PUBLICATIONS L.A. Anchordoqui, M.M. Glenz, and L. Parker, “Black Holes at the IceCube84neutrino telescope,” Phys. Rev. D 75 , 024011 (2007).M.M. Glenz and K. Uryu, “Circular solution of two unequal mass particles inPost-Minkowski approximation,” Phys. Rev. D 76 , 027501 (2007).M.M. Glenz and L. Parker, “Study of the Spectrum of Inflaton Perturbations,”to be submitted.