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COSMOLOGICAL PERTURBATIONS AND THEIR EFFECTS ON THE UNIVERSE: FROM INFLATION TO ACCELERATION By ETHAN R. SIEGEL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Ethan R. Siegel I dedicate this work to everybody who stood by me in my darkest hours ACKNOWLEDGMENTS I extend my deepest thanks to Jim Fry for guidance and sound advice throughout my ventures at the University of Florida. For helpful discussions and conversations on a multitude of topics over the years, I acknowledge Wayne Bomstad, Steve Detweiler, Leanne Duffy, and Lisa Everett. For support in various times of need, I thank HaiPing Cheng and the Quantum Theory Project, the University Women's Club, the University of Florida's Alumni Fellowship Program, the Department of Physics, and the College of Liberal Arts and Sciences. I also thank Filippos Klironomos for his assistance in the incarnation of this electronic dissertation. Extensive use has been made of NASA's Astrophysics Data System's bibliographic services, as well as Spires' highenergy physics database. On the topic of cosmological inhomogeneities, I acknowledge Dan Chung and Uros Seljak for their helpful input. I wholeheartedly thank Ed Bertschinger for making his COSMICS code available for public use, which has proved invaluable in understanding and computing various aspects of cosmological perturbation theory. Jesus Gallego and Jaime Zamorano are thanked for providing unpublished data from their surveys of local HII galaxies, and Rafael Guzman, Chip Kobulnicky, David Koo, and Mariano Moles are thanked for valuable assistance in my work and understanding of starburst galaxies. In addition to the aforementioned people, I also acknowledge Eanna Flanagan, Konstantin Matchev, Pierre Ramond and Bernard Whiting for discussions on gravitational radiation in extra dimensions. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................. ............... 4 LIST OF TABLES .................................... 7 LIST OF FIGURES ...... ........... ............ ...... 8 A BST RA C T . . . . . . . . .. .. 9 CHAPTER 1 INTRODUCTION TO COSMOLOGICAL PERTURBATIONS ........ 10 1.1 Energy Density ............. .... ............. 10 1.2 Theory of Inflation ...... .......... ........... 11 1.3 Cosmological Perturbations from Inflation ......... ......... 12 1.4 Cosmological Evolution in a Perturbed Universe ....... ..... ... 13 1.5 Nonlinear Evolution of Perturbations .......... ....... ... 17 2 THE GRAVITATIONAL WAVE BACKGROUND ......... ........ 21 2.1 Primordial Gravitational Waves ................. .... .. 21 2.2 Extra Dimensions .................. ............. .. 22 2.3 A Thermal Graviton Background ................ .... .. 24 2.4 Detection of Extra Dimensions .................. ...... .. 25 2.5 Alternative Thermalization Mechanisms .................. .. 27 2.6 Problems of Extra Dimensions .................. ..... .. 28 2.7 Summary ................... .. .... .......... 29 3 STRUCTURE FORMATION CREATES MAGNETIC FIELDS ... ..... .30 3.1 Introduction ................. . . ... 30 3.2 Magnetic Fields: Background .................. .... .. 31 3.3 Cosmological Perturbations .................. ..... .. 34 3.3.1 Cold Dark Matter .................. ........ .. 35 3.3.2 Light Neutrinos .................. .......... .. 36 3.3.3 Photons .................. .............. .. 37 3.3.4 Baryons ....... ....... ........... ..... .. 38 3.3.5 Charge Separations .................. ........ 40 3.4 Magnetic Fields .................. .............. .. 45 3.5 Discussion .......... ......... ................. 47 4 EFFECTS ON COSMIC EXPANSION ................ .... .. 52 4.1 Accelerated Expansion .................. .......... .. 52 4.2 Effects of Inhomogeneities .................. ........ .. 53 4.3 Effects on the Expansion Rate .................. ..... .. 56 4.4 Contributions of Nonlinear Inhomogeneities ................ .. 56 4.5 Variance of the Energy in Inhomogeneities ..... . . ..... 60 4.6 Sum mary .................. ............... .. .. 62 5 FULLY EVOLVED COSMOLOGICAL PERTURBATIONS . . ... 65 5.1 Precision Cosmology .................. .......... .65 5.2 Selection of the Data Sample .................. ..... .. 67 5.3 Constraints on Cosmological Parameters .................. .. 71 5.4 Conclusions and Future Prospects ................ .... .. 75 6 CONCLUDING REMARKS ............... .......... .. 77 6.1 Creation of Perturbations .................. ........ .. 77 6.2 Early Evolution of Perturbations ................... . 78 6.3 Final State of Perturbations .................. ...... .. 80 6.4 Fate of the Universe .................. ......... .. .. 82 APPENDIX A ERRORS IN HIGHZ GALAXIES AS DISTANCE INDICATORS . ... 85 A.1 Universality among H II Galaxies ................. .. .. 85 A.2 Systematic Errors .................. ............. .. 87 A.3 Measurement Uncertainties. .................. .. .. 89 A.4 Statistical Errors .................. ............. .. 91 B ON AN ELECTRICALLY CHARGED UNIVERSE . . ...... 93 B.1 Introduction .................. ................ .. 93 B.2 Generating a Net Electric Charge ................ .... .. 94 B.3 N',wtonian Formulation ................... ......... 96 B.4 Relativistic Formulation ................... . .. 101 B.5 Discussion .................. ................ .. 105 REFERENCES ................... ......... ....... 108 BIOGRAPHICAL SKETCH .................. ............. .. 125 LIST OF TABLES Table page 51 Table of Selected HighRedshift Galaxies ......... ....... 69 LIST OF FIGURES Figure 21 Parameter space for a thermal gravitational wave background page 24 5q and Oq/H as a function of redshift .................. ..... 44 Spectral energy dnrity of the magnetic field ....... . . 48 Spectral density of gravitational potential energy ................ ..57 Fractional contributions of energy in inhomogeneities . . 58 Fractional contributions of inhomogeneities at second order . . .... 61 Fluctuation in potential energy vs. cutoff scale . . ....... 63 Distance modulus vs. redshift .................. ......... .. 72 Constraints on Qm and QA . . . . . .. 774 log Mz vs. log LHE for H II and starburst galaxies ................ ..88 Simulation of velocity dispersions for starforming regions . . ..... 90 Expansion factors for positive and negative charges ............... 99 Evolution of a net charge asymmetry .................. ....... 102 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COSMOLOGICAL PERTURBATIONS AND THEIR EFFECTS ON THE UNIVERSE: FROM INFLATION TO ACCELERATION By Ethan R. Siegel December 2006 Chair: James N. Fry Major Department: Physics The universe is, on the largest scales, nearly perfectly isotropic and homogeneous. This degree of smoothness was accentuated in the past, when density inhomogeneities departed from perfect uniformity by only thirty parts per million. These tiny imperfections in the early universe, however, have had enormous impact in causing the universe to evolve into its present state. This dissertation examines the role of these cosmological perturbations throughout various important events during the history of the universe, including inflation, linear and nonlinear structure formation, and the current phase of accelerated expansion. The spectrum of perturbations is calculated in the context of extra dimensions, and shown under which conditions it can be thermal. The effects of gravitational collapse are shown to generate magnetic fields, but not to significantly alter the expansion rate or cause acceleration. Finally, Lymanbreak galaxies are examined as a possible distance indicator, and it is found that they may emerge as a powerful tool to better understand the energy content of the universe. CHAPTER 1 INTRODUCTION TO COSMOLOGICAL PERTURBATIONS 1.1 Energy Density The universe, as observed today, is filled with intricate and complex structure. Looking at length scales on the size of an individual planet ( 1010 pc), the solar system (_ 103 pc), the galaxy ( 101 Mpc), or even clusters of galaxies (~ 10 Mpc), it is evident that there are large departures from the average value of energy density, p. A measure of the departure from homogeneity at any position is given by the quantity 6, where &p f (11) P P On very small (i.e., planetary) scales, density contrasts can be as high as 6 ~ 1030. However, as larger and larger scales are examined, the density contrast of a typical point in space is found to be much smaller. When scales of 0(100 Mpc) or larger are examined, it is found that density contrasts are nearly always small, such that 6 < 1 [1]. What can be learned from this is that the universe is, on its largest scales, very nearly isotropic (the same in all directions) and homogeneous (the same at all positions in space). In the framework of general relativity, a universe that is both isotropic and homogeneous is described by the FriedmannRobertsonWalker metric, dst2dt + a2(t)(dx di), (12) where a(t) is the scale factor of the universe. Because the universe is expanding and has a finite age, it is manifest that the degree of isotropy and homogeneity which is observed today was greater in the past. This is confirmed by observations of the relic radiation from the big bang, known as the cosmic microwave background (CMB), which shows the universe to have an amplitude of density fluctuations of 6 3 x 105 [2]. These fluctuations in density, although insignificant when compared to the homogeneous part at early times, play a vital role in the universe's evolution, bringing it from a state of nearperfect homogeneity to the complex nonlinear structures observed today. 1.2 Theory of Inflation The theory of inflation [3] provides a mechanism to put these primordial cosmological perturbations in place at the time of the big bang. The big bang, as a cosmological theory, is the only compelling theory in the context of Einstein's general relativity that provides an explanation for the presence of the CMB radiation, the observed Hubble expansion of the universe, and the light element abundances (through big bang nucleosynthesis). It is shown in Peebles 1993 [4] that all reasonable alternatives to the big bang scenario either fail to reproduce one of the above three observations or cannot be compatible with general relativity. The big bang, however, is not an origin of the universe, but is rather a set of initial conditions. It is the above theory of inflation that naturally produces these initial conditions. For the big bang theory to evolve into a universe which is compatible with observations, the initial conditions must be that the global curvature of the universe is spatially flat to an accuracy of 2 per cent [5]. The temperature and density of the universe must be uniform across scales far larger than the horizon, and the density of magnetic monopoles in the universe must be very small. These three problems are known as the flatness, horizon, and monopole problem. The method by which inflation solves this problem is to have a de Sitterlike phase of expansion during the very early universe. de Sitter expansion is characterized by the metric ds2 = dt2 + eAt(d di), (13) which is similar to equation (12), except that the scale factor a = et, where A is a constant. A is related to the expansion rate of the universe at that time by the equation A = H2, where H = a/a is the Hubble parameter. From equation (13), it is manifest that the universe is expanding at an exponentially fast rate. Given enough time, the universe can expand by an arbitrary number of efoldings. A universe expanding in this way will have its matter density reduced by a factor of Pi (a(ti) = e (tj t) (14) Pf \a(tf) ) where pi and ti are the density and age of the universe at the start of inflation, and pf and tf are the density and age at the end of inflation. This removes any initial densities or differences in density, solving both the horizon and monopole problems. Additionally, any initial curvature (departure from flatness) will be driven away by a factor of a2(ti)/a2(tj), providing a solution to the flatness problem as well. 1.3 Cosmological Perturbations from Inflation In addition to setting up the initial conditions necessary for the big bang, inflation also predicts very slight departures from homogeneity in the universe, producing fluctuations in both energy density [69] and in gravitational radiation [10, 11]. It is these predictions for the departures from perfect homogeneity, produced by quantum fluctuations, which are then stretched during inflation across all scales, that are the focus of this work. The perturbations produced by inflation are Gaussian in their statistical properties, and are also scaleinvariant. The perturbations in gravitational radiation are constrained to be a very small fraction of the energy density in the universe [12]. Nevertheless, the detection of such gravitational radiation and measurement of its properties would have the capabilities to tell us much about the early universe, as gravitational radiation is expected to be decoupled from the rest of the universe from the time of its creation. If a scaleinvariant spectrum of gravitational waves were observed, it would be a further great confirmation of the inflationary paradigm, and constraints on the model of inflation could be inferred. However, there also exists the possibility that the spectrum of gravitational waves could be thermal. A thermal gravitational radiation spectrum could result from many possibilities, one of which is as a signature of extra dimensions [13]. This is discussed at length in Chapter 2 of this work. When inflation comes to an end, the universe reheats. The process of reheating transfers the energy from the vacuum (which was responsible for the rapid expansion) into matter and energy. This universe is described, in the homogeneous approximation, by the FriedmannRobertsonWalker metric of equation (12). However, it is the density inhomogeneities, or primordial cosmological perturbations, that will lead to the formation of all structure in the universe. Therefore, the equation of interest as a starting point for much of the remainder of this work is that for a perturbed FriedmannRobertsonWalker universe, ds2 = a2(T)[(1 + 20)d2 + (1 20)dx di], (15) where 4 and 4 are the scalarmode perturbations to the gravitational potential. Of course, it makes no difference which gauge is chosen [14], as the physics of cosmological perturbations is the same in all gauges. The preference of the author is to work in the conformal Newtonian gauge (also known as the longitudinal gauge), as chosen in equation (15). 1.4 Cosmological Evolution in a Perturbed Universe As the universe cools from its initial, smooth, hot, dense state, many cosmologically interesting phenomena occur (and many others may occur, dependent upon the reheat temperature of the universe and the physics involved at very high energy scales). A cosmological baryon asymmetry must be generated at very early times. As the observed baryontophoton ratio today is r7 6.1 x 1010, an asymmetry in the number density of baryons over antibaryons of this magnitude must be produced. The process by which this occurs is generically referred to as baryogenesis, and there are many different times at which it may occur, including at the GUTscale, at the electroweak scale, through leptogenesis, or through the AffleckDine scenario (see Dine and Kusenko 2004 [15] for a review). Also, in the early universe, a substantial amount of dark matter must be generated. This can occur either through freezeout of a stable, thermally produced relic, through the misalignment of the vacuum, through the production of a sterile, heavy neutrino, or through a perhaps more exotic mechanism (see Bertone, Hooper and Silk 2004 [16] for a review). These unsolved puzzles, along with the many questions surrounding the early universe physics of electroweak symmetry breaking, the QCD phase transition, and (possibly) supersymmetry breaking, are expected to be unaffected by the presence of cosmological perturbations. As the universe evolves, most of the physics that occurs in the earliest stages is expected to occur exactly as it would in a perfectly homogeneous FriedmannRobertsonWalker universe. One possibility in the very early universe, however, for which cosmological perturbations may play a seminal role is if the universe begins with (or obtains at early times) an asymmetry in its net charge. It has been pointed out that a large net charge in the universe would be ruinous at early times for cosmological 4He synthesis [17] and for the cosmic microwave background [18]. Although a conclusive solution is beyond the scope of this dissertation, cosmological perturbations in an expanding universe may have the capability of driving an electrically charged universe to a neutral state. This possibility, and preliminary work on the subject, can be found in Appendix B. As the universe continues to expand and cool, the building blocks of the universe begin to form. After the QCD phase transition, quarks and gluons become bound into hadrons. Unstable particles decay and/or coannihilate, leaving the universe devoid of exotic particles. Neutrinos freezeout, and decouple from the rest of the universe. Electrons and positrons coannihilate, leaving an electron asymmetry that matches up nearly perfectly with the proton asymmetry. When the universe cools substantially so that stable deuterium can form without being destroyed by the thermal photon bath, nucleosynthesis occurs, producing deuterium, 3He, 4He, and 7Li. Nucleosynthesis is complete roughly four minutes after the big bang. Although there has been work in the past suggesting alternative, complex models of nucleosynthesis (such as inhomogeneous nucleosynthesis) to be slightly favored (see Steigman 2006 [19] for a review of bigbang nucleosynthesis and its alternatives), the standard picture of big bang nucleosynthesis appears to match up perfectly well with observations within the systematic errors [20]. It is generally assumed that no interesting physics occurs until the time of recombination (where electrons and ions combine to form neutral atoms), roughly 380,000 years later. The only things of note which occur over that era are that cosmological perturbations grow according to the M6szdros effect [21], and the universe transitions from a radiationdominated state to a matterdominated one. However, there is a very interesting and subtle effect that occurs during this time. As tr (the baryontophoton) ratio is very small and the universe is still quite hot and dense during this era, every ion and electron is consistently bombarded by these high energy photons. Ions have comparable charges to electrons, but their masses are orders of magnitude greater. The scattering cross sections of charged particles with photons scales (for nonrelativistic scattering) as 8r( )2 (16) 3 Vmc" where q is the charge and m is the mass. As a result of the differences in mass and cross section, electrons are affected by interactions with photons in a much more profound way than ions. While the Coulomb forces keeps the electrons and protons tightly coupled, the momentum transfer from photons works to create charge separations and currents during the radiation era. The tightly coupled component is dominant, and behaves as a baryonic component in cosmological perturbation theory (see Ma and Bertschinger 1995 [22] for a very sophisticated treatment). In contrast, the charge separations and currents created by momentum transfer are very small, but nonetheless are of great import for the generation of magnetic fields at early times [23]. The generation of magnetic fields in the young universe by this mechanism is detailed in Chapter 3, which also discusses the possibility that the origins of presently observed cosmic magnetic fields may lie in this mechanism. As the universe transitions from a radiationdominated state to a matterdominated one, gravitational inhomogeneities begin to substantially grow for the first time. The M6szdros effect dictates that an inhomogeneity 6 grows as 6(t) = (l+ ) 6(to), (17) where Y is given by Pm(t) Y (18) Pr(t) from an initial time to until the time of interest, t, where Pm is the matter density and pr is the radiation density. This approximation is valid throughout the linear regime of gravitational collapse and structure formation, and provides an accurate description for the growth of overdense inhomogeneities. During this epoch of complete ionization, electrons and ions are constantly interacting with one another, and attempting to form neutral atoms. There are two processes that impede the formation of neutral atoms. The first, which delays the onset of neutral atom formation (known as recombination), is that the baryontophoton ratio, r7, is so low. Even though the photon temperature is significantly below the ionization energy of a neutral atom (T, < 13.6 eV), the number of photons per baryon is very great, and their energy follows a Poisson distribution. As a result, there are still enough photons of sufficient energy to keep the universe 100 per cent ionized even when the average temperature of the universe is significantly below the typical atomic ionization energy. The second process that is responsible for impeding the formation of a neutral, transparent universe is the fact that each Lymanseries photon (transition to the ground state of hydrogen) emitted by a recombining atom will encounter and reionize another neutral atom. If, however, the emitted photon has enough time to redshift sufficiently that it cannot reionize another atom, the universe will net one neutral atom. Also, a rare twophoton emission process will allow an atom to recombine without emitting a Lymanseries photon. This process of recombining the majority of the atoms in the universe takes about 105 years to complete [5], and drops the ionization fraction (Xe) of the universe from Xe 1 to Xe 104 [24]. After this point, ions and electrons can no longer efficiently find one another, and the process of recombination freezes out. As the cosmic background of photons is no longer constantly scattering off of electrons, it freely streams from the epoch of recombination until the present day, making the epoch of recombination synonomous with the surface of last scattering. 1.5 Nonlinear Evolution of Perturbations Once recombination has occurred, the universe is in a matterdominated, expanding state, full of small density inhomogeneities on all scales. It is in this postrecombination universe that largescale structure formation begins to occur. The density inhomogeneities, initially, have an amplitude of ~ 2 x 105 pm, with the fluctuations having a Gaussian distribution. As the fluctuations evolve according to linear perturbation theory (with overdensities growing according to the M6szaros effect) initially, and as gravitational perturbations continue to grow, gravitational collapse goes nonlinear, causing a rapid acceleration in structure formation. The overdense regions on small scales go nonlinear first, as they enter the horizon (and thus become causally connected) first. The perturbations, on the other hand, are across all scales, and have a roughly scaleinvariant spectrum (where the power spectrum, P(k), scales as P(k) oc k"), where the spectral index n 1. The structure which arises from this follows scaling solutions, as described in Fry 1984 [25] and Schaeffer 1984 [26], for example. The result of all of this is that an initially smooth universe with only very slight perturbations in energy density becomes a complex web of structure, with substantial power on both small (i.e., galactic) and large (i.e., supercluster) scales. (For a very interesting comparison of numerical simulations of structure formation through the nonlinear regime up to the present day, the reader is referred to O'Shea et al. 2005 [27].) While this largescale structure forms, the universe continues to expand and cool, dropping from a temperature of T 3000 K at recombination to a temperature of T 2.725 K at present. The various epochs of the universe, home to the onset of extremely interesting physics, are tracked most easily by redshift, z, defined by a 1 + z, (19) a(t) where a(t) is the scale factor of the universe at a given time, and ao is the scale factor at present. When gravitational collapse occurs to a sufficient extent on small scales, the mass collected in a small area of space becomes large enough to ignite nuclear fusion. This is the epoch at which the first stars form. Exactly at what epoch star formation begins is very important for understanding the evolution of matter and structure in our universe. A signature of the formation of the first stars would be a surefire signature of nonlinear collapse. The transition from a smooth, linear universe (such as the universe at the time of recombination) to a highly complex, nonlinear one (observed today) is not yet well understood. Recently, many have discussed the possibility that the gravitational energy bound in nonlinear inhomogeneities could backreact, and significantly impact the expansion rate [2830]. It appears that the impact on the expansion rate is insignificant, however [3134]. This physical process and its effects on the universe are discussed in great detail in Chapter 4. The densest regions of nonlinear structure become home to the first stars, as illustrated in adaptive mesh refinement simulations [35]. The data from the WMAP satellite indicate that the optical depth of the universe, r, is quite large [5]. From this information, it appears that the first stars turned on very early, as the presence of a large number of stars will reionize the neutral gas that formed during recombination. From the optical depth, which is measured to be 7 = 0.17 0.04, it appears that reionization occurs at roughly 11 < z < 30. However, the observation of a GunnPeterson trough [36] in quasar spectra around z 6 [37] indicates that reionization is not complete until that epoch. Seemingly bizarre solutions, such as a double epoch of reionization [38], have been proposed to remedy this situation. Future release of data from satellites exploring the cosmic microwave background may yield lower values of r, which would be consistent with a more simplistic explanation of gradual reionization. Once the first complex nonlinear structures form, they continue to evolve, with the densest regions attracting the most matter and forming the most massive structures. Galaxies grow through both monolithic collapse and a series of hierarchical mergers, and via further gravitational collapse on larger scales, the first clusters of galaxies will form as well. The types of objects which can be observed at early times are very bright galaxies (in the optical and infrared) and quasars (primarily in the radio), as well as intervening objects along the line of sight (through absorption and the Lymana forest). For a flat universe that contained a critical density in matter, the expansion rate would continue to decrease as the matter density diluted, following the Hubble law of equation (110), H2 8 p, (110) where Pm is again the matter density (equal to the critical density) and H is the Hubble expansion parameter. However, the universe's expansion rate, as inferred from a combination of many sources of data (see Chapter 5 and references therein) is consistent with about 30 per cent of the energy density in matter and about 70 per cent in some type of vacuum energy. The expansion law, then, appears to obey equation 111 H2 2 8G (Pm+ + P) (111) a 3 where PA is the energy density in vacuum energy, and the sum of matter density and vacuum energy density is equal to the critical density. The data from type Ia supernovae have been used to illustrate and support the fact that the picture of the universe is inconsistent without a vacuum energy term in the equation for the Hubble law [39]. However, due to systematic errors inherent in any single observational method, it is vital to collect data from a large number of methods. Chapter 5 investigates the possibility of using a new method (first detailed in Melnick, Terlevich and Terlevich 2000 [40] and first attempted in Siegel et al. 2005 [41]) to measure the cosmological parameters of matter and vacuum energy density in the universe. As with any type of distance indicator, the method of Chapter 5, to use Lymanbreak galaxies as a distance indicator, is subject to many sources of error, both random and systematic. These errors are detailed in Appendix A. The data sets available are now sufficient to paint a coherent picture of the universe and its energy contents very well on the largest scales, and relatively well on even small scales [12]. There are many interesting problems and phenomena in the universe that are hitherto unexplained, yet physics of the answers may lie in something as simple as departures from the ideal model. The remainder of this work details some instances where cosmological inhomogeneities, whether at early times or late times, on large or small scales, may play a vital role in understanding the universe. Finally, Chapter 6 will summarize the major results of Chapters 2 through 5, and will point towards future avenues of investigation, such as determining the fate of the universe. CHAPTER 2 THE GRAVITATIONAL WAVE BACKGROUND Inflationary cosmology predicts a lowamplitude graviton background across a wide range of frequencies. This chapter shows that if one or more extra dimensions exist, the graviton background may have a thermal spectrum instead, dependent on the fundamental scale of the extra dimensions. The energy density is shown to be significant enough that it can affect nucleosynthesis in a substantial way. The possibility of direct detection of a thermal graviton background using the 21cm hydrogen line is discussed. Alternative explanations for the creation of a thermal graviton background are also examined. 2.1 Primordial Gravitational Waves One of the most powerful windows into the early universe are backgrounds of particles whose interactions have frozenout. The primordial photon background, the primordial baryon background and the primordial neutrino background are all examples of particles that were once in thermal equilibrium. At various times during the history of the universe, the interaction rate of the species in question dropped below the Hubble expansion rate of the universe, causing the species in question to freezeout. The primordial photon background is observed as the cosmic microwave background (CMB), the baryon background is observed as stars, galaxies, and other normal matter, and the neutrino background, although not yet observed, is a standard component of big bang cosmology. In addition to these backgrounds, a primordial background of gravitons (or, equivalently, gravitational waves) is expected to exist as well, although it, too, has yet to be detected. The frequency spectrum and amplitude of this background have the potential to convey much information about the early universe. This chapter focuses on using the cosmic gravitational wave background (CGWB) as a probe of extra dimensions. The success of the inflationary paradigm [3] in resolving many problems associated with the standard bigbang picture [42] has led to its general acceptance. Inflationary big bang cosmology predicts a stochastic background of gravitational waves across all frequencies [10], [11]. The amplitude of this background is dependent upon the specific model of inflation, but the fractional energy density in a stochastic CGWB is constrained [12] to be g, < 0(1010). (21) In inflationary cosmology, the predicted CGWB, unlike the CMB and the neutrino background, is nonthermal. Gravitational interactions are not strong enough to produce a thermal CGWB at temperatures below the Planck scale (mpi & 1.22 x 1019 GeV). As the existing particles in the universe leave the horizon during inflation, the only major contributions to the energy density will be those particles created during or after reheating, following the end of inflation. Unless the reheat temperature (TRH) is greater than mpl, gravitational interactions will be too weak to create a thermal CGWB. The measurement of the magnitude of the primordial anisotropies from missions such as COBE/DMR [43] and WMAP [5] provides an upper limit to the energy scale at which inflation occurs [44]. From this and standard cosmological arguments [45], an upper limit on TRH can be derived to be TRH ~ 6.7 x 1018 ()1/4 i GeV, (22) \ ) where g* is the number of relativistic degrees of freedom at TRH, tpl is the Planck time, and t4 is the lifetime of the inflation. A stronger upper limit on TRH ( 108 1010 GeV) can be obtained from nucleosynthesis [46] if supersymmetry is assumed. In all reasonable cases, however, TRH << mpl, indicating that the CGWB is nonthermal in inflationary cosmology. 2.2 Extra Dimensions If the universe contains extra dimensions, however, predictions about the shape and amplitude of the CGWB may change drastically. Cosmologies involving extra dimensions have been wellmotivated since Kaluza [47] and Klein [48] showed that classical electromagnetism and general relativity could be unified in a 5dimensional framework. More modern scenarios involving extra dimensions are being explored in particle physics, with most models possessing either a large volume [49, 50] or a large curvature [51, 52]. Any spatial dimensions which exist beyond the standard three must be of a sufficiently small scale that they do not conflict with gravitational experiments. The 3+1 dimensional gravitational force law has been verified down to scales of 0.22 mm [53]. Thus, if extra dimensions do exist, they must be smaller than this length scale. Although there exist many different types of models containing extra dimensions, there are some general features and signals common to all of them. In the presence of 6 extra spatial dimensions, the 3+6+1dimensional action for gravity can be written as S = f d4x f d l'y1G +N / G'N = GN 2, (23) where g is the 4dimensional metric, GN is Newton's constant, g', G', and R' denote the higherdimensional counterparts of the metric, Newton's constant, and the Ricci scalar, respectively, and mD is the fundamental scale of the higherdimensional theory. In 3+6 spatial dimensions, the strength of the gravitational interactions scale as (T'/mo)(1/2) If 6 = 0, then mD = mpl, and standard 4dimensional gravity is recovered. When energies in the universe are higher than the fundamental scale mD, the gravitational coupling strength increases significantly, as the gravitational field spreads out into the full spatial volume. Instead of freezing out at r O(mpj), as in 3+1 dimensions, gravitational interactions freezeout at O(mD) [49]. (mD can be much smaller than mpl, and may be as small as TeVscale in some models.) If the gravitational interactions become strong at an energy scale below the reheat temperature (mD < TRH), gravitons will have the opportunity to thermalize, creating a thermal CGWB. Figure 21 illustrates the available parameter space for the creation of a thermal CGWB in the case of large extra dimensions, following the formalism in Giudice, Rattazzi and Wells 1999 [54]. and fractional energy density (f~) of a thermal CGWB are 7r2 (3.91 4/13 P3= (TcM)4, (24) S= P 3.1 x 104 (9)4/3, (25) Pc where pc is the critical energy density today, TCMB is the present temperature of the CMB, and g, is the number of relativistic degrees of freedom at the scale of mD. 9+ is dependent on the particle content of the universe, i.e. whether (and at what scale) the universe is supersymmetric, has a KK tower, etc. Other quantities, such as the temperature (T), peak frequency (v), number density (n), and entropy density (s) of the thermal CGWB can be derived from the CMB if g, is known, as (3.91\ (3.91 1Tg = H  Sg = S.MB 8* M 9* g =TCMB Sg VSCMB \ g , These quantities are not dependent on the number of extra dimensions, as the large discrepancy in size between the three large spatial dimensions and the 6 extra dimensions suppresses those corrections by at least a factor of r 1029. As an example, if mD is just barely above the scale of the standard model, then g, = 106.75. The thermal CGWB then has a temperature of 0.905 Kelvin, a peak frequency of 19 GHz, and a fractional energy density Q, 6.1 x 107 2.4 Detection of Extra Dimensions Although the fractional graviton energy density is expected to be small today, it may be detectable either indirectly or directly. Nucleosynthesis provides an indirect testing ground for a thermal CGWB. Standard bigbang nucleosynthesis predicts a helium4 abundance of Yp = 0.2481 0.0004 [55]. With a thermal CGWB included, the expansion rate of the universe is slightly increased, causing neutronproton interconversion to freezeout slightly earlier. A thermal CGWB can be effectively parameterized as neutrinos, as they serve the same function at that epoch in the universe (as noncollisional radiation). The effective number of neutrino species is increased by N,eff 27.1 (g,)4/3, or 0.054 (for g, = 106.75). This would yield a new prediction of Yp = 0.2489 0.0004 for helium4. Although observations are not yet able to discriminate between these two values, the constraints are tightening with the advent of recent data [56]. An increase in the precision of various measurements, along with an improvement in the systematic uncertainties, may allow for the indirect detection of a thermal CGWB. Direct detection of a thermal CGWB is much more challenging, but would provide quite strong evidence for its existence. Conventional gravitationalwave detectors include cryogenic resonant detectors [57], which have evolved from the bars of Weber [58], doppler spacecraft tracking, and laser interferometers [59]. The maximum frequency that these detectors can probe lies in the kHz regime, whereas a thermal CGWB requires GHzrange detectors. An interesting possibility for detection may lie in the broadening of quantum emission lines due to a thermal CGWB. Individual photons experience a frequency shift due to gravitational waves [60]. For a large sample of radiofrequency photons in a gravitational wave background, the observed line width (W) will broaden by 106.75)I/3 AW ho 10, (27) vto 9 where to is the present age of the universe, v is the peak frequency of the thermal CGWB and ho is the metric perturbation today due to the thermal CGWB [61]. As 0(1031) is a very small broadening, a radio line with a narrow natural width is the preferred candidate to observe this effect. One possibility for this type of observation is the 21cm emission line of atomic hydrogen. So long as the emitting atoms and the detectors are sufficiently cooled, broadening due to thermal noise will be suppressed below AW. Because the lifetime (1/F) of the excited state of hydrogen is large (' 107 yr) and the frequency of the emitted light (v,) is high ( 109 Hz), the natural width (W) is among the smallest known F 2.869 x 1015 1s W 2.02 x 1024 (28) v 1.42040575179 x 109 s1 The width of the 21cm line is regrettably seven orders of magnitude larger than the expected broadening due to a thermal CGWB. Extraordinarily accurate measurements would need to be taken for direct detection of this background. Additionally, temperatures of the atoms and detectors would need to be cryogenically cooled to n 1018 Kelvin to suppress thermal noise below AW. This last criterion is far beyond the reach of current technology, and either a major advance or experimental innovation would be required to measure the desired effect using this technique. 2.5 Alternative Thermalization Mechanisms Extra dimensions are not the only possible explanation for the existence of a thermal CGWB. Currently, there are three known alternative explanations that would also create a thermal CGWB. They are as follows: there was no inflation, there was a spectrum of lowmass primordial black holes that have decayed by the present epoch, or the gravitational constant is timevarying (the Dirac hypothesis). Each alternative is shown below to face difficulties that may make extra dimensions an attractive explanation for the creation of a thermal CGWB. The predictions of inflation are numerous [44], and many have been successfully confirmed by WMAP [5]. The major successes of inflation include providing explanations for the observed homogeneity, isotropy, flatness, absence of magnetic monopoles, and origin of anisotropies in the universe. Additionally, confirmed predictions include a scaleinvariant matter power spectrum, an Q = 1 universe, and the spectrum of CMB anisotropies. To explain a thermal CGWB by eliminating inflation would require alternative explanations for each of the predictions above. Although alternative theories have been proposed, as in Hollands and Wald 2002 [62], they have been shown to face significant difficulties [63]. The successes of inflation appear to suggest that it may likely provide an accurate description of the early universe. Primordial black holes with masses less than 1015 g would have decayed by today, producing thermal photons, gravitons, and other forms of radiation. Density fluctuations in the early universe, in order to produce a large mass fraction of lowmass primordial black holes, and not to produce too large of a mass fraction of highmass ones, favor a spectral index n that is less than or equal to 2/3 [64]. Accepting the observed scaleinvariant (n 1) spectrum of density fluctuations [65] may disfavor primordial black holes as a reasonable candidate for creating a thermal CGWB. The Dirac hypothesis states that the difference in magnitude between the gravitational and electromagnetic coupling strengths arises due to time evolution of the couplings [66]. If true, gravitational coupling would have been stronger in the early universe. At temperatures well below the Planck scale, gravity would have been unified with the other forces, creating a thermal CGWB at that epoch. However, this hypothesis produces consequences for cosmological models that are difficult to reconcile [67], and any time variation is severely constrained by geophysical and astronomical observations [68]. The acceptable limits for variation are small enough that they cannot increase coupling sufficiently to generate a thermal CGWB subsequent to the end of inflation. The difficulties faced by each of these alternative explanations points towards extra dimensions as perhaps the leading candidate for the creation of a thermal CGWB. 2.6 Problems of Extra Dimensions There exist two major obstacles to the construction of a more complete phenomenological model containing extra dimensions with mD < TRH. The first of these is the moduli problem [6'] String moduli interactions with standard model fields are highly suppressed, leading to a long lifetime of the string moduli. String moduli decay, however, must be consistent with astrophysical constraints [70]. To accomplish this, string moduli need either a small production amplitude or very specific decay channels, which both require finetuning. The second problem is the overproduction of longwavelength tensor modes from inflation [71, 72]. While the shortwavelength modes (the modes inside the horizon when gravitational interactions freezeout) will thermalize, gravitational waves of longer wavelengths will be unaffected. As the scale of inflation must be above mD, the amplitude of these waves is expected to be large. This would leave an unacceptable imprint in the CMB. Both problems arise from the fact that at energies above mD, macroscopic gravity breaks down [73]. Although these problems may not be resolved until a quantum theory of gravity is realized, they do not change the fact that a thermal CGWB would arise from extra dimensions with mD < TRH. Furthermore, there is a more fundamental question concerning the nature of extra dimensions. The three observed spatial dimensions are quite large, on the order of _ 1028 cm. On the other hand, any extra spatial dimensions must be, at most, of a length scale less than 0.22 mm. It is very difficult to construct a compelling theory that naturally produces three large spatial dimensions and forces the rest to be small. A possible solution to this puzzle may lie in the work of Chodos and Detweiler 1980 [74], where it was shown that a universe with four spatial dimensions of initially comparable size may naturally evolve to a state with three large, expanding dimensions and one small, contracting one. 2.7 Summary This chapter has attempted to show that extra dimensions may be responsible for the production of a thermal gravitational wave background. A thermal CGWB, as opposed to the stochastic CGWB of standard inflationary cosmology, is a prediction of extra dimensions with a scale below the reheat temperature. The detection of a thermal CGWB, although challenging at present, would provide strong evidence for the existence of extra dimensions. The detected absence of a thermal CGWB would conversely disfavor the existence of extra dimensions up to the energy scale of the reheat temperature. CHAPTER 3 STRUCTURE FORMATION CREATES MAGNETIC FIELDS This chapter examines the generation of seed magnetic fields on all scales due to the growth of cosmological perturbations. In the radiation era, local differences in the ion and electron density and velocity fields are induced by momentum transfer from photons. The currents which flow due to the relative motion of these fluids lead to the generation of magnetic fields. Magnetic fields are created on all cosmological scales, peaking at a magnitude of 0(1023 Gauss) at the epoch of recombination. Magnetic fields generated in this manner provide a promising candidate for the seeds of magnetic fields presently observed on galactic and extragalactic scales. 3.1 Introduction The presence of magnetic fields on galactic and extragalactic scales is a major unsolved problem in modern astrophysics. Although the observational evidence for magnetic fields in largescale structures is overwhelming, there is no consensus as to their origins. The standard paradigm for the creation of these fields is the dynamo mechanism, in which an initial, small seed field is amplified by turbulence and/or differential rotation to account for the fields observed today. In principle, once a seed field is in place, it should be possible to follow its evolution and amplification from the collapse of structure and the effects of any relevant dynamos. In this chapter, a new mechanism for the generation of seed fields is put forward. It is argued that cosmological perturbation theory in the radiation era produces charge separations and currents on all scales, both of which contribute to magnetic fields. These seed fields persist until the onset of gravitational collapse, at which point field amplification and dynamo processes can magnify such seeds, possibly to the O(pG) scales observed today. This chapter illustrates that the generation of magnetic fields in this manner is a necessary consequence of structure formation. The magnitude of these seed fields is calculated, and it is shown that these seed fields may be sufficiently strong to account for all of the observed magnetic fields in largescale structures. The layout of this chapter is as follows: the next section gives an overview of the observational evidence for magnetic fields along with a brief theoretical picture of their generation. After that, there is an explanation of the novel idea that the early stages of structure formation in a perturbed universe generate magnetic fields. Subsequently, a detailed treatment of cosmological perturbations is presented, with a specific view towards the creation and evolution of local charge separations and currents. The magnitudes of the seed magnetic fields which arise via this mechanism as a function of scale and epoch are then calculated. Finally, the results of this mechanism are compared with competing theories. Also included is a discussion of avenues for future investigation of this topic, including possible observational signatures which would arise as predictions of this mechanism. 3.2 Magnetic Fields: Background In all gravitationally bound or collapsing structures in which the appropriate observations are made, magnetic fields with strength pG are seen [75]. The four major methods used to study astrophysical magnetic fields are synchrotron radiation, Faraday rotation, Zeeman splitting, and polarization of starlight. These observational techniques are detailed in depth in Ruzmaikin, Sokolov and Shukurov 1988 [76], with Faraday rotation often proving the most fruitful of the above methods. Magnetic fields have been found in many different types of galaxies, in rich clusters, and in galaxies at high redshifts. Spiral galaxies, including our own, appear to have relatively large magnetic fields of 0(10 pG) on the scale of the galaxy [77], with some (such as M82) containing anomalously strong fields up to _ 50 pG [78]. Elliptical and irregular galaxies possess strong evidence for magnetic fields (of order ~ pG) as well [79], although they are much more difficult to observe due to the paucity of free electrons in these classes of galaxies. Coherence scales for magnetic fields in these galaxies, as opposed to spirals, are much smaller than the scale of the galaxy. Furthermore, galaxies at moderate (z 0.4) and high redshifts (z > 2) have been observed to require significant ( MG) magnetic fields to explain their observed Faraday rotations [80, 81]. Magnetic fields are also observed in structures larger than individual galaxies. The three main types of galaxy clusters are those with cooling flows, those with radiohalos, and those devoid of both. Galaxy clusters with cooling flows are observed to have fields of 0.2to 3 G [82], the Coma cluster (a prime example of a radiohalo cluster) is observed to have a field strength r 2.5 pG [83], while clusters selected to have neither cooling flows nor radio halos still exhibit indications of strong (0.1 1 pG) fields [84]. There even exists evidence for magnetic fields on extracluster scales. An excess of Faraday rotation is observed for galaxies lying along the filament between the Coma cluster and the cluster Abell 1367, consistent with an intercluster magnetic field of 0.2 0.6 pG [85]. On the largest cosmological scales, there exist only upper limits on magnetic fields, arising from observations of the cosmic microwave background [86] and from nucleosynthesis [87], setting limits that on scales > 10 Mpc, field strengths are < 10s G. Observational evidence for magnetic fields is found in galaxies of all types and in galaxy clusters, both locally and at high redshifts, wherever the appropriate observations can be made. A review of observational results can be found in Valle6 1997 [88]. The theoretical picture of the creation of these fields, however, is incomplete. Fields of strength SpiG can be explained by the magnification of an initial, small seed field on galactic (or larger) scales by the dynamo mechanism [8991]. A protogalaxy (or protocluster) containing a magnetic field can have its field strength increased by many orders of magnitude through gravitational collapse [92, 93], and can then be further amplified via various dynamos. Dynamos which can amplify a small seed field into the large fields observed today involve helical turbulence (a) and/or differential rotation (w). Various types of these dynamos include the meanfield dynamo [76, 94, 95], the fluctuation dynamo [79, 96], and mergerdriven dynamos [97], among others. However, the dynamo mechanism does not explain the origin of such seed fields. While the initial seeds that grow into magnetic fields are anticipated to be small, they must still come from somewhere [98], and their existence is not explained by the dynamo mechanism alone. There are many mechanisms that can produce smallstrength magnetic fields on astrophysically interesting scales, either through astrophysical or exotic processes (see Widrow 2002 [75] for a detailed review). Exotic processes generally rely on new physics in the early universe, such as a firstorder QCD phase transition [99, 100], a first order electroweak phase transition [101, 102], broken conformal invariance during inflation [103, 104], specific inflation potentials [105], or the presence of charged scalars during inflation [106108]. Astrophysical mechanisms, in contrast, are generally better grounded in known physics, although they have difficulty generating sufficiently strong fields on sufficiently large scales. The difference in mobility between electrons and ions can lead to seed magnetic fields from radiationera vorticity [109, 110], from vorticity due to gasdynamics in ionized plasma [111115], from stars [116], or from active galactic nuclei [117]. Although there are many candidates for producing the seed magnetic fields required by the dynamo mechanism, none has emerged as a definitive solution to the puzzle of explaining their origins. The novel mechanism proposed in this chapter is that seed magnetic fields are generated by the scattering of photons with charged particles during the radiation era. Unlike the mechanism of [109, 110], which is disfavored [118] due to its requirement of substantial primordial vorticity (although see [119, 120] for an argument that some vorticity is necessary), the fields of interest here are generated by the earliest stages of structure formation, requiring no new physics. Ions (henceforth taken to be protons, for simplicity) and electrons are treated as separate fluids, with opposite charges but significantly different masses. The massweighted sums of their density and velocity fields will determine the evolution of baryons in the universe, and should agree with previous treatments, such as Ma and Bertschinger 1995 [22]. The difference of the ion and electron density and velocity fields, however, will provide a measure of local charge separation and of local current density, both of which contribute to magnetic fields. Since cosmological perturbations, which serve as seeds for structure formation, exist on all scales, it is expected that seed magnetic fields will be generated on all scales by this mechanism. The remainder of this chapter focuses on calculating the magnitude of the magnetic fields generated by this process and discussing their cosmological ramifications. 3.3 Cosmological Perturbations Although the early universe is isotropic and homogeneous to two parts in 105 [2], it is these small density inhomogeneities, predicted by inflation to occur on all scales [8], which lead to all of the structure observed in the universe today. As it is the early epoch of structure formation that is of interest for the creation of magnetic fields, this chapter calculates the evolution of inhomogeneities in the linear regime of structure formation. The most sophisticated treatment of cosmological perturbations in the linear regime to date is that of Ma and Bertschinger 1995 [22], which provides evolution equations for an inhomogeneous universe containing a cosmological constant, dark matter, baryons, photons, and neutrinos. This section extends their treatment to encompass separate proton and electron components. The massweighted sum of protons and electrons will recover the baryon component, whereas the difference of the density fields is representative of a charge separation, and the difference of the velocity fields is that of a net current. The dynamics of any cosmological fluid can be obtained, in general, from the linear Einstein equations (see Peebles and Yu 1970 [121], Silk and Wilkson 1980 [122], and Wilson and Silk 1981 [123] for earlier treatments). Although the choice of gauge does not impact the results, the Conformal Newtonian gauge leads to the most straightforward calculations. The metric is given by ds2 = a2(rT)[(1 + 2)dr2 + (1 2 i)dxidxi], (31) where V) 0 when gravitational fields are weak. The linear Einstein equations are then as follows: k2+ 3a( a) = 47rGa2 To, a a a 2 k 2 4 2 +(0 + 2)+ (2 2)+ ( ) = GGa2 6V a a a 3 3 k2( ) = 127rGa2(p + P)a, (32) where a is the shear term, which is negligible for nonrelativistic matter (but important for photons and neutrinos). For a cosmological fluid that is either uncoupled to the other fluids or massaveraged among uncoupled fluids in the early universe, the following evolution equations hold: S= (1+w)( 3) 3a(C, w)6, a a w c2 0 = (1 3w)O + 1c k26 k +k24 (33) a 1l+w 1+w where 6 is defined as the local density relative to the spatial average (6 Jp/p), 0 ik'v where v is the local peculiar velocity, and cs is the sound speed of the fluid. For individual components with intercomponent interactions, equation (33) must be modified to include these interactions. Examples of such interactions include the momentum transfer between photons and charged particles and the Coulomb interaction between protons and electrons. For protons, electrons, and cold dark matter (CDM), an equation of state w = 0 is assumed, and for radiation and neutrinos, w = 1. The master equations for each component of interest is computed explicitly in subsections 3.3.13.3.5. 3.3.1 Cold Dark Matter As the cold dark matter component (denoted by the subscript c) is collisionless and pressureless, it can be simply read off from equation (33) that the equations which govern its evolution are 6c = 0C + 3q, S= 0o + k24. (34) a Any cold (i.e., nonrelativistic), collisionless component will behave according to the dynamics given by equation (34). 3.3.2 Light Neutrinos For massless (or nearly massless) particles, pressure is nonnegligible. Additionally, the shear term (a) may be important as well. The only accurate way to compute the evolution of such a component of the universe is by integration of the Boltzmann Equation, which is given for light neutrinos (denoted by subscript v) by + ik(k ), = 4[ ik(k. )], (35) 87 in Fourier space. The approximation that neutrinos are massless and uncoupled is very good from an age of the universe of approximately t 1 s until the epoch of recombination. The evolution equations for light neutrinos are then 4 0, + 4 , 3 4 S= 1[l()  (+ 1)u(l+)], (36) 2+ (21+16) where cr, is related to F, by 2a, = FT2, and the index 1 governs the final equation for I > 2. Tlv is defined by the expansion of the perturbations in the distribution function, T,, 00 = (i)(21 + 1),T(k,r)Pi(k h), (37) l=0 where Pl(k n) are the Legendre polynomials. Equations (3537) are valid for any noncollisional species behaving as radiation. 3.3.3 Photons Photons (denoted by subscript 7), although similar to light neutrinos, evolve differently due to their large coupling to charged particles. Thomson scattering describes the interactions of photons with electrons, where the differential crosssection is given by the formula du 3aT S (1 + Cos2 ), (38) dQ 167r where UT is the Thomson crosssection [124]. Photons also scatter with protons, but with a crosssection suppressed by a factor of m /m2 (the masssquared ratio of electrons to protons). ~,, which is the polarizationsummed phasespace distribution for photons, is the same as the distribution function for neutrinos (see equation 37). Photons also contain a nonzero difference between the two linear polarization components, denoted by 9,. The linearized collision operators for Thomson scattering [22, 125127] yield the set of master equations for photons, 4 ^ = 3 + 4, 0 k2 7 + k2V + aneT (Ob 0), 8 3 9 J2 = 15 5 k gan(eTra' 10 k i = 21+ 1 [,(1+1) (I + 1)F,(1)] aneTJyl, yr = 1[m Y(m1) (m + 1)97(m+)] 2m + 1 +anTUT  ff g^YM (39) 11U 53 where F~o = 6,, F~, = 40,/3k, JTF2 = 2y,, the indices 1 and m are valid for 1 > 3 and m > 0, and the subscript b denotes the baryonic component, which is the massweighted sum of the electrons and protons. Electronphoton scattering is so dominant over protonphoton scattering as to render the latter negligible, but the electronproton coupling (via electromagentism) is sufficiently strong that, to leading order, those two fluids move in kinetic equilibrium. 3.3.4 Baryons The net behavior of the baryonic component can be derived from combining the massweighted contributions of the proton fluid and the electron fluid. Both protons and electrons contain all of the terms present in the CDM equations (see section 3.3.1), but additionally contain important soundspeed terms and terms arising from Thomson scattering. Additionally, the Coulomb interaction enters through the contribution of the electric field to the To components of the stressenergy tensor. The coupling of the Coulomb interaction to density inhomogeneities can be calculated through a combination of the electromagnetic Poisson equation, V24 = V = 47rpc, (310) where Pc is the electric charge density, and the Euler equation, l(a + 1(v V) = V ql + C, (311) a at a a ma with q/m as the chargetomass ratio of the particle in question and C the collision operator. The Coulomb contribution appears as 47re(np ne)qi/mi in the evolution equation for Oi, where i denotes a species of particle with a mass mi and charge qi. The evolution equations are therefore 6e = 0e + 3, e = ae + C k ,e + k2 a 47re2 + Fe(0 e)  (n ne), (312) me for electrons (denoted by subscript e), and Jp = Op + 30, O, = aOP,+c k 2 +k24 a 47re2 + FP,(, O) + (n ne), (313) Tmp for protons, denoted by subscript p, where the damping coefficients for electrons (Fe) and protons (Fp) are given by Fe 44p1rnrTa S4ptnfTa ineh 2e 1.6 x 101i0n (314) 3pp \mp} Note the difference in the sign of the final terms in the equations for 9e and Op, which will prove important in the analysis below. From equations (312) and (313) for electrons and protons the dominant gravitational and electromagnetic combinations can be constructed separately. The remainder of this subsection details the evolution of baryons in the linear regime of a perturbed universe. Baryonic matter can be treated as the combination of electrons and protons, thus the mass weighted sum of proton and electron overdensities gives rise to the baryonic perturbations, m mp me mT 6b me +  Ob = Oe + M Op. (315) mb mb Mb Mb By substituting the expressions for equations (312) and (313) into equation (315), a set of equations for the evolution of baryonic matter is obtained. So long as approximations such as np ne and mb mp > me hold, quantities which are obviously small compared to the others (such as n, n,) can be neglected. The evolution equations are then S= b0 + 3, Ob Ob + c k26 + k2k + r(0 b), (316) a where Fb P em/mb. The baryonphoton coupling term in equation (316) is driven by the electronphoton interaction. To the extent that electrons and protons move together (the tightcoupling approximation), the baryonic fluid is dragged by the electronphoton interactions, as has been shown by Harrison 1970 [109] and subsequent authors. Equation (316) is identical to the equations for baryon evolution derived in Ma and Bertschinger 1995 [22]. 3.3.5 Charge Separations From equations (312) and (313), a difference component as well as a sum component can be obtained. As the limits on a net electric charge asymmetry in the universe are very strict [17, 18, 128], any component arising from the differences in densities and/or velocities of protons and electrons will not be strong enough to significantly impact the evolution of the other species of particles in the universe, including the baryon component. The charge difference component (denoted by subscript q) is the difference between the proton and electron components, such that 6q = 6p Se and Oq = 0p 0e. The gravitational potential ought not to enter into these equations, due to the fact that gravity acts equivalently on electrons and protons. However, velocities and number densities may differ, n, n, _ nJq. The master equations for the chargeasymmetric component are as follows: 6q = q a + 47nee2 9, = 9 ^ + c k, a me Fre( 9b + q), (317) where the approximations Fp, < Fe and mb mp have been utilized where applicable. The term 4rrnee2jq/me in equation (317) arises from the Coulomb force acting on charged particles, while the final term, F(07 Ob + Oq), arises from the difference in Thomson scattering between protons and electrons. This final term is a source of charge separation independent of and in addition to any initial charge asymmetry, and will create a local charge asymmetry even when there is none initially. In the evaluation of equation (317), the electromagnetic terms dominate the cosmological terms, such that an excellent approximation in the prerecombination universe is 4rne2 q6 = ,, ,O = e q6 Fe(0 Ob + Oq). (318) me For some purposes, it is useful to express the set of equations found in equation (317) as a single ordinary differential equation. This can be accomplished by setting Oq = 6, and again by neglecting the unimportant cosmological terms Oqa/a and c2k26,. Many of the coefficients in equation (317) are functions of a, but the derivatives in equation (317) are with respect to conformal time, r. A change of variables can be performed, using the relation that 45h3c5 1/2 t = [3273G(kT)4 =Na2to, N 72.2, (319) in the radiation era, where to is the age of the universe today, to express all derivatives as derivatives with respect to a, denoted by primes (instead of dots). The evolution of Jq can be tracked by evolving equation (320) below, S. r 1 16N2 rnr,0 e21 d6 + 2N e,0o ;g + a, a 2 me a = 4N2 rFo(0y Ob), (320) where the subscript 0 denotes the present value of a quantity. This is simply the equation of a damped harmonic oscillator, with coefficients that change slowly with time compared to damping or oscillation times. The behavior can be characterized as overdamped at the earliest times, critically damped when a z 3.9 x 1015, and free at late times. Of all the terms in equation (320), only 07, Ob, and 6q (and derivatives) are functions of a; all other quantities are constant coefficents. Although there does not exist a simple analytic form for (08 Ob) in general, at sufficiently early times there exists the simple approximation 07 Ob 6.0 x 1019 k4 a5, (321) valid when the following condition is met: a < 105 for k < 0.1Mpc1, a 106 (1 ) for k > 0.1 Mpc1. Equation (321) is an approximation for a flat ACDM cosmology with cosmological parameters Ho = 71 km s1 Mpc ,m = 0.27, Qb = 0.044, and a Helium4 mass fraction of Y = 0.248. These parameters are used in all subsequent analyses for the calculation of cosmological quantities. The approximation in equation (321) breaks down at sufficiently late times. When this occurs, numerical methods must be used to obtain the quantity (0 Ob). The software package COSMICS [129] is ideal for performing this computation, as it performs numerical evolution of equations (34,36,39, and 316) concurrently. Computational results for the quantities 07 and Ob are given by COSMICS, which are valid at all times in the linear regime of structure formation. It is found that when the approximation in equation (321) breaks down, the quantity (08 Ob) grows more slowly initially, and proceeds to oscillate at a roughly constant amplitude at later times. These oscillations in the quantity (0, Ob) are closely related to the acoustic oscillations between baryons and photons observed in the cosmic microwave background [5]. Numerical integration of equation (320) can be accomplished in various ways, as illustrated in Press et al. 1992 [130]. At sufficiently late times (when a > 3.9 x 1015), numerical results indicate that the quasiequlibrium solution S Tmb (Po' (0 Ob), (322) 37we2 Pb,o / obtained by neglecting the first two terms in equation (320), is an excellent approximation. With 0, and 0b given by COSMICS in units of Mpc1, the prefactor in equation (322) can be written as (T m6b (Po) 1.64 x 10 Mpc. (323) 3 7re2 \b,o/ The quantity 0q then follows directly from equation (317) to be S TT b (P7,o 324 qo 3 7r e2 pb,o b) (324) The solutions in equations (322) and (324) are valid until gravitational collapse becomes nonlinear, which means that they are still valid at the epoch of recombination (z 1089). The results of numerically integrating the equations for 6q and 0q on various length scales up through recombination are presented in Figure 31. It is worth pointing out that the results obtained in this section can be applied to a situation where a net electric charge is present. In appendix B, the possibility of using the evolution equations derived for 0q and 6q is applied to a universe with a broken U(1) symmetry. The possibility exists that, under the proper circumstances, an initially charged universe may become neutral simply due to the expansion dynamics. 40 / ._ i / ,' 45 S50 55 106 105 104 103 Z z Figure 31. 6q (solid lines) and Oq/H, where H is the Hubble parameter (dashed lines) as a function of redshift (z). The lines shown are for comoving scales of (from top to bottom) 10, 1, 0.1, and 0.01 Mpc1. 6q rises as a a5 initially, then ceases to grow when the scale of interest enters the horizon, and oscillates at an amplitude which first continues to rise slowly, then falls, eventually matching on to the equilibrium solution that 6q oc Ob. Oq can be obtained from q6 through equation (317). This graph uses output from COSMICS, and as such needs to be multiplied by the COBE normalization of Bunn and White [2]. 3.4 Magnetic Fields With the results derived in Section 3 for Oq and 6q, values for the local current densities and local charge separations can be obtained at any time in the prerecombination universe on all scales. Both q6 and Oq will contribute to magnetic fields, as currents create magnetic fields directly, and the bulk motion of a region of net charge will also produce a magnetic field. For each comoving distance scale (given by the value of k) and each timeslice (determined by the scale factor a) of the universe, there will be a unique magnetic field amplitude associated with that scale. This field may serve as the seed for the largescale magnetic fields observed today. An expression for magnetic fields can be derived from the currents arising from the relative motion of the protons and electrons in the universe. Magnetic fields can be derived from Maxwell's equations aE V B = 0, V xB =47rJ+ (325) with the current density J given by J = n, e ne e v ne [6qVb (1 + b)q], (326) where vq = v ve, and the displacement current is neglected. By taking the curl, a direct expression for magnetic fields as a function of a and k is obtained as a convolution S 47neo e f dk' k x k' a2 l 2 (27)3 k [12k + 9q(k') b( k')]. (327) While the magnetic field strength can, in principle, be obtained by solving equation (327), it is more favorable to obtain the power spectrum of the magnetic field. The power spectrum is obtained by examining the second moment of the magnetic field B(k), which is (Bi(k)B(2)} = (2)3D(k + k2) P PB(k), P4 2 1 ( kS k), (328) where 6D is the Dirac delta function and PB(k) is the magnetic field power spectrum. Note that the direction parallel to k does not contribute to magnetic fields, and therefore the direction perpendicular to k is projected out in equation (328). The power spectrum, PB(k), is then given by the expression PB (k)= k2 sin2 a  2 12 (27)3 S1 ,x (1k Pq )(k b)P6b (k k'I) Ik k' 12 1V 12 ; Po ( Ik) P)oqk(I k k') 1k' / 2  b 1 PObq(i q )P8bO, (k V') 2 PoB (k ')Pob, (i k')J, (329) \k k' 12 where the angle A is the angle between the vectors k and k'. The expression for power in any two quantities, q and V), P(I(k), is generically defined by ((k1)(k2 )) = (27r)3JD( l + k2)PPO(k). (330) The solutions obtained for Oq and 6q in equations (322) and (324) can be substituted into the equation for the power spectrum, equation (329). By numerically integrating the resulting expression, the spectral density can be obtained. The spectral density is 4lrk3PB(k)/(27r)3, and provides both a measure of the magnetic field strength on a given scale (k) and a measure of the energy stored in magnetic fields. The results for the spectral density of magnetic field energy on comoving scales ranging from 103 Mpc1 to 102 Mpc1 at the epoch of recombination are shown in Figure 32. The peak of the spectral density corresponds to a typical magnetic field strength of 1023 Gauss on comoving scales of 0.1 Mpc1. 3.5 Discussion The major result of this paper has been to illustrate that seed magnetic fields of cosmologically interesting strengths and scales are necessarily generated by the same processes that cause structure formation. As overdense regions in the early universe slowly grow during the radiation era, photon interactions with both protons and electrons create charge separations and current densities of small magnitudes, but on all scales. These charge separations and currents grow in magnitude as the universe ages, causing magnetic fields to grow as well. Magnetic power peaks at approximately the time of horizon crossing, falling slowly after that. The net result is that, at the epoch of recombination (and hence prior to significant field amplification due to gravitational collapse or dynamo effects), seed magnetic fields of magnitude 0(1023 G) are created by the simple dynamics of charged particles. The results of this paper are very accurate up through the epoch of recombination. At this epoch, however, the universe transitions from a fully ionized state (where the ionization fraction Xe 1) to a state where the ionization fraction is very small, Xe 104 [24]. While the photons are generally decoupled from the baryons at this point, the free electrons continue to interact with the photons, due to the extraordinary number of photons per free electron. In the absence of any interactions with photons, a charge separation would evolve as = 4nne2, = 3 H23 H2 K2 3q, (331) me 2 1041 1042 7 1043 1044 1045 \ 046 \ S047 \ 104  1048 y \ \ \  1049 1050 // 1051 103 102 0.1 1 10 102 k (Mpc1) Figure 32. Spectral energy density of the magnetic field (B, in gauss) generated by cosmological perturbations on a given comoving scale (k, in Mpc1) at the epoch of recombination (z 1089). The line illustrates 47rk3PB(k)/(27r)3, which is the spectral density in units of G2; the peak value is a magnetic field strength B ~ 1023 G. The upper lines are the simple power spectrum, PB(k). I I IIII"I I 111II II illi I I" '' ""I 1111111 c, where K2 is the ratio of the electric to gravitational forces, K 2 (4.77 x 1019)2. Gmpme A charge separation free of external interactions would oscillate (via plasma oscillations) with frequency w & KH. However, as there are many complicated effects that begin to become important after recombination, including gravitational collapse, dynamo effects, and continued electronphoton scattering, it is unlikely that the simple equation (331) accurately describes the evolution of charge separations in the postrecombination universe. While field amplification due to gravitational collapse is negligible at the epoch of recombination, this will not be the case at all times. At recombination, the universe has only been matterdominated for a brief time, and thus density perturbations have only grown by a small amount in that time, leading to an insignificant amplification of the field strength. As magnetic flux gets frozen in, however, nonlinear collapse causes ABi to increase by many orders of magnitude [92, 93]. The major source of amplification of an initial seed field, however, comes from dynamo effects, as discussed in Section 2. The key to solving the puzzle of the origin of cosmic magnetic fields lies in determining whether the seed fields produced by a given mechanism can be successfully amplified into the O(uG) fields observed today. A major problem with many of the astrophysical mechanisms that produce seed fields is that they produce lowmagnitude fields at insufficiently early times for dynamo amplification to produce fields as large as pG. The Biermann mechanism, for instance, can produce seed fields of order 1019 G, but only at a redshift of z 20. Although those initial fields are larger than the 1023 G fields produced by the growth of cosmic structure, the fact that magnetic fields from structure formation are in place at z 1089 makes them an extremely attractive candidate for the seeds of cosmic magnetic fields. As argued by Davis et al. 1999 [131], a seed field as small as 1030 G at recombination could possibly be amplified into a /G field today. Clearly, more work on understanding dynamo amplification is necessary before a definitive solution to the puzzle of cosmic magnetic fields can emerge. One interesting mechanism worth investigating further is for the cosmic seed fields generated by density perturbations to seed supermassive black holes. It is known that the magnetic field energy in active galactic nuclei and quasars is comparable to the magnetic field energy in an entire galaxy. However, these structures cannot generate their own magnetic fields from nothing; they require a preexisting seed field. It therefore appears to be a reasonable possibility that the seed fields generated by cosmic structure formation could provide the necessary fields to seed supermassive black holes. The resultant amplification via collapse and dynamo effects could explain the origin of largescale magnetic structures in the universe. If largescale magnetic fields exist at the epoch of recombination, they may be detectable by upcoming experiments. The results shown in Figure 32 provide a prediction of largescale magnetic fields at the epoch of the cosmic microwave background. Sufficiently large magnetic fields on large scales at recombination may be detectable by PLANCK [132, 133], although current estimates of their sensitivity indicate that the field strengths predicted in this paper (r 1023 G) would be significantly out of range of PLANCK's capabilities ( 1010 G). Nonetheless, a knowledge of the field strengths at recombination allow for predictions of CMB photon polarizations and Faraday rotation, both of which may be, at least in principle, observable. It is also of interest to note that any primordial charge asymmetry or largescale currents (and therefore magnetic fields) created in the very early universe (a < 3.9 x 1015) will be driven away by these dynamics. Equation (320) has an approximate solution for 6q which is critically (exponentially) damped at a 3.9 x 1015, capable of reducing an arbitrarily large charge or current by as much as a factor of e101". Any preexisting 6q or q8 will be driven quickly to the value given by equations (322) and (324) at the epoch of critical damping. This ought to be applicable even to a global asymmetry, which can be treated as a charge anisotropy (6q) on the scale of the horizon. Therefore, the results in this paper for charge separations, currents, and magnetic fields are independent of the initial conditions on 6q, Oq, and IB1 in the universe. There has been other recent work that claims to generate a magnetic field from cosmological perturbations via "baryonphoton slip," photon anisotropic stress, and a second order velocity vorticity [134]. The results of this work do not require a velocity vorticity or anisotropic stress, nor do they require second order quantities. This paper derives magnetic fields from cosmological perturbations in a very straightforward manner, simply by calculating the charge separations and currents which necessarily arise from the differing interactions on protons and electrons, and obtaining magnetic fields directly from those quantities. The motivation behind the methods used in Ichiki et al. 2006[134] are obscure and not easily comprehended, while their results are inconsistent with those obtained in this paper, as their results for magnetic field strength and spectral density are suspiciously large. Furthermore, it is unclear how their results for velocity vorticity are obtained, as it is wellknown that the vorticity vanishes at second and all orders if there is none initially. Overall, the dynamics of ions, electrons, and photons during the radiation era necessarily leads to charge separations and currents on all scales, which in turn generate magnetic fields. These fields supersede any preexisting fields and are in place prior to substantial gravitational collapse. Thus, the dynamics of structure formation from cosmological perturbations emerges as a promising and wellmotivated new candidate to explain the origins of cosmic magnetic fields. CHAPTER 4 EFFECTS ON COSMIC EXPANSION We evaluate the effect of cosmological inhomogeneities on the expansion rate of the universe. Our method is to expand to Newtonian order in potential and velocity but to take into account fully nonlinear density inhomogeneities. To linear order in density, kinetic and gravitational potential energy contribute to the total energy of the universe with the same scaling with expansion factor as spatial curvature. In the strongly nonlinear regime, growth saturates, and the net effect of the energy in inhomogeneities on the expansion rate remains negligible at all times. In particular, inhomogeneity contributions never mimic the effects of dark energy or induce an accelerated expansion. 4.1 Accelerated Expansion Recent observations of typeIa supernovae [135] and the cosmic microwave background [5] in tandem suggest that the cosmological expansion is accelerating. Understanding the source of this accelerated expansion is one of the greatest current unsolved problems in cosmology [136]. Acceleration seems to render inadequate a universe consisting entirely of matter, and appears to require an additional, unknown type of energy (dark energy, perhaps realized as a cosmological constant). An alternative to dark energy is that acceleration arises from a known component of the universe whose effects on the cosmic expansion have not been fully examined. One possibility currently being examined is that inhomogeneities in a matter dominated universe, on either subhorizon [29, 30] or superhorizon scales [28, 137139], may influence the expansion rate at late times. The central idea is that the energy induced by inhomogeneities leads to additional source terms in the Friedmann equations, with effects on the dynamics that leave no need for a separate dark energy component. In their entirety, these proposals present conflicting claims and a general state of much confusion: does the energy in inhomogeneities produce an accelerated expansion, acting in effect as dark energy [138], or does it behave as curvature [31]? Is the magnitude of the effect small, large, or even divergent, on either large scales [138], or on small scales at late times [30]? Part of the confusion arises from the fully relativistic perturbation theory formulation of many of these calculations. Although this is undeniably a valid approach, the number of terms in a perturbation theory calculation can be large and can mask the underlying physics. In this Chapter, taking advantage of phenomenological results that have been derived from a combination of quasilinear perturbation theory, nonlinear theory, and numerical simulations, we compute the potential and kinetic inhomogeneity energies within the horizon to Newtonian order in potential and velocity for fully nonlinear density contrasts. We find these energies to be small at present, and their projected values remain small, even far into the future. The following section considers the effect of inhomogeneities, for weak gravity and slow motions but for arbitrary density perturbations, characterized in terms of the density power spectrum. After that, the results for the kinetic and potential energies in both the linear and the fully nonlinear regimes are presented, as a function of the cosmological expansion factor. Finally, the implications of these results for the present and future expansion history of the universe are discussed. 4.2 Effects of Inhomogeneities The purpose of this chapter is to investigate whether energy in inhomogeneities can mimic the effects of dark energy for a universe containing only matter. To this end, we work in an = 1 Einsteinde Sitter universe, with no curvature or cosmological constant, and compute the effects of inhomogeneities on the cosmic expansion rate. The dynamics of cosmological expansion are governed by the Friedmann equations, )= Gp, G(p + 3p). (41) a 3 a 3 Any mass or energy density that makes up a significant fraction of the total can influence the evolution of the cosmological scale factor a(t). A contribution to the energy density of the universe with equation of state pi = wpi has pi oc a3(1+w), or pi/pm oc a3W; in particular, a component with p oc a2 behaves as w = 1 or curvature, and a component with constant p behaves as a cosmological constant or dark energy. We introduce the effects of inhomogeneities following the formulation of Seljak and Hui 1996 [140]. In the conformal Newtonian gauge, with metric ds2 = a2() [_(1 + 2)dr2 +2 (1 2)dX2], (42) the timetime Einstein equation (G'O) yields 3()2(1 2)+(2 + 6)a V20 + ,(VO)2 = 87rGp(1 + )(1 + v2), (43) Where V' from the spacespace components of G. (Our numerical factors are slightly corrected from those found in Seljak and Hui 2006 [140]; these factors make little difference in the overall results.) The source on the righthand side includes a density perturbation 6 = 6p/p in the material rest frame, with the transformation to the cosmological frame expanded to leading order for small v2. Ignoring 0 V20, (VI)2, and v2, the homogeneous part of this equation reproduces the usual Friedmann equation. The inhomogeneous part reveals that q obeys the Poisson equation with source 47Gpa2j. The volume average of the entire equation then leads to (a2 Gp8 (1 5W+ 2K), (44) a 3 where W and K are the Newtonian potential and kinetic energy per unit mass, 1 1 W = 1 ((1+6) ), K = 1 ((1 + 6)v2) (45) These expressions are correct to first order in q and v2, but neither an assumption nor an approximation in 6. We assume that (V20) = 0; in all other places the Poisson equation is adequate to determine 0. The Newtonian potential and kinetic energies thus can influence cosmological expansion. We can compute both W and also K completely and exactly from knowledge only of the density power spectrum. The potential is related to the density inhomogeneity by the Poisson equation, V24 = 47rGpa2 an expression which holds even for nonlinear inhomogeneities. From this, we obtain W 47rGpa2 f dk P(k) d A2(k), (46) an expression correct in both linear and nonlinear regimes if P(k) is the appropriate linear or nonlinear power spectrum. The last equality defines the dimensionless spectral density A4k). In linear perturbation theory, valid for small inhomogeneities, the density contrast grows as 6 = Jo(x)D(t), where in a matter dominated universe D(t) oc a(t) oc t2/3 [141]. The kinetic energy follows from the linearized equation of continuity, 6 + V v/a = 0 [141], Kn =2 j P(k) (47) 2 (27)3 k2 (the usual factor f(Q) Q .6 = 1 for ,m = 1). The kinetic energy scales with a(t) as &2D2, while the potential energy scales as pa2D2; and so both W and K grow as D2/a oc a(t), or Pu = p(W + K) oc a2. As was noted by Geshnizjani, Chung and Afshordi 2005 [31] for superhorizon inhomogeneities, energy in inhomogeneities has the same effect on the expansion rate as spatial curvature in perturbation theory. We note that Kimn/ljWin = H2/47Gp = 2, a fixed ratio in the linear regime. The full kinetic energy in principle involves higher order correlation functions and is not a simple integral over the power spectrum. Nonetheless, the full kinetic energy can be obtained simply from the potential energy through the cosmic energy equation of Irvine 1961 [142] and Layzer 1963 [143], (d 2dd=a W, (48) dt a dt a + dt with initial conditions set in the linear regime, Kjm = jWi Equations (46) and (48) provide us with expressions sufficient to calculate nonperturbative contributions to the expansion rate for both the gravitational potential perturbation and kinetic energy components. The results of these calculations are given in the next section. 4.3 Effects on the Expansion Rate Equations (46) and (48) determine the energy in inhomogeneities of the universe as a function of epoch, which we characterize by the expansion factor a/ao. For the primordial power spectrum, we use the CDM power spectrum as given by [144], with spectral index n = 1, ~, = 1, and COBE normalized amplitude JH = 1.9 x 105 [2]. To obtain the nonlinear power spectrum we use the linearnonlinear mapping of Peacock and Dodds 1994 & 1996 [145, 146]. The results of these calculations are shown in Figures 41 and 42. Figure 41 shows the dimensionless spectral density of gravitational potential energy A2(k) defined in equation (46), evaluated at the present, plotted as a function of wavenumber k. The dashed curve shows the density in linear perturbation theory, and the solid curve shows its fully nonlinear form. Figure 42 shows the contributions of potential energy and kinetic energy to the energy density of the universe, for past and future expansion factors in an Qm = 1 universe. At early times, perturbation theory gives an accurate result, but at a/ao w 0.05 (redshift z f 20) the behavior starts to change, for an interval growing faster than a1 with the fastest growth as a1'2, and then saturating and growing significantly more slowly, eventually as In a. 4.4 Contributions of Nonlinear Inhomogeneities In this chapter we have evaluated the size and the time evolution of the contribution of inhomogeneities to the expansion dynamics of a matterdominated universe, including the effects of fully nonlinear density inhomogeneities. When density fluctuations are in the linear regime, the ratio of the inhomogeneity contribution to the matter density 106 107 108 1' Figure 41. Spectral density of gravitational potential energy A2v(k) [the integrand of equation (46)], evaluated at the present, plotted as a function of wavenumber k. The dashed line shows A' in linear perturbation theory; the solid line shows the fully nonlinear form. 105 03 102 0.1 1 10 102 k (h Mpc1) 102 I 11111"'I 11"1 1I 11"1 1I 11"11I 11"1 1I ,". I"I 103 104 105 10  107 107 103 102 0.1 1 10 102 103 a /a Figure 42. Fractional contributions of gravitational potential energy W (longdashed line) and kinetic energy K (solid line) to the total energy density of the universe, plotted as a function of past and future expansion factor for an ,m = 1 universe. The shortdashed line is the sum of contributions from inhomogeneities. The dotted lines show results from linear perturbation theory. grows linearly with expansion factor, as does curvature in an open universe, making only a very small contribution to the expansion rate. As density fluctuations begin to go nonlinear, the energy in inhomogeneities grows at a slightly faster rate, at most as a1'2 oc a3w, or w = 0.4. This by itself, even if the dominant energy component, would be only temporarily and only very slightly accelerating, with deceleration parameter q0 = (1 + 3w) = 0.1. Since, at this time, the total fraction of energy in inhomogeneities is Qvu 105 < 1, this has a negligible effect on cosmological expansion dynamics. As the universe further evolves, so that the main contributions to W and K come from deeply nonlinear scales, we compute the potential energy from integration of the nonlinear power spectrum, and obtain kinetic energy from the cosmic energy equation, as detailed in equation (48). In a scaleinvariant model with power spectrum P k" as k > 0, the kinetic and potential energies K and W scale with the expansion factor as a(n)/(3+n) [147] (logarithmically in a as n * 1), with ratio K 4 (49) \W\ 7+n Numerical simulations show that this continues to hold for the CDM spectrum with effective index n = d log P/d log k at an appropriate scale, the basis of the linearnonlinear mapping [145, 146]. For the CDM spectrum, with n  1 on large scales, this means that growth stops, and the ratio tends to the virial value Ki/WI  at late times. We note that aside from the integration of the LayzerIrvine equation, many of these results were obtained by [140]. Our results show that the contributions of the potential and kinetic energies of inhomogeneities has never been strong enough to dominate the expansion dynamics of the universe. For a universe with Qm = 1 today, normalized to the large scale fluctuations in the microwave background, the net effect of inhomogeneities today is that of a slightly open universe, with Qfk Z 104 in curvature. The maximum contribution comes from scales of order 1 Mpc, falling off rapidly for smaller and larger k, as illustrated in Figure 41. The behavior on asymptotically small scales (k > 106 h Mpc1) depends on an extrapolation that ignores such details as star formation, but Fukugita and Peebles 2004 [12] estimate that the net contribution of dissipative gravitational settling from baryondominated parts of galaxies, including main sequence stars and substellar objects, white dwarfs, neutron stars, stellar mass black holes, and galactic nuclei, is in total 104.9 of the critical energy density. The suggestion that nonlinear effects for large inhomogeneities may mimic the effect of dark energy is not the case for the fully nonlinear theory. It is true that higher order terms in perturbation theory grow faster; the general nth order term grows as D"(t). There indeed comes a scale in space or an evolution in time where the behavior of higher order terms appears to diverge. This is illustrated in Figure 43, where it can clearly be seen that, to second order in density contrast, the contributions from potential and kinetic terms appear to diverge. Nevertheless, the fully nonlinear result is well behaved. It is only the perturbation expansion that breaks down, and the actual energy saturates and grows more and more slowly at late times. As illustrated in Figure 42, the nonlinear potential and kinetic energies remain small compared to the total matter density at all times, even an expansion factor of 103 into the future. Inhomogeneity effects do not substantially affect the expansion rate at any epoch. 4.5 Variance of the Energy in Inhomogeneities It has been pointed out that although the average inhomogeneity energy is small, its variance has a logarithmically divergent contribution from the variance of the potential on superhorizon scales [28], ((AW)2) = f3 3x ) ) = (2Gpa P(k)W2(kR), (410) 2 2 (2 )3 k4( windowed over the horizon volume (for calculational convenience we use a Gaussian rolloff rather than a sharp radial edge). For n  1 as k  0, this is indeed logarithmically 102 A /ao 103 / c  104 / n' K  105."" 10 1 107 106 103 102 0.1 1 10 102 103 a /a Figure 43. Fractional contributions of gravitational potential energy W (longdashed line) and kinetic energy K (solid line) to the total energy density of the universe, plotted as a function of past and future expansion factor for an ,m = 1 universe. The shortdashed line is the sum of contributions from inhomogeneities. This graph shows the contributions to second order in density contrast, 6. Note the apparent divergence is a result of perturbation theory breaking down, as the fully nonlinear result in Figure 42 is well behaved. / dependent on the lowk cutoff (and if n < 1 the divergence is worse), but the rest of the integral is finite for the CDM spectrum. The fluctuation in potential energy, ((AW)2) 1', is shown in Figure 44 as a function of the infrared cutoff knn. The integral is dominated by the smallest values of k, where perturbations are deep in the linear regime. For n = 1 the result is very accurately AW = 1.45 x 105 1 In kminRH 1/2 (We note that for n > 1 the units of kn, are unimportant.) The fluctuation is comparable to the mean (W) = 3.1 x 105 when the cutoff is near the scale of the horizon k = Ho/c, and does not become of order 1 until kmin 10170 (for n = 0.95), or knin ~ 1010` (for n + 1), or ever (for n > 1). While such an exponentially vast range of scales may not be beyond the range of possibility in an inflationary universe, it requires a fearless extrapolation well beyond what is known directly from observation. The fluctuation AW is dominated by contributions from modes that are deep in the linear perturbation regime, and scales with expansion factor as AW oc pa2D, constant in time. This contribution to the energy will appear dynamically in the Friedmann equation as another matter component. Furthermore, in the presence of a true dark energy component, any effects on cosmological expansion arising from inhomogeneities quickly becomes unimportant once dark energy becomes dominant [140]. The fact that fluctuations in the potential diverge remains troublesome. It has been recognized for some time that potential fluctuations in the standard model with n  1 are logarithmically divergent, but since for most purposes the value of the potential is unimportant, this has not been perceived as a significant problem. The effect of potential on the expansion dynamics is real, but the weak logarithmic divergence and the fact that it is a feedback of a gravitational energy on gravitational dynamics may lead one to hope that this divergence is alleviated in a renormalized quantum theory of gravity. 4.6 Summary We have found that, to leading order in 4 and v2 but with fully nonlinear density fluctuations, inhomogeneities on subhorizon scales have only a minimal effect on the 103 < 104 105 50 45 40 35 30 25 20 15 10 5 log1 kmin Figure 44. The expected fluctuation in the potential energy per unit mass ((AW)2)1/2 evaluated at the present as a function of infrared cutoff kmn for n = 0.95, n = 1, and n = 1.05 (solid lines, top to bottom). Dashed lines are analytic approximations that asymptotically become k0.025, (log k)1/2, or constant, respectively. The dotted line shows the result for a rolling spectral index that has n = 0.95 on the horizon today but approaches n = 1 as k  0, as predicted by most models of slowroll inflation. The mean value (W) = 3.1 x 105 is shown as the horizontal dashed line. cosmological expansion dynamics, even far into the future, and in particular never result in an accelerated expansion. Other authors have also shown that recent attempts to explain an accelerated expansion through superhorizon perturbations face significant difficulties [3133]. The possibility that a known component of the universe may be responsible for the accelerated expansion remains intriguing. However, we conclude that subhorizon perturbations are not a viable candidate for explaining the accelerated expansion of the universe. CHAPTER 5 FULLY EVOLVED COSMOLOGICAL PERTURBATIONS This chapter investigates the use of a wellknown empirical correlation between the velocity dispersion, metallicity, and luminosity in H/ of nearby H II galaxies to measure the distances to H IIlike starburst galaxies at high redshifts. This correlation is applied to a sample of 15 starburst galaxies with redshifts between z = 2.17 and z = 3.39 to constrain ,m, using data available from the literature. A bestfit value of ,m = 0.2 1+0 in a Adominated universe and of m, = 0.110 7 in an open universe is obtained. A detailed analysis of systematic errors, their causes, and their effects on the values derived for the distance moduli and Qm is carried out. A discussion of how future work will improve constraints on ,m by reducing the errors is also presented. 5.1 Precision Cosmology Precision cosmology, or accurately constraining the parameters describing the universe, has recently become an active field of research due to the precision of available data sets. Stringent contraints have recently been placed on cosmological parameters from measurements of the microwave background [5], type Ia supernovae [135], and galaxy surveys [1, 148]. Although these sources of data are sufficient for generating consistent values for the mass density (2,m), vacuum energy density (QA), the dark energy equation of state parameter (w), and the value of spatial curvature in the universe (2k), these values must be checked via as many independent methods as possible for consistency, accuracy, and avoiding systematic biases. Furthermore, without the data from supernovae, there would be weak evidence at best for stating that w ~ 1, thus it is important to seek another, independent observation supporting the existence of dark energy. The cosmological parameter with the greatest number of observable crosschecks is 0,. It has been derived using many techniques, including the SunyaevZel'dovich effect [149], weak gravitational lensing [150], Xray luminosities [151], large scale clustering [152], peculiar velocities of galaxy pairs [153], and supernovae data [39]. These methods yield results ranging from ,m = 0.13 to ,m = 0.35, and are all consistent with one another at the 2a7 level. However, they all face difficulties when attempting to differentiate between cosmological models, as they are only weakly dependent on QA, 0k, and w. If a reliable standard candle were found at high redshifts, cosmological models could be discriminated between by precise and accurate observations, as the distance modulus becomes sensitive to fA, fk, and w at higher redshifts. It is known that local H II galaxies and giant H II regions in local galaxies are physically similar systems [154]. This chapter extrapolates a link between nearby H II galaxies and H IIlike starburst galaxies at high redshifts to use such objects as standard candles. This is accomplished through the application of the known correlation between the luminosity in the H/3 line (LHQ), the velocity dispersion (oa), and metallicity (O/H) of nearby H II galaxies discovered in Melnick, Terlevich and Moles 1988 [155] to the H IIlike starburst galaxies found at high redshifts. This correlation, when applied to starburst galaxies at z > 2, allows for discrimination between different values of ,m as first suggested in Melnick, Terlevich and Terlevich 2000 [40], and can discern which cosmological model is most favored by the data. H II galaxies (and H II regions of galaxies) are characterized by a large starforming region surrounded by singly ionized hydrogen. The presence of 0 and Btype stars in an H II region causes strong Balmer emission lines in Ha and H/3. The size of a giant H II region was shown to be correlated with its emission line widths inMelnick 1978 [156]. This correlation was improved upon in Terlevich and Melnick 1981 [157], who showed that LHp of giant H II regions is strongly correlated with their a. This basic correlation, its extension to H II galaxies, and its usefulness as a distance indicator have been explored in the past [154, 155, 158]. The empirical correlation for H II galaxies [155] relates their LHg, (7, and O/H. The relationship is a5 log LHf = log Mz + 29.60, Mz (51) where the constant 29.60 is determined by a zeropoint calibration of nearby giant H II regions [40] and from a choice of the Hubble parameter, Ho = 71 kms1 Mpc1 [159, 160]. The 1a rms scatter about this correlation is 0.33 dex on log LHp from the local sample of H II galaxies found in Melnick, Terlevich and Terlevich 1988 [155]. Starburst galaxies observed at high redshifts exhibit the same strong Balmer emission lines and intense star formation properties [161, 162] as nearby H II galaxies. This chapter follows the suggestion of Melnick, Terlevich and Terlevich 2000 [40] that equation 51 applies to the H IIlike starburst galaxies found at high redshifts, and provides evidence to validate that assumption. The remainder of this chapter discusses the constraints that can be placed on f, and the restrictions that can be placed on the choice of cosmology using starburst galaxies. Section 5.2 details how the data set was selected and analyzed. Section 5.3 states the results obtained from the selected data. The random and systematic errors associated with any distance indicator is a fundamental (and often overlooked) problem inherent to observational cosmology. Appendix A discusses the errors specific to the observational method used in this chapter, including a detailed discourse on the assumption of universality between local H II galaxies and high redshift starburst galaxies. Finally, section 5.4 presents the conclusions drawn from this chapter, and points towards useful directions for future work on this topic. 5.2 Selection of the Data Sample The goal of the analysis presented here is to obtain distances for each H IIlike starburst galaxy at high redshift. H II galaxies must first be detected at high redshift. A sample is then selected on the basis of the correlation in equation (51) holding and for which the distance modulus (DM) can be computed from the observed quantities. The quantities required for analysis of these galaxies are a, the flux in H/3 (FHf), O/H, the extinction in H/3 (AHg), and the equivalent width in the H/3 line (EW). Following the analysis in Melnick, Terlevich and Terlevich 2000 [40], the distance modulus of H II galaxies can be derived from: DM = 2.5 log( ) 2.5 log(O/H) A 26.18, (52) FHu where the constant 26.18 is determined by Ho and equation (51). This chapter makes extensive use of equation (52) because it expresses DM purely in terms of observables. DM is insensitive to f,, QA, 0k, and w at low redshifts (z < 0.1), differing by 0.1 magnitudes or fewer for drastic changes in the choice of parameters above. At high redshifts (z > 2), however, DM can vary by up to 3 magnitudes depending on the choice of parameters. Of the four parameters above available for variation, DM is most sensitive to changes in Qm, as has been noted previously [40]. However, for values of ,m < 0.3, DM is sensitive to variations in the other parameters by 0.2 to 0.5 magnitudes. Since other measurements indicate that indeed Q, < 0.3, this chapter also considers variations in QA and fk. Data for starburst galaxies at z > 2 are found in Pettini et al. 2001 [161] and Erb et al. 2003 [162], which contain measurements for many of the desired observables (and related quantities), along with redshift data. Partial measurements exist for 36 starburst galaxies. According to Melnick, Terlevich and Terlevich 2000 [40], the correlation in equation (51) holds true for young H II galaxies whose dynamics are dominated by O and Btype stars and the ionized hydrogen surrounding them. As H II galaxies evolve in time, shortlived O and Bstars burn out quickly. Although some new O and Bstars are formed, eventually the death rate of O and Bstars exceeds their birth rate, causing a galaxy to be underluminous in Ha and H/3 for its mass. This effect can be subtracted out by examining the EW of these galaxies, and cutting out the older, more evolved galaxies (those with smaller equivalent widths). For this chapter, a cutoff of EW > 20 A is adopted, and galaxies with EW < 20 A are not included, similar to the cutoff of 25 A used previously [40]. There are also galaxies with large EW that do not follow the correlation Table 51. Highredshift galaxies selected to be used as standard candles on the basis of their equivalent widths and velocity dispersions. z 7 1 FHf 2 12+log (O/H) AH_ 3 EW 4 DM 5 2.17 62 29 0.9 0.2 8.55 0.013 23 47.49+2_1 2.18 51 22 1.9 0.5 8.55 0.157 72 45.45 +1 2.54 < 42 1.3 0.3 8.55 0.141 21 44.82+0 31 2.44 < 60 1.3 0.3 8.55 0.285 25 46.64+0 31 2.32 75 21 2.4 0.6 8.55 0.735 47 46.72+1_8 2.17 107 15 2.6 0.7 8.55 0.214 31 48.96+.01 3.11 < 63 3.4 1.0 8.55 0.505 28 45.77+0.31 3.23 69 4 < 1.7 8.55 0.237 < 27 47.12+0.44 0.32 3.39 87 + 12 2.7 0.3 8.70 0.08 0.773 37 46.96 +1 3.10 116 + 8 4.1 0.4 8.62 0.07 0.237 43 48.81+.31 3.09 67 + 6 < 2.3 8.55 0.110 < 31 46.76+0.56 3.07 113 + 7 1.3 0.3 8.39 0.16 1.01 25 49.711+43 3.32 100 + 4 3.5 0.4 8.55 0.852 25 47.73'+. 3.09 55 15 3.0 1.0 8.55 0.284 40 45.22 +1. 2.73 81 1.35 0.2 8.49 0.10 1.14 26 47.49+1 of equation (51) within a reasonable scatter. It is wellknown that a large fraction of local H II galaxies contain multiple bursts of star formation [158]. If multiple unresolved starforming regions are present, the observed a will be very large due to the relative motion of the various regions. Such galaxies are not expected to follow the correlation of equation (51) [158]. The simplest way to remove this effect is to test for nongaussianity in the emission lines from this effect, but signaltonoise and resolution are insufficient to observe this effect. Since a for a system of multiple starforming regions will be much higher than for a single H II galaxy, a cut can be placed on a to remove this effect. Monte Carlo simulations (detailed in Appendix A) indicate that if a is observed to be greater than 130 km s1, it is likely due to the presence of multiple starforming regions. To account for this presence, all galaxies with a > 130 km s1 are discarded. Imposing the above cuts on a and EW selects 15 of the 36 original galaxies, creating the data sample used for the analysis presented here. The properties of those selected galaxies are detailed in Table 51, with further information about the galaxies available in the source papers [41, 161, 162]. Once the sample has been selected, the quantities required to calculate DM using equation (52) must be tabulated for each selected galaxy. Not all of the necessary data elements are available in the literature for these galaxies, so assumptions have been made to account for the missing information, z was measured for all galaxies by the vacuum heliocentric redshifts of the nebular emission lines, a was obtained for all galaxies from the broadening of the Balmer emission lines, Ha for the galaxies from Erb et al. 2003 [162] and H3 for the galaxies from Pettini et al. 2001 [161]. FHf is measured directly for the galaxies in Pettini et al. 2001 [161], but Erb et al. 2003 [162] measures FHa instead, thus FHa must be converted to FHP. The conversion for emitted flux is given in Osterbrock 1989 [163] as FHu = 2.75 FHf, but observed fluxes must correct for extinction. Thus, the complete conversion from FHa to FHP will be given by equation (53) below, 1 AHAH,3 FH = FHa 1 0 2( ) (53) 2.75 where AHa and AHP are the extinctions in Ha and H/3, respectively. Obtaining O/H is more difficult, as measurements of metallicity only exist for 5 of the 36 original starburst galaxies. An average value of O/H is used for the galaxies where O/H measurements are unavailable. Values of O/H are obtained through measurement of the [O II] emission line at 3727 A and the [0 III] lines at 4959 A and 5007 A for five of the galaxies in Pettini et al. 2001 [161]. The strong line index R23 [164] is assumed to have its temperaturemetallicity degeneracy broken towards the higher value of O/H, as is shown to be the case for luminous starburst galaxies at intermediate redshifts in Kobulnicky and Koo 2000 [165]. The combination of the oxygen line measurements with this assumption yields values for O/H for these galaxies. The mean value of O/H is then taken to be the average metallicity for each of the other galaxies where such line measurements are unavailable. Recently, measurements of metallicity in high redshift starburst galaxies have been made [166], using the [N II]/Ha ratio as their metallicity indicator. The authors in Shapley et al. 2004 [166] obtain an average O/H of 8.33 for the galaxies previously found in Erb et al. 2003 [162]. This value is noted as a possible improvement to the one chosen here, and is further discussed as a source of error in this analysis in appendix A of this work. AH, is derived from the E(B V) color of the galaxy in question. Extinction laws are known and established for the Milky Way, the Large and Small Magellanic Clouds (LMC and SMC, respectively), and the H II regions of the LMC and SMC [167], but have not been established for starburst galaxies in general (although see Calzetti et al. 1994 & 2000 [168, 169] for an argument to the contrary). This chapter assumes dust in H II galaxies to be comparable to that in giant H II regions, thus AHp for starburst galaxies is taken to be the AHP derived in Gordon et al. 2003 [167] for the H II regions of the LMC and SMC. A best fit applied to the data in Gordon et al. 2003 [167] yields AH~ = (3.28 0.24) E(B V), AHa = (2.14 0.17) E(B V), (54) for starburst galaxies. These results are also applicable to the flux conversion in equation 53. E(B V) is unavailable for the galaxies from Pettini et al. 2001 [161], but can be derived by noticing the correlation between E(B V) and corrected (G R) colors for starburst galaxies in Erb et al. 2003 [162]. The conversion adopted is E(B V) , 0.481 (G R). Finally, EW is measured for all galaxies in Pettini et al. 2001 [161], but Erb et al. 2003 [162] gives only the spectra for the Ha line. EW is estimated for the Erb et al. 2003 [162] galaxies by estimating the continuum height from each spectra and the area under each Ha peak, calculating the equivalent width in Ha, and converting to H/3 using the Balmer decrements of Osterbrock 1989 [163]. The complete data set is listed in Table 51, and is illustrated alongside various cosmologies in Figure 51. 5.3 Constraints on Cosmological Parameters In the previous section, the distance modulus was calculated for each galaxy in the selected sample. The comparison of these values of DM and the predicted values of 1.5 2 2.5 3 3.5 4 Redshift Figure 51. The distance modulus plotted as a function of redshifts for various cosmological models, along with data from the selected galaxies. OpenCDM universes and ACDM universes with f, of 0.05, 0.30, 0.5, 1.0, and 2.0 are shown. The crosshairs represents the 1a constraints on the DM vs. z parameter space from the selected data sample. The best fits to the data are for a ACDM universe with Q, = 0.21 and QA = 0.79, or for an openCDM universe with = 0.11. DM at a given redshift for different cosmological models provides a constraint on the cosmological parameters. DM is most sensitive at high redshifts to the variation of the cosmological parameter Qm, as pointed out by Melnick, Terlevich and Terlevich 2000 [40]. ~, is therefore the parameter which is constrained most tightly by observations of starburst galaxies. Each galaxy yields a measurement for DM and for z. Although there are multiple models consistent with each individual measurement, observations of many galaxies at different redshifts will allow the construction of a bestfit curve, which is unique to the choice of cosmological parameters f,, QA, Qk, and w. The data sample of 15 galaxies in this chapter is insufficient to distinguish between models in this fashion, as the uncertainties in each individual measurement of DM are too large. The method by which the uncertainties can be reduced is to bin the data according to redshift and find a bestfit value of DM at that point. Due to the size of the sample in this chapter, all 15 points are averaged into one point of maximum likelihood to constrain the cosmology, with errors arising from the random errors of the individual points and from the distribution of points. The average value obtained is DM = 47.03+.4 at a redshift z = 2.80 0.11. The different cosmological models, along with the most likely point and the raw data points, are displayed in figure 51, with Ho = 71 km s1 Mpc1 The constraints placed on f, from this analysis are = 0.2 10.30 in a Adominated universe (,m + 2A = 1; 2k = 0) and Q, = 0.110. 3 in an open universe (Qm + 2k = 1; QA = 0). Figure 52 shows the comparison in ,m vs fA parameter space between the preliminary constraints of this chapter and early constraints arising from CMB data and SNIa data, available in de Bernardis et al. 2000 [170]. CMB and SNIa constraints led to the first reliable estimates of ,m and OA. The preliminary constraints presented here are comparable to early constraints from CMB and SNIa data, as illustrated in figure 52. The accuracy in ,m and QA, as determined from the most recent CMB and SNIa data [5] is now 0.04 in each parameter. A similar, and perhaps even superior accuracy can be achieved using starburst galaxies at high redshifts. V^M~ "^.7 The final section below discusses this possibility, and the systematic and statistical errors which must be overcome are discussed in Appendix A. 5.4 Conclusions and Future Prospects This chapter has demonstrated that using H IIlike starburst galaxies at high redshifts as a standard candle is a promising and wellmotivated avenue to explore for precision cosmology. A future survey of high redshift starburst galaxies with measurements of z, a, AH1, FHQ, O/H, and EW will reduce both random and systematic errors dramatically. Since the inherent scatter of the method is large (as can be seen in figure 51), a large sample size is required to obtain meaningful constraints. This chapter contains a sample size of only 15 galaxies, but future surveys should be able to obtain hundreds of starburst galaxies that survive the selection cuts. For a sample of 500 galaxies, this will improve constraints on Qm to a restriction of 0.03 due to random errors. Additionally, all of the systematics specific to this sample due to incomplete data will disappear. Appendix A discusses how these errors may be reduced, and how, with such a sample, the concordance cosmological model can be tested at a redshift of z r 3, something that has not been done to date. If the assumption of universality between local H II galaxies and high redshift starburst galaxies is correct, this method of measuring Qm is capable of providing very tight constraints, independent of any constraints arising from other sources, including CMB and SNIa data. Additionally, if galaxies are obtained at a variety of redshifts between 2 < z < 4, different cosmological models (including vacuumenergy dominated models with different values of w) can be tested for consistency with the future data set. If 1m < 0.3, the differences in DM at various redshifts become quite pronounced, and meaningful results as to the composition of the nonmatter components of the universe can be obtained as well. Future work on this topic has the potential to provide strong independent evidence either supporting or contradicting the concordance cosmological model of Q~ + QA = 1, w = 1, in addition to providing a very stringent constraint on the Qm + QA parameter space. CHAPTER 6 CONCLUDING REMARKS This dissertation has illustrated the invaluable role of cosmological perturbations throughout the evolution of the universe. These perturbations are departures from perfect homogeneity in density and in gravitational radiation. They are created during the epoch of inflation by quantum fluctuations, which are in turn stretched across all length scales by the exponential expansion of the universe. When inflation ends, the cosmological perturbations become overdense and underdense regions in a radiationdominated universe. The universe then expands and cools, and these cosmological perturbations evolve under the influence of gravity, radiation pressure, and all the other forces of the universe. Cosmological perturbations grow linearly at first, and when the overdense regions have reached a sufficient density, they collapse nonlinearly. This nonlinear collapse leads to the rapid growth of complex structure, forming stars, galaxies, clusters, and filaments, among other structures. The structure exhibited on cosmological scales at the present is a direct result of the evolution of primordial cosmological perturbations. The remainder of this chapter summarizes the major findings of this paper, and points towards future directions for research on the topic of cosmological perturbations and their effects on the universe. Also included is a section on how cosmological perturbations are expected to impact the eventual fate of the universe. 6.1 Creation of Perturbations The perturbations produced by inflation are capable of imprinting both primordial scalar modes (density perturbations) and tensor modes (gravitational radiation). Vector modes may also be produced, but these decay over time, and are unimportant for cosmology. The tensor modes that are produced can either be of a comparable amplitude to the scalar modes (as in chaotic inflation [171]), or can be of practically zero amplitude compared to the scalar modes (as in new inflation [172, 173]). In either case, the spectrum of both density perturbations and gravitational radiation are predicted to be nearly scale invariant, with a possible slight tilt preferring either small scales or large scales. If there is a preference towards larger scales, it is possible that, on vastly superhorizon scales, the increased power can affect either the expansion rate or the global spatial curvature of the universe. Differing opinions and many discussions can be found in the literature [3134, 137] as well as in chapter 4. The work presented in this dissertation indicates that although the variance of the potential energy (AW) in these superhorizon perturbations can become very large, it is the potential energy itself (W), not AW, that couples to the expansion rate of the universe. Therefore, it appears at this juncture that cosmological perturbations on superhorizon scales cannot affect the expansion rate of the universe. On the other hand, there could be a preference towards smaller scales, which would lead to early nonlinearity (and faster structure formation) of the smallest structures. The power spectrum is fairly accurately known [1] for values of k up to ~ 100 Mpc1, and matches very well with simulations of a scaleinvariant spectrum [174]. Although there is no reason to believe the density perturbations on scales smaller than this depart significantly from an n 1 spectrum, the perturbations in gravitational radiation may. As shown in Siegel and Fry 2005a [13] and in chapter 2, in the presence of extra dimensions at a fundamental scale mD, primordial gravitational radiation will acquire a thermal spectrum and an energy density given by equation (24) if the reheat temperature, TRH, is sufficiently high. The possibilities for detecting a thermal background of gravitational radiation (and thus indirectly detecting extra dimensions) could be accomplished through a precision measurement of the primordial 4He abundance or through the broadening of the 21cm HI line. 6.2 Early Evolution of Perturbations Once the initial cosmological perturbations of the universe are in place amidst the other initial conditions of the big bang (expanding, dense, radiationdominated universe), their densities evolve in accordance with all of the physical effects acting on them. These include the gravitational force, which provides a gradient towards the overdense perturbations and away from underdense ones, radiation pressure, the Coulomb force, the nuclear (strong and weak) forces, and scattering from particleparticle interactions. The net result of all the interactions is that overdense regions in the universe grow according to the Miszaros effect at sufficiently early (linear) times, as given in equation (17). While the M6sziros effect does a good job describing the overall evolution of the energy density in a region of space, it cannot give any information about the evolution of the different types of components which compose the universe. As an example, at various epochs in the universe, each of baryons, dark matter, photons, neutrinos, and vacuum energy compose at least 10 per cent of the energy density of the universe. While the universe is radiationdominated, photons and neutrinos are most important, while when it is matterdominated, baryons and dark matter are the most important components. In order to understand how cosmological perturbations have evolved into the largescale structures observed today, it is vital to understand the evolution of the matter components of the universe from very early times until the present day. Chapter 3, based heavily on the works of Ma and Bertschinger 1995 [22] and Siegel and Fry 2006 [23], details the evolution of the matter components of the universe. One novel idea of this work is that the evolving cosmological perturbations give rise to seed magnetic fields on all scales. This can also be found in chapter 3 and in Siegel and Fry 2006 [23], but note that there is a competing group that obtains quite different results through a significantly different calculation [120, 134, 175]. The essential idea is that photons have a much larger interaction cross section with electrons than with protons, inducing charge separations and currents. The Coulomb force also plays a major role, acting as a restoring force. The net result obtained in chapter 3 is that magnetic fields on all scales are created, following the spectrum in figure 32. On the most interesting scales (from 1 100 Mpc), fields of 0(1023 Gauss) are produced at the epoch of recombination. These seed fields may provide the seeds for the magnetic fields observed on large scales today. This may be accomplished either directly, by having these seed fields directly grow into the fields observed today, or by these fields seeding the supermassive black holes/AGNs which then amplify the field, and populate the universe with the resultant magnetic field energy. The sets of equations for a difference component between ions and electrons derived in chapter 3 are the first of their kind. Problems which hitherto could not be solved for lack of having equations that describe a charge difference can now be tackled using these new tools. Appendix B presents the possibility that a net electric charge was present at some point in the universe's past. Prior treatments (see Lyttleton and Bondi 1959 [176] and articles which cite it) have found that many of these scenarios would present unacceptable consequences for cosmology. However, Appendix B points towards the possibility that a net charge in the universe, which could arise (for instance) from a broken U(1) symmetry, would be driven away by the simple dynamics of cosmological perturbations. If this proves to be the case, many possibilities for physics in the early universe, where are presently considered to have unacceptable cosmological consequences, may turn out to be quite valid after all. 6.3 Final State of Perturbations Cosmological perturbations continue to grow linearly, with the most substantial growth occurring once the universe has become matterdominated, until a critical density is reached. Once this occurs, density perturbations enter the nonlinear regime, and grow very rapidly. The nonlinear structure formed in this manner collapses to often very large density contrasts (see chapter 1), but the structure formed is eventually stable to further collapse due to the virial theorem. Were the universe completely devoid of angular momentum, or rather, were the Zel'dovich approximation exact, the universe might look vastly different from its present state. If gravitational collapse were to occur exactly along field gradients, then nuclear reactions would be the only interactions in the universe that prevented all structure from collapsing to singularities. Yet, angular momentum is a fact of life in the universe, as evidenced by the rotations of many astronomical systems. As a result, when structure formation goes nonlinear and gravitational collapse becomes a major effect, angular momentum conservation also becomes a major factor in forming a stable, finalstate structure. Complex physics, such as shocks and heating (which can create magnetic fields via the mechanism in Biermann 1950 [111]), star formation, baryonic collisions, and collapse on multiple scales all play a role in the formation of the presentday structure of the universe. One question that has been answered in Chapter 4 of this work is the question of whether this nonlinear structure will backreact sufficiently, and impact the global expansion rate of the universe. While many authors [2830] contend that the effects of nonlinear collapse could substantially impact the expansion rate, it has been calculated (both in chapter 4 and in Siegel and Fry 2005b [34]) what the effect is explicit. The conclusion is that the effect is negligibly small (of 0 ~ 105 the normal expansion rate) at all times. Cosmological perturbations, in an Einsteinde Sitter universe (fQ = 1, no curvature or vacuum energy), will grow nonlinearly only once the scale of interest is inside the horizon. In this scenario, structure in the universe is selfsimilar, with smaller scales at earlier times behaving identically to larger scales at later times. However, observations of structure formation, among other observables [177, 178], do not support this picture of an Einsteinde Sitter universe. The picture which is most consistent with all the observations is known today as the concordance cosmology, which indicates that the universe has roughly 30 per cent of the critical energy density in matter and roughly 70 per cent in a cosmological constant. The consequences of this for largescale structure in the universe are examined in section 6.4 of this Chapter. Once galaxies form (at a redshift of around z ~ 10 in a ACDM universe), they can be used as deep cosmological probes. In order for something to be a useful distance indicator, there must be a relation between observable quantities and a cosmological distance [179]. Many of these techniques involve individual stars or stellar remnants, or other lowluminosity objects. To be used at the largest distance scales, a distance indicator must first be visible at such large distances. There are only a few classes of objects which are visible out at redshifts z > 2, such as the most luminous galaxies, supernovae, gammaray bursts, and quasars. It is only at z > 2 that it becomes reasonable to constrain the various cosmological parameters (such as Qm, QA, and the dark energy equation of state, w). Thus far, only the supernovae (and only the type Ia supernovae, at that) have successfully been used as a distance indicator out at this high of a redshift. Although there has recently been an attempt to use gammaray bursts as a distance indicator [180], the systematic errors inherent to the method are far larger than the uncertainty between different cosmological models. Systematic errors, as illustrated in Appendix A, are a constant source of difficulty for any observational cosmologist. Uncertainties in the universality of a distance indicator (i.e. that it behaves the same at all redshifts), as well as uncertainties in the environment where the distance indicator lies, can all bias results obtained with any one indicator. It is for these reasons that many differing distance indicators at high redshift are desired for probing cosmological parameters. Chapter 5 (and Siegel et al. 2005 [41]) builds upon the work of Melnick, Terlevich and Terlevich 2000 [40], and uses starforming galaxies at z > 2 to constrain the cosmological parameters. Although systematic errors for this method are both numerous and worrisome, they are quantified and given a detailed treatment in both Appendix A and Siegel et al. 2005 [41]. The result obtained is that, despite systematic errors, in a Adominated universe, it can be concluded that 0.09 < f < 0.51, with a bestfit value of fm = 0.21. This is the only known independent estimate for cosmological parameters from distance indicators at such high redshifts other than supernovae or the cosmic microwave background. 6.4 Fate of the Universe One of the most puzzling cosmological discoveries of the last decade has been the discovery that the expansion rate of the universe is accelerating (discovered by Riess et al. 1998 [181] and Perlmutter et al. 1999 [182], independently). These observations have been confirmed and appear very convincing, yet a compelling theoretical mechanism for explaining the existence of this socalled dark energy has not yet been discovered. Understanding the nature of this accelerated expansion and the physics behind it is one of the great challenges for modern cosmology. An interesting avenue for future research would be to predict the future history and evolution of structure in the universe based on the presence of this accelerated expansion. The accelerated expansion is most easily and simply parametrized by a cosmological constant of energy density PA 6.9 x 1030 g cm3. The presence of a nonzero cosmological constant (A) incorporates a builtin scale to the physics of structure formation. The main consequence of this new type of energy density for structure formation is that scales which are not gravitationally bound to one another at the time of matterA equality never become bound to one another. The Hubble expansion parameter, H, in a universe containing matter, radiation, and a cosmological constant, evolves as H2= H2 Pa4 + Pa3 +PA) (61) (Pc Pc Pc where p,, pm, and PA are the energy densities in radiation, matter, and A, respectively. (pc is the critical density.) From equation 61, it is facile to deduce that at late times (when a becomes large), the Hubble expansion parameter, H, is given by the constant H = Ho QA PA (62) Pc Therefore, the final expansion state of the universe will be much like the initial inflationary state, in that there will be an asymptotically exponential expansion. The scale factor of the universe, a, will evolve in the far future as a eo Vt. (63) The net result is that the structure that is bound at the time of matterA equality remains bound, whereas structure that is not yet bound never becomes bound, and will exponentially recede from one another. The local group contains the Andromeda galaxy, the large and small Magellanic clouds, and a few other, small structures. At a time  5 x 1010 years in the future, the local group, gravitationally bound to our galaxy, will be the only matter in our universe within 500 Mpc of the Milky Way! In fact, in twice that time (~ 1011 years), everything outside of our local group will "red out," meaning that objects in our local group will be the only objects causally in contact with our galaxy in the universe. The current behavior of the universe points towards this exact scenario for the energy content of the universe, with 70 per cent in dark energy and 30 per cent in matter. If this is the case, and the universe continues to evolve according to the known laws of physics, then galaxies and the objects bound to them, the children of cosmological perturbations, will be the last remaining objects in the universe. The universe will consist of a few isolated clumps of matter exponentially expanding away from one another, with nothing but empty space in between. In the end, these lonely clumps of matter, having arisen from the growth and collapse of slightly overdense cosmological perturbations, may be the only substantial things in a cold, empty universe. APPENDIX A ERRORS IN HIGHZ GALAXIES AS DISTANCE INDICATORS There have been many assumptions made along the path to obtaining m via the use of starburst galaxies as a distance indicator, as detailed in chapter 5. The major danger in every observational method is that every assumption made carries along with it an associated error. Some of the assumptions made are inherent to the method used, while others affect only the data sample specific to the one selected in chapter 5. Both will lead to systematic errors, although the samplespecific errors will largely be eliminated by improved measurements, to be taken in future observing runs. Additionally, random errors result from both uncertainties in the measurements and from the intrinsic scatter in the distribution of points. An analysis of all three types of errors ensues below. A.1 Universality among HII Galaxies The most important assumption made was the assumption of universality of the distance indicator used for both local H II galaxies and H IIlike starburst galaxies. Support for this assumption is provided by the fact that both galaxy types follow the empirical correlation of equation (51), as shown in figure A1. Although the physics underlying starburst galaxies has been an open question for over thirty years [183], it is fortunately not necessary to uncover the complete answer to establish universality. It is likely (although unproven) that the physics underlying the correlation for H II galaxies is similar to the physics underlying the TullyFisher relation [184] for spiral galaxies. Specifically, it is thought that line widths (a measure of velocity dispersion) and the luminosity in the Ho line are both intimately tied to the amount of mass in the starforming region. A theoretical investigation of exactly what this link is could prove quite fruitful in understanding the underlying physics of the correlation presented in Melnick, Terlevich and Moles 1988 [155]. The validity of the correlation between Lfl and Mz can be tested directly to determine its range of applicability. By assuming a cosmology, log LHp can be written purely in terms of luminosity distance (dL), FHp, and AHfl, which are either measurable or computable from observables for each galaxy, as shown in equations (51) and (52). log Mz can be determined through measured values for a and O/H. Comparing the quantities log LHf and log M, then allows a test of the correlation in equation 51 for all galaxies of interest. All available H II and H IIlike starburst galaxies with appropriately measured quantities are included to test the correlation. Local galaxies are taken from Melnick, Terlevich and Moles 1988 [155] and from the Universidad Complutense de Madrid (UCM) survey [185, 186], intermediate redshift starburst galaxies are taken from Guzman et al. 1997 [187], and high redshift starburst galaxies are from Pettini et al. 2001 [161] and Erb et al. 2003 [162]. The cosmology assumed to test universality is Qm = 0.3, QA = 0.7, and cuts are applied to all samples so that EW > 20 A and a < 130 km s1. The results are shown in figure A1. The major reasons to conclude that the assumption of universality is valid lie in figure A1. There exists an overlap between all four samples in both LHO and M,, from the sample where the correlation is well established (nearby samples, such as Melnick, Terlevich and Moles 1988 [155] and the UCM survey [185, 186]), to intermediate redshift H IIlike starburst galaxies Guzman et al. 1997 [187], to the high redshift sample used in chapter 5, from Erb et al. 2003 [162] and Pettini et al. 2001 [161]. These four samples all follow the same correlation between LHp and M, within the same intrinsic scatter. (However, it is worth noting that the observed scatter broadens at high redshifts due to measurement uncertainties). By performing a statistical analysis of the data points in figure Ai, it can be shown that the data selected from all samples are consistent with the same choice of slope and zeropoint for the empirical correlation. For these reasons, equation (51) appears to hold not just for local H II regions and galaxies, but for all starburst galaxies regardless of redshift. It is important to note that there is an uncertainty in the zeropoint calibration of figure A1 of 0.08 dex, which has not improved since the correlation was first discovered [155]. This corresponds to an uncertainty in DM of 0.20, which is an unacceptably large error for the accuracy desired. If starburst galaxies are to be taken seriously as a distance indicator for precision cosmology, it is essential that the zeropoint be determined to significantly greater accuracy. This can be accomplished via a twofold approach: a comprehensive survey of the nearby (z < 0.1) H II galaxy population, and a survey of the nearby, very lowluminosity, low velocitydispersion H II regions. Accomplishing both of these goals will allow a marked reduction in the zeropoint error by significantly increasing the sample size from the sample used in Melnick, Terlevich and Moles 1988 [155], and will also probe the very low end of the relation between M, and LHf, where data are sparse. A.2 Systematic Errors The other assumptions which are inherent to this method are the choices of where to cut on EW and on a, and the assumption that AHp is the same for starburst galaxies as it is for local H II regions. These two sources of uncertainty (how to select the sample and what the correct extinction law is for high redshift starburst galaxies) are systematics that cannot be removed by improved observations. Each assumption that is made has an associated error. The assumed extinction laws of equations (54), the cut on EW of 20 A, and the cut on a of 130 km s1 all induce inherent systematic errors. Moving the EW cut from EW > 20 A up to EW > 25 A, as suggested in Melnick, Terlevich and Terlevich 2000 [40], would systematically raise the DM by 0.14 mag for all galaxies present in this sample. The EW threshold for the onset of major evolutionary effects is not yet wellestablished, and necessitates further research. The cut on a comes about in order to remove contamination from objects containing multiple unresolved starforming regions. Since the correlation between LHp and Mz is only valid for single H II galaxies and H II regions [158], a cut must be made to remove objects containing multiple starforming regions. Single H II galaxies are observed to have a gaussian distribution in a peaked at  70 km s1, but objects with multiple unresolved regions are expected to have an entirely different distribution. On the basis of Monte Carlo simulations performed to simulate both single and multiple H II galaxies, a cut on a at 130 km s1 retains 95 per cent of the 44 Q 0 " 40 38 1 8 Figure A1. log Mz vs. log LHfP for local H II galaxies and starburst galaxies at intermediate and high redshifts. The solid line is the best fit of the correlation to the local data set, flanked by the dashed lines, which give the 2a rms scatter. The large diamonds represent the selected high redshift data sample; the small diamonds are the data not selected on the basis of either EW or a. The vertical dotted line is the derived cut on a of 130 km s1. The crosshairs represents the typical uncertainty in each selected data point. 10 12 14 log MZ valid, single H II galaxies, while eliminating 75 per cent of the contaminating objects. The results of the Monte Carlo simulations can be seen in figure A2 below. Additionally, it can be shown that the contaminating objects which are not eliminated depart only slightly from the empirical correlation of equation (51). It is therefore essential, for any future survey, that the proper cuts be determined and applied to EW and o, otherwise substantial uncertainties will arise from the selection of the data sample. Finally, the derived extinction law in AHp itself, from equation (54), has an uncertainty of 38 per cent, due to the fact that there are competing extinction laws that give different results [167169]. Both laws are comparably grey, but have different normalizations. The difference between the two laws leads to a systematic uncertainty in the DM of the highredshift galaxies of 0.17 mag. A.3 Measurement Uncertainties There have also been assumptions made specifically to compensate for incomplete data in the data sets of Pettini et al. 2001 [161] and Erb et al. 2003 [162]. The systematic uncertainties that these assumptions induce can be eliminated in future surveys through measurements of all required quantities. The assumption that the temperaturemetallicity degeneracy is most likely broken towards the upper branch of the R23 vs. O/H curve for luminous starburst galaxies at high redshift is based on sound analysis [165], but is still a dangerous one to make. Measurement of the 4363 A oxygen line would break the R23 vs. O/H degeneracy, and yield a definite value for metallicity for each galaxy. Furthermore, O/H had to be assumed for 11 of the 15 galaxies in the sample, inducing a possible systematic which could affect DM if the assumed average O/H differs from the true value. If the value for O/H from Shapley et al. 2004 [166] is used for the galaxies selected from Erb et al. 2003 [162], the average DM is raised by 0.22 mag. This systematic can be removed in future surveys by a measurement of the [O II] line at 3727 A and the [O III] lines at 4959 A and 5007 A for each galaxy. There are also other metallicity indicators (see Kewley and Dopita 2002 [188]) which may prove to be more reliable at 100 200 300 sigma (km / s) Figure A2. Simulation of the distributions of velocity dispersions, as would be measured for a single, isolated starforming region (red curve), for multiple, interacting starforming regions (blue curve, a simulation of multiple unresolved regions), and for a mixture of both types (green curve). The observed data is plotted in black. Note that the actual data appears to be a superposition of the single region data and the multiple unresolved region data. Multiple unresolved regions at high redshifts appear in the form of enclosed galaxy mergers, and must be removed, as they do not follow the correlation of equation (51). From the simulations, removal of most of the enclosed mergers can be accomplished through a derived cut on ar of 130 km s. 250 200 150 100 50 0 high redshifts. Measurement of the [N II] line at 6584 A, along with Ha, can provide another measurement of metallicity [166]. In theory, many abundance indicators are available and may even be practically accessible [188], and future surveys should allow multiple, independent techniques to be used, significantly reducing errors. Note also that it is unsettling that different metallicity indicators appear to give different values of the metallicity for the same galaxy samples; this may be yet another source of inherent error. There is a large uncertainty on the order of 30 per cent in the measurement of EW for the Erb et al. 2003 [162] sample due to the difficulty of establishing the height of the continuum. Some galaxies may have been included which should not have been, and others may have been excluded which should have been present. The effect on the distance modulus is estimated to be 0.16 mag, but this will be removed by measuring equivalent width in H/3 with a higher signaltonoise spectra for all galaxies in future surveys. Finally, E(B V) colors, a substitute for AHp measurements, are unavailable for galaxies from Pettini et al. 2001 [161], and were derived from an approximate correlation noticed between E(B V) and the corrected (G R) colors in Erb et al. 2003 [162]. There is an overall uncertainty in the extinction due to the fact that the average derived extinction for the Erb et al. 2003 [162] and the Pettini et al. 2001 [161] samples differ by 0.34 dex. Thus, there is an induced systematic in DM of 0.17 mag, which will be eliminated when AHP measurements are explicit taken for all galaxies. A.4 Statistical Errors Random errors, due to both uncertainties in measurement and to the large scatter in the distribution of points, are perhaps the best understood of the sources of error. Measurements of AHf are uncertain by 0.04 to 0.11 dex, depending on the galaxy's brightness. Improved measurements, which rely on the Ha/H/3 ratio instead of solely on E(B V) colors, may reduce the uncertainty significantly. Measurements of FHp are uncertain by roughly 20 to 25 per cent on average, and random uncertainties in O/H are of order 0.10 dex. The largest measurement uncertainty comes from measurements of a, which is obtained by the broadening of the Balmer emission lines. Even relatively small uncertainties in a of order 15 per cent can induce uncertainties in DM of 0.8 mag per galaxy. The induced uncertainty is so large because, as seen in equation (52), DM is dependent on a5, whereas it depends only linearly on the other quantities. It is therefore vital to obtain excellent measurements of the velocity dispersion of these galaxies (which is certainly possible, as some of the highredshift galaxies in Pettini et al. 2001 [161] have uncertainties of only 4 km s1). Future work will be able to measure the Ha and HP lines, as well as three oxygen lines, as these are where the three windows in the infrared are in the atmosphere. For galaxies between 2 < z < 4, the Ha, H3, and multiple oxygen lines will appear at the appropriate wavelengths. These observations should improve the measurements of ao, further reducing the random uncertainties. The distribution of points may not improve as statistics improve due to the intrinsic scatter on the Mz vs. LHf relation, but random errors all fall off as the sample size increases. The errors decrease as N1/2, where N is the number of galaxies in the sample. Even if random errors associated with intrinsic properties (such as FHg, a, or O/H) remain large for individual galaxies, increasing the sample size will drive down the overall random errors. Hence, a sample of 500 galaxies, as opposed to 15, will have its random uncertainties reduced by a factor of 6 or better. The new generation of NearIR MultiObject Spectrographs (such as FLAMINGOS and EMIR) in 10 meter class telescopes will be ideal for obtaining all necessary measurements for such a sample. APPENDIX B ON AN ELECTRICALLY CHARGED UNIVERSE The possibility that the universe could have a net electric charge has been investigated offandon by many scientists for the past halfcentury. A net electric charge could arise as a consequence of many different types of earlyuniverse physics, but there are strict limits on a net charge in the universe today from various cosmological constraints. This appendix examines the possibility that the mathematical formalism and equations developed in chapter 3 could remove an initial net charge from the universe through cosmological dynamics. If this successfully occurs, many interesting physical mechanisms, previously thought to be tightly constrained through presentday measurements, may have operated in the early universe. B.1 Introduction This appendix explores the consequences of a cosmological charge asymmetry in cosmology. One interesting effect of this was first presented in Lyttleton and Bondi 1959 [176], where it was pointed out that a sufficiently large electric charge asymmetry, on the order of r 1 e per 1018 baryons, would enable the repulsive Coulomb force to exceed gravitational attraction on large scales. The original motivation for this proposal was to explain the origin of cosmic expansion. It was explained that if the magnitude of the respective charges on electrons and protons differed by r 2 x 10s e, large scale repulsion would follow. With the advent of the big bang theory, which proved to be a necessity to explain the observed cosmic microwave background radiation (CMB, discovered in Penzias and Wilson 1965 [189]), Hubble expansion was explained as a necessary consequence of that theory. It further appears that, to a much higher degree of accuracy than 2 x 1018 e, the proton and electron charges are equal. From the anisotropies of cosmic rays, which can act as a probe of the net charge in the universe at the present day, it is determined that the overall chargeperbaryon (A) is constrained to be A < 1029 e [128]. Furthermore, the degree of isotropy in the CMB provides constraints on the net electric charge in the universe at a redshift of z 1089 [18], where again the restriction is that the net chargeperbaryon is very small, A < 1029 e. A net chargeperbaryon would also impact primordial nucleosynthesis. Cosmological helium synthesis provides a probe of the net charge at very high redshifts (z 4 x 108). Constraints from nucleosynthesis [17] indicate that, in the early universe, A < 1032 e, the most stringent constraints available on the net charge in the universe. If a net chargeperbaryon does exist, there are two straightforward ways to obtain it, both of which were identified in Lyttleton and Bondi [176]. Either, as stated above, the proton charge (q,) differs slightly from the electron charge (q%) in magnitude, or the number density of protons (np) differs from that of electrons (ne). The former possibility is highly disfavored, as terrestrial experiments indicate that the electric field is zero at the Earth's surface. Assuming equal numbers of protons and electrons on Earth ( 1051 of each species) places strict constraints on qp 4 \qel, as do refined versions of the Millikan experiment. It is observed that charge is quantized in units of e for physically observable particles (although quarks are predicted to have fractional charges), as predicted by the standard model of particle physics [53]. Barring exotic scenarios, such as electrically charged neutrinos, photons, or dark matter [190], it is only reasonable to consider unequal number densities of protons and electrons as giving rise to an electric charge in the universe. The creation of an electric charge asymmetry in this fashion is analogous to the creation of a baryon asymmetry in grand unified theories (see Dine and Kusenko 2003 [15] for a review of baryogenesis). An electric charge asymmetry can be generated by a similar mechanism to the baryon asymmetry, and both are expected to have the same types of inhomogeneities [191]. B.2 Generating a Net Electric Charge A global charge asymmetry can be generated via many channels. The most intuitive method is to temporarily break the U(1) electromagnetic gauge symmetry [191193]. If this U(1) symmetry is broken at some point, an electric charge asymmetry must be produced [191]. When the U(1) symmetry is later restored, the charge asymmetry may remain [192]. Care must be taken to ensure that electric charge remains quantized in units of e [193]. The production of a net charge is analogous to baryogenesis in grand unification models, which have a decay process at high energies that is asymmetric, preferring (slightly) to produce baryons over antibaryons. Similar mechanisms could easily prefer the production of one sign of charge over the other, so long as that production symmetry is restored today. Other mechanisms also exist which admit the production of a net electric charge. Examples include KaluzaKlein models with extra dimensions [128, 194], cosmologies with a varying speed of light [195, 196], and effective interactions allowing electric charge nonconservation by units of 2e at a time [197]. Once a net charge has been established in the form of unequal proton and electron densities, previous treatments assume for simplicity that the total charge within a spatial volume is constant [128, 191, 198]. The major purpose of the work in this appendix is to show that this assumption is not true in general. Electromagnetic forces will induce relative motion between oppositely charged species. Charge will be conserved locally (in that there are no charge nonconserving interactions), but the expansion rates of positive and negative charges are found to differ. This allows currents to flow and the net electric charge density to change with time. This appendix examines how charge asymmetries, both local and global, evolve in an expanding universe. As a result of the dynamics of cosmological expansion, an initial net charge can be either removed completely or reduced significantly. So long as the cosmological bounds on a charge asymmetry due to cosmic rays [128], the CMB [18], and nucleosynthesis [17] are satisfied, there is no limit on any initial electric charge. The remainder of this appendix focuses on how cosmological dynamics affect the overall charge density in an expanding universe. Section B.3 presents an intuitive Newtonian formulation of a universe with an initial charge asymmetry, based upon the gravitational and electromagnetic force laws alone. While section B.3 may be useful for gathering an intuitive picture of the cosmological dynamics, it does not capture all of the correct physical behavior. A fully relativistic formulation, based upon the evolution equations derived in chapter 3, is found in section B.4. This section takes into account not only the Coulomb and gravitational forces, but also interactions between photons and charged particles, as well as all of the other interactions associated with structure formation. Although the livY i, of section B.4 is less intuitive, it is far more rigorous, and captures a much greater degree of the essential physical behavior. A discussion of the preliminary results discovered here and their implications for the early universe is presented in section B.5. B.3 Newtonian Formulation The simplest scenario that can be written down to explore a universe with a global charge asymmetry is a Newtonian cosmology containing an additional Coulomb term. The standard Newtonian cosmology is derivable from the Newtonian force law Fi = mi ai, (Bl) where i denotes a test particle of either a proton (p) or an electron (e). The Newtonian gravitational law, GN m I M Fi =  M (B2) where GN is Newton's constant, and M is the total mass enclosed by a sphere with radius equal to the universe's expansion factor, r. As acceleration is defined as a f r, the resultant equation for the evolution of the expansion factor becomes T 47r S Gp, (B3) r 3 where p is the energy density of the universe. Using Einstein's equations instead of Newton's in an isotropic, homogeneous universe would modify equation (B3) by replacing p with (p + 3p/c2), where p is the general relativistic pressure of the universe. A toy model of interest is an Einsteinde Sitter universe composed solely of protons and electrons. In this model, p mpn, + mn, (B4) for the energy density, where n, and nr are the proton and electron number densities, respectively. The Coulomb force is given by 1 qiQ Fc ,0 r (B5) 47c0 r2 where qj is the charge of a proton (qp) or electron (q,), and Q is the total charge enclosed by a sphere of radius r. The physics acting on protons and electrons is different, due to their opposite charges. Therefore, when the expansion factors are derived from the combined force laws for electrons and protons, it is found that they evolve differently than in equation (B3). The proton and electron expansion factors (rp and r6, respectively) are found to be r 47 e2 G(mn, + mene) + (np n,) (B6) rp 3 3com, and e 4 G(mpn + mene) (n, n). (B7) r, 3 3Eom6 To rewrite these two equations in terms of r,, r6, and their time derivatives alone (i.e. to remove n, and n, from the equations), N is defined to be the total number of protons or electrons contained within a sphere of radius rp or re, respectively. 4xr 4xr N 3npr, 3 ner 3. (B8) Let rp,o = re,o = ro = 1, where the subscript 0 indicates the value of a given quantity at present. r, and re are henceforth written as fractions of their value today. The ratio of the Coulomb to the gravitational forces appears in the cosmological evolution equations, defined as the dimensionless parameter K, where K  1.235 x 1036. (B9) 47eo G(m, + me)2 Finally, define quantities f,, fe, and Ho by Sm me 8Hr2 fp = fe H2 = o, (B10) mp + me mp + me 3 where H0 is the Hubble parameter (H) evaluated today. The evolution equations now take the form r? 1 (fp fe K 1 1) HO  Hr3 (B 11) rP 2 r r f rP r f, 2 H 1) + l 1 r 12) 2 + r3 +fe rP3 r3y (B } K is large, but rp is almost equal to re, therefore the final terms in equations (B11) and (B12) will only be important if there is a significant charge asymmetry. In fact, if r, = re exactly, then both equations reduce to equation (B3). The case of an exaggerated net charge in the universe is illustrated in figure B1. The expansion factor exhibits the standard behavior for an Einsteinde Sitter Universe, which is r/ro = (t/to)2/3. Protons and electrons are found not to flow smoothly together, but rather to oscillate about an equilibrium which they never reach. The charge asymmetry is not a constant over proper volume, but itself oscillates with a decreasing frequency and decreasing amplitude. The oscillation frequency is rapid compared to the rate of decrease of both frequency and amplitude. Changing variables to a centerofmass expansion factor (re) and an asymmetry parameter (6r) assists the exploration of equations (B11) and (B12) in the limit of a small charge asymmetry. The massweighted sum of protons and electrons (rem) and the difference between protons and electrons (Jr) are given by rem frp + fere, r =_ rp re, (B13) where 6r is experimentally and theoretically motivated to be much less than rem. The evolution equations are then S 12 1 3 jr2 m O2 r (K + 2pfe) ' (B14) rem 2 ,m 2 rem 102 10 1 0.1  102 10  104 1 11 106 10 104 103 102 0.1 1 t Figure B1. Expansion factors rp (red) and re (blue), for positive and negative charge distributions (rising) and their time derivatives, v, and Ve (falling). The amplitude of the asymmetry is enhanced by a factor of 104 for visibility. r 3KH 1 (B 5) 2(B15) Jr 2f~fe r32' to leading order in Jr. In the limit as Jr > 0, the standard Newtonian cosmological expansion is recovered. Since K is large (K >> 1), the asymmetry (Jr) behaves as a slowly decaying harmonic oscillator over short timescales. It is useful to parametrize Jr = A cos q. Calculation of Sr shows, in conjunction with equation B15, that 3KH2 1 2 A2 constant. (B16) Equation (B16) illustrates that A C< 02A, since 02 KH2, whereas A/A H2. (Recall that K  1.235 x 1036.) Because K > 1, the amplitude of the asymmetry, the change in the amplitude of the asymmetry, and the expansion rate of the universe all vary slowly with respect to the oscillation frequency. From equation (B16) and the definition that 4 = wt, the oscillation frequency at any epoch is w = s eno Ho(1 + z)3/2 (B17) 8VEoGmpme where no is the number density today (no = 1), and z is the redshift of interest. This corresponds to a frequency today (z = 0) of 134 rad s1, or approximately 21 Hz. The oscillation frequency at any epoch in an Einsteinde Sitter universe is therefore 21(1 + z)3/2 Hz. A slightly more sophisticated treatment includes radiation in the universe. For radiation, the pressure (Prad) is given by prad/C2 = jPrd. The overall energy density of the universe is modified by the additional term (Prad + 3Prad) = 2 ((B18) where (E) is the average energy of a photon, nB is the number density of baryons, and rq 6.1 x 1010 is the baryontophoton ratio today. With radiation included, the equations 