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Modern Physics intertwines active learning pedagogy with the material typically covered in an introductory survey, from the basics of relativity and quantum mechanics through recent developments in particle physics and cosmology. The flexible approach taken by the authors allows instructors to easily incorporate as much or as little active learning into their teaching as they choose. Chapters are enhanced by 'Discovery' and 'Active Reading' exercises to guide students through key ideas before or during class, while 'ConcepTests' help check student understanding and stimulate classroom discussions. Each chapter also includes extensive assessment material, with a range of basic comprehension questions, drill and practice calculations, computer-based problems, and explorations of advanced applications. A test bank and interactive animations as well as other support for instructors and students are available online. Students are engaged by an accessible and lively writing style, thorough explanations, 'Math Interludes' which account for varying levels of skill and experience, and advanced topics to further pique their interest in physics.

We study the dynamical equations for extra-dimensional dependence of a warp factor and a bulk scalar in 5d brane world scenarios with induced brane metric of constant curvature. These equations are similar to those for the time dependence of the scale factor and a scalar field in 4d cosmology, but with the sign of the scalar field potential reversed. Based on this analogy, we introduce novel methods for studying the warped geometry. We construct the full phase portraits of the warp factor/scalar system for several examples of the bulk potential. This allows us to view the global properties of the warped geometry. For flat branes, the phase portrait is two dimensional. Moving along typical phase trajectories, the warp factor is initially increasing and finally decreasing. All trajectories have timelike gradient-dominated singularities at one or both of their ends, which are reachable in a finite distance and must be screened by the branes. For curved branes, the phase portrait is three dimensional. However, as the warp factor increases the phase trajectories tend towards the two dimensional surface corresponding to flat branes. We discuss this property as a mechanism that may stretch the curved brane to be almost flat, with a small cosmological constant. Finally, we describe the embedding of branes in the 5d bulk using the phase space geometric methods developed here. In this language the boundary conditions at the branes can be described as a 1d curve in the phase space. We discuss the naturalness of tuning the brane potential to stabilize the brane world system., 25 pages, 6 figures; reference added, minor typos corrected

The Myers-Briggs Type Indicator ® (MBTI) was administered to a group of 116 students taking the introductory chemical engineering course at North Carolina State University. That course and four subsequent chemical engineering courses were taught in a manner that emphasized active and cooperative learning and inductive presentation of course material. Type differences in various academic performance measures and attitudes were noted as the students progressed through the curriculum. The observations were generally consistent with the predictions of type theory, and the experimental instructional approach appeared to improve the performance of MBTI types (extraverts, sensors, and feelers) found in previous studies to be disadvantaged in the engineering curriculum. The conclusion is that the MBTI is a useful tool for helping engineering instructors and advisors to understand their students and to design instruction that can benefit students of all types.

We study the dynamical equations for extra-dimensional dependence of a warp factor and a bulk scalar in 5d brane world scenarios with induced brane metric of constant curvature. These equations are similar to those for the time dependence of the scale factor and a scalar field in 4d cosmology, but with the sign of the scalar field potential reversed. Based on this analogy, we introduce novel methods for studying the warped geometry. We construct the full phase portraits of the warp factor/scalar system for several examples of the bulk potential. This allows us to view the global properties of the warped geometry. For flat branes, the phase portrait is two dimensional. Moving along typical phase trajectories, the warp factor is initially increasing and finally decreasing. All trajectories have timelike gradient-dominated singularities at one or both of their ends, which are reachable in a finite distance and must be screened by the branes. For curved branes, the phase portrait is three dimensional. However, as the warp factor increases the phase trajectories tend towards the two dimensional surface corresponding to flat branes.

In a longitudinal study at North Carolina State University, a cohort of students took five chemical engineering courses taught by the same instructor in five consecutive semesters. The courses made extensive use of active and cooperative learning and a variety of other techniques designed to address a broad spectrum of learning styles. Previous reports on the study summarized the instructional methods used in the experimental course sequence, described the performance of the cohort in the introductory chemical engineering course, and examined performance and attitude differences between students from rural and urban backgrounds and between male and female students.1–4 This paper compares outcomes for the experimental cohort with outcomes for students in a traditionally-taught comparison group. The experimental group outperformed the comparison group on a number of measures, including retention and graduation in chemical engineering, and many more of the graduates in this group chose to pursue advanced study in the field. Since the experimental instructional model did not require small classes (the smallest of the experimental classes had 90 students) or specially equipped classrooms, it should be adaptable to any engineering curriculum at any institution.

In a continuing study under way at North Carolina State University, a cohort of students took five chemical engineering courses taught by the same instructor in five consecutive semesters. This report examines gender differences in the students’ academic performance, persistence in chemical engineering, and attitudes toward their education and themselves. The women in the study on average entered chemical engineering with credentials equal to or better than those of the men, but exhibited erosion relative to the men in both academic performance and confidence as they progressed through the curriculum. Possible causes of the observed disparities are suggested and remedial measures are proposed.

20 pages, 14 figures, 2 appendix.-- Pre-print archive., We study the stochastic background of gravitational waves produced from preheating in hybrid inflation models. We investigate different dynamical regimes of preheating in these models and we compute the resulting gravity wave spectra using analytical estimates and numerical simulations. We discuss the dependence of the gravity wave frequencies and amplitudes on the various potential parameters. We find that large regions of the parameter space leads to gravity waves that may be observable in upcoming interferometric experiments, including Advanced LIGO, but this generally requires very small coupling constants., The work of J.F.D. was supported by the Spanish MEC, via FPA2006-05807 and FPA2006-05423. G.F. was supported by PHY-0456631. L.K. was supported by NSERC and CIFAR.

Preheating after inflation involves large, time-dependent field inhomogeneities, which act as a classical source of gravitational radiation. The resulting spectrum might be probed by direct detection experiments if inflation occurs at a low enough energy scale. In this paper, we develop a theory and algorithm to calculate, analytically and numerically, the spectrum of energy density in gravitational waves produced from an inhomogeneous background of stochastic scalar fields in an expanding universe. We derive some generic analytical results for the emission of gravity waves by stochastic media of random fields, which can test the validity/accuracy of numerical calculations. We contrast our method with other numerical methods in the literature, and then we apply it to preheating after chaotic inflation. In this case, we are able to check analytically our numerical results, which differ significantly from previous works. We discuss how the gravity wave spectrum builds up with time and find that the amplitude and the frequency of its peak depend in a relatively simple way on the characteristic spatial scale amplified during preheating. We then estimate the peak frequency and amplitude of the spectrum produced in two models of preheating after hybrid inflation, which for some parameters may be relevant for gravity wave interferometric experiments., Comment: 28 pages, 10 figures, refs added, published version

We consider the nonlinear dynamics of inflaton fragmentation during and after preheating in the simplest model of chaotic inflation. While the earlier regime of parametric resonant particle production and the later turbulent regime of interacting fields evolving towards equilibrium are well identified and understood, the short intermediate stage of violent nonlinear dynamics remains less explored. Lattice simulations of fully nonlinear preheating dynamics show specific features of this intermediate stage: occupation numbers of the scalar particles are peaked, scalar fields become significantly non-gaussian and the field dynamics become chaotic and irreversible. Visualization of the field dynamics in configuration space reveals that nonlinear interactions generate non-gaussian inflaton inhomogeneities with very fast growing amplitudes. The peaks of the inflaton inhomogeneities coincide with the peaks of the scalar field(s) produced by parametric resonance. When the inflaton peaks reach their maxima, they stop growing and begin to expand. The subsequent dynamics is determined by expansion and superposition of the scalar waves originating from the peaks. Multiple wave superposition results in phase mixing and turbulent wave dynamics. Thus, the short intermediate stage is defined by the formation, expansion and collision of bubble-like field inhomogeneities associated with the peaks of the original gaussian field. This process is qualitatively similar to the bubble-like inflaton fragmentation that occurs during tachyonic preheating after hybrid or new inflation., 9 pages, 6 figs

We investigate the effects of bosonic trilinear interactions in preheating after chaotic inflation. A trilinear interaction term allows for the complete decay of the massive inflaton particles, which is necessary for the transition to radiation domination. We found that typically the trilinear term is subdominant during early stages of preheating, but it actually amplifies parametric resonance driven by the four-legs interaction. In cases where the trilinear term does dominate during preheating, the process occurs through periodic tachyonic amplifications with resonance effects, which is so effective that preheating completes within a few inflaton oscillations. We develop an analytic theory of this process, which we call tachyonic resonance. We also study numerically the influence of trilinear interactions on the dynamics after preheating. The trilinear term eventually comes to dominate after preheating, leading to faster rescattering and thermalization than could occur without it. Finally, we investigate the role of non-renormalizable interaction terms during preheating. We find that if they are present they generally dominate (while still in a controllable regime) in chaotic inflation models. Preheating due to these terms proceeds through a modified form of tachyonic resonance., 19 pages, 10 figures, refs added, published version

We study the out-of-equilibrium nonlinear dynamics of fields after post-inflationary preheating. During preheating, the energy in the homogeneous inflaton is exponentially rapidly transfered into highly occupied out-of-equilibrium inhomogeneous modes, which subsequently evolve towards equilibrium. The infrared modes excited during preheating evolve towards a saturated distribution long before thermalization completes. We compute the equation of state during and immediately after preheating. It rapidly evolves towards radiation domination long before the actual thermal equilibrium is established. The exact time of this transition is a non-monotonic function of the coupling between the inflaton and the decay products, and it varies only very weakly (around 10^(-35) s) as this coupling changes over several orders of magnitude. This result is applied to refine the relation between the number of efoldings N and the physical wavelength of perturbations generated during inflation. We also discuss the implications for the theory of modulated perturbations from preheating. We finally argue that many questions of the thermal history of the universe should be addressed in terms of pre-thermalization, illustrating this point with a calculation of perturbative production of gravitinos immediately after chaotic inflation. We also highlight the effects of three-legs inflaton interactions on the dynamics of preheating and thermalization in an expanding universe., 15 pages, 13 figures

During the last ten years a detailed investigation of preheating was performed for chaotic inflation and for hybrid inflation. However, nonperturbative effects during reheating in the new inflation scenario remained practically unexplored. We do a full analysis of preheating in new inflation, using a combination of analytical and numerical methods. We find that the decay of the homogeneous component of the inflaton field and the resulting process of spontaneous symmetry breaking in the simplest models of new inflation usually occurs almost instantly: for the new inflation on the GUT scale it takes only about 5 oscillations of the field distribution. The decay of the homogeneous inflaton field is so efficient because of a combined effect of tachyonic preheating and parametric resonance. At that stage, the homogeneous oscillating inflaton field decays into a collection of waves of the inflaton field, with a typical wavelength of the order of the inverse inflaton mass. This stage usually is followed by a long stage of decay of the inflaton field into other particles, which can be described by the perturbative approach to reheating after inflation. The resulting reheating temperature typically is rather low., Comment: 21 pages, 7 figures, revtex4

We describe an algorithm and a C++ implementation that we have written and made available for calculating the fully nonlinear evolution of 5D braneworld models with scalar fields. Bulk fields allow for the stabilization of the extra space. However, they complicate the dynamics of the system, so that analytic calculations (performed within an effective 4D theory) are typically only reliable close to stabilized configurations or when the evolution of the extra space is negligible. In the general case, a numerical study of the 5D equations is necessary, and the algorithm and code we describe are the first ones designed for this task. The program and its full documentation are available on the Web at http://www.cita.utoronto.ca/~jmartin/BRANECODE/. In this paper we provide a brief overview of what the program does and how to use it., 5 pages, 2 figures

Dirac-Born-Infeld type effective actions reproduce many aspects of string theory classical tachyon dynamics of unstable Dp-branes. The inhomogeneous tachyon field rolling from the top of its potential forms topological defects of lower codimensions. In between them, as we show, the tachyon energy density fragments into a p-dimensional web-like high density network evolving with time. We present an analytic asymptotic series solution of the non-linear equations for the inhomogeneous tachyon and its stress energy. The generic solution for a tachyon field with a runaway potential in arbitrary dimensions is described by the free streaming of noninteracting massive particles whose initial velocities are defined by the gradients of the initial tachyon profile. Thus, relativistic particle mechanics is a dual picture of the tachyon field effective action. Implications of this picture for inflationary models with a decaying tachyon field are discussed., Comment: 10 pages, 1 figure

We give a full nonlinear numerical treatment of time-dependent 5d braneworld geometry, which is determined self-consistently by potentials for the scalar field in the bulk and at two orbifold branes, supplemented by boundary conditions at the branes. We describe the BraneCode, an algorithm which we designed to solve the dynamical equations numerically. We applied the BraneCode to braneworld models and found several novel phenomena of the brane dynamics. Starting with static warped geometry with de Sitter branes, we found numerically that this configuration is often unstable due to a tachyonic mass of the radion during inflation. If the model admits other static configurations with lower values of de Sitter curvature, this effect causes a violent re-structuring towards them, flattening the branes, which appears as a lowering of the 4d effective cosmological constant. Braneworld dynamics can often lead to brane collisions. We found that in the presence of the bulk scalar field, the 5d geometry between colliding branes approaches a universal, homogeneous, anisotropic strong gravity Kasner-like asymptotic, irrespective of the bulk/brane potentials. The Kasner indices of the brane directions are equal to each other but different from that of the extra dimension., Comment: 38 pages, 10 figures

In the fall of 1990, the authors began a longitudinal study of a cohort of students enrolled in the introductory engineering course. These students were taught in a sequence of five engineering courses in five consecutive semesters, using a variety of instructional methods including extensive collaborative (team-based) learning, routine assignment of open-ended problems and problem formulation exercises, and other techniques designed to address the spectrum of learning styles found in all engineering classes. The students enrolled in the introductory course in the fall 1992 semester were designated as a comparison group. Academic performance and retention data were collected and attitudes and self-confidence levels were assessed for both groups throughout their second, third, and fourth years of college. This paper provides a mostly qualitative summary of the assessment data analyzed so far.

We investigate cosmological evolution in models where the effective potential V(\phi) may become negative for some values of the field \phi. Phase portraits of such theories in space of variables (\phi,\dot\phi,H) have several qualitatively new features as compared with phase portraits in the theories with V(\phi) > 0. Cosmological evolution in models with potentials with a "stable" minimum at V(\phi), Comment: 51 page, 20 figures

Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $\phi = 0$. Naively, one might expect the field $\phi$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $\phi = v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. In the simplest models where $V(\phi) \sim -m^2\phi^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-\lambda\phi^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions., Comment: 40 pages, 14 figures, revtex

We describe a C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe. The program is particularly useful for the study of reheating and thermalization after inflation. The program and its full documentation are available on the Web at http://physics.stanford.edu/gfelder/latticeeasy/ . In this paper we provide a brief overview of what the program does and what it is useful for., 4 pages, 2 figures

We present a fully nonlinear study of the development of equilibrium after preheating. Preheating is the exponentially rapid transfer of energy from the nearly homogeneous inflaton field to fluctuations of other fields and/or the inflaton itself. This rapid transfer leaves these fields in a highly nonthermal state with energy concentrated in infrared modes. We have performed lattice simulations of the evolution of interacting scalar fields during and after preheating for a variety of inflationary models. We have formulated a set of generic rules that govern the thermalization process in all of these models. Notably, we see that once one of the fields is amplified through parametric resonance or other mechanisms it rapidly excites other coupled fields to exponentially large occupation numbers. These fields quickly acquire nearly thermal spectra in the infrared, which gradually propagates into higher momenta. Prior to the formation of total equilibrium, the excited fields group into subsets with almost identical characteristics (e.g. group effective temperature). The way fields form into these groups and the properties of the groups depend on the couplings between them. We also studied the onset of chaos after preheating by calculating the Lyapunov exponent of the scalar fields., 15 pages, 23 figures

Preheating after inflation may lead to nonthermal phase transitions with symmetry restoration. These phase transitions may occur even if the total energy density of fluctuations produced during reheating is relatively small as compared with the vacuum energy in the state with restored symmetry. As a result, in some inflationary models one encounters a secondary, nonthermal stage of inflation due to symmetry restoration after preheating. We review the theory of nonthermal phase transitions and make a prediction about the expansion factor during the secondary inflationary stage. We then present the results of lattice simulations which verify these predictions, and discuss possible implications of our results for the theory of formation of topological defects during nonthermal phase transitions., 22 pages, 7 figures, JHEP

We study inflationary models in which the effective potential of the inflaton field does not have a minimum, but rather gradually decreases at large $\phi$. In such models the inflaton field does not oscillate after inflation, and its effective mass becomes vanishingly small, so the standard theory of reheating based on the decay of the oscillating inflaton field does not apply. For a long time the only mechanism of reheating in such non-oscillatory (NO) models was based on gravitational particle production in an expanding universe. This mechanism is very inefficient. We will show that it may lead to cosmological problems associated with large isocurvature fluctuations and overproduction of dangerous relics such as gravitinos and moduli fields. We also note that the setting of initial conditions for the stage of reheating in these models should be reconsidered. All of these problems can be resolved in the context of the recently proposed scenario of instant preheating if there exists an interaction ${g^2} \phi^2\chi^2$ of the inflaton field $\phi$ with another scalar field $\chi$. We show that the mechanism of instant preheating in NO models is much more efficient than the usual mechanism of gravitational particle production even if the coupling constant $g^2$ is extremely small, $10^{-14} \ll g^2 \ll 1$.

A theory of gravitational production of light scalar particles during and after inflation is investigated. We show that in the most interesting cases where long-wavelength fluctuations of light scalar fields can be generated during inflation, these fluctuations rather than quantum fluctuations produced after inflation give the dominant contribution to particle production. In such cases a simple analytical theory of particle production can be developed. Application of our results to the theory of quantum creation of moduli fields demonstrates that if the moduli mass is smaller than the Hubble constant then these fields are copiously produced during inflation. This gives rise to the cosmological moduli problem even if there is no homogeneous component of the classical moduli field in the universe. To avoid this version of the moduli problem it is necessary for the Hubble constant H during the last stages of inflation and/or the reheating temperature T_R after inflation to be extremely small., Comment: LaTeX,7 pages, 1 figure. Minor corrections

Preheating after inflation may lead to nonthermal phase transitions with symmetry restoration. These phase transitions may occur even if the total energy density of fluctuations produced during reheating is relatively small as compared with the vacuum energy in the state with restored symmetry. As a result, in some inflationary models one may encounter a secondary stage of inflation after preheating.

We describe a new efficient mechanism of reheating. Immediately after rolling down the rapidly moving inflaton field $\phi$ produces particles $\chi$, which may be either bosons or fermions. This is a nonperturbative process which occurs almost instantly; no oscillations or parametric resonance is required. The effective masses of the $\chi$ particles may be very small at the moment when they are produced, but they ``fatten'' when the field $\phi$ increases. When the particles $\chi$ become sufficiently heavy, they rapidly decay to other, lighter particles. This leads to an almost instantaneous reheating accompanied by the production of particles with masses which may be as large as $10^{17}-10^{18}$ GeV. This mechanism works in the usual inflationary models where $V(\phi)$ has a minimum, where it takes only a half of a single oscillation of the inflaton field $\phi$, but it is especially efficient in models with effective potentials slowly decreasing at large $\phi$ as in the theory of quintessence., Comment: 7 pages, revtex, few comments added

Previously, we proposed a model of low energy Affleck-Dine leptogenesis in the context of thermal inflation. The lepton asymmetry is generated at the end of thermal inflation, which occurs at a relatively low energy scale with the Hubble parameter somewhere in the range $1 \keV \lesssim H \lesssim 1 \MeV$. Thus Hubble damping will be ineffective in bringing the Affleck-Dine field into the lepton conserving region near the origin, leaving the possibility that the lepton number could be washed out. Previously, we suggested that preheating could damp the amplitude of the Affleck-Dine field allowing conservation of the lepton number. In this paper, we demonstrate numerically that preheating does efficiently damp the amplitude of the Affleck-Dine field and that the lepton number is conserved as the result. In addition to demonstrating a crucial aspect of our model, it also opens the more general possibility of low energy Affleck-Dine baryogenesis., 38 pages, 17 figures

We use lattice simulations to examine the detailed dynamics of inflaton fragmentation during and after preheating in $\lambda \phi^4$ chaotic inflation. The dynamics are qualitatively similar to preheating after $m^2 \phi^2$ inflation, involving the exponential growth and subsequent expansion and collision of bubble-like inhomogeneities of the inflaton and other scalar fields. During this stage fluctuations of the fields become strongly non-Gaussian. In the quartic theory, the conformal nature of the theory allows us to extend our simulations to much greater times than is possible for the quadratic model. With these longer simulations we have been able to determine the time scale on which Gaussianity is restored, which occurs after a time on the order of a thousand inflaton oscillations.

We consider scalar Born-Infeld type theories with arbitrary potentials V(T) of a scalar field T. We find that for models with runaway potentials V(T) the generic inhomogeneous solutions after a short transient stage can be very well approximated by the solutions of a Hamilton-Jacobi equation that describes free streaming wave front propagation. The analytic solution for this wave propagation shows the formation of caustics with multi-valued regions beyond them. We verified that these caustics appear in numerical solutions of the original scalar BI non-linear equations. Our results include the scalar BI model with an exponential potential, which was recently proposed as an effective action for the string theory tachyon in the approximation where high-order spacetime derivatives of T are truncated. Since the actual string tachyon dynamics contain derivatives of all orders, the tachyon BI model with an exponential potential becomes inadequate when the caustics develop because high order spatial derivatives of T become divergent. BI type tachyon theory with a potential decreasing at large T could have interesting cosmological applications because the tachyon field rolling towards its ground state at infinity acts as pressureless dark matter. We find that inhomogeneous cosmological tachyon fluctuations rapidly grow and develop multiple caustics. Any considerations of the role of the tachyon field in cosmology will have to involve finding a way to predict the behavior of the field at and beyond these caustics., Comment: 21 pages, 5 figures

We reconsider the old problem of the dynamics of spontaneous symmetry breaking using 3d lattice simulations, and develop a theory of tachyonic preheating, which occurs due to the spinodal instability of the scalar field. Tachyonic preheating is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. As an application of this theory we consider preheating in the hybrid inflation scenario, including SUSY-motivated F-term and D-term inflationary models. We show that preheating in hybrid inflation is typically tachyonic and the stage of oscillations of a homogeneous component of the scalar fields driving inflation ends after a single oscillation. Our results may also be relevant for the theory of the formation of disoriented chiral condensates in heavy ion collisions., Comment: 7 pages, 6 figures. Higher quality figures and computer generated movies in gif format illustrating our results can be found at http://physics.stanford.edu/gfelder/hybrid