School of Mathematical Sciences

An upper bound on the amplitude of the primordial gravitational wave spectrum generated during ultra-violet DBI inflation is derived. The bound is insensitive to the form of the inflaton potential and the warp factor of the compactified dimensions and can be expressed entirely in terms of observational parameters once the volume of the five-dimensional sub-manifold of the throat has been specified. For standard type IIB compactification schemes, the bound predicts undetectably small tensor perturbations with a tensor-scalar ratio $r < 10^{-7}$. This is incompatible with a corresponding lower limit of $r > 0.1 (1-n_s)$, which applies to any model that generates a red spectral index $n_s <1$ and a potentially detectable non-Gaussianity in the curvature perturbation. Possible ways of evading these bounds in more general DBI-type scenarios are discussed and a multiple-brane model is investigated as a specific example.

We numerically solve the Klein-Gordon equation at second order in cosmological perturbation theory in closed form for a single scalar field, describing the method employed in detail. We use the slow-roll version of the second order source term and show that our method is extendable to the full equation. We consider two standard single field models and find that the results agree with previous calculations using analytic methods, where comparison is possible. Our procedure allows the evolution of second order perturbations in general and the calculation of the non-linearity parameter f_NL to be examined in cases where there is no analytical solution available.

In this paper we summarise the status of single field models of inflation in light of the WMAP 7 data release. We find little has changed since the 5 year release, and results are consistent with previous findings. The increase in the upper bound on the running of the spectral index impacts on the status of the production of Primordial Black Holes from single field models. The lower bound on the equilateral configuration of the non-gaussianity parameter is reduced and thus the bounds on the theoretical parameters of (UV) DBI single brane models are weakened. In the case of multiple coincident branes the bounds are also weakened and the two, three or four brane cases will produce a tensor-signal that could possibly be observed in the future.

Inflationary cosmology is the leading explanation of the very early universe. Many different models of inflation have been constructed which fit current observational data. In this work theoretical and numerical methods for constraining the parameter space of a wide class of such models are described. First, string-theoretic models with large non-Gaussian signatures are investigated. An upper bound is placed on the amplitude of primordial gravitational waves produced by ultra-violet Dirac-Born-Infeld inflation. In all but the most finely tuned cases, this bound is incompatible with a lower bound derived for inflationary models which exhibit a red spectrum and detectable non-Gaussianity. By analysing general non-canonical actions, a class of models is found which can evade the upper bound when the phase speed of perturbations is small. The multi-coincident brane scenario with a finite number of branes is one such model. For models with a potentially observable gravitational wave spectrum the number of coincident branes is shown to take only small values. The second method of constraining inflationary models is the numerical calculation of second order perturbations for a general class of single field models. The Klein-Gordon equation at second order, written in terms of scalar field variations only, is numerically solved. The slow roll version of the second order source term is used and the method is shown to be extendable to the full equation. This procedure allows the evolution of second order perturbations in general and the calculation of the non-Gaussianity parameter in cases where there is no analytical solution available.

We numerically calculate the evolution of second order cosmological perturbations for an inflationary scalar field without resorting to the slow-roll approximation or assuming large scales. In contrast to previous approaches we therefore use the full non-slow-roll source term for the second order Klein-Gordon equation which is valid on all scales. The numerical results are consistent with the ones obtained previously where slow-roll is a good approximation. We investigate the effect of localised features in the scalar field potential which break slow-roll for some portion of the evolution. The numerical package solving the second order Klein-Gordon equation has been released under an open source license and is available for download.

Abstract We study transient work fluctuation relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise:(i) internal noise where the fluctuation–dissipation relation of the second kind (FDR II) holds, and (ii) external noise without FDR II.

Abstract This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift.

This multi-author reference work provides a unique introduction to the currently emerging, highly interdisciplinary field of those transport processes that cannot be described by using standard methods of statistical mechanics. It comprehensively summarizes topics ranging from mathematical foundations of anomalous dynamics to the most recent experiments in this field.

This allows construction of a periodic orbit expansion for small, but finite holes generalizing simple random walk theory, which is based on an effectively uncorrelated dynamics. Finally, we compare our diffusion results with those obtained previously for the escape rate and discuss the similarities and differences that arise.

We study the deterministic diffusion coefficient of the two-dimensional periodic Lorentz gas as a function of the density of scatterers. Based on computer simulations, and by applying straightforward analytical arguments, we systematically improve the Machta–Zwanzig random walk approximation [Phys. Rev. Lett. 50: 1959 (1983)] by including microscopic correlations. We furthermore, show that, on a fine scale, the diffusion coefficient is a non-trivial function of the density.

Abstract These are easy-to-read lecture notes for a short first-year Ph. D. student course on Applied Dynamical Systems, given at the London Taught Course Centre in Spring 2008, 2009 and 2010. The full course consists of two parts, covering four and six hours of lectures, respectively. The first part taught by Wolfram Just is on Basic Dynamical Systems. The present notes cover only the second part, which leads From Deterministic Chaos to Deterministic Diffusion.

It is in principle possible to develop the full theory of both from either perspective, but for the bulk of this course, we shall follow the latter route. This allows a generally more simple way of introducing the important concepts, which can usually be carried over to a more complex and physically realistic context.

A century ago Einstein developed a theory of diffusion that is based on the assumption of stochasticity for a Brownian particle. On a microscopic level, however, the particle's dynamics is governed by Newton's deterministic equations of motion, which typically are highly nonlinear. This motivates to replace the hypothesis of stochasticity underlying traditional statistical mechanics by the hypothesis of microscopic deterministic chaos in the equations of motion of a Brownian particle.

Abstract: A particle moving deterministically in a chaotic spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here we consider the dependence of the diffusion coefficient on the size and the position of dynamical channels (holes') linking spatial regions.

One approach starts from time-continuous differential equations and leads to time-discrete maps, which are obtained from them by a suitable discretization of time. This path is pursued, eg, in the book by Strogatz [Str94]. 1 The other approach starts from the study of time-discrete maps and then gradually builds up to time-continuous differential equations, see, eg,[Ott93, All97, Dev89, Has03, Rob95].

Diffusion is a fundamental macroscopic transport process in many-particle systems. It is quantifiable by the diffusion coefficient, which describes the linear growth in the meansquare displacement of an ensemble of particles. The source of this growth is often considered to be a Brownian or random process of collisions between particles. However, on a microscopic scale the equations governing these collisions in physical systems are deterministic and typically chaotic.

Abstract. We consider work fluctuation relations (FRs) for generic types of dynamics generating anomalous diffusion: Lévy flights, long-correlated Gaussian processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the probability distributions of mechanical and thermodynamical work in two paradigmatic nonequilibrium situations, respectively: a particle subject to a constant force and a particle in a harmonic potential dragged by a constant force.

Abstract: The stability analysis introduced by Lyapunov and extended by Oseledec is an excellent tool to describe the character of nonlinear n-dimensional flows by n global exponents if these flows are stable in time. However, there are two main shortcomings:(a) The local exponents fail to indicate the origin of instability where trajectories start to diverge. Instead, their time evolution contains a much stronger chaos than the trajectories, which is only eliminated by integrating over a long time.

Abstract: This is an easy-to-read introduction to foundations of deterministic chaos, deterministic diffusion and anomalous diffusion. The first part introduces to deterministic chaos in one-dimensional maps in form of Ljapunov exponents and dynamical entropies. The second part outlines the concept of deterministic diffusion. Then the escape rate formalism for deterministic diffusion, which expresses the diffusion coefficient in terms of the above two chaos quantities, is worked out for a simple map.

Abstract We consider families of dynamics that can be described in terms of Perron–Frobenius operators with exponential mixing properties. For piecewise C2 expanding interval maps we rigorously prove continuity properties of the drift J (λ) and of the diffusion coefficient D (λ) under parameter variation. Our main result is that D (λ) has a modulus of continuity of order O (| δλ|·(log| δλ|) 2), ie D (λ) is Lipschitz continuous up to quadratic logarithmic corrections.