Time Evolution Of Perturbations Research Articles

Relativistic scalar fields and the quasistatic approximation in theories of modified gravity

Relativistic scalar fields are ubiquitous in modified theories of gravity. An important tool in understanding their impact on structure formation, especially in the context of N-body simulations, is the quasi-static approximation in which the time evolution of perturbations in the scalar fields is discarded. We show that this approximation must be used with some care by studying linearly perturbed scalar field cosmologies and quantifying the errors that arise from taking the quasi-static limit. We focus on f(R) and chameleon models and link the accuracy of the quasi-static approximation to the fast/slow-roll behaviour of the background and its proximity to {\Lambda}CDM. Investigating a large range of scales, from super- to sub-horizon, we find that slow-rolling ({\Lambda}CDM-like) backgrounds generically result in good quasi-static behaviour, even on (super-)horizon scales. We also discuss how the approximation might affect studying the non-linear growth of structure in numerical N-body simulations.

Korteveg-de Vries solitons in a cold quark-gluon plasma

The relativistic heavy ion program developed at RHIC and now at LHC motivated a deeper study of the properties of the quark gluon plasma (QGP) and, in particular, the study of perturbations in this kind of plasma. We are interested on the time evolution of perturbations in the baryon and energy densities. If a localized pulse in baryon density could propagate throughout the QGP for long distances preserving its shape and without loosing localization, this could have interesting consequences for relativistic heavy ion physics and for astrophysics. A mathematical way to proove that this can happen is to derive (under certain conditions) from the hydrodynamical equations of the QGP a Korteveg-de Vries (KdV) equation. The solution of this equation describes the propagation of a KdV soliton. The derivation of the KdV equation depends crucially on the equation of state (EOS) of the QGP. The use of the simple MIT bag model EOS does not lead to KdV solitons. Recently we have developed an EOS for the QGP which includes both perturbative and non-perturbative corrections to the MIT one and is still simple enough to allow for analitycal manipulations. With this EOS we were able to derive a KdV equation for the cold QGP.

Stable and unstable periodic orbits in complex networks of spiking neurons with delays

Is a periodic orbit underlying a periodic pattern of spikes in a heterogeneous neural network stable or unstable? We analytically assess this question in neural networks with delayed interactions by explicitly studying the microscopic time evolution of perturbations. We show that in purely inhibitorily coupled networks of neurons with normal dissipation (concave rise function), such as common leaky integrate-and-fire neurons, all orbits underlying non-degenerate periodic spike patterns are stable. In purely inhibitorily coupled networks with strongly connected topology and normal dissipation (strictly concave rise function), they are even asymptotically stable. In contrast, for the same type of individual neurons, all orbits underlying such patterns are unstable if the coupling is excitatory. For networks of neurons with anomalous dissipation ((strictly) convex rise function), the reverse statements hold. For the stable dynamics, we give an analytical lower bound on the local size of the basin of attraction. Numerical simulations of networks with different integrate-and-fire type neurons illustrate our results.

Solitons in a macroscopic traffic model

The steady behavior in the macroscopic Kerner and Konhäuser (1993, 1994, 1997) traffic model in a closed circuit, is studied numerically to see the time evolution of perturbations around the homogeneous steady case[9]. The simulation shows that after a transient time interval, the perturbation develops a traveling wave structure with the characteristics associated to a typical soliton in nonlinear partial differential equations. In traffic flow, the formation of solitons indicate the permanent presence of jams along the road. The solitons characteristics depend on the fundamental diagram and the specific values in the viscosity and the velocity variance introduced in the model. The nonlinearity and dissipation in the model cause the propagation of the traveling wave, Drazin (1990)[3].

Time evolution of density perturbations in accelerating stratified fluids

We consider N superimposed fluids, having arbitrary initial perturbations at their N-1 interfaces, undergoing a Rayleigh-Taylor instability with constant acceleration. The time evolution of these perturbations is described by a sum over normal modes and in general is a combination of oscillation and exponential rise. We derive the equation governing their growth, discuss the case N = 3 analytically, and give two numerical examples where we plot the time evolution of the perturbations at the four interfaces of five superimposed fluids. Our examples illustrate how the evolution depends on the density profile, on the wavelength of the perturbations, and on the initial conditions.