On the nonlinear stability of symplectic integrators

The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially for nonlinear oscillators. We give conditions under which an initial condition on a compact energy surface will remain bounded for exponentially long times for sufficiently small time steps. For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon- Heiles system, while the leapfrog method does not. We construct explicit methods which are nonlinearly stable for all simple mechanical systems for exponentially long times. We also address questions of topological stability, finding conditions under which the original and modified energy surfaces are topologically equivalent

Viscous Dark Energy Models with Variable G and Lambda

We consider a cosmological model with bulk viscosity ($\eta$) and variable cosmological $(\Lambda\propto \rho^{-\alpha}, \alpha=\rm const.$) and gravitational ($G$) constants. The model exhibits many interesting cosmological features. Inflation proceeds du to the presence of bulk viscosity and dark energy without requiring the equation of state $p=-\rho$. During the inflationary era the energy density ($\rho$) does not remain constant, as in the de-Sitter type. Moreover, the cosmological and gravitational constants increase exponentially with time, whereas the energy density and viscosity decrease exponentially with time. The rate of mass creation during inflation is found to be very huge suggesting that all matter in the universe was created during inflation.Comment: 6 Latex page

Phantom Energy Accretion by Stringy Charged Black Hole

We investigate the dynamical behavior of phantom energy near stringy magnetically charged black hole. For this purpose, we derive equations of motion for steady-state spherically symmetric flow of phantom energy onto the stringy magnetically charged black hole. It is found that phantom energy accreting onto black hole decreases its mass. Further, the location of critical points of accretion is explored, which yields mass to charge ratio. This ratio implies that accretion process cannot transform a black hole into an extremal black hole or a naked singularity, hence cosmic censorship hypothesis remains valid here.Comment: 7 pages, no figur

Opportunities for future supernova studies of cosmic acceleration

We investigate the potential of a future supernova dataset, as might be obtained by the proposed SNAP satellite, to discriminate among different ``dark energy'' theories that describe an accelerating Universe. We find that many such models can be distinguished with a fit to the effective pressure-to-density ratio, $w$, of this energy. More models can be distinguished when the effective slope, $dw/dz$, of a changing $w$ is also fit, but only if our knowledge of the current mass density, $\Omega_m$, is improved. We investigate the use of ``fitting functions'' to interpret luminosity distance data from supernova searches, and argue in favor of a particular preferred method, which we use in our analysis.Comment: Four pages including figures. Final published version. No significant changes from v

Geodesic Warps by Conformal Mappings

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g.\ in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations

Search for Tracker Potentials in Quintessence Theory

We report a significant finding in Quintessence theory that the the scalar fields with tracker potentials have a model-independent scaling behaviour in the expanding universe. So far widely discussed exponential,power law or hyperbolic potentials can simply mimic the tracking behaviour over a limited range of redshift. In the small redshift range where the variation of the tracking parameter $\epsilon$ may be taken to be negligible, the differential equation of generic potentials leads to hyperbolic sine and hyperbolic cosine potentials which may approximate tracker field in the present day universe. We have plotted the variation of tracker potential and the equation of state of the tracker field as function of the redshift $z$ for the model-independent relation derived from tracker field theory; we have also plotted the variation of $V(\Phi)$ in terms of the scalar field $\Phi$ for the chosen hyperbolic cosine function and have compared with the curves obtained by reconstruction of $V(\phi)$ from the real observational data from the supernovae.Comment: 11 pages, 3 figures, late

Codimension Two Compactifications and the Cosmological Constant Problem

We consider solutions of six dimensional Einstein equations with two compact dimensions. It is shown that one can introduce 3-branes in this background in such a way that the effective four dimensional cosmological constant is completely independent of the brane tensions. These tensions are completely arbitrary, without requiring any fine tuning. We must, however, fine tune bulk parameters in order to obtain a sufficiently small value for the observable cosmological constant. We comment in the effective four dimensional description of this effect at energies below the compactification scale.Comment: 4 pages, rextex

Naked Singularity in a Modified Gravity Theory

The cosmological constant induced by quantum fluctuation of the graviton on a given background is considered as a tool for building a spectrum of different geometries. In particular, we apply the method to the Schwarzschild background with positive and negative mass parameter. In this way, we put on the same level of comparison the related naked singularity (-M) and the positive mass wormhole. We discuss how to extract information in the context of a f(R) theory. We use the Wheeler-De Witt equation as a basic equation to perform such an analysis regarded as a Sturm-Liouville problem . The application of the same procedure used for the ordinary theory, namely f(R)=R, reveals that to this approximation level, it is not possible to classify the Schwarzschild and its naked partner into a geometry spectrum.Comment: 8 Pages. Contribution given to DICE 2008. To appear in the proceeding

Zeldovich flow on cosmic vacuum background: new exact nonlinear analytical solution

A new exact nonlinear Newtonian solution for a plane matter flow superimposed on the isotropic Hubble expansion is reported. The dynamical effect of cosmic vacuum is taken into account. The solution describes the evolution of nonlinear perturbations via gravitational instability of matter and the termination of the perturbation growth by anti-gravity of vacuum at the epoch of transition from matter domination to vacuum domination. On this basis, an `approximate' 3D solution is suggested as an analog of the Zeldovich ansatz.Comment: 9 pages, 1 figure