Perturbative improvement of the WKB approximation (original) (raw)

Cosmological particle production and the precision of the WKB approximation

Physical Review D, 2005

Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.

Semiclassical Approximations to Cosmological Perturbations

2007

We apply several methods related to the WKB approximation to study cosmological perturbations during inflation, obtaining the full power spectra of scalar and tensor perturbations to first and to second order in the slow-roll parameters. We compare our results with those derived by means of other methods, in particular the Green's function method, and find agreement for the slow-roll structure.

P o S ( I S F T G ) 0 1 5 P o S ( I S F T G ) 0 1 5 New approximation methods in General Relativity

2009

We show how approximate solutions of the two-body problem in General Relativity, and the approximate solutions of Einstein’s equations in vacuo can be constructed using small deformations of geodesics and of Einstein space-times embedded into a pseudo-Euclidean flat space of higher dimension. The method consists in using expansions of equations around a given simple solution (a circular orbit in the case of geodesics, and Minkowskian or Schwarzschild space in the case of Einstein’s equations) in a series of powers of small deformation parameter, and then solving by iteration the corresponding linear systems of differential equations.

Analytic approximations, perturbation methods, and their applications

The paper summarizes the parallel session B3 Analytic approximations, perturbation methods, and their applications of the GR18 conference. The talks in the session reported notably recent advances in black hole perturbations and post-Newtonian approximations as applied to sources of gravitational waves.

The WKB approximation in the deformed space with the minimal length and minimal momentum

Journal of Physics A: Mathematical and Theoretical, 2008

A Bohr-Sommerfeld quantization rule is generalized for the case of the deformed commutation relation leading to minimal uncertainties in both coordinate and momentum operators. The correctness of the rule is verified by comparing obtained results with exact expressions for corresponding spectra.

Improved WKB analysis of cosmological perturbations

Physical Review D - Particles, Fields, Gravitation and Cosmology, 2005

Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from sub-horizon to super-horizon scales, and include the region near the turning point. We employ both a perturbative Green's function technique and an adiabatic (or "semiclassical") expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green's function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful.

Divergence-free WKB theory

Annals of Physics, 2004

We present a divergence-free WKB theory, which is a new semiclassical theory modified by nonperturbative quantum corrections. Conventionally, the WKB theory is constructed upon a trajectory that obeys the bare classical dynamics expressed by a quadratic equation in momentum space. Contrary to this, the divergence-free WKB theory is based on a higher-order algebraic equation in momentum space, which represents a dressed classical dynamics. More precisely, this higher-order algebraic equation is obtained by including quantum corrections to the quadratic equation, which is the bare classical limit. An additional solution of the higher-order algebraic equation enables us to construct a uniformly converging perturbative expansion of the wavefunction. Namely, our theory removes the notorious divergence of wavefunction at a turning point from the WKB theory. Moreover, our theory is able to produce wavefunctions and eigenenergies more accurate than those given by the traditional WKB method. In addition, the divergence-free WKB theory that is based on the cubic equation allows us to construct a uniformly valid wavefunction for the nonlinear Schr€ odinger equation (NLSE). A recent short letter [T. Hyouguchi, S. Adachi, M. Ueda, Phys. Rev. Lett. 88 (2002) 170404] is the opening of the divergencefree WKB theory. This paper presents full formalism of this theory and its several applications concerning wavefunction and eigenenergy to show that our theory is a natural extension of the traditional WKB theory that incorporates nonperturbative quantum corrections.

The CWKB Method of Particle Production Near the Chronology Horizon

2002

In this paper we investigate the phenomenon of particle production of massles scalar field, in a model of spacetime where the chronology horizon could be formrd, using the method of complex time WKB approximation (CWKB). For the purpose, we take two examp les in a model of spacetime, one already discussed by Sushkov, to show that the mode of particle production near chronology horizon possesses the similar characteristic features as are found while discussing particle production in time dependent curved ba ckground. We get identical results as that obtained by Sushkov in this direction. We find, in both the examples studied, that the total number of particles remain finite at the moment of the formation of the chronology horizon.

Temporal contribution to gravitational WKB-like calculations

Physics Letters B, 2008

Recently, it has been shown that the radiation arising from quantum fields placed in a gravitational background (e.g. Hawking radiation) can be derived using a quasi-classical calculation. Here we show that this method has a previously overlooked temporal contribution to the quasi-classical amplitude. The source of this temporal contribution lies in different character of time in general relativity versus quantum mechanics. Only when one takes into account this temporal contribution does one obtain the canonical temperature for the radiation. Although in this letter the specific example of radiation in de Sitter space-time is used, the temporal contribution is a general contribution to the radiation given off by any gravitational background where the time coordinate changes its signature upon crossing a horizon. Thus, the quasi-classical method for gravitational backgrounds contains subtleties not found in the usual quantum mechanical tunneling problem.

The Synergy between Numerical and Perturbative Approaches to Black Holes

Black Holes, Gravitational Radiation and the Universe, 1999

I describe approaches to the study of black hole spacetimes via numerical relativity. After a brief review of the basic formalisms and techniques used in numerical black hole simulations, I discuss a series of calculations from axisymmetry to full 3D that can be seen as stepping stones to simulations of the full 3D coalescence of two black holes. In particular, I emphasize the interplay between perturbation theory and numerical simulation that build both confidence in present results and tools to aid and to interpret results of future simulations of black hole coalescence.

Eulerian Perturbation Theory in Non-Flat Universes: Second-Order Approximation

1994

The problem of solving perturbatively the equations describing the evolution of self-gravitating collisionless matter in an expanding universe considerably simplifies when directly formulated in terms of the gravitational and velocity potentials: the problem can be solved {\it exactly}, rather than approximately, even for cosmological models with arbitrary density parameter Omega\OmegaOmega. The Eulerian approach we present here allows to calculate the

New WKB method in supersymmetry quantum mechanics

In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Schwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.

The CWKB method of particle production in a periodic potential

In this work we study the particle production in time dependent periodic potential using the method of complex time WKB (CWKB) approximation. In the inflationary cosmology at the end of inflationary stage, the potential becomes time dependent as well as periodic. Reheating occurs due to particle production by the oscillating inflaton field. Using CWKB we obtain almost identical results on catastrophic particle production as obtained by others.

WKB Approximation in Noncommutative Gravity

Symmetry, Integrability and Geometry: Methods and Applications, 2007

We consider the quasi-commutative approximation to a noncommutative geometry defined as a generalization of the moving frame formalism. The relation which exists between noncommutativity and geometry is used to study the properties of the highfrequency waves on the flat background.

Analytic Black Hole Perturbation Approach to Gravitational Radiation

Living Reviews in Relativity, 2003

We review the analytic methods used to perform the post-Newtonian expansion of gravitational waves induced by a particle orbiting a massive, compact body, based on black hole perturbation theory. There exist two different methods of performing the post-Newtonian expansion. Both are based on the Teukolsky equation. In one method, the Teukolsky equation is transformed into a Regge-Wheeler type equation that reduces to the standard Klein-Gordon equation in the flat-space limit, while in the other method (which was introduced by Mano, Suzuki, and Takasugi relatively recently), the Teukolsky equation is used directly in its original form. The former's advantage is that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one 16 2 − 243 8 − 785 6)︀ 5 .

A numerical method for perturbative QCD calculations

2004

Standard methods for performing analytic perturbative calculations for the process of e+ e-+ qq up to 0(a3) are explained and the results given. An emphasis is given to the organisation of calculations using the Cutkosky cutting rules and the renormalisation of the massive quark propagator. Methods for numerical integration are presented including those used in VEGAS. The numerical methods used in the Beowulf program for calculating infra-red safe observables for jet events from electron-positron collisions are also explained. Cancellations of singularities required for numerical calculations are demonstrated using an example in 03 theory both numerically and graphically. Renormalisation by subtraction of appropriate integrals is also covered. Adaptations of the Beowulf procedure required for the inclusion of massive fermions are developed and explained. An alternative method for including the quark self energy and its related cuts using scalar decomposition, numerically equivalent integrals and its spinor structure is introduced. The methods are used to calculate the 0(a3) corrections to the process e+ e-+ qi7 using VEGAS. Drawbacks of the smearing function required in the numerical integration due to the corrections dependence on the mass and centre of mass energy are discussed. Results of the 0(a3) cross section using the numerical method verify the procedure. The method will then be used to see the effects of mass on the thrust distribution and when using the Durham and JADE jet algorithms. in Declaration This thesis has been composed by myself. The method described for the inclusion of massive fermions in numerical perturbative calculations and the results obtained from the numerical calculations are all my own work. The computer program used to generate the results uses the VEGAS numerical integration routine created by Peter Lepage and is based upon the Beowulf program created by Davison Soper. This work has not been submitted for any other degree or professional qualification other than for the doctorate of philosophy.

Numerical evolution in time of curvature perturbations in Kerr black holes

American Mathematical Society eBooks, 1999

In this paper I will review the basic features of the theory of curvature perturbations in Kerr spacetime, which is customarily written in terms of gauge invariant components of the Weyl tensor which satisfy a perturbation equation known as the Teukolsky equation. I will describe how to evolve generic perturbations about the Kerr metric and the separable form of the wave solutions that one obtains, and the relation of the Teukolsky function to the energy of gravitational waves emitted by the black hole. A discussion of a numerical scheme to evolve perturbations as a function of time and some preliminary results of our research project implementing it for matter sources falling into the black hole is included.