Quantum dynamics of the slow rollover transition in the linear delta expansion

Jones, H. F.; Parkin, Philip; Winder, D. R.

doi:10.1103/physrevd.63.125013

Cited by 12 publications

(21 citation statements)

References 17 publications

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“…We have chosen λ = 0.01 and w 2 i = w 2 f = 25λ/6 (recall that w 2 f appears with a different sign in the Lagrangian). These parameters coincide with those chosen in [1,2,3,4] in order that we may easily compare our results. Also shown are the exact result, first-order perturbation theory, and the Hartree approximation of Ref.…”

Section: Linear Delta Expansionmentioning

confidence: 95%

“…In previous articles considering the quantum mechanical slow roll [1,2,3,4], the particle begins at a time t = 0 when it is described by a Gaussian wave function, centred at the top of a potential hill. To reproduce this situation here, we consider the particle prepared at t < 0 in the ground state of an harmonic oscillator potential (corresponding to a Gaussian wave function).…”

Section: Inverted Harmonic Oscillatormentioning

confidence: 99%

“…The work presented here is based on an alternative variational approach, the linear delta expansion (LDE), recently applied [4] to the quantum mechanical slow roll. The method was found to reproduce the exact time dependence for longer than any of the alternative methods.…”

Section: Introductionmentioning

confidence: 99%

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Quantum field dynamics of the slow rollover in the linear delta expansion

Bedingham

Jones

2003

Phys. Rev. D

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We show how the linear delta expansion, as applied to the slow-roll transition in quantum mechanics, can be recast in the closed time-path formalism. This results in simpler, explicit expressions than were obtained in the Schrödinger formulation and allows for a straightforward generalization to higher dimensions. Motivated by the success of the method in the quantummechanical problem, where it has been shown to give more accurate results for longer than existing alternatives, we apply the linear delta expansion to four-dimensional field theory.At small times all methods agree. At later times, the first-order linear delta expansion is consistently higher that Hartree-Fock, but does not show any sign of a turnover. A turnover emerges in second-order of the method, but the value of Φ 2 (t) at the turnover is larger that that given by the Hartree-Fock approximation. Based on this calculation, and our experience in the corresponding quantum-mechanical problem, we believe that the Hartree-Fock approximation does indeed underestimate the value of Φ 2 (t) at the turnover. In subsequent applications of the method we hope to implement the calculation in the context of an expanding universe, following the line of earlier calculations by Boyanovsky et al., who used the Hartree-Fock and large-N methods. It seems clear, however, that the method will become unreliable as the system enters the reheating stage. IntroductionA period of inflation in the early universe could have the desirable consequence that a general initial condition will evolve towards the homogeneity, isotropy and flatness which we observe. Basic models require the slow evolution of a scalar field from an initial unstable vacuum state to a final stable state. Without knowing how to perform this inherently non-perturbative calculation exactly, approximation attempts must first prove themselves in the simpler situation of the quantummechanical slow roll. Though this simpler problem cannot be solved analytically, the degrees of freedom are sufficiently few that an exact numerical solution can be found. This allows us to test non-perturbative methods before proceeding to a calculation for the four-dimensional scalar field.The quantum-mechanical slow roll was first treated by Guth and Pi [1], who considered the evolution of a Gaussian wave-packet initially centred at the top of a potential hill V = − 1 2 mωq 2 . Following this, the Dirac time-dependent variational method was used for a potential V = λ(q 2 − a 2 ) 2 /24, first by Cooper et al. [2], who used a Gaussian wave function ansatz, and *

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Section: Linear Delta Expansionmentioning

confidence: 95%

Section: Inverted Harmonic Oscillatormentioning

confidence: 99%

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Quantum field dynamics of the slow rollover in the linear delta expansion

Bedingham

Jones

2003

Phys. Rev. D

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“…In Fig.(1) our formula (8) has been compared with the leading order of eq. (2) by considering the ratio…”

Section: The Method: Asymptotic Seriesmentioning

confidence: 99%

“…Techniques such as the Delta Expansion and Variational Perturbation Theory have been applied by many authors to physical problems of different nature, ranging from classical and quantum mechanics to field theory (see for example [6,7,8,9,10] and references therein). In most cases these methods have allowed to obtain very useful and precise approximate solutions, which are valid even in the non-perturbative regime, where perturbation theory breaks down.…”

Section: The Method: Asymptotic Seriesmentioning

confidence: 99%

Asymptotic and exact series representations for the incomplete Gamma function

Amore

2005

Europhys. Lett.

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Abstract. Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a, x) for any value of a or x. Applications of these formulas are discussed. ‡ paolo@ucol.mx Asymptotic and exact series representations for the incomplete Gamma function 2

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Improved Lindstedt–Poincaré method for the solution of nonlinear problems

Amore

Aranda

2005

Journal of Sound and Vibration

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We apply the Linear Delta Expansion (LDE) to the Lindstedt-Poincaré ("distorted time") method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator (Duffing equation) and of the non-linear pendulum. The approximate solutions found with this method are better behaved and converge more rapidly to the exact ones than in the simple Lindstedt-Poincaré method. * Electronic address: paolo@ucol.mx † Electronic address: fefo@cgic.ucol.mx

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Quantum dynamics of the slow rollover transition in the linear delta expansion

Cited by 12 publications

References 17 publications

Quantum field dynamics of the slow rollover in the linear delta expansion

Quantum field dynamics of the slow rollover in the linear delta expansion

Asymptotic and exact series representations for the incomplete Gamma function

Improved Lindstedt–Poincaré method for the solution of nonlinear problems

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