Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

Yagdjian, Karen; Galstian, Anahit

doi:10.1007/s00220-008-0649-4

Cited by 93 publications

(103 citation statements)

References 30 publications

(40 reference statements)

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“…If Ω = Π is a non-Euclidean space of constant negative curvature and the equation of the problems (1.5) and (1.6) is a non-Euclidean wave equation, then the explicit representation formulas are known (see, e.g., [17,20]) and the Huygens' principle is a consequence of those formulas. Thus, for a non-Euclidean wave equation, due to Theorem 1.1, the functions v f (x, t; b) and v ϕ (x, t) have explicit representations, and the arguments of [29,31] allow us to derive for the solution u(x, t) of the problem (1.7) in the de Sitter metric with hyperbolic spatial geometry the explicit representation, the L p − L q estimates, and to examine the Huygens' principle.…”

Section: Integral Transformmentioning

confidence: 96%

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Global in time existence of self-interacting scalar field in de Sitter spacetimes

Galstian

Yagdjian

2017

Nonlinear Analysis: Real World Applications

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We prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. IntroductionIn this paper we prove the existence of a global in time solution of the semilinear Klein-Gordon equation in the de Sitter space-time. The coefficients of the equation depend on spatial variables as well, that make results applicable to the space-time with the time slices being Riemannian manifolds. In the spatially flat de Sitter model, these slices are R 3 , while in the spatially closed and spatially open cases these slices can be the three-sphere S 3 and the three-hyperboloid H 3 , respectively (see, e.g., [12, p.113]).The metric g in the de Sitter space-time is defined as follows, g 00 = g 00 = −1, g 0j = g 0j = 0, g ij (x, t) = e 2t σ ij (x), i, j = 1, 2, . . . , n, where n j=1 σ ij (x)σ jk (x) = δ ik , and δ ij is Kronecker's delta. The metric σ ij (x) describes the time slices. In the quantum field theory the matter fields are described by a function ψ that must satisfy equations of motion. In the case of a massive scalar field, the equation of motion is the semilinear Klein-Gordon equation generated by the metric g:Here m is a physical mass of the particle. In physical terms this equation describes a local selfinteraction for a scalar particle. A typical example of a potential function would be V (ψ) = ψ 4 . The semilinear equations are also commonly used models for general nonlinear problems.The covariant Klein-Gordon equation in the de Sitter space-time in the coordinates is

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Section: Integral Transformmentioning

confidence: 96%

“…(Comp. with Prop.10.2 [29].) We follow the arguments have been used in the proof of Proposition 8.3 [29].…”

Section: Estimates For Equation Without Sourcementioning

confidence: 99%

Global in time existence of self-interacting scalar field in de Sitter spacetimes

Galstian

Yagdjian

2017

Nonlinear Analysis: Real World Applications

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“…Thus the fundamental solutions are plane gravitational waves with the two linear polarization states "+" and "×".The solutions of the second equation in (18) are then determined by the expression…”

Section: Approximate Solution Of the Linearised Equationsmentioning

confidence: 99%

“…By choosing a particular non Hilbert gauge this leads then to a Klein-Gordon equation and thus to a nontrivial dispersion relation. Furthermore, we refer to some works on the scalar wave equation in dS and Schwarzschild-dS spacetimes [16,17,18,19]. These treatments are, however, not directly connected to the present work, since the equations resulting from the linearisation of Einstein's equations are coupled partial differential equations for six independent variables.…”

Section: Introductionmentioning

confidence: 99%

On gravitational waves in spacetimes with a nonvanishing cosmological constant

2009

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We study the effect of a cosmological constant Λ on the propagation and detection of gravitational waves. To this purpose we investigate the linearised Einstein's equations with terms up to linear order in Λ in a de Sitter and an anti-de Sitter background spacetime. In this framework the cosmological term does not induce changes in the polarization states of the waves, whereas the amplitude gets modified with terms depending on Λ. Moreover, if a source emits a periodic waveform, its periodicity as measured by a distant observer gets modified. These effects are, however, extremely tiny and thus well below the detectability by some twenty orders of magnitude within present gravitational wave detectors such as LIGO or future planned ones such as LISA.

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“…Indeed, the work of Yagdjian and Galstian [27] on de Sitter space contains no such losses. Further evidence for this belief is that we obtain the estimates without a loss when h is independent of t for large t. Removing the loss in the more general setting requires a nontrivial extension of the Littlewood-Paley theory and is left to a future paper.…”

Section: Introductionmentioning

confidence: 99%

Strichartz Estimates on Asymptotically de Sitter Spaces

Baskin

2012

Ann. Henri Poincaré

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Abstract. In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein-Gordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the "large mass" regime. We also provide an application of these estimates to establish small-data global existence for a class of semilinear equations on these spaces.

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Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

Abstract: In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove L p − L q estimates for the solutions of the equation with and without a source term.

Cited by 93 publications

References 30 publications

Global in time existence of self-interacting scalar field in de Sitter spacetimes

Global in time existence of self-interacting scalar field in de Sitter spacetimes

On gravitational waves in spacetimes with a nonvanishing cosmological constant

Strichartz Estimates on Asymptotically de Sitter Spaces

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