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

Abstract

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

Global existence of the self-interacting scalar field in the de Sitter universe

“…The proof of the global existence in [33]- [35] is based on the special integral representations (see Section 1) and L p − L q estimates. Later on, in [20] this result for the same range of the parameters n, m and the same nonlinearity was extended on the equation (0.1), that is, from the spatially flat de Sitter spacetime to the de Sitter spacetime with the time slices being, in particular, the Riemannian manifolds. The case of m ∈ ( √ n 2 − 1/2, n/2) was left open in [20].…”

On the numerical experiments of the Cauchy problem for semi-linear Klein–Gordon equations in the de Sitter spacetime

“…Yagdjian has shown in [28] small global solutions for (1.11) with arbitrary n when the nonlinear term f is of power type p > 1, and the norm of initial data u 0 H s (R n ) + u 1 H s (R n ) is sufficiently small for some s > n/2 ≥ 1 (see also [29] for the system of the equations). Galstian and Yagdjian has extended this result to the case of the Riemann metric space for each time slices in [11]. In [15], the energy solutions (s = 1) in Theorem 1.1 have been shown.…”

“…where the right hand sides of (3.10) are defined as 11) and the second-order central difference operator δ 2 ij is defined as…”

Wave Asymptotics at a Cosmological Time-Singularity: Classical and Quantum Scalar Fields

“…Important results are known for smooth universes, in particular for the De Sitter spacetime. Besides the studies on the stability of the Einsteinnonlinear field systems [47], [49], [52], [58], [62], we can cite for the analysis of non-linear wave equations, the works by Ebert and Reissig [21], Galstian and Yagdjian [30], [31], Nakamura [40]. At our knowledge a lot of hard issues about interacting fields on singular Lorentzian manifolds remain to be investigated.…”