4.3 Cosmological models with a scalar field

The present cosmological observations indicate that the expansion of the universe is accelerating, and this has influenced theoretical studies in the field during the last decade. One way to produce models with accelerated expansion is to choose a positive cosmological constant. Another way is to include a non-linear scalar field among the matter fields, and in this section we review the results for the Einstein–Vlasov system, where a linear or non-linear scalar field have been included into the model.

Lee considers in [107] the case where a non-linear scalar field is coupled to Vlasov matter. The form of the energy momentum tensor then reads

( 1 ) Tαβ = T Vαlβasov+ ∇ α ϕ∇ βϕ − -∇ γϕ∇ γϕ + V (ϕ) gα β. (53 ) 2
Here ϕ is the scalar field and V is a potential, and the Bianchi identities lead to the following equation for the scalar field:
γ ′ ∇ ∇ γϕ = V (ϕ ). (54 )
Under the assumption that V is a non-negative C2 function, global existence to the future is obtained, and if the potential is restricted to the form
−cΦ V (ϕ) = V0e ,

where √ -- 0 < c < 4 π, then future geodesic completeness is proven.

In [187] the Einstein–Vlasov system with a linear scalar field is analyzed in the case of plane, spherical, and hyperbolic symmetry. Here, the potential V in Equations (53View Equation) and (54View Equation) is zero. A local existence theorem and a continuation criterion, involving bounds on derivatives of the scalar field in addition to a bound on the support of one of the moment variables, is proven. For the Einstein scalar field system, i.e., when f = 0, the continuation criterion is shown to be satisfied in the future direction, and global existence follows in that case. The work [186] extends the result in the plane and hyperbolic case to a global result in the future direction. In the plane case when f = 0, the solutions are shown to be future geodesically complete. The past time direction is considered in [188] and global existence is proven. It is also shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached.

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