### 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

Here is the scalar field and is a potential, and the Bianchi identities lead to the following equation
for the scalar field:
Under the assumption that is a non-negative function, global existence to the future is obtained,
and if the potential is restricted to the form
where , 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 in Equations (53) and (54) 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 , 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 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.