As in the previous section we start with the
models with highest degree of symmetry, i.e. the locally
spatially homogeneous models. In the case of a positive
cosmological constant Lee
[63]
has shown global existence as well as future causal geodesic
completeness for initial data which have Bianchi symmetry. She
also obtains the time decay of the components of the energy
momentum tensor as
. The past direction for some spatially homogeneous models is
considered in
[111
]
. Existence back to the initial singularity is proved and the
case with a negative cosmological constant is discussed. In
[64]
Lee considers the case with a nonlinear scalar field coupled to
Vlasov matter. The form of the energy momentum then reads
where
then future geodesic completeness is proved.
In the previous Section 2.3 we discussed the situation when spacetime admits a three-dimensional group of isometries and we distinguished three cases: plane, spherical, and hyperbolic symmetry. In area time coordinates the metric takes the form
where
correspond to the plane, spherical, and hyperbolic case,
respectively, and where
,
, and
. In
[113
]
the Einstein-Vlasov system with a positive cosmological constant
is investigated in the future (expanding) direction in the case
of plane and hyperbolic symmetry. The authors prove global
existence to the future in these coordinates and they also show
future geodesic completeness. The positivity of the cosmological
constant is crucial for the latter result. Recall that in the
case of
, future geodesic completenss has only been established for
hyperbolic symmetry under a smallness condition of the initial
data
[87]
. Finally a form of the cosmic no-hair conjecture is obtained in
[113]
for this class of spacetimes. Indeed, here it is shown that the
de Sitter solution acts as a model for the dynamics of the
solutions by proving that the generalized Kasner exponents tend
to
as
, which in the plane case is the de Sitter solution. The
remaining case of spherical symmetry is analyzed in
[112]
. Recall that when
, Rein
[82]
showed that solutions can only exist for finite time in the
future direction in area time coordinates. By adding a positive
cosmological constant, global existence to the future is shown to
hold true if initial data is given on
, where
. The asymptotic behaviour of the matter terms is also
investigated and slightly stronger decay estimates are obtained
in this case compared to the case of plane and hyperbolic
symmetry.
The results discussed so far in this section
have concerned the future time direction and a positive
cosmological constant. The past direction with a negative
cosmological constant is analyzed in
[111], where it is shown that for plane and spherical symmetry the
areal time coordinate takes all positive values, which is in
analogy with Weaver’s
[118]
result for
. If initial data are restricted by a smallness condition the
theorem is proven also in the hyperbolic case, and for such data
the result of the theorem holds true in all of the three symmetry
classes when the cosmological constant is positive. The
early-time asymptotics in the case of small initial data is also
analyzed and is shown to be Kasner-like.
In
[114]
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 (45
,
46
) 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.
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