Cosmological models with a cosmological constant or a scalar field

2.3 Cosmological models with a cosmological constant or a scalar field

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

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 $t → ∞$ . 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

Here $φ$ is the scalar field and $V$ is a potential, and the Bianchi identities lead to the following equation for the scalar field:

Under the assumption that $V$ is non-negative and $C2$ , global existence to the future is obtained and if the potential is restricted to the form

V (φ) = V0e− cΦ,

where $√ -- 0 < c < 4 π$ 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

ds2 = − eμ(t,r)dt2 + eλ(t,r)dr2 + t2(dθ2 + sin2 θdϕ2 ), κ

where $k = 0,− 1,+1$ correspond to the plane, spherical, and hyperbolic case, respectively, and where $sin0θ = 1$ , $sin1θ = sinθ$ , and $sin−1 θ = sinh θ$ . 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 $Λ = 0$ , 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 $1∕3$ as $t → ∞$ , which in the plane case is the de Sitter solution. The remaining case of spherical symmetry is analyzed in [112 ]. Recall that when $Λ = 0$ , 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 $t = t 0$ , where $2 t0 > 1∕Λ$ . 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 $Λ = 0$ . 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 $V$ 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 $f = 0$ , the continuation criterion is shown to be satisfied in the future direction and global existence follows in that case.