Cosmological solutions

2.2 Cosmological solutions

In cosmology the whole universe is modelled, and the “particles” in the kinetic description are galaxies or even clusters of galaxies. The main goal again is to determine the global properties of the solutions to the Einstein–Vlasov system. In order to do so, a global time coordinate $t$ must be found (global existence) and the asymptotic behaviour of the solutions when $t$ tends to its limiting values has to be analyzed. This might correspond to approaching a singularity (e.g. the big bang singularity) or to a phase of unending expansion. Since the general case is beyond the range of current mathematical techniques, all known results are for cases with symmetry (see however [9 ] where to some extent global properties are established in the case without symmetry).

There are several existing results on global time coordinates for solutions of the Einstein–Vlasov system. In the spatially homogeneous case it is natural to choose a Gaussian time coordinate based on a homogeneous hypersurface. The maximal range of a Gaussian time coordinate in a solution of the Einstein–Vlasov system evolving from homogenous data on a compact manifold was determined in [98 ]. The range is finite for models of Bianchi IX and Kantowski–Sachs types. It is finite in one time direction and infinite in the other for the other Bianchi types. The asymptotic behaviour of solutions in the spatially homogeneous case has been analyzed in [103 ] and [104 ]. In [103 ], the case of massless particles is considered, whereas the massive case is studied in [104 ]. Both the nature of the initial singularity and the phase of unlimited expansion are analyzed. The main concern is the behaviour of Bianchi models I, II, and III. The authors compare their solutions with the solutions to the corresponding perfect fluid models. A general conclusion is that the choice of matter model is very important since for all symmetry classes studied there are differences between the collisionless model and a perfect fluid model, both regarding the initial singularity and the expanding phase. The most striking example is for the Bianchi II models, where they find persistent oscillatory behaviour near the singularity, which is quite different from the known behaviour of Bianchi type II perfect fluid models. In [104 ] it is also shown that solutions for massive particles are asymptotic to solutions with massless particles near the initial singularity. For Bianchi I and II it is also proved that solutions with massive particles are asymptotic to dust solutions at late times. It is conjectured that the same holds true also for Bianchi III. This problem is then settled by Rendall in [102 ].

All other results presently available on the subject concern spacetimes that admit a group of isometries acting on two-dimensional spacelike orbits, at least after passing to a covering manifold. The group may be two-dimensional (local $U (1) × U (1 )$ or $T 2$ symmetry) or three-dimensional (spherical, plane, or hyperbolic symmetry). In all these cases, the quotient of spacetime by the symmetry group has the structure of a two-dimensional Lorentzian manifold $Q$ . The orbits of the group action (or appropriate quotients in the case of a local symmetry) are called surfaces of symmetry. Thus, there is a one-to-one correspondence between surfaces of symmetry and points of $Q$ . There is a major difference between the cases where the symmetry group is two- or three-dimensional. In the three-dimensional case no gravitational waves are admitted, in contrast to the two-dimensional case. In the former case, the field equations reduce to ODEs while in the latter their evolution part consists of nonlinear wave equations. Three types of time coordinates that have been studied in the inhomogeneous case are CMC, areal, and conformal coordinates. A CMC time coordinate $t$ is one where each hypersurface of constant time has constant mean curvature (CMC) and on each hypersurface of this kind the value of $t$ is the mean curvature of that slice. In the case of areal coordinates, the time coordinate is a function of the area of the surfaces of symmetry (e.g. proportional to the area or proportional to the square root of the area). In the case of conformal coordinates, the metric on the quotient manifold $Q$ is conformally flat.

Let us first consider spacetimes $(M, g)$ admitting a three-dimensional group of isometries. The topology of $M$ is assumed to be $ℝ × S1 × F$ , with $F$ a compact two-dimensional manifold. The universal covering $ˆ F$ of $F$ induces a spacetime $ˆ (M ,ˆg)$ by $ˆ 1 ˆ M = ℝ × S × F$ and $∗ ˆg = p g$ , where $ˆ p :M → M$ is the canonical projection. A three-dimensional group $G$ of isometries is assumed to act on $(Mˆ ,ˆg)$ . If $F = S2$ and $G = SO (3)$ , then $(M, g)$ is called spherically symmetric; if $F = T 2$ and $G = E2$ (Euclidean group), then $(M, g)$ is called plane symmetric; and if $F$ has genus greater than one and the connected component of the symmetry group $G$ of the hyperbolic plane $2 H$ acts isometrically on $ˆ 2 F = H$ , then $(M, g)$ is said to have hyperbolic symmetry.

In the case of spherical symmetry the existence of one compact CMC hypersurface implies that the whole spacetime can be covered by a CMC time coordinate that takes all real values [97 , 18 ]. The existence of one compact CMC hypersurface in this case was proved later by Henkel [52 ] using the concept of prescribed mean curvature (PMC) foliation. Accordingly this gives a complete picture in the spherically symmetric case regarding CMC foliations. In the case of areal coordinates, Rein [82 ] has shown, under a size restriction on the initial data, that the past of an initial hypersurface can be covered. In the future direction it is shown that areal coordinates break down in finite time.

In the case of plane and hyperbolic symmetry, Rendall and Rein showed in [97 ] and [82 ], respectively, that the existence results (for CMC time and areal time) in the past direction for spherical symmetry also hold for these symmetry classes. The global CMC foliation results to the past imply that the past singularity is a crushing singularity, since the mean curvature blows up at the singularity. In addition, Rein also proved in his special case with small initial data that the Kretschmann curvature scalar blows up when the singularity is approached. Hence, the singularity is both a crushing and a curvature singularity in this case. In both of these works the question of global existence to the future was left open. This gap was closed in [7 ], and global existence to the future was established in both CMC and areal coordinates. The global existence result for CMC time is partly a consequence of the global existence theorem in areal coordinates, together with a theorem by Henkel [52 ] that shows that there exists at least one hypersurface with (negative) constant mean curvature. Also, the past direction was analyzed in areal coordinates and global existence was shown without any smallness condition on the data. It is, however, not concluded if the past singularity in this more general case without the smallness condition on the data is a curvature singularity as well. The question whether the areal time coordinate, which is positive by definition, takes all values in the range $(0,∞ )$ or only in $(R0, ∞ )$ for some positive $R0$ is also left open. In the special case in [82 ], it is indeed shown that $R0 = 0$ , but there is an example for vacuum spacetimes in the more general case of $U(1) × U (1)$ symmetry where $R0 > 0$ . This question was resolved by Weaver [118 ]. She proves that if spacetime contains Vlasov matter (i.e. $f ⁄= 0$ ) then $R = 0$ . Her result applies to a more general case which we now turn to.

For spacetimes admitting a two-dimensional isometry group, the first study was done by Rendall [100 ] in the case of local $U (1) × U(1)$ symmetry (or local $T 2$ symmetry). For a discussion of the topologies of these spacetimes we refer to the original paper. In the model case the spacetime is topologically of the form $3 ℝ × T$ , and to simplify our discussion later on we write down the metric in areal coordinates for this type of spacetime:

Here the metric coefficients $η$ , $U$ , $α$ , $A$ , $H$ , $L$ , and $M$ depend on $t$ and $θ$ and $1 θ,x, y ∈ S$ . In [100

] CMC coordinates are considered rather than areal coordinates. The CMC and the areal coordinate foliations are both geometrically based time foliations. The advantage with a CMC approach is that the definition of a CMC hypersurface does not depend on any symmetry assumptions and it is possible that CMC foliations will exist for rather general spacetimes. The areal coordinate foliation, on the other hand, is adapted to the symmetry of spacetime but it has analytical advantages that we will see below.

Under the hypothesis that there exists at least one CMC hypersurface, Rendall proves, without any smallness condition on the data, that the past of the given CMC hypersurface can be globally foliated by CMC hypersurfaces and that the mean curvature of these hypersurfaces blows up at the past singularity. Again, the future direction was left open. The result in [100 ] holds for Vlasov matter and for matter described by a wave map (which is not a phenomenological matter model). That the choice of matter model is important was shown by Rendall [99 ] who gives a non-global existence result for dust, which leads to examples of spacetimes [59 ] that are not covered by a CMC foliation.

There are several possible subcases to the $U (1) × U(1)$ symmetry class. The plane case where the symmetry group is three-dimensional is one subcase and the form of the metric in areal coordinates is obtained by letting $A = G = H = L = M = 0$ and $U = log t∕2$ in Equation (44 ). Another subcase, which still admits only two Killing fields (and which includes plane symmetry as a special case), is Gowdy symmetry. It is obtained by letting $G = H = L = M = 0$ in Equation (44 ). In [4 ], the author considers Gowdy symmetric spacetimes with Vlasov matter. It is proved that the entire maximal globally hyperbolic spacetime can be foliated by constant areal time slices for arbitrary (in size) initial data. The areal coordinates are used in a direct way for showing global existence to the future whereas the analysis for the past direction is carried out in conformal coordinates. These coordinates are not fixed to the geometry of spacetime and it is not clear that the entire past has been covered. A chain of geometrical arguments then shows that areal coordinates indeed cover the entire spacetime. This method was applied to the problem on hyperbolic and plane symmetry in [7 ]. The method in [4 ] was in turn inspired by the work [16 ] for vacuum spacetimes where the idea of using conformal coordinates in the past direction was introduced. As pointed out in [7 ], the result by Henkel [53 ] guarantees the existence of one CMC hypersurface in the Gowdy case and, together with the global areal foliation in [4 ], it follows that Gowdy spacetimes with Vlasov matter can be globally covered by CMC hypersurfaces as well (also to the future). The general case of $U (1) × U(1)$ symmetry was considered in [8 ], where it is shown that there exist global CMC and areal time foliations which complete the picture. In this result as well as in the preceeding subcases mentioned above the question whether or not the areal time coordinate takes values in $(0,∞ )$ or in $(R, ∞ )$ , $R > 0$ , was left open. This issue was solved by Weaver in [118 ] where she concludes that $R = 0$ if the distribution function is not identically zero initially.

A number of important questions remain open. To analyze the nature of the initial singularity, which at present is known only for small initial data in the case considered in [82 ], would be very interesting. The question of the asymptotics in the future direction is also an important issue where very little is known. The only situation where a result has been obtained is in the case with hyperbolic symmetry. Under a certain size restriction on the initial data, Rein [87 ] shows future geodesic completeness. However, in models with a positive cosmological constant more can be said.