7.2 Inhomogeneous solutions with 7 Asymptotics of Expanding Cosmological 7 Asymptotics of Expanding Cosmological

7.1 Lessons from homogeneous solutions

Which features should we focus on when thinking about the dynamics of forever expanding cosmological models? Consider for a moment the Kasner solution


where tex2html_wrap_inline1995 and tex2html_wrap_inline1997 . These are the first and second Kasner relations. They imply that not all tex2html_wrap_inline1999 can be strictly positive. Taking the coordinates x, y and z to be periodic, gives a vacuum cosmological model whose spatial topology is that of a three-torus. The volume of the hypersurfaces tex2html_wrap_inline2007 grows monotonically. However, the geometry does not expand in all directions, since not all tex2html_wrap_inline1999 are positive. This can be reformulated in a way which is more helpful when generalizing to inhomogeneous models. In fact the quantities tex2html_wrap_inline2011 are the eigenvalues of the second fundamental form. The statement then is that the second fundamental form is not negative definite. Looking at other homogeneous models indicates that this behaviour of the Kasner solution is not typical of what happens more generally. On the contrary, it seems reasonable to conjecture that in general the second fundamental form eventually becomes negative definite, at least in the presence of matter.

Some examples will now be presented. The following discussion makes use of the Bianchi classification of homogenous cosmological models (see e.g.  [237Jump To The Next Citation Point In The Article]). If we take the Kasner solution and add a perfect fluid with equation of state tex2html_wrap_inline1951, tex2html_wrap_inline1953, maintaining the symmetry (Bianchi type I), then the eigenvalues tex2html_wrap_inline2017 of the second fundamental satisfy tex2html_wrap_inline2019 in the limit of infinite expansion. The solution isotropizes. More generally this does not happen. If we look at models of Bianchi type II with non-tilted perfect fluid, i.e. where the fluid velocity is orthogonal to the homogeneous hypersurfaces, then the quantities tex2html_wrap_inline2021 converge to limits that are positive but differ from 1/3 (see [237Jump To The Next Citation Point In The Article], p. 138.) There is partial but not complete isotropization. The quantities tex2html_wrap_inline1999 just introduced are called generalized Kasner exponents, since in the case of the Kasner solution they reduce to the tex2html_wrap_inline1999 in the metric form (3Popup Equation). This kind of partial isotropization, ensuring the definiteness of the second fundamental form at late times, seems to be typical.

Intuitively, a sufficiently general vacuum spacetime should resemble gravitational waves propagating on some metric describing the large-scale geometry. This could even apply to spatially homogeneous solutions, provided they are sufficiently general. Hence, in that case also there should be partial isotropization. This expectation is confirmed in the case of vacuum spacetimes of Bianchi type VIII [224]. In that case the generalized Kasner exponents converge to non-negative limits different from 1/3. For a vacuum model this can only happen if the quantity tex2html_wrap_inline2031, where R is the spatial scalar curvature, does not tend to zero in the limit of large time.

The Bianchi models of type VIII are the most general indefinitely expanding models of class A. Note, however, that models of class VI tex2html_wrap_inline2035 for all h together are just as general. The latter models with perfect fluid and equation of state tex2html_wrap_inline1951 sometimes tend to the Collins model for an open set of values of h for each fixed tex2html_wrap_inline1973 (cf. [237], p. 160). These models do not in general exhibit partial isotropization. It is interesting to ask whether this is connected to the issue of spatial boundary conditions. General models of class B cannot be spatially compactified in such a way as to be locally spatially homogeneous while models of Bianchi type VIII can. See also the discussion in [20].

Another issue is what assumptions on matter are required in order that it have the effect of (partial) isotropization. Consider the case of Bianchi I. The case of a perfect fluid has already been mentioned. Collisionless matter described by kinetic theory also leads to isotropization (at least under the assumption of reflection symmetry), as do fluids with almost any physically reasonable equation of state [210]. There is, however, one exception. This is the stiff fluid, which has a linear equation of state with tex2html_wrap_inline1955 . In that case the generalized Kasner exponents are time-independent, and may take on negative values. In a model with two non-interacting fluids with linear equation of state the one with the smaller value of tex2html_wrap_inline1973 dominates the dynamics at late times [89], and so the isotropization is restored. Consider now the case of a magnetic field and a perfect fluid with linear equation of state. A variety of cases of Bianchi types I, II and VI tex2html_wrap_inline2049 have been studied in [161Jump To The Next Citation Point In The Article, 162, 163], with a mixture of rigorous results and conjectures being obtained. The general picture seems to be that, apart from very special cases, there is at least partial isotropization. The asymptotic behaviour varies with the parameter tex2html_wrap_inline1973 in the equation of state and with the Bianchi type (only the case tex2html_wrap_inline2053 will be considered here). At one extreme, Bianchi type I models with tex2html_wrap_inline2055 isotropize. At the other extreme, the long time behaviour resembles that of a magnetovacuum model. This occurs for tex2html_wrap_inline2057 in type I, for tex2html_wrap_inline2059 in type II and for all tex2html_wrap_inline2061 in type VI tex2html_wrap_inline2049 . In all these cases there is partial isotropization.

Under what circumstances can a spatially homogeneous spacetime have the property that the generalized Kasner exponents are independent of time? The strong energy condition says that tex2html_wrap_inline2065 for any causal vector tex2html_wrap_inline2067 . It follows from the Hamiltonian constraint and the evolution equation for tex2html_wrap_inline1993 that if the generalized Kasner exponents are constant in time in a spacetime of Bianchi type I, then the normal vector tex2html_wrap_inline2067 to the homogeneous hypersurfaces gives equality in the inequality of the strong energy condition. Hence the matter model is in a sense on the verge of violating the strong energy condition and this is a major restriction on the matter model.

A further question that can be posed concerning the dynamics of expanding cosmological models is whether tex2html_wrap_inline2073 tends to zero. This is of cosmological interest since tex2html_wrap_inline2075 is (up to a constant factor) the density parameter tex2html_wrap_inline2077 used in the cosmology literature. Note that it is not hard to show that tex2html_wrap_inline1993 and tex2html_wrap_inline1703 each tend to zero in the limit for any model with tex2html_wrap_inline1689 which exists globally in the future and where the matter satisfies the dominant and strong energy conditions. First, it can be seen from the evolution equation for tex2html_wrap_inline1993 that this quantity is monotone increasing and tends to zero as tex2html_wrap_inline2087 . Then it follows from the Hamiltonian constraint that tex2html_wrap_inline1703 tends to zero.

A reasonable condition to be demanded of an expanding cosmological model is that it be future geodesically complete. This has been proved for many homogeneous models in [207Jump To The Next Citation Point In The Article].

7.2 Inhomogeneous solutions with 7 Asymptotics of Expanding Cosmological 7 Asymptotics of Expanding Cosmological

image Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de