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7.4 Acceleration due to nonlinear scalar fields

The effect of a cosmological constant can be mimicked by a suitable exotic matter field that violates the strong energy condition: for example, a nonlinear scalar field with exponential potential. In the latter case, an analogue of Wald’s theorem has been proved by Kitada and Maeda in [207208]. For a potential of the form V~ -- e- 8pcf with V~ -- c < 2, the qualitative picture is similar to that in the case of a positive cosmological constant. The difference is that the volume grows like a power of t instead of exponentially and that the asymptotic rate of decay of various quantities is not the same as in the case with positive /\. This is called power-law inflation. A global existence theorem for homogeneous solutions of the Einstein-Vlasov system with a nonlinear scalar field and a positive potential was proved in [225Jump To The Next Citation Point]. This applies in particular to the case of an exponential potential. The detailed asymptotics of geometry and matter for an exponential potential with c < V~ 2- were worked out in [225]. Corresponding global existence results in the case of a perfect fluid with linear equation of state are given in [303Jump To The Next Citation Point]. The behaviour of homogeneous and isotropic models with general c has been investigated in [168].

Our knowledge of the fundamental physics is insufficient to show which potential for the scalar field is most relevant for physics. It therefore makes sense to study the dynamics for large classes of potentials. A useful way of organizing the possibilities uses the ‘rolling’ picture. In a spatially homogeneous spacetime the scalar field satisfies

¨ 1- ' f - 3 trkf = - V (f). (4)
This resembles the equation of motion of a ball rolling on the graph of the potential V with variable friction given by trk. Of course the evolution of trk is coupled back to that of f and so this analogy does not allow immediate conclusions. Nevertheless it gives an intuitive picture of what should happen. The ball should roll down to a minimum of the potential and settle down there, possibility oscillating as it does so.

The simplest case is where the potential has a strictly positive minimum. In [303] it was proved under some technical assumptions that a direct analogue of Wald’s theorem holds. The late time behaviour of the geometry closely resembles that for a cosmological constant. The value of this effective cosmological constant is /\eff = 8pV (f1), where f1 is the value where V has its minimum. The asymptotic behaviour of the matter fields was determined in the case of collisionless matter and perfect fluids with a linear equation of state.

Another important case is where V is everywhere positive and decreasing and tends to zero as f --> oo. The ‘rolling’ picture suggests that f should tend to infinity as t-- > oo. Under suitable technical assumptions this is true and information can be obtained concerning the asymptotics. The exponential potential is a borderline case. An important assumption is that lim V '/V < V~ 2- f--> oo or, more generally ' V~ - lim supf --> oo V /V < 2. Intuitively this says that the potential falls off no faster at infinity than an exponential potential which gives rise to power-law inflation. A theorem in [306Jump To The Next Citation Point] where this assumption is made in a set-up like that in Wald’s theorem shows that there is always accelerated expansion for t sufficiently large. If it is further assumed that V '/V --> 0 as f --> oo then it is possible to say a lot more. It is found that, if sab is the tracefree part of the second fundamental form, R is the spatial scalar curvature, and r is the energy density of matter other than the scalar field then sabsab/(trk)2, R/(tr k)2, and r/(trk)2 tend to zero as t-- > oo. In the limit t --> oo the solution is approximated by one which is isotropic and spatially flat and contains no matter other than the scalar field. This kind of situation is sometimes called intermediate inflation since the potential is intermediate between a constant (corresponding to a cosmological constant) and an exponential (corresponding to power-law inflation).

If V '/V --> 0 as f --> oo and V ''/V ' is bounded for large f then it is possible to get further information. This is related to the ‘slow-roll approximation’. The intuitive idea is that if the slope of the graph of V is not too steep the ball will roll slowly and certain quantities will change gradually. It can be proved that asymptotically the term with second order derivatives in Equation (4View Equation) can be neglected and that the late-time behaviour is described approximately by the resulting first order equation. In fact this can be further simplified to give the equation ' V~ ------ f = - V / 24pV for f alone. This asymptotic description is not only interesting in itself; it gives a powerful method for determining the late time asymptotics when a specific potential has been chosen. For more details see [306].

Both models with a positive cosmological constant and models with a scalar field with exponential potential are called inflationary because the rate of expansion is increasing with time. There is also another kind of inflationary behaviour that arises in the presence of a scalar field with power law potential like f4 or f2. In that case the inflationary property concerns the behaviour of the model at intermediate times rather than at late times. The picture is that at late times the universe resembles a dust model without cosmological constant. This is known as reheating. The dynamics have been analysed heuristically by Belinskii et al. [46]. Part of their conclusions have been proved rigorously in [302]. Calculations analogous to those leading to a proof of isotropization in the case of a positive cosmological constant or an exponential potential have been done for a power law potential in [249]. In that case, the conclusion cannot apply to late time behaviour. Instead, some estimates are obtained for the expansion rate at intermediate times.


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