D. Inflation and fundamental physics
The mechanism of cosmic inflation explores brand new territory, and there are many aspects of this idea that need to be better understood. In some cases simple assumptions have been made that need to be justified; in other cases potentially problematic issues have been ignored, for lack of any concrete way forward. In addition, a lot of details are still missing, because we still do not have a consensus model of physics on the energy scales at which inflation is supposed to have happened. All these issues are linked with fundamental questions in particle physics.
Small parameters: If a given inflationary model is sufficiently well specified, exact predictions can be made for the spectrum of cosmic perturbations that are produced. In most models, in order to achieve a suitable overall amplitude for the cosmic perturbations, a dimensionless parameter in the model must be set to a small value of order 10-12. Although there are claims that in some cases the right amplitude comes out naturally [Freese et al.(1990)], this issue is far from settled. We hope that progress on a fundamental description of physics at high energies will yield a more solid foundation for the inflaton.
Re/Pre-heating: The decay of the inflaton into ordinary matter is a new territory in its own right. To have all the energy in the Universe tied up in the potential energy of a single coherent field and which then decays into ordinary matter is not yet a well-understood process. Much of the analysis has been based on very simple arguments, although some intriguing coherence effects dubbed "pre-heating" have been investigated (for some recent discussions see [Boyanovsky et al.(2001), Kofman(2001)]. No doubt there is room for more progress to be made in this area, which could be crucial in determining which inflation models are really viable. There could be suprises (i.e., super-efficient or inefficient reheating) which would lead to very different inflationary scenarios.
The problem of negative pressure: Inflation depends on the inflaton achieving an equation of state with p < - / 3. While it is easy enough to construct a scalar field which has these properties under the right conditions, until recently it was thought that such states had never been observed in Nature. With the discovery of the cosmic acceleration (see Section III) there is evidence that somehow Nature is able to endow matter with a suitable equation of state, but we are still not sure how. In fact, there is even a threat hanging over inflation related to the cosmological constant problem (discussed in Section III), since the inflaton behaves very much like a cosmological constant during inflation. Whatever mechanism Nature chooses to remove the "vacuum energy" (which naively should exceed observational bounds by 120 orders of magnitude) could just as well kick in to prevent inflation from taking place at all. Alternatively it has been proposed that even ordinary gravity actually has a built-in mechanism that can cancel the vacuum energy with quantum corrections, but that these dynamical corrections happen slowly enough that it is still possible for a suitable period of inflation to take place [Abramo and Woodard(2001), Tsamis and Woodard(1996)]. Whatever the outcome, it is intriguing that this problem is now linked with the observed cosmic acceleration, the understanding of which which there is hope for real progress based on observations.
"Trans-Planckian" modes and inflation: During inflation, quantum field modes are stretched from tiny scales (smaller than the Planck length) to cosmic scales. What do we really know about physics on trans-Planck scales? A simple "Bunch-Davies vacuum" provides the required input in the standard calculation, and it certainly serves the purpose. Interesting recent work [Martin and Brandenberger(2001)] suggests that only extreme deviations from the assumed dispersion relation at these scales could change the predictions for large-scale structure. Ultimately we would like to see this subject on a much firmer footing, especially since the prediction of cosmic perturbations depends on what is input in the first place. (See the discussion in subsection VIID.)
Before Inflation: One of the impressive features of the inflationary picture is how a period of inflation transforms many different possible initial conditions into the kind of state we need to kick off the Standard Big Bang cosmology. It is tempting to think that with that kind of dynamics, we really do not need to think much about what might have happened before inflation. This may in the end turn out to be true, but this is currently an extremely poorly understood subject, and less pleasing results may emerge from a more sophisticated treatment. It is a challenge to treat quantitatively the "space of all pre-inflation states". It has even been argued that fundamental uncertainties to do with placing measures on pre-inflation states make predictions from inflation impossible [Linde et al.(1994)], but few have found these arguments compelling (see for example ref. [Vanchurin et al.(2000)] for an alternative perspective). Recent work has also argued that one cannot have a past described purely by inflation [Borde et al.(2001)], so we are stuck trying to come to grips with the issue of "pre-inflation". Another approach to this issue is to make a specific proposal for the "wavefunction of the Universe" [Hartle and Hawking(1983), Garriga and Vilenkin(1997), Vilenkin(1998), Hawking and Turok(1998)] which at least in principle might address these questions.