There are a number of interesting open questions connected with inflation:
The origin of the Inflaton: It is far from clear what the inflaton actually is and where its potential comes from. This is intimately connected with the question of why the perturbations have the amplitude and spectrum they do. Currently, there is much confusion about physics at the relevant energy scales, and thus there is much speculation about different possible classes of inflaton potentials. One can hope that a clearer picture will eventually appear as some deeper theory (such as string/M theory) emerges to dictate the fundamental laws of physics at the inflation scale.
Physics of the inflaton: Having chosen an inflaton potential, one can calculate the perturbations produced during inflation assuming the relevant field modes for wavelengths much smaller than the Hubble radius are in their ground states. This seems plausible, but it would be nice to understand this issue more clearly. Also, the inflaton field often takes on values O(MP). Will a yet-to-be-determined theory of quantum gravity introduce large corrections to our current calculations?
The cosmological constant problem: A very important open question is linked with the cosmological constant problem [26, 27, 28]. The non-zero potential energy of the inflaton during inflation is very similar to a cosmological constant. Why the cosmological constant is extremely close to zero today (at least from a particle physicist's point of view) is perhaps the deepest problem in theoretical physics. One is left wondering whether a resolution of this problem could make the cosmological constant and similar contributions to Einstein's equations identically zero, thus preventing inflation from every occurring. Interestingly, current data is suggesting that there is a non-zero cosmological constant today (see for example Fig. 11 which shows = 0 is strongly excluded). Cosmic acceleration today is a very confusing idea to a theorist, but it actually is helpful to inflation theory in a number of ways: Firstly it shows that the laws of physics do allow a non-zero cosmological constant (or something that behaves in a similar way). Also, it is only thanks to the non-zero that the current data are consistent with a flat Universe (see Fig. 11).
Wider context and measures: We discussed in Section 3.4 various ideas about how an inflating region might emerge from a chaotic start. This is a very challenging concept to formulate in concrete terms, and not a lot of progress has been made so far. In addition, the fact that so many models are ``eternally inflating'' makes it challenging to define a unique measure for the tiny fraction of the universe where inflation actually ends. It has even been argued [29, 30, 31] that these measure problems lead to ambiguities in the ultimate predictions from inflation.
In fact it is quite possible that the ``chaotic'' picture will extend to the actual inflaton as well [32, 33]. Fundamental physics may provide many different flat directions that could inflate and subsequently reheat, leading to many different versions of the ``Big Bang'' emerging from a chaotic start. One then would have to somehow figure out how to extract concrete predictions out of this apparently less focused space of possibilities.
I am actually pretty optimistic that in the long run these measure issues can be resolved . When there are measure ambiguities it is usually time to look carefully at the actual physics questions being posed... that is, what information we are gathering about the universe and how we are gathering it. It is the observations we actually make that ultimately define a measure for our predictions. Still, we have a long way to go before such optimistic comments can be put to the test. Currently, it is not even very clear just what space we are tying to impose a measure on.
I am a strong opponent of the so-called ``anthropic'' arguments. All of science is ultimately a process of addressing conditional probability questions. One has the set of all observations and uses a subset of these to determine one's theory and fix its parameters. Then one can check if the theoretical predictions match the rest of the observations. No one really expects we can predict all observations, without ``using up'' some of them to determine the theory, but the goal of science is use up as few as possible, thus making as many predictions as possible. I believe that this goal must be the only determining factor in deciding which observations to sacrifice to determine the theory, and which to try and predict. Perhaps the existence of galaxies will be an effective observation to ``use up'' in constraining our theories, perhaps it will be the temperature of the CMB. All that matters in the end is that we predict as much as possible.
Thus far, using ``conditions for life to exist'' has proven an extremely vague and ineffective tool for pinning down cosmology. Some have argued that ``there must be at least one galaxy'' for life to exist [35, 36, 37], but no one really knows what it takes for life to exist, and certainly if I wanted to try and answer such a question I would not ask a cosmologist. Why even mention life, when one could just as well say ``we know at least one galaxy exists'' and see what else we can predict? In many cases (including [35, 36, 37]) the actual research can be re-interpreted that way, and my quibble is really only with the authors' choice of wording. It is these sorts of arguments (carefully phrased in terms of concrete observations) that could ultimately help us resolve the measure issues connected with inflation.