Sean Carroll began to transfer his rudimentary misunderstandings about thermodynamics and statistical physics from the realm of dumb popular books to the physics arXiv:
p = 10-66,000,000Yes, it's ten to the power of minus sixty-six million.
Alan Guth and the rest of the mankind would have to be damn lucky to live in a Universe with inflation, they argue. ;-) Needless to say, this is complete nonsense. Whenever there is an inflaton field, it always has a significant probability of order 100% to appear near the top of its potential at some points and the inflation is then largely inevitable.
Or, if you want to avoid the eternal inflation, the Universe may still decide about the initial states of some spacetime fields and it's damn reasonable to think that the right initial conditions put the inflaton near the top - and inflation is inevitable again.
The kind of "argument" they use - and that can be equally well (mis)applied against Darwin's evolution or any other theory explaining the current observations by the evolution from a simpler beginning - is the following:
In the context of reversible (unitary) evolution, this goal [explanation of flatness etc.] is difficult to satisfy, as Liouville's theorem implies that no dynamical process can evolve a large number of initial states into a small number of final states.Well, all the evolution in the Universe is unitary i.e. microscopically reversible. But it's still true that the entropy always increases, in any system with a macroscopic entropy that is not at equilibrium. And it's still true that inflation solves all the problems that Carroll and Tam claim to be unsolved.
The increase of entropy is completely universal for all physical systems but its "visual" consequences depend on the particular system we study. For example, for two liquids at different temperatures, the increasing entropy will mix the liquids and make the temperature and other "intensive" quantities more uniform.
Whenever the inflaton is near the maximum (and inflation has a chance to begin), the spatial components of the metric tensor are told by Einstein's equations to either exponentially shrink or exponentially expand: so the size of the Universe is doing the same thing. Clearly, unless you make some gigantic, unnatural fine-tuning of the initial state in order to eliminate all the perturbations that would lead to exponentially increasing terms, and unless you do so at all points in spacetime, these exponentially increasing terms win over the shrinking ones after a short time.
The Universe inevitably expands. It follows that the energy density and the density of the magnetic monopoles, gravitational waves, and all kinds of objects is decreasing. This process solves all the usual problems that inflation is said to solve and any opinion that there is a paradox or that inflation can't do it because of statistical physics are completely idiotic. The process continues up to the moment when the inflaton approaches the minimum. Afterwards, the inflation slows down, stops, and the inflaton kinetic energy changes to the latent energy of newly created particles.
The map between the initial microstates is indeed one-to-one, as long as you use a complete theory. But at the beginning of inflation, the potential energy of the inflaton is higher. In other words, the "effective positive cosmological constant" is higher in the initial state. It means that the corresponding de-Sitter-like space is highly curved and the cosmic horizon is small.
The entropy of such a small space is small as well: note that it can't exceed A/4G where A is the surface of the cosmic horizon. So one actually has a pretty small number of initial states at the beginning of inflation. Because the space grows, one may have many more kinds of excitations in the final state - the space as well as the horizon grow and the Hilbert space of possible final states has a much higher dimension.
Exactly because the evolution provides a one-to-one map between the initial microstates and the final microstates and because the number of initial microstates is smaller because the space corresponding to the same degrees of freedom is smaller, it follows that only some final states may be generated by the inflationary evolution. It is not hard to see what is the "general type" of the final states one can get from inflation: they're relatively small perturbations upon the flat space. One can actually watch what a particular quantum of radiation is doing on the inflating background: its wavelength is increasing, the frequency is decreasing, and the energy carried by the perturbation is decreasing as well (note that E=hf).
The final states that can arise from inflation are special exactly in the aspects that are desirable. That's why inflation solves the problems that were correctly claimed to be solved. Sean Carroll and Heywood Tam have just forgotten elementary physics if they ever knew it. Their paper is another example of the plummeting standards that affect even institutions such as Caltech.
Let me also say that their paper is isomorphic to the creationist criticism of Darwin's evolution based on the claim that such evolution would violate the second law because more organized structures are being evolved. You can say that Carroll and Tam have plagiarized these creationists.
It's true that more organized - and lower-entropy - structures evolve but the entropy of the rest of the Universe overcompensates this decrease. We receive the energy from the Sun in the form of visible light and re-emit it back to space in the form of infrared light that has much higher entropy. So we still produce lots of entropy - it's enough for the individual process to be able to lower animals' entropy without violating the growth of the total entropy.
The expanding volume of the Universe plays the role of the Sun. The entropy of the external environment - e.g. the entropy of the cosmic horizon - is quickly increasing during inflation, so it allows particular subsystems of the Universe to lower their entropy without any contradiction to the fact that the total entropy increases.