In this book, Penrose proposes “conformal cyclic cosmology” as the solution to this enigma. Let's pick this apart, word by word. A conformal transformation is a mathematical mapping which preserves angles in infinitesimal figures. It is possible to define a conformal transformation (for example, the hyperbolic transformation illustrated by M. C. Escher's Circle Limit III) which maps an infinite space onto a finite one. The author's own Penrose diagrams map all of (dimension reduced) space-time onto a finite plot via a conformal transformation. Penrose proposes a conformal transformation which maps the distant future of a dead universe undergoing runaway expansion to infinity with the big bang of a successor universe, resulting in a cyclic history consisting of an infinite number of “æons”, each beginning with its own big bang and ending in expansion to infinity. The resulting cosmology is that of a single universe evolving from cycle to cycle, with the end of each cycle producing the seemingly improbable conditions required at the start of the next. There is no need for an inflationary epoch after the big bang, a multitude of unobservable universes in a “multiverse”, or invoking the anthropic principle to explain the apparent fine-tuning of the big bang—in Penrose's cosmology, the physics makes those conditions inevitable.
Now, the conformal rescaling Penrose invokes only works if the universe contains no massive particles, as only massless particles which always travel at the speed of light are invariant under the conformal transformation. Hence for the scheme to work, there must be only massless particles in the universe at the end of the previous æon and immediately after the big bang—the moment dubbed the “crossover”. Penrose argues that at the enormous energies immediately after the big bang, all particles were effectively massless anyway, with mass emerging only through symmetry breaking as the universe expanded and cooled. On the other side of the crossover, he contends that in the distant future of the previous æon almost all mass will have been accreted by black holes which then will evaporate through the Hawking process into particles which will annihilate, yielding a universe containing only massless photons and gravitons. He does acknowledge that some matter may escape the black holes, but then proposes (rather dubiously in my opinion) that all stable massive particles are ultimately unstable on this vast time scale (a hundred orders of magnitude longer than the time since the big bang), or that mass may just “fade away” as the universe ages: kind of like the Higgs particle getting tired (but then most of the mass of stable hadrons doesn't come from the Higgs process, but rather the internal motion of their component quarks and gluons).
Further, Penrose believes that information is lost when it falls to the singularity within a black hole, and is not preserved in some correlation at the event horizon or in the particles emitted as the black hole evaporates. (In this view he is now in a distinct minority of theoretical physicists.) This makes black holes into entropy destroying machines. They devour all of the degrees of freedom of the particles that fall into them and then, when they evaporate with a “pop”, it's all lost and gone away. This allows Penrose to avoid what would otherwise be a gross violation of the second law of thermodynamics. In his scheme the big bang has very low entropy because all of the entropy created in the prior æon has been destroyed by falling into black holes which subsequently evaporate.
All of this is very original, clever, and the mathematics is quite beautiful, but it's nothing more than philosophical speculation unless it makes predictions which can be tested by observation or experiment. Penrose believes that gravitational radiation emitted from the violent merger of galactic-mass black holes in the previous æon may come through the crossover and imprint itself as concentric circles of low temperature variation in the cosmic background radiation we observe today. Further, with a colleague, he argues that precisely such structures have been observed in two separate surveys of the background radiation. Other researchers dispute this claim, and the debate continues.
For the life of me, I cannot figure out to which audience this book is addressed. It starts out discussing the second law of thermodynamics and entropy in language you'd expect in a popularisation aimed at the general public, but before long we're into territory like:
Ahhhh…now I understand! Seriously, much of this book is tough going, as technical in some sections as scholarly publications in the field of general relativity, and readers expecting a popular account of Penrose's proposal may not make it to the payoff at the end. For those who thirst for even more rigour there are two breathtakingly forbidding appendices.We now ask for the analogues of F and J in the case of the gravitational field, as described by Einstein's general theory of relativity. In this theory there is a curvature to space-time (which can be calculated once knows how the metric g varies throughout the space-time), described by a [ ^{0}_{4}]-tensor R, called the Riemann(-Christoffel) tensor, with somewhat complicated symmetries resulting in R having 20 independent components per point. These components can be separated into two parts, constituting a [ ^{0}_{4}]-tensor C, with 10 independent components, called the Weyl conformal tensor, and a symmetric [ ^{0}_{2}]-tensor E, also with 10 independent components, called the Einstein tensor (this being equivalent to a slightly different [ ^{0}_{2}]-tensor referred to as the Ricci tensor^{[2.57]}). According to Einstein's field equations, it is E that provides the source to the gravitational field. (p. 129)
The Kindle edition is excellent, with the table of contents, notes, cross-references, and index linked just as they should be.