Of all the cosmological models proposed in recent years, perhaps the most ingenious is Roger Penrose's conformally cyclic cosmology. Penrose elaborated this idea at length in an outstanding contribution to the 2008 collection, On Space and Time.
Penrose's cyclic cosmological model is a particular application of his conformal compactification construction, which takes any space-time, whether or not it contains singularities, and whether or not it is infinite in time or space, and constructs a finite (compactified) space-time with boundary, whose metric is related to the original metric of space-time by a locally variable scale factor Ω, called the conformal factor. This transformation preserves the causal structure of the original space-time, even if it doesn't preserve lengths and times. The boundary of the conformal compactification contains components which correspond to singularities, and components which correspond to spacelike infinity, timelike infinity, (and null infinity).
Now, in an open or flat Friedmann-Robertson-Walker cosmological model with no cosmological constant, whilst the past 'Big-Bang' singularity corresponds to a spacelike hypersurface in the boundary of the conformal compactification, the future timelike infinity corresponds to a single point. Current astronomical evidence, however, suggests that the expansion of our universe is accelerating due to the presence of a hypothetical dark-energy, which at least mimics the effect of a cosmological constant. The far future of our universe is therefore representable by de Sitter space-time, and in this type of space-time the future timelike boundary is a spacelike hypersurface. Penrose made the simple observation that the future conformal boundary of such a space-time can therefore be joined to the spacelike initial conformal boundary of a Friedmann-Robertson-Walker model to make a cyclic universe.
The fact that the universe is very small and hot at the beginning, and very large and cold in the far future, is not a problem, argues Penrose, because both the early universe and far future universe contain only conformally invariant, massless particles. Without massive particles, there is no way of defining lengths or times, hence the only physically meaningful structure is the conformal structure, i.e., the causal structure. By compressing the conformal factor towards the far future, and expanding it towards the beginning, the geometry of the future conformal boundary can be joined seamlessly to the initial conformal boundary. In other words, the conformal factor Ω must tend to zero as time t tends to ∞, to compress the infinite future into a finite conformal time, and Ω must tend to ∞ as t tends to 0, to stretch the metric as it tends towards the Big-Bang. The conformal metric then matches on the two boundary components, and the components can be identified. The Weyl curvature is zero on both the future boundary and past boundary, hence the Big Bang is still well-defined in the cyclic model as the unique hypersurface on which the Weyl curvature vanishes.
Penrose suggests that cosmic inflation doesn't occur after the Big Bang, but before it! The accelerating expansion of the universe that we currently observe, is identified as the onset of inflation. It is this inflation, proposes Penrose, that generates the scale-invariant spectrum of density perturbations in the post-Big Bang universe.
Penrose proposes that the far future of our universe contains only electromagnetic radiation and gravitational radiation. The electromagnetic radiation comes from the cosmic background radiation of the Big Bang, from stars, and from the eventual evaporation of black holes. The gravitational radiation, meanwhile, comes mostly from the coalescence of black holes. Penrose proposes that massive particles such as electrons, either annihilate with massive particles of opposite charge (positrons), or decay by some as-yet undiscovered mechanism.
The cycle of Penrose's model is one in which the universe 'begins' in a conformally invariant state, with zero Weyl curvature, but in which there is a normal derivative to the Weyl curvature. This seems to trigger the formation of massive particles. The matter then clumps together into stars and galaxies, such clumping increasing the Weyl curvature and decreasing the Ricci curvature. Eventually, much of the matter is swept into black holes, where the Weyl curvature diverges, but the Ricci curvature is zero. The matter which isn't swept into black holes decays or annihilates into radiation, and the black holes eventually evaporate themselves into radiation. The Weyl curvature thereby returns to zero, all particles are massless again, and conformal invariance resumes. Gravitational radiation, however, never thermalizes, and this appears to be responsible for the normal derivative to the Weyl curvature, which triggers the formation of massive particles in the next cycle.
Two potential problems spring to mind. Firstly, following an argument by Gibbons and Hawking, de Sitter space-time is widely believed to possess a minimum temperature due to its cosmological constant. With the value of the cosmological constant we observe, this temperature is about 10-30 Kelvin. A black hole will only evaporate if the temperature of its horizon is greater than the temperature of surrounding space. The temperature of a black hole is inversely proportional to its mass, and a black hole which grows large enough that its temperature drops below 10-30 Kelvin would never evaporate. However, such a black hole would have a mass approximately equal to the current observable universe, so the formation of such a black hole may well be impossible in a universe whose contents are diluted by the accelerating expansion of dark energy.
The second problem is that if the quantum fields in the far future of our universe can be treated as quantum fields in thermal equilibrium in de Sitter space-time, then because such a universe is eternal, quantum fluctuations ensure the spontaneous generation, at a constant rate, of anything you care to name, including massive particles and black holes. This would prevent our universe from ever reaching an exact state of conformal invariance in the far future. However, because gravitational radiation never reaches thermal equilibrium, one could perhaps argue that the quantum fields in the far future of our universe cannot be treated as quantum fields in thermal equilibrium in de Sitter space-time.
Penrose's proposal remains fascinating and elusive.