No explanation on how "expansion of space" fits with it. Did space expansion reverse and collapse back at one point? To restart their expansion after a new Big Bang?
No, the conformal cyclic universe is weird in that there is no contraction that actually happens. Things keep expanding out into forever and that’s part of why the CCU works.
What happens is that the one universe becomes really boring-looking when you look at it from the perspective of one universe, so you use one of your mathematical freedoms (that everything looks the same under a conformal mapping) to shift your perspective. Conformal maps are their own interesting field of mathematics, let me tease you by saying that the usual Mercator map projection has this really nice property that if you zoom in on the map to get a "correct" close-up of any city, you get something that looks angle-for-angle correct at the city scale, you just have to zoom in less on, say, Bogota than Reykjavik because the poles are zoomed so much larger than the equator. By choosing other position-dependent scale factors you can instead get, say, the stereographic projection which has the same basic rule but now which direction is "north" always points to the center of a circle and now Bogota is bigger than Reykjavik, but they still both look "correct" locally, they are just out-of-scale at these longer distances. These angle-preserving deformations of space are called conformal maps. Actually, if you accelerate in any given direction the stars you see all crowd in towards the direction that you accelerate, but they do it conformally. So like the Pleiades will still look to first order like they look now as they squish in towards the direction you are going.
Anyway, there is a subset of physics called quantum field theory and a subset of this subset is called conformal field theory and during the heat death of the universe the QFT becomes a CFT and so you get this freedom to apply conformal mappings, the physics doesn't care and the equations stay the same. Penrose's idea is that if you choose the conformal mapping just right this heat death looks suggestively like a Big Bang for a different universe. So maybe from the perspective of this universe, the universe is ending in a heat death off at infinity, but maybe there is a perspective of a different universe where that same homogeneous ending was actually a homogeneous starting point. You redefine things like "distance" and "time" and stitch together the infinite future of one universe with the Big Bang of another. But like that other universe does not ever itself contract, the contraction comes in this redefinition of everything meaningful. The words of the old eon cease to have meaning and you use the words of the new eon.
If there is no matter and nothing that has any attributes depending on time, then distance and duration becomes meaningless. In conformal geometries scale doesn’t matter so the incredibly immense and microscopically tiny are geometrically equivalent. There’s no contraction, rather a kind of re-scaling.
What happens is that the one universe becomes really boring-looking when you look at it from the perspective of one universe, so you use one of your mathematical freedoms (that everything looks the same under a conformal mapping) to shift your perspective. Conformal maps are their own interesting field of mathematics, let me tease you by saying that the usual Mercator map projection has this really nice property that if you zoom in on the map to get a "correct" close-up of any city, you get something that looks angle-for-angle correct at the city scale, you just have to zoom in less on, say, Bogota than Reykjavik because the poles are zoomed so much larger than the equator. By choosing other position-dependent scale factors you can instead get, say, the stereographic projection which has the same basic rule but now which direction is "north" always points to the center of a circle and now Bogota is bigger than Reykjavik, but they still both look "correct" locally, they are just out-of-scale at these longer distances. These angle-preserving deformations of space are called conformal maps. Actually, if you accelerate in any given direction the stars you see all crowd in towards the direction that you accelerate, but they do it conformally. So like the Pleiades will still look to first order like they look now as they squish in towards the direction you are going.
Anyway, there is a subset of physics called quantum field theory and a subset of this subset is called conformal field theory and during the heat death of the universe the QFT becomes a CFT and so you get this freedom to apply conformal mappings, the physics doesn't care and the equations stay the same. Penrose's idea is that if you choose the conformal mapping just right this heat death looks suggestively like a Big Bang for a different universe. So maybe from the perspective of this universe, the universe is ending in a heat death off at infinity, but maybe there is a perspective of a different universe where that same homogeneous ending was actually a homogeneous starting point. You redefine things like "distance" and "time" and stitch together the infinite future of one universe with the Big Bang of another. But like that other universe does not ever itself contract, the contraction comes in this redefinition of everything meaningful. The words of the old eon cease to have meaning and you use the words of the new eon.