Inflation seems to solve many of the problems of cosmology like horizon problem, flatness problem etc. Now suppose, I am a devil's advocate and tries to find holes in this beautiful theory. I argue that the early universe were having a fractal geometry. For any generic initial conditions suppose it was indeed a fractal. Then obviously no amount of stretching can make it smooth and flat locally. How would one reconcile this picture with inflation?
The universe is a fractal according to inflation, or very nearly. The product of inflation is very close to a scale-invariant spectrum, which is a fractal spectrum, in that rescaling the fluctuations gives almost the same result. This fractal-ness is amplified to the current filamentous fractal structure of galaxies and dark-matter distribution.
When people say "the universe is a fractal" they don't mean nearly scale-invariant mass distribution, which is a consensus view. What they mean is that the distribution is zero density over large distances, to resolve Olber's paradox as suggested by Mandelbrot in "The fractal geometry of nature". This type of idea is not correct, because the universe is homogenous at large scale.
Stretching by inflation will smooth out anything to the special type of inflation initial condition fractal, no matter how differently fractal, because all the structure falls out of the very small cosmological horizon. During inflation, the universe is like an inside-out black hole, we are surrounded by a horizon, and everything falls away from the center into the horizon, leaving a smooth middle together with a fractal (scale invariant) fluctuations.
Roger Penrose discusses this in chapter 27 of his book The Road to Reality. His conclusion is that you can't just assume inflation will smooth out any initial conditions, and that the pre-inflation universe must already have had a very low entropy.
This question is quite speculative. What I will write below is speculative as well, and doubtless there will be down voters on this. However, here goes.
I think there may be a role for fractal or fractal-like geometry in spacetime physics with respect to discrete groups. This might lie with the use of discrete groups with hyperbolic spaces, such as the anti de Sitter spacetime. The AdS in 2-dimensions has a structure similar to the Poincare disk. The Escher print below illustrates a particular discrete group, eg $PSL(2,Z)~\subset~PSL(2,R)$, which tessellates the space. This is a Euclideanized version of the AdS, and the boundary of this disk corresponds to a conformal structure that is equivalent to the isometries of the AdS. $AdS_n~=~SO(n-1,2)/SO(n-1,1)$ with the isometry group $SO(n-1,2)$ for $n~=~2$ this is the $SO(2,1)~\simeq~SL(2,R)$, which is the elementary group of conformal quantum mechanics. For an $AdS_4$ that contains a black hole, where $AdS_4$ is the perfect box to hold a black hole due to its negative Gaussian curvature, the structure of the spacetime becomes $AdS_2\times S^2$ near the horizon.
This is mentioned because it is possible that the tessellation of the $AdS_2$, which may extend to $AdS_3$ or even higher dimensions, has a quasi-periodic structure if there is a fibration over the AdS of a particular nature. This tessellation is a case of quasi-crystals which are five-fold tilings in $2$ and $3$ dimension have pentagonal or isosahedral symmetry. The $AdS_2$ may then exist in as the result of compactification and the near horizon condition on larger $AdS_n$, so that the resulting fibration over the $AdS_2$ gives a quasi-periodic tiling of the hyperbolic space.
There are similarities in behavior between quasi-crystals and the zeros of the Riemann zeta function. It is possible that quasi-crystal structure might play a role in some future proof of the Riemann zeta function conjecture. Where this might play a role in physics is if the quantum states of the universe turn out to have some correspondence to the zeros of the Riemann zeta function. So there is a prospect that quasi-crystals and tessellations are an important part of the understanding the quantum states of the universe.
The one dimensional ‘fractal” or quasi-periodic structure has already played a role in number theory. The integer partition and Ramanujan’s congruencies have been found to be due to a one dimensional fractal or quasi-periodic structure
It is then not unreasonable to think there are even deeper relationships between these types of structures and the quantum states of gravitation and the universe.
The $AdS$ spacetime is of course not the spacetime of the universe. It really is more of a mathematical gadget. So this may be somewhat indirect with respect to this question. Further, there are elements to what I wrote here that are not known.