How much information in the universe?

Suppose the universe is described by a single wavefunction which evolves in time according to Schrodinger dynamics. In this framework remarkably little information is required to describe the universe as a whole.

Consider the sub-volume V, just after the big bang, which evolves into our observable 15 Gy universe. Now suppose there is an ultraviolet cutoff (or minimum resolvable length) given by the Planck length. Then the entropy (log of number of degrees of freedom) is of order V in Planck units. To neglect quantum gravity effects, we need to wait sufficiently long after the big bang that curvatures are small in Planck units, so this entropy is a large number, but still much smaller than the observed entropy of our current universe. Given the wavefunction over this large but finite Hilbert space, and the Hamiltonian, we can evolve this system forward in time until today. Neither the informational complexity (number of bits required to specify the initial state) nor the algorithmic complexity (length of program required to evolve the system) is very large. (Note that memory requirements could grow quite rapidly - especially in an expanding universe.)

If this confuses you, just imagine you had to write a computer program to evolve this finite system (remember, we have both UV and IR cutoffs) forward in time. How much input would your program need, and how long would the code be? Then compare the answer to what would be required, e.g., to simulate to just a small part of planet earth today.

So what is the origin of the apparent complexity of our world? The answer is that the wavefunction described above contains all branches of Everett's many worlds (see previous post). In order to locate your particular branch (the one on which your consciousness resides), you have to specify the outcomes of the branchings in your past (or, at least, the important ones - I think Gell-Mann and Hartle refer to this as decoherent histories). The amount of information required to specify a particular history is related to the number of possible present universes, and is mainly responsible for the complexity we observe.