The Quantum Structure of Space and Time (293 pages)

186 The Quantum Structure of Space and Time

pi(1,ni). It can be shown that the resulting Hilbert space on which the quantum
algebra is represented is the SUGRA Fock space. Massless particles and the overall
scale of null momenta appear in a manner reminiscent of Matrix Theory[4], while
Sg parametrizes the direction of null momenta. The holographic formalism can thus
reproduce the kinematics of M-theory. One would like to show that the dynamical
consistency conditions lead to equations which determine the scattering matrix
uniquely.

5.3.1.2 Holographic cosmology

Only one solution of the dynamical consistency conditions of holographic cosmology

has been found[9]. In this model, the dynamics of a given observer is described,

at each time t by a Hamiltonian which is a random irrelevant perturbation of a

random bilinear Hamiltonian

H = c s,(m)~mnS,(n).

The term irrelevant is used because, for large t x t matrices h, the bilinear dynamics

approaches that of free massless 1 + 1 dimensional fermions.

A simple prescription for the overlap Hilbert spaces, combined with this ansatz

for the Hamiltonian, satisfies all the consistency conditions. One then observes the

emergence, at large times, of a flat FRW geometry, with equation of state p = p.

That is, one can define a distance function on the lattice in terms of which lattice

points are causally disconnected at a given time. This geometry satisfies the scaling

laws of the FRW universe. Moreover, in the large time limit, the exact quantum

dynamics of the system is invariant under the conformal Killing symmetry of this

geometry.

The p = p cosmology was previous introduced by Fischler and the author[8]

in terms of a heuristic picture of a dense black hole fluid. This is a system in

which black holes continuously coalesce to form a single horizon filling black hole.

The random fermion model above is a precise mathematical realization of this idea.

Based on the heuristic picture, one can develop an ansatz for a more normal universe,

with normal regions originally arising as small defects in the black hole fluid. This

leads to a cosmology which can solve all of the standard cosmological puzzles, with

only a small amount of inflation just before nucleosynthesis. Inflation is necessary

only to stretch the scale of fluctuations generated during the p = p era, to the

size of our current horizon. The resulting cosmological model depends on only a

few parameters, and in one parameter range the fluctuations are entirely generated

in the p = p era. They are exactly scale invariant, between sharp infrared and

ultraviolet cutoffs. This fluctuation spectrum is, in principle, distinguishable from

that of inflationary models. It might explain the apparent disagreement between

inflationary models and the data at low L. We do not yet have predictions for the

gravitational wave spectrum in this parameter range.

In other parameter regimes, inflation must generate the observed fluctuations,