The complex plank problem

It is shown that if $(v_{j})$ is a sequence of norm $1$ in a complex Hilbert space and $(t_{j})$ is a sequence of nonnegative numbers satisfying $\sum t_{j}^{2}=1$ then there is a unit vector z for which $|\langle v_{j}, z \rangle|\geq t_{j} for every j. The result is a strong, complex analogue of the author's real plank theorem

CDT and the Big Bang

We describe a CDT-like model where breaking of W3 symmetry will lead to the emergence of time and subsequently of space. Surprisingly the simplest such models which lead to higher dimensional spacetimes are based on the four "magical" Jordan algebras of 3x3 Hermitian matrices with real, complex, quaternion and octonion entries, respectively. The simplest symmetry breaking leads to universes with spacetime dimensions 3, 4, 6, and 10

Cosmological Models and Renormalization Group Flow

We study cosmological solutions of Einstein gravity with a positive cosmological constant in diverse dimensions. These include big-bang models that re-collapse, big-bang models that approach de Sitter acceleration at late times, and bounce models that are both past and future asymptotically de Sitter. The re-collapsing and the bounce geometries are all tall in the sense that entire spatial slices become visible to a comoving observer before the end of conformal time, while the accelerating big-bang geometries can be either short or tall. We consider the interpretation of these cosmological solutions as renormalization group flows in a dual field theory and give a geometric interpretation of the associated c-function as the area of the apparent cosmological horizon in Planck units. The covariant entropy bound requires quantum effects to modify the early causal structure of some of our big-bang solutions.Comment: 26 pages, 11 figures, v2: improved discussion of entropy bounds, references added, v3: minor changes, reference adde

Hot big bang or slow freeze?

We confront the big bang for the beginning of the universe with an equivalent picture of a slow freeze - a very cold and slowly evolving universe. In the freeze picture the masses of elementary particles increase and the gravitational constant decreases with cosmic time, while the Newtonian attraction remains unchanged. The freeze and big bang pictures both describe the same observations or physical reality. We present a simple "crossover model" without a big bang singularity. In the infinite past space-time is flat. Our model is compatible with present observations, describing the generation of primordial density fluctuations during inflation as well as the present transition to a dark energy dominated universe.Comment: new material on absence of singularity, 9 pages, 1 figur