A. FRW Models, Dark Energy and the Late-Time Behavior

It is believed that our universe is an expanding FRW model. However, one misses a lot of (invisible) matter and energy in order to explain the observations.

Submitted: R Slagter and S Pan: New Fate of a Warped 5D FRW Model with a U(1) Scalar Gauge Field [FOOP]

Non Linear Gravitational Waves as Dark Energy in Warped Spacetimes [PRL]

Tangled Up in Spinning Cosmic Strings [RMP]

B. Warped 5D Spacetimes with a U(1) Scalar-Gauge Field

The standard model is extremely successful up to scales 10^3 GeV. The fundamental scale of gravity is the Planck scale 10^19 GeV. It is the scale where quantum gravity will act. The discrepancy between these two scales is called the hierarchy problem. Electro-weak interactions have been tested up to the EW-scale, while gravity, on the other hand, has been tested to several millimeters, 32 orders of magnitude above the Planck-scale.

At high energies, Einstein's GR theory breaks down and a new theory, quantum gravity, is necessary. But even below the fundamental Planck energy scale, significant corrections to GR will arise. Brane world models could overcome the hierarchy problem. The idea originates from string theory.

One of the predictions of string theory is the existence of branes embedded in the full bulk space time. Gravitons can then propagate into the bulk, while other fields are confined to these branes. It also predicts that space time is 10-dimensional, with 6 of them are very compact and small, not verifiable by any experiment. There are many models which attacked the hierarchy problem. Essentially there are globally two categories: flat compact extra dimensions and warped extra dimensions.

Recently, there is growing interest in the second category, i.e., the Randall-Sundrum warped 5-dimensional geometry. We live in a 3+1 dimensional space time embedded in a 5-dimensional space time, with an extra dimension which can be very large compared to the ones predicted in string theory. One estimates that the extra dimension can be as large as 10^-2 cm, which is the under-bound of Newton's law in our world.

The observed 4-dimensional Planck scale is no longer the fundamental scale but an effective one, an important consequence of the extra dimensions, which is now the Planck scale in 5D

C. Causality and Warped Spacetimes

Cosmic strings can be formed in symmetry-breaking phase transitions in the early stages of the universe. They consist of trapped regions of false vacuum in U(1)-gauge theories. They are topological defects, similar to flux tubes in type-II superconductors. When the temperature in the early universe decreased, the scalar field developed a locus of trapped points of false vacuum.

Density perturbations produced by these strings of GUT scale, μ= ε ^2 Mpl^2, where G is Newton's constant, Mpl the Planck mass, μ the mass per unit length of the string and ε the symmetry breaking scale, could have served as seeds for the formation of galaxies and clusters. However, recent observation of the cosmic microwave background (CMB) radiation disfavored this scenario. The WAMP-data proves that cosmic strings cannot contribute more than an insignificant proportion of the primordial density perturbation, Gμ <10^-6 . Recently, brane world scenarios suggest the existence of fundamental strings, predicted by superstring theory. These super-massive cosmic strings, G μ ≈ 1, could be produced when the universe underwent phase transitions at energies much higher than the GUT scale and could play a role very similar to that of cosmic strings.

An other aspect of cosmic strings is the angle deficit they exhibit at finite distance of the core, i.e., the spacetime is there Minkowski minus a wedge. They will produce a very special pattern of lensing effect, not found yet in observations.

Because there is no z-dependency (considered in polar-coordinates), one often studies the related (2+1)-dimensional models, where the stringlike objects are treated as gravitating point particle, or "cosmons". These models are also studied in relation with causality problems and toy-models of quantized gravitating point particles.

An interesting example of the richness of the (2+1)-dimensional gravity, is the Gott spacetime. An isolated pair of cosmic strings, equivalent with two point particles in (2+1)-dimensions, moving in opposite directions with sufficient high velocity, could generate a closed timelike curve (CTC). If an advanced civilization could manage to make a closed loop around this Gott pair, they will be returned to their own past.

However, it is not a surprise that one can prove that the Gott spacetime will also be present at spatial infinity, which constitutes un-physical boundary conditions. Moreover, it turns out that the effective one-particle generator has a tachyonic center of mass. This means that the energy-momentum vector is spacelike.

Even a closed universe will not admit these CTC's, so the chronology protection conjecture is saved. In fact, a configuration of point particles admits a Cauchy formulation within which no CTC's are generated.

One can investigate if chronology protection is still saved in warped 5-dimensional spacetimes.