We propose two methods for obtaining the position space of non-linear relativity, i.e. the dual of its usual momentum space formulation. In the first approach we require that plane waves still be solutions to free field theory. This is equivalent to postulating the invariance of the linear contraction between position and momentum spaces, and dictates a set of energy-dependent spacetime Lorentz transformations. In turn this leads to an energy dependent metric. The second, more problematic approach allows for the position space to acquire a non-linear representation of the Lorentz group independently of the chosen representation in momentum space. This requires a non-linear contraction between momentum and position spaces. We discuss a variety of physical implications of these approaches and show how they point to two rather distinct formulations of theories of gravity with an invariant energy and/or length scale.

Brane Gas Cosmology (BGC) is an approach to M-theory cosmology in which the initial state of the Universe is taken to be small, dense and hot, with all fundamental degrees of freedom near thermal equilibrium. Such a starting point is in close analogy with the Standard Big Bang (SBB) model. The topology of the Universe is assumed to be toroidal in all nine spatial dimensions and is filled with a gas of p-branes. The dynamics of winding modes allow, at most, three spatial dimensions to become large, thus explaining the origin of our macroscopic 3 + 1-dimensional Universe. Here we conduct a detailed analysis of the loitering phase of BGC. We do so by including into the equations of motion that describe the dilaton gravity background some new equations which determine the annihilation of string winding modes into string loops. Specific solutions are found within the model that exhibit loitering, i.e. the Universe experiences a short phase of slow contraction during which the Hubble radius grows larger than the physical extent of the Universe. As a result the brane problem (generalized domain wall problem) in BGC is solved. The initial singularity and horizon problems of the SBB scenario are solved without relying on an inflationary phase. Motivation and IntroductionThe necessity to search for alternatives to the Standard Big Bang (SBB) scenario is driven by a significant number of problems within the theory such as the horizon, flatness, structure formation and cosmological constant problems. Although inflationary models have managed to address many of these issues, all current formulations remain incomplete. In particular, the present models of inflation suffer from the fluctuation, super-Planck scale physics, initial singularity and cosmological constant problems, as discussed in [1]. Furthermore, several theorems have appeared which show that de Sitter spacetime, the simplest example of an inflationary Universe, cannot be a classical solution to supergravity theories [2,3]. As supergravity is the low energy limit of M-theory, it does not seem clear that a pure de Sitter model of inflation will arise naturally within the theory.M-theory is currently our best candidate for a quantum theory of gravity. As such, the theory should provide the correct description of physics in regions of space with high energies and large curvature scales similar to those found in the initial conditions of the Universe. Therefore, it is only natural to incorporate string and M-theory into models of cosmology. According to [6], the initial state of the Universe is small, dense, hot and with all fundamental degrees of freedom in approximate thermal equilibrium.2 For simplicity, the background spatial geometry is assumed to be toroidal, and the Universe is filled with a hot gas of p-branes, the fundamental objects appearing in string theories.These branes may wrap around the cycles of the torus (winding modes), they can have a center-of-mass motion along the cycles (momentum modes) or they may simply fluctuate in the bulk space (...

We investigate the behaviour of exact closed bouncing Friedmann universes in theories with varying constants. We show that the simplest BSBM varying-alpha theory leads to a bouncing universe. The value of alpha increases monotonically, remaining approximately constant during most of each cycle, but increasing significantly around each bounce. When dissipation is introduced we show that in each new cycle the universe expands for longer and to a larger size. We find a similar effect for closed bouncing universes in Brans-Dicke theory, where G also varies monotonically in time from cycle to cycle. Similar behaviour occurs also in varying speed of light theories.

Inspired by recent claims for a varying fine structure constant, alpha, we investigate the effect of "promoting coupling constants to variables" upon various parameters of the standard model. We first consider a toy model: Proca's theory of the massive photon. We then explore the electroweak theory with one and two dilaton fields. We find that a varying alpha unavoidably implies varying W and Z masses. This follows from gauge invariance, and is to be contrasted with Proca' theory. For the two dilaton theory the Weinberg angle is also variable, but Fermi's constant and the tree level fermion masses remain constant unless the Higgs' potential becomes dynamical. We outline some cosmological implications.

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Abstract. These notes summarize a set of lectures on phenomenological quantum gravity which one of us delivered and the other attended with great diligence. They cover an assortment of topics on the border between theoretical quantum gravity and observational anomalies. Specifically, we review non-linear relativity in its relation to loop quantum gravity and high energy cosmic rays. Although we follow a pedagogic approach we include an open section on unsolved problems, presented as exercises for the student. We also review varying constant models: the Brans-Dicke theory, the Bekenstein varying α model, and several more radical ideas. We show how they make contact with strange high-redshift data, and perhaps other cosmological puzzles. We conclude with a few remaining observational puzzles which have failed to make contact with quantum gravity, but who knows... We would like to thank Mario Novello for organizing an excellent school in Mangaratiba, in direct competition with a very fine beach indeed. WHY QUANTUM GRAVITY?The subject of quantum gravity emerged as part of the unification program that led to electromagnetism and the electroweak model. We'd like to unify all forces of Nature. Forces other than gravity are certainly of a quantum nature. Thus we cannot hope to have a fully unified theory before quantizing gravity.To come clean about it right from the start, we should stress that there is no compelling experimental reason for quantizing gravity. For all we know, gravity could stand alone with respect to all other forces, and simply be exactly classical in all regimes. There is no evidence at all that the gravitational field ever becomes quantum 1 . Yet this hasn't deterred a large number of physicists from devoting lifetimes to this pursuit.Assaults on the problem currently follow two main trends: string/M theory [1, 2] and loop quantum gravity [3,4]. Both have merits and deficiencies, commented extensively elsewhere. As a poor third we mention Regge-calculus (and lattice techniques), non-commutative geometry, and several other methods none of which has fared better or worse than the two main strands.This course is not about those theories. Rather it's about the question: Where might experiment fit into these theoretical efforts of quantizing gravity? A middle ground has recently emerged -phenomenological quantum gravity. The requirements are simple: a phenomenological formalism must provide a believable approximation limit for more sophisticated approaches; it must also make clear contact with experimental anomalies that don't fit into our current understanding of the world. The following argument illustrates what we mean by this.When physicists find themselves at a loss they often turn to dimensional analysis. Following this simplistic philosophy we estimate the scales where quantum gravity effects may become relevant by building quantities with dimensions of energy, length and time fromh (the quantum), c (relativity) and G (gravity). These are called the Planck energy E P , the Planck length l P and t...