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Speaker_	Lasenby, Anthony
Talk_	Conformal Geometric Algebra and Cosmology
Abstract_	Conformal geometric algebra is an extension of Geometric Algebra (GA) in which positions in space or spacetime are represented by vectors in a higher-dimensional space, either one or two dimensions larger than the base space. This construction allows conformal transformations, i.e. rotations, translations, dilations and inversions, to be carried out by pure rotations or reflections in the higher space. In particular both rotations and translations can be carried out with `rotors', which are a particularly useful feature of GA. In addition, homogeneous multivectors within the CGA can be used to represent geometrical entities, such as points, lines, planes, spheres, null cones and hyperboloids, and intersections of these can be taken via products in the algebra. Using these techniques, a novel representation of de Sitter space is found, which appears well-adapted for geometrical calculation. More general cosmological models, which evolve from a big-bang to a final asymptotic de Sitter state can also be treated in this approach, but only if a certain physical `eigenvalue' type condition is met. This requirement leads to a novel interplay between conditions at future infinity, and those during the first moments of coming out of the big bang. Applied to the scalar field in the inflationary epoch, they give a link between the number of e-folds of inflation, and the present value of the cosmological constant. In addition, the present universe is predicted to be not exactly flat, but with a small positive curvature. The current status of the observations relevant to this proposal will also be reviewed. As a further application, a 5d conformal geometric algebra approach is taken to the description of Bianchi IX universes containing a cosmological constant. The same requirement for satisfying the `eigenvalue condition' at future infinity as in the previous case, leads now to the selection of a particular model, that goes through a high energy phase corresponding to an effective `big-bang', but which avoids an actual singularity. Such a universe lasts from minus infinity to plus infinity in time, with an asymptotic de Sitter phase at each end, and the effective big bang in the middle.