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The inner product provides a conceptually and algorithmically simple method for calculating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The key to this is that just as a distance between two points ‘on’ the surface of the ordinary 2-sphere S[sup 2] is simply an arc-length (angle multiplied by radius) in ordinary Euclidean 3-space E[sup 3], the distance between two points ‘on’ a 3-sphere S[sup 3] (a 3-hyperboloid H[sup 3]) is simply an ‘arc-length’ in Euclidean 4-space E[sup 4] (Minkowski 4-space M[sup 4]), i.e. an ‘hyper-angle’ multiplied by the curvature radius of the 3-sphere (3-hyperboloid). [ABSTRACT FROM AUTHOR]

The significance to which the cosmic microwave background (CMB) observations by the satellite COBE can be used to refute a specific observationally based hypothesis for the global topology (3-manifold) of the Universe is investigated, by a new method of applying the principle of matched circle pairs. Moreover, it is shown that this can be done without assuming Gaussian distributions for the density perturbation spectrum. The Universe is assumed to correspond to a flat Friedmann–Lemaître model with a zero value of the cosmological constant. The 3-manifold is hypothesized to be a 2-torus in two directions, with a third axis larger than the horizon diameter. The positions and lengths of the axes are determined by the relative positions of the galaxy clusters Coma, RX J1347.5-1145 and CL 09104+4109, assumed to be multiple topological images of a single, physical cluster. If the following two assumptions are valid: (i) that the error estimates in the COBE DMR data are accurate estimates of the total random plus systematic error; and (ii) that the temperature fluctuations are dominated by the naıve Sachs–Wolfe effect; then the distribution of the temperature differences between multiply imaged pixels is significantly wider than the uncertainty in the differences, and the candidate is rejected at the 94 per cent level. This result is valid for either the ‘subtracted’ or ‘combined’ Analysed Science Data Sets, for either 10° or 20° smoothing, and is slightly strengthened if suspected contaminated regions from the galactic centre and the Ophiuchus and Orion complexes are removed. [ABSTRACT FROM AUTHOR]

The study of the kinematics of galaxies within clusters or groups has the limitation that only one of the three velocity components and only two of the three spatial components of a galaxy position in six-dimensional phase-space can normally be measured. However, if multiple topological images of a cluster exist, then the radial positions and sky plane mean velocities of galaxies in the cluster may also be measurable from photometry of the two cluster images. The vector arithmetic and principles of the analysis are presented. These are demonstrated by assuming the suggested topological identification of the clusters RX J1347.5-1145 and CL 09104+4109 to be correct and deducing the sky-plane relative velocity component along the axis common to both images of this would-be single cluster. Three out of four of the inferred transverse velocities are consistent with those expected in a rich cluster. A control sample of random ‘common’ sky-plane axes, independent of the topological hypothesis, implies that this is not surprising. This shows that while galaxy kinematics are deducible from knowledge of cosmological topology, it is not easy to use them to refute a specific candidate manifold. [ABSTRACT FROM AUTHOR]