Boudewijn F. Roukema

Topology of the universe: Theory and observation

“One could imagine that as a result of enormously extended astronomical experience, the entire universe consists of countless identical copies of our Milky Way, that the infinite space can be partitioned into cubes each containing an exactly identical copy of our Milky Way. Would we really cling on to the assumption of infinitely many identical repetitions of the same world?… We would be much happier with the view that these repetitions are illusory, that in reality space has peculiar connection properties so that if we leave any one cube through a side, then we immediately reenter it through the opposite side.” (Schwarzschild 1900 translation 1998)

COBE and global topology: an example of the application of the identified circles principle

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. n n n nMoreover, it is shown that this can be done without assuming Gaussian distributions for the density perturbation spectrum. n n n nThe Universe is assumed to correspond to a flat Friedmann–Lemaitre 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. n n n nIf 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 nai¨ve 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 94xa0per cent level. n n n nThis 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.

A Counterexample to claimed COBE constraints on compact toroidal universe models

It has been suggested that if the Universe satisfies a flat, multiply connected, perturbed Friedmann-Lemaitre model, then cosmic microwave background data from the COBE satellite implies that the minimum size of the injectivity diameter (shortest closed spatial geodesic) must be larger than about (2/5) of the horizon diameter. To show that this claim is misleading, a simple T2×R universe model of injectivity diameter a quarter of this size, i.e. a tenth of the horizon diameter, is shown to be consistent with COBE four-year observational maps of the cosmic microwave background. This is done using the identified circles principle.

Observational Approaches to the Topology of the Universe

Many different and complementary strategies for translating the basic principle of multiple topological imaging into observational analysis are now available, both for three-dimensional and two-dimensional catalogues.

Transverse galaxy velocities from multiple topological images

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. n n n nThe 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. n n n nThree 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.

On the comoving distance as an arc-length in four dimensions

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 2 is simply an arc-length (angle multiplied by radius) in ordinary Euclidean 3-space ℰ3, the distance between two points ‘on’ a 3-sphere 3 (a 3-hyperboloid ℋ3) is simply an ‘arc-length’ in Euclidean 4-space ℰ4 (Minkowski 4-space ℳ4), i.e. an ‘hyper-angle’ multiplied by the curvature radius of the 3-sphere (3-hyperboloid).

Lifting cosmic degeneracy within a single quasar survey

In the almost Friedmann-Lemaitre model of the Universe, the density parameter,

How to avoid the ambiguity in applying the copernican principle for cosmic topology : Take the observational approach

Omega_{{rm m}}

On the Effective Spatial Separations in the Clustering of Faint Galaxies

, and the cosmological constant,

Topology of the Universe: background and recent observational approaches

Omega_Lambda