Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

202 infinity and the nostalgia of the starsof last scatter as seen by a source at cosmological distance from us. The polarizationof the CMB photons scattered by the cluster’s hot electron gas provides a measure ofthe CMB quadrupole moment as seen by the cluster. Therefore, CMB polarizationmeasurements toward several clusters would probe the anisotropy on a variety ofrealizations of primordial fluctuations, thus reducing the cosmic variance uncertainty.However, the feasibility of such observations to the required accuracy is questionable,certainly far from what is currently achievable. More fundamentally, this approachwould lead to a reduction of cosmic variance, not to its suppression. Even assumingthat we can reach the limits of just-formed clusters (z ∼ = 10), we would still explore alimited portion of space. Perhaps future methods, independent of the CMB or BAOs orsupernovae, will measure the curvature to very high precision. However, cosmic variancelimitations will necessarily surface as a consequence of the limited informationwe can obtain from a single location in the universe. We are bound within our cosmichorizon, set by the distance covered by light in the finite age of the universe.Let’s now assume that somehow, in spite of the fundamental limits just described,we know that space is exactly Euclidean, or it has a negative curvature (k = 0ork =−1). A common misconception in cosmology is to state that this directly leads toinfinite spatial sections. However, this is true only if two conditions are met. First, wemust assume that the FLRW metrics – a good approximation in our past light cone – ismaintained indefinitely beyond our cosmic horizon. This is far from obvious, and it is incompetition with other conjectural scenarios. Some inflationary models, for example,postulate a highly inhomogeneous superhorizon distribution with a variety of domains,possibly with different local metrics and sets of fundamental parameters. However, boththe multidomain inflation picture and the all-encompassing FLRW universe ultimatelyelude observational verification.In addition, spaces with null or negative curvature are infinite only if theyhaveasimply connected topology. In more complex topologies, such as a 3-torus, perfectlyEuclidean spaces have spatially finite solutions (Ellis 1971). This can happen in avariety of situations: in three dimensions there are eighteen locally homogeneousand isotropic Euclidean spaces; for negative curvatures the number of possibilities isinfinite. No known theory provides a prediction of the topology of the universe; thus,only observations may be able to decide. A finite dodecahedral Euclidean space wasproposed to explain the apparent lack of power at low multipoles in the CMB anisotropyspectrum (Luminet et al. 2003). 19 Given that the scale of perturbations cannot exceedthe size of the polyhedron itself, a cutoff in wavelengths of density fluctuations isexpected, which would explain the observed low power at large scales. Observationalopportunities to test cosmic topology include the observation of multiple images ofhigh-redshift objects, prediction of total density parameters associated with particularmodels (e.g., 0 = 1.013 for the Poincaré dodecahedral space), and matching featuresin the CMB sky. The latter has not been confirmed by WMAP and will be tested by thePlanck survey. Of course, observational verification may happen only if we are luckyand the fundamental domain has an appropriate size: if such a domain is much largerthan our Hubble volume, then the (whole) universe might be perfectly Euclidean (orhyperbolic) and yet finite, but we would never know.19 The low power at low multipoles (first point at low multipoles in Fig. 9.2) is confirmed also in the most recentWMAP data.