"Cosmology", S. Weinberg, open letter + other errata
page 6(?) - spherical vs hyperspherical
All of "spherical", "elliptical", and "hyperspherical" normally have identical meanings in reference to FLRW cosmological models - these refer to a positively curved covering space of the spatial 3-manifold. The word "hyperpsherical" does not mean negative curvature!
"hyperbolic" is the correct word for a negatively curved covering space of the spatial 3-manifold
page 9 - open letter
"There is no sign of [the same patterns of the distribution of matter and radiation in opposite directions] in the observed distribution of galaxies or cosmic microwave background fluctuations, so any periodicity lengths such as |L_i| must be larger than about 10^10 light years."
It is correct that based on cosmic microwave background constraints (WMAP), it is clear that the in-diameter of the Universe (see e.g. Fig. 10 of Luminet & Roukema 1999 for various definitions of the comoving size of the Universe) is greater than 2 h^-1 Gpc.
However, on a scale of about 12-15 h^-1 Gpc, a large number of papers have been recently published which suggest that, in particular, a Poincare dodecahedral space (for K=+1) or an equi-length 3-torus model (for K=0) better fit the WMAP data than an infinite K=0 model. There is not yet any clear consensus on what the data tell us - some papers argue that the infinite K=0 model cannot be significantly rejected based on the WMAP data. Please see the reference list below for the main papers in this series.
"[Most 3-manifolds] ... seem ill-motivated. In imposing conditions of periodicity we give up the rotation (though not translation) symmetry that led to the Robertson-Walker metric in the first place, so there seems little reason to impose these periodicity conditions while limiting the local spacetime geometry to that described by the Robertson-Walker metric."
This is a little unclear. If by "symmetry" you mean the existence of isometries in the covering space, then there are no global translation symmetries at all for K=-1 or K=+1. Translations only exist for K=0. (Clifford translations exist for some K=+1 spaces, but that's a separate issue.) If you want to retain translational isometries in 3-space, then only K=0 is possible, and your words would seem to imply that K=-1 and K=+1 simply connected spaces are "ill-motivated".
More importantly, the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric only requires local homogeneity and isotropy, not global homogeneity and isotropy. The FLRW metric is intrinsically local, it's about the limit of what happens towards a point. To write this in terms of practical observational cosmology statistics, consider a more realistic model of the Universe, i.e. with a perturbed FLRW metric rather than a perfectly homogeneous one. In this model, "local homogeneity and isotropy" means that various n-point auto-correlation functions of structure tracers within a "neighbourhood" of a Gpc or so should give identical results to within observational uncertainties. This is what is observed, but it does not constrain global homogeneity and isotropy. So it is unclear why a multiply connected 3-manifold with an FLRW metric is in any way "limited" or "ill-motivated".
We are only aware of one paper so far showing that global topology could have an effect (albeit small) on the FLRW metric through the addition of an additional effective acceleration effect (Roukema et al. 2007).
Luminet, J.-P., Roukema, B. F., 1999, Topology of the Universe: Theory and Observation, NATO ASIC Proc. 541: Theoretical and Observational Cosmology , 117, http://cdsads.u-strasbg.fr/abs/1999toc..conf..117L
WMAP: PDS and 3-torus models (and other spherical models) vs K=0 infinite model:
Aurich, R., Lustig, S., Steiner, F., 2006, The circles-in-the-sky signature for three spherical universes, Monthly Notices of the Royal Astronomical Society, 369, 240, http://cdsads.u-strasbg.fr/abs/2006MNRAS.369..240A
Caillerie, S., Lachièze-Rey, M., Luminet, J.-P., Lehoucq, R., Riazuelo, A., Weeks, J., 2007, A new analysis of the Poincaré dodecahedral space model, Astronomy and Astrophysics, 476, 691, http://cdsads.u-strasbg.fr/abs/2007A%26A...476..691C
Key, J. S., Cornish, N. J., Spergel, D. N., Starkman, G. D., 2007, Extending the WMAP bound on the size of the Universe, Physical Review D, 75, 084034, http://cdsads.u-strasbg.fr/abs/2007PhRvD..75h4034K
Luminet, J.-P., Weeks, J. R., Riazuelo, A., Lehoucq, R., Uzan, J.-P., 2003, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature, 425, 593, http://cdsads.u-strasbg.fr/abs/2003Natur.425..593L
Riazuelo, A., Weeks, J., Uzan, J.-P., Lehoucq, R., Luminet, J.-P., 2004, Cosmic microwave background anisotropies in multiconnected flat spaces, Physical Review D, 69, 103518, http://cdsads.u-strasbg.fr/abs/2004PhRvD..69j3518R
Roukema, B. F., Bulinski, Z., Szaniewska, A., Gaudin, N. E., 2008, The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps, Astronomy and Astrophysics, 486, 55, http://cdsads.u-strasbg.fr/abs/2008A%26A...486...55R
Roukema, B. F., Lew, B., Cechowska, M., Marecki, A., Bajtlik, S., 2004, A hint of Poincaré dodecahedral topology in the WMAP first year sky map, Astronomy and Astrophysics, 423, 821, http://cdsads.u-strasbg.fr/abs/2004A%26A...423..821R
effect of global topology on the metric:
Roukema, B. F., Bajtlik, S., Biesiada, M., Szaniewska, A., Jurkiewicz, H., 2007, A weak acceleration effect due to residual gravity in a multiply connected universe, Astronomy and Astrophysics, 463, 861, http://cdsads.u-strasbg.fr/abs/2007A%26A...463..861R