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

hints of infinity in the primordial music 199In this case, however, we do not measure single triangles, as with radio galaxies;rather, we use a statistical measure as defined by the CMB anisotropy power spectrum.The angular scale θ peak of the main peak, or the corresponding multipole l peak ≈ π/θ peak ,is directly related to the total energy density parameter, l peak∼ = 220/√0 , so that CMBanisotropy data yield a remarkably direct evaluation of the spatial curvature (for a reviewsee Bersanelli, Maino, and Mennella 2002). As shown in Figure 9.1, the peak amplitudeof the anisotropy distribution occurs at θ peak ≈ 1 ◦ ,orl peak∼ = 220, corresponding to thecritical density 0∼ = 1 with only a few percent uncertainty.Cosmic background anisotropy data set particularly tight constraints on spatialcurvature when combined with observations at lower redshifts. In particular, supernovadata and statistical studies of matter distribution (the so-called baryonic acousticoscillations [BAOs] 14 ) help us break the degeneracy with the effects of the changingrate of expansion during cosmic history. Recent results from the five-year WMAPdata (Komatsu et al. 2009) combined with BAO results (Percival et al. 2007) andwith supernova data (Astier et al. 2006; Wood-Vasey et al. 2007) yield an estimate of0.9915 < 0 < 1.0175. This is indeed remarkable: even with measurements at ∼1 percentprecision, we still fail to detect a curvature of cosmic space. A very low curvatureis an expectation of inflation, as the initial exponential expansion would have stretchedthe original curvature to a very low level, just as a given portion of the surface ofa balloon flattens as the balloon is inflated. This may just mean that the observableuniverse is far too small compared to the overall curvature radius. However, someanalyses (Dunkley et al. 2005) based on generalized assumptions on the modes ofthe primordial perturbations indicate a slight preference for a closed (finite) universe.Although even the latest WMAP results do not completely dissipate this tendency,these hints are weak and highly tentative. In any case, given that we are so close to acritical universe, high-precision measurements such as those expected from Planck 15are crucial to evaluate whether a measurable deviation from flatness can be detected.We seem to remain persistently on the verge between a finite and an infinite world.This can be translated in lower limits to the curvature radius of the universe, which hasto be greater than 32 Gpc for a positive curvature and 46 Gpc for a negative curvature. 16It is as if Eratosthenes in his famous measurement of the radius of the earth in 250 bcwas not able to measure any curvature: then his conclusion would have been that theearth might be flat and infinite, or that its radius is greater than a given size compatiblewith the accuracy of his observation.14 Baryons generate an oscillatory signature that is visible not only in the CMB power spectrum (Fig. 2) but alsoin the power spectrum of the large-scale structure of the universe at moderate redshifts. These BAOs havebeen seen recently in large galaxy redshift surveys (e.g., Eisenstein et al. 2005; Cole et al. 2005). Just as theoscillations producing CMB anisotropies, BAOs in late-time structures provide a constant comoving lengthscale (a standard ruler) that defines another useful set of “cosmic triangles” to measure global curvature ofspace.15 The ESA Planck satellite was launched on May 14, 2009, and it is designed to obtain a full-sky map ofthe CMB with an unprecedented combination of sensitivity, angular resolution, and spectral coverage. Seehttp://planck.esa.int.16 These curvature radii are derived for a Hubble constant of 72 km s −1 Mpc −1 , as measured by the HST KeyProject (see Freedman et al. 2001). Note also that in a negative curvature space with simply connected topology,space is infinite for any value of the radius of curvature.