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

the impossible proof 201Albrecht and Sorbo 2004; Linde 2007) seem to indicate that Boltzmann’s brain eventswould be far more probable than the quantum appearance of a life-supporting universe,suggesting that right now you are more likely to be a Boltzmann’s brain than what youthink you are. . . .Most cosmologists see these nonsenses as pathological symptoms of some flawssomewhere in the model: these situations are just too antiaesthetic to be taken seriously.Even the infinite repetition paradox, arising for the flat universe model preferred bycurrent data, may be indicative of one such situation. It is worthwhile, therefore, tolook carefully at various hidden assumptions that we make when going from theobservational results that 0 is very close to unity to the conclusion that we actuallylive in an infinite, flat universe.9.4 The Impossible ProofA fundamental problem with the k = 0 universe is that it is unprovable. In fact, assuminga global FLRW metric, one would require an infinitely precise measurement of 0 = 1 (or equivalent quantity) to demonstrate the flatness of space: any uncertaintyaround the (true) critical value can accommodate an infinity of solutions with positiveor negative curvatures. Every physicist knows that no experiment, however precise,can give results without a finite error bar.There are deeper uncertainties built in by nature itself that make a spatially flatuniverse unprovable. Generally, these are related to limitations in the observabilityof the universe by us. A remarkable example applies to CMB data. Even supposing(unrealistically!) an infinite precision in the measurements of the CMB fluctuations,the power spectrum would still be limited in accuracy by a “cosmic variance” becauseof the finite statistics of CMB samples that we can observe from a single location in theuniverse. This ineliminable uncertainty becomes more important when we probe largeangular scales (∝ √ 2/(2l + 1)), where only a few independent sky regions (2l + 1)may be compared to each other to yield the power spectrum coefficients. Currentdata at low l’s are already limited by cosmic variance (see Fig. 9.1). To overcomethis limitation, it would be necessary to gather observations of the CMB from severalobservers distributed at cosmological distances from each other – a possibility thatappears unthinkable. It is possible, therefore, that the answer to our big question on thefiniteness or infinity of the universe is hidden forever inside this kind of fundamentalcosmic uncertainty.In principle, this limitation might be partly overcome with extremely precise measurementsof the Sunyaev-Zel’dovich (SZ) effect 18 on large samples of clusters ofgalaxies at high redshift (Kamionkowski and Loeb 1997). The basic idea is that SZpolarized scattering on distant clusters is sensitive to the CMB field as seen by thecluster; therefore, in principle it can give information on the properties of the surface18 The Sunyaev-Zel’dovich effect is inverse Compton scattering of CMB photons off hot electrons in the gas ofclusters of galaxies. Scattered photons are boosted in energy and their spectrum is distorted so that, in thesolid angle subtended by a cluster, we observe a decrease of the CMB temperature at low frequencies and anincrement at high frequencies (for a review see, e.g., Rephaeli 1995).