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

hints of infinity in the primordial music 197relative contribution of different types of matter-energy components: M = ρ M /ρ C formatter, R = ρ R /ρ C for radiation, = ρ /ρ C for dark energy. 11 These parametersare not fixed by any known theory, and only experiments may be able to pin downtheir values. They are of great importance to our discussion of infinity, because theygovern the dynamics and shape of cosmic space. In particular, the total energy densityparameter 0 = (ρ M + ρ R + ρ )/ρ C is directly related to the adimensional constantk, which controls the global curvature of space: 0 = 1 corresponds to the familiar flat,Euclidean 12 space, 0 < 1 to a hyperbolic space with negative curvature, and 0 > 1to a positively curved spherical space. If we assume that the FLRW metric holds overthe entire spacetime and that the topology of space is simply connected, then the valueof the curvature constant determines finiteness (k =+1) or infinity (k = 0, k =−1)of spatial sections. Under these assumptions, measurements of the total energy densityparameter 0 provide a clear answer to our problem.It is remarkable that modern science has allowed us to speak of the finitenessand infinity of the universe in an elegant, self-consistent, and concise way and, evenmore amazingly, to point at observable parameters that may discriminate between thedifferent scenarios. In the past few years, observations of the CMB (Bennett et al. 2003)and of distant supernovae (Riess et al. 2005) and systematic studies of the large-scaledistribution of galaxies (for a review, see Schindler 2002) have yielded quantitativeestimates of the cosmic density parameters with unprecedented precision. Even moreprecise measurements are expected in the near future. We live in a golden age forcosmology!A most direct way to measure spatial curvature is to measure very large triangles:depending on how the sum of the internal angles compares to the Euclidean value, 180degrees, we can infer the curvature of the underlying space. In particular, a standardlength seen from a given distance will be subtended by different angles depending onthe space curvature. In a two-dimensional analogy, if a given standard rod is subtendedby an angle θ 0 on a flat surface (Euclidean space), it will be subtended by an angleθ > θ 0 on a spherical surface (positive curvature) and by an angle θ < θ 0 on ahyperbolic surface (negative curvature). If we have an a priori estimate of the length ofa source, then a precise measure of the angular diameter (i.e., a measure of θ) carriesinformation on the curvature of space. Of course, the larger the triangles, the moreprecise the measurement we can hope to achieve. Traditionally, a method to measurelarge triangles in the universe has been to exploit gigantic sources like radio galaxies,whose radio lobes extend at megaparsec distances from each other: the triangle is madeby the two lobes and our observing point. Unfortunately, this yields a poor measure ofthe curvature of space because the distance between the two lobes is too small and notsufficiently known.11 In the late 1990s two independent research groups (Perlmutter et al. 1999; Garnavich et al. 1998), usingobservations of Type Ia supernovae, concluded that, contrary to every expectation, the rate of cosmic expansionin recent cosmic epochs has undergone a new era of acceleration. This requires some unknown form of energycharacterized by negative pressure to act as an anti-gravity component. Calculations showed that in order toexplain the observed acceleration, the so-called dark energy must contribute as much as ∼2/3 of the totalenergy density of the universe.12 Hereafter “Euclidean space” refers to the three spatial dimensions of the pseudo-Euclidean Minkowski fourdimensionalspacetime.