Is Space Infinite?
There is a good amount of evidence for the hypothesis that the space is infinite. For example, empirical measurements of the global curvature of spacetime suggest that spacetime is globally flat, which entails that the universe is spatially infinite. (See for example Bahcall et. al. 1999 for details.) Also, the basic theory of inflationary cosmology is now widely accepted, and inflationary cosmology strongly suggests that the universe is spatially infinite. (See for example Guth 2000, p. 571 or Garriga and Vilenkin 2001 for details.) But how compelling is this evidence? To what extent is this evidence compatible with the hypothesis that the spacetime is almost flat? (In an almost flat universe, if one kept going in one direction, one could eventually come back to one’s spatial starting point, but the path travelled would be very long.)
Assuming that space is infinite, one could argue that what exists in distant, unobserved regions of space is not substantially different from what exists in this region of space, and hence one could infer that stars and planets exist in those regions of space as well. With an infinite number of planets, as long as the range of initial conditions varies sufficiently across the different planets, one would fully expect life to exist in an infinite number of locations. (For discussion of this point, see for example Ellis and Brundrit 1979.) How many civilizations would be a lot like ours? How many civilizations would be exactly like ours?
Bahcall, N., J. Ostriker, S. Perlmutter, and P. Steinhardt 1999. The cosmic triangle: Revealing the state of the universe, Science 284: 1481-1488.
Ellis, G. F. R., and Brundrit, G. B. 1979. Life in the infinite universe. Quarterly Journal of the Royal Astronomical Society 20: 37-41.
Garriga, J. and Vilenkin, A. 2001. Many worlds in one. Physical Review D 64, 043511.
Guth, A. 2000. Inflation and eternal inflation. Physics Reports 333-334: 555-74.