© CAMBRIDGE UNIVERSITY PRESS 1999

1.6.6 Inflation with ₀ < 1

Can inflation produce a region of negative curvature larger than our present horizon - for example, a region with ₀ < 1 and = 0? The old approach to this problem was to imagine that there might be just enough inflation to solve the horizon problem, but not quite enough to oversolve the flatness problem, e.g. N ~ 60 (Steinhardt 1990). This requires fine tuning, but the real problem with this approach is that the resulting region will not be smooth enough to agree with the small size of the quadrupole anisotropy Q measured by COBE. According to the Grischuk-Zel'dovich (1978) theorem (cf. Garcia-Bellido et al. 1995), ~ 1 fluctuations on a super-horizon scale L > H₀^-1 imply Q ~ (L H₀)^-2. COBE measured Q_rms < 2 x 10^-5, which implies in turn that the region containing our horizon must be homogeneous on a scale L 500 H₀^-1, i.e. N 70, |1 - ₀| 10^-4.

A new approach was discovered, based on the fact that a bubble created from de Sitter space by quantum tunneling tends to be spherical and homogeneous if the tunneling is sufficiently improbable. The interior of such bubbles are quite empty, i.e., they are a region of negative curvature with -> 0. That was why, in ``old inflation,'' the bubbles must collide to fill the universe with energy; and the fact that this does not happen (because the bubbles grow only at the speed of light while the space between them grows superluminally) was fatal for that approach to inflation (Guth & Weinberg 1983). ⁽³⁾ But now this defect is turned into a virtue by arranging to have a second burst of inflation inside the bubble, to drive the curvature back toward zero, i.e., ₀ -> 1. By tuning the amount of this second period of inflation, it is possible to produce any desired value of ₀ (Sasaki, Tanaka, & Yamamoto 1995; Bucher, Goldhaber, & Turok 1995; Yamamoto, Sasaki, & Tanaka 1995). The old problem of too much inhomogeneity beyond the horizon producing too large a value of the quadrupole anisotropy is presumably solved because the interior of the bubble produced in the first inflation is very homogeneous.

I personally regard this as an existence proof that inflationary models producing ₀ ~ 0.3 (say) can be constructed which are not obviously wrong. But I do not regard such contrived models as being as theoretically attractive as the simpler models in which the universe after inflation is predicted to be flat. (Somewhat simpler two-inflaton models giving ₀ < 1 have been constructed by Linde & Mezhlumian 1995.) Note also that if varying amounts of inflation are possible, much greater volume is occupied by the regions in which more inflation has occurred, i.e., where ₀ 1. But the significance of such arguments is uncertain, since no one knows whether volume is the appropriate measure to apply in calculating the probability of our horizon having any particular property.

The spectra of density fluctuations produced in inflationary models with ₀ < 1 tend to have a lot of power on very large scales. However, when such spectra are normalized to the COBE CMB anisotropy observations, the spherical harmonics with angular wavenumber 8 have the most weight statistically, and all such models have similar normalization (Liddle et al. 1996a).