2. Inflation in a low density universe

We do need an adjustment from the simplest case - an Einstein-de Sitter cosmology - to account for the measurements of the mean mass density. In the two models that lead to Eqs. (40) and (41) the enormous expansion factor during inflation suppresses the curvature of space sections, making _K0 negligibly small. If = 0, this fits the Einstein-de Sitter model (Eq. [35]), which in the absence of data clearly is the elegant choice. But the high mass density in this model was already seriously challenged by the data available in 1983, on the low streaming flow of the nearby galaxies toward the nearest known large mass concentration, in the Virgo cluster of galaxies, and the small relative velocities of galaxies outside the rich clusters of galaxies. ⁽³³⁾ A striking and long familiar example of the latter is that the galaxies immediately outside the Local Group of galaxies, at distances of a few megaparsecs, are moving away from us in a good approximation to Hubble's homogeneous flow, despite the very clumpy distribution of galaxies on this scale. ⁽³⁴⁾ The options (within general relativity) are that the mass density is low, so its clumpy distribution has little gravitational effect, or the mass density is high and the mass is more smoothly distributed than the galaxies. We comment on the first option here, and the second in connection with the cold dark matter model for structure formation in Sec. III.D.

Under the first option we have two choices: introduce a cosmological constant, or space curvature, or maybe even both. In the conventional inflation picture space curvature is unacceptable, but there is another line of thought that leads to a universe with open space sections. Gott's (1982) scenario commences with a large energy density in an inflaton at the top of its potential. This behaves as Einstein's cosmological constant and produces a near de Sitter universe expanding as a exp(Ht), with sufficient inflation to allow for a microphysical explanation of the large-scale homogeneity of the observed universe. As the inflaton gradually rolls down the potential it reaches a point where there is a small bump in the potential. The inflaton tunnels through this bump by nucleating a bubble. Symmetry forces the interior of the bubble to have open spatial sections (Coleman and De Luccia, 1980), and the continuing presence of a non-zero V() inside the bubble acts like , resulting in an open inflating universe. The potential is supposed to steepen, bringing the second limited epoch of inflation to an end before space curvature has been completely redshifted away. The region inside the open bubble at the end of inflation is a radiation-dominated Friedmann-Lemaître open model, with 0 < _K0 < 1 (Eq. [16]). This can fit the dynamical evidence for low _M0 with = 0. ⁽³⁵⁾

The decision on which scenario, spatially-flat or open, is elegant, if either, depends ultimately on which Nature has chosen, if either. ⁽³⁶⁾ But it is natural to make judgments in advance of the evidence. Since the early 1980s there have been occasional explorations of the open case, but the community generally has favored the flat case, _K0 = 0, without or, more recently, with a cosmological constant. The earlier preference for _K0 = 0 and ₀ = 0 led to considerable interest in the picture of biased galaxy formation in the cold dark matter model, as we now describe.

³³ This is discussed in Davis and Peebles (1983a, 1983b) and Peebles (1986). Relative velocities of galaxies in rich clusters are large, but the masses in clusters are known to add up to a modest mean mass density. Thus most of the Einstein-de Sitter mass would have to be outside the dense parts of the clusters, where the relative velocities are small. Back.

³⁴ The situation a half century ago is illustrated by the compilation of galaxy redshifts in Humason, Mayall, and Sandage (1956). In this sample of 806 galaxies, 14 have negative redshifts (after correction for the rotation of the Milky Way galaxy and for the motion of the Milky Way toward the other large galaxy in the Local Group, the Andromeda Nebula), indicating motion toward us. Nine are members of the Local Group, at distances 1 Mpc. Four are in the direction of the Virgo cluster, at redshift ~ 1200 km s^-1 and distance ~ 20 Mpc. Subsequent measurements indicate two of these four really have negative redshifts, and plausibly are members of the Virgo cluster on the tail of the distribution of peculiar velocities of the cluster members. (Astronomers use the term peculiar velocity to denote the deviation from the uniform Hubble expansion velocity.) The last of the 14, NGC 3077, is in the M 81 group of galaxies at 3 Mpc distance. It is now known to have a small positive redshift. Back.

³⁵ Gott's scenario is resurrected by Ratra and Peebles (1994, 1995). See Bucher and Turok (1995), Yamamoto, Sasaki, and Tanaka (1995), and Gott (1997), for further discussions of this model. In this case spatial curvature provides a second cosmologically-relevant length scale (in addition to that set by the Hubble radius H^-1), so there is no natural preference for a power law power spectrum (Ratra, 1994; Ratra and Peebles, 1995). Back.

³⁶ At present, high energy physics considerations do not provide a compelling specific inflation model, but there are strong indications that inflation happens in a broad range of models, so it might not be unreasonable to think that future advances in high energy physics could give us a compelling and observationally successful model of inflation, that will determine whether it is flat or open. Back.