© CAMBRIDGE UNIVERSITY PRESS 1999

1.6.4. Eternal Inflation

Vilenkin (1983) and Linde (1986, 1990) pointed out that if one extrapolates inflation backward to try to imagine what might have preceeded it, in many versions of inflation the answer is ``eternal inflation'': in most of the volume of the universe inflation is still happening, and our part of the expanding universe (a region encompassing far more than our entire cosmic horizon) arose from a tiny part of such a region. To see how eternal inflation works, consider the simple chaotic model with V() = (m² / 2)². During the de Sitter Hubble time H^-1, where as usual H² = (8 G / 3)V, the slow rolling of down the potential will reduce it by

(1.7)

Here m_Pl is the Planck mass (see Table 1.1). But there will also be quantum fluctuations that will change up or down by

(1.8)

These will be equal for _* = m_Pl^3/2 / 2m^1/2, V(_*) = (m / 8m_Pl)m_Pl⁴. If _*, positive quantum fluctuations dominate the evolution: after t ~ H^-1, an initial region becomes ~ e³ regions of size ~ H^-1, in half of which increases to + . Since H , this drives inflation faster in these regions. Various mechanisms probably cut this off as -> m_Pl² / m and V -> m_Pl⁴ - for further discussion and references, see Linde (1995). Thus, although at any given point is likely eventually to roll down the potential and end inflation, in most of the volume of the metauniverse > _* and inflation is proceeding at a very fast rate.

Eternal Inflation is eternal in the sense that, once started, it never ends. But it remains uncertain whether or not it could have begun an infinite length of time ago. Assuming the ``weak energy condition'' T<sub>µ</sub> V<sup>µ</sup> V 0 for all timelike vectors V<sup>µ</sup>, i.e. that any observer will measure a positive energy density, Borde & Vilenkin (1994) proved that a future-eternal inflationary model cannot be extended into the infinite past. However, Borde & Vilenkin (1997) have recently shown that the weak energy condition is quite likely to be violated in inflating spacetimes (except the open universe inflation models discussed below, Section 1.6.6), so a ``steady-state'' eternally inflating universe may be possible after all, with no beginning as well as no end.