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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(phi) = (m2 / 2)phi2. During the de Sitter Hubble time H-1, where as usual H2 = (8 pi G / 3)V, the slow rolling of phi down the potential will reduce it by

Equation 1.7 (1.7)

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

Equation 1.8 (1.8)

These will be equal for phi* = mPl3/2 / 2m1/2, V(phi*) = (m / 8mPl)mPl4. If phi gtapprox phi*, positive quantum fluctuations dominate the evolution: after Deltat ~ H-1, an initial region becomes ~ e3 regions of size ~ H-1, in half of which phi increases to phi + deltaphi. Since H propto phi, this drives inflation faster in these regions. Various mechanisms probably cut this off as phi -> mPl2 / m and V -> mPl4 - for further discussion and references, see Linde (1995). Thus, although phi at any given point is likely eventually to roll down the potential and end inflation, in most of the volume of the metauniverse phi > phi* 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'' Tµnu Vµ Vnu geq 0 for all timelike vectors Vµ, 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.

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