| © 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*() =
(*m*^{2} / 2)^{2}. During the de Sitter Hubble
time *H*^{-1}, where as usual *H*^{2} = (8
*G* / 3)*V*, the slow
rolling of down the potential will
reduce it by

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

These will be equal for _{*} = *m*_{Pl}^{3/2} /
2*m*^{1/2},
*V*(_{*}) = (*m*
/ 8*m*_{Pl})*m*_{Pl}^{4}. If
_{*}, *positive quantum
fluctuations dominate* the
evolution: after *t* ~
*H*^{-1}, an initial region
becomes ~ *e*^{3} 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*_{Pl}^{2} /
*m* and *V* -> *m*_{Pl}^{4} -
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*_{µ}
*V*^{µ} *V*^{} 0 for all timelike
vectors *V*^{µ},
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.