Cosmology 261breaking scale, 5 1 meV, is comparable to but somewhat smaller than M; it
sets the scale of the steps in energy density along the washboard: VN - VN-1 = E.
The model is technically naturally in that the coefficients are not subject to large
quantum corrections.
Fig. 6.3 The washboard potential defined in Eq. 2.The total vacuum energy density isAtotal = Xother + V($) (2)
where Xother incorporates all other contributions from Planck, electroweak, QCD
and other non-axionic physics. The washboard potential (Fig. 6.3) has periodically
spaced minima at $ = 27~ f N + 40 where N is an integer and $0 is the value of q5 at
V = VO. The minima have vacuum density VN = VO + NE where VO is the vacuum
density of the minimum with the smallest non-negative Atotal. The potential also
has minima V-1 , V-2,... with negative vacuum density.
Suppose the universe begins at some minimum with a large positive potentialdensity VN. The universe begins to work its way down the potential by quantum
tunneling through the energy barriers. That is, growing bubbles of vacuum with
V = VN-1 form in a background with V = VN. The process continues from one
minimum to the next until V approaches VO and Atotal 5 c. Since the average
tunneling rate is r - M4e-B where B - M2 f /vN, the relaxation rate decreases
exponentially as the field tunnels downhill. Hence, the universe spends exponen-
tially more time at stages when the cosmological constant is small and positive. At
the last positive minimum Vo, the tunneling time is roughly years. Even-
tually, bubbles nucleate with V = V-1 in their interior, but these are anti-deSitter
minima that undergo gravitational collapse in one Hubble time (about 14 billion
years). The collapsed regions probably form black holes. But since most of the