The Quantum Structure of Space and Time (293 pages)

(^260) The Quantum Structure of Space and Time
cosmological constant, we envisage that it relaxes by lo1'' in years or more.
The essential idea is that A is small but nearly constant (compared to a Hubble
time) today because it has had an exponentially long period (compared to a Hubble
time) to relax.
Of course, to implement this idea, the universe must be much older than 14

billion years. For such an old, expanding universe to have a non-negligible H and

matter density, it had better be that H and the matter density can be reset to large
values at times during the period that A slowly decreases. As one reflects further

upon the idea, it becomes apparent that a cyclic model of the type described by

Neil Turok [2-41 is ideally suited for this purpose - although it is interesting to note
that the model was not designed with this idea in mind.
First, the cyclic model provides more time. Each cycle lasts perhaps a trillion
years, but there is no known limit to how many cycles there may have been in the
past. So, a universe that is years or longer is quite feasible. Second, the
cyclic model provides a mechanism, the periodic bounces between branes, for reg-

ularly replenishing the universe with matter and radiation at regular intervals and,

consequently, regularly restoring H to a large value. Third, the cyclic model does

not include a period of high energy inflation, removing the key roadblock discussed
in the introduction. Finally, the cyclic model includes matter fields that live on the
branes and couple to the brane metric. According to the model, the branes expand
from cycle to cycle; the periodic crunches occur because of a contraction along the
extra dimension. Hence, fields on the branes are redshifted from cycle to cycle but
are not blue shifted during the periods of contraction (of the extra dimension); this
turns out to be useful for maintaining the slow relaxation process for reasons that
are explained in Ref. [l].
As for the slow relaxation mechanism, there are various possibilities. For sim-

plicity, I focus here on a concrete example first introduced twenty years ago by L.

Abbott [5], but in the wrong context. (Another mechanism with similar properties

was introduced by J. Brown and C. Teitelboim a few years later [6].) Abbott pro-

posed relaxing the cosmological constant by adding an axion field $ with a tilted

'washboard' potential

V($) = M4 cos- $6 + -4,

f 27rf

where M N 1 eV, f - 10l6 GeV, and N .1 meV are sample values that serve the

purpose. The gauge interaction provides a natural explanation for the small value

of M, analogous to the explanation for the QCD scale, AQCD. AQ~D - 100 MeV

is generated dynamically and can be expressed in terms of the Planck mass mp as

AQCD - mpexp(-27r/aQco), where ~QCD = 0.13 is not so different from unity.

Here we imagine that $ lives on the hidden brane and is coupled to hidden gauge

fields. A modest difference in the the hidden sector coupling constant, ahidden -

0.09, suffices to obtain the value of M desired for our model. The tilt come from

an interaction that softly breaks the periodic shift-symmetry of the axion. The soft