The Quantum Structure of Space and Time (293 pages)

264 The Quantum Structure of Space and Time

6.5 Discussion

S. Weinberg This is a trivial remark unless, of course, it is wrong. As I understand

it, there is no particular reason in Steinhardt’s scenario why the vacuum energy
density would be related to the matter density that we see now in the present
Universe?
P. Steinhardt No. In the model that I presented, there is nothing that links the
total amount of dark matter to the cosmological constant directly. In fact,
the ratio decreases from cycle to cycle as the cosmological constant relaxes.
However, the universe spends most time when the cosmological constant is
small, as we observe it today.

S. Weinberg The anthropic principle (that is the principle that dares not speak

its name) does require that the vacuum energy density be of the same order of

magnitude, but somewhat larger than the mass energy density that we see now,

and in fact even a little larger than we actually are seeing. But yours does not?

The amount of dark matter produced at the bounce is

presumably the same from cycle to cycle, but the cosmological constant is slowly

relaxing away. The ratio of dark matter to dark energy is, therefore, changing

with time and we do not have a precise prediction for what the value is for this

particular cycle.

A. Linde The equation for the potential which you use for your model contains

a cosine term and a linear term. If you remove the cosine term and leave just

the linear term, this would be exactly the potential of the model which I used

in 1986 to suggest an anthropic solution for the cosmological constant problem,

and pretty recently we repeated the discussion of this model in some details

with Vilenkin and Garriga, and this model is also extremely similar to the

model suggested by Banks at approximately the same time.

G. ’t Hooft I wonder whether you are not introducing another kind of unnatu-

ralness in such a system, which is the very small value for the mass term with

respect to the kinetic term for such a dilaton-like field. After all you have a

variation of the cosmological constant over cosmological scales, so there must

be a very small effective mass in that. Is that not an unnatural feature of such

a model?

P. Steinhardt It was Abbott’s idea that the field be an axion with a mass term
generated by the same kinds of instanton effects as the QCD axion. So, the
coefficient of the cosine term, for example, would be something of order the

Planck scale times exp(-2ra), where a is the strength of the coupling at the

Planck scale. For QCD, a is of order 0.1. Here we imagine that, for an axion

on the other brane coupled to hidden gauge fields, the coupling might be a bit

different. With a value of 0.08, just 20 per cent less, one gets values that work

just fine for our scenario. And, because of the axion’s special symmetries, the

couplings are protected from quantum corrections. So, as Abbott suggested,

P. Steinhardt Right.