The Quantum Structure of Space and Time (293 pages)

244 The Quantum Structure of Space and Time

to really get a handle on what was happening during inflation, which I don’t think
we have now, is to observe the effect of the tensor modes, the gravitational waves.
Many people have said this, it’s hardly an original observation. Fortunately, the
Europeans are going ahead with the Planck satellite which may be able to detect
the effect of tensor modes on the polarization of the microwave background because
they are not wasting their money on manned space flight the way America and
Russia are.
Now, I have talked about what are the necessary conditions for what we see:
adiabatic, scale invariant, Gaussian perturbations. What about sufficient? The
question here comes from the quantum corrections. We normally say that the
quantum corrections are small. Why do we say they are small? Because the measure

of smallness, the factor that you get every time you add a loop to a diagram, is

something like G H2, where G is Newton’s constant and H is the Hubble constant, at

the time the perturbation left the horizon. Experimentally, we know G H2 N
so that is why quantum corrections are small and you don’t have to worry about
them. But, is it really true that the quantum corrections only depend on what was
happening at the time the perturbations left the horizon? The calculations that have
generally been done have been purely classical, for instance Maldacena calculated
corrections, non Gaussian terms, that were corrections to the usual results, but that
corresponded to a tree graph in which you have 3 lines coming into a vertex: that
was not really a specifically quantum effect. When you include quantum effects,
you begin to worry because the Lagrangian, after all, contains terms with positive
powers of the Robertson-Walker scale factor a. For example, for a scalar field with

a potential, you get an a3 just from the square root of the determinant and even

without a potential, just from the (V4)2-term you get a factor of a. Now, there are

lots of complicated cancellations which deal with this and, in fact, you can show that
there are lots of theories in which the same result applies: the quantum corrections

depend only on what was happening at the time of horizon exit and therefore they

are small and therefore we don’t worry about them.

It is clear though that there are other theories where that is not true. In partic-

ular ... A kind of theory where it is true is a minimally coupled massless scalar field

which has zero vacuum expectation value (not the inflaton but an additional scalar

field with zero vev). If it does not have any potential, then the quantum effects

caused by loops of that particle, to any order, do not produce any effects that grow

with a as you go to late times in inflation. But, if you add a potential for the scalar,

V(4), then you get terms that do.

Last week I thought I was going to come here and show you a theory in which

you get positive powers of a so that all bets are off and that the corrections become

very large at late time. And just last week I was able to prove a theorem that, in

fact, in every theory that I am able to think of, the corrections, when they are there,

grow only like loga. So, I am afraid, I don’t have anything exciting to announce.

Although there are quantum effects which do not depend only on what is happening