How to Generate a Nearly Scale-invariant, Adiabatic, Gaussian
Spectrum of Density Perturbations: A Unified Approach
Brief Remarks at the CERCA-Kavli
Meeting
Inflation Panel
I want to focus now on a
pragmatic, empirical issue. Namely, what were the physical conditions when
the fluctuations were produced that seeded the CMB temperature anisotropy and
large scale structure. This is obviously
an important issue for cosmology since it helps us to sort out among
cosmological models. It is also an
important issue for fundamental physics since it determines what we might learn
about microphysics from measurements of the CMB and large scale structure.
So, I wish to address the
question: Is there a mechanism other than inflation that can create a
nearly scale-invariant, adiabatic, Gaussian spectrum of density perturbations
The answer, I would like to
propose, is maybe.
And, to explain my answer, I want to take a unified approach that
enables us to consider all possibilities using the same formalism.
As a starting point, I will
begin with the idea that we can produce
adiabatic and Gaussian density
perturbations beginning with quantum fluctuations, having the exit the horizon
in one phase – let’s call that the exit
phase – and re-enter at some later phase, namely the radiation or matter
dominated phase.
The properties of the
fluctuations can depend, in principle, on the physical properties of the exit
phase and of the re-entry process. For
the exit phase, the key determining feature will be the equation of state w, the ratio of the pressure to the energy
density, or, equivalently, 
. In terms of 
, the scale factor grows as 
and 
.
Now, if you think about it
logically, there are two conceivable ways quantum fluctuations can exit the horizon.
First, if the universe is expanding, the
wavelengths of the modes may stretch faster than the Hubble radius. That is, a grows faster than H-1. From the expression above, we see that this
requires 
The
second possibility occurs if the universe is contracting and the wavelengths
shrink more slowly than H-1. This requires 
Hence,
we see two distinct possibilities emerge.
Next, we want to impose the
condition that the spectrum be nearly scale-invariant. If we compute the spectrum of perturbations
of the Newtonian potential in Newtonian gauge, say, we find for the spectral
index to leading order in 
and its time
derivative,
,
where N is a time-like
variable that labels the number of e-folds before the end of the exit phase
when a given mode kN goes
outside the horizon and ns is
the effective spectral index for modes near kN. We then observe that we can obtain a nearly
scale-invariant 
under two different
conditions: if the universe is expanding and 
or if the universe is
contracting and 
Furthermore, the
expressions in the two limits,
are “dual” in the sense that one transforms into the
other under the transformation 
The duality is important for
two reasons. First, it should help to convince those who have never thought
about the contracting solution that the situation is similar to the
inflationary case they already know. So,
even though the ekpyrotic picture, say, is described in terms of colliding
orbifold planes, the exit phase is contracting and easing to analyze using the
duality. Second, the duality shows that
measurements of the scalar spectral index cannot be used to distinguish the
expanding and contracting cases since there is a precise transformation that
relates one case to the other.
Perhaps it is worth noting
that this duality applies to more than the extreme limits where 
or 
As
recently shown by Boyle et al, the duality exists for general e and relates both the
growing and decaying mode of the perturbations.
To complete our analysis, we
have one other step to consider: namely, how the mode re-enters the
horizon. In the case if inflation, all
that must be done is to increase 
above unity, slowing the down the expansion so that the
Hubble horizon increases faster than the scale factor. This is achieved by having the inflaton
energy decay into matter and radiation. And we know, from the careful
calculations done 20 years ago, that the perturbations re-enter the horizon
with the same amplitude and tilt as when they left.
For the contracting case, we
have to bounce to end the contraction and have the modes re-enter in the
subsequent expansion phase. It is
unclear a priori if
a bounce is possible and, even if it is, whether the perturbations produced in
the contracting phase are affected by the bounce. That is the reason why my answer to the
original question was maybe.
The ekpyrotic and cyclic
proposals have re-ignited interest in bounces and a number of groups have tried
to analyze the propagation of perturbations through a bounce. In those models, the bounce corresponds to a
collision of nearly vacuous, flat, parallel orbifold planes (made homogenous
and flat during a preceding period of accelerated expansion). This means that the bounce occurs under
conditions where the temperature and density are negligible and the geometry is
nearly trivial. Classically, the bounce
seems plausible. Furthermore, although
certain string amplitudes become large, it is plausible that their only effect
is on small scales where, for example, they may lead to the formation of small
black holes. These would play a role in
reheating, but
not prevent the bounce.
Numerous authors have tried
to evaluate the affect on the perturbation spectrum. In most cases, this has been
done by studying the effective 4d theory.
Different authors have reached different conclusions about whether the
perturbations produced during the contracting phase propagate through the
bounce into the expanding phase. Closer
examination shows this because the authors are studying different kinds of
bounces and matching conditions at the bounce.
This is because the bounce is inherently singular and there are
insufficient constraints to uniquely specify the matching conditions.
However, a recent analysis by
Tolley et al (hep-th/0306109) takes the important step of reformulating the
problem in terms of the 5d collision between orbifold planes. Here the planes
provide a continuity because they exist before, during
and after the collision. There appears
to be a unique unitary description for describing the bounce. The construction is done in linear
approximation, but there is a natural non-linear extension. A similar construction is supported by recent
work by string theorists Berkooz et al (hep-th/0212215),
Cornalba and Costa (hep-th/03012137) and Craps and Ovrut (hep-th/0308057). The net result is that the
perturbations produced in the contracting phase propagate through the bounce
and become density perturbation in the expansion phase. The spectral tilt
remains the same as when the modes left the horizon in the contracting phase.
However, the amplitude bears the imprint of the bounce. Its magnitude depends
not only conditions when the modes exited the horizon, but also on the detailed
stringy and extra-dimensional physics at the bounce itself: the Z2
orbifold structure, the warp factor, and the relative velocity of the orbifold
planes at collision.
This analysis deserves
further scrutiny -- and I hope some of
you will so scrutinize -- because it not
only shows that there may be an alternative mechanism for making density
fluctuations, but, in particular, that the amplitude of the spectrum in the
alternative case may provide direct information about the bounce. In the case of string theory, this means
information about extra dimensions and string scale physics. This is not only
important for cosmology, but also for fundamental physics. We are seeking desperately ways of
empirically testing string theory and many have thought that cosmology may be
the best hope. In inflationary cosmology, inflation wipes all information about
physics at the big bang. In the
contracting picture of density perturbations, we have already observed stringy
physics in the amplitude of the microwave background fluctuations.