R. Holman
Physics Department,
Carnegie Mellon University, Pittsburgh PA, 15213
December 21, 1999
This is a summary of a talk given at the INT Non-Equilibrium Quantum Field Theory program with links to the relevant papers. We discuss why the field theory of inflation should be treated in the context of non-equilibrium quantum dynamics. We then show how this can be done in the context of the large N approximation as well as the Hartree approximation and how these results can be quite different than those found in the standard treatments of inflation.
The CMB data seems to point more and more to an inflationary source of the primordial density fluctuations that generated the temperature anisotropies in the CMB as well as large scale structure. Given this, it becomes imperative that we understand the quantum dynamics of inflation. What I would like to argue here is that the standard paradigm of treating the inflaton ``quasi''-classically is, in many interesting circumstances, not the correct way to approach this problem. In particular, when the inflaton evolves in a potential that admits spontaneous symmetry breaking there can arise instabilities due to the existence of a spinodal region, i.e. one where the second derivative of the potential is negative. These instabilities will give rise to non-perturbative growth of quantum fluctuations, driven by modes that ``see'' an inverted harmonic oscillator for their potential. Taming this non-perturbative growth can be done via a resummation of perturbation theory, namely the large N approximation or the Hartree truncation in the case of a single field. In both of these situations, we will see new behavior that went undetected in the standard approach and that could have observational consequences, such as changes in the spectral index.
The work reported here is explained in more detail in the papers: hep-ph/9709232 and hep-ph/9812476
Weinberg and Wu have done a beautiful analysis of the physics of the
spinodal region where 
. The
problems associated with this region first arise when we examine the
effective potential of the theory. The first order quantum correction is
given by
>From eq.(1) we see that modes with 
will contribute to an imaginary part for the effective potential, when 
is in the spinodal
region.
What Weinberg and Wu found is that this imaginary part is related to a decay rate of a state localized near the top of the potential hill at the origin. The wave functional associated with the spinodal modes see an inverted harmonic oscillator and they then spread. This behavior comes about essentially because the act of localization of the wave functional to the spinodal region does not commute with the Hamiltonian action; thus, we do not have a stationary state.
The lesson we should learn from this is that the use of static quantities, such as the effective potential, to describe the physics of the spinodal region is fraught with difficulties. Spinodal physics is inherently dynamical and must be treated as such.
The way to do this is to make use of the Schwinger-Keldysh formalism based on the closed time path integral; this corresponds to evolving the density matrix in time from some initial state.
The spinodally unstable modes will drive the growth of quantum fluctuations
in the field, as represented by the two point function for example, until
they can sample the minima of the potential. If we consider the standard 
theory with spontaneous symmetry breaking and write 
then the minima are
located at 
and it is
clear that, for weak coupling, this is non-perturbatively far from the
origin. The solution to this problems is some sort of resummation of the
perturbative expansion. There are two ways in which this can be done:
We now want to apply the combination of the Schwinger-Keldysh closed time path (CTP) method and the approximation/truncations described above. Both the large N and the Hartree approach have the following steps in common:
via the prescription
is the density matrix representing the state of
the field; we take it to be space-translationally invariant, in keeping with
the assumed FRW form of the metric.
and use the various approximations
above to turn the Hamiltonian H into that of a time dependent harmonic
oscillator. The time dependence comes in not only because the Fourier
coefficients 
are time dependent, but also
because of the time dependence of the scale factor 
.
within the relevant approximation scheme. This will
involve the zero mode 
, the fluctuations as well as the scale factor .
where the
expectation value is again taken in the state .
Putting all this together gives the following set of equations for an
arbitrary potential 
for a single field within the
Hartree truncation. For the details of the large N version, see the papers
refered to above.
where the higher derivative terms are needed for renormalization reasons.
Here we have chosen initial conditions corresponding to the adiabatic vacuum
state in conformal time, which is necessary for a variety of technical
reasons. We have also chosen the field to be in a thermal state initial; it
is most likely that this is not the correct initial state, but we have
chosen this since it is one of the few initial states for which we can solve
the system of equations.
Following the above prescription for the large N case leads to the
following results. Due to the O(N) symmetry, there are Goldstone modes in
the theory and in fact, the physics is dominated by them since there are
N-1 of them, versus the one direction in which the symmetry is broken. The
asymptotic state that the zero mode finally comes to is fixed by a sum rule
which indicates that Goldstone's theorem holds in the broken state:
This is just the statement that the relevant modes obey a massless wave
equation asymptotically. We have rewritten the theory in terms of
renormalized, dimensionless quantities:
What is seen numerically is that when 
starts within the spinodal
region, 
starts small and grows to its asymptotic value. Once it
reaches this value, then executes small amplitude oscillations
about it. While is small, the Hubble parameter H is fixed by
the energy at the top of mexican hat potential (or more carefully, its large
N generalization) which then drives an inflationary period. As the
fluctuations find the minimum of the potential (which is also the spinodal
line where the curvature of the potential goes to zero; this will be
important later) the Hubble parameter drops, ending the inflationary period.
What is more interesting, however, is the late time behavior of various
quantities such as the Hubble parameter, as well as the equation of state.
At late times the equation of state is (after time averaging over fast
(frequency 
) oscillations) 
! This agrees with the
behavior of the Hubble parameter at late times; it behaves as if in a matter
dominated universe.
This is rather unexpected since in the large N limit, the late time
physics is dominated by massless Goldstone modes.These should have an
equation of state 
corresponding to a radiation dominated
universe. Why doesn't this show up in our numerical results?
The reason can be found by analyzing the quantum fluctuations more
carefully. We know that it's the spinodally unstable modes that drive the
non-perturbative growth of the quantum fluctuations. Furthermore, the longer
the comoving wavelength of these modes, the more their mode functions will
contribute to despite being disfavored by phase space.
Furthermore, when the physical size of a mode grows larger than the De
Sitter horizon 
, their mode functions factorize into a function of
wavenumber k and one of time. The time part satisfies the same equations
of motion as the k=0 mode. Putting all this together yields the result
that
>From this we can then show that
where 
denotes the long-wavelength (i.e. larger than
the current horizon size) part of the quantum fluctuations, obeys the
classical scalar field equation:
We can also use the full numerical evolution to backtrack the system and
find out what initial value for 
will give the same time
evolution. We find that 
with 
. Finally we can also show that the classical potential e
nergy for
sources the classical FRW evolution.
What all this really means is that the very long wavelength modes which contribute the most to the spinodal growth of the quantum fluctuations get lumped together with the original zero mode to give rise to an effective zero mode whose evolution gives all the standard classical results of inflation. The metric perturbations are sourced by the quantum fluctuations that have no spinodal instability at the time they re-enter the FRW horizon. We call this result Zero Mode Reassembly (ZMR).
ZMR explains why the non-perturbatively large fluctuations on superhorizon scales do not induce large temperature fluctuations in the CMB: as far as we can observe today, those fluctuations have been subsumed into the effective zero mode that drives inflation and are not available to source metric perturbations.
Due to the continous symmetry of the O(N) the ``line'' where the curvature
becomes zero is all the way down at the minima of the potential; this is
essentially a consequence of having Goldstone modes in the theory. In the
case of a single field, with no continous symmetry, this is not the
case. Let us consider the double well situation with potential:
Then the line of minima connects the minima at 
while the spinodal line connects the spinodal points at 
.
What we find is that there is still zero mode reassembly going on in this situation, but the long-wavelength fluctuations do not join up with the original zero mode to form an effective zero mode. This turns the model into one that looks like two field inflation. Furthermore, the generic situation is that there are two stages of inflation!
The way this happens is the following. The initial stages, where the zero mode is within the spinodal region, give rise to an inflationary period such as was found in the O(N) case discussed above. Once the zero mode reaches the spinodal line however, the quanta become effectively massless and this essentially flattens the potential at this point; this is a dynamical analog of the Maxwell construction of thermodynamics. Because of this flattening at the spinodal, and the fact that the potential is not zero at the spinodal, we have the second phase of inflation. This phase lasts until the zero mode crosses the spinodal line, at which the reassembled fluctuations field is driven to zero and the the zero mode reaches the minima of the potential.
This work is an attempt to understand the system of an inflationary phase transition in the context of non-equilibrium quantum field theory. The initial state is unphysical, described by a concave effective potential, the analog of the equilibrium free energy, with quanta corresponding to imaginary mass states. Such a system must decay into physical states, which we see from the exponential growth of long wavelength fluctuations. This decay continues until a non-perturbative state is reached for which the corresponding quanta are physical with a strictly real or zero mass.
There are two possible ways to reach such a state. The first is simply to provide a significant bias to the state such that the mean value of the field (i.e. the order parameter) reaches its true vacuum value where the field quanta are well defined.
The second way, appropriate for systems with small order parameter, is to allow the system to phase separate into domains for which the field has either positive or negative value. Rather than the order parameter moving along the potential energy diagram, the field drops down into the center of the diagram to the spinodal line. At the spinodal line, the field quanta are massless and physical, and this state of affairs may be relatively long lasting, ending only when one phase becomes so much more prevalent than the other that the system may relax into a definite vacuum state throughout the system.
We therefore have arrived at a consistent physical picture of inflationary phase transitions based upon mean field theory. Already, at this naive level, we have seen new phenomena which impact not only the evolution of the inflaton field but also have important implications for the interpretation of recent and soon to come observational data.
As a final note, we emphasize that the techniques used here are really only a first - or, rather, a second - approximation to a very complicated system of interaction between an unstable scalar field and gravity at very high energies. There are a number of possible avenues which might be taken to improve upon these results for the dynamics and, in particular, for the predictions of observational quantities.
One direction is to move beyond mean field theory. Gravitationally, this means doing something more sophisticated than semi-classical gravity. There has been recent work in this regard within the context of perturbation theory up to two loop order, and interest in this area has grown somewhat due to the possibility of new phenomena in the context of preheating. However, the non-perturbative dynamics of the scalar field studied here corresponds to non-perturbative departures of the gravitational dynamics from that of a purely classical background field so that techniques based upon perturbative expansions are of little help.
In terms of the scalar field dynamics, one possible avenue that has received attention is the 1/N expansion of the O(N) vector model, which includes contributions beyond mean field theory at next to leading order (this approximation is also promising as it might be consistently implemented for gravity as well). Another alternative approach is to use variational methods to compute the dynamics of the system, a technique which might also be combined with the 1/N expansion. However, significant hurdles remain before either of these techniques will be implimentable for interesting field theory problems.