Zero Mode Reassembly, Spinodal Inflation and All That: Applications of Non-Equilibrium QFT to Inflationary Cosmology

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

Spinodal Instabilities

Spinodal Physics is Dynamical

Weinberg and Wu have done a beautiful analysis of the physics of the spinodal region where tex2html_wrap_inline399. 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 tex2html_wrap_inline401 will contribute to an imaginary part for the effective potential, when tex2html_wrap_inline403 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.

Spinodal Physics is Non-Perturbative

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 tex2html_wrap_inline405 theory with spontaneous symmetry breaking and write tex2
html_wrap_inline407 then the minima are located at tex2html_wrap_inline409 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:

Self-Consistent Inflation

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:

Putting all this together gives the following set of equations for an arbitrary potential tex2html_wrap_inline449 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.

Zero Mode Reassembly

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 tex2html_wrap_inline459 starts within the spinodal region, tex2html_wrap_inline461 starts small and grows to its asymptotic value. Once it reaches this value, tex2html_wrap_inline461 then executes small amplitude oscillations about it. While tex2html_wrap_inline461 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 tex2html_wrap_inline471) oscillations) tex2html_wrap_inline473! 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 tex2html_wrap_inline477 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 tex2html_wrap_inline461 despite being disfavored by phase space. Furthermore, when the physical size of a mode grows larger than the De Sitter horizon tex2html_wrap_inline481, 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 tex2html_wrap_inline487 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 tex2html_wrap_inline489 will give the same time evolution. We find that tex2html_wrap_inline491 with tex2html_wrap_inline493. Finally we can also show that the classical potential e nergy for tex2html_wrap_inline489 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.

Spinodal Inflation

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 tex2html_wrap_inline499 while the spinodal line connects the spinodal points at tex2html_wrap_inline501.

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.

Laurence Yaffe
Tue Dec 21 17:03:50 PST 1999