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A nonequilibrium
renormalization group
approach to turbulent
reheating
A case study in nonlinear nonequilibrium
quantum field theory
Part I
J. Zanella, EC
During inflation, the dominant form of matter in
the Universe is a condensate (the inflaton) which
evolves “rolling down the slope” of its effective
potential
d 2
d
 3H
 V '    0
2
dt
dt
When the inflaton nears the bottom of the
potential well, it begins to oscillate and transfers
its energy to ordinary matter (then in its vacuum
state). We call this process reheating
Reheating proceeds through several stages (Felder
and Kofman, hep-ph/0606256):
a) Preheating (Berges, Borsanyi and Wetterich, PRL
93, 142002 (2004)).
b) Nonlinear inflaton fragmentation (Felder and
Kovman, op. cit.).
c) Turbulent thermalization (Micha and Tkachev,
PRD 70, 043538 (2004)).
Generally speaking, the early phases produce an
spectrum with high occupation numbers in a narrow
set of modes.
Turbulent thermalization concerns the spread of the
spectrum over the full momentum space and the
final achievement of a thermal shape.
Felder and Tkachev, hep-ph/0011159
At early times occupation numbers are high and the
process may be described in terms of classical wave
turbulence
As the spectrum spreads occupation numbers fall
and the classical approximation breaks down.
The challenge for us is to provide a quantum
description of turbulent reheating.
Concretely, we shall discuss quantum turbulent
thermalization in a nonlinear scalar field theory in
3+1 flat space-time.
(Since reheating is a relatively fast process in terms
of the Hubble time, this is not such a bad
approximation.)
The basic idea is the same as in Kolmogorov Heisenberg turbulence theory: a mode of the field
with wave number k lives in the environment
provided by all modes with wave number k' > k
The dynamics of the relevant mode is obtained by
tracing over the environment.
This generally leaves the relevant mode in a mixed
state, whose evolution is determined by a
Feynman-Vernon influence functional (IF)
(Polonyi, hep-ph/0605218).
The renormalization group provides a clever way
of computing this influence functional.
Suppose we are given an IF where all modes shorter
than l have been already integrated away
1/l
Already integrated
To be integrated
k (relevant mode)
Instead of integrating
them out in a single
step, we just integrate
out a little bit
1/l
k
(1-s)(1/l)
And then rescale the
theory to restore the
cutoff to its original
value
1/l
k
We iterate the process until all desired modes have
been integrated away
1/l
k
The nonequilibrium renormalization group has two
essential differences with respect to the usual one:
a) Computing the IF
requires doubling the
degrees of freedom, and
so the number of
possible couplings is
much larger. The new
terms are associated with
noise and dissipation.
b) There is a new dimensionful parameter T
which characterizes the lapse between
preparation of the system and observation.
(Although time must be rescaled, we can keep T
constant throughout the process)
T
(1-bs)T
including
this lapse
does not
change the
IF
T
Different T's yield different RG flows, because the
Feynman diagrams depend on it
j
1.2
1.1
1
0.9
j
0.8
0.7
0.6
0
0.2
0.4
0.6
0.8
1
1/T
jj
3.4
3.2
3
1/T times
2.8
j
2.6
2.4
2.2
2
0
0.2
0.4
0.6
0.8
1/T
1
Subtle is the Lord:
We must cope with three possibly complex
functional dependences (on fields, wave number
and time)
But He is not Mean:
In principle we can deal with each of them by
using functional renormalization group techniques
(Wetterich, Phys. Lett. B301, 90 (1993); Dalvit
and Mazzitelli, PRD54, 6338 (1996)). And there
are simpler ways to get results fast.
We are primarily interested in
Long-wavelenght
phenomena
Drop irrelevant
couplings
Slowly-varying
field configurations
Drop time-dependent
coupling constants
T not too small
The RG flow drives
quartic interactions to
zero
but not too large
either
Secular effects
unimportant in the
hard loops
For far IR modes, the IF reduces to




1 T
2
2
2


S   dt  dk   t  t  k  m j  2j ,t  i 
0
2 0
Dissipation
Noise
The coupling constants and the field, time and
wave number scales depend on T and the RG
parameter s=log[/k]
If the dissipation term is not zero, this IF describes
thermalization to an effective temperature given by
the fluctuation-dissipation theorem
Teff


4
Results
Log[Teff]
Log[]
Log[
]
Log[k]
Conclusions:
We did not solve the problem, but we
have a framework for a solution. We
need a self-consistent approach to the
hard loops to be able to extend further
the T range.
Tomorrow we shall see a different
application of the same ideas (one that
actually works better!)