A nonequilibrium renormalization group approach to turbulent reheating

A case study in nonlinear nonequilibrium quantum field theory

J. Zanella, ECPart I

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

0'32

2

V

dt

dH

dt

d

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

k (relevant mode)

Already integrated

To be integrated

1/l

k

Instead of integrating them out in a single step, we just integrate out a little bit

(1-s)(1/l)

1/l

k

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

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)TT

including this lapse does not change the IF

0 0.2 0.4 0.6 0.8 1

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9

1

1.1

1.2

Different T's yield different RG flows, because the Feynman diagrams depend on it

1/T

1/T

1/T times

Subtle is the Lord:

We must cope with three possibly complex functional dependences (on fields, wave number and time)

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.

But He is not Mean:

We are primarily interested in

Long-wavelenght phenomena

Drop irrelevant couplings

Slowly-varying field configurations

Drop time-dependent coupling constants

T not too smallThe 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

2,

22

0 02

21 imkkddtS ttt

T

�

The coupling constants and the field, time and wave number scales depend on T and the RG parameter s=log[/k]

Dissipation

Noise

If the dissipation term is not zero, this IF describes thermalization to an effective temperature given by the fluctuation-dissipation theorem

4

effT

Log[Teff]

Log[]

Log[]

Log[k]

Results

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!)