Quenching preheating by light fields
Olga Czerwin´ska,1 Seishi Enomoto,1, 2 and Zygmunt Lalak1 1Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw ul. Pasteura 5, 02-093 Warsaw, Poland 2University of Florida, Department of Physics, P.O. Box 118440, Gainesville, FL 32611-8440
(Dated: June 22, 2017)
In this paper we investigate the role of additional light fields not directly coupled to the background during preheating. We extend our previous study that proved that the production of particles associated with such fields can be abundant due to quantum corrections, even for the massless states. We also obtain the expression for the occupation number operator in terms of interacting fields which includes the non-linear effects important for non-perturbative particle production. We show that adding too many light degrees of freedom without direct interactions with the background might attenuate or even quench preheating as the result of back-reaction effects and quantum corrections.
PACS numbers: 98.80.-k, 98.80.Cq

arXiv:1701.00015v3 [hep-ph] 20 Jun 2017

I. INTRODUCTION
Post-inflationary particle production is a very complex stage in the evolution of the universe that mixes perturbative and non-perturbative processes [1­5]. Usually it is divided into two main stages:
a) preheating - when exponentially and nonperturbatively produced states typically correspond to the fields directly interacting with the inflaton, they do affect the mass term of the inflaton through back-reaction effects
b) reheating (thermalization) - when the inflaton decays perturbatively and produced particles end up in thermal equilibrium with a well-defined temperature.
Interesting is the question about the impact of the additional fields, especially light ones, on preheating. Their presence in the theory during [6­8] and after inflation is important for multi-field inflation models and for curvaton scenarios [9­11]. For recent reviews of postinflationary particle production see [12] or [13].
In our previous study [14] we showed that light fields which are not coupled directly to the background can be produced due to quantum corrections and their abundance can be sizeable, even for the massless case. In this paper we want to develop these results addressing the problem of additional light degrees of freedom and avoiding at the same time the infinite growth resulting from the approximation used previously. The crucial difference between present considerations at that of [14] lies in the fact that presently the inflaton is massive.
The outline of the paper is as follows. In Section II we develop the formalism necessary to describe the creation of particles in the presence of interactions, with and without time-varying vacuum expectation value (vev) of the considered field. In Section III we apply our formalism to a number of well-motivated cosmological scenarios, including a sector of very light fields. In Section IV we compare our results with the earlier ones [14], discuss the

role of different parameters in the theory, summarize the paper and conclude.

II. PARTICLE PRODUCTION IN TERMS OF INTERACTING FIELDS

Usually the occupation number operator of produced particles is defined in terms of the creation and annihilation operators as Nk  akak. This definition assumes that produced states can be treated as free fields which means that their equations of motion are linear. However, in general fields associated with the produced particles interact with other fields which spoils linearity and results in the non-perturbative production. In that case it is not clear how to define the number operator properly. In this section we address this issue and describe particle number using the theory of interacting fields which takes into account the non-linear effects. To compare these results with a simpler theory of a free field with time-dependent mass term see Appendix A.
For simplicity let us consider a real scalar field  with the Lagrangian of the form:

L

=

1 ()2 2

-

1 2

m202

-

V

[, (other

fields)],

(1)

where m0 is a bare mass of  and V is a general potential. Then equation of motion reads:

0 = (2 + m20) +

V 

= (2 + M 2) + J,

(2)

where M is a physical mass that can depend on time 1

1 In general physical mass can depend not only on time but also on space coordinates. For simplicity we consider only the timedependent case as it is more common in cosmological considerations.

2

and should be a c-number, and

J



(m20

-

M 2)

+

V 

(3)

is a source term that can be an operator. Formal solution of (2) can be presented in a form of
the Yang-Feldman equation

x0

(x) = (t)(x) -

d4y i[(t)(x), (t)(y)]J (y), (4)

t

where the first term describes an asymptotic field defined at x0 = t which satisfies the free field equation of motion

0 = (2 + M 2)(t).

(5)

These relations are equivalent to the Bogoliubov transformation for the wave function

(kt) = kaks - kaks,

(12)

where aks denotes the asymptotic value of the mode k, aks = ikn, okut.
Lagrangian (1) corresponds to the Hamiltonian

H=

d3x

1 2 2

+

1 ()2 2

+

1 2

m202

+

V

.

(13)

Substituting (8) into the above results in a quite complicated expression for the Hamiltonian which can be simplified by choosing the Bogoliubov coefficients of the form

In case that  does not have a vev, the (t) can be decomposed into modes

(t)(x) =

d3k (2)3

eik·x

(kt)a(kt) + (kt)a-(t)k

(6)

fulfilling harmonic oscillator equation

0 = ¨(kt) + k2(kt)

(7)

 with k  |k|, k  k2 + M 2 and obeys the inner product 2 relation: (kt), (kt) = 1.

From (4) also the relation between two asymptotic fields defined at different times x0 = t and x0 = tin can

be derived

t
(t)(x) = in(x) - d4y i[in(x), in(y)]J (y), (8)
tin

where we denoted in(x)  (tin)(x). Evaluating the inner product of the above equation with (kt): (t), (kt) , we can obtain the Bogoliubov transformation for annihilation operators [14]

|k |2

=

ikn 2k

1 +,
2

|k |2

=

ikn 2k

1 -,
2

Arg(k k )

=

Arg ikn

(14)

where 3

ikn  |ikn|2 + k2|ikn|2, ikn  (ikn)2 + k2(ikn)2. (15)

Then the Hamiltonian with diagonalized kinetic terms reads

H=

d3 k (2)3

k

a(kt)a(kt)

+

1 2

(2)3

3(k

=

0)

+ (16)

+

d3x

1 2

(m20

-

M 2)2

+

V

,

which indicates that the operator

Nk(t)  ak(t)a(kt)

(17)

really plays the role of the occupation number. This is because in the system which has a potential energy particle number N would be described classically as

N = H - Veff - V0 ,

(18)

E

t
a(kt) = kaikn +kai-nk - d4y i[kaikn +kai-nk , (y)]J (y),
tin
(9)
where

k = k(t, tin)  ((kt), ikn), k = k(t, tin)  ((kt), ikn) (10)
and the normalization condition reads

|k|2 - |k|2 = 1.

(11)

where H is the total Hamiltonian, Veff an effective potential, V0 a zero-point energy and E is an one-particle energy. Therefore particle number is just the kinetic energy of the system divided by the one-particle energy.
Substituting (9) into (17) and using (14), (15), we can finally obtain the expression for the occupation operator in terms of the interacting fields as

1 Nk(t) = 2

Nk+(t) + Nk-(t)

(19)

2 We use the following definition of the inner product: (A, B)  i(AB - A B).

3 ikn and ikn are constrained by the relation: |ikn|2 - |ikn|2 = k2.

3

with4

Nk+(t)

=

1 k

^k^k + k2^k^k

- (2)33(k = 0),(23)

Nk-(t) = i ^k^k - ^k^k + (2)33(k = 0), (24)

where we defined the Fourier transformation as

^k(t)  d3xe-ik·x(t, x).

(25)

Since Nk±(t) = a(kt)a(kt) ± a(-t)ka(-t)k, Nk+ denotes a total and Nk- a net number of particles with momentum be-
tween k and -k. In the case of a complex scalar  this

expression changes to Nk±(t) = a(kt)a(kt)±b(-t)kb(-t)k, where

bk(t) is an annihilation operator for anti-state, but (23) and

(24) still hold. In such a case

d3 k (2)3

Nk-

corresponds

to

the U (1) Noether charge.

In the case where  has a non-vanishing vev  

0in  0in , we just have to replace   -  in (25) to

obtain the proper expression for the occupation number.

III. NUMERICAL RESULTS FOR MULTI-SCALAR SYSTEMS

To obtain numerical results for some specific models we follow the procedure described in Section II. We are especially interested in time-evolution of particle number density for each considered species:

n(t) =

d3k (2)3

Nk V

,

(26)

where V is the volume of the system, and in timedependence of the background (inflaton). We consider a time range and starting from the initial state we solve equations of motion for all the species and calculate their number density. Then we move to a slightly later time and repeat the procedure taking into account the backreaction of previously produced states on the evolution of the background (given by the induced potential coming from non-zero energy density) and all the species.
Before we present numerical results for specific models, let us focus on a subtlety in calculation of the particle

4 The zero point term can be regarded as the volume of the system because

(2)33(k = 0) = d3xeik·x|k=0 = d3x = V.

(20)

Therefore, we can also find distribution operators:

n+k =

Nk+ V

=

1 k

1 V

^ k ^ k

+ k2

·

1 V

^k ^k

- 1,

n-k =

Nk- V

=i

1 V

^k ^ k

-

1 V

^ k ^k

+ 1.

(21) (22)

number with a general Lagrangian (1). In order to de-
scribe the time evolution of distributions nk = Nk /V for each type of produced states we need to determine
the time evolution of bilinear products of field operators ^k^k , ^k^k and ^k^k . Equations of motion for these operators can be derived by calculating their time
derivatives and using (2)as

^k^k · = ^k^k + ^k^k

(27)

^k^k · = ^k^k + ^k¨^k

= ^k^k - k2 ^k^k - ^kJ^k

(28)

^k^k · = ¨^k^k + ^k¨^k

= -k2( ^k^k + ^k^k ) - ^kJ^k - J^k^k

(29)

where

J^k  d3xe-k·xJ (t, x).

(30)

Physical mass of  is determined by the relation:

0 = ^kJ^k = (m2 - M 2) ^k^k

+

d3xe-ik·x

^k

dV (x) d

(31)

to remove the infinite part of the mass correction.

A. Two scalar system

At first we apply our formalism to the simple theory consisting of two scalar fields

L

=

1 ()2 2

+

1 ()2 2

-

1 2

m22

-

1 2

m22

-

1 g222. 4

(32)

We assume that it is the field  that has time-varying vev

and plays the role of inflaton, 0in  0in = (t) , while

 is another scalar field with vanishing vev that can be,

for instance. a mediator field between the inflaton and

the Standard Model. We also assume m m. The

details of the calculation in this system can be found in

Appendix B.

Asymptotically, when quantum effects can be ne-

glected, we can choose a vacuum solution for (32) of the

form

 = 0 cos(m(t - t0)),

(33)

where 0 denotes the initial amplitude of the oscillations,  (t = t0) = 0. When this trajectory crosses the nonadiabatic area for : |  | < m|0|/g, the mass of  becomes very small and kinetic energy of the background field  is transferred to the field . This results in the creation of  particles with the distribution [5]

n = e , k

-

k2 gm |0

|

(34)

where k is a momentum of a  particle. Once  particles are produced and trajectory of  goes away from the non-adiabatic region, the energy density of  particles can be represented as

d3k

  g|  | (2)3 nk,

(35)

which corresponds to the linear potential acting on  describing the backreaction effects. Then trajectory of  goes back towards the origin and  particles can be produced again both due to the oscillatory behaviour of  and backreaction.
In the Figure 1 we show an example of the numerical results for the Lagrangian (32). According to [5] the first production of  particles results in the number density

n(1)



(gm (0) (2)3

)3/2



4 × 10-9,

(36)

which is consistent with our numerical results. On the
other hand, it is difficult to obtain the analytic results for indirect production products, like ~   -  . But
one can see in the Figure 1 that for the considered Lagrangian energy transfer from the background to ~ and
the production of particles associated with the inflaton is
small for generic choices of parameters. Therefore in this
system it is a good approximation to neglect the quantum part of ~ and the production of its fluctuation.

4



scalar field domination phase this means that gv > 3H,

fdoormminaatttieornd: omgivna>tio2nH: . gv

>

3 2

H

,

while

for

radiation

Following [15] and the analytical method of estimating

the number density of producing particles in the expand-

ing universe presented there for which

n(j)  n(1) · 3j-1

5 3/2 1

2

j5/2 ,

(38)

where j denotes the number of oscillations, we can see

the agreement with our results. If we take j  10 as in

the Figure 1 and n(10)  1 × 10-6, we can see that the

oscillation phase indeed finishes when

1 2

m2

j

2

 (j)



g j n(j).

ÁÑ Ø

Ô½ Ú

Û À ¾
¿´½· µ ¼

ÊØ

ØÒ

Ø¼

ÒÓÒ¹

Ø

n(t)

10-5 10-9 10-13 10-17 10-21
0.

2. × 104

4. × 104

t

n(t)

n(t)

6. × 104

FIG. 1: Time evolution of number density of produced states for g = 0.1, m = 0.001M ,  (t = 0) = M , (t = 0) = 0 in
two scalar system. Scale M  0.04MP L, where MP L denotes the Planck mass MP L  1.22 · 1019 GeV, is chosen to be close
to the unification scale and allows us to stay in agreement
with the observational data.

In our considerations we neglect the expansion of the universe which is valid assuming that the mean time the trajectory spends in the non-adiabatic region is smaller than the Hubble time, see Figure 2. This means that:

1

2

<

,

(37)

gv 3H(w + 1)

where

H

is

a

Hubble parameter

and

w

=

p 

is

a

barotropic

parameter describing the content of the universe. For the

FIG. 2: Time spent by the trajectory in the adiabatic region in comparison with the Hubble time.

The distribution of the produced states is not thermal but, assuming that the whole energy is transferred to the light states which interact with each other and with other particles not present in the simplified Lagrangian, we can naively estimate the maximal reheating temperature as

TRmax 

30R g2

1/4
,

(39)

where R is energy density of the relativistic particles (in our case  or  and ) and g describes the number of relativistic degrees of freedom (g  O(102)). In our system the coupling is big enough to describe energy density as
 = mn and without contradicting our assumptions we
can choose the masses as in Table I. Final estimation of TRmax is also presented in Table I.

B. System with the additional light sector
Usually when describing preheating light fields not coupled directly to the inflaton are neglected. But it

5

TABLE I: Energy densities and upper limits on reheating temperature for two choices of  mass. Mass of  is set to m = 5 · 1014 GeV. Number densities for each state correspond to the results from Figure 1, meaning that n  3.96 · 10-2 GeV3 and n  8.2 · 10-9 GeV3.
m [GeV]  [GeV4] TRmax [GeV]

125

10-6

1.3 · 10-2

700 5.7 · 10-6 2 · 10-2

is important to note that corresponding particles may be produced through an interaction with some other state coupled directly to the background that is produced resonantly. Furthermore, if there are many additional light degrees of freedom, one can expect that energy transfer from the background to the light sector during preheating might be sizeable. In this section we focus on such light fields and discuss the possibility of their production through the indirect interaction with the background field.
We can describe such a situation by extending (32) with n light or massless fields n (m m, m) that are not coupled to the background at the tree-level

L

=

1 2

()2

+

1 2

()2

-

1 2

m2

2

-

1 2

m22

-

1 4

g2

22

+

n

1 2

(

n

)2

-

n

1 2

m2 n2

-

n

1 4

y2

2n2

.

(40)

We assume again that  is time-varying and the other fields do not have a vev:  = n = 0. Then  particles are produced resonantly and as we mentioned before we can expect production of n through the interactions with .
The physical mass of n is given by

M2

=

m2

+

1 2

y2

d3 p (2)3

1 V

^p^p

-

1 2p

+O(y4, y2g2, g4),

(41)

where p  p2 + M2 and V denotes the volume of
the system. We can see that n influence background's evolution via pp operator in their mass term.
We show the results for only one additional field in Figure 3. One can see that all the states are produced and their number density is abundant. If the final number density of  is comparable to the one for  (n  n) its presence may even quench the preheating process by terminating the energy transfer. The reason that  can be produced so efficiently is the strong coupling between  and  that enhances the back-reaction effects.
We would expect that most of the energy would be transferred to n fields as they are very light and the process is energetically favourable. But we can prove that the more light species we include, the larger the final value of |  | becomes and, in other words, the less energy from the background goes to the light fields, see Figure 4.

n(t)
||

10-8
10-13
10-18
10-23 0.

2. × 104 n(t)

4. × 104 t

n(t)

n(t)

6. × 104

FIG. 3: Time evolution of number density of produced states
in the system with additional light sector for g = 0.1, y = 1, n = 1, m = 0.001M , (t = 0) = M , (t = 0) = 0.

1.0

0.8

0.6

0.4

0.2

0.0 0.

1. × 104 2. × 104 3. × 104 4. × 104 5. × 104

t

n=1

n=2

n=5

n=7

n=10

FIG. 4: Envelope of the time evolution of the background  for g = 0.1, y = 1, m = 0.001M , (t = 0) = M , (t = 0) = 0 for different numbers of additional light fields: 1, 2, 5, 7.

The reason why the energy transfer can be stopped in this case can be understood as follows. The physical mass of  in the system is given by

M2

=

m2

+

1 2

g2



2+

1 2

g2

d3 p (2)3

1 V

^p^p

-

1 2p

+

1 2

y2

n

1 V

^n p ^np

-

1 2p

+ O(y4, y2g2, g4). (42)

Considering an approximation X^ p  -iXpX^p for X = , n, one can find that

1 V

X^p X^p

-

1 2Xp



11 2Xp V

NX(+p) .

(43)

Thus, once  or n are produced at the same time they also generate 's effective mass 5 which results in  par-
ticle production area becoming narrower. This leads to

5 These mass correction terms describe a square of plasma frequency discovered by I.Langmuir and L.Tonks in the 1920s which is a critical value for which the wave of  can enter X's plasma

6

TABLE II: Energy densities and upper limits on reheating temperature (both in GeV) for two choices of  and  mass. Mass of  is set to m = 5 · 1014 GeV. Number densities for each state correspond to the results from Figure 3, meaning that n  1.82 · 10-9 GeV3 and n  9.91 · 10-6 GeV3.
m [GeV] m [GeV] n [GeV3]  [GeV4]  [GeV4] TRmax [GeV]

125

100 1.21 · 10-5 1.24 · 10-3 1.21 · 10-3 0.93 · 10-1

700

125 1.21 · 10-5 6.94 · 10-3 1.51 · 10-3 1.26 · 10-1

n  (t)

10-3 10-4 10-5 10-6 10-7
0. 10-7

2. × 104

4. × 104

t

6. × 104

the suppression of  particle production and also spoils the production of other species. Too many n particles produced through indirect coupling to the background prevent the production of particles directly coupled to the background, .
It is interesting to investigate the impact of both couplings - g that couples  to  and the background  and y that couples additional fields n to , on the features of preheating. Varying the coupling y for fixed g leads to the conclusion that the initial stage of preheating does not depend on y coupling for  and  states. It only influences the final abundance of produced  and  states - the bigger y is, the smaller number density of these states we observe, see Figure 5. For  the impact of y is quite opposite - both initial and final stages of  production are strongly influenced by the value of y. This time the bigger y is, the larger number density of  we observe which also results in more effective energy transfer to the background as y coupling drops, see Figure 6. Also, for choices of parameters resulting in n  n we can observe quenching of the energy transfer from the background.
Our study may seem similar to the process of instant preheating [16, 17], where the system of three fields background ,  interacting with the background and some other field  not coupled to , is considered. Instant preheating relies on the fact that  particles produced within one-time oscillation of  decay immediately to  before the next oscillation of . So  states can be

n(t)

10-8

10-9

10-10

10-11 0. 10-5

2. × 104

4. × 104

t

6. × 104

10-7

10-9

10-11

10-13 0.

2. × 104

4. × 104

6. × 104

t

y=0.1

y=0.2

y=0.5

y=0.7

y=1

n(t)

FIG. 5: Time evolution of number density of produced states
,  and  for g = 0.1, n = 1, m = 0.001M , (t = 0) = M , (t = 0) = 0 and different values of y coupling. Values y = 0.7
and y = 1 correspond to quenching of parametric resonance.

or not, because

d3p Nk(+)

(2)3 2XpV

is proportional to

nX MX

if X

particles are massive enough (nX

is X's number density). Moreover, if one considers the massless

thermal equilibrium distribution with the temperature T :

1 V

NX(+p)

1 =2
ep/T - 1

(factor 2 corresponds to the degrees of freedom for momentum

k and -k particles), it corresponds to the thermal mass of the

form:

d3p (2)3

1 V

X^p X^p

1 -
2Xp

T2 .
6

also produced even though there is no direct interaction between  and . In our work the mechanism of production is different - due to the quantum corrections, not the decay, and quenching of the preheating comes from a plasma gas effect here rather than the rapid decay.
Table II presents TRmax and energy densities for each state for the considered model under assumption that  = H or  = H, H being the Higgs field playing the role of the mediator or the light field. We can see that additional light sector that quenches preheating rises TR lowering the number density of  particles at the same time.

7

||

1.0

0.8

0.6

0.4

0.2

0.0 0.

1. × 104 2. × 104 3. × 104 4. × 104 5. × 104

t

y=0.1

y=0.2

y=0.5

y=0.7

y=1

FIG. 6: Envelope of the time evolution the background  for g = 0.1, n = 1, m = 0.001M , (t = 0) = M , (t = 0) = 0 and different values of y coupling. For y = 0.7 and y = 1 we
can observe the quenching of the preheating.

IV. DISCUSSION AND SUMMARY
In our previous work [14] we presented a formalism for describing particle production in a time-dependent background. It turned out it possesses one drawback there exists a secularity in the number density of massless states that can be a product of approximating the fields by their asymptotic values. In this paper we have developed more accurate description by expressing the number operator in terms of interacting fields. Figure 7 compares the two methods for the Lagrangian (32). The new method avoids artificial secularity caused by time integral of the interaction effects with the Green functions seen before. The old method seems to overestimate the production at the late stage because it includes "inverse decay" processes, whereas the new one takes into account mass correction terms. However, the results with secularity are still applicable at the early stages of particle production process.
As the application of the new method in this paper we investigated the role of additional light fields coupled indirectly to the background during resonant particle production processes such as preheating. In particular, we considered models with a scalar field  interacting with the background  through its mass term and with n light fields n. In order to describe particle production in the system, at first we defined number operator in terms of interacting fields and then we solved numerically their equations of motion. In case of a few additional light fields, their production can be also resonant through the quantum correction to their mass term and their final amount can be sizeable. However, many degrees of freedom of these extra light fields can prevent 's and also n's resonant particle production. As a result, energy transfer from the background does

n(t)

10-3 10-5 10-7 10-9 10-11 10-13
0.
1.0 0.8 0.6 0.4 0.2 0.0
0.

1. × 104

2. × 104

t

n(new)

n(old)

n(new)

n(old)

3. × 104

1. × 104 t
new

2. × 104 old

3. × 104

||

FIG. 7: Comparison between time evolution of number density of produced states (upper ) and the background  (lower ) obtained with a new and old methods for g = 1, m = 0.001M , (t = 0) = M , (t = 0) = 0. New denotes the interacting theory described here and old - asymptotic approximation presented in [14].

not work well and this indicates that preheating might be quenched if there are many degrees of freedom of light fields which are connected to the background indirectly.
This work has been supported by the Polish NCN grant DEC-2012/04/A/ST2/00099, OC was also supported by the doctoral scholarship number 2016/20/T/ST2/00175. SE is partially supported by the Heising-Simons Fundation grant No 2015-109. OC thanks Bonn Bethe Centre Theory Group for hospitality during the completion of this paper.

Appendix A: Particle production in free fields theory with time-varying mass terms

Let us consider a real free scalar field  with the timedependent mass term:

L = 1 ()2 - 1 m2(t)2.

(44)

2

2

8

The solution of the equation of motion can be decomposed into

(x) =

d3k (2)3

eik·x

kak + ka-k

(45)

where k = k(x0) is a time-dependent wave function which satisfies

0 = ¨k + k2k (k  k2 + m2),

(46)

and ak, ak are annihilation and creation operators. The vacuum state |0 is defined by the relation ak|0 = 0 and
the commutation relations

[(t, x), (t, x )] = i(x - x ),

(47)

[(t, x), (t, x )] = [(t, x), (t, x )] = 0, (48)

[ak, ak ] = (2)3(k - k ),

(49)

[ak, ak ] = [ak, ak ] = 0

(50)

give an inner product relation of the form: (k, k) = 1. Using (45) we can represent the Hamiltonian as

H= =

d3x 1 2 + 1 ()2 + 1 m22

(51)

22

2

d3k 1 (2)3 2

k (t)

akak + a-ka-k

+k(t)a-kak + k(t)aka-k , (52)

where

k(t)  |k(t)|2 + k2(t)|k(t)|2,

(53)

k(t)  2k(t) + k2(t)2k(t).

(54)

In order to diagonalize the Hamiltonian

H= =

d3k 1 (2)3 2 k(t)

a¯ka¯k + a¯-ka¯-k

(55)

d3k (2)3 k(t)

a¯ka¯k

+

1 (2)33(k 2

=

0)

(56)

we need a set of operators a¯k, a¯k satisfying
[a¯k, a¯k ] = (2)3(k - k ), [a¯k, a¯k ] = [a¯k, a¯k ] = 0, (57)
Then the number operator N¯k  a¯ka¯k is well-defined all the time. Following [18], we can obtain the new operators by the Bogoliubov transformation

a¯k = kak + ka-k

(58)

with the coefficients satisfying

|k |2

=

ikn 2k

+

1 ,
2

|k |2

=

ikn 2k

1 -,
2

Arg(k k )

=

Arg ikn.

(59)

Then the occupation number can be expressed as

Nk(t) =

0|N¯k|0

=

|k |2

=

k 2k

-

1 .
2

(60)

Appendix B: Two scalar system

- details of the calculation

In the system described by the Lagrangian 1, we have a background field  and two quantum fields: ~  -  ,
. The set of differential equations for distributions reads

0 = ¨ + M2 

(61)

^k^k · = ^k^k + ^k^k

(62)

^k^k · = ^k^k - 2k ^k^k

(63)

^k^k · = -2k( ^k^k + ^k^k )

(64)

^k^k · = ^ k^k + ^k^ k

(65)

^k^ k · = ^ k^ k - 2k ^k^k

(66)

^ k^ k · = -2k( ^ k^k + ^k^ k )

(67)

where the source terms are absent because of our choice of physical masses

M2

=

m2

+

1 g2 2

d3p (2)3

1 V

^p^p

1 -
2p

,(68)

M2

=

m2

+

1 g2 2



2

+ 1 g2 2

d3p (2)3

1 V

^p^p

1 -
2p

.

(69)

In order to obtain the above formulae, we applied an approximation

^p1 ^p2 ^p3 ^p4 = ^p1 ^p2 ^p3 ^p4 + O(g2) (70)

and assumed the momentum conservation

X^p X^p

= 1 (2)33(p - p ) · V

X^p X^p

(71)

for the quantum fields X = ~, . Momentum conserva-

tion indicates that X^p X^p = Cp(2)33(p - p ), where

Cp is a proportionality factor. For p = p: X^p X^p =

V

·

Cp,

hence

Cp

=

1 V

X^p X^p .

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