theory of fermionic preheating

A7

PHYSICAL REVIEW D, VOLUME 62, 123516

Theory of fermionic preheating

Patrick B. GreeneDepartment of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7

Lev KofmanCanadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1

~Received 24 April 2000; published 29 November 2000!

In inflationary cosmology, the particles constituting the universe are created after inflation due to theirinteraction with moving inflaton field~s! in the process of preheating. In the fermionic sector, the leadingchannel is out-of-equilibrium particle production in the non-perturbative regime of parametric excitation,which respects Pauli blocking but differs significantly from the perturbative expectation. We develop thetheory of fermionic preheating coupling to the inflaton, without and with expansion of the universe, for lightand massive fermions. We calculate analytically the occupation number of created fermions, focusing on theirspectra and time evolution. In the case of large resonance parameterq we extend for fermions the method ofsuccessive parabolic scattering, earlier developed for bosonic preheating. In an expanding universe, parametricexcitation of fermions is stochastic. Created fermions very quickly, within tens of inflaton oscillations, fill upa sphere of radius.q1/4 in momentum space. We extend our formalism to the production of superheavyfermions and to ‘‘instant’’ fermion creation.

PACS number~s!: 98.80.Cq

ndo

oif

iuth

osric.of

ocin

-he

le

de

ioxin

iove

ner-

ehatine-

rmi

-

on-ser-on-os

the

icr-

fer-

arkos-ca-antg-iedicics

I. INTRODUCTION

During cosmic inflation, it is assumed that the entropy atemperature associated with particles of matter are dilutepractically zero values together with the number densityparticles. After inflation, the inflaton fieldf oscillatesaround the minimum of its effective potentialV(f). Theenergy of inflaton oscillations is converted to the energynewly created particles of matter. Eventually particles of dferent species are settled in a state of thermal equilibrwhich marks the beginning of the conventional epoch ofhot Friedmann radiation domination. The actual processparticle creation from the classical background inflatoncillations occurs very rapidly in the regime of parametresonance@1# before the thermal equilibrium will be settledParticles are created non-perturbatively in the out-equilibrium state. The theory of this process,preheating, iselaborated in detail for the creation of bosons@2#. For bosons~denotedx), the leading effect is the stimulated processparticle creation in the regime of parametric resonanwhere the number density of created particles copiouslycreases with time asnx;exp(*2mdt). Soon, the back reaction of createdx particles becomes important, so that tself-consistent dynamics of interacting bose fieldsf(t,xW )andx(t,xW ), which can be treated classically, can be reveawith lattice simulations@3#.

At the beginning of the preheating investigation, a stuof fermion production did not look very interesting relativto the bosonic case. Indeed, the number density of fermc is bounded by Pauli blocking. Therefore, it was not epected that fermions will influence the dynamics of theflaton and other scalar fields~despite a numerical study@4#which downplayed an observation that fermion productfrom an oscillating scalar is different from the perturbatiprediction!.

0556-2821/2000/62~12!/123516~12!/$15.00 62 1235

dtof

f-meof-

-

fe,-

d

y

ns--

n

However, it was understood with surprise@5,6# that thecreation of fermions from the coherently oscillating inflatofield occurs very differently than what the conventional p

turbation theory of inflaton decayf→cc ~say, due to a

Yukawa couplinghcfc) would predict, and this may havmany interesting cosmological applications. It turns out tthe occupation number of fermions very quickly, withabout ten inflaton oscillations, is saturated at the timaverage value of aboutnc;1/2. Moreover, in momentumspace, fermions are excited within a non-degenerate ‘‘Fesphere’’ of a large radiusk;q1/4m, wherem is the frequencyof inflaton oscillations, andq is the usual dimensionless parameter of parametric excitation,q5h2f0

2/m2. Ironically, itmay be parametric excitations of fermions that are respsible for the most important observable signatures or obvational constraints of preheating. Indeed, in some inflatiary models there is significant production of gravitinduring the preheating stage@7#. Gravitinos are cosmologi-cally dangerous relics; for a mass of;100 GeV their abun-dance relative to that of the relic photons cannot exceedbound n3/2/ng<10215. The theory of gravitino productionfrom preheating is rooted in the theory of spin 1/2 fermionpreheating. Another potentially important application of femionic preheating is a possibility to produce superheavymions with a mass as large as;1018 GeV from inflatons ofmass 1013 GeV, as was noticed in@8,9# and investigated in@9#. Superheavy fermions may be interesting for the dmatter problem and for the problem of ultra-high energy cmic rays. There are other interesting cosmological applitions for the creation of fermions, e.g. The scenario of instpreheating@10# and the creation of massive fermions durininflation @11#. Recently, fermionic preheating in hybrid inflation for some range of parameters was thoroughly stud@12#. At present, it is hard to say how important fermionpreheating will be in the self-consistent non-linear dynam

©2000 The American Physical Society16-1

icas

eretn

th

at

wd

rke

gin

thil

lanticF

olithlitu

thillald

iok-e-

es

a

e

le

-

nl.nith

yspe-

-

pin

oryf

i-

PATRICK B. GREENE AND LEV KOFMAN PHYSICAL REVIEW D62 123516

of Bose and Fermi fields during preheating. The techndifficulty is to incorporate fermions into lattice simulationalongside with bosons. However, in@5# it was reported thateven within the Hartree approximation back reaction of fmions catalyzes bosonic preheating. After all, for compldecay of inflatons one needs a ‘‘three-leg’’ interactiowhich is provided naturally by the Yukawa coupling wifermions.

In this paper we develop the theory of fermionic preheing in an expanding universe, following our short paper@6#,with an emphasis on the analytic results. In particular,will generalize for the fermionic case some of the methowhich we earlier developed for bosonic preheating@2,13#; inparticular, the method of parabolic scattering, which wofor large values ofq, gives us an analytic formula for thoccupation number of fermionsnk(t) as a function of timeand momentum. We extend this method to the productionsuperheavy fermions from a moving scalar field. We beSec. II of this paper with the Dirac equation and differevacuum expectation values~VEVs! for fermions interactingwith a time-dependent background scalar field. In Sec. IIIcreation of fermions without expansion of the universe wbe considered. We mostly will consider scalar fields osciling in a potentialV5m2f2/2, although the methods caeasily be applied to others. We will give a semi-analytreatment of the problem based on some earlier results.the caseq@1 we develop the method of successive parabscatterings. In Sec. IV we take into account expansion ofuniverse, when preheating of fermions acquires new quative features. We extend the theory to describe the prodtion of superheavy fermions. Our formalism also includescase when fermions are created not from inflaton osctions, but from a single instance, when the inflaton fiecrosses a certain level.

II. FORMALISM: FERMIONS COUPLED TO ABACKGROUND SCALAR IN FRW METRICS

We will consider the creation of spin-12 Dirac fermionsc

by a homogeneous, oscillating scalar fieldf in an expand-ing, flat Friedmann-Robertson-Walker~FRW! universe. Ourstrategy will be to solve the Heisenberg equation of motfor the quantumc field in the presence of the classical bacgroundsf and gmn . Furthermore, we will assume that thenergy-momentum of thef field alone determines the expansion rate of the universe. The matter action we use,

SM@f,c,ema#5E d4x eF1

2]mf]mf2V~f!1 i cgmDW mc

2~mc1hf!cc G , ~1!

contains a simple Yukawa coupling between the scalar fiwith effective potentialV(f) and fermion with bare masmc . Here gm is a space-time dependent Dirac gamma mtrix, em

a is the vierbein withe its determinant, andDm5]m

1 14 gabvm

ab is the spin-12 covariant derivative with vierbein

12351

l

-e,

-

es

s

ofnt

elt-

orce

a-c-e-

n

ld

-

dependent spin connection,vmab . Unbarred gammas ar

standard Minkowski space-time Dirac matrices andgab[g [agb] .

A. Classical background fields

We consider flat FRW metricsds25dt22a(t)2dxW2

wherea(t) is the scale factor of the universe. The Hubbparameter H[a/a is determined by the equationH2

5(8p/3Mpl2 )@ 1

2 (f)21V(f)#. The background homoge

neous scalar field obeys the equation of motionf13Hf1]V/]f50. We will be most interested in fermion creatiowhile thef field oscillates about a minimum of its potentiaFor illustration of the methods, we will consider fermiocreation in the context of chaotic inflation scenarios wpotentials of the formV(f)5 1

2 mf2 f21(l/4)f4. After in-

flation, thef field oscillates quasi-periodically with a slowldecreasing amplitude due to the Hubble expansion. Thecific value of the parametermf or l ~or some combinationthereof! will be dictated by cosmology. We mostly will consider the quadratic potential withl50. In this paper weignore back reaction of created particles.

B. Quantum Dirac field

Variation of the action~1! with respect toc leads to thegeneral relativistic generalization of the Dirac equation

$ i gmDm2@mc1hf~ t !#%c~x!50 . ~2!

It can be shown that for an FRW space-time we haveg0

5g0 and g i5@1/a(t)#g i , where thega’s are standard,Minkowski space-time Dirac matrices. Furthermore, the sconnection leads to

1

4gmgabvm

ab53

2S a

aD g0. ~3!

Thus, in our case, the Dirac equation becomes

F ig0]01 i1

agW •¹W 1 i

3

2S a

aD g02~mc1hf!Gc~x!50. ~4!

Only standard gamma matrices now appear. Similarly,c5c†g0 is found to obey the conjugate of Eq.~4!.

In the Heisenberg representation of quantum field the~QFT!, the Dirac equation~2! or ~4! becomes the equation omotion for thec(x) field operator.c(x) may be decomposedinto eigenspinors

c~x,t !5 (s56

E d3k

~2p!3 @ ak,suk,s~ t ! e1 ik•x

1bk,s† vk,s~ t ! e2 ik•x#, ~5!

whereuk,6(t) is a positive-frequency eigenspinor of the Drac equation~4! with helicity 6 1

2 and vk,6(t)5Cuk,6T (t) is

6-2

eu

od

de

(t

n

re-de-o-

on

e-

s-

ite

tion

o-

In

rs iseingre-no

As-uuma-

icedtinged,

THEORY OF FERMIONIC PREHEATING PHYSICAL REVIEW D62 123516

its charge conjugate. HereC5 ig0g2 is the standard chargconjugation matrix. To construct these eigenspinors wethe ansatz

uk,6~ t ! e1 ik•x5aF2 ig0]02 i1

agW •¹W 2 i

3

2S a

aD g0

2~mc1hf!Gxk~ t !R6~k!e1 ik•x, ~6!

whereR6(k) are eigenvectors of the helicity operatork"Ssuch thatk"SR6(k)561 andg0R6(k)511. The time de-pendence of the eigenspinor is contained entirely in the mfunction xk(t). Substituting this ansatz into Eq.~4! leads tothe mode equation

xk14a

axk1F k2

a21Meff2 2 i

~a Meff!.

a1

9

4S a

aD 2

13

2

a

aGxk50,

~7!

wherek25uku2 and the effective mass of the fermions is

Meff[@mc1hf~ t !#. ~8!

The damping term in this equation may be removed byfining a new mode functionXk(t)5a2xk(t). The modeequation becomes

Xk1F k2

a21Meff2 2 i

~a Meff!

a

.

1J~a!GXk50, ~9!

whereJ(a)[@ 14 (a/a)22 1

2 (a/a)#. Whena}tn such as in amatter or radiation dominated universe,J(a);t22 and canbe neglected soon after inflation.

Using the mode equation and ansatz, we find

uk,6~ t !5a21F2 iXk2 i1

2S a

aDXk1~g"k2Meff!XkGR6~k!.

~10!

Taking the charge conjugate ofuk,6 , we find

vk,6~ t !5a21F iXk* 1 i1

2S a

aDXk* 2~g"k1Meff!Xk* G R6~k!,

~11!

where R6(k)[2 ig2R6* (k) is an eigenvector of helicity

such thatk"SR6(k)561 andg0R6(k)521.The energy-momentum tensor is obtained from thec

and c symmetrized! matter action by variation with respecto the vierbein

Tmn5i

2@cg (mDW n)c2cDQ (mgn)c#, ~12!

and the Hamiltonian operator is

HD5E d3x@ ic†c#. ~13!

12351

se

e

-

In general, if this Hamiltonian is diagonal in the annihilatioand creation operatorsak,s , ak,s

† andbk,s , bk,s† at t50, it will

not be for later times. This is the signature of particle cation due to the time dependent background. In order totermine the number of particles produced, we perform a Bgliubov transformation on the creation and annihilatioperators so as to diagonalize the Hamiltonian at timet.

For the problem of particle creation, quite often it is usful to represent the wave function in the adiabatic~semi-classical, WKB! form

Xk~ t !5akN1 expS 2 i E0

t

dt Vk~ t ! D1bkN2 expS 1 i E

0

t

dt Vk~ t ! D , ~14!

where N65@2Vk(Vk7Meff)#21/2 and the coefficientsakandbk correspond to the coefficients of the Bogliubov tranformation.

Once the Bogliubov transformation is done, we may wrthe comoving number density of particlesnk(t)5ubku2 in agiven spin state through the solutions of the mode equa~9!:

nk~ t !5aS Vk2Meff

2VkD @ uXku21Vk

2uXku222VkIm~XkXk* !#,

~15!

where Meff[(mc1hf) as before andVk2[k2/a21Meff

2 .The energy density in these particles is then

rc~ t !52

a3E d3k

~2p!2 Vknk~ t !51

a3pE dk k2Vknk~ t !.

~16!

The normalization of the solutionsXk(t) is such thatXk(t→02)5N1e2 iVkt and nk(0)50. Thus, we find N1

5@2Vk(Vk2Meff)#21/2, as expected. These are the scalled positive frequency initial conditions.

A comment about the regularization of fermion VEVs.principle, fermionic VEVs like Tmn& require regularization,which in the presence of the background metric and scalarather non-trivial; see e.g.@14# and references therein. Wwill consider only the processes of particle creation, ignorvacuum polarization. The creation of particles, which corsponds to the imaginary part of the effective action, hasformal divergences and does not require regularization.suming the particle creation process dominates over vacpolarization, we will not consider the issues of regulariztion.

III. FERMIONIC PREHEATING WITHOUT EXPANSIONOF THE UNIVERSE

It is convenient to begin the investigation of fermionpreheating due to an oscillating scalar field with a simplifisetting neglecting the expansion of the universe. This setmay have not only methodological advantages. Inde

6-3

rtodon

vemrso: aldt

ry

la

.b

ion

-acic

men

r-

f

eiool

nWend

opaal

m-

the

s in

so-er isse

ngesan

r

m-


whenever the frequency off oscillations is much greatethan the rate of cosmic expansionH, it is sensible to neglecthe time dependence of the scale factor in solving the mequation~9!. This can be the case, for example, when sptaneous symmetry breaking occurs rapidly leaving theffield oscillating about a new minimum where the effectimassmf happens to be much larger than the Hubble paraeter at the time of the transition. One such example is hybinflation scenarios which end with TeV scale energy denties. Another example is the inflationary model with the ptential lf4. This theory possesses conformal propertiesthe stage of inflaton oscillations the equations for the fieby means of conformal transformations can be reduced toequations in Minkowskii space-time; see e.g.@6#.

In this paper we will mostly use a chaotic inflationamodel with quadratic potentialV(f)5 1

2 mf2 f2. If we make

the replacementa51 in all the formulas of Sec. II, all theeffects of expansion will be removed. Background osciltions take the form of harmonic oscillationsf(t)5f0f (t)with f (t)5cos(mft) andf0 the time independent amplitudeIt is convenient to define a new, dimensionless time variat[mft. With this change of variables, the mode equat~9! may be written

Xk91@k21~m1Aq f !22 iAq f8#Xk50, ~17!

where we have introduced the dimensionless momentumk

[k/mf , the dimensionless fermion massm[mc /mf , andthe resonance parameterq[h2f0

2/mf2 . These three param

eters completely determine the strength of the effect. In fthis form of the mode equation is valid not only for harmonbackground oscillations, but for genericf oscillations in ageneral potential. For this, we identifymf with the fre-quency of oscillation,fo , with its amplitude, andf (t) withthe periodic background oscillations normalized to unit aplitude. Note that the frequency will be amplitude-dependfor a general, non-quadratic potential.

A. Parametric excitation of fermions

If individual inflatons at rest are decaying into light femions in a processf→cc, perturbative calculations givethe rate of decayGf→cc.h2mf/8p and the spectrum ocreated fermions is sharply peaked aroundmf/2 with thewidth Gf→cc

21 . In Figs. 1 and 2, however, we plot the timdependence and spectrum of occupation number for fermcreated from the coherently oscillating inflaton field, as flows from the numerical solution of Eqs.~17! and ~15!.

The spectrum and time evolution are drastically differefrom what is expected from the perturbative calculations.therefore can talk about a specific phenomena, the paramexcitation of fermions interacting with coherent backgrouoscillations.

It is instructive to compare the spectrum and evolutionfermionic occupation numbers with those of bosonic occution numbers. The mode function of a quantum Bose scfield x coupling asg2x2f2 with an oscillating inflaton obeysthe bosonic oscillator-like equation

12351

e-

-idi--tshe

-

le

t,

-t

ns-

tetric

f-

ar

Xb,k9 1@k21mb21qbf 2#Xb,k50, ~18!

whereqb5g2f02/mf

2 , mb5mx /mf . In Figs. 3 and 4 we plotthe spectrum and time evolution of bosonic occupation nuber calculated from the bosonic oscillator-like equation~18!.We use the familiar model of bosonic resonance due toself-interaction inlf4 inflation, which corresponds tomf

→Alf0 , g253l, qb53, mb50, andf (t) is given by os-cillations in lf4 theory.

In the bosonic case, there are distinct resonance bandwhich bosonic modes are exponentially unstable,nk(t);e2mkt. Particle creation takes place outside of the renance bands as well, although there the occupation numbbounded and oscillates periodically. In the fermionic cank<1 is always bounded by Pauli blocking and oscillatiperiodically with time. The occupation number in both casis changing in time. It is therefore convenient to introduceenvelope functionFk of the particle spectrum@6#, which cor-responds tonk averaged over short-time intervals~order ofthe background oscillation period!. A bosonic envelope func-tion cannot be defined for the resonant bands~where it is

FIG. 1. The occupation numbernk of fermions inmf2 f2 theory

as a function of timet/2p for a range of (q;k): dotted curve for(1.0;0.05), bold curve for (1024;0.5), smooth sine-like curve fo(1022;0.5), jagged sine-like curve for(1;1.3).

FIG. 2. The occupation number of fermions inmf2 f2 theory as

a function ofk2 after 10 inflaton oscillations for resonance paraeterq510. Also shown is the envelope functionFk defined by Eq.~21!.

6-4

o

t ith

e

ey-ec

re

itaprouth

ast

wn

k-ond-

de-

he

e

-

of-

s

ted1a-

e-

s offre-de-

ncyted

m-or

e-thetedmi-

es

lit-ith

e

ci


e2mkt.! This zone corresponds to the gap between almvertical lines in Fig. 4. The fermionic envelope functionFk ,on the contrary, can be defined everywhere. Although ialways bounded by unity, its structure is reminiscent ofresonance-band structure: for some range ofk, the ‘‘reso-nance’’ band,nk is close to unity, while in other ranges, thstable bands, it is significantly smaller if not zero. In@12#different levels ofFk were plotted in the parameter plan(k,q), revealing a structure reminiscent of the stabilitinstability chart of the bosonic parametric resonance. Onthe most important results is that fermionic parametric extations occur very quickly, within about ten~s! backgroundoscillations. Interestingly, fermionic ‘‘resonance bands’’ aexcited last, while non-resonant intervals fill first.

Comparison of bosonic and fermionic parametric exctions is useful to understand some features of bosonicheating. As we have seen, there is production of bosonsside of the resonance band. If we take into accountexpansion of the universe, in the most interesting caselargeqb , the difference between resonant and non-resonexcitations of bosons will be erased, and the regime of

FIG. 3. Time dependence of the occupation number inlf4

theory for q53 for two different modes: one is just inside thresonance band which grows likee2mkt and another outside whichoscillates like sin(nkt).

FIG. 4. Bosonic occupation number after 10 background oslations forlf4 theory whenq5g2/l53. A stripe with sharp edgescorresponds to thee2mkt instability.

12351

st

se

ofi-

-e-t-eofnto-

chastic resonant production of bosons will be settled do@2#.

B. Some generic analytic results

The dynamics of the Fermi field coupled to the bacground homogeneous scalar can be revealed with the secorder oscillator-like equation~17! for the mode functionXk(t). For periodic background oscillationsf (t)5 f (t1T),whereT is a period, some generic analytic results wererived a long time ago@15# in the context of particle creationin a periodic external electromagnetic field. In particular, toccupation number of created particles at instancest5NsT,i.e. exactly afterNs background oscillations, is given by thexpression@15#

nk~NsT!5k2

2Vk2

sin2Nsdk

sin2dk@ Im Xk

(1)~T!#2, ~19!

where cosdk5ReXk(1)(T). To derive this result, one intro

duces two fundamental solutions of Eq.~17!, Xk(1)(t) and

Xk(2)(t), such that initially Xk

(1)(0)51, Xk(1)(0)50 and

Xk(2)(0)50, Xk

(2)(0)51. Expression~19! involves only thevalue of the first fundamental solutionXk

(1)(T) exactly afterthe first oscillation. It says that the occupation numbercreated particles afterNs background oscillations is modulated with a certain frequencynk ~which, as we will see, doesnot coincide withdk /2). However, the practical applicationof the generic formula~19! is rather limited, because it doenot address the full time evolution ofnk(t), and cannotstrictly determine a period of modulationp/nk .

To get an idea of how the occupation number of creaparticlesnk(t) evolves with time, again let us look at Fig.and further at Fig. 7, below, for different values of the prametersk and q. For small and moderateq ~but not toosmallk) the occupation number exhibits high frequency~pe-riod ,T/2) oscillations which are modulated by a long priod behavior. For largeq, the number of fermions jumps ina step-like manner at instances when the effective masthe Fermi field crosses zero, superposed by very highquency oscillations around almost constant values, aspicted in Fig. 7. These jumps are modulated with a frequenk/2: steps in the first half of the cycle up are accumulauntil nk(t) reaches its maximumFk , and then steps down tozero in the second half of the cycle.

However, this picture of high frequency features superiposed over long-period modulation is not universal. Fsmallk, or for one of the most interesting cases ofq@1 andmoderatek, the occupation number of fermions jumps btween 0 and 1 within a time interval much shorter thanperiod of background oscillations, as depicted by the dotcurve in Fig. 1. There are interesting situations when ferons are created in an instant~single kick! process, whereformula~19! is not applicable. Therefore, for different rangof parameters we will develop different approaches.

C. Semi-analytic theory for averaged occupation number

Numerical curves for the mode functions suggest a spting of the time evolution into higher frequency features w

l-

6-5

cheee

a-

e

e-

i

thd

u

s

i-

ith

ing

lic

ortar-

ant

s.ns

e

-


the time scale comparable or less than the period of baground oscillations<T and low-frequency modulations witthe periodp/nk greater thanT. In this case we can utilize thgeneric result~19!. However, it is convenient to use not thvaluesnk(NsT), but rather the smoothed occupation numbnk(t) which is nk(t) averaged over high frequency oscilltions, nk(t)5(1/T)*t

(t1T)dtnk(t). Then we can write thesmoothed occupation number of fermions in a factorizform

nk~t!5Fksin2nkt, ~20!

where we introduce an envelope functionFk . The averageoccupation number of fermions evolvesperiodically withtime. The spectrum ofnk can be characterized by the envlope functionFk and the period of modulationp/nk whichdepend also on the parameterq.

Now we will utilize the result~19!. We found that theenvelope function can be extrapolated from the factorfront of sin2Nsdk in Eq ~19! and given by the expression

Fk51

sin2nkT

k2

2Vk2 @ Im Xk

(1)~T!#2. ~21!

Next we need to determine the frequencynk of the nk modu-lations. The valuedk defined after Eq.~19! cannot be theright answer because it would incorrectly predict thatpeaks ofFk are filled up first, while actually they are fillelast. We tried the combinationnk5p/22dk , because it cor-responds to the correct order of saturation ofFk , and itworks well. Therefore, the modulation frequencynk is givenby the relation cosnkT52ReXk

(1)(T). Thus, to findFk andnk , one need only calculate the complex valueXk

(1)(T) aftera single background oscillation, instead of performing a fnumerical integration of Eq.~17!.

We calculatedXk(1)(T) numerically for1

2 m2f2 theory andconstructed the envelope functionFk plotted in Fig. 5.~Asimilar graph was shown in@6# for 1

4 lf4 theory.! In Fig. 2we show, using Eq.~21!, how the fermionic resonance bandare filled after 10 background oscillations.

FIG. 5. The envelope functionFk for the amplitude of occupation number oscillations inmf

2 f2 theory. Values forq51024, 1022

and 1 are shown, from narrowest to broadest, respectively.

12351

k-

r

d

n

e

ll

The functionnk gives us the time scale for fermion exctation. In Fig. 6 we plot the period of modulationp/nk as afunction of k. This function is peaked whereFk is peaked;i.e., the peaks of the resonance curve are the last to fill.

D. Method of successive scatterings for fermions

It turns out that for large values of the parameterq we cansignificantly advance calculations ofnk(t) beyond the resultsof Sec. III C. One can expect thatq5h2f0

2/mf2 may be much

greater than 1. Indeed, in the context of chaotic inflation wthe potentialV(f)5 1

2 mf2 f2, it follows from the theory of

cosmological perturbations thatf02/mf

2 .1012. On the otherhand, one should certainly admit that the Yukawa couplcan beh@1026, which providesq@1.

We will generalize for fermions the method of paraboscatterings introduced for the bosonic resonance in Ref.@2#.The method is based on the observation that, forq@1, achange of the particle number occurs only during a shtime interval near the zeros of the effective mass of the pticles. This occurs because Eq.~15! for the number of par-ticles in terms of the mode functions is an adiabatic invariof the mode equation.

For largeq, the adiabaticity conditionVk8,Vk2 is violated

only near the timest* when the effective mass vanisheThis leads to a step-like evolution in the number of fermioas illustrated in Fig. 7 which shows the evolution ofnk(t)andMeff(t)5mc1hf(t).

Near the timest* , we may approximate the effectivmass in Eq. ~17! by (m1Aq f)5(1/mf)@mc1hf(t)#'(h/mf)f8(t* )(t2t* )1O„(t2t* )2

…. We may alsowrite this as

~m1Aq f !'Aq f8~t* !~t2t* !56Aq2m2~t2t* !,~22!

where the sign on the left hand side depends on whetherf (t)is increasing (1) or decreasing (2) at t* . Thus, in theneighborhood oft* , the mode equation becomes

FIG. 6. The logarithm of the modulation period~in units of theinflaton periodT) in mf

2 f2 theory, whereq51 corresponds to themiddle curve,q51022 corresponds to the dotted curve, andq51024 corresponds to the remaining.

6-6

a

th

tiohehe

-s

Th

esli-

o-e

ix

lu-

se,

the

m-


Xk91@k21~q2m2!~t2t* !22 i sgn@ f 8~t* !#Aq2m2 #Xk

50, ~23!

which is a Schro¨dinger-like equation for scattering offnegative parabolic potential centered att* and with acom-plex effective frequency. In the bosonic case@2#, the effec-tive frequency is real. Here, the absorptive character ofterm will ensure that the Pauli principle is respected.

The method of parabolic scattering uses the exact soluof Eq. ~23! to provide a connection formula between tadiabatic approximations of the full mode equation on eitside oft* . Suppose thej th zero ofMeff occurs at timet j .For times betweent j 21,t,t j , the general solution of themode equation~17! takes the adiabatic form

Xkj ~t!5ak

j N1 expS 2 i E0

t

dt Vk~t! D1bk

j N2 expS 1 i E0

t

dt Vk~t! D , ~24!

where the coefficientsakj and bk

j are constant fort j 21,t,t j . After the scattering att j , Xk(t), within the intervalt j,t,t j 11, again has the adiabatic form

Xkj 11~t!5ak

j 11N1 expS 2 i E0

t

dt Vk~t! D1bk

j 11N2 expS 1 i E0

t

dt Vk~t! D , ~25!

with new coefficientsakj 11 andbk

j 11 , which are constant fort j,t,t j 11. At t50, our vacuum positive frequency condition requiresak

151 and bk150. Particle creation occur

when, after scattering at the timest j , the initial positivefrequency wave acquires a negative frequency part.number density of produced particles with momentumk isnk

j 115ubkj 11u2 for timest j,t,t j 11. Furthermore, normal-

ization requiresuakj u21ubk

j u251 for all j.

FIG. 7. nk and Meff(t)5mc1hf(t) for q5103, k5Aq'31.6, andmc510mf .

12351

is

n

r

e

The important observation is that the outgoing amplitud(ak

j 11 , bkj 11) can be expressed through the incoming amp

tudes (akj ,bk

j ) by means of a scattering matrix

S akj 11e2 iu k

j

bkj 11e1 iu k

j D 5S A12Dk2e1 iwk 2Dk

Dk A12Dk2e2 iwkD

3S akj e2 iu k

j

bkj e1 iu k

j D . ~26!

Hereu kj 5*0

t jdt Vk(t) is the phase accumulated by the mmentt j . The form of the scattering matrix follows from thsymmetries of Eq.~23! and the~conserved! normalization oftheak andbk coefficients. In particular, the scattering matrdepends on only two, real,k-dependent functions:Dk andwk . We will determineDk below.

If we let x5(q2m2)1/4t, Eq. ~23! may be written

d2Xk

dx21@Lk

22 i ~21! j1x2#Xk50, ~27!

where the parameter

Lk2[

k2

uAq• f 8~t* !u5

k2

Aq2m2. ~28!

A general analytic solution of Eq.~27! is a linear combina-tion of the parabolic cylinder functionsW„2@Lk

22 i(21) j #/2;6A2x…. The scattering functionsDk and wk canbe found from the asymptotic forms of these analytic sotions. In particular,

Dk5~21! je2pLk2/2. ~29!

The momentum dependent phasewk is similar to that derivedin @2# for the bosonic case. As we will not need this phawe don’t write its form here. Substituting Eq.~29! into Eq.~26!, we can determine the change induced in theak andbkcoefficients by a single parabolic scattering in terms ofparameters of the parabolic potential and the phaseu k

j . Spe-cifically, we find

S akj 11

bkj 11D 5S A12e2pLk

2eiwk 2~21! je2pLk

212iu k

j /2

~21! je2pLk222iu k

j /2 A12e2pLk2e2 iwk D

3S akj

bkj D . ~30!

It is now a simple matter to find the change in particle nuber after one scattering. From the normalization ofak andbkwe have the relation

6-7

. I

se

eioti

m

qob

e

teen-on

e

o

ryb

danin

nt.

is

as in

ge

n

tionu-n

pa-


~a* kj 11 b* k

j 11!S 1 0

0 21D S akj 11

bkj 11D

5uaku22ubku25122nkj 11, ~31!

which, when applied to Eq.~30!, gives the desired result

nkj 115e2pLk

21~122e2pLk

2!nk

j

22~21! je2pLk2/2A12e2pLk

2Ankj ~12nk

j !sinu totj ,

~32!

where the phaseu totj 52u k

j 2wk1argbkj 2argak

j . Let usdiscuss this formula, which is the main result of our paperfermions are light,mc!mf , their occupation number ischanging with time only when the background field croszero. Without expansion of the universe, the phasesu accu-mulated between successive zeros are equal. In this cascan try to proceed to find the solution of the matrix equat~26!, as was done for bosons@2#. However, for bosons thawas needed to find the stability-instability bands, whichnot so interesting for fermions. In the case of massive ferons, the time intervals between successive zeros ofMeff arenot equal, and the problem of finding a matrix solution of E~26! becomes even more complicated. However, in the minteresting case of an expanding universe, the phasestween successive zeros ofMeff will become random~see@2#for details!. Then we can just treatu tot as a random phasand use formula~32! as it is.

IV. PARAMETRIC EXCITATION OF FERMIONS WITHEXPANSION OF THE UNIVERSE

To address the problem of fermionic preheating afm2f2 chaotic inflation, we must now deal with the full modequation~9!, which no longer has a periodic time depedence in the complex frequency. Nevertheless, it is still cvenient to work with the form~17! in the time variablet5mft where the parameterq is now understood to be timdependent:

Xk91Fk2

a21Meff

2

mf2 2 i

~a Meff!8

amf1

D~a!

mf2 GXk50, ~33!

where the prime stands for the derivative with respect ttandk is a comoving momentum scaled in units ofmf . Of-ten, another form of the fermionic mode equation~7!, writtenin terms of conformal timeh5*dt/a and mode functionYk5a23/2xk , is used:

]h2Yk1@k21Meff

2 2 i ]h~a Meff!#Yk50. ~34!

This form is useful, for instance, in the conformal theoV(f)5(l/4)f4 when the problem can be reduced to a prolem in Minkowski space-time@6#. In this case, backgrounfield oscillations are periodic with respect to the conformtime, h. However, in the case of a quadratic inflaton potetial, the background field oscillates with a constant periodphysical timet ~or t). Thus, the use of the mode equatio

12351

f

s

onen

si-

.ste-

r

-

-

l-n

~33! is preferable. In this equation,Meff5@mc1Aq f(t)# butthe parameterq is now understood to be time dependeSpecifically, we haveq(t)[h2F2(t)/mf

2 and the scaledphysical momentump[k/a[(1/a)k/mf . Here, F(t) isthe time dependent amplitude of inflaton oscillations. Aswell known ~cf. @2#!, oscillations off in this model quicklyapproach the asymptotic solutionf(t)5F(t) cos(t), F(t)'f0 /a3/2;Mpl /A3p(11t) where time is measured fromthe start of inflaton oscillations,f0'Mpl /A3p, and thescale factor averaged over several oscillations behavesa matter dominated universe:a(t)5(11t)2/3.

A. Stochastic parametric excitation of fermions

Let us first consider the case of light fermions with a larinitial resonance parameter:hf0@mf@mc . Figure 8 showsa numerical solution of Eq.~17! in the absence of expansiowith a resonance parameterq5106. As expected for such alarge q parameter, we see a step-like change in occupanumber with periodic modulation. In Fig. 9, we show a nmerical solution to Eq.~33! in the presence of expansiowith the initial parameterq05106. Here, thecomovingnum-


without expansion. Hereq5106 and the particular mode isk2

'Aq.


with expansion taken into account. Here the initial resonancerameter isq05106 and the mode isk2'Aq.

6-8

ee

-

he

m

ts

u-

sdue-

-m.t

t

ceene

ed

oftricuiter-the

nicto, iffor

ton

in-ess;

aseice-

odermcfor-lyringse-

ion

s.fill

i-

iontionof

ine

ls


ber density of particles~15! is plotted. We see that, as for thnon-expanding case, the evolution between zeros of thefective mass~or, equivalently in this approximation, the inflaton field!, Meff'mf Aq(t)cos(t), is adiabatic. We canthus apply the results of Sec. III D and, in particular, tanalytic formula~32!, mutatis mutandis. The Pauli principleis obviously still obeyed and the typical step in particle nu

ber is still suppressed by a factore2pLk2, which is now time

dependent. The most important qualitative change is thataccumulated phaseu tot is now uncorrelated between succesive zeros of the effective mass~inflaton field!.

This occurs because the effective frequencyVk

5Ap21q2(t)cos2t is no longer periodic and the accumlated phaseu5* j

j 11Vk'2hF(t)/m1O(k2) changes sub-stantially in magnitude within one inflaton oscillation,duk

.Aq/2Ns2 , after theNsth oscillation @2#. The result is that

the sin(utot) term in Eq.~32! becomes a random variable. Ais readily apparent in Fig. 9, this destroys the periodic molation of nk and the parametric excitation of fermions bcomes stochastic, as anticipated in@6# and reported in@8#.

Once the periodic modulation ofnk is destroyed, the construction of Sec. III C is no longer valid: the occupation nuber cannot be characterized by an amplitude and periodfact, the spectrum of created particles is even simpler. Schastic excitation allows a given comoving modek to obtainany amplitude in the range 0<nk<1 if there are a sufficiennumber of parabolic scatterings,Ns . This gives us the pic-ture of stochastically filling a sphere in momentum spaOne might call this a non-degenerate Fermi sphere. Numcal calculations confirm this picture; see Fig. 10. Let us fithe radius,ks , of this sphere. A comoving mode will bexcited ifLk

2(t)<1/p. SinceLk2(t)5k2/a2Aq(t), we have

ks2

a2Aq~t!5

ks2

Aa q0

<1

p. ~35!

Therefore, for light fermions, the comoving radius of excitmodes increases with time asks;a1/4 and scales asks

;q01/4a1/4.

FIG. 10. The comoving occupation number of fermionsmf

2 f2 inflation after 20 inflaton oscillations for initial resonancparameterq05106 andmc50.1mf . Expansion destroys the detaiof the resonance band and leads to a Fermi-sphere of widthq1/4

which grows likea1/4 while q(t).1.

12351

f-

-

he-

-

-Ino-

.ri-d

As a result of the expansion, the amplitudeF(t), andthus the parameterq(t), decreases. Onceq(t) is of orderunity, the excitation is no longer strong, and redshiftingfermion modes will be fast enough to prevent parameexcitation. By this moment, the Fermi sphere will have qexpanding. Eventually, fermions will be produced in the pturbative regime, but if the fermionic mass is non-zero,perturbative decay can only occur for 2mc,mf .

B. Analytic results for the production of supermassive fermions

One of the most significant differences between bosoand fermionic parametric amplification is the possibilitycreate superheavy fermions from much lighter inflatonsthe inflatons are oscillating coherently. As we have seen,largeq, massive fermions are created not when the inflafield itself crosses zero, but when the combinationMeff5mc1hf(t) crosses zero. This is because, at thosestances, even very heavy fermions are effectively massltheir bare mass is ‘‘compensated’’ by the large valuehf, iff can have such a large amplitude. This is exactly the cwhen f is the inflaton field. For example, in the chaotinflationary scenario, the amplitude of its oscillations immdiately after inflation can be as large as 0.1Mpl .

In this case, we again have a situation where the mfunction, now for heavy fermions, has the adiabatic fobetween zeros ofMeff . It can be described with paraboliscattering around these instants. Therefore, our generalmula ~32! works for superheavy fermions as well. The onmodification is that the phase between successive scattewill be defined by the integral over the time intervals btween them. In formula~32! the parameterLk(t) will be

Lk2~t!5

k2

a2Aq0

a3 2m2

. ~36!

In the case of massive fermions, the criteria for excitatwill be

ks2

a2Aq0

a3 2m2

<1

p, ~37!

instead of Eq.~35!. For largeq0, the fermion mass termm2

will be negligible at the start of background oscillationThus, as in the last section, we expect a Fermi sphere towith a radiusks5q0

1/4a1/4. It is easy to see from the denomnator in Eq. ~36! that as the scale factor increases andqdrops,ks

2 will reach a maximum,km . This maximum occurs

when the scale factor isam5(q0 /4m2)1/3. This giveskm2

5(A3p)(q0 /4Am)2/3. Notice thatkm scales asq01/3.

It is tempting to say thatkm will be the final width of theband in comoving momentum space. Numerical solutshows, however, that this is not the case. Particle creacontinues to occur fork.km because there are a numberbackground oscillations for whichq(t)@1 remaining after

6-9

th

re

ern

i-in

d

oa

that

u

asrum

there-

is

ein

n’’:icass

gle

-u-ickr-re

es-ns

fored aswa

in


the time corresponding toa5am . Let af denote the scalefactor when the denominator in Eq.~36! goes to zero. It iseasy to show thataf541/3am . Now let the number of infla-ton oscillations that occur betweena51 anda5am be Nm .Then, one can also show that the number of oscillationsoccur betweenam andaf will be approximately 4Nm . In thistime, even modes suppressed exponentially ase2pLk canachieve significant occupation numbers due to the pexponential factor which accumulates asN. For nk!1, thegeneral result~32! tells us that the change in particle numbafter each kick is given approximately by just the first, spotaneous emission, term. That is, for large momentum,dnk

j

5nkj 112nk

j '1e2pLk!1 for each kick. Thus, we can estmate the range of appreciable particle creation by summthe contribution from all the kicks betweenam and af andrequiring that this beO(1):

(j

dnkj '4NmexpS 2p

k2

km2 D;O~1!, ~38!

where, in the second step, we have approximated thenominator ofLk by its maximum value:km

2 . Note that, con-trary to the light fermion case, there is no perturbative endthe particle creation process. The excitation of superhefermions is abruptly terminated nearaf when q0 /a32m2

approaches zero.Using Eq.~38! with the multiplicative factor and taking

the right-hand side to be exactly 1, we estimate thatspectrum of created supermassive particles after the creprocess terminates is given by the formula

^nk&551

2for k,km,

1

2expS 2p

~k2gkm!2

km2 D for k>km,

~39!

where 2g25 ln(q0 /p2m2). This formula is valid for Nmgreater than a few. In Fig. 11 we plot the numerically calclated final spectrum of supermassive fermions forq05106

FIG. 11. Final spectrum of heavy fermions with massmc

510mf whenq05106. Also shown is the analytic estimate~39!.

12351

at

-

-

g

e-

fvy

eion

-

andmc510mf and the analytic formula~39!, which are in agood agreement. We verified this for several values ofq0

andm and found that, as expected, the formula works welllong as 4m2 is not too close toq0. This result shows that, fosuperheavy particles in an expanding universe, the maximradius of thek sphere scales withq0 asgkm;q0

1/3(ln q0)1/2,

which, up to the logarithmic factor, is in agreement with@9#.However, one should also check that at the moment whenexcitation of superheavy fermions terminates, their backactionh^cc& to the dynamics of the inflaton oscillationsstill negligible.

C. Fermionic production from a single kick

As we have seen, formula~32! can be extended to thcase of an expanding universe. Consider the very first termthat formula. This term describes ‘‘spontaneous emissiothe creation of fermions by the inflaton field in the fermionvacuum state. This is the case when the effective mMeff(t) crosses zero for the first time. Thus, after a sinkick we have

nk5e2pLk2, ~40!

whereLk is as in Eq.~36! with the scale factor in the denominator evaluated at the time of the first kick. The occpation number of fermions generated from such a single kis plotted in Fig. 12. The two curves corresponding to fomula ~40! and to the numerical solution are shown and apractically indistinguishable. One can use this formula totimate the final occupation number of superheavy fermiowhen 4mc

2 is not much smaller thanq0, that is whenNm isonly O(1) and the cumulative effect of kicks afteram , asdescribed in the last section, is insignificant.

V. DISCUSSION

In this paper we developed a non-perturbative theoryfermion production by an oscillating inflaton field. As whave seen, the production of fermions can be characterizeparametric excitation. Even in the simple model of a Yukacoupling between fermions and a background scalar field

FIG. 12. The comoving occupation numbernk of fermions afterMeff crosses its first zero forq5106 andmc550mf . Also shownis the analytic formula~40! which fits perfectly.

6-10

itciar

nc-n-

a-ap

io

t

na-fet i

r-fesiec

i-

e.ex

ivce

th-

i-

the

Thehen

ce

x-pe-fnn-

ekick

r

s.rum

ethe

nderknt


an expanding universe oscillating around the minimum ofquadratic potential, the theory of fermionic parametric extation is rich and leads to important results. Fermionscreated very quickly, within about ten~s! oscillations, in anout-of-equilibrium state. For large values of the resonaparameterq5h2f0

2/mf2 , the occupation number of light fer

mions (mc,mf!hf0) changes in a step-like manner at istancest j when the inflaton amplitudef(t j ) passes throughzero, j 51,2,3, . . . . Wehave developed the method of parbolic scattering for fermions, based closely on a similarproach for the bosonic resonance@2#. It is possible to derivea unified recursive formula, which relates the occupatnumber of fermions or bosonsnk

j 11 at the momentt j 11 tothe earlier valuenk

j :

nkj 115e2pLk

21~162e2pLk

2!nk

j

22~21! je2pLk2/2A16e2pLk

2Ankj ~16nk

j !sinu totj ,

~41!

For bosons, one should use the upper sign and neglecfactor (21) j in the third term@2#, while for fermions, oneshould use the lower sign. For light fermions and bosoLk

25k2/Aqmf2 . For largeq the angleu tot

j can be treated asrandom phase. As a result, formula~41! predicts the stochastic character of parametric excitation of both bosons andmions. In the fermionic case it leads to the conclusion thamomentum space, fermions chaotically fill a broad sphereradiusks;q1/4mf . This formula clearly shows several inteesting features of particle creation. For both bosons andmions, the first term corresponds to spontaneous emisand leads after the first half inflaton oscillation to the sptrum nk5e2pLk. For bosons withnk@1, we see that(nk

j 112nkj )}nk

j , corresponding to stimulated emission. Fnally, for fermions, we find that ifnk

j 51, the next value willalways benk

j 11512e2pLk,1 even in the stochastic casThis prevents the occupation number of fermions fromceeding 1 and protects the Pauli principle.

Formula ~41! can be extended to the case of massbosons and fermions. However, here important differenemerge. For bosons we haveLk

25(k21mb2/Aqmf

2 ), wheremb is thex-boson mass. This leads to the conclusion thatcreation of superheavy,mb.q1/4mf , bosons is exponentially suppressed. However, for fermions we haveLk

2

5k2/(mfAq mf2 2mc

2). Therefore, even superheavy ferm

v.n-

v.

n,

12351

s-e

e

-

n

the

s

r-nof

r-on-

-

es

e

ons with a mass as large asAq mf5hf0 can be created inabundance from the coherent inflaton oscillations@8,9#. Thisoccurs because the effective mass of fermions is given byalgebraic combinationmc1hf(t) and the creation of fermi-ons occurs when this effective mass goes to zero.method of parabolic scatterings, applied to instances wthis happens, leads to both formulas~41! with the appropri-ateLk . The maximum radius in comoving momentum spais found to scale asq0

1/3(ln q0)1/2 for massive fermions.

There are situations where a single instance~single kick!of particle creation may lead to interesting effects. An eample is the scenario of instant preheating, which is escially important for inflationary models without minima othe inflaton potential@10#. Another example is the interactioof the inflaton with superheavy fermions during the inflatioary stage when the combinationmc1hf(t) can go throughzero only once. The generic formula~41! embraces the caswhere bosons and fermions are created by such a single

as well. One takesj 51, nk150, and thennk

25e2pLk2. Rather

than just the parameterq, the effect is described by anothecombination of the coupling constants,f0, andmf . In thiscase the effect is defined by the velocityf* at the momentof creationt* , which is different for bosons and fermionFor bosons, a single instance of creation gives the spect@2,10#

nk5e2p(k21mx2)/gf

* ~42!

where t* corresponds tof(t* )50. For fermions, a singleinstance of creation gives@11#

nk5e2pk2/hf* ~43!

wheref(t* )1mc /h50.We believe that the theory of fermionic preheating will b

an important ingredient of realistic scenarios describingprocess of reheating after inflation.

ACKNOWLEDGMENTS

Collaboration and discussions with Andrei Linde aAlexei Starobinsky significantly influenced this work. Walso thank Ju¨rgen Baacke for useful discussion. This wowas supported by NSERC, CIAR, and NATO Linkage Gra975389.

s..

h

@1# L. A. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. ReLett. 73, 3195 ~1994!; Y. Shtanov, J. Trashen, and R. Bradenberger, Phys. Rev. D51, 5438~1995!; D. Boyanovsky, H.J. deVega, R. Holman, D. S. Lee, and A. Singh,ibid. 51, 4419~1995!.

@2# L. A. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. ReD 56, 3258~1997!.

@3# S. Khlebnikov and I. Tkachev, Phys. Rev. Lett.77, 219~1996!.@4# D. Boyanovsky, M. D’Attanasio, H. J. de Vega, R. Holma

and D. S. Lee, Phys. Rev. D52, 6805~1995!.

@5# J. Baacke, K. Heitmann, and C. Pa¨tzold, Phys. Rev. D58,125013~1998!.

@6# P. Greene and L. Kofman, Phys. Lett. B448, 6 ~1999!.@7# R. Kallosh, L. Kofman, A. Linde, and A. Van Proeyen, Phy

Rev. D61, 103503~2000!; G. F. Giudice, I. Tkachev, and ARiotto, J. High Energy Phys.08, 009 ~1999!.

@8# P. Greene, inCOSMO–98, edited by D. Caldwell, AIP Conf.Proc. No. 478~AIP, Woodbury, New York, 1999!, pp. 72–74;hep-ph/9905256.

@9# G. Giudice, M. Peloso, A. Riotto, and I. Tkachev, J. Hig

6-11

D

gy

y,

g,


Energy Phys.08, 014 ~1999!.@10# G. Felder, L. Kofman, A. Linde, Phys. Rev. D59, 123523

~1999!.@11# D. Chung, E. Kolb, A. Riotto, and I. Tkachev, Phys. Rev.

62, 043508~2000!.@12# J. Garcia-Bellido, S. Mollerach, and E. Roulet, J. High Ener

Phys.02, 034 ~2000!.@13# P. B. Greene, L. Kofman, A. D. Linde, and A. A. Starobinsk

12351

Phys. Rev. D56, 6175~1997!.@14# J. Baacke and C. Patzold, Phys. Rev. D62, 084008~2000!.@15# M. V. Mostepanenko and V. M. Frolov, Yad. Fiz.19, 451

~1974! @Sov. J. Nucl. Phys.19, 885 ~1974!#; A. A. Grib, S. G.Mamayev, and V. M. Mostepanenko,Vacuum Quantum Ef-fects in Strong Fields~Friedmann Laboratory, St. Petersbur1994!.

6-12

theory of fermionic preheating

Documents