Inflation and Preheating in Supergravity with MSSM Flat Directions

0.35 MB·English

by  Anna Kaminska

In(cid:29)ation and Preheating in Supergravity with MSSM Flat
Dire
tions
Anna Kami«ska and Paweª Pa
hoªek
Institute of Theoreti
al Physi
s, Fa
ulty of Physi
s,
University of Warsaw, Ho»a 69, Warsaw, Poland
9
0
0 Abstra
t
2
Motivated by a re
ent dis
ussion about the role of (cid:29)at dire
tions, a typi
al feature of

n
supersymmetri
models, in the pro
essof parti
le produ
tionin the earlyuniversea
onsis-
a
tent model of in(cid:29)ation and preheating in supergravity with MSSM (cid:28)elds has been built. It
J
is based on a model proposed by M. Kawasaki, M. Yamagu
hi and T. Yanagida. In the in-
5
(cid:29)ationarystage,the(cid:29)atdire
tionsa
quirelargeva
uumexpe
tationvalues(VEVs) without

] spoiling the ba
kgroundof slow-roll,high-s
alein(cid:29)ation
onsistent with the latest WMAP5
h
observationaldata.Inthestageofparti
leprodu
tion,naturallyfollowingin(cid:29)ation,therole
p
- of (cid:29)at dire
tion large VEVs depends strongly on e(cid:27)e
ts
onne
ted with the supergravity
p framework and non-renormalizable terms in the superpotential, whi
h have been negle
ted
e
so far in the literature. Su
h e(cid:27)e
ts turn out to be very important,
hanging the previous
h
[ pi
ture of preheating in the presen
e of large (cid:29)at dire
tion VEVs by allowing for e(cid:30)
ient
preheating from the in(cid:29)aton.
1
v
8
7
4 1 Introdu
tion
0
.
1 In(cid:29)ation was introdu
ed asa natural and simple way of solving the problems of
lassi
al
osmol-
0 ogy - the initial
onditions problem (or the (cid:29)atness and horizon problems) and the explanation
9
of the origin of primordial density (cid:29)u
tuations [1, 2, 3, 4, 5, 6, 7, 8℄. The easiest way to obtain
0
: in(cid:29)ationisbyintrodu
ing asingles
alarin(cid:29)aton(cid:28)eldwithaslowlyevolvingva
uumexpe
tation
v
i value [2, 3℄. In order to obtain a proper period of Big Bang Nu
leosynthesis however, one has to
X
end in(cid:29)ation by parti
le produ
tion. The pro
ess of reheating must
onne
t the in(cid:29)aton se
tor
r
a to the observable se
tor [9, 10, 11℄.
Inordertoproperlydes
ribein(cid:29)ationandparti
leprodu
tion onehasto
onsidertheunderlying
theory of parti
les and intera
tions. Supersymmetry is one of the most promising extensions of
the Standard Model (SM) [12, 13, 14, 15, 16, 17℄, and it has triggered a sear
h for supersymmet-
ri
models of in(cid:29)ation and reheating. One of the typi
al features of supersymmetri
extensions
of the SM is the presen
e of (cid:29)at dire
tions [18℄ - dire
tions in (cid:28)eld spa
e, along whi
h the s
alar
potential identi
ally vanishes in the limit of unbroken global supersymmetry. Due to large quan-
tum (cid:29)u
tuations or the
lassi
al evolution of (cid:28)elds during in(cid:29)ation (cid:29)at dire
tions
an easily
a
quire large VEVs [19, 20℄. Therefore, there is a natural question about the role of su
h large
VEVs in the pro
ess of parti
le produ
tion.
It was postulated in ref. [21℄, that large (cid:29)at dire
tion VEVs in(cid:29)uen
e the pro
ess of parti
le pro-
du
tion by blo
king preheating from the in(cid:29)aton - the phase of rapid, non-perturbative in(cid:29)aton
de
ay. In ref. [21℄ a simple toy model was proposed
V Aϕ2χ2+Bmϕχ2+Cα2χ2,
⊃ (1)
1
ϕ α χ
where is the in(cid:29)aton (cid:28)eld, parameterizes the (cid:29)at dire
tion and represents the in(cid:29)aton
de
ay produ
ts (in this model a dire
tion in Higgs (cid:28)elds has been
onsidered). Then, after mode
χ k
de
omposition of the (cid:28)eld , the energy of the mode with momentum is given by:
ω2 = k2+2A ϕ 2+2Bm ϕ +2C α 2.
k
h i h i h i (2)
χ ω
k k
In general, non-adiabati
produ
tion of parti
les is e(cid:30)
ient only when
hanges non-
adiabati
ally
ω˙
τ > 1 preheating,
| | ≡ (cid:12)ω2(cid:12) ↔ (3)
(cid:12) (cid:12)
τ (cid:12) (cid:12) ω
(cid:12) (cid:12) k
where the adiabati
ity parameter is introdu
ed. During
lassi
al preheating is dominated
by the in(cid:29)aton VEV and
hanges non-adiabati
ally due to in(cid:29)aton os
illations. In the presen
e
ω
k
of (cid:29)at dire
tions however,
ould be dominated by the large VEV of the (cid:29)at dire
tion. If this
VEV
hangesveryslowlyin
omparisonwiththeevolutionofthein(cid:29)atonVEV,non-perturbative
χ
produ
tion of parti
les is e(cid:27)e
tively blo
ked.
However, as was pointed out in ref. [22℄, blo
king of preheating from the in(cid:29)aton does not o

ur
when non-perturbative produ
tion of parti
les from the (cid:29)at dire
tion itself is possible. Then
the initially large VEV of the (cid:29)at dire
tion de
reases rapidly, unblo
king preheating from the
in(cid:29)aton. In ref. [22℄ a method was introdu
ed of
al
ulating the amount of parti
les produ
ed
non-perturbatively from the (cid:29)at dire
tion, due to non-adiabati

hanges of the mass matrix
eigenve
tors and eigenvalues related to quantum (cid:29)u
tuations around the (cid:29)at dire
tion. This led
to a dis
ussion (see refs [23, 24, 25, 26℄) about whether non-perturbative de
ay of (cid:29)at dire
tions
andpreheatingfromthein(cid:29)atonispossible.Thedis
ussionwasbasedonsomegeneralproperties
of (cid:29)at dire
tions in a global supersymmetry framework. It did not
onsider any spe
i(cid:28)
model
of in(cid:29)ation and did not propose any model of a
quiring large VEVs by (cid:29)at dire
tions. Therefore
it was di(cid:30)
ult to study the whole issue and determine how the large VEVs of (cid:29)at dire
tions
develop and evolve, and how they impa
t the pro
ess of in(cid:29)ation and parti
le produ
tion.
The goal of our work is to
onstru
t a
onsistent model of in(cid:29)ation and parti
le produ
tion
in a realisti
supersymmetri
extension of the Standard Model, and
onsider in this spe
i(cid:28)
model the behavior of MSSM (cid:29)at dire
tions. Therefore a realisti

haoti
in(cid:29)ation model with
two representative (cid:29)at dire
tions is
onstru
ted. In order to be able to predi
t the evolution
of (cid:29)at dire
tion VEVs it was de
ided to study the produ
tion of large (cid:29)at dire
tion VEVs by

lassi
al evolution during in(cid:29)ation, and therefore a potential for the (cid:29)at dire
tion is required.
Followingrefs[19,20℄weadoptthesupergravityframeworkwithanon-minimalKählerpotential,
whi
h results in a potential for the (cid:29)at dire
tion with a time-evolving minimum at large VEVs
during in(cid:29)ation. It alsoenables usto
al
ulate the previouslynegle
ted in(cid:29)uen
e of supergravity

orre
tionsfor(cid:29)atdire
tionevolutionduringin(cid:29)atonos
illations.Wealso
onsidertheimpa
tof
existen
e of non-renormalizable terms, whi
h has also been negle
ted so far. We (cid:28)nd that these
e(cid:27)e
ts strongly in(cid:29)uen
e the pro
ess of parti
le produ
tion by introdu
ing e(cid:30)
ient
hannels of
non-perturbative parti
le produ
tion both from the (cid:29)at dire
tion and the in(cid:29)aton. As a result
the originally large (cid:29)at dire
tion VEVs are diminished, preheating from the in(cid:29)aton is allowed
and the energy density of the Universe is dominated by the in(cid:29)aton de
ay produ
ts.
2 Building the model
The model
onsidered in this paper, after negle
ting all (cid:28)elds ex
ept for the in(cid:29)aton, redu
es
V = m2ϕ2/2
to the simplest
haoti
in(cid:29)ation model with the in(cid:29)aton potential . This property
together within(cid:29)aton domination provides appropriate slow-roll in(cid:29)ation andavalueofspe
tral
index whi
h is in agreement with the WMAP5 data [27℄. Obtaining su
h a property in a model
whi
h is based on supergravity is not straightforward due to
ompli
ated supergravitational F-
η
terms. A solution to this problem (the so
alled -problem) was proposed by [28℄ and is used in
2
Φ
this paper. A

ording to the solution, ex
ept for the
hiral in(cid:29)aton super(cid:28)eld and the MSSM
X
super(cid:28)elds, the model
ontains one additional
hiral super(cid:28)eld .
Further
onsideration is restri
ted to s
alar (cid:28)elds and the same symbol is used to denote both
the
hiral super(cid:28)eld and its
omplex s
alar
omponent. The following de
omposition in real
(cid:28)elds is used
Φ = (η+iϕ)/√2,
(4)
X = xeiβ.
(5)
ϕ
The (cid:28)eld plays the role of the in(cid:29)aton.
K
We follow [28℄ in
onstru
ting the Kähler Potential and take
1
K (Φ+Φ )2+XX .
∗ ∗
⊃ 2 (6)
The formula (6) for the Kähler potential was obtained by [28℄ as follows. The (cid:28)rst step was to
α
introdu
e a Nambu-Goldstone-like shift symmetry of the in(cid:29)aton
Φ Φ+iCM,
→ (7)
C
where is a dimensionless real parameter. A Kähler potential whi
h is invariant under this
U(1) Z
R 2
symmetry and the additional × symmetry must have the general form [28℄
K(Φ,Φ ,X,X ) = K[(Φ+Φ )2,XX ].
∗ ∗ ∗ ∗
(8)
The formula (6) is just the lowest order term in the general expansion of the formula (8). It

an be easily seen that a theory with an exa
t Nambu-Goldstone-like shift symmetry has no
ϕ
potential for the in(cid:29)aton . Therefore this symmetry has to be broken, but not in the Kähler
η
potentialinordertoavoidthe -problem.Following [28℄,weintrodu
eashiftsymmetrybreaking
W
term in the superpotential
W mXΦ.
⊃ (9)
m ϕ
It gives mass to the in(cid:29)aton .
H H
u d
There are two MSSM-(cid:29)at dire
tions
onsidered in this paper: a dire
tion in Higgs (cid:28)elds
u d d i j k
i j k
(only D-(cid:29)at) and a dire
tion in squark (cid:28)elds, where indexes , and are some family
k = j χ H H
u d
indexes ( 6 ). Let be the
omplex s
alar (cid:28)eld that parametrize the dire
tion
1 χ 1 0
H = , H = .
d u
√2(cid:18) 0 (cid:19) √2 (cid:18) χ (cid:19) (10)
α u d d
i j k
Let be the
omplex s
alar (cid:28)eld whi
h parametrizes the dire
tion
1
uβ = dγ = dδ = α.
i j k √3 (11)
β = γ = δ = β u d d
i j k
In equation (11) 6 6 6 are (cid:28)xed
olor indexes. The
omponents of (cid:28)elds , and
χ
with other
olor indexes are equal to zero. It is
onvenient to de
ompose the
omplex (cid:28)elds
α
and into real (cid:28)elds in the following way
χ =ceiκ,
(12)
α= ρeiσ.
(13)
The full Kähler potential is
1
K = (Φ+Φ )2+XX +K +K ,
∗ ∗ MSSM NM
2 (14)
3
K K
MSSM NM
where is a standard minimal Kähler potential and is a non-minimal part of the
form K = a XX
NM M2 ∗·
Pl
(H+H +H+H +u+u +d+d +d+d ). (15)
· u u d d i i j j k k
M a
Pl
Here is the Plan
k mass and is a dimensionless parameter. The existen
e of
ouplings like
K
NM
is guaranteed in the presen
e of Yukawa
ouplings, sin
e they are ne
essary
ounterterms
K
NM
for operators generated by loop diagrams [19, 29, 30℄. Terms in
ause the existen
e of
M
Pl
minima in the s
alar potential for both (cid:29)at dire
tions, whi
h are of the order of . Therefore
M
Pl
(cid:29)at dire
tions
an naturally aquire large VEVs of the order of by falling into these minima.
For the superpotential we take
W = mXΦ+2hXH H +W +W .
u d MSSM NR
· (16)
2hXH H
u d
The term · is one of only two possible renormalizable
ouplings between the (cid:28)eld
X 2hXH L
′ u
and MSUGRA (cid:28)elds. The se
ond one is · . These two
ouplings
annot
oexist in
the model unless R-parity is broken. We assume that the in(cid:29)aton does not
ouple to MSSM
(cid:28)elds in the Kähler potential and in the superpotential to avoid strong deviations from the
W
MSSM
slow-roll in(cid:29)ation regime with in(cid:29)aton domination. The term is the standard MSSM
superpotential, given by
W = µH H +λlmH Q u +λlmH Q d .
MSSM u· d u u· l m d d· l m (17)
W
NR
The last term is a non-renormalizable part of the superpotential and it has the following
form λ 3√3λ
W = χ (H H )2+ α (u d d ν ).
NR u d i j k R
MPl · MPl (18)
λ λ ν
χ α R
Here and are dimensionless
onstants and is a right-handed neutrino of any given
W
NR
generation. Two non-renormalizable terms
ontained in are the only terms of 4th order in
the (cid:28)elds, whi
h may be relevant for the evolution of VEVs of the two
hosen (cid:29)at dire
tions.
3√3λ (u u d ν )/M
′α i j k R Pl
The term gives no
ontribution to the s
alar potential, unless one ad-
ditionally
onsiders VEVs of some other (cid:29)at dire
tions.
W
NR
Terms in modify minima for (cid:29)at dire
tions, whi
h are shifted away from zero in the s
alar
K M
NM Pl
potential due totermsin .Theminima areno longer
loseto when
oupling
onstants
λ λ
χ α
and are su(cid:30)
iently large. Moreover, they evolve in time during in(cid:29)ation until they rea
h
their (cid:28)nal position at zero at the end of the in(cid:29)aton os
illations.
The soft SUSY-breaking terms in the s
alar potential have negligible e(cid:27)e
ts on our results.
Our initial
onditions are set for the time whi
h
orresponds to about 100 e-folds before the
end of in(cid:29)ation, sin
e only this period is essential for preheating. Initial values for the real (cid:28)elds
ϕ η x ρ c ϕ
, , , and do not require (cid:28)ne tuning. For the in(cid:29)aton (cid:28)eld the only
ondition whi
h
ϕ > M
0 Pl
needs to be satis(cid:28)ed is . It ensures that the number of e-folds is greater than 75, whi
h
is needed for in(cid:29)ation to solve the horizon problem and the (cid:29)atness problem [8℄. We took the
ϕ = 4M
0 Pl
initial value to
onsider only the last 100 e-folds before the end of in(cid:29)ation. Fields
η x ρ c M
Pl
, , and should be initially smaellxepr(thKan) be
ause they are present in Kähler potential,
espe
ially in the exponential fa
tor MP2l in the F-terms. Therefore initially large values of
η x ρ c
any of those (cid:28)elds should de
rease rapidly and the VEVs of the (cid:28)elds , , and should stay
η

on(cid:28)ned naturally below the Plan
k s
ale during in(cid:29)ation. In parti
ular the VEV of (cid:28)eld falls
to zero very qui
kly and, as we have
he
ked numeri
ally, does not have any noti
able in(cid:29)uen
e
η = 0
0
on the evolution of other (cid:28)elds. Therefore in our (cid:28)nal
al
ulations we simply put . The
x
evolution of the (cid:28)eld has also almost no in(cid:29)uen
e on the evolution of other (cid:28)elds, so we set
x = 0.01M ρ c
0 Pl
for it a quite arbitrary initial value . Fields and whi
h are absolute values for
4
α χ

omplex (cid:29)at dire
tion (cid:28)elds and are initially taken to be of the order of the Hubble param-
H
eter , whi
h is also the order of initial quantum (cid:29)u
tuations for those (cid:28)elds during in(cid:29)ation.
H 8πm2ϕ2/6M2
In the slow-roll regime with in(cid:29)aton domination, we have 0 ≈ q 0 Pl. After setting
m = 10 6M
− Pl
as in the simplest model of slow-roll in(cid:29)ation
onsistent with WMAP data we
H 8 10 6M ρ = c = 10 5M
0 − Pl 0 0 − Pl
get ≈ · . Therefore we take .
3 Classi
al evolution of (cid:28)elds
In order to study the pro
ess of a
quiring large VEVs by (cid:29)at dire
tions one has to
onsider the

lassi
alevolution of(cid:28)eldsduringin(cid:29)ation. A
lassi
aldes
riptionispossibledueto theslow-roll

hara
ter of the evolution. At the end of in(cid:29)ation, ex
itations around VEVs of the (cid:28)elds will
be
onsidered in order to determine the impa
t of large (cid:29)at dire
tion VEVs on the pro
ess of
parti
le produ
tion.
The
lassi
al evolution is determined by the equations of motion derived from the supergravity
Lagrangian on
e the initial
onditions have been set. The in(cid:29)aton equation of motion has the
simple form
ϕ¨+3Hϕ˙ +V, = 0.
ϕ
(19)
During in(cid:29)aton domination the main
ontribution to the s
alar potential V is of the form
1/2 m2ϕ2
, whi
h provides a standard
haoti
in(cid:29)ation ba
kground. Due to the shift symme-
ϕ eK
try the Kähler potential does not depend on , and so the supergravity
oe(cid:30)
ient does not
η

ontain the in(cid:29)aton (cid:28)eld. This solves the -problem and allows for high-s
ale in(cid:29)ation. Sin
e
only the last 80-100 e-folds of in(cid:29)ation have any observable
onsequen
es for the evolution of
the Universe, the initial value of the in(cid:29)aton VEV was
hosen in a way that allows the study
H H
u d
of this period of in(cid:29)ation. Due to the absen
e of any
oupling of the in(cid:29)aton with the
udd
and dire
tions in the Kähler potential (be
ause of the shift symmetry) the evolution of the
in(cid:29)aton is largely independent of the evolution of the (cid:29)at dire
tions. Be
ause of this property
udd H H
u d
the pro
ess of a
quiring large VEVs by or dire
tions will not spoil in(cid:29)ation.
udd
The equations of motion for X, and Higgs (cid:28)elds are more
ompli
ated due to the non-
minimal form of the Kähler potential for these (cid:28)elds. One
an expe
t however that the (cid:28)eld
VEVs will evolve toward a minimum of the s
alar potential. The s
alar potential in supergravity
is
ompli
ated as well. However one
an observe that all the (cid:28)eld VEVs ex
ept for the in(cid:29)aton
eK
VEV are naturaly
on(cid:28)ned below the Plan
k s
ale due to the fa
tor in the s
alar potential.
This fa
tor be
omes dominant in the s
alar potential at the Plan
k s
ale. Therefore the s
alar
exp(field2)
potential for all (cid:28)elds ex
ept the in(cid:29)aton rises steeply at the Plan
k s
ale as . As a
x, ρ, c
resultone
anexpe
t tobelessthanunity andexpand thes
alarpotential inthese(cid:28)elds.
X = xeiβ
For (cid:28)eld the s
alar potential has a minimum at zero. The term quadrati
in (cid:28)eld x
is given by the following approximate expression (assuming in(cid:29)aton domination and negle
ting

omplex (cid:28)elds phases)
m2ϕ2
V x2 m2+ f(a)ρ2+f(a)c2+O ρ2c2, ρ3, c3 ,
⊃ (cid:18) 2 (cid:19) (20)
(cid:0) (cid:2) (cid:3)(cid:1)
M = 1 f(a) a > 0 x
Pl
where for simpli
ity we have set and is positive for . The evolution of is
eK
naturally
on(cid:28)ned to low VEVs due to the supergravity term whi
h exponentially steepens
the potential for (cid:28)eld x at the Plan
k s
ale. The evolution of all other (cid:28)elds of the model is
x
independent of the
hoi
e of initial
onditions for and on the evolution of this (cid:28)eld. The only
X
role of (cid:28)eld in the model is providing appropriate s
alar potential for the in(cid:29)aton and both
(cid:29)at dire
tions.
In the global MSSM without
orre
tions
oming from supergravity or non-renormalizable terms
5
α = ρeiσ
the s
alar potential is independent of the (cid:29)at dire
tion . Adding the
orre
tions men-
α
tionedabove
reatesapotentialforthe(cid:29)atdire
tion.Atermquadrati
in inthes
alarpotential
is
reated by supergravity e(cid:27)e
ts and is sensitive to any non-minimal
ouplings in the Kähler
potential. Inthemodel presentedinthispaper thequadrati
termmentioned abovetakesduring
in(cid:29)aton domination (negle
ting the
omplex (cid:28)elds phases) the following approximate form
m2ϕ2
V ρ2 1 a+f(a)c2+f(a)x2+O x2c2, x3, c3 .
⊃ 2 − (21)
(cid:0) (cid:2) (cid:3)(cid:1)
a
In the equation above the parameter des
ribes the in(cid:29)uen
e of the non-minimal
oupling
in the Kähler potential. In supergravity with a minimal Kähler potential the s
alar potential
has a global minimum at zero for the (cid:29)at dire
tion. In this
ase a
quiring large (cid:29)at dire
tion
VEVs due to
lassi
al evolution is impossible. The non-minimal
oupling enables us to shift
a > 1
the minimum toward larger VEVs by an appropriate
hoi
e of . Then the
oe(cid:30)
ient of
ρ x, c < 1
the term quadrati
in in the s
alar potential be
omes negative for . For the purpose
a = 5
of numeri
al
al
ulations we set . The exa
t lo
ation of the minimum is determined by
ρ
higher-order terms in , whi
h
ome from both supergravity and non-renormalizable terms.
M eK
Pl
Supergravity alone stabilizes the minimum around due to the
oe(cid:30)
ient . The presen
e
λ
α
ofanon-renormalizable terms
aledby shiftsthisminimum toward lower VEVs,
hanging the
predi
ted (cid:29)at dire
tion VEVs at the end of in(cid:29)ation toward lower values. In order to study the
λ
α
e(cid:27)e
t of large (cid:29)at dire
tion VEVs on the pro
ess of parti
le produ
tion the value of should
not ex
eed unity.
Supergravity
orre
tions and non-renormalizable terms have a similar e(cid:27)e
t on the potential for
H H λ λ
u d χ α
the dire
tion. The interplay between and determines the di(cid:27)eren
es between the
H H udd
u d
evolution of the two (cid:29)at dire
tions ( and ) under
onsideration. We study two spe
i(cid:28)
s
enarios
λ λ udd H H
α χ u d
1. If ≪ then the dire
tion VEV be
omes large and the dire
tion VEV
drops to zero during in(cid:29)ation, whi
h
orresponds to the s
enario des
ribed in ref. [21℄ in
the limitof globalsupersymmetrywithout non-renormalizable terms.This
asewill enable
us to study the predi
tions of ref. [21℄ in a spe
i(cid:28)
s
enario and determine the impa
t of
supergravity
orre
tions and non-renormalizable terms, whi
h has not been
onsidered so
far.
λ λ
α χ
2. If ∼ then both dire
tions
an a
quire large VEVs during in(cid:29)ation and the impa
t of
non-zeroVEVsofthein(cid:29)aton
lassi
alde
ayprodu
tsonthepro
essofparti
leprodu
tion
and blo
king of preheating by (cid:29)at dire
tion large VEVs
an be studied.
10 4 1M
− Pl
By"large VEV"wemeanava
uumexpe
tationvalueoftheorder − ,whi
hislargein
H 10 7M
− Pl

omparison with the Hubble parameter ∼ at the beginning of in(cid:29)aton os
illations.
uud H H λ
u d α
Sin
e the a
quired value of the or dire
tion VEV is determined by the value of
λ λ λ
χ α χ
and respe
tfully, in order to
reate large VEVs of those dire
tions during in(cid:29)ation and
λ
should not ex
eed unity. For the purpose of numeri
al
al
ulations two sets of -parameters are

onsidered
λ = 10 7 λ = 1
α − χ
1. and
λ = 1 λ = 1
α χ
2. and .
In order to
he
k if
lassi
al evolution
an lead to large (cid:29)at dire
tion VEVs during in(cid:29)ation
we have
hosen small initial VEVs for both dire
tions, whi
h
orrespond to the average size of
δα, δc H
quantum (cid:29)u
tuations typi
al for the
onsidered period of in(cid:29)ation ( ∼ ). In order to
obtain numeri
al predi
tions in spe
i(cid:28)
s
enarios one has still to (cid:28)x two free parameters. The
6
m 10 6
−

hoi
e of the in(cid:29)aton mass ∼ is natural be
ause it implies a spe
tral index of energy
density (cid:29)u
tuations
onsistent with WMAP observations. Following the arguments of ref. [31℄
h h 10 5
−
we
hoose the in(cid:29)aton
oupling parameter to be small and of the order ∼ .
The
lassi
al evolution of (cid:28)elds obtained numeri
ally in the two s
enarios mentioned above is
η
presented below. It turns out that in both
ases the evolution of (cid:28)eld is irrelevant - the VEV
η
of the (cid:28)eld de
reases rapidly to zero and does not in(cid:29)uen
e the further evolution of the other
η = 0
(cid:28)elds. Hen
eforth we simplify our
al
ulations by setting .
λ = 10 7 λ = 1
α − χ
3.1 and
Preliminary numeri
al
al
ulations show that in this
ase some of remaining 7 real VEVs (after
η = 0
setting )
an be also negle
ted. First of all the s
alar potential does not depend on the
σ
phase and any initial velo
ity of this phase will fall to zero due to the Hubble fri
tion term.
udd ρ
Therefore the (cid:29)at dire
tion
an be e(cid:27)e
tively des
ribed by only its absolute value . The
c
VEV of the (cid:28)eld , the absolute value of Higgs (cid:29)at dire
tion, after an initial in
rease, falls to
β κ
zero quite rapidly. Thereafter the s
alar potential
eases to depend on and .
ϕ x c ρ
Hen
eforth we present more a

urate
al
ulations only for 4 real (cid:28)elds: , , and . All three
phases have been set to zero. In this s
enario the (cid:28)nal 100 e-folds of in(cid:29)ation are studied.
The in(cid:29)aton VEV evolves a

ording to standard
haoti
in(cid:29)ation. It smoothly de
reases during
x
in(cid:29)ation and starts to os
illate after its end, as
an be seen in (cid:28)gures (1) and (2). The (cid:28)eld
j
4
j
3
0.006
0.004
2
0.002
Mt
1 6.´107 8.´107 1.´108 1.2´108 1.4´108
-0.002
e -0.004
20 40 60 80 100 120 140
-0.006
ϕ
Figure1:Evolutionofthein(cid:29)aton(cid:28)eld dur-
ing in(cid:29)ation. Values on verti
al axes are ex- Figure2:In(cid:29)atonos
illationsattheendofin-
pressed in Plan
k masses and time on hori- (cid:29)ation. Values on verti
al axes are expressed
zontal axes is expressed in the approximate in Plan
k masses and time on horizontal axes
number of e-folds of in(cid:29)ation. is expressed in Plan
k times.
ϕ
behavesalmostidenti
ally tothein(cid:29)aton(cid:28)eld .Themaindi(cid:27)eren
eisthattheVEVofthe(cid:28)eld
x
is mu
h smaller than the VEV of the in(cid:29)aton. This is illustrated in (cid:28)gures (3) and (4). The
c
VEV of the (cid:28)eld initially rises due to the in(cid:29)uen
e of the non-minimal Kähler
oupling whi
h
shifts the minimum of the s
alar potential for this dire
tion away from zero. After the initial
rise this VEV starts de
reasing and drops to zero rapidly during the (cid:28)rst 35 e-folds of in(cid:29)ation.
λ λ = 1
χ χ
This e(cid:27)e
t is shown in (cid:28)gure (5) and is
aused by the relatively large value of ( ) with
λ udd
α
respe
t to ,whi
h favorsthe
reation oflarge dire
tion VEV.Due to theYukawa
oupling
udd H H udd
u d
between and dire
tions a large dire
tion VEV indu
es an e(cid:27)e
tive mass for the
H H H H
u d u d
dire
tion. This e(cid:27)e
t shifts the minimum of the s
alar potential for the dire
tion
H H
u d
to zero leading to the rapid de
rease of the VEV during in(cid:29)ation.
ρ
The VEV of the (cid:28)eld rises during in(cid:29)ation to a value whi
h is
lose to Plan
k mass due to the
λ
α
small value of (Fig. (6)). At the end of in(cid:29)ation it starts to slowly de
rease, as illustrated in
Fig. (7). The evolution of the spe
tral index in the
ru
ial period 60-50 e-folds before the end
of in(cid:29)ation
an be
al
ulated in the slow-roll regime and is shown in (cid:28)gure (8). The value of the
7
x
0.010
x
0.00001
0.008
0.006 5.´10-6
0.004
Mt
6.´107 8.´107 1.´108 1.2´108 1.4´108
0.002
-5.´10-6
e
20 40 60 80 100 120 140
-0.00001
x
Figure 3: Evolution of the (cid:28)eld during in-
x
(cid:29)ation. Values on verti
al axes are expressed Figure 4: Os
illations of the (cid:28)eld at the
in Plan
k masses and time on horizontal axes end of in(cid:29)ation. Values on verti
al axes are
is expressed in the approximate number of e- expressed in Plan
k masses and time on hor-
folds of in(cid:29)ation. izontal axes is expressed in Plan
k times.
c
0.0015
0.0010
0.0005
e
10 20 30 40
c
Figure 5: Evolution of the (cid:28)eld during during in(cid:29)ation. Values on verti
al axes are expressed
in Plan
k masses and time on horizontal axes is expressed in the approximate number of e-folds
of in(cid:29)ation.
Ρ
Ρ
0.6
0.7
0.6
0.4
0.5
0.2 0.4
0.3
e
20 40 60 80 100 120 140 Mt
6.´107 8.´107 1.´108 1.2´108 1.4´108
ρ
Figure 6: Evolution of the (cid:28)eld during in-
ρ
(cid:29)ation. Values on verti
al axes are expressed Figure 7: Evolution of the (cid:28)eld at the end
in Plan
k masses and time on horizontal axes of in(cid:29)ation. Values on verti
al axes are ex-
is expressed in the approximate number of e- pressed in Plan
k masses and time on hori-
folds of in(cid:29)ation. zontal axes is expressed in Plan
k times.
spe
tral index 50 e-folds before the end of in(cid:29)ation is in agreement with the value derived from
8
0,98
0,97
n
S
0,96
0,95
1#106 3#106 5#106 7#106 1#107
t
n
S
Figure 8: Evolution of the spe
tral index between 100 and 40 e-folds before the end of
in(cid:29)ation. Bla
k lines marks the time of 50 e-folds before the end of in(cid:29)ation and
orresponding
value of the spe
tral index. Time on horizontal axes is expressed in Plan
k times.
n = 0.960+0.014
the WMAP5 data s −0.013 [27℄.
λ = 1 λ = 1
α χ
3.2 and
The in(cid:29)aton VEV evolves a

ording to standard
haoti
in(cid:29)ation as in the previous
ase. Its
evolution in
ludes the slow-roll period whi
h naturally ends with in(cid:29)aton os
illations. Due to
X χ
thedependen
e ofthe s
alarpotential onthephasesof (cid:28)elds and ,both of these(cid:28)eldsevolve
X
non-trivially in the
omplex plane. The evolution of the absolute value of (cid:28)eld is similar to
the previous s
enario and mimi
s the behavior of the in(cid:29)aton. Figure (9) shows the evolution
X λ λ
α χ
of (cid:28)eld in the
omplex plane. Sin
e ∼ now the absolute values of (cid:28)elds
orresponding
ImX
0.00006
0.00004
0.00002
ReX
-0.00005 0.00005 0.00010 0.00015
-0.00002
X
Figure 9: Evolution of (cid:28)eld on the
omplex plane during in(cid:29)aton os
illations.
to both (cid:29)at dire
tions evolve similarly. Numeri
al
al
ulations show that the absolute values of
10 3M
− Pl
both (cid:28)elds grow during in(cid:29)ation, a
hieving a maximum of the order of and then start
to de
rease at the end of in(cid:29)ation due to the time-evolution of the minimum of their potentials.
χ
Be
ause of the dynami
s of the phase of (cid:28)eld the e(cid:27)e
tive mass of the (cid:28)eld is slightly di(cid:27)erent
α χ
to that of the(cid:28)eld . As aresult (cid:28)eld beginsos
illating earlier. Figures(10) and (11)show the
χ χ
evolution of the absolute value of , while (cid:28)gure (12) shows the evolution of in the
omplex
udd
plane. The evolution of the (cid:29)at dire
tion is e(cid:27)e
tively one-dimensional be
ause the s
alar
potential does not depend on the phase of this (cid:28)eld and the Hubble fri
tion qui
kly suppresses
9
c c
0.0012
0.0020
0.0010
0.0015
0.0008
0.0010 0.0006
0.0004
0.0005
0.0002
e
20 40 60 80 100 120
Mt
3.´107 3.5´107 4.´107
Figure 10: Evolution of the absolute value of
theHiggs(cid:28)elddire
tion duringin(cid:29)ation. Val- Figure 11: Evolution of the absolute value of
ues on verti
al axes are expressed in Plan
k the Higgs (cid:28)eld dire
tion during in(cid:29)aton os
il-
masses and time on horizontal axes is ex- lations. Values on verti
al axes are expressed
pressed in the approximate number of e-folds in Plan
k masses and time on horizontal axes
of in(cid:29)ation. is expressed in Plan
k times.
ImΧ
0.0008
0.0006
0.0004
0.0002
ReΧ
-0.0004 -0.0002 0.0002 0.0004 0.0006 0.0008
-0.0002
-0.0004
Figure 12: Evolution of theHiggs (cid:28)eld dire
tion inthe
omplex plane during in(cid:29)aton os
illations
α
any initial phase dynami
s. Figures (13) and (14) show the evolution of the absolute value of .
Using the numeri
ally found evolution the Hubble parameter and slow-roll parameters
an be

al
ulated. They ful(cid:28)ll the slow-roll
onditions. One
an then obtain the spe
tral index in the
slow-roll approximation. The spe
tral index evaluated at the time of about 50-60 e-folds before
the end of in(cid:29)ation is
onsistent with the value of the spe
tral index derived from the WMAP5
observational data [27℄.
4 Ex
itations around VEVs
In this
hapter we introdu
e ex
itations around all MSUGRA (cid:28)elds, whi
h are related to the
(cid:29)at dire
tions under
onsideration. By "related" we mean that either they have large VEVs,
10