Curvature and isocurvature perturbations in two-field inflation
Curvature and iso
urvature perturbations
in two-�eld in�ation
Z. Lalak
, D. Langlois
, S. Pokorski
, K. Turzy«ski
Institute of Theoreti
al Physi
s, Warsaw University,
ul. Ho»a 69, 00-681 Warsaw, Poland;
APC (Astroparti
ules et Cosmologie, CNRS-UMR 7164), Université Paris 7
10, rue Ali
e Domon et Léonie Duquet, 75205 Paris Cedex 13, Fran
e;
GReCO, Institut d'Astrophysique de Paris, CNRS,
98bis Boulevard Arago, 75014 Paris, Fran
e;
Physi
s Department, University of Mi
higan,
450 Chur
h St., Ann Arbor, MI-48109, USA
O
tober 30, 2018
Abstra
t
We study 
osmologi
al perturbations in two-�eld in�ation, allowing for non-
standard kineti
 terms. We 
al
ulate analyti
ally the spe
tra of 
urvature and
iso
urvature modes at Hubble 
rossing, up to �rst order in the slow-roll parameters.
We also 
ompute numeri
ally the evolution of the 
urvature and iso
urvature modes
from well within the Hubble radius until the end of in�ation. We show expli
itly for
a few examples, in
luding the re
ently proposed model of `roulette' in�ation, how
iso
urvature perturbations a�e
t signi�
antly the 
urvature perturbation between
Hubble 
rossing and the end of in�ation.
http://arxiv.org/abs/0704.0212v2
1 Introdu
tion
In�ation provides a simple and elegant s
enario for the early Universe (see e.g. [1℄ for a
re
ent textbook presentation). Although single �eld in�ation models are perfe
tly 
om-
patible with the present 
osmologi
al data, many early universe models based on high
energy physi
s, in parti
ular derived from supergravity or string theory, usually involve
many s
alar �elds whi
h 
an have non-standard kineti
 terms. This is why multi-�eld
in�ationary s
enarios, where several s
alar �elds play a dynami
al role during in�ation,
have re
eived some attention in the literature (see e.g. [2℄-[13℄). However, ex
ept for a
few spe
i�
 models, the predi
tions for the spe
tra of primordial perturbations are, in
general, a nontrivial task, in 
ontrast with single-�eld models.
The main reason is that the 
urvature (or adiabati
) perturbation, whi
h is generated
during in�ation and eventually observed, 
an evolve on super-Hubble s
ales in multi-�eld
in�ation whereas it remains frozen in single-�eld in�ation. This is due to the presen
e of
additional perturbation modes, often 
alled iso
urvature (or entropy) modes, 
orrespond-
ing to relative perturbations between the various s
alar �elds, whi
h a
t as a sour
e term
in the evolution equation for the 
urvature perturbation. This property was �rst pointed
out in [6℄ in the 
ontext of Brans-Di
ke in�ation. This phenomenon o

urs during in�a-
tion and a�e
ts the �nal 
urvature perturbation at the end of in�ation, independently
whether iso
urvature modes survive or not after in�ation.
The purpose of the present work is to study in detail how the iso
urvature perturba-
tions, present during in�ation, a�e
t the 
urvature perturbations, both at Hubble 
rossing
and in the subsequent evolution on super-Hubble s
ales. Sin
e our intention is to stress
some qualitative properties spe
i�
 to multi-�eld in�ation, we have 
hosen to restri
t our
study to the 
ase of two s
alar �elds. Moreover, we 
onsider models where a non-standard
kineti
 term is allowed for one of the s
alar �elds. This in
ludes, in parti
ular, s
enarios
motivated by supergravity and string theory, whi
h have been re
ently proposed (see, e.g.,
[14℄-[18℄).
The produ
tion of adiabati
 and iso
urvature modes for two-�eld in�ation with a
generi
 potential, in the slow-roll approximation, was studied in [19℄ where a de
ompo-
sition into adiabati
 and iso
urvature modes was introdu
ed. Models with non-standard
kineti
 terms for in�atons have been studied in the slow-roll approximation in [9℄ and [11℄,
and the adiabati
-iso
urvature de
omposition te
hnique of [19℄ was later extended to su
h
two-�eld models in [20℄ and [21℄. Very re
ently, two-�eld in�ation with standard kineti
terms was investigated in [22℄ at next-to-leading order 
orre
tion in a slow-roll expansion
and it was also shown that the adiabati
 and iso
urvature modes at Hubble 
rossing are

orrelated at �rst order in slow-roll parameters. In parallel to these analyti
al studies, a
numeri
al study of the evolution of adiabati
 and iso
urvature was presented in [23℄ and
[24℄.
In the present work, we extend the previous analyses in the following dire
tions. First,
we present a detailed analysis of the 
orrelation of adiabati
 and iso
urvature just after
Hubble 
rossing, both analyti
ally and numeri
ally. This 
orrelation was in general sup-
posed to vanish in most previous works, ex
ept in the numeri
al study [23℄ and the
analyti
al work [22℄, both in the 
ontext of 
anoni
al kineti
 terms. Here, we extend
the analysis with non-standard kineti
 terms. We 
ompute analyti
ally the spe
tra and

orrelation at Hubble 
rossing in the slow-roll approximation, by taking spe
ial 
are of
the time-dependen
e shortly after Hubble 
rossing.
Se
ond, we study numeri
ally the whole evolution of adiabati
 and iso
urvature per-
turbations from within the Hubble radius until the end of in�ation. This allows us to go
beyond the slow-roll approximation whi
h is needed to derive analyti
al results. Our nu-
meri
al study enables us to see pre
isely how iso
urvature perturbations 
an be transferred
into adiabati
 perturbations during the in�ationary phase depending on the ba
kground
traje
tory in �eld spa
e. We illustrate this behaviour by studying numeri
ally three mod-
els. The last one is the so-
alled 'roulette' in�ation model, whi
h has been proposed
re
ently [14℄.
The plan of the paper is the following. The next se
tion presents the 
lass of models
we 
onsider and gives the homogeneous equations of motion as well as the equations
governing the perturbations. The third se
tion is devoted to the study of the perturbations
from deep inside the Hubble radius until a few e-folds after Hubble 
rossing. We then
dis
uss, in se
tion 4, analyti
al methods to determine the evolution of the perturbations
on super-Hubble s
ales. Se
tion 5 is devoted to the numeri
al study of the evolution of the
perturbations, whi
h is 
ompared with the analyti
al estimates of the previous se
tions.
We �nally draw our 
on
lusions in the last se
tion.
2 The model
In this paper, we study models with two s
alar �elds, in whi
h one of the s
alar �elds has
a non-standard kineti
 term, des
ribed by an a
tion of the form
(∂µφ)(∂
µφ)− e
2b(φ)
(∂µχ)(∂
µχ)− V (φ, χ)
, (1)
where MP is the redu
ed Plan
k mass, MP ≡ (8πG)−1/2. This type of a
tion usually
appears when χ 
orresponds to an axioni
 
omponent. It is also motivated by generalized
Einstein theories [6, 11℄. When b(φ) = 0 one re
overs standard kineti
 terms for the two
�elds. In this se
tion, we give the equations of motion for the homogeneous �elds and
then for the linear perturbations, following the results (and notation) of [20℄ and [21℄,
where the same type of models was 
onsidered.
2.1 Homogeneous equations
Let us start with the homogeneous equations of motion. We assume a spatially �at FLRW
(Friedmann-Lemaître-Robertson-Walker) geometry, with metri
ds2 = −dt2 + a(t)2dx2, (2)
where t is the 
osmi
 time. One 
an also de�ne the 
omoving time τ =
dt/a(t).
The equations of motion for the s
ale fa
tor and the homogeneous �elds read
φ̈+ 3Hφ̇+ Vφ = bφe
2bχ̇2 , (3)
χ̈+ (3H + 2bφφ̇)χ̇+ e
−2b Vχ = 0 , (4)
φ̇2 +
χ̇2 + V
, (5)
Ḣ = − 1
φ̇2 + e2bχ̇2
, (6)
where H ≡ ȧ/a and a dot stands for a derivative with respe
t to the 
osmi
 time t and a
subs
ript index φ or χ denotes a derivative with respe
t to the 
orresponding �eld.
It is also useful to introdu
e the following slow-roll parameters
ǫφφ =
2M2PH
, ǫφχ = e
b φ̇χ̇
2M2PH
, ǫχχ = e
2b χ̇
2M2PH
, (7)
ηIJ =
ǫ = ǫφφ + ǫχχ = −
. (9)
2.2 Linear perturbations
We now dis
uss the linear perturbations of our model (one 
an �nd a detailed presentation
of the theory of 
osmologi
al perturbations in e.g. [1, 25, 26℄ and a pedagogi
al introdu
-
tion in e.g. [27℄). For simpli
ity, we shall dire
tly work in the longitudinal gauge. In the
absen
e of anistropi
 stress (the o�-diagonal spatial 
omponents of the stress-energy ten-
sor), whi
h is the 
ase when matter 
onsists of s
alar �elds, the metri
 in the longitudinal
gauge is of the form
ds2 = −(1 + 2Φ)dt2 + a2(1− 2Φ)dx2 . (10)
where only s
alar perturbations are taken into a

ount.
We now de
ompose the s
alar �elds into their homogeneous (ba
kground) parts and
the perturbations:
φ(t,x) = φ(t) + δφ(t,x) and χ(t,x) = χ(t) + δχ(t,x) . (11)
We shall work with the Fourier 
omponents of the perturbations, δφk(t) and δχk(t),
routinely omitting the subs
ript k to shorten the expressions. The perturbed Klein-
Gordon equations read
δ̈φ + 3H ˙δφ+
+ Vφφ − (bφφ + 2b2φ)χ̇2e2b
δφ+ Vφχδχ− 2bφe2bχ̇ ˙δχ
= 4φ̇Φ̇− 2VφΦ (12)
δ̈χ + (3H + 2bφφ̇) ˙δχ+
+ e−2bVχχ
δχ+ 2bφχ̇ ˙δφ+
e−2b (Vχφ − 2bφVχ) + 2bφφφ̇χ̇
δφ = 4χ̇Φ̇− 2e−2bVχΦ . (13)
The energy and the momentum 
onstraints, given by Einstein equations, are, respe
-
tively:
3H(Φ̇ +HΦ) + ḢΦ+
Φ = − 1
φ̇ ˙δφ+ e2bχ̇ δ̇χ+ bφe
2bχ̇2δφ+ Vφδφ+ Vχδχ
Φ̇ +HΦ =
φ̇ δϕ+ e2bχ̇ δχ
. (15)
It is 
onvenient, instead of using the perturbations δφ and δχ, de�ned here in the
longitudinal gauge, to introdu
e the so-
alled gauge-invariant Mukhanov-Sasaki variables:
Qφ ≡ δφ+
Φ and Qχ ≡ δχ+
Φ , (16)
whi
h 
an be identi�ed with the s
alar �eld perturbations in the �at gauge.
Substituting (16) into (12)-(13) and using the ba
kground equations of motion as well
as the energy and momentum 
onstraints, one �nds
Q̈φ + 3HQ̇φ − 2e2bbφ χ̇ Q̇χ +
+ Cφφ
Qφ + CφχQχ = 0 (17)
Q̈χ + 3HQ̇χ + 2bφ φ̇ Q̇χ + 2bφ χ̇ Q̇φ +
+ Cχχ
Qχ + CχφQφ = 0 , (18)
where we have de�ned the following ba
kground-dependent 
oe�
ients:
Cφφ = −2e2bb2φχ̇2 +
e2bφ̇2χ̇2
2M4PH
2M4PH
− e2bbφφχ̇2 +
2φ̇Vφ
+ Vφφ (19)
Cφχ =
3e2bφ̇χ̇
4bφ̇χ̇3
2M4PH
2bφ̇3χ̇
2M4PH
e2bχ̇Vφ
+ Vφχ (20)
Cχχ =
3e2bχ̇2
4bχ̇4
2M4PH
2bφ̇2χ̇2
2M4PH
2χ̇Vχ
+ e−2bVχχ (21)
Cχφ =
3φ̇χ̇
e2bφ̇χ̇3
2M4PH
φ̇3χ̇
2M4PH
+ 2bφφφ̇χ̇− 2e−2bbφVχ +
e−2bφ̇Vχ
+ e−2bVφχ (22)
The two equations (17) and (18) form a 
losed system for the two gauge-invariant quan-
tities Qφ and Qχ.
2.3 De
omposition into adiabati
 and entropy 
omponents
As originally proposed in [19℄, in order to fa
ilitate the interpretation of the evolution of

osmologi
al perturbations, it 
an be useful to de
ompose the s
alar �eld perturbations
along the two dire
tions respe
tively parallel and orthogonal to the homogeneous traje
-
tory in �eld spa
e. The proje
tion parallel to the traje
tory is usually 
alled the adiabati
,
or 
urvature, 
omponent while the orthogonal proje
tion 
orresponds to the entropy, or
iso
urvature, 
omponent. Note that there was a semanti
 shift in the terminology sin
e
one used to 
all adiabati
 and entropy modes during in�ation the two parti
ular solutions
for the perturbations that would mat
h after in�ation, respe
tively, to the adiabati
 and
iso
urvature modes de�ned in the radiation era. This terminology is used for example in
the papers on double in�ation su
h that [5℄ and [10℄.
This de
omposition into instantaneous adiabati
 and entropy 
omponents, introdu
ed
in [19℄, has re
ently been extended [28℄ to fully nonlinear perturbations in the 
ontext
of the 
ovariant nonlinear formalism introdu
ed in [29, 30℄. Here, we 
onsider only the
de
omposition at the linear level, but sin
e we allow for non-standard kineti
 terms, we
will need to generalize the equations to su
h a 
ase, as was done in [20℄. Let us re
all here
the main results.
The essential idea is to introdu
e the linear 
ombinations
δσ ≡ cos θ δφ+ sin θ eb δχ and δs ≡ − sin θ δφ+ cos θ eb δχ , (23)
where
cos θ ≡ φ̇
, sin θ ≡ χ̇ e
with σ̇ ≡
φ̇2 + e2bχ̇2 . (24)
The notations σ̇ and δσ are just used for 
onvenien
e; they do not refer to any s
alar �eld
Instead of δσ, it is in fa
t more 
onvenient to work dire
tly with the Mukhanov-Sasaki
variables and therefore to de�ne
Qσ ≡ cos θ Qφ + sin θ ebQχ and δs ≡ − sin θ Qφ + cos θ ebQχ , (25)
by noting that
Qσ ≡ δσ +
Φ. (26)
In the so-
alled 
omoving gauge, the perturbation Qσ is dire
tly related to the three-
dimensional 
urvature of the 
onstant time spa
e-like sli
es. This gives the gauge-invariant
quantity refered to as the 
omoving 
urvature perturbation:
Qσ. (27)
The perturbation δs, 
alled the iso
urvature perturbation, is automati
ally gauge-invariant.
It is sometimes 
onvenient, by analogy with the 
urvature perturbation, to introdu
e a
renormalized entropy perturbation whi
h is de�ned as
S ≡ H
δs. (28)
In �eld spa
e, Qσ 
orresponds to perturbations parallel to the velo
ity ve
tor (φ̇, e
bχ̇),
while δs to the orthogonal ones.
Introdu
ing the adiabati
 and entropy �ve
tors� in �eld spa
e, respe
tively
EIσ =
cos θ, e−b sin θ
, EIs =
− sin θ, e−b cos θ
, I = {φ, χ} , (29)
one 
an de�ne various derivatives of the potential with respe
t to the adiabati
 and
entropy dire
tions. Assuming an impli
it summation on the indi
es I (and J), the �rst
order derivatives are de�ned as
Vσ = E
σVI , Vs = E
sVI , (30)
whereas the se
ond order derivatives are
Vσσ = E
σVIJ , Vσs = E
s VIJ , Vss = E
s VIJ . (31)
By 
ombining the two Klein-Gordon equations for the ba
kground �elds, (3) and (4),
one gets the ba
kground equations of motion along the adiabati
 and entropy dire
tions,
respe
tively,
σ̈ + 3Hσ̇ + Vσ = 0, (32)
θ̇ = −Vs
− bφσ̇ sin θ, (33)
while the equations of motion for the perturbations read:
Q̈σ + 3HQ̇σ +
+ Cσσ
δ̇s+ Cσs δs = 0 (34)
δ̈s+ 3Hδ̇s+
+ Css
δs− 2Vs
Q̇σ + CsσQσ = 0 , (35)
Cσσ = Vσσ −
2M4PH
s2θcθVσ + (c
θ + 1)sθVs
Cσs = 6H
2VσVs
+ 2Vσs +
+ 2bφ(s
θVσ − c3θVs) (37)
Css = Vss −
+ bφ(1 + s
θ)cθVσ + bφc
θsθVs − σ̇2(bφφ + b2φ) (38)
Csσ = −6H
− 2VσVs
where sθ ≡ sin θ and cθ ≡ cos θ.
The above system 
onsists (34-35) of two 
oupled se
ond order di�erential equations
involving only Qσ and δs. In order to relate these variables to the metri
 perturbation
Φ de�ned in the longitudinal gauge, it is useful to use the Poisson-like 
onstraint, whi
h
follows from the energy and momentum 
onstraints (14) and (15),
Φ = − 1
ǫm (40)
where ǫm is the 
omoving energy density and 
an be expressed as
ǫm = σ̇Q̇σ +
σ̇Qσ + VσQσ + 2Vsδs. (41)
2.4 Perturbation spe
tra
The in�ationary observables are 
ustomarily expressed in terms of power spe
tra and

orrelation fun
tions. Given their origin as quantum �u
tuations, the perturbations 
an
be represented as random variables. We introdu
e the power spe
tra of the adiabati
 and
entropy perturbations, respe
tively
〈Qσ∗kQσk′〉 =
PQσ(k)δ(k− k′), 〈δs∗k δsk′〉 =
Pδs(k)δ(k− k′), (42)
as well as the 
orrelation spe
trum
〈Qσ∗k δsk′〉 =
CQσ δs(k)δ(k− k′). (43)
3 Evolution of perturbations inside the Hubble radius
In order to study the generation of perturbations from va
uum �u
tuations, we start, for a
given 
omoving wave number k, at an instant ti (or τi) during in�ation when the physi
al
wave number k/a is mu
h bigger than the Hubble parameter H . Our initial 
onditions
are given, as usual, by the Minkowski-like va
uum at τi,
Qσ(τi) ≃
e−ıkτi
a(τi)
and δs(τi) ≃
e−ıkτi
a(τi)
for initial adiabati
 and iso
urvature �u
tuations, respe
tively. These two initial �u
tu-
ations are statisti
ally independent be
ause the 
orresponding equations of motion are
de
oupled in the limit k ≫ aH .
Although the adiabati
 and entropy �u
tuations are initially, i.e. deep inside the Hub-
ble radius, statisti
ally independent, this is, in general, no longer the 
ase at Hubble

rossing. In the 
ontext of two-�eld in�ation with 
anoni
al kineti
 terms, this point has
been stressed in the numeri
al analysis of [23℄ and studied analyti
ally in [22℄.
In the following, we extend the analysis of [22℄ to non-
anoni
al kineti
 terms.
3.1 Equations in the slow-roll approximation
We start with the perturbations Qσ and δs, whose dynami
s is des
ribed by eqs. (34) and
(35). Using the 
onformal time τ and introdu
ing the variables
uσ = aQσ, us = a δs, (45)
these equations 
an be rewritten in the form
u′′σ +
au′s +
k2 − a
+ a2Cσσ
a′ + a2Cσs
us = 0, (46)
u′′s −
au′σ +
+ a2Css
a′ + a2Csσ
uσ = 0, (47)
where the four 
oe�
ients CIJ are given in eqs. (36)-(39) and a prime denotes a derivative
with respe
t to the 
onformal time τ .
Let us now dis
uss the slow-roll approximation. The only di�eren
e with respe
t to
the 
ase with 
anoni
al kineti
 terms will arise from some of the terms depending on the
derivatives of b. In the slow-roll approximation, one 
an use the relation
= Hησs − bφσ̇s3θ , (48)
and the various 
oe�
ients in the above system of equations simplify to yield
+ k2 − 2 + 3ǫ
1+ 2E
= 0 (49)
where the matri
es E and M are given by
0 −ησs
ησs 0
0 ξs3θ
−ξs3θ 0
−6ǫ+ 3ησσ 4ησs
2ησs 3ηss
3ξs2θcθ −4ξs3θ
−2ξs3θ −3ξcθ(1 + s2θ)
2bφMP
ǫ. (52)
In eqs. (50) and (51), we have kept only the terms linear in bφ, i.e. proportional to ξ,
and negle
ted the terms quadrati
 in bφ as well as the terms proportional to bφφ. We thus
treat ξ on the same footing as the other slow-roll parameters. In order to emphasize the
di�eren
e between generalized kineti
 terms and 
anoni
al kineti
 terms, we have however
separated the terms proportional ξ from the others.
The system of equations (49) that we have obtained is of the form
u′′ + 2Lu′ +Qu = 0 . (53)
The matrix 
oe�
ient for the �rst order time derivative is 2L = 2E/τ , where E is an an-
tisymmetri
 matrix, linear in the slow-roll parameters. Let us introdu
e a time-dependent
orthogonal matrix R whi
h satis�es R′ = −LR. Note that this is possible only if L is an
antisymmetri
 matrix. Reexpressing the above equation (53) in terms of a new matrix
ve
tor v, de�ned by u = Rv, one 
an eliminate the terms proportional to the �rst order
time derivative and we obtain the following equation
v′′ +R−1
−L2 − L′ +Q
Rv = 0. (54)
At linear order in the slow-roll parameters, one �nds
− L2 − L′ ≃
E. (55)
Therefore, apart from the trivial part proportional to the identity matrix, the 
ombination
−L2 − L′ +Q 
ontains
(E+M) =
−2ǫ+ ησσ + ξs2θcθ ησs − ξs3θ
ησs − ξs3θ ηss − ξcθ(1 + s2θ)
, (56)
whi
h is a symmetri
 matrix.
We now assume that the slow-roll parameters vary su�
iently slowly during the few
e-folds when the given s
ale 
rosses out the Hubble radius. We thus repla
e the time-
dependent matrix on the right hand side of (56) by the same matrix evaluated at Hubble

rossing, i.e. for k = aH , and the only remaining time dependen
e appears in the global

oe�
ient 3/τ 2. One 
an now diagonalize this matrix by introdu
ing the time-independent
rotation matrix
R̃∗ =
cosΘ∗ − sin Θ∗
sinΘ∗ cosΘ∗
, (57)
so that
∗ (M+ E) R̃∗ = Diag(λ̃1, λ̃2). (58)
In parti
ular, one 
an easily 
ompute the following linear 
ombinations, whi
h will be
useful later:
λ̃1 + λ̃2 = 3 (ησσ + ηss − 2ǫ− ξcθ) , (59)
(λ̃1 − λ̃2) sin 2Θ∗ = 6
ησs − ξs3θ
, (60)
(λ̃1 − λ̃2) cos 2Θ∗ = 3
ησσ − ηss − 2ǫ+ ξcθ(1 + 2s2θ)
, (61)
where the right hand sides of the three above equations are evaluated at k = aH .
Similarly, the rotation matrix R is slowly varying per efold, sin
e (dR/d ln a)RT = E,
where E is linear in slow-roll parameters. Around Hubble 
rossing, one 
an thus repla
e
R by its value R∗ at Hubble 
rossing.
By introdu
ing
w = R̃−1∗ R∗v, (62)
the system of equations 
an be written as two independent equations of the form
w′′A +
(2 + 3λA)
wA = 0, (A = 1, 2) (63)
λA = ǫ−
λ̃A. (64)
De�ning
+ 3λA (65)
the solution of (63) with the proper asymptoti
 behaviour 
an be written as
ei(µA+1/2)π/2
−τH(1)µA (−kτ)eA(k), (66)
where H
µ is the Hankel fun
tion of the �rst kind of order µ and the eA are two indepen-
dent normalized Gaussian random variables so that
〈eA(k)〉 = 0, 〈eA(k)e∗B(k′)〉 = δABδ(3)(k − k′). (67)
Using the independen
e of the variables w1 and w2, one 
an express the 
orrelations
for the variables uσ and us around Hubble 
rossing time as
a2〈Q†σQσ〉 = cos2Θ∗〈w
1w1〉+ sin2Θ∗〈w
2w2〉 (68)
a2〈δs†Qσ〉 =
sin 2Θ∗
〈w†1w1〉 − 〈w
a2〈δs†δs〉 = sin2Θ〈w†1w1〉+ cos2Θ〈w
2w2〉 (70)
where one 
an substitute
〈w†AwA〉 =
(−τ)|H(1)µA (−kτ)|
2 ≡ 1
(kτ)2
FA(−kτ). (71)
This yields
PQσ =
(1− 2ǫ∗)
cos2Θ∗ F1(−kτ) + sin2Θ∗ F2(−kτ)
CQσδs =
(1− 2ǫ∗)
sin 2Θ∗
[F1(−kτ)− F2(−kτ)] (73)
Pδs =
(1− 2ǫ∗)
sin2 Θ∗ F1(−kτ) + cos2Θ∗ F2(−kτ)
, (74)
where we have used
a ≃ −1 + ǫ∗
. (75)
At this stage, it is worth noting that our derivation is still valid if the parameter ηss, whi
h

orresponds to the 
urvature of the potential along the dire
tion orthogonal to the �eld
traje
tory, is not small. In this 
ase, the entropy �u
tuations are e�e
tively suppressed
and only adiabati
 �u
tuations are generated. As far as the perturbations are 
on
erned,
this parti
ular situation is similar to the single �eld 
ase.
A further simpli�
ation o

urs when λA ≪ 1, in whi
h 
ase µA ≃ 32+λA. The fun
tions
FA(x) 
an be expanded as
FA(x) =
x3|H3/2(x)|2 (1 + 2λAf(x)) = (1 + x2) (1 + 2λAf(x)) , (76)
f(x) = Re
µ (x)
µ=3/2
 . (77)
Using the relations (59-61) and (64), we �nally get for the 
urvature and entropy
perturbations de�ned in (27) and (28), the following expressions
2πσ̇∗
(1 + k2τ 2)
1− 2ǫ∗ +
6ǫ∗ − 2ησσ∗ − 2ξ∗s2θ∗cθ∗
CRS =
2πσ̇∗
(1 + k2τ 2)
θ∗ − 2ησs∗
2πσ̇∗
(1 + k2τ 2)
1− 2ǫ∗ +
2ǫ∗ − 2ηss∗ + 2ξ∗(1 + s2θ∗)cθ∗
.(80)
Let us 
omment these results. First, one 
an verify that, for 
anoni
al kineti
 terms
(i.e. ξ = 0), we re
over the results of [22℄ if we repla
e the fa
tor (1 + k2τ 2) by 1 and
the fun
tion f(−kτ) by the number C = 2 − ln2 − γ ≃ 0.7296, where γ ≃ 0.5772 is
the Euler-Mas
heroni 
onstant. Our �nal expression depends expli
itly on τ and allows
us a more pre
ise estimate of the spe
tra around the time of Hubble 
rossing. As we
will see expli
itly later, there are some in�ationary s
enarios where the amplitude of the

urvature perturbation spe
trum evolves very qui
kly after Hubble 
rossing and never
rea
hes its asymptoti
 value (
orresponding to the limit kτ → 0). In these 
ases, one
needs to evaluate more pre
isely the amplitude around Hubble 
rossing, whi
h our more
detailed formula enables to do.
Another di�eren
e with [22℄ is that we derived the adiabati
 and iso
urvature spe
tra
by working dire
tly with the equations for the adiabati
 and entropy 
omponents, instead
of working with the initial s
alar �elds.
4 Evolution of perturbations on super-Hubble s
ales
When the iso
urvature modes are suppressed, for instan
e if the e�e
tive mass along the
iso
urvature dire
tion is large with respe
t to the Hubble parameter the �nal adiabati
spe
trum 
an be 
omputed simply by taking the usual single-�eld result applied to the
adiabati
 dire
tion:
PsfR(k) ≃
4π2σ̇2
8π2Lkin
, (81)
where all the quantities are evaluated at Hubble 
rossing. The simpli�
ation works be-

ause, in this parti
ular situation where iso
urvature �u
tuations are absent, the 
urvature
perturbation remains frozen on super-Hubble s
ales.
If iso
urvature modes are present however, they will a�e
t the super-Hubble evolution
of the adiabati
 perturbations be
ause they will a
t as a sour
e term on the right hand
side of the equation governing the evolution of the 
urvature perturbation [7℄ (and [28℄
for the non-linear generalisation).
In order to obtain the �nal power spe
tra and 
orrelations, and to 
ompare the pre-
di
tions of a multi-in�aton model with observations, one must then solve the 
oupled
system of di�erential equations (34-35). In general, a numeri
al approa
h is ne
essary
and will be 
onsidered in the next se
tion. In some parti
ular 
ases, within the slow-roll
approximation, one 
an solve analyti
ally the equations of motion on super-Hubble s
ales.
We now dis
uss these 
ases in the rest of this se
tion.
Following [21℄, we 
an then write eqs. (34) and (35) in the slow-roll approximation as:
Q̇σ ≃ AHQσ +BHδs and δ̇s ≃ DHδs , (82)
where:
A = −ησσ + 2ǫ− ξcθs2θ (83)
B = −2ησs + 2ξs3θ ≃ 2
− 2ξsθ (84)
D = −ηss + ξcθ(1 + s2θ) . (85)
Qualitatively, it is 
lear that if the iso
urvature perturbations do not de
ay very fast, there
is a strong intera
tion between the adiabati
 and iso
urvature perturbations, whenever
the 
oe�
ient B be
omes large, i.e. when the 
lassi
al traje
tory makes a sharp turn in
the �eld spa
e or when ξ is relatively large and in�ation is driven at least partially by the
�eld χ (sθ 6= 0). For 
onstant A,B,D, the equations (82) 
an be solved expli
itly to give
Qσ(N) ≃ eANQσ∗ +
(eDN − eAN )δs∗ and δs(N) ≃ eDNδs∗ (86)
where N stands for the number of efolds after Hubble 
rossing. Taking into a

ount that
(H/σ̇) ≃ (H∗/σ̇∗)e−AN , we 
an express the power spe
tra and 
orrelations as:
P(a)R (N) ≃ P̄R∗ + P̄S∗
eγN − 1
+ 2C̄RS∗
eγN − 1
C(a)RS(N) ≃ C̄RS∗ e
γN + P̄S∗
eγN − 1
P(a)S (N) ≃ P̄S∗ e
2γ∗N , (89)
where γ = D−A. The quantities P̄R∗, C̄RS∗ and P̄S∗ 
orrespond to the asymptoti
 limit,
i.e. when kτ → 0, of the expressions (78-80).
The use of this approximation is in pra
ti
e rather limited be
ause it relies on the
assumption that the slow-roll parameters are time-independent between Hubble 
rossing
and the �nal time. In most 
ases, this approximation, whi
h we 
all 
onstant slow-
roll approximation, holds only for a few e-folds and breaks down long before the end of
in�ation.
In some simple in�ationary models, there exists an analyti
al approa
h to 
ompute
analyti
ally the evolution of the perturbations on super-Hubble s
ales. This is the 
ase
for double in�ation with 
anoni
al kineti
 terms (i.e. b = 0) and potential
V (φ, χ) =
2 , (90)
where the equations of motion for the metri
 perturbation Φ and the two s
alar �eld
perturbations 
an be integrated expli
itly in the slow-roll approximation and on super-
Hubble s
ales [5℄ (this 
an be seen as a parti
ular 
ase within a more general 
ontext
dis
ussed in [9℄). One �nds
Φ ≃ −C1Ḣ
+ 2C3
(m2χ −m2φ)m2χχ2m2φφ2
3(m2χχ
2 +m2φφ
, (91)
− 2C3
Hm2χχ
2 +m2φφ
+ 2C3
Hm2φφ
2 +m2φφ
, (92)
where C1 and C3 are time-independent 
onstants of integration.
The 
urvature and iso
urvature perturbations during in�ation are respe
tively
R = Φ +H χ̇ δχ + φ̇ δφ
χ̇2 + φ̇2
, S = H φ̇ δχ− χ̇ δφ
χ̇2 + φ̇2
. (93)
By plugging the solutions (91-92) into the above expressions, one obtains the expli
it
evolution of the adiabati
 and iso
urvature perturbations, knowing that the ba
kground
evolution is given by
χ = 2MP
s sinα, φ = 2MP
s cosα, s = s0
(sinα)2/(R
(cosα)2R
2/(R2−1)
where s = − ln(a/ae) is the number of e-folds between a given instant and the end of
in�ation, and R ≡ mχ/mφ.
Note that the iso
urvature perturbation S that we have de�ned during in�ation, fol-
lowing other works, is proportional but does not 
oin
ide with the iso
urvature pertur-
bation Srad de�ned during the radiation era. In the s
enario dis
ussed in [10℄, where the
heavy �eld χ de
ays into dark matter, the iso
urvature perturbation Srad = δcdm−(3/4)δγ
is related to the perturbations during in�ation via the relation
Srad = −
m2χC3 = −
. (95)
Another method to 
al
ulate the �nal 
urvature perturbations is the so-
alled δN
formalism [31, 32, 33℄. In pra
ti
e however, this method requires the expression of the
number of e-folds as a fun
tion of the initial values of the s
alar �elds and ex
ept in a few
simple 
ases where this expression 
an be determined analyti
ally, a numeri
al approa
h is
also needed in the general 
ase. Moreover, the approa
h we have adopted allows to follow
not only the evolution of the 
urvature perturbation but also that of the iso
urvature
perturbation. This is important if some iso
urvature perturbations survive after the end
of in�ation. Their evolution then depends on the details of the pro
esses whi
h o

ur at
the end in�ation and after, in parti
ular the reheating (or preheating) pro
esses, whi
h
goes beyond the s
ope of this study.
5 Numeri
al analysis
In Se
tion 3, we studied the spe
tra and 
orrelations of the perturbations in the vi
inity
of the Hubble 
rossing and we obtained analyti
 approximations (78)-(80). The aim
of the present se
tion is to 
onfront these expressions with the result of a numeri
al
integration of the equations of motion (34)-(35). We would also like to study numeri
ally
the super-Hubble dynami
s of the perturbations, whi
h is often the only way to 
al
ulate
the in�ationary observables su
h as the spe
tral index ns with a pre
ision required by the
present and forth
oming observations.
5.1 Numeri
al pro
edure
Our numeri
al pro
edure, similar to that of [23℄, is the following. In order to take into
a

ount the statisti
al independen
e of the adiabati
 and iso
urvature perturbations deep
inside the Hubble radius, we integrate eqs. (34)-(35) twi
e: �rst with the initial value
of Qσ 
orresponding to the Minkowski-like va
uum and δs = 0, then with the initial
value of δs 
orresponding to the Minkowski-like va
uum and Qσ = 0. Unless stated
otherwise, we impose the initial 
onditions 8 efolds before the Hubble 
rossing. The initial

onditions also in
lude the slow-roll for the ba
kground �elds. The evolution pro
eeds
along a ba
kground traje
tory whi
h provides a su�
ient number of efolds before the end
of in�ation (50-60, depending on the model). We identify the end of in�ation, at whi
h we
terminate the evolution, when ǫ = 1. As the out
ome of the �rst (se
ond) run, we obtain
the 
urvature and entropy perturbations, R1 and S1 (R2 and S2). We then 
al
ulate the
spe
tra and 
orrelations as:
|R1|2 + |R2|2
|S1|2 + |S2|2
CRS =
R†1S1 +R
. (98)
We shall sometimes des
ribe the 
orrelations using the relative 
orrelation 
oe�
ient:
C̃ = |CRS |√
The value of C̃ lies between 0 and 1, and it indi
ates to what extent the �nal 
urvature
perturbations result from the intera
tions with the iso
urvature perturbations.
Figure 1: Examples of 
lassi
al in�ationary traje
tories for double in�ation with 
anoni
al
kineti
 terms (left), double in�ation with non-
anoni
al kineti
 terms (
enter) and roulette
in�ation (right). The details of the models are des
ribed in Se
tion 5.2. Subsequent tens
of efolds are indi
ated along the 
urves.
5.2 Examples of in�ationary models
There is an enormous number of examples of in�ationary models. Here we restri
t our
analysis to just three 
ases des
ribed below. We will use these examples to 
he
k the
analyti
al results of the pre
eding Se
tions and to illustrate some generi
 features in the
evolution of adiabati
 and iso
urvature perturbations.
5.2.1 Double in�ation with 
anoni
al kineti
 terms
Double in�ation (with b = 0) is 
ertainly the most thoroughly studied example of multi-
�eld in�ation. It employs the potential
V (φ, χ) =
2 . (100)
In order to make de�nite 
al
ulations, we set 7mφ = mχ and we 
hoose the initial 
ondi-
tions φi = χi (later we shall also 
omment on the 
ase φi = 50χi), 8 efolds before the s
ale
we 
onsider leaves the Hubble radius, after whi
h in�ation goes on for about ∼ 60 efolds.
The 
lassi
al traje
tory in �eld spa
e is shown in Figure 1. In our example, the traje
tory
is strongly bent roughly 35th efolds after the Hubble 
rossing. As we shall see, it is at
this moment when the adiabati
 perturbations are strongly fed by the iso
urvature ones.
5.2.2 Double in�ation with non-
anoni
al kineti
 terms
In order to 
on
entrate just on the e�e
ts due to the non-
anoni
al nature of the kineti
terms, we 
onsider a very simple generalization of the previous example by taking b(φ) =
−φ/MP and mφ = mχ in (100). We 
hoose the initial 
onditions so that φi = 0. Then it
is almost exlusively the �eld χ whi
h slides down to the minimum of the potential during
in�ation, but due to non-
anoni
ality of the kineti
 terms, the intera
tion of χ with φ
drives the latter slightly away from zero. The 
lassi
al traje
tory in the �eld spa
e is
shown in Figure 1.
5.2.3 Roulette in�ation
Re
ently, in�ation in the large volume 
ompa
ti�
ation s
heme in the type IIB string
theory model has been investigated in [14℄ (see also [34℄). In our notation, this model 
an
be e�e
tively des
ribed by
b(φ) = b0 −
(101)
V (φ, χ) = V0 + V1
ψ(φ)e−2β1ψ(φ) + V2ψ(φ)e
−β1ψ(φ) cos(β2χ) , (102)
where ψ(φ) = (φ/MP )
and b0, Vi, βi are fun
tions of the parameters of the underlying
string model. A generi
 feature of the potential (102) is that it has an in�nite number
of minima arranged periodi
ally in χ and a plateau for large values of φ, admitting a
large variety of in�ationary traje
tories, whi
h may end at di�erent minima even if they
originate from neighboring points in the �eld spa
e � hen
e the model has been dubbed
roulette in�ation. In this work, we adopt the parameter set no. 1 (in Plan
k units:
b0 = −11; V0 = 9.0 × 10−14; V1 = 3.2 × 10−4; V2 = 1.1 × 10−5; and β1 = 9.4 × 105;
β2 = 2π/3) from [14℄ and 
hoose the parti
ular in�ationary traje
tory shown in Figure 1.
For this traje
tory, the fa
tor bφMP is rather large, of the order 10
, but the e�e
t of the
non-
anoni
al kineti
 terms is strongly suppressed by a very small value of ǫ on the plateau
of the potential. The smallness of ǫ also suppresses the energy s
ale of in�ation and one
needs a smaller number of efolds than in the models des
ribed above. For de�niteness, we
assumed that there are ∼ 50 efolds between the moment that the s
ale of interest 
rosses
the Hubble radius and the end of in�ation.
5.3 Numeri
al results for the perturbations
For the three in�ationary models des
ribed in Se
tion 5.2, we performed the numeri
al
analysis, as des
ribed in Se
tion 5.1. Here, we dis
uss the out
ome for the spe
tra and

orrelations in Figures 2-4. In the right panel of ea
h Figure we plot the evolution of PR
and PS , normalized to the single-�eld result PsfR given in eq. (81), as well as the evolution of
the 
orrelation 
oe�
ient C̃ de�ned in (99) and the parameter B, de�ned in (84), whi
h is
the 
oupling between the iso
urvature and the 
urvature perturbations. These quantities
are plotted as fun
tions of the number of efolds N after Hubble 
rossing. Left panels of
ea
h Figures 2-4 are basi
ally 
lose-ups of the right ones to the vi
inity of the Hubble

rossing. There, we plot the evolution of PR, PS and CRS , normalized to the single-
�eld result PsfR given in eq. (81). These are shown as fun
tions of the variable (k/aH)−1
whi
h allows us to 
ompare dire
tly the numeri
al results with the predi
tions of the eqs.
(78)-(80). Note that in the leading order in the slow-roll parameters ln(aH/k) = N ,
hen
e the logarithmi
 s
ale in the left panels dire
tly 
orresponds to the linear s
ale in
the right panels. In Figures 2-4, we also plot the evolution of the spe
tra, denoted by
the supers
ript (a), when one assumes the 
onstant slow-roll approximation after Hubble

rossing, i.e. when one uses eqs. (87), (88) and (89).
For 
ompleteness, we also dis
uss brie�y the parti
ular 
ase where the generation of
iso
urvature modes is e�e
tively suppressed, situation whi
h applies to some of the models
dis
ussed in the literature for spe
i�
 parameters and/or initial 
onditions.
5.3.1 Double in�ation with 
anoni
al kineti
 terms
In this example, the �eld χ initially dominates the energy density of the Universe and
drives the �rst part of in�ation, and only when it is almost at its minimum, in�ation is
further driven by φ. All the slow-roll parameters are small at the Hubble 
rossing, whi
h
makes eqs. (78)-(80) an ex
ellent approximation of the numeri
al solutions of the equations
of motion. Due to the smallness of B = −2ησs ≈ 2dθ/dN right after the Hubble 
rossing,
the 
urvature perturbations be
ome pra
ti
ally 
onstant during the χ-domination. At the
transition to φ-dominated in�ation B be
omes large, whi
h leads to a sizable in
rement
in PR be
ause of intera
tion with the iso
urvature perturbations. During φ-dominated
in�ation the traje
tory is almost straight again, the iso
urvature perturbations de
ay
qui
kly and the 
urvature perturbations are frozen at the value a
quired at the transition.
This model has the advantage that the numeri
al results 
an be dire
tly 
ompared to
analyti
al ones, as the time evolution of the perturbations 
an be solved without assuming

onstan
y of the slow-roll parameters [10℄, and we �nd a good agreement between two
approa
hes.
Figure 2: Predi
tions for the spe
tra and 
orrelations of the perturbations in double in-
�ation with 
anoni
al kineti
 terms. Thi
k lines show the numeri
al results for PR, PS
and CRS or C̃ normalized to the single-�eld result (81), respe
tively. Cir
les, stars and
squares indi
ate the predi
tions of eqs. (78), (79) and (80), respe
tively. Thin dashed lines
indi
ate the predi
tions of eqs. (87), (88) and (89), respe
tively. The 
oupling B between
the 
urvature and iso
urvature perturbations is also shown.
5.3.2 Double in�ation with non-
anoni
al kineti
 terms
In this example, the ba
kground traje
tory is almost straight. However, the slow-roll
parameter ǫ is around 0.1, whi
h makes the 
oupling B large throughout the entire in�a-
tionary era. Figure 3 shows that eqs. (78)-(80) are a good approximation for the spe
tra
and 
orrelations at the Hubble 
rossing, k/aH = 1, but it is no longer true at super-
Hubble s
ales, k/aH = 0.1 or 0.01, be
ause the iso
urvature perturbations already start
feeding the 
urvature ones sizably. As a result, the �nal 
urvature perturbations originate
almost ex
lusively from the intera
tions with the iso
urvature modes, not from the �u
-
tuations along the in�ationary traje
tory, whi
h makes the relative 
orrelation 
oe�
ient
very 
lose to 1.
Figure 3: Predi
tions for the spe
tra and 
orrelations of the perturbations in double in-
�ation with non-
anoni
al kineti
 terms. Thi
k lines show the numeri
al results for PR,
PS and CRS or C̃ normalized to the single-�eld result (81), respe
tively. Cir
les, stars and
squares indi
ate the predi
tions of eqs. (78), (79) and (80), respe
tively. Thin dashed lines
indi
ate the predi
tions of eqs. (87), (88) and (89), respe
tively. The 
oupling B between
the 
urvature and iso
urvature perturbations is also shown.
5.3.3 Roulette in�ation
As we already argued in Se
tion 5.2, most of the in�ationary traje
tory in this example lies
on the plateau of the potential (102), the slow-roll parameter ǫ is very small, whi
h makes
the dire
t impa
t of the non-
anoni
ality negligible. The traje
tory is, however, strongly

urved in the �eld spa
e and the intera
tion between the iso
urvature and 
urvature modes
is still important. Again, eqs. (78)-(80) a

urately predi
t the spe
tra and 
orrelations in
the vi
inity of the Hubble 
rossing, with deviations on super-Hubble s
ales resulting from
the sour
ing of the 
urvature perturbations by the iso
urvature ones. Eventually, most of
the 
urvature perturbations arise through this e�e
t.
Figure 4: Predi
tions for the spe
tra and 
orrelations of the perturbations in roulette
in�ation. Thi
k lines show the numeri
al results for PR, PS and CRS or C̃ normalized to
the single-�eld result (81), respe
tively. Cir
les, stars and squares indi
ate the predi
tions
of eqs. (78), (79) and (80), respe
tively. Thin dashed lines indi
ate the predi
tions of eqs.
(87), (88) and (89), respe
tively. The 
oupling B between the 
urvature and iso
urvature
perturbations is also shown.
5.3.4 Impa
t of the non-
anoni
al terms
In order to estimate the impa
t of the non 
anoni
al kineti
 terms, one 
an separate ea
h of
the 
oe�
ients (36-39) into two parts: the terms that depend expli
itly on the derivatives
of non-
anoni
al fun
tion b and those whi
h do not. For example, the 
oe�
ient that is
dire
tly responsible for the transfer of iso
urvature modes into 
urvature modes, Cσs is
de
omposed into a `
anoni
al' 
omponent C
σs and a 'non-
anoni
al' 
omponent C
Cσs = C
σs + C
σs , (103)
C(c)σs = 6H
2VσVs
+ 2Vσs +
, C(nc)σs = 2bφ(s
θVσ − c3θVs). (104)
Figure 5: Evolution of the 
oe�
ients C
σs and C
σs de�ned in (104), parametrizing the

oupling between the 
urvature and iso
urvature modes: (a) for double in�ation with non-

anoni
al kineti
 terms and (b) for roulette in�ation.
The other 
oe�
ients 
an be de
omposed similarly
For the two models with non-
anoni
al kineti
 terms, we have 
ompared in Fig. 5 the

anoni
al and non-
anoni
al 
ontributions of the various 
oe�
ients. For double in�ation
with non-
anoni
al kineti
 terms, the non-
anoni
al 
ontribution in Cσs is dominant and
therefore plays a 
ru
ial role in the evolution of the 
urvature mode. For roulette in�ation,
as already noti
ed earlier, the non-
anoni
al 
ontributions turn out to be negligible.
5.3.5 E�e
tively single-�eld 
ases
Many supergravity- or string-inspired models aim at des
ribing supersymmetry breaking
and in�ation in a uni�ed framework. Often, despite the presen
e of many s
alar �elds, one

an �nd model parameters and initial 
onditions su
h that only for one 
ombination of the
Note that there is some arbitrariness in this de
omposition: the de
omposition would be di�erent if
one expresses V
in terms of θ̇ via the ba
kground equation (33).
Figure 6: Predi
tions for the spe
tra and 
orrelations of the perturbations in the e�e
tive
single-�eld 
ase. The lines show the numeri
al results for PR and PS , 
ir
les 
orrespond
to predi
tion of eq. (78) and stars 
orrespond to the analyti
 
al
ulation outlined in Se
tion
5.3.5.
�elds a potential is su�
iently �at to support in�ation, e.g. in pseudo-Goldstone in�ation
[15℄, �better ra
etra
k� s
enario [16℄, no-s
ale supergravity models with moduli stabilised
through D-terms [17℄ or in D-term uplifted supergravity of [18℄. In these works, small
values of the �eld velo
ity (i.e. ǫ ≪ 1) have been ensured by setting the initial 
onditions
in the vi
inity of the saddle point of the potential, while small 
urvature of the potential
along one dire
tion has been obtained by a �ne-tuning of the parameters. It has been
assumed that the iso
urvature perturbations de
ay fast after the Hubble radius 
rossing
and do not a�e
t the 
urvature perturbations even though the traje
tory in the �eld spa
e

an have sharp turns at later stages of the in�ationary evolution. Consequently, the single
�eld approximation (81) has been used in these works to a

ount for the spe
tra of the

urvature perturbation.
In the two-�eld language developed in the present paper, the situation des
ribed above

orresponds to ηss
∼ O(1) ≫ |ησσ|, |ησs|, ǫ. This justi�es setting Θ∗ = 0 in eqs. (68)-(70),
whi
h is equivalent to the assumption that the 
urvature and iso
urvature modes evolve
independently. While we 
an still apply the slow-roll expansion that led to eq. (78) for the
power spe
trum of the 
urvature perturbations, we now have to use the full result (66) to
des
ribe the spe
trum of the iso
urvature modes. For ηss
∼ O(1), the latter result de
ays
very fast for k|τ | → 0 and we 
on
lude that the iso
urvature modes be
ome irrelevant for
the evolution of the 
urvature perturbations soon after the Hubble radius 
rossing.
We 
he
ked numeri
ally that the above 
on
lusion applies for the models des
ribed
in [17℄, whi
h are easily redu
ed to the two-�eld 
ase and their in�ationary traje
tories
are 
urved in the �eld spa
e. For simpli
ity, we would like, however, to illustrate this
point with the model of double in�ation with standard kineti
 terms, des
ribed in Se
tion
5.2.1, for whi
h we set the initial 
ondition φi = 50χi. Then the heavy �eld χ 
ontributes
negligibly to the potential energy and ηss∗ ≃ 0.4. In Figure 6, we plot the numeri
ally

al
ulated spe
tra of the 
urvature and iso
urvature perturbations and 
ompare them with
the analyti
 approximations outlined above. The de
ay of the iso
urvature modes agrees
with the solution (66), for whi
h we show two 
ases: the small solid stars 
orrespond to
the 
onstant value of ηss, whereas the large empty stars show the result 
orresponding to
adjusting the index of the Hankel fun
tion in eq. (66) to the value of ηss at a given instant,
i.e. ηss = 0.40, 0.42, 0.44 for (k/aH)
−1 = 1, 10, 100, respe
tively. With the iso
urvature
modes absent, the 
urvature perturbations are ex
ellently des
ribed by the single-�eld
result, whi
h justi�es the use of the single-�eld approximations in the situations des
ribed
above.
5.3.6 Closing dis
ussion
The three examples presented here show that in multi-�eld in�ationary models, a large
part of the 
urvature perturbations 
an originate from intera
tions between the 
urvature
and iso
urvature perturbations on super-Hubble s
ales, not only from quantum �u
tua-
tions along the traje
tory at the Hubble exit. In su
h 
ases, the single-�eld result (81)
does not provide a 
orre
t predi
tion either for the normalization of the power spe
trum
or for its spe
tral index
ns = 1 + d lnPR/d ln k . (105)
There are te
hniques whi
h allow relating the spe
tral index of the 
urvature perturba-
tions, ns, to the spe
tral indi
es of the entropy perturbations and the 
urvature-entropy

orrelations through a set of 
onsisten
y relations [19, 22, 13℄, but all these quantities
separately depend on the super-Hubble evoulution of the perturbations. Again, we �nd it
the most straightforward to 
al
ulate the spe
tral index ns for ea
h model numeri
ally. In
Table 1, we 
ompare naive estimate ns ∼ 1−6ǫ∗+2ησσ∗ and the predi
tions of eq. (81) for
the spe
tral index ns with the numeri
al results of Se
tion 5.3. In our three examples, one

an see that the single-�eld result signi�
antly overestimates the 
orre
t spe
tral index.
The dis
repan
ies that we �nd follow from the fa
t that the two types of perturbations
ns 1− 6ǫ∗ + 2ησσ∗ single-�eld result full result
double in�ation (
anoni
al) 0.929 0.982 0.967
double in�ation (non-
anoni
al) 0.953 0.968 0.934
roulette in�ation 1.017 1.019 0.932
Table 1: A 
omparison between the predi
tions for the spe
tral index ns in the three
examples of in�ationary models des
ribed in Se
tion 5.2. The third 
olumn 
ontains result
derived from the single-�eld approximation (81); the result of full numeri
al 
al
ulations
are shown in the fourth 
olumn.
experien
e the slow-roll of the ba
kground �elds and the 
urvature of the in�ationary
potential in a di�erent way. Then, if the �nal 
urvature perturbations originate mainly
from the iso
urvature ones, they inherit the features of the iso
urvature power spe
tra at
the Hubble 
rossing.
6 Con
lusion
In this paper, we have studied two-�eld in�ation and extended several previous results on

urvature and iso
urvature perturbations to the 
ase of non-standard kineti
 terms.
First, we have 
al
ulated analyti
ally the 
urvature and iso
urvature spe
tra, as well as
the 
orrelation, just after Hubble 
rossing for two-�eld in�ation models, in
luding next-to-
leading order 
orre
tions in the slow-roll approximation. Our results (78)-(80) generalize
those of Byrnes and Wands, who assumed only standard kineti
 terms. We have also given
a re�ned analyti
al treatment of the spe
tra around Hubble 
rossing, whi
h is important
when the perturbations still evolve after Hubble 
rossing.
Se
ond, we have studied numeri
ally the evolution of the 
urvature and iso
urvature
perturbations after the Hubble 
rossing. This type of analysis is important sin
e in multi-
�eld in�ation, in 
ontrast with single-�eld in�ation, the 
urvature perturbation spe
trum
after in�ation is in general di�erent from the 
urvature perturbation spe
tra at Hubble

rossing be
ause of iso
urvature perturbations, as �rst emphasized in [6℄. This well-
known result applies to multi-�eld in�ation with either standard or non-standard kineti
terms. This e�e
t has been studied numeri
ally for standard kineti
 terms, using the
de
omposition into instantaneous 
urvature and iso
urvature, in the analysis of [23℄. We
have done a similar analysis here for non-standard kineti
 terms. In parti
ular, we have

ompared the numeri
al evolution of the perturbations with an analyti
al approximation,
whi
h we denoted the 
onstant slow-roll approximation: this approximation assumes not
only that the slow-roll approximation is valid, but also that the slow-roll parameters
remain almost 
onstant during the subsequent evolution where iso
urvature perturbations
are signi�
ant. This approximation has been used in [21℄ to 
ompute the ��nal� spe
tra in
the 
ontext of non-standard kineti
 terms. In most 
ases, this approximation is however
not very realisti
 as we 
learly show in our numeri
al study. We have also estimated
the impa
t of non-standard kineti
 terms on the 
oupled evolution of the 
urvature and
iso
urvature modes.
There has been a re
ent interest in 
onstru
ting in�ationary models in the 
ontext
of string theory. These models naturally lead to s
alar �elds with non-standard kineti
terms, to whi
h our analysis 
an apply. In this work, we have studied a very re
ent model,

alled `roulette� in�ation, and 
omputed quantitatively the �nal 
urvature spe
trum, thus
showing that the in�uen
e of iso
urvature modes on super-Hubble s
ales turns out to be
very important. Note that the authors of [14℄ 
omputed the �nal 
urvature spe
trum
by using a �single in�aton approximation�. They stress however that iso
urvature e�e
ts
�
ould produ
e big e�e
ts�, whi
h we indeed 
on�rm in our analysis.
As a message of 
aution for the readers who are not familiar with the e�e
t of iso
urva-
ture modes in multi-�eld in�ation, we have also 
omputed the spe
tral index for 
urvature
perturbations in a few models and 
ontrasted it with the value that one would naively
obtain by using the 
urvature perturbation at Hubble radius like in single �eld in�ation.
Finally, an interesting question, whi
h goes beyond the s
ope of this paper, would be
to investigate in whi
h 
ir
umstan
es these multi-�eld models 
ould produ
e iso
urvature
perturbations, after in�ation and the reheating phase. These �primordial� iso
urvature
perturbations are today severely 
onstrained by CMB data.
A
knowledgements We would like to thank V. Mukhanov for stimulating dis
us-
sions. D.L. would like to thank the Institute of Theoreti
al Physi
s of Warsaw for their
warm hospitaly and for their �nan
ial support via a �Marie Curie Host Fellowship for
Transfer of Knowledge� , proje
t MTKD-CT-2005-029466. Z.L. was partially supported
by TOK proje
t MTKD-CT-2005-029466, by the EC 6th Framework Programme MRTN-
CT-2006-035863, and by the grant MEiN 1 P03D 014 26. S.P. was partially supported by
TOK proje
t MTKD-CT-2005-029466 and by the grant MEiN 1 P03B 099 29. The work
of K.T. is partially supported by the US Department of Energy. K.T. would also like
to a
knowledge support from the Foundation for Polish S
ien
e through its programme
START.
Referen
es
[1℄ V. Mukhanov, �Physi
al foundations of 
osmology,� Cambridge, UK: Univ. Pr.
(2005) 421 p
[2℄ L. A. Kofman and A. D. Linde, Nu
l. Phys. B 282 (1987) 555.
[3℄ V. F. Mukhanov and M. I. Zelnikov, Phys. Lett. B 263 (1991) 169.
[4℄ N. Deruelle, C. Gundla
h and D. Langlois, Phys. Rev. D 45, 3301 (1992); Phys. Rev.
D 46 (1992) 5337.
[5℄ D. Polarski and A. A. Starobinsky, Nu
l. Phys. B 385 (1992) 623.
[6℄ A. A. Starobinsky and J. Yokoyama, arXiv:gr-q
/9502002
[7℄ D. Wands and J. Gar
ia-Bellido, Helv. Phys. A
ta 69 (1996) 211
[arXiv:astro-ph/9608042℄.
[8℄ J. Gar
ia-Bellido and D. Wands, Phys. Rev. D 54 (1996) 7181
[arXiv:astro-ph/9606047℄.
[9℄ V. F. Mukhanov and P. J. Steinhardt, Phys. Lett. B 422 (1998) 52
[arXiv:astro-ph/9710038℄.
[10℄ D. Langlois, Phys. Rev. D 59, 123512 (1999) [arXiv:astro-ph/9906080℄.
[11℄ A. A. Starobinsky, S. Tsujikawa and J. Yokoyama, Nu
l. Phys. B 610, 383 (2001)
[arXiv:astro-ph/0107555℄.
[12℄ N. Bartolo, S. Matarrese and A. Riotto, Phys. Rev. D 64 (2001) 083514
[arXiv:astro-ph/0106022℄.
[13℄ B. van Tent, Class. Quant. Grav. 21 (2004) 349 [arXiv:astro-ph/0307048℄.
[14℄ J. R. Bond, L. Kofman, S. Prokushkin and P. M. Vaudrevange,
arXiv:hep-th/0612197.
[15℄ Z. Lalak, G. G. Ross and S. Sarkar, Nu
l. Phys. B 766 (2007) 1
[arXiv:hep-th/0503178℄.
[16℄ J. J. Blan
o-Pillado et al., JHEP 0609, 002 (2006) [arXiv:hep-th/0603129℄.
[17℄ J. Ellis, Z. Lalak, S. Pokorski and K. Turzynski, JCAP 0610, 005 (2006)
[arXiv:hep-th/0606133℄.
http://arxiv.org/abs/gr-qc/9502002
http://arxiv.org/abs/astro-ph/9608042
http://arxiv.org/abs/astro-ph/9606047
http://arxiv.org/abs/astro-ph/9710038
http://arxiv.org/abs/astro-ph/9906080
http://arxiv.org/abs/astro-ph/0107555
http://arxiv.org/abs/astro-ph/0106022
http://arxiv.org/abs/astro-ph/0307048
http://arxiv.org/abs/hep-th/0612197
http://arxiv.org/abs/hep-th/0503178
http://arxiv.org/abs/hep-th/0603129
http://arxiv.org/abs/hep-th/0606133
[18℄ B. de Carlos, J. A. Casas, A. Guarino, J. M. Moreno and O. Seto,
arXiv:hep-th/0702103.
[19℄ C. Gordon, D. Wands, B. A. Bassett and R. Maartens, Phys. Rev. D 63, 023506
(2001) [arXiv:astro-ph/0009131℄.
[20℄ F. Di Mar
o, F. Finelli and R. Brandenberger, Phys. Rev. D 67, 063512 (2003)
[arXiv:astro-ph/0211276℄.
[21℄ F. Di Mar
o and F. Finelli, Phys. Rev. D 71, 123502 (2005) [arXiv:astro-ph/0505198℄.
[22℄ C. T. Byrnes and D. Wands, Phys. Rev. D 74, 043529 (2006)
[arXiv:astro-ph/0605679℄.
[23℄ S. Tsujikawa, D. Parkinson and B. A. Bassett, Phys. Rev. D 67, 083516 (2003)
[arXiv:astro-ph/0210322℄.
[24℄ K. Y. Choi, L. M. H. Hall and C. van de Bru
k, JCAP 0702 (2007) 029
[arXiv:astro-ph/0701247℄.
[25℄ H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984).
[26℄ V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203
(1992).
[27℄ D. Langlois, �In�ation, quantum �u
tuations and 
osmologi
al perturbations,� Le
-
tures given at Cargese S
hool of Parti
le Physi
s and Cosmology: the Interfa
e,
Cargese (2003). Published in �Cargese 2003, Parti
le physi
s and 
osmology�,p 235-
278. arXiv:hep-th/0405053.
[28℄ D. Langlois and F. Vernizzi, JCAP 0702, 017 (2007) [arXiv:astro-ph/0610064℄.
[29℄ D. Langlois and F. Vernizzi, Phys. Rev. Lett. 95, 091303 (2005)
[arXiv:astro-ph/0503416℄.
[30℄ D. Langlois and F. Vernizzi, Phys. Rev. D 72, 103501 (2005)
[arXiv:astro-ph/0509078℄.
[31℄ A. A. Starobinsky, JETP Lett. 42, 152 (1985) [Pis. Hz. Esp. Tor. Fizz. 42, 124
(1985)℄.
[32℄ M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71 (1996)
[arXiv:astro-ph/9507001℄.
[33℄ M. Sasaki and T. Tanaka, Prog. Theor. Phys. 99, 763 (1998) [arXiv:gr-q
/9801017℄.
[34℄ J. P. Conlon and F. Quevedo, JHEP 0601, 146 (2006) [arXiv:hep-th/0509012℄.
http://arxiv.org/abs/hep-th/0702103
http://arxiv.org/abs/astro-ph/0009131
http://arxiv.org/abs/astro-ph/0211276
http://arxiv.org/abs/astro-ph/0505198
http://arxiv.org/abs/astro-ph/0605679
http://arxiv.org/abs/astro-ph/0210322
http://arxiv.org/abs/astro-ph/0701247
http://arxiv.org/abs/hep-th/0405053
http://arxiv.org/abs/astro-ph/0610064
http://arxiv.org/abs/astro-ph/0503416
http://arxiv.org/abs/astro-ph/0509078
http://arxiv.org/abs/astro-ph/9507001
http://arxiv.org/abs/gr-qc/9801017
http://arxiv.org/abs/hep-th/0509012
	Introduction
	The model
	Homogeneous equations
	Linear perturbations
	Decomposition into adiabatic and entropy components
	Perturbation spectra
	Evolution of perturbations inside the Hubble radius
	Equations in the slow-roll approximation
	Evolution of perturbations on super-Hubble scales
	Numerical analysis
	Numerical procedure
	Examples of inflationary models
	Double inflation with canonical kinetic terms
	Double inflation with non-canonical kinetic terms
	Roulette inflation
	Numerical results for the perturbations
	Double inflation with canonical kinetic terms
	Double inflation with non-canonical kinetic terms
	Roulette inflation
	Impact of the non-canonical terms
	Effectively single-field cases
	Closing discussion
	Conclusion
