Loop Corrections to Cosmological Perturbations in Multi-field Inflationary Models

Abstract

We investigate one-loop quantum corrections to the power spectrum of adiabatic perturbation from entropy modes/adiabatic mode cross-interactions in multiple DBI inflationary models. We find that due to the non-canonical kinetic term in DBI models, the loop corrections are enhanced by slow-varying parameter $\epsilon$ and small sound speed $c_s$. Thus, in general the loop-corrections in multi-DBI models can be large. Moreover, we find that the loop-corrections from adiabatic/entropy cross-interaction vertices are IR finite. Comment: 21 pages, 7 figures; v2, typos corrected, ref added; v3 typos corrected, version for publishing in jcap

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arXiv:0905.0405v3 [hep-th] 14 Jul 2009
Preprint typeset in JHEP style - HYPER VERSION
CAS-KITPC/ITP-109
Loop Corrections to Cosmological Perturbations in
Multi-field Inflationary Models
Xian Gao
Key Laboratory of Frontiers in Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P.R.China
E-mail: gaoxian@itp.ac.cn
Fanrong Xu
Key Laboratory of Frontiers in Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P.R.China,
and
Maryland Center for Fundamental Physics and Department of Physics ,
University of Maryland, College Park, MD 20742, USA
E-mail: frxu@itp.ac.cn
ABSTRACT: We investigate one-loop quantum corrections to the power spectrum of adiabatic perturbation from
entropy modes/adiabatic mode cross-interactions in multiple DBI inflationary models. We find that due to the
non-canonical kinetic term in DBI models, the loop corrections are enhanced by slow-varying parameter ǫ and
small sound speed cs. Thus, in general the loop-corrections in multi-DBI models can be large. Moreover, we
find that the loop-corrections from adiabatic/entropy cross-interaction vertices are IR finite.
KEYWORDS: Cosmological perturbation theory, Inflation, Cosmology of theories beyond the SM, Physics of
the early universe.

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Contents
1. Introduction1
2. Basic Setup
2.1 The model and the background
2.2 Brief review of linear perturbation
2.3Adiabatic-entropy modes three-point cross-interactions
3
3
4
5
3.One-loop Entropy Mode Corrections
3.1Calculating the Loop Corrections
3.1.1 Diagonal contributions
3.1.2 Off-diagonal contributions
3.2 Final Results
6
7
7
11
13
4. Conclusion and Discussion13
A. A Brief Review of “In-in” Formalism
A.1 Preliminaries
A.2 “In” vacuum
A.3 Expectation values in “in-in” formalism
A.4 Perturbation theory
A.5 Positive/negative-fields notation
14
14
14
15
15
16
B. Momentum Integral17
C. Finite parts18
1. Introduction
Current observational data support standard ΛCDM model greatly [1], in which primordial perturbations which
are assumed to be responsible for Cosmic Microwave Background anisotropies and large-scale structure for-
mation are generated from quantum fluctuations and stretched to superhorizon scales during inflation (see e.g.
[2] for a review). Standard single-field slow-roll inflationary models predict almost scale-invariant, Gaussian
and adiabatic primordial fluctuation.
Actually, inflation itself is not a single model, but rather a theoretical framework. From the point of view
of power spectrum, many inflation models are “degenerate”. Power spectrum, or strictly speaking, tree-level
two-point correlation functions of cosmological perturbation do not give us an unique theory of inflation. In
fact, phenomena beyond linear-order have been extensively investigated over the past several years.
The most significant progress beyond power spectrum is the investigation of statistical non-gaussianities of
CMB anisotropies and primordial fluctuations (see e.g. [4] for a nice review). From the field theoretical point
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of view, non-gaussianity describes the interactions of perturbations, which will cause non-vanishing higher-
order correlation functions. Such interactions are mandatory in any realistic inflationary models, which come
from both the non-linear nature of gravity and the self-interactions of inflation field(s). In standard slow-
roll inflation scenario, however, the non-gaussianities have been proved too small to be detected [3], even
with PLANCK. Thus, any detection of non-gaussianties would not only rule out the simplest inflation models
but also give us valuable insight of fundamental physics of inflation. Various models have been investigated
to generate large non-gaussianties by introducing more complicated kinetic terms [5, 6, 7, 8] or more fields
[9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 22, 23].
Interactions not only cause non-vanishing higher-order correlation functions which are responsible for non-
gaussianities of primordial perturbations, they also cause quantum loop-corrections. It is possible that such loop
corrections may be large in their own right. Especially, we could expect large loop corrections in models which
have also large non-gaussianties, since both of them describe the interactions among perturbations. Moreover,
by collecting signatures of both quantum loop corrections and non-gaussianties, we will obtain a more sensitive
test of the physics of inflation.
Early estimations of loop effects in cosmological perturbations were pioneered by Mukhanov, Abramo and
Brandenberger [24, 25], Abramo and Woodard [26], and Unruh [27]. Subsequently, loop effects have been in-
vestigated by many authors [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49].
Recently, stimulated by works of S. Weinberg [50, 51], loop corrections to primordial fluctuations have been
re-investigated by several authors [52, 53, 54, 55, 56, 57, 58, 59]. It has been found that cosmological loop
corrections seem to have some infrared divergences, which scale as ln(kℓ), where ℓ−1is some IR comov-
ing momentum cut-off [52, 53, 54, 55]. Other discussions on cosmological loop corrections and the infrared
divergences can be found in [63, 64, 65, 66, 67, 68, 69]. The presence of IR divergences implies that loop-
contributions may potentially give large corrections to tree-level results, although it was argued in [63, 64, 65]
that these IR divergent corrections do not contribute in quantities that are directly observable.
Actually, now there are two types of so-called “loop corrections” in the literatures. One type is the quantum
loops (q-loop) which originate from the interactions among fields and are evaluated using standard quantum
field theory, while the other type is the classical loops (c-loop) which arise mathematically from some “loop-
type integrals” due to the non-linear map from δφ to ζ [60, 61, 62]. In this work, we focus on the quantum
loops. However, it is well-known that the observational variable is the curvature perturbation ζ and there
has non-linear relation between ζ and δ during superhorizon evolution (e.g. in terms of the “δN-formalism”
[73, 74, 75]). Thus, these “loop” effects also have to be treated seriously ,and especially they can be significant,
if the higher-order derivatives of e-folding number N with respect to the inflaton fields are extremely large.
While in [62] it was shown that the loop corrections for ζ are always suppressed during inflation, if one assume
that the fields follow a classical background trajectory.
Up to now, one-loop corrections to the power spectrum of the cosmological perturbations have been inves-
tigated in single scalar-field loops [52, 53, 54, 55, 50, 51], concerning effects from many light scalar fields [57],
and from graviton loops [56]. Actually, it has been recognized that it is hard to generate large “local-type” non-
gaussianities in single-field models, unless we abandon the slow-roll-type conditions. Thus, the investigation
of multiple field models has particular importance. In multiple field inflationary models, perturbations can be
decomposed instantaneously into one adiabatic mode and several entropy modes [20] (see [21] for a review).
There are interactions between adiabatic mode and entropy modes.
In this work, we investigate the one-loop corrections to adiabatic power spectrum due to the interactions
between entropy mode and adiabatic mode, at the horizon exiting. We find that due to the non-canonical kinetic
term in DBI models, the loop corrections are enhanced by slow-varying parameter ǫ and small sound speed
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cs, as the enhancement of non-gaussianties in such models. Thus, in general, the loop-corrections in multi-
DBI models can be large. Moreover, we find that the loop-corrections from adiabatic/entropy cross-interaction
vertices are IR finite. This is because in the limit of speed sound cs, the leading-order three-point interactions
are totally “derivatively interacted”. While the derivatives (both with respect to time and space) give momentum
factors which cancel the momentum factors in the denominator from the tree-level correlation functions to make
the loop-integrals IR safe.
Quantum cosmological correlation functions are evaluated by using “in-in formalism” (also dubbed as
“Schwinger-Keldysh formalism” ) [70, 71, 72] (also see e.g. [50] for a nice review), which we also give a brief
review in Appendix A. In [58] the application of “in-in” formalism on cosmological perturbation, especially on
loop corrections, has been investigated in details.
This paper is organized as follows. In section 2, firstly we briefly review the linear perturbation of multiple-
DBI models, then we describe the third-order perturbation action, especially the three-point adiabatic/entropy
modes cross-interaction vertices. In section 3, we calculate the one-loop corrections to the adiabatic power
spectrum from the adiabatic/entropy modes cross-interaction vertices, by using the “in-in” formalism. Finally,
we make a conclusion and discuss the related issues.
In this paper, we set ? = c = 8πG = 1.
2. Basic Setup
Perturbation theory of general multiple field modes with non-canonical kinetic terms up to third-order action
has been calculated by several authors [10, 11, 12, 13, 14, 15, 16, 17, 22]. In this work, we focus on multi-field
DBI models [10, 11, 12, 13, 14, 15, 16]. In this section, we briefly review previous known results and setup our
conventions.
2.1 The model and the background
In this work, we focus on two-field DBI inflationary models, with action of the form
S =
?
d4x
?1
2R + Lφ
?
,
(2.1)
with1
Lφ≡ −
1
?f(φI)
??
1 − 2fX + 2f2(X2− XIJXIJ)
?
− V (φI),
(2.2)
where
XIJ≡ −1
2∂µφI∂µφJ,
(2.3)
and X ≡ GIJXIJ.
The slow-rolling parameter is defined as usual
ǫ = −
˙H
H2.
(2.4)
1In general, the full DBI-type action takes the form L = −1
f
“√D − 1
”
− V , with
D = det(δIJ − 2fXIJ) ≡ 1 − 2fX + 4f2X[I
However, it is shown that higher-order terms proportional to f3and f4do not contribute to the calculation of power spectra and
leading order bispectra. Thus, in this work we neglect these terms for simplicity. One can refer [10, 11, 12, 13, 14, 15, 16] for detailed
investigations of multi-DBI inflation models.
IXJ]
J− 8f3X[I
IXJ
JXK]
K+ 16f4X[I
IXJ
JXK
KXL]
L.
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The background “inflaton velocity” ˙ σ can be related with ǫ as
˙ σ ≡
√2X = H√2ǫcs,
(2.5)
where csplays the role of the propagating speed of the perturbation. In DBI inflation model,
c2
s≡ 1 − 2fX ,
(2.6)
where X is evaluated with the background quantities.
2.2 Brief review of linear perturbation
In multi-field inflation models, perturbations can be instantaneously decomposed into one adiabatic mode and
several entropy modes [20]. Restricting to two-field case, we introduce the basis {eσ,es} with eI
where ˙ σ is the background inflaton velocity defined in (2.5), and eI
decomposition
QI≡ δφI= QσeI
Qσand Qsare adiabatic mode and entropy mode respectively.
After the adiabatic/entropy modes decomposition, the second-order perturbation action for two-field DBI
inflation model up to leading-order was derived in [10, 11, 14] in spatially-flat gauge:
σ≡˙φI/˙ σ,
sis orthogonal to eI
σ. Then we introduce the
σ+ QseI
s,I = 1,2.
(2.7)
S(2)=
?
dηd3xa2
2c3
s
??(Q′
σ)2− c2
s(∂iQσ)2?+ c2
s
?(Q′
s)2− c2
s(∂iQs)2??,
(2.8)
where H ≡ ˙ a/a is the Hubble parameter, η is the comoving time, and a prime denotes the derivative with
respect to η. In deriving (2.8), we assume that csis approximately constant and neglect coupling between
adiabatic and entropy modes which may give significant effect in some cases, and keep only the leading order
terms. Note that in general multi-field models, adiabatic mode and entropy modes can propagate with different
(and arbitrary) speeds of sound, which was first pointed out in [12, 13] (see also [14, 15, 16, 17, 22]). Multi-
field DBI model is a special case, where adiabatic mode and entropy modes propagate with the same speed of
sound cs.
In canonical quantization procedure, we write
Qσ(η,k) = uk(η)ak+ u∗
Qs(η,k) = vk(η)ak+ v∗
k(η)a†
k(η)a†
−k,
−k.
(2.9)
In de Sitter approximation a(η) = −1/Hη, the mode solutions can be easily solved
uk(η) =
H
√2k3(1 + icskη)e−icskη,
H
√2k3(1 + icskη)e−icskη.
vk(η) =
cs
(2.10)
Note that this choice of mode functions corresponds to the Bunch-Davis vacuum. Now it is straightforward to
write down the tree-level two-point correlation functions (see Appendix A for more details),
?0??Qσ(η,k1)Qσ(η′,k2)??0?= (2π)3δ3(k1+ k2)Gk1(η,η′) ,
?0??Qs(η,k1)Qs(η′,k2)??0?= (2π)3δ3(k1+ k2)Fk1(η,η′) ,
(2.11)
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