IPMU 09-XXXXICRR-Report-543

Cosmological constraints on rapid roll inflation
Takeshi Kobayashi∗
Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan

Shinji Mukohyama† and Brian A. Powell‡
Institute for the Physics and Mathematics of the Universe (IPMU),The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan(Dated: April 29, 2009)
We obtain cosmological constraints on models of inflation which exhibit rapid roll solutions. Rapidroll attractors exist for potentials with large mass terms and are thus of interest for inflationarymodel building within string theory. We constrain a general ansatz for the power spectrum arisingfrom rapid roll inflation that, in the small field limit, can be associated with tree level hybrid
potentials with variable mass terms and nonminimal gravitational coupling ξRφ2 . We consider
perturbations generated through modulated reheating and/or curvaton mechanisms in place of theobservationally unacceptable primary spectra generated by inflaton fluctuations in these models.
The lack of a hierarchy amongst higher-order k-dependencies of the power spectrum results in models
with potentially large runnings, allowing us to impose tight constraints on such models using CMB
and LSS data. In particular, we find ns < 1 and |α| < 0.01. We conclude with a concrete realizationof rapid roll inflation within warped throat brane inflation that is in good agreement with current
data.
I.

INTRODUCTION

Recent years have seen steady progress in the development of inflation within string theory [1–5]. Inflation in string theory is beset by the η-problem [6, 7],
in which dimension 6, Planck-suppressed operators induce corrections to the inflaton mass of order the Hubble
scale, δm ∼ O(H). Such models are characterized by the2 V ′′
parameter η = MPl
V ∼ O(1), preventing slow roll inflation. These corrections generically arise in the volume
stabilization of brane inflation [7], and a compelling solution to this problem remains an active area of research.
This problem is compounded in models based on warped
throat compactications in which the length of the throat
restricts the maximal field range [8]. It was earlier realized by Linde that it is possible to achieve significant
efoldings of expansion even in the presence of large massterms [9] if the field enters a stage of rapid roll inflation.
This choice of terminology will become clear below.
In this paper, we obtain observational constraints onmodels of inflation that undergo a period of rapid roll inflation. Typical examples of such potentials are those appearing in models of tree-level hybrid inflation, in which
the field evolves along the rapid roll attractor, asymptotically approaching the minimum. These potentials giveobservationally unacceptable density perturbations, andso we consider perturbations generated through othermechanisms, such as modulated reheating or the curvaton scenario. The light fields necessary for generating the∗ Electronic address: takeshi.kobayashi@ipmu.jp† Electronic address: shinji.mukohyama@ipmu.jp‡ Electronic address: brian.powell@ipmu.jp

primordial perturbations in these scenarios are abundantin string theory, making such mechanisms natural in thissetting.
We derive an analytical expression for the power spectrum, P (k), arising from potentials of the form
 4 
φ21
φ(1)
V = V0 1 + v2 2 + O
4
2 MPl
MPl
in which the field is restricted to the range φ ≪ MPl .
The formalism we develop is applicable to both minimally and nonminimally coupled scalar fields. The key
result is that the power spectrum arising from modulated
reheating and/or curvatons does not possess a hierarchy amongst low-order and high-order k-dependencies,in contrast to inflaton-generated perturbations. In fact,
the higher-order runnings of the spectrum are completely
determined by the spectral index, ns , and first-order running, α = dns /dlnk, in the regime where Eq. (1) is justified. Using the latest CMB and LSS data, we performa Bayesian analysis on this power spectrum. We compare these constraints with those obtained on a general1kansatz for the power spectrum, P (k) ∝ k (ns −1)+ 2 αln k0 ,where the parameters ns and α are allowed to vary freely.
The spectrum arising from rapid roll inflation is highly
constrained relative to the generic parameterization, favoring ns < 1 and small running, |α| < 0.01.
Lastly, we present a concrete realization of rapid rollinflation within the context of warped throat brane inflation. We find that it is possible to obtain sufficient
inflation in the presence of a range of mass terms, andthe predictions of the model are in good agreement withthe observational constraints we obtain in this paper.
In Section II, we review the concept of rapid roll inflation. In section III, we discuss the generation of perturbations in these models through either modulated reheating

2or curvatons, and derive an analytic expression for P (k).
In section IV we present the details of the Bayesian analysis, and report our constraints on the power spectrum.
In Section V, we present a specific realization of rapidroll inflation within string theory.
II.

RAPID ROLL INFLATIONIn this section we review the phenomenon of rapid rollinflation. To illustrate this concept, consider a potentialof the form
1
V (φ) = V0 + m2 φ2 .
2(2)This is a prototypical hybrid-type potential, in which inflation ends when an auxiliary field develops a tachyonicinstability around φ ≈ 0. The equation of motion of ahomogeneous scalar field in an FRW universe with thispotential isφ̈ + 3H φ̇ + m2 φ = 0.

(3)

When H ≈ const. we obtain the solution
φ(t) = φ+ er+ Ht + φ− er− Ht
r
3
9 m2
− ,
−
r± = ±
4 H2
2

(4)
(5)

2

9
where mH 2 < 4 . This is true for the potential Eq. (2) inthe limit φ ≪ MPl . The general solution is a superposition of both branches of Eq. (4) but at late times the fieldevolves along the r+ branch. If we choose m2 = 2H 2 , weobtainφ(t) =

φ0
,
a(t)

(6)

or, equivalently,
φ̇(t) = −Hφ(t).

(7)

The dynamics defined by the solution Eq. (7) is termed
rapid roll inflation. It tells us that the logarithmic variation of φ is set by the Hubble scale, or, ∆φ ≈ φ in a
Hubble time, ∆t = H −1 . This is in contrast to slow rollinflation, for which |φ̇| ≪ |Hφ|. The number of efolds ofexpansion obtained during rapid roll is
Z
Z
φi
dφ
(8)
= ln ,
N = Hdt = −φφfwhere φi and φf are the field values at the beginning andend of inflation. The spectral index of perturbations isgiven by ns − 1 = 2r+ , and is therefore blue for this classof models [10].1

1 The horizon crossing formalism breaks down here, and care must

be taken when evaluating the spectrum [10, 11].

Rapid roll solutions exist for more general potentials.
The inflationary dynamics can be described bycH φ̇(t) ≃ −V ′ (φ)(9)

2H 2 = V whenever the conditions ǫ ≪ 1 andwith 3MPl|η̄| ≪ 1 are satisfied. Here ′ 2
2
V
MPl
,
(10)
ǫ =
2
V
c2
η̄ = η +
− c,
(11)
3
′′
2 V
(12)
η = MPlV

where c is a constant of order unity. The solution Eq.
(9) reduces to Eq. (7) for the potential Eq. (2) with
m2 = 2H 2 . In fact, the rapid roll solution Eq. (9) is a
dynamical attractor [12], the stability of which we provein the appendix.
Rapid roll inflation can also be manifested in models
of inflation driven by nonminimally coupled scalar fields.
Consider the action of a nonminimally coupled scalar,


Z
R
1 µ
1
4 √
2
S = d x −g.
2 − 2 ∂ φ∂µ φ − V (φ) − 2 ξRφ
2MPl
(13)
In an FRW universe with ds2 = −dt2 + a(t)2 dx2 , theresulting equations of motion are
1 2
φ̇ + 6ξHφφ̇ + 3ξφ2 H 2 + V (φ), (14)
2
1
2
MPl
Ḣ = − (1 − 2ξ)φ̇2 − 4ξHφφ̇ − 12ξ 2 H 2 φ2
2
− ξφV ′ (φ) + ξ(1 − 6ξ)φ2 Ḣ,
(15)

2
3MPl
H2 =

φ̈ + 3H φ̇ + 6ξ(Ḣ + 2H 2 )φ + V ′ (φ) = 0.

(16)

The value ξ = 16 is known as conformal coupling. In thislimit, the non-potential terms on the right hand side ofEq. (14) can be written as a perfect square,23MPl
H 2 = (φ̇ + Hφ)2 + V (φ).

(17)

Then, for constant potential, V (φ) = V0 , the rapid rollsolution φ̇ = −Hφ gives a de Sitter universe,23MPl
H 2 = V0 .

(18)

The existence of rapid roll solutions in this model are
best understood by recalling that for Ḣ ≈ 0, the Ricci
scalar R ≈ 12H 2 . Then for ξ = 61 , the coupling term in
Eq. (13) forms an effective mass term ∝ H 2 φ2 , and werecover the potential Eq. (2). This model was first investigated in [13], where the conformal term was interpreted
as the Hubble scale mass correction arising from volume
stabilization in the KKLMMT [7] model,
"
4 #

MPl ∆
,
(19)
V (φ) = V0 1 −φ

3where ∆ is a dimensionless constant. This potential isflat except for very small values of φ, and so the de Sitter
solution holds across much of the inflationary trajectory.
As in the minimally coupled case, rapid roll attractor
solutions exist for more general potentials [13]. In thispaper, we seek constraints on rapid roll inflation arisingfrom potentials of the form Eq. (2) with arbitrary nonminimal coupling strengths.
III.

PRIMORDIAL PERTURBATIONSIn this section we derive an analytic expression for the
power spectrum generated by the rapid roll models introduced in the last section. An important aspect of thiswork is that we consider perturbations not generated by
the inflaton itself, but rather from a mechanism such as
modulated reheating [14, 15] or curvatons [16–18]. We do
this for several reasons. The first is that spectra arisingfrom the minimally coupled inflaton with potential Eq.
(2) or flat potentials with nonminimal couplings ξ ≈ 1/6
are blue (ns > 1) with small tensors and thus not observationally viable [23]. In addition to phenomenologicalmotivations, such mechanisms are also attractive froma theoretical point of view. In particular, string theory
provides natural candidates for the light moduli fields
that generate perturbations in these alternative scenarios. The low energy scales required to suppress the inflaton perturbations on CMB scales are well tolerated
by string inflation models, particularly brane inflation in
warped throat compactifications. String inflation is alsoa pertinent place to realize the rapid roll inflation models
considered in this paper.
A.

Modulated Reheating and Curvatons

In order to reheat the universe, the inflaton field must
undergo decays whose end products are the particles of
the Standard Model. In supersymmetric and string theories, the couplings that facilitate these decays are notconstants, but functions of scalar fields in the theory. If
these fields are light during inflation (m ≪ H), they willfluctuate in space with the result that the coupling constants (and hence decay rates) will become functions ofspace. Different parts of the universe thus decay and reheat at different times – the resulting spatial modulationsin the reheat temperature give rise to energy density perturbations,
δTRH
δρ
∝
.
TRH
ρ

(20)

Pδ ∝

1
π2



H
σ

2

.

(21)

where M is some mass scale. The key result is thatthe power spectrum of the resulting fluctuations is then(22)

If the curvaton is sufficiently light during inflation, σ =const as a result of the Hubble drag and Pδ ∼ H 2 . As in
the case of modulated reheating by decaying light particles, the power spectrum is completely determined by
the evolution of the Hubble parameter.
B.

Power Spectrum

We now obtain an analytic expression for the power
spectrum of curvature perturbations of a nonminimallycoupled scalar field with a potential of the form
#
"
∞
X
v2n n
(23)
x ,
V = V0 1 +
(2n)!
n=1
x ≡

φ2
2 ,
MPl

(24)

where V0 and v2n are constants, and we suppose thatx ≪ 1. The scalar field satisfies the equations of motion
Eqs. (14)-(16). Let us define ∆1 and ∆2 by
#
"
N
+1
2
X
3MPl
H2
∆1 ≡
− 1+
α2n xn ,
V0
n=1
∆2 ≡

N
X
φ̇
β2n xn ,+Hφ n=0(25)

where N is non-negative integer, and α2n and β2n are
constants to be determined. We shall determine α2n with(n = 1, · · · , N + 1) by demanding that
∆1 = O(xN +2 ).

(26)

On the other hand, we demand that ∆2 satisfies
˙2∆+ γ∆2 = O(xN +1 )H

For a light modulus field χ, one finds [14]
δχ
δρ
∝
,
ρ
M

∝ H 2 /M 2 , where M can be taken sufficiently smaller
than MPl to ensure that the inflaton perturbations are
subdominant.
The primordial density perturbation may also be generated via the curvaton mechanism [16–18]. The curvaton, σ, is a light scalar during inflation with an associatedvacuum fluctuation, δσ. These fluctuations constitute a
superhorizon isocurvature perturbation that grows during a post-inflationary radiation dominated phase. After
curvaton decay, the perturbation is converted into an adiabatic mode with amplitude [16](27)

for a positive constant γ. This condition uniquely determines β2n (n = 1, · · · , N ) and γ (> 0). Note that thepositivity of γ is required by the stability of Eq. (27):∆2 → O(xN +1 ) as a → ∞.

4
We now set N = 0 to obtain the lowest order result.
We follow the three steps to complete the program: First,
we eliminate Ḣ, H 2 and φ̈ in Eqs. (26) and (27) byusing Eqs. (15), (14) and (16), respectively. Next, we
determine γ by demanding that the coefficient of φ̇ in
Eq. (27) after the first step should vanish. Lastly, we
expand Eqs. (26) and (27) with respect to x and solve
them order by order.
The result is
1− 2ξ β0 − ξ,
α2 =2p3 ± 9 − 12(v2 + 4ξ)
β0 =,2
γ = 3 − 2β0 .
(28)
Since γ is required to be positive, we need to choose the“−” sign for β0 . Thus,hip1− ξ 3 − 9 − 12(v2 + 4ξ) − ξ, (29)α2 =4p
3 − 9 − 12(v2 + 4ξ)
β0 =
,
(30)
2
p
γ =
9 − 12(v2 + 4ξ).
(31)

with
φ2
A = α2 02
M 2
 Pl  hipφ01
− ξ 3 − 9 − 12(v2 + 4ξ) − ξ
=
2 ,
4
MPl
(36)
p
B = 2β0 = 3 − 9 − 12(v2 + 4ξ).
(37)

A unique aspect of power spectra arising from rapidroll inflation is the lack of a hierarchy amongst higherorder time derivatives of the Hubble parameter andhigher-order k-dependencies of the power spectrum.
Such hierarchies do exist in slow roll inflation, with theresult that higher-order runnings are suppressed. For
example, in the case of nearly conformal inflation with
ξ = 1/6 + δξ on a flat potential (v2n = 0), we haveThe attractor equation Eq. (27) with N = 0 and γ > 0
implies that ∆2 → O(ǫ1 ) as a → ∞. Thus, we haveφ̇ + β0 Hφ ≃ 0. This gives
φ ≃ φ0



a
a0

−β0

,

(32)

where φ0 and a0 are the values of φ and a at t = t0 .
Then, Eq. (26) with N = 0 implies that
−2β0

2
φ2
3MPl
H2
P (k) ∝
≃ 1 + α2 02
V0
MPl
k=aH



k2
a20 H 2

φ2
≃ 1 + α2 02
MPl



2 2
3MPl
k
2
a0 V0

−2β0

(33)

By choosing a0 to be the scale factor at the pivot scalek = k0 , we obtain
P (k) ≃

P (k0 )
φ2

1 + α2 M02

Pl

"

φ2
1 + α2 02
MPl



k
k0

−2β0 #,

(34)

where φ0 is the field value at the pivot scale. This canbe written more concisely in the form
P (k) ="

P (k0 )
1+A
1+A



k
k0

−B #

,

(35)

φ20
2 ,
MPl

(38)

B = 2(1 + 12δξ) + O(δξ 2 ).

(39)

A = −[δξ + O(δξ 2 )]Then, the spectral index and runnings of the power spectrum Eq. (35) arens − 1 ≃ 2δξ



d
dlnk

d
dlnk

2

m

lnP (k)
k=k0

φ20
2 ,
MPl

≃ −4δξ

(40)

φ20
2 ,
MPl

(41)

..
.
lnP (k)k=k0≃ −(−2)m δξ

φ20
2 .
MPl

(42)

Note that all the derivatives are of the same order in δξ
φ2
and M02 , with higher-order runnings increasing in magPl
nitude. It is therefore not possible to model these spectra
as a Taylor expansion in lnk, necessitating the use of Eq.
(35) for imposing constraints.
In the case of perturbations generated through modulated reheating/curvatons, it is important to mention
that minimally coupled models, have power spectra proportional to the Hubble parameter formulated in the Einstein frame, P (k) ∝ HE2 . In contrast, nonminimally coupled models have spectra related to the Jordan frame
Hubble parameter, P (k) ∝ HJ2 . While this distinction
is not very important when determining the duration of2
inflation (HE and HJ differ by a factor φ2 /MPl), it is ofimportance for the observables, which are themselves of2
the order φ2 /MPl.
IV.

COSMOLOGICAL CONSTRAINTS

We now obtain cosmological constraints on the power
spectrum of curvature perturbations derived in the last

5
section, Eq. (35). In order to ensure that the inflaton
perturbations are suppressed on CMB scales, and for additional reasons that will become apparent in Section V,
we consider low-scale inflation. We therefore do not include a gravitational wave contribution to the temperature anisotropy. We utilize Markov Chain Monte Carloto obtain Bayesian parameter constraints from the 5th year WMAP cosmic microwave background data [19–23]
and the 4th release SDSS Luminous Red Galaxy (LRG)
galaxy power spectrum data [24]. We use the CosmoMC[25] code with a modified version of CAMB [26] in order tocompute the Cℓ -spectra arising from Eq. (35). MCMC
techniques [27–30] generate samples from the likelihoodof model parameters, L(d|θ), where d represents the ndimensional data and θ the n-dimensional parameter vector. Via Bayes Theorem, the likelihood function relates
any prior knowledge about the parameter values, π(θ),
to the posterior probability distribution,p(θ|d) =L(d|θ)π(θ)
L(d|θ)π(θ)
= R
.
P (d)
L(d|θ)π(θ)dθ(43)

The posterior probability distribution function of a single
parameter θi is obtained by marginalizing p(θ|d) over theremaining parameters,Zp(θi |d) = p(θ|d)dθ1 · · · dθi−1 dθi+1 · · · dθn−1 . (44)We adopt a four parameter base cosmology: the baryon
density Ωb h2 , the cold dark matter density Ωc h2 , theratio of the sound horizon to the angular diameter distance at decoupling, θs , and the optical depth to reionization, τ . The power spectrum Eq. (35) is constrained byvarying the parameters P (k0 ), A, and B directly in theMarkov chains, at the pivot scale k0 = 0.002 hMpc−1 .
We also marginalize over the contribution from theSunyaev-Zeldovich effect by varying the amplitude ASZ[23, 31], assume purely adiabatic initial fluctuations, impose spatial flatness, and adopt a top-hat prior on theage of the universe: t0 ∈ [10, 20] Gyrs.
The posterior probability distribution in the directionsof A and B is expected to be non-Gaussian. This isbecause the lines A = 0 and B = 0 give exactly scale
invariant spectra, providing good fits to the data. The
posterior thus has a high likelihood spine across the fullprior of A along B = 0, and conversely across the fullprior of B along A = 0. These spines are very narrow,since, for example, as one moves to larger values of B
along the A = 0 line, a small step δA gives a strongly
non-power law spectrum. Therefore, while the posterior along these directions has high mean likelihood, themarginalized likelihood tends to be small. Sampling fromthis distribution thus poses a challenge, and we therefore utilize multicanonical sampling [32, 33] in lieu ofthe more standard Metropolis-Hastings algorithm. Multicanonical sampling is useful for probing into the tailsof non-Gaussian distributions. We measure convergenceacross eight chains using the Gelman-Rubin R statistic(R < 1.1).

The power spectrum Eq. (35) is not a freely tunable
function. One requirement on the form of the spectrum isthat the spacetime must be inflationary while astrophysical scales are generated. Otherwise, causally generated
quantum fluctuations would not be stretched to superhorizon scales. Defining the parameter ǫ = −Ḣ/H 2 , andwith P (k) ∝ H 2 , we have−1
1 dlnP (k) 1 dlnP (k)−1ǫ=,
2 dlnk2 dlnk(45)

withdlnP (k)
= −AB
dlnk



k
k0

−B "

1+A



k
k0

−B #−1. (46)

This is enforced by simply requiring that ǫ < 1 across therange of scales probed by cosmological observations, corresponding to wavenumbers ∆lnk ≈ 10. Combinations of
A and B which violate this condition are rejected fromthe sample by assigning such models very low likelihood.
As a means of comparison with the conventional parameterization, we transform the parameters A and
B into the spectral parameters ns (k0 ) and α(k0 ) =dns (k0 )/dlnk,
AB
,
1+A
AB 2
α(k0 ) =.
(1 + A)2

ns (k0 ) = 1 −

(47)
(48)

While the prior π(A, B) is chosen to be flat, the prior distribution π(ns , α) that results from this transformation
is not. The Jacobian of the transformation
π(A, B) =∂(A, B)π(ns , α),∂(ns , α)(49)

is in fact zero at the point (ns = 1, α = 0), and so theprior is not well-defined at this point. While this presents
a formal difficulty, it is of little consequence in practicesince the Markov chains do not sample the neighborhoodof this point with sufficient precision for the results to beaffected by the prior in this area. We have verified thatthe posterior distribution is dominated by the likelihood
across the prior range of ns and α, and so π(ns , α) is
effectively flat in this region.
Figure 1 displays the 1-σ (68% CL) and 2-σ (95% CL)
error contours for the rapid roll model compared to theconstraints obtained on ns and α when they are allowed
to vary as free parameters (hereafter referred to as theplr model), as done in standard analyses by adopting the
form
ln

1
k
k
P (k)
= (ns − 1)ln + αln2 .
P (k0 )
k0
2
k0

(50)

The rapid roll model is significantly more predictive thanthe model-independent, plr spectrum with a preference

6
0.04
0.03

5000

WMAP
ACBAR

0.02
4000

l(l+1)Cl/2π µK

0.01

α

0
−0.01
−0.02
−0.03

3000

2000

1000

−0.04
0
300

−0.05
−0.06
0.9

1000

2000

l

0.95

1
ns

1.05

1.1

FIG. 1: Marginalized constraints on rapid roll inflation ascompared to constraints obtained on the spectral parameters in the standard definition (plr) Eq. (50). The blue
contours denote 68% and 95% CL for the plr model with
WMAP5+SDSS+ACBAR. The filled (orange-yellow) contours denote the same for the rapid roll model. The black
dashed contours are constraints on rapid roll inflation with
only WMAP5+SDSS.
Model
ns (k0 )rapid roll 0.972+0.022−0.031plr
1.04+0.16−0.21

α(k0 )−2lnL̂
+0.01−5.01 × 10−3 −0.001 2693.5−0.035+0.024
2691.2−0.025TABLE I: Marginalized 2-σ errors on the spectral parameters
for the rapid roll model and the standard parameterization
for WMAP5+SDSS+ACBAR.

for ns < 1 and small running. The marginalized constraints are given in Table 1. The tight constraints on
the rapid roll parameters are a result of the correlations
between ns , α, and the infinite tower of higher-order kdependent terms (e.g. Eq. (40)). The running of therunning, β = dα/dlnk, and all terms of higher-order, arecompletely determined by ns and α. For example,β=α2,ns − 1(51)

and so as the running is increased (for ns 6= 1), β likewise increases in magnitude. The tight constraints on α
are therefore a direct result of the data’s intolerance to alarge β. The constraints on the upper (lower) value of ns
are a result of the data’s requirement that blue spectra
be accompanied by significant negative (positive) running, as indicated by the black contours in Figure 1. The
magnitude of running permitted in the rapid roll modelis insufficient to accommodate very tilted spectra.
Of the spectra lying within the 2-σ envelope, the mostdistinctive are those that exhibit a relatively abrupt in-

FIG. 2: Best-fit plr model (red solid) as compared to a rapidroll spectrum lying at the edge of the 95% CL (red dashed).
WMAP 5-year data (black points) and the 2008 ACBAR data
(purple points) are included. The inclusion of ACBAR data
imposes additional constraints on rapid roll models which exhibit higher-order running on small scales.

crease in power on small scales at around 0.02 hMpc−1 .
This k-dependence is a result of higher-order runnings.
Since the WMAP data becomes noise-dominated ataround the second peak, the addition of small-scale CMB
measurements might have an effect on constraints. In
Figure 2 we present the CℓT T -spectrum of the best fit
plr model (solid line) along with a model lying on theoutskirts of the 2-σ envelope (dashed line) when only
WMAP5+SDSS are used. As suggested by Figure 2 andconfirmed in Figure 1, the inclusion of the latest ACBAR
data-set [34] yields a distinctive improvement over the
WMAP+SDSS only constraints, ruling-out such spectra
at greater than 95% CL.
We also note that the rapid roll model does a poorer
job of fitting the data than the plr model, with a difference in effective chi-square, ∆χ2 = −2∆lnL ≈ 2, between them. Of course, this is not to say that the plr
model is necessarily preferred by the data, since a propercomparison must also incorporate the number of degrees
of freedom available to each model. However, this is notas simple as just counting the tunable parameters (bothmodels have 8 – with 3 defining the power spectrum),
since this number is not representative of model complexity, the quantity of interest when performing Bayesianmodel selection [35]. Rather, one should identify the
number of parameters that are sufficiently constrainedby the data, giving an effective number of parameters, C[35, 36],
C = χ2 (θ) − χ2 (θ̂),(52)

where θ, is the parameter vector, the bar indicates anaverage, and the hat denotes the best-fit value. Using thisprescription, we find that the plr model has 7 effective

7degrees of freedom, while the rapid roll model has 6.7.2
The fact that the rapid roll model has fewer effective
parameters can be understood by examining the ansatz
Eq. (35). As mentioned, both A and B give high mean
likelihood across the full range of their respective priors.
While the marginalized distribution of these parameters
is well localized, there exist directions in the prior volume
along which these parameters are unconstrained. The
effective complexity criterion therefore does not count A
and B as two free parameters, but rather a little lessthan this. It is therefore appropriate to conclude thatcurrent data does not single-out a preferred model, andit will be necessary for future experiments to improve ourunderstanding of the power spectrum.

V.

RAPID ROLL D-BRANE INFLATIONAs a simple example of minimally coupled rapid-roll
inflation (see also the appendix), let us consider D3/D3
brane inflation in a warped throat [7]. In this model,
inflation is driven while a D3-brane is attracted towards
an D3 sitting at the throat tip. The radial position√ r ofthe D3 plays the role of the inflaton through φ ≡ T3 r,
and the warped tension of the D3 sources inflation. It is
known that the inflaton receives a large mass from moduli stabilization effects, preventing slow-roll. Here, we
also take into account corrections due to the throat coupling to the bulk Calabi-Yau space, which can be analysed using gauge/gravity duality [41] (see also [42–44])3 .
We consider the case where the leading bulk effect arises
from a non-chiral operator of dimension ∆ = 2 in the
dual CFT, where the inflaton potential takes the form
V (φ) = 2h40 T3(

3h4 T3
1 − 20 4 + β
8π φ



φ
MPl

2 )

,

(53)

2 The Sunyaev-Zeldovich amplitude A

2
2
3MPl
Hinf
≃ 2h40 T3 .

(54)

SZ is poorly constrained by
the data and therefore does not contribute to this figure.
3 There can also be additional contributions from moduli stabilization, as was investigated in [37–40] in order to fine-tune theinflaton potential.

(55)

From Eqs. (10) and (12) one can estimate
ǫ ≃ 2β 2

φ2
2 ,
MPl

η ≃ 2β.

(56)

Then, setting
c=

3+

√
9 − 24β,2(57)

the inflationary attractor Eq. (9) gives√H
3 + 9 − 24β 1
≃−
.
12β
φ
φ̇

(58)

We note that the coefficient of 1/φ agrees with −1/β0from Eq. (30) by setting ξ = 0 and v2 = 2β. Upon
estimating the number of efoldings that can be obtained,
one should note that the inflaton field range is boundedby the length of the throat [8]. Specifically, for throats
supported by√N units of D3-brane charge, the boundgives φUV ∼ T3 R, where R4 ∼ gs N α′2 [46]. Also, theend of inflation can be identified with the time when Eq.
(54) breaks down, which gives
φend ∼

2 4
MPlh0 T 3
β

1/6

.

(59)

Hence the maximum number of efoldings that can beobtained is
Nmax =Z φendφUVwhere h0 is the warping at the tip of the throat, T3 is thebrane tension, and the second term denotes the D3-D3
Coulombic interaction. When the bulk effects are absent, β = 1/3 as was derived in the original KKLMMT
paper [7]. The bulk effect introduces a negative contribution to the inflaton mass-squared, i.e. makes β smaller
than 1/3.
2
We restrict ourselves to the region φ2 ≪ MPl. Whenthe Coulombic attraction is negligible compared to themass term:
2 4
MPlh0 T 3≪ 1,
βφ6

we obtain rapid-roll inflation with

≃

3+

H

dφ
φ̇

√
9 − 24β
ln
12β

1/4

β 1/6 N 1/4 gs α′1/2
2/3

h0 L

!

.

(60)

Here we have used T3 ∼ 1/(gs α′2 ) and MPl ∼ L3 /(gs α′2 )where L is the typical length scale of the six-dimensionalbulk. The efolding number Eq. (60) depends sensitively
on the mass parameter β, hence, one sees that even aslight effect from the bulk can change the duration of
inflation drastically.
On the other hand, inflation should have started before
the present Hubble scale exited the horizon. Therefore
the required number of efoldings is at least!
!
1/4
3/4 ′3/2
29 Vinf
29 h0 gs α
N ≃ ln 10, (61)≃ ln 10MPl
L3where we have assumed instantaneous (p)reheating, i.e.
the universe was dominated by radiation right after inflation ended until matter-radiation equality. (This isrealized in the case where there are D-branes left after

8
0.015
β = 0.001
β = 0.3
0.01

α

0.005

0

−0.005−0.01−0.015
0.9

0.95

n

1

1.05

s

FIG. 3: Contour plot of log 10 (φH /φU V ), where φH is the
inflaton field value when the present Hubble scale exited thehorizon. φH is estimated by assuming that the inflaton potential Eq. (53) is valid throughout. The horizontal and verticalaxes denote, respectively, β and log 10 h0 . The other parameters are set to gs = 0.1, N = 1000, and L/α′1/2 = 10.
Necessary number of efoldings can be obtained in the shaded
region.

inflation, since closed strings generated from D3-D3 annihilation soon decay into lighter states such as gravitons, KK modes, and open string modes on the residual
D-branes, and then thermalize. See e.g. [45].)
In Figure 3, we present the ratio between the field
range required to obtain sufficient inflation and the allowed range in the β - log10 h0 plane. For example,
if we take gs = 0.1, N = 103 , L/α′1/2 = 10, and
1/4
h0 = 10−11 (which set the inflation energy scale Vinf andthe local string scale h0 /α′1/2 to be of order 1 TeV, andMPl α′1/2 ∼ 104 ), then the necessary number of efoldings
can be obtained when β . 0.2 is satisfied. As can be seenfrom Eqs. (60) and (61), smaller h0 and β is preferred
for enough inflation to occur.
The inflaton field value when the pivot scale exited thehorizon can similarly be estimated from Eqs. (60) and
(61),
φ0
∼
MPl



Vinf
4
βMPl

1/6

10

1/4
28 Vinf

MPl

!

3+

12β
√
9−24β

.

(62)

Note that for the pivot scale, the term inside the log in
Eq. (61) differs by an order of magnitude.
The spectral index and its running of the Hubblesquared during inflation can be estimated from Eq. (35)with ξ = 0 and v2 = 2β, or from Eqs. (A20) and (A21),18
φ2ns − 1 ≃ − 2 · 2β 2 2 ,
c
MPl

(63)

FIG. 4: Prediction of the rapid roll brane model Eq. (65) for
the values β = 0.001 and β = 0.3 compared to the constraints
obtained in Section IV.

dns
φ2
36
≃ 2 (3 − c) · 2β 2 2 ,d ln kc
MPl(64)

where the right hand sides are given at the moment ofhorizon crossing Eq. (62). One clearly sees a linear relation between Eqs. (63) and (64),


p
dns
(65)
≃ − 3 − 9 − 24β (ns − 1).
d ln kIn Figure 4, we compare this prediction to the constraints obtained in Section IV. We plot the prediction
Eq. (65) for the two values β = 0.3 and β = 0.001 to
illustrate the range of values of α and ns consistent withcurrent data. For fixed β, each prediction is a function of
φ0 which depends on the inflationary energy scale. The
rapid roll brane inflation model is in good agreement withcurrent data for a wide range of β.
Finally, we mention that even though we have treatedD-brane inflation as a minimally coupled model with
Hubble scale mass, interpreting the mass as arising fromthe inflaton’s conformal coupling to gravity does notmake much difference for the inflaton dynamics and theefolding number that can be obtained. However, it does
make distinct differences for the spectral index and its
running of the Hubble-squared, for reasons discussed at
the end of Section III. One can explicitly see this for the
D-brane inflation case by comparing the above resultswith that of [12].
VI.

CONCLUSIONS

We have obtained cosmological constraints on rapidroll models of inflation. Rapid roll solutions exist as attractors for tree-level hybrid-type potentials with a range

9of mass terms, with both minimal and nonminimal gravitational couplings. Such solutions are therefore relevant
to model building in string theory. We considered perturbations generated through modulated reheating and/or
the curvaton scenario instead of the observationally unacceptable inflaton-generated perturbations in these models. We obtained an analytic expression for the power
spectrum in this case," −B #
P (k0 )k
P (k) =
1+A
,
(66)
1+A
k0
observing a lack of a hierarchy amongst lower-orderand higher-order k-dependencies. We have performed a
Bayesian analysis on this power spectrum using the latest CMB and LSS data. The higher-order runnings are
fully determined by ns and α, and so these parameters
are tightly constrained relative to general power-law +running spectra. We find that rapid roll models are constrained to have ns < 1 and small running, |α| < 0.01.
These tight predictions make possible the falsifiability of
rapid roll inflation with upcoming CMB missions. The
higher-order runnings that occur in these spectra mightalso be further constrained via upcoming astrophysicalprobes, such as the 21-cm line of neutral hydrogen clouds.
We also construct a realistic model of rapid roll brane
inflation. We find that it is possible to obtain sufficient inflation even in the presence of the large massterms that arise from moduli stabalization in these models. The power spectrum generated by this model is ingood agreement with the constraints obtained in thiswork, across a range of mass scales. Therefore, while
the inflaton-generated perturbations of such models are
not observationally viable, the spectra generated throughother means, such as modulated reheating or curvatons,
breathe new life into these constructions. Additionally,
the tight constraints imposed on these spectra suggestthat future CMB data has the potential to further restrict them, or rule such models out entirely.
VII.

ACKNOWLEDGEMENTS

The calculations were performed by the EUP cluster
system installed at Graduate School of Frontier Sciences,
University of Tokyo. The work of S.M. was supported inpart by MEXT through a Grant-in-Aid for Young Scientists (B) No. 17740134, and by JSPS through a Grant-inAid for Creative Scientific Research No. 19GS0219. This
work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Appendix A: Discussions on Minimally Coupled
Rapid-Roll Inflation

We investigate the dynamics of a rapid-rolling inflaton which minimally couples to gravity. The calculations

carried out in this appendix is analogous to that of [12]where a conformally coupled inflaton was studied.

1.

Conditions for Rapid-Roll Inflation

We consider a minimally coupled inflaton with the Lagrangian

 2
√
1
MPl
L = −gR − g µν ∂µ φ∂ν φ − V (φ) .
(A1)
2
2
Choosing a flat FRW background, we obtain the Friedmann equation
2
3MPl
H2 =

1 2
φ̇ + V (φ),2(A2)

and the equation of motion of φ,φ̈ + 3H φ̇ + V ′ (φ) = 0.

(A3)

We claim that during inflation, the inflaton dynamics iswell described by the following approximations
23MPl
H 2 ≃ V,(A4)

cH φ̇ ≃ −V ′ ,(A5)where c is a dimensionless constant.
Let us define the following parameters,
ǫ≡

2
MPl
2



V′
V

2

,

η̄ ≡ η +

c2
− c,
3

(A6)

where
2
η ≡ MPl

V ′′
.
V

(A7)

Then the necessary conditions for the approximate relations (A4) and (A5) to hold can be derived, respectively,as3
ǫ ≪ 1,c2

3|3ǫ − η̄| ≪ 1.
c2

(A8)

Note that we have taken a time-derivative of both sidesof the approximate relation (A5) and then substituted it
into (A3) in order to derive the necessary condition for(A5). When c ∼ O(1), the necessary conditions (A8)reduce to simple formsǫ ≪ 1,|η̄| ≪ 1.

(A9)This shows that the parameters which should be minimized in order to realize inflation are ǫ and η̄ (instead of
ǫ and η). The constant c in (A5) is the largest constantthat minimizes η̄, as we will show in the next section.

102.

Case Study: non-negligible constant η

Stability of the Attractor

Let us define β to parametrize the validness of theattractor (A5) as′

cH φ̇ = −V (1 + β).

(A10)

We derive an evolution equation of β in this section.
Some useful relations are
2
3MPl
H2
V
(
)
r
12(1 + β)2
1
=
1+ 1+
ǫ = 1 + O(ǫ), (A11)2c2

η̄/c2 = 0 is solved by√
3 ± 9 − 12η≡ c± .
(A16)c=
2√
Since 3 − 2c± = ∓ 9 − 12η ≶ 0, one sees from (A14)that the larger solution c+ is chosen for the inflationary
attractor,Σ≡φ̇2= 2 (Σ − 1) ,VḢ
3(Σ − 1)
=−
.
2
H
Σ



1+β
1 − 2Σ +η (1 + β) + cc

3 − 2c < 0,(A14)

β damps as the universe expands, and eventually its value
settles down to that corresponding to the source terms,
β = O(ǫ, η̄).

(A15)

Hence the the condition (A14) is required for the relations(A4) and (A5) to be an inflationary attractor.

(A17)

By using the relations in the previous section and considering β to be damped to (A15), one can compute thetime differentiations of the Hubble parameter,Ḣ9= − 2 ǫ + O(η̄ǫ, ǫ2 ),
2
H
c

1
H

Ḣ
H2

!·

18
= − 2 (c − 3)ǫ + O(η̄ǫ, ǫ2 ).
c

(A18)

(A19)

Especially, the spectral index and its running of the
Hubble-squared are (k = aH),d ln H 2Ḣns − 1 ==2 2
d ln k
H

Ḣ
1+ 2
H

!−1

18
= − 2 ǫ + O(η̄ǫ, ǫ2 ), (A20)cNext we study the values of c chosen for inflation. As
we stressed in the previous section, c is a constant which
minimizes η̄/c2 .
dnsḢ=2 1+ 2d ln kH

Case Study: negligible η√
9 − 12η
.
2

Scale Dependence of H 2

3.

= β(3 − 2c) + β 2 (3 − c) + O(ǫ, η̄). (A13)Once β becomes small, the far right hand side is dominated by the linear term (i.e. β(3 − 2c)). One can seethat as long as

3+

It should also be noted that η ≤ 3/4 needs to be satisfied.

(A12)

Then by time-differentiating both sides of (A10), one obtains
β̇
3
=
H
Σ

c=

!−3
=

1
H

Ḣ
H2

!·

36
(3 − c)ǫ + O(η̄ǫ, ǫ2 ). (A21)c2This corresponds to the familiar slow-roll inflation.
Here η̄/c2 = 0 gives c = 3, which realizes the slowroll approximations. It is clear that the stability condition (A14) is satisfied.

These expressions can alternatively be derived directlyfrom Eq. (35).

[1] J. M. Cline, arXiv:hep-th/0612129.
[2] R. Kallosh, Lect. Notes Phys. 738,
[arXiv:hep-th/0702059].

[3] C. P. Burgess, PoS P2GC, 008 (2006) [Class. Quant.
Grav. 24, S795 (2007)] [arXiv:0708.2865 [hep-th]].
[4] L. McAllister and E. Silverstein, Gen. Rel. Grav. 40, 565

119 (2008)

11(2008) [arXiv:0710.2951 [hep-th]].
[5] D. Baumann and L. McAllister, arXiv:0901.0265 [hepth].
[6] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, Phys. Rev. D 49, 6410 (1994)[arXiv:astro-ph/9401011].
[7] S. Kachru, R. Kallosh, A. Linde, J. M. Maldacena,
L. P. McAllister and S. P. Trivedi, JCAP 0310, 013(2003) [arXiv:hep-th/0308055].
[8] D. Baumann and L. McAllister, Phys. Rev. D 75, 123508(2007) [arXiv:hep-th/0610285].
[9] A. Linde, JHEP 0111, 052 (2001) [arXiv:hepth/0110195].
[10] W. H. Kinney, Phys. Rev. D 72, 023515 (2005) [arXiv:grqc/0503017].
[11] K. Tzirakis and W. H. Kinney, Phys. Rev. D 75, 123510(2007) [arXiv:astro-ph/0701432].
[12] T. Kobayashi and S. Mukohyama, arXiv:0810.0810 [hepth].
[13] L. Kofman and S. Mukohyama, Phys. Rev. D 77, 043519(2008) [arXiv:0709.1952 [hep-th]].
[14] G. Dvali, A. Gruzinov and M. Zaldarriaga, Phys. Rev. D
69, 023505 (2004) [arXiv:astro-ph/0303591].
[15] L. Kofman, arXiv:astro-ph/0303614.
[16] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002)[arXiv:hep-ph/0110002].
[17] T. Moroi and T. Takahashi, Phys. Lett. B 522, 215(2001) [Erratum-ibid. B 539, 303 (2002)] [arXiv:hepph/0110096].
[18] D. H. Lyth, C. Ungarelli and D. Wands, Phys. Rev. D
67, 023503 (2003) [arXiv:astro-ph/0208055].
[19] G.
Hinshaw
et
al.
[WMAP
Collaboration],
arXiv:0803.0732 [astro-ph].
[20] B. Gold et al. [WMAP Collaboration], arXiv:0803.0715[astro-ph].
[21] M. R. Nolta et al. [WMAP Collaboration],
arXiv:0803.0593 [astro-ph].
[22] J.
Dunkley
et
al.
[WMAP
Collaboration],
arXiv:0803.0586 [astro-ph].
[23] E.
Komatsu
et
al.
[WMAP
Collaboration],
arXiv:0803.0547 [astro-ph].
[24] J. K. Adelman-McCarthy et al. [SDSS Collaboration],
Astrophys. J. Suppl. 162, 38 (2006) [arXiv:astroph/0507711].
[25] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002)[arXiv:astro-ph/0205436].

[26] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J.
538, 473 (2000) [arXiv:astro-ph/9911177].
[27] N. Christensen and R. Meyer, arXiv:astro-ph/0006401.
[28] N. Christensen, R. Meyer, L. Knox and B. Luey, Class.
Quant. Grav. 18, 2677 (2001) [arXiv:astro-ph/0103134].
[29] L. Verde et al. [WMAP Collaboration], Astrophys. J.
Suppl. 148, 195 (2003) [arXiv:astro-ph/0302218].
[30] D. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambride,
U.K. (2003)[31] E. Komatsu and U. Seljak, Mon. Not. Roy. Astron. Soc.
336, 1256 (2002) [arXiv:astro-ph/0205468].
[32] B. A. Berg, Fields Inst. Commun. 26, 1 (2000)[33] J. Gubernatis and N. Hatano, Computing in Science &Engineering 2, 95 (2002)[34] C. L. Reichardt et al., Astrophys. J. 694, 1200 (2009)[arXiv:0801.1491 [astro-ph]].
[35] M. Kunz, R. Trotta and D. Parkinson, Phys. Rev. D 74,
023503 (2006) [arXiv:astro-ph/0602378].
[36] A. R. Liddle, Mon. Not. Roy. Astron. Soc. Lett. 377, L74(2007) [arXiv:astro-ph/0701113].
[37] D. Baumann, A. Dymarsky, I. R. Klebanov, J. M. Maldacena, L. P. McAllister and A. Murugan, JHEP 0611,
031 (2006) [arXiv:hep-th/0607050].
[38] D. Baumann, A. Dymarsky, I. R. Klebanov, L. McAllister
and P. J. Steinhardt, Phys. Rev. Lett. 99, 141601 (2007)[arXiv:0705.3837 [hep-th]].
[39] A. Krause and E. Pajer, JCAP 0807, 023 (2008)[arXiv:0705.4682 [hep-th]].
[40] D. Baumann, A. Dymarsky, I. R. Klebanov and L. McAllister, JCAP 0801, 024 (2008) [arXiv:0706.0360 [hep-th]].
[41] D. Baumann, A. Dymarsky, S. Kachru, I. R. Klebanov and L. McAllister, JHEP 0903, 093 (2009)[arXiv:0808.2811 [hep-th]].
[42] A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara,
Phys. Rev. D 61, 066001 (2000) [arXiv:hep-th/9905226].
[43] A. Ceresole, G. Dall’Agata and R. D’Auria, JHEP 9911,
009 (1999) [arXiv:hep-th/9907216].
[44] O. Aharony, Y. E. Antebi and M. Berkooz, Phys. Rev. D
72, 106009 (2005) [arXiv:hep-th/0508080].
[45] X. Chen and S. H. Tye, JCAP 0606, 011 (2006)[arXiv:hep-th/0602136].
[46] S. S. Gubser, Phys. Rev. D 59, 025006 (1999) [arXiv:hepth/9807164].