Title Effects of local features of the inflaton potential on thespectrum and bispectrum of primordial perturbations

Author(s) Cadavid, Alexander Gallego; Romano, Antonio Enea;Gariazzo, Stefano

Citation The European Physical Journal C (2016), 76

Issue Date 2016-07

URL http://hdl.handle.net/2433/226920

Right

© The Author(s) 2016; This article is distributed under theterms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), whichpermits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made. Fundedby SCOAP3.

Type Journal Article

Textversion publisher

Kyoto University

Eur. Phys. J. C (2016) 76:385DOI 10.1140/epjc/s10052-016-4232-4

Regular Article - Theoretical Physics

Effects of local features of the inflaton potential on the spectrumand bispectrum of primordial perturbations

Alexander Gallego Cadavid1,3, Antonio Enea Romano1,2,3,4,5,a, Stefano Gariazzo4,5

1 Instituto de Fisica, Universidad de Antioquia, A.A.1226, Medellin, Colombia2 Department of Physics, University of Crete, 71003 Heraklion, Greece3 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan4 Department of Physics, University of Torino, Via P. Giuria 1, 10125 Torino, Italy5 INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

Received: 6 September 2015 / Accepted: 24 June 2016 / Published online: 8 July 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We study the effects of a class of features of thepotential of slow-roll inflationary models corresponding toa step symmetrically dumped by an even power negativeexponential factor, which we call local features. Local-typefeatures differ from other branch-type features consideredpreviously, because the potential is only affected in a limitedrange of the scalar field value, and they are symmetric withrespect to the location of the feature. This type of featureonly affects the spectrum and bispectrum in a narrow rangeof scales which leave the horizon during the time interval cor-responding to the modification of the potential. On the con-trary branch-type features have effects on all the perturbationmodes leaving the horizon when the field value is within theinterval defining the branch, introducing for example differ-ences in the power spectrum between large and small scalewhich are absent in the case of local-type features. The spec-trum and bispectrum of primordial curvature perturbationsare affected by oscillations around the scale k0 exiting thehorizon at the time τ0 corresponding to the feature. We alsocompute the effects of the features on the CMB tempera-ture and polarization spectra, showing the effects of differentchoices of parameters.

1 Introduction

Theoretical cosmology has entered in the last decades in anew era in which different models can be compared directly tohigh-precision observations [1–4]. One fundamental sourceof information as regards the early Universe is the cosmicmicrowave background (CMB) radiation, which accordingto the standard cosmological model consists of the photons

a e-mail: aer@physics.uoc.gr

that decoupled from the primordial plasma when the neutralhydrogen atoms started to form.

According to the inflation theory [5,6] the CMB tem-perature anisotropies arose from primordial curvature per-turbation, whose spectrum is approximately scale invari-ant. An approximately scale invariant spectrum of curva-ture perturbation, with a small tilt, provides a good fit ofCMB data [4], but recent analyses of the WMAP and Planckdata have shown evidence of a feature around the scalek = 0.002 Mpc−1 in the power spectrum of primordial scalarfluctuations [7–23] that corresponds to a dip in the CMB tem-perature spectrum at l � 20. This kind of feature of the cur-vature perturbations spectrum provides an important obser-vational motivation to find theoretical models able to explainit. In particular in this paper we will consider the effects offeatures of the inflaton potential in single field inflationarymodel as possible explanation of the features of the powerspectrum.

The effects of features of the inflaton potential were firststudied by Starobinsky [24], and CMB data have shown someglitches of the power spectrum [25,26] compatible with thesefeatures [27–31].

The Starobinsky model and its generalizations [32–36]belong to a class of branch features (BFs) which involve astep function or a smoothed version of the latter [37], andconsequently introduce a distinction between a left and rightbranch of the potential. In this paper instead we will considerthe effects of local features (LF) [38] which only modify thepotential locally in field space, while leaving it unaffectedsufficiently far from the feature. The important consequenceis that also the effects of LFs on the spectrum and bispectrumare local, while BFs modify the spectrum in a wider range ofscales. Features of the inflaton potential could be producedby different sources [39,40], for example particle production

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385 Page 2 of 14 Eur. Phys. J. C (2016) 76 :385

1.0 1.1 1.2 1.3 1.4 1.5

1.0002

1.0004

1.0006

1.0008

1.0010

1 2 3 4 5

3. 10 8

2. 10 8

1. 10 8

1. 10 8

2. 10 8

3. 10 8

1 2 3 4

0.0150

0.0155

0.0160

0.0165

0.0170

0.0175

0.0180

1 2 3 4

0.5

0.0

0.5

VF

V0

Fig. 1 From left to right and top to bottom the numerically computed VF/V0,�H, ε, and η are plotted for λ = 10−11, σ = 0.05, and n = 1 (blue)and n = 2 (red). The dashed black lines correspond to the featureless behavior

[41] or phase transitions [42], but here we will only studytheir effects adopting a phenomenological approach, as doneoriginally by Starobinsky in his seminal work (Fig. 1).

2 Single field slow-roll inflation

We consider inflationary models with a single scalar field anda standard kinetic term with action [43,44]

S =∫

d4x√−g

[1

2M2

Pl R − 1

2gμν∂μφ∂νφ − V (φ)

], (1)

where MPl = (8πG)−1/2 is the reduced Planck mass. Thevariation of the action with respect to the metric and thescalar field gives the Friedmann equation and the equation ofmotion of the inflaton,

H2 ≡(a

a

)2

= 1

3M2Pl

(1

2φ2 + V (φ)

), (2)

φ + 3H φ + ∂φV = 0, (3)

where H is the Hubble parameter and we denote with dotsand ∂φ the derivatives with respect to time and scalar field,

respectively. The definitions we use for the slow-roll param-eters are

ε ≡ − H

H2 , η ≡ ε

εH. (4)

3 Local features versus branch features

We consider a single scalar field inflationary model withpotential [38]

V (φ) = V0(φ) + VF (φ), (5)

VF (φ) = λe−(φ−φ0

σ)2n ; n > 0, (6)

where V0 is the featureless potential, and we call this type ofmodification of the slow-roll potential local features (LFs).Many of the features previously studied belong to the cate-gory of branch features, which differ from LFs because theirdefinition involves step functions or their smoothed version,which effectively divide the potential in separate branches(Fig. 2).

Some examples of BFs are given by the Starobinsky model[27] or its generalizations [33,35]

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1.0 1.1 1.2 1.3 1.4 1.5

1.0002

1.0004

1.0006

1.0008

1.0010

1 2 3 4 5

2. 10 8

2. 10 8

4. 10 8

1 2 3 4

0.0150

0.0155

0.0160

0.0165

0.0170

0.0175

1 2 3 4

0.2

0.0

0.2

0.4

0.6

VF

V0

Fig. 2 From left to right and top to bottom the numerically computed VF/V0,�H, ε, and η are plotted for λ = 10−11, σ = 0.05 (blue), σ = 0.1(red), and n = 1. The dashed black lines correspond to the featureless behavior

VF (φ) = λ(φ − φ0)nθ(φ − φ0). (7)

The Starobinsky model corresponds to the case n = 1, forwhich an analytical solution was first found in [24,27], andmore recently these studies were followed by [32,36]. Thetype of feature we study in this paper differs from [33] in thefact that here we study only local features, which only affectthe potential in a limited range of the field values, while in[33] the features are not local and modify the potential foran entire branch, due to the absence of any dumping factor(Fig. 3).

Some other smoothed versions involving hyperbolictrigonometric function instead of the Heaviside function havealso been studied [37]. They can schematically be expressedas

VF (φ) = λ tanh

(φ − φ0

�

). (8)

For these models the potential is not only modified aroundthe feature, but for any value in the branch defined by thefeature. The direct consequence is that the effects of BFs onperturbation modes are not only visible around the scale k0

leaving the horizon around the time of the feature, defined

as φ(τ0) = φ0, but for any scale leaving the horizon whenφ has a value within the feature branch. This is evident forexample from the fact that the spectrum of a BF shows astep around k0 [45], i.e. the affected scales are all the oneslarger (or smaller, depending on the feature) than k0. Onthe contrary LFs only affect the perturbation modes whichleave the horizon around τ0, and consequently the spectrumdoes not show a step, but a local dumped oscillation and itapproaches the featureless spectrum for sufficiently smallerand larger scales. This is very important because it couldallow one to model local features of the observed spectrumwithout affecting other scales. The different effects of LFsand BFs on the power spectrum are shown in Fig. 13, where itcan be seen that they both produce oscillations qualitativelysimilar around k0, but in the case of BFs there is also a stepbetween large and small scales, which is absent for LFs. ForBFs if the branches of the feature definition were invertedthe role of small and large scales would also be accordinglyinverted (Figs. 4 and 5).

In this paper we will consider the case of power-law infla-tion (PLI) to model the featureless behavior,

V0(φ) = Ae−

√2q

φMPl . (9)

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385 Page 4 of 14 Eur. Phys. J. C (2016) 76 :385

1.0 1.1 1.2 1.3 1.4 1.5

0.9995

1.0000

1.0005

1.0010

1 2 3 4 5

2. 10 8

1. 10 8

1. 10 8

2. 10 8

3. 10 8

0.5 1.0 1.5 2.0 2.5 3.0

0.015

0.016

0.017

0.018

0.5 1.0 1.5 2.0 2.5 3.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

VF

V0

Fig. 3 From left to right and top to bottom the numerically computed VF/V0,�H, ε, and η are plotted for λ = 10−11 (blue) and −λ = 10−11

(red), σ = 0.05, and n = 1. The dashed black lines correspond to the featureless behavior

While PLI is not in good agreement with CMB data due tothe high value of the predicted tensor-to-scalar ratio r , it canbe used as a good toy model to show qualitatively the generaltype of effect produced by LFs. Future work may be devotedto the test of different potentials V0(φ), for direct comparisonwith the data (Figs. 6, 7, 8, 9, and 10).

4 Local versus branch features effects

One important difference between the effects of BFs and LFsas mentioned earlier is that BFs introduce a step betweenlarge and small scales in the spectrum, whose size dependson the featureless model and on the choice of the parameters,in particular λ. The effects of LFs and BFs producing oscil-lations of comparable size are shown for different potentialsin Fig. 13. As it can be seen, depending on the featurelesspotential V0(φ), there can be steps of different size in thespectrum. If λ is very small the effects of BFs and LFs couldbe phenomenologically indistinguishable, but only a detailedand systematic data analysis can allow one to establish whichone of the two categories is observationally preferred, or ifthey are both compatible with observations (Figs. 11, 12).

Another important difference is that an LF produces twoslow-roll violation phases, associated to the increasing anddecreasing part of the potential feature, while BFs have onlyone. In this sense a LF can be thought as the superposition oftwo BFs: for example two smoothed steps with different tran-sition points φ0 and inverted branches are equivalent to onelocal smoothed step. The distance between the two transitionpoints is related to the LF σ parameter, but each BF has alsohis own σ parameter controlling the smoothness of each BFtransition, so the equivalence is not complete, and there canstill be differences between the effects of a LF and appropri-ate combinations of BFs, as shown in Fig. 14. Schematically,based on general symmetry arguments, we can write

VLF(φ) ≈ V 1BF(φ) + V 2

BF(φ). (10)

For example we can approximate a LF as

VLF(φ) = λe−(φ−φ0

σ)2n ≈ V 1

BF(φ) + V 2BF(φ) (11)

V 1BF(φ) = λ1 tanh

(φ − φ1

σ1

),

V 2BF(φ) = λ1 tanh

(φ2 − φ

σ1

)(12)

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1.0 1.1 1.2 1.3 1.4 1.5

1.002

1.004

1.006

1.008

1.010

VF

V0

1 2 3 4 5

2. 10 8

1. 10 8

1. 10 8

2. 10 8

3. 10 8

0.5 1.0 1.5 2.0 2.5 3.0

0.0150

0.0155

0.0160

0.0165

0.0170

0.0175

0.5 1.0 1.5 2.0 2.5 3.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

Fig. 4 From left to right and top to bottom the numerically computed VF/V0,�H, ε, and η are plotted for λ = 10−11 (blue) and λ = 10−12 (red),σ = 0.05, and n = 1. The dashed black lines correspond to the featureless behavior

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

1.8 10 9

2. 10 9

2.2 10 9

2.4 10 9

2.6 10 9

2.8 10 9

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

0.24

0.26

0.28

0.3

0.32

0.34

0.36

r

Fig. 5 The numerically computed Pζ and r are plotted for λ = 10−11, σ = 0.05, and n = 1 (blue) and n = 2 (red). The dashed black linescorrespond to the featureless behavior

φ0 = φ2 + φ1

2, λ = 2λ1, σ = φ2 − φ1

2, σ1 = σ

2(13)

Only a systematic analysis of observational data can deter-mine which is the phenomenologically preferred type of fea-ture, but in general a single BF is expected to produce effectsdifferent from a single LF, since this is approximately equiv-alent to the combination of two single BFs (Fig. 13).

5 Curvature perturbations

From now on we adopt a system of units in which c = h =MPl = 1. The study of curvature perturbations is attainedby expanding perturbatively the action with respect to thebackground FRLW solution [46,47]. In the comoving gaugethe second and third order actions for scalar perturbations are

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0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

2. 10 9

2.2 10 9

2.4 10 9

2.6 10 9

2.8 10 9

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

0.24

0.26

0.28

0.3

0.32

0.34

r

Fig. 6 From left to right the numerically computed Pζ and r are plotted for λ = 10−11, σ = 0.05 (blue), σ = 0.1 (red), and n = 1. The dashedblack lines correspond to the featureless behavior

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

2.0 10 9

3.0 10 9

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

0.22

0.24

0.26

0.28

0.3

0.32

0.34

r

Fig. 7 The numerically computed Pζ and r are plotted for λ = 10−11 (blue) and −λ = 10−11 (red), σ = 0.05, and n = 1. The dashed black linescorrespond to the featureless behavior

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

2. 10 9

2.2 10 9

2.4 10 9

2.6 10 9

2.8 10 9

0.1 0.5 1.0 5.0 10.0 50.0 100.0

k

k0

0.24

0.26

0.28

0.3

0.32

0.34

r

Fig. 8 The numerically computed Pζ and r are plotted for λ = 10−11 (blue) and λ = 10−12 (red), σ = 0.05, and n = 1. The dashed black linescorrespond to the featureless behavior

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2 1 0 1 2 3 4log

k

k0

4

2

0

2

4

FNL

Fig. 9 The numerically computed equilateral shape bispectrum is plot-ted for λ = 10−11, σ = 0.05, and n = 1 (blue) and n = 2 (red)

2 1 0 1 2 3 4log

k

k01.0

0.5

0.0

0.5

1.0

FNL

Fig. 10 The numerically computed equilateral shape bispectrum isplotted for λ = 10−11, σ = 0.05 (blue) σ = 0.1 (red), and n = 1

2 1 0 1 2 3 4log

k

k 0

0.5

0.0

0.5

1.0

1.5

2.0

FNL

Fig. 11 The numerically computed equilateral shape bispectrum isplotted for λ = 10−11 (blue) and −λ = 10−11 (red), σ = 0.05, andn = 1

S2 =∫

dtd3x[a3εζ 2 − aε(∂ζ )2

], (14)

S3 =∫

dtd3x

[a3ε2ζ ζ 2 + aε2ζ(∂ζ )2 − 2aεζ (∂ζ )(∂χ) + a3ε

2ηζ 2ζ

+ ε

2a(∂ζ )(∂χ)∂2χ + ε

4a(∂2ζ )(∂χ)2 + f (ζ )

δL

δζ

∣∣∣∣1

], (15)

2 1 0 1 2 3 4log

k

k 0

1.0

0.5

0.0

0.5

1.0

FNL

Fig. 12 The numerically computed equilateral shape bispectrum isplotted for λ = 10−11 (blue) and λ = 10−12 (red), σ = 0.05, andn = 1

where

δL

δζ

∣∣∣∣1

= 2a

(d∂2χ

dt+ H∂2χ − ε∂2ζ

), (16)

f (ζ ) = η

4ζ + terms with derivatives on ζ, (17)

and we denote with δL/δζ |1 the variation of the quadraticaction with respect to ζ [46]. The Lagrange equations for thesecond order action give

∂

∂t

(a3ε

∂ζ

∂t

)− aεδi j

∂2ζ

∂xi∂x j= 0. (18)

Taking the Fourier transform and using the conformal timedτ ≡ dt/a we obtain

ζ ′′k + 2

z′

zζ ′k + k2ζk = 0, (19)

where z ≡ a√

2ε, k is the comoving wave number, andprimes denote derivatives with respect to the conformal time.

A similar approach can be adopted for the perturbationsof the tensor modes hk , which satisfy the equation

h′′k + 2

a′

ah′k + k2hk = 0. (20)

The power spectrum of scalar perturbations is the Fouriertransform of the two-point correlation function of ζ . For thepower spectrum of scalar perturbations we adopt the defini-tion

Pζ (k) ≡ 2k3

(2π)2 |ζk |2, (21)

and for the power spectrum of tensor perturbations

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Fig. 13 The numerically computed power spectra are plotted as a func-tion of k/k0 for different potentials. The dashed lines are the spectrafor the featureless potential V0(φ), the blue lines are for the spectrumcorresponding to a LF, and the red lines to a BF. On the left we haveV0(φ) = Vvac + 1

2m2φ2, where Vvac = 3.3 × 10−13M3

pl and m =6 × 10−9Mpl . For a LF the potential is V (φ) = V0(φ) + λe−(

φ−φ0σ

)2,

with λ = 6.5 × 10−21, σ = 10−4, while the BF is of the type studiedbefore in [33] corresponding to V (φ) = V0(φ) + λθ(φ0 − φ), with

λ = −10−16.On the right we have V0(φ) = Ae−

√2q

φMPl , the power-

law potential. For a LF the potential is V (φ) = V0(φ) + λe−(φ−φ0

σ)2

,

with λ = −10−11 and σ = 0.05, while the BF corresponds toV (φ) = V0(φ) + λ

2 [1 + tanh(φ0−φ

σ)], with λ = −3 × 10−11 and

σ = 0.05. The parameters have been chosen so that the different typesof features produce oscillations of similar size, so that the effects can becompared consistently. As it can be seen, depending on the featurelesspotential V0, there are cases in which the step between the large andsmall scale spectrum produced by BFs can be important. The oscillationpatterns can also be different, since a single BF has only one phase ofslow-roll violation, while in a single LF there are two slow-roll viola-tion phases, corresponding to the increasing and decreasing part of thefeature

0.5 1.0 1.5 0

0.5

1.0

1.5

2.0

V

Fig. 14 The local feature potential VLF in Eq. (11) and its approxima-tion as the sum of two BF potentials V 1

BF(φ) + V 2BF(φ) defined in Eq.

(12) are plotted, respectively, in blue and in light brown around φ0 forφ1 = 1, φ2 = 1.2, λ1 = 1, and n = 1. This is an example of the factthat appropriate combinations of two BFs can produce modifications ofthe potential similar to a local feature

Ph(k) ≡ 2k3

π2 |hk |2. (22)

The tensor-to-scalar ratio is defined as the ratio between thespectrum of tensor and scalar perturbations

r ≡ PhPζ

. (23)

6 Calculation of the bispectrum of curvatureperturbation

The bispectrum Bζ is defined as the Fourier transform of thethree-point correlation function as

〈ζ(�k1, t)ζ(�k2, t)ζ(�k3, t)〉 = (2π)3Bζ (k1, k2, k3)δ(3)(�k1 + �k2, �k3).

(24)

After a field redefinition, we can re-write the third orderaction as

S3 =∫

dtd3x

[−a3εηζ ζ 2 − 1

2aεηζ∂2ζ

], (25)

from which the interaction Hamiltonian can be written interms of the conformal time as

Hint (τ ) =∫

d3x εηa

[ζ ζ ′2 + 1

2ζ 2∂2ζ

]. (26)

Finally, the three-point correlation function is given by [46,48]

〈�|ζ(τe, �k1)ζ(τe, �k2)ζ(τe, �k3)|�〉= −i

∫ τe

−∞

⟨0

∣∣∣[ζ(τe, �k1)ζ(τe, �k2)ζ(τe, �k3), Hint

]∣∣∣ 0⟩.

(27)

After substitution we get

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Eur. Phys. J. C (2016) 76 :385 Page 9 of 14 385

Bζ (k1, k2, k3) = 2�[ζ(τe, k1)ζ(τe, k2)ζ(τe, k3)

×∫ τe

τi

dτηεa2(

2ζ ∗(τ, k1)ζ′∗(τ, k2)ζ

′∗(τ, k3)

−k21ζ ∗(τ, k1)ζ

∗(τ, k2)ζ∗(τ, k3)

)

+ two permutations of k1, k2, and k3

], (28)

where � is the imaginary part. The integral is computed fromτi to τe, where τi is some time before τ0 when the featureeffects on the background and perturbations evolution start tobe important, and τe is some time after the horizon crossing,when the modes have frozen [49–53] (Fig. 14).

A commonly used quantity introduced to study non-Gaussianity is the parameter fNL

6

5fNL(k1, k2, k3) ≡ Bζ

Pζ (k1)Pζ (k2) + Pζ (k1)Pζ (k3) + Pζ (k2)Pζ (k3),

(29)

where

Pζ ≡ 2π2

k3 Pζ . (30)

After replacing Pζ in Eq. (29) we can obtain fNL in terms ofour definition of the spectrum Pζ (k)

fNL(k1, k2, k3)

= 10

3

(k1k2k3)3

(2π)4

Bζ

Pζ (k1)Pζ (k2)k33 + Pζ (k1)Pζ (k3)k3

2 + Pζ (k2)Pζ (k3)k31

.

(31)

In this paper we will use a different quantity to study non-Gaussianity

FNL(k1, k2, k3; k∗) ≡ 10

3(2π)4

(k1k2k3)3

k31 + k3

2 + k33

Bζ (k1, k2, k3)

P2ζ (k∗)

,

(32)

where k∗ is the pivot scale at which the power spectrum isnormalized, i.e. Pζ (k∗) ≈ 2.2 ×10−9. When the spectrum isapproximately scale invariant our definition of FNL reducesto fNL in the equilateral limit, but in general fNL and FNL

are not the same. For example in the squeezed limit they aredifferent, but FNL still provides useful information as regardsthe non-Gaussian behavior of Bζ .

7 Effects of the parameter n

7.1 Background

The parameter n is related to the dumping of the feature, andlarger values are associated to a steeper change of the poten-tial, as shown in Fig. 1. The slow-roll parameters show an

oscillation around the feature time τ0 with a larger amplitudefor larger n, since a steeper potential change is also associ-ated to larger derivatives of the Hubble parameter as shownin Fig. 1. To better understand the effects on the slow-rollparameter we define the quantity

�H = HF − H0, (33)

where HF is the Hubble parameter for the model with a fea-ture, and H0 is for the featureless model. From the definitionin Eq. (4) we can easily see that at leading order in �H wehave

εF = ε0 + �ε, (34)

�ε ≈ −∂t�H

H20

, (35)

where we have defined

εF = − HF

H2F

, (36)

ε0 = − H0

H20

. (37)

The temporary violation of slow-roll conditions comes fromthe time derivative of �H , so even small changes in theexpansion history of the Universe can produce important non-Gaussianities if they happen sufficiently fast. In the limit ofvery large n the feature of the potential tends to a local bumpcharacterized by a very steep transition.

7.2 Perturbations

As shown in Fig. 5 the tensor-to-scalar ratio r and the spec-trum of primordial curvature perturbations show oscillationsaround the scale k0 = −1/τ0 with an amplitude whichincreases for larger n. We can understand this from the behav-ior of �H , which has a larger time derivative for larger n,because the transition for the potential is also steeper. Asseen in Fig. 9 the equilateral limit of the bispectrum alsoshows oscillations around k0, which are larger for larger n,for the same reason given above. It is important to observethat both the spectrum and the bispectrum are only affectedin a limited range of scales, since this is a LF. For a BF,instead, the change affects all the scales before of after thefeature [33,45], because the potential is modified in the entirebranch.

8 Effects of the parameter σ

8.1 Background

The parameter σ determines the size of the range of fieldvalues where the potential is affected by the feature, as shown

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in Fig. 2. The slow-roll parameters are smaller for larger σ

since a larger width of the feature tend to reduce the timederivative of the Hubble parameter. In this case, in fact, themodification of the potential is also associated to smallerderivatives with respect to the field, since the shape of thepotential is less steep.

8.2 Perturbations

As shown in Fig. 6 the spectrum of primordial curvatureperturbations and the tensor-to-scalar ratio r have oscillationsaround k0, whose amplitude is larger for smaller σ , becausein this case the potential changes faster and consequently theslow-roll parameters are larger. In Fig. 10 we can see that theequilateral limit of the bispectrum also presents oscillationsaround k0 with larger amplitude for smaller σ . Both for thespectrum and bispectrum the effects are confined in a limitedrange of the scales, differently from what we would havewith BFs.

9 Effects of the parameter λ

9.1 Background

The parameter λ controls the magnitude of the potential mod-ification, as shown in Figs. 3 and 4. For larger absolute valuesof λ, the slow-roll parameters are larger in absolute value,since a larger feature of the potential induces a larger timederivative of the Hubble parameter. The sign of λ produceopposite and symmetric effects, since it implies an oppo-site sign for the derivative of the potential with respect to thefield, and consequently of the Hubble parameter with respectto time.

9.2 Perturbations

As shown in Figs. 7, 8, 11, 12, larger absolute values of λ

produce oscillations with a larger amplitude for the tensor-to-scalar ratio r , the spectrum and bispectrum around k0.Features with the same absolute value and opposite sign ofλ correspond to oscillations that are symmetric with respectto the featureless spectrum and bispectrum.

10 Effects on the CMB temperature and polarizationspectrum

In this section we present the effects of the feature on theCMB temperature and polarization spectrum. To study howthe feature impacts on the CMB spectra we modified theBoltzmann equations solver Code for Anisotropies in theMicrowave Background (CAMB) [54] to use the modified

primordial power spectrum instead of the usual power-lawexpression Ps(k) = As (k/0.05 Mpc−1)ns−1, where As is thenormalization and ns is the tilt of the power-law spectrum.

The CMB spectrum is the convolution of the power spec-trum of initial perturbations with the transfer function, whichis calculated by CAMB assuming the standard cosmologicalmodel. We fix all the cosmological parameters to the Planck2015 best-fit values [4].

In Figs. 15, 16, 17, and 18 we show the CMB spectraobtained with different combinations of the parameters λ, σ ,and n: in Table 1 we list the values of the feature parametersfor each combination. We show the spectra in terms of thequantity D� = �(� + 1)C�/(2π). For each of the tempera-ture (TT, Fig. 15), E-mode polarization (EE, Fig. 17), andB-mode polarization (BB, Fig. 18) auto-correlation powerspectra we plot the D� spectra (top) and the relative differencewith respect to the featureless spectrum (bottom) for a largerange of multipoles. In Fig. 16 we show also the TE cross-correlation power spectra. The spectra are compared withthe most recent experimental data: we plot the Planck data[55] for the TT, TE, and EE spectra and the points obtainedby the SPTPol experiment [56] and the Bicep–Keck (BK)collaboration [57] data for the BB spectrum. The SPTPoldata provide the first detection of the B modes generated bythe lensing of E-mode perturbations. We recall that the BKdata plotted here contain a significant contamination from Bmodes emitted by dust [58].

From the various plots it is possible to see how the fea-ture can change the predicted CMB spectra. In particular,the most significant variations are in the TT and in the EEspectra, where relative differences of the order of 10–15 %are visible, while the presence of the feature has a very smallimpact on the B-mode spectrum. Choosing the values listedin Table 1 for the parameters describing the feature, andk0 = 5×10−4 Mpc−1, As = 2.2×10−9, and ns = 0.967 wecan see from Fig. 15 that the effects of some feature can par-tially reproduce the dip at � � 20 in the TT spectrum. Furtherstudies involving data fitting can determine more accuratelythe values of the parameters which provide the best explana-tion of the observed deviation of the power spectrum fromthe power-law form, but they go beyond the scope of thispaper, and they will be reported separately. From Figs. 15,16, 17, and 18 it is also clear that the presence of the LFaffects only a part of the CMB spectra, while far from themultipoles corresponding to the feature scale the spectra areequal to those obtained in the featureless case.

11 Conclusions

We have studied the effects of local features of the inflationpotential on the spectrum and bispectrum of single field infla-tionary models with a canonical kinetic term. These features

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Fig. 15 We plot the DTTl = �(�+ 1)CTT

� /(2π) spectrum in unitsof μK 2 with respect to the multipole l, and the relative dif-ference with respect to the featureless behavior. The solid black

lines correspond to the featureless behavior. The values of λ, σ ,and n we used for the different curves are listed in Table 1

Fig. 16 We plot the DT El = �(� + 1)CT E

� /(2π) spectrum in units of μK 2 with respect to the multipole l. The solid black lines correspond to thefeatureless behavior. The values of λ, σ , and n we used for the different curves are listed in Table 1

only modify the potential in a limited range of the scalarfield values, and consequently only affect the spectrum andbispectrum in a narrow range of scales, which leave the hori-zon during the time interval corresponding to the modifi-cation of the potential. This is different from branch-type

features which effectively divide the potential into separatebranches, because they involve a step-like function in theirdefinition. Some examples of branch-type features are theStarobinsky model [27] and its generalizations [33,35]. InBF models the spectrum can, for example, exhibit a step,

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385 Page 12 of 14 Eur. Phys. J. C (2016) 76 :385

Fig. 17 We plot the DEEl = �(� + 1)CEE

� /(2π) spectrum inunits of μK 2 with respect to the multipole l, and the relative dif-ference with respect to the featureless behavior. The solid black

lines correspond to the featureless behavior. The values of λ, σ ,and n we used for the different curves are listed in Table 1

Fig. 18 We plot the DBBl = �(� + 1)CBB

� /(2π) spectrum inunits of μK 2 with respect to the multipole l, and the relative dif-ference with respect to the featureless behavior. The solid black

lines correspond to the featureless behavior. The values of λ, σ ,and n we used for the different curves are listed in Table 1

reminiscent of the branch-type potential modifications. Forlocal features there is no step, and the spectrum returns to thefeatureless form for scales sufficiently larger or smaller thank0.

The tensor-to-scalar ratio r , the spectrum, and the bispec-trum of primordial curvature perturbations are affected by thefeature, showing modulated oscillations which are dumpedfor scales larger or smaller than k0. The amplitude of the oscil-

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Table 1 The values of λ, σ , and n we used for the different spectra inFigs. 15, 16, 17, 18

Label λ σ n

Feature A 10−11 0.05 1

Feature B 10−11 0.05 2

Feature C −10−11 0.05 1

Feature D 10−11 0.1 1

Feature E 10−12 0.05 1

lations depends on the parameters defining the local feature,and the effects are larger when the potential modification issteeper, since in this case there is a stronger violation of theslow-roll conditions.

We have also computed the effects of the features on theCMB temperature and polarization spectra, showing how anappropriate choice of parameters can produce effects in qual-itative agreement with the observational CMB data. Due tothis local-type effect these features could be used to modelphenomenologically local glitches of the spectrum, withoutaffecting other scales, and it will be interesting in the futureto perform a detailed observational data fitting analysis usingthe new CMB data of the Planck mission.

Acknowledgments This work was supported by the European Union(European Social Fund, ESF) and Greek national funds under the ARIS-TEIA II Action. and the Dedicacion exclusica and Sostenibilidad pro-grams at UDEA, the UDEA CODI projects IN10219CE and 2015-4044. The work of S. G. was supported by the Theoretical Astroparti-cle Physics research Grant No. 2012CPPYP7 under the Program PRIN2012 funded by the Ministero dell’Istruzione, Università e della Ricerca(MIUR). We thank the anonymous Referee for the useful comments andsuggestions which have helped to improve the manuscript.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

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