Helical phase inflation and its observational constraints

Mudassar Sabira, Waqas Ahmedb, Yungui Gonga, Tianjun Lic,d, Jiong Lina

aSchool of Physics, Huazhong University of Science and Technology, Wuhan, Hubei430074, China

bSchool of Physics, Nankai University, No.94 Weijin Road, Nankai District, Tianjin,China

cSchool of Physical Sciences, University of Chinese Academy of Sciences, No.19AYuquan Road, Beijing 100049, China

dCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, Beijing 100190, China

Abstract

We consider a class of helical phase inflation models from the N = 1supergravity where the phase component of a complex field acts as an infla-ton. This class of models avoids both the η problem and the trans-Planckianfield excursion problem due to the phase monodromy of the superpotentialand the phase rotations evolving to trans-Planckian domain rather than thephysical field. We study the inflationary predictions of this class of modelsin the context of large extra dimensional brane cosmology, which can easilyaccommodate the 2018 Planck constraints on ns − r plane. The predictedrange of primordial gravitational waves will be within the reach of the futureLiteBIRD satellite experiment.

1. Introduction

Our observable Universe has a finite age of 14 billion years, while it has al-ready expanded to about 46 billion light years. And it appears spectacularlyhomogenous and isotropic with cosmic microwave background temperatureanisotropies only at the order of 10−5 or less. The current far off regions ofthe Universe were in causal contact, and the Universe may have gone through

Email addresses: msabir@hust.edu.cn (Mudassar Sabir), waqasmit@itp.ac.cn(Waqas Ahmed), yggong@mail.hust.edu.cn (Yungui Gong), tli@itp.ac.cn (TianjunLi), jionglin@hust.edu.cn (Jiong Lin)

Preprint submitted to Elsevier August 15, 2019

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an era of exponential expansion which provides a solution to these puzzles instandard cosmology [1, 2, 3, 4, 5, 6]. The inflationary models have specificpredictions that can be tested experimentally. A plethora of inflation modelsare available in the literatures, see Ref. [7] for a detailed list. For slow-rollinflation, field excursions are related to the primordial gravitational wavethat has not been detected so far. However, the future satellite experimentswill have the required sensitivity to measure the tensor-to-scalar ratio up to∼0.001.

Helical phase inflation from N = 1 supergravity was proposed where thephase of a complex field acts as an inflaton while the radial component isstrongly stabilized [8, 9, 10]. The phase field rolls down along the deformedhelicoid shaped potential. It is a generically difficult problem to generate asufficiently flat scalar potential in supergravity due to the exponential kahlerpotential factor in the scalar potential. To circumvent this eta problem usu-ally additional symmetries are imposed. In helical phase inflation, we havethe U(1) phase monodromy of superpotential to circumvent this eta problemautomatically. Moreover, the helical model can easily interpolate betweenthe natural inflation [11] and the Starobinski-like inflation [2] in a singlepotential.

The four-dimensional single field chaotic inflationary models have beenfound to have clear tension with the condition ∆θ < 1. However, in brane-world scenario this tension is significantly reduced due to the modified Fried-mann equation with a ρ2 correction [12, 13, 14, 15, 16, 17, 18, 19, 20]. Thismakes brane inflation in large extra dimensional scenario an interesting pos-sibility to explore further.

In this paper, we briefly review the helical phase inflation from the N = 1supergravity. By varying the parameters, we can interpolate from naturalinflation to Starobinsky-like inflation. We study the observational constraintson the model parameters, and present the viable parameter space whichis consistent with the Planck 2018 data and BICEP2 results. Also, thenatural inflation is marginally consistent with the observations at the 2σlevel. Next, we consider the helical phase inflation in the setup of the largeextra dimensional scenario where our four-dimensional world is embeddedin a five-dimensional space-time [21]. we discuss the modified Friedmannequation, and then study the observational constraints. Similarly, we presentthe viable parameter space which is consistent with the Planck 2018 data andBICEP2 results as well. In particular, the natural inflation is excluded bythe observations. While the Starobinksi-like inflation provides the favorable

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central value for spectral index ns, and the range of values of tensor-to-scalarratio r spans full experimental and theoretical estimate.

2. Helical phase inflation

In helical phase inflation, the inflaton θ, i.e., the phase component of acomplex field, is a pseudo Nambu-Goldstone boson (PNGB) [8, 9, 10]. Thepotential of a complex field admits helicoid structure and the inflaton evolvesalong a local valley, tracing a helical trajectory. The Kahler potential K andthe holomorphic superpotential W are

K = ΦΦ +XX − g(XX)2, (1)

W = aX

Φ(Φχ − 1), χ = b+ ic . (2)

It was thoroughly shown in Ref. [8] that the exponent χ has a geometricalorigin associated with non-geometric flux compactification of Type IIB stringtheory.

The η problem of supergravity theory is solved due to a global U(1)symmetry of the Kahler potential that introduces phase monodromy in thesuperpotential.

Φ→ e2πiΦ, K → K, W → W + aX

ΦΦχ(e2πiχ − 1). (3)

The scalar potential is determined by the Kahler potential K and superpo-tential W

V = eK(KijDiWDjW − 3WW ), (4)

where Kij = ∂i∂jK and DiW = ∂iW + KiW . During inflation the field Xis strongly fixed at its vacuum expectation value 〈X〉 = 0. Hence the scalarpotential can be written as follows

V (r, θ) = a2 er2

r2

(r2be−2cθ − 2rb cos(c log r + b θ)e−cθ + 1

). (5)

in reduced Planck units (MP ≡ 1/(8πG) = 1), with Φ ≡ r eiθ. With stabi-lized field norm |Φ| = 1, but no constraint on c, the scalar potential V (r, θ)becomes

V (θ) = a2(e−2cθ − 2 cos(b θ)e−cθ + 1

). (6)

3

Natural

1σ Starobinski-like

2σ Starobinski-like

0.960 0.965 0.970 0.975

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

ns

log10(r)

(a)

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

b

c

(b)

Figure 1: The constraints on helical phase inflation with N = 60 e-folds in GR case.The blue dots correspond to the 1σ constraint and the brown dots correspond to the 2σconstraint. The shaded regions are the marginalized 1σ and 2σ contours from Planck 2018data and BICEP2 results [22, 23].

By varying b and c we can interpolate from natural inflation (c = 0) toStarobinsky-like inflation (b = 0, c > 0).

Fig. 1 shows helical phase inflation results in the framework of Einstein’sgeneral relativity (GR). For the Starobinski-like inflation the number of e-folds, spectral index and tensor-to-scalar ratio are

N∗ =ecθ∗ − ecθe − c(θ∗ − θe)

2c2, (7)

ns = 1−4c2(ecθ∗ + 1

)(ecθ∗ − 1)2 , (8)

r =32 c2

(ec θ∗ − 1)2. (9)

The condition at the end of inflation ε(θe) = 1 is used to solve for θe

cθe = ln(1 +√

2c) . (10)

In the limit c � 1 limit, N∗ ' (ecθ∗ − ecθe)/(2c2), we get the α-attractorresult [24]

ns ' 1− 4c2

ecθ∗' 1− 2/N∗ , (11)

r ' 32c2

e2cθ∗' 8

c2N2∗. (12)

4

Similarly, for natural inflation, the corresponding analytic expressions for thenumber of e-folds, spectral index and tensor-to-scalar ratio are

N∗ =2

b2ln

[cos (b θe/2)

cos (b θ∗/2)

], (13)

ns = −2b2 csc2

(b θ∗2

)+ b2 + 1 , (14)

r = 8b2 cot2

(b θ∗2

), (15)

and the end of inflation condition ε(θe) = 1 determines the field θe as

bθe = arccos

(2− b2

2 + b2

). (16)

The field excursions in four-dimensional case are typically of the order of 10∼ O(10) in reduced Planck units as elaborated in Ref. [8]. This is fine sinceinflaton is a phase.

3. Brane inflation

In the braneworld cosmology, our four-dimensional world is a 3-braneembedded in a higher-dimensional bulk. The Friedmann equation is modifieddue to the high-energy corrections (Randall-Sundrum terms) to the Einsteinequations on the brane [25, 26, 27, 28, 29]

H2 =ρ

3M2P

(1 +

ρ

2λ

), (17)

where MP = M4/√

8π is the reduced Planck mass, and λ is the brane tensionthat relates four-dimensional Planck scale M4 and five-dimensional Planckscale M5 as below

λ =3

4π

M65

M24

. (18)

The nucleosynthesis limit implies that λ & (1 MeV)4 ∼ (10−21)4 in the re-duced Planck unit. A more stringent constraint can be obtained by requiringthe theory to be reduced to Newtonian gravity on scales larger than 1 mmcorresponding to λ & 5× 10−53, i.e., M5 & 105 TeV [27]. Notice that in thelimit λ→∞, we recover the standard Friedman equation in four dimensions.

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The ρ2 correction term in the modified Friedmann equation makes theslow-roll parameters small for a given potential such that [27, 21]

εH = εV1 + V/λ

(1 + V/2λ)2 , (19)

ηH = ηV1

(1 + V/2λ), (20)

where V is the scalar potential. During the slow-roll phase, ρ ≈ V , εV andηV are the standard slow-roll parameters for the canonical scalar field. Inthe high-energy limit, we obtain V � λ and εH/ηH � 1 even if εV and ηVare large due to the large slope of the potential.

Accordingly the formula for the number of e-folds becomes

N(θ) = −∫ θend

θ

V

V ′

(1 +

V

2λ

)dθ . (21)

The spectral tilt for scalar perturbations can be rewritten in terms of theslow-roll parameters as following

ns − 1 = −6εH + 2ηH . (22)

In the Randall-Sundrum model II [30], the tensor-to-scalar ratio is mod-ified in the high-energy limit as below 1,

r = 16A2t

A2s

= 24εH . (23)

where the tensor and scalar perturbation amplitudes are given by [27, 21, 31]

A2t =

1

150π2V

(1 +

V

2λ

)F 2 , (24)

A2s =

1

75π2

V 3

V ′2

(1 +

V

2λ

)3

, (25)

F 2 =

[√1 + x2 − x2 sinh−1

(1

x

)]−1

, (26)

1In the usual four-dimensional case, r ' 16 εH .

6

and

x ≡(

3H2

4π λ

)1/2

=

[2V

λ

(1 +

V

2λ

)]1/2

. (27)

In the low-energy limit (V/λ� 1), F 2 ≈ 1, we recover the result r = 16εH . Inthe high-energy limit (V/λ� 1), F 2 ≈ 3V/2λ. Note that the right-handedsides of Eqs.(24) and (25) should be evaluated at the horizon crossing.

3.1. Numerical results for helical phase inflation on a brane

It is clear from the helical phase inflaton potential (6) and the slow-rolldefinitions in the brane cosmology the spectral index ns and the tensor-to-scalar ratio r essentially depend on three parameters viz. b, c, and a2/λ forany fixed number of e-foldings say N = 60.

Natural

1σ Starobinski-like

2σ Starobinski-like

0.960 0.965 0.970 0.975-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

ns

log10(r)

(a)

0 2 4 6 8 10 120

20

40

60

80

100

b

c

(b)

Figure 2: The observational constraints on helical phase inflation on a brane taking N =60 e-folds and a2/λ = 1000. Apart from natural inflation case the field excursions forparameters within 2σ constraints generally have ∆θ < 1. The 1σ and 2σ constraints onthe parameters b and c are shown with blue and brown colors respectively in the rightpanel.

In Fig. 2 we show the numerical results of the parametric scan for thefixed ratio a2/λ = 1000. The results for two special cases viz. naturalinflation (c = 0) and starobinsky-like inflation (b = 0, c > 0) are representedby red and green curves respectively. We have used the latest Planck 2018and BICEP2 results [22, 23] for 1σ and 2σ contours.

The experimental value of curvature perturbation can always be satisfiedin the model that in turns fixes the value of brane tension λ and a2 relatedto the scale of inflation.

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Similar numerical study was performed by fixing the value of the ratioa2/λ = 10, 100 but no qualitative difference was found as long as we remainin the high energy regime of a2/λ� 1.

3.2. Analytic approximationsFor the starobinski-like case we set b = 0 and the analytic expressions for

ns and r in the limit of large a2/λ are

ns ≈ 1− 8λ c2

a2

1

(ec θ∗ − 1), (28)

r ≈ 192λ c2

a2

1

(ec θ∗ − 1)2. (29)

For the general potential (6), we get some extra factors of sin(b θ) in theexpressions of ns and r that explain the spread of blue region in Fig. 2surrounding the green Starobinski-like curve. However, the parameters a2/λand c2 again appear as a product λ c2/a2.

4. Conclusions

We have considered the helical phase inflation models from the N = 1supergravity where the phase component of a complex field is inflaton. Suchkind of models escapes both the η problem and the trans-Planckian field ex-cursion problem because of the phase monodromy of the superpotential andthe phase rotations evolving to trans-Planckian domain rather than the phys-ical field. We studied the observational constraints on the model parameters,and present the viable parameter space which is consistent with the Planck2018 data and BICEP2 results. The allowed parameter b is smaller, andthe natural inflation is marginally consistent with the observations at the 2σlevel. In the brane case, we discussed the modified Friedmann equation, andthen studied the observational constraints as well. Similarly, we present theviable parameter space which is consistent with the Planck 2018 data andBICEP2 results. The natural inflation lies well outside the 2σ region in ns-rplane as evident from Fig. 2, so it was excluded by the observations. Whilethe Starobinksi-like inflation provides the favorable central value for spec-tral index ns, and the range of values of tensor-to-scalar ratio r spans fullexperimental and theoretical estimate. For example, the future LiteBIRDexperiment, an experiment designed for the detection of B-mode polariza-tion pattern embedded in the Cosmic Microwave Background anisotropies, issensitive enough to detect primordial gravitational waves up to r ∼ 10−3 [32].

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Acknowledgments

MS would like to thank Higher Education Commission Pakistan for Ph.D.scholar- ship. This research was supported by the Projects 11647601, 11875062,and 11875136 supported by the National Natural Science Foundation ofChina, by the Major Program of the National Natural Science Foundationof China under Grant No. 11690021, and by the Key Research Program ofFrontier Science, CAS.

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