Simple single-eld ination models with arbitrarily small tensorscalar ratio Nina K. Stein

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Simple single-ﬁeld inﬂation models with arbitrarily small

tensor/scalar ratio

Nina K. Stein

Dept. of Physics, Univ. at Buﬀalo, SUNY

William H. Kinney

Dept. of Physics, Univ. at Buﬀalo, SUNY and

Dept. of Physics, Indian Institute of Technology, Madras

(Dated: December 27, 2022)

Abstract

We construct a family of simple single-ﬁeld inﬂation models consistent with Planck / BICEP

Keck bounds which have a parametrically small tensor amplitude and no running of the scalar

spectral index. The construction consists of a constant-roll hilltop inﬂaton potential with the end

of inﬂation left as a free parameter induced by higher-order operators which become dominant

late in inﬂation. This construction directly demonstrates that there is no lower bound on the

tensor/scalar ratio for simple single-ﬁeld inﬂation models.

arXiv:2210.05757v2 [astro-ph.CO] 27 Dec 2022

I. INTRODUCTION

Inﬂationary cosmology [1–7] remains a uniquely successful phenomenological framework

for understanding the origin of the universe, making quantitative predictions which cur-

rent data strongly support [8–12]. Inﬂation relates the evolution of the universe to one or

more scalar inﬂaton ﬁelds, the properties of which dictate the dynamics of the period of

rapidly accelerating expansion that terminates locally in a period of reheating, followed by

radiation-dominated expansion. The speciﬁc form of the potential of the inﬂaton ﬁeld or

ﬁelds is unknown, but diﬀerent choices of potential result in diﬀerent values for cosmological

parameters, which are distinguishable by observation [13, 14]. Recent data, in particular

the Planck measurement of Cosmic Microwave Background (CMB) anisotropy and polariza-

tion [15–17] and the BICEP/Keck measurement of CMB polarization [18] now place strong

constraints on the inﬂationary parameter space. Consequently, many previously viable inﬂa-

tionary potentials, including some of the simplest and most theoretically attractive models,

are in conﬂict with these observations, in particular the upper bound on the tensor scalar ra-

tio r. Furthermore, near-future measurements could reduce this upper bound from r≤10−2

to r≤10−3. [19, 20]

This raises the question: is there an upper bound on rwhich would rule out all

simple single-ﬁeld inﬂation models? This question presupposes a deﬁnition of “simple”,

which is inherently subjective. Recent work has explored this question in the context of

supersymmetry-inspired α-attractor models of inﬂation [21, 22]. In this paper, we will adopt

a deﬁnition of “simple” that consists of a single scalar ﬁeld, with canonical Lagrangian, and

a potential that can be approximated during the epoch of inﬂation by a single leading-order

operator. This has the advantage of being entirely generic, applying to any hilltop-type

potential. Any arbitrary single-ﬁeld potential can be represented by an eﬀective operator

expansion,

V=V0+dV

dφ φ=0

φ+1

d2V

dφ2φ=0

φ2+1

d3V

dφ3φ=0

φ3+··· .(1)

Diﬀerent choices of coeﬃcients result in diﬀerent inﬂationary dynamics and diﬀerent predic-

tions for observables, such as the tensor/scalar ratio rand scalar spectral index nS. This can

be used to falsify particular inﬂationary models [13]. For example, the simplest monomial

potentials of the form

V(φ)∝φp, p > 0,(2)

are ruled out, since they overproduce tensor perturbations, violating the upper bound set

by the BICEP/Keck measurement [23]. Hilltop models, of the general form

V(φ) = V0−λφ

µp

(3)

fare better, where λis a dimensionless coupling constant, and µis a mass scale determining

the range of validity of the eﬀective expansion, with ∆φ∼µduring inﬂation. The ﬁeld

excursion ∆φis directly related to the tensor/scalar ratio by the Lyth bound [24],

∆φ≥MP√2r, (4)

where MPis the reduced Planck mass. “Swampland” conjectures, motivated by string theory

[25], suggest that the ﬁeld excursion is bounded in any eﬀective ﬁeld theory which can be

completed in the ultraviolet,

∆φ<MP,(5)

with the consequence that there may be a corresponding upper bound on rin viable inﬂa-

tionary models. A well-known example of a model which can accommodate arbitrarily small

ris one for which the mass term for the inﬂaton ﬁeld φis suppressed, for example by a shift

symmetry, with the leading behavior

V(φ) = V0"1−φ

µ4

+···#.(6)

In this case, the scalar normalization depends only on the ratio of the height of the potential

V1/4

0to the width µ[26], and taking µ<MPresults in an upper bound on the tensor/scalar

ratio rof [11]

r < 1

4N3.(7)

The scalar spectral index nSin the limit µMPsimilarly depends only on the number of

e-folds Nof inﬂation,

nS= 1 −3

N.(8)

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Simplesingle-eldinationmodelswitharbitrarilysmalltensor/scalarratioNinaK.SteinDept.ofPhysics,Univ.atBualo,SUNYWilliamH.KinneyDept.ofPhysics,Univ.atBualo,SUNYandDept.ofPhysics,IndianInstituteofTechnology,Madras(Dated:December27,2022)AbstractWeconstructafamilyofsimplesingle-eldinationmodelsconsist...

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Simple single-eld ination models with arbitrarily small tensorscalar ratio Nina K. Stein

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