2.4 Inflationary observables
3.1.1 Laurent expansion
poles of increasing order, we continue to describe corrections to the aforementioned formalism and derive leading order corrections to the inflationary observables.
Assuming the corrections to follow the pattern of shift symmetry in an EFT sense, we then study an infinite tower of corrections and demonstrate that the leading order corrections coincide with the structure obtained before. After outlining phenomenological fingerprints deriving from the corrections, we attempt to embed the previous considerations into some UV theory. Following a generic argument as to what the coarse structure of the UV candidate K¨ahler potential might be, we give perturbative and exact examples which reproduce the kinetic functions under study in this work. We then turn our attention to String Theory and argue that the necessary terms may be obtained. Specifically, we recall that the more general form of string loop corrections to the volume moduli K¨ahler potential in string compactifications spoil the log-structure of the K¨ahler potential, and we hence expect them to break the shift symmetry at large field ranges. We conclude by discussing our results, and point out that our steepening corrections generically produce a moderate loss of CMB power at large angular scales for which we give an analytical estimate.
3.1 Pole inflation
We begin with a quick recapitulation of the formalism presented in [52] and provide extensions in order to translate the results presented to canonically normalised fields.
3.1. POLE INFLATION 39 where we assume the kinetic function KE(ρ) to be given by a Laurent series with a pole of order patρ=ρ0 = 0 (without loss of generality) plus sub-leading terms, which are higher-order in ρ (not higher order inρ−1) and are thus irrelevant close to the pole.20 In principle, higher order terms inρ−1, i.e. higher orders in the pole, are of increasing importance when ρ→0. We will neglect those terms for now to ease our analysis of the first pole and give a condition which has to be satisfied in order to do so in (3.13).
The location of the pole corresponds to a fixed point of the inflationary tra-jectory, which is therefore characterised almost completely by this point. Upon canonical normalisation, the fixed point translates into a nearly shift-symmetric plateau in the potential. As the inflationary behaviour will be determined by the trajectory of the non-canonical field in the vicinity of the pole, one may approxi-mate VE(ρ) to be
VE =V0(1 +cρ+. . .), (3.2) where we may leave the coefficient c unspecified. Our results will not depend on any choice of cas we will later demonstrate. All higher order terms may also have arbitrary coefficients as they will be sub-dominant close to the pole ρ0 = 0.
The crucial assumption in Lagrangian (3.1) is that kinetic pole and potential minimum do not coincide.21 In other words, scenario (3.1) and (3.2) may also be recast - by means of a field redefinition ρ→ρ+ρ0 - to read
KE(ρ) = 1
(ρ−ρ0)p +. . . , VE(ρ) = V0(ρn+. . .). (3.3) with n≥1. Going back to ρ→ρ−ρ0, the above will becomeVE ∼1 +cρ . . .with c=−n/ρ0 in the vicinity of the pole ρ0 and hence results in the same inflationary dynamics as scenario (3.1) and (3.2). For n = 2, this argument may also be understood in the following sense; if the specifics of a scalar field potential are unknown, as long as the scalar field is stabilised at ρmin and the kinetic function has a pole ρ0 6=ρmin, inflation compatible with PLANCK will be realised.
20We can move any poleρ0toρ0→ρ0= 0 by means of a field redefinitionρ→ρ−ρ0 without changing the dynamics of the system.
21If the potential also had a minimum atρmin=ρ0= 0, expansion (3.2) would obviously not be suitable. This issue was not mentioned explicitly in the original work. We will revisit this when turning to perturbations of the pole structure in section 3.2.
For simplicity, we now assume the pole to be located atρ0 = 0 and the potential hence to be given by (3.2). For this non-canonical set-up, the slow-roll parameters (1.22) are
V = 1 2KE(ρ)
1 VE(ρ)
∂VE(ρ)
∂ρ 2
, and
ηV = 1 KE(ρ)V(ρ)
∂2V(ρ)
∂ρ2 − ∂VE(ρ)
∂ρ
1 2KE(ρ)
∂KE(ρ)
∂ρ
, (3.4)
where ϕ is the canonically normalised inflaton and KE(ρ) is the kinetic function of Lagrangian (3.1). An explicit calculation then yields
V = 1
2apρp, ηV =− p
2apρp−1. (3.5)
The number of e-folds Ne is obtained as Ne =
Z 1
√2V dϕ= Z
KE(ρ)VE(ρ)
∂VE(ρ)
∂ρ −1
dρ . (3.6)
Sufficiently close to the pole atρ= 0, i.e. at largeNe, the number of e-folds hence evaluates to
Ne = ap
(p−1)ρp−1 , and thus ρ=
ap (p−1)Ne
p−11
. (3.7)
Since we assume p > 1, indeed the number of e-folds increases as the field ρ approaches the pole. At lowest order in 1/Ne, the inflationary predictions for this model are therefore given by
ns = 1− p p−1
1
Ne, r = 8a
1 p−1
p
(p−1)p−1p 1 N
p p−1
e
, (3.8)
whereap is the leading coefficient of the Laurent expansion as in (3.1). The above derivation is indeed independent of the linear coefficient cof (3.2).
Putting all of this together, we observe that the presence of a fixed point of the kinetic function that does not coincide with the minimum of the potential
3.1. POLE INFLATION 41 translates to an effective shift symmetry of the canonically normalised inflaton at large field values, provided that all higher-order poles in the kinetic function beyond the leading-order pole defining the fixed point have successively suppressed coefficients as in equation (3.13) below. This provides us with a new handle on finding regimes where inflaton potentials show an effective shift symmetry via analysing the local structure of the non-canonical kinetic function. In other words, vastly enhancing the kinetic term such that it becomes dominant with regard to the potential - e.g. with a pole in the kinetic function as above - enters the canonical normalisation such that, given reasonable assumptions about the specific potential, the canonically normalised field will slowly roll down its effective potential which will be of plateau type.
The case p= 2 is special for a number of reasons. First of all, this gives rise to a value of the spectral index that agrees exceedingly well with PLANCK. Secondly, from a theoretical perspective, large classes of models with different interactions actually give rise to nearly identical predictions (3.8) with p= 2. In what follows, we will therefore focus on corrections to p= 2 poles.
To finish the discussion, we now turn to canonical variables. We begin by demonstrating that the coefficient c may indeed be kept arbitrary when studying the inflationary dynamics also in canonical fields. Recalling potentialVE =V0(1 + cρ . . .) and the pole ρ0 = 0, we observe that for
c <0, ρ >0 and thus Ne → ∞ for ρ→0+,
c >0, ρ <0 and thus Ne → ∞ for ρ→0−, (3.9) where inflation occurs for decreasing (c < 0) or increasing (c > 0) fieldρ. For the exemplary case p= 2, canonical normalisation introduces
ρ∝e±ϕ/√ap and ρ∝ −e±ϕ/√ap (3.10) for c < 0 and c > 0 respectively.22 The sign of the exponent is arbitrary and
22Forp= 2, canonical normalisation∂ρ/ρ=∂ϕevaluates to log|ρ|=ϕ+ϕ0. Forρ >0, this is solved by the first term of (3.10) while forρ <0, it is solved by the second. The constant of integrationϕ0 may be tuned to absorb any value|c|.
simply reflects whether inflation occurs for increasing (plus sign) or decreasing (minus sign) canonical fieldϕ. Moreover, any value|c|can be absorbed by means of a field redefinition, i.e. choosing a suitable integration constant ϕ0. We hence see that also the potential in canonically normalised fields is independent of the linear coefficientc and universally reads
V0(ϕ) =
V0
1−A ϕ2−p2
, p6= 2
V0
1− e−
√ϕ ap
, p= 2
(3.11)
where A = 2−p
2√ ap
2−p2
. This shows the plateau at ρ → 0 occurring for ϕ→ ∞ if p≥2 and for ϕ→0 otherwise. The higher powers in ρof V0(ρ) beyond the linear term are irrelevant due to the fact that the pole structure has inflation taking place for ρ→0. Higher powers in the Laurent expansion of KE(ρ)
KE(ρ) = ap
ρp +X
q>p
aq
ρq (3.12)
will perturb V0(ρ) → V(ρ) = V0(ρ) + ∆V(ρ). Therefore, an extended plateau in the potential equation (3.11) requires us to restrict to the regime where the following condition holds
aq ρq ap
ρp ∀ q > p . (3.13)
Similar to the suppression of higher-dimension operators in some scalar potential, there is a priori no reason why condition (3.13) should hold. We hence propose con-dition (3.13) as a statement dual to the requirement to suppress higher-dimension operators in the canonical picture and will give a toy model realisation of (3.13) in section 3.2 and specifically via expression (3.32). This suppression pattern of the residues of the Laurent expansion dictated by the approximate shift symmetry on the plateau forms a complete analogue of the two known requirement to suppress higher order terms in the canonical formulation.
3.1. POLE INFLATION 43