2.4 Inflationary observables
2.4.2 Higher order analysis
We now reconsider the inflationary part of the potential, but do not require the inflaton to be placed sufficiently far away from the stationary point (2.28) such that only one of the terms of (2.27) dominates the inflationary dynamics. Now, we allow for both terms to contribute.
Considering inflation corresponding to increasing fibre, the inflationary poten-tial with theC1-term included reads
VinfR ∼V0
−C1 τ1 − C2
√τ1+C2loopτ1
. (2.48)
The inflaton inflates to the right while the C1-term may break the plateau at smallerτ1 and induce a local maximum. We consider theC1-term arising fromV(1) to be of importance while we have still omitted the string loop inducedC1loop-term.
The reasoning is that first the string loop term is τ1−1 suppressed with regard to the higher derivative term. Second, for typical values of C1KK as given in (2.20), the C1loop-term will be additionally suppressed. At last, while the string loop term scales with gs, the higher derivative term is g−3/2s enhanced. The canonically
2.4. INFLATIONARY OBSERVABLES 33 normalised and uplifted potential hence receives a falling correction at large e-foldings, i.e.
VinfR (ϕ)∼V˜0R 1−2eκ2ϕ−ε2e−κϕ
, ε2 = C1
4C2βR3 , (2.49) where we have already expanded the inflationary potential and have omitted the string loop induced term for notational ease.17 From the above we may already infer that ε2 1 for the inflationary plateau not to be spoiled. The slow-roll parameters (1.22) then receive ε2 dependent corrections of the form
V = 1 2
V0 V
2
= 1
2 −κ eκ2ϕ+κ ε2e−κϕ
, (2.50)
ηV = V00
V =−1
2κ2eκ2ϕ−κ2ε2e−κϕ, (2.51) where we have taken the potential to be slowly varying during inflation, i.e.VinfR ∼ const. Recalling (1.23) dNe = (2V)−1/2dϕ, i.e.
Ne ∼2κ−2e−κ2ϕ+O(ε2), (2.52) we hence arrive at the expression for the spectral index ns including higher order corrections
ns= 1− 2
Ne −3ε2κ4Ne+ε2κ6
2 Ne2 +. . . . (2.53) The above suggests that there is a further phenomenological fingerprint in the form of running of the spectral index. Considering the next-to-leading correction to the spectral index
δns =−3ε2κ4Ne+ ε2κ6
2 Ne2, (2.54)
and requiringδns .0.008 for Ne = 55, which is the 2-σ range for the ns measure-ment from Planck, we find an upper bound on ε2 to be
ε2 = |C1| 4C2
βR3 . 2.4×10−6. (2.55)
17During inflation, the term ensuring the existence of the minimum is negligible.
This is in agreement with earlier works employing exponential corrections to the inflationary plateau [14, 30–33, 35, 50, 51].18 The bound on ε2 also restricts
|C1|
4C2βR3 ∼ λ−3Vgs15/2Π1Π−42 (C2KK)6 . 2.4×10−6 (2.56) and hence gives a constraint that has to be fulfilled in the first place when consid-ering observationally viable inflation to the right.19
Considering the scenario where inflation occurs for the inflaton rolling to the left, we now start with the potential
VinfL ∼V0 −C1 τ1
+C1loop
τ12 +C2√ τ1
!
. (2.57)
TheC2-term destroys the plateau. Again, we have omitted the string loop induced term as it is suppressed with regard to the higher derivative C2 term by similar reasoning as was employed when justifying the omission of the C1loop-term when studying inflation to the right. The canonically normalised and uplifted potential hence receives a falling correction at large e-foldings, i.e.
VinfL (ϕ)∼V˜0L 1−2e−κϕ−ε2eκ2ϕ
, ε2 = C2
√2C1βL3/2, (2.58) where we have again expanded the inflationary potential and have omitted the string loop induced term as it plays no role on the inflationary plateau. Similarly to the rolling to the right case above, we can derive ns and obtain
ns= 1− 2
Ne − 3√ 2ε2κ
√Ne +ε2κ3√ Ne
√2 − 3
2ε4κ4Ne+. . . . (2.59) Considering the 2-σ bounds by PLANCK, i.e. requiring δns . 0.008 at Ne = 55,
18Note that potential (2.49) may not account for power loss at low-`in the CMB temperature spectrum as the correction comes with a minus sign and hence induces a local maximum and not an inflection point as would be required for power suppression.
19ForV ∼103and Π1∼Π2the bound requiresβR.10−3, thus placing a stronger constraint onβR than the minimal required length of the plateau.
2.4. INFLATIONARY OBSERVABLES 35 we obtain the upper bound
ε2 ∼λ−3/2V−1 gs5/2V3/2
Π2Π−5/21 (C1KK)3/2 .10−3. (2.60) Given the bounds on ε2 induced by the data-compatible range of δns, we also find that the contribution of the δns to the magnitude of running dns/dlnk =
−dns/dNe is typically.O(10−4) and hence at least an order of magnitude smaller than the contribution to the running from n(0)s − 1 = −2/Ne which is about dn(0)s /dlnk ∼ −10−3. Finally, let us provide one example of parameter choices for a model of inflation with shrinking fibre τ1 in Table 2.2.
W0 gs V τ1min Π1 Π2 C1KK C2KK ns L1 2 0.3 460 3 100 1 0.163 0.0288 0.966
Table 2.2: Example of compactification parameters and inflationary observables for inflation to the left (L1).
In this chapter, we have presented a way to realise an effective shift symme-try by tuning and balancing different perturbative corrections against each other.
While a part of the effective shift symmetry derives from an intrinsic one at para-metrically large volume, this could not serve to protect the inflaton potential against higher orders in the parameter regime required to drive observationally viable slow-roll inflation Hence balancing and tuning terms had to be invoked.
Chapter 3
The non-canonical point of view
Embedding inflation in some UV theory usually yields a non-canonical kinetic term in the effective Lagrangian in an intermediate step (as an example, recall the kinetic Lagrangian (2.13) of chapter 2). Conventionally, inflationary dynamics are studied once the kinetic term is canonically normalised. However, obtaining a non-canonical kinetic term as an intermediate step suggests to analyse the kinetic function directly and hence to - independently of the specifics of the potential - use the non-canonical kinetic term as a short cut to the inflationary dynamics. In this non-canonical language, recent work established a reformulation of plateau-type inflaton potentials in terms of a certain pole structure of the kinetic function [52].
In this chapter we enhance these ideas by establishing an extended duality between a kinetic function with a certain pole structure and shift symmetry of the Einstein frame canonically normalised inflaton potential [16]. Moreover, we study the breaking of the shift symmetry at large fields. Since non-canonical kinetic terms are a generic consequence of compactifications of higher-dimensional models such as string theory, this may provide a new avenue of constructing this set of phenomenologically promising models from more fundamental embeddings. One of the main aims of this chapter is to provide the analogue formulation of this shift symmetry for non-canonical models of inflation, to which we turn next.
The rest of the chapter is structured as follows. First, we recall the formulation of inflation where the inflationary dynamics’ complexity has been shifted in parts to the kinetic term rather than the potential. Given a suppression hierarchy for
37
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.