A Kinetic terms for the moduli

Modulated Reheating and Large Non-Gaussianity in String Cosmology

Abstract:

A generic feature of the known string inflationary models is that the same physics that makes the inflaton lighter than the Hubble scale during inflation often also makes other scalars this light. These scalars can acquire isocurvature fluctuations during inflation, and given that their VEVs determine the mass spectrum and the coupling constants of the effective low-energy field theory, these fluctuations give rise to couplings and masses that are modulated from one Hubble patch to another. These seem just what is required to obtain primordial adiabatic fluctuations through conversion into density perturbations through the ‘modulation mechanism,’ wherein reheating takes place with different efficiency in different regions of our Universe. Fluctuations generated in this way can generically produce non-gaussianity larger than obtained in single-field slow-roll inflation; potentially observable in the near future. We provide here the first explicit example of the modulation mechanism at work in string cosmology, within the framework of LARGE Volume Type-IIB string flux compactifications. The inflationary dynamics involves two light Kähler moduli: a fibre divisor plays the rôle of the inflaton whose decay rate to visible sector degrees of freedom is modulated by the primordial fluctuations of a blow-up mode (which is made light by the use of poly-instanton corrections). We find the challenges of embedding the mechanism into a concrete UV completion constrains the properties of the non-gaussianity that is found, since for generic values of the underlying parameters, the model predicts a local bi-spectrum with of order ‘a few’. However, a moderate tuning of the parameters gives also rise to explicit examples with potentially observable by the Planck satellite.

1

1 Introduction

Although initially motivated as a solution to the initial-condition problems of standard Big-Bang cosmology, inflationary models [1] turn out also to provide a successful phenomenological description of the properties of primordial density fluctuations (which are predicted to arise as late-time consequences of quantum fluctuations of the inflaton).

Until relatively recently most of these predictions were made using single-field, slow-roll inflationary models, for which the inflaton is the only light scalar relevant during inflation [2, 3]. The popularity of these models can be traced to several of their attractive features: () simplicity; () effectiveness (they describe primordial fluctuations very well); () predictivity (near, but not exact, scale invariance, scale-independent spectral tilt, gaussian fluctuation statistics, and so on); and () robustness (that is, predictions can be made without knowing the details of the cosmological history between the end of inflation and the beginning of the late-time Big-Bang epoch.)

Why defy Occam?

Despite the simplicity and success of the single-field framework, two independent lines of reasoning suggest exploring multi-field models [4, 5] in more detail. These two lines of reasoning have their roots both in observational developments as well as theoretical considerations, as we now describe.

The first reason to explore more complicated models arises from ongoing improvements in sensitivity of modern observations. In particular, besides testing the consistency of concordance cosmology through improved precision on cosmological parameters [6], new instruments like the Planck satellite [7] will soon probe the existence of non-gaussianity down to a level that can be produced by many types of multi-field models [8, 9] (though not by vanilla single-field slow-roll scenarios). It behooves us as a field to be alive to the kinds of physics to which these measurements might be sensitive, particularly if single-field slow-roll models were to be falsified by the discovery of non-gaussianity.

The second reason to explore more complicated models becomes evident once one tries to integrate inflationary cosmology into our broader understanding of physical law. This raises several independent issues. Most famously, inflationary models require the inflaton to be much lighter than the slowly changing Hubble scale during inflation, . This is a difficult thing to arrange once the inflaton is coupled to any other fields present at the energies of interest, even if these other fields do not play a direct rôle in cosmology [5, 10].

Furthermore, there are very likely to be a good number of other fields to which a putative inflaton might couple, although to make this precise requires specifying a concrete theory. The challenge when doing so is that we do not know the laws of physics at the high energies usually required [11], however the proximity of to and the relevance of quantum fluctuations to primordial fluctuations strongly suggests that it should involve whatever makes sense of quantum gravity at high energies. Since string theory is presently our best-developed framework for understanding quantum gravity, it provides a natural framework for exploring these issues in a crisp way.

And it is in string cosmology that we meet the second reason for exploring multi-field inflationary models in more detail. This is because close to a decade of exploration reveals that multi-field dynamics is the norm for string inflation and not the exception [12]. Although it remains difficult in string theory to achieve a sub-Hubble inflaton mass (and so also slow-roll inflation), once this has been done the same mechanism tends also to drag down the masses of several other fields as well, making them also potentially cosmologically active. This observation — that multi-field models arise fairly generically in this way once embedded into microscopic models — has recently motivated a more detailed study of the systematics of inflationary multi-field dynamics in general [13].

Non-gaussianity and post-inflationary dynamics

One of the main differences between single-field and multi-field inflationary models is the wide variety of fluctuation-generation mechanisms available in the multi-field case. Whereas single-field models must generate adiabatic curvature fluctuations, multi-field models can allow a variety of fluctuations in which the various fields need not alter the gravitational curvature. And although these isocurvature fluctuations would pose an observational problem for Cosmic Microwave Background (CMB) observations if they were to persist to the present epoch, they need not be bad during inflation if they can be converted to adiabatic fluctuations sometime between the epoch of horizon exit and more recent times. This could easily happen, such as if thermal equilibrium were established sufficiently early before the present epoch.

Indeed, generating adiabatic fluctuations through such a roundabout route might even be a virtue rather than a complication, since alternative mechanisms predict primordial fluctuations to depend differently on inflationary properties than the standard single-field mechanism. In particular, the interactions inherent amongst the fields in any multi-field mechanism can often generate observable amounts of non-gaussianity in the CMB, making any discovery of non-gaussianity into a new window into the physics responsible for generating fluctuations, and possibly into some aspects of the post-inflationary regime. Most interestingly, non-gaussianity produced in this way after inflation can often be distinguished observably from non-gaussianity generated directly during inflation, such as by non-standard single-field models [3, 14].

The two best-developed mechanisms for converting, after inflation, inflationary isocurvature perturbations into late-epoch adiabatic fluctuations are the curvaton mechanism [15] and modulation mechanism [16, 17, 18, 19]. One might ask whether these ever arise in string-inflationary models, given the preponderance of multi-field scenarios these models have. Interestingly, they largely do not: for most string-inflation models explored to date only adiabatic fluctuations are appreciably generated at horizon exit, allowing their observational implications to be simply understood in terms of effective single-field models [12, 20].

Isocurvature fluctuations from string vacua

That is not to say that other means for generating fluctuations cannot arise in string theory, and ref. [21] provides a construction of an inflationary cosmology that uses the curvaton mechanism, within an explicit string vacuum in the LARGE Volume Scenario (LVS) [22, 23, 24]. What is perhaps remarkable is that no instances of the modulation mechanism have so far arisen, given that this mechanism was designed with string theory in mind. (The modulation mechanism starts with the observation that couplings in string theory are often given as the values of fields, and so can vary from one Hubble patch to another if these fields so vary. If, in particular, the couplings that vary this way are involved in the transfer of energy from the inflaton into the heat of the later hot Big-Bang epoch, then this reheating would be modulated from one Hubble patch to another, in a way that might describe the observed primordial fluctuations. In particular, such a modulation has been argued generically to produce non-gaussianity at the soon-to-be observable level of [19].)

The goal of this paper is to provide an explicit string realisation of this modulation mechanism, much as ref. [21] did for the curvaton mechanism. We are able to do so, but it proves to be unexpectedly difficult to do, for a variety of reasons. The main reason for this is the difficulty in assembling in one model all of the things that are assumed when building a modulated reheating model: the requirement that inflation takes place in the first place; the presence of a modulating field that is light during inflation, modulates the inflaton decay rate and is almost decoupled from the inflaton; and the conditions required for the modulation mechanism to beat the standard one. Finally, we ask that the modulating field decays before the inflaton, ensuring the absence of any curvaton-like contributions to the generation of the density perturbations.

Here we provide the first explicit string model for which all these conditions apply. By doing so within the LVS we also avoid the need for fine-tuning. What must be chosen carefully, however, is the underlying extra-dimensional manifold, including a particular brane set-up and a particular choice of world-volume fluxes. In particular we make use of novel ingredients in string model building, such as poly-instanton [25] contributions to the superpotential, that allow the exploration of new kinds of phenomenology and cosmology within the LVS [26, 27].

LVS string vacua have several features that make them particularly attractive from the point of view of model building in cosmology. One of these is what makes the LVS attractive for other applications as well: its explicit incorporation of moduli stabilisation within a systematic expansion of the low-energy potential in inverse powers of the extra-dimensional volume. That is, if the extra-dimensional volume (in string units) is , then the LVS scenario starts with the observation that for large the low-energy scalar potential has the generic form of a sum of terms of order , and , where is the dimensionless volume, , of a comparatively small four-cycle in the geometry. For some choices of parameters this has a minimum with , which makes very large volumes natural to obtain.

The low-energy potential, , has another property that is of more specific interest for inflationary applications, however. This is that the leading, , contributions to the potential can be independent of many of the other extra-dimensional moduli, . These moduli therefore can only enter at subdominant order in , in practice arising at one string loop [24]:

(1)

where and are calculable functions. For large this allows these moduli to be parametrically light compared with others [28], and in particular to be small compared with the Hubble scale, . This both explains why the inflaton is light [29] and why many other fields are also equally light [21].

In the particular construction explored here, inflation is driven naturally by a fibre modulus — similar to [29] — which acquires a potential from loop corrections, as above. A particular blow-up mode also remains light during inflation since it is stabilised as in [26] due to subleading poly-instanton corrections [25]. The absence of single-instanton contributions to the superpotential for this rigid four-cycle is guaranteed by the presence of extra fermionic zero modes which are Wilson line modulini. Because this blow-up mode is lighter than the Hubble parameter during inflation, it acquires significant isocurvature fluctuations.

The key observation is that this field can play the rôle of a modulation field because its VEV controls the size of the inflaton coupling to visible sector degrees of freedom, resulting in a modulated decay rate due to the fluctuations of the blow-up mode. Reheating takes place as in [30, 31, 32] due to the perturbative decay of the inflaton, converting the initial isocurvature perturbations of the light blow-up mode into adiabatic perturbations that swamp the standard contributions coming from inflationary fluctuations of the inflaton field itself. Finally, couplings of the modulating field also ensure that it decays before the inflaton, ensuring the absence of curvaton-like contributions to primordial density perturbations.

Predictions for non-gaussianity

There are several advantages to having such an explicit UV completion of the modulation mechanism. First, as is often the case, string theory provides a transparent geometrical interpretation of the modulation mechanism in terms of the properties of the internal manifold under consideration. Second, the consistency of the construction sets strong constraints on the final prediction for the non-gaussianities, whose properties we analyse in some detail. In particular, we find non-gaussianities of local form with a bi-spectrum parameter, . Although this is potentially observable by the Planck satellite, we find small tri-spectrum parameters, and .2

The paper is organised as follows. §2 briefly reviews the framework of LVS compactifications on K3 or fibred Calabi-Yau three-folds. With later applications to the modulation mechanism in mind, we choose a manifold with five moduli: the fibre divisor, (which plays the rôle of the inflaton); the modulus of the base of the fibration, (which controls the overall volume ); the first blow-up mode, (whose non-perturbative effects fix the volume at exponentially large values); a second blow-up mode, (which supports the visible sector); and a third blow-up mode, (which intersects and plays the rôle of the modulating field).3 We seek a regime with moduli stabilised with the hierarchy which, because of the LV ‘magic,’ can be done using hierarchies among the input fluxes that are at most .

§3 then describes the inflationary scenario, including a cartoon of how reheating takes place, in order to set up a viable modulation mechanism. In §4 we then describe how primordial fluctuations are generated by the post-inflationary dynamics, showing that the -dependence of the masses and couplings can produce acceptable adiabatic fluctuations from the modulation mechanism. Our conclusions are briefly summarised in §5.

2 The LARGE-volume string inflationary model

In this section, we identify a set of moduli fields with potentially interesting cosmological applications. We derive the leading order contributions to the potential and kinetic terms for these fields, to be used in the next sections for developing a model of inflation accompanied by modulated reheating. We will keep concise in this section, since we mainly apply techniques developed in other works to which we refer.

2.1 Field content

The model we develop requires a compactification based on a Calabi-Yau manifold with a K3 or -fibred structure, characterised by at least five Kähler moduli, two of which are light during inflation and so are relevant for cosmology. We list them here:

  • A fibre modulus, , playing the rôle of inflaton field as in [29] and wrapped by a stack of D7-branes.4 The low-energy scalar potential acquires a dependence on through string loop contributions sourced by the D7s [24, 35]. The potential is naturally flat enough to drive inflation.

  • A base modulus, , that mainly controls the overall extra-dimensional volume stabilised at exponentially large values. It is wrapped by the same stack of D7-branes needed to generate the string loop potential for . This modulus is heavy during inflation, and lies on its minimum throughout this phase.

  • A blow-up mode , an assisting field required to stabilise the volume at its minimum. Its presence is usual in LARGE volume set-ups. The potential depends on through non-perturbative contributions generated by a stack of D7-branes wrapping and supporting a hidden sector that undergoes double gaugino condensation in a race-track way.5 It is heavy during inflation.

  • Two intersecting blow-up modes, and . supports a GUT or MSSM-like construction with D7-branes and it is stabilised in the geometric regime by -terms, which render this field heavy during inflation. On the other hand is light during inflation, and plays the rôle of modulating field in our cosmological application. In fact, is wrapped by an Euclidean D3-brane (E3) instanton which generates tiny poly-instantons corrections to the superpotential that fix this modulus at subleading order. Single instanton contributions associated with are absent due to the presence of Wilson line modulini which give rise to extra fermionic zero modes.

2.2 Compactification

In order to realise our scenario, we consider an orientifold of a Calabi-Yau three-fold with a K3 or a fibration structure and .6 Hence we have five smooth divisors , , whose volumes are given by the real part of the Kähler moduli . The Kähler form can be expanded in a basis of dual two-forms as7 .

The volume is expressed in terms of the two-cycle moduli as [37, 38]:

(2)

where and given that every Calabi-Yau manifold is characterised by the fact that the signature of the matrix is [39], so with 1 positive and 4 negative signs in our case. The triple intersection numbers are given by 8:

(3)

and are related to the parameters in (2) as follows: , , , , , .

This particular structure of the intersection numbers implies that the is a K3 or divisor fibred over a base whose volume is , is a diagonal del Pezzo divisor, whereas and are two rigid four-cycles intersecting over a two-cycle given by:

(4)

We shall also assume that is a rigid four-cycle with Wilson lines, i.e.  while (for explicit examples of this kind of divisors in this context see [37]).

The four-cycle moduli are defined as:

(5)

and so they take the form:

(6)
(7)
(8)

It is convenient to rewrite the volume in terms of the four-cycle moduli as:

(9)

where we set , , , and .

2.3 Brane set-up and fluxes

In this section we shall perform an explicit choice of brane set-up and fluxes that can give rise to the desired phenomenological features of a modulated reheating scenario. We consider the visible sector to be realised by two stacks of D7-branes wrapped around the divisors and 9 together with an E3 instanton wrapped around 10:

(10)

We also turn on the following world-volume fluxes:

(11)

where the half-integer contributions come from the cancellation of the Freed-Witten anomalies. Let us fix the -field in order to cancel the total flux on so that the instanton contributes to the superpotential. Hence we choose which gives rise to the following total world-volume fluxes: 11

The presence of non-vanishing gauge fluxes has four implications:

  1. They induce -charges for the Kähler moduli. The charges of the blow-up moduli and under the diagonal of the D7-stacks wrapping and read:

    (12)

    These charges can be written one in terms of the other as:

    (13)
  2. The gauge coupling of the field theory on the D7-stack wrapping the divisor acquires a flux-dependent shift. In fact, the gauge coupling is given by:

    (14)

    where is the real part of the axio-dilaton, while the flux-dependent factor is:

  3. The world-volume fluxes generate moduli-dependent Fayet-Iliopoulos (FI) terms which take the form:

    (15)
    (16)

    where we denoted with the Kähler potential of the 4D effective theory.

  4. The chiral intersections between different stacks of D7-branes depend on the gauge fluxes in the following way:

In order to have an instanton contributing to the superpotential we need to kill its chiral intersections with the visible sector (and preferably also with any other sector of the theory). These chiral intersections, if non-vanishing, tend to destroy the instanton contribution, since they would lead to a superpotential of the form , where the correspond to open string matter fields, charged under the visible sector gauge group. In order to preserve the visible sector group, these fields must have zero VEVs resulting in a vanishing [36]. Hence we need to set which implies . On the other hand, we need to have non-vanishing chiral intersections between the D7-stacks on and . This is guaranteed by the relations (13) among the moduli charges which ensure that we can have and while .

Moreover the condition is crucial in order to keep the modulating field light. In fact, the FI terms (15) and (16), generically introduce a dependence on both and the combination

(17)

since

(18)

Therefore both of these fields would get a very large mass. However, the condition implies that and introduce a dependence in the -term potential only on the combination , leaving as a flat direction. This discussion allows to appreciate the essential rôle of the intersection between the two cycles and : thanks to this structure we are left with a flat direction that we can then use for cosmological purposes.

In order to simplify the system under consideration, we shall also require which implies and . The final condition written in terms of flux quanta and wrapping numbers reads:

(19)

which leads to:

Illustrative flux choice

An illustrative choice of fluxes that leads to and gives rise to three families of chiral matter (i.e. ) is:

(20)

The corresponding divisors (which we assume to be smooth) wrapped by the visible sector and the intersecting D7-stack are:

(21)

and

(22)

which, from the condition (19), implies . The resulting charge , flux-dependent shift and FI-term for the visible sector become:

(23)

2.4 Supergravity effective action

Let us now outline the main features of the effective low-energy 4D supergravity model derived by compactifying over the manifold described in section 2.2. After including the leading perturbative corrections, the Kähler potential is (we work throughout in the 4D Einstein frame):

(24)

The corrections are controlled by the quantity , where is the Euler number of the compact manifold while the one-loop open string corrections take the form studied in [35].

The non-perturbative superpotential is assumed to have a racetrack dependence on , with additional poly-instantons corrections on :12

(25)

The superpotential is characterised by the constant , associated with the tree-level flux stabilisation of the dilaton and the complex structure moduli, and by the non-perturbative corrections weighted by the constants and . The parameters and are given by and and arise due to gaugino condensation on D7-branes (with and being the rank of the associated gauge group). The rôle of the racetrack form for the superpotential will become clear in section 2.6.

Even though is a rigid cycle (), single instanton contributions to the superpotential are absent, due to the presence of extra fermionic zero modes which are Wilson line modulini (). However, this zero mode structure can give rise to instanton corrections to the gauge kinetic functions of the field theories living on : this is the origin of the so-called ‘poly-instanton’ corrections which depend on . These tiny non-perturbative effects could be killed by the presence of chiral intersections between the E3 instanton on and visible sector fields living on the cycle that intersects . We have however chosen the gauge fluxes appropriately so to cancel these chiral intersections. Moreover, this set-up would normally generate -dependent loop corrections, which can in principle render this blow-up mode heavy. However, for appropriate choices of fluxes, the loops depend on the same combination of four-cycles fixed by the -terms, and so a flat direction remains in the (, ) plane. We discuss this issue in the next subsection.

2.5 D-term stabilisation

We recall that is the rigid four-cycle supporting a GUT or MSSM-like model in terms of wrapped D7-branes. It has been shown in [36] that the cycle supporting chiral matter cannot get any non-perturbative correction, since an instanton wrapped around would generically have chiral intersections with visible sector fields. Therefore the corresponding modulus has to be fixed using different effects [38]. This is what we are going to analyse here, exploiting the field dependence of the -term potential, and of subleading (but crucial) string loop contributions.

The world-volume flux generates a modulus-dependent FI-term . The resulting -term potential provides the leading order effect that depends on and :

(26)

where the gauge coupling is given by . In the expression (26) we include also the possible presence of canonically normalised visible sector singlets (open string states) with corresponding charges given by .

We focus on supersymmetric minima where , for the following reason. The total scalar potential includes also -term contributions from the matter fields:

(27)

where the are positive numbers, and denotes the scalar potential for the remaining Kähler moduli that we will analyse in detail in the next sections. Given that in a LARGE volume expansion whereas (as we shall see in the next section), a non-vanishing would give rise to a dangerous run-away behavior for the volume mode, that must be avoided.

Hence we shall look for non-supersymmetric minima where is vanishing up to corrections. The leading order cancellation of the FI-term can be achieved in two ways. First, the case in which and ; second, the case where and , where is a visible sector singlet (like a right-handed sneutrino for example) and . The first case would fix the combination without forcing any divisor to shrink to zero size. However, the two-cycle would reach the wall of the Kähler cone, i.e. , resulting in a lack of control over the effective field theory. We need therefore to focus on the second case where a singlet acquires a non-zero VEV. The minimum for is located at (for ):

(28)

Substituting this value in the total scalar potential we find:

(29)

As we shall see in the next section, the Hubble constant during inflation scales as (setting ), whereas the mass-squared of the canonically normalised modulating field scales as . This implies that in order to have a light modulating field, the potential for this field has to be developed at order . However the potential (29) introduces a dependence on the combination at order which would render the modulating field too heavy if . Hence we have to perform a very moderate tuning of of the order . 13

After having analysed -term and matter field -term contributions to the potential for , we examine how it is influenced by string loop corrections, showing that the latter are able to fix this combination. Given that each cycle wrapped by a stack of D7-branes receives one-loop open string corrections [24, 35], one would expect effects of order which depend on both and . However, the coefficients of these corrections can be appropriately tuned so that at leading order they depend only on the combination , whereas all the other terms are very suppressed. The leading order loop correction then reads [35]:

(30)

These loop effects can indeed compete with the -dependent -term piece in (29). The complete scalar potential for the combination is then constituted by the sum of all contributions:

(31)

and it admits a minimum at . If we integrate out and place it to its minimum we are left with a volume dependent potential of the form:

(32)

that is proportional to . This shows that, as anticipated, -term, -term, and string loop contributions to the moduli potential stabilise the linear combination , leaving a leading order flat direction which will be softly lifted by polyinstanton corrections, and will play the rôle or our modulating field.

2.6 F-term stabilisation

We now study the structure of the -term scalar potential for the Kähler moduli using the Kähler potential and the superpotential of eqs. (24) and (25). We follow the discussion of ref. [29] and [26]. The scalar potential can be organised in an expansion in inverse powers of the volume:

(33)

where is a number that depends on the parameters of the model (more on this later). We now discuss the previous potential order by order in the expansion, showing that at each order it can provide stable minima for some of the moduli.

Moduli stabilisation at

After minimising the axionic direction, as usual in models based on LVS, and trading for the volume (in the limit in which for ), the leading order contribution to the -term scalar potential can be readily calculated and results: 14

(34)

The scalar potential, at this order in the expansion, contains terms with different signs associated with the racetrack structure of the superpotential. Eq. (34) does not depend on the fibre modulus [29], nor on the blow-up mode supporting the poly-instanton corrections [26]. It does not do so because the dominant contribution to the potential of arises via string loops [24] at order . As we show later, develops a potential at even more subleading order, , where is a parameter that assumes values larger than .

On the other hand, as we learned in the previous section, combined effects of -term, -term, and string loop contributions do stabilise the combination : upon integrating it out, we are left with a volume dependent contribution to the potential summarised in eq. (32), which can be simply regarded as a shift of the correction term:

(35)

We now focus on the potential (34), replacing with . In order to express the conditions for the stabilisation of and in a compact way, it is convenient to introduce the quantity . The vanishing of the first derivatives of with respect to and provides the following extrema for and (keeping the first order contributions in an expansion in and ):

(36)

where

(37)

and

(38)

with for large . We also impose that is positive in order to have a minimum in the -axion direction. The extrema associated with equation (36) correspond to stable minima both for the volume (or alternatively the field ) and the field . These formulae show that the volume is LARGE thanks to an exponential dependence on the parameters of the model.

Notice that in the case of single exponential (), (38) reduces to , implying that for (the limit of large volume we are interested in) the corrections to due to are always subleading. As we will discuss in what follows, the consistency of our approximations requires to avoid this, and so we set . We finally point out that this minimum is AdS and so additional uplifting terms are needed to uplift the potential to a nearly Minkowski vacuum (for explicit de Sitter examples see [40]).

Moduli stabilisation at order

Each cycle wrapped by a stack of D7-branes receives one-loop open string corrections [24, 35] which, as pointed out in [29], generate a subleading potential for . This potential, for a certain range of , results flat enough to be suitable for inflation. In this subsection we focus only on the and -dependent loop corrections which can be estimated using a procedure identical to [29]:

(39)

where , , are given by:

(40)

where , , and are constants that depend on the VEV of the complex structure moduli which have been fixed at tree level (see [29] for more details). In what follows we regard these constants as free, to be fixed using consistency or phenomenological requirements. When , the minimum for is at:

(41)

justifying the scaling of (39). In the following, for definiteness, we consider the case .

Moduli stabilisation at

The leading order contribution to the scalar potential from the -dependent poly-instanton corrections turns out to be more suppressed than the ones analysed in the previous subsection since it scales as [26]:

(42)

where we have neglected -independent pieces in , since they are not relevant for moduli stabilisation. Substituting (36) in (42), we obtain a very compact expression

(43)

where