Modulated Reheating and Large NonGaussianity 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 lowenergy 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 nongaussianity larger than obtained in singlefield slowroll 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 TypeIIB 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 blowup mode (which is made light by the use of polyinstanton corrections). We find the challenges of embedding the mechanism into a concrete UV completion constrains the properties of the nongaussianity that is found, since for generic values of the underlying parameters, the model predicts a local bispectrum 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 Introduction
Although initially motivated as a solution to the initialcondition problems of standard BigBang 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 latetime consequences of quantum fluctuations of the inflaton).
Until relatively recently most of these predictions were made using singlefield, slowroll 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, scaleindependent 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 latetime BigBang epoch.)
Why defy Occam?
Despite the simplicity and success of the singlefield framework, two independent lines of reasoning suggest exploring multifield 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 nongaussianity down to a level that can be produced by many types of multifield models [8, 9] (though not by vanilla singlefield slowroll scenarios). It behooves us as a field to be alive to the kinds of physics to which these measurements might be sensitive, particularly if singlefield slowroll models were to be falsified by the discovery of nongaussianity.
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 bestdeveloped 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 multifield inflationary models in more detail. This is because close to a decade of exploration reveals that multifield dynamics is the norm for string inflation and not the exception [12]. Although it remains difficult in string theory to achieve a subHubble inflaton mass (and so also slowroll 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 multifield models arise fairly generically in this way once embedded into microscopic models — has recently motivated a more detailed study of the systematics of inflationary multifield dynamics in general [13].
Nongaussianity and postinflationary dynamics
One of the main differences between singlefield and multifield inflationary models is the wide variety of fluctuationgeneration mechanisms available in the multifield case. Whereas singlefield models must generate adiabatic curvature fluctuations, multifield 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 singlefield mechanism. In particular, the interactions inherent amongst the fields in any multifield mechanism can often generate observable amounts of nongaussianity in the CMB, making any discovery of nongaussianity into a new window into the physics responsible for generating fluctuations, and possibly into some aspects of the postinflationary regime. Most interestingly, nongaussianity produced in this way after inflation can often be distinguished observably from nongaussianity generated directly during inflation, such as by nonstandard singlefield models [3, 14].
The two bestdeveloped mechanisms for converting, after inflation, inflationary isocurvature perturbations into lateepoch adiabatic fluctuations are the curvaton mechanism [15] and modulation mechanism [16, 17, 18, 19]. One might ask whether these ever arise in stringinflationary models, given the preponderance of multifield scenarios these models have. Interestingly, they largely do not: for most stringinflation 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 singlefield 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 BigBang 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 nongaussianity at the soontobe 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 curvatonlike 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 finetuning. What must be chosen carefully, however, is the underlying extradimensional manifold, including a particular brane setup and a particular choice of worldvolume fluxes. In particular we make use of novel ingredients in string model building, such as polyinstanton [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 lowenergy potential in inverse powers of the extradimensional volume. That is, if the extradimensional volume (in string units) is , then the LVS scenario starts with the observation that for large the lowenergy scalar potential has the generic form of a sum of terms of order , and , where is the dimensionless volume, , of a comparatively small fourcycle in the geometry. For some choices of parameters this has a minimum with , which makes very large volumes natural to obtain.
The lowenergy 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 extradimensional 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 blowup mode also remains light during inflation since it is stabilised as in [26] due to subleading polyinstanton corrections [25]. The absence of singleinstanton contributions to the superpotential for this rigid fourcycle is guaranteed by the presence of extra fermionic zero modes which are Wilson line modulini. Because this blowup 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 blowup 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 blowup 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 curvatonlike contributions to primordial density perturbations.
Predictions for nongaussianity
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 nongaussianities, whose properties we analyse in some detail. In particular, we find nongaussianities of local form with a bispectrum parameter, . Although this is potentially observable by the Planck satellite, we find small trispectrum parameters, and .
The paper is organised as follows. §2 briefly reviews the framework of LVS compactifications on K3 or fibred CalabiYau threefolds. 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 blowup mode, (whose nonperturbative effects fix the volume at exponentially large values);
a second blowup mode, (which supports the visible sector);
and a third blowup mode, (which intersects and plays the rôle of the modulating field).
§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 postinflationary 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 LARGEvolume 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 CalabiYau 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 base modulus, , that mainly controls the overall extradimensional volume stabilised at exponentially large values. It is wrapped by the same stack of D7branes needed to generate the string loop potential for . This modulus is heavy during inflation, and lies on its minimum throughout this phase.

A blowup mode , an assisting field required to stabilise the volume at its minimum. Its presence is usual in LARGE volume setups. The potential depends on through nonperturbative contributions generated by a stack of D7branes wrapping and supporting a hidden sector that undergoes double gaugino condensation in a racetrack way.
^{5} It is heavy during inflation. 
Two intersecting blowup modes, and . supports a GUT or MSSMlike construction with D7branes 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 D3brane (E3) instanton which generates tiny polyinstantons 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 CalabiYau threefold
with a K3 or a fibration structure and .
The volume is expressed in terms of the twocycle moduli as [37, 38]:
(2) 
where and given that every CalabiYau 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
(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 fourcycles intersecting over a twocycle given by:
(4) 
We shall also assume that is a rigid fourcycle with Wilson lines, i.e. while (for explicit examples of this kind of divisors in this context see [37]).
The fourcycle 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 fourcycle moduli as:
(9) 
where we set , , , and .
2.3 Brane setup and fluxes
In this section we shall perform an explicit choice of brane setup 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 D7branes
wrapped around the divisors and
(10) 
We also turn on the following worldvolume fluxes:
(11) 
where the halfinteger contributions come from the cancellation of the FreedWitten 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 worldvolume fluxes:
The presence of nonvanishing gauge fluxes has four implications:

They induce charges for the Kähler moduli. The charges of the blowup moduli and under the diagonal of the D7stacks wrapping and read:
(12) These charges can be written one in terms of the other as:
(13) 
The gauge coupling of the field theory on the D7stack wrapping the divisor acquires a fluxdependent shift. In fact, the gauge coupling is given by:
(14) where is the real part of the axiodilaton, while the fluxdependent factor is:

The worldvolume fluxes generate modulidependent FayetIliopoulos (FI) terms which take the form:
(15) (16) where we denoted with the Kähler potential of the 4D effective theory.

The chiral intersections between different stacks of D7branes 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 nonvanishing, 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 nonvanishing chiral intersections between the D7stacks 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 D7stack are:
(21) 
and
(22) 
which, from the condition (19), implies . The resulting charge , fluxdependent shift and FIterm for the visible sector become:
(23) 
2.4 Supergravity effective action
Let us now outline the main features of the effective lowenergy 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 oneloop open string corrections take the form studied in [35].
The nonperturbative superpotential is assumed to have a racetrack dependence on ,
with additional polyinstantons corrections on :
(25) 
The superpotential is characterised by the constant , associated with the treelevel flux stabilisation of the dilaton and the complex structure moduli, and by the nonperturbative corrections weighted by the constants and . The parameters and are given by and and arise due to gaugino condensation on D7branes (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 socalled ‘polyinstanton’ corrections which depend on . These tiny nonperturbative 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 setup would normally generate dependent loop corrections, which can in principle render this blowup mode heavy. However, for appropriate choices of fluxes, the loops depend on the same combination of fourcycles fixed by the terms, and so a flat direction remains in the (, ) plane. We discuss this issue in the next subsection.
2.5 Dterm stabilisation
We recall that is the rigid fourcycle supporting a GUT or MSSMlike model in terms of wrapped D7branes. It has been shown in [36] that the cycle supporting chiral matter cannot get any nonperturbative 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 worldvolume flux generates a modulusdependent FIterm . 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 nonvanishing would give rise to a dangerous runaway behavior for the volume mode, that must be avoided.
Hence we shall look for nonsupersymmetric minima where is vanishing up to corrections. The leading order cancellation of the FIterm 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 righthanded sneutrino for example) and . The first case would fix the combination without forcing any divisor to shrink to zero size. However, the twocycle 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 nonzero 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 masssquared 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 .
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 D7branes receives oneloop 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 Fterm 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:
(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 blowup mode supporting the polyinstanton 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 D7branes receives oneloop 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 polyinstanton 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