Towards Natural Inflation in String Theory
We provide type IIB string embeddings of two axion variants of natural inflation. We use a combination of RR 2 form axions as the inflaton field and have its potential generated by non perturbative effects in the superpotential. Besides giving rise to inflation, the models developed take into account the stabilization of the compact space, both in the KKLT and large volume scenario regimes, an essential condition for any semi-realistic model of string inflation.
Recent observational progress has drastically transformed cosmology into a quantitative science Hinshaw:2012fq ; Ade:2013zuv ; Hou:2012xq ; Sievers:2013wk . From these measurements we derive increasingly strong evidence for cosmological inflation, a very early epoch of accelerated expansion lasting for about 60 e-foldings of the scale factor.
Recently, the BICEP2 collaboration reported the first measurement of B-mode polarization of the CMB at large angular scales Ade:2014xna . If this result stands after further corroboration and turns out to be primordial, then in the context of inflation it corresponds to a detection of primordial gravitational waves with a tensor-to-scalar ratio . Further results quantifying the polarized foreground emissions from e.g. galactic dust along the lines of Ade:2014gna ; Flauger:2014qra ; Mortonson:2014bja will help settle this question in future.
The amount of e-folds of slow-roll inflation
can be related to the tensor-to-scalar ratio, , via the Lyth bound
If we take the B-mode detection for the time being as signalling a primordial inflationary tensor modes with , then the Lyth bound Lyth:1996im ; Boubekeur:2005zm tells us that the field excursion during inflation was super-Planckian: 111The field excursion can be reduced to sub-Planckian, if we allow for a non-monotonic , BenDayan:2009kv . These types of models have received revived interest due to the BICEP2 result. We wish to note that there is no problem with having enough e-folds even with and at the pivot scale per se. The actual limitation is that a smaller field excursion implies a faster change in which in turn implies a larger deviation from nearly scale invariance, which is harder to accommodate given the now e-folds measured by PLANCK BenDayan:2009kv ; Hotchkiss:2011gz ; Hebecker:2013zda .
Inflation is known to be sensitive to the high-scale effects of a possible UV completion of the low energy theory, however large and small field models are affected to different extents by this UV sensitivity. In small-field inflation clearly requiring the tuning of dimension-6 operators to avoid contributions to the slow-roll parameter . To see this note that a generic inflationary model will contain dimension-6 operators of the type . In a small-field model the starting-point inflaton potential necessarily has the form
at small field values . Hence, corrects by an value, destroying slow-roll.
Large-field inflation in contrast is UV sensitive to an infinite series of dangerously irrelevant operators. Clearly, we must appeal to a protective symmetry to save the inflation direction in the scalar potential. This will almost by definition amount to an effective shift symmetry which is broken at leading order by the inflationary scalar potential itself. While this notion of a protective shift symmetry in large-field inflation is certainly natural in the bottom-up Wilsonian sense (corrections from self-interactions of inflaton fluctuations die out at large field values, and quantum Einstein gravity correction scale ), realizing such a shift symmetry and establishing control over its breaking clearly requires high-scale information mandating the embedding into a theory of quantum gravity.
At present there is not a unique and well understood theory of quantum gravity. However, string theory constitutes the most prominent candidate, with many non-trivial intricate results concerning its mathematical structure, the right low-energy field content to potentially accommodate our local universe, and a successful microscopic description of a large part of black hole physics. This provides a clear motivation to study inflation and in particular its large-field varieties in string theory. The requirement of realizing a well-respected shift symmetry typically leads us to consider the many string theory axions from higher-dimensional -form gauge fields, or their mirror-dual partners of complex structure moduli, as good inflaton candidates. However, many sectors of the theory display a periodicity under shift of the -form axionic fields while the kinetic terms of these axions
for 10d to 4d compactifications with volume in string units. The periodicity range of the canonically normalized axion field
is sub-Planckian in controllable compactifications requiring large volume and weak string coupling Banks:2003sx . This discrete shift symmetry with sub-Planckian period is broken by the presence of quantized -form fluxes or (dual) brane configuration, which unwrap the discrete shift symmetry into a system of multiple non-periodic branches: the full system with fluxes or branes shows monodromy in the potential energy of the axion on each branch, while periodicity is retained when summing over multiple branches Silverstein:2008sg ; McAllister:2008hb ; Kaloper:2008fb ; Kaloper:2011jz ; Palti:2014kza ; Kaloper:2014zba ; Marchesano:2014mla ; Blumenhagen:2014gta ; Hebecker:2014eua ; Grimm:2014vva ; Dine:2014hwa ; McAllister:2014mpa .
An alternative to monodromy proper arises in the presence of at least 2 axions Kim:2004rp ; Berg:2009tg ; Choi:2014rja ; Higaki:2014pja ; Kappl:2014lra ; Ben-Dayan:2014zsa ; Tye:2014tja ; Long:2014dta ; Gao:2014uha ; Li:2014lpa . Non-perturbative effects provide cosine potentials with typical sub-Planckian periodicities and
An alignment of the Kim:2004rp or a hierarchy like e.g. with Ben-Dayan:2014zsa ; Tye:2014tja (a fully non-perturbative variant of Berg:2009tg ) drives the emergence of a mass hierarchy in the axionic sector, which is translated into the appearance of an effective super-Planckian axion decay constant . The arising effective single-field inflaton potential realizes the original idea of natural inflation Freese:1990rb .
A crucial step for any construction of large-field inflation in string theory consists of showing the compatibility of the shift symmetry and field-range extension mechanism with the process of moduli stabilization. The moduli potential tends to back react on the inflationary vacuum energy. This generates corrections to the inflaton potentials, which energetically lead generically to flattening of a naive potential shape as inferred from the pure large-field mechanism itself Dong:2010in . Generically, the inflationary vacuum energy may very well participate in moduli stabilization which often can enhance the stability of the compactification while showing the flattening effect Dong:2010in ; McAllister:2014mpa . If we restrict ourselves to the supersymmetric setup provided by Calabi-Yau flux compactification, stabilization of the volume moduli often requires non-perturbative effects. For this reason, CY compactifications often lead to problems with having the inflationary vacuum energy participating in moduli stabilization, requiring a (moderate) separation of scales between the moduli potential and the inflaton sector. This, however, is an artifact of our restricting to CY compactifications in the first place.
In this paper we are discussing several methods of embedding an aligned or hierarchical axion potential of the type (7) into type IIB string theory compactified on CY manifolds with 3-form flux fixing the complex structure moduli and the axio-dilaton Giddings:2001yu . Due to the arguments discussed above, and those expressed in section 4.1, we will restrict ourselves to the use 2-form R-R sector axions which provide a rather well-protected shift symmetry in the context of type IIB on CY orientifolds with O7 planes and D3/D7 branes Grimm:2004uq ; Grimm:2007xm ; McAllister:2008hb .
The paper is organized as follows. In section , we discuss natural inflation models in the two axions case. We will demonstrate that Kim:2004rp ; Ben-Dayan:2014zsa ; Tye:2014tja ; Berg:2009tg actually stem from the same origin. We embed such models in a supergravity (SUGRA) framework in section . In section we discuss their string theoretic derivation. We will study both the KKLT mechanism Kachru:2003aw and the large-volume scenario Balasubramanian:2005zx for combining Kähler moduli stabilization with non-perturbative effects from gaugino condensation on -branes or instanton from Euclidean D3-branes providing the axion potential for the 2 R-R sector axions. Finally we conclude in section .
2 Natural inflation from two axions
A common origin
In this section we discuss the two mechanisms that generate effective super-Planckian decay constants from fundamental sub-Planckian ones, following the works Kim:2004rp ; Ben-Dayan:2014zsa ; Tye:2014tja . We show that actually both models (as well as Berg:2009tg ) come from the same origin and only correspond to different deformations of the underlying potential, or different breaking of the same shift symmetry.
Consider a two axion potential of the form
where the axions and have canonical kinetic terms and all decay constants are sub-Planckian: .
We start our discussion by imposing an alignment condition on the above scalar potential which ensures the presence of flat direction by arranging for . Then the determinant of the second derivative matrix vanishes everywhere, signalling the possibility of a flat direction 222In Mathematics is called the Monge-Ampère equation. Starting from such an equation is much more general than our discussion here. The Monge-Ampère equation provides an excellent starting point for model building and model classification. We intend to return to this in future work.. Hence all that is left is to find a region where the gradient is small, and we have the flat region desired for inflation, provided that the other eigenvalues of are positive. This happens naturally at the origin of field space in (8). Slow-roll inflation requires deforming the flat direction slightly. We can achieve this in two non-equivalent directions away from the flat limit – either we relax the alignment condition in the spirit of Kim-Nilles-Peloso, or we introduce a subdominant scalar potential providing the slowly varying deformation away from flatness as in Dante’s inferno and Hierarchical Axion inflation. We will now discuss both possibilities in more detail.
We start by discussing the Kim-Nilles-Peloso (KNP) mechanism. As was mentioned above, in the perfect alignment limit the mass matrix is singular. With inflation model building in mind one deforms this condition to allow for small misalignment:
This has the effect of lifting the flat direction in a way suitable for slow-roll inflation provided is sufficiently small. One must note that for sufficiently small , the alignment mechanism holds for generic values of the various parameters in Eq. (8), as can be seen from the fact that the determinant of the mass matrix
becomes singular in the limit of perfect alignment (), signalling the presence of a flat direction.
To leading order in , the mass eigenvalues are
which keeping in mind that the axions’ masses are of the form implies that the large effective decay constant, corresponding to the eigenvalue , scales as
And so, by considering two almost aligned axions with originally sub-Planckian decay constants, one generates an effective super-Planckian decay constant as is required for natural inflation for the price of tuning an alignment of the original axion decay constants. This extension of the field range without requiring a super-Planckian fundamental domain constitutes the main advantage of this model, when compared with the original single–cosine realization of natural inflation Freese:1990rb .
The Hierarchical Axions (HA) model (and its close cousin Dante’s Inferno) corresponds to setting in (8). By definition there is no alignment whatsoever and the deformation of the potential that will allow for inflation corresponds to introducing a hierarchy in the decay constants by having . The mass matrix at the global minimum is such that
so in the limit an exact shift symmetry is recovered and for , the symmetry is broken such that inflation ends at a stable minimum at the origin, provided that
The effective axion masses in the limit are
and so we see that the mass spectrum is hierarchical, with . Integrating out the heavy mode results in a single effective axion potential with an effective decay constant:
The model generates super-Planckian effective decay constants similarly to KNP, but replaces the tuned alignment with a hierarchy between the decay constants. It further keeps the entire inflationary analysis in a sub-Planckian domain avoiding the functional fine-tuning necessary in large field models. The advantages of the model as laid out in Ben-Dayan:2014zsa are threefold: i) utilizing only non-perturbative effects, ii) the smallest number of axions, iii) and the least amount of tuning of the input parameters. These advantages make the model specifically tractable from the string theoretic point of view.
Both the KNP alignment and the Hierarchical Axions mechanism produce an effectively single-field inflaton potential of the form with an effective enhanced field range . Hence, its observational predictions for the spectral index of curvature perturbation, and the tensor-to-scalar ratio agree with those of natural inflation itself Kappl:2014lra . In particular, for the scalar potential in the 60 e-fold range approaches that of -inflation with and , while for the model becomes of small-field type with growing more red and leaving the Planck 95% region, while the tensor-to-scalar ratio drops to (currently) unobservable levels ( for ).
3 Supergravity embeddings
In this section we provide explicit ways to build the field theory models Kim:2004rp ; Ben-Dayan:2014zsa ; Tye:2014tja described above into supergravity. The goal of this approach is to provide a stepping stone to the realisation of these inflationary models in string compactifications of type IIB string theory which we present in section 4.
The minimal model requires two chiral multiplets whose dynamics are determined by the canonical Kähler potential
and the non-perturbatively generated superpotential
In the case one has the KNP alignment mechanism giving rise to a super-Planckian axion, while if one chooses from the start inflation will proceed through hierarchies in the parameters . The structure of the scalar potential will be similar in both cases and so we analyse them simultaneously whenever possible.
The scalar potential can be made to have a hierarchy between the terms stabilising the real parts of and and those generating the inflationary potential, thereby decoupling the heavy fields from the inflationary dynamics.
The F term potential
can be written as
For the sake of short formulae we have defined the and dependent quantities
Both in the KNP and the hierarchical regimes we focus on regions of parameter space where so that one can stabilise at high scale before analysing the inflationary potential. This hierarchy in the superpotential descends into the scalar potential: . In these simple supergravity models the need for this hierarchy derives from the desire to analytically minimise the potential in a controlled way and it is not a fundamental requirement of these models since we expect the alignment/hierarchical mechanisms to produce a super-Planckian direction even in the absence of this hierarchy in . This situation will change once we consider stringy embeddings of this idea, as we will see in section 4, since this tuning will be related to parametric decoupling of the moduli vacuum.
We note in passing, that more generally parametric decoupling is an overly conservative criterion which may me relaxed in concrete string embeddings. The moduli potential may back react appreciably during the inflaton evolution, which due to energetic reasons generically leads to a flattening of the inflaton potential expected from the parametrically decoupled limit Dong:2010in . In particular, if perturbative high-scale mechanisms serve to fix all the moduli, inflation may even participate and help with moduli stabilization allowing for significant yet controllable flattening effects McAllister:2014mpa . The non-perturbative mechanisms of volume stabilization we use here are more sensitive to backreaction effects. This limits the amount of controllable flattening achievable, and motivates us to restrict ourselves to the limit of parametric decoupling for the sake of explicitness.
Note, that the combined minimum of the three cosine terms above and the moduli potential usually is an AdS vacuum. As discussed in many of the dS vacuum constructions in string theory in recent years, we need to add an uplifting contribution from e.g. an anti D3-brane Kachru:2003aw , D-terms with field-dependent FI terms Burgess:2003ic ; Cicoli:2012vw or dilaton dependent non perturbative effects Cicoli:2012fh
to the scalar potential to lift the AdS minimum to a near Minkowski state . Provided, we already arranged for sufficient hierarchy between the moduli masses and the axion mass scales arising from the three cosine terms in in the prior AdS vacuum, this will survive an uplifting term of the above type. Hence, we will from now on tacitly assume the presence of such an uplifting term in our setups, which justifies the form for the three cosine terms arising from the moduli potential.
Even though this will alter the expressions for the mass eigenvalues, which will receive dependent contributions, the existence of a mass hierarchy remains and the large effective decay constants are still given by Eqs. (12) and (16) for the KNP and HA cases respectively.
Noting that depends only on the combination we see that the tree level action stabilises at
and so must lie in a circle of radius centred at the origin of the plane. This is illustrated in figure 1. At this level there is still one flat direction left in the plane as well as two in the . Since is independent from this result applies to both the KNP and the hierarchical axion mechanism. The flat directions will be lifted by the next-to-leading order contribution to the potential, , which we analyze separately in the two regimes.
3.1 KNP alignment mechanism
We start by studying the vacua of in KNP in the limit of perfect alignment, , where we know there will be one unfixed direction in the -plane. We will then follow Kim:2004rp and allow for a slight misalignment which will lift the remaining flat direction in a way suitable for inflation. This procedure is more subtle here than in the field theory case since the leading contribution to the potential, leaves the angular direction in the plane unfixed. One therefore has to ensure that by allowing for a slight misalignment in the decay constants in order to realise inflation, one is not simultaneously destabilising the angular direction in the .
In the alignment limit we have
where for algebraic simplicity we have set and defined . This potential will simultaneously stabilise and the axionic combination . The minimum is located at and . This implies that the vacuum in the -plane lies at the intersection of the circle and the straight line passing through the origin . By construction, in this limit, the same linear combination of appears in the potential and so the orthogonal combination is exactly flat.
The introduction of a slight misalignment in the axionic decay constants,
which will generate the inflationary potential, will also perturb the b-vacuum. This can in extreme cases lead to the destruction of the b-minimum or in more mild ones lead to a shift in the minimum’s position. Since for inflationary purposes , the b-vacuum survives the introduction of a misalignment, with its position shifting by a small factor proportional to the smallness of the misalignment of the decay constants :
This fixes the remaining flat direction in the plane in the misaligned regime in a similar way to what was described before. The only difference is that the straight line intersecting the circle in the -plane no longer passes through the origin. The effect of the misalignment in the stabilisation of the real part of the fields and is therefore negligible.
The potential for the axions then admits the following expansion
which coincides with the field theory model of Eq. (8), with the identifications and .
3.2 Hierarchical axions mechanism
Just like in the KNP case, for the hierarchical axion inflation model, the potential is generated by Eq. (22), but now with . Anticipating that inflationary model building will require we can expand as
is simultaneously responsible for stabilising the angular direction in the b-plane (or equivalently stabilising ) and giving rise to inflation. To leading order in a expansion, we find that , implying that the b-vacuum is the same as in the aligned KNP regime. In this case the potential then simplifies to the desired form
One then concludes that the model of Eqs. (17), (18) does indeed provide a supergravity description of the two axion versions of natural inflation of Kim:2004rp and Ben-Dayan:2014zsa ; Tye:2014tja . Generically this did not have to be the case since the structure of supergravity requires a model that initially involves four fields: 2 axions and 2 saxions/moduli, whose masses are closely linked. What we have shown is that the mechanism that allows for the generation of a potential suitable for inflation simultaneously guarantees that there is a mass hierarchy between the lightest axion and all the saxions/moduli. When trying to embed these models in string compactifications we will therefore be looking for the generic structures of (17), (18), the adequate mass hierarchies and tuneable parameters.
4 Natural inflation in string compactifications
4.1 Axions in string compactifications
We confine our discussion to GKP-type compactifications of type IIB string theory on O3/O7 orientifolds of warped Calabi-Yau 3-folds with 3-form NS-NS and R-R flux. We assume a choice of 3-form fluxes such that they stabilize the complex structure moduli and the type IIB axio-dilaton supersymmetrically at a high mass scale while generating an effectively constant superpotential Giddings:2001yu .
For the stabilization of the Kähler moduli we consider non-perturbative stabilization a la KKLT Kachru:2003aw involving gaugino condensation on D7-brane or D5-brane stacks or Euclidean D3-brane (ED3) instantons, or the Large Volume Scenario (LVS) involving a combination of the leading -correction to the CY 3-volume and an ED3-brane instanton or D7-brane stack.
At the level prior to imposing the O7 projection, the Kähler moduli sector consists of 2-cycle moduli . Here the denotes the NS-NS 2-form axions arising from the NS 2-form -field on the 2-cycles , while the denote the 2-cycle geometric volumes in string units. The tree-level Calabi-Yau volume is then given by
Imposing the O7 projections projects the Käbler moduli sector into an O7-even and odd subspaces with respective dimensions and . Moreover, this forces a rearrangement of the real scalars into 4-cycle moduli
and 2-form axion multiplets
where the type IIB axio-dilaton is and we have O7-odd 2-form R-R and NS-NS axions
For the case of a single Kähler modulus and odd 2-form axion moduli we can invert the relation between and and acquire . This gives a 4d Kähler potential of the Kähler moduli (see e.g. Lust:2006zg )
Guided by this example, we conjecture that for CY manifolds with a volume of swiss-cheese form with Kähler moduli we may find cases where the Kähler potential takes the form (see e.g. Gao:2013rra )
The swiss-cheese type 4-cycle intersection numbers then are positive . Due to the mixing between the and in the results for the full -sector Kähler metric and its inverse are bit lengthy. The full expressions are given in eq.s (C.11) and (C.12) of Appendix C in Grimm:2004uq .
We can now see from the form of e.g. Eq. (38) that the projection dictated by the O7-action leads to a breaking of the shift symmetry which the -axions enjoyed at the 4D level. If moduli stabilization proceeds via terms in the superpotential, then must be a holomorphic function of the chiral superfields. Hence, superpotential stabilization of the moduli generically stabilizes the and separately. The manifest dependence of on the -axions being hence breaks the shift symmetry of the -axions. This fact renders the NS-NS 2-form axions unsuitable for large-field inflation in these constructions.
If volume moduli stabilization had proceeded directly via stabilizing some of the geometric 4-cycle volumes e.g. by corrections to , this could have restored the shift symmetry for the involved. However, we will not pursue this opportunity here, and in our constructions the will always appear in non-perturbatively.
This leaves the and R-R axions as potential inflaton candidates. Non-perturbative stabilization a la KKLT of the Kähler moduli then implies that the -axions acquire the same mass scales as their moduli partners . In the Large Volume Scenario we require only a fraction of the to appear in via non-perturbative effects. For those which do not appear in but get fixed by Kähler corrections, this allows to split the mass scale of their -axion partners from their moduli mass scale Cicoli:2014sva . In such cases, we may use -axions as inflaton candidates.
Generically, however, we find the -axions are least coupled to the process of volume stabilization regardless whether this proceeds non-perturbatively in or perturbatively in . Hence, we focus on 2-axion models of large-field inflation driven by the R-R axions of two multiplets.
4.2 Embedding into string compactifications
In this section we try to build explicit models of the KNP and HA mechanisms using the orientifold odd axions of string compactifications. This analysis is a little more involved than the supergravity case since not only do we have to generate a suitable potential for the axions, but we must also stabilise the geometry of the compact space. To achieve this we combine the KKLT or LVS setups with superpotential terms originating from gaugino condensation on D5 branes Grimm:2007xm ; Grimm:2007hs :
where can be chosen as needed between the even moduli in each setup.
4.2.1 Inflating in KKLT
In the simplest case of KKLT moduli stabilisation, we consider a single even modulus model, such that the volume of the compact space is given by , where is the 2-cycle volume. Allowing for non-zero intersection between the orientifold even and the odd sectors, one can write the Kähler potential in terms of the Kähler coordinates as
The kinetic part of the scalar field Lagrangian then reads
where denotes the chiral superfield basis, , and the real scalar field basis of moduli space . The kinetic matrix for the real degrees of freedom, ignoring the dilaton for the moment, admits the following expansion
where we keep only the leading terms in in each diagonal entry and neglected off-diagonal terms333 The full expression of the kinetic matrix up to order is somewhat complicated and not very illuminating. By presenting only the leading diagonal terms one can understand the behaviour of the eigenvalues of in terms of the intersection numbers and and get a simple estimate of their order of magnitude.. We then see that requiring positivity of leads us to consider compactifications with . If these conditions are not met, the multiplets become ghosts.
A successful string inflation model must not only give rise to inflation but also be able to keep the non-inflationary moduli fixed. In the current context this requires a separation between the physical mass scales of the , and fields and the and axions. This scale separation is also a prerequisite for the classical stability of the vacuum: after uplifting the KKLT vacuum of is separated from decompactification by a barrier with a height . Since any inflationary energy density constitutes an extra form of uplifting, one must have . The twin requirements of scale separation and vacuum stability therefore impose the following hierarchy on the scalar potential
Provided this is met one can minimise and separately, which constitutes a considerable simplification in the search for the F-term potential’s vacuum.
The leading contribution to the potential, , depends only on the volume modulus and on the quadratic combination , and is essentially the generalisation of the KKLT potential for compactifications with orientifold odd axions intersecting the volume modulus. Extremising we find that the KKLT minimum is approximately located at
As in the supergravity versions of these two axion models, the real partners of the axionic fields are Kähler stabilised at leading order in a circle of fixed radius, determined by the solution to 444Due to the structure of the potential we have not been able to find an analytic expression for the vev of that gave a good agreement with the numerical results while still being compact enough to be spelled out here. We therefore proceed with the analysis numerically, keeping in mind that in the cases of interest one finds , in accordance with the requirement , derived from the abscence of ghosts., with the angular direction unfixed.
The component of the scalar potential depends both on and as well as on various linear combinations of and and on the axions, in both the KNP and hierarchical axions scenarios, and so will lift the remaining flat direction in the -plane. Unfortunately, the structure of the potential complicates the minimisation process as soon as one moves away from the (tachyonic) origin of the b-plane. This renders our efforts to find analytic expressions for the location of the vacuum futile and forces us to resort to numeric methods. In any case, the qualitative picture is identical to that of the supregravity models described in the previous section, and at the end of the process one ends up with all the moduli, volume included, stabilised in a consistent way.
KNP alignment mechanism
At the KKLT minimum, the dominant contribution to is
which together with the cosine term from , Eq. (46), constitute the inflationary potential .
Defining the misallignemt parameter in terms of the superpotential parameters as
one can map this stringy model onto the field theory analysis of section 2. The potential for the canonically normalised -axions
can then be written, upon uplifting, as
From Eq. (43) one also defines
and so one finds that the large effective decay constant is then written in terms of the compactification parameters as