Kähler moduli inflation and WMAP7
Abstract:
Inflationary potentials are investigated for specific models in type IIB string theory via flux compactification. As concrete models, we investigate several cases where the internal spaces are weighted projective spaces. The models we consider have two, three, or four Kähler moduli. The Kähler moduli play a role of inflaton fields and we consider the cases where only one of the moduli behaves as the inflaton field. For the cases with more than two moduli, we choose the diagonal basis for the expression of the CalabiYau volume, which can be written down as a function of fourcycle. With the combination of multiple moduli, we can express the multidimensional problem as an effective onedimensional problem. In the large volume scenario, the potentials of these three models turn out to be of the same type. By taking the specific limit of the relation between the moduli and the volume, the potentials are reduced to simpler ones which induce inflation. As a toy model we first consider the simple potential. We calculate the slow roll parameters , and for each inflationary potential. Then, we check whether the potentials give reasonable spectral indices and their running ’s by comparing with the recently released sevenyear WMAP data. For both models, we see reasonable spectral indices for the number of efolding . Conversely, by inserting the observed sevenyear WMAP data, we see that the potential of the toy model gives requisite number of efolds while the potential of the Kähler moduli gives much smaller number of efolding. Finally, we see that two models do not produce reasonable values of the running of the spectral index.
1 Introduction
What has been a big challenge in theoretical physics in recent decades is to explain our present universe. It has been observed that the universe is expanding, especially with acceleration. Thus, the accepted standard big bang theory should be supplemented by inflationary period [1, 2] to extrapolate from the observed current universe to the early universe. Therefore, the inflation has been one of the most important paradigm in modern cosmology. The inflationary model was first introduced in Ref.[3] and developed by many others [4]. In the absence of quantum gravity, these models are effective field theories minimally coupled to scalar fields with Einstein gravity. The problem is that the choice of inflaton and its potential is rather ad hoc. Solving this problem requires studying some fundamental theory such as string theory. It seems very reasonable to investigate the inflationary model from the point of a fundamental theory such as string theory. In this top down approach, we have to find the inflaton field and have a nice potential such that the slow roll parameters , , and are small enough to allow enough number of efolding. In general, it is complicated to work out the potential. However, luckily, in type IIB string theory, several examples including nonperturbative potential were worked out in great detail.
The string theory, which is the leading candidate for the quantum gravity, has been an active research area recently for the application to cosmology [5, 6]. However, the four dimensional theory compactified on six dimensional CalabiYau manifold accompanies lots of massless fields called moduli. Recent study resolves this problem by turning on fluxes appropriately [7]. However, by turning on the fluxes alone, the vacuum of the effective four dimensional theory just becomes supersymmetric anti de Sitter (AdS) spacetime. Based on this approach, the metastable de Sitter vacua was constructed by Kachru, Kallosh, Linde and Trivedi (called KKLT scenario) in Ref.[8]. They obtained de Sitter vacuum by breaking supersymmetry with the addition of antibrane and by including a uplift term. Both of these effects uplift the negative vacuum energy to positive vacuum energy. Along the KKLT scenario, the application to inflationary cosmology was triggered by the work of Kachru et al.[9]. Even though we have the quite successful de Sitter vacuum, the effective fourdimensional theory compactified on CalabiYau space may give rise to many degenerate vacua often called landscape. In other words, different inflationary scenarios can be realized in different regions of the string theory landscape. Therefore, finding the de Sitter vacua is not unique, and there can be many attempts to do it.
In string theory there are various moduli that can be interpreted as inflaton: the first one is the distance between the branes (Dbrane inflation [10]) or between brane and antibrane [11]. In this case the effective mass is often too big to give enough efolds. The second one is the one from geometric moduli such as Kähler moduli which are the fourcycle inside the CalabiYau. In Ref.[12], the Kähler moduli inflation in type IIB string theory was studied based on so called large volume scenario [13]. It is required that at least two moduli be needed for the inflationary model. The role of axionic field in Kähler inflation was studied in Ref.[14]. Any realistic string theoretic model of the inflation, formulated in terms of an effective four dimensional supergravity, must have all moduli (axiondilaton, complex structure, Kähler moduli, and brane positions) stabilized, and have at least one inflaton, with a potential flat enough to provide a slowroll evolution. There are other models in string theory discussing cosmological scenarios: brane gas model [15], prebig bang scenario [16], rolling tachyon [17], ekpyrotic [18], Dterm [19, 20], and racetrack inflation [21].
The reason we focus on the Kähler moduli is that the potential has almost flat direction and both moduli stabilization and flatness of the potential are achieved by the same mechanism. In the case of the large volume scenario of the flux compactifications, these are valid for a very large class of models. The problem in studying models with multiple moduli is that the field space is multidimensional, and the trajectory of the inflaton field during inflation can be quite complicated. In the particular model with four moduli, we study model [22]. This model is one of Swisscheese models. The volume of the CalabiYau manifold is determined by the biggest cycle of the manifold and the small fourcycles make holes inside of CalabiYau manifold: the volume of the CalabiYau manifold has a form like ’s are fourcycles. We deal with the multidimensional problem by choosing the diagonal basis and then pick one of the moduli as an inflaton. where
This paper is organized as follows: In section 2, we review flux compactification in type IIB string theory including the large volume scenario and also discuss the string one loop correction to the potnetial. The flux stabilizes the axion field, the complex structure, and the dilaton moduli. The nonperturbative effect stabilizes Kähler structure moduli. In section 3, the derivation of the inflationary potential from large volume scenario is reviewed. The potential obtained in large volume scenario is reduced to simpler one when the Kähler moduli has a special limit. In section 4, we apply the paradigm of section 3 to Swisscheese models with two, three, or four Kähler moduli. We first consider the simplest model, , which has two Kähler moduli. Next, we study two models with three moduli, Fano [23] and [24]. Finally, the model with four moduli, [22], is investigated. For more than two moduli, we take the diagonal basis for the volume of the the CalabiYau manifold. This makes it simpler to handle the analysis for finding the inflaton field and studying the inflation. When we consider the inflationary potential, all the above models are reduced to the same form of potential. In section 5, we study two cases as inflationary models. In the first case, we study the inflationary potential starting with a toy model. In the second case, we study Kähler moduli inflation with canonically normalized inflaton field. We calculate the number of efolding, , and slow roll parameters, , , and , for each model which make it possible to find the spectral index and its running. For both cases, we find that the slow roll parameters can be expressed as a number of efolding. For the values of the allowed number of efolding , the slow roll parameters have reasonable values. Then, we compare the spectral index and its running with the sevenyear WMAP data. We see that the spectral indices for both cases fit the sevenyear WMAP data while the results of the running do not fit the sevenyear WMAP data. We conclude with the discussion.
2 A short review of the type IIB flux compactification
When we compactify the tendimensional string theory on real sixdimensional CalabiYau manifold, the effective fourdimensional theory has lots of massless fields called moduli. In nature, however, since these moduli fields have not been detected observationally, we need to resolve this problem. One way to give them masses is to induce the potential. In recent study [7], it was shown that the moduli fields can have masses by turning on fluxes. In the present work, we will work in type IIB string theory compactified on CalabiYau orientifolds, with RR and NSNS 3form fluxes by and , respectively [7]. See [25, 26, 27] for a review. The 3form fluxes are quantized as
(1) 
where and are 3cycles in the internal CalabiYau space and is related to string length such that . Ignoring the gauge sectors, the theory is specified by the Kähler potential and superpotential . The superpotential does not depend on the Kähler moduli and takes the form
(2) 
where 3form is given by . Here, the axiondilaton field is given by where and are dilaton and RR 0from, respectively. The is the holomorphic (3,0) form of the internal CalabiYau manifold. The Kähler potential up to leading order in string coupling and is given by
(3) 
where the volume of CalabiYau, , is given by . In supergravity, the scalar potential is known as
(4) 
where the Kähler metric is defined by and run over all moduli. Moreover, the derivative is defined by . The superpotential is independent of the Kähler miduli, and the term is canceled [7]. Hence, this scalar potential is reduced to
(5) 
where and run over dilaton and complex structure moduli. The minimum of this potential is found by solving
(6) 
The correction to Kähler potential was obtained in [28] :
(7) 
where and is Euler number of . We require and the hodge numbers and should have the relation such that .
The superpotential receives no corrections. However, it receives nonperturbative correction of Kähler moduli through D3brane instantons or gaugino condensation from wrapped D7branes. It takes the form
(8) 
where is a oneloop determinant. The Kähler moduli is defined by where is the volume of a 4cycle in , and is given by where is RR fourform. For D3brane instanton, only depends on the complex structure moduli and with and for D3instantons. In the following, we consider .
After fixing or integrating out the dilaton and complex structure moduli, the Kähler potential becomes
(9) 
Here, and where and are dimensionless numbers. Substituting two equations in (9) into (5), we get
(10) 
There are two stages for large volume AdS minimum of the potential (10):

The fourcycle volumes .

for large and the potential approaches zero from below.
At large volume , the correction term proportional to will dominate over the nonperturbative terms and . The nonperturbative terms which are exponentially suppressed can compete with perturbative terms. When we consider the limit , the moduli are taken to have a limit except the smallest one denoted by with the relation . As a result we get the potential in the large volume limit:
(11) 
where is an intersection number and are twocycle volumes of the CalabiYau manifold.
In [29, 30, 31], the stringy one loop correction in string coupling was considered. There are two types of contributions to the Kähler potential, coming from winding mode, , and KaluzaKlein mode, . The correction term to the potential can be written as
(12) 
where and are unknown functions on the complex structure moduli. After stabilizing the complex structure moduli, these become unknown constants. This one loop correction plays an important role in stabilizing the moduli for the cases with three Kähler moduli in the following section.
3 The Kähler moduli inflation
In this section, we consider inflationary model with the potential arising from the flux compactification of type IIB string theory. The various Kähler moduli play a role of inflaton field. We focus on the model that only one of Kähler moduli plays as an inflaton field. During the inflation, the size of the bulk six dimensional CalabiYau volume is supposed to be fixed while the variation of the internal small fourcycle induces an inflation. We are assuming that the inflaton field rolls from a point which is far from the the stable point. Inflation will ends when the inflaton reaches the stable point where the moduli get fixed.
In the following, we will consider the model of which the volume of the CalabiYau can be calculated. First, as an illustration, we will consider simplified CalabiYau volume,
(1) 
where controls the overall volume and are blowups whose only nonvanishing triple intersections are with themselves. and are positive constants and depend on particular model. Later, we consider the model with the number of moduli fields, two, three, or four. Then, we see that by diagonalization of the volume the three models arrive at the form of Eq.(1).
The dilaton and complex structure moduli are stabilized by fluxes and the Kähler moduli are stabilized by superpotential in Eq. (9) and the Kähler potential with correction is given by
(2) 
For large volume scenario and , the scalar potential becomes [12]
(3) 
We will discuss the inflationary potential derived from the above potential in Eq.(3). One of Kähler moduli will play as an inflaton field. By taking the limit , the above potential is simplified to
(4) 
where the constant is with .
The kinetic term couples to Kähler metric and we need to redefine the field to consider ordinary inflationary potential. Then, the canonically normalized field is obtained through straightforward calculation:
(5) 
In terms of , the inflationary potential becomes
(6) 
The plot of this inflationary potential is shown in Fig. (1).
In order to see if the potential gives enough inflation, we have to compute the so called slow roll parameters. The slow roll parameters are defined by
(7) 
By straightforward calculation with the given potential, we express the parameters as original Kähler moduli:
(8) 
The number of efolding can be read with from the potential:
(9) 
Providing , we may estimate each slow parameter: In this limit the efolding goes to
(10) 
and the three slow parameters , , and goes to
(11) 
It is required that
(12) 
to match the COBE normalization for the density fluctuations . We have used the relation efoldings before the end of inflation. If we use the relation with . The left hand side is evaluated at horizon exit,
(13) 
we get the following relation:
(14) 
From Eq. (14), we can estimate the volume. If we set , , and , then this equation is reduced to
(15) 
From the analysis of WMAP data, for , we have . We see then that for specific number of efolding, say , the volume is roughly given by
(16) 
By numerical study of Eq. (10), the volume for is roughly .
With the help of the expression of and we can see
(17) 
When we consider and with the above relations, the rough estimate for slow roll parameters , and become
(18) 
This estimation is checked explicitly by numerical evaluation in figure (2). Note that these values of and are independent of the parameters of the theory (e.g. , , and , etc.) and are dependent only on the number of efolding. On the other hand, is dependent on the particular value of and , etc. Later on, in the model with the number of Kähler moduli more than two, we will find similar behavior for the slow roll parameters and . Finally, we plot the spectral index as a function of in figure (3).
4 Examples of the Swisscheese model
4.1 Two Kähler moduli model:
In this section, we consider concrete CalabiYau manifolds characterized by weighted projective space called Swisscheese model. One of the advantages using this model is that we can explicitly express the volume of the CalabiYau manifold and give a chance to calculate many things. A Swisscheese model is a real 6manifold with 4cycles. Of these 4cycles, one is large and controls the size of the cheese, while the others are small and controls the size of the holes. The volume of the cheese can be expressed as in Eq. (1). The small cycles may be thought of as local blowup effects; if the bulk cycle is large, the overall volume is largely insensitive to the small cycles.
In this subsection, the case is the example of CalabiYau of degree 18 hypersurface embedded in the complex weighted projective space [13, 32]. The overall volume in terms of 2cycle volumes ’s is given by
(1) 
With the fourcycle volumes such as , , the of volume CalabiYau manifold takes the form
(2) 
The large volume claims that and remains small. The superpotential is then given by
(3) 
We take the limit with . Then, the potential becomes
(4) 
In order to discuss the stabilization we have to solve
(5) 
and by taking the limit we get the following:
(6) 
So we have seen the stabilization of the potential with the given and the volume. For the discussion of the numerical analysis, we have
(7) 
If we set , and , we have
(8) 
Finally, let us consider inflationary potential. Following the procedure of the previous section, we also get the same form of the potential:
(9) 
4.2 The three Kähler moduli model: and
4.2.1 Case I:
In the case of three Kähler moduli, the potentials were calculated [23]. In this section, we get the inflationary potential by the same procedure with section 3.
We start with an example, the Fano threefold [23] which is a quotiont of a real sixdimensional CalabiYau manifold with hodge numbers and . Then, the volume can be expressed in terms of 4cycle moduli as
(10) 
With the diagonal basis
(11) 
we can rewrite the volume simpler as
(12) 
There is another type of CalabiYau manifold which has the same number of Kähler moduli but different number of complex structure. We study it in more detail in following section.
4.2.2 Case II:
This CalabiYau manifold has and and has been studied in [24]. For this case, we have two stacks of D7branes wrapping rigid fourcycles and where on the first one a nontrivial line bundle is turned on. We get MSSM matter from intersections and where the prime denotes the orientifold image. The stacks of D7branes wrap the rigid fourcycles and with line bundles and . With this choice, there are no chiral zero modes on the D7E3 brane intersections. The E3brane wraps . For this case, the volume in terms of ’s is given by
(13) 
Again in terms of diagonal basis
(14) 
we have the volume:
(15) 
For the model , [24] we write the scalar potential as
(16) 
where are numerical factors. Changing of coordinates, using Euclidean D3brane cycle and standard model cycle with the relations and , gives the potential:
(17) 
From this potential, we can get inflationary potential:
(18) 
which has the same form with the previous potentials.
The potential has a critical point at but this is not a minimum but a saddle point along at fixed and . The inclusion of the oneloop correction stabilizes this direction. So the one loop corrected scalar potential [31] has the form
(19)  
Figure (4) depicts the stabilization of the direction .
The model with D7brane induces Dterm. The Dterm contribution to the potential becomes[24]
(20) 
Note that because of Dterm the large volume minimum is destabilized: For the large volume scenario, we need to set . Now, let us consider the inflation with this potential. As we have seen, the inflationary potential is obtained by neglecting the term. Moreover, we set and . The inflaton is assumed to be and this scalar field is supposed to be rolling from off the minimum. Then Dterm plays a role in this case. However, if we compute the slow parameters and with the inclusion of Dterms the potential is not flat. Dterm dominates in the potential and it behaves linearly in .
4.3 Four moduli Kähler model:
In order to accommodate two MSSMlike D7stacks and E3brane, all on different ‘small’ cycles, we need CalabiYau manifolds with at least four moduli. Since they will generically intersect the E3branes, the E3D7 strings will correspond to chiral zeromodes of the instanton. We have E3brane placed on a small fourcycle and two MSSMlike D7branes with unitary gauge groups placed on the two remaining fourcycles. In order to cancel the total D7 tadpole by the O7plane, a hidden D7brane is needed. Here, three out of four moduli come from blowups. There are four Swisscheese models, and we focus here on just one model: [22]. The volume can be expressed as
(21) 
For positive Kähler cone , the following conditions should be satisfied:
(22) 
We choose the diagonal basis:
(23) 
In this basis, the volume can be rewritten as
(24) 
We can divide our model into two scenarios. It depends on which divisor , and branes are placed. For scenario I, , and are placed on , , and , respectively. While for scenario II, , and are placed , , and , respectively.
The term potential for scenario I is given by
(25) 
By varying with , , we get
(26) 
Note that if we demand a large volume while at the same time fulfilling the Kähler cone constraints, the term in the square root becomes negative. We cannot realize the large volume in scenario I. Therefore, it is inappropriate to discuss the stabilization of this model.
For scenario II, we see that the potential is written as
(27) 
This potential is also similar to scenario I whereas it has a large volume limit. We can proceed the same procedure to find the inflationary potential: