COMPLETACIONES SIN FILTROS

setup). Now the idea is that if the potential generated is flat enough, before reaching the minimum a period of inflation will be induced.

The mechanism of graceful exit and reheating in this class of models is the standard one: once the minimum is reached the expansion stops, the inhomogeneities of the inflaton then grow, condense and finally decay into Standard Model particles.

An example in this class is K¨ahler moduli inflation [58] where the inflaton is the volume of some 4-cycle of the Calabi-Yau manifold. This is realized in the large volume scenario (LVS) that we will review in chapter 6. In other models, e.g. the racetrack models [59, 60, 61], the inflaton is the axion complexifying the K¨ahler modulus and the potential is generated by non-perturbative effects. Another proposal involving axions is N-flation [62] where a large number of them realize the assisted inflation mechanism [63] (further comments will be given in section 5.5).

3.4 Discussion

Let us finally comment on some features common to string inflationary models and discuss what we can learn from string cosmology.

The presence of many moduli in the 4-dimensional effective action is a very common feature in flux compactifications. Closed string moduli arise from complex structure and K¨ahler deformations of the compact space. For a typical Calabi-Yau manifold there are

O(100) independent deformations. If D-branes are present, there will be additional open string moduli in 4-dimensions. Assuming that a stage of inflation is realized, the most natural expectation is a multifield inflationary dynamics, where many or all of the moduli evolve at the same time. Useless to say that this introduces enormous technical difficulties. It is therefore a widespread technical assumption that it is possible to take just one dynamical field and keep all others at their stabilized values. This is a good assumption when e.g. one of the moduli is much lighter than all the others. The hierarchy guaranties that the heavy moduli relax quite fast to their minima while the inflaton candidate is just at the beginning of its inflationary trajectory. Already in the single-field case, embedding inflation into string theory is a hard task; therefore, we will not attempt here to look at more complicated situations. Apart from their simplicity, another motivation to focus our attention on single-field models is that they can successfully reproduce observations. A typical feature of multifield models, that distinguishes them from single-field ones, is that part of the perturbations are isocurvature as opposed to adiabatic [32, 33]. Up to now, CMB measurements are consistent with exclusively adiabatic perturbations.

Let us go back to the single-field case. As we reviewed in section 3.2.1, for slow-roll models all the relevant observables are determined by the two slow-roll parameters. In this sense the slow-roll approximation erases much of the information about the high energy physics

responsible for inflation. Therefore, constructing a consistent model of slow-roll inflation in string theory does not provide an indication that string theory correctly describes the laws of nature. On the other hand discovering when and how inflation can take place in the framework of string theory gives us a useful tool to find regions of the landscape suitable to describe our universe. Also, slow-roll inflation is one of the simplest possible cases and it is natural to start with it and see which obstacles we are confronted with by the consistency of string theory. Once we learn how to overcome these obstacles, e.g. moduli stabilization or the η−problem, we can consider more complicated models. An even more exciting possibility is that string theory will guide us to non-slow-roll models with distinctive signatures. An example is DBI inflation [50, 51] that we will review in section 5.5.3. The presence of higher order derivative terms typically induces large non-Gaussianities and makes the model in principle falsifiable.

Every phenomenological application of string theory, e.g. to cosmology (see chapter 4 and 5) or particle physics (section 6), is based on an effective action and it is therefore limited by the validity of the action itself. For example we look at the massless spectrum neglect- ing α0 excitations and we disregard the whole KK tower. Also, almost every analysis is performed at classical level neglecting loop corrections. The effects that all these correc- tions can induce in the tree level approximated analysis should be carefully considered. This is what we will do in section 6 for the soft terms produced by the LVS scenario. In the string inflationary literature, loop andα0 corrections are often neglected. The philos- ophy is that they can be considered in a second step, once a successful tree-level model has been found. This will be our attitude in chapters 4 and 5. Eventually, one would like to ensure that none of the possible corrections spoils the phenomenological successes of the model. In this regard, we are still at the beginning of the path that leads to a bona fide string theory model of inflation.

4 Radial brane inflation

The material presented in this chapter is mainly based on [64]. We investigate the possi- bility, for a concrete type IIB flux compactification setup, to obtain a phenomenologically successful model of brane inflation. We consider a D3-brane moving in a warped (de- formed) conifold. The potential for this mobile D3-brane is generated by the attraction of an anti D3-brane at the tip of the conifold plus the effects induced by the stabilization of closed string moduli. We study the motion along the radial direction of the warped conifold. We show that, for fine tuned values of the parameters, this dynamics induces a prolonged stage of inflation and generates the correct CMB perturbations.

One of the purposes of this case study is to understand how the consistency of string theory constrains or obstructs the explicit construction of an inflationary model. Beyond constructing a successful model of slow-roll inflation1, we will learn some generic facts about the dynamics of a D3-brane in the KKLT setup (reviewed in section 2.6). The study in this chapter will provide several motivations to consider alternative models of brane inflation. This will be widely discussed in chapter 5 where we propose and analyze a model of inflation at the tip of the conifold, where no anti D3-branes are present and the motion takes place in the angular directions.

The structure of this chapter is as follows: in section 4.1 we briefly review the state of the art about brane inflation and explain which improvements are provided by the present work; in section 4.2 we describe the structure of the effective superpotential and its relation to the D7-brane embedding. Section 4.3 explains the type IIB setup and reviews the η-problem for brane inflation, which forms a main issue in this chapter; the effective potential for the K¨ahler and open string moduli in the warped conifold background is obtained. In section 4.4 we perform the minimization of the potential in all but the radial directions for a generic D7-brane embedding. In section 5 we apply these formulae to two particular cases: the Ouyang and the Kuperstein embeddings. Only the latter, in a fine tuned case, gives rise to a prolonged stage of inflation. In section 4.6 we analyze how the stringy parameters determine the cosmological evolution and how they control when inflation takes place. A series of technical details are discussed in appendix B. In appendix B.5 we summarize the various forces acting on the D3- and anti D3-brane and comment on their relative importance.

1_{Up to perturbative corrections that we do not take into account here. We postpone the discussion to} chapter 6

4.1 Preliminaries

One of the key steps to obtain a viable inflation scenario in string theory is to stabilize all massless moduli, except for the inflaton. As we reviewed in section 2.5 and 2.6, in the framework of type IIB flux compactifications, it is possible in principle to fix all closed string moduli.

If D-branes are present, an additional open string moduli sector is included. An interesting possibility (that we reviewed in 3.3) is to investigate if one of the open string moduli can play the role of the inflaton. If we have a pair of brane-anti brane, the Coulomb (plus gravitational) attraction could in principle drive inflation, where the distance between the branes would corresponds to the inflaton. For spacetime-filling D3-branes, this distance is just the separation along the compact directions. Clearly the distance between two points can not be larger than the size of the manifold itself. This geometrical bound on the range of variation of the inflaton prevents a prolonged stage of inflation. A way out was proposed in [65] (KKLMMT): if the branes are located in a region with strong warping, their reciprocal Coulomb attraction gets redshifted, the potential becomes flatter and inflation can last much longer.

Now, one has to ensure that no other forces spoil the achieved flatness of the Coulomb potential. Unfortunately, the stabilization of the K¨ahler moduli induces a force on the D3-brane. This is due to a non-trivial interplay between the volume and the D3-brane position. Once the K¨ahler moduli (among which the volume) are stabilized, the inflaton is endowed with a large mass. As a result, the second slow-roll parameter (3.5) turns out to be too large, η_&2/3, showing the break-down of slow-roll inflation [65].

The idea of exactly canceling this moduli stabilization effect by fine tuning some other effect has received a certain amount of attention [57, 64, 66, 67, 68]. In this chapter, based on [64, 57], we show two ways in which this can be achieved: one is by using the uplifting term to cancel the inflaton mass and will be discussed in section 4.7; the other is by means of threshold corrections to the non-perturbative superpotential. We will concentrate on the second possibility and construct explicitly a model where slow-roll inflation can take place.

Recently, threshold corrections to the non-perturbative superpotential Wnp, introduced

in section 2.6, became available for the warped conifold background [69] (previously such effects had been calculated in [70] for toroidal orientifolds). This means, as we will describe in detail in section 4.2, that we know how the prefactor A in (2.33) depends on D3-brane open string moduli. We would like to use this effect to cancel the large mass induced by moduli stabilization (an example of the so called η-problem that will be review in 4.3). Unfortunately, this is not quite possible because the term we want to cancel is proportional2 _toφ2 _{while threshold corrections arising from two large classes of}

4.1 Preliminaries 41

f

V

f

V

Figure 4.1: The plots display the inflaton potentialV(φ) for the Kuperstein embedding for two different values of the uplifting parameter β = 1.21 (left), β = 1.4 (right). The left plot shows that via fine tuning an inflection point suitable for inflation can be obtained.

D7-embedding functions g(w) (ACR [71, 72] and Kuperstein embeddings [73]) give rise only to terms proportional to φ and to integer powers of φ3/2_{; as we will see, this can be}

traced back to the holomorphicity of these supersymmetric embeddings. This is crucial because terms with a different φdependence can cancel only locally in a small φ interval, rather than globally. Outside this small interval the inflaton potential is not of the slow- roll type. In fact in general, the potential possesses a maximum and a minimum plus an inflection point in between, as shown in the right part of figure 4.1. With suitable fine- tuning, displayed in the left figure, it can be arranged that the maximum and minimum coincide with the inflection point, the potential hill at smallφdisappears, and the potential becomes flat enough for slow-roll inflation.

We then study the cosmological evolution induced by the inflaton potentialV(φ). In the fine tuned case in the left part of figure 4.1, a slow-roll inflation phase takes place (in particular η ' 0), but this happens only in a small region around the inflection point, whereη= 0 because the potential switches from concave to convex. We investigate when it is possible for the D3-brane to fall all the way into the throat and when the inflaton gets stuck somewhere in the throat. This is relevant both in the fine-tuned and in the more general case in which a maximum and a minimum are present. In fact, if one wants to end inflation with D3- anti D3-brane annihilation, the D3-brane has to go all the way down to the tip (φ∼0). If it gets stuck somewhere before, inflation lasts forever and the

us to approximate the deformed conifold with the singular conifold. Close to the tip, the effects of the deformation become relevant; we study this situation in appendix B.4. The result is that, close to the tip, the moduli stabilization produces a term proportional to φ3 _{which could be, contrary to the}

φ2 term for the singular conifold, cancelled by threshold corrections, at least in principle. Anyways, the cancellation can be achieved only for a short range of values ofφwhich is qualitatively analogous to the singular conifold situation. This is why we do not pursue this direction of investigation any further.

universe becomes large and empty.