Inflation near a metastable de Sitter vacuum from moduli stabilisation

We study the cosmological properties of a metastable de Sitter vacuum obtained recently in the framework of type IIB flux compactifications in the presence of three D7-brane stacks, based on perturbative quantum corrections at both world-sheet and string loop level that are dominant at large volume and weak coupling. In the simplest case, the model has one effective parameter controlling the shape of the potential of the inflaton which is identified with the volume modulus. The model provides a phenomenological successful small-field inflation for a value of the parameter that makes the minimum very shallow and near the maximum. The horizon exit is close to the inflection point while most of the required e-folds of the Universe expansion are generated near the minimum, with a prediction for the ratio of tensor-to-scalar primordial fluctuations $r \sim 4 \times 10^{-4}$. Despite its shallowness, the minimum turns out to be practically stable. We show that it can decay only through the Hawking-Moss instanton leading to an extremely long decay rate. Obviously, in order to end inflation and obtain a realistic model, new low-energy physics is needed around the minimum, at intermediate energy scales of order $10^{12}$ GeV. An attractive possibility is by introducing a"waterfall"' field within the framework of hybrid inflation.


Introduction
Inflation has been the most prevalent scenario for solving the horizon and flatness problems [1,2]. It is usually realised using a scalar field coupled to gravity, which rolls down towards a (local) minimum of the potential (which is not necessarily the true minimum). In the ensuing years several elaborated versions appeared, where more than one fields are included to drive inflation. For example, in the case of the hybrid scenario [3] inflation ends when a second 'waterfall' scalar field takes over and rolls rapidly down to the slope towards the true minimum that should describe our observable Universe. In many effective theories with ultraviolet (UV) completion this is a common way to implement the inflationary scenario, as the scalar potential of the theory involves several scalar fields defining trajectories with metastable minima before a true minimum has entailed. In a wide class of string theory constructions, the inflaton field is materialised by some modulus, since there are always plenty of them in generic compactifications, parametrising in particular the internal manifold. However, an important step for a successful implementation of the inflationary scenario in string theory is the existence of a (metastabe) de Sitter vacuum of the corresponding effective field theory [4,5]. Furthermore, in the broader picture the viability of such a construction is intimately related to the important issue of moduli stabilisation, providing positive square-masses for all the associated scalar fields.
In the present work, we investigate the issue of cosmological inflation in a class of effective models emerging in the framework of type-IIB string theory and its extended geometric picture, namely the F-theory. Within this context, in Ref [6,7] it was shown that when the internal geometry is equipped with an appropriate configuration of intersecting D7-branes and orbifold O7-planes, string loop effects induce logarithmic corrections due to closed strings propagating effectively along the two-dimensional space [8] transverse to each D7-brane towards points of localised gravity [9]. Working in the large volume limit, it has been demonstrated how the associated Kähler moduli can be stabilised and an uplift of the vacuum energy can be ensured by D-terms related to anomalous U(1) symmetries on the D7-branes world-volume [10].
A first attempt to implement inflation in this framework using the volume modulus as inflaton was done in [11], in the presence of a new Fayet-Iliopoulos term written recently in supergravity [12,13] which provides an extra uplift of the vacuum energy and allows a tuning of its value at the minimum. The origin of such a term is however unclear in string theory making questionable the applicability of the stabilisation mechanism to inflation. In the present work, instead, we follow an alternative path, focusing only on the minimal amount of ingredients which were already used to ensure moduli stabilisation at a metastable de Sitter vacuum. Our first goal is to investigate the possibility of realistic inflation without imposing the present tiny value of the vacuum energy at the minimum of the scalar potential. We will then address this issue together with the question of ending inflation by introducing a "waterfall" field in the context of hybrid inflation.
We recall first briefly the main ingredients of the stabilisation mechanism. All complex structure moduli and the axion-dilaton field are fixed in a supersymmetric way (imposing vanishing F-terms) on the fluxed induced superpotential [14,15]. We end up in general with a constant superpotential but vanishing scalar potential for the Kähler moduli due to the no-scale structure of their kinetic terms at the leading order [16][17][18]. Radiative corrections though break the no-scale structure and bring a logarithmic dependence on the co-dimension two volumes transverse to the D7-branes worldvolumes [6,7]. Moreover, D-term contributions from anomalous U(1) factors associated with the intersecting D7-branes provide an uplift mechanism [10] so that a de Sitter (dS) vacuum is naturally achieved.
In the simplest case, the scalar potential of the effective four dimensional theory can be expressed in terms of the volume modulus and two other (orthogonal combinations) Kähler moduli fields. It turns out that essentially only one free parameter, dubbed x in the following, controls the shape of the potential and in particular delimits its two extrema (minimum and maximum). More precisely, the requirement of a dS minimum confines x in a very small region where the potential stays almost flat, and the two extrema of the potential are very close to each other. In effect, x regulates the measurable parameters related to inflation.
In this restrictive context, we start our investigations by examining in some detail possible implementations of various existing inflationary scenarios including slow roll and in particular the hilltop inflation. Varying x, we adjust the value of the slow-roll parameter η so that inflation starts near the maximum with the correct value of the spectral index. However in this case the slow-roll parameters remain small all the way to the minimum and inflation doesn't stop, producing much more than the required 60 e-folds. Alternatively, imposing the correct number of e-folds, the resulting spectral index does not reproduce the observable value. Thus, the hilltop possibility is ruled out.
We then proceed with a novel proposal where the horizon exit occurs near the inflection point of the potential where the η parameter vanishes and inflation takes place essentially near the minimum where the required number of e-folds are produced. This is a reasonable and completely justifiable assumption at the other end, however, in close analogy with the concept of hilltop inflation. Indeed, both are characterised with the property that the slow-roll parameter ε is negligible at both extrema and consequently, short interval of the inflaton trajectory is enough to accumulate the required number of e-folds. Remarkably, we find that an inflationary phase is feasible near the minimum and the desired number of e-folds can readily be achieved. It also predicts a ratio of tensor-to-scalar primordial fluctuations r 4 × 10 −4 . Moreover, because of the proximity of the two extrema of the potential, the inflaton is restricted to a short range of values ensuring small field inflation, compatible with the validity of the effective field theory.
On the other hand, since the minimum is generated from quantum corrections, it is metastable and is expected to decay to the true minimum in the runaway direction of large volume. We then perform an estimate of its lifetime due to either tunnelling by the Coleman-de Lucia instanton [19,20], or passing over the barrier by the Hawking-Moss instanton [21]. Our analysis shows that in the x-region where inflation is viable the false vacuum decay is due to the latter, leading to an extremely long lifetime.
The paper is organised as follows. In Section 2, we present a short review of the framework and the main features of the mechanism of moduli stabilisation (subsection 2.1), as well as of the minimisation of the potential that depends on one effective parameter. In Section 3, we start with the basic equations for inflationary solutions (subsection 3.1) and perform the analysis for hilltop inflation (subsection 3.2) and inflation around the minimum from the inflection point (subsection 3.3). We then work out the observable quantities and compare the parameters of the model with those of the underlying string theory (subsection 3.4). In Section 4, we compute the vacuum decay (subsection 4.1) and discuss the issue of ending inflation in the context of the hybrid proposal by introducing a "waterfall" field (subsection 4.2). We conclude in Section 5 with a summary of the main results.

.1 Type IIB model of intersecting D7-branes and moduli stabilisation
In this paper we consider the model developed in [6] within the type IIB string framework where complexe structure moduli and the dilaton are stabilised in the standard supersymmetric way by turning on 3-form fluxes. This model takes into account the quantum corrections of a three intersecting D7-branes configuration [7]. These corrections break the no-scale structure of the effective theory and give a non-zero contribution to the F-part of the supergravity scalar potential. If one also considers the U(1) anomalous symmetries of the D7-branes, Fayet-Iliopoulos D-terms must be introduced in the scalar potential [10]. The latter can be used to uplift the scalar potential to a de Sitter minimum with all Kähler moduli stabilised. Denoting by τ i for i = 1, 2, 3 the imaginary parts of the D7-branes world-volume Kähler moduli, the Kähler potential of the model is [6] where κ = √ 8πG is the inverse of the reduced Planck mass (we work with c =h = 1) and the compactification volume V is expressed (in Planck units) as The ξ term stands for α 3 corrections [18] and is proportional to the Euler number χ CY of the Calabi-Yau manifold In the large volume limit, it induces four-dimensional kinetic terms localised in the internal manifold [9]. In the case of orbifolds, it is generated at one string loop order and reads: ξ orb = −π 2 χ orb g 2 s /12 where g s is the string coupling and χ orb counts the difference between the numbers of closed string N = 2 hyper and vector multiplets, χ orb = 4(n H − n V ). The γ i are model dependant parameters for the logarithmic quantum corrections associated with the 7-branes [7]. They are induced from massless closed strings emitted from the localised vertices towards the 7-brane sources, thus propagating in two dimensions. Taking identical γ i for simplicity, they are given by 5 where T 0 /g s is the effective 7-brane tension. Note the minus sign in the last equality of (4). One can extract the F-part of the scalar potential from (1). It depends only on the volume V and after defining µ = e ξ 2γ , its exact expression is where W 0 is the constant superpotential contribution left over from the 3-form fluxes upon the complex structure moduli stabilisation. In the large volume limit, V F takes the much simpler form The D-part of the scalar potential coming from the D7-branes can also be expressed very simply in the large world-volume limit where the d i for i = 1, 2, 3 are model dependent constants related to the U(1) anomalies. Contrary to the F-part, this D-part depends on the three τ i fields. Instead of these Kähler moduli we will rather work with the canonically normalised fields from which we obtain the following base, after isolating the volume from the two other perpendicular directions In terms of these fields, the D-part of the potential (7) reads so that the total scalar potential is

Local de Sitter minimum
We study now the minimum of the scalar potential (13). The field φ will be associated with the inflaton and its evolution will determine the inflation era. We must then stabilise the two other canonically normalised fields u, v at their values u 0 and v 0 dictated by the minimisation of V D in (7). Their values at the minimum read for which the potential V D becomes with d ≡ 3(d 1 d 2 d 3 ) 1 3 . Hence after stabilisation of the two transverse moduli, the total scalar potential reduces in the large volume limit to where we defined The last inequality is obtained for γ < 0, which is a condition for a dS vacuum to exist at large volume. The parameter q essentially shifts the local extrema towards large volumes. It is essential in the string context but does not play a role for inflation. Thus, for simplicity, we will take it zero in the numerical study of section 3, before coming back to its significance in section 3.4. C is an overall constant which plays no role in the minimisation but will be related to the observed spectral amplitude, when the model is considered as a candidate for an inflationary scenario. Thus, σ is the only effective parameter of the model. In section 3 we will study the inflationary possibilities from the above model. The inflaton will be identified with the canonically normalised φ , thus we express here the potential (16) in terms of the inflaton φ Note that φ is dimensionless. In order to minimise and study the slow-roll parameters we compute the first two derivatives of V . From (18) we get Solving V (φ ) = 0 leads to the two solutions with φ − the local minimum and φ + the local maximum, with φ − < φ + . W 0/−1 are the two branches of the Lambert function (or product logarithm) and x is defined through the relation As mentioned above, it is clear from (21) and (22) that when x is kept constant, varying q shifts the local extrema. The critical value x c 0.072132 gives a Minkowski minimum, i.e. with V (φ − ) = 0. The region 0 < x < x c gives a de Sitter (dS) minimum and x > x c gives an anti-de Sitter (AdS) one.
The region x < 0 corresponds to the case where the two branches of the Lambert function join and the potential loses its local extrema. The shape of the potential in the three regimes is shown in Figure 1.
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions we obtain the following base plicity and defining µ = e x 2g , one can extract the Ff the scalar potential from (12). It depends only of the e V (or equivalently the modulus t) and reads e D-part of the scalar potential coming from the D7 branes reads in the large volume limit ary to the F-part V F , the D-part V D depends on the three li t, u and v. When considering the volume as the possiflaton, we place the two other moduli at their minimal u 0 and v 0 dictated by the minimisation of V D in (19). minimal values read ich the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions e two other perpendicular direcbase = e x 2g , one can extract the Fom (12). It depends only of the e modulus t) and reads . (18) potential coming from the D7 ge volume limit D-part V D depends on the three idering the volume as the possio other moduli at their minimal the minimisation of V D in (19).
In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . In order to minimize and study the slow-roll param we compute the first two derivatives of V . From (25) w Solving V 0 (f ) = 0 leads to the two solutions µ = e x 2g , one can extract the Ffrom (12). It depends only of the the modulus t) and reads r potential coming from the D7 arge volume limit e D-part V D depends on the three sidering the volume as the possiwo other moduli at their minimal y the minimisation of V D in (19).
The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions endicular direc- (15) n extract the Fends only of the nd reads µV ) . (18) ng from the D7 t nds on the three me as the possiat their minimal n of V D in (19).
The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes .
In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we ge Solving V 0 (f ) = 0 leads to the two solutions (15) .
ning µ = e x 2g , one can extract the Fntial from (12). It depends only of the ently the modulus t) and reads 2(g+2V )+(4g V ) ln(µV ) (µV )) 2 (6g 2 +V 2 +8gV +g(4g V ) ln(µV )) . (18) scalar potential coming from the D7 the large volume limit V F , the D-part V D depends on the three n considering the volume as the possithe two other moduli at their minimal ted by the minimisation of V D in (19). read ntial becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions e can extract the Fdepends only of the t) and reads . (18) oming from the D7 limit epends on the three olume as the possiuli at their minimal ation of V D in (19).
The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads . (18) The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v.
When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essenti large volumes. C is an o in the model but is given tion.

Potential minimum, parameters
In the following section sibilities from the abov tified to the canonically denote f from now on.
of the inflaton In order to minimize we compute the first two The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes tion.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions tions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation. 4 Potential minimum, maximum and slow-roll parameters In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions tions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationar sibilities from the above model. The inflaton will be tified to the canonically normalised modulus t, whi denote f from now on. Hence we can express (23) in of the inflaton f (which again, is the total volume m t).
It reads In order to minimize and study the slow-roll para we compute the first two derivatives of V . From (25) Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions he volume from the two other perpendicular direcbtain the following base city and defining µ = e x 2g , one can extract the Fscalar potential from (12). It depends only of the (or equivalently the modulus t) and reads -part of the scalar potential coming from the D7 nes reads in the large volume limit o the F-part V F , the D-part V D depends on the three and v. When considering the volume as the possin, we place the two other moduli at their minimal and v 0 dictated by the minimisation of V D in (19).
the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . in the model but is given by the amplitude spectrum oberva tion.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary pos sibilities from the above model. The inflaton will be iden tified to the canonically normalised modulus t, which w denote f from now on. Hence we can express (23) in term of the inflaton f (which again, is the total volume modulu t).
It reads In order to minimize and study the slow-roll parameter we compute the first two derivatives of V . From (25) we ge Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions 2 i olating the volume from the two other perpendicular direcons we obtain the following base r simplicity and defining µ = e x 2g , one can extract the Fart of the scalar potential from (12). It depends only of the olume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 uxed branes reads in the large volume limit ontrary to the F-part V F , the D-part V D depends on the three oduli t, u and v. When considering the volume as the possile inflaton, we place the two other moduli at their minimal alues u 0 and v 0 dictated by the minimisation of V D in (19). heir minimal values read r which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . in the model but is given by the amplitude spectrum oberva tion.
4 Potential minimum, maximum and slow-roll parameters In the following sections we will study the inflationary pos sibilities from the above model. The inflaton will be iden tified to the canonically normalised modulus t, which w denote f from now on. Hence we can express (23) in term of the inflaton f (which again, is the total volume modulu t).
It reads In order to minimize and study the slow-roll parameter we compute the first two derivatives of V . From (25) we ge Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions 2 i olating the volume from the two other perpendicular direcons we obtain the following base r simplicity and defining µ = e x 2g , one can extract the Fart of the scalar potential from (12). It depends only of the lume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 uxed branes reads in the large volume limit ontrary to the F-part V F , the D-part V D depends on the three oduli t, u and v. When considering the volume as the possile inflaton, we place the two other moduli at their minimal lues u 0 and v 0 dictated by the minimisation of V D in (19). heir minimal values read r which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v.
When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes .

parameters
In the following sections we will study the inflationar sibilities from the above model. The inflaton will be tified to the canonically normalised modulus t, whi denote f from now on. Hence we can express (23) in of the inflaton f (which again, is the total volume m t).
It reads In order to minimize and study the slow-roll para we compute the first two derivatives of V . From (25) V Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions olume from the two other perpendicular direcin the following base and defining µ = e x 2g , one can extract the Falar potential from (12). It depends only of the r equivalently the modulus t) and reads rt of the scalar potential coming from the D7 reads in the large volume limit e F-part V F , the D-part V D depends on the three d v. When considering the volume as the possie place the two other moduli at their minimal v 0 dictated by the minimisation of V D in (19). l values read V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v.
When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary pos sibilities from the above model. The inflaton will be iden tified to the canonically normalised modulus t, which w denote f from now on. Hence we can express (23) in term of the inflaton f (which again, is the total volume modulu t).
It reads In order to minimize and study the slow-roll parameter we compute the first two derivatives of V . From (25) we ge Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions 2 i ing the volume from the two other perpendicular direcwe obtain the following base g mplicity and defining µ = e x 2g , one can extract the Ff the scalar potential from (12). It depends only of the e V (or equivalently the modulus t) and reads he D-part of the scalar potential coming from the D7 branes reads in the large volume limit ary to the F-part V F , the D-part V D depends on the three li t, u and v. When considering the volume as the possiflaton, we place the two other moduli at their minimal s u 0 and v 0 dictated by the minimisation of V D in (19). minimal values read hich the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get V 00 (f ) = 27 2 Solving V 0 (f ) = 0 leads to the two solutions he volume from the two other perpendicular direcbtain the following base city and defining µ = e x 2g , one can extract the Fscalar potential from (12). It depends only of the (or equivalently the modulus t) and reads -part of the scalar potential coming from the D7 nes reads in the large volume limit to the F-part V F , the D-part V D depends on the three u and v. When considering the volume as the possin, we place the two other moduli at their minimal and v 0 dictated by the minimisation of V D in (19). imal values read ✓ d 1 the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V  (19).
Their minimal values read for which the V D potential becomes .

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary sibilities from the above model. The inflaton will be i tified to the canonically normalised modulus t, which denote f from now on. Hence we can express (23) in te of the inflaton f (which again, is the total volume mod t).
It reads In order to minimize and study the slow-roll parame we compute the first two derivatives of V . From (25) we Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t). It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions 2 3 Scalar potential from D7-branes moduli stabilisation The Kähler potential of the model is [3] K = 2 k 2 ln (t 1 t 2 t 3 ) Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes Hence after stabilisation of the tw the total scalar potential reduces to In the large volume limit we obtai sion The q parameter essentially shifts the large volumes. C is an overall constan in the model but is given by the amplit tion.

Potential minimum, maximum and parameters
In the following sections we will study sibilities from the above model. The tified to the canonically normalised m denote f from now on. Hence we can of the inflaton f (which again, is the t t). It reads In order to minimize and study the we compute the first two derivatives of Solving V 0 (f ) = 0 leads to the two sol 3 Scalar potential from D7-branes moduli stabilisation The Kähler potential of the model is [ Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . Hence after stabilisation of the two transverse moduli, the total scalar potential reduces to In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions 2 3 Scalar potential from D7-branes moduli stabilisation The Kähler potential of the model is [3] K = 2 k 2 ln (t 1 t 2 t 3 ) Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes Hence after stabilisation of the two tran the total scalar potential reduces to In the large volume limit we obtain the sion The q parameter essentially shifts the local large volumes. C is an overall constant whic in the model but is given by the amplitude sp tion.

Potential minimum, maximum and slow parameters
In the following sections we will study the i sibilities from the above model. The inflato tified to the canonically normalised modul denote f from now on. Hence we can expre of the inflaton f (which again, is the total v t). It reads In order to minimize and study the slowwe compute the first two derivatives of V . Fr Solving V 0 (f ) = 0 leads to the two solutions Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we sion The q parameter essentially shif large volumes. C is an overall c in the model but is given by the tion.

Potential minimum, maximu parameters
In the following sections we wil sibilities from the above model tified to the canonically norma denote f from now on. Hence w of the inflaton f (which again, i t). It reads In order to minimize and stu we compute the first two derivat Solving V 0 (f ) = 0 leads to the t Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads (V +2g ln(µV )) 2 (6g 2 +V 2 +8gV +g(4g V ) ln(µV )) .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit w sion The q parameter essentially sh large volumes. C is an overall in the model but is given by the tion.

Potential minimum, maxim parameters
In the following sections we w sibilities from the above mode tified to the canonically norm denote f from now on. Hence of the inflaton f (which again t). It reads In order to minimize and s we compute the first two deriv Solving V 0 (f ) = 0 leads to the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we ob sion The q parameter essentially shifts th large volumes. C is an overall const in the model but is given by the amp tion.

Potential minimum, maximum a parameters
In the following sections we will stu sibilities from the above model. Th tified to the canonically normalised denote f from now on. Hence we ca of the inflaton f (which again, is th t). It reads In order to minimize and study t we compute the first two derivatives The Kähler potential of the model is [3] K = 2 k 2 ln (t 1 t 2 t 3 ) Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes Hence after stabilisation of t the total scalar potential reduces t In the large volume limit we sion The q parameter essentially shift large volumes. C is an overall con in the model but is given by the am tion.

Potential minimum, maximum parameters
In the following sections we will sibilities from the above model. tified to the canonically normali denote f from now on. Hence we of the inflaton f (which again, is t). It reads In order to minimize and stud we compute the first two derivativ Solving V 0 (f ) = 0 leads to the tw +W 1 e three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get Solving V 0 (f ) = 0 leads to the two solutions Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we obt sion The q parameter essentially shifts th large volumes. C is an overall const in the model but is given by the amp tion.

Potential minimum, maximum a parameters
In the following sections we will stu sibilities from the above model. Th tified to the canonically normalised denote f from now on. Hence we ca of the inflaton f (which again, is th t). It reads In order to minimize and study t we compute the first two derivatives The Kähler potential of the model is [3] K = 2 k 2 ln (t 1 t 2 t 3 ) Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes Hence after stabilisation of th the total scalar potential reduces t In the large volume limit we sion The q parameter essentially shifts large volumes. C is an overall con in the model but is given by the am tion.

Potential minimum, maximum parameters
In the following sections we will sibilities from the above model. tified to the canonically normali denote f from now on. Hence we of the inflaton f (which again, is t). It reads In order to minimize and stud we compute the first two derivativ Solving V 0 (f ) = 0 leads to the tw

Scalar potential from D7-branes moduli stabilisation
The Kähler potential of the model is [ Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes . Hence after stabilisation of the two transverse moduli, the total scalar potential reduces to V = In the large volume limit we obtain the simpler expression The q parameter essentially shifts the local extrema towars large volumes. C is an overall constant which plays no role in the model but is given by the amplitude spectrum obervation.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationary possibilities from the above model. The inflaton will be identified to the canonically normalised modulus t, which we denote f from now on. Hence we can express (23) in terms of the inflaton f (which again, is the total volume modulus t).
It reads In order to minimize and study the slow-roll parameters we compute the first two derivatives of V . From (25) we get V 00 (f ) = Solving V 0 (f ) = 0 leads to the two solutions The Kähler potential of the model is [3] K = 2 k 2 ln (t 1 t 2 t 3 ) Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we obtain the sion The q parameter essentially shifts the local large volumes. C is an overall constant whi in the model but is given by the amplitude s tion.

Potential minimum, maximum and slow parameters
In the following sections we will study the sibilities from the above model. The inflat tified to the canonically normalised modu denote f from now on. Hence we can expre of the inflaton f (which again, is the total v t). It reads In order to minimize and study the slow we compute the first two derivatives of V . F Solving V 0 (f ) = 0 leads to the two solution Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local e large volumes. C is an overall constant which in the model but is given by the amplitude spe tion.

Potential minimum, maximum and slowparameters
In the following sections we will study the in sibilities from the above model. The inflaton tified to the canonically normalised modulu denote f from now on. Hence we can express of the inflaton f (which again, is the total vo t). It reads In order to minimize and study the slow-r we compute the first two derivatives of V . Fro Solving V 0 (f ) = 0 leads to the two solutions Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we ob sion The q parameter essentially shifts t large volumes. C is an overall cons in the model but is given by the amp tion.

Potential minimum, maximum parameters
In the following sections we will stu sibilities from the above model. Th tified to the canonically normalise denote f from now on. Hence we c of the inflaton f (which again, is th t). It reads In order to minimize and study we compute the first two derivatives Solving V 0 (f ) = 0 leads to the two tions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema large volumes. C is an overall constant which plays n in the model but is given by the amplitude spectrum o tion.

Potential minimum, maximum and slow-roll parameters
In the following sections we will study the inflationar sibilities from the above model. The inflaton will be tified to the canonically normalised modulus t, whi denote f from now on. Hence we can express (23) in of the inflaton f (which again, is the total volume m t).
It reads In order to minimize and study the slow-roll para we compute the first two derivatives of V . From (25) V 0 (f ) = Solving V 0 (f ) = 0 leads to the two solutions Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads (V +2g ln(µV )) 2 (6g 2 +V 2 +8gV +g(4g V ) ln(µV )) .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes large volumes. C is an overall constan in the model but is given by the amplit tion.

Potential minimum, maximum an parameters
In the following sections we will study sibilities from the above model. The tified to the canonically normalised m denote f from now on. Hence we can of the inflaton f (which again, is the t t). It reads In order to minimize and study the we compute the first two derivatives o Solving V 0 (f ) = 0 leads to the two sol Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts large volumes. C is an overall cons in the model but is given by the am tion.

Potential minimum, maximum parameters
In the following sections we will st sibilities from the above model. T tified to the canonically normalise denote f from now on. Hence we c of the inflaton f (which again, is t t). It reads In order to minimize and study we compute the first two derivative Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes (24 The q parameter essentially shifts the local extrema towar large volumes. C is an overall constant which plays no rol in the model but is given by the amplitude spectrum oberva tion. 4 Potential minimum, maximum and slow-roll parameters In the following sections we will study the inflationary pos sibilities from the above model. The inflaton will be iden tified to the canonically normalised modulus t, which w denote f from now on. Hence we can express (23) in term of the inflaton f (which again, is the total volume modulu t).
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the loca large volumes. C is an overall constant wh in the model but is given by the amplitude s tion.

Potential minimum, maximum and slo parameters
In the following sections we will study the sibilities from the above model. The infla tified to the canonically normalised modu denote f from now on. Hence we can expr of the inflaton f (which again, is the total t). It reads In order to minimize and study the slow we compute the first two derivatives of V . F Solving V 0 (f ) = 0 leads to the two solution Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the large volumes. C is an overall constan in the model but is given by the amplit tion.

Potential minimum, maximum and parameters
In the following sections we will study sibilities from the above model. The tified to the canonically normalised m denote f from now on. Hence we can of the inflaton f (which again, is the t t). It reads In order to minimize and study the we compute the first two derivatives of Solving V 0 (f ) = 0 leads to the two sol Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shifts the local extrema tow large volumes. C is an overall constant which plays no in the model but is given by the amplitude spectrum obe tion. 4 Potential minimum, maximum and slow-roll parameters In the following sections we will study the inflationary p sibilities from the above model. The inflaton will be id tified to the canonically normalised modulus t, which denote f from now on. Hence we can express (23) in te of the inflaton f (which again, is the total volume mod t).
It reads In order to minimize and study the slow-roll parame we compute the first two derivatives of V . From (25) we Starting from the real parts t i of the Kähler moduli for the three magnetised D7 branes, we can define the normalised fields Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes In the large volume limit we obtain sion The q parameter essentially shifts the large volumes. C is an overall constant in the model but is given by the amplitu tion.

Potential minimum, maximum and parameters
In the following sections we will study sibilities from the above model. The i tified to the canonically normalised m denote f from now on. Hence we can e of the inflaton f (which again, is the to t). It reads In order to minimize and study the we compute the first two derivatives of Solving V 0 (f ) = 0 leads to the two solu Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads (V +2g ln(µV )) 2 (6g 2 +V 2 +8gV +g(4g V ) ln(µV )) .
The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes large volumes. C is an overall constant w in the model but is given by the amplitud tion.

Potential minimum, maximum and s parameters
In the following sections we will study th sibilities from the above model. The infl tified to the canonically normalised mo denote f from now on. Hence we can ex of the inflaton f (which again, is the tota t). It reads In order to minimize and study the sl we compute the first two derivatives of V Solving V 0 (f ) = 0 leads to the two soluti Isolating the volume from the two other perpendicular directions we obtain the following base Taking for simplicity and defining µ = e x 2g , one can extract the Fpart of the scalar potential from (12). It depends only of the volume V (or equivalently the modulus t) and reads The D-part of the scalar potential coming from the D7 fluxed branes reads in the large volume limit Contrary to the F-part V F , the D-part V D depends on the three moduli t, u and v. When considering the volume as the possible inflaton, we place the two other moduli at their minimal values u 0 and v 0 dictated by the minimisation of V D in (19).
Their minimal values read for which the V D potential becomes The q parameter essentially shif large volumes. C is an overall c in the model but is given by the tion.

Potential minimum, maximu parameters
In the following sections we wil sibilities from the above model tified to the canonically norma denote f from now on. Hence w of the inflaton f (which again, i t). It reads In order to minimize and stu we compute the first two derivat Solving V 0 (f ) = 0 leads to the t The values of the potential and its derivatives at the extrema can be derived from (21) and (22): Firstly, from (24) we see that only the parameter x determines the ratio between the values of the potential at the extrema. Indeed we get This ratio is plotted in the left panel of Figure 2. Secondly, we see that at the two extrema, the slow-roll parameter η V = V V also depends on x only. It reads Again, from (28) and (27) we see that x is the only important parameter of the model for the shape of the potential. From Figure 2 we see that as soon as x 0.05 there is no scale separation anymore between the values of the potential at the two local extrema. 3 Inflation possibilities from the model

Generalities about inflation
Inflation is characterised by an accelerated expansion of the Universe. In the standard Friedmann-Lemaître-Robertson-Walker (FLRW) metric parametrized by a scale factor a, an inflationary phase occurs whenä > 0. The dots denote derivatives with respect to the FLRW time. We define the Hubble parameter and recall the Friedmann equations for an expanding Universe filled with a single scalar field as well as the Klein-Gordon equation for the scalar field in FLRW background We recall that our φ is dimensionless. Following the Hamilton-Jacobi method [22], we make a change of variable to take the inflaton φ as the time variable. This change of variable is given by rewritinġ H = dH dφφ in equation (31), leading to 2. The observed spectral index n S measuring the deviation from a scale-invariant power spectrum, is related to the slow-roll parameters at horizon exit by 3. The spectral amplitude A S induced by observations is

Hilltop inflation
The hilltop inflation scenario [1,23] emerged more than thirty years ago. The idea is the following: the inflaton starts rolling from a local maximum down to the minimum of the potential. In the vicinity of the local maximum, the slow-roll parameter ε is negligible while η is determined by the observed spectral index. The horizon exit occurs near the maximum, and the 60 remaining e-folds are obtained from there to a point before the minimum, where ε = 1 and inflation stops. As ε → 0 at the maximum, one can generate an infinite number of e-folds. In reality, the number of e-folds is dictated by the initial condition. The closest to the maximum the inflaton starts rolling down, the largest the number of e-folds is.
The fact that the inflaton starts rolling from the maximum of the potential may be motivated if one considers that this maximum was related to a symmetry restoration point. At higher temperatures this point could have been a symmetric minimum of the potential, which became a maximum after spontaneous symmetry breaking occurring when temperature cooled down. Hence if the inflaton sits at a symmetric point at higher temperatures, it is natural to take initial conditions near the maximum.
In our model, according to (28) the values of the η slow-roll parameter at the maximum only depends on x. Solving (28) in order to have η(t + ) −0.02 we find x 2.753 × 10 −7 . This value can also be obtained graphically from the right panel of Figure 2.
In order to study the possibility of hilltop inflation we then take x 2.753 × 10 −7 , q = 0 and C = 1.0 × 10 8 and solve (34) numerically. 7 This allows to find the Hubble parameter solution all along the inflation trajectory. Due to the shape of the potential (containing exponentials and linear terms), we had to use very high precision data types. This was achieved through a C++ code using cpp_dec_float data types provided by the multiprecision package of the boost library.
Plots of all the interesting parameters are shown in Figure 3. The horizontal axes show the values of the inflaton φ . As the field starts from the maximum and goes towards the minimum, the arrow of time is from right to left (i.e. for decreasing φ . ) We see from the bottom panel of Figure 3 that the slow-roll parameters ε and η stay small, hence the slow-roll regime holds all the way from the maximum down the minimum. Since ε 1, inflation continues until the minimum and there is no natural criterium marking the end of inflation. Of course, the value of the potential at the minimum being of order of the inflation scale, some new physics should be added to lower the potential near the actual cosmological constant. Nevertheless we see from Figure 3 that there is a huge number of e-folds all along the inflationary trajectory, i.e. between the would be horizon exit at η * = −0.02 (near the maximum) and the minimum. Hence, this model cannot accommodate hilltop-inflation scenario because the constraints N * 60 and η * = −0.02 cannot be satisfied together by adjusting x, the only relevant parameter here.

Inflation around the minimum from the inflection point
General idea. We now consider the case where the e-folds are obtained only near the minimum. This allows to alleviate the constraint η(φ + ) −0.02. We start with initial conditions near the maximum with no initial speed. We come back to this point at the end of the section. The inflationary phase corresponds to the inflaton rolling down its potential. As it goes from the maximum to the minimum, the second derivative V (φ ) changes sign and if η(φ + ) < −0.02, it will pass through the value η(φ * ) = −0.02 before the inflection point. The x parameter of the model can then be chosen so that 60 e-folds are obtained from this point to the end of inflation. From the above argument we see that in order this scenario to correctly match the observational data, the initial position of the inflaton has to be higher than the inflection point, where η is negative, so that η = −0.02 is taken at the horizon exit.
As in the hilltop case of section 3.2, we solved the evolution equation (34) numerically starting near the maximum with vanishing inflaton initial speed. We got N * 60 for x 3.3 10 −4 . In that case, φ − = 4.334 and φ + = 4.376. The e-folds are computed from the horizon exit φ * 4.354 at which η(φ * ) = −0.02, to the minimum φ − . Is should be observed that the corresponding inflaton field displacement ∆φ 0.02 , is much less than one in Planck units, corresponding to small field inflation, compatible with the validity of the effective field theory. We show the numerical solution in Figure 4. Again, the horizontal axes correspond to the inflaton φ . As the field starts from the maximum and goes towards the minimum, the arrow of time is from right to left (i.e. for decreasing φ .) In Figure 4 we see that at the minimum neither ε nor η is bigger than one (η is however close to 1). It is easy to understand from the plot of the ε parameter and formula (39) that almost all e-folds are obtained near the minimum because ε is very tiny there. The vertical line shows the value of φ * for which η(φ * ) = −0.02. It is very close to the inflection point, hence the modes exit the horizon near (a bit before) the inflection point. Study near the minimum. We wish to describe carefully what happens close to the minimum. Indeed, one has to check that the field goes on the other side of the minimum (φ < φ − ). We then expect that the field stops and goes back towards the minimum, starting its oscillation phase, usually related to the reheating period and the inflaton decay [24][25][26]. Nevertheless, one usually assumes tha the inflaton potential (almost) vanishes at the minimum. The inflaton decays into other particles and the cosmological constant stays then neglible in front of the radiation and matter densities for a sufficiently long period of time (assuming a solution to the cosmological constant problem).
In our model, for the parameter x 3.3 × 10 −4 chosen here, there is no scale separation between the inflation scale H * (at horizon exit) and the scale at the minimum, see Figure 2. Therefore the standard reheating scenario cannot occur, because the potential energy of the inflaton (or equivalently the cosmological constant) remains important at the minimum. Hence, if nothing is modified in the model the energy density of the created particles stays small compared to the cosmological constant. We come back to this discussion in section 4.2.
In the C++ program used to solve numerically (34), the field values are stored in cpp_dec_float types of variable, available in the multiprecision package of the boost library. These variables allow to store numbers of the form a × 10 n with a 100 digits precision on the coefficient a. When we reach the minimum, the field evolves very slowly and if φ stop − φ − is small, so is H stop − H min and the 100 digits precision is not enough to determine when the inflaton stops (i.e. when ∂ H/∂ φ = 0). In order to bypass this difficulty, we expand the Hubble parameter H around the slow-roll solution H sr by defining a new variable δ H through Replacing the first derivative of δ H, in (34), one finds the new form of equation (34) δ where we neglected the δ H 2 term. The advantage of this new formulation is that now, even if δ H is small compared to H sr , their values are not stored using the same coeffiicient and we do not have to store precisely H = H sr + δ H. The numerical solution of the evolution equation (46) confirms that the inflaton indeed reaches φ stop < φ − , but stays very close. Due to the small values of the slowroll parameters (see Figure 4) the inflaton is still in a slow-roll regime near the minimum and the oscillations are very slow. In fact we checked numerically that the number of e-folds during the first oscillations is greater than the one from φ * to φ − . As mentioned earlier, this is easily understandable considering the fact that due to the large value of V − = V (φ − ) (same scale as the inflation scale) the kinetic energy of the inflaton is not big enough to produce particles which would change significantly the equation of state of the Universe. We come back to this point in section 4.2.
Initial conditions. In the above study we started with the inflaton near the maximum with vanishing speed. In fact, one can change these initial conditions without altering the conclusions of the study as long as the inflaton starts between the maximum and the inflection point with a relatively small speed. Indeed, the constraint that the inflaton starts higher that the inflection point comes from the fact that it has to cross the η = −0.02 point when rolling towards the minimum. Nevertheless, an argument of symmetry restoration similar to the one explained in section 3.2 for the hilltop scenario motivates that the field starts near the maximum. In that case, if the initial speed stays relatively small, the inflaton is damped sufficiently near the maximum, such that the study does not change with respect with the case with vanishing initial speed. A solution for a non-zero initial speed is shown in Figure 5. If the initial speed is too large, equation (30) shows that the major contribution to the Hubble parameter comes from the inflaton kinetic energy. As V does not vary much from the maximum to the minimum, the inflaton only sees a flat potential until it reaches the wall at small φ . In that case the previous study does not hold, and slow-roll inflation is not obtained.

Physical observables and theoretical parameters
In this section we study the implications of the inflationary scenario described in section 3.3 to physical observables and we discuss the relation of the parameters of the model to those of the fundamental string theory.
Inflation scale. We see from Figure 4 that when the modes exit the horizon, the value of the slowroll parameter related to the amplitude of primordial fluctuations is ε * 2.5 × 10 −5 , implying a value for the ratio r of tensor to scalar perturbations From (42) we deduce that κ 4 V * = 24π 2 ε * A S 1.48 × 10 −11 .
This constraint fixes the overall amplitude of the scalar potential. Indeed for the x value of interest V * V (φ − ) and we see from (24) that the value of the potential at the minimum reads with w(x) = − 1 6 e −13+3W 0 (−e −x−1 ) 2 + 3W 0 (−e −x−1 ) .
For the value of x 3.3 × 10 −4 realising the inflationary scenario described in section 3.3, we obtain w(x) 1.87 × 10 −8 . Together with the constraint (48), equation (49) fixes the overall constant to Note that for q = 0 the value of C is different from the one used in the plots of Figure 4, but as we explained in section 2.2, the overall constant just scales the potential and has no implication in the study of the inflation phase dynamics (in particular, it does not appear in the slow-roll parameters computation). From (48) we deduce that the inflation scale is H * κ V * 3 2.2 × 10 −6 κ −1 5.28 × 10 9 TeV .
In the last equality we used the value of the Planck scale that we recall here: κ −1 2.4 × 10 18 GeV. As mentioned earlier, we observe from Figure 4 that the value of the potential at the horizon exit and at the minimum are almost identical. Hence the positive value of the potential at the de Sitter minimum is given by κ 4 V dS κ 4 V * and is way above the observed cosmological constant today.
String parameters. We now relate the parameters of the model to those of the underlying string theory and examine the constraints implied by the inflationary scenario described above. The string parameters are: ξ , γ, related to the quantum corrections, d associated with the anomalous U(1) charges of the 7-branes, and W 0 the constant superpotential remaining after complex structure moduli and axion-dilaton stabilisation. For the sake of clarity, we write again the expressions for the first two in (3) and (4), resulting from string computations of quantum corrections [7]: It follows that the parameters entering the scalar potential, introduced in (17), read: As already mentioned, note that in order to have C > 0 we need γ < 0 and hence a negative Euler number χ CY . We have also defined the x parameter by From (21) we deduce that the volume at the minimum is a function of q and x only: Thus, for a given value of x, one obtains large volume for large (negative) q. In fact from (54), q is indeed negative for positive T 0 , implying a surplus (locally) of D7-branes relative to orientifold O7planes [7]. Then large values of q are reached as long as g s is small. Hence the weak coupling and large volume limits are related in a simple way. We now turn back to the string parameters W 0 and d, which are partially fixed by the observational constraint through (51). From the expressions (54), the superpotential reads whereas the d parameter from the U(1) D-terms of D7 branes is We see from (57) that for values of −γ around 10 −2 , the value W 0 ∼ 1 is reached as soon as −q > ∼ 5.
On the other hand, from (58) we see that d ∼ 1 is reached for −q > ∼ 7. We conclude that for −q = 1/(g s T 0 ) not much greater than a few units, our inflationary model can be accommodated in the weak string coupling and large volume limits. This justifies that the large volume limit could safely be taken in the expressions (6) -(7) of the the scalar potential contributions V F and V D . Moreover, the superpotential W 0 and D-term coefficient d take values of order one. In fact, W 0 around unity can be naturally obtained from combinations of integer fluxes.

New physics around the minimum
In the section 3 we studied the inflation possibilities for the scalar potential of the type IIB string model described in section 2. We have seen in section 3.3 that it is possible to realise an inflationary period near the minimum of the potential, with horizon exit near the inflection point, by adjusting the parameter x introduced in (23). We now address two important remaining questions concerning the stability of the minimum and the scenario for the end of the inflationary era.

Stability of the minimum
For the value of x considered in section 3.3 in order to get an inflationary period, the values of the potential at the minimum and maximum are very close. Hence it is important to know if the inflaton can escape from the local minimum, or the false vacuum, and tunnel through the barrier of the potential before evolving classically towards the true minimum in the runaway direction at large field values. We recall that the shape of the potential for the value of the parameter x giving an inflationary epoch is similar to the one shown in the right panel of Figure 1.
To evaluate the false vacuum stability we use the methods developed by Coleman et al. [19,20]. In order to keep their conventions, we will use the dimension-full inflaton inflaton going above the potential barrier instead of properly tunnelling. As mentioned after (66) such solutions always exist, but when H − < H c their action is higher than the CdL solutions and give thus negligible contribution to the tunnelling rate (60). In Figure 8, we see that H − > H c holds for x = 3.3 × 10 −4 , which explains that we are not able to find the standard CdL instanton. Hence in this case only the HM instanton contribute to the tunnelling rate. The tunnelling coefficient B introduced in (60) is then computed from (67) and reads In the last equality, ∆V is the height of the barrier and H * is the inflation scale, i.e. the Hubble parameter when the modes exit the horizon. We recall that since in our model the potential is almost flat along the inflationary trajectory, we have H − H + H * . For x = 3.3 × 10 −4 we find ∆V 2.0 × 10 −4 V * and from (48) -(52) we deduce B 3.3 × 10 9 , i.e. Γ = Ae −3.3×10 9 .
It follows that the decay rate of the local minimum is extremely large and the vacuum is practically stable.

End of inflation: new physics around the minimum
As established in the section 3.3, having 60 e-folds near the minimum constrains the energy of the dS vacuum. This value is way much greater than the observed value today, hence this dS vacuum cannot be the true vacuum of the theory. Indeed, with such a big value, the Universe would continue expanding and never reach the standard cosmology with radiation and matter domination eras. Hence we see that in our model the cosmological constant problem arises naturally coupled to inflation. Of course we will not tackle this problem here, but show that taking it into account through the introduction of new physics near the minimum of our potential brings in a natural scenario for the end of the inflation epoch. This relates our model to the hybrid inflation proposal [3], where a second field is added to the model. This so-called "waterfall" field adds another direction to the scalar potential. If falling into this direction becomes favorable at a certain point of the inflaton trajectory, this immediately ends the inflation era and the theory reaches another minimum at a different energy scale. Below we describe a toy model of hybrid inflation adapted to our model. Consider the following Lagrangian for the inflaton φ and the extra waterfall field S S can be seen as a Higgs-like field undergoing symmetry breaking at an intermediate energy scale located between the inflation scale and the minimum of the potential. In equation (73), V (φ ) is the inflaton potential studied previously and V S (φ , S) an additional part containing the dependence in S and its coupling with φ . We express this second contribution in the following way with f (S) some function of S. Depending on the sign of its effective squared mass m S 2 = −M 2 + f (φ ), the waterfall field S stays in two separate phases. When m S 2 > 0, i.e. M 2 < f (φ ), the minimum in the S-field direction is at the origin and the extra contribution to the scalar potential vanishes When the mass of S becomes tachyonic, a phase transition occurs and the new vacuum is obtained at a non-vanishing S vacuum expectation value S = ± |m S | √ λ ≡ ±v, when m S 2 = −M 2 + f (φ ) < 0.
The value of the potential V S at the minimum of this broken phase is We can choose f (φ ) such that during the phase of inflation, when the field φ rolls down the potential, we stay in the symmetric phase and the S field is stabilised with a vanishing vacuum expectation value and a large positive mass. The inflation phase is then equivalent to the one field model studied in section 3.3.
If near the minimum m S 2 < 0, a phase transition occurs and the S field goes to its value (77) at the new minimum. This amounts to change the potential V (φ ) near the minimum, by a constant V down = V S (v) < 0. The effect of such a downlift is double: it decreases the value of the cosmological constant and if the waterfall direction is steep enough, it gives a natural criterium to stop inflation (ε > 1). We show in Figure 9 a schematic plot of such a potential.

Conclusion
Reconciling moduli stabilisation and de Sitter vacua is a key issue in the quest of an effective potential appropriate for cosmological inflation. In the type IIB string theory framework of the present work, it is shown that a (suitable) non-vanishing potential can be generated with the internal volume modulus playing the role of the inflaton φ , triggering exponential growth of the Universe. Intersecting spacefilling D7-branes on the other hand, are the cornerstones of the stabilisation mechanism for Kähler moduli, and the creation of a positive cosmological constant in agreement with Universe's accelerated expansion today. Then, Kähler moduli stabilisation is achieved thanks to the logarithmic radiative corrections induced when effectively massless closed strings traverse their codimension-two bulk towards localised gravity sources. Moreover, the dS vacuum is obtained due to the positive D-term contributions whose origin comes from U(1) factors related to the intersecting D7-branes.
In the large volume limit, the induced effective potential for Kähler moduli receives a minimalist structure where its shape and in particular the volume separation, ∆φ , of its two local extrema can be parametrised in terms of a single non-negative parameter, x. The largest, albeit rather small ∆φ separation occurs at a critical value x c > 0 where beyond this point only AdS solutions are admissible. As x attains smaller values, the distance between the two extrema diminishes and at the final admissible point x = 0, it collapses to zero too. The upshot of the above picture is that there exists a non-zero value x < x c at which a new inflationary small-field scenario is successfully implemented.
It is shown that this novel scenario is quite distinct from other well known solutions, such as hilltop inflation. Its main benchmarks are: 1. Most of the required number of e-folds (∼ 60) are collected in the vicinity of the minimum of the potential, while the horizon exit arises near (from above) the inflection point. 22 2. The corresponding inflaton field displacement is of order 10 −2 compatible with the validity of the effective field theory in small-field inflation.
3. The prediction for the tensor-to-scalar ratio of primordial density fluctuations in the early universe is r ≈ 4 × 10 −4 .
4. As explained above, the potential is induced by radiative corrections which yield a false vacuum expected to decay to the true one towards the direction of large φ values. Implementing well established methods for the possible tunnelling [20] or its passing over the potential barrier [21], a detailed study of the decay rate is performed and found that the false vacuum has an extremely long lifetime.
While inflation is successfully described close in on a sufficiently long-lived minimum of the potential, yet the cosmological constant acquires a rather large value compared to that observed today. We describe how this problem can be evaded within the context of hybrid inflation which can be realised when a second field creates a new "waterfall" direction in the potential and inflation stops as soon as the slow-roll parameter ε exceeds unity. In closing, it is worth emphasising that the successful implementation of the cosmological inflation in the above analysis is based only on a few simple characteristics occurring in generic type IIB string vacua. The few coefficients involved [7] depend on well defined topological properties such as the Euler characteristic of the compactification manifold, and the coefficients of the D-terms determined by the geometric configuration of the intersecting D7-brane stacks. The robustness of the results is further corroborated by the fact that, for all the measurable inflationary observables, the implications of the logarithmic corrections and the D-terms is conveyed just through the parameter x which depends only on the ratio of their coefficients and the fluxed superpotential. Consequently, the present analysis can in principle apply to an ample class of vacua in the string landscape.