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Physics Letters B

www.elsevier.com/locate/physletb

No-scale D-term inflation with stabilized moduli

Wilfried Buchmüller, Valerie Domcke, Clemens Wieck *

Deutsches Elektronen-Synchrotron, Hamburg, Germany

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ARTICLE INFO

Article history:

Received 19 September 2013

Received in revised form 13 December 2013

Accepted 16 January 2014

Available online 23 January 2014

Editor: G.F. Giudice

ABSTRACT

We study the consistency of hybrid inflation and moduli stabilization, using the Kallosh-Linde model as an example for the latter. We find that F-term hybrid inflation is not viable since inflationary trajectories are destabilized by tachyonic modes. On the other hand, D-term hybrid inflation is naturally compatible with moduli stabilization due to the absence of a large superpotential term during the inflationary phase. Our model turns out to be equivalent to superconformal D-term inflation and it therefore successfully accounts for the CMB data in the large-field regime. Supersymmetry breaking can be incorporated via an O'Raifeartaigh model. For GUT-scale inflation one obtains stringent bounds on the gravitino mass. A rough estimate yields 105 GeV < m3/2 < 1010 GeV, contrary to naive expectation.

© 2014 The Authors. Published by Elsevier B.V. All rights reserved.

1. Introduction

Hybrid inflation [1] is an attractive mechanism for generating the cosmological density perturbations. It is naturally realized in the framework of grand unified theories (GUTs) and string theory, as F-term [2,3] or D-term inflation [4,5] where the GUT scale emerges through the Fayet-Iliopoulos (FI) term of an anomalous U (1) symmetry. However, the embedding of hybrid inflation in a UV-complete theory, which has significant effects on GUT-scale inflation, remains an open question.

The probably best-motivated UV-complete theory for this embedding is string theory. In this framework, six dimensions have to be compactified on a Calabi-Yau manifold to obtain a four-dimensional effective theory with N = 1 supersymmetry. In the classical perturbative four-dimensional theory massless scalar fields, so-called moduli, arise as remnants of the internal manifold. The stabilization of these moduli has been a widely discussed subject for many years. In type IIB string compactifications on Calabi-Yau manifolds with D-branes and fluxes, it has been shown that all complex structure moduli and the axio-dilaton can be stabilized by fluxes [6]. Kahler moduli, on the other hand, can be stabilized by non-perturbative contributions to the superpotential, such as gaugino condensates on stacks of D-branes [7]. The latter have been used in a model by Kallosh and Linde (KL) [8], where a single Kahler modulus is stabilized in a racetrack poten-

* This is an open access article under the CC BY license (http://creativecommons. org/licenses/by/3.0/). Funded by SCOAP3.

* Corresponding author.

E-mail addresses: wilfried.buchmueller@desy.de (W. Buchmuller), valerie.domcke@desy.de (V. Domcke), clemens.wieck@desy.de (C. Wieck).

tial with vanishing vacuum energy in a local minimum. This setup has the appealing feature of scale separation between the Hubble scale Hinf during inflation and the gravitino mass, which can be very small compared to Hjnf.

In this Letter, we study the effects of stabilizing the Kahler modulus in such a racetrack potential on the dynamics of hybrid inflation. As was pointed out in [9], even a tiny displacement of the modulus field due to its gravitational coupling to the inflaton field can be fatal for a potential inflationary trajectory, as can be seen explicitly by integrating out the modulus field. Our work is related to earlier attempts of combining hybrid inflation and moduli stabilization in F-term [9-11] and D-term inflation [12] as well as in chaotic inflation [13]. Here, we use a specific form of Kahler potential, motivated by the no-scale Kahler potential of the modulus field and an approximate superconformal symmetry. Similar to [9,10], we find that F-term hybrid inflation is spoiled by corrections induced by the modulus sector. In particular, whenever one direction of the complex inflaton is flat, the other one is tachyonic. However, we find that D-term hybrid inflation can be successfully combined with moduli stabilization.

The resulting no-scale D-term inflation model has a number of interesting features. Along the inflationary trajectory it is actually equivalent to the superconformal D-term inflation model proposed in [14]. As shown in [15], in the large field regime it asymptotically yields the Starobinsky model [16], which agrees remarkably well with the recently released Planck data [17]. Supersymme-try breaking can be accomplished by adding a quantum corrected O'Raifeartaigh model [18] without spoiling moduli stabilization or inflation. For GUT-scale inflation one obtains stringent bounds on the gravitino mass.

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This Letter is organized as follows. Our scheme of racetrack moduli stabilization and its coupling to F-term hybrid inflation is discussed in Section 2. Turning to D-term hybrid inflation in Section 3, we calculate all relevant corrections to the inflationary dynamics arising from moduli stabilization, summarize the inflationary predictions, and discuss supersymmetry breaking in this context. We conclude in Section 4.

2. F-term hybrid inflation

In its simplest form, the superpotential of F-term hybrid inflation in terms of the chiral superfields S, 0+, and 0— can be written as [2]

Whi = AS(0+0-- v2). (2.1)

In this setup, S contains the inflaton and 0± are the so-called waterfall fields, carrying charge ±q under some local U (1) symmetry, which are responsible for ending inflation. Moreover, v is of the same order as the GUT scale and the coupling X is chosen to be real.

The slow-roll potential for the inflaton is typically generated by supergravity interactions and the one-loop Coleman-Weinberg potential. At a critical field value Sc = v the waterfall fields obtain a tachyonic mass and inflation ends with spontaneous symmetry breaking of the U (1) symmetry. For a more detailed account of the dynamics and phenomenology of F-term hybrid inflation in super-gravity, see e.g. [19,20].

2.1. KL moduli stabilization

When hybrid inflation is embedded in a higher-dimensional theory, the question of moduli stabilization has to be addressed. For simplicity, we consider a scenario in which the overall volume of the compactified dimensions is parameterized by a single Kahler modulus p = a + ifi. This case is well understood in type IIB string theory. In particular, it is assumed that the dilaton and all complex structure moduli have been stabilized by fluxes [6] and only one Kahler modulus remains massless. This Kahler modulus can be stabilized by non-perturbative contributions to the superpotential [7,8] in combination with a no-scale Kahler potential,

K = —3ln (p + p). (2.2)

In case of two non-perturbative terms, the superpotential reads

W KL = W 0 + Ae—ap + Be—bp.

Here, Wo, A, and B are determined by fluxes, and the non-perturbative terms in Eq. (2.3) are generated by gaugino condensates on stacks of D-branes. The parameters a and b are given by 2r, i e {a, b}, where Ni are the ranks of the condensed gauge

groupis.

In the model of Kallosh and Linde [8], W0 is adjusted to produce a supersymmetric Minkowski vacuum. The minimum of V occurs at /) = 0 and

a = ao =

a — b

This is achieved by choosing

/aA\b—a /aA\b—a

Wo=—a U — b U '

such that WKL(a0) = Dp WKL(a0) = V(a0) = 0. In this setup, the modulus is generically very heavy,

mp = - (a — b)abAB ln — —

aA\faA\

so that mp ~ 0(10 3-10 in Planck units, for typical parameter values. Note that the compactified dimensions have to be stabilized

at large enough volume V = O(a0' ) to satisfy both the supergrav-ity approximation and the single-instanton approximation of this analysis. In particular, it is required that a0 ^ 1 and aa0, ba0 ^ 1. In the following, we assume a0 = 0(10-100) for typical values of the racetrack parameters.

2.2. Effective scalar potential

Combining the two sectors discussed above to a model with superpotential

W = W kl + W HI ,

with unspoiled inflation turns out to be a non-trivial task. As pointed out in [9,10], even when the modulus mass is larger than the inflationary Hubble scale, supergravity corrections from the modulus sector generically ruin inflation. During the slow-roll phase the minimum of the modulus potential is slightly shifted, causing the modulus to move by an amount Sp during inflation. The back-reaction of this shift generates a large mass for the infla-ton so that n = 0(1). This problem persists when using a no-scale Kahler potential with or without a shift symmetry for the inflaton [10].

However, this n-problem can be overcome using a particular Kahler potential,

K = —3ln

p + p — 3 (IS |2 +10+|2 +10— I2) — X (s 2 + s 2)

= —3ln X,

with x e R, which has approximate no-scale form [21] with an SU(1, 3) symmetry broken only by the term proportional to x. As discussed in Section 3.2 this type of Kahler potential is also well motivated from the underlying superconformal symmetry of super-gravity (see, e.g. [22]). Note that for x = 1 Eq. (2.8) reproduces the standard form of a shift symmetric no-scale Kahler potential. Using Eq. (2.7) and Eq. (2.8) the scalar potential during inflation, i.e., at S > v and 0+ = 0- = 0, reads

v = ¿{ x2 v4 + K x + 31 s + x SI2

W — 3 Av2(S + x S) JWKl + c.c.

where primes denote derivatives with respect to p. To take the shift of the modulus during inflation into account we expand Eq. (2.9) in the displacement Sp = p — a0, where a0 denotes the minimum of the pure KL potential, i.e., the minimum after inflation. Thus, we compute the effective potential

Veff = V + ( dp VSp + -dzp VSp2 + c.c.

+ dpdp V SpSp + O (Sp3

(2.10)

and eliminate Sp demanding that Veff be minimized, i.e., dSp Veff = dSp Veff = 0. At second order in S this yields

V eff =

2a0 + x( S2 + S2) —

X 2 + 2

+ O |S|3 .

(2.11)

Evidently, there are two possible values of x which allow for a vanishing mass of Re S and Im S, respectively, and hence for flat directions suitable for inflation,

= ±(3 ± V7 ).

(2.12)

However, it turns out that for any value of x, either Re S or Im S has a tachyonic mass, since

mRe S = -

mIm S = -

(x2 - 6x + 2, (x2 + 6x + 2).

(2.13a) (2.13b)

Thus, any possible inflationary trajectory is destabilized. Note that tachyonic masses of this order cannot be canceled by masses stemming from the Coleman-Weinberg one-loop potential. Therefore, minimal F-term hybrid inflation appears impossible in this simple setup of moduli stabilization. This conclusion leads us to consider a model of D-term hybrid inflation, where the moduli corrections to the inflationary sector are negligible.

3. D-term hybrid inflation

In D-term inflation the picture is quite different from the previously discussed case. It has the appealing feature that a GUT-scale Fayet-Iliopoulos term1 can be naturally generated from anomalous U(1) symmetries in certain string compactifications [23,26]. This FI-term, together with quantum corrections to the scalar potential, drives inflation. Although D-term inflation is well motivated from string theory, it is necessary to check whether a consistent stabilization of all moduli is possible.2

The superpotential of D-term hybrid inflation reads

W di = XS0+0-.

In pure D-term inflation without moduli stabilization, using a no-scale Kahler potential for the relevant fields results in an F-term potential equivalent to the one of F-term hybrid inflation with v = 0. The inflationary trajectory corresponds to a flat direction along 0± = 0. The D-term potential is generated by the FI-term Ç and the waterfall fields which have non-zero charges under a U (1)

gauge symmetry with coupling g. During inflation, it induces

a vacuum energy V0 = . For a detailed description of D-term inflation with canonical Kahler potential, see [4,5].

3.1. Moduli corrections

In our model the superpotential is given by

W = W kl + W DI, (3.2)

and the Kahler potential is the same as in Eq. (2.8). To determine the influence of the modulus sector on the inflation sector we proceed as in the F-term case, i.e., we expand the potential in the displacement Sp, minimize it, and investigate the resulting effective potential for S and 0±. Before integrating out the modulus, the scalar potential is given by V = VF + VD, with

1 The consistency of a constant FI-term in supergravity is a subtle issue [23-25], which we do not address in this Letter. In this context, an interesting approach was used in [12], generating an effective FI-term from vacuum expectation values in the modulus sector.

2 Note that the coupling to a KKLT-type modulus sector using a different Kahler

potential has been investigated in [12] along similar lines. For a recent discussion

and further references, see [27].

V F = X21S |2 (|0+12 + |0-12) + X2|0+0-12

p + p + X (s2 + S2) + - x 2| s !2

lw k l|-

WKL - 3XS0+0- )WKl + c.c.

(|0+12 - !0-|2) - Ï

(3.3a) (3.3b)

with X as defined in Eq. (2.8). Since Veff is much more complicated than the compact expression in the F-term scenario, cf. Eq. (2.10), we restrict ourselves to providing the moduli corrections to the most important quantities. These are, in particular, the scalar masses in the inflation sector.

The inflaton receives a non-zero mass contribution not only from the non-vanishing derivative of WKL in Eq. (3.3a), but also from terms which arise after performing the expansion Eq. (2.10), i.e., from integrating out the modulus. However, the resulting correction is zero to first order in WKL and WKL and can thus be neglected since WKL, W^L < 0(10-6) for typical values of the racetrack parameters, which renders the corrections much smaller than the contributions from the Coleman-Weinberg potential. Remember that WKL and its derivative have to be evaluated at values of p slightly shifted from a0, thus yielding non-zero results. The same order of suppression applies to the correction of the first derivative of the scalar potential, proportional to the slow-roll parameter e. This justifies treating S as a flat direction of the tree-level scalar potential of the combined theory, as in the pure D-term case.

Corrections to the masses of the waterfall fields are small as well. The end of inflation occurs when one of the waterfall fields obtains a tachyonic mass. Thus, large corrections to the waterfall masses can have grave consequences for the inflationary dynamics. Following the same procedure as for the inflaton mass, we obtain

m0± = m0±,0 + Am0± (Wk^

l0±,0 ■

|S |2 2 fc t g q^.

The latter, with X0 = 2a0 - 11S |2 - £ (S2 + S2) after integrating out the modulus, is the standard result from pure D-term inflation. The leading order corrections are of the form

Am0± =

2Y 0 W K L - 6W kl / 2X2|S |2

Y 0 X0 W K'L V X0 2 ,,,1 2

t g2q£

+ O (W KL2, W K L2, W KL W K L,..),

with Y0 = X0 + 11S + xS|2. Note that these corrections are para-metrically larger than the ones found in [12], due to effective mass terms stemming from the expansion in Sp. However, since Wkl, WKl ^ WKl ~ mp, the correction is still negligibly

small and does not influence the dynamics of inflation significantly. Moreover, there are no corrections which cause 0± to be stabilized away from the origin.

3.2. Superconformal symmetry and the Starobinsky model

Having identified a promising D-term hybrid inflation model with stabilized moduli, we now turn to the phenomenological consequences of this model. Interestingly, during inflation this model

is actually equivalent to a model based on a superconformal symmetry [14]. There, the superpotential is identical to the one in Eq. (3.1) and the Kahler potential reads

Ksc = -3ln( -1 where

0 = -3 + |0+|2 + |0-|2 + |S |2 + I (S2 + S2)

is the so-called frame function. This type of frame function characterizes a large class of models, dubbed canonical superconformal supergravity models in [22]. They feature a remarkably simple structure in the Jordan frame with canonical kinetic terms and a scalar potential which closely resembles that of global supersym-metry. The superconformal symmetry, which is the starting point in constructing these models, is explicitly broken by gauge fixing the so-called compensator field, resulting in the appearance of the Planck scale and the Fl-term, and by the term proportional to x in Eq. (3.8). This particular symmetry breaking structure allows to keep the attractive features implied by the superconformal symmetry, cf. [14,22] for details.

In [14] the D-term scalar potential is found to be

p-2 2 VD = y ß2q(|0+|2 -|0-|2) - £]

with Q2 = — 3. It is straightforward to verify that this is identical to Eq. (3.3b) after rescaling

S = V P + p S0± = V P + p0± •

(3.10)

The F-term scalar potential is determined by the Kahler function K + ln |W|2, which is invariant under the transformation (3.10) since

K sc (S ,0±) = —3ln (VP + P) + ln Q—2(S' ),

ln | W(S,0±) |2 = +3ln {Jp + p) + ln | W(S)|2. (3.11)

Hence, even after rescaling the discussion of Section 3.1 remains valid and the F-term potential vanishes along the inflationary trajectory, as it does in the model of [14].

Along the inflationary trajectory, the two models thus feature the same scalar potential, allowing us to apply the analysis of [14] to the model presented here. Here we merely summarize the most important results: We find a two-field inflation model with an at-tractor solution along the real (imaginary) axis for negative (positive) values of x . At the end of hybrid inflation, cosmic strings are formed. The spectral index can be as low as ns & 0.96. However, for generic values of the gauge coupling g and the U (1) charges ±q of the waterfall fields, this leads to a too large cosmic string tension, violating the bound obtained from the recent Planck results [28].

This problem can be circumvented by choosing a relatively large value for gq, i.e., 10 > gq > X, cf. [15]. In this case agreement with all Planck results can be achieved, including the cosmic string bound [17,28]. Remarkably, in the large-field regime and for an inflationary trajectory along the attractor solution, the model is asymptotically equivalent to the Starobinsky model [16]. In particular, to leading order in 1/N*, with N* the number of e-folds elapsed after the reference scale of the CMB fluctuations exited the horizon, the scalar spectral index, the tensor-to-scalar ratio, and the running of the spectral index are given by

: 1--,

dns dlnk

(3.12)

which, for N* & 55, describes the Planck data very well [17].3 For g2 & 1, as expected for a GUT gauge coupling, requiring the correct normalization of the scalar contribution to the primordial fluctuations fixes the Fl-term at roughly the GUT scale, Vf & 7.7 x 1015 GeV. For example, for q = 8 this implies a cosmic string tension of G¡1 & 3.16 x 10—7, very close the recent Planck limit G¡l < 3.2 x 10—7 [28]. This large value of q is problematic from the point of view of GUT model building, which suggests to explore alternative ways to satisfy the cosmic string bound, cf. [14].

3.3. Low-energy supersymmetry breaking

During inflation the D-term inflation model under consideration

exhibits a positive vacuum energy V0 = and thus, supersym-metry is broken. After inflation has ended, however, one of the waterfall fields receives a vacuum expectation value which causes VD to vanish identically, while the other one and the inflaton are stabilized at the origin. lt then follows that VF = VD = m3/2 = 0 after inflation, i.e., supersymmetry is restored. In view of low-energy phenomenology, it is thus necessary to check whether the presented model can be combined with a separate sector of su-persymmetry breaking without spoiling either inflation or moduli stabilization.

A simple way of breaking supersymmetry is adding a quantum corrected O'Raifeartaigh model with the following Kahler potential and superpotential for a chiral 'Polonyi' field P [18],

2 IP I4 2

Kp =|P I2 — ' Wp = ¡2 P. (3.13)

Here, heavy fields of mass A ^ 1 have been integrated out, and ¡2 is the scale of supersymmetry breaking. In addition, to allow for a small or vanishing cosmological constant we tune the value of W0 away from the KL-value Eq. (2.5) by an amount AW0. In an underlying string compactification this is achieved by slightly tuning the flux quanta which determine the vacuum expectation value of the Gukov-Vafa-Witten potential. As a result, a complete model with broken supersymmetry can be defined by

K = —3ln X + Kp, W = W kl + W di + W p + AW 0. (3.14)

Note that the supersymmetry breaking sector is not of no-scale form. This is phenomenologically required for low-energy super-symmetry breaking [29]. The derivation of this Kahler potential from a higher-dimensional theory remains an open problem.

The compatibility of this supersymmetry breaking mechanism with moduli stabilization has been studied in [18,29,30]. The constant AW0 shifts the Minkowski minimum of the potential to an AdS minimum with VAdS & — ¿iM^aL at roughly the same value

of ct0. The uplift due to the Polonyi field raises the value of V in the minimum to zero if

(3.15)

resulting in a Minkowski vacuum with broken supersymmetry. In this vacuum the gravitino mass is given by

(3.16)

at leading order in ß2 and A.

3 Note that inflation terminates at Sn & 1, cf. [15], so that corrections to the Kah-

ler potential Eq. (2.8), suppressed by inverse powers of the Planck mass, may be relevant.

In this minimum the Polonyi field is stabilized on the real axis at P0 « A2. Moreover, it is possible to decouple the Polonyi field before the beginning of inflation, i.e., at masses larger than the inflationary Hubble scale. We can achieve a mass hierarchy

mp> mp > Hinf» m3/2,

(3.17)

by appropriately choosing A, and the parameters in Wkl. Specifically, the mass of the Polonyi field m2p in the Minkowski minimum reads

/ 2 2^ » ^

(3.18)

Notice that we have used a0 in all of the above expressions because the back-reaction of the shift Sp on the dynamics of the Polonyi field is negligible. However, it is important to keep in mind that p is still slightly shifted away from a0 due to the presence of P and AW0, so that WKL, WKL = 0. Requiring that mP > Hjnf and demanding A > i in the effective theory (3.13) leads to the lower bounds ¡,A > 10—5 for typical values of the racetrack parameters.4

Remarkably, as a consequence GUT-scale inflation implies a stringent lower bound on the gravitino mass. From Eqs. (3.16)-(3.18) one obtains

m2/2 > 0.1 A2H2nf > 10—25. (3.19)

Starting from the KL model for moduli stabilization, one may have expected that an arbitrarily small value of the gravitino mass is possible. However, since both mP and the mass scale A are constrained by the GUT scale, one is driven to a regime of 'high-scale supersymmetry' with m3/2 > 105 GeV. Even if the Polonyi field is allowed to be lighter than Hinf but heavier than the inflaton, thus taking part in the dynamics of inflation, this bound is not significantly relaxed.

Notice that the choice of parameters in the Polonyi sector only slightly influences the modulus sector and vice versa. Therefore, in a large portion of parameter space the proposed mechanism of su-persymmetry breaking does not interfere with moduli stabilization. Especially, even if ¡i is chosen to be very large compared to the GUT scale, additional tuning of AW0 will always prevent destabi-lization of the modulus.

Quantifying the impact of the Polonyi field on the inflationary dynamics is slightly more involved. As in our previous discussion of moduli corrections to the inflaton sector, the impact on e, the inflaton mass, and the waterfall masses has to be evaluated. In order to consider all possible terms, we proceed along the lines of Section 3.1 and take a possible shift SP during inflation into account, as well as corrections resulting from integrating out the modulus. This results in the following corrections:

• The mass of the inflaton, which can be chosen to be the real part of S, receives the correction

Ami S = m2V2 0 + X)2'

(3.20)

at leading order in ¡ 2 and S2.5 Note that this term is present even before integrating out p, which yields small corrections of higher order in S2. Eq. (3.20) implies that successful inflation also puts an upper bound on the gravitino mass,6 unless

4 Here we have used Hinf ~ 0.1MGUT, with MGUT > 10—3.

5 We have assumed that 2a0 > | S |2 towards the end of inflation, which is satisfied even in the large field regime discussed in [15].

6 We thank the referee for pointing this out.

X = —1, which corresponds to a shift-symmetric Kahler potential. For x = — 1, demanding that the correction to the slow-roll parameter n does not alter the prediction for ns by more than 1a, cf. the recent Planck data [17], leads to m3/2 < 1010 GeV/|1 + x|. The bound resulting from the correction to the slow-roll parameter e is less severe.

The leading order mass correction to the waterfall fields originates solely from the effective potential Veff where the modulus has been integrated out, analog to the corrections in Eq. (3.6). Specifically,

Am4>± =

/2(S2 + S2 + 2x |S|2)

(3.21)

with m^±,0 defined by Eq. (3.5). Depending on the size of i these corrections can be parametrically larger than the ones from the modulus sector, cf. Eq. (3.6). However, since ¡2 is smaller than W¡KL for typical racetrack parameter values, is still suppressed by at least three orders of magnitude compared to m2^± 0.

We conclude that our model can be extended by a simple su-persymmetry breaking sector without spoiling any of its features. In this setup, the gravitino mass has to satisfy lower and upper bounds,

105 GeV < m3/2 < 1010 GeV,

(3.22)

which are due to the high scale of inflation and the slow roll conditions, respectively.

4. Conclusion

In light of the recent Planck data, slow-roll inflation remains a very successful paradigm for the earliest stages of our universe. Realizing this paradigm in a concrete UV-completed particle physics theory, however, faces a number of challenges, including the identification of the particle physics nature of the inflaton, a possible embedding in string theory and the connection to su-persymmetry breaking after inflation. Here, we propose a model of supersymmetric hybrid inflation which allows for racetrack moduli stabilization, as employed in certain type IIB string compact-ifications, as well as for supersymmetry breaking by means of a quantum corrected O'Raifeartaigh model, while simultaneously explaining the cosmological parameters measured by the Planck satellite.

Using the standard no-scale Kahler potential, augmented by a symmetry breaking term, we find that F-term hybrid inflation is unfeasible. Generically, the inflaton mass receives large corrections, spoiling slow-roll inflation. While this can be remedied by tuning the symmetry breaking parameter x, the presence of a large tachyonic mass destabilizing any potential inflationary trajectory is unavoidable. However, supersymmetric D-term hybrid inflation is not plagued by this problem. Tracking the evolution of the modulus field during inflation and integrating out the modulus, we find that the corrections to the inflationary dynamics induced by the modulus sector are small. If the modulus is stabilized before the onset of inflation, i.e., mp > Hjnf, we obtain an effective inflation model which, along the inflationary trajectory, is identical to su-perconformal D-term inflation.

Concerning the inflationary predictions, i.e., amplitude and spectral indices of the CMB power spectrum, we find very good agreement with the recent Planck data. Generically, cosmic strings produced at the end of D-term inflation exhibit a string tension exceeding current bounds. However, viable regions of parameter

space remain, for large values of the waterfall U(1) charge q. In the large-field regime the scalar potential of the inflaton field is identical to that of the Starobinsky model.

In order to account for supersymmetry breaking in the Minkowski vacuum after inflation, we add a quantum corrected O'Raifeartaigh model. We calculate possible interactions between the inflation, modulus, and Polonyi field sector. We find that the only displacement of the modulus minimum resulting in relevant corrections is the one stemming from the slow-roll of the inflaton. Generically, however, all these corrections turn out to be small, allowing for an effectively decoupled supersymmetry breaking sector. The gravitino mass is constrained to the range 105 GeV < тз/2 < 1010 GeV.

In summary, we present a working model of inflation, successfully combined with KL moduli stabilization and supersymmetry breaking and in accordance with experimental data. Further interesting questions concern the embedding of our model into a higher-dimensional GUT or string model, and the implications for low-energy particle phenomenology.

Acknowledgements

The authors thank Ido Ben-Dayan and Alexander Westphal for helpful discussions. This work has been supported by the German Science Foundation (DFG) within the Collaborative Research Center 676 "Particles, Strings and the Early Universe". The work of C.W. is supported by a scholarship of the Joachim Herz Stiftung.

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