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Minimal In
ation |
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Luis Alvarez-Gaum ea, C esar G omezb,a, Raul Jimenezc,a |
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aTheory Group, Physics Department, CERN, CH-1211, Geneva 23, Switzerland. |
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bInstituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, E-28049 Madrid, Spain. |
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cICREA & Institute of Sciences of the Cosmos (ICC), University of Barcelona, 08028 Barcelona, Spain. |
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Abstract |
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Using the universal Xsupereld that measures in the UV the violation of conformal invariance we build up a model |
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of multield in
ation. The underlying dynamics is the one controlling the natural
ow of this eld in the IR to the |
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Goldstino supereld once SUSY is broken. We show that
at directions satisfying the slow roll conditions exist only if |
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R-symmetry is broken. Naturalness of our model leads to scales of SUSY breaking of the order of 1011 13Gev, a nearly |
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scale-invariant spectrum of the initial perturbations and negligible gravitational waves. We obtain that the in
aton eld |
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is lighter than the gravitino by an amount determined by the slow roll parameter . The existence of slow-roll conditions |
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is directly linked to the values of supersymmetry and R-symmetry breaking scales. We make cosmological predictions |
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of our model and compare them to current data. |
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Key words: SUSY; cosmology; in
ation |
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1. Introduction |
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In spite of the enormous success of in
ationary cosmol- |
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ogy [1, 2, 3, 4, 5, 6, 7, 8, 9] at describing the observed |
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properties of the Universe, we are still missing a deriva- |
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tion from rst principles where the in
aton eld is iden- |
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tied with one, or several, fundamental elds in particle |
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physics. This manifests itself the in the fact that we still |
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do not count with a natural way of identifying the in
aton |
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eld and the properties of its potential required to satisfy |
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experimental constraints [10, 11]. |
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It was quickly realized after the in
ationary scenario |
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was proposed more than 30 years ago, that supersymmetry |
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could provide a natural scenario with plenty of
at direc- |
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tions which could lead to in
ation [18, 19, 20, 21, 22, 23]. |
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When the theory couples to supergravity, there are a num- |
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Email addresses: [email protected] (Luis |
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Alvarez-Gaum e), [email protected] (C esar G omez), |
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[email protected] (Raul Jimenez)ber of new problems that appear [24], and we will discuss |
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some of them later on. |
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Current observational constraints from CMB tempera- |
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ture and polarization experiments and large-scale struc- |
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ture limit the amount the in
aton eld has moved to ap- |
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proximately <2Mpl[14], where Mplis the reduced Planck |
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mass. Therefore, in
ationary models that search for the |
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in
aton at very large energies, like for example chaotic |
|
in
ation, are severely constrained already by current ob- |
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servations. With the current new generation of CMB ex- |
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periments (Planck, EBEX, Spider, SPUDS etc...) it will |
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be possible to further constraint how much the in
aton |
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eld has displaced during the in
ationary period that gave |
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rise to our current casual horizon. It is therefore useful to |
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revisit again the problem of steep directions in SUGRA |
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models to understand if a
at direction can be obtained at |
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all. |
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In this paper we will suggest a natural embedding of in- |
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ationary dynamics in the eective low-energy Lagrangian |
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Preprint submitted to Physics Letters B October 26, 2018arXiv:1001.0010v1 [hep-th] 30 Dec 2009describing supersymmetry breaking. Our approach will |
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be quite independent of the microphysics underlying su- |
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persymmetry breaking, and will only rely on universal |
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properties of this symmetry. Since we are not commit- |
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ting ourselves to any particular microscopic realization of |
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supersymmetry breaking, some of our comments about re- |
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heating for instance will be rather sketchy. A more de- |
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tailed and precise presentations of our ideas will appear |
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elsewhere [25]. Like most in
ationary theories containing |
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supersymmetry, we present a simple model of multield |
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in
ation (sometimes called hybrid) [26], identify naturally |
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the in
aton eld and its potential, and then t a few obser- |
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vational data to estimate the few parameters of our model. |
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We compute, in particular, the number of e-folding and the |
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amplitude of density
uctuations at horizon crossing. It is |
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surprising to nd that the scale of supersymmetry break- |
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ing indicated by this analysis is between 1011 1014GeV. |
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An interesting spin-o of our model is that the in
aton is |
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lighter than the gravitino by an amountp, whereis |
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one of the slow roll parameters (see below). |
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We would like to stress that in this paper we are always |
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assuming F-breaking of supersymmetry. In D-breaking |
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scenarios our arguments do not apply, at least as presented |
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here1. |
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2. General framework |
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Supersymmetry is a natural framework to dene in- |
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ationary scenarios for two main reasons. First of all, |
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SUSY naturally leads to the existence of
at, or nearly |
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at directions (pseudomoduli), allowing for slow roll sce- |
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narios. Second, and more important, the order parameter |
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of supersymmetry breaking is the vacuum energy density. |
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Hence, naturally associated with its breaking, supersym- |
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metry contains two main ingredients necessary in in
a- |
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tionary scenarios: vacuum energy and reasonably
at di- |
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rections. |
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1We thank Gia Dvali for raising this point. See for instance the |
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last entry in [21]In a remarkable recent work, Komargodski and Seiberg |
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[27] have presented a new formalism to understand super- |
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symmetry breaking, its general properties, its non-linear |
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realizations [28], and a systematic way to understand the |
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low-energy couplings of goldstinos to other elds. Al- |
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though many things were known before (see references in |
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[27]) this work, the presentation is quite insightful, and it |
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played a major part in the inspiration of this work. |
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The basic starting point in [27] is the Ferrara-Zumino |
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multiplets of currents [29]. A vector supereld composed |
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of the R-symmetry current, the supercurrent, and the en- |
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ergy momentum tensor. This vector supereld satises the |
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general relation: |
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D_J;_=DX: (1) |
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The chiral supereld Xis essentially dened uniquely2 |
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in the ultraviolet. Following [27] the supereld Xhas the |
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following properties: |
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In the UV description of the theory, it appears in the |
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right hand side of 1, where it represents a measure of |
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the violation of conformal invariance. |
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The expectation value of its 2component is the or- |
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der parameter of supersymmetry breaking. In this |
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work we are only considering F-breaking of super- |
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symmetry. We denote by fthe expectation value of |
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theF-component of X. It will sometimes be useful |
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to writef=2, whereis the microscopic scale of |
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supersymmetry breaking. |
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When supersymmetry is spontaneously broken, we |
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can follow the
ow of Xto the infrared (IR). In the IR |
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this eld satises a non-linear constraint and becomes |
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2The ambiguities in the supercurrent multiplet and Xare related |
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to improvement terms in the various currents. |
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2the \goldstino" supereld3. |
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X2 |
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NL= 0; (2) |
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XNL=G2 |
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2F+p |
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2G+2F: (3) |
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The scalar component xofXbecomes a goldstino |
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bilinear. Its fermionic component is the goldstino |
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fermionG, andFis the auxiliary eld that gets the |
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vacuum expectation value. A major part in the anal- |
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ysis in [27] is based on this novel nonlinear constraint |
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satised by the supereld Xin the IR. As shown |
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there, the correct normalization of the goldstino su- |
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pereld to derive all relevant low-energy theorems of |
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broken supersymmetry is XNL=3 |
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8fX. |
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Finally,Xgeneralizes the usual spurion couplings ap- |
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pearing in the description of low-energy supersymmet- |
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ric lagrangians. If msoftdescribes the soft supersym- |
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metry breaking masses at low energies, the standard |
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spurion in the lagrangian is replaced bymsoft |
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fXNL. |
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This allows one to write the leading low-energy cou- |
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plings of the goldtino to other matter elds. |
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Since we are going to consider goldstino couplings, we will |
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work with a eld whose expectation values are well below |
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the Planck scale. |
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Our proposal is to identify in the UV the in
aton eld |
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with the scalar component of the supereld X. SinceXis |
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dened uniquely (up to the ambiguity mentioned in foot- |
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note one) in the UV, this provides a well dened prescrip- |
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tion. Furthermore, we will identify the in
ationary period |
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precisely with the
ow of Xfrom the UV to the IR i.e. |
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X!XNL. Note that by making this assumption we do |
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not need to think of the in
aton as any extra fundamental |
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eld. In fact, independently of how SUSY is broken, and |
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3A modied version of the nonlinear constraint (2) appears when |
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one considers spontaneous R-symmetry breaking. In that case, the |
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goldstino and the corresponding axion will be part of the same mul- |
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tiplet.what is the underlying fundamental theory we can always |
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identify the X supereld as well as its scalar component |
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x. More importantly, by making this assumption we are |
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identifying the vacuum energy driven in
ation with the |
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actual SUSY breaking order parameter. |
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In the supergravity context, once we have the K ahler |
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potentialK(X;X) and the superpotential W(X), the full |
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scalar potential is given by [30]: |
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V=eK |
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M2(K 1 |
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X;XDW DW 3 |
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M2jWj2) (4) |
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with |
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DW =@XW+1 |
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M2@XKW: (5) |
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Mis the high energy scale below which we can write the ef- |
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fective action describing the dynamics of the X-supereld. |
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It could be the Planck scale, or a GUT scale depending on |
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the microscopic theory. We will work well below the scale |
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M, and for simplicity take M=MplIn equation (4) we |
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can see one of the basic problems in supergravity in
a- |
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tion [24]. As we will see later on, to satisfy the slow roll |
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conditions, a necessary condition is that the -parameter, |
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dened by: |
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=M2 |
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plV00 |
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V; (6) |
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be much smaller than one. If we choose a K ahler potential |
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K(X;X) with R-symmetry, for instance the canonical one |
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K(X;X) =XX+:::, where the :::represents a function |
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ofXX, it is easy to see that from the exponent of (4) we |
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always get a contribution to equal to 1: = 1 +:::, no |
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matter which component of Xis taken as the in
aton eld. |
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This of course violates the slow roll conditions. Since we |
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are considering a situation with supersymmetry breaking |
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and gravity (early universe), we cannot exclude supergrav- |
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ity from the picture, and this leads to the -problem in |
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these theories. |
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The simplest way out of this problem without unreason- |
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able ne tuning, is to have explicit R-symmetry breaking |
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3in the K ahler potential4. If we have explicit R-breaking, |
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the expansion of Vfor small elds takes the form: |
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X=M(+i) (7) |
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V=f2(1 +A1(2+2) +B1(2 2) +:::)(8) |
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fis the supersymmetry breaking parameter representing |
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the expectation value of an F-term, and hence with square |
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mass dimensions. We assume that Vis locally stable at |
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least during in
ation. Hence A1B1>0. We express the |
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potential in terms of the dimensionless elds ;. Their |
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masses can be read o from (8): |
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m2 |
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=2f2 |
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M2(A1+B1); m2 |
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=2f2 |
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M2(A1 B1):(9) |
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The numbers A1;B1are taken to be O(1). |
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One could be more explicit, and choose some super- |
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symmetry breaking superpotential, like W=fX, and |
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K ahler potential explicitly breaking R-symmetry, like: |
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K=XX+ (c=M2)(X3X+XX3) +:::as in [27] lead- |
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ing to an eective action description of Xfor scales well |
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belowM. At this stage, we prefer not to consider explicit |
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examples of UV-completions of the theory. |
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We consider the beginning of in
ation well below M, |
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hence the initial conditions are such that ; << 1. In |
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fact, sinceis the lighter eld, we take this one to be the |
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in
aton, and consider that initially ;pf=M . For us |
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the in
ationary period goes from this scale until the value |
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of the eld is close to the typical soft breaking scale of the |
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problemmsoft, where the eld X!XNL(2), at this scale |
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XNLbehaves like a spurion [27] and as shown in Ref. [27], |
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the leading couplings to low-energy supersymmetric mat- |
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ter can be computed as spurion couplings, for instance5, |
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ifQ;V represent respectively low energy chiral and vector |
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4R-symmetry is a well-known problem in phenological applica- |
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tions of supersymmetry. R-symmetry does not allow soft breaking |
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masses for the gauginos; and spontaneous breaking of the symmetry |
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may lead to axions with unacceptable couplings. Often one wants to |
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preserve R-parity to avoid other possible phenomenological disasters. |
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5The details can be found in[27] section 4, in particular around |
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equations (4.3,4).superelds, we can have the couplings: |
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L= Z |
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d4XNL |
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f2 |
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m2QeVQ (10) |
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+Z |
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d2XNL |
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f1 |
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2BijQiQj+:::+h:c: |
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plus gauge couplings. |
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Once we reach the end of in
ation, the eld Xbecomes |
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nonlinear, its scalar component is a goldstino bilinear and |
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the period of reheating begins. The details of reheating de- |
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pend very much on the microscopic model. At this stage |
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one should provide details of the \waterfall" that turns the |
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huge amount of energy f2into low energy particles. Part of |
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this energy will be depleted and converted into low energy |
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particles through the soft couplings in (10), and hence we |
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can in principle compute a lower bound on the reheating |
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temperature. Before making some comments on the re- |
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heating period, we analyze the cosmological consequences |
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of a potential as simple as (8), as well as the assumptions |
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we have made earlier about the in
aton and its range as |
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in
ation takes place. |
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3. The In
aton Potential and Slow Roll Conditions |
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To study the conditions under which our potential pro- |
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vides in
ation consistent with the latest cosmological con- |
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straints, we examine the slow-roll parameters, dened as |
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[13]: |
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=M2 |
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pl |
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2V0 |
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V2 |
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; (11) |
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=M2 |
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plV00 |
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V; (12) |
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whereMplis the reduced Planck mass and ' denotes deriva- |
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tive with respect to the in
aton eld. The observables are |
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then expressed in terms of the above slow roll parameters |
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as: |
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nS= 1 6+ 2; (13) |
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r= 16 (14) |
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4nt= 2; (15) |
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2 |
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R=VM4 |
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pl |
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242: (16) |
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nSis the slope of the scalar primordial power spectrum, |
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ntis the corresponding tensor one, ris the scalar to tensor |
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ratio and 2 |
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Ris the amplitude of the initial perturbations. |
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All these numbers are constrained by current cosmological |
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observations [10, 11, 12]. We will use their constraints to |
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explore the naturalness of our in
ationary trajectories. In- |
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ation takes place when the slow-roll parameters are much |
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smaller than 1. |
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We will use the amplitude of initial perturbations and |
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the number of efoldings to t some of the paramenters |
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of the toy model in the previous section. Recall that the |
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potential in the range of interest is: |
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V=f2(1 +A1(2+2) +B1(2 2) +:::);(17) |
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which appears in gure 1. We can compute ;while |
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rolling in the direction: |
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= 2 ( (A1 B1))2+::: (18) |
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= 2 (A1 B1) +:::; (19) |
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since << 1,is naturally small. We can make small |
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by a slight ne tuning of the dierence A1 B1. We will |
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writelater as a ratio of the in
aton and gravitino masses. |
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Once the slow roll conditions are satised, we can compute |
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the number of efoldings (see for instance [16, 17]): |
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N=1 |
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MZdxp |
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2=Zf |
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id |
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2p(20) |
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From (19) we get: |
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N=1p |
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2jA1 B1jlogf |
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i: (21) |
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In most models of supersymmetry breaking, the gravitino |
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mass is given by: |
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m3=2=f |
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M; (22) |
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hence, we can rewrite the parameters and masses in (9) |
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as: |
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jA1 B1j=1 |
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2m2 |
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|
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m2 |
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3=2;jA1+B1j=1 |
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2m2 |
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|
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m2 |
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3=2;(23)thus: |
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N=p |
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2m3=2 |
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m2logf |
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i(24) |
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The number of efoldings is considered normally to be be- |
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tween 50 100. Finally we will use the amplitude of initial |
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perturbations to get one extra condition in the parameters |
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of our potential. Using [11] (16) can be written as: |
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V |
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1=4 |
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=f1=2 |
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21=4(jA1 B1j)1=2=:027M; (25) |
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whereis taken atN-efoldings before the end of in
ation. |
|
Summarizing, the two cosmological constraints we get on |
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the parameters of our potential can be written as: |
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N=p |
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2m3=2 |
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m2logf |
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i; (26) |
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21=4m3=2 |
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mpf |
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M1=2 |
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= 0:027; (27) |
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and theparameter can be written as: |
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=m |
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m3=22 |
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: (28) |
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We takeiabove the supersymmetry breaking scale |
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pf=M ==M , andfclose tomsoft=M, therefore we |
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can easily get values for Nbetween 50 100 for moderate |
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values of, which is expressed here as the square of the |
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ratio of the in
aton to the gravitino mass. It is interesting |
|
to notice that from (27), we can write the supersymmetry |
|
breaking scale in terms of the -parameter: |
|
|
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M5:2 10 4: (29) |
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Hence for a value of :1 we can get 1013GeV. |
|
Lower values of the supersymmetry breaking scale can be |
|
obtained by reducing . However, since the in
aton mass |
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is |
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m=m3=2p; (30) |
|
we may end up with an in
aton whose mass is substantially |
|
lighter than the gravitino. For these values of ;, we |
|
have thati1013=M;f103=M, and the number of |
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efoldings is110. |
|
5We conclude then that with moderate values of be- |
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tween:1 :01 we can get supersymmetry breaking scales |
|
between 1011 1013without major ne tunings. We eas- |
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ily get enough efoldings, and furthermore, the in
aton is |
|
lighter than the gravitino by an amount given byp. |
|
For the above range of parameters we can compare the |
|
predicted value of nSin our model with observational con- |
|
straints. This is shown in the right panel of Fig. 1. The |
|
yellow region is the current cosmological constraints from |
|
WMAP5 [11] and the other colored areas are the predic- |
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tions for our model with minimal ne tuning for an stable |
|
(unstable)Xpotential, i.e. the eld is concave (convex) re- |
|
spectively. The constraints will improve greatly when the |
|
Planck satellite releases its results next year, and therefore |
|
our model can be tested much more accurately. |
|
Reheating can proceed in many ways, since we have not |
|
provided a detailed microscopic model. Once in the non- |
|
linear regime, the XNLeld (whose scalar component is |
|
made of a goldstino bilinear) could eciently convert the |
|
f2-energy density into radiation. We can calculate the |
|
amount of entropy and particle density by using the Boltz- |
|
man equation and assuming that the pair of Goldstinos |
|
will have an out-of-equilibrium decay[16]. Using that |
|
TRH= 10 10p |
|
f=GeV3=2 |
|
GeV (31) |
|
we obtain a range 107< TRH<109. This produces a |
|
particle abundance of n1070 90which are standard |
|
values. We can also compute the amount of entropy gen- |
|
erated by the out-of-equilibrium decay as |
|
Sf=Si= 107(p |
|
f=GeV ) 1=2(32) |
|
which yields values in the range 10 to 1, and assures that |
|
there is no entropy overproduction. We could also compute |
|
the depletion of this energy through the soft couplings (10) |
|
yielding very similar values as above. In both cases, we |
|
can get sucient reheating with temperatures betweenpf |
|
and a fraction of m3=2. The true value depends very much |
|
on the details of the microscopic model. However, thereseems to be no obstruction to reheating the universe to |
|
and acceptable value of temperature, particle abundances |
|
and entropy. We are currently working in a more detailed |
|
theory incorporating our scenario [25]. |
|
4. Conclusions |
|
In this short note we have studied the possibility of hav- |
|
ing supersymmetry breaking as the driving force of in
a- |
|
tion. We have used the unique chiral supereld Xwhich |
|
represents the breaking of conformal invariance in the UV, |
|
and whose fermionic component becomes the goldstino at |
|
low energies. Its auxiliary eld is the F-term which gets |
|
the vacuum expectation value breaking supersymmetry. |
|
It is crucial in our analysis to have explicit R-symmetry |
|
breaking along with supersymmetry breaking. This allows |
|
us to avoid the problem in supergravity and to take the |
|
supersymmetric limit. The simplest model we obtain de- |
|
scribes the components of Xwell below the Planck scale. |
|
It is written in terms of three parameters: the supersym- |
|
metry breaking parameter fand the masses of the real and |
|
imaginary components of the eld x(the scalar component |
|
of X). In our analysis the imaginary part of xplays the role |
|
of the in
aton, and its mass was shown to be smaller than |
|
the gravitino mass by an amount given byp. This imag- |
|
inary component represents a pseudo-goldstone boson, or |
|
rather, a pseudomoduli. In supersymmetric theories such |
|
elds abound, and any of them could be used to construct |
|
some form of hybrid in
ation. In our case, however, we |
|
want to use the minimal choice that is naturally provided |
|
by the universal supereld Xthat must exist in any su- |
|
persymmetric theory. |
|
Since we have not presented any detailed model, the cos- |
|
mological consequences are a bit rudimentary, especially |
|
concerning reheating at the end of in
ation. However, the |
|
comparison of the simplest model with present data, yields |
|
very interesting values for the supersymmetry breaking |
|
scale, and the ratio of the in
aton and gravitino masses. |
|
6Figure 1: Left panel: The potential as a function of ( ) and () components of the eld X. Note the nearly
at direction ( ) that we use for our |
|
in
ationary trajectories. Graceful exit and particle creation occurs in the non-linear part of the Xeld. Right panel: WMAP5 cosmological |
|
constraints (yellow region) in the r nSplane. For no-ne-tuned minimal in
ation models the green and red area show our predictions for |
|
both cases of a stable (concave) potential and unstable (convex) potential. The Planck satellite will be able to provide signicantly tigther |
|
constraints on rand especially nS(at the<0:5% level) thus further constraining our model. The dashed line is the limit in rthat can be |
|
achieved with an ideal CMB polarization experiment [14] |
|
These are bonuses which come directly from the observa- |
|
tions of the initial density perturbations from WMAP data |
|
[11]. The fact that the in
aton is lighter than the gravitino |
|
may have interesting low-energy phenomenological impli- |
|
cations. Furthermore in this simple model it is easy to |
|
obtain sucient number of efoldings with moderate values |
|
of theparameter. |
|
To explore our proposal in more detail, it is important |
|
to construct an explicit model, even if not very realistic, |
|
in order to understand in more detail the end of in
ation, |
|
the reheating mechanisms, and also the ne structure of |
|
the in
aton potential. We hope to report on this in the |
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near future [25]. |
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Acknowledgements |
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We would like to thank G. Dvali, G. Giudice, J. Les- |
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gourgues, S. Matarrese, G. Ross, Nathan Seiberg, M.A. |
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V azquez Mozo, and L. Verde for useful discussion. C.G. |
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and R.J. would like to thank the CERN Theory Group for |
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