aa r X i v : . [ h e p - t h ] J a n Preprint typeset in JHEP style - HYPER VERSION
ISS-flation
Nathaniel J. Craig a,b a Theory Group,Stanford Linear Accelerator Center,Menlo Park, CA 94025 b Institute for Theoretical Physics,Stanford University,Stanford, CA 94306
Abstract:
Inflation may occur while rolling into the metastable supersymmetry-breakingvacuum of massive supersymmetric QCD. We explore the range of parameters in which slow-roll inflation and long-lived metastable supersymmetry breaking may be simultaneously real-ized. The end of slow-roll inflation in this context coincides with the spontaneous breaking ofa global symmetry, which may give rise to significant curvature perturbations via inhomoge-nous preheating. Such spontaneous symmetry breaking at the end of inflation may give riseto observable non-gaussianities, distinguishing this scenario from more conventional modelsof supersymmetric hybrid inflation. ontents
1. Introduction 12. Supersymmetric Hybrid Inflation 2
3. Supersymmetric Hybrid Inflation from Strongly Coupled Theories 54. Metastable SUSY breaking in Supersymmetric QCD 7
5. Mesonic inflation in SQCD: Rolling into the metastable vacuum 10 η problem 13
6. Inflationary predictions for ISS-flation 157. Curvature perturbations from broken flavor symmetries 16
8. Conclusions 22
1. Introduction
Spontaneously broken supersymmetry (SUSY) has long been one of the most attractive pos-sibilities for physics beyond the Standard Model [1]. Considerable effort has been devoted toelucidating dynamical mechanisms of SUSY-breaking, in which strong-coupling effects giverise to a naturally small SUSY-breaking scale [2]. The SUSY-breaking vacua of such theoriesneed not be global minima of the potential, and metastability of phenomenologically-viablevacua appears to be generic when embedding the MSSM into a larger setting. Indeed, the– 1 –tudy of metastable dynamical SUSY-breaking has recently undergone something of a renais-sance, catalyzed by the observation that massive supersymmetric QCD (SQCD) possessesSUSY-breaking local minima whose lifetimes can be longer than the present age of the uni-verse [3] (for an excellent review, see [4]). The simplicity and genericness of such theoriesmakes them particularly well-suited to supersymmetric model-building.Far from serving only as a possible resolution of the hierarchy problem, spontaneously-broken supersymmetry emerges frequently in inflationary settings as well [5–8]. There exist aplethora of possible models aimed at realizing inflation in a natural context [9], many of whichexploit supersymmetry and supersymmetry breaking [10–14]. Based on the moderate successof such theories, it is tempting to suppose that inflation may be built into a supersymmetry-breaking sector. Indeed, it has already been demonstrated that inflation may arise in simplestrongly-coupled gauge theories [13]. This suggests that it may be possible to inflate whilerolling into the metastable vacuum of massive SQCD from sub-Planckian initial field values.The appeal of successful inflation in this scenario is twofold: first, it provides a generic andtechnically natural means of simultaneously realizing inflation and supersymmetry breaking;second, it furnishes a robust UV completion for the inflationary sector. Moreover, the spon-taneous global symmetry-breaking that accompanies the end of inflation in these models maygive rise to additional curvature perturbations exhibiting observable non-gaussianities. Inthis note we address the prospects for, and signatures of, slow-roll inflation in models withmetastable dynamical supersymmetry breaking.The organization of the paper is as follows. In Sec. 2 we review the traditional scenarioof supersymmetric hybrid inflation proposed in [12], wherein one-loop corrections drive in-flation along a flat direction of a theory with a relatively simple superpotential. In Sec. 3we review the means by which supersymmetric hybrid inflation may be realized in strongly-coupled gauge theories [13]. Subsequently, in Sec. 4 we turn to the Intriligator, Seiberg, andShih (ISS) model [3] of metastable supersymmetry breaking in supersymmetric QCD andexplore its implications. In the vicinity of the supersymmetry breaking metastable vacuum,features of the ISS model are reminiscent of supersymmetric hybrid inflation. With this mo-tivation, in Sec. 5 we explore the prospects for, and constraints on, inflation while rollinginto the metastable supersymmetry breaking vacuum of the ISS model; in Sec. 6 we discussthe subsequent inflationary predictions assuming the inflaton is primarily responsible for theprimordial curvature perturbation. However, the spontaneous symmetry breaking that ac-companies the end of slow-roll inflation may give rise to additional curvature perturbationsthrough inhomogenous preheating [15]. In Sec. 7 we explore this possibility and its conse-quences, including the prospects for observable non-gaussianities. In Sec. 8 we conclude theanalysis and consider directions for further work.
2. Supersymmetric Hybrid Inflation
Hybrid inflation provides a compelling alternative to traditional chaotic inflation, in that asecond non-inflationary field provides the bulk of the inflationary energy density and a natural– 2 –nd to inflation without recourse to trans-Planckian field values [16, 17]. Such theories werefound to arise naturally in the context of supersymmetry [14]. One particularly effectivemodel, due to [12], obtains the scalar potential of hybrid inflation from a simple globally-supersymmetric superpotential involving three chiral superfields: a superfield ψ transformingas a fundamental of the gauge symmetry; a superfield ψ c transforming as an anti-fundamental;and a gauge singlet superfield X. The superpotential of this theory is W X = hψ c Xψ − µ X, (2.1)which exhibits an unbroken R symmetry. In the D-flat directions, the scalar potential of thetheory obtains the form V = h | X | ( | ψ c | + | ψ | ) + | hψ c ψ − µ | . (2.2)For X ≤ X c = µ/ √ h, the minimum for the ψ, ψ c is at h ψ, ψ c i = µ/ √ h, and there exists asupersymmetric vacuum at h X i = 0 h ψ, ψ c i = µ/ √ h. (2.3)However, for X > X c , the minimum is at h ψ, ψ c i = 0 , thanks to the effective mass termarising from h X i . For
X > X c , there arises a µ contribution to the vacuum energy density that drives inflation;inflation ends when X < X c and the scalars ψ, ψ c roll off into the supersymmetric vacuum.At tree level, there’s nothing to drive X to its minimum and end inflation; an adequatepotential is obtained only when corrections at one loop are taken into consideration. Indeed,when X > X c the F -term for X is nonzero and supersymmetry is broken; the scalar potentialaccumulates one-loop corrections of the form V ( X ) = X i ( − F π M i ( X ) log (cid:18) M i ( X ) Λ (cid:19) (2.4)where Λ is a cutoff scale. The contributions to the effective potential from X > X c comefrom splittings in the ψ, ψ c superfields; the ψ scalars have masses m s = h X ± hµ , while the ψ fermions have masses m f = h X . The one-loop effective potential along the inflationarytrajectory is then given by V eff ( X ) = µ + N c h π (cid:20) µ log (cid:18) h | X | Λ (cid:19) + ( hX − µ ) log (cid:18) − µ hX (cid:19) (2.5)+( hX + µ ) log (cid:18) µ hX (cid:19)(cid:21) . We will henceforth abuse notation by using the same notation for superfields and their scalar components. – 3 –his contribution from the one-loop effective potential drives X to the origin. When X ≃ X c , the ψ, ψ c become tachyonic and roll off to their supersymmetric minima. Below X c , theinflaton X is efficiently driven to the origin by the ψ, ψ c vevs. Inflation ends as the ψ, ψ c vevscancel the effective contribution to the vacuum energy density and the slow-roll conditionsare violated. Defining, for convenience, the parametrization X = xX c , i.e., X = µ √ h x, theeffective potential is of the form V eff ( x ) = µ + N c h π (cid:20) µ log (cid:18) hx µ Λ (cid:19) + ( x µ − µ ) log (cid:18) − x (cid:19) (2.6)+( x µ + µ ) log (cid:18) x (cid:19)(cid:21) . The slow-roll parameters in this scenario are (in reduced Planck units M P = 1 / √ πG =2 . × GeV) ǫ = M P (cid:18) V ′ V (cid:19) ≃ h N c M P π µ x (cid:20) ( x −
1) log (cid:18) − x (cid:19) + ( x + 1) log (cid:18) x (cid:19)(cid:21) (2.7) η = M P V ′′ V ≃ h N c M P π µ (cid:20) (3 x −
1) log (cid:18) − x (cid:19) + (3 x + 1) log (cid:18) x (cid:19)(cid:21) . (2.8)The slow-roll conditions η ≪ , ǫ ≪ x ∼ , where both ǫ, η growrapidly and inflation comes to an end. Notice that | ǫ | ≪ | η | and generally η < , therebyguaranteeing a suitably red spectrum. The initial displacement X e required to obtain a sufficient number of e-foldings, N e ∼ ± , is given by N e ≃ M P Z X e X c (cid:18) VV ′ (cid:19) dX ≃ π N c (cid:18) µM P (cid:19) ( x e − h . (2.9)This suggests the field value of X at N e e-foldings prior to the end of inflation is X e ≃ √ N e N c π hM P . (2.10)This corresponds to a naturally sub-Planckian initial condition, provided h √ N c < ∼ . It is natural to ask whether such a theory is compatible with observation. The vac-uum fluctuation of the inflaton generates a time-independent curvature perturbation ζ withspectrum [18–21] P / ζ ≃ VV ′ (cid:18) H πM P (cid:19) ≃ √ ǫ (cid:18) H πM P (cid:19) . (2.11)– 4 –ith H = q V / M P = hµ √ M P , this corresponds to P / ζ ≃ r N e N c (cid:18) µM P (cid:19) . (2.12)The WMAP normalization is P / ζ = 4 . × − , as taken at the comoving scale k = 0 . − . Matching the observed curvature perturbation thus entails µM P ∼ × − · N / c (2.13)and hence the assumption of GUT-scale µ ∼ GeV.The spectral index, for ǫ ≪ η, is given simply by n s ≈ − | η | ≈ − N e ≈ . , (2.14)somewhat higher than the central value n s ≈ .
95 given by WMAP data [22]. The tensor-to-scalar ratio is negligible, r < ∼ − , as is the running in the spectral index, dn s /d log k < ∼ − . Taken together, this simple scenario of globally supersymmetric hybrid inflation accu-mulates a number of successes: reasonable correspondence with the observed spectral index,sub-Planckian initial conditions, ready provision of N e ∼
54 e-foldings, and a graceful exitfrom inflation. Moreover, as we shall see in subsequent sections, the requisite superpotentialarises naturally in the context of supersymmetric gauge theories.
3. Supersymmetric Hybrid Inflation from Strongly Coupled Theories
Let us now consider scenarios wherein the superpotential (2.1) is generated in the context ofstrongly-coupled supersymmetric gauge theories, as was demonstrated in [13]. Consider an SU (2) gauge theory whose matter content consists of four doublet chiral superfields Q I , ¯ Q J (with flavor indices I, J = 1 ,
2) and a singlet superfield S, with the classical superpotential W cl = gS ( Q ¯ Q + Q ¯ Q ) (3.1)with coupling constant g. In the classical theory without any superpotential, the moduli spaceof D-flat directions is parametrized by the following SU (2) invariants: S, M JI = Q I ¯ Q J , B = ǫ IJ Q I Q J , ¯ B = ǫ IJ ¯ Q I ¯ Q J (3.2)These invariants are subject to the constraintdet M − ¯ BB = 0 . (3.3)With the superpotential turned on, the classical moduli space consists of two branches – onewith S = 0 , M = B = ¯ B = 0 , where the quarks are massive and the gauge symmetry is– 5 –nbroken; and one with S = 0 , where the mesons and baryons satisfy the above constraintand the gauge group is broken.It is particularly interesting to consider the S = 0 branch, where a non-zero vacuumenergy may drive inflation. For S = 0 , far from the origin, the ‘quarks’ Q, ¯ Q become massiveand decouple; the theory consists of a free singlet S and pure supersymmetric SU (2) . The SU (2) sector has an effective scale Λ L given by the one-loop matching to the quarks at thescale Λ of the original theory, Λ L = gS Λ . (3.4)In the pure SU (2) sector, gaugino condensation generates an effective superpotential W eff = Λ L = gS Λ (3.5)This effective superpotential gives rise to a non-zero F -term, F S = g Λ , and supersymmetryis broken with a vacuum energy density ∼ g Λ . While this is a convenient heuristic, it is useful to consider the full theory in detail. Thequantum modified constraint for the confined theory is given bydet M − ¯ BB − Λ = 0 , (3.6)so that the full quantum superpotential is given by W = A (det M − ¯ BB − Λ ) + gS Tr M. (3.7)For S = 0 , the F -terms are F A = det M − B ¯ B − Λ F M JI = Aǫ IK ǫ JL M KL + gδ JI SF S = g Tr M (3.8) F B = A ¯ BF ¯ B = AB For S = 0 , all F -terms save F S may be set to zero via¯ B = B = 0 , M JI = δ JI Λ , A Λ /g = − S, F S = g Λ , (3.9)which defines a natural D-flat trajectory for the rolling of S. The inflationary behavior of the theory maps onto that of conventional supersymmetrichybrid inflation. For
S > S c , the quarks are integrated out, providing a non-zero vacuumenergy ∼ g Λ . The one-loop effective potential for S drives the inflaton along its D-flattrajectory towards S c . Inflationary predictions are the same as before, albeit without theneed to introduce any additional dimensionful parameters. By exploiting the dynamics ofstrongly-coupled theories, the hybrid inflation mass scale is set by the strong coupling scaleΛ . – 6 – . Metastable SUSY breaking in Supersymmetric QCD In the preceding sections, we have seen how slow-roll inflation may arise naturally in thecontext of supersymmetric gauge theories. Consider now the case of massive supersymmetricQCD (SQCD). In [3] it was shown that for sufficiently small mass of the electric quarks in SU ( N c ) SQCD, there is a long lived metastable vacuum in which supersymmetry is broken.Given the close resemblance of the ISS model to the inflationary theories considered above, it istempting to consider scenarios in which inflation occurs during the transition to a metastableSUSY-breaking vacuum. In this section we shall briefly review the key features of the ISSmodel before turning to inflationary dynamics in subsequent sections. In the ultraviolet, the microscopic (electric) ISS theory consists of asymptotically free N =1 supersymmetric SU ( N c ) QCD with N f massive flavors Q, Q c ; the theory exhibits an SU ( N f ) L × SU ( N f ) R approximate global flavor symmetry. The quarks transform underthe symmetries of the theory as Q ∼ ( (cid:3) N c , (cid:3) N F L ) Q c ∼ ( (cid:3) N c , (cid:3) N F R ) . (4.1)The nonzero quark masses break the global symmetry to SU ( N f ) L × SU ( N f ) R → SU ( N f )with a superpotential W e = m Tr QQ c . (4.2)For the sake of simplicity, the masses m are taken to be degenerate. This theory with small m is technically natural. This electric theory becomes strongly coupled at a scale Λ. In order to retain full control of thedynamics, we assume m ≪ Λ . Below this strong-coupling scale the system may be described byan IR-free dual gauge theory provided N c < N f < N c [23]. The dual macroscopic (magnetic)theory is an SU ( N ) gauge theory (where N = N f − N c ) with N f magnetic ‘quarks’ q and q c . This dual theory possesses a Landau pole at a scale Λ and runs free in the IR. Themagnetic quarks have the same approximate SU ( N f ) L × SU ( N f ) R flavor symmetry as theelectric theory, transforming under the symmetries of the magnetic theory as q ∼ ( (cid:3) N , (cid:3) N F L ) q c ∼ ( (cid:3) N , (cid:3) N F R ) . (4.3) In principle the strong-coupling scale ˜Λ of the IR theory is different from the strong-coupling scale Λof the UV theory, related only through an intermediate scale ˆΛ by the relation Λ N c − N f ˜Λ N f − N c ) − N f =( − N f − N c ˆΛ N f , with ˜Λ , ˆΛ not uniquely determined by the content of the electric theory. Naturalness andconceptual simplicity, however, suggest Λ ∼ ˜Λ , which will remain our convention throughout. – 7 –here is also an additional gauge singlet superfield, M, that is a bi-fundamental of the flavorsymmetry: M ∼ ( (cid:3) N F L , (cid:3) N F R ) . (4.4)The meson superfield M may be thought of as a composite of electric quarks, with M ij = Q i Q cj , whereas the magnetic quarks q, q c have no obvious expression in terms of the electricvariables.This infrared theory is IR free with N f > N. The metric on the moduli space is smoothabout the origin, so that the K¨ahler potential is regular and can be expanded as K = 1 β Tr ( | q | + | q c | ) + 1 α | Λ | Tr | M | + ... (4.5)The coefficients α, β are O (1) positive real numbers that are not precisely determined in termsof the parameters of the electric theory, and at best may be estimated by na¨ıve dimensionalanalysis [24–26]; the size of these couplings will be relevant in our subsequent inflationaryanalysis.After appropriate rescalings to obtain canonically-normalized fields M, q, q c , the tree-levelsuperpotential in the magnetic theory is given by W m = y Tr qM q c − µ Tr M, (4.6)where µ ∼ m Λ and y ∼ √ α. Supersymmetry is spontaneously broken at tree level in thistheory; the F -terms of M are F † M ij = y q a i q c ja − µ δ ij (4.7)which cannot vanish uniformly since δ ij has rank N f but q a i q c ja has rank N f − N c < N f . Hence supersymmetry is spontaneously broken in the magnetic theory by the rank condition.This SUSY-breaking vacuum lies along the quark direction, h M i ssb = 0 h q i ssb = h q c i ssb ∼ µ N (4.8)with h F M i ∼ √ N c µ . A careful analysis of the one-loop contribution to the effective potentialaround this vacuum reveals that there are no tachyonic directions, and all classical pseudo-moduli are stabilized at one loop. The remaining fields not stabilized at one loop are goldstonebosons of the broken flavor symmetry, and remain exactly massless to all orders.The theory also possesses supersymmetric vacua, in accordance with the non-vanishingWitten index of SQCD. For large values of the meson vev, the quarks become massive andmay be integrated out below the scale h M i ; here the magnetic theory becomes pure SU ( N )super-Yang Mills. This theory has a dynamically generated strong coupling scale, Λ m ( M ),given by Λ m ( M ) = M (cid:18) M Λ (cid:19) a , (4.9)– 8 –here a = N f N − , a strictly positive quantity when the magnetic theory is IR free. Here,for simplicity, we have taken the meson vev to be proportional to the identity, i.e. h M i ∼ M N f . Gaugino condensation at this scale leads to an ADS superpotential [27] for M . Belowthe mass of the quarks this additional nonrenormalizable contribution obtains the form [27] W det = N (cid:16) y N f det M Λ Nf − N (cid:17) N . The complete low-energy superpotential in the magnetic theoryis then given by W = N (cid:0) y N f det M (cid:1) N Λ − a − µ Tr M + y Tr qM q c . (4.10)Interpreted physically, the parameter a characterizes the irrelevance of the determinant super-potential. Larger values of a correspond to N f N ≫ F M = 0 . However, this vacuum is very distant from the origin (and also the metastablevacuum) due to the irrelevance of the SUSY-restoring gaugino contribution to the superpo-tential; it lies in the meson direction at h M i SUSY = µy (cid:18) Λ µ (cid:19) a a N f h q i SUSY = h q c i SUSY = 0 . (4.11)The existence of supersymmetric vacua indicates the SUSY-breaking vacuum is metastable;in [3] it was shown that the metastable vacuum may be made parametrically long-lived.Explicitly, the bounce action corresponding to the nucleation of a bubble of true vacuum wasfound to be S ∼ (cid:16) Λ µ (cid:17) a a , which can be made arbitrarily large – and the false vacuum longlived – by taking µ ≪ Λ. Ensuring that no transition to the supersymmetric minimum hasoccurred during the lifetime of the Universe (i.e., that the lifetime of the nonsupersymmetricuniverse exceeds 14 Gyr) places a constraint on the theory [28] aa + 2 log Λ µ > ∼ .
73 + 0 .
003 log µ TeV + 0 .
25 log N. (4.12)This is a very weak constraint (amounting to (Λ /µ ) a/ ( a +2) > ∼
2) , and one that is naturallysatisfied by the hierarchy µ/ Λ ≪ µ ≪ Λ , notice also that the longevitybound also becomes weaker for large a (correspondingly, N f ≫ N ) since the operator creatingthe SUSY vacuum becomes increasingly irrelevant in this limit.The parametric longevity of the metastable SUSY-breaking vacuum persists at finitetemperature, and in fact finite-temperature effects have been shown to lead to preferentialselection of the metastable-vacuum after high-scale reheating [28–30]. Here we wish to con-sider a somewhat different scenario – namely, that inflation itself may result from populationof the metastable vacuum, with the mesonic scalar M playing the role of the inflaton and thesquarks q, q c serving as the waterfall fields of hybrid inflation. While it bears considerableresemblance to a multi-component version of supersymmetric hybrid inflation, this scenariois distinguished observationally by the unique features of the ISS model.– 9 – . Mesonic inflation in SQCD: Rolling into the metastable vacuum Let us now consider the inflationary dynamics resulting from massive SQCD with N c < N f < N c . At high energies the theory is characterized by the electric description, with superpo-tential (4.2) and strong-coupling scale Λ . Below the scale Λ , the dynamics are described interms of the IR-free magnetic variables with superpotential (4.6). Given a random initialvev for the mesonic scalar M, inflation may occur as the theory settles into the metastableSUSY-breaking vacuum. To analyze the inflationary dynamics, let us parametrize the inflaton trajectory by M ∼ ϕ √ N f N f , which maximally respects the SU ( N f ) global flavor symmetry of the theory. Thenthe tree-level scalar potential near the origin (neglecting the determinant superpotential re-sponsible for creating the SUSY vacuum) is given by V = N f µ − yµ Tr( qq c ) − yµ Tr( q ∗ q c ∗ ) + y N f | Tr( qq c ) | (5.1)+ y N f | ϕ | (cid:2) Tr( | q c | ) + Tr( | q | ) (cid:3) . After diagonalizing the resultant mass matrix, the masses of the squarks along this tra-jectory are given by m s = y | ϕ | N f ± yµ , while those of the quarks are m f = y | ϕ | N f . The one-loopeffective potential for ϕ is then V eff ( ϕ ) = N f µ + N N f y π " µ log (cid:18) y ϕ N f Λ (cid:19) + (cid:18) yϕ N f − µ (cid:19) log (cid:18) − N f µ yϕ (cid:19) (5.2)+ (cid:18) yϕ N f + µ (cid:19) log (cid:18) N f µ yϕ (cid:19) . The squarks in this theory become tachyonic and roll off into the SUSY-breaking vacuumbelow ϕ c = q N f y µ. Taking the parametrization ϕ = xϕ c = q N f y µx, we have V eff ( x ) = N f µ (cid:26) N y π (cid:20) (cid:18) yx µ Λ (cid:19) + (cid:0) x − (cid:1) log (cid:18) − x (cid:19) (5.3) (cid:0) x + 1 (cid:1) log (cid:18) x (cid:19)(cid:21)(cid:27) . The slow-roll parameters for this trajectory are the same as those of supersymmetric hybridinflation, namely ǫ ≃ y N M P π µ x (cid:20) ( x −
1) log (cid:18) − x (cid:19) + ( x + 1) log (cid:18) x (cid:19)(cid:21) (5.4) η ≃ y N M P π µ (cid:20) (3 x −
1) log (cid:18) − x (cid:19) + (3 x + 1) log (cid:18) x (cid:19)(cid:21) . (5.5)– 10 –hese slow-roll parameters naturally satisfy the slow-roll constraints ǫ, | η | ≪ x > . Up to flavor rotations, the inflationary story is as follows: Assuming arbitrary initialvevs for the scalar components of the meson superfield M, with h M i > µ √ y , the scalar com-ponents of q, q c roll rapidly to h q i = h q c i = 0 due to the effective mass terms arising from themeson vevs. Since SUSY is broken away from the supersymmetric vacuum, mass splittingsbetween the squarks and quarks provides a gently-sloping potential for M at one loop, drivingits slow-roll evolution towards the origin.For arbitrary meson vevs, the scalar potential contains a constant contribution to theenergy density of order N f µ ; however, this is only reduced to N c µ in the metastablesupersymmetry-breaking vacuum. In the context of supergravity, we assume some mecha-nism arises in the present era to cancel the cosmological constant in this vacuum. While M is slowly rolling, the effective energy density N f µ drives inflation. Inflation continues untilthe diagonal components of M reach the critical value, i.e., h M c i = µ √ y N f . (5.6)Here the squarks q, q c become tachyonic, but only N of the N f flavors of q, q c obtain nonzerovevs in rolling off to the supersymmetry-breaking vacuum. This is a consequence of the rank-condition breaking of SUSY in ISS models, wherein there are not enough independent degreesof freedom to cancel all the F-terms of M. When h M i ≤ M c , N flavors of the squarks q, q c become tachyonic and roll off into the SUSY-breaking vacuum, with h q i = h q c i T = µ √ y N ! . (5.7)In this vacuum the remaining N c flavors are stabilized at one loop, and η ∼
1; thus inflationcomes to an end. Notice that η ∼ M that do notobtain large masses from the squark vev; the one-loop effective masses for the pseudo-moduliat this stage are sufficient to terminate slow-roll inflation. Here the SU ( N f ) flavor symmetryof the magnetic theory is broken by the squark vev to SU ( N ) × SU ( N c ) . It is natural to consider whether this inflationary trajectory can produce sufficient e-foldingsof inflation. As with supersymmetric hybrid inflation, the value of ϕ at N e e-foldings fromthe end of inflation is ϕ e = √ N e N π yM P . (5.8) As was pointed out in [3], adding a constant term to the magnetic superpotential so that the metastablevacuum has our observed vacuum energy makes the SUSY vacua anti-deSitter; this may lead to a suppressedtunneling rate due to quantum gravity effects, thereby preserving the metastability of the SUSY-breakingvacuum against first-order transitions to the AdS SUSY vacua. – 11 –hereas conventional supersymmetric hybrid inflation enjoys significantly sub-Planckian ini-tial vevs with suitably small values of the yukawa coupling, one might worry here that y is notnaturally small at energies close to Λ . We may estimate the size of y at the strong-couplingscale using na¨ıve dimensional analysis; from [31] we have y < ∼ π √ NN f near Λ . For sufficientlylarge N f , y may be made naturally O (10 − ) or smaller near the strong-coupling scale, andruns free in the IR.The total displacement of the inflaton (the sum of displacements of the N f diagonal com-ponents of M, defining the inflationary trajectory) is ϕ. In order to obtain a total inflationarydisplacement √ N e N π yM P , each component of M need only be displaced by a distinctly sub-Planckian amount ∼ q N e N π N f yM P . With the NDA estimate for the size of y below the scaleΛ , this corresponds to a required displacement of each individual field by ∼ √ N e N f M P , whichmay be made sufficiently small for large N f . Moreover, this guarantees h M i ≪ Λ , renderingthe magnetic description valid throughout the inflationary trajectory.Unlike the case of supersymmetric hybrid inflation, recall that there exist additionalvacua in the ISS theory – supersymmetric minima created by gaugino condensation far fromthe origin. A supersymmetric minimum lies along the inflaton trajectory at h ϕ i SUSY ∼ p N f µ (cid:18) Λ µ (cid:19) aa +2 y − a +3 a +2 . (5.9)For initial field values of O ( M P ) , it is plausible for the inflaton to roll away from the origin andinto the supersymmetric vacuum, rather than towards the origin along the slow logarithmicpotential generated by one-loop effects. It is doubtful whether inflation will occur while rollinginto the SUSY vacuum, since large corrections to the K¨ahler potential so far from the originmake slow roll implausible in that direction.Naturally, one would like to check the field value above which rolling into the SUSYvacuum is preferred. For ϕ ≫ ϕ c , the logarithmic effective potential for an inflaton componentrolling toward the origin goes as V eff ( ϕ ) ≈ N f µ (cid:20) N y π (cid:18) log (cid:18) y ϕ Λ (cid:19) + 32 (cid:19)(cid:21) , (5.10)whereas the tree-level contribution from gaugino condensation that gives rise to the SUSYvacuum is V susy ( ϕ ) = N f Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y yϕ p N f Λ ! a − µ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.11) It is important to note here that the effective yukawa coupling y appearing in the scalar potential (5.1)contains factors of wavefunction renormalization from the K¨ahler potential, and thus depends logarithmicallyon the energy scale. – 12 –he maximum of this potential V eff + V susy lies around ϕ max ≈ p N f µ (cid:18) Λ µ (cid:19) aa +2 y − a +1 a +2 (cid:18) Nπ (cid:19) a a a /a . (5.12)The requirement that the initial value of the inflaton lie within the basin of attraction of themetastable SUSY-breaking vacuum, rather than the SUSY-preserving one, amounts to thecondition ϕ e ≪ ϕ max . For large a (i.e., N f /N ≫ s N e N π N f y M P Λ ≪ M P . However, it is instrumental to consider how large Λ may be before imperiling the radiativestability of the Planck scale. As noted in [32, 33], in effective theories with a large number ofspecies, radiative stability of Newton’s constant gives rise to a constraint on the size of thecutoff. Coupling the magnetic theory to N = 1 supergravity, the contribution to the effectivePlanck mass from light fields is of the form δM P ∼ Λ π ∂ K∂φ † α ∂φ β ! − ∂ K∂φ † α ∂φ β ! = Λ π (cid:0) N f + 2 N N f (cid:1) . (5.14)Radiative stability of Newton’s constant therefore suggests Λ < ∼ πN f M P . Thus the naturalvalue of Λ – as well as the viable value most likely to favor slow roll towards the metastablevacuum – is Λ ∼ πM P /N f . With this value of Λ and the NDA estimate of y, the condition(4.10) becomes s N e N N f ≪ . (5.15)The late turnover in the potential owes largely to the considerable irrelevance of thedeterminant superpotential. This suggests that, for large N f and naturally Planck-scale Λ , the turnover may be pushed sufficiently close to M P to guarantee that ϕ rolls towards theorigin rather than the SUSY vacuum. Notice this is compatible with a ≫ , a condition thatlikewise supports the parametric longevity of the metastable SUSY-breaking vacuum. η problem Thus far our discussion has been restricted to the case of global supersymmetry; naturally,one would like to extend the analysis to the case of local supersymmetry. As in the case ofsupersymmetric hybrid inflation, there is no η problem at leading order provided an exactlycanonical K¨ahler potential K = Tr | M | + Tr | q | + Tr | q c | . The supergravity scalar potentialis of the general form V s = e K/M P ∂ K∂φ † α ∂φ β ! − (cid:18) ∂W∂φ α + WM P ∂K∂φ α (cid:19) ∂W † ∂φ † β + W † M P ∂K∂φ † β ! − M P | W | . (5.16)– 13 –long the inflationary trajectory of the ISS model ( M = ϕ √ N f N f , q = q c = 0), coupling thetheory to supergravity thus yields the scalar potential V s ( ϕ † , ϕ ) = N f µ exp (cid:20) ϕ † ϕM P (cid:21) "(cid:18) ϕ † ϕM P + ... (cid:19) − ϕ † ϕM P (5.17)= N f µ (cid:18) ϕ † ϕ ) M P + ... (cid:19) provided an exactly canonical K¨ahler potential. All terms proportional to | ϕ | cancel directly,preserving the small mass of the inflaton; the leading supergravity correction is O (cid:0) | ϕ | (cid:1) . However, this convenient cancellation should be considered a fine-tuning. Any additionalcontributions to the K¨ahler potential beyond canonical terms – which one would anticipatein the context of effective field theory – might be expected to generate the usual SUSY η problem. A K¨ahler potential for the inflationary trajectory of the form K ( ϕ † , ϕ ) = ϕ † ϕ + b ( ϕ † ϕ ) Λ + ..., (5.18)where b is a dimensionless coefficient, yields a scalar potential for the inflaton with termsquadratic in ϕ, V ( ϕ ) = N f µ (cid:18) bϕ † ϕ Λ + ... (cid:19) . (5.19)These corrections arise both in the context of global and local supersymmetry, leading to aninflaton mass m ϕ = bN f µ Λ > ∼ bH , which kills inflation for b ∼ O (1).It is natural to consider how finely-tuned the coefficients of higher-order terms in theK¨ahler potential must be in order to preserve slow-roll inflation. Consider the magneticK¨ahler potential in terms of the rescaled variables. Generalizing the large- N f argumentsof [31] to the case of N ≥ , higher-order corrections to the K¨ahler potential should obtainthe form δK ( M † M ) = b (4 π ) N N f Λ Tr (cid:16) M † M (cid:17) Tr (cid:16) M † M (cid:17) + b (4 π ) N N f Λ Tr (cid:16) M † M M † M (cid:17) + ... (5.20)where b , b are O (1) coefficients. Upon rescaling to obtain canonically-normalized fields, thistranslates to a K¨ahler potential along the inflationary trajectory of K ( ϕ † ϕ ) = | ϕ | + 16 π b N N f Λ (cid:0) | ϕ | (cid:1) + 16 π b N N f Λ (cid:0) | ϕ | (cid:1) + ... (5.21)These corrections to the K¨ahler potential give rise to an inflaton mass term of the form m ϕ = 16 π ( b + b ) µ N N f Λ , and thus correction to the slow-roll parameter η of order δη ≈ π ( b + b ) M P N N f Λ ∼ b + b N . In order for this contribution to the slow-roll parameters notto interfere with slow-roll inflation, the coefficients b , b must satisfy b , b ≪ N . (5.22)– 14 –ere the condition on b , b may be reasonably satisfied for N > ∼ . Although quadratic cor-rections to the inflationary potential arise as expected, the coefficients may naturally be smallenough to preserve slow-roll inflation without additional fine-tuning. It is worth emphasizingthat this statement relies upon na¨ıve dimensional analysis estimates of corrections to theK¨ahler potential; such corrections are not explicitly known in terms of microscopic variables.Yet it is suggestive that the SUSY η problem for inflationary theories involving supersym-metric gauge dynamics may be resolved by carefully considering the strong-coupling behaviorof the K¨ahler potential near the cutoff.
6. Inflationary predictions for ISS-flation
In order to obtain inflationary predictions, it is useful to employ the formalism of [34] fora multi-component inflaton. There arise contributions to the spectral index and densityperturbations from fluctuations both along the inflationary trajectory (parameterized by ϕ )and orthogonal to the inflationary trajectory. In order to account for orthogonal contributions,consider parameterizing M by M = ϕ p N f N f + φ (6.1)where φ is an adjoint of the flavor symmetry, corresponding to directions orthogonal to theinflationary trajectory. The amplitude of density perturbations in this case is given by P ζ ( ϕ, φ ) / = 12 √ πM P V / p V φ V φ + V ϕ V ϕ (6.2)where, e.g., V ϕ = ∂V∂ϕ . Assuming initial fluctuations are all roughly of the same order, thecontributions from orthogonal directions are equal to those from the inflationary trajectory,hence P ζ ( ϕ, φ ) / ≃ √ P ζ ( ϕ ) / ≃ r N e N f N y (cid:18) µM P (cid:19) ≃ r N e π N f (cid:18) µM P (cid:19) . (6.3)Notice that the constraint guaranteeing the validity of slow-roll inflation in the ISS theory, N f /N ≫ µ, M P . Recall that the WMAP normalization is P / ζ = 4 . × − ; this corresponds to a constraint µM P ∼ × − N − / f . (6.4)Neglecting the dependence on N f , we would again obtain the inference of GUT-scale µ ∼ GeV. This is somewhat unfortunate for low-scale gauge mediation of supersymmetry breaking.Even in gravity-mediated SUSY-breaking scenarios, this would naively lead to gaugino andgravitino masses of order m / ∼ m / ∼ µ /M P ∼ GeV, far too high for weak-scale– 15 –USY. The situation is helped somewhat by N f ≫ , but only by a few orders of magnitudefor even an absurdly large value of N f ; obtaining m / ∼ m / ∼ M weak (and thus µ ∼ )in a gravity-mediated scenario would require N f ∼ . Indeed, Maintaining the hierarchy µ ≪ Λ ≃ πM P /N f in this scenario instead constrains N f < ∼ . The spectral index of the density perturbations in this case is given by n s − − M P V ( V ϕ V ϕ + V φ V φ ) + 2 M P V ( V ϕ V ϕ + V φ V φ ) (cid:18) V ϕϕ V ϕ V ϕ + 2 V φϕ V φ V ϕ + V φφ V φ V φ (cid:19) (6.5) ≈ − y N M P π ϕ e = − N e (6.6)Thus we find the spectral index of density perturbations to again be n s ≃ − N e ≈ . . (6.7)While this appropriately red spectrum is appealing, the high scale of µ required to match theobserved amplitude of density perturbations – even provided N f ≫ m / is far too large for anomaly-mediated split SUSY, which requires m / < ∼
50 TeV for TeV-scale gauginos. However, if we assume cancellation of the cosmologicalconstant in the SUSY-breaking metastable vacuum in a post-inflationary era, it is reasonableto consider generation of gaugino masses directly from R- and SUSY-breaking as in [35].Provided the F-term of the chiral compensator is engineered so that F φ ≪ m / (as mightbe obtained in a suitable extra-dimensional construction), one readily obtains a gravitinomass m / ∼ √ N c µ /M P ∼ GeV with gaugino/Higgsino mass m / ∼ m / /M P ∼ TeV . Alternately, one might imagine the persistence of an unbroken R symmetry in the low-energytheory, whose breaking by additional dynamics generates suitably small gaugino and Higgsinomasses.A more attractive scenario for conventional gauge or gravity mediation would requirefurther separation of the scales involved in SUSY-breaking and the generation of densityperturbations. Such a scenario may arise naturally in the context of ISS models, whereinhomogenous preheating results from the breaking of the global flavor symmetry at the endof slow-roll.
7. Curvature perturbations from broken flavor symmetries
In [15, 38, 39] it was observed that curvature perturbations may be generated by inhomoge-nous preheating due to the breaking of an underlying global symmetry at the end of slow-rollinflation. Quantum fluctuations generated during the inflationary era correspond to fluctu-ations in the initial conditions of the preheating phase. Whereas in the case of unbroken– 16 –lobal symmetry these fluctuations in the initial conditions lead to background evolutionsthat are related by time translation, in the case of broken global symmetry they give riseto inhomogeneities in preheating efficiency and thereby generating curvature perturbations.The scale of these curvature perturbations depends on the dynamics of both the inflationaryand preheating phases, and may readily constitute the main source of perturbations to thebackground metric.The ISS model provides a natural context for the realization of this scenario, since the SU ( N f ) global flavor symmetry of the theory is broken at the end of slow-roll inflation inthe SUSY-breaking vacuum. Contributions to the power spectrum of curvature perturbationsarising from inhomogenous preheating may be significant and, moreover, admit a lower scaleof µ suitable for weak-scale SUSY breaking. Let us first briefly review the general mechanism elucidated in [15]. In a multi-componentinflationary scenario where the inflaton ~φ consists of many background fields φ i related by aglobal symmetry, there may arise fluctuations of the φ i both parallel and perpendicular tothe inflaton trajectory. Fluctuations parallel to the direction of motion in field space corre-spond to adiabatic curvature perturbations of the sort generated in single-field inflationaryscenarios, while fluctuations orthogonal to the direction of motion correspond to isocurvatureperturbations. At the time t between the end of the slow-roll inflationary era and decay ofthe inflaton, the value of the background inflaton will have acquired a spatial dependence dueto quantum fluctuations, ~φ ( t , x ) = ~φ ( t ) + δ~φ ( t , x ) . (7.1)The values of the background inflaton fields φ i at the end of slow-roll inflation serve as initialconditions for the comoving number density n χ of particles produced during preheating [40].If the global symmetry of ~φ is unbroken, these fluctuations are all related by symmetry trans-formations, and do not lead to fluctuations in preheating efficiency. If the global symmetry isbroken before the preheating phase, however, these fluctuations are no longer related by sim-ple transformations, and inhomogeneities in preheating may ensue. In this case, fluctuationsin ~φ ( t , x ) result in fluctuations of n χ . Since the energy density generated during preheatingis proportional to the comoving number density, ρ χ ∝ n χ . Assuming non-adiabatic pressureperturbations during the preheating phase may be neglected, curvature perturbations duringpreheating can thus be expressed in terms of the number density perturbation, ζ ≡ ψ − H δρ χ ˙ ρ χ ≈ α δn χ n χ . (7.2)Here we have taken the spatially flat gauge; the proportionality constant α depends on theredshifting of the preheating particle. Now fluctuations in ~φ parallel to the direction of mo-tion in field space lead to initial conditions for n χ related by time-translation; the resultantadiabatic curvature perturbations could be gauged away by a suitable choice of slicing and– 17 –hreading. Fluctuations in ~φ perpendicular to the direction of motion may not be gauged awayin an analogous manner, and lead to observable fluctuations in n χ . In this manner, isocur-vature perturbations are converted into adiabatic perturbations through inhomogeneities inpreheating efficiency. From Eqn. (7.2) we may estimate the curvature perturbations arisingfrom inhomogenous preheating efficiency, ζ ≈ α ∂ ln( n χ ) ∂φ ⊥ δφ ⊥ (7.3)where δφ ⊥ denotes fluctuations perpendicular to the inflationary trajectory during preheating .It is crucial to note that the amplitude of quantum fluctuations δφ ⊥ is determined duringthe slow-roll inflationary phase, while α ∂ ln( n χ ) ∂φ ⊥ comes from the details of the preheatingprocess. The resultant power spectrum and spectral index of the curvature perturbationsfrom inhomogenous preheating are P ζ ( k ) = (cid:20) α ∂ ln( n χ ) ∂φ ⊥ (cid:21) P δφ ⊥ ( k ) (7.4) n s − d P ζ d ln k = d ln P δφ ⊥ d ln k . (7.5)The key feature is that the power spectrum of these curvature perturbations is the productof orthogonal perturbations P δφ ⊥ ( k ) , determined during the slow-roll inflationary process, andan amplifying factor h α ∂ ln( n χ ) ∂φ ⊥ i from the inhomogenous preheating process after slow-rollinflation has ended.The fluctuations in n χ may be readily calculated in models of instant preheating [41]. Herethe inflaton φ is coupled to the scalar preheating field χ via an interaction L φχ = − g | φ | χ , and χ is coupled to fermions ψ by an interaction L χψ = h ¯ ψψχ. The process φ → χ → ψ leadsto efficient conversion of the energy stored in the inflaton into fermions.If the inflaton trajectory doesn’t pass exactly through the minimum of the potential,but instead at a minimum distance | φ ∗ | , the preheat particles will be characterized by aneffective mass m χ ( φ ∗ ) = g | φ ∗ | . Then the comoving number density of preheat particles isgiven by [41] n χ = (cid:16) g | ˙ φ ∗ | (cid:17) / π exp " − π m χ ( φ ∗ ) g | ˙ φ ∗ | . (7.6)The power spectrum of curvature perturbations may then be calculated from the dependenceof | φ ∗ | on φ ⊥ . Let us now consider the explicit realization of inhomogenous preheating inISS-flation.
The ISS model possesses a large SU ( N f ) flavor symmetry that is broken to SU ( N f ) → SU ( N c ) × SU ( N ) in the metastable SUSY-breaking vacuum. As this global symmetry is– 18 –roken at the end of slow-roll inflation, prior to any preheating or reheating effects, it pro-vides a natural context for the realization of inhomogenous preheating and the generation ofcurvature perturbations.Consider the fate of components of the inflaton trajectory after slow-roll inflation hasceased, ϕ < ∼ ϕ c . N components obtain masses m ∼ yµ from the squark vev, while theremaining N c components are pseudo-moduli with vanishing tree-level masses. These pseudo-moduli instead obtain positive masses at one loop, which in [3] were determined to be m ∼ log 4 − π N y µ . Once the squarks roll off into the SUSY-breaking vacuum, the components of the inflatonfeel a quadratic potential near the origin, but one that is much steeper for the N componentswith tree-level masses than for the N c pseudo-moduli. Consider now the inflationary trajec-tory after slow-roll, parameterized by massive components ϕ and pseudo-moduli ϕ : M = ϕ √ N N ϕ √ N c N c ! (7.7)The potential near the origin is essentially quadratic, given by V ( ϕ , ϕ ) ≈ m ϕ + m ϕ (7.8)with m , m as above. Under the influence of the quadratic potential, the components ϕ , ϕ roll to the origin. Recall that the inflaton is coupled to the squarks q, q c via an interaction L Mqq c = − y N f | ϕ | (cid:0) | q | + | q c | (cid:1) (7.9)and that there also exists a coupling of squarks to fermions of the form L qq c ψ = y p N f qψ ϕ ψ q c + y p N f q c ψ ϕ ψ q . (7.10)These couplings give rise to preheating as the inflaton components roll through the origin, withthe light components of the squarks q, q c playing the role of the preheat fields. Oscillationsof the inflaton about the origin begin when | M | < µ √ y , i.e., the amplitudes of oscillation are | ϕ | = ϕ ,c = q Ny µ, | ϕ | = ϕ ,c = q N c y µ. The velocities of the fields as they pass throughthe minimum of the potential are given by | ˙ ϕ | ≈ m ϕ ,c and | ˙ ϕ | ≈ m ϕ ,c , respectively.As the fields begin to oscillate, ϕ rolls much faster than ϕ , with | ˙ ϕ || ˙ ϕ | ∝ m m = (cid:18) log 4 − π N y (cid:19) − / ≫ . (7.11)As such, when ϕ rolls through the minimum of its potential and initiates preheating, ϕ isstill close to its initial amplitude. The velocity entering into the number density of preheatparticles, Eqn. (7.6), is dominated by | ˙ ϕ | , while the displacement at the minimum is set– 19 –y the amplitude of | ϕ | . To good approximation, then, | ˙ φ ∗ | ≈ m ϕ ,c , | φ ∗ | ≈ ϕ ,c , m χ ≈ y N f (cid:0) ϕ + ϕ (cid:1) , and the comoving number density of preheat particles is then given by n χ ≈ (cid:0)p N/N f yµ (cid:1) / π exp − πyϕ p N N f µ ! . (7.12)The fluctuations in n χ are dominated by those of ϕ , so we have to leading order δn χ n χ ≈ − πyϕ p N N f µ δϕ ≈ − πµ r yN δϕ . (7.13)The ensuing power spectrum of curvature perturbations arising from inhomogenous preheat-ing in the ISS model is given by (see Appendix A for details) P / ζ ≈ r yN f N µM P . (7.14)Whereas the adiabatic curvature perturbations arising during slow-roll inflation go as ( µ/M P ) , those from inhomogenous preheating go as µ/M P , allowing a greater separation of the in-flationary and SUSY-breaking scale µ from M P . Matching the observed spectrum due tocurvature perturbations of this type would entail µM P ∼ × − · N / N − / f (7.15)This separation of scales is nearly two orders of magnitude greater than considered previ-ously; neglecting factors of N, N f it suggests µ ∼ GeV, and further separation of scalesis obtained at large N f . However, obtaining m / ∼ M weak in this scenario would still re-quire N f ∼ , far too large to maintain the hierarchy µ ≪ Λ ≃ πM P /N f . Assumingpreheating is relatively efficient, the curvature perturbations arising from inhomogenous pre-heating should dominate over those arising from the inflaton alone provided N / f < ∼ M P µ . For µ/M P ∼ − , this is the case for N f < ∼ . The spectral index of these curvature perturbations is n s ≈ . , (7.16)preserving the red spectral index prediction from supersymmetric hybrid inflation. Alternatively, one might imagine coupling the inflaton to a separate preheating sector notitself embedded in the ISS model. Assuming ϕ preferentially decays to preheat particles ς viaan interaction of the form L ϕς = − λ | ϕ | ς , the resultant density perturbations are of theform P / ζ ≈ λ √ s N f N c N y µM P . (7.17)– 20 –atching the observed curvature perturbations in this scenario would entail µM P ∼ × − · λ − N − / f . (7.18)Assuming λ is perturbative, one may obtain weak-scale SUSY-breaking in this scenario from N f ∼ ; the hierarchy µ ≪ Λ is automatically satisfied in this scenario. In this case,curvature perturbations from inhomogenous preheating would certainly dominate over thosefrom slow-roll inflation. The spectral index is again n s ∼ . , independent of the preheatingmechanism. This is perhaps the most compelling setting for inflation and weak-scale SUSYbreaking in the ISS model. There is no a priori reason to expect the curvature perturbations arising from inhomogenouspreheating to be entirely gaussian [42]. Indeed, such perturbations may exhibit a significantdegree of non-gaussianity and may serve to discriminate ISS-flation from more conventionalmodels of supersymmetric hybrid inflation.The degree of non-gaussianity may be characterized by the non-linearity parameter f NL parameterizing the non-gaussian contribution to the Bardeen potential Φ , Φ = Φ G + f NL Φ G . (7.19)Here Φ G denotes the gaussian part. Assuming | f NL | > , Φ and ζ may be accurately relatedvia Φ = − ζ = − α δn χ n χ . (7.20)We may estimate f NL in ISS-flation by expanding δn χ /n χ to second order in δϕ ; for instan-taneous preheating within the ISS sector this yields δn χ n χ ≃ − πyϕ p N N f µ δϕ − πy p N N f µ − π y ϕ N N f µ ! ( δϕ ) (7.21)As such, the non-linearity parameter is given by f NL ≃ π s NN f − ! (7.22)Given N/N f ≪ , this suggests that the non-gaussianities of the inhomogenous preheatingcurvature perturbations in this model are well approximated by | f NL | ≈ . (7.23)– 21 – similar estimate is obtained for the case of a separate preheating sector, although in bothcases it should be emphasized that this is only a rough approximation. This result corre-sponds well with intuition developed for non-gaussianities in inhomogenous preheating sce-narios with small global symmetry breaking [42]; O (1) global symmetry breaking, as in thecase of ISS-flation, leads to a relatively small degree of non-gaussianity. It is interesting tonote, however, that this degree of non-gaussianity is above the observational limit f NL ∼ | f NL | ∼
8. Conclusions
We have seen that inflation may naturally occur while rolling into the supersymmetry-breaking metastable vacuum of massive supersymmetric QCD. Although the combinationof supersymmetry-breaking and inflation is more a matter of novelty than profundity, thisscenario is particularly attractive in that it contains a concrete UV completion of the infla-tionary sector and the potential for distinctive observational signatures. Successful slow-rollinflation in the ISS model requires a large number of flavors and relatively small magneticgauge symmetry, as well as a natural hierarchy m ≪ µ ≪ Λ < ∼ M P . Although quadraticcorrections to the inflationary potential arise in the presence of a non-canonical K¨ahler po-tential, the ensuing contribution to slow-roll parameters may be small enough to forestall theconventional SUSY η problem. Moreover, the spontaneous global symmetry breaking that ac-companies the end of slow-roll inflation in the ISS model may give rise to dominant curvatureperturbations through inhomogenous preheating. Such perturbations may possess observablenon-gaussianities, further distinguishing ISS-flation from its more conventional cousins.It is difficult to simultaneously obtain weak-scale SUSY breaking and the observed infla-tionary spectrum strictly from the dynamics of the ISS model; standalone ISS-flation wouldseem to favor split SUSY or other high-scale mediation. However, weak-scale SUSY-breakingusing conventional gauge- or gravity-mediation is feasible if the primary contribution to pri-mordial curvature perturbations arises from coupling to a separate preheating sector.It would be interesting to consider concrete realizations of split supersymmetry using ISSSUSY-breaking dynamics, given the high scale of SUSY-breaking required to match inflation-ary observables in the absence of inhomogenous preheating. Likewise, a more careful analysisof non-gaussianities arising from inhomogenous preheating may be useful in light of recentevidence for significant non-gaussianities in the WMAP 3-year data [45].– 22 – cknowledgements I am especially indebted to Shamit Kachru for helpful discussions and comments on themanuscript. I would also like to thank Savas Dimopoulos and Jay Wacker for useful conver-sations. I acknowledge the hospitality of the International Centre for Theoretical Physics andthe Simons Workshop at Stony Brook, where parts of this work were completed. This researchis supported by an NDSEG Fellowship, National Science Foundation grant PHY-9870115, andthe Stanford Institute for Theoretical Physics.
Appendix A. Power spectrum from inhomogenous preheating
The power spectrum of the curvature perturbations arising from inhomogenous preheating inthe ISS theory is given by P ζ ( k ) = (cid:20) α ∂ ln( n χ ) ∂φ (cid:21) P δϕ ( k ) = π y N µ P δϕ ( k ) . (A.1)The power spectrum of δϕ is set during inflation, and may be calculated accordingly. Onsuperhorizon scales the amplitude of quantum fluctuations is given by | δϕ ( k ) | ≃ H k √ k (cid:18) kaH (cid:19) η (A.2)and the resultant power spectrum is P δϕ ( k ) ≡ k π | δϕ ( k ) | = (cid:18) H k π (cid:19) (cid:18) kaH (cid:19) η (A.3)Evaluated at horizon exit, k = aH, we arrive at the observable power spectrum P δϕ = (cid:18) H π (cid:19) ≈ N f µ π M P (A.4)Thus the power spectrum of curvature perturbations arising from inhomogenous preheatingin the ISS model is given by P / ζ ≈ r yN f N µM P (A.5)As for the spectral index of these curvature perturbation, we have n s − ≡ d log P ζ d log k = d log P δϕ d log k ≈ η ≈ − N e (A.6)– 23 – eferences [1] S. Dimopoulos and H. Georgi, Softly broken supersymmetry and su(5) , Nucl. Phys.
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