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LuisAlvarez-Gaume,BW26April20131Minimal InﬂationSupersymmetry Breakingand Non-gaussianitiesWork in progress with C. Gómez and Raúl JiménezFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April20132Originally Inﬂation was related to the horizon, ﬂatnessand relic problemsNowadays, its major claim to fame is seeds ofstructure. There is more and more evidence that thegeneral philosophy has some elements of truth, and it isremarkably robust…Many of the features that agree with observation arerather model independentLately Planck and WMAP9 have provided interestingmeasurements on Non-GaussianityInﬂation is over 30 years oldFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April20133˙aa⇥2+ka2=8 G3⇥¨aa=4 G3(⇥ + 3p)˙ + 3H( + p) = 0S =116 G⇤⇥g(R 2 ) +⇤⇥g12gab⇤a⇥⇤b⇥ V (⇥)⇥=12˙⇥2+ V (⇥) p =12˙2V ( )p = w˙ + 3˙a(t)a(t)(1 + w) = 0= 0a0a⇥3(1+w)H˙a(t)a(t)⇥c =3H28 Gc1 =ka2H2FRLWSingle ﬁeld construction, natural insome cases, as we will see not in thesupersymmetric caseFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April20134VV<< 1,VV<< 1Slow roll paradigmHybrid inﬂation˙2<< V ( )Accelerating the Universe¨aa=4 G3(⇥ + 3p)p <13 ¨a(t) > 0Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April20135Origins of InﬂationThe number of models trying generating inﬂation islarge. Frequently they are not very compelling andwith large ﬁne tunings.Different UV completions of the SM providealternative scenarios for cosmology, and it makessense to explore their cosmic consequences. Henceany such theories lead to some variations onCosmology and/or inﬂation.Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April20136Basic propertiesEnough slow-roll to generate the necessary numberof e-foldings and the necessary seeds for structure.A (not so-) graceful exit from inﬂation, otherwise weare left with nothing.A way of converting “CC” into useful energy:reheating.Everyone tries to ﬁnd “natural” mechanisms withinits favourite theory.Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April20137Supersymmetry is our choiceIn the standard treatment of global supersymmetry the order parameter ofsupersymmetry breaking is associated with the vacuum energy density.More precisely, in local Susy, the gravitino mass is the true order parameter.Having a vacuum energy density will also break scale and conformalinvariance.When supergravity is included the breaking mechanism is more subtle, andthe scalar potential far more complicated.Needless to say, all this assumes that supersymmetry exists in NatureIt provides naturally an inﬂaton and a graceful exitFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April20138SSB ScenariosObservable SectorHidden SectorMEDIATORIt is normally assumed that SSB takes places at scales well below thePlanck scale. The universal prediction is then the existence of amassless goldstino that is eaten by the gravitino. However in thescenario considered, the low-energy gravitino couplings are dominatedby its goldstino component and can be analyzed also in the global limit.This often goes under the name of the Akulov-Volkov lagrangian, orthe non-linear realization of SUSYm3/2 =fMp=µ2Mpµ ! 1M ! 1 m3/2 ﬁxedFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April20139Flat directionsOne reason to use SUSY in inﬂationary theories is the abundance of ﬂatdirections. Once SUSY breaks most ﬂat directions are lifted, sometime by non-perturbative effects. However, the slopes in the potential can be maintainedreasonably gentle without excessive ﬁne-tuning. The symmetries of thesuperpotential become complexiﬁed:For ﬂat Kahler potentials, and F-term breaking, there is always a complex ﬂatdirection in the potential. A general way of getting PSGB, the key to most susymodels.The property below holds for any W breaking SUSY.Most models of supersymmetric inﬂation are hybrid models (multi-ﬁeld models,chaotic, waterfall...)F = @ W( ).V = @ W( ) @ W( )V ( + z h F i) = V ( )G ! GcFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201310R-symmetry explicitly brokenThe R-symmetry is explicitly broken to a discrete subgroupAn explicit R-symmetry does not allow a soft mass for the gluinosIf it is spontaneously broken, the light axion ﬁeld generated hasgenerically unacceptable couplings.Furthermore, since we need to include (super)gravity, it is anecessary condition to solve the -problem that plagues many ofthese theories⌘(x, ✓) ! eiq ↵(x, ei↵✓)Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201311Arbey et al arXiv:1207.1348Acharya, Kane & Kumar: arXiv:1204.2795News from SUSYFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201312Important properties of SSBThe starting point of their analysis is the Ferrara-Zumino (FZ) multiplet of currentsthat contains the energy-momentum tensor, the supercurrent and the R-symmetrycurrent (we follow the presentation of Komargodski and Seiberg)Jµ = jµ + Sµ + ˙ S˙µ + ( ⇥⇤) 2T⇤µ + . . .D˙J ˙ = D XX = x(y) + 2 ⇥(y) + 2F(y)⇥ =23µ˙ S˙µ, F =23T + i⌅µjµFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201313S= d4K( i, ¯¯i) + d2W( i) + d2 ¯ ¯W(¯¯i)J ˙ = 2gi(D i)( ¯D ˙¯ ) 23 [D , ¯D ˙ ]K + i⇥ (Y ( ) ¯Y (¯))General Local LagrangianX = 4 W13D2K12D2Y ( )X is a chiral superﬁeld, microscopically it contains the conformal anomaly (the anomalymultiplet), hence it contains the order parameter for SUSY breaking as well as thegoldstino ﬁeld. It may be elementary in the UV, but composite in the IR. Generically itsscalar component is a PSGB in the UV. This is our inﬂaton. The difﬁculty with thisapproach is that WE WANT TO BREAK SUSY ONLY ONCE! unlike other scenarios in theliterature, and cancelling the cosmological constant today yields very strong constraints onthe inﬂationary parameters. This is why we call this scenario minimal inﬂation.The key observation is: X is essentially unique, and:X XNLUV IRX2NL = 0SPoincare/PoincareFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201314Some IR consequencesL = d4XNLXNL + d2f XNL + c.c.XNL =G22F+ 2 G + 2FThis is precisely the Akulov-Volkov LagrangianFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201315Coupling goldstinos to otherﬁelds: reheatingWe can have two regimes of interest. Recall that a useful way toexpress SUSY breaking effects in Lagrangians is the use of spurionﬁelds. The gluino mass can also be included...msoft << E <<E << msoftThe goldstino superﬁeld is the spurionIntegrate out the massive superpartnersadding extra non-linear constraints⇥d4 XNLfm2QeVQ +⇥d2 XNLf(B Q Q + AQ Q Q ) + c.c.X2NL = 0, XNL QNL = 0 For light fermions, and similar conditionsfor scalars, gauge ﬁelds,...Reheating depends very much on the details of the model, as does CPviolation, baryogenesis...Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201316An important part of our analysis is the fact that the graceful exit is provided by the Fermipressure in the Landau liquid in which the state of the X-ﬁeld converts once we reach theNL-regime. This is a little crazy, but very minimal however...Some detailsW(X) = f0 + f XV = eKM2(K 1X, ¯XDW ¯DW3M2|W|2) D W = X W +1M2 XK WK(X, ¯X) = X ¯X 1 +a(X + ¯X)2MbX ¯X6M2c X2+ ¯X29M2+ . . .!2M2log✓X + ¯XM+ 1◆Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201318ds2= 2gz¯zdsd¯z = @z@¯zK(↵, )M2(d↵2+ d 2)S = L3Zdta3✓12g(↵, )M2( ˙↵2+ ˙2) f2V (↵, )◆t = ⌧M/f S = L3f2m 13/2Zd⌧a3✓12g(↵, )(↵02+ 02) V (↵, )◆z = M(↵ + i )/p2Choosing useful variablesFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201319↵00+ 3a0a↵0+12@↵ log g(↵02 02) + @ log g↵0 0+ g 1V 0↵ = 000+ 3a0a0+12@ log g( 02↵02) + @↵ log g↵0 0+ g 1V 0= 0a0a=HM3/2=1p3✓12g(↵02+ 02) + V (↵, )◆H =r118⇣3V +p6V 0 + 9V 2⌘D ˙ i/dt ⇠ 0Cosmological equationsThe full equations of motion, without fermionsLooking for the attractor and slow roll implies that the geodesic equation on the target manifold issatisﬁed for a particular set of initial conditions. This determines the attractor trajectories in generalfor any model of hybrid inﬂation. Numerical integration shows how it works. We have not tried toprove “theorems’ but there should be general ways of showing how the attractor is obtained this wayFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201320Trajectories with more than the observed 12 e-folding a=0, b=1, c=-1.5Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201321One more exampleFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201323Epsilon and etaFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April2013240.2 0.4 0.6 0.8a0.10.20.30.40.5b0.05 0.10 0.50 1.00 5.00t0.020.040.060.080.10V0.05 0.10 0.15 0.20 0.25 0.30a0.20.40.60.81.0bAttractor and inﬂationary trajectoriesNearly a textbook example of inﬂationary potentialFriday, 26 April 13
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LuisAlvarez-Gaume,BW26April201325Decoupling and the Fermi sphere⌘ = (mINFm3/2)2The energy density in the universe (f^2) contained in the coherent X-ﬁeld quicklytransforms into a Fermi sea whose level is not difﬁcult to compute, we match the highenergy theory dominated by the X-ﬁeld and the Goldstino Fock vacuum into a theorywhere effectively the scalar has disappeared and we get a Fermi sea, whose Fermimomentum isTo produce the observed number of particles in the universe leads to gravitino masses inthe 10-100 TeV region.qF =sf⌘Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201326Non-GaussianitiesSee Raúl’s talk for details on observations and basic deﬁnitions.In our theory we naturally have two ﬁelds, and before weenter the inﬂationary regime, there is some slashing back andforth. Regardless of the initial velocity (within limits) thenonlinear friction slows down the ﬁeld rapidly, entering into aneffective single ﬁeld regime. From Chen (2006):Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201327Parameter distributionsFor most trajectories which can generate 40-50 efoldings. Nongaussianities aregenerated at large scales, comparable to the horizon scale today, and scales of the sizeof dwarf galaxies. This is the typical behaviour for most trajectories.pf =pfNL105MOnce again we get relatively high values for supersymmetry breaking but below GeV1014Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201328Summary of our scenarioWe take as the basic object the X ﬁeld containing the Goldstino. Its scalar componentabove SSB behaves like a PSGB and drives inﬂationIts non-linear conversion into a Landau liquid in the NL regime provides an originalgraceful exit, in our case the conversion is not complete and we get a dark universewith goldstinos and inﬂatons. The conversion is not complete because the massrelations are such that the inﬂaton is not much heavier than the goldstinos. We get ahybrid universe populated by dark objects. The next step would be to work out somesimple scenario for reheating.Reheating can be obtained through the usual Goldstino coupling to low energy matterIn the simplest of all possible such scenarios, the Susy breaking scale is ﬁtted to be ofthe order of 10^{13-14} GeV, m of the order of 10-100 TeV (the plot from Kane et al).It is interesting that for a range of parameters, the conclusions we draw from NG agreewith the ﬁt to ﬂuctuations.We believe that our ideas show that the general properties of supersymmetry breakingin a supergravity context provide a natural description of inﬂation in terms of a hybridﬁeld that avoid the standard no-go theorems on non-gaussianity. It seems a robustscenario as required by inﬂation.Friday, 26 April 13
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LuisAlvarez-Gaume,BW26April201329Thank youFriday, 26 April 13
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