Renata Kallosh
with Linde and with Ceco , Ferrara, Porra , Van Proeyen, Bercnocke, Roest, Westphal, Wrase We discuss infla onary models which are flexible enough to fit the data (Planck 2013 or BICEP2 or in between), which can be implemented in string theory/supergravity, and which may tell us something interes ng and instruc ve about the fundamental theory from the sky
We describe new models of infla on and dark energy/cc. New results on de Si er Landscape: how to avoid tachyons in string theory mo vated supergravi es
Recent new tools have allowed us to construct new simple models of infla on with dS upli ing in the context of spontaneously broken supersymmetry
A nilpotent chiral mul plet, Volkov-Akulov golds no and A New Toy In Town! D-brane physics BICEP: Pretty Swirly Things r 0.2 ⇠
BICEP2 - Planck Drama IsIs ItIt Dust?Dust? IsIs ItIt Dust?Dust?
Flauger, Hill, Spergel June 2014 r =0
(Flauger, Hill, & Spergel, arXiv:1405:7351)
(Flauger, Hill, & Spergel, arXiv:1405:7351) Genus Topology and Cross-Correla on of BICEP2 and Planck 353 GHz B-Modes: Further Evidence Favoring Gravity Wave Detec on
Genus TopologyWesley N. Colley and J. Richard and Cross-Correlation of BICEP2 with PlanckGo 353, III, September 2014 GHz B-modes: Further Evidence for Gravity Wave Detection 11
Table 1. Relative contributions to BICEP map, for di↵erent anal- yses of the dust. The error on each measured correlation, CPB, is estimated to be 0.039. C⇤ refers to the correlation necessary to imply a r of 0. In all cases, the correlation measured is at least 2.5 below this level (for Maps I–V, these levels are 2.6 ,4.9 , 3.1 ,4.1 and 3.1 ).
Map (µK) C r C xyr0.1 353 PB ⇤ Map I 4.97 0.101 0.202⇠ 0.292 0.259 0.106 Map II 3.31 0.112 0.302 0.370 0.181 0.134 Map III 3.12 0.202 0.321 0.219 0.333 0.079 Map IV 2.96 0.181 0.337 0.274 0.278 0.099 Map V 3.03 0.161 0.331 0.301 0.250 0.109 Do not take it too seriously as it is based on redigi zed plots of both BICEP2 and Planck. 0), x =0.552 and we would find r =0.2, as calibrated by BICEP2’s power spectrum analysis. We present in Table 1 those results for Planck Maps I – V. However the basic idea to check Cross-The mean value of r from the 5 methods is r =0.106. A simple estimate of the uncertainty associated with our valuesCorrela on of BICEP2 and Planck is a good of r is the direct standard deviation of the above val- ues from the di↵erent methods; that computes to 0.020. ± Thisone. This is what BICEP2 and Planck are doing is a reasonable estimate of the error associated with our varied mapping processes. (For comparison, one could applynow, using the actual combined data. And we median statistics [c.f. Gott et al. 2001] to our 5 r val- ues; the median value is r =0.106, while the chance the true valueare wai ng… is between 0.99 and 0.109 is 62.5%, roughly 1 .) If we used the independent 353D =23.1 BD estimate from a sim- ple power-law interpolation between Planck at 353 GHz and Planck at 143 GHz to estimate the dust amplitude at 150 GHz, we would have gotten a mean value of r =11.4. Thus Figure 14. At top, the BICEP2 map (as in Fig. 1); in the mid- the uncertainty in r to do the uncertainty in this ratio is dle, our Planck 353 GHz Map V (as in Fig. 7). At bottom is 0.008. However, there is still some additional error in the the correlation of these two maps. All maps are in Mercator pro- ± r =? jection in the region RA 30 , 65 Dec 50 .Red estimate of z. The BICEP2 team reports that the gravita- | | 6 6 6 shows positive-positive correlations; blue shows negative-negative tional lensing power can vary by about 45.5%. As such, we correlations; green shows anti-correlations (negative-positive or recomputed our x, y and r values with z increased and de- positive-negative). creased by 45.5% (zmax was adjusted by the same resulting addends on z); this introduces an additional error of 0.016. Note that Equation (12) shows that adding or subtracting degree patches. We can measure the theoretical Cl for gravi- (0.455 0.1955) = 0.089 from z changes y not at all, but adds ⇥ tational lensing at l = 119 from Fig. 2 in the BICEP2 paper or subtracts 0.089 from x with consequent changes of r of (BICEP2 2014a). The theoretical Cl amplitude is of course 0.016. We also have the error in r introduced by the 0.039 ± based on lensing data from Planck and elsewhere. Taking error in the correlation measurements; this translates to an the ratio of the Cl’s from lensing and noise at l =119allows error of 0.029 in r. Each individual map has an uncertainty us to calculate that the ratio z/w =0.774. The noise and in its correlation coe cient CPB of 0.039 determined as we ± gravitational lensing power spectra are proportional to each have described, by cross-correlating BICEP2 with random other (both have Cl const over the range 50 Details on dust in the BICEP2 patch of the sky. Current conclusion Extrapola on of the Planck 353 GHz data to 150 GHz gives a dust power DBB ≡ l(l + 1)CBB/(2π) of 1.32 × 10−2 μK2 over the mul pole range of the primordial recombina on bump (40 < l < 120); l l CMB the sta s cal uncertainty is ±0.29 × 10−2 μK2CMB and there is an addi onal uncertainty (+0.28, −0.24) × 10−2 μK2CMB from the extrapola on. This level is the same magnitude as reported by BICEP2 over this l range, which highlights the need for assessment of the polarized dust signal even in the cleanest windows of the sky. The present uncertain es are large and will be reduced through an ongoing, joint analysis of the Planck and BICEP2 data sets. Chaotic Inflation in Supergravity: shift symmetry Kawasaki, Yamaguchi, Yanagida 2000 Kahler poten al and superpoten al W = mS The poten al is very curved with respect to S and Im Φ, so these fields vanish. But Kahler poten al does not depend on = p2Re =( + ¯)/p2 The poten al of this field has the simplest form, as in chao c infla on, without any exponen al terms: RK, Linde, Rube 2010 W = Sf( ) Superpoten al must be a REAL holomorphic func on. (We must be sure that the poten al is symmetric with respect to Im , so that Im = 0 is an extremum (then we will check that it is a minimum). The Kahler poten al is any func on of the type (( ¯)2,SS¯) The poten al as a func on of the K real part of at S = 0 is V = f( ) 2 | | FUNCTIONAL FREEDOM in choosing infla onary poten al Here S is a golds no mul plet: supersymmetry is broken only in the golds no direc on FUNCTIONAL FREEDOM in choosing infla onary poten al in supergravity allows us to fit any set of ns and r. 10 Planck Collaboration: Constraints on inflation Model Parameter Planck+WP Planck+WP+lensing Planck + WP+high- Planck+WP+BAO n 0.9624 0.0075 0.9653 0.0069 0.9600 0.0071 0.9643 + 0.0059 CDM + tensor s ± ± ± r0.002 < 0.12 < 0.13 < 0.11 < 0.12 2 ln 0 0 0 -0.31 Lmax Table 4. Constraints on the primordial perturbation parameters in the CDM+r model from Planck combined with other data sets. 1 The constraints are given at the pivot scale k = 0.002 Mpc . 5 potential V ( ),takeasquarerootofit,makeitsTaylor series expansion and thus construct a real holomorphic function which approximate V ( ) with great accuracy. However, one should be careful to obey the rules of the game as formulated above. For example, suppose one wants to obtain a fourth de- m2 2 gree polynomial potential of the type of V ( )= 2 (1+ a + b 2) in supergravity. One may try to do it by tak- ing K =( + ¯)2/2+SS¯) and f( )=m (1 + cei✓ ) [52]. For general ✓,thischoiceviolatesourconditionsfor f( ). In this case, the potential will be a fourth degree polynomial with respect to Im if Re =0.However, in this model the flat direction of the potential V ( ) (and, correspondingly, the inflationary trajectory) devi- ate from Re =0.(Also,inadditiontotheminimumat Sunday, March 31, 13 =0, the potential will develop an extra minimum at 1 i✓ Fig. 1. Marginalized joint 68% and 95% CL regions for ns and r0.002 from Planck in combination with= otherc datae sets compared.) As a to result, the potential along the infla- the theoretical predictions of selected inflationary models. FIG. 1: The green area describes observational consequences of tionary trajectory is not exactly polynomial, contrary to inflation in the Higgs model (10) with v 1 (a 1), for the the expectations of [52, 53]. Moreover, the kinetic terms reheating priors allowing N < 50 could reconcile this model lead to⌧ inflation with a(t) exp(At f ), with A > 0 and 0 < f < 1, withinflationary the Planck data. regime when the field rolls down fromwhere thef maximum= 4/(4 + ) ofand >of0. In the intermediate fields will inflation be there non-canonical and non-diagonal. the potential. The continuation of this area upwardsis no natural corresponds end to inflation, but if the exit mechanism leaves to the prediction of inflation which begins when the inflationary field initially predictions on cosmological perturbations un- Exponential potential and power law inflation This may not be a big problem, since the potential in is at the slope of the potential at v v.Inthelimitmodified, thisv class of models, predicts r 8 (ns 1)/( 2) | | !1 the direction orthogonal to the inflationary trajectory is Inflationwhich with corresponds an exponential potential to a 0, the predictions( coincideBarrow & Liddle with, the1993). It is disfavoured, being outside the ! joint 95% CL contour for any exponentially. steep. Therefore the deviation of this field predictions of the simplest chaotic inflation model with a quadratic 4 m2V( 2) = exp (35) from Re =0will not be large, and for sufficiently large potential . M 2 pl Hill-top models values of the inflaton field the potential will be ap- is called power law inflation (Lucchin & Matarrese, 1985), In another interesting class ofproximately potentials, the inflaton given rolls away by the simple polynomial expression because the exact solution for the scale factor is given by from an unstable equilibrium as in the first new inflationary mod- 2 2 a(t) t2/ . This model is incomplete, since inflation would els (Albrecht & Steinhardt, 1982f;(Linde /p, 19822) )..Butinordertomakeafullinvestigationof We consider | | not end without an additional mechanism to stop it. Assuming inflation p in such models one would need to study evo- an inflationary regime when the field rolls fromV the( ) 4 1 + ... , (37) such a mechanism exists and leaves predictions for cosmo- µp logicalmaximum perturbations of unmodified, the potential this class of at models =0 predicts,asinnewinflation lution of all fields numerically, and make sure that all r = 8(ns 1) and is now outside the joint 99.7% CL contour. where the ellipsis indicates higherstability order terms conditions negligible during are satisfied. An advantage of the scenario. Natural initial conditions for inflationinflation, but needed in this to ensure the positiveness of the potential model are easily set by tunneling fromlater nothing on. An exponent into a of p = methods2 is allowed only developed as a large field in [38, 39] is that all fields but one Inverse power law potential 3 inflationary model and predicts n 1 4M2 /µ2 + 3r/8 and T vanishs duringpl inflation, all kinetic terms are canonical universe with spatial topology ,seee.g.[24]andthe2 2 4 Intermediate models (Barrow, 1990; Muslimov, 1990) with in- r 32 Mpl/µ . This potential leads to predictions in agree- and diagonal along the inflationary trajectory, and in- versediscussion power law potentials in [51]. The results of investigationment with ofPlanck the+WP+BAO ob- joint 95% CL contours for super- Planckian values of µ, i.e., µ vestigation9 M . of stability is straightforward. servational consequences of this model [38, 42, 48] are & pl Models with p 3 predict n 1 (2/N)(p 1)/(p 2) = 4 s described byV( the) green area in Figure(36) 1.when Predictionsr 0. The hill-top of potentialFortunately, with p = 3 lies outside one can the obtain an exactly polynomial po- Mpl this model are in good agreement with observational data tential V ( ) in the theories with K = K(( ¯)2,SS¯) for a certain range of values of the parameter a 1. ⌧ using the methods of [38, 39], if the polynomial can be However, this does not mean that absolutely any po- represented as a square of a polynomial function f( ) tential V ( ) can be obtained in this simple context, or with real coefficients. As a simplest example, one may consider f( )=m 1 c + d 2 . The resulting po- that one has a full freedom of choice of the functions f( ) tential of the inflaton field can be represented as . It is important to understand the significance of 2 2 the restrictions on the form of the Kähler potential and m 2 superpotential described above. According to [39], in the V ( )= 1 a + a2b 2) . (11) ¯ 2 ¯ 2 theory with the Kähler potential K = K(( ) ,SS) RK, Linde and Westphal, 1405.0270 2 the symmetry of the Kähler potential ¯,aswellas Here a =0.25c/p2 and a b = d/2.Weusetheparametriza- ! ± the condition that f( ) is a real holomorphic function are tion in terms of a and b because it allows us to see what required to ensure that the inflationary trajectory, along happens0.20 with the potential if one changes a: If one de- which the Kähler potential vanishes is an extremum of creases a,theoverallshapeofthepotentialdoesnot the potential in the direction orthogonal to the inflati- change, but0.15 it becomes stretched. The same potential S =Im =0 r onary trajectory .Afterthat,theproper can be also0.10 obtained in supergravity with vector or ten- choice of the Kähler potential can make it not only an sor multiplets [36]. extremum, but a minimum, thus stabilizing the inflation- 0.05 Inflation in this theory may begin under the same ini- ary regime [38, 39, 44]. tial conditions0.00 as in the simplest large field chaotic in- The requirement that f( ) is a real holomorphic func- flation models0.93 0.94n.Thedi0.95 0.96fference0.97 0.98 is that0.99 in1.00 the small a nS tion does not affect much the flexibility of choice of the limit, the last2 60 e-foldings of inflation are described2 by (( ¯)2 ,SS¯) , W = Sf( ) , V = f( ) inflaton potential: One can take any positively defined theK theory . Meanwhile for large a one| has the| same Natural Infla on in Supergravity Natural infla on in theories with axion poten als is known for nearly 25 years (Freese et al 1990), but un l now it did not have any stable supergravity generaliza on. Invariably, there was an instability with respect to some moduli, or we needed some assump ons about string theory upli ing. The problem was solved only recently: RK, Linde, Vercnocke, 1404.6204 All non-inflaton moduli stabilized 3 and perform the following change of variables T iT V and S iS.Wefind ! ! 2 ⇤ aT aT + 8 W = S(e e ,K= K , (18) p2 ⌘ 6 where the inflaton is now the imaginary part of a scalar T . It leads to exactly the same physics as Model 2, and 4 very similar physics compared to Model 1. The relevant potential is, therefore, given again (approximately) by Fig. 1. 2 Model 4 2000 4000 6000 8000 10 000f Finally we give a supergravity model reminiscent of a potential with a sum of several cosines as in [4, 5]. The superpotential and K¨ahler potential are FIG. 2. Irrational axion potential for A = B =1,a =0.01p3, b =0.005p7. The field is shown in Planck units, from 0 to 10000. This may create an impression that the potential is very steep, but 2 aT bT W = p2⇤ S A sin + B sin ,K= K . in fact the potential is very flat and allows chaotic inflation. Just 2 2 ✓ ◆ as in the string landscape scenario [12–17], inflation may end in any (19) of the infinitely many metastable dS vacua with di↵erent values of The inflaton potential is the cosmological constant [18]. a b 2 V =2⇤4 A sin + B sin . (20) 2p2 2p2 vicinity of each of these dS vacua. As a result, one can ⇣ ⌘ have a broad spectrum of possibilities which allows to fit a large variety of observational data within the context III. IRRATIONAL AXION LANDSCAPE of a single model with a small number of parameters. Now we will make what could seem a minor modifica- tion of the previous model, but we will find a dramat- IV. INFLATION FOR a, b & 1 ically di↵erent potential. The K¨ahler potential now is K = K+, and the superpotential slightly di↵ers from Until now, we discussed the scenario with a, b 1. (18) However, string theory suggests that the parameters⌧ a, b 1. Can we still have natural inflation in that case? 2 aT bT & W = ⇤ S(1 Ae Be ) . (21) Let us consider a model with The potential at S = T + T¯ =0is 2 aT bT + W = ⇤ S(e e ),K= K . (23) a V = ⇤4 1+A2 + B2 2A cos We will assume that a, b & 1, a b 1. One can show p2 that in this case inflation is indeed possible.⌧ ⇣ (a b) b Large field natural infla on with a single axion +2AB cos 2B cos . (22) The absolute minimum of the potential in this theory p2 p2 is at T = 0. However, one can showRK, Linde, Vercnocke, 1404.6204 that inflation occurs 2 ⌘ in the regime of a slow roll from( theT + saddleT¯) point of the K = + SS¯ g(SS¯)2 This potential has an interesting behavior discussed potential with Re T = a/2. Re T remains2 very close to in [4, 5], but now we have its explicit supergravity im-Natural infla on occurs even in theories with a/2 during inflation, and only in the verya, b > 1, end itas starts plementation without any need for an uplifting. As dis-suggested by string theory. For moving towards the global|a – b| << 1, minimum with one can have T = 0. The cussed long ago by Banks, Dine and Seiberg [18], a par-natural infla on even in the theory inflaton potential in this theorywith is well a single axion field approximated by. ticularly rich behavior is possible if the ratio a = q is irra- b 4 a2/2 (a b) tional. This leads to a landscape-type structure of the po- V =2⇤ e 1 cos . (24) p2 tential with infinite number of di↵erent stable Minkowski ⇣ ⌘ vacua and metastable dS vacua with di↵erent values of This potential allows inflation even for a, b 1ifthe the cosmological constant, see Fig. 2. If one of the con- di↵erence between a and b is small, a b 1. A similar stants a and b in this scenario is irrational, we have an idea, in a di↵erent context, was used| in| ⌧ the racetrack infinite number of possible dS minima, which allows to inflation model [19], and then applied to natural inflation solve the cosmological constant problem using anthropic in [4]. In this way, one can bring natural inflation one considerations. step closer towards its implementation in string theory. Moreover, inflationary predictions in this scenario de- Note that we were able to do it in the theory of a single pend on the behavior of the inflaton potential in the axion field. Models described above can easily explain large r. Good for BICEP2. They can also describe r << 1, but not without tuning. Can we do it naturally? Let us return to Planck and some mysteries related to its results. Cosmological a ractors RK, Linde 2013 We found a new class of chao c infla on models with spontaneously broken conformal or superconformal invariance. Observa onal consequences of such models are stable with respect to strong deforma ons of the scalar poten al. Planck 2013 2 4 ⇠ 2 R+ R and R Universal T-Model and beyond 4 2 Miracles to be explained: MANY apparently unrelated theories make same predic on 2 12 1 n = ,r= s N N 2 This point is at the sweet spot of the Planck allowed region. Here N = O(60) is the required number of e-foldings of infla on corresponding to perturba ons on the scale of the observable part of the universe. Why predic ons of different theories converge at the same point? Why convergence is so fast? What is the rela on to non-minimal coupling to gravity? Anything related to broken conformal invariance? Any way to explain it or at least account for it in supergravity? Significant sensi vity of infla onary models on a choice of the frame 1 1 1 K(z,z¯) ⌦(z,z¯) R = e 3 R 2 2 The superconformal theory underlying supergravity requires the choice of a Jordan frame e (X, X¯) R N The superconformal theory is defined by the Kahler poten al of the embedding manifold, including the conformon. 2n 2n ' F ( / )= ( / ) V (')= n tanh p6 V 1.0 0.8 2n Looks like le er T 0.6 Func on of tanh 0.4 0.2 -20 -10 0 10 20 j A universal attractor for inflation at strong coupling Renata Kallosh1, Andrei Linde1 and Diederik Roest2 1Department of Physics and SITP, Stanford University, Stanford, California 94305 USA, [email protected], [email protected] and 2Centre for Theoretical Physics, University of Groningen Nijenborgh 4, 9747 AG Groningen, The Netherlands, [email protected] We introduce a novel non-minimal coupling between gravity and the inflaton sector. Remarkably, for large values of this coupling all models asymptote to a universal attractor. This behaviour is independent of the original scalar potential and generalises the attractor of new Higgs inflation to all models. The attractor is located in the ‘sweet spot’ of Planck’s recent results. Introduction Due to the non-minimal coupling, we will refer to this form The last few years have seen exciting experimental advances, of the theory as Jordan frame. In order to transform to the both in particle physics and in cosmology. The LHC has de- canonical Einstein frame, one needs to redefine the metric: tected the first scalar field ever, whose characteristics are 1 g ⌦( ) g , ⌦( )=1+⇣ V ( ) . (2) compatible with being the Higgs particle. The Planck satel- µ⌫ ! µ⌫ lite has measured the CMB temperature anisotropies with This bring the Lagrangian to the following,p Einstein-frame unprecented precision, the results of which are fully con- form: sistent with inflation. Remarkably, it has moreover proven 1 1 1 3 2 2 V ( ) possible to forge a link between these two fields. Whereas = p g R (⌦( ) + (log ⌦( ))0 )(@ ) . LE 2 2 2 ⌦( )2 it was previously thought that the self-coupling of the Higgs field was too large to allow for inflation, the introduction of Note that in the absence of non-minimal coupling, ⇠ = 0, 2 a non-minimal coupling ⇠ R between the inflationary and the distinction between Einstein and Jordan frame vanishes. gravitational sectors ameliorates this situation [? ]. Inter- In this case the inflationary dynamics is fully determined by estingly, it was found that for couplings of order one and the properties of the scalar potential V ( ). In the presence higher, new Higgs inflation asymptotes to identical predic- of a non-minimal coupling, however, one has to analyse the tions as the Starobinsky model of inflation [? ]. interplay between the di↵erent contributions to the inflation- Until presently, it has only been possible to modify the ary dynamics due to V ( ) and ⇠. Remarkably, in the strong 2 4 scalar potential of new Higgs inflation in this way. In coupling limit, we will demonstrate that these dynamics are this letter we will introduce a similar, but distinct, non- solely determined by ⇠ and independent of V ( ). minimal coupling between the inflationary and gravitational sector. Schematically it is of the form ⇠ V ( )R. For 2 4 A universal attractor theory, this coupling coincides with a non-minimal coupling In this section we consider the strong coupling limit of infla- 2 p ⇠R that is more commonly considered, but for di↵erent tion, where ⇠ becomes very large. We will later quantify how scalar potentials the two non-minimal couplings are distinct. large ⇠ needs to be for this limit. First we will present two Also note that this operator is dimension-4 for any choice of arguments for a universal attractor behaviour in the limit of the scalar potential, not just for the quartic one; the param- infinite ⇠. eter ⇠ is always dimensionless. In an e↵ective field theory The first, heuristic argument involves a comparison of the one would thus generically expect such a coupling, if it is three terms that involve in the Jordan frame Lagrangian. 2 not ruled out by any symmetry arguments. We will find Slow-roll inflation always requires the potential energy to that this novel non-minimal coupling allows for a interest- A second, more robust argument starts from the kinetic dominate over0.25 the kinetic terms. Moreover, in the large-⇠ ing modification of all scalar potentials: for su ciently large term in Einstein frame. In the large-⇠ limit, the two contri- limit the non-minimal coupling term will also dominate over coupling all choices for V ( ) asymptote to Starobinsky in- butions to the kinetic terms scale di↵erently under ⇠.Re-the latter. Therefore the kinetic terms plays a subdominant flation. 0.20 taining only the leading term, the Lagrangian becomes role in this limit. Neglecting this term, the field equation for becomes algebraic and implies Non-minimal coupling 1 3 2 2 V ( ) = p g R (@ log(⌦( ))) . (5) 0.15 The startingLE point of2 many 4 inflationary models ⌦( is)2 a La- ⇠R =4 2 V ( ) . (3) grangian consisting of the Einstein-Hilbert term for gravity p plusRemarkably, a kinetic term the and canonically scalar potential normalised for the field inflaton' involves field. the Substituting this back into the Jordan frame Lagrangian function ⌦( ) of the scalar potential itself, 0.10 We add a non-minimalNon-minimal coupling to gravity coupling between these two sectors yields of a form that (to our knowledge) hasRK, Linde, not beenRoest 1310.3950 considered 2 ' = 3/2 log(⌦( )) . (6) ⇠ = p g[ 1 R + R2] , (4) before. The LagrangianJordan frame, reads arbitrary V(φ) 0.05 J 2 2 p L 16 Therefore, in terms1 of1 ', the theory1 has2 lost all reference to J = p g[ R + ⇣R V ( ) (@ ) V ( )] . (1) theL original scalar2 potential.2 It has 2 the universal form and hence the Starobinsky model of inflation [? ] appears. Einstein frame, p large ζ 0.950 0.955 0.960 0.965 0.970 0.975 0.980 1 1 2 2 p2/3' 2 0.0 = p g R (@') ⇣ (1 e ) , (7) LE 2 2 0.955 0.960 0.965 0.970 0.975 0.980 V h V i which coincides with the scalar formulation of the Starobin- -0.5 Just as before, with the same skyobserva onal consequences, model. The spectral index and tensor-to-scalar ratio are independently of V(φ) 2 12 -1.0 j j ns =1 ,r= 2 . (8) (a) N N (b) But at small ζ it reduces to the original theory V(φ), so by Figure 9. The potential ( 2 v2)2 as a function of the canonically normalized field ' in the 4 -1.5 in thechanging large-EinsteinζN we can interpolate between the original theory frame.limit. (a) ⇠ < 0, These 1 ⇠ v2 1. are (b) ⇠ > located0. Notice the similarity in the between ‘sweet these two spot’ of potentials. However, physical interpretation | | ⌧ of these two potentials is very di↵erent. The value of the and the universal a ractor point at large canonically normalized field ' in the model with ⇠ < 0 typically isζ very. large. Meanwhile the value of Planck’sthis recent field for the modelresults. with ⇠ > 0 can be very small, which is the basis of the Higgs inflation scenario [9]. The crucial assumption in the above derivation was that -2.0 statement, one may compare the potential ( 2 v2)2 with ⇠ < 0 in the limit ⇠ v2 1 4 | | ! the kineticwith term the potential is in dominated the Higgs inflation model by [9]with the⇠ > 0 second and v<1, for the contribution. same values In of parameters and ⇠ , see Figs. 9(a) and 9(b). The results of our calculations of the | | other words,parameters wens and requirer and of the amplitude of perturbations of metric in this model coincide with the corresponding results of Ref. [9]. -2.5 The relation between the inflationary regime with n ' = 3/2 log(a + b⇠ V ( )) . (11) V ( )= . (13) The resulting scalarp potential readsp Without non-minimal couplings, these have the following cosmological observables: 2 (a bep2/3')2 2+n 4n 2 . (12) ⇠ (1 a + bep2/3')2 ns =1 ,r= , (14) 2N N Kallosh, AL, Roest 2013 30 . 0 φ4 0.25 Planck+WP 3 Planck+WP+highL ) φ Convex 0.20 Planck+WP+BAO 0.002 r Concave Natural Inflation 2 Power law inflation 0.15 φ Low Scale SSB SUSY R2 Inflation 2/3 0.10 V / φ V / 2/3 V 2 Tensor-to-Scalar Ratio ( φ 0.05 / V 3 / N =50 ⇤ 0.00 N =60 0.94 0.96 0.98 1.00 ⇤ Primordial Tilt (ns) interpolate between the standard predictions of the chaotic inflation scenario with various potentials V ( )and⇠ =0[6],andtheattractorpoint(1.1). Recently we considered models with a large family of potentials V ( )andintroduceda generalized version of non-minimal coupling to gravity, such as ⇠ V ( )R,or⇠ R [12]. For small ⇠, these models have the same predictions as the usual inflationp models with minimal coupling to gravity, but in the large ⇠ limit their predictions converge to (1.1). The inflaton potential of the canonically normalized inflaton field ' in all models yielding the universal result (1.1) can be represented as p 2 ' V (')=V 1 e 3 + ... (1.2) 0 ⇣ ⌘ b' in the limit ' . More general potentials V (1 e + ...)havebeenconsideredinmany !1 0 inflationary theories, starting from [21], with di↵erent values of the parameter b. However, in the context of the cosmological attractors discussed so far, the constant b was always equal 2/3. In this paper we will consider two classes of supergravity models, where the potentialsp at large values of the inflaton field ' are given by p 2 ' V (')=V 1 e 3↵ + ... . (1.3) 0 ⇣ ⌘ A striking feature is that the new parameter ↵ in both classes of models is related in the same way to the K¨ahler curvature R of the inflaton’s scalar manifold: R = 2 . The scalar K K 3↵ curvature thus that plays a crucial role in the generalized attractors that we put forward in this paper. SU(1,1) One of such models is an U(1) ↵- model found in [2] in a particular version of super- ↵ model in supergravity gravity where the inflaton field is a part of a vector multiplet rather than a chiral multiplet. The potential of theModels with one real scalar from a vector mul plet model is Ferrara, RK, Linde, Porra p 2 ' 2 V ↵e 3↵ . (1.4) ⇠ The values of ns and r forIn this model, n this models and r do not depend on do not dependβ. onFor α , βand = 1, in the limit of large N and the poten al is the same as in Starobinsky model. For small ↵ are given by small α 2 12 n =1 ,r= ↵ s N 2 N 2 12 ns =1 ,r= ↵ 2 . (1.5) For large α the predic ons are the same as in the simplest N N In Section 2 of this paperchao c infla on with a quadra c poten al: we analyze observational consequences of this model for generic 2 8 values of ↵ > 0 and find thatns =1 in the large,r↵ (small= curvature) limit, the observational N N predictions for ns and r of this class of models coincide with the predictions of the simplest chaotic inflation model with a quadratic potential, which are given by 2 8 n =1 ,r= , (1.6) s N N 2 Kallosh, AL, Roest 2013 Superconformal α-a ractors RK, Linde, Roest 1311.0472 Another class of cosmological a ractors naturally appears in superconformal theory and supergravity. This class includes, in par cular, models 1 1 µ ' L = p g R @µ'@ ' F (tanh ) 2 2 p6↵ At these models have universal predic on ↵ . 1 2 12 n =1 ,r= ↵ s N N 2 However, for predic ons depend on the choice of F(x). For ↵ 1 the simplest choice F = x n , the predic ons coincide with those of the simplest chao c infla on models V 'n ⇠ ϕ4" ϕ3" ϕ2" ϕ" ϕ2/3" The curvature of the Kahler manifold is inversely propor onal to α Small α means high curvature, small r - good for Planck Large α means small curvature, large r - good for BICEP2 Thus, finding ns and r may tell us something important about the nature of gravity and geometry of superspace. 1 1 µ ' L = p g R @µ'@ ' F (tanh ) 2 2 p6↵ What if instead of the monomial func ons F = xn one considers F = x +x2 +x3+… ? Then in the large α limit, the predic ons approach a single a ractor point: The simplest chao c infla on with a quadra c poten al. Kallosh, AL, Roest 1405.3646 Double a ractor 0.25 0.20 0.15 r 0.10 0.05 0.00 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 nS By changing the strength of non-minimal coupling to gravity and the curvature of the Kahler manifold in supergravity, one can con nuously interpolate between two a ractor points for these classes of large field models: the predic ons of the simplest chao c infla on models with large r, favored by BICEP2, and the lowest part of the ns-r plane favored by Planck. Renata: Bert, please, decide if you want to explain this case here and what the limit to Minkowski is Solution in Class II in eq. (3.17) clearly admit the Minkowski (AdS) non-supersymmetric limit at vanishing (negative ) . Even for =0thelimitto✏ =0issingular,thereforehereagainthe Minkowski and AdS vacua have a rich dependence on ✏ which should not be taken to zero. There is a solution 2 in the Class II, described in Appendix B.2, for the case with a3 =0 choice which corresponds also to WUUU =0,whichhasalimittoboth as well as ✏ vanishing, still supersymmetry remains broken and also flat directions are recovered. The mass eigenvalues at a =1expandednearsmallvaluesof⇤ and ✏ are: 1 ✏2 13 ✏2 1+3a2 ✏2 3 ⇤ m2 = ,m2 = ⇤ ,m2 = + ⇤ ,m2 = ,m2 = + 1,2 48 96 3 192 2 4 192 5 16 96 6 8 96 8 ✏2 ⇥ ⇥ ⇥ (5.11) If we first send ⇤ to zero at fixed ✏ we find few small masses ✏2 ✏2 ✏2 m2 = ,m2 = ,m2 = (5.12) 3 192 4 192 6 8 96 ⇥ If we send now ✏ to zero, we find 3 flat directions in the non-supersymmetric Minkowski minimum. Thus all our dS examples which we studied agree with the predictions of the no-go theorem in the beginning of this section. 6 Discussion In this paper we have given a general recipe for constructing abundant families of de Sitter vacua in supergravity models that are motivated by compactifications of string theory. The reason there are many locally stable supersymmetric Minkowski vacua in supergravity is that at DaW =0and W =0themassmatrixattheminimumisoftheform eK D2W 2 0 M 2 = a¯b , (6.1) Minkowski | 0 | eK D2W 2 ✓ | |ab¯ ◆ so that all eigenvalues are either positive or zero, but never negative. The arrangement for the de Sitter vacua made inAnaly c Classes of Metastable de Si er Vacua this paper is to impose a hierarchy for the supersymmetry breaking and use the no-scale T modulus in such a way that the approximateRK, Linde, Vercnocke no-scale, Wrase condition, 1406.4866 e↵ectively removes the negative 3 W 2 term from the potential From 2003 ll recently: | | De Si er hun ng in the landscape D W 2 3 W 2 (✏2) . (6.2) No underlying principle was known with broken supersymmetry as to how to achieve local | T | | | ⇠ O stability for dS vacua: absence of tachyons The remaining part of the potential for the Xi fields is positive and small We have found how to generate analy cally mul ple dS vacua in supergravity which are locally stable. We use computers only to test the new principle, and to find examples, like the D W 2 (✏2) . (6.3) ones in STU models. | i | ⇠ O Our new strategy is to use the advantage of one of the fields in supergravity to be a no- We evaluatescale one, as in KKLT, and to solve the following equa ons the full mass matrix under the conditions that V = ⇤ , @aV =0,DaW = Fa(✏) , @a@bV = Vab(⇤, ✏) , (6.4) We have found certain condi ons when it is possible to prove that all de Si er solu ons of 18 these equa ons are locally stable! Hence, we can produce analy c de Si er vacua in the landscape in abundance. Our examples in STU models confirm the generic condi ons required for the proof of stability. To relate this new dS supergravity landscape to specific string theory compac fica ons is the next step. First a empt: let us deform slightly the locally stable susy Minkowski vacua to get W W + W K K + K ! ! locally stable de Si er vacua DW 2 3 W 2 > 0 | | | | It did not work We proved a no-go theorem that small deforma on of a W and/or K of a locally stable susy Minkowski vacua will not produce a locally stable de Si er vacua Therefore a part of the supergravity landscape with locally stable de Si er vacua must be disconnected from the supersymmetric locally stable Minkowski vacua The no-go theorem, therefore, predicts, that the limit from a locally stable de Si er vacuum cannot be a supersymmetric locally stable Minkowski vacuum This predic on is confirmed in our examples: the limits are either stable Minkowski with broken susy, or have flat direc ons. A New Toy In Town! A nilpotent chiral mul plet, Volkov-Akulov golds no and D-brane physics Observa on: not a single version of string theory monodromy supergravity model is known as of today. There is always a problem of non-inflaton moduli stabiliza on or a consistency problem. Meanwhile, numerous well working cosmological supergravity models of infla on and/or upli ing involving a golds no mul plet have not been associated with string theory. See examples in earlier part of the talk. All earlier a empts to associate string theory models with effec ve supergravity are based on standard unconstrained chiral mul plets: spin 0 <-> spin 1/2 Fermionic VA golds no live on D-branes, have only spin 1/2 Unconstrained chiral superfield S(x, ✓)=s(x)+p2 ✓ G(x)+ ✓2F S(x) 2 Nilpotent chiral superfield S (x, ✓)=0 GG S = + p2 ✓ G + ✓2F S 2 F S G(x)G(x) s(x) ) 2 F S(x) D-p-branes and Volkov-Akulov golds no 1 1 S = d10x det G = Em0 ... Em9 , DBI+WZ ↵2 µ⌫ ↵2 ^ ^ Z Z p Em =dxm + ↵2 ¯ md Too many fields, some of them do not directly par cipate in infla on. We must add higher correc ons to the Kahler poten al to stabilize these fields near their zero values. The cure: Volkov-Akulov nonlinear realiza on of supersymmetry does not require fundamental scalar degrees of freedom, they are replaced by bilinear fermionic combina ons with zero vev. Procedure: Replace standard unconstrained chiral superfields by nilpotent superfields, which do not have dynamical scalar components. Ferrara, RK, Linde 1408.4096 2 RK, Linde 1408.5950 S (x, ✓)=0 Antoniadis, Dudas, Ferrara, Sagno 1403.3269 Supergravity with nilpotent mul plets is a new theory requiring a very sophis cated analysis, especially when fermion interac ons are involved. However, the bosonic sector is much simpler than before. Calculate poten als as func ons of all superfields as usual, and then DECLARE that S =0 for the scalar part of the nilpotent superfield. No need to stabilize and study evolu on of the S field. So much simpler!!! De Si er Vacua and Dark Energy We proposed a systema c procedure for building locally stable dS vacua without tachyons in stringy theory STU models, or using the Polonyi type superfield. Polonyi type superfield was successfully used in the past in supergravity models, in par cular to provide a supersymmetric version of an F-term upli ing for the KKLT De Si er vacua. However, it was not known how to relate Polonyi to string theory NEW: replace Polonyi by a nilpotent superfield Nilpotent golds no superfied S2=0 is associated with the D-brane physics in string theory, via Volkov-Akulov theory. This leads to a supersymmetric KKLT upli ing in string theory. W = W M 2S, K = K + SS¯ KKLT KKLT M 4 V = V (⇢, ⇢¯)+ KKLT (⇢ +¯⇢)3 Ferrara, RK, Linde 1408.4096 For the first me using nilpotent superfield S2=0 we have simple string theory mo vated supergravity models for infla on with de Si er vacua exit. Example RK, Linde 1408.5950 ( ¯)2 K = + SS,¯ W = SM2(1 + c 2)+W 2 0 The field ImΦ is heavy and quickly vanishes, during infla on we find a quadra c poten al for Re Φ No cosmological At the minimum Φ vanishes and the poten al is Polonyi model 4 2 120 V = M 3W 10 0 0 ⇠ SuperHiggs effect: gravi no eats golds no from the S-mul plet and becomes fat! 2 D W =0 DSW = M Generaliza on to generic infla onary models with V= f2(Φ) during infla on is straigh orward H2 −35 V = V E4 Planck length : 10 m infl 3 infl ⇠ 10−33 m 16 Einfl 10 GeV −30 ⇠ r 0.15 B-modes from 10 m ⇠ infla on −27 H 10 m T = 1013 GeV H 2⇡ ⇠ Hawking temperature GUTs −24 of gravita onal radia on 10 m LHC 10−21 m −18 126 Gev 10 m 10−15 m