Renata Kallosh
DESY Workshop, September 2014 with Linde and with Cecotti, Ferrara, Porrati, Van Proeyen, Bercnocke, Roest, Westphal, Wrase
We discuss inflationary 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 interesting and instructive about the fundamental theory from the sky
We describe new models of inflation and dark energy/cc. New results on de Sitter Landscape: how to avoid tachyons in string theory motivated supergravities
Recent new tools have allowed us to construct new simple models of inflation with dS uplifting in the context of spontaneously broken supersymmetry
A nilpotent chiral multiplet, Volkov-Akulov goldstino A New Game In Town! and D-brane physics BICEP2 - Planck Drama IsIs ItIt DDuusst?t? IsIs ItIt DDuusst?t? Mortonson, Seljak Flauger, Hill, Spergel May-June 2014
(Flauger, Hill, & Spergel, arXiv:1405:7351)
(Flauger, Hill, & Spergel, arXiv:1405:7351) Genus Topology and Cross-Correlation of BICEP2 and Planck 353 GHz B- Modes: Further Evidence Favoring Gravity Wave Detection
Colley and Gott, September 15, 2014
Still preliminary as it is based on redigitized plots of both BICEP2 and Planck.
However the basic idea to check Cross- Correlation of BICEP2 and Planck is a good one. This is what BICEP2 and Planck are doing now, using the actual combined data. And we are waiting… Planck intermediate results. XXX. The angular power spectrum of polarized dust emission September 19 at intermediate and high Galactic latitudes
Details on dust in the BICEP2 patch of the sky. Talk by Bouchet here on September 22.
Current conclusion: there is a lot of dust, however, we need a joint analysis with BICEP
Sean Carroll blog Sep 21 Planck Speaks: Bad News for Primordial Gravitational Waves? Results in from Planck: seems likely that the famous BICEP2 results were dust, not gravitational waves.
George Efstathiou comment September 21, 2014
As a member of the Planck Science Team, I would urge caution concerning the interpretation. What we are saying is that polarised dust emission in the BICEP2 field is high. But it may be that there is something left in the BICEP2 signal that can be attributed to gravitational waves. We need to cross-correlate the Planck maps with the BICEP2 maps. This analysis is underway. r ??? new version of Mortonson and Seljak
the updated contours on r including the new Planck dust 353GHz prior: the new limits are r<0.09 (95% cl), r>0.14 is excluded at 99.5%
Chaotic Inflation in Supergravity: shift symmetry Kawasaki, Yamaguchi, Yanagida 2000
Kahler potential and superpotential The potential is very curved with respect to S and Im F, so these fields vanish. But Kahler potential does not depend on
The potential of this field has the simplest form, as in chaotic inflation, without any exponential terms: RK, Linde, Rube 2010
Superpotential must be a REAL holomorphic function. (We must be sure that the potential is symmetric with respect to Im , so that Im = 0 is an extremum (then we will check that it is a minimum). The Kahler potential is any function of the type
The potential as a function of the real part of at S = 0 is
FUNCTIONAL FREEDOM in choosing inflationary potential
Here S is a goldstino multiplet: supersymmetry is broken only in the goldstino direction FUNCTIONAL FREEDOM in choosing inflationary potential in supergravity allows us to fit any set of ns and r.
Why there is so much discussion of a large field inflation, and fundamental theory?
Large r means large field, but not the other way around. Starobinsky model, Higgs inflation, and all recently discovered supersymmetric attractors are large field models
(inflation at canonical field > 5 Mp), and yet they lead to r of the order 0.003, not to be ruled out in the next 10 years. 10 Planck Collaboration: Constraintson 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 ΛCDM+tensor s r0.002 < 0.12 < 0.13 < 0.11 < 0.12 −2∆lnLmax 0 0 0 -0.31
Table4. Constraintson theprimordial perturbationparametersintheΛCDM+r model fromPlanckcombinedwithother datasets. −1
Theconstraintsaregiven at thepivot scalek∗ = 0.002 Mpc . 5
5 2
. potential V(φ),take a square root of it,makeits Taylor 0 series ex pansion and thus construct a real holomorphic
function which approximate V(φ) with great accuracy.
0
)
2
2 .
0 C 0 0 However, one should be careful to obey the rules of the
. onvex
0 r
( Concave game as formulated above.
o
i
5
t
1 a
. For example, suppose one wants to obtain a fourth de- 0
R m 2 φ2 r
a gree polynomial potential of thetype of V(φ)= (1+ l 2
a 2 c
0 aφ + bφ ) in supergravity. One may try to do it by tak-
S
1 .
- 2 i ✓ 0 o ¯ ¯
t ing K =(Φ + Φ) / 2+ SS) and f (Φ) = mΦ(1 + ce Φ)
- r
o [52]. For general ✓,this choice violates our conditions for
s
n
5 e
0 f (Φ). In this case, the potential will be a fourth degree
. T 0 polynomial with respect to ImΦ if ReΦ = 0.However,
in this model the flat direction of the potential V(Φ)
0 0 . (and, correspondingly, the inflationary trajectory) devi- 0 0.94 0.96 0.98 1.00 ate from ReΦ = 0.(Also, in addition to the minimum at Primordial Tilt (ns)
Sunday, March 31, 13 Φ = 0, the potential will develop an extra minimum at − 1 − i ✓ Fig.1. Marginalizedjoint 68% and95% CL regionsfor ns andr0.002 fromPlanckincombinationΦwith=other− cdataesetscompared.) Asatoresult, thepotential along theinfla- thetheoretical predictionsof selectedinflationary models. FIG. 1: T he 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 thef expectations of [52, 53]. Moreover, the kinetic terms reheating priors allowing N∗ < 50 could reconcile thismodel leadtoinflationwitha(t) ∝ exp(At ), withA> 0and0< f < 1, withintheflaPlanctionkadata.ry regime when the field rolls down fromwherethefm=a4x/i(4mu+mβ)oandf β >o0f. Inthintermediatee fields wiinflationll betherenon-canonical and non-diagonal. the potential. T he continuation of this area upwisanordnaturals correendsp otoninflation,ds but if theexit mechanismleaves to the prediction of inflation which begins when ttheheinflationaryfield φ initiallypredictions on cosmological perturbations un- Exponential potential and power lawinflation This may not be a big problem, since the potential in is at the sl ope of the potential at |v − φ| v.Inmodified,the limthisit vclass! of1 models, thpredictse direr ≈c−tion8β(nsorth− 1)/(βogon− 2) al to the inflationary trajectory is Inflationwhicwithh canorreexponentialsp onds potentialto a ! 0, the predictions (cBarrooincwid&e Liddlewith,t1993he ). It is disfavoured, being outside the joint 95% CL contour for anyβe.xponentially steep. Therefore the deviation of this field predictions of the si mplest cφhaotic inflation model with a quadratic 2 4 from ReΦ = 0 will not be large, and for sufficiently large potential m Vφ(φ2).= Λ exp −λ (35) 2 Mpl Hill-top models values of the inflaton field χ the potential will be ap- Inanother interestingclassof potentials, theinflatonrollsaway is called power law inflation (Lucchin & Matarrese, 1985), proximap tel y given by the si mple polynomial ex pression because the exact solution for the scale factor is given by froman unstableequilibriumas inthefirst new2inflationarymod- a(t) ∝ t2/λ2. This model is incomplete, since inflation would els(Albrecht & Steinhardt, 1982|f;(Lindeχ/ , 19822)|)..BWeconsiderut in order to make a full investigation of not end without an additional mechanismto stop it. Assuming inflφationp in such models one would need to study evo- an inflationary regime when the field φ rolls from Vth(φe) ≈ Λ4 1− + ... , (37) such a mechanism exists and leaves predictions for cosmo- µp logicalmaximperturbationsum ofunmodified,the potenthis clastialsofatmodelsφ =0predicts,as in new inflation lution of all fields numerically, and make sure that all r = −8(ns − 1) andisnowoutsidethejoint 99.7%CL contour. wheretheellipsisindicateshigherst abiorderlittermsy condinegligibletionsduringare satisfied. An advantage of the scenario. Nat ural initial conditions for iinflation,nflationbut neededin thitosensurethepositivenessof thepotential model are easily set by tunneling from laternothingon. An exponentinto aof p = metho2isalloweddsonlydevaselopalargeedfieldin [38, 39] is that all fields but one Inverse power lawpotential 3 inflationary model and predicts n −1≈ −4M2/µ2 + 3r/8 and universe with spatial topology T ,see e.g. [24] and the vans ish duriplng inflation, all kinetic terms are canonical r ≈ 32φ2M2/µ4. This potential leads to predictions in agree- Intermediatemodels(Barrow, 1990; Muslimov, 1990) with in- ∗ pl and diagonal along the inflationary trajectory, and in- versedpoiswercuslaswiopotentialsn in [51]. The results of investigation of the ob- ment withPlanck+WP+BAOjointvest95%igaCLtiocontoursn of stforabsuperility- is straightforward. servat ional consequences of this model [Planckian38, 42,values48]ofarµ,ei.e., µ& 9Mpl . φ −β Modelswith p ≥ 3predict n − 1 ≈ −(2/N)(p− 1)/(p− 2) described byV(φt)he= Λg4reen area in Figur(36)e 1. Predict ions of s Mpl when r ∼ 0. Thehill-top potentialFowithrtupn=at3eliesly,outsideone cthean obtain an exactly polynomial po- thismodel arein good agreement with observational data tential V(φ) in the theories with K = K ((Φ − Φ¯)2,SS¯) for a certain range of values of the parameter a ⌧ 1. using the met hods 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 (φ) tential V(φ) can be obtained in this si mple contex t, or with real coefficients. As a simplest example, one may 2 that one has a full freedom of choice of the functions consider f (Φ) = mΦ 1 − cΦ + dΦ . The resulting po- f (Φ). It is important to understand the significance of tential of the inflaton field can be represented as the restrictions on the form of the Kähler potential and 2 2 m φ 2 super pot ential described above. According to [39], in the V(φ)= 1 − aφ + a2bφ2) . (11) ¯ 2 ¯ 2 theory with the Kähler potential K = K ((Φ − Φ) ,SS) p RK, Linde and Westphal, 1405.0270 2 the symmetry of the Kähler potential Φ ! ± Φ¯,as well as Here a = 0.25c/ 2 and a b = d/ 2.We use the parametriza- thecondition that f (Φ) isa real holomorphicfunction are tion in terms of a and b because it allows us to see what required to ensure that the inflationary trajectory, along happens 0.20with the potential if one changes a: If one de- which the Kähler potential vanishes is an extremum of creases a,the overall shape of the potential does not 0.15 the potential in the direction orthogonal to the inflati- changer, but it becomes stretched. The same potential onary trajectory S =ImΦ =0.After that, the proper can be al0.10so obtained in supergravity with vector or ten- choice of the Kähler potent ial can make it not only an sor multiplet s [36]. ex tremum, but a minimum, thus st abilizing the inflation- 0.05 Inflation in this theory may begin under the same ini- ary regime [38, 39, 44]. 0.00 tial condition0.93 s 0.94as in0.95the0.96simp0.97lest0.98large0.99fie1.00ld chaotic in- n The requirement that f (Φ) is a real holomorphic func- flation models φ .The difnfeS rence is that in the small a tion does not affect much the flexibility of choice of the limit, the last 60 e-f,old ings of inflation, are described by inflaton potential: One can take any positively defined the theory φ2. Meanwhile for large a one has the same Natural Inflation in Supergravity Natural inflation in theories with axion potentials is known for nearly 25 years (Freese et al 1990), but until now it did not have any stable supergravity generalization. Invariably, there was an instability with respect to some moduli, or we needed some assumptions about string theory uplifting. The problem was solved only recently: RK, Linde, Vercnocke, 1404.6204
All non-inflaton moduli stabilized Large field natural inflation with a single axion RK, Linde, Vercnocke, 1404.6204
Natural inflation occurs even in theories with a, b > 1, as suggested by string theory. For |a – b| << 1, one can have natural inflation even in the theory with a single 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 attractors RK, Linde 2013 We found a new class of chaotic inflation models with spontaneously broken conformal or superconformal invariance. Observational consequences of such models are stable with respect to strong deformations of the scalar potential.
Planck 2013
2 R+ R and Universal T-Model and beyond
Miracles to be explained: MANY apparently unrelated theories make same prediction
This point is at the sweet spot of the Planck allowed region.
Here N = O(60) is the required number of e-foldings of inflation corresponding to perturbations on the scale of the observable part of the universe.
Why predictions of different theories converge at the same point? Why convergence is so fast? What is the relation 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 sensitivity of inflationary models to the choice of the frame
The superconformal theory underlying supergravity requires the choice of a Jordan frame
The superconformal theory is defined by the Kahler potential of the embedding manifold, including the conformon. 2n Looks like letter T Function of tanh Non-minimal coupling to gravity RK, Linde, Roest 1310.3950 Jordan frame, arbitrary V(f)
Einstein frame, large z
Just as before, with the same observational consequences, independently of V(f)
But at small z it reduces to the original theory V(f), so by changing z we can interpolate between the original theory and the universal attractor point at large z.
RK, Linde, Roest 2013
0
3
. 0
5
2 . 0 Planck+WP
3 Planck+WP+highL 0
) f
2
2 .
0 C 0 0 Planck+WP+BAO
. onv 0
r ex
( C Natural Inflation
onca
o i 5 v
t e Power lawinflation
1 a
. 2 0
R f LowScaleSSB SUSY
r a l 2
a R Inflation
c
0
S
1 .
- 2/ 3 0
o V / φ t
-
r f V / φ
o
s n 5 2
e 2/3 0 V / φ
. f T 0 V / φ3
N⇤=50
0
0 .
0 0.94 0.96 0.98 1.00 N⇤=60 Primordial Tilt (ns) model in supergravity
Models with one real scalar from Ferrara, RK, Linde, Porrati a vector multiplet
In this model, ns and r do not depend on b. For a, b = 1, the potential is the same as in Starobinsky model. For small a
For large a the predictions are the same as in the simplest chaotic inflation with a quadratic potential: RK, Linde, Roest 2013 Superconformal a-attractors RK, Linde, Roest 1311.0472 Another class of cosmological attractors naturally appears in superconformal theory and supergravity. This class includes, in particular, models
At these models have universal prediction
However, for predictions depend on the choice of F(x). For the simplest choice F = xn , the predictions coincide with those of the simplest chaotic inflation models j4²
j3²
j2²
j² j2/3² The curvature of the Kahler manifold is inversely proportional to a
Small a means high curvature, small r - good for Planck
Large a 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. What if instead of the monomial functions F = xn one considers F = x +x2 +x3+… ?
Then in the large a limit, the predictions approach a single attractor point: The simplest chaotic inflation with a quadratic potential.
RK, Linde, Roest 1405.3646 Double attractor
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 continuously interpolate between two attractor points for these classes of large field models: the predictions of the simplest chaotic inflation models with large r, favored by BICEP2, and the lowest part of the ns-r plane favored by Planck. Dark Energy in 2014 We need a de Sitter Landscape even more than before
The cosmological constant with w=-1 appears to be even a better fit to data
astro-ph 1401.4064 Betoule et al Recent Progress in Building de Sitter Landscape The goal is to understand the origin of de Sitter vacua in string theory/supergravity and to find a constructive way to generate multiple dS vacua which are locally stable.
By 2011 the conclusion was rather pessimistic concerning local stability of the majority of the supergravity solutions with de Sitter vacua.
In 2013 new work where some solutions without tachyons were constructed. Blaback, Roest, Zavala, type IIB Danielsson, Dibitetto, type IIA
Our work in 2014: how to construct de Sitter solutions in supergravity without tachyons. String Theory Landscape may explain the incredibly small value of the cosmological constant
if the total number of such vacua is huge, In string theory, genetic code is written in properties of compactification of extra dimensions
Up to 10500 different combinations
Sakharov 1984; Bousso, Polchinski 2000; Kachru, RK, Linde, Trivedi, 2003; Douglas 2003 Stable supersymmetric anti de Sitter vacua (under water, negative cosmological constant) and Minkowski vacua (sea level, vanishing cosmological constant) can be easily constructed. However, de Sitter vacua (over water, positive cosmological constant) break supersymmetry, very difficult to avoid tachyons and achieve stability.
String Theory Landscape Analytic Classes of Metastable de Sitter Vacua RK, Linde, Vercnocke, Wrase, 1406.4866
From 2003 till recently: De Sitter hunting in the landscape
No underlying principle was known with broken supersymmetry as to how to achieve local stability for dS vacua: absence of tachyons We have found how to generate analytically multiple dS vacua in supergravity which are locally stable. We use computers only to test the new principle, and to find examples, like the ones in STU models. Our new strategy is to use the advantage of one of the fields in supergravity to be a no- scale one, as in KKLT, and to solve the following equations
We have found certain conditions when it is possible to prove that all de Sitter solutions of these equations are locally stable! Hence, we can produce analytic de Sitter vacua in the landscape in abundance. Our examples in STU models confirm the generic conditions required for the proof of stability. To relate this new dS supergravity landscape to specific string theory compactifications is the next step. First attempt: let us deform slightly the locally stable susy Minkowski vacua to get
locally stable de Sitter vacua
It did not work
We proved a no-go theorem that small deformation of a W and/or K of a locally stable susy Minkowski vacua will not produce a locally stable de Sitter vacua
Therefore a part of the supergravity landscape with locally stable de Sitter 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 Sitter vacuum cannot be a supersymmetric locally stable Minkowski vacuum
This prediction is confirmed in our examples: the limits are either stable Minkowski with broken susy, or have flat directions. A New Toy In Town! A nilpotent chiral multiplet, Volkov-Akulov goldstino and D-brane physics
Observation: it is hard to present explicit string theory monodromy supergravity model. There is always a problem of non-inflaton moduli stabilization or some kind of a consistency problem. Work in progress continues!
Meanwhile, numerous well working cosmological supergravity models of inflation and/or uplifting involving a goldstino multiplet have not been associated with string theory. See examples in earlier part of the talk.
All earlier attempts to associate string theory models with effective supergravity are based on standard unconstrained chiral multiplets: spin 0 <-> spin 1/2
Fermionic VA goldstino live on D-branes, have only spin 1/2 Too many fields, some of them do not directly participate in inflation. We must add higher corrections to the Kahler potential to stabilize these fields near their zero values.
The cure: Volkov-Akulov nonlinear realization of supersymmetry does not require fundamental scalar degrees of freedom, they are replaced by bilinear fermionic combinations with zero vev.
Procedure: Replace standard unconstrained chiral superfields by nilpotent superfields, which do not have dynamical scalar components.
Ferrara, RK, Linde 1408.4096 RK, Linde 1408.5950 Antoniadis, Dudas, Ferrara, Sagnotti 1403.3269 Supergravity with nilpotent multiplets is a new theory requiring a very sophisticated analysis, especially when fermion interactions are involved. However, the bosonic sector is much simpler than before.
Calculate potentials as functions 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 the evolution of the S field.
So much simpler!!! De Sitter Vacua and Dark Energy
We proposed a systematic procedure for building locally stable dS vacua without tachyons in stringy theory STU models, or using the Polonyi (stabilizer) type superfield.
Polonyi type superfield was successfully used in the past in supergravity models, in particular to provide a supersymmetric version of an F-term uplifting for the KKLT De Sitter vacua. However, it was not known how to relate Polonyi to string theory
NEW: replace Polonyi by a nilpotent superfield
Nilpotent goldstino superfied S2=0 is associated with the D-brane physics in string theory, via Volkov-Akulov theory. This leads to a supersymmetric KKLT uplifting in string theory.
Ferrara, RK, Linde 1408.4096 For the first time using nilpotent superfield S2=0 we have simple string theory motivated supergravity models for inflation with de Sitter vacua exit. Example RK, Linde 1408.5950
The field ImF is heavy and quickly vanishes, during inflation we find a quadratic potential for Re F
No cosmological At the minimum F vanishes and the potential is Polonyi model problem
SuperHiggs effect: gravitino eats goldstino from the S-multiplet and becomes fat!
Generalization to generic inflationary models with V= f2(F) during inflation is straightforward Anthropic approach to L in string theory: Why we need many de Sitter vacua
Anthropic bound: |L|= 3 H2< 10-120
Before quantum corrections After quantum corrections Planck length : 1035 m
1033 m
30 B-modes from 10 m inflation 1027 m
Hawking temperature GUTs 24 of gravitational radiation 10 m
LHC 1021 m
18 126 Gev 10 m 1015 m If r instead of 0.15 is 0.05, H2 is 1/3 of the BICEP2 value
Hawking temperature of gravitational radiation
Detection of r at this level would be extremely valuable! Might take the next decade… Back up slides Unconstrained chiral superfield
Nilpotent chiral superfield D-p-branes and Volkov-Akulov goldstino