Andrei Linde Based on work with Kallosh, Roest, Wrase, Carrasco, Senatore, East, Kleban, Yamada and Scalisi Cambridge, 2017
me Hawking Sakharov
The Very Early Universe Proceedings, Nuffield Workshop, Cambridge, UK June 21 - July 9, 1982 G.W. Gibbons, S.W. Hawking, S.T.C. Siklos
"It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch". Alan Guth 1981
Now we know that the universe is not just a free lunch: It is an eternal feast were ALL possible types of dishes are served. A.L.
Inflation Starobinsky, 1980 – modified gravity, R + R 2 . Original motivation was opposite to inflation: Instead of explaining uniformity of the universe, assumed that the universe was homogeneous from the very beginning. Observational predictions (Mukhanov and Chibisov 1981) are great. Guth, 1981 - old inflation. Beautiful idea, first outline of the new paradigm, but did not quite work. A.L. , 1982 - new inflation 1983 - chaotic inflation (also Albrecht, Steinhardt)
V = m 2 φ 2 2 Inflation can start at the Planck density if there is a single Planck size domain with a potential energy V of the same order as kinetic and gradient density; no need in hot Big Bang. This is the minimal requirement, compared to standard Big Bang, where initial homogeneity is requires across 10 90 Planck size domains . � 1.2 1.0 0.8 0.6 0.4 0.2 2 φ - 2 - 1 0 1
1 ) The universe is flat, W = 1. (In the mid-90’s, the consensus was that W = 0.3, until the discovery of dark energy.) 2) The observable part of the universe is uniform. 3) It is isotropic. In particular, it does not rotate. (Back in the 80’s we did not know that it is uniform and isotropic at such an incredible level.) 4) Perturbations produced by inflation are adiabatic 5) Unlike perturbations produced by cosmic strings, inflationary perturbations lead to many peaks in the spectrum 6) The large angle TE anti-correlation (WMAP, Planck) is a distinctive signature of superhorizon fluctuations (Spergel, Zaldarriaga 1997), ruling out many alternative possibilities
7) Perturbations should have a nearly flat (but not exactly flat) spectrum (Mukhanov, Chibisov 1981). A small deviation from flatness is one of the distinguishing features of inflation. It is as significant for inflationary theory as the asymptotic freedom for the theory of strong interactions. 8) Inflation produces scalar perturbations, but it also produces tensor perturbations with nearly flat spectrum, and it does not produce vector perturbations. There are certain relations between the properties of scalar and tensor perturbations. 9) Scalar perturbations are Gaussian. In non-inflationary models, the local describing the level of local non-Gaussianity can be as parameter f NL large as 10 4 , but it is predicted to be O(1) in all single-field inflationary models. Prior to the Planck2013 data release, there were rumors that local >> O(1), which would rule out all single field inflationary models. f NL ± Planck2015 result confirms predictions with accuracy 0.03% NL obtain f local = 0 . 8 ± 5 . 0, NL these estimators on Gaussian
Planck2015
Destri, de Vega, Sanchez, 2007 V = m 2 φ 2 Nakayama, Takahashi and Yanagida, 2013 1 − a φ + b φ 2 � � Kallosh, AL, Westphal 2014 2 Kallosh, AL, Roest, Yamada 1705.09247 V 3 observables: A s , n s , r 3 parameters: m, a, b Φ But the best fit is provided by models with plateau potentials
Kallosh, AL 2013; Ferrara, Kallosh, AL, Porrati, 2013; Kallosh, AL, Roest 2013; Galante, Kallosh, AL, Roest 2014 Start with the simplest chaotic inflation model √− g L = 1 1 2 R − 1 2 ∂φ 2 − 1 2 m 2 φ 2 Modify its kinetic term ∂φ 2 √− g L = 1 1 2 R − 1 6 α ) 2 − 1 2 m 2 φ 2 (1 − φ 2 2 ϕ √ Switch to canonical variables φ = 6 α tanh √ 6 α The potential becomes V = 3 α m 2 tanh 2 ϕ √ 6 α
(for a complex field) 3 α ds 2 = Z ) 2 dZd ¯ Hyperbolic geometry Z (1 − Z ¯ of a Poincaré disk 3 α = R 2 Escher ≈ 10 3 r
A projection of the Escher disk of the radius on the √ 3 α quadratic inflationary potential
General chaotic inflation model √− g L = 1 1 2 R − 1 2 ∂φ 2 − V ( φ ) Modify its kinetic term ∂φ 2 √− g L = 1 1 2 R − 1 6 α ) 2 − V ( φ ) (1 − φ 2 2 ϕ √ Switch to canonical variables φ = 6 α tanh √ 6 α The potential becomes ϕ V = V (tanh 6 α ) √ This is a plateau potential for any nonsingular V ( φ )
V 6 Potential in the original 4 variables of the conformal 2 theory 2 f - 2 - 1 1 - 2 V Potential in canonical 6 variables 4 2 j - 15 - 10 - 5 5 10 15 - 2 Inflation in the landscape is facilitated by inflation of the landscape
Galante, Kallosh, AL, Roest 1412.3797 1 2 R − 3 ⇣ ∂ t ⌘ 2 − V ( t ) 4 α t Suppose inflation takes place near the pole at t = 0 , and V(0) > 0 , V’(0) >0, and V has a minimum nearby. Then in canonical variables 1 2 R − 1 2( ∂ϕ ) 2 − V 0 (1 − e − √ 3 α ϕ + ... ) 2 Then in the leading approximation in 1/N, for any non-singular V n s = 1 − 2 r = α 12 N , N 2
Galante, Kallosh, AL, Roest 1412.3797 THE BASIC RULE: For a broad class of cosmological attractors, the spectral index n s depends mostly on the order of the pole in the kinetic term, while the tensor-to-scalar ratio r depends on the residue. Choice of the potential almost does not matter , as long as it is non-singular at the pole of the kinetic term. Geometry of the moduli space, not the potential, determines much of the answer. An often discussed concern about higher order corrections to the potential for large field inflation does not apply to these models.
( ∂φ ) 2 2 M 2 σ 2 − g 2 √− g L = 1 1 2 R − 1 6 α ) 2 − 1 2 m 2 φ 2 − 1 2( ∂σ ) 2 − 1 2 φ 2 σ 2 (1 − φ 2 2 Potential in canonical variables has a plateau at large values of the inflaton field, and it is quadratic with respect to s .
Kallosh, AL, 1604.00444 ( ∂φ ) 2 √− g L = 1 1 2 R − 1 6 α ) 2 − 1 2( ∂σ ) 2 − V ( φ , σ ) (1 − φ 2 2 Couplings of the canonically normalized fields are determined by derivatives such as 3 α e − √ r 2 3 α ϕ ∂ φ ∂ 2 2 λ ϕ , σ , σ = ∂ ϕ ∂ 2 σ V ( φ , σ ) = 2 σ V ( φ , σ ) | φ → √ 6 α (3.12) As a result, couplings of the inflaton field to all other fields are exponentially suppressed during inflation . The asymptotic shape of the plateau potential of the inflaton is not affected by quantum corrections.
AL 1612.04505 ( ∂ µ φ ) 2 6 α ) 2 � ( ∂ µ σ ) 2 1 p� g L = R � V ( φ , σ ) . 2 � 2(1 � φ 2 2 Can we have inflation in such potentials?
� � In terms of canonical fields ϕ with the kinetic term ( ∂ µ ϕ ) 2 , the potential is 2 p ϕ V ( ϕ , σ ) = V ( 6 α tanh 6 α , σ ) . p Many inflationary valleys representing alpha-attractors
( ∂ µ φ ) 2 ( ∂ µ σ ) 2 1 √− g L = R 6 β ) 2 − V ( φ , σ ) . 2 − 6 α ) 2 − 2(1 − φ 2 2(1 − σ 2 √ ϕ χ In terms of canonical fields p V ( ϕ , χ ) = V ( 6 α tanh 6 β tanh √ 6 β ) . 6 α , √ Two families of attractors, related to the valleys along the two different inflaton directions: 1 − n s ≈ 2 r ≈ 12 α N , N 2 . or 1 − n s ≈ 2 r ≈ 12 β N , N 2 .
Up to now, we discussed bosonic models of cosmological attractors, but most of them have supergravity versions. Construction of models of SUGRA inflation is especially simple now, using the new methods described in the talk by Kallosh. These methods can provide SUGRA versions of any bosonic inflationary potential, and describe arbitrary values of the cosmological constant and the gravitino mass.
Kallosh, AL, Wrase, Yamada 1704.04829, Kallosh, AL, Roest, Yamada 1705.09247 We will study it in SUGRA, by methods described in the talk by Kallosh 2 (1 − Z i Z i ) 2 0 − 1 X G = log W 2 log + S + S + g SS SS, 2 2 (1 − Z 2 i )(1 − Z i ) i =1 where ✓ g SS = W − 2 V + 3 0 and the scalar potential is V = Λ + m 2 2 ( | Z 1 | 2 + | Z 2 | 2 ) + M 2 � 2 � ( Z 1 + Z 1 ) − ( Z 2 + Z 2 ) 4 he last term gives th as Z i = tanh φ i +i θ i 2 . √ For M >> m, the last term in the potential forces the two inflaton fields to coincide, φ 1 = φ 2
Two strongly interacting attractors with a = 1/3 merge into one attractor with a = 2/3. a = 1/3 a = 1/3 a = 2/3 This figure shows only the lower part of the potential, cutting the upper part. Now look at the full potential
The minimum corresponds to the attractor merger shown at the previous slide. This is where inflation ends. But it begins at the infinitely long upper plateau of height O(M 2 ).
At large fields, the a -attractor potential remains 10 orders of magnitude below Planck density. Can we have inflation with natural initial conditions here? The same question applies for the Starobinsky model and Higgs inflation. � 1.2 1.0 0.8 Carrasco, Kallosh, AL 1506.00936 East, Kleban, AL, Senatore 1511.05143 0.6 Kleban, Senatore 1602.03520 Clough, Lim, DiNunno, Fischler, Flauger, 0.4 Paban 1608.04408 0.2 100 φ - 100 - 50 0 50
To explain the main idea, note that this potential coincides with the cosmological constant almost everywhere. � 1.2 1.0 0.8 0.6 0.4 0.2 100 φ - 100 - 50 0 50
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