RenataKallosh,StanfordCosmologyWorshop

March2017a

October2016CMB-S4 Science Book

First Edition

CMB-S4 Collaboration

August 1, 2016

arX

iv:1

610.

0274

3v1

[astr

o-ph

.CO

] 10

Oct

201

6

4 Exhortations

2000 2005 2010 2015 2020

10−4

10−3

10−2

10−1

WM

AP

Planck

CMB−S4

Year

Appro

xim

ate

raw

exp

erim

enta

l sensi

tivity

(µ

K)

Space based experiments

Stage−I − ≈ 100 detectors

Stage−II − ≈ 1,000 detectors

Stage−III − ≈ 10,000 detectors

Stage−IV − ≈ 100,000 detectors

Figure 2. Plot illustrating the evolution of the raw sensitivity of CMB experiments, which scales asthe total number of bolometers. Ground-based CMB experiments are classified into Stages with Stage IIexperiments having O(1000) detectors, Stage III experiments having O(10,000) detectors, and a Stage IVexperiment (such as CMB-S4) having O(100,000) detectors. Figure from Snowmass CF5 Neutrino planningdocument.

1.2.1 Raw sensitivity considerations and detector count

The sensitivity of CMB measurements has increased enormously since Penzias and Wilson’s discovery in1965, following a Moore’s Law like scaling, doubling every roughly 2.3 years. Fig. 2 shows the sensitivity ofrecent experiments, expectations for upcoming Stage-3 experiments, characterized by order 10,000 detectorson the sky, and the projection for a Stage 4 experiment with order 100,000 detectors. To obtain many of theCMB-S4 science goals requires of order 1 µK arcminute sensitivity over roughly half of the sky, which for afour-year survey requires of order 500,000 CMB-sensitive detectors.

To maintain the Moore’s Law-like scaling requires a major leap forward, a phase change in the mode ofoperation of the ground based CMB program. Two constraints drive the change: 1) CMB detectors arebackground-limited, so more pixels are needed on the sky to increase sensitivity; and 2) the pixel count forexisting CMB telescopes are nearing saturation. Even using multichroic pixels and wide field of view optics,these CMB telescopes are expected to field only tens of thousands of polarization detectors, far fewer thanneeded to meet the CMB-S4 science goals.

CMB-S4 thus requires multiple telescopes, each with a maximally outfitted focal plane of pixels utilizingsuperconducting, background limited, CMB detectors. To achieve the large sky coverage and to takeadvantage of the best atmospheric conditions, the South Pole and the Chilean Atacama sites are baselined,with the possibility of adding a new northern site to increase sky coverage to the entire sky not contaminatedby prohibitively strong Galactic emission.

CMB-S4 Science Book

T

T

E

Figure 1: This Figure is taken from [16], it represents a forecast of CMB-S4 constraints in the ns � r planefor a fiducial model with r = 0.01. Here the grey band shows predictions of the sub-class of ↵-attractor models[2, 3, 4]. We have added to this figure a blue circle with the letter T inside it corresponding to a highestpreferred value 3↵ = 7 and the purple one corresponding to the lowest preferred value 3↵ = 1 in a seven-diskgeometry. All intermediate cases 3↵ = {1, 2, 3, 4, 5, 6, 7} are between these two. They all describe the classof ↵-attractor models with V ⇠ tanh2('/

p6↵), so-called quadratic T -models. The quadratic E-models with

V ⇠ (1 � ep

2/3↵')2 tend to be slightly to the right of the T -models, see [2]. We show them as a navy circlewith the letter E inside it.

by requiring that

3↵ = 7 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 ⌘ ⌧3↵ = 6 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 ⌘ ⌧ , ⌧7 = const3↵ = 5 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 ⌘ ⌧ , ⌧6 = ⌧7 = const3↵ = 4 : ⌧1 = ⌧2 = ⌧3 = ⌧4 ⌘ ⌧ , ⌧5 = ⌧6 = ⌧7 = const3↵ = 3 : ⌧1 = ⌧2 = ⌧3 ⌘ ⌧ , ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 2 : ⌧1 = ⌧2 ⌘ ⌧ , ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 1 : ⌧1 ⌘ ⌧ , ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const (4.17)

We illustrate in Fig. 1 the features of ↵-attractor models [2, 3, 4] with the seven-disk geometryusing the recent discussion of B-modes in the CMB-S4 Science Book [16]. We show in Fig. 1predictions of ↵-attractor models with seven-disk geometry in the ns � r plane for N ⇠ 55, forthe minimal value 3↵ = 1 and for the maximal value 3↵ = 7.

5 Values of 3↵ in string theory

Here we will show how to derive the 7-disk geometry (4.13) in string theory. We start withthe derivation of non-compact symmetries in string theory following [17], [18]. The toroidal

7

a-attractor models

FutureB-modesatellitemissions

Wellmotivatednewmodelsoriginatinginstringtheory,M-theory,maximalsupergravity

Ferrara,RK,2016,RK,Linde,Wrase,Yamada,Workinprogress

0.960 0.962 0.964 0.966 0.968 0.970-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

a-attractorslog10 r- ns plane

ns

3a=7,6,5,4,3,2,1

B-modetargets:maximalsupersymmetry

• a-attractorcosmologicalmodels,ns- inplace,r–waitingfordata

• Caseofminimalsupersymmetry,1Majorana spinor

• Hyperbolicgeometryofthemodulispace:p=2 witharbitrarya• ind=4N=1supergravity,arbitraryr,with somespecialchoices

• Recentwork:Caseofmaximalsupersymmetry,8Majorana spinors

• Hyperbolicgeometryofthemodulispacewithfavoritevaluesof• 3a=7,6,5,4,3,2,1originatingin• d=10stringtheory,d=11M-theory,d=4N=8supergravity

• 7 valuesofrbetween 10-2 and10-3

• RecentprogressinN=8supergravity

Outline

RelatedtalksbyLinde,Wrase,Yamada

1-p/N

Yanghui triangle,13thcentury

B-modes • Thomson scattering within local quadrupole

anisotropies generates linear polarization • Scalar modes Æ T, E • Tensor modes Æ T, E, B • Ratio r = ΔT / ΔS • Gravitational waves at LSS

create B-mode polarization • Probes Lyth bound of Inflation • Ekpyrotic models Æ r = 0

Lorenzo Moncelsi

Planck 2015

BICEP2 2014

W. Hu

B>0 B<0

Moriond 22/3/16

Planck XX 2015

BK14 w / 95GHz 2016

r<0.07

a-attractors,2yellowlinesinthesweetspotofPlanckdata

RK,Linde,Roest,2013

α-attractorsSimpleFannedT-models

⇣1� e

p23↵'

⌘2n⇣tanh

'p6↵

⌘2n

SimpleFannedE-models

↵ =1

3, r = 10�3

MaximalsupergravityMaximalsuperconformal theory

�2�6

� �6 �2

�IfB-modeswillbediscoveredsoon,r>10-2naturalinflationmodels,axion monodromymodels,a-attractormodels,…,willbevalidatedNoneedtoworryaboutlogscaler

Otherwise,weswitchtologr

a-attractormodelssitandwaitforthedata,onr,ns isalreadyinagreementwiththedata

r ⇡ 3↵4

N2ns ⇡ 1� 2

N

N=60

Supposeinflationtakesplacenearthepoleatf=0.Theinflationarypredictionsaredeterminedbytheorder andtheresidue ofthepole.

Intheleadingapproximationin1/N

Galante,RK,Linde,Roest 2014

Theexperimentalvalueofthescalartiltsuggests ns = 1� p

N

p ⇡ 2

ns = 1� 2

N, r = a2

8

N2

Hyperbolicgeometryofa-attractorsrequiressecondorderpole,p=2,theresidue ofthepolea2isaninversecurvature.Thereisnodependenceonc andon… inV

a2 = � 1

RK=

3↵

2

p ⇡ 2Zaldarriaga etal.2015

Incasethat

1

2R� 1

2

ap�p

(@�)2 � V0(1 + c�+ . . . )

ns#

log10r#

↵ = 1, r ⇠ 0.13

↵ = 1, r = 3⇥ 10�3

↵ = 1/9, r = 4⇥ 10�4

↵ = 1/3, r = 10�3

conformal)a*ractors,)Higgs)infla2on)))

N=4)supergravity,)unit)size)Poincare)disk)

GoncharovALinde)model)

Any a <20GenericN=1supergravity

Anya

3↵ = R2Escher ⇡ 103r

↵ = 1, r ⇠ 0.13

↵ = 1, r = 3⇥ 10�3

↵ = 1/9, r = 4⇥ 10�4

↵ = 1/3, r = 10�3N=4$supergravity,$unit$size$Poincare$disk$

Goncharov:Linde$model$

ns#

log10r#

Starobinsky$model,$conformal$aAractors$$

↵ = 1, r = 3⇥ 10�3

a=2fiberinflationCicoli,Burgess,Quevedo

r < 0.07

HyperbolicgeometryofaPoincaré disk

NMSSM

2015

Specialchoicesofa andfuturedata

a = 1 r ⇡ 3⇥ 10�3

a = 1/3 r ⇡ 10�3

CriticalpointofsuperconformalN=1 attractors,Higgsinflation,R2 …

MaximalsuperconformalN=4model,maximalsupergravityN=8

a = 1/9 r ⇡ 3⇥ 10�4 1984modelofGoncharov-Linde

Any a <20 GenericN=1supergravity

Allofthesemodelsfitthecurrentdata

r < 0.07

↵ = 2 r ⇡ 6⇥ 10�3 Fiberinflation

2015

7/36/3

3/3

2/3

1/3

5/34/3

Newin2016

FerraraandRK

Newin2017RK,Linde,Wrase,Yamada

Workingdynamicalmodelsfor7-diskmanifold

Meaningofthemeasurementofthecurvatureofthe3dspace

k=0,k=-1,k=+1 Spatialcurvatureparameter

Inthecontextofnewsupergravitycosmologicalmodels,measuringrmeansmeasuringtheKahlerspacecurvature

ns = 1� 2

N, r = ↵

12

N2

⌦K = �0.0004± 0.00036

scalarfieldsarecoordinatesofthe Kahler geometry

ds

2 = �dt

2 + a(t)2�ijdxidx

j

RK = � 2

3↵

Decreasingr,decreasinga,increasingcurvatureRK

LesHouches,July,2013

N=1supersymmetryrequiresthatscalarfieldsarecoordinatesoftheKahler geometry

Reminder:Riemanngeometrywithrealcoordinatesdoesnotworkforsupersymmetry

276 13 Complex manifolds

We then take za to be the set of 2n complex coordinates where the index a runs firstthrough the n unbarred or ‘holomorphic’ coordinates and then through the n barredor ‘anti-holomorphic’ coordinates. We consider the map �i ! za defined in (13.1) asa coordinate transformation of the type normally considered in di↵erential geometry,and we use the standard transformation formulas of Chapter 7. Complexity causesno di�culty.We assume that M possesses a Euclidean signature metric structure, with

Christo↵el connection, and curvature tensor, all as defined in Chapter 7. For con-venience we record the relevant formulas again:

ds2 = gijd�id�j , (13.2)

�ijk = 1

2gim(@jgkm + @kgjm � @mgjk) , (13.3)

Rijkl = @i�

kjl � @j�

kil + �k

im�mjl � �k

jm�mil . (13.4)

The transformation rules of vector fields under a real coordinate change �i ! �0iare

V 0i(�0) =@�0i

@�jV j(�) ,

V 0i (�

0) =@�j

@�0iVj(�) . (13.5)

Covariant derivatives are defined (see (7.78)) by

rjVi = @jV

i + �ijkV

k ,

rjVi = @jVi � �kjiVk . (13.6)

The transformation rules and covariant derivatives extend to higher rank tensors asdiscussed in Chapter 7.

All quantities above can be expressed in complex coordinates by applying thetransformation formulas to the map �i ! za. For example,

V i(�) ! V a(z) =@za

@�jV j(�) . (13.7)

We then separate the unbarred and barred components

V ↵ = V ↵ + iV ↵+n , V ↵ = V ↵ � iV ↵+n , ↵ = 1, . . . n . (13.8)

The ‘splitting’ of an index a into ↵ and ↵ is not preserved by general transformationof complex coordinates z0↵ = f↵(z, z), but it is preserved under the special classof ‘holomorphic’ coordinate transformations z0↵ = f↵(z). Under this subgroup ofdi↵eomorphisms the holomorphic indices ↵ of any tensor transform into holomorphicindices ↵0 and anti-holomorphic ↵ into ↵0.

Kahler geometryhascomplexcoordinates

ds2 = gik(z, z)dzidzk = @i@kK(z, z)

TheRicciandthescalarcurvatureoftheKahler manifoldare

RK = gikRik = gikRimkngmn

Geometricmeaningofa-parameter

1 Introduction

During the next few years we might expect some dramatic new information from B-mode experiments

either detecting primordial gravity waves or establishing a new upper bound on r, and from LHC

discovery/non-discovery of low scale supersymmetry. A theoretical framework to discuss both of

these important factors in cosmology and particle physics has been proposed recently. It is based on

the construction of new models of chaotic inflation [1] in supergravity compatible with the current

cosmological data [2] as well as involving a controllable supersymmetry breaking at the minimum

of the potential [3–7]. In this paper we will develop supergravity models of inflation motivated by

either string theory or extended supergravity consderations, known as cosmological ↵-attractors [8–16].

Here we will enhance them with a controllable supersymmetry breaking and cosmological constant at

the minimum. We find this to be a compelling framework for the discussion of the crucial new data

on cosmology and particle physics expected during the next few years. Some models of this type were

already discussed in [14].

The paper is organized as follows. We begin in Section 2 with a brief review of key vocabulary and

features of these and related models with references to more in-depth treatments. In Section 3 we

present the ↵-attractor supergravity models that make manifest an inflaton shift-symmetry by virtue

of having the Kahler potential inflaton independent – which we will refer to as Killing-adapted form.

Section 4 presents a universal rule: given a bosonic inflationary potential of the form F2(') one can

reconstruct the superpotential W =⇣S+ 1

b

⌘f(�) for the Kahler potentials described in Section 3. The

resulting models with f 0(') = F(') have a cosmological constant ⇤ and an arbitrary SUSY breaking

M at the minimum. In Section 5 we study more general class of models with W = g(') + Sf((')

and the same Kahler potential. For these models it is also possible to get agreement with the Planck

data as well as dark energy and SUSY breaking. Moreover, these models have nice properties with

regard to initial conditions for inflation, analogous to the ones studied in [28] for models without SUSY

breaking and dark energy. We close in Section 6 with a summary of what we have accomplished.

2 Review

2.1 ↵, and attraction

There is a key parameter ↵ in these models, for which the Kahler potential K = �3↵ ln(T + T ). It

describes the moduli space curvature [9] given by RK = � 2

3↵ . Another, also geometric, interpretation

of this parameter is in terms of the Poincare disk model of a hyperbolic geometry with the radiusp3↵, illustrated by the Escher’s picture Circle Limit IV [15, 16]. As clarified in these references,

from the fundamental point of view, there are particularly interesting values of ↵ depending on the

original theory. From the maximal N = 4 superconformal theory, [17], one would expect ↵ = 1/3

with r ⇡ 10�3. This corresponds to the unit radius Escher disk [15], as well as a target of the

future space mission for B-mode detection, as specified in CORE (Cosmic ORigins Explorer). Some

interesting simplifications occur for ↵ = 1/9, which corresponds to the GL model [18,19]. From N = 1

1

ds2 =3↵

(1� ZZ)2dZdZ ds2 =

3↵

(T + T )2dTdT

EscherintheSky,RK,Linde2015

Disk orhalf-plane

CurvatureofthemodulispaceinKahler geometry

3↵ = R2Escher ⇡ 103r

HyperbolicgeometryofaPoincaré disk

http://mathworld.wolfram.com/PoincareHyperbolicDisk.html

ZZ < 1

T + T > 0

SL(2,R) symmetry

-20 -10 10 20φ

0.2

0.4

0.6

0.8

1.0

�

0 5 10 15 20φ

0.2

0.4

0.6

0.8

1.0

1.2�

1

2R� 1

2@'2 � ↵µ2

⇣tanh

'p6↵

⌘2 1

2R� 1

2@'2 � ↵µ2

⇣1� e

p23↵'

⌘2

SimplestT-model SimplestE-model

Plateaupotentialsa-attractors

1

2R� 3↵

@Z@Z

(1� ZZ)2� V0ZZ

Ingeometricvariables

incanonicalvariables

1

2R� 3↵

@T@T

(T + T )2� V0(T � 1)2

Howtoachievethesesimplecosmologicalmodelswithasinglelightinflaton fromfundamentaltheories?Modulistabilizationrequired

Frommaximalsupersymmetry:stringtheoryD-branephysics,M-theory,N=8supergravity,tominimalN=1supergravityphenomenologicalmodels

ConsistencyofdeSittervacua withN=1supergravitywithnon-linearlyrealizedsupersymmetry Bergshoeff,Freedman,RK,VanProeyen,Jul 29,2015

Hasegawa,Yamada,Jul 30,2015

TalkbyWraseKKLT

Non-linearlyrealizedsupersymmetry(nottheonewhichwedonotseeatLHC)

Morestudiesofnon-linearlyrealizedpartnerless supersymmetryandcosmology

Baumann,Bergshoeff,Carrasco,Dall’Agata,Delacretaz,Dudas,Farakos,Ferrara,Gorbenko,Green,RK,Karlsson,Kuzenko,Linde,Murli,Thaler,Quevedo,Sagnotti,Senatore,Sorokin,Terada,Zwirner,Uranga,Vernocke,Wrase,Yamada

TalksbyLinde,Senatore,Wrase,Yamada

Volkov-Akulov,1972,Komargodski-Seiberg 2009,

Newinterestintheoreticalcosmology,2014

1.Nilpotent(Volkov-Akulov goldstino)scalar-lessmultiplet:cosmologicalstabilizer

2.Nilpotentorthogonalfermion-lessmultiplet

• MaximalsupersymmetryandB-modes• M-theoryind=11• Superstringtheoryind=10• N=8supergravityind=4

2016,FerraraandRK

Scalarsarecoordinatesofthecoset spaceinN=8supergravityind=4G

H=

E7(7)

SU(8)

ds2 = kdTdT

(T + T )2, k = 1, 2, 3, 4, 5, 6, 7

OnecanarguethatgeometrieswithdiscreetnumberofunitsizePoincaré disksarepossiblewhenconsistentreductionofsupersymmetryisperformed.Uponidentificationoftheirmodulionefinds

=3a

Atleastonediskandnotmorethanseven

N=55e-foldings

E7(7)(R) � [SL(2,R)]7

r ⇡ {1.3, 2.6, 3.9, 5.2, 6.5, 7.8, 9.1}⇥ 10�3

ns ⇡ 0.963

Detailsandassumptionsunderlying3a=1,2,3,4,5,6,7 prediction

M-theorycompactified ona7-manifoldwithG2 holonomyspecialchoiceofBetti numbersonecanobtaind=4N=1supergravitywithrank7scalarcoset

so that70 ! 2(1 + 6) = 2⇥ 7 = 14 . (3.5)

This truncation has a Kahler structure supporting N = 1 supersymmetry. One can identify 7Poincare disks due to the decomposition

E7(7)(R) � [SL(2,R)]7 . (3.6)

The original kinetic term is reduced to a form with the Kahler potential of the form

K = �7X

i=1

ln(�i(⌧i � ⌧i)) (3.7)

with 7 pairs of independent scalars and the [SL(2,R)]7 symmetry, a seven-disk manifold. Thefact that the disk of the SL(2) commuting with SO(6, 6) has the same Kahler curvature ofthe other six SL(2)/SO(2) (each separately corresponding to ↵ = 1/3) can be understood bystring triality arguments [14] and by the underlying special geometry of the N=2 truncation[15].

4 M theory on a 7-manifold with G2 holonomy

Instead of a compactification on a 7-torus, one can compactify M theory on a 7-manifold withG2 holonomy. The early investigation of G2 holonomy in physics was performed in [16], withreview of the first 20 years in [17]. One of the most recent application of this compactificationcan be found in [18] and, of course, many more studies of M theory on G2 were performed overthe years.

Here we will focus on the model studied in [19, 20], it requires the following choice of theBetti numbers

(b0, b1, b2, b3) = (1, 0, 0, 7) . (4.1)

This theory is identified with the maximal rank reduction on the seven torus and leads directlyto d=4 N = 1 ‘curious supergravity’ where 7 complex scalars are coordinates of the coset space

hSL(2,R)SO(2)

i7. (4.2)

The corresponding Kahler potential describing the scalar sector of this theory is the one ineq. (3.7) with 7 pairs of independent scalars and the [SL(2,R)]7 symmetry. This model is oneof the ‘Four curious supergravities’ defined in [20]. The other 3 have N = 2, N = 4, N = 8supersymmetries, we are interested only in N = 1 ‘curious supergravity’. It has the fieldcontent defined by Betti numbers : the d=4 fields originating from the d=11 metric gMN are

gµ⌫ ! b0 = 1Aµ ! b1 = 0A ! b1 + b3 = 7 (4.3)

5

so that70 ! 2(1 + 6) = 2⇥ 7 = 14 . (3.5)

This truncation has a Kahler structure supporting N = 1 supersymmetry. One can identify 7Poincare disks due to the decomposition

E7(7)(R) � [SL(2,R)]7 . (3.6)

The original kinetic term is reduced to a form with the Kahler potential of the form

K = �7X

i=1

ln(�i(⌧i � ⌧i)) (3.7)

with 7 pairs of independent scalars and the [SL(2,R)]7 symmetry, a seven-disk manifold. Thefact that the disk of the SL(2) commuting with SO(6, 6) has the same Kahler curvature ofthe other six SL(2)/SO(2) (each separately corresponding to ↵ = 1/3) can be understood bystring triality arguments [14] and by the underlying special geometry of the N=2 truncation[15].

4 M theory on a 7-manifold with G2 holonomy

Instead of a compactification on a 7-torus, one can compactify M theory on a 7-manifold withG2 holonomy. The early investigation of G2 holonomy in physics was performed in [16], withreview of the first 20 years in [17]. One of the most recent application of this compactificationcan be found in [18] and, of course, many more studies of M theory on G2 were performed overthe years.

Here we will focus on the model studied in [19, 20], it requires the following choice of theBetti numbers

(b0, b1, b2, b3) = (1, 0, 0, 7) . (4.1)

This theory is identified with the maximal rank reduction on the seven torus and leads directlyto d=4 N = 1 ‘curious supergravity’ where 7 complex scalars are coordinates of the coset space

hSL(2,R)SO(2)

i7. (4.2)

The corresponding Kahler potential describing the scalar sector of this theory is the one ineq. (3.7) with 7 pairs of independent scalars and the [SL(2,R)]7 symmetry. This model is oneof the ‘Four curious supergravities’ defined in [20]. The other 3 have N = 2, N = 4, N = 8supersymmetries, we are interested only in N = 1 ‘curious supergravity’. It has the fieldcontent defined by Betti numbers : the d=4 fields originating from the d=11 metric gMN are

gµ⌫ ! b0 = 1Aµ ! b1 = 0A ! b1 + b3 = 7 (4.3)

5

Stringtheorycompactified on T2 ⇥ T2 ⇥ T2 ⇢ T6

K = �7X

i=1

ln(⌧i + ⌧i)) ) �7 ln(⌧ + ⌧)

K = � ln(S + S)� 3 ln(U + U)� 3 ln(T + T ) ) �7 ln(⌧ + ⌧)

N=8supergravity:consistentreductiontoN=1 E7(7)(R) � [SL(2,R)]7

?

?

7-diskcosmologicalmodelFerrara,RK,2016,RK,Linde,Wrase,Yamada,Workinprogress

1. StartwithM-theory,orStringtheory,orN=8supergravity2. PerformaconsistenttruncationtoN=1supergravityind=4witha7-diskmanifold

3. Addasuperpotential preservingN=1supersymmetrywithaminimumat

4. AddacosmologicalpartwithanilpotentstabilizerS

5. RunMathematica:themodelworksbeautifully

An analogous identification was performed in [22], where an early dimensional reduction ofsuperstring theories was studied. The resulting d=4 N = 1 supergravity, neglecting the matterfields C in [22], has the following Kahler manifold:

K = � ln(�i(s� s))� 3 ln(�i(t� t)) . (6.12)

We will show in the next section that using string theory compactification on a product of 3tori T2 ⇥ T2 ⇥ T2 ⇢ T6 one can get the seven-disk geometry.

K = � ln(�i(s� s))� ln(�i(t1 � t1))� ln(�i(t2 � t2))� ln(�i(t3 � t3))

� ln(�i(u1 � u1))� ln(�i(u2 � u2))� ln(�i(u3 � u3)) (6.13)

Thus, the model (6.12) in [22] corresponds to the one in (6.13) under condition that

t1 = t2 = t3 = t , u1 = u2 = u3 = const . (6.14)

This means that some fields of higher-dimensional geometry were discarded, for example all ui

fields and the di↵erence between ti fields. If instead we would impose on (6.13) the condition

s = t1 = t2 = t3 = u1 = u2 = u3 = ⌧ (6.15)

we would reproduce the Kahler geometry (6.10) of the single Poincare disk of the radius square3↵ = 7. In analogous manner we can get other values

3↵ = {1, 2, 3, 4, 5, 6, 7} (6.16)

by requiring that

3↵ = 7 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 ⌘ ⌧3↵ = 6 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 ⌘ ⌧ , ⌧7 = const3↵ = 5 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 ⌘ ⌧ , ⌧6 = ⌧7 = const3↵ = 4 : ⌧1 = ⌧2 = ⌧3 = ⌧4 ⌘ ⌧ , ⌧5 = ⌧6 = ⌧7 = const3↵ = 3 : ⌧1 = ⌧2 = ⌧3 ⌘ ⌧ , ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 2 : ⌧1 = ⌧2 ⌘ ⌧ , ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 1 : ⌧1 ⌘ ⌧ , ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const (6.17)

We illustrate in Fig. 1 the features of ↵-attractor models [3, 4, 7] with the seven-disk geometryusing the recent discussion of B-modes in the CMB-S4 Science Book [23]. We show in Fig. 1predictions of ↵-attractor models with seven-disk geometry in the ns � r plane for N ⇠ 55, forthe minimal value 3↵ = 1 and for the maximal value 3↵ = 7.

The constraints on the fields presented in eqs. (6.14), (6.17) are expected to emerge as aconsequence of specifically designed potentials, which aline the fields in the desirable direction.We have not yet presented such potentials here, these are still to be constructed. Exampleswhen the structure of the potential was designed to put constraints on the field of the theory areknown in cosmological literature. A relevant mechanism was invented in [24] in models known

8

W0 = M⇣ X

1i<jp

(⌧i � ⌧j)2 +

7X

i=p+1

(⌧i � c)2⌘

K0 = �1

2

7X

i=1

ln⇣ (⌧i + ⌧i)2

4⌧i⌧i

⌘

K1 = SS W1 = Sf(X

i

⌧i) + g

DiW = 0

r ⇡ 3↵4

N2ns ⇡ 1� 2

N3a=7,6,5,4,3,2,1