Cosmological Controversies

Andrei Lindein collaboration with Renata Kallosh

According to the standard hot Big Bang universe, the total
number of particles during its expansion did not changemuch, so the universe at the Planck time was supposed tocontain about 1090 particles. At the Planck time t =O(1),there was one particle per Planck length ct =O(1).
Thus, at the Planck time t = 1, the universe consisted of
1090 causally disconnected parts of size ct =O(1). These
parts did not know about each other. If someone wanted tocreate the universe at the Planck time, he/she could only
make a Very Small Bang in his/her own tiny part of theuniverse of a Planck size ct = O(1). Everything else wasbeyond causal control.

m2 2
V =
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. This is the minimal requirement, compared
to standard Big Bang, where initial homogeneity is requires across1090 Planck size domains.

�

1.2
1.0
0.8
0.6
0.4
0.2

-2

-1

0

1

2

φ

m2 2V =1
2

Destri, de Vega, Sanchez, 2007
Nakayama, Takahashi and Yanagida, 2013
Kallosh, AL, Westphal 2014
Kallosh, AL, Roest, Yamada 1705.09247

a +b 2

3 observables: As, ns, r

VNo problem with initial conditions

3 parameters: m, a, b
Example: m =10-5, a = 0.12,
b=0.29f
But it is better to have models which require no morethan 1 or 2 free parametersKallosh, AL, Roest 2014

Start with the simplest chaotic inflation model

1
1
p L= R
g
2

1 2
@
2

1 2 2
m
2

Modify its kinetic term

1
1
p L= R
g
2

1 @ 2
2
2 (1
)2

1 2 2
m
2

6↵

Switch to canonical variables=

The potential becomesp'6↵ tanh p
6↵'
V = 3↵ m tanh p6↵22All of these models haveferent potentials.
They have
V
Kallosh, AL 2013
all, they have two attractordel corresponding to Fig. 1 in a form
blue stars in Figs. 2 and 3, dPotential in the originalsts variables
isfor ↵ ! 1 and ↵ ! 0. M
with kinetic term
small ↵ (i.e. in the limit wh
2
1
1 @
1 2 2
R
m
,
(2.1) f becomes small) their cosmol
2
2
2
2 (1
2
)
ble
respect
to evenvery
Allwithof thesemodelspredict
6↵
potentials. 2
12
n
=
1
,
r
=
↵
. (1.1). Here (x) is the scalar field,Vthe
s
2
N
N
2
This
propertywas
explain
Potential invalue,canonical
ny positive
andvariables< 6↵, so that
lated in a particularly gener
flattens becausethen kinetic
term isofpositive.
The kinetic
stretchingnear theandboundaryin this class of models, as w
is notcanonical
has a geometriccosmological attractors, has
h a moduli space geometry. At ↵ ! 1
moduli space. If inflation oc
aotic inflation model with a quadraticj2(which happens for sufficien
ical field. At present themodel ofnear the pole can be well rd by the data, which implies that thefirst derivative near the poleat.

ODELS OF ↵-ATTRACTORS

6

4

2

-2

-1

1

2

-2

6

4

2

-15

-10

-5

5

-2

10

15

1
1
p L= Rg2

1 (@ )2
2
2 (1
)2
6↵

1 2 2
m
2

1
(@ )2
2

1 2 2
M
2

Potential in canonical variables has a plateau at large values of the inflaton field,
and it is quadratic with respect to s.

Chaotic inflation with a parabolic potential goes first, starting at
nearly Planckian density. When the field s rolls down, the plateau
inflation begins.

No problem with initial conditions

There is another, simple and general way to solve theproblem of initial conditions for low energy scale inflation,
without using additional fields, to be discussed by Flauger.
East, Kleban, AL, Senatore 1511.05143
Kleban, Senatore 1602.53520
Clough, Lim, DiNunno, Fischler, Flauger, Paban 1608.04408

One way to formulate our results
Consider the smallest possible universe starting in the Planck
size domain with Planck density in an inflationary theory witha plateau potential. Suppose that the universe is flat or open
but compact (e.g. a torus). It can be grossly inhomogeneous,with the inhomogeneities much greater than the value of an
inflationary potential. If the universe begins with the fieldsufficiently far away from the minimum of the potential, andif it does not immediately collapse as a whole within the
Planck time, i.e. if the universe is not just a quantum
fluctuation, then it continues expanding until inflation begins
and makes the universe flat and homogeneous.
East, Kleban, AL, Senatore 1511.05143
AL 1710.04278

Recent work with Renata Kallosh and Yusuke Yamada,1811.01023, 1906.02156, 1906.04729, 1909.04687

The main goal is to use Planck results andidentify possible CMB targets for future
observational missions

Astro2020 Science White Paper“Probing the origin of our Universe through cosmic
microwave background constraints on gravitational waves”“If these thresholds (r =10-3) are passed without a detection, most
textbook models of inflation will be ruled out; and, while the possibilityof an early inflationary phase would still remain viable, the data would
then force a significant change in our understanding of the primordial
Universe.”
Recently this statement was repeated in“PICO: Probe of Inflation and Cosmic Origins”

Fortunately, as we will see from this talk, the models with r << 10-3
are well known for quite some time, they are among the models
favored by Planck 2015 and 2018, and they do not require any
significant changes in our understanding of the primordial Universe.

contours).

Planck 2018
Inflationary model
R + R2 /(6M 2 )Power-law potential
Power-law potential
Power-law potential
Power-law potential
Power-law potential
Power-law potential
Non-minimal coupling
Natural inflation
Hilltop quadratic model
Hilltop quartic model
D-brane inflation (p = 2)
D-brane inflation (p = 4)
Potential with exponential tails
Spontaneously broken SUSY
E-model (n = 1)
E-model (n = 2)
T-model (m = 1)
T-model (m = 2)

Potential V( )⇣⌘2p⇤4 1 e 2/3 /MPl
MPl10/3 2/3
MPl3
MPl8/3 4/3
MPl2 2MPl 3
44 4+ ⇠ 2 R/2⇥⇤
⇤4 1 + cos ( / f )
⌘
⇣
2 2
⇤4 1
/µ2 + . . .
⇣
⌘
4
4 4
⇤ 1
/µ4 + . . .
⇣
⌘
4
2
p
⇤ 1 µD 2 / + . . .
⇣⌘⇤4 1 µ4D 4 / p + . . .
⇥⇤⇤4 1 exp ( q /MPl ) + . . .
⇥⇤⇤4 1 + ↵h log ( /MPl ) + . . .
("✓q◆ 1 #)2np⇤4 1 exp2
3↵E1 MPl("✓q◆ 1 #)2np⇤4 1 exp2
3↵E2 MPl
" ✓q
◆ 1#
2m
4
T
⇤ tanh6↵1 MPl
" ✓q
◆ 1#
2m
4
T
⇤ tanh6↵2 MPlParameter range

2

ln B

...

...

...

...
...
...
...
...
...
4 < log10 ⇠ < 4
0.3 < log10 ( f /MPl ) < 2.5

2.8
2.5
10.4
22.3
40.9
89.1
3.1
9.4

2.6
1.9
4.5
7.1
19.2
33.3
1.6
4.2

0.3 < log10 (µ2 /MPl ) < 4.85

1.7

2.0

2 < log10 (µ4 /MPl ) < 2

0.3

1.4

6 < log10 (µD 2 /MPl ) < 0.3

2.3

1.6

6 < log10 (µD 4 /MPl ) < 0.3

2.2

0.8

3 < log10 q < 3
2.5 < log10 ↵h < 1

0.5
9.0

1.0
5.0

2 < log10 ↵E1 < 4

0.2

1.0

2 < log10 ↵E2 < 4

0.1

0.7

2 < log10 ↵T1 < 4

0.1

0.1

2 < log10 ↵T2 < 4

0.4

0.1

on inflation [1], is performed for the simplest models✓◆nRK, Linde, 1906.02156
V = V0 1,
mn

potential
is very non-linear,
but the predictions,by thepopulargreen area,
g theThe
terms
indicatedby .... However,for m . 1,shown
the most
hilltoin theconverge
to 0.94
the predictions
theorya linear= 4 showninlargeFig.m1limit
predictns =for N = of
50a andnswith= 0.95for N =
potential, for any n. What is going on?
are ruled out by observational data.

hese predictions change for m & 1, but they approach safer values ns & 0.9
nck 2018 only for m & 10. In the large m limit the green lines describing predodel in Fig. 1 converge at the red circles corresponding to the predictions oftheory with a linear potential V ⇠ . Moreover, a similar result is correct= 4, but for all hilltop potentials (3.3) [24]. How can the complicated theoriege m limit give the same prediction as the theory with a simple potentia
s going on?

answer this question, let us look at the the potential (3.3), which is show
ine in Fig. 3. This potential has a maximum (hilltop) at = 0, and th

0.25
Tensor-to-Scalar Ratio (r0.002 )
0.05
0.10
0.15
0.20

Co
nv
Co ex
nca
ve

R 2 Inflation
V /
V /
V /

Tensor-to-scalar ratio (r0.002 )
0.05
0.10
0.15
0.20
0.00

Version 2, after thereferee report…

2/3

2

V / 3
N⇤ =50
0.94

0.96
0.98
Primordial Tilt (ns )

1.00

0.25

0.00

Planck 2013 version 1

Planck+WP
Planck+WP+highL
Planck+WP+BAO
Natural Inflation
Power law inflation
Low Scale SSB SUSY

N⇤ =60Planck+WP+BAO
Planck+WP+highL
Planck+WP
Natural Inflation
Hilltop quartic modelPower law inflation
Low scale SSB SUSY

Co
nv
Co ex
nca
ve

R 2 Inflation
V /
V /
V /

2
2/3

V / 3
N⇤ =50N⇤ =60

0.94

0.96
0.98
Primordial tilt (ns )

1.00

V

✓

4

V = V0 1

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m4

1

Last 60 e-foldings begin here

0.5

Inflation ends here
10

-0.5

◆

20

30

During the last 60 e-foldings thepotential is approximately linear,which explains the results of thecalculations

40

50

ϕ

The universe collapses here

The green hilltop area in Planck, CMB-S4 and PICO
figures is not predicted by simple consistent
inflationary models with 1 or 2 parameters.
Remember that with 3 parameters, a simple polynomial
potential can describe all Planck data.

T-models (yellow) and E-models (red)2 'VT = V0 tanh p
6↵<latexit sha1_base64="MPjLsB4matmby9HGi36wv7FX7hQ=">AAACF3icbVC7SgNBFJ31GeMramkzKIJV2I1gbMSAjaWCeUA2LncnE3dwdnaduRsISwr/wcZfsbFQxFY7/8bJo/B1YOBwzj3cuSdMpTDoup/OzOzc/MJiYam4vLK6tl7a2GyYJNOM11kiE90KwXApFK+jQMlbqeYQh5I3w5vTkd/sc21Eoi5xkPJODNdK9AQDtFJQKjeCS3pMG4FLfQQVXVVyvw86jYSf2Bz1za3G/NAHmUYwHAalXbfsjkH/Em9Kdmv1u+qJp6rnQenD7yYsi7lCJsGYtuem2MlBo2CSD4t+ZngK7AauedtSBTE3nXx815DuWaVLe4m2TyEdq98TOcTGDOLQTsaAkfntjcT/vHaGvaNOLlSaIVdssqiXSYoJHZVEu0JzhnJgCTAt7F8pi0ADQ1tl0Zbg/T75L2lUyt5BuXJh22iSCQpkm+yQfeKRKqmRM3JO6oSRe/JInsmL8+A8Oa/O22R0xplmtsgPOO9fduKh3A==</latexit>⇣

V E = V0 1
<latexit sha1_base64="kJbMb0eJAyeim2HYxW5Fqi4WtVI=">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</latexit>ep 2 ⌘23↵ 'T-models (yellow) and E-models(red)⇣p 2 ⌘22 'VT = V0 tanh p<latexit sha1_base64="MPjLsB4matmby9HGi36wv7FX7hQ=">AAACF3icbVC7SgNBFJ31GeMramkzKIJV2I1gbMSAjaWCeUA2LncnE3dwdnaduRsISwr/wcZfsbFQxFY7/8bJo/B1YOBwzj3cuSdMpTDoup/OzOzc/MJiYam4vLK6tl7a2GyYJNOM11kiE90KwXApFK+jQMlbqeYQh5I3w5vTkd/sc21Eoi5xkPJODNdK9AQDtFJQKjeCS3pMG4FLfQQVXVVyvw86jYSf2Bz1za3G/NAHmUYwHAalXbfsjkH/Em9Kdmv1u+qJp6rnQenD7yYsi7lCJsGYtuem2MlBo2CSD4t+ZngK7AauedtSBTE3nXx815DuWaVLe4m2TyEdq98TOcTGDOLQTsaAkfntjcT/vHaGvaNOLlSaIVdssqiXSYoJHZVEu0JzhnJgCTAt7F8pi0ADQ1tl0Zbg/T75L2lUyt5BuXJh22iSCQpkm+yQfeKRKqmRM3JO6oSRe/JInsmL8+A8Oa/O22R0xplmtsgPOO9fduKh3A==</latexit>

6↵

V E = V0 1

e

3↵ '

<latexit sha1_base64="kJbMb0eJAyeim2HYxW5Fqi4WtVI=">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</latexit>

Efstathiou and Gratton, 1910.00483 (133 pages)

U-duality symmetry benchmarks for a-attractors
Maximal supersymmetry

E7(7) (R)Special cases:[SL(2, R)]7
a = 2, orange, also fibre
inflation, Cicoli et al
a = 1, blue, also Higgs,
Starobinsky and conformal
attractors
a = 1/3, black, also
maximal superconformal
theoryBenchmarks for T-models and E-models
T-models

E-models

ns

ns<latexit sha1_base64="fhU2agvCoNeyHG1PPJREc4f0e+4=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4CkkabHsrePFY0X5AG8pmu2mXbjZhdyOU0p/gxYMiXv1F3vw3btoKKvpg4PHeDDPzwpQzpR3nw1pb39jc2i7sFHf39g8OS0fHbZVkktAWSXgiuyFWlDNBW5ppTruppDgOOe2Ek6vc79xTqVgi7vQ0pUGMR4JFjGBtpFsxUINS2bF91/PrVZSTWrXu56RyWfM85NrOAmVYoTkovfeHCcliKjThWKme66Q6mGGpGeF0XuxniqaYTPCI9gwVOKYqmC1OnaNzowxRlEhTQqOF+n1ihmOlpnFoOmOsx+q3l4t/eb1MR7VgxkSaaSrIclGUcaQTlP+NhkxSovnUEEwkM7ciMsYSE23SKZoQvj5F/5O2Z7sV27vxy43OKo4CnMIZXIALVWjANTShBQRG8ABP8Gxx69F6sV6XrWvWauYEfsB6+wTjF45G</latexit><latexit sha1_base64="fhU2agvCoNeyHG1PPJREc4f0e+4=">AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4CkkabHsrePFY0X5AG8pmu2mXbjZhdyOU0p/gxYMiXv1F3vw3btoKKvpg4PHeDDPzwpQzpR3nw1pb39jc2i7sFHf39g8OS0fHbZVkktAWSXgiuyFWlDNBW5ppTruppDgOOe2Ek6vc79xTqVgi7vQ0pUGMR4JFjGBtpFsxUINS2bF91/PrVZSTWrXu56RyWfM85NrOAmVYoTkovfeHCcliKjThWKme66Q6mGGpGeF0XuxniqaYTPCI9gwVOKYqmC1OnaNzowxRlEhTQqOF+n1ihmOlpnFoOmOsx+q3l4t/eb1MR7VgxkSaaSrIclGUcaQTlP+NhkxSovnUEEwkM7ciMsYSE23SKZoQvj5F/5O2Z7sV27vxy43OKo4CnMIZXIALVWjANTShBQRG8ABP8Gxx69F6sV6XrWvWauYEfsB6+wTjF45G</latexit>

2 'VT = V0 tanh p
6↵<latexit sha1_base64="MPjLsB4matmby9HGi36wv7FX7hQ=">AAACF3icbVC7SgNBFJ31GeMramkzKIJV2I1gbMSAjaWCeUA2LncnE3dwdnaduRsISwr/wcZfsbFQxFY7/8bJo/B1YOBwzj3cuSdMpTDoup/OzOzc/MJiYam4vLK6tl7a2GyYJNOM11kiE90KwXApFK+jQMlbqeYQh5I3w5vTkd/sc21Eoi5xkPJODNdK9AQDtFJQKjeCS3pMG4FLfQQVXVVyvw86jYSf2Bz1za3G/NAHmUYwHAalXbfsjkH/Em9Kdmv1u+qJp6rnQenD7yYsi7lCJsGYtuem2MlBo2CSD4t+ZngK7AauedtSBTE3nXx815DuWaVLe4m2TyEdq98TOcTGDOLQTsaAkfntjcT/vHaGvaNOLlSaIVdssqiXSYoJHZVEu0JzhnJgCTAt7F8pi0ADQ1tl0Zbg/T75L2lUyt5BuXJh22iSCQpkm+yQfeKRKqmRM3JO6oSRe/JInsmL8+A8Oa/O22R0xplmtsgPOO9fduKh3A==</latexit>⇣

V E = V0 1
<latexit sha1_base64="kJbMb0eJAyeim2HYxW5Fqi4WtVI=">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</latexit>ep 2 ⌘23↵ 'In classification of Encyclopaedia Inflationaris, there are two maintypes of such models. The simplest ones are BI (brane inflation)models. They have potentials which look like inverted hilltopnHilltop:

1

'
mn

1

BI:

mn
'n

Just like the simplest hilltop inflation, the BI models are ruled out
since they lead to immediate collapse of the universe after inflation.
The second class is called KKLTI models (from KKLT Inflation).

KKLTI:VKKLT I ⇠

✓

n

m
1+ n
'

◆ 1

They describe Dp-brane inflation, which (in the small r limit) predict(12 8ns )|r!0 =
N 9

p
p

mk
+...
'k

cludes Dp-brane inflation models with V ⇠ 1
y T-models,modelsE-models
and
KKLTI
modelson
Logsimplestr scale:
[16,
50,
51],
where
k
=
7
p,
see
sectionIV.
Their
M-theory,kn the left,
versions with V ⇠ 1 m, which were called BI (brane in'k4] allows
flation) in [16], are inconsistent for the same reason as the(orange),
simplest hilltop models [9]. Consistent generalizations of(purple)'kmk
1
minimalthese models with potentials V ⇠ (1 + 'k ) = 'k +mk
ses showwere proposed in [52] in the context of D3 brane inflated data.
tion. These models were generalized and called KKLTI
uation of(KKLT inflation) in [16], and further developed in [51].

Predictions of ↵-attractors and four D-brane models
e examwith p = 3, 4, 5, 6 (i.e. with k = 4, 3, 2, 1) can be
re inflaa-attractors
and KKLTIby
modelsof Dp-brane
inflationstripes
with p =with
3, 4,r5,⌧
6 1
represented
five verticalattractor
obinskyform a setof b – stripes, which become vertical at small r:
andas well
5 8 4 33↵ = 1
1 ns =,= 2, , , ,.
(2)
N
3 5 3 2
nformalyet AanAs one
cansimplest
see from
Fig. 5, they
covermodels
most of D-brane
the 2
combinationof thea-attractors
and KKLTI
esponds
area inmost
the of(nthespacefavoredby Planck
2018.
Moreinflation covers
area
favoredby Planck
2018, allthe ways , r)= 0. to cover most of the 1 area favored by Planck 2018
n in down
fig- to rover,ationary
it is sufficient to consider ↵-attractors and two D-brane

(2) showninFig.
5mkappearnot
only
e -stripes
left,
the
versions
with
V
⇠
1
,
which
were
calledBI(branein'inmaractor-stripes
(2)
shown
in
Fig.
5
appearnot
only
allowsors and flation)
D-branemodels,but also forin thea geninasm
b
in
[16],
are
inconsistentsame
reasonattractors
models, but
a gen- ↵symmange),
ThereandareD-brane
many di↵erent
waysalso
to inintroduce
s
sym
ation
context
introduced
in
[53],
see
also
[54]
simplest
hilltopmodels[9].
Consistent
generalizations
urple)
ole inflation
context
in [53],seeitalso[54]
attractors.
In theintroduced
context of thispaper,
is useful
to
k
k
p
'
m
1
a
-attractors
and
KKLTI
modelsofD-braneinflationformaphysicallynimal
isII, start
V of
thispaper.
Polepotentials
inflation
describes
thesemodelswithV⇠ (1+ 'k [53]) = It
with
the
pole
interpretationofthesemodelskItections
II,
V
of
this
paper.
Pole
inflation
describes' +ismotivated (in SUGRA and string theory) subclass of models of poleshow
inflatigicalattractors
with with
the
pole
order
q inq thewere
proposedin [52]
inpolethe context
of
D3
brane
in
infla
smological
the
order
in
the
inflation
withattractors
data.
1 aq
2
tion.
These
modelswere
generalized
calledoftermthe ofinflaton
see
(3).
In
particular,= Lfield,V=(@⇢)(⇢)and.
(3) KKLof
cion
theL inflaton
field,see(3).
In Vparticular,
kin
q

t
2
⇢
(KKLT
inflation)in [16],andfurtherdevelopedin [51
actorsarethepoleinflatonmodelswithq=2,
area-attractors
thepole
inflatonmodels
with
q
=
2,
m
describe pole inflation with q = 2 (supported by SUGRA)
Herethepolepotentials
of orderis(bothat ⇢(both=KKLTI
0 and
and
the
residueasD-braneinflation
potentials
KKLTI
and at moderane
inflation
andPredictionsof q↵-attractors
four
D-branexamthe=pole
is pa2,.- 1
If
the
potential
is regularnear
D-branestringtheorymotivatedmodelskofofpole
inflationq=k4,inflation
3,
the
class
the
pole
with
3,belong
4,to
5,the
6to
(i.e.
with
=
4,the
3, pole,2, with1) can↵
=ith
4,
3,
2,
1
belong
class
of
the
pole
nfla8
4
3
o
5 =
85
4
3
on
potentials
with
q
,
,
,
respectively.
representedbyfivevertical
attractor
stripes
with
r
⌧
2 Th
3
3 respectively.
2 c>0 ,
entialswith Vq ==V03(1, 5c⇢, +35,. . .),
(4)
nsky
2 The
2
and
modelsdescribe
cosmologicalwhich
All of thesemodels at
small r describe aattractors
set of b – stripes
with in60
wells describe cosmological attractors whichq in
60 ing
(o
t
mall
r
limit
predict
1
n
=
,
where
=
.
5
8
4
3
one
finds
that
inflation
occursinasmall
vicinity
of
the
s
q
N
q
1
↵imit
= 1 predict 1 1 ns n=, where, n and==2,depend,1 . , on, q, on. ingtia
s =
o
in
NN
q
pole.
Inflationary
predictions
r
s
3
5
3
2
rmal
se resultscannumberbe compared
withNthephenomenoaq , on theof e-foldings, and,
in general,2 on tialoft↵
an-can
As
one
can
see
from
Fig.
5,
they
cover
most
The pole
residueaqof
definesthe characteristic
scale ofinflation
M = ofaqoftheults
becomparedwiththephenomenolt parametrizationinflationarymodels
based
on
th
p
mo
the constant (somec in thepotential.
2 as 4a or 2a )authors
defineM
onds
inininflationary
the(ns , r) spacefavoredby qPlanckMo
metrization
modelsonone2018.mode
byumptionarea
thatof“natural”
modelsof qbased
inflationrAs
anexample,
let
us
first
considerthesimplest
andnxpectfigcovermostarea
favored
20
bythe
ab
on
that
in
modelsof1 inflation
oneby Planck
1 over,ns“natural”=top+1, wherepofisthesomephenomenologc
3↵Figure 1: The figure from the latest CMB-S4 Science Case paper [5]. The gray area shows predictions of'
the simplest ↵-attractor model V ⇠ tanh2 M
for 47 < N < 57. The green area is for the hilltop model with4
V ⇠ 1 ('/M ) . This model is theoretically inconsistent for M
1, which is the only range of M where it
could match observational data [9].

Renata Kallosh and AL 1909.04687• A combination of the simplest ↵-attractors and D-brane inflation models covers
most of the area in the (ns , r) space favored by Planck 2018. For ↵-attractor
models, there are discrete targets 3↵ = 1, 2, ..., 7, predicting 7 diﬀerent values ofr = 12↵/N 2 in the range 10 2 & r & 10 3 .
• In the small r limit, ↵-attractors and D-brane inflation models describe vertical -stripes in the (ns , r) space, with ns = 1/N , = 2, 53 , 85 , 32 , 43 . A
phenomenological description of these models and their generalizations can beachieved in the context of pole inflation.
• Future precision data on ns may help to discriminate between these models
even if the precision of the measurement of r is insuﬃcient for the discovery ofgravitational waves produced during inflation.

Of course, one may argue that it is premature to plan for the long journey when thegoal is nearby, and the B-mode detection at r & 10 2 is possible. For example, power-law