Download Large-Field Inflation - Naturalness and String Theory

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Large-Field Inflation - Naturalness and String Theory
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Declination [deg.]
BICEP2: B signal
−50
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−60
Planck Collaboration: The Planck mission
−65
Angular scale
90◦
6000
18◦
1◦
0.2◦
50
0.1◦
0.07◦
0
Right ascension [deg.]
5000
D![µK2]
4000
3000
2000
1000
0
2
10
50
500
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Multipole moment, !
1500
2000
2500
• Why use string theory in cosmology?
singularity at the beginning! — need quantum gravity
to resolve
inflation is UV-sensitive — needs a high-scale theory
what are dark energy & dark matter?
• if we try to use string theory for cosmology — better build on
generic consequences of string theory!
string vacuum structure seems to be a “landscape” of
extremely many vacua — this can accommodate a tiny C.C.
false-vacuum eternal inflation & tunneling set cosmological
initial conditions — hint from large-scale CMB anomalies?
• if we try to use string theory for cosmology — better build on
generic consequences of string theory!
string compactification produces moduli:
- moduli stabilization & moduli phenomenology (reheating,
baryogenesis ...) - spectrum of massive scalars
and light axions:
- axions driving large-field inflation in string theory - ---> monodromy
horizon problem of the hot Big Bang
time t
why so many causally
disconnected regions with
-5
ΔT/T ~ 10 ?
today: 13.7 billion years
CMB: 370,000 years
Big Bang:
t=0
Inflation ...
•
inflation: period quasi-exponential expansion of the
very early universe
(solves horizon, flatness problems of hot big bang ...)
[Guth ‘80]
•
driven by the vacuum energy of a slowly rolling light
scalar field:
e.o.m.:
φ̈ + 3H φ̇ + V = 0
!
[Linde; Albrecht & Steinhardt ‘82]
Inflation ...
•
slow-roll inflation:
[Linde; Albrecht & Steinhardt ‘82]
scale factor grows exponentially : a ∼ e
Ht
⇒
Ḣ
1
!≡− 2 $
H
2
!
V
V
"
! 2
%1
,
 2
 φ̇ ! V
if : 
|φ̈| ! |3H φ̇|
V
!˙
$
%1
η≡
!H
V
!!
2
ȧ
with the Hubble parameter: H = 2 ! const. ∼ V
a
φE!+∆φ
!
dφ
Ne
e-folds Ne in a ~ e : Ne = Hdt =
√
2"
2
φE
Inflation ...
•
inflation generates metric perturbations:
scalar (us) & tensor
2
PS
∼
∼
•
H
∼
!
k
!
δρ
ρ
"2
nS −1
scalar spectral index:
nS = 1 − 6! + 2η
and
2
PT ∼ H ∼ V
window to GUT scale &
direct measurement of inflation scale
•
tensor-to-scalar ratio:
PT
r≡
= 16!
PS
shades of difficulty ...
•
•
tensor-to-scalar ratio links levels of difficulty:
PT
r ≡
= 16! ≤ 0.003
PS
2)
r << O(1/Ne models:
∆φ ! O(MP )
•
50
Ne
"2 !
∆φ
MP
"2
[Lyth ’97]
Small-Field inflation ... needs control of
leading dim-6 operators
➥ enumeration & fine-tuning reasonable
⇒
2)
r = O(1/Ne models:
∆φ ∼ O(MP )
•
!
needs severe fine-tuning of all dim-6
operators, or accidental cancellations
⇒
r = O(1/Ne) models:
!
∆φ ∼ Ne MP " MP
⇒
Large-Field inflation ... needs
suppression of all-order corrections
➥ symmetry is essential!
why strings?
•
We need to understand generic dim ≥ 6 operators Op≥6
⇒
∆η
!
"p−4
φ
∼ V (φ)
MP
!
"p−6
φ
∼
!1
MP
∀p ≥ 6
if
φ > MP
•
requires UV-completion, e.g. string theory: need to
know string and α‘-corrections, backreaction effects, ...
•
detailed information about moduli stabilization
necessary!
•
string theory manifestation of the supergravity eta
problem
shift symmetry
•
effective theory of large-field inflation:
1
1
2
4−p p
L = R + (∂µ φ) − µ φ
2
2
•
the last term — the potential — spoils the shift
symmetry …
•
However, if:
•
quantum GR only couples to Tµν :
δV
(n)
∼ V0
!
V0
MP4
"n
V0 = µ
4−p p
, V0
!
"
!! n
V0
MP2
" V0
φ !
4
MP
not
δV
(n)
n
φ
∼ cn n
MP
•
shift symmetry
while field fluctuation interactions:
4−p
µ
(φ! + δφ) ∼
p
4−p p
µ φ!
!
1+
"
n
n
δφ
cn n
φ!
#
die out with increasing field displacement …
•
if the inflaton potential breaks the shift symmetry
weakly & smoothly (means: with falling derivatives)
—
it does not matter, whether the shift symmetry is
periodically broken, or secularly
•
a shift symmetry itself does not guarantee smoothness
of breaking — need UV theory as input, for all models!
axion monodromy
[Silverstein & AW ’08]
[McAllister, Silverstein & AW ’08]
[Kaloper & Sorbo ’08]
[Dong, Horn, Silverstein & AW ’10]
[Lawrence, Kaloper & Sorbo ’11]
[Marchesano, Shiu & Uranga ’14]
[Blumenhagen & Plauschinn ’14]
[Hebecker, Kraus & Witkowski ’14]
[McAllister, Silverstein, AW & Wrase ’14]
axion inflation in string theory ...
•
shift symmetry dictates use of string theory axions for large-field inflation
- periodic, e.g.
b=
!
B2
b → b + (2π)
,
Σ2
2
since
Sstring
1
⊃
2πα!
!
B2
- field range from kinetic terms f < MP :
S∼
!
10
√
2
d x −g|H3 | ⊃
B2 = bω2
⇒
φ = fb
,
!
√
1
2
d x −g4 4 (∂µ b)
L
4
MP
f = 2 < MP
L
[Banks, Dine & Gorbatov ’03]
however, maybe not strict: [Grimm; Blumenhagen & Plauschinn ’14]
axion inflation in string theory ...
•
large field-range from assistance effects of many fields - N-flation ...
•
[Dimopoulos, Kachru, McGreevy & Wacker ’05]
[Easther & McAllister ’05]
or monodromy - generic presence from fluxes & branes !
[Silverstein & AW ’08]
[McAllister, Silverstein & AW ’08]
[Dong, Horn, Silverstein & AW ’10]
- cos-potential for 2 axions can align/tune for
large-field direction as well
[Kim, Nilles & Peloso ’04]
[Berg, Pajer & Sjörs ’09]
[Ben-Dayan, Pedro & AW ‘14]
[Tye & Wong ‘14]
M N to thep+1
M
analogous
gaugeMinvariance
under A → A + dΛ0 in electro
milarly, there are other potential fields denoted Cp+1 sourced by p-dimensiona
axion
monodromy
—
the
general
story
→
A
+
∂
Λ
,
the
field
C
transforms
as
C
→
gauge
transformation
A
M
M
0
bjects (Dp-branes) [23].
a one-form,
1 EM
et term
action
contains
the
gauge-invariant
terms
is the Maxwell action, written in terms of the field strength F
•
= dA.
In electromagnetism,
the action
contains
the gauge-invariant terms
EM
Stueckelberg
gauge
symmetry:
m, known as a Stueckelberg term, can arise from spontaneous symmetry
"
#
!
√
4
4 vacuum expectation
M N value2√
2
"
#
the
of
a
charged
field.
d x −g F F
−4ρ (A + ∂ M
C)
,
(2.1)
N + 2. . .
2
MSNEM
=
d x −gM FM NM
F
− ρ (AM + ∂M C) + . . .
,
ring theory, one finds generalizations of these Maxwell and Stueckelberg
gauge transformation B → B + dΛ1 accompanied by appropriate shifts of
−
Λ
→field
A +CC
∂→
Λ0C, the
field
C
transform
here under the A
gauge
transformation
Athe
→
A
+
∂
Λ
,
as
C
→
nsformation
0
M ⇒
Mtransforms
M
M
0
though we will focus on specific examples in type IIB string theory below,
−
Λ
.
The
first
term
is
the
Maxwell
action,
written
in
terms
of
the
field
streng
0
e Maxwell
action,terms
written
the
F = dA.
nsidering
the relevant
arisingininterms
D = 10of
type
IIAfield
stringstrength
theory. There
he
second
term,
known
as
a
Stueckelberg
term,
can
arise
from
spontaneous
string
theory
contains
analogous
gauge
symmetries
for
al
fields
C
with
odd
p,
and
it
is
useful
to
define
the
following
generalized
p
as a Stueckelberg
term, can arise from spontaneous4 symmetry
reaking,
with
ρ
the
vacuum
expectation
value
of
a
charged
field.
hat
respect
all
the
gauge
symmetries
of
the
theory:
NSNS and
RR axions
- e.g.transformation
IIA:4
extended
to a combined
•
m In
expectation
value
of
a
charged
field.
type II string theory, one finds generalizations of these Maxwell and S
H =
dB , transformation
δBand
= dΛStueckelberg
y,rms,
onewith
finds
generalizations
of Bthese
the
gauge
→ B Maxwell
+ dΛ1 accompanied
1 , by appropria
F
=
Q
,
⇒
0
0
he
C
fields.
Although
we
will
focus
on
specific
in −F
type
IIB
string
th
p
δC1 =
Λ
,
sformation B → B + dΛ1 accompaniedexamples
by appropriate
shifts
of
0 1
B , relevant terms arising in D
t us start byF̃considering
=
10
type
IIA
string
th
2 = dC1 + F0the
δC
.
will focus on specific examples
in
type
IIB
string
theory
below,
3 = −F
0 Λ1 ∧ B
1
e have potential
fields
C
with
odd
p,
and
it
is
useful
to
define
the
following
p
F
B
∧
B
,
(2.2)
=
dC
+
C
∧
H
+
F̃
4
3
1
3
0
he
relevant
terms
arising
in
D
=
10
type
IIA
string
theory.
There
2
eld strengths that
respect
all the
gauge
symmetries
the dimensionality
theory:
The
effective
action
starting
from aoftotal
D=
with
odd
p,
and
it
is
useful
to
define
the
following
generalized
5
type
IIB
similar
nteger.
These
are
gauge-invariant,
with
the
transformation
B
→
B
+
dΛ
•
to
1
!
"
#
√
1
1
H
=
dB
,
10
2
all
the
gauge
symmetries
of
the
theory:
d into
−to breaking
x open
−G strings,
|H|but+
|F̃
the type I string, in which closed strings are unstable
2
4
3
%
%
dimensional
spacetime,
thebydual
6-form
flux Qin6Shiu
≡
#10 F4’14]
= M F6th)
[Marchesano,
Uranga
In F̃5 we do
not find anorFequivalently
working
directly
the&M
ten-dimensional
1 ∧ B ∧ B term
[Blumenhagen
&
Plauschinn
’14]
models
in
§3,
we
will
incorporate
the
analogue
in
type
IIB
string
theory
of
these
ad
However, T-duality
on
a
circle,
including
the
duality
between
D7-branes
and
D8-br
flux monodromy
[Hebecker, Kraus & Witkowski ’14]
requires
thiswill
coupling
to be present
upon dimensional
reduction.
This
indeed
fluxes,
which
yield interesting
behavior
in some
cases,
but
for now
will works
focus
[McAllister,
Silverstein,
AWwe
& Wrase
’14]
precisely
[25]. Specifically,
consider reducing
typethis,
IIB we
theory
leading
contributions
to the potential
at large ten-dimensional
field range. Given
haveon
anaac
fluxes
a
potential
for
the
axions:
9generate
9 ∼
9
6
the x form
direction, x = x + 2π), with
the(along
schematic
&
'
!
9
C 0 = x Q 0 + C0 ,
√
1
1
10
2
2
2
2
4
− ! 4 d x −G
|H|
+
|Q
B|
+
|Q
B
∧
B|
+
γ
g
|Q
B
∧
B|
+
.
.
.
.
0
0
4
0
9
s
C2 = x Q0 B + C2 ,
gs2
α
•
are
fluctuations
of
the
potential
fields
about
the
background.
Substituting
where
C
p
Here in the last term and the ellipses we have allowed for corrections that could be
produces
periodically
spaced
set
of
multiple
branches
into
(2.7),
we
find
an
effective
F
∧
B
∧
B
contribution
to
F̃
,
and
an
effective
F
∧
B
1
5
1
from the tree-level four-point and higher-point functions (γ4 being an order 1 numb
in F̃
In the four-dimensional
effective theory, 2there are many contributions of this
3 . large-field
2
of
potentials:
have also set to zero the contribution from |F̃6 | = |C3 ∧ H + Q0 B ∧ B ∧ B/6| , ha
leading to axion potentials of the schematic form
•
mind situations where H flux is present in order to contribute to moduli stabilization,
(n) n
2 a + Q(n−1) an−1 + · · · + Q(0) )2
(Q
p0
minimizes
f (χ, the
. . . ) |F̃6 | term at zero. ! More generally,+there
· · · ∼should
f˜(χ, . . be
. ) ainteresting
for a %configu
1,
L2n
5
Similar
comments
apply
in
the
more
generic
cases
with
D
>
10
[24].
/2
is
a
positive
integer, and “χ, . . . ” r
where
we have denoted the axion field by a, n = p0
6
See e.g. equation 12.1.25 of [23]. However, we caution the reader that we follow the sign conve
(i) as additional scalar fields, whose important effects we
to
the
moduli
fields
χ,
as
well
for
given
flux
quanta
Q
potential is non-periodic – a given branch
[25], not those of [23].
Q(i) change by brane-flux tunneling – 8Q(i) shift absorbed by a-shift – many
7
branches, full theory has periodicity
flux monodromy
•
p-form axions get non-periodic potentials from
coupling to branes or fluxes/field-strengths
•
produces periodically spaces set of multiple
branches of large-field potentials:
V (φ) ∼ µ
φ + Λ cos(φ/f )
4−p p
4
φ
with
the
Lagrangian
density
1
1
µ
1
2
2
2
We∗ present
in Figure 1. In what follows we will focus on the axionm
(∂φ)
F
φ
R
−
−
+
F
,
L
=
(4)
pl which we dub “natural chaotic
(4)
scale
string
4-form models
of
[12,
21],
2
2
48
24
1
1 2
µ picture
1 monodromy
flux
2
24D effective
∗
T
mpl
(∂φ)
F(4)which
φ F
−[Kaloper
− ’08]from
+ [Dubovsky,
,
(1)
L = as
inflation”,
aR
benchmark
to
discuss
Lawrence
&
Roberts
’11]
& theory
Sorbo
(4)
2
2
48
24
[Lawrence,
Kaloper
&
Sorbo
’11]
[Kaloper
&
Lawrence
’14]
thewhere
general F
phenomenon
of
axion
monodromy.
We
start
tribu
= dA
is a four-form field strength,
(4)
(3)
…
see
talk
with
the
Lagrangian
density
4D
effective
action:
F
field
strength,
axion
ɸ
:
•
4
dA(3) is a gauge
four-form
fieldThe
strength,
with mome
where
F(4)a =
three-form
field.
canonical
A(3)
peri
gauge
field.
The
canonical
momentum
A(3) pa three-form
∗
1
1
µ
1
2
2
2
∗
=
F
−
µφ
is
quantized
in, units
of The
thetheo
el
A123
− (∂φ) − F(4) + φ F(4)
(1)
L∗= mpl R(4)
pA123 = F(4)
− µφ is
2 2
2 quantized
48 in units
24 of the electric
tributions
;
using
this,
one
may
write
the
Hamiltoni
charge
e
2
;
using
this,
one
may
write
the
Hamiltonian
as
charge
e
= dA
is a four-form field strength, with
F
4D effective
Hamiltonian:
• where
Natural Chaotic Inflation and UV Sensitivity
Nemanja Kaloper
(4)
(3)
Department of Physics, University of California, Davis, Davis, CA 95616
periodicity
"2for n
gauge
field.
The
canonical
momentum
A(3) a three-form
!
1
1
1
!
"
1
1
1
2
2
2
2
2
2
2
∗
ne
) + +in (∇φ)
µφ
H(p
pA123 H
= =F(4)
−=
is (p
quantized
units
the
ne of++
) +
µφelectric
. +(2)
φ(∇φ)
φµφ
2
2
2
2
2
2
2 as
charge e ; using this, one may write the Hamiltonian
Albion Lawrence
Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454
If the recent measurement of B-mode polarization by BICEP2 is due to primordial gravitational
waves, it implies that inflation was driven by energy densities at the GUT scale MGU T ∼ 2 ×
1016 GeV . This favors single-field chaotic inflation models. These models require transplanckian
excursions of the inflaton, forcing one to address the UV completion of the theory. We use a
benchmark 4d effective field theory of axion-4-form inflation to argue that inflation driven by a
quadratic potential (with small corrections) is well motivated in the context of high-scale string
theory models; that it presents an interesting incitement for string model building; and the dynamics
of the UV completion can have observable consequences.
22
.
four
04.2912v1 [hep-th] 10 Apr 2014
2
!
"
If φ If
→ φφ +
f
,
we
must
have
µf
=
e
for
consistency.
1
1
1
2
2
2
for
n
≥
1
⇒
µf
,
n
→
n
+
1
2= e µf
→
φ
+
f
,
we
must
have
=
e
for
consist
ne + µφ .
(2)
H = (pφ ) + (∇φ) +
The quantum2number 2n can be shifted
by the nucleation
2
four-form,
V(
The quantum number n can be shifted by the nucle
of If
membranes,
leading
to
the
multivalued
potential
for
φ
2
φ
→
φ
+
f
,
we
must
have
µf
=
e
for
consistency.
again,
axion
unwound
into
multiple
•
of
membranes,
leading
to
the
multivalued
potential
shown
in Figurenumber
1. Atntree
level,
if membrane
nucleation
If M
The
quantum
can
be
shifted
by
the
nucleation
n=1
n=2
branches,
n
jumps
by
flux
tunneling,
shown
in
Figure
1.
At
tree
level,
if
membrane
nucle
is suppressed,
one
has
a
model
of
chaotic
inflation
with
a
port
of membranes, leading to the multivalued potential for φ
periodicity
by
summing
over
branches
n=0
shown
in potential.
Figure 1. AtThe
tree fundamental
level,
quadratic
of inflation
the
is suppressed,
one
has aif membrane
modelperiodicity
ofnucleation
chaotic
&w
M
If Muvn=3Not
1 2 2
is
suppressed,
one
has
a
model
of
chaotic
inflation
with
a
portant
if
scalar
implies
that
corrections
to
V
=
µ
φ
take
the
erat
quadratic potential. The fundamental
periodicity
o
2
BRX-TH-676
INTRODUCTION
The detection of B-mode polarization by the BICEP2
experiment [1, 2] gives tantalizing evidence that quantum
gravity may be directly relevant for observational cosmology. If the future checks confirm that the BICEP2 Bmodes are generated by primordial gravitational waves,
this will give support to the idea that these waves are induced by quantum fluctuations of the graviton. Furthermore, simple chaotic inflation models such as V = 12 m2 φ2
[3] are in excellent current agreement with the data. To
generate the observed density fluctuations in the CMB
and large scale structure, chaotic inflation models occur
16
< M4
at energy densities ∼
GeV )4 . Any
GU T ∼ (2 × 10
models generating observable primordial gravitational radiation require transplanckian excursions of the inflaton
in field space of order ∆φ ∼ 10mpl [4, 5] over the course
of inflation.1 Therefore all such models are sensitive to
the UV completion at the Planck scale, requiring a good
4 n
quadratic potential.
The fundamental periodicity of the
picture taken from:
[Kaloper, Lawrence ‘14]
flux axion monodromy with moduli stabilization
[Hebecker, Kraus & Witkowski ’14]
[McAllister, Silverstein, AW & Wrase ’14]
•
type IIB string theory:
!
d x
10
"
$
#
#
2
2
|dB|
# #
2
2
+
|F
|
+
|F
|
+
F̃
#
#
1
3
5
gs2
with:
•
F̃5 = dC4 − B2 ∧ F3 + C2 ∧ H3 + F1 ∧ B2 ∧ B2
ɸ2 , ɸ3, ɸ4 terms …
- generically flattening of the potential from adjusting moduli
and/or flux rearranging its distribution on its cycle - ‘sloshing’,
while preserving flux quantization
[Dong, Horn, Silverstein & AW ’10]
uct
of
three
two-tori,
(T
)
(perhaps
later
we
will
generalize
to
higher
.1 IIB example on product manifold and complex st
aces). For
simplicity
take
them
to
be
rectangular
tori,
y
≡
y
+
L
,
y
≡
1
1
1
2
flux axion monodromy with moduli
stabilization
justment
6
= L1 L2 , so the total internal volume V is L . [McAllister,
Put 3-form
fluxAW & Wrase ’14]
Silverstein,
2
et us work (1)
in type(2)IIB string
theory,(1)including
the
|F
∧
B
∧
B|
term
(se
1
(3)
(2)
(3)
(i) 2
(i) 2
2
2
2
F3 =•Q31simple
dy1 ∧torus
dy1 example:
∧ dy1 + Q32ds
dy2= ∧ dy
∧
dy
(26)
L12(dy
)
+
L
(dy
)
2
2
1
2
2 3
Consider a product of three two-tori,
i=1 (T ) (perhaps later we will gen
2
3 For simplicity take them to be rectangular tori, y
nus
Riemann
surfaces).
!
1
pt labels
the three T ’s. Thatb is, (i)
we have(i)Q31 units of flux on the product
axion
(i)
2B =
6
dy
∧
dy
1
2
+L
L of=flux
L1 Lon
total internal
volume
V is L . Put 3-for
2 the
les
and
Q32 units
the
product
of the three
y2 cycles.
2 . Denote
2 ,Lso
i=1
d 1-form flux in the symmetric
(1) configuration
(2)
(3)
(1)
(2)
(3)
F3 = Q31 dy1 ∧ dy1 ∧ dy1 + Q32 dy2 ∧ dy2 ∧ dy2
fluxes
3
!
Q1
(i)
2
F
=
dy
(27)
here the superscript1 labels the three
T ’s. That is, we have Q31 units
of flu
1
L
1 i=1
(i)
(i
the three y1 cycles and Q32 units of flux on the product of the three y2
L
φ
b
2
3
u
=
,
=
i) Include
effective
4d
action
gives
ɸ
-potential:
•
quantized 1-form flux in the symmetric configuration
F1 .
L1
MP
L2
!
$
"
#
ons
3
4
2
4
2
!
3
b
Q32
Q
(i)
1
(i)
s
!
2 ḃ
4 g
2
4
2
3
2
3
b
L ∼ MP 4 + MP 1 Q(i)1 L (i)
u
+
Q
u
+
+
µ
φ
∼
φ̇
F
=
dy
1
31
1
3
(28)
LB =
L 22 dy1 ∧ dy2L2
u
L
1 i=1
L
i=1
6
2
2
"
use Riemann
•
(i) surfaces: can fix Vol = L as well & get m ɸ
hat is, Q1 = dy1 F1 .
8
3
!
10
Planck Collaboration: Constraints on inflation
Model
ΛCDM + tensor
Parameter
ns
r0.002
−2∆ ln Lmax
Planck+WP
0.9624 ± 0.0075
< 0.12
0
Planck+WP+lensing
0.9653 ± 0.0069
< 0.13
0
Planck + WP+high-!
0.9600 ± 0.0071
< 0.11
0
0.00
Tensor-to-Scalar Ratio (r0.002)
0.05 0.10 0.15 0.20
0.25
Table 4. Constraints on the primordial perturbation parameters in the ΛCDM+r model from Planck c
The constraints are given at the pivot scale k∗ = 0.002 Mpc−1 .
Pla
Pla
Pla
Na
Po
Lo
Con
vex
Con
cav
e
R2
V
V
V
V
N∗
0.94
0.96
0.98
Primordial Tilt (ns )
1.00
N∗
Fig. 1. Marginalized joint 68% and 95% CL regions for ns and r0.002 from Planck in combination wit
the theoretical predictions of selected inflationary models.
open questions ...
•
BICEP2 may provide evidence for primordial
tensor modes with r = 0.05 … 0.16 — if so, only
large-field inflation survives …
•
axion monodromy provides one avenue for large
field inflation in string theory - technically natural &
distinctive predictions ...
•
many powers
… possible ; we need
generalizations ... harder look at universality, generic
distinctiveness from field theory models & potential
backreaction issues in string models
2/3
ɸ
4
ɸ