Inflation, B-mode, and String Theory Gary Shiu - University of Wisconsin & HKUST
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Inflation, B-mode, and String Theory
Gary Shiu
University of Wisconsin & HKUST
Inflation, B-mode, and String Theory
Gary Shiu
University of Wisconsin & HKUST
KEK-CPWS The 4th UTQuest Workshop
KEK Theory Center Cosmophysics Group Workshop
“B-mode Cosmology”
Based on:
• F. Marchesano, GS, A. Uranga, “F-term Axion Monodromy
Inflation,’' JHEP 1409, 184 (2014) [arXiv:1404.3040 [hep-th]].
and some forthcoming work:
• GS, W. Staessens, F. Ye, “Widening the Axion Window via Kinetic
and Stuckelberg Mixings”, 1502.xxxxx [hep-th], to appear.
• GS, W. Staessens, F. Ye, “Large Field Inflation from Axion
Mixings”, 1502.xxxxx [hep-th], to appear.
• J. Brown, I. Garcia-Etxebarria, F. Marchesano, GS, “Tunneling in
Axion Monodromy Inflation”, in progress.
Temperature Fluctuations
Planck Collaboration: The Planck mission
Planck Collaboration: The Planck mission
Angular scale
90 18 1 0.2 0.1 0.07
6000
5000 PLANCK
4000
D [µK2]
3000
2000
1000
0
2 10 50 500 1000 1500 2000 250
Multipole moment,
14. The SMICA CMB map (with 3 % of the sky replaced by a constrained Gaussian realization). Fig. 19. The temperature angular power spectrum of the primary CMB from Planck, showing a precise measurement of seven acoustic peaks,
are well fit by a simple six-parameter ⇤CDM theoretical model (the model plotted is the one labelled [Planck+WP+highL] in Planck Collabora
XVI (2013)). The shaded area around the best-fit curve represents cosmic variance, including the sky cut used. The error bars on individual po
also include cosmic variance. The horizontal axis is logarithmic up to ` = 50, and linear beyond. The vertical scale is `(` + 1)Cl /2⇡. The measu
spectrum shown here is exactly the same as the one shown in Fig. 1 of Planck Collaboration XVI (2013), but it has been rebinned to show be
the low-` region.
A nearly scale-invariant, adiabatic, Gaussian temperature 90 18 1
Angular scale
0.2 0.1
tected by Planck over the entire sky, and which therefore c
tains both Galactic and extragalactic objects. No polarization
fluctuation spectrum as generically predicted by inflation
6000
formation is provided for the sources at this time. The PC
5000
di↵ers from the ERCSC in its extraction philosophy: more e↵
has been made on the completeness of the catalogue, without
4000 ducing notably the reliability of the detected sources, wher
D [µK2]
seems to be in excellent agreement with data.
the ERCSC was built in the spirit of releasing a reliable cata
3000 suitable for quick follow-up (in particular with the short-liv
Herschel telescope). The greater amount of data, di↵erent sel
2000
15. Spatial distribution of the noise RMS on a color scale of 25 µK tion process and the improvements in the calibration and m
the SMICA CMB map. It has been estimated from the noise map making processing (references) help the PCCS to improve
1000
ained by running SMICA through the half-ring maps and taking the performance (in depth and numbers) with respect to the pre
-di↵erence. The average noise RMS is 17 µK. SMICA does not 0 ous ERCSC.
duce CMB values in the blanked pixels. They are replaced by a con- 2 10 50 500 1000 1500 2000
ined Gaussian realization. Fig. 16. Angular spectra for the SMICA CMB products, evaluated over Multipole moment, The sources were extracted from the 2013 Planck frequen
Two massless fields that are guaranteed to exist are:
ζ Goldstone boson
of broken time translations
hij graviton
Two massless fields that are guaranteed to exist are:
ζ Goldstone boson
of broken time translations
hij graviton
expansion
symmetry breaking
( for slow-roll inflation)
Two massless fields that are guaranteed to exist are:
ζ Goldstone boson
of broken time translations
hij graviton
expansion
symmetry breaking
( for slow-roll inflation)
E-modes:
B-modes:
A distinguishing parameter is the tensor-to-scalar ratio r.
Many experiments including BICEP/KECK, PLANCK, ACT,
PolarBeaR, SPT, SPIDER, QUEIT, Clover, EBEX, QUaD…
can potentially detect such primordial B-mode if r≾10-2.
LiteBIRD may even have the sensitivity to detect r ~ 10-3.
M. Hazumi’s take home message:
r ~ 10-2 in ~ 5 years, r~ 10-3 in 2020s
B-mode and Inflation
If primordial B-mode is detected, natural interpretations:
✦ Inflation took place
✦ The energy scale of inflation is the GUT scale
⇣ r ⌘1/4
Einf ' 0.75 ⇥ ⇥ 10 2 MPl
0.1
✦ The inflaton field excursion was super-Planckian
⇣ r ⌘1/2
& MPl Lyth ’96
0.01
✦ Great news for string theory due to strong UV sensitivity!
!Assumptions in the Lyth Bound single field slow-roll Bunch-Davies initial conditions vacuum fluctuations
Assumptions in the Lyth Bound
single field
slow-roll
Ashoorioon, Dimopoulos, Sheikh-Jabbari, GS
Collins, Holman, Vardanyan
Bunch-Davies initial conditions Aravind, Lorshbough, Paban
vacuum fluctuationsAssumptions in the Lyth Bound
single field
slow-roll
Ashoorioon, Dimopoulos, Sheikh-Jabbari, GS
Collins, Holman, Vardanyan
Bunch-Davies initial conditions Aravind, Lorshbough, Paban
Particle production during inflation
vacuum fluctuations
can be a source of GWs
Cook and Sorbo
Senatore, Silverstein, Zaldarriaga
Barnaby, Moxon, Namba, Peloso, GS, Zhou
Mukohyama, Namba, Peloso, GSAssumptions in the Lyth Bound
single field
slow-roll
Ashoorioon, Dimopoulos, Sheikh-Jabbari, GS
Collins, Holman, Vardanyan
Bunch-Davies initial conditions Aravind, Lorshbough, Paban
Particle production during inflation
vacuum fluctuations
can be a source of GWs
Cook and Sorbo
1
= 10-5 Senatore, Silverstein, Zaldarriaga
P
Barnaby, Moxon, Namba, Peloso, GS, Zhou
S
Mukohyama, Namba, Peloso, GS
C
Only known model of particle production that:
Detectable tensors w/o too large non-Gaussianity
0.1 = 10-4 = 10-3
CV Chiral, non-Gaussian tensor spectrum
Can accommodate a blue tensor tilt
0.001 0.01 0.1 due to an axionic a F∧ F coupling
rSuper-Planckian Fields in Effective Field Theory
Chaotic Inflation Linde ’86
Chaotic Inflation Linde ’86 Classical backreaction is under control.
Chaotic Inflation Linde ’86
graviton
inflaton
Quantum corrections are small.Concerns arise if we consider coupling the theory to the UV degrees of freedom of a putative theory of quantum gravity. Light fields can become heavy; Heavy fields can become light
Large Field Inflation in String Theory
Large corrections, unless the inflaton couples weaker
than gravitationally to everything else.
This even stronger UV sensitivity intrinsic to large field inflation
should be explained in the UV-completion.
Finding concrete models of Large Field Inflation in String Theory
is the goal of our work. We present two classes of models:
Axion Monodromy: F. Marchesano, GS, A. Uranga
Axion Mixings: GS, W. Staessens, F. YeSuper-Planckian Fields in String Theory
Natural Inflation
Freese, Frieman, Olinto
Pseudo-Nambu-Goldstone bosons
are natural inflaton candidates:Natural Inflation
Freese, Frieman, Olinto
Pseudo-Nambu-Goldstone bosons
are natural inflaton candidates:
They satisfy a shift symmetry that is only
broken by non-perturbative effects:
decay constantNatural Inflation
Freese, Frieman, Olinto
Pseudo-Nambu-Goldstone bosons
are natural inflaton candidates:
They satisfy a shift symmetry that is only
broken by non-perturbative effects:
decay constant
Successful inflation requires a
super-Planckian decay constant:Axions in String Theory
String theory has many higher-dimensional form-fields:
e.g.
3-form flux 2-form gauge potential:
gauge symmetry:
Integrating the 2-form over a 2-cycle gives an axion:
The gauge symmetry becomes a shift symmetry.Axion Decay Constants
Svrcek and Witten
Banks et al.
The axion decay constant follows from dimensional reduction:
Axions with super-Planckian decay constants don’t
seem to exist in controlled limits of string theory.Multiple Axions N-flation Dimopoulos, Kachru,McGreevy,Wacker ‘05 Aligned natural inflation Kim, Nilles, Peloso ’04 But quantum corrections to Mp also grow with # dof.
Axion Monodromy
= a combination of chaotic inflation and natural inflation
The axion periodicity is lifted, but the symmetry still
constrains corrections to the potential:Axion Monodromy Inflation
Siverstein & Westphal ’08
!
Combine chaotic inflation and
! Idea: natural inflation
!
2⇡f
The axion periodicity is lifted, allowing for super-Planckian
displacements. The UV corrections to the potential should
still be constrained by the underlying symmetry.Axion Monodromy Inflation
Siverstein & Westphal ’08
!
Combine chaotic inflation and
! Idea: natural inflation
!
2⇡f
The axion periodicity is lifted, allowing for super-Planckian
displacements. The UV corrections to the potential should
still be constrained by the underlying symmetryAxion Monodromy
The axion shift symmetry in string theory is broken only by non-perturbative
effects (instantons) or boundaries [Wen, Witten];[Dine, Seiberg]
Non-perturbative effects → discrete symmetry (no monodromy)
A 5-brane wrapping the 2-cycle breaks 5-brane
the axion shift symmetry:
anti-5-brane
McAllister, Silverstein and WestphalAxion Monodromy Inflation
Siverstein & Westphal ’08
!
Combine chaotic inflation and
! Idea: natural inflation
Early developments:
✦ McAllister, Silverstein, Westphal → String scenarios
✦ Kaloper, Lawrence, Sorbo → 4d framework
5B ! anti
C (2)
=c 5B
figure taken from McAllister, Silverstein, Westphal ‘08Axion Monodromy Inflation
Siverstein & Westphal ’08
!
Combine chaotic inflation and
! Idea: natural inflation
Early developments:
✦ McAllister, Silverstein, Westphal → String scenarios
exceedingly complicated, uncontrollable ingredients, backreaction, …
✦ Kaloper, Lawrence, Sorbo → 4d framework
5B ! anti
C (2)
=c 5B
figure taken from McAllister, Silverstein, Westphal ‘08Axion Monodromy Inflation
Siverstein & Westphal ’08
!
Combine chaotic inflation and
! Idea: natural inflation
Early developments:
✦ McAllister, Silverstein, Westphal → String scenarios
exceedingly complicated, uncontrollable ingredients, backreaction, …
✦ Kaloper, Lawrence, Sorbo → 4d framework
UV completion?
5B ! anti
C (2)
=c 5B
figure taken from McAllister, Silverstein, Westphal ‘08F-term Axion Monodromy Inflation
!
Giving a mass to an
Axion Monodromy
Obs:
!
~ axion
✦ Done in string theory within the moduli stabilization
program: adding ingredients like background fluxes
generate superpotentials in the effective 4d theory
figure taken from Ibañez & Uranga ‘12F-term Axion Monodromy Inflation
!
Giving a mass to an
Axion Monodromy
Obs:
!
~ axion
✦ Done in string theory within the moduli stabilization
program: adding ingredients like background fluxes
generate superpotentials in the effective 4d theory
Use same techniques to
Idea: generate an inflation potential
figure taken from Ibañez & Uranga ‘12F-term Axion Monodromy Inflation
!
Giving a mass to an
Axion Monodromy
Obs:
!
~ axion
✦ Done in string theory within the moduli stabilization
program: adding ingredients like background fluxes
generate superpotentials in the effective 4d theory
Use same techniques to
Idea: generate an inflation potential
• Simpler models, all sectors understood at weak coupling
• Spontaneous SUSY breaking, no need for brane-anti-brane
• Clear endpoint of inflation, allows to address reheatingToy Example: Massive Wilson line
✤ Simple example of axion: (4+d)-dimensional gauge field
integrated over a circle in a compact space Πd
Z
= A1 or A1 = (x) ⌘1 (y)
S1
✦ φ massless if ∆η1 = 0 S1 is a non-trivial circle in Πd
exact periodicity and (pert.) shift symmetry
✦ φ massive if ∆η1 = -μ2 η1 kS1 homologically trivial in Πd
(non-trivial fibration)Toy Example: Massive Wilson line
✤ Simple example of axion: (4+d)-dimensional gauge field
integrated over a circle in a compact space Πd
Z
= A1 or A1 = (x) ⌘1 (y)
S1
✦ φ massless if ∆η1 = 0 S1 is a non-trivial circle in Πd
exact periodicity and (pert.) shift symmetry
✦ φ massive if ∆η1 = -μ2 η1 kS1 homologically trivial in Πd
(non-trivial fibration)
F2 = dA1 = d⌘1 ⇠ µ !2 shifts in φ increase energy
via the induced flux F2
periodicity is broken and shift symmetry approximateMWL and twisted tori
✤ Simple way to construct massive Wilson lines: consider
compact extra dimensions Πd with circles fibered over a base,
like the twisted tori that appear in flux compactifications
✤ There are circles that are not contractible but do not
correspond to any harmonic 1-form. Instead, they correspond
to torsional elements in homology and cohomology groups
Tor H1 (⇧d , Z) = Tor H 2 (⇧d , Z) = ZkMWL and twisted tori
✤ Simple way to construct massive Wilson lines: consider
compact extra dimensions Πd with circles fibered over a base,
like the twisted tori that appear in flux compactifications
✤ There are circles that are not contractible but do not
correspond to any harmonic 1-form. Instead, they correspond
to torsional elements in homology and cohomology groups
Tor H1 (⇧d , Z) = Tor H 2 (⇧d , Z) = Zk
✤ Simplest example: twisted 3-torus T̃3
H1 (T̃3 , Z) = Z ⇥ Z ⇥ Zk
d⌘1 = kdx2 ^ dx3 F = k dx2 ^ dx3 two normal
one torsional
1-cycles 1-cycle
µ=
kR1 under a shift φ → φ +1
R2 R3 F2 increases by k unitsMWL and monodromy
V ( ) ⇠ |F |2
k 2k 3k 4k 5k F
How does monodromy and
Question: approximate shift symmetry help
prevent wild UV corrections?Torsion and gauge invariance
✤ Twisted tori torsional invariants are not just a fancy way of
detecting non-harmonic forms, but are related to a hidden
gauge invariance of these axion-monodromy models
✤ Let us again consider a 7d gauge theory on M1,3 x T̃3
✦ Instead of A1 we consider its magnetic dual V4
d⌘1 = k 2
V 4 = C 3 ^ ⌘ 1 + b2 ^ 2 dV4 = dC3 ^ ⌘1 + (db2 kC3 ) ^ 2Torsion and gauge invariance
✤ Twisted tori torsional invariants are not just a fancy way of
detecting non-harmonic forms, but are related to a hidden
gauge invariance of these axion-monodromy models
✤ Let us again consider a 7d gauge theory on M1,3 x T̃3
✦ Instead of A1 we consider its magnetic dual V4
d⌘1 = k 2
V 4 = C 3 ^ ⌘ 1 + b2 ^ 2 dV4 = dC3 ^ ⌘1 + (db2 kC3 ) ^ 2
✦ From dimensional reduction of the kinetic term:
Z Z 2
7 2 µ
d x |dV4 | d4 x |dC3 |2 + 2 |db2 kC3 |2
k
• Gauge invariance C3 ! C3 + d⇤2 b2 ! b2 + k⇤2
• Generalization of the Stückelberg Lagrangian
Quevedo & Trugenberger ’96Effective 4d theory
✤ The effective 4d Lagrangian
Z 2
µ
d4 x |dC3 |2 + 2 |db2 kC3 |2
k
describes a massive axion, has been applied to Kallosh et al.’95
QCD axion generalized to arbitrary V(φ) Dvali, Jackiw, Pi ’05
Dvali, Folkerts, Franca ‘13
✤ Reproduces the axion-four-form Lagrangian proposed by
Kaloper and Sorbo as 4d model of axion-monodromy inflation
with mild UV corrections
Z
4 2 2 F4 = dC3
d x |F4 | + |d | + F4 Kaloper & Sorbo ‘08
d = ⇤4 db2
✤ Can be realized in SUGRA as an F-term Groh, Louis, Sommerfeld ’12
✤ Reminiscent of Natural Chaotic Inflation in SUGRA
Kawasaki, Yamaguchi, Yanagida ’00Effective 4d theory
✤ Effective 4d Lagrangian
Z 2
µ F4 = dC3
d4 x |dC3 |2 + 2 |db2 kC3 | 2
k d = ⇤4 db2
✤ Gauge symmetry UV corrections only depend on F4
1
X 2i
1 1 2
Le↵ [ ] = (@ )2 µ 2
+ ⇤4 ci
2 2 i=1
⇤2i
X 2n X ✓ 2 2
◆n
F 2 2 µ
cn 4n µ cn
n
⇤
n
⇤4
suppressed corrections up to the scale where V(φ) ~ Λ4
effective scale for corrections Λ → Λeff = Λ2/μEffective 4d theory
✤ Effective 4d Lagrangian
Z 2
µ F4 = dC3
d4 x |dC3 |2 + 2 |db2 kC3 | 2
k d = ⇤4 db2
✤ Gauge symmetry UV corrections only depend on F4
✓ ◆
⇤
⇤ ! ⇤e↵ =⇤
µDiscrete symmetries and domain walls
✤ The integer k in the Lagrangian
Z 2
µ
d4 x |F4 |2 + 2 |db2 kC3 |2
k
corresponds to a discrete symmetry of the theory broken
spontaneously once a choice of four-form flux is made.
This amounts to choose a branch of the scalar potential
k=4
2⇡f figure taken from Kaloper & Lawrence ‘14Tunneling between branches
Discrete symmetries and domain walls
Gary Shiu
✤ The integer kTunneling
in the Lagrangian between branches
Z 2
4 Gary2 µShiu 2
d x |F |
The tunneling formulae of Coleman quoted+ |db kC |
k 2 in Kaloper, Lawrence, and Sorbo:
4 2 3
27⇡ 2 4
corresponds to a discrete P symmetry
=e 2( V )3 of the theory broken (0.1)
The spontaneously
tunneling formulaeonce a choice
of Coleman of four-form
quoted in Kaloper,flux is made.
Lawrence, and Sorbo:
can be This amounts
understood to choose
in terms a branch
of the action of a of the
domainscalar
wall (Ipotential
do not keep track of
27⇡ 2 4
numerical factors carefully here): P =e 2( V )3 (0.1
✤ Branch jumps are made via nucleation of domain walls that
couple to C 3 , and this
can be understood in terms of theputs
P = aS maximum
⇥R03 to wall
e =ofe a domain
action the inflaton
(I do notrange (0.2)
keep track o
numerical
✤ factors carefully
Tunneling rate here): branches w/ Brown,Marchesano,Garcia-Etxebarria
between
where is the domain wall tension and R0 is the bubble radius. The bubble radius
3
can be fixed by demanding that the P = e S=
energy e ⇥R0 between two sides of the domain
di↵erence (0.2
walls to where
be balanced by the energy
σ = domain wall of the domain
tension, R0 =wall:
bubble radius
where is the domain wall tension and R0 is the bubble radius. The bubble radiu
R03 ( the
can be fixed by demanding that = R02 di↵erence
V ) energy ) R0 = /between
V (0.3)
two sides of the domain
walls to be balanced by the energy of the domain wall:
To estimate V , we can expressDiscrete symmetries and domain walls
unneling
✤ Thisformulae of Coleman quoted in Kaloper, Lawrence, and S
gives the usual Coleman formula for 4D field theory:
27⇡ 2 4
P =e 2( V )3
✤ In string theory models, σ = tension of branes wrapping an
nderstood in cycle,
internal terms ΔV
of ~the action
V/N, of aindomain
we found a single wall (I do
modulus not kee
case:
factors carefully here): gs8
✓
N MP4
◆3
S=
(Ms L)24 V
S ⇥R03
P =e =e
✤ Even with the high inflation scale suggested by BICEP,
s the domain wall tension and
V R0 is 8 the bubble radius. The bub
. 10
MP4
ed by demanding that the energy di↵erence between two sides of t
✤ Tunneling is (marginally) suppressed for Ms L ≳ 10 and gs ≲ 1.
e balanced by the energy of the domain wall:
✤ Constraints from other tunneling channels in string theory.
R03 ( V ) = R02 ) R0 = / VMassive Wilson lines in string theory
✤ Simple example of MWL in string theory: D6-brane on M1,3 x T̃3
✤ An inflaton vev induces a non-trivial flux F2 proportional to φ
but now this flux enters the DBI action
p
det (G + 2⇡↵0 F2 ) = dvolM 1,3 |F2 |2 + correctionsMassive Wilson lines in string theory
✤ Simple example of MWL in string theory: D6-brane on M1,3 x T̃3
✤ An inflaton vev induces a non-trivial flux F2 proportional to φ
but now this flux enters the DBI action
p
det (G + 2⇡↵0 F2 ) = dvolM 1,3 |F2 |2 + corrections
✤ For small values of φ we recover chaotic inflation, but for
large values the corrections are important and we have a
potential of the form
p
V = L4 + h i 2 L2
Similar to the D4-brane model of Silverstein and Westphal
except for the inflation endpointMassive Wilson lines in string theory
✤ Simple example of MWL in string theory: D6-brane on M1,3 x T̃3
✤ An inflaton vev induces a non-trivial flux F2 proportional to φ
but now this flux enters the DBI action
p
det (G + 2⇡↵0 F2 ) = dvolM 1,3 |F2 |2 + corrections
✤ For small values of φ we recover chaotic inflation, but for
large values the corrections are important and we have a
potential of the form
p
V = L4 + h i 2 L2
Similar to the D4-brane model of Silverstein and Westphal
except for the inflation endpointMassive Wilson lines and flattening
✤ The DBI modification
2
p
h i ! L4 + h i 2 L2
can be interpreted as corrections due to UV completion
✤ E.g., integrating out moduli such that H < mmod < MGUT
will correct the potential, although not destabilise it
Kaloper, Lawrence, Sorbo ‘11
✤ In the DBI case the potential is flattened: argued general effect
due to couplings to heavy fields Dong, Horn, Silverstein, Westphal ‘10
✤ Large vev flattening also observed in examples of confining
gauge theories whose gravity dual is known [Witten’98]
Dubovsky, Lawrence, Roberts ’11Other string examples
✤ We can integrate a bulk p-form potential Cp over a p-cycle to
get an axion
Z
Fp+1 = dCp , Cp ! Cp + d⇤p 1 c= Cp
⇡p
✤ If the p-cycle is torsional we will get the same effective action
Z Z
10 2 4 2 µ2
d x|F9 p| d x |dC3 | + 2 |db2 kC3 |2
kOther string examples
✤ We can integrate a bulk p-form potential Cp over a p-cycle to
get an axion
Z
Fp+1 = dCp , Cp ! Cp + d⇤p 1 c= Cp
⇡p
✤ If the p-cycle is torsional we will get the same effective action
Z Z
10 2 4 µ2
2
d x|F9 p| d x |dC3 | + 2 |db2 kC3 |2
k
✤ The topological groups that detect this possibility are
Tor Hp (X6 , Z) = Tor H p+1 (X6 , Z) = Tor H 6 p
(X6 , Z) = Tor H5 p (X6 , Z)
one should make sure that the corresponding axion mass is
well below the compactification scale (e.g., using warping)
Franco, Galloni, Retolaza, Uranga ’14Other string examples
✤ Axions also obtain a mass with background fluxes
✤ Simplest example: φ = C0 in the presence of NSNS flux H3
Z
W = (F3 ⌧ H3 ) ^ ⌦ ⌧ = C0 + i/gs
X6
✤ We also recover the axion-four-form potential
Z Z Z
C 0 H 3 ^ F7 = C 0 F4 F4 = F7
M 1,3 ⇥X6 M 1,3 PD[H3 ]Other string examples
✤ Axions also obtain a mass with background fluxes
✤ Simplest example: φ = C0 in the presence of NSNS flux H3
Z
W = (F3 ⌧ H3 ) ^ ⌦ ⌧ = C0 + i/gs
X6
✤ We also recover the axion-four-form potential
Z Z Z
C 0 H 3 ^ F7 = C 0 F4 F4 = F7
M 1,3 ⇥X6 M 1,3 PD[H3 ]
✤ M-theory version: Beasley, Witten ’02
✤ A rich set of superpotentials obtained with type IIA fluxes
Z
eJc ^ (F0 + F2 + F4 ) Jc = J + iB
X6
potentials higher than quadratic
✤ Massive axions detected by torsion groups in K-theoryLarge Field Inflation
from Axion Mixings
GS, W. Staessens, F. Ye, 1502.xxxxx [hep-th], to appear.Mixing Axions
Large Field Inflation from Axion Mixings
• String Theory compactifications ; e↵ective action with mixing
✤ Axions can
axions (seemix kinetically and via Stuckelberg couplings:
later)
Z 2 ! 3
XN N
X
1 1
e↵
Saxion = 4 Gij (dai i
k A) ^ ?4 (da j j
k A) ri a i Tr(G ^ G ) + Lgauge 5
2 i,j=1 8⇡ 2 i=1
1
✤ Such
we are left withmixings
one
• 2 types
axion can
ã
of kinetic lead
, one to an effective
non-Abelian
mixing
gauge groupsuper-Planckian
and a set of chiraldecay
ions charged under the non-Abelian gauge group. By integrating out the massive
constant evenmixing:
for N=2 is& not a modestly small non-Abelian group.
(1) metric
gauge boson, four-point couplings G among
ij thediagonal
chiral fermions emerge, suppressed
he squared mass (2)
of U(1)
the mixing:
gauge boson. k i Integrating
6= 0 for some 2 {1,
outi the N} fermions and
. . . ,chiral
heavy
✤ For example, gauged N=2,
axions: onea i
!linear
a i
+ k combination
i
, A ! A + of axions is eaten by
d
on-Abelian gauge bosons, for which the procedure is briefly outline in appendix C,
the Stuckelberg U(1) while the 1 orthogonal combination
s a cosine-potential for the remaining axion ã :
• Tr(G ^ G )-term associated to non-Abelian gauge group
PN ✓i PN
◆
; collective periodicity: i=1 ri a ã'1 i=1 ri ai + 2⇡
Vaxion (ã1 ) = ⇤4 1 cos . with fã1 > M (2.84)
P
• axions couple to D-brane instantons f ã 1
; individual periodicity: a i
a i
+ 2⇡⌫ i
h provides an explicit realisation of natural inflation with a 2
! , ⌫ i
Z axion field. In
single
✤ Explicit
note: string models realizing
e↵ective contributions this scenario
of D-brane instantons e↵ can
to Saxion be constructed
is model-dependent
ndix F we propose a method to identify a proper spectrum of chiral fermions
see e.g. Ibáñez-Uranga (2007, 2012), Blumenhagen-Cvetič-Kachru-Weigand (2009)
fying the anomaly constraints. more in GS, Staessens, Ye, to appear.
Generic Kinetic mixing
h the insights gathered in sections 2.1 and 2.2 we can now tackle the most genericere the continuous param- ing angle ✓ and ratios " or % of metric entries) in p the p
G 2 G 2
Regionaxion3: formoduli space
intermediary which
kinetic leavethethe
mixing lowofenergy spectrum
range 11 12 G
f⇠ =
1
r2
.
Large Field Inflation from Axion Mixings
the decay
(10)
constant
intact. This (7)decoupling
can be represented
of thethrough
tour plots as functions of the continuous parameters "
ment from the low energy spectrum holds where
con- range enhance-
axion field
generally
r1 (G11 + G12 )
for
2r and ✓ as in fig. 1 upon fixing the U (1) charges k i and the numerator d
1
✤ Axion decaythe constant
parameters
p can
the multi-axion be
ri . Regions enhanced
system moduli to
in thedescribed super-Planckian:
by
space eq. (2)limit
with and(12). not
10) andtheG11
p just
Considering m
⇠ O(1015
opy between the diagonal f⇠ > 10the
2
minimal
G11 setup inconsidered
are highlighted white. here. In contrast, en-
pwithin the axion
hancement in the axion constant
field range scales as ⇠ N in
= 1, and a non-negligible p obtained for moduli space
N
⇡
✓ ⇡ 2 , the U (1) charges N-flation, and as ⇠ N ! n [13] in aligned natural
ing e↵ects,infla-
i.e. 1 ⇢2 ⇠ O(1
If we assume r1 = r2 and tion, with N the number of axions and n 2eigenvalue Z the coeffi- repulsion can b
ecay constant can be sim-" cients for the axion-instanton
" couplings. The presence
stant, similar ofto the Z 0 ma
these light fields generically renormalize the Planck mass
2
p p and we expect on general grounds [14] that M P l ⇠ N .THEORY
STRING
G11 1 + %2 Thus our scenario is minimal in that parametrically fewer
= p , (11) ✓ of freedom are needed ✓ to achieve the
r 1 1 %2 degrees It issame
naturalen-to ask if o
FIG. 1. Contour plots of decay constant f⇠ (✓, ") for 2r1 =
hancement
1 2 and so their associated 1 2quantum string theory where axion
corrections
2r 2 = 2k = k = 2 (left) and r1 = 2r2 = k = 2k = 2
ter %2 ⌘ G12 /G11 measures(right). to
Thethe Planck
f⇠ -values rangemass are (purple)
from small less severe.
to large (red) ion models with a super-P
tive to Planck scale physic
. In this moduli space re-following the rainbow contour colors. Unphysical regions with
Let us end this section by briefly discussing theimplementation
possi-
complex f⇠ are located in the black band. theory is n
hes trans-Planckian values bility to lower the e↵ective axion decay constant Here to we within
lay out the criteria
ntries in the metric are of Whilethe dark matter
we exploit multiple window. If we
axions to obtain an consider
e↵ective the needssame to satisfy
con- in order
and can
l ones, namely for: be lowered
super-Planckian todecay
figuration
the desired
constant,
as in regionour
dark
2, mechanism
matter
but assume di↵ers window
that rwe =proposed
r
w/o above. Clos
1 2 and
lowering thefundamentally
fundamental
k 1 = kfrom2
, the energy
earlier
axion scale:
approaches.
decay Unlike N-flation
constant
rally from the dimensiona
instead reads
p-forms: as summarised in
1. (12)
[11] and aligned natural inflation [12], the enhancement
in the physical axionpfield range we
2 2
p found here p is p not ness we restricted to four
2 ) orientifold compacti
inetic mixing the range of
tied to the number of degrees G11 ofGfreedom
12 G11introduced G11 (in-1 % (CY 3
f ⇠ = = p ,
theory [21].(13)
The backgrou
represented through con-cluding axions, gauge fields, and any
r1 (G11 + G12 ) additional fields
r2 1 + % 2
(1,1)
needed to ensure consistency of the theory). This can the Hodge-numbers h± ,
e continuous parameters be " seen already in the minimal setup above as an en- number of orientifold-even
g the U (1) charges k i and where the numerator more
hancement in neither (9) nor (11) requires in GS,significantly
decreases
adjusting Staessens,
the forms Ye,
in tothe appear.
respectively, and thConclusions ✤ Presented two broad ways to realize large field inflation in string theory: axion monodromy and axion mixings. ✤ F-term axion monodromy inflation provided a new, concrete implementation of monodromy into supergravity in a way compatible with spontaneous supersymmetry breaking. ✤ UV corrections are under control because the shift symmetry is spontaneously broken. ✤ α’ corrections to EFT [Garcia-Etxebarria,Hayashi,Savelli,GS,’12; Junghans,GS,‘14] important for inflation & moduli stabilization. ✤ Large field inflation can also be realized via axion mixings w/o the need of large # axions or bi non-Abelian gauge groups.
Conclusions ✤ A broad class of large field inflationary scenarios that can be implemented in any limit of string theory w/ rich pheno: ✤ Moduli stabilization needs to be addressed in detailed models
Gordon Research Conferences
Conference Program
String Theory & Cosmology
New Ideas Meet New Experimental Data
May 31 - June 5, 2015
The Hong Kong University of Science and Technology
Hong Kong, China
Chair:
Gary Shiu
Vice Chair:
Ulf Danielsson
Application Deadline
Applications for this meeting must be submitted by May 3, 2015. Please apply early, as some meetings become
oversubscribed (full) before this deadline. If the meeting is oversubscribed, it will be stated here. Note: Applications for
oversubscribed meetings will only be considered by the Conference Chair if more seats become available due to
cancellations.
Check out the website: http://www.grc.org/programs.aspx?id=16938
This is the golden age of cosmology. Once a philosophical subject, cosmology has burgeoned into a precision science as
ground and space-based astronomical observations supply a wealth of unprecedently precise cosmological measurements.Hong Kong Institute for Advanced Study
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