Cosmo 2014
Chicago, IL
August 25th, 2014
M-flation after BICEP2
Amjad Ashoorioon (Lancaster University)
Mainly in collaboration with
Shahin Sheikh-Jabbari (IPM)
Based on
A.A, M.M. Sheikh-Jabbari, arXiv:1405.1685 [hep-th]]
and
A.A., H. Firouzjahi, M.M. Sheikh-Jabbari JCAP 0906:018,2009, arXiv:0903.1481 [hep-th],
A.A., H. Firouzjahi, M.M. Sheikh-Jabbari JCAP 1005 (2010) 002, arXiv:0911.4284 [hep-th]
A.A.,M.M. Sheikh-Jabbari, JCAP 1106 (2011) 014, arXiv:1101.0048 [hep-th]
A.A., U.Danielsson, M. M. Sheikh-Jabbari, Phys.Lett. B713 (2012) 353, arXiv:1112.2272 [hep-th]
Introduction
The increasingly precise CMB measurements by Planck mission in combination
with other cosmological date have ushered us into a precision early Universe
cosmology era:
𝑑 log 𝑃𝑆
𝑛𝑠 ≡
+ 1 = 0.9603 ± 0.0073
Planck 2013
𝑑 log 𝑘
𝑟 ≤ 0.11;
Introduction
 BICEP2 surprise: claims that have observed the B-modes with 𝑟 = 0.2+0.07
−0.05 at
ℓ ≃ 80.
 Detection of 𝑟 ≥ 0.01 poses theoretical model-building challenges:
o To embed such a model in supergravity, one has to insure the flatness of the
theory on scales
Δ𝜙
𝑟 1/2
Lyth (1997)
≥ 1.06
𝑀𝑝𝑙
0.01
o In stringy models, due to geometric origin of inflation in higher dimensions, Δ𝜙 ≲
𝑀𝑝𝑙 .
McAllister & Baumann (2007)
 From Planck experiment: 𝑟 ≤ 0.11 at 𝑘∗ = 0.002 𝑀𝑝𝑐 −1 , ℓ ≃ 28
 A priori these two experiments are not mutually-exclusive and can be reconciled
A.A., K. Dimopoulos, M.M. Sheikh-Jabbari, G. Shiu, JCAP 1402 (2014) 025, arXiv:1306.4914
A.A., K. Dimopoulos, M.M. Sheikh-Jabbari, G. Shiu, arXiv:1403.6099 [hep-th]], to appear in PLB
Realization of Large-Field Models in String Theory
 Single-Field approach (aka Individualistic approach!):
o An individual axionic field, whose potential is shift symmetric. in presence of fluxes
spirals super-Planckian distances
Monodromy Inflation
Silverstein & Westphal (2008)
McAllister, Silverstein, Westphal
(2009)
 Many Field approach (aka Socialistic approach!):
See Gary’s and Eva’s
Talks
o Many moduli, which could be axions or not, cooperate to cause inflation.
o Even though the effective field excursion is larger than 𝑀𝑝𝑙 , individual field displacement
is less!
N-flation, Kachru et. al (2006)
M-flation, Ashoorioon & Sheikh-Jabbari (2009)
Multiple M5 brane Inflation, A. Krause, M. Becker, K, Becker (2005)
A. Ashoorioon & A. Krause (2006)
• Gauged M-flation
N
𝑅4 × 𝐶𝑌3
2 ˆ
C  123 ij 
 ijk x k
3
D3
PP-wave background
i , j  1, 2 , 3 parameterize 3 out
6 dim to the D3-branes and
10-d IIB supergravity background
ds
S 
2

 2 dx dx

g ab  G MN  a X
M
b X
8
i 1
K 1
K
x denotes 3 spatial dim along
and five transverse to the D3-branes.
2
i 2
 2
 mˆ  ( x ) ( dx )   dx K dx K

4
1 
d
x
STr
3 4


( 2 ) l s g s

1
3
 g ab
N
| QJ | 
I
ig s
4 l s
2
X
I
,X
J
C
Q

IJ

i
2 l s
2
X
IJ 0 12 3

1
4
F F





Myers (1999)
I , J  4 ,5 ,..., 9
M , N  0 , 1, ..., 9
IJ
(6)
I
,X
J

a , b  0 , 1, 2, 3
Matrix Inflation from String Theory
4 g s2ˆ 2 the above background with constant dilaton is solution to the SUGRA
With mˆ 
9
2
V  
1

4 2 l s
2

2
Upon the field redefinition
X
,X
i
i 
  8 g s
,X
i
j

ig s ˆ
3 . 2 l
2
s
 ijk X i  X j , X k  
1
2
ˆ 2 X i2
m
Xi
( 2 ) g s l s
3
 
V  Tr  
i,
 4

j
 X
j

2
,
i
j

i

3
  ˆ g s . 8 g s
 k ,  l  
jkl
j

2 
 i 
2

m
2
2
2
mˆ  m
From the brane-theory perspective, it is necessary to choose mˆ and ˆ such that
mˆ 
2
2 2
4 g s ˆ
9
N D3-branes are blown up into a single giant D5-brane under the influence of RR
6-form. The inflaton corresponds to the radius of this two sphere.
Truncation to the SU(2) Sector:
i
are N X N matrices and therefore we have 3 N 2 scalars. It makes the analysis very
difficult
However, one may show that there is a consistent classical truncation to a sector with
single scalar field:
 i  ˆ ( t ) J i ,
i  1, 2 , 3
J i are N dim. irreducible representation of the SU(2) algebra:
J
i
,J
j
  i
ijk
Jk
Tr  J i J
j

N
12
N

 1  ij
2
Plugging these to the action, we have:
S 
d x
4
Defining
M P
g
R  Tr J
2

  Tr J
V 0 ( ) 
 eff
4

2 1/ 2
 
4
ˆ
2 eff
3
2
2
 1
 ˆ 4 2 ˆ 3 m ˆ 2  
 ˆ
ˆ
        
 
  
 2
2
3
2


3

Tr J
2
   Tr  J 
2
i
i 1
to make the kinetic term canonical, the potential takes the form
 
3
m
2
2

2
 eff 
2
Tr J
2

8
N ( N  1)
2
,
 eff 

Tr J

2
2
N ( N  1)
2
,
Analysis of the Gauged M-flation around the Single-Block Vacuum
V ( ) 
 eff
 (   )
2
4
2
 
2m
 eff
Hill-top or Symmetry-Breaking
inflation, Linde (1992)
Lyth & Boubekeur (2005)
In the stringy picture, we have N D3-branes that are
blown up into a giant D5-brane under the influence of RR
6-form.
 1
(a)
N  5  10
(c)
4
(b)
   10
6
M
p
Mass Spectrum of  Spectators
(a) ( N  1) 2 - 1  -modes
1
M  ,l 
2
2
l
M
 ,l

Degeneracy of each
l -mode is 2 l  1
 eff ( l  2 )( l  3 )  2  eff ( l  2 )  m
2
(b) ( N  1) 2 - 1  -modes
2
0l N 2
1
2
l
2
1 l  N
 eff ( l  2 )( l  1)  2  eff ( l  1)  m
2
Degeneracy of each
2
l-mode is 2 l  1
(c) 3N 2  1 vector modes
M
( N  1)

2
A ,l

𝑚χ2
 eff
Degeneracy of each
 l ( l  1)
2
l-mode is 2 l  1
4
 

 1  ( N  1) 2  1  3 N 2  1  5 N  1
  modes   modes vector - field modes
2
2
𝐻2
Solving the model parameters based on Observables
2
𝑀𝑝𝑙
𝑁𝑒
=
1 𝑉′
𝜖≡
2 𝑉
𝜙𝑖
𝜙𝑓
𝑑𝜙 𝑉(𝜙) 1
𝜇2
𝜇2 + 4𝑦
= 𝑦−𝑥 −
ln 2
𝑉′(𝜙)
8
32
𝜇 + 4𝑥
2
=1
𝑛𝑆 − 1 = 2𝜂 − 6𝜖
𝜙𝑖
2 ′′
𝑀𝑝𝑙
𝑉
𝜂≡
2𝑉
2
2
𝑥 = 4𝑀𝑝𝑙
+ 𝑀𝑝𝑙 16𝑀𝑝𝑙
+ 2𝜇2
𝑦
2 =
𝑀𝑝𝑙
(1)
𝑥 ≡ 𝜙𝑓 𝜙𝑓 − 𝜇
𝑦 ≡ 𝜙𝑖 𝜙𝑖 − 𝜇
(2)
2
12 + 144 + 8 1 − 𝑛𝑆 𝜇2 /𝑀𝑝𝑙
1 − 𝑛𝑆
(3)
Plugging (2) and (3) in (1) one can find solve 𝜇/𝑀𝑝𝑙 in terms of 𝑛𝑆 numerically.
𝐴𝑆 =
𝑉(𝜙𝑖 )
≃ 2.195 × 10−9
4
2
24𝜋 𝑀𝑝𝑙 𝜖(𝜙𝑖 )
One can read off 𝜆𝑒𝑓𝑓
(a) Symmetry-Breaking Region   
 For 𝑛𝑆 = 0.9603, and 𝑁𝑒 = 60
𝑟 = 0.1991 ≃ 0.2
Right at the BICEP2
sweet spot
 From Planck experiment, within 2𝜎
0.9457 ≤ 𝑛𝑆 ≤ 0.9749
However not all this interval is covered by this branch of the model!
if 𝜇 → 0, 𝑛𝑆 𝑚𝑖𝑛 = 𝑛𝑆
𝑞𝑢𝑎𝑟𝑡𝑖𝑐
if 𝜇 → ∞, 𝑛𝑆 𝑚𝑎𝑥 = 𝑛𝑆
0.9457 ≤ 𝑛𝑆
60
(𝑁𝑒 = 60) =
𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
58
≃ 0.9508
61
(𝑁𝑒 = 60) =
≤ 0.9749
0.1322 ≤ 𝑟60 ≤ 0.2623
 If 𝜎 𝑛𝑆 = 0.0029 as promised by CMBPOL
𝑟 ∈ [0.1983,0.2204]
117
≃ 0.9669
121
(a) Symmetry-Breaking Region   
 Spectra of the Isocurvature modes:
o The lightest mode is 𝑗 = 0 gauge mode. For 𝑁𝑒 = 60
≲ 1.64 × 10−2 (𝜇 → 0)
8.27 × 10−3 (𝜇 → ∞) ≲
For 𝑛𝑆 = 0.9603,
= 1.24 × 10−2
𝑗 = 0, is the massless mode 𝑈 1 ⊂ 𝑈 𝑁
generates cosmic magnetic fields?!
seed for dynamo mechanism that
Hilltop Regions (b) and (c)
 
 Due to symmetry 𝜙 → 𝜙 − 𝜇 at the level of background these two regions predict
the same
 For 𝑛𝑆 = 0.9603, and 𝑁𝑒 = 60
𝑟 = 0.0379
 From Planck experiment, within 2𝜎
117
0.9457 ≤ 𝑛𝑆 ≤ 121 ≃ 0.9669 (when 𝜇 → ∞)
0.0155 ≤ 𝑟60 ≤ 0.1322
𝜆𝑒𝑓𝑓
60
≲ 7.9948 × 10−14
 If 𝜎 𝑛𝑆 = 0.0029 as promised by CMBPOL
25.43 𝑀𝑝𝑙 ≲ 𝜇60
𝑟 ∈ [0.0310,0.0475]
Hilltop Regions (b) and (c)  / 2  
 
&
0    /2
 Symmetry 𝜙 → 𝜙 − 𝜇 breaks down at the quantum level.
• In region (b), the lightest mode is 𝑗 = 0 gauge mode
9.83 × 10−4 ≲
≲ 8.27 × 10−3 (𝜇 → ∞)
• In region (c), the lightest mode is 𝑗 = 1 𝛼 −mode
2.84 × 10−4 ≲
≲ 2.91 × 10−2 (𝜇 → ∞)
 Around 𝜙 = 0, the isocurvature modes can act as preheat fields. The couplings
of preheat fields to the inflaton are known.
Ω𝐺𝑊 ℎ2 ∝ 10−16 at the peak frequency 1 𝐺𝐻𝑧
Which can be observed at Chongqin HFGW detector
or Birmingham HFGW experiment.
Conclusions & Future Directions
• M-flation solves the fine-tunings associated with chaotic inflation couplings.
• It produces super-Planckian effective field excursions from many individual subPlanckian ones which yield large tensor/scalar ratio compatible with Planck.
• M-flation which is qualitatively new third venue within string theory inflationary
model-building.
• Matrix nature of the fields results in the production of isocurvature productions at the
CMB scales.
• Due to hierarchical mass structure of the isocurvature modes, one can avoid the
𝑀𝑝𝑙
“beyond-the-cutoff” problem, exists in N-flation, even if Λ = 𝑁
A.A., M.M. Sheikh-Jabbari, JCAP 1106 (2011) 014, arXiv:1101.0048 [hep-th]
Conclusions & Future Directions
• The loop corrections from interactions of the graviton with the scalar field create the
term
Λ2
𝑀𝑝2
𝑅𝜙 2 , if Λ = 𝑀𝑝𝑙 . In M-flation and many field models such induced terms is
naturally suppressed.
A.A., U.Danielsson, M. M. Sheikh-Jabbari, Phys.Lett. B713 (2012) 353, arXiv:1112.2272 [hep-th]
• M-flation has a natural built-in mechanism of preheating to end inflation around the
𝜙 = 0 vacuum which can produces large GHz frequency gravitational wave spectrum
which could be seen by ultra-high frequency gravitational probes.
• Open Issue I: Reheating around the 𝜙 = 𝜇
• Open Issue II: Building a full-fledged stringy setup with all moduli fixed.
Works in progress
Thank you