University of Sussex
MArch 10th, 2014
M-flation, Signatures
and Advantages
Amjad Ashoorioon (Lancaster University)
Based on
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]
A.A., B. Fung, R. B. Mann, M. Oltean, M. M. Sheikh-Jabbari, arXiv:1312.2284 [hep-th], to appear in JCAP
Introduction
Planck data strongly supports the idea of inflation
≤ 0.11 %95
which puts some favourite models like in trouble,
considering Bunch-Davies vacuum.
c.f. Ashoorioon, Dimopoulos, Sheikh-Jabbari & Shiu (2013)
Still any detection of ≥ 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
Lyth (1997)
Δ
/
> 1.06
0.01
o In supergravity and stringy models, one usually finds the size of the region in
which inflation can happen to be much smaller than M pl
McAllister & Baumann (2007)
I focus on M-flation that uses Matrices as inflaton.
Embedded preheating in some regions
high frequency gravitational waves.
• Gauged M-flation
4 × 3
N
C + 123 ij =
2 κˆ
ε ijk x k
3
D3
PP-wave background
parameterize 3 out
6 dim ⊥ to the D3-branes and
10-d IIB supergravity background
3
2
+
−
ds = 2 dx dx − mˆ
2
∑ (x )
i 2
8
( dx ) + ∑ dx K dx K
and five transverse to the D3-branes.
K =1

1
ig s
1
4
I
I
J
( 6)
µν 


d
x
STr
1
−
−
g
|
Q
|
+
X
,
X
C
−
F
F
ab
J
µν
3 4
2
∫
IJ 0123

(2π ) l s g s
4π l s
4


[
]
M, N = 0,1, ...,9
g ab = G MN ∂ a X M ∂ b X N
Q IJ = δ
•
x K denotes 3 spatial dim along
+ 2
i =1
S=
i , j = 1, 2 ,3
IJ
+
i
2π l
2
s
[X
I
,X
J
]
Myers (1999)
I , J = 4,5,..., 9
a , b = 0, 1, 2, 3
We assume there is a hierarchy between three of the extra-dimensions and the other three, so basically we
have deal with the large three of the dimensions perpendicular to the D3 branes.
Matrix Inflation from String Theory
4 g s2κˆ 2
the above background with constant dilaton is solution to the SUGRA
With mˆ =
9
2
V =−
1
(
4 2 π l s2
Upon the field redefinition
)
2
[X
Φi ≡
i
,X
j
λ = 8π g s
i
]+ 3 .ig2 π κˆl
s
,X
j
2
s
ε ijk X i [X j , X k ] +
1 2 2
mˆ X i
2
Xi
( 2π ) 3 g s l s2
 λ
V = Tr  −
Φi,Φ
 4
[
][X
j
][Φ , Φ ]
i
j
iκ
m2 2 
+
ε jkl [Φ k , Φ l ] Φ j +
Φ i 
3
2

κ = κˆ g s . 8π g s
From the brane-theory perspective, it is necessary to choose
mˆ 2 = m2
m̂
and κˆ
such that
4gs2κˆ 2
ˆ =
m
9
In JCAP 0906:018,2009, arXiv:0903.1481 [hep-th], we relaxed this condition and took , and 2
as independent parameters. In this talk, I will mainly focus on the SUSY case.
•
In the stringy picture, We have N D3-branes that are blown up into a single giant D5brane under the influence of RR 6-form. The inflaton corresponds to the radius of this
two sphere.
Truncation to the SU(2) Sector:
Φ i are N X N matrices and therefore we have 3N 2 scalars. It makes the analysis very
difficult
However from the specific form of the potential and since we have three Φ i , it is possible
to show that one can consistently restrict the classical dynamics to a sector with single
scalar field:
Φ i = φˆ ( t ) J i ,
i = 1, 2 ,3
J i are N dim. irreducible representation of the SU(2) algebra:
[J , J ] = iε
i
j
ijk
Jk
Tr (J i J
j
)=
N
N 2 − 1 δ ij
12
(
)
Plugging these to the action, we have:
2
M P

λ
κ
m
1
2
µ
2
4
3
S = ∫d x − g
R + Tr J  − ∂ µ φˆ ∂ φˆ − φˆ +
φˆ −
φˆ 2  
2
3
2
 2

 2
4
3
( ) ≡ ∑ Tr (J )
Tr J
2
2
i
i =1
1/ 2
Definingφ ≡ (Tr J 2 ) φˆ to make the kinetic term canonical, the potential takes the form
λeff
2κ eff
m2 2
φ −
φ +
φ
V0 (φ ) =
4
3
2
4
3
λeff ≡
2λ
8λ
=
,
Tr J 2 N(N 2 −1)
κeff ≡
κ
Tr J
2
=
2κ
2
N(N −1)
,
Consistency of the Truncation to the SU(2) Sector
• SU(2) sector is a sector in which the computations are tractable. But is it consistent?
To see that let us defines
φˆ =
Ψi = Φi −φ̂ Ji
4
Tr(Φi Ji )
N ( N 2 −1)
V = V 0 (φ ) + V ( 2 ) (φˆ , Ψ i )
If we start with the initial consitionsΨi
Tr (Ψi J i ) = 0
 δ V(2) 

 = 0
δ
Ψ
i Ψi =0

V( 2) (φˆ, Ψi = 0) = 0
= Ψ& i = 0
and φˆ ≠ 0 , Ψi will remain zero.
• What is the special role of SU(2) generators among other N X N matrices?
Tr (Γi Ξi ) = 0
Φ i = Γi − Ξ i
V = V0 ( Γi ) + V(1) ( Γi , Ξ i )
[
[
] ]
V(1) = Tr (- λ [Γi ,[Γi ,Γk ]] + i εijk Γi ,Γ j ) Ξk + Ο (Γ 2 )
To have Γ i -sector decoupled
a) fijk = i εijk
[Γ ,Γ ] = f
i
j
ijk
Three Γ i should form a Lie-Algebra
Γk
Γi are forming a SU ( 2) algebra Φi = ∑φα Jiα ,
α
b) f ijk = 0
Γi are three Abelian subgroups of U(N)
i = 1,2,3 N = ∑ Nα
A.A., H. Firouzjahi, M.M. Sheikh-Jabbari,
arXiv:0911.4284 [hep-th
α
No interesting inflationary dynamics.
Analysis of the Gauged M-flation around the Single-Block Vacuum
V (φ ) =
λ eff
4
φ 2 (φ − µ ) 2
µ≡
2m
λ eff
(a)
Hill-top or Symmetry-Breaking
inflation, Linde (1992)
Lyth & Boubekeur (2005)
(a) φ i > µ
(c)
φi ≈ 43.5 M P
φ f ≈ 27 . 1 M P
λeff ≈ 4.9 × 10−14
µ ≈ 26 M P
(b)
λ ≈1
(b) µ / 2 < φi < µ
(c)
0 < ! <
φi ≈ 23.5 M P
φf ≈ 35 MP
λeff ≈ 7.2 × 10−14
µ ≈ 36MP
"
2
N ≈ 5×104
∆φ ≤ 10−6 M p
φi ≈ 12.5 M P
λeff ≈ 7.2 × 10−14
φf ≈ 1 MP
µ ≈ 36MP
Mass Spectrum of Spectators
We initially have 3N 2 and 4 N 2 gauge fields. Truncating to the SU(2) sector and using the
EOM and the gauge symmetry of the action leaves us with 5 N 2 − 1 isocurvature modes,
or “spectators”.
The other 5 N 2 − 1 even though classically frozen, have quantum fluctuations. To compute
these effects, let us calculate the mass spectrum of these modes.
• Spectrum of scalar spectators
Expanding the action up to second order,
V(2)
where
we have:
 λ ˆ2

m2
 λ

= Tr  φ Ω i Ω i +
Ψ i Ψ i +  − φˆ 2 + κ φˆ  Ψ i Ω i 
2
 2

2

[
Ωk ≡i εijk Ji , Ψj
]
If we have the eigenvectors of the Ω i
Ω i = ω Ψi
It turns out that
 λ eff 2 2
m2 
 Tr Ψ i Ψ i
V 2 = 
φ (ω − ω ) + κ eff ωφ +
4
2


finding the eigenvectors of Ω i is mathematically
the vector spherical harmonics:
the same as finding the
Dasgupta, Sheikh-Jabbari &
Von Raamsdonk (2002)
Mass Spectrum of Spectators
(a) & ' 1 zero modes with ω = −1
M 2 = λ eff φ 2 − 2κ eff φ + m 2 =
V′
φ
These modes are unphysical as they correspond to gauge transformation over the background
solution
To see that, recall under an infinitesimal gauge transformation Φ i → Φ i + ig[Φ i , Λ ]
where Λ is an arbitrary traceless Hermitian matrix.
$-modes with ω = −(l + 2),
(b) ( N − 1) 2
l ∈Ζ 0 ≤ l ≤ N − 2
1
M = λ eff ( l + 2 )( l + 3)φ 2 − 2κ eff ( l + 2) + m 2
2
2
l
Degeneracy of each
l -mode is 2 l + 1
l = 0 α − mode is nothing more than the adiabatic mode. Therefore we have & ' 1
isocurvature α − mode.
(c) & ( 1 ' 1 %-modes with
ω = l −1,
l ∈Ζ 1≤ l ≤ N
Degeneracy of each
l-mode is 2 l + 1
1
M = λ eff ( l − 2 )( l − 1)φ 2 + 2κ eff ( l − 1) + m 2
2
2
l
'1
Mass Spectrum of Spectators
• Spectrum of scalar spectators
Expanding the action in gauge fields up to second order,
we can read the mass spectrum, solving for the eigenvalue problem
with degeneracy
which has eigenvalues
for each mode. Therefore we
we have a system of vector fields with the mass parameter
is massless and corresponds to the
sector in the
matrices and has two
polarizations. Other vector field modes are massive and have three d.o.f each. Therefore
in total we have 3& ' 1 vector d.o.f.
Mass Spectrum of χ Spectators
(a) ( N − 1) 2 - 1 α -modes
M α2 ,l =
l ∈Ζ 0 ≤ l ≤ N − 2
1
λeff (l + 2)(l + 3)φ 2 − 2κ eff (l + 2) + m 2
2
(b) ( N + 1) 2 - 1 β -modes
M β2 ,l =
l ∈Ζ 1≤ l ≤ N
1
λeff (l − 2 )(l − 1)φ 2 + 2κ eff (l − 1) + m 2
2
(c) 3N 2 − 1 vector modes
M A2 ,l =
[( N − 1)
][
] [
λ eff
4
*
φ 2 l (l + 1)
]
2
− 1 + ( N + 1) 2 − 1 + 3 N 2 − 1 = 5N − 1
α − modes β − modes vector - field modes
2
)
Ψr,lm Modes
Power Spectra in the Presence of
1
1
1
L = − ∂ µφ ∂ µφ − ∂ µ Ψ ∗ r,lm ∂ µ Ψr,lm - V 0 (φ ) − M r2,lm (φ ) Ψ ∗ r,lm Ψr,lm
2
2
2
r =α,β,A
& =0 , they remain zero. Therefore the
If you start from the initial condition Ψr,lm =Ψ
r,lm
inflationary trajectory is a straight line in the field space and there is no cross-correlation
between adiabatic and entropy spectra.
Mukhanov-Sasaki

k
1

Q&&φ + 3 H Q& φ + 2 Q φ +  V 0 ,φφ − 3
a
a M P2


2

k
2
&&
&
+
M
(
)
δΨ
φ
r , lm + 3 H δ Ψ r , lm + 
r
,
l
 a2

2
variable
.
 a &2  

φ   Q φ = 0 ; Q ≡ δφ +
φ
 H
 

H

ℜ
=
Qφ
S r ,lm =
 δ Ψ r,lm = 0
φ&

3
H k
ℜ& =
H& a
k'
) Q ∗φ Ψr,lm = 0
k'
PΨr ,lm
Φ
H
Ψ
φ& r ,lm
k3 3
= 2 δ ( − ) Ψ∗r ,lm Ψ∗r ,lm
2π
kkkk
Q
k3 3
=
δ ( −
2π 2
2
H
scalar metric perturbations
in longitudinal gauge
Φ
k'
kkkk
Qφ
k'
kkkk
Cψ i Q
φ
k'
)Q
*
kkkk
k'
kkkk
k3 3
PQφ =
δ ( −
2
2π
2
φ&
(a) Power Spectra in Symmetry-Breaking Inflation φ > µ
nℜ ≈ 0.96
µ ≈ 26 M P
λeff ≈ 4.9 × 10−14
PT (k 60) ≈ 4.8 ×10−10
l =1
β
m
2
/01,2
1.1
−12
×10
0.978
3
/3
r ≈ 0.2
n T ≈ − 0 . 025
≃ 4.7 × 1056
This region of parameter space is ruled out with Planck
if one considers Bunch-Davies (BD) vacuum.
•
It is possible to reconcile this model and other high energy models of inflation with
Planck considering non-BD vacua for tensor and scalar fluctuations, with ≃ few × 10)
Ashoorioon, Dimopoulos, Sheikh-Jabbari & Shiu, arXiv:1306.4914
µ /2 <φ < µ
Power Spectra in Symmetry-Breaking Inflation
λeff ≈ 7.2 × 10−14
l =1
β
m
2
6.6
×10−16
1 .05
µ ≈ 36 M P
3
nℜ ≈ 0.961
≃ 0.048
PT ( k 60) ≈ 1.3 × 10 −11
/01,2
/3
n T ≈ − 0 . 006
≃ 2.7 × 1058
CMBPOL or QUIET
should be able to verify
this scenario.
Gauged M-flation
C) Power Spectra in Symmetry-Breaking Inflation 0 < φ < µ / 2
λeff ≈ 7.2 × 10−14 & µ ≈ 36 M P
2
l = 1 6λeff φ − 1.2
α
6κ eff φ
×10−11
nℜ ≈ 0.96 & Pℜ ≈ 2 ×10−9
PSα ,1
0 .95
3
PR
*
= 5.12 ×10 −3
+ m2
)
r ≈ 0.048
9 : ≃ '0.006
CMBPOL or QUIET
should be able to verify
this scenario.
Gauged M-flation
Particle Creation and Preheating Scenario around ; = = vacuum
The backreaction of the spectator modes on the inflaton dynamics can become
large when ε ,η ≈ 1
• This could be the bonus of our model, as spectator modes help to drain the energy
of the inflaton, since their masses change very fast.
• One can show that if inflation ends in the susy-breaking vacuum, this process is
not effective to produce spectator particles through parametric resonance:
M α2 ,β
φ =µ
=
λeff µ 2
2
(ω + 1) 2
ωα = −(l + 2)
ωβ = (l − 1)
M
2
A φ =µ
• For$ and % modes:
χ&&k + 3Hχ& k + Ω 2k χ k = 0
• For the gauge mode
&& + HA& + Ω 2 A = 0
A
k
k
k k
for example for α and β modes:
k2
2
Ω k = 2 + M χ2 + g 3ϕ + g 42ϕ 2
a
∀?,
&
Ω
k
Ω 2k
@ ≡'"
<< 1
φ ≈µ
2
4
g =
=
λeff µ 2
4
l (l + 1)
λeff (ω 2 − ω )
2
rest masses
are large around
susy-breaking
vacuum.
g3 =
λeff µ
2
(2ω 2 + ω )
No parametric resonance around
the susy-breaking vacuum
Particle Creation and Preheating Scenario around ; = B
•
The situation is quite different around the SUSY vacuum
2
M α ,β
φ =0
M A2
•
=
φ =0
λeff µ 2
2
=0
For large values of ω for α and β modes and for all values of C for the gauge
modes
&
Ω
k
Ω 2k
>> 1
parametric resonance happens.
φ ≈0
•
The mass of $-modes and %-modes become tachyonic for ℓ > ℓEFG , where ℓH I!J = 94 and ℓK I!J = 16.
•
We have to find the corrections up to quartic order which stabilizes this instability
(N)
L* = ∫ Q 6 R 'S{'U* Tr(X N )}
(6)
L* = ∫ Q 6 R 'S{'Λ* Tr(X 6 )}
U*
]^^
≪ Λ* ≃ 1.0069 × 10
[
4
χ&&k + 3Hχ& k + Ω 2k χ k + 4Λ χ χ k3 = 0
&& + HA& + Ω 2 A = 0
A
k
k
k k
•
X l = a 3 / 2 χ l & A l = a1 / 2 Al
X l′′ + Ωω2 X l +
Ωω2 ≡
a 3µ 2
λeff
2
t
'≡
&
t ′→0
Ω
nl = ω
2
l
=
&
l2 ≡
2k 2
λeff µ 2
=0
exp( −iΩω t ′)
2Ω ω
 µ 2λ X l′
1
2


+
X
−
l
 2 Ωω2
 2


2
d
dt ′
A l′′ + Ωl2 A l = 0
l2 ϕ 2 2
1 a′2 a′′
+
(
l
+
l
)
+
−
a 2 2µ 2
4 a 2 2a
exp( −iΩl t ′)
Al =
lim
2Ω l
t ′→0
Ωl2 ≡
l2 ϕ 2 2
3ϕω
3 a′2 3 a′′
+
(
−
)
+
+
1
−
−
ω
ω
a2 µ 2
µ
4 a2 2 a
lim X
ω
2qX l3
& t′ ≡ µ
Ω
n = l
 2

l
l
 µ 2λ Al′ 2
1  1
2


+
A
−
l
 2  a2
 2 Ω l2



GW production from Preheating
• Parametric resonance could be a source of gravitational waves.
• Exponential particle production for some momenta
2
&

a
&h& − 2  + 2 a&& h + 3a& h& − 1 ∇2h = 16π Gδ S TT
ij
ij
2
 a2 a  ij a ij a ij
a


π k3
dΩGW
1 dρ
=
=
d ln k ρ crit d ln k 3H 2 L2
large inhomogeneities
δ S ij = δ T ij −
where
∑ hij,0 (k )
δ ij
3
Tkk
2
i, j
• This is in addition to the stochastic background of GW produced during inflation
• Such GW is a probe of the inflaton potential and its couplings at the end of inflation.
• Universe is transparent to GW
useful source of information from early universe.
9_
10 7
Largest j Gauge modes
10 4
ℓ=0
10
0.01
10 -5
20
40
60
80
100
120
140
`′
9_
1021
1016
1011
Largest j beta mode with
quartic interaction
106
10
ℓ=0
10-4
0
5
10
15
20
`′
GW production from Preheating: Single Mode
• We used HLattice (developed by Zhiqi Huang (2007)) to compute the GW
spectrum produced by individual highest j modes as the preheat field
GW production from Preheating: Single Mode
The gravitational wave from
the gauge modes dominates
over the ones from $ and %
modes.
GW production from Preheating: More large j Gauge Mode
Time Evolution:
• Linear Period: while inflaton oscillates coherently around its minimum, the effect
of multi-preheat modes is larger than a single mode.
• Non-Linear Period: inhomogeneities of the inflaton grow, gravitational radiation is
counteracted by the backreaction. Nonlinear effects suppress the degeneracy
effects.
Frequnecy Dependence:
• Our current data already shows that the GWs of our model are in the 1−3 GHz band
and they are almost flat with amplitudes around 105b .
• The signal may be seen in Birmingham HFGW resonant antenna or the one at
Chongqin University
Conclusions
• M-flation solves the fine-tunings associated with chaotic inflation couplings and
produce super-Planckian effective field excursions during inflation.
• M-flation which is qualitatively new third venue within string theory inflationary
model-building using the internal matrix degrees of freedom.
• Matrix nature of the fields suggests isocurvature productions at the CMB scales.
• Hierarchical mass structure of the isocurvature modes, one can avoid the
“beyond-the-cutoff” problem.
A.A., M.M. Sheikh-Jabbari, JCAP 1106 (2011) 014, arXiv:1101.0048 [hep-th]
Conclusions
• Interactions of the graviton with the scalar field
•
cd
g
d
ef
'problem if Λ = In many-field models like M-flation, the problem can be avoided
Λ=
&h
Ashoorioon, Danielsson, Sheikh-Jabbari,
Phys.Lett. B713 (2012)
• M-flation has a natural built-in mechanism of preheating around the SUSY vacuum.
• The couplings of the preheat fields are related to self couplings of inflaton, thus known.
• The parametric resonance produces large GHz frequency GW which could be seen
by ultra-high frequency gravitational probes like Birmingham or the one at Chognqing
University.
• Other signatures in this inflationary region:
1. Observable GW at cosmological scales with = 0.048.
2. Iscocurvature perturbations with
[i
[j
≃ 5 × 105N .
Thank you