Matrix Inflation

Amjad Ashoorioon (Uppsala University)In collaboration with

Hassan Firouzjahi (IPM)

&

Shahin Sheikh-Jabbari (IPM)

Based on

JCAP 0906:018,2009, arXiv:0903.1481 [hep-th]

and JCAP 1005:002,2010, arXiv:0911.4284 [hep-th]

October 21st, 2010

Non-thermal Cosmology WorkshopAnn Arbor, MI

Introduction

• WMAP7 strongly supports the idea of inflation as the theory of early universe and

structure formation.

• Inflation is a paradigm and one can construct many inflationary models compatible

with the current data.

• In its simplest form, inflation is driven by a scalar field minimally coupled to gravity.

• There has been lots of efforts to realize inflation in string theory, mainly using D-Dbar

interactions in a warped throat.KKLMMT (2003)

• On the other hand, general compactifications with fluxes naturally posses several

branes to satisfy constraints like tadpole cancellation.

• In these theories, more than one scalar field is present.

• In multi-field inflationary models, one can usually perform a rotation in the field space,

where the inflaton field evolves along the trajectory and the other fields are orthogonal

to it.Gordon, Wands, Bassett & Maartens (2001)

and their commutators .

Introduction

• In this talk, we promote the inflaton fields to general N X N hermitian matrices, and

hence the name Matrix Inflation, or M-flation for brevity.

• Working with matrices, we are able to use their commutators in the potential, besides

their simple products.

• In our class of Matrix inflation models, we consider three N X N matrices, 3 ,2 ,1 , =Φ ii

We consider the potential which is quadratic in iΦ [ ]ji ΦΦ ,

Therefore, we have three types of terms in the potential: [ ] 2 ,Tr ji ΦΦ , [ ]kjiijkε ΦΦΦ ,Tr ,

2Tr iΦ

• As we will see, the model is motivated from string theory and brane dynamics.

• Despite its simple form, Matrix inflation has a rich dynamics:

- It can solve the fine-tuning associated with standard chaotic inflationary models.

- Besides the adiabatic perturbations, we have isocurvature ones.

- The model has an embedded preheating mechanism that uses the isocurvature fields

as preheat fields.

Outline

• Matrix Inflation Setup

• Matrix Inflation from String Theory

• Truncation to the SU(2) Sector

• Consistency of the Truncation to the SU(2) Sector

• Analysis of the Matrix Inflation around the Single-Block Vacuum

• Mass and power spectra of isocurvature modes in Matrix Inflation

• Particle Creation and Preheat Scenario

• Analysis of the potential for n-block vacua

• Conclusion

• UV behavior of Matrix Inflation in presence of many species

Matrix Inflation Setup

( ) [ ]( )∫ ∑

ΦΦΦ−Φ∂Φ∂−−= jiiii

i

P VRM

gxdS ,, Tr2

1

2

4 µµ

As we will show momentarily, the full potential can be motivated from dynamics of branes

In string theory and takes the form:

[ ][ ] [ ]

Φ+ΦΦΦ+ΦΦΦΦ−= 2

2

2 ,

3, ,

4Tr ijlkjkljiji

miV ε

κλ

The potential is invariant under )( NU group (acting on the matrices) and )2(SU group

acting on ji , indices.

( ) [ ]( )

ΦΦΦ+Φ∂Φ∂−= ,,Tr

2

1

3

12

2

jiiii

P

VM

Hµ

µ

The EOMs are:

[ ][ ] [ ] 0,, ,32 =Φ+ΦΦ+ΦΦΦ+Φ+Φ lkjljkjljll miH κελ&&&

∑ Φ∂Φ∂−=i

ii

PMH µ

µTr2

12

&

Matrix Inflation from String Theory

In the context of string theory, it is known that the world-volume of N coincident

3-branes is described by supersymmetric U(N) gauge theory. In this system the

transverse position of the branes, are scalars in the adjoint representation

of U(N), and hence N X N matrices. The DBI action for the system in the background

of RR six form flux (sourced by distribution of D5 branes) is given by:

9,...,4 , =Φ II

[ ]∫

+−−= )6(

2

4

43 0123 ,

4||1STr

)2(

1IJ

CXXl

igQgxd

glS

JI

s

sI

Jab

ss ππMyers (1999)

N

b

M

aMNab XXGg ∂∂= 9 ..., ,1 ,0, =NM9,...,5,4, =JI

3 2, 1, ,0, =ba

[ ]JI

s

IJIJXX

l

iQ ,

22π

δ +=

We consider the the 10-d IIB supergravity background:

∑∑==

+−+ +−=8

1

3

1

2222)()(ˆ2

K

KK

i

idxdxdxxmdxdxds

k

ijkij xC εκ

3

ˆ2123

=+

3,2,1, =ji parameterize 3 out

6 dim ⊥ to the D3-branes andK

x denotes 3 spatial dim along

and five transverse to D3-branes.

Matrix Inflation from String Theory

With9

ˆ4ˆ

222 κsg

m = the above background with constant dilaton is solution to the SUGRA

EOM. We compactify the transverse dimensions on a 6d CY manifold with two 3d

cycles, one of which is very long and the other one is quite small. In the

light-cone gauge on the D3-branes, expanding the action up to fourth order in yields

( )∫

−∂∂−= iii

ss

XVXXxdgl

Sµ

µπ 2

1Tr

)2(

1 4

43

( )[ ][ ] [ ] 22

222ˆ

2

1,

2.3

ˆ, ,

24

1ikji

ijk

s

sjiji

s

XmXXXl

igXXXX

lV ++−= ε

π

κ

π

23)2(

ss

ii

lg

X

π≡ΦUpon the field redefinition

[ ][ ] [ ]

Φ+ΦΦΦ+ΦΦΦΦ−= 2

2

2 ,

3, ,

4Tr ijlkjkljiji

miV ε

κλ

sgπλ 2= ss gg 2. ˆ πκκ = 22ˆ mm =

From the brane-theory perspective, it is necessary to choose m and κ such that

9

ˆ4ˆ

222 κsg

m = . However we may also relax this condition and take κλ , and2m as

IX

independent parameters.

Truncation to the SU(2) Sector:

iΦ are N X N matrices and therefore we have 23N scalars. It makes the analysis very

difficult

However from the specific form of the potential and since we have three iΦ , it is possible

to show that one can consistently restrict the classical dynamics to a sector with single

scalar field:3,2,1 ,)(ˆ ==Φ iJt ii φ

iJ are N dim. irreducible representation of the SU(2) algebra:

[ ] kijkji JiJJ ε=, ( ) ( )ijji N

NJJ δ 1

12Tr

2 −=

Plugging these to the action, we have:

∫

−+−∂∂−+−= 2

23424 ˆ

2ˆ

3

2ˆ2

ˆ ˆ2

1Tr

2φφ

κφ

λφφ µ

µ

mJR

MgxdS P

( ) φφ ˆTr 2/1 2

J≡Defining to make the kinetic term canonical, the potential takes the form

22

34

023

2

4)( φφ

κφ

λφ

mV

effeff+−= ,

)1(

2

Tr ,

)1(

8

Tr

2

2222

−=≡

−=≡

NNJNNJeffeff

κκκκκκκκκκκκ

λλλλλλλλλλλλ

( ) ( )∑=

≡3

1

22TrTr

i

iJJ

are forming a

Three should form a Lie-Algebra

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

iii J φ−Φ=Ψ ( )iiJNN

Φ−

= Tr)1(

4ˆ2

φ ( ) 0Tr =Ψ ii J

),ˆ()()2(0 iVVV Ψ+= φφ 0)0,ˆ()2( ==ΨiV φ 0

0

)2(=

Ψ=Ψi

i

V

δ

δ

If we start with the initial consitions 0=Ψ=Ψ ii& and 0ˆ ≠φ , iΨ will remain zero.

• What is the special role of SU(2) generators among other N X N matrices?

iii Ξ−Γ=Φ ( ) 0Tr =ΞΓ ii

),()()1(0 iii VVV ΞΓ+Γ=

[ ][ ] [ ]( )[ ] )Γ( ΓΓ Γ ΓΓ -Tr 2

)1( Ο,εi,,V kjiijkkii +Ξ+= λ

To haveiΓ -sector decoupled [ ] kijkji f, ΓΓΓ =

iΓ

ijkijk if ε =a)

b) 0=ijk

f

)2(SUiΓ algebra

iΓ are three Abelian subgroups of U(N)No interesting inflationarydynamics

3,2,1 , ==Φ ∑ iJii

α

ααφ ∑=

ααNN

22eff )(4

)( µφφλ

φ −=Veff

0

2

λµφ

m≡=

(a) µφ >i

Hill-top or Symmetry-Breaking

inflation, Linde (1992)Lyth & Boubekeur (2005)

Pi M 57.43≈φ Pf M 07.27≈φ PM 26≈µ

(b) µφµ << i2/

Pi M 5.23≈φ Pf M 03.35≈φ PM 36≈µ

141091.4

−×≈eff

λ

(a)

(b)

PMm 1007.46−×≈

Peff M 1057.913−×≈κ

141018.7

−×≈eff

λPMm 1082.6

6−×≈ Peff M 1057.913−×≈κ

Note:This is exactly the same condition we had to satisfy to

have our 10-d background be a supergravity solution.

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.

(c)

(c) 2/0 µφ << i

Due to symmetry µφφ +−→ this inflationary region has the same properties as µφµ << i2/

PM 10ˆ 7−≈∆φ

510≈N

Analysis of the Matrix Inflation around the Single-Block Vacuum

ϕ

V(ϕ)

To fit the WMAP data

Chaotic Inflationary scenarios with

a) 22

2

1φmV =

PM 10≈∆φFrom EFT perspective

super-Planckian excursions areproblematic!

φφφφφφφφ ∆≈∆ − 2/3 ˆ N If 1>>N-1ˆpM<<∆φ

b) 4

4

1φλ effV =

To fit the WMAP data14-

10 ≈effλ and PM 10≈∆φ Such value of quartic

coupling and field displacement are

unnatural from EFT

perspective!

Assuming 1 ≈λ & )1(

82 −

=NN

eff

λλ , one needs 5

10≈N

PM710ˆ −≈∆φ

0=κ falls into this category:

λκ 222m<

Mass Spectrum ofι

Ψ Modes in Matrix Inflation

The other 132 −N even though classically frozen, have quantum fluctuations. To compute

these effects, let us calculate the mass spectrum of these modes. Expanding the action

up to second order, we have:

ΩΨ

+−+ΨΨ+ΩΩ= iiiiii

mV φκφ

λφ

λ ˆˆ22

ˆ2

Tr2

22

)2(

where[ ]

jiijkk Jεi Ψ≡Ω ,

If we have the eigenvectors of the iΩ

iieff

eff mV ΨΨ

++−= Tr

2)(

4

222

2 ωφκωωφλ

iii Ψ=ΨΩ ωωωω

It turns out that finding the eigenvectors of iΩ is mathematically the same as finding the

the vector spherical harmonics:Dasgupta, Sheikh-Jabbari & Von Raamsdonk (2002)

zero modes with

Mass Spectrum ofι

Ψ Modes in Matrix Inflation

2

002

0

2222

)2)(34()1(

2)(2

φωω

φω

ωφκωωφλ

VVV

mM eff

eff

+++′

−+′′=

++−=

1−=ω(a) 12 −N

φφφφφφφφκκκκφφφφλλλλ

VmM effeff

′=+−= 222

2

[ ]Λ=Ψ ,ii J

Λ is an arbitrary traceless matrix

(b) -modes:αααα 1-)1(2−N

2

eff

2

eff

2)2(2)3)(2(

2

1mlllM l ++−++= κκκκφφφφλλλλ

21 ),1( −≤≤Ζ∈+−= Nlllωωωω Degeneracy of each

l -mode is 12 +l

(c) -modes:ββββ 1-)1(2+N

2

eff

2

eff

2)1(2)1)(2(

2

1mlllM l +−+−−= κκκκφφφφλλλλ

Nlll ≤≤Ζ∈= 1 ,ωωωω Degeneracy of each

l-mode is 12 +l

Power Spectra in the Presence of lmr ,Ψ Modes

r,lmr,lmlmrr,lm

µr,lmµ

µ

µ M- V L ΨΨ−Ψ∂Ψ∂−∂∂−= ∗∗)(

2

1)(

2

1

2

1 2

,0 φφφφ

0,, =Ψ=Ψ lmrlmr&If you start from the initial condition , they remain zero. Therefore the

inflationary trajectory is a straight line in the field space and there is no cross-correlationbetween adiabatic and entropy spectra.

βα , ,0=r

0 )( 3

;0

.1

3

2

,2

2

,,

23

23,02

2

=Ψ

++Ψ+Ψ

=

−+++

r,lmlrlmrlmr

P

Ma

kH

QH

a

MaVQ

a

kQHQ

δδδδφφφφδδδδδδδδ

φφφφ φφφφφφφφφφφφφφφφφφφφφφφφ

&&&

&&&&

k'k'k'k'

kkkkk'k'k'k'

kkkk *3

2

3

)(2

φφδπφ

QQk

PQ −= k'k'k'k'

k'k'k'k'

k'k'k'k'

kkkk , ,3

2

3

)(2,

lmrlmr

kP

lmr

∗∗Ψ ΨΨ−= δ

π

0)(2

3

2

3

=Ψ−= ∗

r,lmQQ

kC

Qi kkkkk

'k'k'k'

kkkk φψ

δπ

Φ+≡H

Qφ

δφφ

&

Mukhanov-Sasaki

variable

φφ

QH

&=ℜ

Φ=ℜ2

2

a

k

H

H

&&

lmrlmr

HS

,,Ψ=

φ&

scalar metric perturbations

in longitudinal gauge

Power Spectra in Symmetry-Breaking Inflation µφ >

959.0≈ℜn

1110

162.1

−×981.0 2

N

ββββ

1=l

2

2

2

m

eff

eff

+

−

φκ

φλ

978.0 3

ββββ

2=l

2m12

10

131.1

−×

718

10

842.8

−×002.12

2

m

eff

+

φφφφκκκκ

141091.4

−×≈eff

λ PMm 1007.46−×≈

Peff M 1057.913−×≈κ

Zero

mode

10

601084.4)(

−×≈kPT

025.0−≈Tn

2.0≈r

Planck should be able to verify this model.

2N zero modes can be

removed by gauge transformations

Power Spectra in Symmetry-Breaking Inflation µφµ <<2/

961.0≈ℜn

1110

46.1

−×987.0 2

N

ββββ

1=l

2

2

2

m

eff

eff

+

−

φφφφκκκκ

φφφφλλλλ

0545.1 32

2

m

eff

+

φφφφκκκκ

1610

55.6

−×

αααα

1=l

1910

69.4

−×

007.1 3

2

2

6

6

m

eff

eff

+

−

φφφφκκκκ

φφφφλλλλ

1410187.7

−×≈eff

λ PMm 10824.66−×≈

Peff M 10940.112−×≈κ

Zero

mode

11

6010307.1)(

−×≈kPT

006.0−≈Tn

048.0≈r

CMBPOL or QUIET should be able to verify this

scenario.

2N zero modes can be

removed by gauge transformations

Power Spectra in Symmetry-Breaking Inflation 2/0 µµµµφφφφ <<

961.0≈ℜn

1410

84.3

−×006.1 2

N

αααα

1=l

2

2

2

m

eff

eff

+

−

φκ

φλ

953.0 32

2

6

6

m

eff

eff

+

−

φφφφκκκκ

φφφφλλλλ

1110

23.1

−×

β

1=l

1610

558.6

−×

054.1 32

2

m

eff

+

φκ

1410187.7

−×≈eff

λ PMm 10824.66−×≈

Peff M 10940.112−×≈κ

Zero

mode

11

6010307.1)(

−×≈kPT

006.0−≈Tn

048.0≈r

CMBPOL or QUIET should be able to verify this scenario.

The potential is invariant under symmetry

2N zero modes can b removed by gauge

transformations

006.0−≈Tn

10 20 30 40 50 60 70Ne

-0.2

0.2

0.4

0.6

0.8

1.0

M2Α,1

H2

k= expH-60L ae He

10 20 30 40 50 60Ne

0.2

0.4

0.6

0.8

1.0

1.2

PS

PR

Particle Creation and Preheat Scenario

Even though one can show that backreaction of lmr ,Ψ on the inflaton dynamics is not large,

It can become important when 1, ≈ηε

This could be the bonus of our model, as modes help to drain the energy of the

inflaton, since their masses are changing very fast.lmr,Ψ

We will focus on theory, as its preheating has already been worked out by

Greene, Kofman, Linde & Starobinsky (1997)

The structure of parametric resonance is completely determined by

For we have an enhancement in the parametric resonance leading

to creation of χ particles

12 +∈ Zn particle creation is peaked around 0=k

Zn 2∈ particle creation is peaked aroundλ

ε22

32

2

inf

2 gHk =

~0.15

~0.5

Preheating in the case of symmetry-breaking inflation is in progress!Ashoorioon & J.T. Giblin

in preperation

Particle Creation and Preheat Scenario

In this case with respectively for zero, and modes

For zero mode, odd α l -mode and even −β l mode. 0=k ~0.15

For even α l -mode and odd −β l mode. )1(4

3 2

inf

2 += llHk ε ~0.5

This means that large l even α -modeand odd −β l mode makes the biggest

contribution to preheating. Also one should note that their preheating will be more

effective here due to their 2l+1 degeneracy

UV behavior of Matrix Inflation in presence of many species

In a theory with many particles the scale where quantum gravity effects become large is lowered

toDvali (2007)

: species with mass below the cutoff,

In matrix inflation:

Let’s compare the amount of excursions of the “physical” inflaton is less than when inflation

happens in region

To find the number of species in our case, we assume that .

14

21091.4

)1(

8 −×=−

≡NN

eff

λλλλλλλλ 54618=N

P

ifM

NN 1058.2

)1(

ˆ 6

2

−×=−

−=∆

φφφφφφφφφφφφ PM 1005.1

5−×=Λ<

Pi M 57.43≈φ Pf M 07.27≈φ PM 26≈µ

141091.4

−×≈eff

λ PMm 1007.46−×≈

Peff M 1057.913−×≈κ

Also, one should notice that the mass parameter of the inflaton turns out to be about 40% of the

cutoff.

Analysis of the potential for n-block vacua:

In this case, we have n-field inflationary, n..1 , =αφα that are only gravitationally coupled

The classical (inflationary) dynamics around the “multi-giant vacua” decouple from each

other and one may build an inflationary model around either of these.

If we start with a field which is initially in the sector specified by a given set of , then

remains a conserved quantity by the classical trajectory of the system. In general

various fields in the same sector specified by a set of can mix with each other.

αN

αN

αN

That is, in general the inflationary trajectory in the space of is curved.αφ

Various scenarios can occur in such a n-field inflationary model. Let us consider one two-block example that

can arise in Matrix Inflation with

Analysis of the potential for n-block vacua:

198 200 202 204 206 208

Φ

MP

26

28

30

32

34

Χ

MP

17

60104.4)( −×≈kP

S 987.0≈Sn 228.0≈C

Conclusions

• Matrix inflation is an interesting realization of inflation which is strongly supported

from string theory. Matrix inflation can solve the fine-tunings associated with chaotic

inflation and produce super-Planckian effective field excursions during inflation.

• Due to Matrix nature of the fields there would be many scalar fields in the model. This

leads to the production of isocurvature productions at the CMB scales.

• Matrix inflation has a natural built-in mechanism of preheating to end inflation.

• In particular, if there is an isocurvature component (at a level still allowed by present

data) but it is ignored in the CMB analysis, the sound horizon and cosmological

parameters determination is biased, and, as a consequence, future surveys may

incorrectly suggest deviations from a cosmological constant.

A. Milligen, L. Verde, M. BeltranarXiv:1006.3806 [astro-ph.CO]

Take Isocurvature Perturbations Seriously!

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