Non-minimal k-inflation

Wade Naylor High Energy Theory GroupOsaka University (HETOU)

APS2012, Kyoto, 2nd March 2012

In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

Nonminimal k-inflation

Nomination for shortest title at APS2012

APS2012, Kyoto, 2nd March 2012

In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

/~20Contents

• Comment (my non-Gaussianity)

• Introduction

• Model

• Motivation

• Jordan ≣ Einstein

• Details on non-minimal k-inflation

• Inflationary attractors even for K(φ)<0

• Power spectrum and tilt

• Final comments (classical stability + non-Gaussianity)

3

/~20My non-Gaussianity

• Prof. Moss (Ph.D.)

• Prof. Sasaki (Post Doc)

• Prof. Kubota (HETOU)

• Kubota-Moss-Sasaki scale invariant spectrum ⇒ I should know everything they know

• Non-Gaussianity ⇒ I don’t know and need students like Misumi and Okuda

4

/~20Het Camp 2011

• Misumi and Okuda are nowhere to be seen?

5

/~20Introduction

• k-inflation can be motivated from effective field theory + curvature coupling terms, ξφ2R

• Jordan or Einstein frame?

• Single field models give full agreement between two frames including non-Gaussianity

• Qiu and Yang, Non-Gaussianities of single field inflation with non-minimal coupling, Phys. Rev. D 83 (2011) 084022.

• non-minimal coupled DBI (ξ=0, 1/6) [see Easson et al., Phys. Rev. D 80 (2009); ibid. 81(2010)]

6

/~20Model

• For example in a two field Lagrangian:

• Then below some H<<M we can integrate out ρ to get

where M → M - ξR/2

• Other examples might be DBI with more general ξRφ2

term

7

S =�

d4x��g

�f(�)R + 2P (�, X)

�

P (�, X) = K(�)X + L(�)X2 + · · ·

Non-minimal k-inflation

X = �12gµ�⇥µ�⇥��

Leff =1

2(@�)2 +

(@�)4

M4+ ...+ V (�)� 1

2⇠R�2

L =1

2(@�)2 +

1

2(@⇢)2 +

⇢

M(@�)2 +

1

2M2⇢2 + V (�)� 1

2⇠R(�2 + ⇢2)

/~20Motivation

• Obtain constraints on inflation models via observational data

• k-inflation with non-minimal coupling

• Most general single field theory with up to 1st order derivatives in φ (cf. Horndeski/G-inflation at 2nd order)

• Can we get constraints on conformal coupling ξ?

• Due to variable sound speed, k-inflation can generate large non-Gaussianity (e.g., equilateral limit):

8

fNL / 1

c2s

/~20Comment on E and J frames

Einstein frame

action action SS

mode function mode functionuk uk

power spectrum power spectrum PR

non-Gaussianity non-Gaussianity�0|uk1uk2uk3 |0⇥�0|uk1 uk2 uk3 |0⇥

Jordan frame(minimal) (non-minimal)

9

T. Kubota, N. Misumi, W. Naylor and N. Okuda, JCAP 02 (2012) 034, arXiv:1112.5233 [gr-qc]

bP bR

bR = R

/~20Standard k-inflation

• Consider FLRW universe described by line element

• Friedmann equation and the continuity equation are

• K(φ) changing from <0 to >0 essentially leads to inflationary attractor solutions

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(Armendariz-Picon, Damour, Mukhanov, 1999 )

E(�, X) = 2XP,X � P

S =⇤

d4x⇥�g

� 12�2

R + P (⇥, X)⇥

X = �12gµ⇥⇥µ�⇥⇥�

Tµ⇥ =⇥P

⇥X⇥µ�⇥⇥�� Pgµ⇥

E = �3⇥

E(E + P )

ds2 = �dt2 + a2(t)�ijdxidxj

3H2 = Emaster equation

P (�, X) = K(�)X +X2 + · · ·

/~20Attractors in k-inflationIn k-inflation, the universe can expand exponentially using only master equation E = �3

⇥E(E + P )

P=E 

P=‐E 

E 

P 

11

(Armendariz-Picon, et al., PLB 458 1999)Solid lines are stable attractors

/~20k-inflation phase diagram

We obtain similar attractor/slow-roll solutions in the non-minimal case (see later if time permits)

12

(Armendariz-Picon, et al., PLB 458 1999)

/~20Non-minimal k-inflation 13

S =⇤

d4x⇥�g

� 12�2

R� 12⇥⇤2R + P (⇤, X)

⇥

non-minimal coupling

X = �12�2gµ⇥⇥µ�⇥⇥� = �2X

gµ⇥ = �2gµ⇥ �2 ⇥ |1� ⇥�2⇤2|

Conformal transformation

S =⇤

d4x⌅�g

� 12�2

R + P (⇥, X)⇥

P (�, X) = K(�)X + L(�)X2

K(⇤) =K(⇤)� ⇥(K(⇤) � 6⇥)�2⇤2

(1� ⇥�2⇤2)2L(�) = L(�) = 1

ξ<0 obtained by substituting ξ→ -|ξ|

via field redefinition

/~20E.g., setting K(φ)=-1

• Actually even for K(φ)<0 effect of ξ>0 coupling leads to a sign change in K(φ)

• ξ＝1/100,1/10 and 1/3

14

-4 -2 2 4f

2

4

6

8

10

K` HfL

ˆ

/~20

K(φ)=-1

• Speed of sound cs stays small until dφ/dt (∝X) drops

• Note that level of non-Gaussianity

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fNL / 1

c2s

bc2s =bP, bXbE, bX

=bP, bX

bP, bX + 2 bX bP, bX bX

X =1

2gµ⌫@µ�@⌫�

0.5 1.0 1.5f

0.2

0.4

0.6

0.8

1.0c2 X`

/~20Example attractor solution

• We see an attractor for E=p and E=-p (here for ξ=1/3)

• More plots in progress ...

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0.05 0.10 0.15 0.20 0.25R

-0.25

-0.20

-0.15

-0.10

-0.05

0.05

E

/~20Power spectrum & spectral index

• Slow roll parameters:

(Note that ĉs is a function of time ⇒ dĉs/dt <<1)

• Standard quantization leads to:

plots currently in progress (including e-foldings)...

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S(2) =�

dtd3x�a3 �

c2s

ˆR2 � a�(⇥R)2�

new variablesv � c2

s⇥2v � z

zv = 0

b✏ = �bHbH2

=bX bP, bXbH2

, b⌘ =b✏b✏ bH

, bs = bcsbcs bH

v = z bR , z2 =2ba2b✏bc2s

bP bRbk =

1

36⇡2

bE2

bE + bP=

1

8⇡2

bH2

bcsb✏, bns � 1 =

d lnPbRbk

d lnbk= �2b✏� b⌘ � bs

vbk +⇣c2sbk2 �

z

z

⌘vbk = 0

/~20

Appendix: Conformal properties

a = �a d� = d�

R = �2�R + 3

� (�2)�

�2

�2X

�

K =K

�2+

32

� (�2)�

�2

�2

P =P

�4+

34

� �2

�2

�2

H =1�

�H +

12

�2

�2

�

ˆH =1

�2

�H � 1

2H

�2

�2� 3

4

� �2

�2

�+

12

�2

�2

�

dt = �dt

18

�2 ⇥ |1� ⇥�2⇤2|

bR = R

/~20Final Comments

• In k-inflation with non-minimal coupling, is there a characteristic signal, different from other models?

• Classical stability (backreaction) needs further investigation, even for DBI non-minimal models (next slide)

• Non-canonical models lead to non-Gaussianity & non-minimal coupling broadens allowed parameter space (to be confirmed)

• Shape of non-Gaussianity is important and has contribution coming from ξ (later slide, time permitting)

• Preheating in non-minimal K-inflation interesting?

19

/~20Classical stability?

• In Jordan frame easy to see , where φc2 = (κ2ξ)-1=Mpl2/(8πξ)

• However another instability can be found from EOM:

Taking trace of Tµν and subbing back into EOM ⇒

where (Px=1, standard result)

• ℋ constraints ⇒ only for anistropic spacetimes; cf.

Futamase et al., Phys.Rev. D 39 (1989) 405-411

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PX2�+ P� + (PXXrµX + PX�rµ�)rµ� = ⇠�R Rµ⌫ � 1

2gµ⌫R =

1

M2pl

Tµ⌫

Tµ⌫ = PXrµ�r⌫�+ gµ⌫P + ⇠hgµ⌫2(�

2)�rµr⌫(�2) + (Rµ⌫ � 1

2gµ⌫R)�2

Ge↵ =G

1� �2/�2c

R =�2c

M2pl(�

2 � �2c)

h⇣PX + 6⇠ � 6⇠�

PX�

PX

⌘rµ�rµ�� 6⇠�

PXX

PXrµXrµ�+ 4P � 6⇠�

P�

PX

i

�2c =

M2pl

⇠(1� 6⇠/PX)

� > �c

/~20k-inflation with non-minimal coupling

• Shape of non-Gaussianity from ξ-contribution

• In all limits non-minimal part has nonzero value

• Result from N. Misumi’s Master’s thesis (cf. Qiu & Yang)

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squeezed limit

equilateral limit F(1,x2,x3)x22x32

x3=k3/k1

x2=k2/k1

F (k1,k2,k3)

k1k2k3/ 3k1k2k3

2k3 +1

k2

X

i>j

kikj

/~20Appendix: Other shapes 22

The shape of non-Gaussianity depends on model of inflation; however, current data not sensitive to shape; only overall amplitude fNL

(Babich et al. 2004)

F(1,x2,x3)x22x32

x3=k3/k1, x2=k2/k1

/~20Appendix: ADM Decomposition• In what follows drop the hat and for Einstein frame

• Hamiltonian and momentum constraints equations are given by

• Solve constraints equations to first order only for N and Ni (Chen et al. JCAP 01); in unitary gauge

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R(3) + 2P � 2N�2P,X(� �N i⇥i�)2 �N�2(EijEij � E2) = 0

⇥j(N�1Eji)�⇥i(N�1E) = P,XN�1⇥i�(��N i⇥i�)

N i�t

N�t

t + �t

t

dxiS =12

�d4x⇥

hN [R(3) + 2P + N�2(EijEij � E2)]

Eij =12(hij �⇥iNj �⇥jNi)

ds2 = �N2dt2 + hij(dxi + N idt)(dxj + N jdt)

�� = 0 , hij = a2e2R�ij

/~20Appendix: Invariance of curvature perturbation R

• In the Einstein frame fluctuations around FLRW imply

which via a conformal transformation leads to

• Comparison between the metrics in each frame implies:

• Thus, scalar curvature (R) and tensor perturbations are conformally invariant

• More details see: Chiba & Yamaguchi JCAP 0810 (slow roll), and Gong et al. JCAP 1109 (Tautology & δN formalism)

dbs2 = �(dbt)2 + ba(bt)2e2 bR (�ij + b�ij) , (@ib�ij = 0, �ijb�ij = 0) .

ds2 = �(dt)2 + a(t)2e2R (�ij + �ij) , (@i�ij = 0, �ij�ij = 0) ,

=1

⌦2

��(dbt)2 + ba(bt)2e2R (�ij + �ij)

bR = R, b�ij = �ij

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