Non-minimal k-inflation Wade Naylor High Energy Theory Group - - PowerPoint PPT Presentation

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Non-minimal k-inflation Wade Naylor High Energy Theory Group - - PowerPoint PPT Presentation

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Non-minimal k-inflation Wade Naylor High Energy Theory Group Osaka 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

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Non-minimal k-inflation

Wade Naylor High Energy Theory Group Osaka University (HETOU)

APS2012, Kyoto, 2nd March 2012 In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

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Nonminimal k-inflation

Nomination for shortest title at APS2012

APS2012, Kyoto, 2nd March 2012 In collaboration with T. Kubota, M. Nobuhiko and N. Okuda

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Contents

  • 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

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My 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

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Het Camp 2011

  • Misumi and Okuda are nowhere to be seen?

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Introduction

  • 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

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Model

  • 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(ϕ)R + 2P(ϕ, X)
  • P(ϕ, X) = K(ϕ)X + L(ϕ)X2 + · · ·

Non-minimal k-inflation

X = −1 2gµν∂µϕ∂νϕ

Leff = 1 2(∂φ)2 + (∂φ)4 M 4 + ... + V (φ) − 1 2ξRφ2 L = 1 2(∂φ)2 + 1 2(∂ρ)2 + ρ M (∂φ)2 + 1 2M 2ρ2 + V (φ) − 1 2ξR(φ2 + ρ2)

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Motivation

  • Obtain constraints on inflation models via
  • bservational 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 c2

s

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Comment on E and J frames

Einstein frame action action S

ˆ S

mode function mode function

ˆ uˆ

k

uk

power spectrum power spectrum PR non-Gaussianity non-Gaussianity

0|uk1uk2uk3|0⇥ 0|ˆ uk1 ˆ uk2 ˆ 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]

b P b

R

b R = R

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Standard 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

10 (Armendariz-Picon, Damour, Mukhanov, 1999 )

E(φ, X) = 2XP,X − P

S = ⇤ d4x√−g 1 2κ2 R + P(φ, X) ⇥

X = −1 2gµν∂µφ∂νφ Tµν = ∂P ∂X ⇥µφ⇥νφ Pgµν

˙ E = −3 √ E(E + P)

ds2 = −dt2 + a2(t)δijdxidxj

3H2 = E

master equation P(φ, X) = K(φ)X + X2 + · · ·

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Attractors in k-inflation

In k-inflation, the universe can expand exponentially using

  • nly 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

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k-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)

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Non-minimal k-inflation

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S = ⇤ d4x√−g 1 2κ2 R − 1 2ξφ2R + P(φ, X) ⇥ non-minimal coupling

X = −1 2Ω2ˆ gµν∂µφ∂νφ = Ω2 ˆ X

ˆ gµν = Ω2gµν

Ω2 ≡ |1 − ξκ2φ2|

Conformal transformation

ˆ S = ⇤ d4x ⌅ −ˆ g 1 2κ2 ˆ R + ˆ P(φ, X) ⇥

ˆ P(φ, X) = ˆ K(φ) ˆ X + ˆ L(φ) ˆ X2

ˆ K(φ) = K(φ) − ξ(K(φ) − 6ξ)κ2φ2 (1 − ξκ2φ2)2

ˆ L(φ) = L(φ) = 1

ξ<0 obtained by substituting ξ→ -|ξ| via field redefinition

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E.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

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  • 4
  • 2

2 4 f 2 4 6 8 10 K ` HfL

ˆ

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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 c2

s

b c2

s =

b P, b

X

b E, b

X

= b P, b

X

b P, b

X + 2 b

X b P, b

X b X

ˆ X = 1 2 ˆ gµν∂µφ∂νφ

0.5 1.0 1.5 f 0.2 0.4 0.6 0.8 1.0 c `2 X `

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Example 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 0.25 R

  • 0.25
  • 0.20
  • 0.15
  • 0.10
  • 0.05

0.05 E

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Power 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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ˆ S(2) =

  • dˆ

td3x

  • ˆ

a3 ˆ

  • ˆ

c2

s

ˆ ˙ R2 − ˆ aˆ (⇥R)2

new variables ¨ v c2

s⇥2v ¨

z z v = 0

b ✏ = − ˙ b H b H2 = b X b P, b

X

b H2 , b ⌘ = ˙ b ✏ b ✏ b H , b s = ˙ b cs b cs b H

v = z b R , z2 = 2b a2b ✏ b c2

s

b P

b R b k =

1 36⇡2 b E2 b E + b P = 1 8⇡2 b H2 b csb ✏ , b ns − 1 = d ln P b

R b k

d ln b k = −2b ✏ − b ⌘ − b s

¨ vb

k +

⇣ c2

sb

k2 − ¨ z z ⌘ vb

k = 0

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Appendix: Conformal properties

ˆ a = Ωa dˆ τ = dτ

R = Ω2 ˆ R + 3 (Ω2) Ω2 2 ˆ X

  • ˆ

K = K Ω2 + 3 2 (Ω2) Ω2 2

ˆ P = P Ω4 + 3 4 ˙ Ω2 Ω2 2

ˆ H = 1 Ω

  • H + 1

2 ˙ Ω2 Ω2

  • ˆ

˙ H = 1 Ω2

  • ˙

H − 1 2H ˙ Ω2 Ω2 − 3 4 ˙ Ω2 Ω2

  • + 1

2 ¨ Ω2 Ω2

  • dˆ

t = Ωdt

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Ω2 ≡ |1 − ξκ2φ2| b R = R

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Final 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?

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Classical 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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PX2φ + Pφ + (PXXrµX + PXφrµφ)rµφ = ξφR

Rµν − 1 2gµνR = 1 M 2

pl

Tµν Tµν = PXrµφrνφ + gµνP + ξ h gµν2(φ2) rµrν(φ2) + (Rµν 1 2gµνR)φ2

Geff = G 1 − φ2/φ2

c

R = ˜ φ2

c

M 2

pl(φ2 ˜

φ2

c)

h⇣ PX + 6ξ 6ξφPXφ PX ⌘ rµφrµφ 6ξφPXX PX rµXrµφ + 4P 6ξφ Pφ PX i

˜ φ2

c =

M 2

pl

ξ(1 − 6ξ/PX) φ > ˜ φc

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k-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

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Appendix: Other shapes

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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

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Appendix: 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( ˙ φ − N i∂iφ)2 − N −2(EijEij − E2) = 0

⇥j(N −1Ej

i) ⇥i(N −1E) = P,XN −1∂iφ( ˙

φ N i∂iφ)

N i∆t

N∆t

t + ∆t t

dxi

S = 1 2

  • d4x

√ hN[R(3) + 2P + N −2(EijEij − E2)]

Eij = 1 2(˙ hij ⇥iNj ⇥jNi)

ds2 = −N 2dt2 + hij(dxi + N idt)(dxj + N jdt)

δφ = 0 , hij = a2e2Rδij

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Appendix: 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)

db s2 = −(db t)2 + b a(b t)2e2 b

R (δij + b

γij) , (∂ib γij = 0, δijb γij = 0) . ds2 = −(dt)2 + a(t)2e2R (δij + γij) , (∂iγij = 0, δijγij = 0) , = 1 Ω2

  • −(db

t)2 + b a(b t)2e2R (δij + γij)

b R = R, b γij = γij

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