G-inflation: models and perturbations

MASAHIDE YAMAGUCHI(Tokyo Institute of Technology)

06/08/11 @takehara理論物理学の展望

arXiv:1008.0603, PRL 105, 231302 (2010), T.Kobayashi, MY, J.YokoyamaarXiv:1012.4238, PRD 83, 083515 (2011), K. Kamada, T. Kobayashi, MY, J. YokoyamaarXiv:1103.1740, PRD in press, T. Kobayashi, MY, J. YokoyamaarXiv:1105.5723, T. Kobayashi, MY, J. Yokoyama

Contents

IntroductionWhat is G ? What is G-inflation ?

Powerspectrum of primordial perturbationsTensor perturbationsDensity perturbations

Summary

Introduction

LagrangianWhy does the Lagrangian generally depend on only

a position q and its velocity dot{q} ?

The Euler-Lagrange equation gives an equation of motion up to thesecond time derivative if the Lagrangian is given by L = L(q,dot{q},t).

Newton recognized that an acceleration, which is given by the second time derivative of a position, is related to the Force :

What happens if the Lagrangian depends on higher derivative terms ?

Ostrogradski’s theoremAssume that L = L(q, dot{q},ddot{q}) and depends on ddot{q} :

(Non-degeneracy)

Canonical variables :

Non-degeneracy ⇔

there is a function a=a(Q1,Q2,P2) such that

These canonical variables really satisfy the canonical EOM :

Hamiltonian:

P1 depends linearly on H so that no system of this form can be stable !!

Loophole of Ostrogradski’s theorem

We can break the non-degeneracy condition, which states depends on ddot{q} :

This equation is really up to the second order.No Ostrogradski’s instability !!

e.g. (This Lagrangian is degenerate.)

G = Galileon fieldField equations have Galilean shift symmetry in flat space :

Nicolis et al. 2009Deffayet et al. 2009

Lagrangian has higher order derivatives, but EOM are second order.

Galileon cosmology

What happens when Galileon field is present ?

It can behave like dark energy.

It can drive inflation and was named G-inflation by us.

Chow & Khoury 2009, Silva & Koyama 2009,Kobayashi et al. 2010, De Felice, Mukohyama, Tsujikawa 2010,Many others …

Field equations cannot have Galilean shift symmetry in curved space :

The extension to curved space is necessary.

is not invariant under

Extend it the most generally as long as the equations of motions are up to second order.

Covariantization of Galileon fieldDeffayet et al. 2009, 2011

This (& the Gauss-Bonnet term) is the most general non-canonical and non- minimally coupled single-field model which yields second-order equations.

NB : ● G4 = MG2 / 2 yields the Einstein-Hilbert action● G4 = f(φ) yields a non-minimal coupling of the form f(φ)R● The new Higgs inflation with comes from G5 ∝φ

after integration by parts.

Equations of motion

Gravitational EOM under the Friedmann background

Under the homogeneous and isotropic background:

Scalar field EOM under the Friedmann background

Under the homogeneous and isotropic background:

NB : Pφ

vanishes if all of K & Gi depend only on X.

Exact de Sitter inflation

We would like to look for the exact de Sitter solution :

Assume that the model has a shift symmetry :

J = 0 is an attractor solution.

If these equations have a non-trivial solution with H≠0 & dot{φ}≠0, exact de Sitter inflation can be realized.

Exact de Sitter inflation II

For the exact de Sitter solution :

This model has a shift symmetry :

e.g.

x (0 < x < 1) is a constant satisfying

For μ< MG,

Note, however, that shift symmetry must be broken to terminate inflation.

Powerspectrum of primordial fluctuations

Primordial tensor perturbationsPerturbed metric :

Expand the action up to the second orderto evaluate the powerspectrum of tensor perturbations.

does not contain hij up to the second order.

Quadratic action for tensor perturbations

No ghost instabilities ⇔No gradient instabilities ⇔

For G4X≠0 or G5φ≠0 or G5X≠0, the sound velocity squared cT2 can deviate from unity.

Quadratic action for tensor perturbations II

New variables :

Sound horizon crossing ⇔

Superhorizon solutions :

Decaying mode

Assuming

Slow-roll (slow varying) parameters

EOM in momentum space :

Khoury & Piazza 2009,Noller & Magueijo 2011.

yT runs from -∞ to 0 as the Universe expands.

The decaying mode really decays.We impose

Powerspectrum of tensor perturbations

Mode functions :

Commutation relations :

Note that the blue spectrum nT > 0 can be easily obtained as long as 4ε+ 3fT - gT < 0.

polarization tensor

Primordial density fluctuationsPerturbed metric :

Unitary gauge :

Expand the action up to the second order

Eliminate αand βby use of the constraint equations Obtain the quadratic action for R

Prescription:

Note that this gauge does not coincide with the comoving gauge because

, different from the k-inflation model.

Expansion of the action up to the second order and constraint equation

Hamiltonian constraint :

Momentum constraint :

Quadratic action for scalar perturbations

No ghost instabilities ⇔No gradient instabilities ⇔

NB : In case of k-inflation with G3 = G5 = 0 and G4 = MG2 / 2, FS = MG2ε= - MG2 dot{H} / H2, which means thatdot{H} > 0 is prohibitted by the stability condition.

Quadratic action for scalar perturbations II

New variables :

Sound horizon crossing ⇔

Superhorizon solutions :

Decaying mode

Assuming

Slow-roll (slow varying) parameters

EOM in momentum space :

Khoury & Piazza 2009,Noller & Magueijo 2011.

yS runs from -∞ to 0 as the Universe expands.

The decaying mode really decays.We impose

Powerspectrum of scalar perturbations

Mode functions :

Commutation relations :

Note that almost scale invariance requires 2ε+ 3sS + gS << 1, while each slow-roll parameter can be large.

Tensor-to-scalar ratio :

Gauss-Bonnet term

Background gravitational equations :

Background field equations :

Tensor and scalar perturbations :

Our formulae apply for the Gauss-Bonnet case by the above replacements.

Summary

We have proposed a new inflation model named G-inflation,which is driven by a Galileon field.

G-inflation predicts new consistency relations between r and nT.

Kinetically driven G-inflation can predict large tensor-to-scalar ratio and large non-Gaussianity.Scalar fluctuations are generated even in exact de Sitter background.