G-inflation: models and perturbations MASAHIDE YAMAGUCHI (Tokyo - PowerPoint PPT Presentation

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.Yokoyama arXiv:1012.4238, PRD 83, 083515 (2011), K. Kamada, T. Kobayashi, MY, J. Yokoyama arXiv:1103.1740, PRD in press, T. Kobayashi, MY, J. Yokoyama arXiv:1105.5723, T. Kobayashi, MY, J. Yokoyama

Contents  Introduction What is G ? What is G-inflation ?  Powerspectrum of primordial perturbations Tensor perturbations Density perturbations  Summary

Introduction

Lagrangian Why does the Lagrangian generally depend on only a position q and its velocity dot{q} ? Newton recognized that an acceleration, which is given by the second time derivative of a position, is related to the Force : The Euler-Lagrange equation gives an equation of motion up to the second time derivative if the Lagrangian is given by L = L(q,dot{q},t). What happens if the Lagrangian depends on higher derivative terms ?

Ostrogradski’s theorem Assume 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(Q 1 ,Q 2 ,P 2 ) such that Hamiltonian: These canonical variables really satisfy the canonical EOM : P 1 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} : e.g. (This Lagrangian is degenerate.) This equation is really up to the second order. No Ostrogradski’s instability !!

G = Galileon field Nicolis et al. 2009 Deffayet et al. 2009 Field equations have Galilean shift symmetry in flat space : Lagrangian has higher order derivatives, but EOM are second order.

Galileon cosmology What happens when Galileon field is present ? Chow & Khoury 2009, Silva & Koyama 2009, Kobayashi et al. 2010,  It can behave like dark energy. De Felice, Mukohyama, Tsujikawa 2010, Many others …  It can drive inflation and was named G-inflation by us. The extension to curved space is necessary. Field equations cannot have Galilean shift symmetry in curved space : is not invariant under Extend it the most generally as long as the equations of motions are up to second order.

Covariantization of Galileon field Deffayet 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 : ● G 4 = M G 2 / 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 G 5 ∝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 : P φ vanishes if all of K & G i depend only on X.

Exact de Sitter inflation Assume that the model has a shift symmetry : J = 0 is an attractor solution . We would like to look for the exact de Sitter 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 This model has a shift symmetry : For the exact de Sitter solution : e.g. x (0 < x < 1) is a constant satisfying For μ < M G , Note, however, that shift symmetry must be broken to terminate inflation.

Powerspectrum of primordial fluctuations

Primordial tensor perturbations Perturbed metric : does not contain h ij up to the second order. Expand the action up to the second order to evaluate the powerspectrum of tensor perturbations.

Quadratic action for tensor perturbations For G 4X ≠ 0 or G 5 φ ≠ 0 or G 5X ≠ 0, the sound velocity squared c T 2 can deviate from unity. No ghost instabilities ⇔ No gradient instabilities ⇔

Quadratic action for tensor perturbations II New variables : Sound horizon crossing ⇔ Superhorizon solutions : Decaying mode

Slow-roll (slow varying) parameters Khoury & Piazza 2009, Noller & Magueijo 2011. Assuming y T runs from - ∞ to 0 as the Universe expands. We impose The decaying mode really decays. EOM in momentum space :

Powerspectrum of tensor perturbations Mode functions : polarization tensor Commutation relations : Note that the blue spectrum n T > 0 can be easily obtained as long as 4 ε + 3f T - g T < 0.

Primordial density fluctuations Perturbed metric : Unitary gauge : Note that this gauge does not coincide with the comoving gauge because , different from the k-inflation model. Prescription:  Expand the action up to the second order  Eliminate α and β by use of the constraint equations  Obtain the quadratic action for R

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 G 3 = G 5 = 0 and G 4 = M G 2 / 2, F S = M G 2 ε = - M G 2 dot{H} / H 2 , which means that dot{H} > 0 is prohibitted by the stability condition.

Quadratic action for scalar perturbations II New variables : Sound horizon crossing ⇔ Superhorizon solutions : Decaying mode

Slow-roll (slow varying) parameters Khoury & Piazza 2009, Noller & Magueijo 2011. Assuming y S runs from - ∞ to 0 as the Universe expands. We impose The decaying mode really decays. EOM in momentum space :

Powerspectrum of scalar perturbations Mode functions : Commutation relations : Note that almost scale invariance requires 2 ε + 3s S + g S << 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 n T .  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.