インフレーションによる重力波偏光の生成

インフレーションによる重力波偏光の生成 小玉英雄 @CPGS 2010.4.15

R2 modification of the 4D Einstein Gravity

Topological Riem2 Terms • For a 4D Riemannian manifolds M, Characteristic classes of T(M) ) two topological invariants • Gauss-Bonnet term • Chern-Simons term

Coupling to Scalar Fields • Action • Field Equations • Unless ¾(Á) is an odd function, ¾CCS break P.

String Theory motivation

Origin of Riem2 Terms in SST • Heterotic SST • Field members • Action

Anomaly Cancellation Condition • 4D effective action Ref: SvrcekP, Witten E: Axions in string theory, JEHP06 (2006) 051

String corrections • Type II (& M): O(R£ (R®’)3 ) =®’3O(R4 ) • Cosmology taking account of O(R4) corrections. [Bento, Bertolami 1996; Maeda, Ohta 2004, 2005; Akune, Maeda, Ohta 2006; Elizalde et al 2007] • Some inflationary/DE solutions were found, but most models are not realistic. • Hetrotic: O(R£R®’)=®’ O(R2 ) • O(R2 ) correction S /®’ CGB compactification)¾(Á) CGB • Dynamics of the Gauss-Bonnet cosmology (D>4) • Flat and AdS solutions. The latter is unstable. [Boulware, Deser 1985] • Transiently inflationary solutions with contracting internal space. The solutions are asymptotically Kasner. [Ishihara H 1986] • Vast work recently.

Inflation model

Cosmological Solution • Spatially homogeneous and flat model: • EOM: The Chern-Simons int. does not affect the FLRW cosmology. During the slow roll phase, the R^2 terms have small effects.

Super-inflation • Super-inflation by Gauss-Bonnet term • Antoniadis I, Rizos J, Tamvakis K: NPB415, 497(1994). • Kawai S, Soda J: PD59, 063506 (1999); arXiv:gr-qc/9906046; PLB 460, 41 (1999). • Cartier C, Hwang Jc, and Copeland EJ: PRD64, 103504 (2001). Under the condition the EOM is approximately given by

For example, for »=»0 (f/m)2n, we obtain the super-inflation solution: for this solution, Hence, the super-inflation is realised only during a finite period: n=2 case Satoh, Kanno, Soda: PRD77 (2008) 023526

Polarisation of GWs • Tensor perturbation • Polarisation

Production of GW Polarisation • Slow-Roll Inflation Case • Wave equation • WKB solution Cf. Lue, Wang, Kamionkowski: PRL83, 1506 (1999)

Super-inflating Case • Subhorizon perturbation ( -kh<1/6) ) the R-mode grows and the L-mode decays for 3/8<-kh <1/6: • Superhorizon • Difficulty The amplitude of L modes diverges at around the horizon crossing!! Satoh, Kanno, Soda: PRD77 (2008) 023526

Influence on CMB

CMP Polarisation • Stokes Parameters for the Linear Polarisation • Polarisation Tensor Pure E-mode b E cosb+ B sinb Pure E-mode Pure B-mode Lue, Wang, Kamionkowski 1999

Polarised Radiation Transfer • Flux intensity tensor I¹º where e¹kp (p=1,2) is the polarisation basis: e¹¢k¹=0. • The Stokes parameters are expressed as Linear Polarisation CircularPolarisation

Boltzmann Equation • When the WKB approximation is valid for radiation fields, I¹º satisfies the Boltzmann equation: GW + mode )d T, E-mode GW £ mode )B-mode

Observational Effect on CMB • CMB anisotropy due to GWs • Temperature • B-mode Hence, • For unpolarised GWs: <TB>=0 • For circular polarised GWs: <TB>0 Lue, Wang, Kamionkowski 1999

インフレーションによる 重力波偏光の生成