Inflationary Massive Gravity

Transcription

1 New perspectives on cosmology APCTP, 15 Feb., 017 Inflationary Massive Gravity Misao Sasaki Yukawa Institute for Theoretical Physics, Kyoto University C. Lin & MS, PLB 75, 84 (016) [arxiv: ] G. Domenech, T. Hiramatsu, C. Lin, MS, M. Shiraishi, Y. Wang, arxiv:

2 Introduction

3 Inflation: the origin of Big Bang Brout, Englert & Gunzig 77, Starobinsky 79, Guth 81, Sato 81, Inflation is a quasi-exponential expansion of the Universe at its very early stage; perhaps at t~10-36 sec. It was meant to solve the initial condition (singularity, horizon & flatness, etc.) problems in Big-Bang Cosmology: if any of them can be said to be solved depends on precise definitions of the problems. Quantum vacuum fluctuations during inflation turn out to play the most important role. They give the initial condition for all the structures in the Universe. Cosmic gravitational wave background is also generated. 3

4 4 温故知新 (learning from the past) ds dt a ( t) dh ; ( 3) 1 a( t) H sinh Ht Creation of Open Universe! Now in the context of String Landscape

5 length scales of the inflationary universe log L L=H 0-1 Size of the Observable Universe 10 8 cm ~ 46 e-folds 10 8 cm LIGO band L=H -1 Inflationary Universe Bigbang Universe log a(t) 5

6 Planck constraints on inflation Planck 015 XX V scalar spectral index: n s ~ 0.96 tensor-to-scalar ratio: r < 0.1 simplest V model is almost excluded n s 1 d 3 log[ k PS ( k)] dlog k PT ( k) r P ( k) S 6

7 Current status scalar spectral index: n s <1 at ~ 5 s tensor/scalar ratio: r < 0.1 implies E inflation < GeV simple, canonical models are on verge of extinction (m model excluded at > s) R (Starobinsky) model seems to fit best. But why? (large R correction but negligible higher order terms) f NL local <O(1) suggests (effectively) single-field slow-roll (but non-slow-roll models with f NL local =O(1) not excluded) some element of non-canonicality is needed 7

8 Massive Gravity? 8

9 Gauge theory: The idea of massive gravity Higgs VEV spontaneously breaks gauge symmetry Gravity: massive gauge field Spontaneous broken general covariance massive gravitons This assumes however Poincare symmetry on flat background. If no background, covariance should NOT be violated. E Spontaneous broken local Lorentz invariance = existence of a preferred frame drgt gravity Boulware-Deser (BD) ghost must be removed spin = ++1 (+1) dof (tensor+vector+scalar) massive tensor modes! (helicity ) 9

10 4 scalar (Stuckelberg) fields: Dubovsky s model a 0 i (, ) Dubovsky 004 ( a) ( b) ( a) ( a) a ( b) a ds abe e dx dx e e b x e b SO : ( ), ( 31, ) VEV spontaneously breaks local SO(3,1) symmetry a ( a) a a ( a) a e : x e x required symmetry (Poincare symmetry is not imposed) i i j i i i 0 i, ( ); global ( 3) action: j j SO MP 4 ij S d x R mg f ( X, Z ) X g g N Lapse fcn. g g Z g h X 0 i 0 j ij i j ij only 0 +1 dof (tensor+scalar) 3-metric becomes dynamical 10

11 Inflationary massive gravity: minimal model 0 Identify with inflaton: 0 ij ij 4 M 1 9 ( ) ( ) Z Z P S d x R g V MPmg 8 Z ik kj Z Z Z Z 3 ; Z Z Z mg H during inflation assumption: 1 mg during reheating Symmetry: ij ij ii i i j i i i j, ; j global SO( 3), const. These symmetries guarantees i to be non-dynamical. x : w ( t) x v( t) x O( 1 ), w w i i i i ij j i ij ji at leading order in gradient expansion 11

12 notes on non-dynamical modes helicity 1 and traceless helicity 0 modes (=3 NG bosons) at leading order on spatially flat slicing (~ decoupling limit) 9 4 j 1 i 4 i S m ( ) gm P d x i i O M P k 4 3 i i i 4 x mm : cutoff scale become dynamical at higher orders S d x g g Z Z ~ k ( ) 4 ij ij i rescaling: i i k 4 i i S d x ( ) ( ) g P H massive enough: can be integrated out 1

13 massive tensor perturbation tensor nd order action S M ( ) P 3 3 T d x dt a ( ij ) m ( ) g ij 8 quantization eom k a 3 dk ij 3 ij k s ( ) k k () t k ikx a(, s) e (, s) ( t) e h. c. / a( k, s) : e ij ( k, s) : : positive frequency fcn. annihilation operator polarization tensor e ( k, s) e ij ( k, s) k a * * k 3H k mg 0 : k k k k k 3 ij ss i a KG normalization 13

14 during inflation P T ( k) tensor spectrum H M P spectral index: Lyth bound r if r if m g 3H 0 mg H k k f n T g / H m 3 at the end of inflation k a( t ) H( t ) f f f mg ; 3H, blue-tilted! H H 1 / 15 MPr : distance traveled by during inflation P P T S ( k) ( k) , 16 for standard slow-roll inflation travels more than Planck distance beyond validity of QFT? 14

15 Observational Signatures? 13

16 resonant GW amplification Lin & MS 15 3/ 1 a f Mt mg ; f sin( Mt ) e after inflation a M : inflaton mass M : decay (reheating) rate Mathiew-like eq. for k/a << H < m g k a 3 ( ) k H k mg 0 k 0 dx x dx x x d d e sin ( x) k x Mt, broad parametric resonance for f 3 M Mx e 1 x 0 16

17 broad parametric resonance amplified by a factor ~ 10 6! f 3 M 6 10,

18 parameter dependence 18

19 evading Lyth bound tensor perturbation can be exponentially amplified by broad parametric resonance: P ( k) AP ( k); T T,0 A 1 scalar (curvature) perturbation remains the same: P ( k) P ( k) S r S,0 P P T S ( k) 16 A ( k) Lyth bound is modified as 15M P r A tensor perturbation can be large enough to be detected without invalidating low-energy EFT 19

20 non-gaussianity? Domenech, Hiramatsu, Lin, MS, Shiraishi & Wang 17 3 rd order Hamiltonian in =0 spatially isotropic gauge: H L M d x a 3-pt fcn: 3 int int SST P ij i c SST m g 4 H coupling term that could appear in lowenergy EFT from (unknown) UV physics i c ij i j Z 4 s s 3 ( 3) S ST H ij i 3j i j i j k1k k3 ( ) k1 k k 3 ki k 4 3 t 4 M P k i kt kt e k k dominates CMB 3-pt fcn if SST >> 1 kk k k k k t 1 3 0

21 scale-dependent non-gaussianity TTT ij a ij i c j c 1 after horizon re-entry small scale modes re-enters horizon earlier large l multipoles are suppressed max dependence of f local NL SST 100,

22 WMAP 010/Planck 015 f f local NL local NL 3 1 for 500 WMAP 010 max for 500 Planck 015 max non-g from SST (arbitrary scale)

23 more from Planck 015/other shapes a hint of non-g due to SST but not yet conclusive non-g from SST 3

24 Summary Inflation is a natural platform for modified gravity Inflation = scalar-tensor theory GW (tensor mode) can become massive during inflation without encountering BD ghost problem symmetry: i i j, i i ; i j j SO( 3), const. GW can be parametrically amplified during reheating evading Lyth bound even if r > GW spectrum may be blue-tilted primordial GW may be detectable by LIGO/Virgo/KAGRA 3 rd order interaction can give rise to sizable scale-dependent non-gaussianity E already a hint in WMAP/Planck data needs further tests! 4