Spontaneous Compactification of Bimetric Theory based on - PowerPoint PPT Presentation

1 Spontaneous Compactification of Bimetric Theory based on arXiv:1405.0064 [hep-th] N a h o mi K a n ( G i f u N a t i o n a l C o l l e g e o f T e c h n o l o g y ) , T a k u y a M a k i ( J a p a n Wo me n ' s C o l l e g e o f P h y s i c a l E d u c a t i o n ) K i y o s h i S h i r a i s h i ( Y a ma g u c h i U n i v e r s i t y ) S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

2 In a six-dimensional model of bimetric theory, massless and massive gravitons emerge with a spontaneous compactification to four dimensions. S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

3 Why massive gravity? ・H o w d o e s g r a v i t y b e h a v e a t a c o s mo l o g i c a l s c a l e ? ・ I n t h e v e r y e a r l y u n i v e r s e , i s t h e e v o l u t i o n o f s c a l e f a c t o r s d i f f e r e n t f r o m t h e p r e s e n t b e h a v i o r ? ・A s a n e x e r c i s e f o r c o v a r i a n t a p p r o a c h t o q u a n t u m g r a v i t y Problems in massive gravity ・ vDVZ discontinuity ( v a n D a m & V e l t ma n ; Z a k h a r o v , 1 9 7 0 ) t h e l i mi t m →0 i s d i f f e r e n t f r o m G R ・ Vainshtein Mechanism ( 1 9 7 2 P L B 3 9 ) n o n l i n e a r e f f e c t s a t s t r o n g g r a v i t y i s i mp o r t a n t ( ! ? ) ・ Boulware-Deser Ghost ( 1 9 7 2 P R D 6 ) A g h o s t d e g r e e o f f r e e d o m a p p e a r s i n n o n l i n e a r ma s s i v e g r a v i t y , b e c a u s e i t i s i mp o s s i b l e t o ma k e a s c a l a r c o n s t r a i n t w i t h o u t d e r i v a t i v e s ( f r o m a s i n g l e me t r i c ) . S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

4 B i g r a v i t y ( B i me t r i c g r a v i t y ) ma s s l e s s a n d ma s s i v e g r a v i t o n s G h o s t - f r e e N o n - l i n e a r ma s s i v e g r a v i t y ( o n e me t r i c i s n o n - d y n a mi c a l ) d R G T ( d e R h a m, G a b a d a d z e , T o l l e y ) 2 0 1 0 H a s s a n , R o s e n 2 0 1 1 G h o s t - f r e e N o n - l i n e a r B i g r a v i t y H a s s a n , R o s e n , S c h mi d t - M a y , 2 0 1 2 ( v e r y s o r r y t o ma n y o t h e r a u t h o r s f o r o mi t t i n g t h e i r w o r k . . . ) S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

5 μ 1 μ 2 . . . μ n ν 1 ν 2 ν n ～ - Σ n C n δ ν n K μ 1 K μ 2 … K mass term g ν 1 ν 2 . . . μ n g f two metrics μν , μν ν νρ K μ ＝ g f ρμ T h e g e n e r a l i z e d K r o n e c h e r d e l t a S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

6 vierbein massive gravity A A 1 D M M . . . M 1 2 n ～Σ n c n δ D e … e e e mass term ( n g ' s , D - n f ' s ) A A . . . A g f 1 2 M M 1 D A c c o r d i n g t o A l e x a n d r o v ( G R G 4 6 ( 2 0 1 4 ) 1 6 3 9 ) , v i e r b e i n b i g r a v i t y i s a l s o G h o s t f r e e . S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

7 Six-dimensional Model of BimetricTheory In models of Bigravity, Massive gravity mass of massive graviton < - - ' n e w ' s c a l e , b y h a n d ( ? ) we wish to consider connections to some other scales and/or mechanisms. We n e e d mi x i n g p a r t i n t h e a c t i o n o f b i me t r i c t h e o r y . g a u g e f i e l d mi x i n g ( H o l d o m, 1 9 8 6 ) ( t h i s i s a l s o u s e d i n r e c e n t mo d e l s o f d a r k ma t t e r ) S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

8 i d e a : C o n s i d e r f l u x mi x i n g a n d me t r i c mi x i n g a t t h e s a me t i me = a t t h e s t a g e o f c o mp a c t i f i c a t i o n o f e x t r a s p a c e ! T h e s i x - d i me n s i o n a l mo d e l ： α : a d i me n s i o n l e s s p a r a me t e r S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

9 Compactification e x t r a s p a c e ＝S 2 ( t w o me t r i c s o f t w o r a d i i i n g e n e r a l ) curvatures: fluxes: ， S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 0 Compactification of Six-dimensional Einstein-Maxwell theory as in Randjbar-Daemi, Salam, Strathdee NPB214(1983) a static solution is given as a minimum point of the effective potential for scales of extra dimensions with fine-tuning to obtain flat four-dimensional spacetime or, if we set , S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 1 V i s mi n i mi z e d a t 0 : y ＝y t h e p a r a me t e r t u n i n g mu s t b e d o n e a s t h e a b o v e t a k e s a mi n i mu m v a l u e a t x ＝x 0 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 2 masses of gravitons The four-dimensional effective action for gravitons after the compactification I n t h e w e a k f i e l d l i mi t : , w e f i n d S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 3 a n d h h N o w w e g e t t h e L a g r a n g i a n f o r t w o g r a v i t o n s g , f . h h T o e l i mi n a t e t h e l i n e a r t e r m i n g , f , we s h o u l d c h o o s e t h e p a r a me t e r s a s S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 4 These come from Two constraints from variations of two lapse functions. Then, the Lagrangian for two gravitons reads: w h e r e ma s s l e s s ma s s i v e S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 5 A s i mp l e c a s e : ・ ma s s - s q u a r e o f ma s s i v e g r a v i t o n : ( t u n i n g : e t c . ) S i n c e ， ma s s o f ma s s i v e g r a v i t o n i s l e s s t h a n t h a t o f a K K mo d e g e n e r a l i z a t i o n t o mu l t i g r a v i t y - - > s p e c t r a w i t h h i e r a r c h y S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4

1 6 Summary and outlook In a Six-dimensional Bimetric model, massless and massive gravitons appear at the spontaneous compactification. ・Cosmology (many constraints!) ・Spectrum of spin2, spin1, spin0 ・Tuning-free theory? (extension of models of Salam-Sezgin-Nishino-Maeda?) S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 0 1 4