Accelerating cosmologies in an integrable model with noncommutative - PowerPoint PPT Presentation

1 Accelerating cosmologies in an integrable model with noncommutative minisuperspace variables arXiv:1903.07895 [gr-qc] N a h o mi K a n ( N I T , G i f u C o l l e g e ) M a s a s h i K u n i y a s u , K i y o s h i S h i r a i s h i , a n d K o h j i r o h T a k i mo t o ( 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 2 0 1 9

2 §1. Introduction *We study classical and quantum noncommutative cosmology with a Liouville-type scalar degree of freedom. *The noncommutativity is imposed on the minisuperspace variables through a deformation of the Poisson algebra. *We investigate the effects of noncommutativity of minisuperspace variables on the accelerating behavior of the cosmic scale factor. *The probability distribution in noncommutative quantum cosmology is also studied and we propose a novel candidate for interpretation of the probability distribution in terms of noncommutative arguments. S t r i n g s a n d F i e l d s 2 0 1 9

3 §1 . I n t r o d u c t i o n §2 . T h e mo d e l §3 . C l a s s i c a l d y n a mi c s §4 . A c c e l e r a t i n g u n i v e r s e §5 . Wa v e f u n c t i o n o f t h e U n i v e r s e §6 . Wi g n e r f u n c t i o n o f t h e U n i v e r s e §7 . D i s c u s s i o n a n d O u t l o o k S t r i n g s a n d F i e l d s 2 0 1 9

4 §2 . T h e mo d e l T h e L i o u v i l l e s c a l a r mo d e l . A s s u mi n g , , Φ i s a f n c . o f t ↓ " C o s mo l o g i c a l " e f f e c t i v e L a g r a n g i a n S t r i n g s a n d F i e l d s 2 0 1 9

5 where , , , , . This "Cosmological" effective Lagrangian can also be obtained from various theories, including f(R) theory, higher-dim. theory with compactification (with flux, or cosmological const., or Ricci-non-flat int. space,). S t r i n g s a n d F i e l d s 2 0 1 9

6 §3 . C l a s s i c a l d y n a mi c s C o mmu t a t i v e C a s e L a g r a n g i a n ➡ H a mi l t o n i a n ➡ ➡ S t r i n g s a n d F i e l d s 2 0 1 9

7 We a r e c o n s i d e r i n g a " c o s mo l o g i c a l " mo d e l , s o R e me mb e r t h e H a mi l t o n i a n c o n s t r a i n t ! H ＝0 * s o l u t i o n * ( P , t , y a r e c o n s t a n t s ) 0 0 S t r i n g s a n d F i e l d s 2 0 1 9

8 Noncommutative Case H a mi l t o n i a n : P o i s s o n b r a c k e t s : H a mi l t o n ' s e q u a t i o n s : S t r i n g s a n d F i e l d s 2 0 1 9

9 * s o l u t i o n * H θ ＝0 s a t i s f y i n g t h e c o n s t r a i n t w h i c h i s o r i g i n a l l y f o u n d b y S t r i n g s a n d F i e l d s 2 0 1 9

1 0 Noncommtativity from Commutative variables Let us identify: ρ: a n a r b i t r a r y c o n s t a n t . T h e n , Hamilton's equations r e c o v e r s t h e s a me e q u a t i o n s f o r X , Y , Π X , Π Y , a n d t h e s a me s o l u t i o n s , f o r a n y ρ. S t r i n g s a n d F i e l d s 2 0 1 9

1 1 §4 . A c c e l e r a t i n g u n i v e r s e , I f > 0 , e x p a n s i o n i s a c c e l e r a t i n g . U > 0 U < 0 r e d c u r v e s : , b l u e c u r v e s : S t r i n g s a n d F i e l d s 2 0 1 9

1 2 §5 . Wa v e f u n c t i o n o f t h e U n i v e r s e T o o b t a i n Wh e e l e r - D e Wi t t e q u a t i o n ( f o r t h e mi n i s u p e r s p a c e ) , r e p l a c e mo me n t a a s a n d . E x p r e s s WD W e q . i n N o n c o mmu t a t i v e c a s e b y c o mmu t a t i v e v a r i a b l e s : E X : C o n f i r m ! N o t e t h a t i f ρ ＝－ θ , Y ! ＝ y S t r i n g s a n d F i e l d s 2 0 1 9

1 3 Now, WDW eq. of Noncommutative Quantum Cosmology becomes: T h e s o l u t i o n o f t h e WD W e q . w h e r e w i t h a n d We a r e i n t e r e s t e d i n , i n s t e a d o f ! ( We w a n t t o s e e s o me c o r r e s p o n d e n c e w i t h c l a s s . s o l . ) S t r i n g s a n d F i e l d s 2 0 1 9

1 4 if ρ＝－θ , ← common variable both for C & NC Y ＝y then . Thus, for a wave packet peaking around , ν～P we can regard approximately. H e r e a f t e r , w e c o n s i d e r ν ↑ ↑ ↑ (rectangular amplitude) S t r i n g s a n d F i e l d s 2 0 1 9

1 5 U>0 θ=0 θ=0.1 θ=-0.1 bold curves indicate classical solutions! S t r i n g s a n d F i e l d s 2 0 1 9

1 6 U<0 θ=0 θ=0.1 θ=-0.1 bold curves indicate classical solutions! S t r i n g s a n d F i e l d s 2 0 1 9

1 7 §6 . Wi g n e r f u n c t i o n o f t h e U n i v e r s e F o r a w a v e f u n c t i o n , t h e Wi g n e r f u n c t i o n i s d e f i n e d a s : p r o p e r t i e s : w h e r e i s t h e F o u r i e r t r a n s f o r m o f . F o r o u r w a v e f u n c t i o n : S t r i n g s a n d F i e l d s 2 0 1 9

1 8 I t s F o u r i e r t r a n s f o r m: O u r i d e a : d e f i n e a n d i n t e g r a t e o u t Χ . w h e r e S t r i n g s a n d F i e l d s 2 0 1 9

1 9 Check and Confirm our idea C o mp a r e w i t h t h e F o u r i e r t r a n s f o r m o f t h e d e n s i t y o b t a i n e d i n t h e p r e v i o u s s e c t i o n : S t r i n g s a n d F i e l d s 2 0 1 9

2 0 U>0 θ=0 θ=0.1 θ=-0.1 a l mo s t i n d i s t i n g u i s h a b l e S t r i n g s a n d F i e l d s 2 0 1 9

2 1 U<0 θ=0 θ=0.1 θ=-0.1 a l mo s t i n d i s t i n g u i s h a b l e S t r i n g s a n d F i e l d s 2 0 1 9

2 2 §7 . D i s c u s s i o n a n d O u t l o o k ● A N o n C o mmu t a t i v e ( N C ) d e f o r ma t i o n o f t h e mi n i s u p e r s p a c e v a r i a b l e s i s s t u d i e d b y me a n s o f a n i n t e g r a b l e mo d e l . I t s a n a l y t i c a l s o l u t i o n s a r e o b t a i n e d i n c l a s s i c a l a n d q u a n t u m c o s mo l o g y . ● We s h o w e d t h a t t h e p e a k o f t h e w a v e p a c k e t r e p r o d u c e s t h e c l a s s i c a l t r a j e c t o r y b y u s i n g e x a c t s o l u t i o n s w i t h a n i n t e r p r e t a t i o n o f t h e N C v a r i a b l e s i n t h e p r e s e n t mo d e l . ● We p r o p o s e d a n e w p r o b a b i l i t y d i s t r i b u t i o n i n N C q u a n t u m c o s mo l o g y c o n s t r u c t e d f r o m t h e Wi g n e r f u n c t i o n . I t s v a l i d i t y i n t h e p r e s e n t s o l v a b l e mo d e l i s c o n f i r me d n u me r i c a l l y . S t r i n g s a n d F i e l d s 2 0 1 9

2 3 ● I n f u t u r e s t u d y , w e w i l l i n v e s t i g a t e g e n e r a l N C c o s mo l o g y b y u s i n g t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n . G e n e r a l d e f o r ma t i o n s o f mi n i s u p e r s p a c e v a r i a b l e s s h o u l d b e s t u d i e d f u r t h e r . ● T h e mo d e l w i t h a p h a n t o m s c a l a r f i e l d a n d / o r a p h a n t o m g a u g e f i e l d ma y a l s o b e w o r t h s t u d y i n g i n t h e c o n t e x t o f N C c o s mo l o g y . S t r i n g s a n d F i e l d s 2 0 1 9