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

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

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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 ,

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Accelerating cosmologies in an integrable model with noncommutative minisuperspace variables

arXiv:1903.07895 [gr-qc] N a h

  • mi

K a n ( N I T , G i f u C

  • l

l e g e ) M a s a s h i K u n i y a s u , K i y

  • s

h i S h i r a i s h i , a n d K

  • h

j i r

  • h

T a k i mo t

  • (

Y a ma g u c h i U n i v e r s i t y )

1 S t r i n g s a n d F i e l d s 2 1 9

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§1. Introduction

*We study classical and quantum noncommutative cosmology with a Liouville-type scalar degree of freedom. *The noncommutativity is imposed

  • n

the minisuperspace variables through a deformation of the Poisson algebra. *We investigate the effects

  • f noncommutativity
  • f

minisuperspace variables

  • n

the accelerating behavior

  • f

the cosmic scale factor. *The probability distribution in noncommutative quantum cosmology is also studied and we propose a novel candidate for interpretation

  • f the

probability distribution in terms

  • f

noncommutative arguments.

2 S t r i n g s a n d F i e l d s 2 1 9

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§1 . I n t r

  • d

u c t i

  • 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

  • n
  • f

t h e U n i v e r s e §6 . Wi g n e r f u n c t i

  • n
  • f

t h e U n i v e r s e §7 . D i s c u s s i

  • n

a n d O u t l

  • k

3 S t r i n g s a n d F i e l d s 2 1 9

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§2 . T h e mo d e l

T h e L i

  • 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 .

  • f

t ↓ " C

  • s

mo l

  • g

i c a l " e f f e c t i v e L a g r a n g i a n

4 S t r i n g s a n d F i e l d s 2 1 9

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SLIDE 5

where , , , , . This "Cosmological" effective Lagrangian can also be

  • btained from various theories, including f(R) theory,

higher-dim. theory with compactification (with flux, or cosmological const., or Ricci-non-flat int. space,).

5 S t r i n g s a n d F i e l d s 2 1 9

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§3 . C l a s s i c a l d y n a mi c s

C

  • 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

  • n

i a n ➡ ➡

6 S t r i n g s a n d F i e l d s 2 1 9

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We a r e c

  • n

s i d e r i n g a " c

  • s

mo l

  • g

i c a l " mo d e l , s

  • R

e me mb e r t h e H a mi l t

  • n

i a n c

  • n

s t r a i n t ! H ＝0 * s

  • l

u t i

  • n

* ( P , t , y a r e c

  • n

s t a n t s )

7 S t r i n g s a n d F i e l d s 2 1 9

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Noncommutative Case H a mi l t

  • n

i a n : P

  • i

s s

  • n

b r a c k e t s : H a mi l t

  • n

' s e q u a t i

  • n

s :

8 S t r i n g s a n d F i e l d s 2 1 9

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* s

  • l

u t i

  • n

*

s a t i s f y i n g t h e c

  • n

s t r a i n t

H

θ＝0

w h i c h i s

  • r

i g i n a l l y f

  • u

n d b y

9 S t r i n g s a n d F i e l d s 2 1 9

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Noncommtativity from Commutative variables

Let us identify:

ρ: a n a r b i t r a r y c

  • n

s t a n t . T h e n ,

Hamilton's equations

r e c

  • v

e r s t h e s a me e q u a t i

  • n

s f

  • r

X , Y , Π X , Π Y , a n d t h e s a me s

  • l

u t i

  • n

s , f

  • r

a n y ρ.

1 S t r i n g s a n d F i e l d s 2 1 9

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§4 . A c c e l e r a t i n g u n i v e r s e

, I f > , e x p a n s i

  • n

i s a c c e l e r a t i n g . U > U <

r e d c u r v e s : , b l u e c u r v e s :

1 1 S t r i n g s a n d F i e l d s 2 1 9

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§5 . Wa v e f u n c t i

  • n
  • f

t h e U n i v e r s e

T

  • b

t a i n Wh e e l e r

  • D

e Wi t t e q u a t i

  • n

( f

  • 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

  • n

c

  • mmu

t a t i v e c a s e b y c

  • mmu

t a t i v e v a r i a b l e s :

E X : C

  • n

f i r m !

N

  • t

e t h a t i f ρ＝－θ, Y ＝y !

1 2 S t r i n g s a n d F i e l d s 2 1 9

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Now, WDW eq. of Noncommutative Quantum Cosmology becomes:

T h e s

  • l

u t i

  • n
  • 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

  • f

! ( We w a n t t

  • s

e e s

  • me

c

  • r

r e s p

  • n

d e n c e w i t h c l a s s . s

  • l

. )

1 3 S t r i n g s a n d F i e l d s 2 1 9

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if ρ＝－θ, Y ＝y ←common variable both for C & NC 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

  • n

s i d e r ↑ ↑ ↑ (rectangular amplitude)

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ν

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U>0

θ=0 θ=0.1 θ=-0.1 bold curves indicate classical solutions!

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U<0

θ=0 θ=0.1 θ=-0.1 bold curves indicate classical solutions!

1 6 S t r i n g s a n d F i e l d s 2 1 9

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§6 . Wi g n e r f u n c t i

  • n
  • f

t h e U n i v e r s e

F

  • r

a w a v e f u n c t i

  • n

, t h e Wi g n e r f u n c t i

  • n

i s d e f i n e d a s : p r

  • p

e r t i e s : w h e r e i s t h e F

  • u

r i e r t r a n s f

  • r

m

  • f

. F

  • r
  • u

r w a v e f u n c t i

  • n

:

1 7 S t r i n g s a n d F i e l d s 2 1 9

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I t s F

  • u

r i e r t r a n s f

  • 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

  • u

t Χ. w h e r e

1 8 S t r i n g s a n d F i e l d s 2 1 9

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Check and Confirm our idea

C

  • mp

a r e w i t h t h e F

  • u

r i e r t r a n s f

  • r

m

  • f

t h e d e n s i t y

  • b

t a i n e d i n t h e p r e v i

  • u

s s e c t i

  • n

:

1 9 S t r i n g s a n d F i e l d s 2 1 9

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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

2 S t r i n g s a n d F i e l d s 2 1 9

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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

2 1 S t r i n g s a n d F i e l d s 2 1 9

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§7 . D i s c u s s i

  • n

a n d O u t l

  • k
  • A

N

  • n

C

  • mmu

t a t i v e ( N C ) d e f

  • r

ma t i

  • n
  • 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

  • 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

  • l

u t i

  • n

s a r e

  • 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

  • s

mo l

  • g

y .

  • We

s h

  • w

e d t h a t t h e p e a k

  • f

t h e w a v e p a c k e t r e p r

  • d

u c e s t h e c l a s s i c a l t r a j e c t

  • r

y b y u s i n g e x a c t s

  • l

u t i

  • n

s w i t h a n i n t e r p r e t a t i

  • n
  • 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

  • p
  • s

e d a n e w p r

  • b

a b i l i t y d i s t r i b u t i

  • n

i n N C q u a n t u m c

  • s

mo l

  • g

y c

  • n

s t r u c t e d f r

  • m

t h e Wi g n e r f u n c t i

  • 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

  • l

v a b l e mo d e l i s c

  • n

f i r me d n u me r i c a l l y .

2 2 S t r i n g s a n d F i e l d s 2 1 9

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  • 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

  • s

mo l

  • g

y b y u s i n g t h e p r

  • b

a b i l i t y d i s t r i b u t i

  • n

f u n c t i

  • n

. G e n e r a l d e f

  • r

ma t i

  • n

s

  • 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

  • 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

  • m

s c a l a r f i e l d a n d /

  • r

a p h a n t

  • m

g a u g e f i e l d ma y a l s

  • b

e w

  • r

t h s t u d y i n g i n t h e c

  • n

t e x t

  • f

N C c

  • s

mo l

  • g

y .

2 3 S t r i n g s a n d F i e l d s 2 1 9