1

QFT ON CURVED SPACETIMES &

ITS APPLICATIONS IN COSMOLOGY

Anna Ijjas

Columbia University

2

T H I S T A L K : A P R A C T I O N E R ’ S A C C O U N T F R O M A P R A G M A T I C P E R S P E C T I V E

priority to QFT: cosmology is `EFT’

QFT enters on small scales only, background is driven by classical

dynamics (may or may not be Einstein gravity)

The cosmologist’s dilemma

WE HAVE NO FULL-BLOWN THEORY OF QUANTUM GRAVITY [YET]

S a v e y o u r p h i l o s o p h y q u e s t i o n s f o r t h e d i s c u s s i o n !

two `schools:’

WE HAVE EXPERIMENTAL ACCESS TO ENERGIES WHERE EINSTEIN GRAVITY IS NOT ENOUGH [more later]

3

PLAN

Classical Friedman-Robertson-Walker dynamics

T H E B A C K G R O U N D

What goes well and what goes very wrong.

I N F L A T I O N A N D E T E R N A L I N F L A T I O N

A VERY brief introduction

Q F T O N C U R V E D S P A C E - T I M E

When all goes well.

Q F T O N C O N T R A C T I N G B A C K G R O U N D S

4

The classical backgrounda r X i v : 1 8 0 3 . 0 1 9 6 1

P A T C H S I Z E : R ( t ) ~ a ( t ) x R ( t 0 )

H O R I Z O N S I Z E :

H - 1 ~ a ( t ) e x H - 1 ( t 0 )

E Q U A T I O N O F S T A T E :

e = ( 3 / 2 ) ( 1 + p / r )

D U S T : e = 3 / 2

R A D I A T I O N : e = 2 ,

a(t)

a(t)2 R(t0)a(t0) = 1

a(tLSS) = 1/1000

vs.

5

QFT on curved spacetimeB a r d e e n , S t e i n h a r d t , T u r n e r 1 9 8 3

h i j : t e n s o r m o d e s a . k . a .

g r a v i t a t i o n a l w a v e s

c l a s s i c a l e q . o f m o t i o n :

F o u r i e r t r a n s f o r m :

i n t r o d u c e c o n j u g a t e

m o m e n t u m :

p r o m o t e b o t h h kr a n d π k

r t o o p e r a t o r s t h a t s a t i s f y e q u a l - t i m e c o m m u t a t i o n r e l a t i o n s

s i n c e b o t h h kr & π k

r

h e r m i t i a n :w h e r e

C A N O N I C A L Q U N A T I Z A T I O N

m o d e f u n c t i o n s

s a t i s f y :

w i t h b a s i s

6

Inflation and Eternal Inflationa r X i v : 1 8 0 3 . 0 1 9 6 1 ; S t e i n h a r d t 1 9 8 3 ; V i l e n k i n 1 9 8 3 ; G u t h 2 0 0 0

H O R I Z O N S I Z E :

H - 1 ~ a ( t ) e x H - 1 ( t 0 )

w i t h i n u n i t H u b b l e t i m e :

q u a n t u m r u n a w a y / e t e r n a l i n f l a t i o n / m u l t i v e r s e :

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background evolution field picture

p��2 ⇠ H

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vs.

7

QFT on contracting backgroundsa r X i v : 1 8 0 3 . 0 1 9 6 1

H O R I Z O N S I Z E :

H - 1 ~ a ( t ) e x H - 1 ( t 0 )

N O q u a n t u m r u n a w a y /e t e r n a l i n f l a t i o n /

m u l t i v e r s e !

V

φ ✏ = 312 �̇

2

12 �̇

2 + V' 1

2

✓V,�

V

◆2

< 1<latexit sha1_base64="BfeO4KKwgWidNq+nmlnKjO1mQGM=">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</latexit><latexit sha1_base64="BfeO4KKwgWidNq+nmlnKjO1mQGM=">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</latexit><latexit sha1_base64="BfeO4KKwgWidNq+nmlnKjO1mQGM=">AAACX3icbVFdS+QwFE2r7o7jV1efxJfgICguQzsKrqAg+uKjgjMK03FIM7czwTTtJrfKUPonfduHfdlfsplaxK8LCYdzzs1NTqJMCoO+/8dx5+YXvn1vLDaXlldW17wf6z2T5ppDl6cy1XcRMyCFgi4KlHCXaWBJJOE2eriY6bePoI1I1Q1OMxgkbKxELDhDSw29pxAyI2Sq6Ck9oGGsGS+qPejQcJRiEWYTUd53yi/Z/V5JQyMS+E1fZQkx7tYn9YbFz8paFr0y1GI8wb37Dj2hwdBr+W2/KvoZBDVokbquht6zHczzBBRyyYzpB36Gg4JpFFxC2QxzAxnjD2wMfQsVS8AMiiqgku5YZkTjVNulkFbs246CJcZMk8g6E4YT81GbkV9p/RzjX4NCqCxHUPxlUJxLiimdpU1HQgNHObWAcS3sXSmfMBsN2j9p2hCCj0/+DLqd9nE7uD5snZ3XaTTIFtkmuyQgR+SMXJIr0iWc/HXmnGVnxfnnNtxV13uxuk7ds0Helbv5Hwd7tFo=</latexit><latexit sha1_base64="BfeO4KKwgWidNq+nmlnKjO1mQGM=">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</latexit>

> 3

a(t)

a(t)

D(t0)a(t0) = 1

e ≥ 3