GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT

GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT

Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University

Munchen, Oct 9 2008

Based onBased on::Tomas Janssen & Tomislav Prokopec, arXiv:0707.3919 [gr-qc] (2007) Tomas Janssen & Tomislav Prokopec, arXiv:0707.3919 [gr-qc] (2007) Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, arXiv:0807.0439 [gr-qc]Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, arXiv:0807.0439 [gr-qc]Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008)Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008)

˚ 1˚

THE COSMOLOGICAL CONSTANT PROBLEM

μν μν μν2 4

(vacuum matter)gravitationalgeometry energy momentumcoupling tensor

8 G ˆG (g) g = Tc c

(μ,ν =0,1,2,3)

˚ 2˚

Vacuum fluctuates and thereby contributes to the stress-energy tensor of the vacuum (Casimir 1948):

vac vac geom vacobs 2

8 G(T ) g

c

THE COSMOLOGICAL CONSTANT PROBLEM: The expected energy density of the vacuum

A finite volume V = L³ in momentum space constitutes reciprocal lattice: each point of the lattice is a harmonic oscillator with the ground state energy E/2, where E²=(cp)²+(mc²)².

Through Einstein’s equation this vacuum energy curves space-time such that it induces an accelerated expansion:

4 76 4vac Pl~m ~10 GeV

2 -46 40obs Pl~(H m ) ~10 GeV

is about 122 orders of magnitude larger than the observed value:

Q: H²Λ/3 is a classical attractor. Does it remain so in quantum theory?

Plmax ~ mk

BACKGROUND SPACE TIME

LINE ELEMENT (METRIC TENSOR):

˚ 3˚

● for power law expansion the scale factor reads:

),..1,1,1(,)()(1

22222

D

diagagorxdtadtds

aHHH

t

t01

1

0

/1

0

,)1(a

p

wwH

H .,const)1(

2

32

FRIEDMANN (FLRW) EQUATIONS (=0):

,3

82M

NGH

)(4 MMN pGH

˚ 4˚

(MASSLESS) SCALAR FIELD ACTION

SCALAR EOM

In momentum space (=0, V=0):

220

20

22

,0)(''

)2(1

)('1

VR

a

aD

aVR

g

)(

2

1

2

1 2

VRggxdS D

00ˆ,ˆ)(ˆ)()2(

)(ˆ *1

1

kk

xkikk

xkikD

D

aaeaekd

x

SCALAR THEORY

)1(2

3,)()(

4

||1)( )2()1(

kHkH

a kkk

Scalar field spectrum Pφ in de Sitter (ε=0)

sin( )ˆ ˆ0 ( , ) ( ', ) 0 ( , ) , '

dk k xx x k x x x

k k x

P

2

2

2

2

)(1

4),(

aH

kHkP

CONTAINS IR SINGULARITY

˚ 5˚

coincident 2 point function (propagator) in dS limit:

SCALAR THEORY: SINGULARITIES

►we find:

)ln(1

10)(0);( 2

0)(02 aHxxxi E

dS

● when =constant, the

1/term can be subtracted

● when =ε(t), but slowly changing in time, s.t. dε/dt<<Hε

close to matter era: =3/2+ε:

)2/3()ln(2

1

2

1

)2/3(3

10)(0 22/32

OaHx E

3/2,3

2,0,

2

3ta

tHHww MMM

22

2

~1

~)2/3(

tH

H

implying a secular growth of vacuum fluctuations that can compensate a

cosmological term

2~

3

32 ||

)2(0)(0

max

min

k

aHk

kIR

kdx

the IR singularity of a coincident 2 point function:

is IR singular for 0 ≤ ε ≤ 3/2 large quantum backreaction expected

● singularities occur when = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 & 0

CLASSICAL ATTRACTOR IN FLRW SPACES

Q: can quantum vacuum fluctuations change the late time de Sitter attractor behaviour?

Einstein’s equations in FLRW spaces (0):

˚ 6˚

► CLASSICAL SOLUTION

M

MMM

pw

Hw

H

H

,3

1)1(2

322

,

33

82 M

NGH

twH M 3

)1(2

3coth

3

Classical (de Sitter) attractor

Quantum behaviour (?)

3/H

t3/

0

3)1(

23

cosh

)1(23

2

tw

w

M

M

SCALAR PROPAGATOR IN FLRW SPACES

This propagator allows for determination of the quantum backreaction and more generally effects of quantum scalar fields in dynamical FLRW spaces

˚ 7˚

SCALAR PROPAGATOR

Janssen & Prokopec 2007Janssen, Miao & Prokopec 2008

aHHyDDD

FD

DDHH

xxi DD

DD

D

D

D0122/

12

2 ,4

1;2

;2

1,

2

1

2/2

12

1

)4(

)'(|1|)';(

,)1(

)2(41

)2)(1(21

)2)(1(

2

12

22

2

DDDDDDD

D

HOPE: THAT THIS SCALAR PROPAGATOR RESUMS THE LOGS OF a:

2sin4,

'

||||)|'(|)';( 2

22 lHy

xxixxy

1const.)],()ln(1[2

200

H

HOaHaHH

)'(),;( xxixxiRgg D ► EOM

► Ansatz: ),()'(),;( 2/1 yaaxxi D

►

l = geodesic distance in de Sitter space

LAGRANGIAN FOR PERTURBATIONS˚ 8˚

Graviton: lagrangian to second order in h

► PERTURBATIONS ,16,)()(ˆ22

N

aa

Gggxg

►GAUGE: graviton propagator in exact gauge is not known. We added a gauge fixing term:

)()()(ˆ xx

► GRAVITON-SCALAR MIXING

shellon0

2200

2)2( )(')2(''2

1''

VaaHDaaL DD

● lagrangian must be diagonalized w.r.t. the scalar fields 00 &

02 '2

1,

2

1

gaFFFggLGF

● upon a suitable rotation tensor, vector and 2 scalar fields decouple on shell

GRAVITON PROPAGATOR IN FLRW SPACES˚ 9˚

Janssen, Miao & Prokopec 2008

aHHyDDD

FD

DDHH

xxi nDnD

nDnD

D

D

Dn 0,,12

,,

2/

12

2 ,4

1;2

;2

1,

2

1

2/2

12

1

)4(

)'(|1|)';(

,)1(

4)2(

2)2)(1(

)1(2

)1(1

2

12

22

2,

DDDDnnnDn

DnD

EOM (symbolic) DiiD

GRAVITON PROPAGATORS

► VECTOR DOFs: 1vector1vector , iiDD lj

lj

ijij

► GHOST DOFs: 001

0001

000shellon

ghost ,,

iiiDDD ghost

)2,1,0(,)1(2

)1(1 2

nHnn

nDngggDn

Dnn iiD

GRAVITON PROPAGATORS˚10˚

► SCALAR AND TENSOR DOFs (G=3x3 operator matrix):

,

)()(0

)()(0

00

22

02

20

2022

022

isiciisc

iiscisic

i

iMklrs

DiIiMG

0)( 3

22

i

Di klrsslkrklrs

,2

)3(2

D

D )sin(),cos(,3

)3(2)2tan(

scD

D

SCALAR 1 LOOP EFFECTIVE ACTION

When the determinant is evaluated in a FLRW space, it leads to a backreaction that can compensate Λ.

ONE LOOP (MASSLESS) SCALAR FIELD EFFECTIVE ACTION:

˚11˚

DIAGRAMMATICALLY 1 LOOP(vacuum bubble):

NB: Can be calculated by knowing the relevant propagator.

NB2: Propagators are not known for general spaces; now known for FLRW spaces with constant ε.

Janssen, Miao & Prokopec 2008

[ ] [ ] [ ]

1/ 2

1loop contribution

1[ ] [ ] ln

2[ ]

i iS iS ie D e e S Tr g

Det g

..

2

1][ 4

ggxdS

GRAVITON 1 LOOP EFFECTIVE ACTION

☀ When renormalized, one gets the one loop effective action:

˚12˚

Janssen, Miao & Prokopec 2008

]det[]det[

]det[(fields)

tensort

ghost)0()2()0(

ijklorvecij

iSiSiSi

DD

DeeDe

L

ijklij DiTrDTr

iDTr

iS

1

ghosttensor

vector)0( )][ln()][ln(2

)][ln(2

)3(

1

11

1

1ln2)1(ln)1011149186(

16

11

0

23244

33

221023

QQ p

H

H

aVa

► i: renormalization dependent constants

► H0: a Hubble parameter scale

► (z)=dln[(z)]/dz: digamma function

► can be expanded around the poles of (z):

0,2

1,

3

2,

4

3,

5

4,..1,..,

4

5,

3

4,

2

3,2p

►EFFECTIVE ACTION:

● the poles 0, 1, 2 (dS, curv, rad) are not relevant.

NB: Q & pQ can be obtained from the conservation law: )3(44QQQ pHaa

dt

d

DYNAMICS NEAR THE DE SITTER POLE

►near the de Sitter pole (ε=0): small quantum effect

˚13˚

● Late time dynamics: asymptotes a nearly-classical de Sitter attractor:

0ln2

93

1

23 2

0

42

HHG

H

HB

w

HHAGH N

MN

(A,B: undetermined constants)

Classical (de Sitter) attractor

Quantum corrected attractor

3/H

t3/

t3/

DYNAMICS NEAR THE MATTER ERA POLE

►near the matter era pole (ε=3/2): secular growth and large quantum effects

˚14˚

● Late time dynamics: asymptotes the near-pole classical attractor:

)1(341011149186|1|,0)1(241

23 32

42

MpppppppM

N

M

ww

HG

w

HH

Classical branch

1. Quantum branch

3/H

INSET

3/H

2

3

DYNAMICS NEAR THE MATTER ERA POLE II

►Hubble parameter vs time (εp~3/2, <0):

˚15˚

● Late time dynamics: asymptotes the near-pole classical attractor:

Classical (de Sitter) attractor Quantum corrected attractor

3/H

t3/

tH

3

2

DYNAMICS NEAR THE 5/4 POLE

►near the ε=5/4 pole (>0): secular growth and large quantum effects

˚16˚

● Late time dynamics: asymptotes the near-pole classical attractor:

0)(

)1(241

23

42

p

p

M

N

M w

HG

w

HH

Classical branch

Quantum branch

3/H

4

5

INSET

3/H

DYNAMICS NEAR THE 5/4 POLE II

►Hubble parameter vs time (εp~5/4, >0):

˚17˚

● Late time dynamics: asymptotes the near-pole classical attractor:

Classical (de Sitter) attractor Quantum attractor

3/H

t3/

tH

5

4

DARK ENERGY AND COSMOLOGICAL CONSTANT

Dark energy has the characteristics of a cosmological constant Λeff, yet its origin is not known

˚18˚

But why is Λeff so small?

UNKNOWN SYMMETRY?

GRAVITATIONAL BACKREACTION!?

EXPLANATION?

This work suggests that it may be the gravitational backreaction of gravitons (plus matter).

SUMMARY AND DISCUSSION

Scalar matter and graviton VACUUM fluctuations in a near de Sitter universe induce a weak quantum backreaction at 1 loop order (also at 2 loops?).

˚19˚

We considered the quantum backreaction from massless scalar and graviton 1 loop vacuum fluctuations in expanding backgrounds

► What is the quantum backreaction of other quantum fields (fermions, photons)?

1 loop backreaction can be strong when = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 (-2/3≤w≤0)

OPEN QUESTIONS:► we calculated in the approximation ε=(dH/dt)/H²=const. What is the effect of dε/dt 0 (mode mixing)? ► is the backreaction gauge dependent? (Exact gauge?)

Janssen & Prokopec 2008

Koksma & Prokopec 2008

Miao & Woodard 2008, ..► what happens at 2 loop?