˚ 1˚
QUANTUM SCALAR CORRECTIONS TO THE
GRAVITATIONAL POTENTIALS ON DE SITTER
Tomislav Prokopec, ITP Utrecht University
S. Park, T. Prokopec, R.P. Woodard, ``Quantum Scalar Corrections to the Gravitational Potentials on
de Sitter Background,'' arXiv:1510.03352 [gr-qc]
Leonard, Park, Prokopec, Woodard, Phys.Rev. D90 (2014) 2, 024032 [arXiv:1403.0896 [gr-qc]]
Marunovic, Prokopec, Phys.Rev. D83 (2011) 104039 [arXiv:1101.5059 [gr-qc]]
Marunovic, Prokopec, Phys.Rev. D87 (2013) 10, 104027 [arXiv:1209.4779 [hep-th]]
D. Glavan, S.P. Miao, T. Prokopec and R.P. Woodard, ``Graviton Loop Corrections to Vacuum
Polarization in de Sitter in a General Covariant Gauge,” arxiv:1504.00894, Class.Quant.Grav. 32
(2015) 19, 195014
D. Glavan S.P. Miao, T. Prokopec and R.P. Woodard, Class.Quant.Grav 31 (2014) 175002,
[arXiv:1308.3453 [gr-qc]]
CERN, 18 Aug 2016
CONTENTS
˚ 2˚
1) QUANTUM GRAVITY IS AN EFFECTIVE THEORY
2) QUANTUM GRAVITATIONAL EFFECTS ON FLAT MINKOWSKI SPACE
3) QUANTUM EFFECTS DURING INFLATION (DE SITTER AS TOY MODEL)
4A) AT BOUNDARY OF STABILITY: MASSLESS SCALAR ON DE SITTER
4B) STABLE THEORY: SCALAR ELECTRODYNAMICS
4C) WHEN STABILITY FAILS.. YUKAWA THEORY
4D) PERTURBATIVE GRAVITY ON DE SITTER: VACUUM POLARIZATION
4) GRAVITON SELF-ENERGY FROM SCALAR FIELDS
5A) DYNAMICAL GRAVITONS
5B) GRAVITATIONAL POTENTIALS ON DE SITTER
5) CONCLUSIONS AND OUTLOOK
˚ 3˚
QUANTUM GRAVITY IS
AN EFFECTIVE THEORY
GRAVITY AS EFFECTIVE THEORY
˚ 4˚
● GRAVITY BECOMES STRONGLY COUPLED AT THE PLANCK SCALE
● AT PLANCK ENERGY ALL LOOPS CONTRIBUTE EQUALLY. For example


2
2
²=16G
IN COMPARISON TO TREE LEVEL EXCHANGE: SUPPRESSED AS
● EFFECTIVE THEORY IS OF THE FORM (²=16G)

R
 2 2


k12 2  1
𝜉
− 𝑅𝜙 2
2
● THIS THEORY CAN BE PERT QUANTIZED. AS LONG AS ci‘s ARE NOT
HUMANGOUSLY LARGE, THE FIRST TWO TERMS DETERMINE
LOW ENERGY GRAVITY
John F. Donaghue, Phys.Rev. D50 (1994) 3874-3888 [gr-qc/9405057]
GRAVITY AS EFFECTIVE THEORY 2
˚ 5˚
● E.G. AT ONE LOOP, TWO HEAVY MASSES (in nonrel limit) ATTRACT
EACH OTHER BY A FORCE DETERMINED BY THE POTENTIAL
Bjerrum-Bohr, Donaghue, Holstein, Phys.Rev. D67 (2003) 084033 [hep-th/0211072]
CONTRIBUTING DIAGRAMS:
BOX & CROSSED BOX
TRIANGLE DIAGRAMS
VACUUM POLARIZATION DIAGRAMS
TECHNIQUE: IN-OUT SCATTERING AMPLITUDE.
DOUBLE SEAGULL
VERTEX CORRECTIONS
˚ 6˚
QUANTUM GRAVITATIONAL
EFFECTS ON MINKOWSKI
˚ 7˚
1 LOOP GRAVITON SELF-ENERGY: MINKOWSKI
Marunovic, Prokopec, Phys.Rev. D83 (2011) 104039 [arXiv:1101.5059 [gr-qc]]
● ALL MATTER COUPLES TO GRAVITY: SCALARS, VECTORS, FERMIONS
● CONSIDER MASSLESS NONMINIMALLY COUPLED SCALARS
● ACTION:
● GRAVITATIONAL ACTION (quadratic in perturbations around Minkowski):
²=16G
- LICHNEROWICZ OPERATOR (on Minkowski):
- CONTRIBUTING 1 LOOP DIAGRAMS
RENORMALIZATION
● DIM REG REQUIRES 2 COUNTERTERMS:
Minimal subtraction scheme [when =0 agrees with ‘t Hooft, Veltman 1974]:
˚ 8˚
RENORMALIZED SELF-ENERGY
˚ 9˚
● we work in Schwinger-Keldysh formalism, suitable for non-equil. problems
● PERTURBING THE METRIC AROUND MINKOWSKI
● 1PI EFFECTIVE EQUATION OF MOTION FOR THE GRAVITON:
RETARDED SELF-ENERGY: CAUSAL
PERTURBATIVE SOLUTION TO 1PI EFFECTIVE EOM
˚10˚
● TREE LEVEL SOLUTION FOR POINT PARTICLE MASS M at r=0:
NEWTONIAN POTENTIALS
● METRIC PERTURBATION:
where ²=16GN IS LOOP COUNT. PARAMETER OF QG
● PERTURBED 1PI EOM
● SOLUTION: PERT CORRECTED BARDEEN POTENTIALS
- agrees with Park+Woodard (2010) in their gauge
BARDEEN POTENTIALS
˚11˚
● GAUGE INV SCALAR POTENTIALS: inv. under infinitesimal coord. transforms
ARE BARDEEN POTENTIALS
WHERE
NB: , REDUCE TO USUAL GRAV POTENTIALS N, N IN LONGITUDINAL GAUGE
RESUMMATION
˚12˚
● SOLVING 1PI EQ RESUMS 1 LOOP
(BUBBLE & DAISY) DIAGRAMS
● SCHWINGER-DYSON EQUATION:
RESUMMED DIAGRAMS INCLUDE..
RECALL: GAP EQUATIONS IN COND MATTER PRODUCED FAMOUS RESULTS: SC,..
RESUMMED 1LOOP POTENTIALS
˚13˚
● TIME-LIKE BARDEEN POTENTIAL  (=0, 1/3, rs/lp=10): LARGE MASS
RESUMMED 1LOOP POTENTIALS 2
˚14˚
● SPATIAL BARDEEN POTENTIAL  (=0, 1/3, rs/lp=10): LARGE MASS
● RESUMMATION TELLS US THAT AT SMALL DISTANCES QUANTUM
SCALAR PERTURBATIONS ANTISCREEN NEWTONIAN POTENTIALS
RESUMMED 1LOOP POTENTIALS 3
˚15˚
● TIME-LIKE & SPATIAL BARDEEN POTENTIALS ,  (=0,1/3 rs/lp=0.05),
SMALL MASS:
● RESUMMATION TELLS US THAT AT SMALL DISTANCES QUANTUM
SCALAR PERTURBATIONS SCREEN NEWTONIAN POTENTIALS
● SMALL MASS FIELDS ARE WEAK EVERYWHERE,
NB: CURVATURE STILL DIVERGES at r0: Kretchmann:
GRAVITON LIGHT CONE
● ONE-LOOP SCALAR QUANTUM FLUCTUATIONS AFFECT
PROPAGATION OF GRAVITONS ON MINKOWSKI
● LIGHT CONE GETS MODIFIED AS:
● PROPAGATION VELOCITY OF GRAVITONS ∞ AS ct0.
˚16˚
˚17˚
QUANTUM EFFECTS
DURING INFLATION
˚18˚
AT BOUNDARY OF STABILITY:
MASSLESS SCALARS ON DE SITTER
MASSLESS SCALARS ON DE SITTER
˚19˚
● EXAMPLE 1: MASSLESS SCALAR ON DE SITTER (D=4):
ACTION:
 1

S   d 4 x  g      g  ,
 2

 EOM for : g     ˆ( x) 
 g  a 4 , g   a  2  ,    diag (1,1,1,1)
1
( 2η  2aH  η   i2 )ˆ( x)  0
2
a
 In a dS invariant state, the propagator must be a function of dS inv distance l(x;x’):
y ( x; x ' )  4 sin 2 Hl ( x; x ' ) / 2,

  2
2
y ( x; x ' )  H 2a ( )a ( ' )  |    ' |    x  x 

 then the propagator obeys a dS inv equation (in D dim):
g    
H2

d2
d 
 D ( x  x' )
i( x; x' )   y(4  y) 2  D(2  y) i( x; x' )  i 2
, ( D  4)
dy
dy
H

g


 This equation has no solution!
Allen, Folacci, PRD35 (1987)
˚20˚
MASSLESS SCALAR ON DE SITTER 2
● MASSLESS MINIMALLY COUPLED SCALAR (MMCS) D=4:

 d
d
 4 ( x  x' )
 y (4  y )  4(2  y )  i( x; x' )  i 2
dy
H g

 dy
(*)
● THE NAIVE SOLUTION (in D=4) is (up to an irrelevant constant C):

H 2  1 1  y    1
1
i( x; x' )  F ( y)  2   log    
 log 4  y  
4  y 2  4   4  y 2

 This UNIQUE dS inv solution solves the wrong equation:

 d
d
 4 ( x  x' )  4 ( x  ~x ' )
 y (4  y )  4(2  y )  i( x; x' )  i 2
i 2
dy
dy
H g
H g


(**)
 Here ~x   ( , xi ) denotes the antipodal point of x   ( , xi ):
 From y ( x; ~x ' )  [4  y ( x; x' )] it follows that the terms in 2nd square brackets [.]
source the second -function in Eq. (**).
 The solution (**) follows from:
 1

  2
1
4
2

2
2


 2  2

i

(
x

x
'
),





,

x
(
x
;
x
'
)


(|
t

t
|

i

)

x
 x
 
2

4


x
(
x
;
x
'
)


MASSLESS SCALAR ON DE SITTER 3
˚21˚
● CONCLUSION: MMCS cannot be quatized in a dS inv way.
● ONE POSSIBLE SOLUTION (that breaks dS sym, respects spatial
homogeneity, but solves the right equation) is:
H2
i( x; x' )  2
4
  1 1  y  1





log

ln
a
(

)
a
(

)

const





  y 2  4  2

 This dS breaking solution solves the right equation, but
(when dim reg is applied) the coincident propagator grows (linearly) with time.
 This has physical consequences. For example, when one introduces a quartic

interaction L quartic    4 at one loop one generates a Hartree mass:
4!
(m ) Hartree 
2

H 2
 i( x; x) 
ln( a), ln( a)  Ht
2
4 2
 In a self-interacting scalar theory, the mass squared thus grows linearly in time.
This is a consequence of abundant particle creation in dS. Precisely this type
of particle creation generates scalar cosmological perturbations, which in turn
induce CMB temperature fluctuations and seed the Universe’s large scale structure.
☺ QUESTION: IS THERE A dS INV SCALAR MASS?
˚22˚
MASSIVE SCALAR PROPAGATOR ON dS
● dS INVARIANT SCALAR FIELD PROPAGATOR


 g m2 ( x; x' )   D ( x  x' )
Chernikov, Tagirov, Annales Poincaré Phys.Theor. A9 (1968)
H D 2  D21  D  D21  D 
( x; x' ) 
2 F1
D/2
D
(4 )
 2 
2
 D 1  m
D  
  2,
2

 H
2

D 1
2
 D , D21  D ; D2 ;1  y ( x4; x ')

  2
2
y( x; x' )  |    ' |    x  x 
● COINCIDENT SCALAR PROPAGATOR
H D 2  D21  D  D21  D 
D

( x; x) 

1

2   ( x; x) div  ( x; x) fin
D/2
1
1
(4 )
 2  D  2  D 
H D2 D21  
1 






( x; x)div 

D
/
2


D

1


1

D
/
2



E
4 ( D 1) / 2 
D  1
( x; x) fin
 D21  H D
 ( D 1) / 2 2
2
m
The m→0 LIMIT IS SINGULAR;
EXPLAINS WHY THERE IS NO dS INV SOLUTION.
˚23˚
SCALAR FIELD: MASS GENERATION
● in a self-interacting scalar theory, scalar has a growing mass.
Is there a dS inv late time limit?
● RESUMMATION: SELF-CONSISTENT HARTREE:
1
 
 1
S [ ]   d D x  g        m02 2   4 
2
4! 
 2
 
● ACTION
● GAP EQUATION
2
MF
m
● SOLUTION
2
MF
m


 m  ( x; x) fin , if   0, m  m  ( x; x) div  0
2
2
2
2
2
2
0
Ford,Vilenkin, PRD 26 (1982); Serreau; Garbrecht, Rigopoulos (2011+)
Prokopec; Prokopec, Lazzari (2012+)
m2
m 4 mcr4



 0, ( 2  0), | m 2 | mcr2 ,
2
4
2
H D  D21  D 4
3 H 2
m 


2 ( D 1) / 2
2 2
2
cr
● THERE ARE STUDIES OF 2 and 3 LOOPS
(primarily to check Stochastic inflation; also w<-1 found)
Kahya, Onemli, Woodard (2010);
Onemli, Woodard (2004)
˚24˚
SCALAR FIELD: MASS GENERATION
● STOCHASTIC INFLATION RESULT
Starobinsky, Yokoyama 1996
 QUALITATIVELY THE SAME AS THE GAP EQUATION,
BUT QUANTITATIVE DIFFERENCE:
2
mSY

27 (3 / 4) 2
2
H


H
2 2 (1 / 4)

˚25˚
STABLE THEORY:
SCALAR QED ON DE SITTER
SCALAR ELECTRODYNAMICS
˚26˚
● (BARE) LAGRANGIAN
1
1
 1

S SQED[ , A ]   d D x  g      g   m02 2  g  g F F   S ct [ , A ]
2
4
 2

● ONE RELEASES SYSTEM FROM A FREE VACUUM STATE
 Initial boundary divergences lead to exponentially decaying tails,
can be removed by chosing a suitable (pertubative) interacting state
● THE PHOTON GAINS A (PERTURBATIVE) MASS
Prokopec, Tornkvist, Woodard (2002,2003); Prokopec, Puchwein, (2004)
e2 H 2
m 
ln a(t )  
2 2
2
3e 2 H 4
m 
4 2 (m2  R)
2
● THE SCALAR CAN BE KEPT (PERTURBATIVELY) MASSLESS
● QUANTUM BACKREACTION: 2 LOOP STRESS-ENERGY TENSOR
T 

 2 loop
 diag  ρ,-p,-p,-p 2 loop
3e 2 H 4
  p 2 loop   2 loop  cons t.  
ln a(t )   const.;
4
64
Prokopec, Tsamis, Woodard (2007,2008)
e2 H 4
  p 2 loop 
 cons t.
4
64
● SOME OF THE CONTRIBUTING DIAGRAMS
˚27˚
˚28˚
WHAT HAPPENS AT LATE TIMES?
DOES ONE REACH A DE SITTER INVARIANT STATE?
IF YES, WHICH ONE?
TO ANSWER THESE QUESTIONS WE NEED NON-PERTURBATIVE METHOD
 STOCHASTIC INFLATION
STOCHASTIC SCALAR QED
˚29˚
Prokopec, Tsamis, Woodard, Ann.Phys.323 (2008) [0707.0847 [gr-qc]]




STRATEGY: identify active and passive fields.
Active fields are amplified in IR and can produce leading log(a).
Passive fields (gauge fields, fermions): NEED TO BE INTEGRATED OUT
Stochasticize the resulting EFFECTIVE SCALAR THEORY, with Veff().
☺ RESULTS:
♣ the scalar and photon acquire a (nonperturbative) mass:
m  0.034e H , m  3.3H
2
2
2
2
2
NB: AGREE WITH
3e 2 H 4
m 
4 2 (m2  R)
2
♣ fluctuations give constant contributions to the density invariants & c.c.:
F ( x) F ( x)  0.12 H 4 ( g  g  g  g );
Veff ( )  0.027 H 4 ;   0.2083GH 4
NB: These results are fully nonperturbative (not suppressed by the coupling const. e)
CONCLUSION: IN SQED ASYMPTOTICALLY ONE REACHES A
dS INVARIANT, STABLE STATE OF LOWER ENERGY
AFTER INTEGRATING THE (PASSIVE) PHOTONS
ONE GETS THE EFFECTIVE SCALAR POTENTIAL
8 2
V ( z)
4 eff
3H
Veff SMALL FIELD
z  e 2  / H 2
˚30˚
CONTRIBUTION TO ENERGY DENSITY FROM
THE EFFECTIVE SCALAR THEORY
8 2
V ( z)
4 eff
3H
8 2
V ( z)
4 
3H
z  e 2  / H 2
˚31˚
˚32˚
WHEN STABILITY FAILS:
YUKAWA ON DE SITTER
FERMIONS IN YUKAWA
● AT ONE LOOP IN YUKAWA THEORY
PHOTONS ACQUIRE A MASS:
˚33˚
Garbrecht, Prokopec, PRD73 (2006)
2
4
y2H 2
3
y
H
m 
ln a(t )   m2 
2

4
8 2 (m2  R)
2
► m=scalar mass; y=Yukawa, ξ=nonminimal coupling, R=Ricci scalar
STOCHASTIC THEORY IS OBTAINED BY INTEGRATING OUT
THE FERMIONS. ONE OBTAINS THE FOLLOWING EFFECTIVE POTENTIAL:
Miao, Woodard, PRD (2006), gr-qc/0602110
..AND THE CONTRIBUTION TO THE ENERGY DENSITY IS:
˚34˚
EFFECTIVE POTENTIAL IS UNSTABLE
● EFFECTIVE POTENTIAL
8 2
V (m / H )
4 eff
3H
CONTRIBUTION TO
THE ENERGY DENSITY  unstable evolution
8 2
V (m / H )
4 
3H
TWO DIFFERENT VALUES OF V0
˚35˚
PERTURBATIVE GRAVITY ON DE SITTER:
VACUUM POLARIZATION
GRAVITONS ON DE SITTER
NON-COVARIANT APPROACH
˚36˚
● WHEN ONE FULLY FIXES THE GAUGE, GRAVITONS ON dS
CAN BE THOUGHT AS TWO POLARIZATIONS OF A MMCS:
 EOM for GRAVITONS :
1
( 2η  2aH  η   i2 )hˆ ( x)  0, hijTT     , x  ij h ,  i hijTT  0  hiiTT
2
a
● IDENTICAL PROBLEM AS MMCS: NO dS INV PROPAGATOR!?
D
i j

d2
d 
1
2
  ( x  x' )
 y(4  y) 2  D(2  y) i ij  kl ( x; x' )   Pik Pjl  Pil Pjk 
Pij Pkl i 2
, Pij   ij  2
dy
dy 
2
D2

 H g



y ( x; x' )  4 sin 2 Hl ( x; x' ) / 2,

  2
2
y ( x; x' )  H 2 a( )a( ' )  |    ' |    x  x 

 LHS: THE SAME OPERATOR AS FOR MCMS!  no solution!
 BUT the RHS PROJECTORS Pij BREAK dS INV!
 No general proof as yet whether dS inv propagator exist!
 We do have now graviton propagator in a covariant exact (de Donder) gauge!
Mora, Tsamis, Woodard, J.Math.Phys.53(2012); Miao, Tsamis, Woodard, J.Math.Phys.52 (2011)
Non-cov propagator cannot be used to study de Sitter breaking
(gauge dependence?) of physical quantities.
˚37˚
COVARIANT GRAVITON PROPAGATOR
Mora, Tsamis, Woodard, J.Math.Phys.53(2012); Miao, Tsamis, Woodard, J.Math.Phys.52 (2011)
 THE GRAVITON PROPAGATOR CONTAINS SPIN 2 AND SPIN 0 PARTS
● EXACT COVARIANT DE DONDER GAUGE (operator):
b


g~  g    h , g     h   h   0
2




● SPIN 2 PART P ( x), P ( x' )
{b=gauge parameter)
(TT on both index groups)
  ( 2)  
2 ( D  2) 2


( x; x' )  4
P
(
x
)
P
 ( x' )  ' y ( x; x' )  ' y ( x; x' ) AAABB ( x; x' )
2 
H ( D  3)
 g    
 AAABB ( x; x' )   AABB ( x; x' )
( D  2) H 
2
AABB
( x; x' )   AAB ( x; x' )
 AAB ( x; x' )   AB ( x; x' )
( D  2) H 
2
AB
( x; x' )   A ( x; x' )
 D ( x  x' )
 A ( x; x' ) 
g
MMCS: EXPECT dS BREAKING
˚38˚
COVARIANT GRAVITON PROPAGATOR 2
● GAUGE PARAMETER

Db  2
2( D  1)
 D
b2
b2
g 



( D   ) H 2 , P ( x' )  (spin 0 projectors)
● SPIN 0 PART  P ( x)     




  
( 0)


 2 2
( x; x' ) 
P ( x) P ( x' )W NN ( x; x' )
( D  1)( D  2)

(  D) H  ( x; x' )   ( x; x' )
(  D) H  ( x; x' )   ( x; x' )
DH  ( x; x' )   ( x  x' )
2
W NN
WN
2
WN
TACHYONIC WHEN <D (b<2,D>1)
W
D
2
W
g
TACHYONIC WHEN Re[D]>0
• FOR SIMPLICITY WE CONSIDER A CLASS OF GAUGES b>2,
(>D) FOR WHICH THE N-PROPAGATOR IS dS INVARIANT
● APPLICATIONS: coinc. graviton propagator (in de Donder gauge): dS breaking
Kahya, Miao Woodard (2011), 1112.4442
ONE-LOOP VACUUM POLARIZATION
˚39˚
Glavan, Miao, TP, Woodard (2015)
 WE USED THE COVARIANT GRAVITON AND PHOTON PROPAGATORS
DIAGRAMS
+ counter-terms
 MIAO & WOODARD CALCULATED SPIN=2 CONTRIBUTION
(in a heroic calculation they included the dS breaking contribution)
 GLAVAN & TP CALCULATED THE SPIN=0 CONTRIBUTION
(sheepishly they studied only the b>2 (>D) CASE, when
the gauge dependent part of the s=0 propagator is dS invariant)
 Miao & Woodard found a v=dependence in vac polarisation
v=ln[a()/a(’)]
not present in the vac pol from Leondard & Woodard paper.
Suggests a gauge dependence.
˚40˚
VACUUM POLARIZATION ON DE SITTER
Glavan, Miao, TP, Woodard (2015)
 USE THE COVARIANT GRAVITON AND PHOTON PROPAGATORS TO CALC
GRAVITON INDUCED 1 LOOP VAC POL:
   ( x; x' ) 
+ COUNTER-TERMS
 USEFUL (F-G/EL-MAG) REPRESENTATION:
   ( x; x' )      T  ( x; x' ),










     ( x; x' )  0  '    ( x; x' )

T  ( x; x' )        F ( x; x' )         G ( x; x' ),        0 0
Prokopec, Tornkvist, Woodard; Leonard, Prokopec, Woodard 1304.7265, 1210.6968
 ON MINKOWSKI: LORENTZ INVARIANT (G=0) BUT GAUGE DEPENDENT RESULT
 2  D2  D2  1


  Mink ( x; x' )  16( D  1) D C2  C0 (b)  '  '   1x 2 


D2
( D  2) 2  Db  2 
C2  O( D  4), C0 (b)  

  O( D  4)
4
 b2 
2
NEGATIVE SEMIDEFINITE FOR b>2:
VACUUM POLARIZATION: SPIN 2
˚41˚
 SPIN 2 PART:
F
( 2)
85 2
 2H 2
4
( x; x' ) 
ln( a ) ( x  x' ) 
72 2
16 4
1

 2 1 
ln(
aa
'
)



E  
2 

3

x

 

5 2 H 2 2  ln(  2 x 2 )  5 2 H 6 (aa ' ) 2  L( y ) 2(2  y ) ln( y / 4)
 

 

 2 
4
2
144 4  x 2
144

4
y

y


5 2 H 2
 2H 2
4
G ( x; x' )  
ln( a) ( x  x' ) 
2
4
24 4
( 2)
2
y 
1

 2 1 
ln(
aa
'
)



E 
2 

3

  x 
 2 H 2 aa' 2
5 2 H 6 (aa' ) 2  (1  y ) L( y ) (3  y ) ln( y / 4) 
2  1 


 0    2  



4
4
96
72
4
4 y
 x 


 y
 y  y 1  y
L( y )  Li 2    ln   ln 1    ln 2  
4
4  4 2 4
CONTAINS QUALITATIVELY DIFFERENT TERMS when compared with the vac pol
of Leonard & Woodard obtained in a fixed (de Donder) gauge.
VACUUM POLARIZATION: SPIN 0
˚42˚
 SPIN 0 PART:
2
2
 2 1

 2H 2 4
 b 8  H
4
4


'

ln(
a
)

(
x

x
'
)


(
x

x
'
)


(
x

x
'
)



2
2
2
48 2
 b  2  72

 2b  1   48 aa'
( 0)
F ( x; x' )  
 

 2  '4  ln(  2 x 2 )   b  8   2 H 2 2  ln(  2 x 2 )   2 H 6
 b2  

  
 

 
(aa ' ) 2 N F ( y )

4
2
4
2
4
 384 aa '  x

  b  2  576
 x
 6




2
2
2

 2 H 2 2  ln(  2 x 2 )   2 H 6
 2b  1   H
4
2




G ( x; x' )  
1

ln(
a
)

(
x

x
'
)



(
aa
'
)
N
(
y
)
 

G
2
 12 4
192 4  x 2
 b  2   24


(0)
N F ( y), NG ( y) : COMPLICATED FUNCTIONS CONTAINING GENERALISED
HYPERGEOMETRIC FUNCTIONS AND THEIR DERIVATIVES
YOU DO NOT WANT TO SEE THEM! THEY STRONGLY DEPEND ON b.
 THE SPIN 0 & 2 STRUCTURE FUNCTIONS CAN BE USED TO CALCULATE
THE 1-LOOP EFFECT ON DYNAMICAL PHOTONS AND COULOMB POTENTIAL
ON DE SITTER IN PRESENCE OF GRAVITON LOOPS. UNLIKE Leonard, Woodard,
WE CAN STUDY GAUGE DEPENDENCE (we have a gauge parameter b) [in progress]
RENORMALIZATION
˚43˚
 WE USE DIM REG (RESPECTS ALL SYMMETRIES)
• SURPRISING RESULT: NEED A NON-COVARIANT COUNTER TERM C
(THAT ALSO VIOLATES dS SYMMETRY)
L( A, g )  C4 ( D) F   F g  g  g  g  C ( D) H 2 F F g  g  g
 C ( D) H 2 Fij Fkl g ik g jl  g
F  4a
D 4
C4 2 ( D  4)C4 H  D

2
 
 0  ( x  x' )
 C  (3D  8)C4 H 
aa'
a




G  4a D4 C  ( D  6)C4 H 2 D ( x  x' )
 FROM FLAT SPACE RESULT:
 D4
( D / 2)
C2  C0 (b)
C4 
D/2
2
64
( D  1)( D  2) ( D  3)( D  4)
( D  2) 2  Db  2 
C2  O( D  4), C0 (b)  

  O( D  4)
4
 b2 
2
RENORMALIZATION 2
˚44˚
2
ik jl
 THE NONCOVARIANT COUNTER TERM: L( A, g ) non cov  C ( D) H Fij Fkl g g  g
C
( 2)

 2 H D  4[8?]  ( D  2) ( D 2  8)( D  1)( D  2)
1



FINITE


(4 ) D / 2  ( D / 2)
2( D  2)( D  1)
D4


 2 H D  4 ( D  1) ( D  1)[( D  3)  ( D  2)]   D  1 
1

2


4


2
ln(
2
)
  2 

(4 ) D / 2 ( D / 2)
4( D  1)
D 1

 

C ( 0)
 ( D / 2) D23  bN  D23  bN   D 

1


1






1
1





(
D

2
)


b


b
2


N
N
 2 H D4
(3D  8)( D  2)
2
2





D/2
(4 ) 16( D  1)( D  2)(2  D / 2)  
   D3
  D3
 1
 1
  
 1

 bN   
 bN     bN     bN  
  2bN   2
  2
 2
 2
  

( D  1) 2 2( D  1)
3(3b  14)
bN 


4
b2
4(b  2)
THE ORIGIN OF NON-COVARIANT
RENORMALIZATION
˚45˚
 THE ORIGIN OF THE NONCOVARIANT (dS BREAKING) COUNTER TERM:
☼ GRAVITATIONAL INTERACTIONS HAVE TWO DERIVATIVES:
- increases superficial degree of divergence (gravity non-renormalizable)
☼ THE COINCIDENT GRAVITON PROPAGATOR DIVERGES ON dS AS:
1/(D-4) [fluctuations at all scales contribute: primarily UV effect]
☼ IN LORENTZIAN SIGNATURE: INTERACTIONS TIME ORDERED,
[exemplified by the (t-t’) and (t’-t) functions in the propagator]
IS THERE ANY WAY OF GETTING RID OF IT?
☼ INCOMPLETE (albeit covariant) GAUGE FIXING, thus an average gauge.
SINCE THE PHOTON AND GRAVITON ARE NOT GAUGE INVARIANT,
WRITING THE ACTION FOR A GAUGE INV COMBINATION MIGHT CANCEL
THE NON-COV CT. CRITICISM: CAN BE REALLY DONE ONLY AT
LINEAR ORDER IN GRAVITATIONAL PERTURBATIONS.
˚46˚
DYNAMICAL PHOTONS
 1LOOP EQUATION FOR DYNAMICAL PHOTONS:
 1 LOOP RETARDED RENORMALIZED VACUUM
POLARIZATION INDUCED BY 1 LOOP GRAVITONS
=
SOLVE THE ABOVE EQUATION PERTURBATIVELY [J=0]:
𝐴0 = 0, 𝐴𝑖 = 𝑖
(𝑘)𝑢(𝜂, 𝑘)𝑒 𝑖𝑘∙ 𝑥 ,
𝑢(𝜂, 𝑘)= 𝑢(0) (𝜂, 𝑘)+𝑢(1) (𝜂, 𝑘),
 RESULT (to leading order at late times, a):
 NONCOVARIANT GRAV.PROP.: 𝑢
 COVARIANT GRAV.PROP.: 𝑢
1
1
0
1
−𝑖𝑘𝜂
𝑒
2𝑘
Leonard, Woodard (2012,13)
𝜂, 𝑘 = 𝑢
𝜂, 𝑘 = 𝑢
𝑢(0) (𝜂, 𝑘)=
0
𝜂, 𝑘
𝜂, 𝑘
𝐺𝐻 2
 3𝜋
𝐺𝐻 2

3𝜋


𝑖𝑘 ln 𝑎

𝐻𝑎
𝑖𝑘 ln 𝑎

𝐻𝑎
6
45 −
2𝑖𝑘
𝐻
+ 5𝑒 2𝑖𝑘/𝐻
Glavan, Miao, Prokopec, Woodard (16)
UNEXPECTED: RESULT DEPENDS ON GRAVITON GAUGE!?:
˚47˚
PHYSICAL EFFECTS: POINT CHARGE
Glavan, Prokopec, Miao, Woodard 1308.3453(2013)
 RESPONSE TO POINT CHARGE q (at the origin) ON dS
 CLASSICAL
 ONE LOOP

C( 0) (r ) 
q
4 r

, r r

 2H 2
( 0) 
 (r )  C (r ) 
8 2
(1)
C
NB: SAME RESULTS
OBTAINED IN PHYSICAL
RADIAL COORDINATES


1


(  0 )  2 2 2  ln( aHr )  IRRELEVANT 
 3a H r




8 2 C(1) (r )

 2 H 2 C( 0) (r )
NB2: EFFECT GROWS
WITH DISTANCE
PHYSICAL DISTANCE/RH
NB3: POINT MAGNETIC DIPOLE: GIVES ANALOGOUS RESULTS
˚48˚
GRAVITON SELFENERGY FROM
MMC SCALARS ON DE SITTER
˚49˚
GRAVITON SELF-ENERGY: SCALARS
S. Park and R.P. Woodard (2011)
 AT 1 LOOP ON DE SITTER WE HAVE
=


2
COUNTER-TERMS
 AT 1 LOOP ON DE SITTER WE HAVE (F0,2: spin=0, 2 structure functions)
where:
LINEARISED WEYL
TENSOR:
˚50˚
SPIN 0 STRUCTURE FUNCTION
 DE SITTER INVARIANT SPIN=0 STRUCUTRE FUNCTION:
- here Li2 is dilogarithm function:
SPIN 2 STRUCTURE FUNCTION
˚51˚
 DE SITTER INVARIANT SPIN=2 STRUCUTRE FUNCTION:
MESSAGE: DE SITTER
INVARIANT, BUT
COMPLICATED!
˚52˚
EFFECT ON DYNAMICAL GRAVITONS
Park, Woodard, PRD, arXiv:1101.5804, 1109.4187 (2011)
Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014)
 RESULT: AT 1 LOOP SCALARS DO NOT AFFECT DYNAMICAL
GRAVITONS, i.e. NO TERMS THAT GROW SECULARLY IN TIME.
LICHNEROWICZ
ON DE SITTER
ᴥ PARK&WOODARD [in 1109.4187] SHOWED THAT THE EFFECT CAN BE
REDUCED TO A TIME-LIKE BD TERM. IS IT UNPHYSICAL?!?
ᴥ IT IS MORE CONVENIENT TO RECAST THE SELF-ENERGY IN THE
NON-COV. REPR. ANALOGOUS TO THE E-F REP FOR QED [Prokopec et al]
Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014)
GRAVITON SELF-ENERGY:
NON-COVARIANT REPRESENTATION
˚53˚
Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014)
 GENERAL STRUCTURE ON DE FLRW SPACES:
ᴥ where:
-SPIN=0 OBEY CONSERVATION IDENT’s:
-SPIN=2 STRUCTURE FUNCTIONS ARE TRANSVERSE AND TRACELESS:
AND CAN BE OBTAINED e.g. BY CONTRACTING LINEARIZED WEYL TENSORS:
NON-COV. STRUCTURE FUNCTIONS
˚54˚
Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014)
 RENORMALIZED SPIN 0 & 2 STRUCTURE FUNCTIONS (G0=0) :
NOTE: THE NON DS INVARIANT REPRESENTATION IS MUCH SIMPLER!
˚55˚
EFFECT ON DYNAMICAL GRAVITONS 2
 SOLVE THE 1PI 1 LOOP EQUATION PERTURBATIVELY:
GRAVITON PLANE WAVE ON DE SITTER:
POLARIZATION TENSOR IS TRANSV. &TRACELESS (in Lifshitz gauge):
PERTURBED METRIC HAS THE SAME FORM,
˚56˚
EFFECT ON DYNAMICAL GRAVITONS 3
ONE LOOP DE SITTER CONTRIBUTION TO THE RHS [
]
SINCE NO TERM GROWS
AS a², THERE ARE NO
GROWING SECULAR
TERMS IN TIME,
CONFIRMING THE
RESULT OF
PARK&WOODARD
˚57˚
PERTURBATIVE SOLUTION TO 1PI
EFFECTIVE
EOM
Park, Prokopec, Woodard, 1510.03352 [gr-qc]
● TREE LEVEL SOLUTION FOR POINT PARTICLE MASS M at r=0:
NEWTONIAN POTENTIALS
● METRIC PERTURBATION:
● PERTURBED 1PI EOM
𝐷𝜇𝜈𝜚𝜎 ℎ𝜚𝜎(1) (x)
● LICHNEROVICZ OPERATOR ON DE SITTER:
● SOLUTION: PERT CORRECTED GRAVITAT. POTENTIALS (long. gauge):
CONCLUSIONS AND OUTLOOK
˚58˚
MMC SCALARS DO NOT CAUSE SECULAR GROWTH OF
DYNAMICAL GRAVITON WAVE FUNCTION ON DE SITTER SPACE.
- PROBABLE REASON IS DERIVATIVE COUPLING OF GRAVITONS TO MMC
SCALARS. NON-MINIMAL COUPLING OR MASS COULD CHANGE THAT.
MMC SCALARS DO GENERATE SECULAR GROWTH (~ln(a))
OF GRAVITATIONAL POTENTIALS ON DE SITTER.
CONFORMAL CONTR
NON-CONFORMAL CONTRIBUTIONS:CAN BE
REINTERPRETED AS TIME-DEPENDENT
RESCALING OF MASS OR EQUIV NEWTON
CONST. (CORRESPONDING TO SCREENING)
OUTLOOK
˚59˚
WHAT WE WOULD REALLY LIKE TO KNOW IS HOW MATTER AND
GRAVITATIONAL LOOPS AFFECT DYNAMICAL TENSOR AND SCALAR
GRAVITON PERTURBS IN INFLATION. FOR THAT WE NEED GRAVITON
PROPAGATOR ON QUASI-DE SITTER (ACCEL.) SPACES, =d[1/H]/dt0.
STILL A LOT OF WORK TO BE DONE ON PERTURBATIVE
QUANTUM GRAVITY ON DE SITTER, EVEN AT 1 LOOP LEVEL!
.. LET ALONE RESUMMATIONS..
LATE TIME EFFECTS FROM QUANTUM INFLATIONARY PERTURB.: DE
D. Glavan, T. Takahashi, T. Prokopec (2016)