1. DE SITTER BREAKING FROM GRAVITONS AT THE ONE LOOP LEVEL ˚ 1˚ Tomislav Prokopec, ITP Utrecht University Mainz, 25 Jun 2015 D. Glavan S.P. Miao, T. Prokopec and R.P. Woodard, Class.Quant.Grav 31 (2014) 175002, [arXiv:1308.3453 [gr-qc]] 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, CQG (2015?)

2. CONTENTS ˚ 2˚ 4) STABLE THEORY: SCALAR ELECTRODYNAMICS 6) CONCLUSIONS AND OPEN PROBLEMS 1) QUANTUM EFFECTS ON DE SITTER 2) REVIEW OF THE MASSLESS SCALAR FIELD 3) WHEN STABILITY FAILS.. YUKAWA THEORY 5) PERT QUANTUM GRAVITY ON DE SITTER: VACUUM POLARIZATION

3. ˚ 3˚ QUANTUM EFFECTS ON DE SITTER: SCALARS

4. ˚ 4˚ QUANTUM FIELDS ON DE SITTER ● EXAMPLE 1: MASSLESS SCALAR ON DE SITTER (D=4): ● ONE CAN QUANTIZE: SCALAR, FERMIONIC, VECTOR AND TENSOR (GRAVITON) FIELDS ON DE SITTER, AND CONSIDER THEIR (NON-) PERTURBATIVE EVOLUTION AND BACKREACTION ON dS BACKGROUND ACTION:  EOM for  :  In a dS invariant state, the propagator must be a function of dS inv distance l(x;x’) :  then the propagator obeys a dS inv equation:  This equation has no solution! Allen, Folacci, PRD35 (1987)

5. ˚ 5˚ MASSLESS SCALAR ON DE SITTER ● THE NAIVE SOLUTION (in D=4) is (up to an irrelevant constant):  This UNIQUE dS inv solution solves the wrong equation:  Here denotes the antipodal point of : ● MASSLESS MINIMALLY COUPLED SCALAR (MMCS) D=4:  From it follows that the terms in 2nd square brackets [.] source the second  -function in Eq. (**).  The solution (**) follows from:

6. ˚ 6˚ MASSLESS SCALAR ON DE SITTER 2 ● ONE POSSIBLE SOLUTION (that breaks dS sym, respects spatial homogeneity, but solves the right equation) is:  This dS breaking solution solves the right equation, but (when dim reg is applied) the coincident propagator grows (linearly) with time. ● CONCLUSION: MMCS cannot be quatized in a dS inv way.  This has physical consequences. For example, when one introduces a quartic interaction at one loop one generates a Hartree mass:  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?

7. ˚ 7˚ MASSIVE SCALAR FIELD PROPAGATOR ● dS INVARIANT SCALAR FIELD PROPAGATOR ● COINCIDENT SCALAR PROPAGATOR Chernikov, Tagirov, Annales Poincaré Phys.Theor. A9 (1968)  The m→0 LIMIT IS SINGULAR; EXPLAINS WHY THERE IS NO dS INV SOLUTION.

8. ˚ 8˚ SCALAR FIELD: MASS GENERATION ● RESUMMATION: SELF-CONSISTENT HARTREE: ● in a self-interacting scalar theory, scalar has a growing mass. Is there a dS inv late time limit? ● ACTION ● GAP EQUATION ● SOLUTION Ford,Vilenkin, PRD 26 (1982); Serreau; Garbrecht, Rigopoulos (2011+) ● THERE ARE STUDIES OF 2 and 3 LOOPS (primarily to check Stochastic inflation; also w<-1 found) Kahya, Onemli, Woodard (2010); Onemli, Woodard (2004) Prokopec; Prokopec, Lazzari (2012+)

9. ˚ 9˚ SCALAR FIELD: MASS GENERATION ● STOCHASTIC INFLATION RESULT  QUALITATIVELY THE SAME AS THE GAP EQUATION, BUT QUANTITATIVE DIFFERENCE: Starobinsky, Yokoyama 1996

10. ˚10˚ WHEN STABILITY FAILS: YUKAWA THEORY

11. ˚11˚ FERMIONS IN YUKAWA ● AT ONE LOOP IN YUKAWA THEORY PHOTONS ACQUIRE A MASS: ► m=scalar mass; y=Yukawa, ξ=nonminimal coupling, R=Ricci scalar Garbrecht, Prokopec, PRD73 (2006) STOCHASTIC THEORY IS OBTAINED BY INTEGRATING OUT THE FERMIONS. ONE OBTAINS THE FOLLOWING EFFECTIVE POTENTIAL:..AND THE CONTRIBUTION TO THE ENERGY DENSITY IS: Miao, Woodard, PRD (2006), gr-qc/0602110

12. ˚12˚ EFFECTIVE POTENTIAL IS UNSTABLE ● EFFECTIVE POTENTIAL TWO DIFFERENT VALUES OF V 0 CONTRIBUTION TO THE ENERGY DENSITY  unstable evolution

13. ˚13˚ STABLE THEORY: SCALAR ELECTRODYNAMICS

14. ˚14˚ SCALAR ELECTRODYNAMICS ● ONE RELEASES SYSTEM FROM A FREE VACUUM STATE ● (BARE) LAGRANGIAN ● THE PHOTON GAINS A (PERTURBATIVE) MASS ● THE SCALAR CAN BE KEPT (PERTURBATIVELY) MASSLESS Prokopec, Tornkvist, Woodard (2002,2003); Prokopec, Puchwein, (2004) ● QUANTUM BACKREACTION: 2 LOOP STRESS-ENERGY TENSOR Prokopec, Tsamis, Woodard (2007,2008)  Initial boundary divergences lead to exponentially decaying tails, can be removed by chosing a suitable (pertubative) interacting state

15. ˚15˚ ● SOME OF THE CONTRIBUTING DIAGRAMS

16. ˚16˚ 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

17. ˚17˚ STOCHASTIC SCALAR QED  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 V eff (  ). Prokopec, Tsamis, Woodard, Ann.Phys.323 (2008) [0707.0847 [gr-qc]] ♣ the scalar and photon acquire a (nonperturbative) mass: ♣ fluctuations give constant contributions to the density invariants & c.c.: NB: These results are fully nonperturbative (not suppressed by the coupling const. e) ☺ RESULTS: CONCLUSION: IN SQED ASYMPTOTICALLY ONE REACHES A dS INVARIANT, STABLE STATE OF LOWER ENERGY NB: AGREE WITH

18. ˚18˚ AFTER INTEGRATING THE (PASSIVE) PHOTONS ONE GETS THE EFFECTIVE SCALAR POTENTIAL V eff SMALL FIELD

19. ˚19˚ CONTRIBUTION TO ENERGY DENSITY FROM THE EFFECTIVE SCALAR THEORY

20. QUANTUM GRAVITY ON DE SITTER: STABLE OR UNSTABLE? ˚20˚

21. ˚21˚ GRAVITONS ON DE SITTER NON-COVARIANT APPROACH ● IDENTICAL PROBLEM AS MMCS: NO dS INV PROPAGATOR!? ● WHEN ONE FULLY FIXES THE GAUGE, GRAVITONS ON dS CAN BE THOUGHT AS TWO POLARIZATIONS OF A MMCS:  EOM for GRAVITONS :  BUT the RHS PROJECTORS P ij 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!  LHS: THE SAME OPERATOR AS FOR MCMS!  no solution! Mora, Tsamis, Woodard, J.Math.Phys.53(2012); Miao, Tsamis, Woodard, J.Math.Phys.52 (2011) Domazet, Prokopec, in prograss This propagator cannot be used to study de Sitter breaking (gauge dependence?) of physical quantities.

22. ˚22˚ COVARIANT GRAVITON PROPAGATOR ● EXACT COVARIANT DE DONDER GAUGE (operator):  THE GRAVITON PROPAGATOR CONTAINS SPIN 2 AND SPIN 0 PARTS Mora, Tsamis, Woodard, J.Math.Phys.53(2012); Miao, Tsamis, Woodard, J.Math.Phys.52 (2011) ● SPIN 2 PART (TT on both index groups) MMCS: EXPECT dS BREAKING {b=gauge parameter)

23. ˚23˚ COVARIANT GRAVITON PROPAGATOR 2 ● SPIN 0 PART (spin 0 projectors) ● GAUGE PARAMETER TACHYONIC WHEN  1) 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

24. ˚24˚ ONE LOOP VACUUM POLARIZATION  WE USED THE COVARIANT GRAVITON AND PHOTON PROPAGATORS Glavan, Miao, TP, Woodard (2015)  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. DIAGRAMS

25. ˚25˚ VACUUM POLARIZATION ON dS  USE THE COVARIANT GRAVITON AND PHOTON PROPAGATORS TO CALC Glavan, Miao, TP, Woodard (2015)  GRAVITON INDUCED 1 LOOP VAC POL: + COUNTER TERMS  USEFUL (F-G/EL-MAG) REPRESENTATION:  ON MINKOWSKI: LORENTZ INVARIANT (G=0) BUT GAUGE DEPENDENT RESULT NEGATIVE SEMIDEFINITE FOR b>2: Prokopec, Tornkvist, Woodard; Leonard, Prokopec, Woodard 1304.7265, 1210.6968

26. ˚26˚ VACUUM POLARIZATION: SPIN 2  SPIN 2 PART: CONTAINS QUALITATIVELY DIFFERENT TERMS when compared with the vac pol of Leonard & Woodard obtained in a fixed (de Donder) gauge.

27. ˚27˚ VACUUM POLARIZATION: SPIN 0  SPIN 0 PART: COMPLICATED FUNCTIONS CONTAINING GENERALISED HYPERGEOMETRIC FUNCTIONS AND THEIR DERIVATIVES YOU DO NOT WANT TO SEE THEM! THEY STRONGLY DEPEND ON b.

28. ˚28˚ RENORMALIZATION  WE USE DIM REG (RESPECTS ALL SYMMETRIES) SURPRISING RESULT: NEED A NON-COVARIANT COUNTER TERM  C (THAT ALSO VIOLATES dS SYMMETRY)  FROM FLAT SPACE RESULT:

29. ˚29˚ RENORMALIZATION 2  THE NONCOVARIANT COUNTER TERM:

30. ˚30˚ THE ORIGIN OF NON-COVARIANT RENORMALIZATION  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 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.

31. ˚31˚  LOOP EFFECTS ON PHOTONS ON dS: 1PI EOM PHYSICAL EFFECTS OF GRAVITONS  CAN STUDY EFFECTS WITH CLASSICAL SOURCES: point charge, dipoles OR ON DYNAMICAL PHOTONS (such as vacuum fluctuations) NB: SO FAR THE EFFECTS WERE STUDIED BY USING THE Leonard, Woodard VAC POL calculated in a fixed (de Donder gauge): NOT (YET) POSSIBLE TO STUDY GAUGE (IN-)DEPENDENCE. Leonard, Woodard 1202.5080, 1304.7265 (2012,’13) NB: THERE IS ALSO 2PI: EFFECTS ON THE PHOTON CORRELATOR NB2: TO GET A GAUGE INDEPENDENT ANSWER, ONE WOULD HAVE TO DO AN ANALOGOUS CALCULATION AS DONE BY BJERRUM BOHR. HARD!

32. ˚32˚  RESPONSE TO POINT CHARGE q (at the origin) ON dS PHYSICAL EFFECTS: POINT CHARGE  CLASSICAL  ONE LOOP NB: SAME RESULTS OBTAINED IN PHYSICAL RADIAL COORDINATES PHYSICAL DISTANCE/R H NB2: EFFECT GROWS WITH DISTANCE NB3: POINT MAGNETIC DIPOLE: GIVES ANALOGOUS RESULTS Glavan, Prokopec, Miao, Woodard 1308.3453(2013)

33. ˚33˚  1 LOOP MODIFICATION OF DYNAMICAL PHOTON PHYSICAL EFFECT: DYNAMICAL PHOTON  CLASSICAL  ONE LOOP Wang, Woodard 1408.1448 (2014)

34. ˚33˚ GRAVITON SELF-ENERGY: SCALARS  USE MMC SCALAR PROPAGATOR AND 3 and 4 GRAVITON-SCALAR VERTICES Park, Woodard, PRD, arXiv:1101.5804, 1109.4187 (2011) Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014) COUNTER TERMS  AT 1 LOOP SCALARS DO NOT AFFECT GRAVITONS SIGNIFICANTLY ON dS, i.e. NO TERMS THAT GROW SECULARLY IN TIME Leonard, Park, Prokopec, Woodard, PRD, 1403.0896 (2014)  FOR DETAILS SEE  NEWTONIAN (BARDEEN) POTENTIALS on dS COULD BE AFFECTED AS

35. DISCUSSION WHILE IT IS NOT KNOWN HOW TO STUDY EFFECTS OF QUANTUM GRAVITON FLUCTUATIONS, PROGRESS HAS BEEN MADE FOR SCALAR, VECTOR AND FERMIONIC FIELDS (both using perturbative methods and stochastic inflation). ˚35˚ PHYSICS OF DE SITTER IS ESSENTIALLY NONPERTURBATIVE (for massless and light scalars, gravitons, and other fields that couple to them). SCALAR AND VECTOR FIELDS ACQUIRE/YIELD COMPUTABLE CORRECTIONS DURING INFLATION; FERMIONIC FIELDS TEND TO DESTABILIZE DE SITTER. QUANTUM FLUCTUATIONS OF SCALARS ARE STRONG ENOUGH TO RESTORE A BROKEN SYMMETRY IN THE CASE OF REAL AND N-COMPONENT SCALAR FIELD WITH AN O(N) SYMMETRY. LATE TIME STATE CAN BE COMPUTED: FLUCTUATIONS ARE LIGHT AROUND THE ORIGIN, HEAVY AT LARGE FIELD VALUES.

36. DISCUSSION ON GRAVITONS ˚36˚ WE HAVE THE OLD RESULT OF TSAMIS AND WOODARD ON 2 LOOP GRAVITON QBR ON dS obtained using cut-off regularization. Need to be checked by using covariant graviton propagator & dim reg. Gauge dependence ill understood. No scattering matrix on dS available. PROGRESS ON UNDERSTANDING PERTURBATIVE AND NON- PERTURBATIVE EFFECTS IN GRAVITATIONAL SECTOR HAS BEEN SLOW BUT STEADY. SOME (IMPARTIAL) RESULTS ON 1 LOOP VACUUM POLARIZATION AVAILABLE. MORE PROGRESS NEEDED TO UNDERSTAND GAUGE DEPENDENCE OF: Coulomb Force, effects on dynamical photons, etc.

37. SYMMETRY RESTORATION ON DE SITTER ˚37˚ Lazzari, Prokopec, arXiv:1304.0404 [hep-th] (2013) Prokopec, JCAP 1212 (2012) [arXiv:1110.3187 [gr-qc]]

38. ˚38˚ MEAN FIELD MASS ● MEAN FIELD MASS (self-consistent Hartree approximation) ● BROKEN SYMMETRY CASE: m²<0: ◙ SOLVING THIS YIELDS (BROKEN SYMMETRY CASE, m²<0): ◙ QUESTION: HOW ACCURATE IS THIS RESULT/METHOD? ◙ ANSWER: ONE CAN CHECK IT BY USING STOCHASTIC THEORY. NB: ANALOGOUS ANALYSIS CAN BE DONE FOR O(N) MODEL  MASSIVE WOULD-BE GOLDSTONE BOSONS (??)

39. ˚39˚ EFFECTIVE ACTION APPROACH ● USE THE FOKKER PLANCK EQUATION: ◙ USE A LEGENDRE TRANSFORM  PARTITION FUNCTION NB: This method integrates out all super-Hubble fluctuations and yields a PDF for deep infrared fields at asymptotically late times. ◙ ASSUME FOR SIMPLICITY AN INITIAL STATE ◙ CONFORMAL DIAGRAM OF dS: flat and global coordinates Lazzari, Prokopec, arXiv:1304.0404 [hep-th] (2013)

40. ˚40˚ PROOF OF SYMMETRY RESTORATION By Cauchy-Schwarz Thm (f= , g=1): ( averaging taken w.r.t. PDF  ≥0; equality holds when f  g ) ◙ LEGENDRE TRANSFORM

41. ˚41˚ ASYMPTOTIC STATE FOR REAL SCALAR ● EFFECTIVE ACTION AND ITS PDF: SYM BREAKING CASE ◙ CONCLUSION: SYMMETRY GETS RESTORED FOR ARBITRARY TREE POTENTIAL! SAME CONCLUSION REACHED IN THE O(N) CASE: GOLDSTONE THM RESPECTED (GOLDSTONE BOSONS REMAIN MASSLESS) ◙ HENCE : quantum fluctuations in dS strong enough to restore symmetry! ◙ For large  (weak ) at  ~  0 : a sudden turn in V eff (a `Maxwell construction’) invert  labeling!

42. ˚42˚

43. ˚43˚ ASYMPTOTIC STATE REAL SCALAR 2 ● ANALYTIC APPROXIMATIONS FOR EFFECTIVE ACTION  FLUCTUATIONS GENERATE A LARGE AMOUNT OF ENERGY ♥ FOR SMALL FIELD VALUES: ♥ FOR LARGE FIELD VALUES: ◙ FLUCTUATIONS ARE LIGHT FOR SMALL BACKGROUND FIELD; HEAVY FOR LARGE VALUES OF BACKGROUND FIELD

44. ˚44˚ ASYMPTOTIC STATE SCALAR O(N) ● similar as in O(1) case: symmetry gets restored ◙ MASS FOR SMALL/LARGE SELF-COUPLING ♥ FOR SMALL FIELD VALUES: ◙ AS IN O(1) MODEL: FLUCTUATIONS ARE LIGHT FOR SMALL BACKGROUND FIELD; HEAVY FOR LARGE VALUES OF BKG FIELD  agrees with the O(1) case (with N  1,  /6)