1. Infrared Phenomena of Field Theory in de Sitter space Takahiro Tanaka

2. Non-Gaussianity  Free field approximation is not sufficient. But once we take into account interaction, we may feel uneasy to continue to neglect loop corrections,  CMB non-Gaussianity might be measurable! Non-linear dynamics (Bennet et al, arXiv:1212.5225 ) arXiv:1212.5225 although it is a completely separate issue whether loop corrections during inflation are under our control or not.

3. Various IR issues IR divergence coming from k -integral Secular growth in time ∝ (Ht) n Adiabatic perturbation, which can be locally absorbed by the choice of time slicing. Isocurvature perturbation ≈ field theory on a fixed curved background Tensor perturbation Background trajectory in field space isocurvature perturbation adiabatic perturbation

4. Basics of IR behavior in de Sitter space Toy model for inflation Pathology of massless minimally coupled scalar field – No de Sitter invariant vacuum = Euclidean vacuum  : a minimally coupled scalar field with a mass m 2 in dS. :embedding of de Sitter in 5 dim Minkowski Scale inv. ~ de Sitter inv. Linear growth ≠ de Sitter inv. There is no O(4,1)/de Sitter invariant vacuum state Lorentz boost invariance in 5 dimension

5. To see what is going on in the massless limit, it is useful to consider in the closed chart. Euclidean vacuum spherical harmonics

6. EOM: positive frequency functions: Euclidean vacuum is determined by requiring regularity on a hemisphere when the positive frequency functions are continued to Euclidean.

7. Two point functions after summation is manifestly de Sitter invariance. angle between  and  ’. We again find that two point function is ill defined in the massless minimally coupled case ( m =  =0)

8. – Divergence comes from L =0 mode u 0 becomes const.×∞, and so it becomes effectively real.  Replacing the mode u 0 (  ) by another one at the expense of violating de Sitter invariance. (Allen-Folacci vacuum) regular at  →  i ∞ c.f. Klein-Gordon norm: Green function is finite, but not de Sitter invariant.  →   =   →  i ∞  /2

9. – Two point function in the new vacuum excluding L =0 mode To obtain this expression and solve with the boundary condition reflecting the source in the coincidence limit regularity at the anti-podal point No divergence in the massless limit

10. – Kirsten Garriga vacuum Problem is in the presence of O-mode ( L =0). Same approach to the O-mode: Instead, we should quantize this degrees of freedom like a free particle.  →0 limit is ill-defined Define vacuum state by In Schrodinger picture Kirsten & Garriga(1993) H.O.) Wave function does not depend on  0 The wave function itself looks regular and well-defined. But,. Hence, de Sitter inv.

11. – Kirsten Garriga vacuum is well-defined and de Sitter invariant. But the wave function is infinitely broad in the direction 1) Exactly massless limit is an artificial setup. If we introduce interaction, effective mass term will arise. In this case, is not an appropriate observable because it is not invariant under. So, is divergent Then observable will be 2) Exact masslessness is guaranteed by symmetry. (Shift-symmetry) After UV regularization for large separation Linear growth in time

12. Various IR issues IR divergence coming from k -integral Secular growth in time ∝ (Ht) n Adiabatic perturbation, which can be locally absorbed by the choice of time slicing. Isocurvature perturbation ≈ field theory on a fixed curved background Tensor perturbation Background trajectory in field space isocurvature perturbation adiabatic perturbation

13. § Isocurvature perturbation m 2 > 0 : de Sitter invariant vacuum state even with interaction exists. (rather trivial) If we choose de Sitter invariant vacuum at the beginning, the state remains unchanged. (trivial) So, there is no secular time evolution in this case! However, if the initial state is different, secular time evolution will happen. So maybe the secular growth is not an instability but just a relaxation process to the de Sitter invariant vacuum state. ≈ field theory in de Sitter space (Marolf and Morrison ( 1010.5327 )) Clustering property: when {x 1,…,x m } are far apart from {x m+1,…,x n } Perturbative stability of de Sitter invariant state

14. potential summing up only long wavelength modes beyond the Horizon scale distribution m2⇒0m2⇒0 Large vacuum fluctuation Subtle issue arises in the small mass limit. If the field fluctuation is too large, it is easy to imagine that a naïve perturbative analysis will break down once interaction is turned on. De Sitter inv. vac. state does not exist in the massless limit. Allen & Folacci(1987) Kirsten & Garriga(1993)

15. Stochastic interpretation Let’s consider local average of  : Equation of motion for  : More and more short wavelength modes participate in  as time goes on. Newly participating modes act as random fluctuation In the case of massless  4 :   2  → Namely, in the end, thermal equilibrium is realized : V ≈ T 4 ≈ H 4 (Starobinsky & Yokoyama (1994))

16. Wave function of the universe ~ parallel universes Distant universe is quite different from ours. Each small region in the above picture gives one representation of many parallel universes. However: wave function of the universe = “a superposition of all the possible parallel universes” Do “simple expectation values give really observables for us?” Our observable universe must be so to keep translational invariance of the wave fn. of the universe No!

17.  If we subtract the local average Identifying the dominant component of IR fluctuation O(k) unless |x| ≫ L size of our observable universe. Window function W(|x|)W(|x|) |x| |x| L Dominant IR fluctuation is concentrated on  then, with 

18. Decoherence of the wave function of the universe Decoherence Cosmic expansion Statistical ensemble Various interactions Superposition of wave packets Correlated BeforeAfter Un-correlated Coarse graining Unseen d.o.f. Our classical observation picks up one of the decohered wave packets. Sorry, but this process is too complicated. |a＞|a＞ |b＞|b＞ |c＞|c＞

19. Tensor perturbation Graviton ~ massless minimally coupled scalar field Phase space reduction = method that keeps only physical degrees of freedom. Difference is in the absence of k=0,1 modes – Origin of divergence is absent. – Existence of well behaved de Sitter invariant vacuum state at the linear level The same equation as in the scalar case However, de Sitter invariance is not manifest because the phase space reduction is non-covariantly.

20.  Covariant approach  Covariant gauge fixing term:  Manifestly de Sitter invariant.  In Euclidean, there seems no IR divergence even if we consider higher loop effects because of finiteness of the Euclidean sphere. Analytic continuation  However, not all Euclidean modes have positive eigenvalues. So even at the tree level the existence of de Sitter invariant graviton Green function is not so obvious.

21. Loop corrections Original claim by Tsamis and Woodard: → H Reduction of the effective   Even if there is de Sitter invariant vacuum, stability of dS invariant state is not guaranteed.  However, the evaluating expansion rate in pure gravity setup is very subtle.

22. Gauge issue H  H ≈  R, so you may think  H is gauge invariant. – However, ≠  R (1) ( ) =  R (1) ( )+  R (2) ( )+… is gauge invariant, but  R (1) ( ) varies under gauge transformation when we consider the second order perturbation of h . (At linear order there is no back reaction.) – This criticism was made by Unruh already in 1998.  is guage invariant ☺, but in fact we can formally show that the quadratic or higher order terms exactly cancel the tadpole term at each order of loop expansion.  This would be rather expected because R  =0 is EOM. (Garriga & Tanaka (2008))

23. Observable close to observation A conformally coupled field in pure de Sitter space:  = a  Abramo and Woodard found null result at one loop level by evaluating the expectation value of this operator. Non-local effect from inside of the past light cone  Another observable was proposed by Abramo and Woodard (2001) Similarly if we use minimally coupled scalar as a probe and ask the reversed question what is the source J that gives  =1, (Garriga & Tanaka (2008))

24. §IR divergence in single field inflation Factor coming from this loop: scale invariant spectrum curvature perturbation in co-moving gauge. - no typical mass scale Transverse traceless Setup: 4D Einstein gravity + minimally coupled scalar field Broadening of averaged field can be absorbed by the proper choice of time coordinate.

25. Gauge issue in single field inflation – In conventional cosmological gauge invariant perturbation theory, gauge is not completely fixed. Yuko Urakawa and T.T., PTP122: 779 arXiv:0902.3209 Time slicing can be uniquely specified:  =0 OK! but spatial coordinates are not. Residual gauge: Elliptic-type differential equation for   i. Not unique locally!  To solve the equation for   i, by imposing boundary condition at infinity, we need information about un-observable region. observable region time direction

26. Basic idea why we expect IR finiteness in single field inflation The local spatial average of  can be set to 0 identically by an appropriate gauge choice (=time-dependent scale transformation. Even if we choose such a local gauge, the evolution equation for  stays hyperbolic. So, the interaction vertices are localized inside the past light cone. Therefore, IR divergence does not appear as long as we compute  in this local gauge. But here we assumed that the initial quantum state is free from IR divergence.

27. Complete gauge fixing vs. Genuine gauge-invariant quantities Local gauge conditions. Imposing boundary conditions on the boundary of the observable region But unsatisfactory? The results depend on the choice of boundary conditions. Translation invariance is lost.  Genuine coordinate-independent quantities. No influence from outside Complete gauge fixing ☺ Correlation functions for 3-d scalar curvature on  =constant slice. R(x1) R(x2)R(x1) R(x2) Coordinates do not have gauge invariant meaning. x origin x(X A  =1) =X A +  x A x Specify the position by solving geodesic eq. with initial condition XAXA g R(X A ) := R(x(X A  =1)) = R(X A ) +   x A  R(X A ) + …  g R(X 1 ) g R(X 2 )  should be truly coordinate independent. (Giddings & Sloth 1005.1056) (Byrnes et al. 1005.33307) Use of geodesic coordinates:

28. In  =0 gauge, EOM is very simple IR regularity may require Non-linearity is concentrated on this term. Formal solution in IR limit can be obtained as with L -1 being the formal inverse of. Only relevant terms in the IR limit were kept. IR divergent factor ∋ Extra requirement for IR regularity

29. Then, satisfying is impossible, However, L -1 should be defined for each Fourier component. because for    ≡  d  k (e ikx v k (t) a k + h.c.), for arbitrary function f ( t, x ) while with IR regularity may require

30. However, Euclidean vacuum (selected by i  -prescription) and its excited states satisfy the IR regular condition. We still find the conditions required for the absence of IR divergences. “Wave function must be homogeneous in the direction of background scale transformation” If we naively set initial condition by choosing a free field vacuum at a finite time, these IR regularity conditions are not satisfied. Putting aside further complicated discussions, my current understanding is…

31. With this choice, IR divergence disappears. ・ extension to the higher order: IR divergent factor total derivative Instead, one can impose,, which reduces to conditions on the mode functions. with

32. What is the physical meaning of these conditions? Background gauge: Quadratic part in  and s is identical to s = 0 case. Interaction Hamiltonian is obtained just by replacing the argument  with  s. In addition to considering g R, we need additional conditions Physical meaning of IR regularity condition and its higher order extension. ~ ~ Therefore, one can use 1) common mode functions for   and   ~    ≡  d  k (e ikx v k (t) a k + h.c.)    ≡  d  k (e ikx v k (t) a k + h.c.) ~ ~ 2) common iteration scheme.

33. However, the Euclidean vacuum state (defined by the regularity  0 →± i ∞ ) satisfies this condition. (Proof will be given in our coming paper) It looks quite non-trivial to find consistent IR regular states. D k v k (  0 )=0 : incompatible with the normalization condition. We may require the previous condition compatible with Fourier decomposition Retarded integral with  (  0 )=  I (  0 ) guarantees the commutation relation of 

34. Isocurvature mode Potentially large IR fluctuation of isocurvature mode is physical. - Stochastic inflation - But what we really measure is not a simple expectation value. We need to develop an efficient method to compute “conditional probability” in field theory. Tensor perturbation There seem to be no IR cumulative effect of tensor modes. But rigorous proof is lacking. Adiabatic mode Adiabatic perturbation in the long wavelength limit is locally “gauge”. Even if we compute in global gauge, true “gauge”-invariant observables should be free from large IR effects. However, possible quantum state is restricted since the residual “gauge” is not completely fixed.

35. §Summary about Graviton loop (Tsamis & Woodard (1996,1997)) 2-loop order computation of in pure gravity with cosmological constant. There are several issues: 1)Initial vacuum is dS inv. free vacuum, so it might be just a relaxation process toward the true interacting dS inv. vac. 2)The expansion rate of the universe is not gauge invariant when there is no marker to specify the hypersurface. 3)If we evaluate the scalar curvature R instead of H, R is really constant. Screening of  ? Graviton is often analogous to a massless minimally coupled scalar field, but dS inv. vacuum exists for gravition. (Allen & Turyn (1987)) (Garriga & Tanaka (2008)) (Unruh (1998)) Graviton in the long wavelength limit is locally gauge, isn’t it? But, proving that there is no IR effect from graviton is not trivial. Without reference fields one cannot specify a spacetime point gauge invariantly. But only gauge invariant quantities are observables. (See Kitamoto & Kitazawa (2012))

36. This derivation already indicates that the leading term in the squeezed limit given by the consistency relation vanishes once we consider “genuine coordinate independent quantities”. Super-horizon long wavelength mode k 1 should be irrelevant for the short wavelength modes k 2, k 3. In the squeezed limit, 3pt fn is given by power and index of spectrum. Standard consistency relation for k 1 << k 2, k 3 The only possible effect of k 1 mode is to modify the proper wave numbers corresponding to k 2, k 3., neglecting tensor modes Let’s consider  in geodesic normal coordinates X ~ e  k 1 x ∵ g  k 2 and g  k 3 are not correlated with the long-wavelength mode  k 1. ≈ 0 If we consider a state:IR divergences!

37. §IR divergence in single field inflation Factor coming from this loop: scale invariant spectrum curvature perturbation in co-moving gauge. - no typical mass scale Transverse traceless = IR divergence caused by adiabatic perturbation

38. Special property of single field inflation – In conventional cosmological perturbation theory, gauge is not completely fixed. Yuko Urakawa and T.T., PTP122: 779 arXiv:0902.3209 Time slicing can be uniquely specified:  =0 OK! but spatial coordinates are not. Residual gauge d.o.f. Elliptic-type differential equation for   i. Not unique locally!  To solve the equation for   i, by imposing boundary condition at infinity, we need information about un- observable region. observable region time direction

39. Basic idea of the proof of IR finiteness in single field inflation The local spatial average of  can be set to 0 identically by an appropriate (but non-standard) gauge choice. Even if we choose such a local gauge, the evolution equation for  formally does not change, and it is hyperbolic. So, the interaction vertices are localized inside the past light cone. Therefore, IR effect is suppressed as long as we compute  in this local gauge.

40. Complete gauge fixing vs. Genuine gauge-invariant quantities Local gauge conditions. Imposing boundary conditions on the boundary of the observable region But unsatisfactory? The results depend on the choice of boundary conditions  Genuine gauge-invariant quantities. No influence from outside Complete gauge fixing ☺ Correlation functions for 3-d scalar curvature on  =constant slice. R(x1) R(x2)R(x1) R(x2) Coordinates do not have gauge invariant meaning. x origin x(X A  =1) =X A +  x A x Specify the position by solving geodesic eq. with initial condition XAXA g R(X A ) := R(x(X A  =1)) = R(X A ) +   x A  R(X A ) + …  g R(X 1 ) g R(X 2 )  should be genuine gauge invariant. Translation invariance of the vacuum state takes care of the ambiguity in the choice of the origin. (Giddings & Sloth 1005.1056) (Byrnes et al. 1005.33307)

41. One-loop 2-point function at leading slow-roll exp. No interaction term in the evolution equation at O(  0 ) in flat gauge. ◎ R(X A ) ~ e -2    g R(X 1 ) g R(X 2 )     g R  (X 1 )  g R  (X 2 )  g R  (X 1 )  g R  (X 2 )  g R  (X 1 )  g R  (X 2 )  ∝ ∝     d(logk)k    (D  u k (X 1 ))  (u * k (X 2 )) + 2  (Du k (X 1 ))  (Du * k (X 2 )) +  (u k (X 1 ))  (D  u * k (X 2 ))  + c.c. + (manifestly finite pieces)     u k a k + u * k a † k & D  loga   (x  ) u k  k  (1-ik/aH)e ik/aH where ◎ flat gauge  synchronous gauge  IR divergence from     in general. However, the integral vanishes for the Bunch-Davies vacuum state. Du k  k   logk (k  u k ) ∵  g R(X 1 ) g R(X 2 )       d(logk)  logk  (k  u k (X 1 ))  (k  u * k (X 2 ))  + c.c.   To remove IR divergence, the positive frequency function corresponding to the vacuum state is required to satisfy Du k  k   logk (k  u k ). ◎R  gR◎R  gR ∝

42. 3) Giddings and Sloth’s computation assumed adiabatic vacuum and they found no IR divergence. This means that our generalized condition of scale invariance should be compatible with the adiabatic vacuum choice. 1) To avoid IR divergence, the initial quantum state must be “scale invariant/Bunch Davies” in the slow roll limit. 2) To the second order of slow roll, a generalized condition of “scale invariance” to avoid IR divergence was obtained, and found to be consistent with the EOM and normalization. “Wave function must be homogeneous in the residual gauge direction” Summary of what we found

43. In  =0 gauge, we find Story was not so simple.  There is no divergence for any vacuum of free field. If we choose this solution, curvature perturbation in the geodesic normal coordinates becomes  However, this does not mean that the initial quantum state is arbitrary  in the sense a special non-linear solution was selected. is a solution in the approximation that keeps only the terms potentially related to IR divergences. Interaction picture field In flat gauge we trivially have  =   at the leading order of slow roll expansion. gauge transform which is different from Use of ( x ・∇ ) is acceptable? In flat gauge strange homogeneous solution must be added to get.

44. Tree level 2-point function 2-point function of the usual curvature perturbation is divergent even at the tree level.  (X 1 )  (X 2 )      (X 1 )   (X 2 )    d(logk)k   u k (X 1 ) u * k (X 2 )   c.c.     u k a k + u * k a † k u k  k  (1-ik/aH)e ik/aH for Bunch Davies vacuum where Logarithmically divergent!  Why there remains IR divergence even in BD vacuum?  is not gauge invariant, but g R(X) ≈R(X) is. Local gauge-invariant quantities do not diverge for the Bunch- Davies vacuum state.  g R(X 1 ) g R(X 2 )   ≈  d(logk)k    u k (X 1 )  u * k (X 2 )   d(log k) k   d(logk)k   u k (X 1 ) u * k (X 2 )   d(logk)  Of course, artificial IR cutoff removes IR divergence  d(log k)k   u k (X 1 ) u * k (X 2 )   d(log k)P k but very artificial!

45. One-loop 2-point function at the next leading order of slow-roll. (YU and TT, in preparation) At the lowest order in , Du k  (  log a  x i  i ) u k  k   logk (k  u k ) was requested. Some extension of this relation to O(  ) is necessary: Natural extension is Notice that u k  e ikx should have the same coefficient IR divergence can be removed by an appropriate choice of the initial vacuum even if we consider the next leading order of slow roll. EOM for f : consistent with the above requirement !!

46. Candidates of observable As a substitute for  k :  is free from the divergence of log k min ? size of our observable universe. Window function W(r)W(r) r L No! No!  Recall the diagram IR divergence is not removed Do you like these ? or y

47. Introduction Are there de Sitter invariant vacuum for graviton? – Graviton ~ massless minimally coupled scalar does not have dS invariant Fock vacuum Relaxing cosmological constant? – Woodard gr-qc/0408002 – Sec.3 What Bothers People String theorist, Loop space gravity people, phenomenologist, mathematical relativists – Sec.4 You Can’t Always Get What You Want chukan-happyo September 200447

48. Conclusion There is no evidence that supports the presence of significant back reaction. But still it is an open question if there is such an IR cumulative effect or not. – No-go theorem is expected. System including inflaton field must be in our scope. chukan-happyo September 200448

49. How to pick up one decohered history Discussing quantum decoherence is annoying. – Which d.o.f. to coarse-grain? – What is the criterion of classicality? To avoid subtle issues about decoherence, we propose to introduce a “projection operator”. Picking up one history is difficult. Instead, we throw away the other histories presumably uncorrelated with ours. We compute with over-estimate of fluctuations (Urakawa & Tanaka PTP122:1207) One un-satisfactory proposal

50. IR finiteness ∵ Expansion in terms of interaction-picture fields: Integration over the vertex y is restricted to the region within the past light-cone. One of Green fns. is retarded, G R. x y Window fn. For each  y, IR fluctuation of  int ( y ) is suppressed since  is restricted. OK to any order of loop expansion! y x x' time x’ Projection acts only on the external lines. How the contribution from the IR modes at k ≈ k min is suppressed?

51. IR finiteness  ≈  r :past light cone Then, G ( y, y ) ≈ becomes large.  y -integral looks divergent, but homogeneous part of  is constrained by the projection.  x G R ( x, y ) → 0 faster than G R ( x, y ) for  y →  ∞ looks OK, at least, at one-loop level ! looks OK, at least, at one-loop level ! Past light cone during inflation shrinks down to horizon size. y  G R ( x, y ) →constant for  y →  ∞ However, for  y →  ∞, the suppression due to constraint on  gets weaker. ∞∞∞∞∞ Window fn. time  ~ secular growth in time