Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 2015 Background: 1 / 34

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1 Quantum Scalar Corrections to the Gravitational Potentials on de Sitter Background: Sohyun Park Korea Astronomy and Space Science Institute (KASI) APCTP FRP Gravitation and Numerical Relativity Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 1 / 34

2 Outline Quantum Scalar Corrections to the Gravitational Potentials on de Sitter Background arxiv: SP, Prokopec and Woodard Question: How quantum fluctuations during inflation affect gravity? Why it is interesting to see quantum fluctuations during inflation? A three-step procedure to study the effect Summary and Discussion Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: / 34

3 Why quantum fluctuations during inflation? A generic prediction of inflation: particle production Quantum fluctuations are amplified and preserved to late times so that they seed large scale structure formation. Expansion of spacetime can lead to particle creation by delaying the annihilation of virtual pairs ripped out of the vacuum: Schrödinger, Physica 6 (1939) 899 The effect is maximized if the expansion is accelerated: (the cosmological patch of) de Sitter a(t) = e Ht, H = const. ɛ = 0 Quantum effects during inflation Quantum field theory in de Sitter the virtual particles are massless The persistent time gets longer if their mass is smaller, so massless particles live the longest the virtual particles have no conformal symmetry. The emergence rate of conformally invariant particles is suppressed as 1/a. Parker Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 3 / 34

4 Energy-time uncertainty principle and lifetime Energy-time uncertainty principle in flat space: E t 1 would NOT notice a violation of energy conservation provided t 1/ E A virtual pair emerged with mass m, wavenumber k: before after lifetime flat space E = 0 E = m + k 1 t In expanding universe: k phys = inflation E = 0 E = m + k a (t) m +k k with time-dependent a(t) a(t) t+ t t dt m + k a (t ) 1 To get the maximum life take m = 0 and de Sitter a(t) = a I e Ht k t+ t t dt 1 a(t ) 1 k Ha(t) (1 e H t ) 1 Massless particles created during inflation can live forever if they emerge with k Ha(t). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 4 / 34

5 Conformal invariance and the emergence rate In flat space: the emergence rate Γ flat = const. (Poincare invariance) In an expanding universe: Γ(t) (no time translation invariance) For particles with conformal invariance the rate gets reduced with time: 1 Take dt = a(t)dη then the FRW metric becomes: ds = dt + a (t)d x d x = a (t)( dη + d x d x) Do conformal transf. for a theory possessing conformal symmetry with Ω = 1 a(t) : in D = 4, g µν Ω (x)g µν, A µ A µ, ψ ψω 3/, φ φω 1 In (η, x) coords. the theory and FRW metric: locally identical to flat space, dn dη = Γ flat Back to physical time dn dt = dn dη = Γ flat dη dt a(t) In an expanding universe, the emergence rate of virtual particles possessing conformal symmetry is suppressed by a factor of 1/a. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 5 / 34

6 Quantum fluctuations of MMC scalars and gravitons Only two types of particles with no mass & no conformal symmetry: gravitons massless, minimally coupled (MMC) scalar Occupation number of these two particles grows with time: N(k, t) = ( Ha(t) k ) significant for infrared wave numbers k Ha Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 6 / 34

7 Particle Production during Inflation Consider the massless, minimally coupled scalar ϕ(t, x): L = 1 µϕ νϕg µν g In the FRW universe, after Fourier transf. to momentum space L = d 3 xl becomes L = ] d 3 k 1 a3 ϕ(t, k) ϕ (t, k) 1 ak ϕ(t, k) ϕ (t, k) (π) 3 [ No coupling between different k s: pick one mode with k and call it q(t) L = 1 a3 q 1 k aq : a harmonic oscillator with m(t) a 3 (t) and ω(t) = k a(t) Hamiltonian: H(t) = 1 m(t) q + 1 m(t)ω (t)q with E.O.M: q + 3H q + k a q = 0 solution for de Sitter (a(t) = e Ht, H = const.) q(t) = u(t)α + u (t)α, u(t, k) = H k 3 [ 1 ik Ha(t) ] e ik Ha(t) The expectation value of the energy of the Bunch-Davis vacuum Ω (which is defined as the state with minimum energy in the distant past, α Ω = 0) is [ ( ) ] [ ] < Ω H(t) Ω >= k 1 a + Ha = k 1 k a + N(t) Number of particles with wave number k grows significantly for k Ha: ( ) Ha(t) N(k, t) = k Number grows: significant quantum effects Growth is for infrared k: computed reliably in the sense of low energy effective field theory Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 7 / 34

8 Using Quantum Gravity as an Effective Field Theory Quantum gravity is not perturbatively renormalizable. Completely reliable predictions about long range quantum gravitational phenomena can be made by using perturbative general relativity as a low energy effective field theory. The UV divergences can be absorbed by BPHZ counterterms. The finite, nonlocal terms from primitive diagrams dominate over the finite part of local counterterms. General Relativity as an effective field theory: The leading quantum corrections, John F. Donughue gr-qc/ Counterterms are distinct from primitive terms: BPHZ counterterms are guaranteed to be local Massless particles make nonlocal corrections For infrared regime the finite, nonlocal terms dominate over the local counterterms: Local Counterterm Effects p 4 Nonlocal Primitive Effects p 4 ln(p ) Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 8 / 34

9 Using Quantum Gravity as an Effective Field Theory Quantum gravity is not perturbatively renormalizable. Completely reliable predictions about long range quantum gravitational phenomena can be made by using perturbative general relativity as a low energy effective field theory. The UV divergences can be absorbed by BPHZ counterterms. The finite, nonlocal terms from primitive diagrams dominate over the finite part of local counterterms. General Relativity as an effective field theory: The leading quantum corrections, John F. Donughue gr-qc/ Counterterms are distinct from primitive terms: BPHZ counterterms are guaranteed to be local Massless particles make nonlocal corrections For infrared regime the finite, nonlocal terms dominate over the local counterterms: Local Counterterm Effects p 4 Nonlocal Primitive Effects p 4 ln(p ) Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 8 / 34

10 How a loop of MMC scalars changes dynamical gravitons and the force of gravity? Lagrangian of gravity plus a MMC scalar Define the graviton field h µν as L = 1 [ ] g 1 R Λ 16πG µϕ νϕg µν g g µν = g µν + κh µν, g µν = a η µν, κ = 16πG, a = 1 Hη, (the open conformal patch of de Sitter space) Vary the one-particle irreducible (1PI) effective action w.r.t h µν to get the quantum corrected, linearized Einstein field equation D µνρσ h ρσ(x) H = 1 3 Λ d 4 x [ µν Σ ρσ] (x; x )h ρσ(x ) = κ T µν lin (x) D µνρσ : Lichnerowicz operator defined such that D µνρσ h µν is the linearized Einstein tensor, R µν 1 g µν (R Λ) [µν ] i Σ ρσ (x; x ): graviton self-energy = 1PI graviton -point function : quantum correction to the Lichnerowicz operator the self-energy of any particle: the quantum correction to that particle s kinetic operator e.g. For photons, the vacuum polarization: quantum correction to Maxwell s eqns Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 9 / 34

11 A three-step procedure To learn how a loop of MMC scalars changes dynamical gravitons and the force of gravity 1 Compute and renormalize the one-loop contribution to the graviton self-energy from a MMC scalar on de Sitter background i[ µν Σ ρσ] (x; x ) Convert the in-out self-energy to the retarded one of the Schwinger-Keldysh formalism [ µν Σ ρσ] (x; x ) [ µν Σ ρσ Ret] (x; x ) 3 Solve the quantum corrected, linearized Einstein field equation D µνρσ h ρσ(x) d 4 x [ µν Σ ρσ] (x; x )h ρσ(x ) = κ T µν lin (x) Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 10 / 34

12 One Loop Scalar Contributions to Graviton Self-Energy One loop contribution to the graviton self-energy from consists of 3 Feynman diagrams: i[ µν Σ ρσ ](x; x ) = 1 T µναβ (x) T ρσγδ (x ) I J α γ i (x; x ) β δ i (x; x ) I =1 J=1 x x + x F µνρσαβ (x) I α β i (x; x ) δ D (x x ) I =1 + x + I =1 C µνρσ (x) δ D (x x ). I Interaction vertices between MMC scalars and gravitons derive from expanding the Lagrangian: L = 1 µϕ ν ϕg µν g = 1 µϕ ν ϕg µν g κ µϕ ν ϕ ( 1 hg µν h µν ) g { κ [ 1 µϕ ν ϕ 8 h 1 ] 4 hρσ h ρσ g µν 1 } g hhµν +h µ ρ hρν 3 + O(κ ). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 11 / 34

13 MMC scalar propagator The MMC scalar propagator obeys µ [ g g µν ν ]i (x; x ) = iδ D (x x ) No de Sitter invariant solution for the propagator (Allen and Follaci 1987) A solution preserving the homogeneity and isotropy: ( ) i (x; x ) = A y(x; x ) + HD (4π) D Γ(D 1) Γ( D ) ln(aa ) = de Sitter invariant function of y + de Sitter breaking term where { A(y) HD Γ( D ) ( 4 ) D 1 Γ( D (4π) D D + +1) ( 4 ) D ( πd ) Γ(D 1) 1 y D π cot y Γ( D ) [ ]} 1 Γ(n+D 1) ( y ) n 1 Γ(n+ D + +1) ( y ) n D + n=1 n Γ(n+ D ) 4 n D +. Γ(n+) 4 The de Sitter breaking term drops differentiated by α β y(x; x ) aa H x : the de Sitter invariant length function Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 1 / 34

14 Two Primitive Diagrams Contribution from 4-point vertices [ µν ρσ ] i Σ (x; x ) 1 4pt 4 I =1 ( D 4 ) iκ H D { Γ(D) 1 = g 4 (4π) D Γ( D +1) Contribution from 3-point vertices [ µν ρσ ] i Σ (x; x ) = g { g 3pt + x µ x ν x ρ x σ 1 y 4 1 in D = 4 x8 F µνρσαβ (x) I α β i (x; x ) δ D (x x ) y x µ x (ρ g µν g ρσ g µ(ρ g σ)ν y x σ) α(y) + xν x (µ γ(y) + g µν g ρσ H 4 µν δ(y) + [g x ρ d 4 x 1 x 8 quartically divergent } δ D (x x ) = 0 for D = 4 y x ν) x (ρ x σ) x σ + x µ β(y) x ν g ρσ] H ɛ(y) } Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 13 / 34

15 Correspondence with flat space limit Flat space limit H 0: x 0 t t, y(x; x ) H x x µ H x µ, x ν H x ν, x µ x ν H η µν gives [ µν ρσ ] i Σ (x; x ) = κ Γ ( D ) { flat 16π D η µ(ρ η σ)ν [ ] x D + x (µ η ν)(ρ x σ) [ 4D ] x D+ + x µ x ν x ρ x σ [ D ] x D+4 + η µν η ρσ [ 1 (D D 4) ] x D } [ + η µν x ρ x σ + x µ x ν η ρσ] [ D(D ) x D+ ] This agrees with G. t Hooft and M. Veltman, Ann. Inst. Henri Poincaré XX (1974) 69. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 14 / 34

16 Correspondence with stress tensor correlators The graviton self-energy is related to the -point correlator of the stress tensor as µν ρσ i[ ] (x; x ) = 1 4 κ g(x) g(x ) Ω δt µν (x)δt ρσ (x ) Ω + O(κ 4 ) The stress tensor correlator obtained by Perez-Nadal, Roura and Verdaguer (JCAP 1005 (010) 036, arxiv: ) agrees with out result. Ω δt µν (x)δt ρσ (x ) Ω = F µνρσ = P(µ)n µn ν n ρn σ + Q(µ)(n µn ν g ρσ + n ρn σg µν ) +R(µ)(n µn ρg νσ + n ν n σg µρ + n µn σg νρ + n ν n ρg µσ ) + S(µ)(g µρ g νσ + g νρ g µσ ) + T (µ)g µν g ρσ Note: their 5 basis tensors are converted into ours as n an b n c n d = n an b g c d + n c n d g ab = 1 H 4 (4y y ) x a x b x c [ 1 H (4y y g ab ) x c x d x d, + x a x b g c d y 4n (a g b)(c n d ) = H 4 (4y y ) x (a x b) x (c x d ) H 4 (4y y )(4 y) g a(c g d )b = 1 H 4 x a x (c g ab g c d = g ab g c d y + 1 H 4 1 (4 y) x a y x d ) x b + 1 x b x c ], y H 4 (4 y) x (a x b) x (c x d ) x d, x a x b x c x d, Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 15 / 34

17 One Loop Counterterms For quantum gravity at one loop order the necessary counterterms are R and C first derived by t Hooft and Veltman, 1974 Graviton -point function graviton fields S.D.D. = 4 4 s, with general coord. invariance 3 possibilities: R, R µν R µν, R µνρσ R µνρσ with the Gauss-Bonet relation, only are linearly indep. For calculational convenience, reorganize R as R = [ R D(D 1)H ] + D(D 1)H R D (D 1) H 4 So we employ four counterterms: L 1 c 1 [R D(D 1)H ] g, L3 c 3 H [ R (D 1)(D )H ] g, L4 c 4 H 4 g. L c C αβγδ C αβγδ g, Note: the divergences can really be eliminated with just L and the particular linear combination of L 1, L 3 and L 4 which is proportional to R g. We define two nd order differential operators by expanding the scalar and Weyl curvatures around de Sitter background R D(D 1)H P µν κh µν + O(κ h ), C αβγδ P µν αβγδ κhµν + O(κ h ). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 16 / 34

18 Counterterms in terms of two projection operators Spin zero projection operator: P µν = D µ D ν g µν [ D + (D 1)H ], Spin two projection operator P µν αβγδ = Dµν αβγδ + 1 [ g αδ D µν ] βγ D g βδ Dµν αγ g αγ Dµν βδ +g βγ Dµν αδ 1 ] + [g αγ g βδ g αδ g βγ D µν, (D 1)(D ) where we define, D µν αβγδ D µν βδ 1 [ δ (µ α δν) δ Dγ D β δ (µ β δν) δ Dγ Dα δ(µ α δν) γ D δd β +δ (µ ] β δν) γ D δd α, g αγ D µν αβγδ = 1 [ δ (µ δ Dν) D β δ (µ β δν) δ D g µν D δ D β +δ (µ β D δd ν)], D µν g αγ g βδ D µν αβγδ = D(µ D ν) g µν D. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 17 / 34

19 Counterterms in terms of two projection operators The counterterms are expressed in terms of these two operators: iδ S 1 δh µν (x)δh ρσ(x ) h=0 = c 1 κ g P µν P ρσ iδ D (x x ) c 1 κ Π µν Π ρσ iδ D (x x ), iδ S δh µν (x)δh ρσ(x ) h=0 = c κ g g ακ g βλ g γθ g δφ P µν αβγδ Pρσ κλθφ iδd (x x ) c κ ( D 3 )[Π µ(ρ Π σ)ν Πµν Π ρσ ] iδ D (x x ) D D 1 iδ S 3 δh µν (x)δh ρσ(x = c ) 3 κ H g D µνρσ iδ D (x x ) 0 h=0 iδ S 4 δh µν (x)δh ρσ(x = c ) 4 κ H 4 [ 1 g h=0 4 g µν g ρσ 1 g µ(ρ g σ)ν ] iδ D (x x ) 0 where we define Π µν µ ν η µν in flat space limit. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 18 / 34

20 Renormalizing the Flat Space Result: a guide for de Sitter Reorganize the primitive terms in the terms of two projection operators so as to be in the form of counterterms: [ µν ρσ ] i Σ (x; x ) = Π µν Π ρσ F 0 ( x ) + flat [Π µ(ρ Π σ)ν Πµν Π ρσ D 1 ] F ( x ). Find the structure functions F 0 and F comparing this with the previous primitive result: F 0 ( x ) = F ( x ) = κ Γ ( D ) 1 ( 1 ) D 16π D 8(D 1) x κ Γ ( D ) 1 16π D 4(D ) (D 1)(D +1) ( 1 x ) D Note: Π µν Π ρσ 4 are w.r.t x Extract these outside the integral w.r.t x. Now the factor of 1/ x D 4 is logarithmically divergent. Then extract one more d Alembertian ( 1 x ) D = (D 3)(D 4) ( 1 x ) D 3. Now the integrand converges, however, we still cannot take the D = 4 limit owing to the factor of 1/(D 4). The solution is to add zero in the form of the identity D ( 1 ) D 1 4π iδ D (x x ) x Γ( D 1) = 0. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 19 / 34

21 Renormalizing the Flat Space Result: a guide for de Sitter Rewrite it by adding zero: ( 1 ) D { 1 x = (D 3)(D 4) x D 6 µ } D 4 x D + 4π D µ D 4 iδ D (x x ) (D 3)(D 4)Γ( D 1) = 1 { } ln(µ x ) 4 x + O(D 4) + 4π D µ D 4 iδ D (x x ) (D 3)(D 4)Γ( D 1). : nonlocal finite term : local divergent term The divergence now segregated on the delta function: remove them with counterterms: [ µν ρσ ] } i Σ (x; x ) = Π µν Π {c ρσ 1 κ iδ D (x x ) + [Π µ(ρ Π σ)ν Πµν Π ρσ ] { } D 3 ) ( c κ iδ D (x x ) flat D 1 D by choosing the constants c 1 and c as, c 1 = µd 4 Γ( D ) 8 π D (D ) (D 1) (D 3)(D 4), c = µ D 4 Γ( D ) 8 π D (D +1)(D 1)(D 3) (D 4). The fully renormalized { graviton self-energy for flat space} background is, µν ρσ i[ ] Σ ren = lim flat D 4 i [ µν ρσ ] Σ (x; x ) i flat { = Π µν Π ρσ κ ln(µ x ) 9 3 π 4 x Again, this agrees with t Hooft and Veltman [ µν ρσ ] Σ (x; x ) flat, } [ + Π µ(ρ Π σ)ν 1 { 3 Πµν Π ρσ] κ } ln(µ x ) π 4 x Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 0 / 34

22 Renormalizing de Sitter result Reorganize the primitive result in terms of the projection operators as for flat space: µν ρσ i[ ] Σ (x; x ) = g(x) P µν (x) g(x ) P ρσ (x { } ) F 0 (y) { + g(x) P µν αβγδ (x) g(x ) P ρσ κλθφ (x ) T ακ T βλ T γθ T δφ( D D 3 } ) F (y), where the bitensor is T ακ (x; x ) 1 y(x;x ) H xα x κ. Note: T ακ (x; x ) η ακ in flat space Find the structure functions F 0 and F comparing this with the previous primitive result: κ H D 4 Γ ( D ) { } 1 ) D +... F 0 (y) = F (y) = ( 4 (4π) D 8(D 1) y κ H D 4 Γ ( D ) { 1 (4π) D Add zero in the form of the identity Then [ ( 4 y ) D = [ D 4(D 3)(D )(D 1)(D +1) ( 4 y ] ( D ) ( 1 H 4 ) D 1 (4π) D iδ D (x x ) y Γ( D 1)HD g = 0. H ]{ 4 y ln ( y 4 ) D +... } )} 4 y + O(D 4) + (4π) D iδ D (x x )/ g (D 4)(D 3)Γ( D 1)HD nonlocal finite term local divergent term Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 1 / 34

23 Renormalizing de Sitter result Add the counterterms to subtract the divergences off: i[ µν Σ ρσ ] (x; x ) = g [ c 1 κ P µν P ρσ + c κ g ακ g βλ g γθ g δφ P µν αβγδ Pρσ κλθφ c 3 κ H D µνρσ + c 4 κ H 4 [ 1 g 4 g µν g ρσ 1 g µ(ρ g σ)ν ]] iδ D (x x ). The fully renormalized graviton self-energy for de Sitter is : i [ ] µν ρσ Σ ren (x; x ) = = g(x) P µν (x) { lim µν ρσ i[ ] Σ (x; x µν ρσ ) i[ ] Σ (x; x ) D 4 [ ] g(x ) P ρσ (x ) F 0R (y) + g(x) P µν αβγδ (x) g(x )P ρσ [ κλθφ (x ) T ακ T βλ T γθ T δφ ] F R (y). { [ where F 0R = κ H 4 ( ) ] } { [ (4π) 4 H 1 7 y 4 ln y +, F 4 R = κ H 4 ( ) ] } (4π) 4 H 1 40 y 4 ln( y + 4 }, Note 1: The leading terms agree with the corresponding flat results. Note : F 0R and F R are the first fully renormalized results for the graviton structure functions on de Sitter. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: / 34

24 Spin zero structure function F 0R = κ H 4 { [ 1 (4π) 4 H 7 4 ( y ) ] y ln ( y )+ y ln y ln( y ) y ln( y 4 ) 1 45 ln( y 4 ) y 5 6 y 4 ln(1 y 4 ) y ln(1 y 4 ) 1 0 ln(1 y 4 ) 7(1π + 65) y π y 4 ln( y 4 ) y ( y ) 4 ln ( y) [7Li (1 y 4 ) Li ( y 4 ) + 5 ln(1 y 4 ) ln( y 4 ) ] }. Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 3 / 34

25 Spin two structure function F R = κ H 4 { [ 1 (4π) 4 H 40 4 ( y )] y ln( ( y ) y ln y ln( y ) 119 ( y ) 4 60 ln 4 [ [ (4y y 8) 4 47 ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) 399 ( y ) ] y [ 193 ( y ) ( y ) 3 7 ( y ) 379 ( y ) ] ln( y ) [ + 14 ( y ) 5 1 ( y ) 4 19 ( y ) ( y ) 143 ( y ) ( 4 ) ] ln(1 y 60 y 4 ) [ ( y ) 9 ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) 1951 ( y ) ] y ln( y 4 ) [ ( y ) 7 ( y ) 6 ( y ) 5 ( y ) 4 ( y ) 3 ( y ) ( y ) ] 4 y ln ( y ) [ 367 ( y ) ( y ) 3 37 ( y ) 1751 ( y ) ] ln( y ) (y [ 8) 4( y) (4y y ] [ 1 ) 5 Li (1 y 4 ) Li ( y ] ]} 4 ). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 4 / 34

26 Solving the quantum-corrected linearized Einstein equation Use the renormalized self-energy for the quantum correction term: µνρσ g D hρσ(x) d 4 x [ ] µν ρσ Σ ren (x; x )h ρσ(x ) = 1 κ g T µν lin (x), Only know the self-energy at one loop order (at order κ = 16πG), solve it perturbatively: h µν (x) = h (0) µν (x) + κ h (1) µν (x) + O(κ4 ), [ ] µν ρσ Σ ren (x; x ) = κ [ µν Σ ρσ 1 ] (x; x ) + O(κ 4 ). The corresponding one loop correction is d 4 x [ ] µν ρσ Σ 1 (x; x )h (0) ρσ (x ) = i d 4 x g(x) P µν (x) g(x ) P ρσ (x } ) {F 0 h (0) ρσ (x ) } {T ακ T βλ T γθ T δφ F +i d 4 x g(x) P µν αβγδ (x) g(x ) P ρσ κλθφ (x ) h (0) ρσ (x ). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 5 / 34

27 Solving the quantum-corrected linearized field eqn for dynamical gravitons For dynamical gravitons, that is for zero stress-energy T µν lin (x) = 0: h (0) ρσ (x) = ɛρσ( k)a u(η, k)e i k x H, u(η, k) = [1 ik ] [ ik ] exp, 0 = ɛ 0µ = k i ɛ ij = ɛ jj and ɛ ij ɛ k 3 ij = 1 Ha Ha The result is zero! d 4 x [ ] µν ρσ Σ 1 (x; x )h (0) ρσ (x ) = 0 Inflationary Scalars Don t Affect Gravitons at One Loop, SP and Woodard, arxiv: Gravitons interact with MMC scalar only through their kinetic energies which are redshifted. (Gravitons couple minimally only to differentiated scalars.) A little Doubt about this result: ignoring surface terms is really legitimate? Ignored surface terms based on the assumption that either they fall off or can be absorbed into corrections of the initial state. Found a counterexample to this assumption: A certain surface terms cannot be ignored. Leonard, Prokopec and Woodard, arxiv: Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 6 / 34

28 Noncovariant representation of the graviton self-energy A noncovariant representation of the conformally rescaled graviton field Noncovariant Rep includes de Sitter breaking basis vectors in terms of u(x; x ) ln(aa ) Covariant Rep: g µν = g µν + κχ µν vs Noncovariant Rep: g µν = a (η µν + κh µν ) Covariant Rep: 5 basis tensors - 3 relations = structure ftns vs Noncovariant Rep: = 4 µν ρσ i[ ] Σ (x; x ) = F µν (x) F ρσ (x [ ) F 0 (x; x ] ) +G µν (x) G ρσ (x [ ) G 0 (x; x ] ) + F µνρσ[ F (x; x ] ) + G µνρσ[ G (x; x ] ) Leonard, SP, Prokopec and Woodard, arxiv: Checked that surface terms really fall off like powers of the scale factor Confirmed the previous result: no effect on dynamical gravitons from MMC scalars Much simpler than the de Sitter covariant representation, so can be easily employed to study the force of gravity Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 7 / 34

29 Structure functions in the noncovariant representation F 0R (x; x ) = F R (x; x ) = κ (aa H ) { [ ln(µ x ) ] 304π 4 (aa H ) x κ (H aa ) { [ ln(µ x ) (4π) 4 30(H aa ) x ] + 3 G 0 (x; x ) = 0 G (x; x ) = κ (H aa ) { (4π) ln( y 4 ) 3 (4 y) + } 3 Ψ(y) 6 y [ y +6 [ 1 y 1 4 y 4 y ] ( y ) ln 1 } 4 3 Ψ(y) ] ( y ) ln + 3 } 4 ( y)ψ(y) where Ψ(y) 1 ln( y 4 ) ( ln 1 y ) ( y ) ( y ) ln Li Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 8 / 34

30 Schwinger-Keldysh (= in-in = closed-time path) formalism The perturbative effective field eqn seems to be ready for use, but if one were to interpret it in the spirit of the in-out formalism it would possess Acausality: The in-out effective field equation at x µ receives influence from points x µ which lie in the future of x µ, and at spacelike separation from it. Imaginary parts: The in-out effective field develops an imaginary part if there is particle production. Neither of these features prevents one from describing flat space scattering problems, but problematic for cosmological settings in which we do not know the asymptotic future. The more natural question: how the fields evolve when released at finite time in a prepared state. That question is answered by the Schwinger-Keldysh formalism, which produces true expectation values, rather than in-out matrix elements, so the effective field equations at x µ depend only on points on or within its past light-cone, the effective fields associated with Hermitian operators are real. The linearized Schwinger-Keldysh effective field equation is obtained by replacing the in-out self-energy with its retarded counterpart, [ µν Σ ρσ] (x; x ) [ µν Σ ρσ Ret ] (x; x ) [ µν Σ ρσ ++ ] (x; x ) + [ µν Σ ρσ + ] (x; x ). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 9 / 34

31 Structure functions in the Schwinger-Keldysh formalism This converts the nonzero structure functions in to the retarded ones of the Schwinger-Keldysh formalism, { F (1) 0 (x; x ) = iκ 576π 3 { F (1) (x; x ) = iκ 64π 3 4 4H aa H aa 40 [ [ ( y ) ] ] ln 4aa 1 Θ 1 4 H aa ln(aa ) Θ [ +H 4 a a y + 3 ( y 4 ( y) ln 4 y [ [ ] ] ln Θ ( y 4aa ) 1 { [ 4 G (1) (x; x ) = iκ 64π 3 H 4 a a 3 4 y + 1 ( y ) ] } 3 ln Θ. 4 y ) ] Θ + H aa ln(aa ) 1 +H 4 a a [ 1 3 }, Θ 4 y 1 ( y ) ] } 6 ln Θ, 4 y (Note that G (1) 0 (x; x ) is zero for the MMC scalar at one loop.) Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 30 / 34

32 One loop corrections from MMC scalar to the Newtonian potential For the linearized response to a stationary point mass M h (0) GM 00 (x) = a x = Φ(0), h (0) 0i (x) = 0, h (0) (x) = GM ij a x δ ij = Ψ (0) δ ij, T µν = Maδ 3 ( x)δ 0 µ δ0 ν in flat space h (0) 00 (x) = GM x One loop corrections The solutions are, h(0) 0i (x) = 0, h (0) (x) = ij GM x δ ij, T µν = Mδ 3 ( x)δ 0 µ δ0 ν h (1) 00 (x) f 1, h (1) (x) = 0, h (1) (x) f 0i ij 3 δ ij f 1 (x) = κ M a S1 0 (x) + κ M [ a 3 + ( ] 0 ah 0) S 1 (x) = Φ(1) f 3 (x) = κ M a S1 0 (x) + κ M [ a 1 ] 3 ah 0 S 1 (x) = Ψ(1) where f (η, x) = [1/(4π)] d 3 x f (η, x )/ x x S 1 0 (x) = dη S 1 (x) = dη a(η ) [if 1 0 (x, x )] x =0, a(η ) [ F 1 (x; x ) + 1 G 1 ] (x; x ) x =0 Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 31 / 34

33 One loop corrections from MMC scalar to the Newtonian potential In flat space Φ flat = GM r Ψ flat = GM r { 1 + G ( 0πc 3 r + O G )} { 1 G ( 60πc 3 r + O G )} SP and Woodard, arxiv: , Marunovic and Prokopec, arxiv: Not the first for this result, but the first to solve the effective field eqns using the Schwinger-Keldysh or in-in formalism. The previous calculations e.g. Radkowski, 1970, Donoghue 1993,... were done by the scattering amplitude technique. In de Sitter space Φ ds = GM { 1 + G ar 0πc 3 (ar) + GH [ πc 5 Ψ ds = GM { 1 G ar 60πc 3 (ar) + GH [ πc ln(a) 3 ( Har )] ( 10 ln + O G H 4)} c 1 30 ln(a) 3 ( Har ) 10 ln + c 3 Har c ] ( + O G H 4)} the de Sitterized version of the flat space correction + intrinsic de Sitter correction SP, Prokopec and Woodard, arxiv: Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 3 / 34

34 Summary and Discussion Derived one loop contributions to the graviton self-energy from MMC scalar on de Sitter background: covariant and noncovariant representations for the tensor structure. Used these representations to quantum correct the linearized Einstein field equations for dynamical gravitons and the force of gravity. The noncovariant representation is much easier to use in the effective field equations than the covariant one. Inflationary production of MMC scalars has no effect on dynamical gravitons. Inflationary production of MMC scalars gives quantum corrections to the force of gravity: a time dependent renormalization of the mass term, [ M M 1 GH ] c 5 30π ln(a), or equivalently a time dependent renormalization of the Newton s constant, [ G G 1 GH ] c 5 30π ln(a). Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 33 / 34

35 THE END Thank you for your attention! Quantum Scalar Corrections to the Gravitational Potentials November on de Sitter 13, 015 Background: 34 / 34