S. P. Miao
National Cheng Kung University, Taiwan
FRW Geometry
Homogeneous, isotropic and spatial flat:
𝑑𝑑𝑆𝑆 2 = −𝑑𝑑𝑡𝑡 2 + 𝑎𝑎 2 𝑡𝑡 𝑑𝑑𝑥𝑥⃑ ∙ 𝑑𝑑𝑥𝑥⃑
Physical distance: 𝑎𝑎 𝑡𝑡 𝑑𝑑𝑥𝑥⃑
𝑘𝑘
Physical momentum: 𝑎𝑎(𝑡𝑡)
Max. accelerated: 𝐻𝐻 𝑡𝑡 = 𝐻𝐻 = const.  𝑎𝑎(𝑡𝑡)~𝑒𝑒 𝐻𝐻𝐻𝐻
𝐺𝐺𝐻𝐻 2 < 5 × 10−11  𝐻𝐻 < 1014 Gev
QG still perturbative, not negligible
Particles almost effectively massless
Locally de Sitter Background
Spacetime Expansion Strengthens
Loop Effects
 How to think about QFT Effects?
 Classical response to virtual particles
 Maximum IR Enhancements:
 PERSISTENCE TIME  m = 0 + inflation
 EMERGENCE RATE  no conformal invariance
 Realized by
 Massless, minimally coupled scalars
 Gravitons
E-Time Uncertainty Principle
 ∆𝑡𝑡∆𝐸𝐸 > 1 to resolve ∆𝐸𝐸  Hence ∆𝑡𝑡∆𝐸𝐸 < 1 to NOT resolve
 Flat Space: Virtual pair has ∆E = 2 𝑚𝑚2 + 𝑘𝑘2
 Hence can last ∆t < 2
1
𝑚𝑚2+𝑘𝑘2
 Eg: Vacuum polarization
 Most for e± because smallest m
 Smallest k’s live longest  EM stronger at shorter distance (less
polarization)
 FRW: E(t) = 2


𝑚𝑚2
+
𝑘𝑘 2
𝑎𝑎 2 (𝑡𝑡)
𝑡𝑡+∆𝑡𝑡
∆𝑡𝑡∆𝐸𝐸  2 ∫𝑡𝑡
𝑑𝑑𝑡𝑡 ′ 𝐸𝐸 𝑡𝑡 ′ < 1
𝑡𝑡+Δ𝑡𝑡
𝑘𝑘
𝑚𝑚 = 0  2k ∫𝑡𝑡
𝑑𝑑𝑡𝑡 ′ 𝑎𝑎 𝑡𝑡 ′ < 1
 Any m = 0 virtual with k < Ha(t) live forever (in de Sitter)
Killer Symmetry Suppresses
Emergence Rate
 Conformal invariance is a killer symmetry:
𝑔𝑔𝑔µν = Ω2 𝑔𝑔µν  ℒ → ℒ′
 re-defined fields  ℒ′ → ℒ
1
 EM: ℒ ′ = − 4 𝐹𝐹µρ 𝐹𝐹νσ 𝑔𝑔µν 𝑔𝑔ρσ −𝑔𝑔Ω𝐷𝐷−4 = ℒ in D =4
 −𝑑𝑑𝑡𝑡 2 + 𝑎𝑎 2 𝑑𝑑𝑥𝑥 2 = 𝑎𝑎 2 (−𝑑𝑑η2 + 𝑑𝑑𝑥𝑥 2 )  𝑎𝑎𝑎𝑎η = 𝑑𝑑𝑑𝑑
 Conformal invariance  same locally (in conformal
coordinates) as flat
 Hence
 Hence
𝑑𝑑𝑑𝑑
𝑑𝑑η
𝑑𝑑𝑑𝑑
𝑑𝑑𝑡𝑡
= Γ𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
=
Γ𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
𝑎𝑎(𝑡𝑡)
 Any m=0, conformal virtuals that emerge Do live forever, but
few emerge
MMC + Fermions – Fermion self energy
ℒ1 = ─
1
2
� Ψφ −𝑔𝑔
−𝑔𝑔 𝜕𝜕µφ𝜕𝜕ν φ𝑔𝑔 µν − f Ψ
� 𝑒𝑒 µ (𝑖𝑖𝜕𝜕µ – 1 𝐴𝐴µ𝑐𝑐𝑐𝑐 𝐽𝐽 𝑐𝑐𝑐𝑐 )Ψ
+ −𝑔𝑔 Ψ
𝑏𝑏
2
(astro-ph/0309593 by Prokopec, Woodard)
1
1
� Ψφ −𝑔𝑔
ℒ 2 = − 2 −𝑔𝑔 𝜕𝜕µ φ𝜕𝜕ν φ𝑔𝑔 µν − 2 𝑚𝑚 2 φ2 − ξ𝑅𝑅φ2 − fΨ
� 𝑒𝑒 µ (𝑖𝑖𝜕𝜕µ –
+ −𝑔𝑔 Ψ
𝑏𝑏
1
2
𝐴𝐴µ𝑐𝑐𝑐𝑐 𝐽𝐽 𝑐𝑐𝑐𝑐 )Ψ
(gr-qc/0602011 by Garbrecht. Prokopec)
 Compute i [𝑖𝑖 Σ𝑗𝑗 ] (x; x’) from mmc at 1-loop in dim. Reg.
MMC + Fermions – consequence I
 Absorb UV using field strength renormalization
iδ𝑧𝑧2 (𝑎𝑎𝑎𝑎𝑎)
𝐷𝐷−1
2
µ
γ𝑖𝑖𝑖𝑖
i𝜕𝜕µ δ𝐷𝐷 (x-x’) ; δ𝑧𝑧2 =
𝐷𝐷
Γ( −1)
𝑓𝑓2
2
𝐷𝐷/2
(𝐷𝐷−4)(𝐷𝐷−3)
16 π
 Quantum-correct Dirac Equation using S-K formalism
−𝑔𝑔𝑖𝑖(γµ 𝐷𝐷µ )𝑖𝑖𝑖𝑖 Ψ𝑗𝑗 (x)− ∫ 𝑑𝑑 4 𝑥𝑥 ′ [𝑖𝑖 Σ𝑗𝑗 ] 𝑥𝑥; 𝑥𝑥 ′ Ψ𝑗𝑗 (x′) = 0
 particle production  |φ| grow  develop mass for ℒ1 & ℒ 2

2
E.g. 𝑚𝑚 Ψ
≅
3𝑓𝑓2 𝐻𝐻 4
8π2 (𝑚𝑚2 + ξ𝑅𝑅)
for ℒ 2
 ℒ 1 : Ψ (late time)  (𝑁𝑁 −
π
 ℒ 2 : Ψ± = ∓ exp [ i 2 ν± ] [
3 −1
)4
4
exp [ ±𝑖𝑖
π𝑘𝑘 1/2
]
4𝐻𝐻𝐻𝐻
(1)
𝑓𝑓
3π
𝐻𝐻± (
3
4
(𝑁𝑁 − ) 3/2 ] ; 𝑁𝑁 ≡ ln (𝑎𝑎)
𝑘𝑘
)
𝐻𝐻𝐻𝐻
1
2
; ν± = ∓ 𝑖𝑖𝑚𝑚 Ψ /𝐻𝐻
 Fermion mode fun. ~ declining oscillatory behavior
MMC + Fermions – Consequence IIa
ℒ3 = −
1
2
−𝑔𝑔 𝜕𝜕µ φ𝜕𝜕ν
φ𝑔𝑔 µν
� 𝑒𝑒 µ (𝑖𝑖𝜕𝜕µ –
+ −𝑔𝑔 Ψ
𝑏𝑏
1
2
−
δξ𝑅𝑅φ2
𝐴𝐴µ𝑐𝑐𝑐𝑐 𝐽𝐽 𝑐𝑐𝑐𝑐 )Ψ
1
− 4! δλφ4
� Ψφ −𝑔𝑔
−𝑔𝑔 − f Ψ
(gr-qc/0602110 by Miao, Woodard)
 Integrate out fermions at leading log. order  𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 (φ)
δξ =
𝐷𝐷
2
4𝑓𝑓 2 𝐻𝐻 𝐷𝐷−4 Γ(1− )
(4π)𝐷𝐷/2 𝐷𝐷(𝐷𝐷−1)
𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 (φ) = −
=
𝐻𝐻 4
+
𝑓𝑓 2
24π 2
{2γ𝑧𝑧
8π 2
(1 − γ) ; δλ =
𝐷𝐷
2
24𝑓𝑓 4 𝐻𝐻 𝐷𝐷−4 Γ(1− )
𝑧𝑧
(4π)𝐷𝐷/2
+
3𝑓𝑓 4
(ξ(3)
π2
− γ)
− ζ 3 − γ 𝑧𝑧 2 + 2 � 𝑑𝑑𝑑𝑑 𝑥𝑥 + 𝑥𝑥 3 𝜓𝜓 1 + 𝑖𝑖𝑖𝑖 + 𝜓𝜓 1 − 𝑖𝑖𝑖𝑖
𝐻𝐻 4 ∞
−1 𝑛𝑛
− 4π2 ∑𝑛𝑛=2 𝑛𝑛+1
0
𝑓𝑓φ
𝜁𝜁 2𝑛𝑛 − 1 − 𝜁𝜁 2𝑛𝑛 + 1 𝑧𝑧 𝑛𝑛+1 ; 𝑧𝑧 ≡ ( 𝐻𝐻 )2 ; 𝑧𝑧 ≪ 1
 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 (φ) unbounded below (always negative)
MMC + Fermions – Consequence IIb
 Compute 𝑇𝑇µν at leading log. order  𝑇𝑇µν → −𝑉𝑉𝑠𝑠 (φ)𝑔𝑔µν
𝑉𝑉𝑠𝑠 (φ) =
=
𝐻𝐻 4
− 8π2
{
𝐻𝐻 4 1
{ 𝑧𝑧
8π2 2
1
2
1
1
− γ 𝑧𝑧 +[ 4 − γ + ζ 3 ]𝑧𝑧 2 − 2 [𝑧𝑧 + 𝑧𝑧 2 ] 𝜓𝜓 1 + 𝑖𝑖𝑖𝑖 + 𝜓𝜓 1 − 𝑖𝑖𝑖𝑖 ]
1
𝑛𝑛
𝑛𝑛+1 } ; 𝑧𝑧 ≪ 1
+ 4 𝑧𝑧 2 − ∑∞
𝑛𝑛=2 (−1) 𝜁𝜁 2𝑛𝑛 − 1 − 𝜁𝜁 2𝑛𝑛 + 1 𝑧𝑧
 unbounded below but positive first, then negative
 𝑧𝑧 ≫ 1, 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 (φ) & 𝑉𝑉𝑠𝑠 (φ) both  Coleman-Weinberg form

𝐻𝐻 4 1 2
− 8π2 {2 𝑧𝑧 ln(|𝑧𝑧|) −
1
ζ 3 + 4 − γ 𝑧𝑧 2 + zln 𝑧𝑧 −
 Flat space Standard Model  stable
5
6
− 2γ z + O(ln z )}
∵ negative V + positive V of gauge boson  constraint on Higgs mass
𝑘𝑘
 Negative growing fermion vacuum energy: 𝐸𝐸 = − 𝑚𝑚 2 + ( 𝑎𝑎) 2
 particle production  |φ| grow  develop mass
𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 & 𝑉𝑉𝑠𝑠 for Yukawa
𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 =
𝐻𝐻 4
− 8π2 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒
z
; 𝑉𝑉𝑠𝑠 =
𝐻𝐻 4
− 8π2 𝑓𝑓𝑠𝑠
z
𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 falls for Yukawa & grows for SQED
Gravitons + Massless Fermion
arXiv:0511140 by Miao, Woodard
ℒ=
1
𝑅𝑅
16π𝐺𝐺
� 𝑒𝑒 µ γ𝑏𝑏 𝑖𝑖𝐷𝐷µ Ψ −𝑔𝑔 + c-terms
−𝑔𝑔 +Ψ
𝑏𝑏
 Compute i[𝑖𝑖 Σ𝑗𝑗 ](𝑥𝑥; 𝑥𝑥 ′ )from gravitons at 1 loop in dim. reg.
 Renormalize with BPHZ counter terms
� [𝑖𝑖(𝛾𝛾 µ 𝐷𝐷µ)2 +
∆ℒ = α1κ2 Ψ
𝐷𝐷
𝑅𝑅
𝐷𝐷−1
� 𝑖𝑖(𝛾𝛾 µ 𝐷𝐷µ)Ψ −𝑔𝑔
]𝑖𝑖(𝛾𝛾 µ𝐷𝐷µ )Ψ −𝑔𝑔 +α 2κ2 𝑅𝑅Ψ
� 𝐻𝐻 2𝑖𝑖(γ𝑘𝑘 𝐷𝐷𝑘𝑘 )Ψ −𝑔𝑔 (non-invariant c-term: comment later)
+α 3κ2 Ψ
 Quantum-correct Dirac equation using S-K formalism
−𝑔𝑔𝑖𝑖(γµ𝐷𝐷µ )𝑖𝑖𝑖𝑖 Ψ𝑗𝑗 (x)− ∫ 𝑑𝑑 4 𝑥𝑥 ′ [ 𝑖𝑖 Σ𝑗𝑗 ] 𝑥𝑥; 𝑥𝑥 ′ Ψ𝑗𝑗 (x′) = 0
Gravitons + Massless Fermion
 1st IR log using dimensional regularization in de Sitter
 Fermion mode fun. ~#𝐺𝐺𝐻𝐻 2 ln[𝑎𝑎 𝑡𝑡 ] at 1-loop
 Perturbation breaks down at ln [𝑎𝑎 𝑡𝑡
1
]~ 2
𝐺𝐺𝐻𝐻
 fermions propagate through the sea of IR gravitons 
buffed by random walking of them  mode fun. grows
 secular effects from spin-spin interactions




No #𝐺𝐺𝐻𝐻2 ln[𝑎𝑎 𝑡𝑡 ] in ``QG + MMC’’ (Kahya & Woodard)
IR gravitons only couple to mmc through red-shift K.E.
But fermions has extra SPIN in addition (0803.2377)
simple rules for catching leading log. has derived
 It differs from Starobinsky’s IR truncation
Gravitons + Massless Fermion – Consequences
 QG + light fermions ( m ≪ H )
 Suppress: how fermions propagate
ω
 𝑢𝑢(𝑡𝑡, 𝑘𝑘)~𝑓𝑓(𝐻𝐻 )𝑒𝑒 −𝑖𝑖ω𝑡𝑡 , ω =
𝑚𝑚2 + 𝑘𝑘2/𝑎𝑎2(𝑡𝑡)
 u oscillates  interactions at different times cancel
 Enhance: how they interact with gravity
 New (mass) interaction does not fall
 Expect field strength ~#𝐺𝐺𝐻𝐻2 𝑎𝑎 𝑡𝑡 ln[𝑎𝑎 𝑡𝑡 ]
 Change 𝐸𝐸𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 : < 𝑇𝑇µν >≠ 0
 B-mode polarization but it is order of (𝐺𝐺𝐻𝐻2 )2
 Toy model for the inflationary baryogenesis
 QG + EM ( Leonard, Woodard ) : show secular effects
 QG + QG ( Mora, Woodard ) : expect secular effects
Gravitons + Massive Fermion
arXiv: 1207.5241 by Miao
ℒ=
1
𝑅𝑅
16π𝐺𝐺
� 𝑒𝑒 µ γ𝑏𝑏 𝑖𝑖𝐷𝐷µ Ψ −𝑔𝑔 − 𝑚𝑚Ψ
� Ψ + c-terms
−𝑔𝑔 +Ψ
𝑏𝑏
 Massive fermion propagator is not that simple in dS
 From the solution of Candelas and Raine

𝑖𝑖𝑖𝑖 𝑚𝑚 𝑥𝑥; 𝑥𝑥
𝐶𝐶.𝑅𝑅
=
𝐷𝐷−1
𝑎𝑎𝑎𝑎′ − 2
𝑖𝑖𝑖𝑖 𝑚𝑚 (𝑥𝑥; 𝑥𝑥 ′ )
Gravitons + Massive Fermion—Massive
Fermion Propagator
 Infinite series can’t be reduced to any elementary function
 Finite from the infinite series even in 𝐷𝐷 = 4
 In 𝐷𝐷 = 4, the series tend to cancel
𝐷𝐷
2
 But they both are multiplied by Γ(2 − )
 𝑚𝑚 ≪ 𝐻𝐻 (order m,1-loop)
 Massless fermion prop.  𝑖𝑖𝑖𝑖 𝑥𝑥; 𝑥𝑥 ′ =
𝐷𝐷
Γ( 2 −1)
4π𝐷𝐷/2
1
𝑖𝑖γµ 𝜕𝜕µ ∆𝑥𝑥 𝐷𝐷−2
Old GR. Prop. & Non-inv. C-terms
 Old gauge noncovariant but SIMPLE
𝑔𝑔𝜇𝜇ν ≡ 𝑎𝑎 2 [η 𝜇𝜇ν + κℎ𝜇𝜇𝜇 ]
𝐼𝐼
𝑖𝑖[𝜇𝜇ν ∆𝜌𝜌𝜌𝜌 ] 𝑥𝑥; 𝑥𝑥 ′ = ∑𝐼𝐼=𝐴𝐴,𝐵𝐵,𝐶𝐶 [𝜇𝜇ν 𝑇𝑇𝜇𝜇ν
] × 𝑖𝑖∆𝐼𝐼 𝑥𝑥; 𝑥𝑥 ′
𝐼𝐼 ] are constants
 [𝜇𝜇ν 𝑇𝑇𝜌𝜌𝜌𝜌
 𝑖𝑖∆𝐼𝐼 (𝑥𝑥; 𝑥𝑥 ′ ) are simple functions in D=4
 𝑦𝑦(𝑥𝑥; 𝑥𝑥 ′ ) ≡ 𝐻𝐻2 𝑎𝑎𝑎𝑎𝑎 𝑥𝑥⃗ − 𝑥𝑥⃗′
 𝑖𝑖∆𝐴𝐴 ~
1
𝑦𝑦
2
− η − η′ − 𝑖𝑖𝑖𝑖
− ln 𝑦𝑦 + ln(𝑎𝑎𝑎𝑎′ ) ,
𝑖𝑖∆𝐵𝐵 ~ 𝑖𝑖∆𝐶𝐶 ~
2
1
𝑦𝑦
 C-terms: regulation tech. + gauge fixing (break special spatial conf.)
 non-inv. C-term
 Old GR. Prop. breaks time dilatation
Gauge Issue of Graviton Prop.
 Average gauge fixing versus exact gauge fixing
 Covariant gauge fixing term to ℒ 
 Divergent response for a point source (GR) (Mottola et al.)
 On-shell singularity for M2(x; x′) of SQED in the de Sitter invariant
analogue of Feynman gauge (Woodard & Kahya)
 linearization instability
 Gauge theory, background isometries, spatial Tn
 Non de Sitter invariant gauge (old graviton prop.)
 imposing exact gauge condition (new graviton prop.)
(arXiv:1106.0925 by Miao,Tsamis & Woodard)
(arXiv:1205.4468 by Mora, Tsamis & Woodard)
 The coefficient of ln[𝑎𝑎 𝑡𝑡 ] for the old GR. Prop. the same as that
in the new GR. Prop.
Our Conjecture, Inv. Op. & observables
arXiv:1204.1784 by Miao & Woodard
Throw away gauge-fixed Green’s fun. ?
 We construct the flat space S-matrix from them!
 Need to separate physical information from unphysical
 Inv. Op. ≠ observable

Every gauge fixed GF represents some invariant
𝑡𝑡
𝐴𝐴 0 𝑡𝑡, 𝑥𝑥 = 0 = 𝐴𝐴1(0, 𝑥𝑥)  𝐴𝐴1 𝑡𝑡, 𝑥𝑥 = ∫0 𝑑𝑑𝑑𝑑 𝐹𝐹01 (𝑠𝑠, 𝑥𝑥)
 No cosmological S-matrix  what is being measured?
 Maybe leading secular effects gauge independent
 The coefficient of ln[𝑎𝑎 𝑡𝑡 ] for the old GR. Prop. the same as that
in the new GR. Prop.
 NB these effects aren’t present in flat space
 Cf. poles terms of gauge fixed GF’s in flat space QFT
 How to check  re-compute in other gauges
Why some QG Results are Reliable
 QG not renomalizable
 No physical principle fixes the finite part
𝑖𝑖κ2 { ∆α1
𝑚𝑚 2
𝜕𝜕
𝑎𝑎
+ ∆α2 𝑚𝑚𝑚𝑚𝜕𝜕0 + ∆α3 𝑚𝑚𝑚𝑚γ0 γ𝑘𝑘 𝜕𝜕𝑘𝑘 + ∆α4 𝐻𝐻2 𝑚𝑚𝑚𝑚 }δ4 (𝑥𝑥 − 𝑥𝑥 ′ )
But loops of massless  non-analytic contribution can’t be affected by
local counterterms
 Nonlocal terms dominated over Δα at late time
𝑖𝑖𝑖𝑖2
16π2
{ 3ln[𝑎𝑎] 𝜕𝜕 2 +
97ln [𝑎𝑎]
𝑚𝑚𝑚𝑚𝜕𝜕0
16
+
9ln[𝑎𝑎]
0 γ𝑘𝑘 𝜕𝜕
𝑚𝑚𝑚𝑚γ
𝑘𝑘
16
 Low energy effective theory
+
 Fermi theory versus Standard Model etc.
 The UV completion of QG cannot
 Add new massless particles
 Change the behavior of long range forces
95ln[𝑎𝑎] 2
𝐻𝐻 𝑚𝑚𝑚𝑚
8
} δ4 (𝑥𝑥 − 𝑥𝑥 ′ )
Massive −𝑖𝑖[𝑖𝑖 Σ𝑗𝑗 ] 𝑥𝑥; 𝑥𝑥 ′ at 1-loop(still on going)
+ 2-page long tabulated nonlocal terms!
 Some 𝑎𝑎 3 ln[𝑎𝑎 𝑡𝑡 ] (or 𝑎𝑎 2 ln[𝑎𝑎 𝑡𝑡 ]) might get cancelled!
Non-Pertuabative Technique I
 Starobinsky’s formalism for a scalar model
 Truncate scalars in IR & set in D = 4 (No UV div. at leading log)
 Re-sum IR logs for 𝑉𝑉(φ) from below
ℒ=─
Dφ +
1
2
−𝑔𝑔 𝜕𝜕µ φ𝜕𝜕ν φ𝑔𝑔µν − V(φ) −𝑔𝑔
𝑉𝑉𝑉(φ)
1+δ𝑧𝑧
= 0  φ = φ0 −
φ 𝑡𝑡, 𝑥𝑥⃑ = φ0 𝑡𝑡, 𝑥𝑥⃑ −
1 𝑉𝑉 ′ φ0
𝐷𝐷 1+δ𝑧𝑧
1 𝑉𝑉𝑉(φ0 )
𝐷𝐷 1+δ𝑧𝑧
+
;
1 𝑉𝑉 ′′ φ0
𝐷𝐷 2 1+δ𝑧𝑧
1
𝐷𝐷
≡ 𝐺𝐺(𝑥𝑥; 𝑥𝑥𝑥) retarded GF
+⋯
 Each and only scalars produces IR Log (no derivative)
 Actives: produce IR Log
 Passives: propagate through IR Log
Non-Pertuabative Technique II
 actives + passives (∂′s passives), Eg: Yukawa, SQED
 integrate out passives & evaluate the effective action with
constant actives (effective potential)
 UV div. at leading Log  turn D on
 ∂′s actives + passives or ∂′s passives, Eg: all QG model
(0803.2377 by Miao & Woodard)




Can’t infrared truncate the field
can’t either ignore or integrate out ∂′s actives
Keep dimensional regulation on
Simple rule:
Questions
 How to construct non-perturbative Tech. for QG models?
 Check simple rules for other QG models
 Eg: QG + massive fermions serves as ``data’’
 Can QFT effects play a role for a modification of gravity on
galactic (& larger) scales ?
 Effective potential from Yukawa, 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 =
𝐻𝐻 4
𝑓𝑓2 φ2
− 8π2 𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 𝐻𝐻 2
 𝑅𝑅 = 12𝐻𝐻2 in de Sitter  what are those factors of H for a
general metric?
 Is the leading IR effects a gauge artifact?
 Conjecture: leading IR might be gauge-independent
 No S matrix or invariant rates in FRW
 Invariant op. doesn’t guarantee ``physical observables’’