de Sitter Inv. & Quantum Gravity
(arXiv:0907.4930, 1002.4037, 1106.0925, 1107.4733)
S. P. Miao (Utrecht)
N. C. Tsamis (Crete)
R. P. Woodard (Florida)
Primordial Inflation was nearly
²
de Sitter with small GH
ds2 = -dt2 + a²(t) dx.dx
²
∆²R(k) ≈ GH²(tk)/πε(tk) & ∆ h(k) ≈ 16GH²(tk)/π
WMAP data for k = .002/Mpc
& ε(t) = -Ḣ/H²
For single-scalar inflation with k = H(tk)a(tk)
H(t) = á/a
∆²R = 2.441 x 10-9
&
r = ∆²h/∆²R < 0.22
Hence
ε ≈ r/16 < 0.014 (even smaller before tk!)
GH² ≈ π/16 x r x ∆²R < 10-10
de Sitter Inv. QFT Is BORING
Zero back-reaction
<hµν(x)> = #ĝµν(x) 0 with right δΛ
Coincident Scalar VEV’s are constant
<Cρσµν(x) Cρσµν(x)> = constant
Correlators same late as early
<φ(x) φ(x+∆x)> = F(∆x)
IR Divergences Break dS Inv.
m2 < 0 not Poincare Inv. for flat space
k2 barrier protects m = 0 in flat space
2
<φ2(t,x)> depends upon when released
E ~ (∂tφ) + k2 φ2 u(t,k) = e-ikt/√2k
But barrier redshifts with a(t)
E ~ a3 [(∂tφ)2 + (k/a)2 φ2]
u(t,k) = [1 – ik/Ha] eik/Ha H/√2k3
<φ2(t,x)> = UV + (H/2π)2 ln[a(t)]
Observable for MMC Scalars
i∆(x;x’) = (dS inv) + k ln[a a’]
Drops from free <∂µφ∂νφ>
But not in 2 loop <Tµν> for φ4
Or in 1 loop –i[µΠν](x;x’) for SQED
gr-qc/0204065 with Onemli
gr-qc/0205130 with Prokopec & Tornkvist
Or in 1 loop –i[iΣj](x;x’) for Yukawa
astro-ph/0309593 with Prokopec
How about Quantum Gravity?
i[µν∆ρσ](x;x’) has ln[a a’] terms
Drops from free <Cµνρσ(x) Cαβγδ(x’)>
But probably not at 2 loops (any bets?)
Not in 1 loop –i[iΣj](x;x’) for QG + Dirac
gr-qc/0511140 with Miao
Or in <ζ(t,x) ζ(t,0)> in QG + Scalar
arXiv:1006.3999 with Kahya & Onemli
Killing IR Logs Requires dS Inv
Observables are important (& unsolved)
IR logs in <Cµνρσ(x)Cµνρσ(x)√-g(x)>
L=1 loop 0
L=2 loops 1
L=3 loops 2, etc.
More & more cancellations needed
Which don’t happen for MMC scalars
Do you believe in the Tooth Fairy?
Some People HATE IR Logs
(they want QFT to be boring)
Previous Position
Manifestly dS inv. i[µν∆ρσ](x;x’) exist
ln[a a’] in my solution is artifact or error
Current Position
No manifestly dS inv. solution
But free dyn. gravitons physically dS inv.
Doubt current position, but
not relevant for back-reaction
Dynamical gravitons typically weaker
Constrained gravitons typically stronger
Tests of GR only binary pulsar
Inflation ∆h2(k) ~ 16GH2(tk)/π
Tests of GR Solar system
Inflation ∆R2(k) ~ GH2(tk)/πε(tk)
Both in cov. i∆’s but NOT in physical i∆’s
Back-Reaction requires both
Sourced by dynamical (at 1 loop)
Grav. response by constrained (at 2 loops)
Methodology behind “Previous
Position”
Add α(Dνhνµ + βDµhνν)2
Solve in Euclidean space & continue
Ok except few “singular” choices of α and β
Burden of my Talk:
Obstacle to adding gauge fixing term
Obstacle to analytic continuation
Origin of “singular” gauges
“Exact” vs “Average” Gauges
Illustrate with EM in flat space
Exact: ∂iAi= 0 (Coulomb)
Average: L L - ½ (∂µAµ)2 (Feynman)
Derive Average from Exact
Start in canonical functional formalism
∫ [dET] [dAT] eiSfixed
S. Coleman, Erice 1973
Coleman’s Seven Steps
1.
2.
3.
4.
5.
6.
7.
Integrate out ET
Use ∫[dAT] = ∫ [dA] δ[∂iAi] √det[∂i∂i]
Undo A0 constraint
Write integrand as invariant
δ[∂µAµ] w field dependent gauge trans
δ[∂µAµ–f(x)] w C-number gauge trans
Multiply by ∫[df] Exp[-½i∫f2]
Obstacle on T3 × R
Invariant: ∂iFi0 = J0 Q = 0
Feynman: [-∂t2+∂i∂i]A0 = J0 Q ≠ 0 ok
Problems at Coleman’s steps 2 & 3
No 0-modes for δ[∂iAi] and A0
Hence no 0-mode for gauge fixing term
Same Obstacle on de Sitter
IR ∞ of φφ* self-energy (gr-qc/0508015)
Analytic Continuation Sees Only
Logarithmic IR Divergences
[□-M2]i∆(x;x’) = iδ4(x-x’)/√-g
i∆(x;x’) =∫d3k/(2π)3 eik—(x-x’)
x[θ(t-t’)u(t,k)u*(t’,k)+θ(t’-t)u*(t,k)u(t’,k)]
u(t,k) = [π/(4Ha3)]½ Hν(1)[k/Ha]
ν = [9/4 – M2/H2]½
uu* ~ k-2ν [1 + O(k2)]
IR ∞’s for 2ν ≥ 3 M2 ≤ 0
But only logarithmic for M2 = -N(3+N) H2
IR ∞’s Signal Wrong Physics
DON’T subtract them, fix the physics
Exclusive Incl. in flat QED, QCD & QG
Physical Problem: E ~ a3 [ẋ2+(k/a)2x2]
k2 barrier redshifts away
Super-horizon modes diffuse out
How far depends upon when released
Bunch-Davies released at t -∞
Standard Fixes Release at
finite time
No change if Bunch-Davies for all k
But can’t enforce BD for k < Hainitial
Vilenkin (NPB:226,527,1983)
Change Bunch-Davies for k < Hainitial
NCT and RPW (CQG:11,2969,1994)
Bunch-Davies on T3xR with no k < Hainitial
Exact de Donder Gauge:
Dνhνµ = ½ Dµhνν
i[µν∆ρσ](x;x’)
= (Inv. Spin 0) S0 + (Inv. Spin 2) S2
¾[□+4H²][□+6H²]² S0 = -iδ4(x-x’)/√-g
M² = -4H² is Log ∞
M² = -6H² is power law ∞
□³[□-2H²]² S2 = 128 H4 iδ4(x-x’)/√-g
M2 = 0 is Log ∞
M2 = 2 H2 is IR finite
Final Comments
i[µν∆ρσ](x;x’) not dS invariant
IR divergences for all 0 ≤ ε ≤ 3/2
Constants are observable! (E.g., vacuum E)
Beware dropping constrained gravitons
Renormalizing nonlocal operators
Beware myopic focus on power spectra
Nothing special about de Sitter
Observables crucial & unresolved
IR logs in some diagrams
They keep our feet on the ground!
Beware approximations
Only 3 dim. reg. & renorm. QG loops on dS