Int. Quantum Fields
in de Sitter Space
Richard Woodard
University of Florida
Why de Sitter?
No sex to disrupt study!
A paradigm for inflation
1.
2.
3.
FRW: ds² = -dt² + a²(t) dx²
De Sitter: a(t) = exp[Ht]
Cool stuff happens!
Spacetime Exp. Strengthens QFT
Why?
Maximum Effect for
Loops classical physics of virtuals
Expansion holds virtuals apart longer
Inflation
M=0
No conformal invariance (classically)
Two Particles
MMC scalars
gravitons
MMC Scalar Models
λφ^4 (Brunier,
1.
M²(x;x') ∆u(t,k) & <Tµν>
Growing scalar mass & pos. vac. Energy
SQED (Kahya,
2.
Kahya, Onemli)
Prokopec, Tornkvist, Tsamis)
M²(x;x') ∆u & Π(x;x') ∆ε
<φ*φ>, <(Dµφ)*Dνφ>, <FµνFρσ> & <Tµν>
Growing photon mass & neg. vac. Energy
Yukawa (Duffy,
3.
Prokopec, Miao)
M²(x;x') ∆u, Σ(x;x') ∆u & <φψψ>
Growing fermion mass & neg. vac. Energy
Quantum Gravity Models
QG + Dirac (Miao)
1.
Σ(x;x') ∆u(t,k)
Growing fermion field strength
QG + MMC Scalar (Kahya)
2.
M²(x;x') ∆u(t,k)
Possible tilt in Power Spectrum
QG (Tsamis)
3.
Σ(x;x') & <hµν>
Consistent with relaxation of Λ
Infrared Logarithms
WHAT: factors of ln(a) = Ht in QFT results
EG: λφ^4
1.
2.
P = λ H^4[-2 ln²(a) -
4/3
ln(a)]/(4π)^4 + ...
WHY:
3.
<Ω‫׀‬φ²‫׀‬Ω> = H² [ln(a) + UV]/(4π²)
(Ford & Vilenkin, Linde, Starobinsky, 1982)
∫_0^t dt’ 1 = ln(a)/H
NB: ln(a)’s even in Power Spectrum
4.
Weinberg, hep-th/0605244.
How many ln(a)’s per coupling
constant?
∆₤ = (c.c)(φ, hµν)^N (other fields)
up to N ln(a)’s for each (c.c.)²
1.
2.
λφ^4 ln²(a) for each λ
SQED (eφ*∂φA & e²φ*φAA)
ln(a) for each e²
3.
4.
Yukawa (gφψψ) ln(a) for each g²
QG (κh∂h∂h) ln(a) for each G H²
The Perturbative Conundrum
When lowest ln(a) loops go order 1 …
1.
2.
3.
λφ^4 ln(a) ≈ 1/√λ
SQED ln(a) ≈ 1/e²
Yukawa ln(a) ≈ 1/g²
QG ln(a) ≈ 1/GH²
… ALL ln(a) loops go order 1!
And perturbation theory breaks down.
Leading Log Approximation
Starobinskii’s Rules (astro-ph/9407016)
Proof (with Tsamis,
gr-qc/0505115)
Generic Free Field Expansion
φ= φ0 + λVφ³
= φ0 + λVφ0³ + λ²V²φ0^5 + ...
= φ0 [1+λVφ0² + (λVφ0²)²+...]
φ^N=φ0^N [1+λVφ0²+...]
Hence each extra λ brings V φ0²
One ln(a) from V
One ln(a) from φ0²
Infrared Truncations
IR Truncated Yang-Feldman
Non-Perturbative Solution
Physics of competition
Inflationary particle production pushes φ up
Classical force pushes it back down
Expect a static limit for ρ(t,φ)
Cosmology is NOT an RG Flow
Renorm. Group: xµ → A xµ
Cosmology: dxµ → a(t) dxµ
RG Cosmology: Replace A with a(t)
WRONG for MMC scalar + λφ4
1.
2.
3.
4.
RG for A ∞ is free
a(t) ∞ is NOT free
More General Theories
Cannot Ignore Passives
Passives can carry
IR logs
Eg. < F(x) F(x) >
Actives interact
through passives
Eg. Quartic coupling
Cannot Ignore the UV
IR (H<k<Ha) vs. UV (Ha<k)
IR logs only from IR of actives
Effective ints from IR+UV of passives
What to do?
Integrate out passive
Complicated Effective Action!
IR truncate and simplify
Reduces to Effective Potential
Evaluate for φ(x) = const
At L.L.O. each φ MUST contr. to a ln(a)
So forget about space dependence
Because passives are passive . . .
Green’s functions give pos. powers of a’/a
Nonperturbative Results for SQED
<φ*φ> ≈ 1.6495 H²/e²
M²γ ≈ 3.32133 H²
M²φ ≈ .8961 ٠ 3e²H²/8π²
ρvac ≈ -.6551 ٠ 3H^4/8π²
1.
2.
3.
4.
Cf. a dielectric slab in a charged capacitor
≈-.2085 ٠ Λ/8πG ٠ GH²
Small wrt Λ/8πG but HUGE wrt ρcrit
And DYNAMICAL
Conclusions
Infl. + m=0 + no conf. enhanced QFT
Starobinskii’s stochastic formalism
MMC scalars and gravitons
Effects manifest as factors of ln[a(t)]
Gives leading IR logs for V(φ) models
Can be summed for V(φ) bounded below
UV doesn’t matter
General models have passive fields
Eg ψ in Yukawa
Don’t cause IR logs but can carry them
Also mediate interactions between actives
Conclusions II
Integrate out passives, then IR truncate
Same as computing effective potential!
Done for Yukawa and SQED
V(φ) unbounded below for Yukawa!
SQED φ reaches ~ H/e!
IR logs DO NOT always sum to a constant
Non-pert. confirmation of Davis, Dimopoulos, Prokopec
and Tornkvist (PLB501 (2001) 165)
Next up: Quantum Gravity