Interacting Quantum Fields
on de Sitter Space
R. P. Woodard
Academia Sinica
December 30, 2011
(Spatially
Flat)
FRW Cosmology
ds² = -dt² + a²(t)dx·dx
H(t) = å/a
ε(t) = -Ḣ/H2 = 1 – aä/å2
Current values
H0 = (73.8 ± 2.4) km/s-Mpc ~ 2.4·10-18 Hz
ε0 = 0.33 ± 0.13
Inflation: å > 0 & ä > 0 H > 0 & ε < 1
Happening now! (but with GH02 ~ 10-122)
Primordial inflation (with GH2 ~ 10-10)
Various power law expansions
Radiation domination
Matter domination
a(t) ~ t²⁄³ H(t) = 2/(3t) ε(t) = 3/2
Curvature domination
a(t) ~ t½ H(t) = 1/(2t) ε(t) = 2
a(t) ~ t H(t) = 1/t ε(t) = 1
Vacuum energy domination
a(t) ~ eKt H(t) = K ε(t) = 0
Single Scalar Inflation
L = -½∂µφ∂νφgµν√-g – V(φ)√-g
ds2 = -dt2 + a2(t)dx·dx & φ = f(t)
Reconstruction: given a(t) H(t) ε(t)
∂tḟ + 3Hḟ + V’(f) = 0
3H2 = 8πG [½ḟ2 + V(f)]
(2ε-3)H2 = 8πG [½ḟ2 – V(f)]
8πGḟ2 = 2εH2 f(t) & t(f)
8πGV = (3–ε) H2 V(φ)
Conclusion
Can get any 0 < ε(t), including 0 < ε < 1
Long inflation requires flat V(φ)
What the CMB data says
Two power spectra (k0 = 0.002 Mpc-1)
∆R2(k) = (2.43±0.082)▪10-9 (k/k0)-0.0332±0.0093
r = ∆h2(k0)/∆R2(k0) < 0.17 (95% conf.)
For single-scalar inflation (with k = H(tk)a(tk))
Comparing theory to observation
∆R2(k) ~ GH2(tk)/πε(tk)
∆h2(t) ~ 16GH2(tk)/π
ε(tk0) ~ r/16 < 0.011
GH2(tk0) ~ π/16 ▪ ∆R2(k0) ▪ r < 8.1▪10-11
Conclusion
Primordial inflation very close to de Sitter
Quantum gravity effects small but not negligible
de Sitter Space constant H
Maximally symmetric solution with Λ> 0
5-dimensional embedding: xµ XA
0 2
ds2
1 2
2 2
3 2
4 2
-(X ) + (X ) + (X ) + (X ) + (X ) = 1/H2
2
2
Ht
= -dt + a (t)dx·dx with a(t) = e
0
-2
2
X = a/2H (1 – a + H x·x)
i
Xi = a x
4
-2
2
X = a/2H (1 + a – H x·x)
Conformal coordinates (dt = a dη)
2
2
ds2 = a [-dη + dx·dx] with a = -1/Hη
Conformal time runs from -∞ < η < 0
2
N.B. gµν = a ηµν
Life Cycle of a Mode
ds2 = -dt2 + a2(t)dx·dx translation inv.
k = 2π/λ a good quantum number
But physics depends on k/a(t)
Behavior of H(t)·a(t)
Oscillations for k/a(t) >> H(t)
No oscillations for k/a(t) << H(t)
inflation Ha ~ K·eKt grows
deceleration Ha ~ (p/t)·(t)p ~ (t)p-1 falls
Horizon crossing
1st crossing: k = H(tk)·a(tk) during inflation
2nd crossing: k = H·a after inflation
MMC Scalar φ(t,x)
L = -½ ∂µφ∂νφgµν√-g
= ½ a3[(∂tφ)2 – ∂iφ∂iφ/a2]
L = ∫d3x L
f(t,k) = ∫d3x e-ik·x φ(t,x)
L = ½ a3 ∫d3k/(2π)3 [|∂tf|2 – k2/a2 |f|2]
Each f(t,k) a harmonic oscillator
With m(t) ~ a3(t) and ω(t) = k/a(t)
Ground state at time t has E = ½ k/a(t)
But no stationary states
Quantum Mechanics of f(t,k)
Mode Eqn: ∂tḟ + 3Hḟ + k2/a2 f = 0
Bunch-Davies vacuum
α |Ω> = 0
Minimum energy for t -∞
E(t,k) = ½ a3(t) [|ḟ(t,k)|2 + k2/a2(t) |f(t,k)|2]
u(t,k) = H/(2k3)½ [1 – ik/Ha(t)] eik/Ha(t) (de Sitter)
f(t,k) = u(t,k) α + u*(t,k) α† with [α,α†] = 1
<Ω|E(t,k)|Ω> = ½ a3|ů|2 + ½ k2a|u|2 = k/2a + H2a/4k
<Ω|E(t,k)|Ω> = k/a(t) [½ + N(t,k)]
N(t,k) = [Ha(t)/2k]2 for a single mode
UV (k>>Ha) N ~ 0
IR (k<<Ha) explosive growth (CMB anisotropies)
Compare with MCC φ(t,x)
L = -½ ∂µφ∂νφgµν√-g –
1/12
φ2R√-g
= ½ a3(∂tφ)2 -½ a(∂iφ)2 -½ a3Ḣφ2 –a3H2φ2
= 1/(2a) {[∂η(aφ)]2 – [∂i(aφ)]2 - ∂η(Ha3φ2)}
a(t) φ(t,x) a flat space m=0 scalar
v(t,k) = exp[-ik∫tdt’/a(t’)]/[(2k)½a(t)]
<Ω|E(t,k)|Ω> = ½ k/a(t)
cf u(t,k) = exp[-ik∫tdt’/a(t’)]/(2k)½·[1/a(t) – iH/k]
Both oscillate and fall off for k >> Ha
Both stop oscillating for k << Ha
u(t,k) const, whereas v(t,k) const/a(t)
Summary so far
Primordial Inflation approximately de Sitter
QFT strengthened because
GH2 ~ 10-10 vs GH02 ~ 10-122 K
Inflation produces MMC scalars & gravitons
This is what caused the CMB anisotropies
Will interact with themselves & other fields
MMC scalar & graviton perturbations fossilize
QG perturbative but not negligible
Preserve effects from high scales
Cf. arXiv:1005.4530 with Ford, Miao, Ng & Wu
Many interesting things besides CMB
Changing Kinematics & Forces
Inflationary scalars & gravitons can change
Free particle wave functions (mass & field strength)
Force laws (for EM & GR)
How to check
Compute φ &/or hµν corrections to particle’s 1PI 2-pt
Solve linearized effective field eqns
1PI 2-point functions
Scalars: -iM2(x;x’) (scalar self-mass-squared)
Fermions: -i[iΣj](x;x’) (fermion self-energy)
Photons: +i[µΠν](x;x’) (vacuum polarization)
Gravitons: -i[µνΣρσ](x;x’) (graviton self-energy)
Quantum-Correcting Maxwell’s
Eqns with Vacuum Polarization
∂ν[√-g gνρgµσ Fρσ(x)] + ∫d4x’[µΠν](x;x’)Aν(x’) = Jµ(x)
Jµ = 0 kinematics of photons
Jµ ≠ 0 EM forces
Important Simplifications
Fµν = ∂µAν - ∂νAµ
√-g gνρgµσ = ηνρηµσ in conformal coordinates for D=4
Solve Perturbatively
[µΠν] = [µΠν]1 + [µΠν]2 + . . .
Aµ = A0µ + A1µ + A2µ + . . .
∂ν(F0)νµ = Jµ
∂ν(F1)νµ = -∫d4x’ [µΠν]1(x;x’) A0ν(x’)
One loop [µΠν] from SQED
(PRL 89 (2002) 101301, astro-ph/0205331 with Prokopec & Tornkvist)
Transverse tensor structure
(ηµνηρσ – ηµρηνσ) ∂’ρ∂σ [ΠF + ΠC + ΠG](x;x’)
(δmnδrs- δmrδns) ∂’r∂s ΠK(x;x’) (purely spatial)
∂2 = ηµν∂µ∂ν , ∆η = η-η’ , ∆x = |x-x’|
ΠF(x;x’) = -α/96π ∂4[θ(∆η-∆x) {ln[µ2(∆η2-∆x2)] - 1}]
ΠC(x;x’) = -α/6π ln(a) δ4(x-x’)
ΠG(x;x’) = -αH2a/8π ∂2[a’ θ(∆η-∆x) {ln[H2(∆η2-∆x2)] + 1}]
ΠK(x;x’) = +αH4(aa’)2/4π θ(∆η-∆x) {ln[H2(∆η2-∆x2] + 2}
Points to note
Each Π(x;x’) causal – from Θ(∆η-∆x) -- & real
ΠF(x;x’) same as in flat space
ΠG(x;x’) & ΠK(x;x’) contains powers of a and a’
One loop results
Long wavelength photons develop mass
EM force screened after N~6 e-foldings
Might this give magnetogenesis?
Certainly reached during primordial
inflation IF there are charged MMC scalars
Vacuum energy decreases
Perturbation theory breaks down
Nonperturbative Results for SQED
<φ*φ> ≈ 1.6495 H²/e²
Mγ2≈ 3.32133 H2
Mφ2 ≈ .8961 ٠ 3e²H2/8π²
ρvac ≈ -.6551 ٠ 3H4/8π²
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
MMC Scalar Models
φ4 (Brunier,
1.
Kahya, Onemli)
M²(x;x') ∆u(t,k) & <Tµν>
Growing scalar mass & pos. vac. Energy
SQED (Kahya,
2.
Prokopec, Tornkvist, Tsamis)
M²(x;x') ∆u(t,k) & [µΠν](x;x') ∆εµ(t,k)
<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.
[iΣj](x;x') ∆u(t,k)
Growing fermion field strength
QG + MMC Scalar (Kahya, Park)
2.
M²(x;x') ∆u(t,k)
[µνΣρσ](x;x’) ∆εµν(t,k) & ∆Φ(t,r)
QG (Tsamis, Mora)
3.
[µνΣρσ](x;x') & <hµν>
Consistent with relaxation of Λ
A Chance to Play at being
Feynman & Schwinger
Flat space quantum field theorists had to
Cosmological quantum field theorists have to
Quantum correct the ground state
Identify observables
Resolve the IR problem
Resolve the UV problem
Quantum correct the ground state
Identify observables
Resolve the IR problem
Resolve the UV problem
Important problems to solve & data to check
Conclusions
Primordial inflation approximately de Sitter
Primordial inflation enhances QFT because
Effects from two species of particles
MMC scalars (if any)
Gravitons
Corrections to kinematics & forces
GH2 ~ 10-10 >> GH02 ~ 10-122
Particle production from accelerated expansion
Fossilized effects preserved to late times
Compute corrections to 1PI 2-point functions
Solve linearized effective field eqns
Many effects studied, many left to study