Is Weinberg’s Theorem violated?
Larry Ford(Tufts U.),
C-H Wu, K-W Ng (Academia Sinica)
Richard Woodard (University of Florida)
Shun-Pei Miao (University of Utrecht)
Weinberg’s Theorem
hep-th/0602442
Gravity + Inflaton + free scalars +
free spinors + free vector fields
Quantum corrections to the primordial
power spectum
Effects down by GH2
Largest possible IR enhancement is
powers of ln[a(t)]
Result of Ford, Wu and Ng
gr-qc/0608002
Gravity + Inflaton + Conformal Matter
Extra contribution due to conf. goes
like [1/abeg]2
Because aend = 1 (for them) we all
GUESS effect ∼ [a(t)/abeg]2
TWO PUZZLES:
1.
2.
How can Weinberg’s bound be violated?
How can conformal matter get a big IR
enhancement?
Strength of Quantum Gravity
Coupling constant G = 1/MP2
Corrections go like GE2 = (E/MP)2
TRIVIALLY large if E > MP but
Don’t know physics at those scales
Perturbation theory isn’t valid
Can SOMETIMES get IR enhancement
for E << MP
Strength of Quantum Loops
Classical response to virtual particles
More virtuals = larger response
TWO FACTORS give density of virtuals
1.
2.
Time virtuals persist (controlled by
Energy-Time Uncertainty Principle)
Rate at which they emerge (controlled by
kinetic term in Lagrangian)
How does it work for FRW?
ds2 = -dt2 + a2(t) dxi dxi
PERSISTENCE TIME
H(t) > 0 increases it
Maximum effect for q(t) < 0 (inflation)
EMERGENCE RATE
Conformal invariance reduces it
E-Time Unc. Principle for Flat Space
∆t ∆E > 1 to resolve ∆ E
Virtual pair has ∆E = 2(m2 + k2)1/2
Hence ∆t ∆E < 1 to NOT resolve
Hence can last ∆t < 1/2(m2 + k2)1/2
Eg: Vacuum polarization
Most for e± because smallest m
Smallest k’s live longest EM stronger at
shorter distances (less polarization)
E-Time Unc. Principle for FRW
ds2 = -dt2 + a2(t) dxi dxi
k = 2π/λ still conserved
But physics depends on k/a(t)
E(t) = [m2 + k2/a2(t)]1/2
∆t ∆E ∫tt+∆ t dt’ E(t’) < 1
m=0 ∫tt+∆ t dt’/a(t’)
Bounded wrt ∆t for inflation!
Any m=0 virtuals live forever!
What about the EMERGENCE RATE?
Conformal invariance, a killer symmetry:
L g’µν=Ω2gµν L’ field redefinition L
EM in 4D
L = -1/4 FµρFνσgµνgρσg1/2
D-4
L’= -1/4 FµρFνσgµνgρσg1/2Ω , F’µρ=Fµρ
Massless conformal coupled scalar (mcc)
√
1
1 D − 2 2√
− ∂µ φ∂ν φ −g −
Rφ −g
2
4D−1
Killer Symmetry Suppresses Emergence
Rate
-dt2+a2dx2=a2(-dη2+dx2) adη=dt
Conformal invariance same locally
(in conformal coordinates) as flat
Hence dN/dη = Γflat
Hence dN/dt = Γflat/a(t)
Any m=0, conf. virtuals that emerge
DO live forever, but few emerge!
Maximum IR Enhancements
Requirements:
Realized by:
PERSISTENCE TIME m=0 + inflation
EMERGENCE RATE no conf. invariance
Massless, minimally coupled scalars
Gravitons
NOT by conformal matter
How did Wu, Ng and Ford get anything?
They initially used Raychaudhuri Eqn.
dθ/dt=-Rµνuµuν- 1/3θ2_σµνσµν+ωµνωµν+(uν;µuµ);ν
θ(t)=#∫ # dt [Rµνuµuν]q
Rµν=8π G(Tµν-1/2gµνT)
(Tµν)RW=a-2(Tµν)flat
dρ/dt+θ(p+ρ)=0
δρ/ρ0= -(1+w) ∫ dη θ/a(η)
Cannot ignore (uν;µuµ);ν because it cancels parts of
-Rµνuµuν
Cannot Ignore the Acceleration Term
Co-moving gauge : uµ=δµ0
g00=-1
ds2=-dt2+2ah0idxidt+a2(δij+hij)dxidxj
θ=uµ;µ=d/dt ln(g1/2)=3H+1/2 d/dt hii
δθ=1/2 d/dt hii due to conformal matter
µν
g
∂ν ϕ
µ
u =−
.
αβ
−g ∂α ϕ∂β ϕ
h0i
∂i δϕ
u0 = 1 + O(∆ 2 ) , ui =
+ 2
+ O(∆ 2 ) .
a
a ϕ̇0
h0i (t, x) = −∂i
R00
δϕ(t, x)
a(t)ϕ̇0 (t)
+ O(∆ 2 ) .
1
1
= −3Ḣ − 3H 2 − ḧii + 2H ḣii + ḣ0i,i + Hh0i,i + O(∆ 2 ) .
2
a
(uν;µ uµ );ν
√
1
1
−→ √ ∂ν
−gg νk ġk0 =
ḣ0i,i + Hh0i,i + O(∆ 2 ) .
−g
a
Really hii Mixes with δϕ
First order perturbation :
1
δ ϕ̈ + 3Hδ ϕ̇ + V0′′ δϕ + ϕ̇0 ḣii = 0
2
1 2 ′
∇2 ∂ δϕ 1 ∂ 2 1 2 conf
2
−κ ϕ̇0 δ ϕ̇ + κ V0 δϕ − 2
− 2
a ḣii =
κ T00
2
a ∂t ϕ̇0
2a ∂t
2
(δϕ)1 = ϕ̇0
,
(ḣii )1 = 6Ḣ
δϕ(x) ≡ ϕ̇0 (t)Φ(x)
ϕ̈
0
ḣii = 6ḢΦ − 4
+ 6H Φ̇ − 2Φ̈
ϕ̇0
···
ϕ̈0 ϕ̈0
ϕ̈20
ϕ0
∇2
2
2
∂t + 5H + 2
∂t + 4Ḣ + 6H + 4H
−2 2 +2
− 2
ϕ̇0
ϕ̇0
ϕ̇0
ϕ̇0
a
Φ̇ =
1 2 conf
κ T00
2
Green’s Function Solution
The formal solution:
dΦ/dt ≡ Ψ
′
t′ , k) = θ(t−t ) Ψ2 (t, k)Ψ 1 (t′ , k) − Ψ1 (t, k)Ψ 2 (t′ , k)
G(t,
W (t′ , k)
2
D ′
′ κ
conf
Φ̇(x) =
d x G(x; x ) T00
(x′ )
2
It cannot be solved exactly but can be related to
solution of Mukhanov equation
k
θ̈ +H θ̇
2
∂t +H∂t + 2 −
Qi = 0 , θ(t) ≡ H/(a −Ḣ )
a
θ
2
Qi (t, k)
a2 (t) −Ḣ(t)
Before the 1st horizon crossing :
After the 1st horizon crossing :
Ψi (t, k) =
Qs (t) = θ(t)
,
±ik
Q± (t, k) = e
Qℓ (t) = θ(t)
t
t
dt′ /a(t′ )
dt′
a(t′ )θ2 (t′ )
Asymptotic Results for dΦ/dt
dΦ/dt is BIG at early times
dΦ/dt is SMALL at late times
Φ(t) = Φ0 + ∫0t dt’ dΦ/dt’
THIS EXPLAINS THEIR BIG Φ(end)!
Not from late times when k/a(t) << MP
From EARLY times when k/a(t) >> MP
Explaining the Two Puzzles
No violation of Weinberg’s Thm
Their [1/a(beg)]2 is NOT [a(t)/a(beg)]2
It’s really [k/a(beg)]2
No IR enhancement from conf. matter
Effect from early times when k/a >> MP
All modes give big effects in this regime!
Trans-Planckian Issue
No paradoxes, but is it RIGHT?
Problems with k/a(beg) >> MP
We don’t know fundamental physics there
Perturbation theory isn’t valid
We agree to disagree
Wu, Ng & Ford: start at beginning of inflation &
use pert. theory big corrections from early
times
Miao & Woodard: assume modes emerge from
trans-Planck regime in vacuum only small
corrections from late times
Bua’s Bound
Bua Chaicherdsukul (hep-th/0611352) :
No large effects to power spectrum
from fermions or gauge particles, no
matter how they couple
Counter example to Bua’s bound --Yukawa theory
1
1
1
αβ √
2 √
4√
L = − ∂α ϕ∂β ϕg
−g − δξϕ R −g − λϕ −g
2
2
4!
√
√
+iψeβb γ b Dβ ψ −g − f ϕψψ −g
Integrate out fermions at leading Log. Order
and compute Veff
Scalar induces Fermion mass
Fermion vacuum E < 0
Unbounded below!
When can we get big QFT effects?
Conclusions
No Violation to Weinberg’s theorem
Big quantum corrections due to
conformal matter come from highly
Trans-Planckian regime
Only MMC scalars and gravitons get big IR
enhancements
Agree to disagree about significance
QFT corrections CAN be big
No general rule yet