Quantum Gravity & Inflation
N.C. Tsamis & R.P. Woodard
arXiv: 0904.2368, 1001.4929,
1006.4834 & 1103.5134 and (with M.
G. Romania) 1006.5150 & 1108.1696
FRW Cosmology & Inflation
ds2 = -dt2 + a2(t) dx.dx
Parameters:
H(t) = d/dt ln[a(t)]
ε(t) = -Ḣ/H2
-18 Hz
H0 ~ 71 km/(s-Mpc) ~ 3.2 x 10
ε0 ~ 0.4
Inflation H > 0 with ε < 1
Primordial Inflation (t ~ 10-33 s)
HI ~ 1037 Hz with εI < 0.014
The Horizon Problem
ds2 = -dt2 + a2 dr2 = 0 dr = dt/a(t)
Rpast = ∫dt/a(t) for tpast < t < tnow
Rfuture = ∫dt/a(t) for tinitial < t < tpast
Const ε = -Ḣ/H2 H = 1/εt a ~ t1/ε
∫dt/a(t) = 1/[(ε-1)Ha]
Deceleration ε > 1 Ha ~ t1-1/ε falls
Rpast ~ 1/[(ε-1)Ha]now
Rfuture ~ 1/[(ε-1)Ha]past << R
past
Causality (Rpast/Rfuture)2 < 1
With deceleration late and early
Recombination (Rpast/Rfuture)2~ 2000
Nucleosynthesis (R {past}/Rfuture)2 ~ 109
Early acceleration Ha ~t1-1/ε grows
Rfuture ~1[(1-ε)Ha]initial
For tinitial 0
Rfuture >> Rpast
Single-Scalar Inflation
L= [-½ ∂µφ∂νφgµν – V(φ) + R/16πG]√-g
1.
2.
3H2 = 8πG [½(dφ/dt)2 + V(φ)]
-2Ḣ - 3H2 = 8πG [½(dφ/dt)2 - V(φ)]
Reconstruction (for ANY a(t) with Ḣ ≤ 0)
(1+2) -2Ḣ = 8π G (dφ/dt)2
φ(t) = φI + ∫t dt’ [-Ḣ(t’)/4πG]½
Invert t = t(φ)
(1-2) 2Ḣ+6H2 = 16πG V
V(φ) = 1/8πG [3H2 + Ḣ][t(φ)]
Problems of φ-Driven Inflation
Must have (dφ/dt)2 << V(φ) initially
Also spatially homogenous over V > H-3
V(φ) must be flat for enough inflation
V(φ) must give δρ/ρ ~ 10-5
Vmin ~ 0
Need ∆L = g φψψ for reheating
Must then RE-tune Veff(φ) for above
“Fundamental theorists need
to provide some guidance …”
3 Simple Ideas:
Λ isn’t small (GΛ ~ 10-6)
No scalars, no fine tuning
Quantum IR from L = (16πG)-1 (R-2Λ)√-g
3 Simple Consequences:
Λ starts inflation
IR gravitons eventually stop it
Long Inflation because gravity is weak
Each mode k is a harmonic osc.
with m ~ a3(t) and ω = k/a(t)
LMMC = ½a3 [(dφ/dt)2 – (∇φ)2/a2]
3
-ik.x φ(t,x)
f(t,k) = ∫d x e
3
L = ∫d x L
=∫d3k/(2π)3 ½a3 [|ḟ|2 – (k/a)2|f|2]
f(t,k) x(t)
E(t,k) = ½a3(t) [ẋ2 + (k/a)2x2]
<ψ|E(t,k)|ψ> = k/a(t) [½ + N(t,k)]
x(t) = u(t) α + u*(t) α† <E(t)>
= ½ m [|ů(t)|2 + ω2|u(t)|2]
Constant m & ω ẍ + ω2 x = 0
x(t) = x0 cos(ωt) + ẋ0/ω sin(ωt)
= e-iωt ½[x0+iẋ0/ω] + eiωt ½[x0–iẋ0/ω]
u(t) = e-iωt/√2ω
m(t) = a3(t) & ω(t) = k/a(t)
ẍ + 3H ẋ + (k/a)2 x = 0
a = eHt u(t,k) = H/√2k3 [1 – ik/Ha] eik/Ha
2
3
2
|u(t,k)|2 = H /2k [1 + (k/Ha) ]
2
2
3
2
2 2
|ů(t,k)| = H /2k (k /Ha )
2
N(t,k) = (Ha/2k)
Many Gravitons N(t,k) = (Ha/2k)2
Not Much Energy k/a(t)
Silly view of redshifting IR gravitons
k > H(t) a(t) physical
k < H(t) a(t) pure gauge
k > H(t) a(t) physical
ρIR= a-3(t) ∫Ha d3k/(2π)3 N(t,k) k/a(t) = H4/16π
Cf ρΛ = 3H2/8π G ρIR/ρΛ = GH2/6π < 10-11
Big volume can beat small ρ
Const. ρ over R M ~ ρR3
U ~ -GM2/R ~ -Gρ2R5
ρU ~ U/R3 ~ -Gρ2R2
Perturbative Results
Perturb around
3[Heff(t)]2 = Λ + 8πGρ(t)
ds2 = -dt2 + a2(t) dx2 with a(t) = eHt
ρ ∼ +Λ2
ρ ∼ –GΛ3 ln[a(t)]
ρL ∼ –Λ2 [GΛln(a)]L-1
dρ/dt = -3Heff(ρ+p) p(t) ~ -ρ(t)
Hence p ∼ -ρ ∼ Λ2 f[GΛln(a)]
Pert. theory breaks down for ln(a) ~ 1/GΛ
Can we guess the effective
field equations?
Classical: Gµν = -Λgµν
Quantum: Gµν = -Λgµν + 8πGTµν[g]
No hope to guess full Tµν[g]
All scattering amplitudes to all orders!
Quantum GR may not even be right!
Only need most cosmo. significant part
for ds2 = -dt2 + a2(t) dx.dx
Quantum GR should be right for this
Guidance in Model-Building
Locality Conundrum
Tµν[g] must be nonlocal
Causality Conundrum
Nonlocality must be restricted to the past
Action-Reaction Conundrum
Effective field equations
Locality Conundrum
de Sitter Rρσµν = H2[gρµgσν - gρνgσµ]
DαRρσµν = 0
Tµν[gde Sitter] = #H4 gµν
Gµν = -3H2 gµν + #8πGH4 gµν
= -3H2 [1 – 8/3 πGH2] gµν
Λ’ = [1 – 8/3 πGH2] Λ
Stays de Sitter forever with Λ’
Therefore Tµν[g] must be nonlocal
Causality Conundrum
Tµν(x) = -2/√-g δΓ[g]/δgµν(x)
guarantees conservation
But if gµν at t-∆t then also at t+∆t
q(t) in Γ[q] = ∫ds q(s) q(s-∆t)
δΓ/δq(t) = ∫ds[δ(s-t)q(s-∆t)+q(s)δ(s-∆t-t)]
= q(t-∆t) + q(t+∆t)
Acausality!
Three ways to avoid acausality
Schwinger-Keldysh two fields
Partial Integration Trick
g±µν forward/backwards sources
Tµν[g] = -2/√-g δΓ[g+,g-]/δg+µν |g± = g
Γ[φ] = ∫d4y φ(y)∫d4z Gret(y;z) φ(z)
δΓ/δφ(x) = ∫d4zGret(x;z) φ(z) + ∫d4yφ(y)Gret(y;x)
∫d4z Gret(x;z) φ(z) + ∫d4y Gret(x;y) φ(y)
Only need DG(x;y) = δ4(x-y)
Give Field Eqns & enforce conservation
Action-Reaction Conundrum
(for partial integration trick)
Want: nonlocal distortion of Λ term
But L = √-g Λ f[□-1R]
gµν(x) Λ f[□-1 R]
R safely in the past of xµ
-2/√-g δS/δgµν(x) = gµν(x) f(□-1R) +
[variation of □-1] + ΛGµν(x) □-1f’(□-1R)
Don’t want nonlocal distortion of Gµν term!
Solution: Try for field eqns directly
Perfect Fluid Ansatz
Tµν[g] =p[g] gµν + (ρ+p) uµuν
Includes general FRW
DµTµν = 0 4 eqns
p, ρ uµ (gµν uµuν = -1) 5 variables
Pick p[g] ρ[g] & uµ[g] for DµTµν=0
Enforcing conservation about FRW + ∆gµν
0th order uµ = δµ
Get ∆u0 from gµν uµuν = -1
Dµ[(ρ+p)uµ] = u.∂p ∂t [a3(ρ+p)] = Known
(ρ+p) u.D uν = -(∂ν+uν u.∂) p ∂t(ui/a) = Known
Tools for Building p[g]
Inverse covariant d’Alembertian □-1
Act it on lowest dim curvature R
Specialization to FRW
√-g □ f = ∂µ [√-g gµν ∂ν f]
F = □-1 f solve □ F = f (retarded BC)
Lowest dimension scalar differential operator
R = 6Ḣ + 12 H2
□F = -a-3(d/dt) [a3 Ḟ] [□-1f](t) = ∫tdt’ a-3∫t’dt’’ a3 f
de Sitter (a(t) = eHt) □-1R = -4 ln(a) + small
Try p[g] = Λ2 f[-GΛ □-1R]
Spatially Homogeneous Case
Gµν = (p-Λ)gµν + (ρ+p) uµuν
Two Equations
X =1/ R = -∫tdt’ a-3∫t’dt’’ a3 [12H2+6Ḣ]
p = Λ2 f(-GΛ X)
ρ+p = a-3∫tdt’ a3 ṗ and uµ = δµ
3H2 = Λ + 8πG ρ
-2Ḣ–3H2 = -Λ + 8πG p
(easier)
One Number: GΛ (nominally ∼ 10-6)
One Function: f(x) (grows w/o bound)
Numerical Results for
GΛ=1/300 and f(x) = ex-1
X= -∫tdt’ a-3∫t’dt’’ a3R
Criticality
p = Λ2f(-GΛX) = Λ/8πG
Evolution of X(t)
Falls steadily to Xcr
Then oscillates with
constant period and
decreasing amplitude
Generic for any f(x)
growing w/o bound
Inflation Ends, H(t) goes < 0,
R(t) oscillates about 0
Analytic Treatment
2Ḣ + 3H2 = Λ[1 - 8πGΛf(-GΛX)]
X(t) = Xcr + ∆X(t) f ≈ fcr - GΛ∆X f’cr
Use R = 6Ḣ + 12H2 L.H.S. = R/3 – H2
2Ḣ + 3H2 ≈ 24π(GΛ)2 f’cr ∆X
∆X = □-1 R – Xcr
Act
= -[∂t + 3H] ∂t to localize
[∂t2 + 2H ∂t + ω]R ≈ 0
R(t) ≈ sin(ωt)/a(t)
ω = 24πG2Λ3 f’cr
(agrees with plots!)
Approximating X = □-1 R
=-∫t dt’ a-3∫t’ dt’’ a3 (6Ḣ + 12 H2)
Use ∂t(Ha3) = a3 (Ḣ + 3 H2)
During Inflation |Ḣ| << H2
Also ∫tdt’ a-3 [(Ha3) – (Ha3)initial] ≈ ∫tdt’ H
And ∫t dt’ H = ln[a/ainitial]
a3 (6Ḣ + 12H2) = ∂t(4Ha3) + small
Hence X ≈ -4ln[a/ainitial] = Xcr – 4ln[a/acr]
During Oscillations |Ḣ| << H2
a3 (6Ḣ + 12H2) = ∂t(6Ha3) + small
Hence X ≈ Xcr - 6ln[a/acr]
Generic Expansion Histories
with only ω = 24πfcr’(GΛ)2Λ
Near End of Inflation
a(t) = acr e-N
H2(t) ≈ ω/9 (4N + 4/3)
During Oscillations, with ∆t = t–tcr
a(t) ≈ acr C2[C1 + ω∆t + √2 cos( ω∆t+φ)]
H(t) ≈ ω [1 - √2 sin(ω∆t+φ)]/[a(t)/acr]
With φ ≈-½π, C1 ≈ 3, C2 ≈ 1/6
Origin of Scalar Perturbations
In Fundamental QG Inflation
1.
L = 1/16πG (R - 2Λ)(-g)½
Two hij’s can make a scalar!
E.g. Graviton KE: ḣij ḣij + ∇hij ∇hij
Usually negligible but if IR logs make
homogeneous ~ O(1) maybe perts ~ O(GΛ)
In Phenomenological Model
2.
Tµν[g] = p gµν + (ρ+p) uµuν
p = Λ2 f(-GΛ/ R) fixed by retarded BC
But ρ and ui at t=0 not fixed by DµTµν = 0
Analysis (in conformal coords)
0th: 2a’’/a3 - a’2/a4 = Λ[1 – 8πGΛf(-GΛX0)]
hµνdxµdxν = -2φdη-2B,idxidη-2[ψδij+E,ij]dxidxj
Φ = φ – a’/a (B-E’) – (B’-E’’)
Ψ = ψ + a’/a (B-E’)
Gij Eqn Ψ = Φ
and
2/a2 Φ’’+6a’/a3 Φ’+[4a’’/a3-2a’2/a4]Φ
= -8πG2Λ3 f’(-GΛX0)
x 0-1[∇2/a2 Φ - 6/a2 Φ’’ - 24 a’/a3 Φ’ - 4/a2 X0’ Φ’]
∂t2Φ + 4H∂tΦ + (2Ḣ+3H2)Φ
= -8πG2Λ3 f’[-GΛX(t)] NL
Early f’(-εX(t)) << 1
+ de Sitter Φ = 1/a and Φ = 1/a3
Same for all k’s
Late f’(-εX(t)) ≈ fcr’
Oscillates with constant frequency ω
d2Φ/dt2 ≈ -ω 1/□ [d2Φ/dt2]
Amplitude seems constant (numerically)
Energy transfer to matter crucial
Perturbations Summary
(arXiv:1006.5150, 1006.5681)
Scalars exotic
Need action to fix normalization
Sub-horizon redshift
Super-hor. during infl. approx. const.
Super-hor. after inflation oscillate at ω
Rapid reheating from n ~ ω 101000000 modes
Tensors normal for exotic a(t)
Oscillations too late for ∆h2
Leave bump at fnow ~ 1010 Hz
Tensor Perturbations
No change from usual eqn
ẍ + 3 H ẋ + k2/a2 x = 0
Of course a(t) is unusual . . .
Oscillations in H(t)
And H(t) drops below zero!
But this happens at the end of inflation
Little effect on far super-horizon modes
Equation for M(t,k) = |u(t,k)|2
M(t,k) = u u*
Now use ∂t2u + 3H∂tu + (k/a)2u = 0
∂t2M = -3H ∂tM + 2(k/a)2 M + 2 (∂tu) ∂tu*
Now use u ∂tu* - (∂tu) u* = i/a3
∂tM = (∂tu) u* + u ∂tu*
∂t2M = (∂t2u) u* + 2 (∂tu) ∂tu* + u ∂t2u*
Implies (∂tu) ∂tu* = [(∂tM)2 + a-6]/4M
∂t2M + 3H ∂tM + 2(k/a)2 M
= [(∂tM)2 + a-6]/2M
Bump in the gravity wave
background
After Inflation
Model driven by X = 1/ R
Oscillations & H < 0 efficient reheating
H = 1/2t R = 6Ḣ + 12H2 = 0
QG ends inflation, reheats & then turns
off for most of cosmological history
X(t) = -∫tdt’ a-3∫t’dt’’ a3 R Xcr
Changes at Matter Domination
H(t): 1/(2t) 2/(3t)
R(t): 0
3/(4t2)
X(t): Xcr Xcr - 4/3 ln(t/teq)
The Sign Problem: further screening!
The Magnitude Problem:
1.
2.
p ≈ –Λ/G (GΛ)2 fcr’ ∆X ≈ -1086 p0 x fcr’ ∆X
A Soln: Change X to 1/□ [R×-R00/Λ]
-R00 = +3qH2 ≈ Λ during inflation
Why Late Acceleration
from ptot ≈ -ω∆X/(24πG)
Why Acceleration with w = -1?
∆X = 1/□ [R × –R00/Λ]
1/□ = -∫dt’ a-3 ∫dt’’ a3
R × –R00 ~ -1/t’’4
Hence ∆X dominated by const. lower limit
Why Late?
∆X ~ (Hm)2/Λ
2 
2
Λnow ~ (Hm) ω /Λ << (Hm)
Tµν[g] = p gµν + (ρ+p) uµuν
Guess p[g] = Λ2 f(-GΛ X)
Homogeneous evolution: (generic f)
X falls to make p cancel –Λ/8πG
Then oscillates with const. period & decreasing amp.
Reheats to radiation dom. (R=0)
X1 = 1/ R
Infer ρ and ui from conservation
Matter dom. R≠0
X2 = 1/□ [R×-R00/Λ] gives late acceleration
Perturbations
No change to cosmo. tensors, bump at f ~ 1010 Hz
Scalar norm. not predicted but ok time dependence