Nonlocal Cosmology
S. Deser (arXiv:0705.0153)
N. C. Tsamis (arXiv:0904.1151)
C. Deffayet and G. Esposito-Farese
(arXiv:1106.4989)
Modifications of Gravity
Only local, stable, metric-based is f(R)
Nonlocal modifications proposed for
Summing quantum IR effects from inflation
Explaining late time acceleration w/o DE
Explaining galaxies & clusters w/o DM
Isaac Newton in 1692/3
“that one body may act upon another at a
distance thro' a Vacuum, without the
Mediation of any thing else, by and
through which their Action and Force
may be conveyed from one to another,
is to me so great an Absurdity that I
believe no Man who has in philosophical
Matters a competent Faculty of thinking
can ever fall into it.”
Was Newton wrong about
action-at-a-distance?
We don’t think so
Primordial inflation IR gravitons
Fundamental theory is local
But quantum effective field eqns are not
M=0 loops could give big IR corrections
N(t,k) ~ [Ha(t)/2kc]2 for every k
Perhaps their attraction stops inflation
Late modifications from vacuum polarization
Would affect large scales most
But for now, just model-building
Late-Time Acceleration
(arXiv:0705.0153 with Deser)
Nonlocality via
-½
for □ = (-g) ∂µ(√-g gµν∂ν)
Retarded BC {□-1 & ∂t□-1} = 0 at t=0
Act it on R □-1R dimensionless
L = √-g R[1 + f(□-1R)]/16πG
□-1
f(X) the “Nonlocal distortion function”
Gµν + ∆Gµν = 8πGTµν
∆Gµν = [Gµν + gµν□ - DµDν] (f + □-1[Rf’])
(ρ
σ)
ρσ
+ (δµ δν - ½gµνg ) ∂ρ(□-1R) ∂σ(□-1[R f’])
Specialization to FRW
ds2 = -dt2 + a2(t) dx.dx
R = 6Ḣ + 12 H2
t
t’
-1
3
□ = -∫0 dt’/a ∫0 dt’’a3
Two Built-In Delays
R=0 during Radiation Dom. (H = 1/2t)
5
No modification until t ~ 10 years
□-1R ~ -4/3 ln(t/teq) during Matter Dom.
10
□-1R ~ -15 at t = 10
years
Reconstructing f(X) for ΛCDM
(arXiv:0904.0961 with Deffayet)
How It Works
for slowly varying a(t)
Gµν + ∆Gµν = 8πGTµν ,∆Gµν ~ Gµν(f + □-1[Rf’])
Just rescale G! Geff = G/(1 + f + □-1[Rf’])
Friedman Eqn: 3H2 ~ 8πGeff ρ0/a3(t)
But Geff strengthens gravity!
Growth of Geff balances 1/a3(t)
Not relevant for solar system
But should increase structure formation
Dodelson & Park (arXiv:1209.0836)
Not purely Geff(t) when space dependent
Delayed so late that only ~10-30% effect
Local Version Is Haunted
(Nojiri & Odintsov, arXiv:0708.0924)
R[1+f(□-1R)] R[1+f(Φ)] + Ψ[□Φ–R]
Varying wrt Ψ enforces □Φ = R
NB both scalars have 2 pieces of IVD
Ψ□Φ -∂µΨ∂νΦgµν
-½∂µ(Ψ+Φ)∂ν(Ψ+Φ)gµν
+ ½∂µ(Ψ-Φ)∂ν(Ψ-Φ)gµν
Ψ-Φ has negative KE
No new initial value data for
the original nonlocal version
Synch. gauge: ds2 = -dt2 + hij(t,x) dxidxj
IVD for GR: hij(0,x) & ḣij(0,x) = 6 + 6
4+4 for constrained fields
2+2 for dynamical gravitons
IVD in nonlocal cosmo count the ∂t‘s
2
-2
0
R ~ ∂t & □-1 ~ ∂t □-1R ~ ∂t
2
∆Gµν has up to ∂t □-1
Hence hij(0,x) & ḣij(0,x), but what are they?
t=0 Constraints same as GR
Recall Gµν + ∆Gµν = 8πGTµν
∆Gµν = [Gµν + gµν□ - DµDν] (f + □-1[Rf’])
(ρ
σ)
ρσ
+ (δµ δν - ½gµνg ) ∂ρ(□-1R) ∂σ(□-1[Rf’])
Retarded BC [□-1 & ∂t□-1] = 0 at t=0
f(X) also vanishes at X=0
Only [gµν□ - DµDν] {f(□-1R) + □-1[Rf’(□-1R)]} ≠ 0
Synchronous constraints ∆G00 & ∆G0i
g00□ - D0D0 = ½hijḣij∂t - ∆ 0 at t=0
g0i□ - D0Di = -∂0∂i + ½hjkḣki∂j 0 at t=0
No Ghosts at t = 0
Recall Gµν + ∆Gµν = 8πGTµν
∆Gµν = [Gµν + gµν□ - DµDν] (f + □-1[Rf’])
ρσ
(ρ
σ)
+ (δµ δν - ½gµνg ) ∂ρ(□-1R) ∂σ(□-1[Rf’])
Dynamical eqns Gij + ∆Gij = 8πGTij
2
gij□ - DiDj -hij∂t + O(∂t)
At t = 0 ∆Gij = 2f’(0) hijR
kl
Rij = ½ḧij + O(∂t) & R = h ḧkl + O(∂t)
kl
Gij + ∆Gij ½ḧij – [½-f’(0)]hijh ḧkl + O(∂t)
0 < f’(0) « 1 No graviton becomes a ghost!
Avoid Geff with
Tµν[g] =p[g] gµν + (ρ+p) uµuν
(arXiv:0904.1151 with Tsamis)
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
Λ-Driven Inflation with QG backreaction from p = Λ2 f(-GΛ□-1R)
Gµν = (p-Λ)gµν + (ρ+p) uµuν
= -∫tdt’ a-3∫t’dt’’ a3 [12H2+6Ḣ]
µ
µ
-3
t
3
ρ+p = a ∫ dt’ a ṗ and u = δ 
Two Equations
-1R
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
Dark Matter vs Mod. Gravity
Gµν = 8πGTµν works
for solar system
But not for galaxies
Theory: v² = GM⁄r
Obser: v2 ~ (a0GM)1/2
Maybe missing Mass
Or modified gravity
MOND (Milgrom 1983)
ρ(x,y,z) ≡ mass in stars and gas
gNi ≡ Newtonian acceleration
gi ≡ actual acceleration
gi µ(|g|/a0) = gNi
a0 ∼ 10-10 m/s2
GR regime: µ(x) = 1 for x >> 1
MOND regime: µ(x) = x for x << 1
Eg. µ(x) = x/(1+x), or tanh(x), . . .
Good agreement with galaxies
but need relativistic model for
Gravitational Lensing
Recently disturbed systems
The Bullet Cluster!
Cosmology
Previous models have new fields
TeVeS (Bekenstein 2004)
Another form of dark matter?
Our Goal: A purely metric version
Metric potentials for static,
spherically symmetric
ds2 = -B(r)c2dt2 + A(r)dr2 + r2dΩ2
b(r) = B(r) - 1 Rotation curves
rb’(r) = 2v2/c2 [4GMa0/c4]½
a(r) = A(r) – 1 Lensing
Data a(r) ~ + rb’(r)
GR vs MOND
for a MONDian ρ(r)
M(r) = 4π/c2 ∫r dr’ r’2ρ(r’)
GR a(r) = rb’(r) = 2GM(r)/c2r
MONDian GM(r)/r2 « a0
δSGR/δb = (c4/16πG)[(ra)’ + O(h2)] - ½r2ρ
4
2
δSGR/δa = (c /16πG)[-rb’ + a + O(h )]
MOND a(r) = rb’(r) = [4GM(r)a0/c4]½
∂r(a2) = ∂r(rb’)2 = (16πGa0/c4) r2ρ
2
h
LMOND to cancel
from GR
& add h3 for MOND
LGR= -½r2ρb + (c4/16πG)[-rab’ + ½a2]
LMOND= r2(c4/16πG)[ab’/r - ½(a/r)2
+ c2/a0 [-1/6 (b’)3 + k(b’ – a/r)3 + . . . ]
h3/r2 of GR « c2/a0 (h/r)3 of MOND for r « rH
S = ∫dr [LGR + LMOND]
∂r(rb’)2 – 6k∂r(rb’-a)2 = (16πGa0/c4) r2ρ
-6k/r (rb’-a)2 = 0
Conclusions
Last chance for modified gravity based on gµν
Not fundamental (we think)
Models devised for
From QG corrections during inflation
Purely phenomenological for now
Summing QG corrections from inflation
Producing late acceleration w/o Dark Energy
Describing galaxies & clusters w/o Dark Matter
Tools for nonlocal model building
Inverse covariant d’Alembertian
Invariant volume of past light-cone