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 Fundamental theory is local But quantum effective field eqns are not M=0 loops could give big IR corrections

Primordial inflation IR gravitons 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 -1 for = (-g)-½∂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 f(X) the “Nonlocal distortion function”

Gµν + ∆Gµν = 8πGTµν∆Gµν = [Gµν + gµν - DµDν] (f + -1[Rf’])

+ (δµ(ρ δν

σ)- ½gµνgρσ) ∂ρ(-1R) ∂σ(-1[R f’])

Specialization to FRWds2 = -dt2 + a2(t) dx.dx

R = 6Ḣ + 12 H2

-1 = -∫0tdt’/a3∫0

t’dt’’a3

Two Built-In Delays R=0 during Radiation Dom. (H = 1/2t)

No modification until t ~ 105 years

-1R ~ -4/3 ln(t/teq) during Matter Dom. -1R ~ -15 at t = 1010 years

Reconstructing f(X) for ΛCDM(arXiv:0904.0961 with Deffayet)

How It Worksfor 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)

Growth of Geff balances 1/a3(t) But Geff strengthens gravity!

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

ΨΦ -∂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 R ~ ∂t

2 & -1 ~ ∂t-2 -1R ~ ∂t

0

∆Gµν has up to ∂t2-1

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’])

+ (δµ(ρ δν

σ) - ½gµνgρσ) ∂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’])

+ (δµ(ρ δν

σ) - ½gµνgρσ) ∂ρ(-1R) ∂σ(-1[Rf’])

Dynamical eqns Gij + ∆Gij = 8πGTij gij - DiDj -hij∂t

2 + O(∂t) At t = 0 ∆Gij = 2f’(0) hijR Rij = ½ḧij + O(∂t) & R = hklḧkl + O(∂t)

Gij + ∆Gij ½ḧij – [½-f’(0)]hijhklḧkl + O(∂t)

0 < f’(0) « 1 No graviton becomes a ghost!

Avoid Geff withTµν[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 back-reaction from p = Λ2 f(-GΛ-1R)

Gµν = (p-Λ)gµν + (ρ+p) uµuν -1R = -∫tdt’ a-3∫t’dt’’ a3 [12H2+6Ḣ] ρ+p = a-3∫tdt’ a3 ṗ and uµ = δµ

Two Equations 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 forGΛ=1/300 and f(x) = ex-1

X= -∫tdt’ a-3∫t’dt’’ a3R

Criticalityp = 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 gN

i ≡ Newtonian acceleration

gi ≡ actual acceleration gi µ(|g|/a0) = gN

i

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!

CosmologyPrevious 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’) MONDian GM(r)/r2 « a0

GR a(r) = rb’(r) = 2GM(r)/c2r δSGR/δb = (c4/16πG)[(ra)’ + O(h2)] - ½r2ρ

δSGR/δa = (c4/16πG)[-rb’ + a + O(h2)]

MOND a(r) = rb’(r) = [4GM(r)a0/c4]½

∂r(a2) = ∂r(rb’)2 = (16πGa0/c4) r2ρ

LMOND to cancel h2 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)

From QG corrections during inflation Purely phenomenological for now

Models devised for 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