Cosmological Expansion from Nonlocal Gravity Correction

Tomi Koivisto, ITP Heidelberg

1. Outline

Introduction2. Nonlocalities in physics 3. The gravity model4. Scalar-tensor formulation

Dynamics 5. Radiation domination 6. Matter domination7. Acceleration 8. Singularity 9. Summary

Constraints10. Solar system 11. Perturbations12. Ghosts

Conclusions

3rd Kosmologietag at IBZ, Bielefeld, May 8-9, 2008

e-Print: arXiv:0803.3399, to appear in PRD

Nonlocalities in physics

-Nonlocality <-> Infinite number of derivatives Interactions at x not ~ δ(x)

-String field theory is nonlocal Since strings are extended objects

-BH information paradox requires nonlocal physics?

- Gravity as an effective theory: Leading quantum corrections nonlocal!

t’ Hooft & Veltman: Annales Poincare Phys.Theor.A20:69-94,1974

Susskind: J.Math.Phys.36:6377-6396,1995

Nonlocal gravity modification

4 11 ( / )

2S dx g f R R

S. Deser & R.P. Woodard: Phys.Rev.Lett.99:111301,2007.

- Like a variable G

- When f(x)=cx, could stabilize the Euclidean actionC. Wetterich: Gen.Rel.Grav.30:159-172,1998

- Thus, consider the class of simple modications:

- Recent suggestion: could provide dark energy

...then f should be about ~-1. It’s argument is dimensionless -> fine tuning alleviated ?

scalar-tensor formulation 4 11 ( / )

2S dx g f R R

Introduce a field and a Lagrange multiplier:

Define : ( )f

4 21(1 ) '( )( )

2S dx g R f

- Equivalent to a local model with two extra d.o.f !- Massless fields with a nonlinear sigma -type (kinetic) interaction

Bi-

4 1(1 ( )) ( )

2S dx g f R R

Cosmology: Radiation domination

In the very early universe the correction vanishes:

As matter becomes non-relativistic:

- BBN constrains the corrections during RD - The possible effects are a consequence of the onset MD

0 / 0R R

' 33

' 33

'( ) ( '') ( '') ''

( ')

'( ) ( '') ( '') '( ( '')) ''

( ')

t t

t t

dtt a t R t dt

a t

dtt a t R t f t dt

a t

Dust dominated era Approximate solution:

- If n>0, the coupling grows - If N(-1)^n<0, the nonlocal contribution to energy grows

/Dust Radiationr

( ) 2 log( / 4)a ra

Assume f(x) = Nx^n :

( ) 2 [2 log(1/ )]na N ra 1

12( ) [2 log(1/ )]

3

nn

effw a N n ra

Solutions

- For larger |n| the evolution is steeper (here n=3,n=6) - N is roughly of the order (0.1)^(n+1) in Planck units

One may reconstruct f(x) which gives the assumed expansion! But, assuming power-law f(x)=Nx^n, the expansion goes like:

Singularity

1) Simply reconstruct different f(x) resulting in finite w 2) Regularize the inverse d’Alembertian! 3) Consider higher curvature terms

Power-law and exponential f(x) which result in acceleration lead to a sudden future singularity at t=t_s>t_0:

Barrow, Class.Quant.Grav. 21 (2004) L79-L82

- Density (and expansion rate) remain finite at t_s - Pressure (and acceleration rate) diverge at t_sPossible resolutions:

/ /( ...)abcdabcdR R R R R

Summary of cosmic evolutions

f(x)=Nx^n

AccelerationSlows down expansion

Matter domination

Nonlocal effect

Singularity

n<0

n>0

N(-1)^n>0 N(-1)^n<0

?

Regularized

Solar system constraints- If the fields are constant:

- Where the corrections to the Schwarzschild metric are

2 2 2 2 2* *2 21 1

G M G Mds dt dr r d

r r

1 4 '( )

1 8 '( )

f

f

*

1 8 '( )

1 6 '( ) 1

f GG

f

- Exact Schwarzschild solution: R=0, fields vanish - They are second order in GM/r < 10^(-6) - Seems they escape the constraints on |G_*/G|, |γ-1| ~ 10^(-5)

Perturbation constraints- In the cosmological Newtonian gauge:

- Effective anisotropic stress appears: (relevant for weak lensing?)

- Poisson equation is different too: (detectable in the ISW?)

2 2 2 2( ) 1 2 1ds a d d x

1

2

2(1 )

- Matter growth is given by the G_*: (constraints from LSS !)

*2 4.. .

H G

Ghost constraints-From

one sees that graviton is not ghost when (1+ψ)>0.

4 21(1 ) '( )( )

2S dx g R f

-The Einstein frame,

one may use the general result for quadratic kinetic Lagrangians L (valid when L>0): two decoupled perturbation d.o.f propagating at c

4 2 21 3( ) '( )( )

2 2S dx g R e f

- Thus if L>0 & (1+ψ)>0 ,no ghosts, instabilities or acausalities.

Langlois & Renaux-Petel: JCAP 0804:017,2008

Conclusions-Effective gravity could help with the cosmological constant problems:

- Coincidence: (Delayed) response to the universe becoming nonrelativistic - Fine tuning: Only Planck scale involved

( / )f R

- Simplest models feature a sudden future singularity - Seems to have reasonable LSS, could avoid ghosts and Solar system constraints...

Whereas f(R) gravity does not help with the fine tunings in the first place andin addition is ruled out (or severely constrained) by ghosts, LSS and Solar system.