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3rd Kosmologietag at IBZ, Bielefeld, May 8-9, 2008. Cosmological Expansion from Nonlocal Gravity Correction. Tomi Koivisto, ITP Heidelberg. e-Print: arXiv:0803.3399, to appear in PRD. 1. Outline Introduction 2. Nonlocalities in physics 3. The gravity model 4. Scalar-tensor formulation

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cosmological expansion from nonlocal gravity correction

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

Cosmological Expansion from Nonlocal Gravity Correction

Tomi Koivisto, ITP Heidelberg

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

1. Outline

Introduction

2. Nonlocalities in physics 3. The gravity model

4. Scalar-tensor formulation

Dynamics

5. Radiation domination 6. Matter domination

7. Acceleration 8. Singularity 9. Summary

Constraints

10. Solar system 11. Perturbations

12. Ghosts

Conclusions

nonlocalities in physics
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?

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

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

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

nonlocal gravity modification
Nonlocal gravity modification

- Thus, consider the class of simple modications:

- Like a variable G

- When f(x)=cx, could stabilize the Euclidean action

C. Wetterich: Gen.Rel.Grav.30:159-172,1998

- Recent suggestion: could provide dark energy

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

...then f should be about ~-1. It’s argument is dimensionless

-> fine tuning alleviated ?

scalar tensor formulation
scalar-tensor formulation

Bi-

Introduce a field and a Lagrange multiplier:

Define :

- Equivalent to a local model with two extra d.o.f !

- Massless fields with a nonlinear sigma -type (kinetic) interaction

cosmology radiation domination
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

dust dominated era
Dust dominated era

Approximate solution:

Assume f(x) = Nx^n :

- If n>0, the coupling grows

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

solutions
Solutions

One may reconstruct f(x) which gives the assumed expansion!

But, assuming power-law f(x)=Nx^n, the expansion goes like:

- 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

singularity
Singularity

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_s

Possible resolutions:

1) Simply reconstruct different f(x) resulting in finite w

2) Regularize the inverse d’Alembertian!

3) Consider higher curvature terms

summary of cosmic evolutions
Summary of cosmic evolutions

n>0

f(x)=Nx^n

Nonlocal effect

N(-1)^n>0

N(-1)^n<0

n<0

Slows down expansion

Acceleration

Matter domination

Regularized

Singularity

?

solar system constraints
Solar system constraints

- If the fields are constant:

- Where the corrections to the Schwarzschild metric are

- 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
Perturbation constraints

- In the cosmological Newtonian gauge:

- Effective anisotropic stress appears:

(relevant for weak lensing?)

- Poisson equation is different too:

(detectable in the ISW?)

- Matter growth is given by the G_*:

(constraints from LSS !)

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

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

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

conclusions
Conclusions
  • Effective gravity could help with the
  • cosmological constant problems:

- Coincidence: (Delayed) response to the universe

becoming nonrelativistic

- Fine tuning: Only Planck scale involved

- 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 and

in addition is ruled out (or severely constrained) by ghosts, LSS and Solar system.