Accelerated expansion from structure formation

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Accelerated expansionfrom structure formation

• astro-ph/0605632, astro-ph/0607626
• Syksy Räsänen
• CERN

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Dark energy

• Interpreted in the context of the homogeneous and
  isotropic FRW model

• Observations (SNIa) dL(z).

• For dust, p 0 ? ?m ? a-3 decreases too fast
• ? add a component which decays more slowly than
  dust
• ? p lt 0

• The problem is not p lt 0 but z 1 why??m ?de
  today.
• This is the coincidence problem.

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Backreaction

• The average behaviour of an inhomogeneous
  spacetime is not the same as the behaviour of the
  corresponding smooth spacetime.

• Applying the field equations does not commute
  with averaging
• ltGmn(gab)gt ? Gmn(ltgab?gt)
• ? average quantities (lt ? gt, lt ? gt, ) do not
  satisfy the Einstein equation.

• This is the fitting problem (Ellis 1983) how do
  we find the homogeneous model that best fits the
  inhomogeneous universe?

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FRW assumptions
1) FRW scale factor Observables can be computed
from an overall scale factor.
2) FRW dynamics The overall scale factor evolves
according to the FRW equations.
3) FRW perturbations The inhomogeneities
evolve according to linear perturbation theory
around the average.
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Backreaction, exactly

• Let us take a universe with irrotational dust.
  The metric is

• The Einstein equation gives the following exact,
  local, covariant scalar equations

• Here ??is the expansion rate of the local volume
  element, s? 0 is the shear and (3)R is the
  spatial curvature.

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• The Buchert equations

• The FRW equations

• The average expansion can accelerate, even though
  the local expansion rate decelerates everywhere.
• What is the physical meaning of this?

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Acceleration from collapse

• Linear perturbation theory around the FRW
  equations breaks down when d 1.
• A simple treatment of a forming structure the
  spherical collapse model.
• The FRW equations themselves break down when
  perturbations with d 1 occupy a large
  fraction of space.
• A toy model of structure formation the union of
  an underdense and an overdense spherical region.
• For an empty void we have a1 ? t and for an
  overdensity we have a2 ? 1-cosf, t ? f-sinf.
• The overall scale factor is a (a13 a23)1/3.

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Collapse and coincidence

• Perturbations are nested inside each other
  hierarchically, so part of the universe is always
  collapsing.
• First structures collapse around z 50.
• The size of the structures which are about to
  collapse relative to the horizon size grows,
  saturating at (RNL)2/(aH)-2 10-5 around 10-100
  billion years.
• The effects of small collapsing regions and voids
  add up.

• One would expect the departure from the FRW
  equations to be largest when the collapsing
  structures have reached their maximum size.

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Observations

• ?m0 0.150.35, q0 -1.2-0.3
• ? ?R0 0.91.5, ?Q0 -0.7-0.2
• H0t0 0.700.97
• 2dF Galaxy Redshift Survey (astro-ph/0312533)
  voids (with d -0.9) occupy 40 of the volume
• A lower limit on the variance in a two-region
  toy model, this would give ?Q0 -0.04.

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• Lyman-a forest observations and simulations
  (astro-ph/0509262)

z 2.0 ?Q -0.26
z 3.4 ?Q -0.07
z 3.8 ?Q -0.05
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Conclusion

• The FRW equations do not describe the expansion
  of an inhomogeneous space.
• The Buchert equations show that even when the
  local expansion decelerates everywhere, the
  average expansion can accelerate.
• Acceleration is intimately related to collapse,
  and structure formation has a preferred time
  around the acceleration era.
• The next step is to build a quantitative model.
• The scale factor assumption should also be
  checked.