Finnish-Japanese Workshop, Helsinki, March 8, Accelerated expansion from structure formation astro-ph/ , astro-ph/ Syksy Räsänen CERN

Finnish-Japanese Workshop, Helsinki, March 8, Backreaction The average behaviour of an inhomogeneous spacetime is not the same as the behaviour of the corresponding smooth spacetime. The average behaviour of an inhomogeneous spacetime is not the same as the behaviour of the corresponding smooth spacetime. This is the fitting problem (Ellis 1983): how do we find the homogeneous model that best fits the inhomogeneous universe? This is the fitting problem (Ellis 1983): how do we find the homogeneous model that best fits the inhomogeneous universe? Applying the field equations does not commute with averaging: Applying the field equations does not commute with averaging: ≠ G  ( ) ≠ G  ( ) ⇔ average quantities  , … ) do not satisfy the Einstein equation.

Finnish-Japanese Workshop, Helsinki, March 8, Backreaction, exactly Take a universe with irrotational dust. The Einstein eq. is Take a universe with irrotational dust. The Einstein eq. is Projecting gives the following exact, local, covariant scalar equations: Projecting gives the following exact, local, covariant scalar equations: Here θ  is the expansion rate of the local volume element, σ  ≥ 0 is the shear and (3) R is the spatial curvature. Here θ  is the expansion rate of the local volume element, σ  ≥ 0 is the shear and (3) R is the spatial curvature.

Finnish-Japanese Workshop, Helsinki, March 8, The average expansion can accelerate, even though the local expansion rate decelerates everywhere. The average expansion can accelerate, even though the local expansion rate decelerates everywhere. What is the physical meaning of this? What is the physical meaning of this? The Buchert equations: The Buchert equations: The FRW equations: The FRW equations: Here. The backreaction is contained in Here. The backreaction is contained in.

Finnish-Japanese Workshop, Helsinki, March 8, Acceleration from collapse Linear perturbation theory around the FRW equations breaks down when |δ| ~ 1. Linear perturbation theory around the FRW equations breaks down when |δ| ~ 1. A simple treatment of a forming structure: the spherical collapse model. A simple treatment of a forming structure: the spherical collapse model. The FRW equations themselves break down when perturbations with |δ| ~ 1 occupy a large fraction of space. The FRW equations themselves break down when perturbations with |δ| ~ 1 occupy a large fraction of space. A toy model of structure formation: the union of an underdense and an overdense spherical region. A toy model of structure formation: the union of an underdense and an overdense spherical region. For an empty void we have a 1 ∝ t and for an overdensity we have a 2 ∝ 1-cosφ, t ∝ φ-sinφ. For an empty void we have a 1 ∝ t and for an overdensity we have a 2 ∝ 1-cosφ, t ∝ φ-sinφ. The overall scale factor is a = (a 1 3 +a 2 3 ) 1/3. The overall scale factor is a = (a 1 3 +a 2 3 ) 1/3.

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7 One would expect the departure from the FRW equations to be largest when the collapsing structures have reached their maximum relative size. One would expect the departure from the FRW equations to be largest when the collapsing structures have reached their maximum relative size. Collapse and coincidence Perturbations are nested inside each other hierarchically, so part of the universe is always collapsing. Perturbations are nested inside each other hierarchically, so part of the universe is always collapsing. First structures collapse around z  50. First structures collapse around z  50. The size of the structures which are about to collapse relative to the horizon size grows, saturating at (R NL ) 2 /(aH) -2 ≈ around billion years. The size of the structures which are about to collapse relative to the horizon size grows, saturating at (R NL ) 2 /(aH) -2 ≈ around billion years. The effects of small collapsing regions and voids add up. The effects of small collapsing regions and voids add up.

Finnish-Japanese Workshop, Helsinki, March 8, Conclusion The FRW equations do not describe the expansion of an inhomogeneous space. 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. 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. 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 next step is to build a quantitative model.