Finnish-Japanese Workshop, Helsinki, March 8, 2007 1

Accelerated expansionfrom structure formation

astro-ph/0605632, astro-ph/0607626astro-ph/0605632, astro-ph/0607626

Syksy RäsänenSyksy Räsänen

CERNCERN

Finnish-Japanese Workshop, Helsinki, March 8, 2007 2

Backreaction The average behaviour of an inhomogeneous spacetime The average behaviour of an inhomogeneous spacetime

is not the same as the behaviour of the corresponding is not the same as the behaviour of the corresponding smooth spacetime.smooth spacetime.

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

Applying the field equations does not commute with Applying the field equations does not commute with averaging:averaging:

<<GG((gg))> ≠ > ≠ GG((<<gg>>))

⇔ ⇔ average quantities average quantities < < ρ ρ >><< θ θ >>, … ) do not satisfy , … ) do not satisfy the Einstein equation.the Einstein equation.

Finnish-Japanese Workshop, Helsinki, March 8, 2007 3

Backreaction, exactly

Take a universe with irrotational dust. The Einstein eq. isTake a universe with irrotational dust. The Einstein eq. is

Projecting gives the following exact, local, covariant scalar Projecting gives the following exact, local, covariant scalar equations:equations:

Here Here θθis the expansion rate of the local volume element, is the expansion rate of the local volume element,

σσ ≥ 0 is the shear and ≥ 0 is the shear and (3)(3)RR is the spatial curvature. is the spatial curvature.

€

Gμν = 8πGNρuμ uν

€

˙ θ +1

3θ 2 = −4πGNρ − 2σ 2

1

3θ 2 = 8πGNρ −

1

2(3)R + σ 2

€

˙ ρ + θρ = 0

Finnish-Japanese Workshop, Helsinki, March 8, 2007 4

The average expansion can accelerate, even though The average expansion can accelerate, even though the the local expansion rate decelerates everywhere.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:

€

3˙ ̇ a

a= −4πGN ρ + Q

3˙ a 2

a2= 8πGN ρ −

1

2(3)R −

1

2Q

∂ t ρ + 3˙ a

aρ = 0

€

3˙ ̇ a

a= −4πGNρ

3˙ a 2

a2= 8πGNρ − 3

k

a2

∂ tρ + 3˙ a

aρ = 0

The FRW equations:The FRW equations:

HereHere .. The backreaction is contained in The backreaction is contained in

..

€

Q ≡2

3( θ 2 − θ

2) − 2 σ 2

€

f ≡d3∫ x (3)g f

d3∫ x (3)g

€

a(t) ∝ d3∫ x (3)g( )1/ 3

€

˙ θ +1

3θ 2 = −4πGNρ − 2σ 2

1

3θ 2 = 8πGNρ −

1

2(3)R + σ 2

˙ ρ + θρ = 0

Finnish-Japanese Workshop, Helsinki, March 8, 2007 5

Acceleration from collapse Linear perturbation theory around the FRW equations Linear perturbation theory around the FRW equations

breaks down when |δ| breaks down when |δ| ~~ 1. 1. A simple treatment of a forming structure: the spherical A simple treatment of a forming structure: the spherical

collapse model.collapse model.

The FRW equations themselves break down when The FRW equations themselves break down when perturbations with |δ| perturbations with |δ| ~~ 1 occupy a large fraction of space. 1 occupy a large fraction of space.

A toy model of structure formation: the union of an A toy model of structure formation: the union of an underdense and an overdense spherical region.underdense and an overdense spherical region.

For an empty void we have For an empty void we have aa11 t ∝ t ∝ and for an overdensity and for an overdensity we have we have aa22 ∝ ∝ 1-cos1-cosφφ, , t φ-∝t φ-∝ sinsinφφ..

The overall scale factor is The overall scale factor is a a = (= (aa1133

+a+a2233))1/31/3..

Finnish-Japanese Workshop, Helsinki, March 8, 2007 6

€

H ≡˙ a

a=

a13

a13 + a2

3 H1 +a2

3

a13 + a2

3 H2 = v1H1 + v2H2

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q ≡ −1

H 2

˙ ̇ a

a=

H12

H 2v1q1 +

H22

H 2v2q2 − 2v1v2

(H1 − H2)2

H 2

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

Collapse and coincidence

Perturbations are nested inside each other hierarchically, Perturbations are nested inside each other hierarchically, so part of the universe is always collapsing.so part of the universe is always collapsing.

First structures collapse around First structures collapse around z z 50.50. The size of the structures which are about to collapse The size of the structures which are about to collapse

relative to the horizon sizerelative to the horizon size grows, saturating at grows, saturating at ((RRNLNL))22/(/(aHaH))-2-2 ≈≈ 1010-5 -5 around 10-100 billion years.around 10-100 billion years.

The effects of small collapsing regions and voids add up.The effects of small collapsing regions and voids add up.

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Conclusion The FRW equations do not describe the expansion of an The FRW equations do not describe the expansion of an

inhomogeneous space.inhomogeneous space. The Buchert equations show that even when the local The Buchert equations show that even when the local

expansion decelerates everywhere, the average expansion expansion decelerates everywhere, the average expansion can accelerate. can accelerate.

Acceleration is intimately related to collapse, and structure Acceleration is intimately related to collapse, and structure formation has a preferred time around the acceleration era.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.