ix ème ecole de cosmologie, cargese, november 3, 2008 1 1 structure formation as an alternative to...

IXème Ecole de Cosmologie, Cargese, November 3, 2008IXème Ecole de Cosmologie, Cargese, November 3, 2008

11

Structure formation as an alternative to dark

energy

Structure formation as an alternative to dark

energy

Syksy Räsänen

University of Geneva

Syksy Räsänen

University of Geneva


22

ContentsContents

1)Brief overview of the observations

2)The effect of clumpiness on the expansion rate

3)Understanding acceleration due to structure formation physically

4)Evaluating the magnitude of the effect

5)Conclusions

1)Brief overview of the observations

2)The effect of clumpiness on the expansion rate

3)Understanding acceleration due to structure formation physically

4)Evaluating the magnitude of the effect

5)Conclusions


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1) A factor of 2 in distance1) A factor of 2 in distance The early universe is well described by a model which is homogeneous and isotropic, contains only ordinary matter and evolves according to general relativity.

However, such a model underpredicts the distances measured in the late universe by a factor of 2.

This is interpreted as faster expansion. There are three possibilities:

1) There is matter with negative pressure.

2) General relativity does not hold.

3) The universe is not homogeneous and isotropic.

The early universe is well described by a model which is homogeneous and isotropic, contains only ordinary matter and evolves according to general relativity.

However, such a model underpredicts the distances measured in the late universe by a factor of 2.

This is interpreted as faster expansion. There are three possibilities:

1) There is matter with negative pressure.

2) General relativity does not hold.

3) The universe is not homogeneous and isotropic.


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PossibilitiesPossibilities

By introducing exotic matter or modified gravity it is possible to explain the distance observations.

Such models suffer from the coincidence problem. Why has the acceleration started recently, i.e. why ρde∼ρm today?

More importantly, linearly perturbed FRW models do not include non-linear structures.

Before concluding that new physics is needed, we should take into account the known breakdown of homogeneity and isotropy.

By introducing exotic matter or modified gravity it is possible to explain the distance observations.

Such models suffer from the coincidence problem. Why has the acceleration started recently, i.e. why ρde∼ρm today?

More importantly, linearly perturbed FRW models do not include non-linear structures.

Before concluding that new physics is needed, we should take into account the known breakdown of homogeneity and isotropy.


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2) Backreaction2) Backreaction

The average evolution of an inhomogeneous and/or anisotropic spacetime is not the same as the evolution of the corresponding smooth spacetime.

At late times, non-linear structures form, and the universe is only statistically homogeneous and isotropic, on scales above 100 Mpc.

Finding the model that describes the average evolution of the clumpy universe was termed the fitting problem by George Ellis in 1983.

The average evolution of an inhomogeneous and/or anisotropic spacetime is not the same as the evolution of the corresponding smooth spacetime.

At late times, non-linear structures form, and the universe is only statistically homogeneous and isotropic, on scales above 100 Mpc.

Finding the model that describes the average evolution of the clumpy universe was termed the fitting problem by George Ellis in 1983.


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Backreaction, exactlyBackreaction, exactly

Consider a dust universe. The Einstein equation is

. The (exact, local, covariant) scalar part is:

Here θ is the expansion rate, ρ is the energy density, σ2≥0 is the shear, ω2≥0 is the vorticity and (3)R is the spatial curvature.

We take ω2=0.

Consider a dust universe. The Einstein equation is

. The (exact, local, covariant) scalar part is:

Here θ is the expansion rate, ρ is the energy density, σ2≥0 is the shear, ω2≥0 is the vorticity and (3)R is the spatial curvature.

We take ω2=0.

€

Gαβ = 8πGNρuα uβ

€

˙ θ +1

3θ 2 = −4πGρ − 2σ 2 + 2ω2

1

3θ 2 = 8πGρ −

1

2(3)R + σ 2 −ω2

€

˙ ρ + θρ = 0


77

€

3˙ ̇ a

a= −4πGρ

3˙ a 2

a2= 8πGρ − 3

k

a2

˙ ρ + 3˙ a

aρ = 0

€

3˙ ̇ a

a= −4πG ρ + Q

3˙ a 2

a2= 8πG ρ −

1

2(3)R −

1

2Q

∂ t ρ + 3˙ a

aρ = 0

The Buchert equations (1999):

Here . The backreaction variable is

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

The Buchert equations (1999):

Here . The backreaction variable is

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

The FRW equations: The FRW equations:

€

˙ θ +1

3θ 2 = −4πGρ − 2σ 2

1

3θ 2 = 8πGρ −

1

2(3)R + σ 2

˙ ρ + θρ = 0

€

a(t) ∝ d3x (3)g∫( )1/ 3

€

Q ≡2

3θ 2 − θ

2

( ) − 2 σ 2

€

f ≡d3x (3)g∫ f

d3x (3)g∫


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3) Demonstrating acceleration3) Demonstrating acceleration The average expansion rate can increase, because the fraction of volume occupied by faster expanding regions grows.

Structure formation involves overdense regions slowing down and underdense regions speeding up.

Consider a toy model with one overdense and one underdense region, both described by the spherical collapse model.

For an empty void we have a1 ∝ t, for an overdense region we have a2 ∝ 1-cosφ, t ∝ φ-sinφ.

The overall scale factor is a = (a13 +a2

3)1/3.

The average expansion rate can increase, because the fraction of volume occupied by faster expanding regions grows.

Structure formation involves overdense regions slowing down and underdense regions speeding up.

Consider a toy model with one overdense and one underdense region, both described by the spherical collapse model.

For an empty void we have a1 ∝ t, for an overdense region we have a2 ∝ 1-cosφ, t ∝ φ-sinφ.

The overall scale factor is a = (a13 +a2

3)1/3.


99

€

H ≡˙ a

a=

a13

a13 + a2

3 H1 +a2

3

a13 + a2

3 H2 = v1H1 + v2H2

€

q ≡ −1

H 2

˙ ̇ a

a=

H12

H 2v1q1 +

H22

H 2v2q2 − 2v1v2

(H1 − H2)2

H 2


1010

4) Towards reality4) Towards reality

Acceleration due to structures is possible: is it realised in the universe?

The non-linear evolution is too complex to follow exactly.

Because the universe is statistically homogeneous and isotropic, a statistical treatment is sufficient.

We can evaluate the expansion rate with an evolving ensemble of regions.

Acceleration due to structures is possible: is it realised in the universe?

The non-linear evolution is too complex to follow exactly.

Because the universe is statistically homogeneous and isotropic, a statistical treatment is sufficient.

We can evaluate the expansion rate with an evolving ensemble of regions.


1111

The peak modelThe peak model

We start from a FRW background of dust with an initial Gaussian linear density field.

We identify structures with spherical isolated peaks of the smoothed density field. (BBKS 1986)

We keep the smoothing threshold fixed at σ(t,R)=1, which gives the time evolution R(t).

Each peak expands like a separate FRW universe.

The peak number density as a function of time is determined by the primordial power spectrum and the transfer function.

We take a scale-invariant spectrum and the cold dark matter transfer function.

We start from a FRW background of dust with an initial Gaussian linear density field.

We identify structures with spherical isolated peaks of the smoothed density field. (BBKS 1986)

We keep the smoothing threshold fixed at σ(t,R)=1, which gives the time evolution R(t).

Each peak expands like a separate FRW universe.

The peak number density as a function of time is determined by the primordial power spectrum and the transfer function.

We take a scale-invariant spectrum and the cold dark matter transfer function.


1212

Ht as a function of time (in Gyr, with teq=50 000 yr)Ht as a function of time (in Gyr, with teq=50 000 yr)

The expansion rate is .

There are no parameters to adjust.

Consider two approximate transfer functions.Bonvin and Durrer BBKS

The expansion rate is .

There are no parameters to adjust.

Consider two approximate transfer functions.Bonvin and Durrer BBKS

€

H(t) = dδ−∞

∞

∫ vδ (t)Hδ (t)

Ht as a function of R/ReqHt as a function of R/Req


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5) Conclusion5) Conclusion Observations of the late universe are inconsistent with homogeneous and isotropic models with ordinary matter and gravity.

FRW models do not include non-linear structures. The Buchert equations show that the average expansion of a clumpy space can accelerate.

Modelling evolving structures with the peak model, the expansion rate Ht rises by 10-30% around 105 teq.

Many things are missing: trough-in-a-peak problem, correct transfer function, non-spherical evolution, ...

The average geometry has to be related to light propagation.

Observations of the late universe are inconsistent with homogeneous and isotropic models with ordinary matter and gravity.

FRW models do not include non-linear structures. The Buchert equations show that the average expansion of a clumpy space can accelerate.

Modelling evolving structures with the peak model, the expansion rate Ht rises by 10-30% around 105 teq.

Many things are missing: trough-in-a-peak problem, correct transfer function, non-spherical evolution, ...

The average geometry has to be related to light propagation.

ix ème ecole de cosmologie, cargese, november 3, 2008 1 1 structure formation as an alternative to...

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