Backreaction in the relativistic Zel'dovich approximation · Backreaction in the relativistic Zel'dovich approximation Alexander Wiegand 1 with Thomas Buchert 2and Charly Nayet 1Fakultät

Backreaction in the relativistic Zel'dovich

approximation

Alexander Wiegand1

with Thomas Buchert2 and Charly Nayet2

1Fakultät für Physik, Universität Bielefeld

2CRAL Université Lyon 1

7. Kosmologietag,

Bielefeld

04.05.2012

A. Wiegand (Universität Bielefeld) Zel'dovich backreaction Kosmologietag 2012 1 / 32

Outline

1 Averaging inhomogeneous universes

2 Newtonian vs. Relativistic Zel'dovich approximation

3 Results

4 Conclusion


Outline



3 Results

4 Conclusion


Why averaging?

Many measurements are averages

H0 ≡1N

N∑i=1

vi

di−→ H0 =

1V

∫vd

dV

Inhomogeneous evolution too complex

to follow ⇒ Average description

Standard cosmological approach: Implicit averaging ⇒Average is identied with the single exact homogeneous

and isotropic solution of the Einstein equations

Gµν(〈gµν〉) 6= 〈Gµν(gµν)〉

Friedmann Averaged


Why averaging?


H0 ≡1N

N∑i=1

vi

di−→ H0 =

1V

∫vd

dV






Friedmann Averaged


Why averaging?


H0 ≡1N

N∑i=1

vi

di−→ H0 =

1V

∫vd

dV






Friedmann Averaged


Why averaging?


H0 ≡1N

N∑i=1

vi

di−→ H0 =

1V

∫vd

dV






Friedmann Averaged


Averages of scalar quantities

Considered here: Averages of three-scalars in the ADM

3+1 split of the underlying dust universe

〈f 〉D (t) :=

∫D f (t,X)dµg∫D dµg

; dµg :=√

(3)g(t,X)d3X

Three metric of spatial slice (3)g(t,X) general andtherefore also local expansion rate θ (t,X) inhomogeneous

θ (t,X) =√

(3)g(t,X)−1∂t

(√(3)g(t,X)

)Time evolution and averaging do not commute

∂t 〈f 〉D = 〈∂t f 〉D+ 〈f θ〉D−〈f 〉D 〈θ〉D





〈f 〉D (t) :=


; dµg :=√

(3)g(t,X)d3X


θ (t,X) =√

(3)g(t,X)−1∂t

(√(3)g(t,X)


∂t 〈f 〉D = 〈∂t f 〉D+ 〈f θ〉D−〈f 〉D 〈θ〉D





〈f 〉D (t) :=


; dµg :=√

(3)g(t,X)d3X


θ (t,X) =√

(3)g(t,X)−1∂t

(√(3)g(t,X)


∂t 〈f 〉D = 〈∂t f 〉D+ 〈f θ〉D−〈f 〉D 〈θ〉D


The Buchert equations

Averaging the local equations directly yields

3H2D = 8πG〈%〉D−

12〈R〉D−

12QD+ Λ

3aDaD

= −4πG〈%〉D+QD+ Λ

0 = ∂t 〈%〉D+ 3HD 〈%〉DThese are evolution equations for the average scale

factor or Hubble rate

aD (t) :=

(VDVDi

) 13

; HD :=aDaD

=13〈θ〉D

New component: Kinematical backreaction QD

QD :=23

(⟨θ2⟩D−〈θ〉

2D

)−2⟨σ2⟩

D =⟨

K2−KijKij⟩D−

23〈K〉2D




3H2D = 8πG〈%〉D−

12〈R〉D−

12QD+ Λ

3aDaD

= −4πG〈%〉D+QD+ Λ



aD (t) :=

(VDVDi

) 13

; HD :=aDaD

=13〈θ〉D


QD :=23

(⟨θ2⟩D−〈θ〉

2D

)−2⟨σ2⟩

D =⟨

K2−KijKij⟩D−

23〈K〉2D




3H2D = 8πG〈%〉D−

12〈R〉D−

12QD+ Λ

3aDaD

= −4πG〈%〉D+QD+ Λ



aD (t) :=

(VDVDi

) 13

; HD :=aDaD

=13〈θ〉D


QD :=23

(⟨θ2⟩D−〈θ〉

2D

)−2⟨σ2⟩

D =⟨

K2−KijKij⟩D−

23〈K〉2D


Uncommon properties

1 Backreaction can lead to accelerated expansion in a dust

universe

3aDaD

=−4πG〈%〉D+QD+ Λ ⇒aDaD

> 0 if 4πG〈%〉D <QD

Physical explanation: Volume fraction of fasterexpanding regions rises so that there can be volumeacceleration without positive Λ

2 Backreaction triggers dynamical curvature

a−2D ∂t

(a2D〈R〉D

)=−a−6

D ∂t(

a6DQD

)⇒ Additional degree of freedom. Coupling is generic.Substantial scalar curvature today consistent with CMB(see Räsänen arXiv:0812.2872)


Uncommon properties

1 Backreaction can lead to accelerated expansion in a dust

universe

3aDaD

=−4πG〈%〉D+QD+ Λ ⇒aDaD

> 0 if 4πG〈%〉D <QD

Physical explanation: Volume fraction of fasterexpanding regions rises so that there can be volumeacceleration without positive Λ

2 Backreaction triggers dynamical curvature

a−2D ∂t

(a2D〈R〉D

)=−a−6

D ∂t(

a6DQD

)⇒ Additional degree of freedom. Coupling is generic.Substantial scalar curvature today consistent with CMB(see Räsänen arXiv:0812.2872)


Cosmological implications

Standard cosmology

described by 3 parameters

ΩDm :=8πG3H2D〈%〉D ΩDΛ :=

Λ

3H2D

ΩDk :=−k

a2H2ΩDQ :=−

QD6H2D

Cosmic triangle

ΩDm + ΩDΛ + ΩDk +ΩDQ = 1


Cosmological implications

Average cosmology described

by 4 parameters

ΩDm :=8πG3H2D〈%〉D ΩDΛ :=

Λ

3H2D

ΩDR :=−〈R〉D6H2D

ΩDQ :=−QD6H2D

Cosmic quartet (Roy et al. 2011)

ΩDm + ΩDΛ + ΩDR+ ΩDQ = 1


Outline



3 Results

4 Conclusion


Newtonian backreaction

Backreaction in terms of kinematical variables

QD = 23

(⟨θ2⟩D−〈θ〉

2D

)+ 2⟨ω2−σ2⟩

D

becomes dependent on derivatives of the peculiar

velocity eld

QD = 2〈II(vi,j)〉D−23 〈I(vi,j)〉2D

which means that it is a surface term (Buchert&Ehlers 1997)

QD=1a2

21

Vq

∫∂Dq

dS · (u(∇q ·u)− (u ·∇q)u)−23

(1

Vq

∫∂Dq

dS ·u)2


Newtonian Zel'dovich approximation

Lagrangian picture: express the Eularian position and

velocity of a uid element by f(·, t) : X 7→ xThe Zel'dovich approximation implies perturbing this

function by the gravitational potential

fZ(X, t) = a(t)(

X + ξ(t)∇0ψ(X))

QD may be expressed by this function f, if we replace theinvariants of the gradient of the velocity eld in

BKSQD = 21〈J〉Di

〈J II(vi,j)〉Di−

23

(1〈J〉Di

〈J I(vi,j)〉Di

)2

by

I(vi,j) = 12J εabcε

ijk f |ai f |bj f |ck

II(vi,j) = 12J εabcε







fZ(X, t) = a(t)(

X + ξ(t)∇0ψ(X))


BKSQD = 21〈J〉Di


23

(1〈J〉Di

〈J I(vi,j)〉Di

)2

by










fZ(X, t) = a(t)(

X + ξ(t)∇0ψ(X))


BKSQD = 21〈J〉Di


23

(1〈J〉Di

〈J I(vi,j)〉Di

)2

by






Backreaction in the NZA

Result for the backreaction term in the NZA

NZAQD =ξ2 (t)

(γ1 + ξ (t)γ2 + ξ2 (t)γ3

)(1 + ξ (t)〈Ii〉Di + ξ2 (t)〈IIi〉Di + ξ3 (t)〈IIIi〉Di)

2

γ1 := 2〈IIi〉Di−

23〈Ii〉2Di

γ2 := 6〈IIIi〉Di−23〈IIi〉Di〈Ii〉Di

γ3 := 2〈Ii〉Di〈IIIi〉Di−23〈IIi〉2Di

with the initial values

Ii := I(ψ|i|j), IIi := II(ψ|i|j), IIIi := III(ψ|i|j)

For an EdS background ξ (t) = a(t)−1 and therefore

ξ2 (t)∝ 1/a(t)


Backreaction in the NZA

Result for the backreaction term in the NZA

NZAQD =ξ2 (t)

(γ1 + ξ (t)γ2 + ξ2 (t)γ3

)(1 + ξ (t)〈Ii〉Di + ξ2 (t)〈IIi〉Di + ξ3 (t)〈IIIi〉Di)

2

γ1 := 2〈IIi〉Di−

23〈Ii〉2Di

γ2 := 6〈IIIi〉Di−23〈IIi〉Di〈Ii〉Di

γ3 := 2〈Ii〉Di〈IIIi〉Di−23〈IIi〉2Di

with the initial values


For an EdS background ξ (t) = a(t)−1 and therefore

ξ2 (t)∝ 1/a(t)


Relativistic Zel'dovich approximation (RZA)

To evaluate backreaction eects on void and cluster

scales use Relativistic Zel'dovich approximation (RZA)

Motivation

Quantitative estimate of the importance ofinhomogeneitiesInitialization of Nbody simulationsComparison to Newtonian theorySuccessful interpolation between exact solutions in theNewtonian case


Relativistic Zel'dovich approximation (RZA)

To evaluate backreaction eects on void and cluster

scales use Relativistic Zel'dovich approximation (RZA)

Motivation

Quantitative estimate of the importance ofinhomogeneitiesInitialization of Nbody simulationsComparison to Newtonian theorySuccessful interpolation between exact solutions in theNewtonian case


Relativistic case

The coframe decomposition

gij = Gabηa

iηb

j

allows for a generalization of the Zel'dovich approxima-

tion to GR (Kasai 1995,Matarrese&Terranova 1996, Buchert&Ostermann 2012)

RZAηai(t,Xk) := a(t)

(δa

i + ξ(t)Pai

)Pa

i = Pai(

ti,Xk) ; ξ (ti) = 0 ; a(ti) = 1

Time evolution given by

ξ(t) + 2a(t)a(t)

ξ(t) +

(3

a(t)a(t)−Λ

)(ξ(t) + 1) = 0

Initial values of the perturbation one form

Ii := I(

Pai

); IIi := II

(Pa

i

); IIIi := III

(Pa

i

)A. Wiegand (Universität Bielefeld) Zel'dovich backreaction Kosmologietag 2012 14 / 32

Relativistic case

The coframe decomposition

gij = Gabηa

iηb

j

allows for a generalization of the Zel'dovich approxima-

tion to GR (Kasai 1995,Matarrese&Terranova 1996, Buchert&Ostermann 2012)

RZAηai(t,Xk) := a(t)

(δa

i + ξ(t)Pai

)Pa

i = Pai(

ti,Xk) ; ξ (ti) = 0 ; a(ti) = 1

Time evolution given by

ξ(t) + 2a(t)a(t)

ξ(t) +

(3

a(t)a(t)−Λ

)(ξ(t) + 1) = 0

Initial values of the perturbation one form

Ii := I(

Pai

); IIi := II

(Pa

i

); IIIi := III

(Pa

i

)A. Wiegand (Universität Bielefeld) Zel'dovich backreaction Kosmologietag 2012 14 / 32

Outline



3 Results

4 Conclusion


Backreaction in the RZA

Result for the backreaction term in the RZA

RZAQD =ξ2(γ1 + ξγ2 + ξ2γ3

)(1 + ξ〈Ii〉CD + ξ2〈IIi〉CD + ξ3〈IIIi〉CD)2

γ1 := 2〈IIi〉CD −

23〈Ii〉2CD

γ2 := 6〈IIIi〉CD −23〈IIi〉CD〈Ii〉CD

γ3 := 2〈Ii〉CD〈IIIi〉CD −23〈IIi〉2CD

Expression has the same form as in Newtonian theory

⇒If one starts with at initial conditions, the evolution

of QD is basically Newtonian

However: In GR QD triggers nontrivial curvature






γ1 := 2〈IIi〉CD −

23〈Ii〉2CD












γ1 := 2〈IIi〉CD −

23〈Ii〉2CD








LTB solutions

Evaluation of the backreaction for an LTB metric

ds2 =−dt2 +R′2(t, r)

1 + 2E(r)dr2 + R2(t, r)dΩ2

shows that the RZA is an exact solution for the case

E (r) = E

⟨I(Θi

j)⟩

LTB=

4πVLTB

∫ rD

0

∂r(

RR2)

√1 + 2E

dr

⟨II(Θi

j)⟩

LTB=

4πVLTB

∫ rD

0

∂r(

R2R)

√1 + 2E

dr =13

⟨I(Θi

j)⟩2

LTB

because in both cases QD is exactly zero

QD := 2〈II〉D−23〈I〉2D⇒QLTB = 0 , WLTB = 0


LTB solutions

Evaluation of the backreaction for an LTB metric

ds2 =−dt2 +R′2(t, r)

1 + 2E(r)dr2 + R2(t, r)dΩ2

shows that the RZA is an exact solution for the case

E (r) = E

⟨I(Θi

j)⟩

LTB=

4πVLTB

∫ rD

0

∂r(

RR2)

√1 + 2E

dr

⟨II(Θi

j)⟩

LTB=

4πVLTB

∫ rD

0

∂r(

R2R)

√1 + 2E

dr =13

⟨I(Θi

j)⟩2

LTB

because in both cases QD is exactly zero

QD := 2〈II〉D−23〈I〉2D⇒QLTB = 0 , WLTB = 0


Deviations from background parameters

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Scale in Mpc

Var

HWQ

L12

in%

EdS, h=0.5

L CDM, h=0.7

∆WRD

∆WmD

∆HD H0

0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.1

0.5

1.0

5.0

10

redshift z

∆in

%

2dFSDSS

fullsky

Fluctuations of QD only interesting below 100 Mpc

Other parameters uctuate more strongly for even

bigger domains


Collapsing domains

0 2 4 6 8 10 12

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

time t in Gyrs

W

0 2 4 6 8 10 12 14-0.5

0.0

0.5

1.0

time t in Gyrs

W

On scales of 50 Mpc in regions with an overdensity,

backreaction enhances the collapse

For even smaller scales we nd the analogue of the

pancake collapse in Newtonian structure formation


Outline



3 Results

4 Conclusion


Conclusion

In the relativistic Zel'dovich approximation, backreaction

is close to its Newtonian counterpart

Therefore only small scale modication of cosmic

evolution

However: breakdown of the approximation for late times

Gradient expansion (Enqvist et al. 2011) and nonperturbative toy

models (Räsänen 2008) point to a larger eect


Conclusion




evolution





Conclusion




evolution





Conclusion




evolution





Thank you for your attention

Questions?

Remarques?

Objections?


Newtonian limit

Sending the coframes to integrable ones

ηai→ Nηa

i = f a|i

leads to the Newtonian equivalent of the perturbation

one forms which are second derivatives of the Newtonian

potentialNPa

i = ψ|a|i

and therefore the invariants become



Newtonian limit


ηai→ Nηa

i = f a|i



potentialNPa

i = ψ|a|i




Newtonian limit


ηai→ Nηa

i = f a|i



potentialNPa

i = ψ|a|i




Plane symmetric solutions

Example for dierences in the Newtonian vs. GR

description: Plane symmetry

ds2 =−dt2 + a(t)2(

dx2 + dy2 + (1 + P(z, t))2 dz2)

In the Newtonian case only the Raychaudhuri equation

Θ +Θkl Θ

lk =−4πG%+ Λ⇒ P(z, t) = aC1 (z) +

C2 (z)a3/2

In the GR case also the Hamilton constraint

⇒ P(z, t) =C (z)a3/2





ds2 =−dt2 + a(t)2(

dx2 + dy2 + (1 + P(z, t))2 dz2)


Θ +Θkl Θ

lk =−4πG%+ Λ⇒ P(z, t) = aC1 (z) +

C2 (z)a3/2


⇒ P(z, t) =C (z)a3/2





ds2 =−dt2 + a(t)2(

dx2 + dy2 + (1 + P(z, t))2 dz2)


Θ +Θkl Θ

lk =−4πG%+ Λ⇒ P(z, t) = aC1 (z) +

C2 (z)a3/2


⇒ P(z, t) =C (z)a3/2


Template metrics

(Picture by Mauro Carfora)

RW metric incompatible with the

average properties of the universe ⇒use template metric as average metric

3gD = γDij dXi⊗dXj =

(dr2

1−κD (t) r2+ dΩ2

)Testable by Clarkson's C test

(arXiv:0712.3457): D = (1 + z)dA

C (z) = 1 + H2 (DD′′−D′2)

+ HH′DD′

C (z)≡ 0 for FLRW, but for template:

C (zD) =−HD (zD) r (zD)κ′D (zD)

2HD0

√1−κD (zD) r2 (zD)


Template metrics

(Picture by Mauro Carfora)

RW metric incompatible with the

average properties of the universe ⇒use template metric as average metric

3gD = γDij dXi⊗dXj =

(dr2

1−κD (t) r2+ dΩ2

)Testable by Clarkson's C test

(arXiv:0712.3457): D = (1 + z)dA

C (z) = 1 + H2 (DD′′−D′2)

+ HH′DD′

C (z)≡ 0 for FLRW, but for template:

C (zD) =−HD (zD) r (zD)κ′D (zD)

2HD0

√1−κD (zD) r2 (zD)


Perturbative corrections to the background

100

101

102

103

104

z

10-8

10-7

10-6

10-5

10-4

Ωe

ffΛCDM, n

s=0.96

ΛCDM, ns=1.04

EdS, h=0.45

EdS, h=0.70

Brown, Robbers, Behrend arXiv:0811.4495 Clarkson, Ananda, Larena arXiv:0907.3377

Perturbation theory in Newtonian gauge gives a globally

modied background of about 10−5 today

On smaller scales where the Newtonian terms are

important the contribution is higher


Why perturbation theory might not be the full

answer

General form of corrections to the average expansion

rate (Räsänen 2010)

〈θ〉= 3Hτ

(1 +

1

(aH)2 〈∂iΦ∂iΦ〉0∞∑

n=0

λn⟨δ2⟩n

0 + . . .

)

If perturbations become of order unity on any scale the

sum∑∞

n=0λn may be important

Has been found to be large in LTB models even if Φ is

small and one would expect the perturbed FRW metric

to apply



answer



〈θ〉= 3Hτ

(1 +

1

(aH)2 〈∂iΦ∂iΦ〉0∞∑

n=0

λn⟨δ2⟩n

0 + . . .

)


sum∑∞




to apply



answer



〈θ〉= 3Hτ

(1 +

1

(aH)2 〈∂iΦ∂iΦ〉0∞∑

n=0

λn⟨δ2⟩n

0 + . . .

)


sum∑∞




to apply


The rst coecient seems to be large

Already second order terms comparably in magnitude

with the rst order terms (Clarkson, Umeh 2011)

∂2Φ∂2Φ∼∆2R

(keq

kH

)2

ln3(

kUV

keq

)because the suppression of the second order by ∆R is

overcome by the particular size of the equality scale

∆2R

(keq

kH

)2

≈ 2.4Ω2mh2⇒O (1)

This seems to continue and leads to important fourth

order corrections


The rst coecient seems to be large

Already second order terms comparably in magnitude

with the rst order terms (Clarkson, Umeh 2011)

∂2Φ∂2Φ∼∆2R

(keq

kH

)2

ln3(

kUV

keq

)because the suppression of the second order by ∆R is

overcome by the particular size of the equality scale

∆2R

(keq

kH

)2

≈ 2.4Ω2mh2⇒O (1)

This seems to continue and leads to important fourth

order corrections


Inhomogeneous metric

If we want to describe the

inuence of structures on the

expansion we rst of all need

to describe structure.

Therefore more general

metric.

In cosmology we are interested in the temporal evolution

of the universe. ⇒ 3+1 split into spatial hypersurfaces

4g =−dt2 + 3g ; 3g = gab dXa⊗dXb

⇒ Foliation into space and time similar to RW, but

general spatial metric


Inhomogeneous metric

If we want to describe the

inuence of structures on the

expansion we rst of all need

to describe structure.

Therefore more general

metric.

In cosmology we are interested in the temporal evolution

of the universe. ⇒ 3+1 split into spatial hypersurfaces

4g =−dt2 + 3g ; 3g = gab dXa⊗dXb

⇒ Foliation into space and time similar to RW, but

general spatial metric


Rewriting Einstein's equations

For dust source choose comoving frame:nµ = uµ = (−1,0,0,0). Split symmetric part of the expansiontensor uµ;ν = u(µ;ν) + u[µ;ν] into trace and tracefree part:

Kij :=−uµ;νhµihν j → −Kij = Θij = σij +13θgij

The system of ADM equations becomes

12R+

13θ2−σ2 = 8πG%+ Λ

σij‖i =

23θ|j

% = −θ%

∂tgij = 2gikσk

j +23θgij

∂tσij = −θσi

j−Rij +

(4πG%−

13θ2−

13θ+ Λ

)δi

j

⇒ ADM system completely equivalent to Einsteins

equations (for comoving dust)A. Wiegand (Universität Bielefeld) Zel'dovich backreaction Kosmologietag 2012 30 / 32

Rewriting Einstein's equations

For dust source choose comoving frame:nµ = uµ = (−1,0,0,0). Split symmetric part of the expansiontensor uµ;ν = u(µ;ν) + u[µ;ν] into trace and tracefree part:

Kij :=−uµ;νhµihν j → −Kij = Θij = σij +13θgij

The system of ADM equations becomes

12R+

13θ2−σ2 = 8πG%+ Λ

σij‖i =

23θ|j

% = −θ%

∂tgij = 2gikσk

j +23θgij

∂tσij = −θσi

j−Rij +

(4πG%−

13θ2−

13θ+ Λ

)δi

j

⇒ ADM system completely equivalent to Einsteins

equations (for comoving dust)A. Wiegand (Universität Bielefeld) Zel'dovich backreaction Kosmologietag 2012 30 / 32

Literature

T. Buchert: Dark Energy from structure a status report.Gen. Rel. Grav. 40, 467 (2008)

Buchert, T.: On average properties of inhomogeneous uids ingeneral relativity: 1. dust cosmologies. Gen. Rel. Grav. 32, 105(2000)

Räsänen, S.: Dark Energy from backreaction. JCAP 0402,003 (2004)

Räsänen, S.: Accelerated expansion from structure formation.JCAP 0611, 003 (2006)

Räsänen, S.: Applicability of the linearly perturbed FRWmetric and Newtonian cosmology. arXiv:1002.4779 (2010)

G.F.R. Ellis: Relativistic cosmology: its nature, aims andproblems. (D. Reidel Publishing Company, Dordrecht, 1984),pp. 215288


Literature

Clarkson, C., Bassett, B. and Lu, T.~H.-C.: A General Test ofthe Copernican Principle. PRL 101, 001 (2008)

T. Buchert, M. Kerscher and C. Sicka: Back reaction ofinhomogeneities on the expansion: The evolution ofcosmological parameters. Phys. Rev. D 62, 043525 (2000)

N. Li and D. J. Schwarz: Scale dependence of cosmologicalbackreaction. Phys. Rev. D 78, 083531 (2008)

S. Räsänen: Light propagation in statistically homogeneousand isotropic dust universes. JCAP 2, 011 (2009)

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Backreaction in the relativistic Zel'dovich approximation · Backreaction in the relativistic Zel'dovich approximation Alexander Wiegand 1 with Thomas Buchert 2and Charly Nayet 1Fakultät

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