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Lecture 2: Linear Perturbation Theory

Wayne Hu

Trieste, June 2002

Structure Formation

and the

Dark Sector


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Outline Covariant Perturbation Theory

Scalar, Vector, Tensor Decomposition

Linearized Einstein-Conservation Equations

Dark (Multi) Components

Gauge Applications:

Bardeen Curvature Baryonic wiggles

Scalar Fields Parameterizing dark components

Transfer function Massive neutrinos

Sachs-Wolfe Effect Dark energy

COBE normalization


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Covariant Perturbation Theory Covariant= takes sameformin all coordinate systems

Invariant= takes the samevaluein all coordinate systems Fundamental equations:Einstein equations, covariantconservation

of stress-energy tensor:

G = 8GT

T = 0 Preserve general covariance by keeping alldegrees of freedom: 10

for each symmetric 4

4 tensor

1 2 3 4

5 6 7

8 9

10


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Metric Tensor Expand the metric tensor around thegeneral FRW metric

g00= a2, gij =a2ij.where the 0 component isconformal time =dt/aandij is a

spatial metric of constant curvatureK=H20(tot 1). Add in a general perturbation (Bardeen 1980)

g00 = a2(1 2A) ,g0i = a2Bi ,gij = a2(ij 2HLij 2HijT) .

(1)A a scalarpotential;(3)B i a vectorshift,(1)HLaperturbation to the spatialcurvature;(6)HijT atrace-freedistortion

to spatial metric =(10)


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Matter Tensor Likewise expand the matterstress energytensor around a

homogeneous density and pressurep:

T00 = ,T0i = (+p)(vi Bi) ,

T i

0 = (+p)vi

,Tij = (p+p)

ij+p

ij,

(1)adensity perturbation; (3)vi a vectorvelocity, (1)papressure perturbation; (5)ij ananisotropic stressperturbation

So far this isfully generaland applies to any type of matter orcoordinate choice including non-linearities in the matter, e.g.

cosmological defects.


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Counting DOFs

20 Variables (10 metric; 10 matter)

10 Einstein equations4 Conservation equations+4 Bianchi identities

4 Gauge (coordinate choice 1 time, 3 space)

6 Degrees of freedom

Without loss of generality these can be taken to be the6componentsof thematter stress tensor

For the background, specifyp(a)or equivalentlyw(a)

p(a)/(a)theequation of stateparameter.


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Scalar, Vector, Tensor Inlinear perturbation theory, perturbations may be separated by

theirtransformation propertiesunder rotation and translation. The eigenfunctions of theLaplacian operatorform a complete set

2Q(0) = k2Q(0) S ,

2

Q

(1)

i = k2

Q

(1)

i V ,2Q(2)ij = k2Q(2)ij T ,and functions built out of covariant derivatives and the metric

Q(0)

i = k1

iQ(0)

,Q

(0)ij = (k

2ij 13

ij)Q(0) ,

Q(1)ij =

1

2k[iQ(1)j + jQ(1)i ] ,


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Spatially Flat Case For a spatially flat background metric, harmonics are related to

plane waves:

Q(0) = exp(ik x)Q

(1)i =

i2

(e1 ie2)iexp(ik x)

Q(2)ij =

3

8(e1 ie2)i(e1 ie2)jexp(ik x)

wheree3 k.

For vectors, the harmonic points in a direction orthogonal to ksuitable for thevortical componentof a vector

For tensors, the harmonic is transverse and traceless as appropriatefor the decompositon ofgravitational waves


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Perturbationk-Modes For thekth eigenmode, thescalar componentsbecome

A(x) = A(k) Q(0) , HL(x) = HL(k) Q(0) ,

(x) = (k) Q(0) , p(x) = p(k) Q(0) ,

thevectors componentsbecome

Bi(x) =1

m=1

B(m)(k) Q(m)i , vi(x) =

1m=1

v(m)(k) Q(m)i ,

and thetensors components

HT ij(x) =2

m=2

H(m)T (k) Q(m)ij ,

ij(x) =2

m=2(m)(k) Q

(m)ij ,


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Homogeneous Einstein Equations Einstein (Friedmann) equations:

1

a

da

dt

2

= 8G

3

1

a

d2a

dt2 =

4G

3 (+ 3p)

so thatw p/ < 1/3for acceleration

Conservation equation T= 0implies

= 3(1 + w)

a

a


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Homogeneous Einstein Equations Counting exercise:


17 Homogeneity and Isotropy

2 Einstein equations

1 Conservation equations+1 Bianchi identities

1 Degree of freedom

without loss of generality choose ratio of homogeneous & isotropic

component of thestress tensorto the densityw(a) =p(a)/(a).


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Covariant Scalar Equations Einstein equations(suppressing0) superscripts (Hu & Eisenstein 1999):

(k2 3K)[HL+ 13HT+

aa

1k2

(kB HT)]

= 4Ga2+ 3

a

a(+p)(v B)/k

, Poisson Equation

k2

(A+HL+

1

3HT) + dd + 2

a

a

(kB HT)

= 8Ga2p ,

a

aA HL 1

3HT K

k2(kB HT)

= 4Ga2(+p)(vB)/k ,

2a

a 2

a

a

2+

a

a

d

d k

2

3

A

d

d+

a

a

(HL+

1

3kB)

= 4Ga2(p+1

3) .


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Covariant Scalar Equations Conservation equations:continuityandNavier Stokes

dd

+ 3aa

+ 3

aap = (+p)(kv+ 3 HL) ,

d

d+ 4

a

a

(+p)

(v B)k

= p 2

3(1 3K

k2)p+ (+p)A ,

Equations are not independent since G

= 0via theBianchiidentities.

Related to the ability to choose acoordinate systemor gauge torepresent the perturbations.


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Covariant Scalar Equations DOF counting exercise


4 Einstein equations2 Conservation equations

+2 Bianchi identities2 Gauge (coordinate choice 1 time, 1 space)


without loss of generality choose scalar components of thestresstensorp, .


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Covariant Vector Equations Einstein equations

(1 2K/k2)(kB(1) H(1)T )= 16Ga2(+p)(v(1) B(1))/k ,

d

d+ 2

a

a

(kB(1) H(1)T )

= 8Ga2p(1) . Conservation Equations

d

d

+ 4a

a [(+p)(v(1)

B(1))/k]

= 12

(1 2K/k2)p(1) ,

Gravity providesno sourceto vorticity decay


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Covariant Vector Equations DOF counting exercise


4 Einstein equations2 Conservation equations

+2 Bianchi identities2 Gauge (coordinate choice 1 time, 1 space)


without loss of generality choose vector components of thestresstensor(1).


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Covariant Tensor Equation Einstein equation

d2d2

+ 2aadd

+ (k2 + 2K)H(2)T = 8Ga

2p(2) .

DOF counting exercise


2 Einstein equations0 Conservation equations+0 Bianchi identities

0 Gauge (coordinate choice 1 time, 1 space)


wlog choose tensor components of thestress tensor(2).


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Arbitrary Dark Components Total stress energy tensor can be broken up intoindividual pieces

Dark componentsinteract only through gravity and so satisfyseparate conservation equations

Einstein equation source remains the sum of components.

To specify an arbitrary dark component, give the behavior of the

stress tensor:6 components:p,(i), wherei = 2, ..., 2. Many types of dark components (dark matter, scalar fields,

massive neutrinos,..) havesimple formsfor their stress tensor in

terms of the energy density, i.e. described byequations of state.

An equation of state for the backgroundw =p/is notsufficientto determine the behavior of the perturbations.


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Gauge Metric and matter fluctuations take ondifferent valuesin different

coordinate system No such thing as a gauge invariant density perturbation! Generalcoordinate transformation:

= +Txi = xi +Li

free to choose(T, Li) to simplify equations or physics.

Decompose these into scalar and vector harmonics.

G andTtransform astensors, so components in differentframes can be related


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Gauge Transformation Scalar Metric:

A = A T aaT,

B = B + L+kT,

HL = HL k

3L

a

aT,

HT =

HT+kL ,

Scalar Matter (Jth component):

J = J JT,

pJ = pJ pJT,

vJ = vJ+ L,

Vector:

B(1)

= B(1)

+ L(1)

, H(1)T

=H(1)T

+kL(1)

, v(1)J

=v(1)J

+ L(1)

,


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Gauge Dependence of Density

Background evolution of the density induces a density fluctuation

from a shift in the time coordinate


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Common Scalar Gauge Choices A coordinate system isfully specifiedif there is an explicit

prescription for(T, L

i

) or for scalars(T, L) Newtonian:

B = HT = 0

A (Newtonian potential)

HL (Newtonian curvature)L = HT/kT = B/k + HT/k2

Good:intuitive Newtonian like gravity; matter and metricalgebraically related; commonly chosen foranalytic CMBand

lensingwork

Bad:numericallyunstable


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Example: Newtonian Reduction In the general equations, setB =HT = 0:

(k2 3K) = 4Ga2+ 3a

a(+p)v/k

k2(+) = 8Ga2p

so = if anisotropic stress = 0and

d

d+ 3

a

a

+ 3

a

ap = (+p)(kv+ 3) ,

d

d+ 4

a

a

(+p)v = kp

2

3(1 3

K

k2)p k+ (+p) k ,

Competition betweenstress(pressure and viscosity) andpotential

gradients


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Relativistic Term in Continuity Continuity equation contains relativistic termfrom changes in the spatial curvature perturbation to the scale factor

For w=0 (matter), simply density dilution; for w=1/3 (radiation) density dilution plus (cosmological) redshift

a.k.a.ISW effect photon redshift from change in grav. potential


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Common Scalar Gauge Choices Comoving:

B = v (T0i = 0)

HT = 0

= A

= HL (Bardeen curvature)

T = (v B)/kL = HT/k

Good:is conservedif stress fluctuations negligible, e.g. above

the horizon if|K| H2

+Kv/k=a

a

p+p

+2

3

1 3K

k2

p

+p

0

Bad:explicitly relativistic choice


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Common Scalar Gauge Choices Synchronous:

A = B= 0

L HL 13HT

hT = HT or h= 6HL

T = a

1 daA+c1a

1

L = d(B+kT) +c2

Good:stable, the choice of numerical codes

Bad:residualgauge freedomin constantsc1,c2must bespecified as an initial condition, intrinsically relativistic.


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Common Scalar Gauge Choices Spatially Unperturbed:

HL = HT = 0

L = HT/kA , B = metric perturbations

T = aa

1

HL

+1

3H

T

Good:eliminates spatial metric in evolution equations; useful in

inflationary calculations(Mukhanov et al)

Bad:intrinsically relativistic.

Caution:perturbation evolution is governed by the behavior ofstress fluctuations and an isotropic stress fluctuationpis gauge

dependent.


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Hybrid Gauge Invariant Approach With the gauge transformation relations, express variables ofone

gaugein terms of those inanother allows a mixture in the

equations of motion

Example:Newtonian curvature above the horizon. Conservation ofthe Bardeen-curvature=const. implies:

=3 + 3w

5 + 3w

e.g. calculatefrom inflation determines for any choice of

matter content or causal evolution.

Example:Scalar field (quintessence dark energy) equations incomoving gauge imply asound speedp/= 1independent of

potentialV(). Solve in synchronous gauge (Hu 1998).


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Transfer Function Example Example:Transfer functiontransfers the initial Newtonian

curvature to its value today (linear response theory)

T(k)=(k, a= 1)

(k, ainit)

Conservation of Bardeen curvature: Newtonian curvature is a

constantwhenstress perturbations are negligible: above thehorizon during radiation and dark energy domination, on all scales

during matter domination

When stress fluctuations dominate, perturbations are stabilized by

theJeans mechanism

HybridPoisson equation: Newtonian curvature, comoving density

perturbation (/)comimplies decays

(k2 3K)= 4G 2


30/52

Transfer Function Example Freezingof stops ateq

(keq)2H (keq)2init Transfer function has ak2 fall-offbeyondkeq 1eq

1

0.1

0.0001 0.001 0.01 0.1 1

0.01

T(k)

k (h1Mpc)

wiggles

k2


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Gauge and the Sachs-Wolfe Effect Going fromcomoving gauge, where the CMB temperature

perturbation is initially negligible by the Poisson equation, to the

Newtoniangauge involves a temporal shift

t

t =

Temporal shift implies a shift in thescale factorduring matterdomination

a

a =

2

3

t

t

CMB temperature iscoolingasT a Inducedtemperature fluctuation

T

T =

a

a =

2

3


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Gauge and the Sachs-Wolfe Effect Add thegravitational redshifta photon suffers climbing out of the

gravitational potential

T

T

obs

= +T

T =

1

3


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COBE Normalization

Sachs-WolfeEffect relates the COBEdetection to the gravitational

potential on the last scattering surface

Decompose the angular and spatial information into normal modes: spherical harmonicsfor angular, plane wavesfor spatial

Gm (n,x,k) = (i) 4

2+ 1Ym

(n)eikx .

x,

D

jl(kD)Yl0

[+ ](n,x

) =

1

3 x

+Dn,

)D=0

(

Last Scattering Surface

0


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COBE Normalization cont.

Multipole momentdecomposition for each k

Power spectrumis the integral overkmodes

Fourier transformSachs-Wolfe source

Decompose plane wave

(n,x) = d3k

(2)3m

(m)

(k)Gm (x,k,n)

C = 4 d

3

k(2)3

m

(m)

(m)

(2+ 1)2

[+](n,x) =1

3 d3k

(2)3 k, )e

ik(D n+x)(

exp(ikDn) =

(i)

4(2+ 1)j(kD)Y0(n) ,


35/52


Extract multipole moment,assume a constantpotential

Construct angular power spectrum

For scale invariant potential(n=1), integral reduces to

Log power spectrum = Log potential spectrum / 9

(0)2+ 1

= 13

k, )j(kD)

= 1

3 k, 0)j(kD)

(

(

C= 4 dk

kj2

(kD)

1

92

0

dxxj2

(x) = 1

2(+ 1)

(+ 1)

2 C

=

1

9

2

(n= 1)


36/52


Relate to density fluctuations: Poisson equation and Friedmann eqn.

Power spectra relation

In terms of density fluctuation at horizon and transfer function

For scale invariant potential

k2 4Ga2

= 3

2H20

2

m

2=94

H0

k

42m

2

2

2

H

k

H0

n+3

T2

(k)

(+ 1)

2

C = 1

4

2m

2

H (n= 1)

=


37/52


Some numbers

Detailed calculation from Bunn & White (1997)including decay of

potential in low density universe and tilt

(+ 1)2

C = 14

2m2H

(n= 1)=

28K

2.726 106K

2 1010

H (2 105)1

m

H= 1.94 1050.7850.05lnm

m e0.95(n1)0.169(n1)

2


38/52

Matter Power Spectrum

Combine the transfer functionwith the COBE normalization

k (hMpc-1)

2(k)[=k3P(k)/22]

0.1 10.01

1

10-2

101

10-1

10-3

non-linear

scale


39/52

Matter Power Spectrum

Usually plotted asthepower spectrum, not log-power spectrum

k (hMpc-1)

P(k)(h-1Mp

c)3

0.1 10.01

103

104

105


40/52

Galaxy Power Spectrum Data

Galaxy clusteringtracks the dark matter but as a biased tracer

k (hMpc-1)

P(k)(h-1Mp

c)3

0.1 10.01

103

104

105

2dF

PSCz


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Galaxy Power Spectrum Data

Each galaxy population has different linear biasin linear regime

k (hMpc-1)

P(k)(h-1Mp

c)3

0.1 10.01

103

104

105

2dF

PSCz


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Acoustic Oscillations Example:Stabilization accompanied byacoustic oscillations

Photon-baryonsystem - under rapid scatteringd

d+ 3

a

a

b + 3

a

apb = (b +pb)(kvb + 3) ,

d

d+ 4

a

a [(b +pb)vb/k] = pb + (+p) ,or with =/andR = 3b/

= k3vb

vb = R

1 +Rvb +

1

1 +Rk+k

Forced oscillatorequation see Zaldarriagas talks Anisotropic stressesand entropy generation throughnon-adiabatic

stressSilkdampsfluctuations Silk 1968

Acoustic Peaks in the Matter


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Acoustic Peaks in the Matter

Baryon density & velocity oscillateswith CMB

Baryons decouple at /R ~ 1, the end of Compton drag epoch

Decoupling: b(drag) Vb(drag), but not frozen

Continuity: b=kVb

Velocity Overshoot Dominates: b Vb(drag) k>> b(drag)

Oscillations /2 out of phasewith CMB

Infall into potential wells (DC component)

.

End of Drag Epoch Velocity Overshoot + Infall

Hu & Sugiyama (1996)

Features in the Power Spectrum


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Features in the Power Spectrum

Featuresin the linear power spectrum

Breakat sound horizon

Oscillationsat small scales; washed out by nonlinearities

k (hMpc-1)

Eisenstein & Hu (1998)

numerical

P(k)

(arbitrarynor

m.)

0.01 0.1

0.1

1

nonlinear

scale

Combining Features in LSS + CMB


45/52


Consistency check on thermal history and photonbaryon ratio

Inferphysical scale lpeak(CMB) kpeak(LSS) in Mpc1

Eisenstein, Hu & Tegmark (1998)Hu, Eisenstein, Tegmark & White (1998)

k3P(k)

k (Mpc1)0.05

2

4

6

8

0.1

Powe

r(arbitrarynorm.)

CDM



46/52


Consistency check on thermal history and photonbaryon ratio

Inferphysical scale lpeak(CMB) kpeak(LSS) in Mpc1

Measure in redshift survey kpeak(LSS) in hMpc1 h

Eisenstein, Hu & Tegmark (1998)Hu, Eisenstein, Tegmark & White (1998)

Pm(k)

k3P(k)

k (Mpc1)0.05

2

4

6

8

0.1

Powe

r(arbitrarynorm.)

h

CDM

P i i D k C


47/52

Parameterizing Dark Components

Prototypes: Cold dark matter 0 0 0(WIMPs)

Hot dark matter 1/30(light neutrinos)

Cosmological constant 1 arbitrary arbitrary

(vacuum energy)

Exotica:

Quintessence 1 0(slowly-rolling scalar field)

Decaying dark matter 1/301/3(massive neutrinos)

Radiation backgrounds 1/3

0 0

1/3

scale

dependent

01/3

(rapidly-rolling scalar field, NBR)

Ultra-light fuzzy dark mat.

equation of state

wg

sound speed

ceff2

viscosity

cvis2

variable

Hu (1998)

M i N t i


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Massive Neutrinos Relativisticstressesof a light neutrinoslowthegrowthof structure

Neutrino species withcosmological abundancecontribute to matterash

2 =m/94eV, suppressing power asP/P 8/m

M i N t i


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Massive NeutrinosCurrent data from 2dF galaxy survey indicates m


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Dark Energy Stress & Smoothness

Raising equation of stateincreases redshift of dark energy

domination and raises large scale anisotropies Lowering the sound speedincreases clustering and reduces

ISW effect at large angles

w=1

w=2/3

ceff=1

ceff=

1/3

1010

1011

1012

Power

10 100l

ISW Effect

Total

Hu (1998); Hu (2001)

Coble et al. (1997)

Caldwell et al. (1998)

L i CMB T C l i


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LensingCMB TemperatureCorrelation

Any correlation is a direct detectionof a smooth energy

densitycomponent through the ISW effect Show dark energy smooth >5-6 Gpcscale, test quintesence

Hu (2001); Hu & Okamoto (2001)

10 100 1000

109

1010

1011

"Perfect"

Planck

crosspower

L

S


52/52

Summary In linear theory, evolution of fluctuations is completely defined

once thestressesin the matter fields are specified.

Stresses and their effects take on simple forms in particular

coordinate orgauge choices, e.g. the comoving gauge.

Gaugecovariant equationscan be used to take advantage of these

simplifications in an arbitrary frame.

Curvature(potential) fluctuations remainconstantin the absence

of stresses.

Evolution can be used to test the nature of thedark components,

e.g.massive neutrinosand thedark energyby measuring the

matter power spectrum.

Problem:luminous tracers of the matter clustering arebiased

next lecture.

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